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1,116,691,497,515 | arxiv | \section{Introduction}
One of the main ingredients of the direct method of the Calculus of Variations (\cite{DacorognaH}) to show existence of minimizers for an integral functional of the kind
$$
I(\mathbf{u})=\int_\Omega\psi(\nabla \mathbf{u}(\mathbf{x}))\,d\mathbf{x}
$$
is its weak lower semicontinuity. Here $\Omega\subset\mathbb{R}^N$ is a regular (Lipschitz), bounded domain, and feasible mappings $\mathbf{u}:\Omega\to\mathbb{R}^m$ are smooth or Lipschitz, so that $\nabla \mathbf{u}$ is a $m\times N$-matrix at each point $\mathbf{x}\in\Omega$. The weak lower semicontinuity property is in turn equivalent to suitable convexity properties of the continuous integrand $\psi:\mathbf{M}^{m\times N}\to\mathbb{R}$. Morrey (\cite{Morrey}, \cite{MorreyB}) proved that this weak lower semicontinuity (in $W^{1, \infty}(\Omega; \mathbb{R}^m)$) is equivalent to the quasiconvexity of the integrand $\psi$, namely,
$$
\psi(\mathbf F)\le\frac1{|D|}\int_D\psi(\mathbf F+\nabla \mathbf{v}(\mathbf{x}))\,d\mathbf{x}
$$
for every $\mathbf F\in\mathbf{M}^{m\times N}$, and every test map $\mathbf{v}$ in $D$. This concept does not depend on the domain $D$, and can, equivalently, be formulated in terms of periodic mappings (\cite{Sverak}) so that such a density $\psi$ is quasiconvex when
$$
\psi(\mathbf F)\le\int_Q\psi(\mathbf F+\nabla \mathbf{v}(\mathbf{y}))\,d\mathbf{y}
$$
for all $\mathbf F\in\mathbf{M}^{m\times N}$, and every periodic mapping $\mathbf{v}:Q\to\mathbb{R}^m$. Here $Q\subset\mathbb{R}^N$ is the unit cube.
Unfortunately, the issue is far from settled by simply saying this, since even Morrey realized that it is not at all easy to decide when a given density $\psi$ enjoys this property. For the scalar case, when either of the two dimensions $N$ or $m$ is unity, quasiconvexity reduces to usual convexity. But for genuine vector situations, it is not so. As a matter of fact, necessary and sufficient conditions for quasiconvexity in the vector case ($N, m>1$) were immediately sought, and important new convexity conditions were introduced:
\begin{itemize}
\item Rank-one convexity. A continuous integrand $\psi:\mathbf{M}^{m\times N}\to\mathbb{R}$ is said to be rank-one convex if
$$
\psi(t_1\mathbf F_1+t_2\mathbf F_2)\le t_1\psi(\mathbf F_1)+t_2\psi(\mathbf F_2),\quad t_1+t_2=1, t_1, t_2\ge0,
$$
whenever the difference $\mathbf F_1-\mathbf F_2$ is a rank-one matrix.
\item Polyconvexity. Such an integrand $\psi$ is polyconvex if it can be rewritten in the form $\psi(\mathbf F)=g(\mathbb{M}(\mathbf F))$ where $\mathbb{M}(\mathbf F)$ is the vector of all minors of $\mathbf F$, and $g$ is a convex (in the usual sense) function of all its arguments.
\end{itemize}
It was very soon recognized that quasiconvexity implies rank-one convexity (by using a special class of test fields), and that polyconvexity is a sufficient condition for quasiconvexity. The task suggested itself as trying to prove or disprove the equivalence of these various kinds of convexity. In the scalar case, all three coincide with usual convexity, so that we are facing a purely vector phenomenon. It turns out that these three notions of convexity are different, and counterexamples of various sorts have been found over the years. See \cite{AlibertDacorogna}, \cite{DacorognaDouchetGangboRappaz}, \cite{SerreA}, \cite{Terpstra}.
If we focus on the equivalence of rank-one convexity and quasiconvexity, Morrey conjectured that they are not equivalent (\cite{Morrey}), though later he simply stated it as an unsolved problem (\cite{MorreyB}). The issue remained undecided until the surprising counterexamble by V. Sverak (\cite{Sverak}) after some other additional and very interesting results (\cite{SverakB}, \cite{SverakC}, \cite{SverakE}). What is quite remarkable is that the original counterexample is only valid when $m\ge3$, and later attempts to extend it for $m=2$ failed (\cite{bandeiraornelas}, \cite{PedregalH}, \cite{PedregalSverakB}). Other counterexamples have not been found despite insistent efforts of the author that were definitely discarded in \cite{sebsze}. References \cite{faracolaszlo}, and \cite{KristensenB} are also relevant here.
The situation for two-component maps has, therefore, stayed unsolved, though some evidence in favor of the equivalence has been gathered throughout the years. See \cite{ChaudMuller}, \cite{Muller}, \cite{MullerB}, \cite{ParryA}. It is also interesting to point out that for quadratic densities, rank-one convexity and quasiconvexity are equivalent regardless of dimensions. This has been known for a long time (\cite{Ball}, \cite{MorreyB}), and it is not difficult to prove it by using Plancherel's formula. A different point of view is taken in \cite{BandeiraPedregal}. Another field where the resolution of this equivalence for two components maps would have an important impact is the theory of quasiconformal maps in the plane. There is a large number of references for this topic. See \cite{astalaiwaniecmartin} for a rather recent account. In particular, if the equivalence between rank-one convexity and quasiconvexity for two component maps turns out to be true, then the norm of the corresponding Beurling-Ahlfors transform equals $p^*-1$ (\cite{iwaniec}).
In this note, we prove that indeed for $m=2$ rank-one convexity is equivalent to quasiconvexity. The way in which we are going to think about the problem is by using the dual formulation of this equivalence through Jensen's inequality. What we will actually show is that, when $m=2$, every homogeneous gradient Young measure is a laminate. See Chapter 9 in \cite{PedregalI}.
What is essential or special about $m=2$? This is a question that one has to understand, as it seems quite relevant to a final resolution of the problem. The answer turns out to be quite enlightening: for two component maps, one can define an appropriate map going from one component to the other, and show the existence of a fixed point for such a map that translates into a rank-one decomposition for any such two-component gradient. For more than two components, more than one map would be involved, and fixed points for every couple of components may not match. This fixed point result (Kakutani's) is classical and nothing but a natural generalization of the usual Brower fixed point theorem.
More specifically, suppose we are given a periodic gradient $(\nabla u, \nabla v)$ with two components $(u, v):Q\to\mathbb{R}^2$ where $Q$ is the unit cube in $\mathbb{R}^N$, which is piecewise-affine with respect to some finite, arbitrary triangulation $\Gamma$. By a standard density argument about approximation by piece-wise affine mappings, it suffices, to reach our goal, to show that the corresponding discrete underlying gradient Young measure is a laminate.
This two-component map establishes a very clear way of moving from operations on the gradient of the first component $\nabla u$ to the same operations on the gradient $\nabla v$ of the second component by simply replacing $\mathbf{u}_i$ by the corresponding $\mathbf{v}_i$ in the same element of the triangulation $\Gamma$, if the finite support of $(\nabla u, \nabla v)$ is the set of pairs $\{(\mathbf{u}_i, \mathbf{v}_i)\}_i$.
The procedure is incorporated in the definition of a certain mapping. In addition, such map keeps track of decomposition directions as in the definition of laminates and $(H_n)$-conditions (\cite{DacorognaE}). See Appendix I for a reminder of main facts. Given a probability measure supported in the discrete set of vectors $\{\mathbf{u}_i\}_i$ of the first gradient, that is decomposed in the form of a $(H_n)$-condition along a set of successive directions, we focus on those decompositions, performed in the same way for the second gradient $\nabla v$, that preserve such family of decomposition directions coming from the first component. Intuitively, a fixed-point for such a map would respect:
\begin{enumerate}
\item decomposition directions for both components; and
\item equal volume fractions for the two components jointly, because the passage from one component to the other through the above identification $\mathbf{u}_i\mapsto\mathbf{v}_i$ respects such volume fractions for a fixed point.
\end{enumerate}
Therefore fixed points for such a map are identified with joint, i.e. simultaneously in the two components, $(H_n)$-conditions
whose decomposition directions are parallel, i.e. with laminates. Our claim, then, reduces to proving the existence of at least one fixed point for such a map.
Most of the technicalities are related to showing that a suitable framework can be set up so that the appropriate assumptions hold for the fixed-point result to be applied.
One crucial issue, though, is to understand what is special about a probability measure associated with a gradient $(\nabla u, \nabla v)$, since we know that not every such probability measure should allow the treatment through such fixed point argument. Indeed, this crucial ingredient is related to the fact that such mapping, together with its domain, is well-defined for the probability measure associated with such a gradient, and the assumptions for the existence of a fixed point are met,
while it would not be so for an arbitrary probability measure supported in $\mathbf{M}^{2\times N}$.
The proof of our main result is divided in three parts:
\begin{enumerate}
\item Section \ref{dominio}: we determine in a suitable way the domain of our underlying map $\mathbf T$.
\item Section \ref{tres}: the set-valued map $\mathbf T$ is defined, and Kakutani's fixed-point theorem is stated.
\item Section \ref{cuatro}: the required hypotheses for the fixed-point theorem to be applied are proved.
\end{enumerate}
Since the $(H_n)$-condition formalism will play a fundamental role, we have included a final appendix about it for the convenience of readers.
One of the main applied fields where vector variational problems are relevant is non-linear elasticity (\cite{Ball}). In particular, polyconvexity has played a major role in existence results. See also \cite{Ciarlet}. A main hypothesis to be assumed in this area is the rotationally invariance, as well as the behavior for large deformations. See \cite{DacorognaKoshigoe} for a discussion on all these notions of convexity under this invariance. Higher-order theories have also been explored, at least from an abstract point of view (\cite{Dal Maso et al.}, \cite{Meyers}). More general concepts of quasiconvexity have been introduced in \cite{Fonseca-Muller}. Recent interesting results about approximation by polynomials are worth mentioning \cite{Heinz}.
Explicit examples of rank-one convex functions can be found in various works: \cite{BandeiraPedregal}, \cite{DacorognaDouchetGangboRappaz}, \cite{SverakB}, among others. See also \cite{VanHove1}, \cite{VanHove2}. The recent book \cite{rindler} is to be considered.
\section{The domain}\label{dominio}
We will be working with piecewise affine, two-component maps with respect to a specific family of triangulations of the unit cube $Q$ of $\mathbb{R}^N$. This unit cube $Q$ can be decomposed in a finite number of simplexes and with a finite number $d(N)$ of normals to the flat faces of those simplexes. By making small copies of this decomposition, we can build a family of triangulations that provide uniform approximations of Lipschitz functions by piecewise affine maps. This is standard and well-known (see, for instance, \cite{ekelandtemam}).
For $N=2$, three normals suffice, while for dimension $N=3$, seven are necessary, and so on. Because of this approximation argument, we can focus on piecewise affine, two-component maps with respect to such families of triangulations.
Let $\Gamma$ be an arbitrary, regular, finite triangulation of $Q$, as indicated in the previous paragraph, with elements $\{T_i\}_i$, $\lambda_i=|T_i|>0$, nodes $\{\mathbf P_p\}_p$, and planar interfaces $\gamma_{ij}$ if $T_i$ and $T_j$ share a flat boundary. For $N=2$, the triangulation $\Gamma$ can be clearly chosen so that $|T_i|$ is the same positive number for all the elements of $\Gamma$.
Let $(u, v):Q\to\mathbb{R}^2$ be a $Q$-periodic map, piece-wise affine with respect to $\Gamma$ so that
\begin{equation}\label{gradiente}
(\nabla u(\mathbf{x}), \nabla v(\mathbf{x}))=\sum_i\chi_{T_i}(\mathbf{x})(\mathbf{u}_i, \mathbf{v}_i),\quad\mathbf{x}\in Q,
\end{equation}
and let
\begin{equation}\label{margi}
\nu=\sum_i\lambda_i\delta_{(\mathbf{u}_i, \mathbf{v}_i)},\quad \nu_u=\sum_i\lambda_i\delta_{\mathbf{u}_i},\quad
\nu_v=\sum_i\lambda_i\delta_{\mathbf{v}_i},
\end{equation}
be the underlying probability measure with vanishing first moment, and its two marginals, respectively. Put
\begin{equation}\label{subconjuntos}
\mathbb{U}=\{\mathbf{u}_i\}\subset\mathbb{R}^N,\quad \mathbb{V}=\{\mathbf{v}_i\}\subset\mathbb{R}^N.
\end{equation}
There is definitely something special about $\nu$ in (\ref{margi}).
Indeed, we well know that not every discrete probability measure supported in $\mathbf{M}^{2\times N}$ may come from a gradient as in \eqref{gradiente}. Because both components $u$ and $v$ are piecewise affine with respect to the same triangulation $\Gamma$, the set of $d(N)$ normals across planar interfaces for both components is the same. This is a fundamental fact
that will be used in a crucial way below. In addition, we will also make use of nodal values of vector $(u, v)$ on nodes $\{\mathbf P_p\}$, and again this would not be possible if the probability measure supported in $\mathbf{M}^{2\times N}$ is not coming from a gradient.
For each value of $m\in\mathbb{N}$, select two concrete $2^m$-tuples
\begin{gather}
\mathbb{X}_m=(\mathbf{x}_1, \mathbf{x}_2, \dots, \mathbf{x}_{2^m})\in\mathbb{U}^{2^m}\subset(\mathbb{R}^N)^{2^m},\nonumber\\
\mathbb{Y}_m=(\mathbf{y}_1, \mathbf{y}_2, \dots, \mathbf{y}_{2^m})\in\mathbb{V}^{2^m}\subset(\mathbb{R}^N)^{2^m},\nonumber
\end{gather}
complying with the following fundamental properties:
\begin{enumerate}
\item Compatibility. For every $j$, $1\le j\le 2^m$, there is always $i$ with
$$
(\mathbf{x}_j, \mathbf{y}_j)=(\mathbf{u}_i, \mathbf{v}_i)
$$
corresponding to the same element $T_i$ of $\Gamma$.
\item Adjacency. For every $k$, $1\le k\le 2^{m-1}$, vectors
$\mathbf{x}_{2k}, \mathbf{x}_{2k-1}$ in $\mathbb{X}_m$, on the one hand, and
$\mathbf{y}_{2k}, \mathbf{y}_{2k-1}$ in $\mathbb{Y}_m$, on the other,
are adjacent with respect to the given triangulation $\Gamma$ corresponding to interface $\gamma_{ij}$ if
\begin{gather}
\mathbf{x}_{2k}=\mathbf{u}_i,\quad \mathbf{x}_{2k-1}=\mathbf{u}_j,\nonumber\\
\mathbf{y}_{2k}=\mathbf{v}_i,\quad \mathbf{y}_{2k-1}=\mathbf{v}_j,\nonumber
\end{gather}
in such a way that
$$
\mathbf{x}_{2k}-\mathbf{x}_{2k-1}\parallel \mathbf{y}_{2k}-\mathbf{y}_{2k-1}
$$
and this direction is one of the $d(N)$ normals indicated above.
\item Nodal organization. Both $\mathbb{X}_m$ and $\mathbb{Y}_m$ can be organized according to a partition in pairwise-disjoint $2^{m-n}$-tuples $\mathbb{X}_{m, p}$ (and $\mathbb{Y}_{m, p}$), one for each node $\mathbf P_p$ of $\Gamma$,
$$
\mathbb{X}_{m, p}=(\mathbf{x}_{j_p+1}, \mathbf{x}_{j_p+2}, \dots, \mathbf{x}_{j_p+2^{m-n}})
$$
if there are $2^n$ nodes in $\Gamma$, in such a way that each $\mathbb{X}_{m, p}$ only contains values of the gradient of $u$ (or of $v$) corresponding to elements $T_{i, p}$ of $\Gamma$ having $\mathbf P_p$ as one of its nodes, and all planar interfaces $\gamma_{ij}$ having $\mathbf P_p$ as one of its nodes are represented in $\mathbb{X}_{m, p}$ through the adjacency condition;
notice that $\{\mathbb{X}_{m, p}\}_p$ will not induce, in general, a partition of the set of elements $\{T_i\}$ of $\Gamma$, i.e. of the set $\mathbb{U}$, or of the full set of flat interfaces $\{\gamma_{ij}\}$, because each triangle $T_i$ of $\Gamma$, and each planar interface $\gamma_{ij}$, has various vertices $\mathbf P_p$, and each one of these is shared by various simplexes of $\Gamma$, and various different interfaces. In other words, the sets of vectors in each $\mathbb{X}_{m, p}$ will not be disjoint, but the sets of indices $\{j_p+1, j_p+2, \dots, j_p+2^{m-n}\}$, varying with $p$, are. There are always sequences of triangulations of increasing fineness with a number of nodes which is a power of $2$. Insisting in that sets $\mathbb{X}_{m, p}$ have the same number of elements, and a power of $2$, forces us to admit that interfaces having a node in $\mathbf P_p$ cannot be equally, i.e. occurring the same number of times, represented in $\mathbb{X}_{m, p}$. It is as if some interfaces were counted more than once.
\item Representation. There is one specific collection of weights $\{\overline t_j\}\in[0, 1]^{2^m}$, such that
\begin{equation}\label{nuu}
\nu_u=\sum_j \overline t_j\delta_{\mathbf{x}_j};
\end{equation}
as a consequence,
\begin{equation}\label{nuv}
\nu_v=\sum_j \overline t_j\delta_{\mathbf{y}_j}.
\end{equation}
\end{enumerate}
It is clear that these tuples can be chosen, in a non-unique way, at least for each value of $m$ large. In addition, by allowing $m$ to be larger if necessary, we can assume, without loss of generality and through a standard perturbation argument, that
$$
\overline t_{2i}+\overline t_{2i-1}=2^{1-m},\quad 1\le i\le 2^{m-1},
$$
by choosing a finer representation of both $\nu_u$ and $\nu_v$ in \eqref{nuu} and \eqref{nuv}, respectively. This property is not necessary, but it will make things a bit simpler.
Once $\mathbb{X}_m$ and $\mathbb{Y}_m$ have been selected as just indicated, define the set
\begin{gather}
\Theta_m=\{\mathbf{t}=(t_1, t_2, \dots, t_{2^m})\in\mathbb{R}^{2^m}: t_j\ge0, t_{2i}+t_{2i-1}=2^{1-m}, 1\le i\le 2^{m-1}, \nonumber\\
\sum_it_i\delta_{\mathbf{x}_i}=\nu_u\}.\nonumber
\end{gather}
Note that, because of the way in which $\mathbb{X}_m$ and $\mathbb{Y}_m$ have been chosen, we also have
\begin{gather}
\Theta_m=\{\mathbf{t}=(t_1, t_2, \dots, t_{2^m})\in\mathbb{R}^{2^m}: t_j\ge0, t_{2i}+t_{2i-1}=2^{1-m}, 1\le i\le 2^{m-1},\nonumber\\
\sum_it_i\delta_{\mathbf{y}_i}=\nu_v\}.\nonumber
\end{gather}
It is interesting to realize that each element $\mathbf{t}\in\Theta_m$ represents a certain partition of the unit cube $Q$, organized in a suitable way bearing in mind the $(H_n)$-formalism for laminates. The full set $\Theta_m$ is a collection of such partitions with a particular structure that is very convenient for our purposes.
\begin{proposition}
For $m$ sufficiently large, the set $\Theta_m$ so selected, is non-empty, compact, and convex.
\end{proposition}
\begin{proof}
The compactness and the convexity of $\Theta_m$ are straightforward.
It is non-empty because, by construction,
$$
\overline\mathbf{t}=(\overline t_1, \dots, \overline t_{2^m})
$$
coming from the representation condition above
belongs to $\Theta_m$. For this, we may have to take $m$ large enough.
\end{proof}
\section{The map and its role}\label{tres}
Each element $\mathbf{t}\in\Theta_m$ gives rise to a whole structure according to the $(H_n)$-formalism that is defined recursively as follows (check the final Appendix):
\begin{enumerate}
\item Initialization. Put
$$
t_i^{(m)}=t_i,\quad \mathbf{x}_i^{(m)}=\mathbf{x}_i,\quad \mathbf{y}_i^{(m)}=\mathbf{y}_i,
$$
for $1\le i\le 2^m$.
\item Recursion.
\begin{enumerate}
\item Relative weights. For
$$
k=m-1, m-2, \dots, 1,\quad 1\le i\le 2^k,
$$
put
$$
t_i^{(k)}=t_{2i-1}^{(k+1)}+t_{2i}^{(k+1)},
$$
and
$$
\lambda_i^{(k)}=\begin{cases}
\frac{t_{2i}^{(k+1)}}{t_i^{(k)}},& t_i^{(k)}>0\\
1/2,& t_i^{(k)}=0\end{cases}.
$$
In this way
\begin{equation}\label{relabs}
t_{2i}^{(k+1)}=t_i^{(k)}\lambda_i^{(k)},\quad t_{2i-1}^{(k+1)}=t_i^{(k)}(1-\lambda_i^{(k)}),
\end{equation}
and $t_1^{(0)}=1$. Note however that, because the way in which the set $\Theta_m$ has been chosen,
$$
t_i^{(m-1)}=2^{1-m}, \quad 1\le i\le 2^{m-1},
$$
and then
\begin{equation}\label{constante}
\lambda_i^{(k)}=1/2,\quad 1\le k\le m-2, 1\le i\le 2^k.
\end{equation}
\item Decomposition direction. For
$$
k=m-1, m-2, \dots, 1,\quad 1\le i\le 2^k,
$$
define
\begin{equation}\label{dirdes}
\mathbf{X}_i^{(k)}=\mathbf{x}_{2i-1}^{(k+1)}-\mathbf{x}_{2i}^{(k+1)},\quad
\mathbf{Y}_i^{(k)}=\mathbf{y}_{2i-1}^{(k+1)}-\mathbf{y}_{2i}^{(k+1)}.
\end{equation}
\item New level. For
$$
k=m-1, m-2, \dots, 1,\quad 1\le i\le 2^k,
$$
set
\begin{gather}
\mathbf{x}_i^{(k)}=(1-\lambda_i^{(k)})\mathbf{x}_{2i-1}^{(k+1)}+\lambda_i^{(k)}\mathbf{x}_{2i}^{(k+1)},\label{equis}\\
\mathbf{y}_i^{(k)}=(1-\lambda_i^{(k)})\mathbf{y}_{2i-1}^{(k+1)}+\lambda_i^{(k)}\mathbf{y}_{2i}^{(k+1)}.\label{ygriega}
\end{gather}
\end{enumerate}
\end{enumerate}
Decomposition directions $\mathbf{X}_i^{(k)}$ and $\mathbf{Y}_i^{(k)}$, vectors $\mathbf{x}_i^{(k)}$ and $\mathbf{y}_i^{(k)}$,
and relative weights $\lambda_i^{(m-1)}$ as well, depend upon $\mathbf{t}$. To make this dependence explicit we will simply put
\begin{equation}\label{elementos}
\mathbf{X}_i^{(k)}(\mathbf{t}),\quad \mathbf{Y}_i^{(k)}(\mathbf{t}), \quad\mathbf{x}_i^{(k)}(\mathbf{t}),\quad \mathbf{y}_i^{(k)}(\mathbf{t}), \quad\lambda_i^{(m-1)}(\mathbf{t}).
\end{equation}
Recall that $\lambda_i^{(k)}=1/2$ for all
$$
0\le k\le m-2,\quad 1\le i\le 2^k,
$$
according to \eqref{constante}.
A joint, simultaneous rank-one decomposition of $\nu$ in \eqref{margi}
demands that decomposition directions $\mathbf{X}_i^{(k)}$ and $\mathbf{Y}_i^{(k)}$ are proportional to each other for all
$$
0\le k\le m-1,\quad 1\le i\le 2^k.
$$
This is guaranteed for $k=m-1$ because of the way sets $\mathbb{X}_m$ and $\mathbb{Y}_m$ have been selected (recall the adjacency condition in Section \ref{dominio}).
This fundamental property sought motivates the definition of our set-valued mapping
$$
\mathbf T:\Theta_m\mapsto 2^{\Theta_m},
$$
by putting
$$
\mathbf T(\mathbf{t})=\{\mathbf{s}\in\Theta_m: \mathbf{Y}_i^{(k)}(\mathbf{s})\parallel\mathbf{X}_i^{(k)}(\mathbf{t})\hbox{ for all }0\le k\le m-2, 1\le i\le 2^k\}.
$$
Note again how $\mathbf{Y}_i^{(m-1)}(\mathbf{s})$ is always parallel to $\mathbf{X}_i^{(m-1)}(\mathbf{s})$ precisely because decomposition directions at the level $k=m-1$ correspond to interfaces between two adjacent elements of the triangulation $\Gamma$.
The whole point or our concern is the following.
\begin{proposition}\label{facil}
The gradient measure $\nu$ in \eqref{margi} is a laminate if, for some $m$, there is a fixed point for $\mathbf T$, i.e. there is $\mathbf{t}\in\Theta_m$ such that $\mathbf{t}\in\mathbf T(\mathbf{t})$.
\end{proposition}
\begin{proof}
The proof is immediate given the way in which both the set $\Theta_m$ and the map $\mathbf T$ have been defined through sets $\mathbb{X}_m$ and $\mathbb{Y}_m$.
\end{proof}
We will be using the following classic result to show the existence of a fixed-point of $\mathbf T$ for some large $m$.
\begin{theorem}\label{kakutani} (Kakutani's fixed point theorem)
Let $\mathcal A\subset\mathbb{R}^d$ be a non-empty, compact, convex set, and let $\mathbf F:\mathcal A\mapsto \mathcal A$ be an upper semicontinuous, set-valued map with non-empty, convex, compact values. Then $\mathbf F$ has a fixed point; that is, there is $\hat x\in \mathcal A$ with $\hat x\in\mathbf F(\hat x)$.
\end{theorem}
This is a classical theorem on fixed-points for set-valued maps, which is but a generalization of the classic Brower's fixed point theorem. It is well-known, and can be found in many places, for instance in \cite{smirnov}.
The fundamental properties that the application of this result to our framework requires are the non-emptiness, compactness and convexity of $\mathbf T(\mathbf{t})$ for each $\mathbf{t}\in\Theta_m$, in addition to the upper semicontinuity.
\section{Main properties of the map $\mathbf T$}\label{cuatro}
We start with the upper semicontinuity required by Theorem \ref{kakutani}.
This property is, as a matter of fact, elementary, since if
$$
\mathbf{s}_j\in\mathbf T(\mathbf{t}_j),\quad \mathbf{s}_j\to \mathbf{s}, \mathbf{t}_j\to \mathbf{t},
$$
then, we must necessarily have $\mathbf{s}\in\mathbf T(\mathbf{t})$. This is straightforward because the dependence of elements in \eqref{elementos} on $\mathbf{t}$ is continuous.
On the other hand, the compactness of each subset $\mathbf T(\mathbf{t})$ is also clear since all these images are closed subsets of the compact set $[0, 1]^{2^m}$.
\subsection{Convexity of images}
Ensuring this convexity property is responsible for the precise definition of the set $\Theta_m$ we have adopted. It is pretty clear after the following statement.
\begin{proposition}
\begin{enumerate}
\item For
$$
k=m-1, m-2, \dots, 1,\quad 1\le i\le 2^k,
$$
vectors
$$
\mathbf{x}_i^{(k)}(\mathbf{t}),\quad \mathbf{y}_i^{(k)}(\mathbf{t})
$$
in \eqref{equis} and \eqref{ygriega}, respectively, depend linearly on $\mathbf{t}$ for $\mathbf{t}\in\Theta_m$, and consequently, so do decomposition directions
$$
\mathbf{X}_i^{(k)}(\mathbf{t}),\quad \mathbf{Y}_i^{(k)}(\mathbf{t})
$$
in \eqref{dirdes}.
\item For each $\mathbf{t}\in\Theta_m$, the set $\mathbf T(\mathbf{t})$ is convex.
\end{enumerate}
\end{proposition}
\begin{proof}
For the first part, note that
if we resort to \eqref{equis} and \eqref{ygriega}, we realize that for $k=m-1$, because
$$
t_i^{(m-1)}=2^{1-m},\quad 1\le i\le 2^{m-1},
$$
and vectors
$$
\mathbf{x}_j^{(m)},\quad \mathbf{y}_j^{(m)}
$$
are given and fixed (taken, respectively, from the sets $\mathbb{X}_m$ and $\mathbb{Y}_m$, for all $1\le j\le 2^m$ once these have been chosen), those formulas are linear in the components of $\mathbf{t}$ because weights $\lambda_i^{(k-1)}$ are. On the other hand, for
$$
k=m-2, m-3, \dots, 2, 1,
$$
those same formulas indicate that
$$
\mathbf{x}_i^{(k)},\quad \mathbf{y}_i^{(k)}
$$
depend linearly on
$$
\mathbf{x}_j^{(k+1)},\quad \mathbf{y}_j^{(k+1)}
$$
precisely because those relative weights $\lambda_i^{(k)}$, for $\mathbf{t}\in\Theta_m$, are exactly $1/2$. By the recursive nature of $(H_n)$-conditions, we have the claimed linear dependence.
The first statement immediately yields the second. If
$$
\mathbf{s}_i\in\mathbf T(\mathbf{t}),\quad i=0, 1,
$$
and $r\in(0, 1)$, then, for
$$
\mathbf{s}=r\mathbf{s}_1+(1-r)\mathbf{s}_0,
$$
we will have
$$
\mathbf{Y}_i^{(k)}(\mathbf{s})=r\mathbf{Y}_i^{(k)}(\mathbf{s}_1)+(1-r)\mathbf{Y}_i^{(k)}(\mathbf{s}_0)
$$
precisely by the previous fact. Hence, if
$$
\mathbf{Y}_i^{(k)}(\mathbf{s}_i)\parallel \mathbf{X}_i^{(k)}(\mathbf{t}),\quad i=0, 1,
$$
so will $\mathbf{Y}_i^{(k)}(\mathbf{s})$ be. This means that $\mathbf{s}\in\mathbf T(\mathbf{t})$.
\end{proof}
\subsection{Non-emptiness of images}This is the most delicate issue of our proof.
We regard the first component $u$ of our two-component map
$$
(u, v):Q\to\mathbb{R}^2
$$
as fixed but arbitrary, and allow the second component $v$ to vary. Recall that $\{\mathbf P_p\}_p$ is an enumeration of the nodes of the triangulation $\Gamma$. The $Q$-periodic function $v$, piecewise-affine with respect to $\Gamma$, is uniquely determined by the set of its nodal values $\{v(\mathbf P_p)\}$, and hence it can be identified in a natural way with $\mathbb{R}^q$ if $q$ is the finite number, depending on dimension $N$ and the fineness of $\Gamma$, of nodes of $\Gamma$ (in fact $q$ was chosen as $2^n$ in Section \ref{dominio}).
By a natural abuse of language, we will say that $v\in\mathbb{R}^q$. For an arbitrary $\mathbf{t}\in\Theta_m$, determined through the first-component $u$, regarded as given and fixed but otherwise arbitrary, consider the following subset of $\mathbb{R}^q$
\begin{equation}\label{subconjunto}
\Upsilon(\mathbf{t})=\{v\in\mathbb{R}^q: \mathbf T(\mathbf{t})\neq\emptyset\}.
\end{equation}
The non-emptiness of every image $\mathbf T(\mathbf{t})$ amounts to showing the following fact.
\begin{lemma}
For every $\mathbf{t}\in\Theta_m$, the set $\Upsilon(\mathbf{t})$ in \eqref{subconjunto} is always the full $\mathbb{R}^q$.
\end{lemma}
\begin{proof}
The proof proceeds after a typical connectedness argument. We will show that $\Upsilon(\mathbf{t})$ is non-empty, closed and open, and so it will be the full set $\mathbb{R}^q$.
The non-emptiness of $\Upsilon(\mathbf{t})$ is clear because $u$ itself, through its nodal values $u(\mathbf P_p)$, belongs to $\Upsilon(\mathbf{t})$. Note that when we take $v=u$, so that our two-component map becomes $(u, u)$, then $\mathbf{t}\in\mathbf T(\mathbf{t})$, and so $\Upsilon(\mathbf{t})$ is non-empty.
It is elementary to realize that $\Upsilon(\mathbf{t})$ is closed, given that $\Theta_m$ is compact, and $\mathbf T(\mathbf{t})$ is closed. There is no difficulty here.
The crucial step is to show the openness of $\Upsilon(\mathbf{t})$. To this end, if we put
$$
\mathbf{V}_i^{(k)}=\mathbf{X}_i^{(k)}(u, \mathbf{t}),\quad 0\le k\le m-1, 1\le i\le 2^k,
$$
a fixed collection of decomposition directions, some of which could be null, determined through the first-component $u$ (we have explicitly indicated so in the above notation),
Lemma \ref{crucial} below directly shows that $\Upsilon(\mathbf{t})$ is also open.
If this lemma is correct, our statement is proved, and so is the non-emptiness of every image $\mathbf T(\mathbf{t})$.
\end{proof}
As just indicated, the fundamental step necessary to show the non-emptiness of images $\mathbf T(\mathbf{t})$ for each $\mathbf{t}\in\Theta_m$ is the following.
Assume decomposition vectors
$$
\{\mathbf{V}_i^{(k)}\}_{0\le k\le m-1, 1\le i\le 2^k}\subset\mathbb{R}^N
$$
are given in such a way that there is a $Q$-periodic function $v:Q\to\mathbb{R}$, piecewise affine with respect to a triangulation $\Gamma$ of $Q$ for which there is $\mathbf{s}\in\Theta_m$ with
$$
\mathbf{Y}_i^{(k)}(v, \mathbf{s})\parallel \mathbf{V}_i^{(k)}
$$
for all
$$
0\le k\le m-1,\quad 1\le i\le 2^k.
$$
We are using here, as already indicated, the more complex notation $\mathbf{Y}_i^{(k)}(v, \mathbf{s})$ for decomposition direction to stress that these depend on the underlying function $v$, which is changing.
In particular,
for the last level $k=m-1$,
$$
\mathbf{V}_i^{(m-1)},\quad 1\le i\le 2^{m-1},
$$
is one of the finite number $d(N)$ of normals used in the triangulation $\Gamma$. Some of those decomposition directions $\mathbf{V}_i^{(k)}$ could vanish, but this is even more advantageous as then we are free to select a parallel direction without any restriction.
\begin{lemma}\label{crucial}
There is a neighborhood $\mathcal{V}$ of $v$ in $\mathbb{R}^q$, through the above identification, such that for every $\overline v\in\mathcal{V}$ there is $\overline\mathbf{s}\in\Theta_m$ with
$$
\mathbf{Y}_i^{(k)}(\overline v, \overline\mathbf{s})\parallel \mathbf{V}_i^{(k)}
$$
for all
$$
0\le k\le m-1,\quad 1\le i\le 2^k.
$$
\end{lemma}
\begin{proof}
If $\{\mathbf P_p\}_p$ is an enumeration of the nodes of the triangulation $\Gamma$, it suffices to focus on perturbations of the function $v$ produced by changing the nodal value $v_l$ of $v$ at a certain fixed, but otherwise arbitrary, node $\mathbf P_l$ while retaining the value $v_p$ of $v$ at the other nodes $\mathbf P_p$, $p\neq l$. Let $I(l)$ indicate the set of indices $i$ of those values $\mathbf{v}_i\in\mathbb{V}$ (recall \eqref{subconjuntos}) of the gradient $\nabla v$ in elements of $\Gamma$ affected by the value $v_l$ of $v$ at the node $\mathbf P_l$, i.e. $I(l)$ is the set of those indices of elements of $\Gamma$ one of whose nodes is $\mathbf P_l$. Notice how this is closely related to the nodal organization property of Section \ref{dominio}.
Our claim is then that for some small positive $\epsilon$, the piecewise affine function $\overline v$ that shares the nodal values $\overline v_p=v_p$ with $v$ for $p\neq l$, but
\begin{equation}\label{margen}
|\overline v_l-v_l|<\epsilon,
\end{equation}
is such that its gradient $\nabla \overline v$ is the result of a $(H_n)$-condition with the same decomposition directions $\mathbf{V}_i^{(k)}$, preserving relative weights $1/2$ at all levels except the last one (as required in the definition of $\Theta_m$). The value of $\overline\mathbf{s}$ is then the result of final weights coming from the top-to-bottom description of $(H_n)$-conditions (see the final Appendix).
Recall that
$$
\nabla v(\mathbf{x})=\sum_i\chi_{T_i}(\mathbf{x})\mathbf{v}_i,\, \mathbf{x}\in Q,\quad \nu_v=\sum_i\lambda_i\delta_{\mathbf{v}_i}.
$$
Our hypothesis for $v$ implies that
$$
\nu_v=\sum_i s_i\delta_{\mathbf{y}_i^{(m)}}
$$
and each vector $\mathbf{y}_i^{(m)}$ (which is one of the $\{\mathbf{v}_i\}$ taken on by $\nabla v$ over the elements of the triangulation $\Gamma$) can be written in terms of the decomposition directions $\mathbf{V}_i^{(k)}$, and
of the set of scalars
\begin{equation}\label{variables}
\mathbf{S}=\{S_i^{(k)}\}_{k=0, 1, \dots, m-2, i=1, \dots, 2^k}\cup\{S_{i, +}^{(m-1)}\}_{i=1, \dots, 2^{m-1}}\cup\{S_{i, -}^{(m-1)}\}_{i=1, \dots, 2^{m-1}}
\end{equation}
generated along the process through $(H_n)$-conditions of the discrete probability measure associated with $\nabla v$ by using decomposition directions $\mathbf{V}_i^{(k)}$. This exactly means that
\begin{gather}
\mathbf{y}_{2i}^{(k+1)}=\mathbf{y}_i^{(k)}+S_i^{(k)}\mathbf{V}_i^{(k)},\quad
\mathbf{y}_{2i-1}^{(k+1)}=\mathbf{y}_i^{(k)}-S_i^{(k)}\mathbf{V}_i^{(k)},\quad 0\le k\le m-2, 1\le i\le 2^k,
\label{ecuanime}\\
\mathbf{y}_{2i}^{(m)}=\mathbf{y}_i^{(m-1)}+S_{i, +}^{(m-1)}\mathbf{V}_i^{(m-1)},\quad
\mathbf{y}_{2i-1}^{(m)}=\mathbf{y}_i^{(m-1)}-S_{i, -}^{(m-1)}\mathbf{V}_i^{(m-1)}.\label{ecuanimeu}
\end{gather}
Note how \eqref{ecuanime} means that relative volume fractions up to level $m-2$ are exactly $1/2$, so that the corresponding vector $\mathbf{s}$ of final weights belongs to $\Theta_m$. This is the reason why we do not have to consider two families of numbers
$$
S_{i, +}^{(k)},\quad S_{i, -}^{(k)}
$$
for $k=m-2, \dots, 2, 1, 0$.
Nodal values $v_p$ of $v$ at nodes $\mathbf P_p$ can also be understood as functions of $\mathbf{S}$ in \eqref{variables}.
The important point is to realize that this dependence of $v_p(\mathbf{S})$ on each individual independent variable $S_i^{(k)}$ in \eqref{variables} is affine (eventually constant) when all other components are kept fixed. This is so because of the recursive linear way in which $(H_n)$-conditions are built (check the Appendix at the end of the paper) if decomposition directions $\mathbf{V}_i^{(k)}$, regarded as constant vectors, are to be respected: vector values $\mathbf{y}_i^{(m)}$ of $\nabla v$ depend linearly on each individual $S_i^{(k)}$, as indicated above, and the dependence of the values of $\mathbf{v}_j$ on nodal values $v_p$ is also linear. We can conclude that functions $v_p(\mathbf{S})$ are multilinear.
If we now fix our attention on a specific, but arbitrary nodal value $v_l=v(\mathbf P_l)$, and assume that \eqref{margen} does not hold, then it is elementary to realize that the gradient of $v_l(\mathbf{S})$ with respect to $\mathbf{S}$ at the precise value of $\mathbf{S}$ furnishing the nodal value $\overline v_l$ must vanish: this value of $\mathbf{S}$ with $v(\mathbf{S})=\overline v_l$ ought to be a local extreme for $v_l(\mathbf{S})$, either a local maximum or a local minimum. But because of this multilinear dependence on $\mathbf{S}$, it is a fact that the gradient of $v_l(\mathbf{S})$ with respect to $\mathbf{S}$ can never vanish even if we restrict these values $\mathbf{S}$ by demanding that
$v_p(\mathbf{S})$, the nodal values of $v$ at $\mathbf P_p$ all of which are also multilinear functions of $\mathbf{S}$, be given, fixed numbers for all $p\neq l$. As a matter of fact, the nodal organization property in Section \ref{dominio} has been enforced so that there is a specific subset $\mathbf{S}_l$ of independent variables from \eqref{variables} which only affect the nodal value $v_l$, but all other nodal values $v_p$ for $p\neq l$ are independent of those precise variables in $\mathbf{S}_l$. In this way, if we keep constant the variables of $\mathbf{S}\setminus\mathbf{S}_l$, all nodal values $v_p$, $p\neq l$, will stay constant, but the value $v_l$ will depend, in a multilinear fashion, on the variables of $\mathbf{S}_l$. Since a non-constant, multilinear function cannot have extreme points,
our claim \eqref{margen} is then correct, and the arbitrariness of $l$ implies our statement.
\end{proof}
Theorem \ref{kakutani} can then be applied, and, through Proposition \ref{facil}, we conclude that $(\nabla u, \nabla v)$ is indeed a laminate. The arbitrariness of the triangulation $\Gamma$ implies that every two-component gradient is a laminate, and hence rank-one convexity implies quasiconvexity in the case $2\times N$.
\section{Appendix}
We include here, for the convenience of our readers, a short discussion about the notion of $(H_n)$-condition with respect to a given cone $\Lambda$ of admissible directions, as introduced in \cite{DacorognaE}.
We start with a given, discrete probability measure supported in $\mathbf{M}^{m\times N}$
$$
\nu=\sum_i\lambda_i\delta_{\mathbf{u}_i},\quad \sum_i\lambda_i\mathbf{u}_i=\mathbf 0,\quad \lambda_i>0, \sum_i\lambda_i=1, \mathbf{u}_i\in\mathbf{M}^{m\times N},
$$
and put
$$
\nu^{(1)}=\delta_0,\quad\operatorname{supp}(\nu^{(1)})\subset\mathbf{M}^{m\times N}.
$$
Given
\begin{equation}\label{pasok}
\nu^{(k)}=\sum_i\lambda_i^{(k)}\delta_{\mathbf{u}_i^{(k)}},\quad\operatorname{supp}(\nu^{(k)})\subset\mathbf{M}^{m\times N}
\end{equation}
we recursively split the delta measure supported at each $\mathbf{u}_i^{(k)}$ along a certain direction $\mathbf{U}_i^{(k)}$ (which eventually could be the null vector) taken from a selected cone of feasible directions $\Lambda\subset\mathbf{M}^{m\times N}$ , and with relative weights $t_i^{(k)}$ and $1-t_i^{(k)}$, so that
\begin{equation
\delta_{\mathbf{u}_i^{(k)}}\mapsto t_i^{(k)}\delta_{\mathbf{u}_i^{(k)}+(1-t_i^{(k)})\mathbf{U}_i^{(k)}}+(1-t_i^{(k)})\delta_{\mathbf{u}_i^{(k)}-t_i^{(k)}\mathbf{U}_i^{(k)}}.
\end{equation}
Note that weights $t_i^{(k)}$ are given by the various mass points involved, provided decomposition vector $\mathbf{U}_i^{(k)}$ is not zero. Indeed
\begin{gather}
t_i^{(k)}=\frac{|(\mathbf{u}_i^{(k)}-t_i^{(k)}\mathbf{U}_i^{(k)})-\mathbf{u}_i^{(k)}|}{|(\mathbf{u}_i^{(k)}+(1-t_i^{(k)})\mathbf{U}_i^{(k)})-(\mathbf{u}_i^{(k)}-t_i^{(k)}\mathbf{U}_i^{(k)})|},\nonumber\\
1-t_i^{(k)}=\frac{|(\mathbf{u}_i^{(k)}+(1-t_i^{(k)})\mathbf{U}_i^{(k)})-\mathbf{u}_i^{(k)}|}{|(\mathbf{u}_i^{(k)}+(1-t_i^{(k)})\mathbf{U}_i^{(k)})-(\mathbf{u}_i^{(k)}-t_i^{(k)}\mathbf{U}_i^{(k)})|}.\nonumber
\end{gather}
The new probability measure is obtained by replacing each such decomposition back into $\nu^{(k)}$ in (\ref{pasok})
$$
\nu^{(k+1)}=\sum_i\lambda_i^{(k)}\left(t_i^{(k)}\delta_{\mathbf{u}_i^{(k)}+(1-t_i^{(k)})\mathbf{U}_i^{(k)}}+(1-t_i^{(k)})\delta_{\mathbf{u}_i^{(k)}-t_i^{(k)}\mathbf{U}_i^{(k)}}\right),
$$
and reorganizing such representation.
One same vector in the support of $\nu^{(k+1)}$ may come from several decompositions in the previous step. Note that if
$\mathbf{U}_i^{(k)}=\mathbf 0$ for all $i$ and some fixed $k$, then $\nu^{(k+1)}=\nu^{(k)}$, and if only some $\mathbf{U}_i^{(k)}=\mathbf 0$ then the matrix $\mathbf{u}_i^{(k)}$ is passed intact onto the next level.
The final measure $\nu^{(m)}$, after a finite number $m$ of steps, should be the one we started with $\nu=\nu^{(m)}$. This is the top-to-bottom procedure.
It is important to stress that the fundamental cone for vector variational problems is the rank-one cone
$$
\Lambda=\{\mathbf{U}\in\mathbf{M}^{m\times N}: \operatorname{rank}(\mathbf{U})\le1\},
$$
and that such a cone is the full set of directions if either dimension $m$ or $N$ is unity.
It is enlightening to describe such $(H_n)$-conditions exclusively in terms of vectors and weights. The most direct way of doing this is by keeping record of weights and mass points for the successive probability measures (recall that weights are given and determined by such mass points as indicated above unless denominators vanish), namely
\begin{equation
\{\{(\lambda_i^{(k)}, \mathbf{u}_i^{(k)})\}_{1\le i\le 2^k}\}_{0\le k\le m}
\end{equation}
where
\begin{gather}
\lambda_i^{(k)}=\lambda_{2i-1}^{(k+1)}+\lambda_{2i}^{(k+1)}, \nonumber\\
\mathbf{u}_i^{(k)}=\frac{\lambda_{2i-1}^{(k+1)}}{\lambda_{2i-1}^{(k+1)}+\lambda_{2i}^{(k+1)}}\mathbf{u}_{2i-1}^{(k+1)}+
\frac{\lambda_{2i}^{(k+1)}}{\lambda_{2i-1}^{(k+1)}+\lambda_{2i}^{(k+1)}}\mathbf{u}_{2i}^{(k+1)},\nonumber
\end{gather}
for all $0\le k\le m-1$, $1\le i\le 2^k$.
Relative weights are given by
\begin{gather}
t_i^{(k)}=\frac{\lambda_{2i-1}^{(k+1)}}{\lambda_{2i-1}^{(k+1)}+\lambda_{2i}^{(k+1)}}=\frac{|\mathbf{u}_i^{(k)}-\mathbf{u}_{2i-1}^{(k+1)}|}{|\mathbf{u}_{2i}^{(k+1)}-\mathbf{u}_{2i-1}^{(k+1)}|},\nonumber\\
1-t_i^{(k)}=\frac{\lambda_{2i}^{(k+1)}}{\lambda_{2i-1}^{(k+1)}+\lambda_{2i}^{(k+1)}}=\frac{|\mathbf{u}_i^{(k)}-\mathbf{u}_{2i}^{(k+1)}|}{|\mathbf{u}_{2i}^{(k+1)}-\mathbf{u}_{2i-1}^{(k+1)}|}.\nonumbe
\end{gather}
When $\mathbf{u}_{2i}^{(k+1)}=\mathbf{u}_{2i-1}^{(k+1)}$, the weight $t_i^{(k)}$ can be chosen in any way in the interval $[0, 1]$.
Note that for fixed $k$, several of the $\mathbf{u}_i^{(k)}$'s may be the same vector, that $\mathbf{u}_1^{(0)}=\mathbf 0$, and that $\mathbf{u}_i^{(k)}$ have to be vectors in the convex hull of the support of $\nu$.
The differences
$
\mathbf{U}_i^{(k)}=\mathbf{u}^{(k+1)}_{2i}-\mathbf{u}^{(k+1)}_{2i-1} \hbox{ or rather }\mathbf{U}_i^{(k)}\| \mathbf{u}^{(k+1)}_{2i}-\mathbf{u}^{(k+1)}_{2i-1}
$$
furnish decomposition directions on each step.
The set of vectors
$$
\{(\mathbf{U}_i^{(k)})\}_{0\le k\le m-1, 1\le i\le 2^k}
$$
is the (complete) set of decomposition directions of the $(H_n)$-condition.
Notice that each vector $\mathbf{u}_i^{(k)}$ goes with a weight $s_i^{(k)}$ which is the product of $k$ of the decomposition weights $t_i^{(j)}$ for $0\le j\le k-1$, in such a way that $s_1^{(0)}=1$, and
$$
\nu^{(m)}\equiv\nu=\sum_{i=1}^{2^m}s_i^{(m)}\delta_{\mathbf{u}_i^{(m)}}
$$
There is a whole bunch of intermediate probability measures for fixed $k$
$$
\nu^{(k)}=\sum_{i=1}^{2^k}s_i^{(k)}\delta_{\mathbf{u}_i^{(k)}}.
$$
These are the same as in (\ref{pasok}). Weights $\lambda_i^{(k)}$ there are obtained by adding together several of the weights $s_i^{(k)}$ when corresponding vectors $\mathbf{u}_i^{(k)}$ are identical.
In addition, each $\mathbf{u}_i^{(k)}$ in the support of $\nu^{(k)}$ is the first-moment of, at least, one precise (sub)probability measure associated with the $(H_n)$-condition. Namely,
$$
\nu_i^{(k)}=\sum_j r_{i, j}^{(k)}\delta_{\mathbf{u}_j}.
$$
These are such that
$$
\nu=\sum_i s_i^{(k)}\left(\sum_j r_{i, j}^{(k)}\delta_{\mathbf{u}_j}\right)
$$
for all $k$. Note that these probability measures $\nu_i^{(k)}$ are associated with a certain sub-$(H_n)$-condition of the original $(H_n)$-condition, starting from $\mathbf{u}_i^{(k)}$ as the barycenter.
There is nothing special about the zero vector being the initial vector. The same construction can be made in exactly the same way, had we started out with a different vector $\mathbf F\in\mathbf{M}^{m\times N}$, since the basic operation involved in $(H_n)$-conditions is translation-invariant.
One fundamental observation, after the discussion above, is that the whole $(H_n)$-condition is completely determined once weights and mass points in the final level have been chosen. This would correspond to the bottom-to-top scheme. Namely, suppose we have
$
\{(\lambda_i, \mathbf{u}_i)\}_{1\le i\le 2^m},\quad \nu=\sum_i\lambda_i\delta_{\mathbf{u}_i}, \sum_i\lambda_i\mathbf{u}_i=\mathbf 0,
$$
where some of the $\mathbf{u}_i$'s may be repeated, but they are given.
Put
$$
\lambda_i^{(m)}=\lambda_i,\quad \mathbf{u}_i^{(m)}=\mathbf{u}_i,
$$
and define recursively
$$
\lambda_i^{(k)}=\lambda_{2i-1}^{(k+1)}+\lambda_{2i}^{(k+1)},
$$
and
$$
\mathbf{u}_i^{(k)}=\frac{\lambda_{2i-1}^{(k+1)}}{\lambda_i^{(k)}}\mathbf{u}_{2i-1}^{(k+1)}+\frac{\lambda_{2i}^{(k+1)}}{\lambda_i^{(k)}}\mathbf{u}_{2i}^{(k+1)}
$$
when $\lambda_i^{(k)}>0$, but $\mathbf{u}_i^{(k)}$ chosen in any way in the segment $[\mathbf{u}_{2i-1}^{(k+1)}, \mathbf{u}_{2i}^{(k+1)}]$ if $\lambda_i^{(k)}=0$. Despite this ambiguity when $\lambda_i^{(k)}=0$, the $(H_n)$-condition is determined in a unique way because the total mass is carried when there is no such ambiguity. Notice that $\lambda_i^{(k)}=0$ implies that both $\lambda_{2i-1}^{(k+1)}$ and $\lambda_{2i}^{(k+1)}$ vanish.
According to our discussion above, given relative weights $t_i^{(k)}$ and $1-t_i^{(k)}$, for $0\le k\le m-1$, $1\le i\le 2^k$, in all steps, final weights $\lambda_i^{(m)}$ are given through appropriate products of $m$ relative weights.
|
1,116,691,497,516 | arxiv | \section{Introduction}
The paper is dedicated to the problem of the existence of
universal acting objects - actors in the category of alternative algebras (see Section \ref{def}
for the definitions). Actions in algebraic categories were
studied in \cite{Ho}, \cite{Mc}, \cite{LS}, \cite{Lu}, \cite{No},
\cite{Lo}, \cite{LL} and in other papers. The authors were looking
for the analogs of the group of automorphisms of a group for
associative algebras, rings, commutative associative algebras, Lie
algebras, crossed modules and Leibniz algebras. They gave the
constructions of universal objects (universal in different senses)
in the corresponding categories and studied their properties.
These objects are: the ring of bimultiplications of a ring, the
associative algebras of multiplications (or bimultipliers) of an
associative algebra and multiplications (or multipliers) of a
commutative associative algebra, the actor of a crossed module,
the Lie algebra of derivations and the Leibniz algebra of
biderivations of a Lie and a Leibniz algebras, respectively. In
\cite{BJ} was proposed a categorical approach to this problem,
which was continued in \cite{BJK}, \cite{BJK1}, \cite{BBJ},
\cite{Bou} and \cite{BB}. In particular, representable internal
object action was defined and necessary and sufficient conditions
of its existence in a semi-abelian category was established. In
\cite{CDL2} for any category of interest $\mathbb{C}$, in the sense of
\cite{Orz}, we defined the corresponding category of groups with
operations $\mathbb{C}_G$, $\mathbb{C}\subseteq \mathbb{C}_G$; for any object $A\in
\mathbb{C}$ we defined an actor $\act(A)$ of $A$; this notion is
equivalent to the one of split extension classifier for $A$,
defined in a semi-abelian category in \cite{BJK}. In the same
paper we gave a definition and a construction of a universal
strict general actor $\mathfrak{B}(A)$ of $A$, which is an object in
$\mathbb{C}_G$ in general and has all universal properties of the objects
listed above. We proved that there exists an actor of $A$ in $\mathbb{C}$
if and only if the semidirect product $\mathfrak{B}(A)\ltimes A$ is an
object in $\mathbb{C}$ and, in this case, $\mathfrak{B}(A)$ is an actor of $A$ in
$\mathbb{C}$. Applying this result we considered examples of groups,
associative algebras, Lie and Leibniz algebras, crossed modules
and precrossed modules. In \cite{CDL3} we gave the construction of
an actor of a precrossed module, where we introduced the notion of
a generalized Whitehead's derivation. In the present paper we
define a category of general alternative (g-alternative) algebras
over a field $F$ (denoted by $\galt$), which is a category of
interest. We present the category of alternative algebras (denoted
by $\alt$) as a full subcategory of $\galt$. In the case, where
$\ch F \neq 2$, we have $\alt=\galt$. Applying the results of
\cite{CDL2}, for any g-alternative algebra $A$ we give a
construction of a universal strict general actor of $A$ and obtain
sufficient conditions for the existence of an actor of $A$ in
$\galt$. From this we easily deduce analogous results for the case
of alternative algebras.
In Section \ref{def} we recall the definitions of a category of
interest, the general category of groups with operations of a
category of interest, and give examples. We recall the definitions
of structure, derived action and actor in a category of interest.
We state a necessary and sufficient condition for the existence of
an actor in a category of interest in terms of a universal strict
general actor. The construction of this object for the case of the
category $\galt$ is given in Section \ref{acting}. In the
beginning of this section we define derived actions in the
categories of g-alternative and alternative algebras. We state
some properties of g-alternative algebras in terms of the axioms
of the corresponding category of interest, which are essentially
well-known for the case of alternative algebras. Then we give a
construction of a universal strict general actor $\mathfrak{B}(A)$ of a
g-alternative algebra $A$; as special cases of known definitions
we define a semidirect product and the center of an object in
$\galt$ (equivalently, in $\alt$). In Section \ref{properties} for
any g-alternative algebra $A$ we define the $\mathfrak{B}(A)$-substructure
$\soci(A)$ of $A$, due to which we define certain subset
$\asoci(A)$, which turned out to be a $\mathfrak{B}(A)$-substructure of
$A$. We study the properties of this object, which are applied in
what follows. In Section \ref{sufficient} we give an example of
algebra, which shows that the object $\mathfrak{B}(A)$ is not a
g-alternative algebra in general; we state sufficient conditions
for $\mathfrak{B}(A)\in \galt$. In Section \ref{existence_galt} we give
examples of algebras for which the action of $\mathfrak{B}(A)$ on $A$ is
not a derived action in general. The final result states that if
$\asoci(A)=0$, then $\mathfrak{B}(A)$ is a g-alternative algebra, the
action of $\mathfrak{B}(A)$ on $A$ is a derived action in $\galt$ and
according to the general result, given in \cite{CDL3},
$\mathfrak{B}(A)=\act(A)$. In particular, if $A$ is anticommutative (i.e.
$aa'=-a'a$, for any $a,a' \in A$) and $\ann(A)=0$, then there
exists an actor of $A$ in $\galt$. At the end of the section we
give another description of an actor of a g-alternative algebra in
terms of a certain algebra of bimultiplications, defined in
analogous way as for the case of rings or associative algebras. In
Section \ref{existence_alt} we investigate more properties of the
object $\mathfrak{B}(A)$ under the conditions that $A$ is anticommutative
with $\ann(A)=0$; in particular, it is proved that $\mathfrak{B}(A)\in
\alt$ and its action on $A$ is a derived action in $\alt$, which
give the sufficient conditions for the existence of an actor in
the category of alternative algebras. Note that in the cases of
associative algebras and rings the algebras of bimultiplications
played an important role in the extension and obstruction theories
\cite{Ho,Mc}. From this point of view the results obtained in this
paper could be applied to a cohomology and the corresponding
extension and obstruction theories of alternative algebras.
\section{Preliminary definitions and results} \label{def}
This section contains some well-known and new definitions and
results which will be used in what follows.
Let $\mathbb{C}$ be a category of groups with a set of operations $\Omega$ and
with a set of identities $\mathbb{E}$, such that $\mathbb{E}$ includes the group
identities and the following conditions hold. If $\Omega_i$ is the set
of $i$-ary operations in $\Omega$, then:
\begin{itemize}
\item[(a)]
$\Omega=\Omega_0\cup\Omega_1\cup\Omega_2$;
\item[(b)] the group operations
(written additively: ($0,-,+$) are elements of $\Omega_0$, $\Omega_1$ and
$\Omega_2$ respectively. Let $\Omega_2'=\Omega_2\setminus\{+\}$,
$\Omega_1'=\Omega_1\setminus\{-\}$ and assume that if $*\in\Omega_2$, then
$\Omega_2'$ contains $*^\circ$ defined by $x*^{\circ}y=y*x$. Assume
further that $\Omega_0=\{0\}$;
\item[(c)] for each $*\in\Omega_2'$, $\mathbb{E}$
includes the identity $x*(y+z)=x*y+x*z$;
\item[(d)] for each
$\omega\in\Omega_1'$ and $*\in\Omega_2'$, $\mathbb{E}$ includes the identities
$\omega(x+y)=\omega(x)+\omega(y)$ and $\omega(x)*y=\omega(x*y)$.
\end{itemize}
Note that the group operation is denoted additively, but it is not
commutative in general. A category $\mathbb{C}$ defined above is called a
\emph{category of groups with operations}. The idea of the
definition comes from \cite{Hig} and axioms are from \cite{Orz}
and \cite{Por}. We formulate two more axioms on $\mathbb{C}$ (Axiom~(7)
and Axiom~(8) in \cite{Orz}).
If $C$ is an object of $\mathbb{C}$ and $x_1,x_2,x_3\in C$:
\begin{axiom}
$x_1+(x_2*x_3)=(x_2*x_3)+x_1$, for each
$*\in\Omega_2'$.
\end{axiom}
\begin{axiom} For each ordered pair
$(*,\overline{*})\in\Omega_2'\times\Omega_2'$ there is a word $W$ such that
\begin{gather*}
(x_1*x_2)\overline{*}x_3=W(x_1(x_2x_3),x_1(x_3x_2),(x_2x_3)x_1,\\
(x_3x_2)x_1,x_2(x_1x_3),x_2(x_3x_1),(x_1x_3)x_2,(x_3x_1)x_2),
\end{gather*}
where each juxtaposition represents an operation in $\Omega_2'$.
\end{axiom}
A category of groups with operations satisfying Axiom~1 and Axiom~2
is called a {\it category of interest} in \cite{Orz}.
Denote by $\mathbb{E}_G$ the subset of identities of $\mathbb{E}$ which includes
the group identities and the identities (c) and (d). We denote by
$\mathbb{C}_G$ the corresponding category of groups with operations. Thus
we have $\mathbb{E}_G \hookrightarrow \mathbb{E}$, $\mathbb{C}=(\Omega,\mathbb{E})$,
$\mathbb{C}_G=(\Omega,\mathbb{E}_G)$ and there is a full inclusion functor
$\mathbb{C}\hookrightarrow\mathbb{C}_G$. The category $\mathbb{C}_G$ is called a
\emph{general category of groups with operations} of a category of
interest $\mathbb{C}$ in \cite{CDL1} and \cite{CDL2}.
\noindent {\bf Examples of categories of interest.} In the case of
the category of associative algebras with multiplication
represented by $*$, we have $\Omega_2'=\{*,*^\circ\}$. For Lie
algebras take $\Omega_2'=([\;,\;],[\;,\;]^\circ)$ (where
$[a,b]^\circ=[b,a]=-[a,b]$). For Leibniz algebras (see \cite{Lo}),
take $\Omega_2'=([\;,\;],[\;,\;]^\circ)$, (here $[a,b]^\circ=[b,a]$).
It is easy to see that the categories of all these algebras are
categories of interest. In the cases of the categories of groups,
abelian groups and modules over a ring we have
$\Omega_2'=\varnothing$. As it is noted in \cite{Orz} Jordan algebras
do not satisfy Axiom~2. Below we will see that the category of
alternative algebras is a category of interest as well.
\begin{defi}\cite{Orz}
Let $A$, $B\in\mathbb{C}$. An \emph{extension} of $B$ by $A$ is a sequence
\begin{equation} \label{extension}
\xymatrix{0\ar[r]&A\ar[r]^-{i}&E\ar[r]^-{p}&B\ar[r]&0}
\end{equation}
in which $p$ is surjective and $i$ is the kernel of $p$. We say
that an extension is \emph{split} if there is a morphism $s \colon B\longrightarrow E$ such
that $ps=1_B$.
\end{defi}
\begin{defi}\cite{Orz}
A split extension of $B$ by $A$ is called a $B$-\emph{structure} on
$A$.
\end{defi}
According to \cite{Orz}, for $A,B\in\mathbb{C}$ ``a set of actions of $B$
on $A$'', means that there is a map $f_* \colon B\times A\longrightarrow A$, for
each $*\in\Omega_2$. Instead of ``set of actions'' often is used for
simplicity ``an action'', and it means that there is a set of
actions $\{f_*\}_ {*\in\Omega_2}$. Note that a set of actions has a
different meaning in \cite{BJK}.
A $B$-structure on $A$ induces a set of actions of $B$ on $A$
corresponding to the operations in $\mathbb{C}$. If \eqref{extension} is a split
extension, then for $b\in B$, $a\in A$ and $*\in\Omega_2{}'$ we have
\begin{align}
b\cdot a &=s(b)+a-s(b), \label{derived_dot}\\
b*a&=s(b)*a. \label{derived_star}
\end{align}
Actions defined by \eqref{derived_dot} and \eqref{derived_star} are called \emph{derived actions} of
$B$ on $A$ in \cite{Orz}. Under $B$-\emph{substructure} of $A$
naturally we will mean a subobject $A'$ of $A$ which is closed under
all derived actions of $B$ on $A'$, i.e. left and right derived
actions.
In the case of associative algebras over a ring $R$ a derived
action of $B$ on $A$ is a pair of $R$-bilinear maps
\begin{equation}\label{bilinear}
B\times A\longrightarrow A,\quad A\times B\longrightarrow A
\end{equation}
which we denote respectively as $(b,a)\mapsto ba$, $(a,b)\mapsto
ab$, with conditions
\begin{align*}
&\quad(b_1b_2)a=b_1(b_2a),\\
&\quad a(b_1b_2)=(ab_1)b_2,\\
&\quad(b_1a)b_2=b_1(ab_2),\\
&\quad(a_1a_2)b=a_1(a_2b),\\
&\quad b(a_1a_2)=(ba_1)a_2,\\
&\quad (a_1b)a_2=a_1(ba_2),
\end{align*}
for any $a_1, a_2\in A$ and $b_1, b_2\in B$. Note that these
identities are well-known, and they can be obtained easily from
the definition of a derived action \eqref{derived_star} and the
associativity axiom.
According to \cite{Orz}, in any category of interest, given a set
of actions of $B$ on $A$, the \emph{semidirect product} $B\ltimes
A$ is the universal algebra, whose underlying set is $B\times A$
and the operations are defined by
\begin{align*}
(b',a')+(b,a)&=(b'+b,a'+b'\cdot a),\\
(b',a')*(b,a)&=(b'*b,a'*a+a'*b+b'*a).
\end{align*}
\begin{theo}\cite{Orz} \label{derived_semi}
A set of actions of $B$ on $A$ is a set of derived actions if and
only if $B\ltimes A$ is an object of $\mathbb{C}$.
\end{theo}
\begin{defi}\cite{CDL1,CDL2}\label{actor}
For any object $A$ in $\mathbb{C}$, an actor of $A$ is an object
$\act(A)$ in $\mathbb{C}$, which has a derived action on $A$ in $\mathbb{C}$,
and for any object $C$ of $\mathbb{C}$ and a derived action of $C$ on $A$
there is a unique morphism $\varphi \colon C\longrightarrow \act(A)$ with $c\cdot
a=\varphi(c)\cdot a$, $c*a=\varphi(c)*a$ for any $*\in\Omega_2{}'$, $a\in A$
and $c\in C$.
\end{defi}
In \cite{CDL2}, for any object $A$ of a category of interest
$\mathbb{C}$, we define a universal strict general actor $\mathfrak{B}(A)$ of $A$,
which is an object of $\mathbb{C}_G$, and give the corresponding
construction. We present this construction for the case of
g-alternative algebras (see below the definition) in Section
\ref{acting}.
\begin{theo}\cite{CDL2} \label{actor_semi}
Let $\mathbb{C}$ be a category of interest and $A\in\mathbb{C}$. $A$ has an actor
if and only if the semidirect product $\mathfrak{B}(A)\ltimes A$ is an object
of $\mathbb{C}$. If it is the case, then $\act(A)=\mathfrak{B}(A)$.
\end{theo}
From Theorems \ref{derived_semi} and \ref{actor_semi} we have
\begin{coro} \label{actor_interest}An object $A$ of a category of interest $\mathbb{C}$ has an
actor if and only if $\mathfrak{B}(A)\in \mathbb{C}$ and the action of $\mathfrak{B}(A)$ on
$A$ is a derived action.
\end{coro}
Recall that an \emph{alternative algebra} $A$ over a field $F$ is
an algebra which satisfies the identities $x^2y = x(xy)$ and $yx^2
= (yx)x,$ for all $x,y \in A$. These identities are known
respectively as the left and right alternative laws. We denote the
corresponding category of alternative algebras by $\alt$. Clearly
any associative algebra is alternative. The class of 8-dimensional
Cayley algebras is an important class of alternative algebras
which are not associative \cite{Sch}; commutative alternative
algebras are Jordan algebras.
Note that in all categories of algebras, considered in this paper
as categories of interest, algebras are generally without unit,
but, of course, algebras with unit are also included in these
categories. We will always note when we deal with algebras over a
field with characteristic 2, but nevertheless in such cases we
will use the sign ``$-$'' before the elements of the algebra, in
order to denote the additive inverse elements, e.g. $-a$, for the
element inverse to $a$ of the algebra $A$; this we do essentially
in order not to confuse a reader during the computations while we
apply different axioms or certain assumptions on algebras.
Here we introduce the notion of a general alternative algebra.
\begin{defi} A \emph{general alternative algebra} (shortly
\emph{g-alternative algebra}) $A$ over a field $F$ is an algebra,
which satisfies the following two axioms for any $x, y, z \in A$
\begin{enumerate}[\hspace*{0.5cm} {A}x{i}om~${2}_1$.]
\item $ x(yz) = (xy)z + (yx)z - y(xz)$;
\item $(xy)z = x(yz) + x(zy) - (xz)y $.
\end{enumerate}
\end{defi}
These axioms are dual to each other in the sense, that if $x \circ
y=yx$, then Axiom~$2_1$ for the operation $\circ$ gives
Axiom~$2_2$ for the original operation, and obviously, Axiom~$2_2$
for the $\circ$ operation gives Axiom~$2_1$. We consider these
conditions as Axiom~2 and consequently, the category of
g-alternative algebras can be interpreted as a category of
interest, which will be denoted by $\galt$. According to the
definition of a general category of groups with operations for a
given category of interest, we obtain that $\alt_G=\galt_G$ and it
is a category of groups together with the multiplication
operation, which satisfies the identity $x(y+z)=xy+xz$.
Denote by $\overline{\galt}$ the full subcategory of
$\galt$ of those objects and homomorphisms between them,
which satisfy the following condition
E$_1$. \quad $(xy)x=x(yx)$, for any $x,y \in A$.
This identity is called the \emph{flexible law} \cite{Sch} or
\emph{flexible identity} \cite{ZSSS}.
\begin{Prop}\label{galt}
\begin{enumerate}
\item[(i)] For any field F, we have the equality $\alt=
\overline{\galt}$;
\item[(ii)] If $\ch F \neq 2$, then
$\galt=\overline{\galt}=\alt$.
\end{enumerate}
\end{Prop}
\begin{proof} (i) Let A be an alternative algebra, then we have
\[(x+y)^2z=(x+y)((x+y)z), \ \mbox{for any}\ x,y,z\in A \, ,\]
from which follows Axiom~$2_1$. Analogously, the identity
$x(y+z)^2= (x(y+z))(y+z)$ gives Axiom~$2_2$. Thus we have
$\alt \subseteq \galt$. It is well-known fact that
every alternative algebra satisfies the condition E$_1$;
see the equivalent definition of alternative algebras and the
proof of Artin's theorem, e.g. in \cite{Kur}. We include here the
proof based on the fact that alternative algebras are
g-alternative algebras. From Axiom~$2_1$, for $y=z$ we have
\begin{equation} \label{25}
xy^2=(xy)y+(yx)y-y(xy)
\end{equation}
and since $xy^2=(xy)y$, we obtain $(yx)y=y(xy)$, which is the
condition E$_1$. Now we shall show that $\overline{\galt}\subseteq
\alt$. Let $A\in \overline{\galt}$, again from \eqref{25},
applying the condition E$_1$ we obtain $xy^2=(xy)y$. Analogously,
from Axiom~$2_2$ for $x=y$ we have $x^2z=x(xz)+x(zx)-(xz)x$; this
by E$_1$ implies $x^2z=x(xz)$, which ends the proof of ($i$).
(ii) We have only to show that if $\ch F \neq 2$, then every
g-alternative algebra satisfies the condition E$_1$. From
Axiom~$2_2$ for $y=z$, we have $(xy)y=xy^2+xy^2-(xy)y$. From this
and \eqref{25} we have $2y(xy)=2(yx)y$, applying the fact that
$\ch F\neq2$ we obtain the condition E$_1$, which ends the proof.
\end{proof}
The following example shows that in the case $\ch F=2$ we have the
strict inclusion $\alt \subset \galt$.
\begin{ex} Let $A$ be the free g-alternative algebra generated by the one element set $\{x\}$ over a field
$F$, and $\ch F=2$. Then, according to Axiom~$2_1$, since
$(xx)x+(xx)x=0$, we only obtain that $x(xx)=-x(xx)$, as it is for
any element $a\in A$, i.e. $a=-a$, since $\ch F=2$. Analogously,
from Axiom~$2_2$, we will have $(xx)x=-(xx)x$, which shows that
$A$ is not an alternative algebra.
\end{ex}
\section{Derived actions and universal acting objects in the
categories of g-alternative and alternative algebras} \label{acting}
According to the definition of derived action in a category of
interest, in the category of g-alternative algebras over a field
$F$ we obtain the following: a derived action of $B$ on $A$ in
$\galt$ is a pair of $F$-bilinear maps \eqref{bilinear}, which we
denote by $(b,a)\mapsto b a$, $(a,b)\mapsto a b$ with the
conditions
\begin{align*}
\mbox{I}_1. \quad b (a_1 a_2)& = (b a_1) a_2 + (a_1 b)a_2 - a_1(b a_2), \\
\mbox{I}_2. \quad (a_1 a_2) b & = a_1 (a_2 b) + a_1 (b a_2) - (a_1 b) a_2 , \\
\mbox{I}_3. \quad (b a_1) a_2 & = b (a_1 a_2) + b (a_2 a_1) - (b a_2) a_1 , \\
\mbox{I}_4. \quad a_1(a_2 b)&=(a_1 a_2)b + (a_2 a_1) b - a_2(a_1 b),\\
\mbox{II}_1. \quad (b_1 b_2) a & = b_1 (b_2 a) + b_1 (a b_2) - (b_1 a) b_2 , \\
\mbox{II}_2. \quad a (b_1 b_2) &= (a b_1) b_2 + (b_1 a) b_2 - b_1(a b_2), \\
\mbox{II}_3. \quad (a b_1) b_2& = a(b_1 b_2) + a(b_2 b_1) - (a b_2) b_1 , \\
\mbox{II}_4. \quad b_1(b_2a)&=(b_1 b_2) a + (b_2 b_1) a - b_2(b_1 a),
\end{align*}
for any $a_1, a_2\in A$ and $b_1, b_2\in B$. These identities are
obtained according to \eqref{derived_star} and Axiom~2 for
g-alternative algebras.
For the case of alternative algebras for derived actions we will
have $\mbox{I}_1-\mbox{I}_4, \mbox{II}_1-\mbox{II}_4$, and two more identities obtained from
the condition E$_1$:
\begin{align*}
& \mbox{III}_1. \quad a(ba)=(ab)a, \\
& \mbox{III}_2. \quad b(ab)=(ba)b,
\end{align*}
for any $a\in A, b\in B.$ In the case where $A$ is a vector space
over a field $F$ we obtain the well-known definition of an
alternative bimodule or, equivalently, of a representation of an
alternative algebra \cite{Sch}.
From Axiom~2 we easily deduce the following useful identities in
terms of Axiom~2 of the corresponding category of interest, which
express the well-known properties of alternative algebras (see
e.g. \cite{Sch}, \cite{ZSSS}). We shall use the notation like
$(1\leftrightarrow 2,-)$ in order to denote that right side of the
identity is obtained by the permutation of the first and the second
elements from the left side, and the signs of the summands are
changed; the notation like $(1\rightarrow 3\rightarrow
2\rightarrow 1)$ denotes that the right side of the identity is
obtained by the following changes in the left side: the first
element takes the place of the third one, the third element the
place of the second one and the second element takes the place of
the first one.
\begin{align}
(1\leftrightarrow 2,-) \qquad \qquad \qquad (xy)z-x(yz)=-(yx)z+y(xz) \label{31}\\
(1\leftrightarrow 3,-) \qquad \qquad \qquad (xy)z-x(yz)=-(zy)x+z(yx) \label{32}\\
(2\leftrightarrow
3,-) \qquad \qquad \qquad (xy)z-x(yz)=-(xz)y+x(zy) \label{33}\\
(1\rightarrow3\rightarrow2\rightarrow
1)\qquad \qquad \qquad (xy)z-x(yz)=(yz)x-y(zx) \label{34}\\
(1\rightarrow 2 \rightarrow 3\rightarrow 1)\qquad \qquad \qquad (xy)z-x(yz)=(zx)y-z(xy) \label{35}\\
(xy)z+z(xy)=x(yz)+(zx)y \label{36}\\
(xy)z+(yx)z=x(yz)+y(xz) \label{37}\\
z(xy)+z(yx)=(zx)y+(zy)x \label{38}
\end{align}
for any $x,y,z\in A$. Note that we have \eqref{31}
$\Leftrightarrow$ Axiom~$2_1$, \eqref{33} $\Leftrightarrow$
Axiom~$ 2_1$, and \eqref{34} from right to the left is the same as
\eqref{35}. In the case $\ch F\neq 2$, \eqref{32} is equivalent
to the flexible law E$_1$.
Let $\{B_i\}_{i\in I}$ be the set of all g-alternative algebras,
which have a derived action on $A$ in $\galt$. From the properties
$\mbox{I}_1-\mbox{I}_4$ it follows that identities
\eqref{31}--\eqref{38} are true for any $x,y,z\in A$, where one of
the elements from $x,y,z$ is in $B_i$ for any $i\in I$.
Let $A$ and $B$ be g-alternative (resp. alternative) algebras and
$B$ has a derived action on $A$ in $\galt$ (resp. $\alt$).
According to the general definition of a semidirect product in a
category of interest given in Section \ref{def}, $B\ltimes A$ is a
g-alternative (resp. an alternative) algebra, whose underlying set
is $B\times A$ and whose operations are given by
\begin{align*}
(b',a')+(b,a)&=(b'+b,a'+ a),\\
(b',a')(b,a)&=(b'b,a'a+a'b+b'a).
\end{align*}
According to Definition \ref{actor}, for any g-alternative (resp.
alternative) algebra $A$, an \emph{actor} of $A$ is an object
$\act(A)\in \galt$ (resp. $\act(A)\in \alt$),
which has a derived action on $A$ in $\galt$ (resp.
$\alt$) and for any g-alternative (resp. alternative)
algebra $C$ and a derived action of $C$ on $A$, there is a unique
homomorphism $\varphi \colon C\longrightarrow \act(A)$ with $c a=\varphi(c) a$, for
any $a\in A$ and $c\in C$.
Here we give a construction of a universal strict general actor
$\mathfrak{B}(A)$ of a g-alternative algebra $A$, which is a special case
of the construction given in \cite{CDL2} for categories of
interest. In this case we have only two binary operations: the
addition, denoted by `` + '', and the multiplication, which will
be denoted here by dot ``$\cdot$''. Note that this sign was
omitted usually in above expressions for g-alternative algebras,
and it will be so in the next sections as well, when there is no
confusion. Since the addition is commutative, the action
corresponding to this operation is trivial. Thus we will deal only
with actions, which are defined by multiplication according to
\eqref{derived_star}, and this action will be denoted by $\cdot$
as well
\begin{align*}
b\cdot a&=s(b)\cdot a.
\end{align*}
Consider all split extensions of $A$ in $\galt$
\[ \xymatrix{E_j \colon 0\ar[r]&A\ar[r]^-{i_j}&C_j\ar[r]^-{p_j}&B_j\ar[r]&0},\quad j\in\mathbb{J} \, . \]
Note that it may happen that $B_j=B_k=B$, for $j\neq k$, in this
case the corresponding extensions derive different actions of $B$ on
$A$.
Let $\{(b_j\cdot, \cdot b_j)|b_j\in
B_j\}$ be the set of all pairs of $F$-linear maps $A\rightarrow
A$, defined by the action of $B_j$ on $A$. For any element $b_j\in
B_j$ denote ${{\bf b}}_j=(b_j\cdot,\cdot b_j)$. Let
$\mathbb{B}=\{{{\bf b}}_j|b_j\in B_j,\;j\in\mathbb{J}\}$.
According to Axiom~2 from the definition of $\galt$ as a category
of interest, we define the multiplication,
${{\bf b}}_i\cdot{{\bf b}}_k$, for the elements of $\mathbb{B}$ by the
equalities
\begin{align*}
&({{\bf b}}_i\cdot{{\bf b}}_k)\cdot(a)=b_i\cdot(b_k\cdot
a)+b_i\cdot(a\cdot b_k)-(b_i\cdot a)\cdot b_k \, ,\\
&(a)\cdot({{\bf b}}_i\cdot{{\bf b}}_k)=(a\cdot b_i)\cdot b_k+(b_i\cdot
a)\cdot b_k-b_i\cdot(a\cdot b_k) \, .
\end{align*}
For $b={\bf b}_{i_1}\cdot \ldots \cdot{\bf b}_{i_n}$ with certain brackets,
$b\cdot a$ is defined in a natural way step by step according to
brackets and to above defined equalities.
We define the operation of addition by
\begin{align*}
&({{\bf b}}_i+{{\bf b}}_k)\cdot(a)=b_i\cdot a+b_k\cdot a.
\end{align*}
For the unary operation ``-'' we define
\begin{align*}
&(-{{\bf b}}_k)\cdot(a)=-(b_k\cdot a),\\
&(-b)\cdot(a)=-(b\cdot(a)),\\
&- (b_1+\dots +b_n)=-b_n-\dots-b_1,
\end{align*}
where $b,b_1,\dots,b_n$ are certain combinations of the dot
operation on the elements of $\mathbb{B}$, i.e. the elements of the type
${\bf b}_{i_1}\cdot \ldots \cdot{\bf b}_{i_n}$, where we mean certain
brackets, $n>1$. Obviously, the addition is commutative.
Denote by $\mathfrak{B}'(A)$ the set of all pairs of $F$-linear maps
obtained by performing all kinds of above defined operations on
the elements of $\mathbb{B}$. We define the following relation: we will
write $b\sim b'$, for $b,b'\in \mathfrak{B}'(A),$ if and only if
$b\cdot(a)=b'\cdot(a)$, for any $a\in A$. This relation is a
congruence relation on $\mathfrak{B}'(A)$, i.e. it is compatible with the
operations we have defined in $\mathfrak{B}'(A)$. We define
$\mathfrak{B}(A)=\mathfrak{B}'(A)/\sim$. The operations defined on $\mathfrak{B}'(A)$ induce
the corresponding operations on $\mathfrak{B}(A)$. For simplicity we will
denote the elements of $\mathfrak{B}(A)$ by the same letters $b, b'$, etc.
instead of the classes $clb, clb'$, etc.
According to the general result \cite[Proposition 4.1]{CDL2}
$\mathfrak{B}(A)$ is an object of $\galt_G$; moreover, it is obvious, that
$\mathfrak{B}(A)$ is an $F$-algebra.
Define the set of actions of $\mathfrak{B}(A)$ on $A$ in a natural way:
for $b\in \mathfrak{B}(A)$ we define $b\cdot a=b\cdot(a)$. Thus if
$b={\bf b}_{i_1}\cdot_1 \ldots \cdot_{n-1}{\bf b}_{i_n}$, where we mean
certain brackets, we have
\begin{align*}
&b\cdot\,a=({\bf b}_{i_1}\cdot_1 \ldots \cdot_{n-1}{\bf b}_{i_n})\,\cdot\,(a),\\
\end{align*}
where the right side of the equality is defined according to the
brackets and Axiom~2. For $b_k\in B_k$, $k\in\mathbb{J}$, we have
\begin{align*}
{\bf b}_k\cdot a={\bf b}_k\cdot(a)=b_k\cdot a.
\end{align*}
Also
\[(b_1+b_2+ \dots +b_n)\cdot
a=b_1\cdot(a)+ \dots +b_n\cdot(a), \; \mbox{for} \; b_i\in \mathfrak{B}(A),\;
i=1, \ldots, n \, .\]
It is easy to check (see \cite[Proposition 4.2]{CDL2} for the
general case) that the set of actions of $\mathfrak{B}(A)$ on $A$ is a set
of derived actions in $\galt_G$.
The construction of a universal strict general actor of an
alternative algebra $A$ in $\alt$ is analogous to the one of
$\mathfrak{B}(A)$; in this case we consider all split extensions $E_j$ of
$A$ in $\alt$, i.e we will deal with the family $\{B_i\}_{i\in I}$
of all alternative algebras, which have a derived action on $A$ in
$\alt$. The corresponding object will be denoted by
$\mathfrak{B}_{\alt}(A)$, and it is an object of $\alt_G$.
For any $A\in \galt$, define the map $d \colon A\longrightarrow \mathfrak{B}(A)$ by
$d(a)={\bf{\mathfrak{a}}}$, where ${\bf{\mathfrak{a}}}=\{(a \cdot ,\cdot a)\}$. Thus we have by
definition
\begin{equation*}
d(a)\cdot a'=a\cdot a', \quad a'\cdot d(a)=a'\cdot a, \ \mbox{for any} \
a,a'\in A.
\end{equation*}\
According to the general results (see the case of a category of
interest, \cite[Lemma 4.5 and Proposition 4.6]{CDL2}) $d$ is a
homomorphism in $\galt_G$ and moreover $d \colon A\longrightarrow \mathfrak{B}(A)$ is a
crossed module in $\galt_G$ with certain universal property.
From the general definition of a center in categories of interest
\cite{CDL2}, (cf. \cite{Orz}), for any g-alternative algebra $A$,
we obtain that the \emph{center} of $A$ is defined by
\[Z(A)=\Ker d=\{z\in A \mid \mbox{for all}\ a\in A,
az=za=0\} \, .\]
Therefore the center in this case is a left-right annulator of $A$,
denoted by $\ann(A)$ (see \cite{Mc} for the case of rings).
\section{$\soci(A)$, $\asoci(A)$ and their properties}\label{properties}
For any $x, y, z \in A\bigcup\mathfrak{B}(A)$, $A\in \alt$, consider
the following types of elements:
\begin{itemize}
\item[(sac)] $x(yz)+x(zy), \, (yz)x+(zy)x$ ;
\item[(as)] $x(yz)-(xy)z$;
\item[(aas)] $x(yz)+(xy)z$;
\item[(ap)] $x(yz)+y(xz)$, \, $(yz)x+(yx)z$.
\end{itemize}
Let ${\{B_i\}}_{i\in I}$ be the family of g-alternative algebras which have a
derived action on $A$. According to the above notation we define
the following sets in $A$.
\begin{itemize}
\item $S_1^{\rm sac}$ (resp. $S_2^{\rm sac}$)\,: the set of elements of $A$
of the type (sac), where one element (resp. two elements) from the triple
$(x,y,z)$ is (resp. are) in $\bigcup_{i\in I}{B_i}$.
\item $\bar{S}_1^{\rm sac}$ (resp. $\bar{S}_2^{\rm sac}$)\,: the set of elements of $A$
of the type (sac), where one element (resp. two elements) from the triple
$(x,y,z)$ is (resp. are) in $\mathfrak{B}(A)$.
\item $S_1^{\rm as}$ (resp. $S_2^{\rm as}$)\,: the set of elements of $A$
of the type (as), where one element (resp. two elements) from the triple
$(x,y,z)$ is (resp. are) in $\bigcup_{i\in I}{B_i}$.
\item $\bar{S}_1^{\rm as}$ (resp. $\bar{S}_2^{\rm as}$)\,: the set of elements of $A$
of the type (as), where one element (resp. two elements) from the triple
$(x,y,z)$ is (resp. are) in $\mathfrak{B}(A)$.
\item $S_1^{\rm aas}$ (resp. $S_2^{\rm aas}$)\,: the set of elements of $A$
of the type (aas), where one element (resp. two elements) from the triple
$(x,y,z)$ is (resp. are) in $\bigcup_{i\in I}{B_i}$.
\item $\bar{S}_1^{\rm aas}$ (resp. $\bar{S}_2^{\rm aas}$)\,: the set of elements of $A$
of the type (aas), where one element (resp. two elements) from the triple
$(x,y,z)$ is (resp. are) in $\mathfrak{B}(A)$.
\item $S_1^{\rm ap}$ (resp. $S_2^{\rm ap}$)\,: the set of elements of $A$
of the type (ap), where one element (resp. two elements) from the triple
$(x,y,z)$ is (resp. are) in $\bigcup_{i\in I}{B_i}$.
\item $\bar{S}_1^{\rm ap}$ (resp. $\bar{S}_2^{\rm ap}$)\,: the set of elements of $A$
of the type (ap), where one element (resp. two elements) from the triple
$(x,y,z)$ is (resp. are) in $\mathfrak{B}(A)$.
\end{itemize}
\begin{defi} For any $A\in \galt$ define $\soci(A)$ as
the $\mathfrak{B}(A)$-substructure of $A$ generated by the set $S_1^{\rm sac}$.
\end{defi}
\begin{defi}
For any $n\geq1$ denote $\asoci^n(A)=\{x\in
A|(\dots((xa_1)a_2)\dots a_n)\in \soci(A),\, \text{for any} \,\, a_1,a_2,\dots,a_n\in A\}$. \
Define $\asoci(A) = \bigcup _{n\geq 1}\asoci^n(A)$.
\end{defi}
From the definition it follows that $\soci(A)\subseteq
\asoci^1(A)$ and $\asoci^n(A)\subseteq
\asoci^{n+1}(A)$ for any $n$.
\begin{Lem}\label{substructure}
Let $A'$ be a $B_i$-substructure of $A$ for any $i\in I$. Then $A'$
is a $\mathfrak{B}(A)$-substructure of $A$ in $\galt_G$.
\end{Lem}
\begin{proof} Let $b=b_ib_j$ be an element of $\mathfrak{B}(A)$. For any $x\in A'$ we have
\[(b_ib_j)x=b_i(b_jx)+b_i(xb_j)-(b_ix)b_j \, .\]
By the condition of the lemma, every element on the right side is an
element of $A'$, therefore $(b_ib_j)x\in A'$. By induction, in
analogous way, it can be proved that $bx\in A'$ for any $b\in \mathfrak{B}(A)$.
\end{proof}
\begin{Lem}\label{asoci}
$\asoci(A)$ is a two sided ideal of $A$.
\end{Lem}
\begin{proof} It is obvious that if $x\in \asoci(A)$, then
$xa\in \asoci(A)$ for any $a\in A$. Now we shall show that $ax\in
\asoci(A)$. Since $x\in \asoci(A)$, there exists a natural number
$k$, such that for any $a_1,a_2,\dots,a_k \in A$, $(\dots
((-xa_1)a_2)\dots )a_k\in \soci(A)$; in particular, $(\dots
((-xa)a_1)\dots )a_{k-1}\in \soci(A)$. We have $(ax)a_1\simeq
(-xa)a_1$. By the definition $\soci(A)$ is an ideal of $A$,
therefore $(\dots ((-xa)a_1)\dots)a_{k-1}\simeq
(\dots((ax)a_1)\dots)a_{k-1}$. From this it follows that
$(\dots((ax)a_1)\dots)a_{k-1}\in \soci(A)$, which ends the proof.
\end{proof}
\begin{Prop}\label{asoci_sub} For any $A\in \galt$, $\asoci(A)$ is a
$\mathfrak{B}(A)$-substructure of $A$.
\end{Prop}
\begin{proof} By Lemma \ref{substructure} we have only to prove that for any $b_i\in
B_i$, $i\in I$, if $x\in \asoci(A)$, then $xb_i$, $b_ix\in
\asoci(A)$. First consider the case, where $x\in
\asoci^1(A)$. For any $a\in A$ we have
\[(xb_i)a=x(b_ia)+x(ab_i)-(xa)b_i \, .\]
The sum of the first two summands on the right side is in
$\soci(A)$; since $xa_1\in \soci(A)$ and $\soci(A)$, by
definition, is a $\mathfrak{B}(A)$-substructure in $A$, the third summand
$(xa)b_i$ is in $\soci(A)$ as well. From this we conclude that
$(xb_i)a\in \soci(A)$, which means that $(xb_i)\in \asoci(A)$.
Consider the case where $x\in \asoci^2(A)$. We have
$(xa_1)a_2\in \soci(A)$, for any $a_1, a_2\in A$. We compute
\[((xb_i)a_1)a_2=(x(b_ia_1)+x(a_1b_i))a_2-((xa_1)b_i)a_2\, .\]
By Lemma \ref{asoci}, $xa_1\in \asoci^1(A)$, thus from the above
proved case we obtain $(xa_1)b_i\in \soci(A)$, and therefore
$((xa_1)b_i)a_2$ is in $\soci(A)$ as well. Moreover
$(x(b_ia_1)+x(a_1b_i))a_2\in \soci(A)$ since
$\soci(A)$ is an ideal of $A$. From this we conclude that
$((xb_i)a_1)a_2\in \soci(A)$. In this way by induction we
show that for any natural number $n$ and $x\in \asoci^n(A)$,
$xb_i\in \asoci(A)$. In analogous way is proved that
$b_ix\in \asoci(A)$, which ends the proof.
\end{proof}
\noindent
\textbf{Notation.} For any set $X$ of elements of $A$, $A\in \galt$, we will
write $X\simeq 0$ if and only if $X\subseteq \asoci(A)$.
We will write $x_1\simeq x_2$ (resp.$x_1\sim x_2$), for $x_1, x_2 \in A$, if and only if
$x_1-x_2 \in \asoci(A)$ (resp. $x_1-x_2 \in \soci(A)$).
Therefore $x_1\sim x_2$ implies $x_1\simeq x_2$.
By Proposition \ref{asoci_sub} and the definition of $\soci(A)$, \,$\simeq$ \, and \,$\sim$ \, are congruence relations for the elements
in $A$.
\begin{Lem}\label{aas1}
\begin{enumerate}
\item[(i)] $S_1^{aas}\sim0$ and $S_1^{as}\simeq0$.
\item[(ii)] If $A$ is an anticommutative g-alternative algebra, then $A$
is antiassociative and second level associative, i.e.
$((xy)z)a=(x(yz))a$, for any $a,x,y,z\in A$.
\item[(iii)] If $A$ is an anticommutative g-alternative algebra
and $\ann(A)=0$, then $A$ is antiassociative, associative and
$2x(yz)=2(xy)z=0$, for any $x,y,z\in A$.
\item[(iv)] If $A$ is anticommutative g-alternative algebra over a
field $F$ with $\ch F \neq 2$ and $\ann(A)=0$, then
$A=0$.
\end{enumerate}
\end{Lem}
\begin{proof} (i) First we show that $S_1^{aas}\sim0$. We have
$(xy)z+x(yz)\sim -(xz)y-y(xz)\sim -(xz)y+(xz)y=0$.
For $S_1^{as}\simeq0$, from Axiom $2_1$ we obtain $(x(yz))a\sim
-(xa)(yz)$.
Applying antiassociativity up to congruence relation (i.e. the
fact that $S_1^{aas}\sim0$) and the definition of $\soci(A)$ we
obtain $((xy)z)a\sim -((xy)a)z\sim ((xa)y)z\sim -(xa)(yz)$.
Therefore we have $(x(yz))a-((xy)z)a\sim 0,$ which proves that
$S_1^{as}\simeq0$.
(ii) Antiassociativity of $A$ is a special case of
$S_1^{aas}\sim0$ in (i), where ``$\sim$'' is replaced by ``='',
since $A$ is anticommutative. The proof of the second level
associativity of $A$ is a special case of the proof of
$S_1^{as}\simeq0$ in (i), where $x, y,z\in A$ and ``$\sim$'' is
replaced by ``=''again.
(iii) Since $\ann(A)=0$, second level associativity of $A$ implies
associativity and therefore, applying (ii) we have
$x(yz)=(xy)z=-x(yz)$, from which follows the result.
(iv) Since from (iii) $(2\cdot 1_F)x(yz)=0$ and $\ch F
\neq 2$, it follows that $x(yz)=0$ for any $x,y,z\in A$, which
implies that $z=0$, for any $z\in A$, since $\ann(A)=0$.
\end{proof}
\begin{Prop}\label{bar_sac2}
$\bar{S}_2^{\rm sac}\simeq 0$.
\end{Prop}
The proof is based on several lemmas.
\begin{Lem}\label{sac2}
$S_2^{\rm sac}\simeq 0$.
\end{Lem}
\begin{proof} We shall prove the congruence relation for the
elements of the type $x(yz)+x(zy)$, and the case $(yz)x+(zy)x$ is
left to the reader.
Consider the case where $x=b_i, y=b_j$. For any $a\in A$ we have
\[(b_i(b_jz)+b_i(zb_j))a\sim -(b_ia)(b_jz)-(b_ia)(zb_j) \, .\]
The right side is an element of $S_1^{\rm sac}$ and therefore it
is an element of $\soci(A)$, which proves that
$(b_i(b_jz)+b_i(zb_j))\sim 0$. Analogously, it can be proved that
$((xb_i)b_j+(b_ix)b_j)a\sim0$ and therefore
$(xb_i)b_j+(b_ix)b_j\simeq0$.
Consider the case $y=b_i, z=b_j$. Thus we have to show that
$x(b_ib_j)+x(b_jb_i)\in \asoci(A)$. For any $a\in A$ we have
$(x(b_ib_j)+x(b_jb_i))a=((xb_i)b_j)a+((b_ix)b_j)a-(b_i(xb_j))a+((xb_j)b_i)a+
((b_jx)b_i)a-(b_j(xb_i))a$. Applying the case noted above (i.e.,
$((xb_i)b_j+(b_ix)b_j)a\sim0$) we obtain that the right side is
$\sim$-congruent to the following one $-(b_i(xb_j))a-(b_j(xb_i))a=
-b_i((xb_j)a)-b_i(a(xb_j))+(b_ia)(xb_j)-b_j((xb_i)a)-b_j(a(xb_i))+(b_ja)(xb_i)\sim
(b_ia)(xb_j)+(b_ja)(xb_i)$.
For any $a'\in A$ we have
$((b_ia)(xb_j))a'\sim-((b_ia)a')(xb_j)\sim(((b_ia')a)xb_j)\sim
-(b_ia')(a(xb_j))\sim(b_ia')(x(ab_j))\sim-(b_ia')(x(b_ja))\sim
(b_ia')((b_ja)x)\sim-((b_ia')(b_ja))x\sim((b_ja)(b_ia'))x\sim-(b_ja)((b_ia')x)\sim
(b_ja)((b_ix)a')\simeq((b_ja)(b_ix))a'\sim((b_ja)(xb_i))a'$.
Here we applied the fact that $\soci(A)\sim0$ and Lemma \ref{aas1} (i).
Thus we obtain that $((b_ia)(xb_j)+(b_ja)(xb_i))a'\simeq0$, which
gives that $((x(b_ib_j)+x(b_jb_i))a)a'\in \asoci(A)$, for any $a,
a'\in A$ and therefore $x(b_ib_j)+x(b_jb_i)\in \asoci(A)$.
\end{proof}
\begin{Lem}\label{bar sac1}
$\bar{S}_1^{\rm sac}\simeq 0$.
\end{Lem}
\begin{proof} We shall show that $(b_ib_j)(yz)+(b_ib_j)(zy)\in
\asoci(A)$ and other cases are left to the reader.
$(b_ib_j)(yz)+(b_ib_j)(zy)=b_i(b_j(yz))+b_i((yz)b_j)-(b_i(yz))b_j+b_i(b_j(zy))+
b_i((zy)b_j)-(b_i(zy))b_j\simeq-(b_i(yz))b_j-(b_i(zy))b_j\simeq(b_i(zy))b_j-(b_i(zy))b_j\simeq0$.
Here we applied Proposition \ref{asoci_sub} and Lemma \ref{sac2}.
\end{proof}
\begin{Lem}\label{aas2}
$S_2^{\rm as}\simeq0$, \quad $S_2^{\rm aas}\simeq 0$.
\end{Lem}
\begin{proof} We shall prove that $b_k(yb_l)-(b_ky)b_l\simeq0$, for
any $y\in A$, $b_k\in B_k$ and $b_l\in B_l$ $k, l\in I$; other
cases are proved in analogous ways applying Lemma \ref{bar sac1}. We have
$(b_k(yb_l))a-((b_ky)b_l)a\simeq-(b_ka)(yb_l)+((b_ky)a)b_l\simeq
((b_ka)y)b_l+((b_ky)a)b_l\simeq-((b_ky)a)b_l+((b_ky)a)b_l\simeq0$.
\end{proof}
Applying these lemmas we prove Proposition \ref{bar_sac2}.
\begin{proof}(Proposition \ref{bar_sac2}) We shall show that
$(b_ib_j)(y(b_kb_l))+(b_ib_j)((b_kb_l)y)\in \asoci(A)$.
We have
$(b_ib_j)(y(b_kb_l))+(b_ib_j)((b_kb_l)y)=(b_ib_j)((yb_k)b_l+(b_ky)b_l-b_k(yb_l))+
(b_ib_j)(b_k(b_ly)+b_k(yb_l)-(b_ky)b_l)$.
Applying Proposition \ref{asoci_sub} and Lemma \ref{sac2} we
obtain that this expression is $\simeq$-congruent to the following
$-(b_ib_j)(b_k(yb_l))-(b_ib_j)((b_ky)b_l)$, which by Lemma
\ref{aas2} is $\simeq$-congruent to $0$.
\end{proof}
\begin{Lem}\label{bar_aas1}
$\bar{S}_1^{\rm as}\simeq 0$, \quad $\bar{S}_1^{\rm aas}\simeq 0$.
\end{Lem}
\begin{proof} We shall prove that $((b_ib_j)y)z-(b_ib_j)(yz)\simeq0$, $y, z\in A$.
We have
$((b_ib_j)y)z-(b_ib_j)(yz)=(b_i(b_jy)+b_i(yb_j)-(b_iy)b_j)z-b_i(b_j(yz))-
b_i((yz)b_j)+(b_i(yz))b_j\simeq -((b_iy)b_j)z+(b_i(yz))b_j\simeq
-((b_iy)b_j)z-(y(b_iz))b_j\simeq((b_iy)z)b_j-(y(b_iz))b_j\simeq
-((yb_i)z)b_j-(y(b_iz))b_j\simeq-(y(b_iz))b_j-(y(b_iz))b_j=0$.
Other cases are proved in similar ways applying Proposition \ref{bar_sac2}.
\end{proof}
\begin{Prop}\label{bar_as2}
$\bar{S}_2^{\rm as}\simeq 0$.
\end{Prop}
{\begin{proof} We shall show that
$(b_ib_j)(y(b_kb_l))-((b_ib_j)y)(b_kb_l)\simeq0$. The general case
can be proved by application Lemmas \ref{aas2} and \ref{bar_aas1}. We apply Lemma
\ref{aas2} and obtain
$(b_ib_j)(y(b_kb_l))-((b_ib_j)y)(b_kb_l)\simeq(b_ib_j)((yb_k)b_l)-(((b_ib_j)y)b_k)b_l\simeq
b_i(b_j((yb_k)b_l))-((b_i(b_jy))b_k)b_l\simeq
b_i((b_j(yb_k))b_l)-((b_i(b_jy))b_k)b_l\simeq
(b_i(b_j(yb_k)))b_l-((b_i(b_jy))b_k)b_l\simeq(b_i(b_j(yb_k)))b_l-((b_i(b_jy))b_k)b_l\simeq
((b_i(b_jy))b_k)b_l-((b_i(b_jy))b_k)b_l\simeq0$.
\end{proof}
In analogous ways are proved the following statements.
\begin{Prop}\label{bar_aas2}
$\bar{S}_2^{ \rm aas}\simeq 0$.
\end{Prop}
\begin{Lem}\label{bar_ap1}
$\bar{S}_1^{\rm ap}\simeq 0$.
\end{Lem}
\begin{Prop}\label{bar_ap2}
$\bar{S}_2^{\rm ap}\simeq 0$.
\end{Prop}
\section{Sufficient conditions for $\mathfrak{B}(A)\in \galt$} \label{sufficient}
We know from Section \ref{def} that $\mathfrak{B}(A)$ is an object in $\galt_G
$. We begin
this section with an example, which shows that the elements of $\mathfrak{B}(A)$ generally
do not satisfy Axiom~$2_1$ and Axiom~$2_2$, and therefore $\mathfrak{B}(A)$ is not a
g-alternative algebra in general.
\begin{ex}\label{ex_prod} Let $A$ and $\Lambda$ be associative algebras over a field $F$ with $\ch F \neq 2, 3$; let $A$
be anticommutative, $\ann(A)\neq 0$, and $A$
has a
derived action of $\Lambda$ in the category of associative
algebras, such that $\lambda aa'=0, \lambda\lambda ' a=0$ and
$\lambda a=-a\lambda$, for any $\lambda, \lambda'\in \Lambda,$ and $a,a'\in
A$. For example one can take $\Lambda=A$, since
anticommutativity of $A$ implies antiassociativity, and together
with $\ch F \neq 2$ it gives that $aa'a''=0$ for any $a,a',
a''\in A$. Let $R$ be a g-alternative algebra with unit 1, which acts on $A$
in $\galt$, in such a way that $1a=a1=a$ for any $a\in A$.
Let $A\times A$ be the product associative algebra. Consider the
following actions of $R$ and $\Lambda$ on $A\times A$:
\[r(a,a')=(ra,0), \quad
(a,a')r=(ar,0), \qquad
\lambda(a,a')=(0,\lambda a), \quad
(a,a')\lambda=(0,a\lambda),\]
for any $r\in R, \lambda\in \Lambda$ and $(a,a')\in A\times A$.
It is obvious that the action of $R$ on $A\times A$ is a derived action
and, it can be easily checked, that the action of $\Lambda$ on
$A\times
A$ is a derived action as well.
We will show that the equality
\[b_i(b_jb_k)=(b_ib_j)b_k+(b_jb_i)b_k-b_j(b_ib_k) \, ,\]
where $b_i\in B_i$, $b_j\in B_j$, $b_k\in B_k$ doesn't hold in
general.
Consider the case, where $b_i=\lambda$, $b_j=b_k=1$,
$\lambda\in \Lambda$, and $1$ is unit of $R$.
We first compute the results of the following actions and obtain:
\begin{align*}
(1 \lambda)(a,a') &=-(0,a\lambda), &
(\lambda 1)(a,a') &=(0, 2\lambda a), \\
(a,a')(1\lambda) &=(0,2a\lambda), &
(a,a')(\lambda 1) &=-(0, \lambda a), \\
\lambda (1(a,a')) &=(0,\lambda a), &
\lambda((a,a')1) &=(0,\lambda a), \\
(1(a,a'))\lambda &=(0,a\lambda), &
((a,a')1)\lambda &=(0,a\lambda), \\
r((a,a')\lambda) &=(0,0), &
r(\lambda(a,a')) &=(0,0), \\
((a,a')\lambda)r &=(0,0), &
(\lambda(a,a'))r &=(0,0),
\end{align*}
for any $(a,a`)\in A\times A, r\in R$ and unit $1$ of $R$.
We shall show that the following equality is not true in general
\[(\lambda(1 \cdot 1))(a,a')= ((\lambda
1)1)(a,a')+((1\lambda)1)(a,a')-(1(\lambda 1))(a,a') \, .\]
The computations of both sides give that this equality is
equivalent to the following one \quad
$2\lambda a=4\lambda a -2 a\lambda -\lambda a=0$.
Since $\lambda a=-a\lambda$ and $\ch F\neq 3$, this equality gives
$\lambda a=0$, which is not true in general.
This shows that in the case of this example Axiom~$2_1$ is not
true. The same example Axiom~$2_2$ is not true as well.
This can be checked by analogous computations or
we can apply the duality in the following way. Define in $A$ the
dual operation by $x \circ y=yx$. Axiom~$2_2$ for the original dot
operation is equivalent to the Axiom~$2_1$ for the dual
``$\circ$'' operation. But since both operations have the same
properties, we can conclude from the above prove that Axiom~$2_1$
is not true for the ``$\circ$'' operation.
\end{ex}
We are looking for the sufficient conditions for $\mathfrak{B}(A)$ to be a
g-alternative algebra, i.e. for the conditions under which the
elements of $\mathfrak{B}(A)$ satisfy Axioms~$2_1$ and $2_2$. For any $b_1,
b_2, b_3 \in \mathfrak{B}(A)$ we must have the following identities
\begin{enumerate}
\item[] B1. \quad $-(b_1(b_2b_3))a+((b_1b_2)b_3)a+((b_2b_1)b_3)a-(b_2(b_1b_3))a=0$ ,
\item[] B2. \quad $-a(b_1(b_2b_3))+a((b_1b_2)b_3)+a((b_2b_1)b_3)-a(b_2(b_1b_3)=0$;
\end{enumerate}
and the dual identities
\begin{enumerate}
\item[] B3= B2$^\circ$. \quad $-((b_1b_2)b_3)a+(b_1(b_2b_3))a+(b_1(b_3b_2))a-((b_1b_3)b_2)a=0$,
\item[] B4= B1$^\circ$. \quad $-a((b_1b_2)b_3)+a(b_1(b_2b_3)+a(b_1(b_3b_2))-a((b_1b_3)b_2)=0$.
\end{enumerate}
First we compute the left side of identity B1. By the definition of the multiplication
in $\mathfrak{B}(A)$ we obtain:
\begin{multline} \label{B1}
-(b_1(b_2b_3))a+((b_1b_2)b_3)a+((b_2b_1)b_3)a-(b_2(b_1b_3))a\\
= -b_1((b_2b_3)a)-b_1(a(b_2b_3))
+(b_1a)(b_2b_3) + (b_1b_2)(b_3a)
+(b_1b_2)(ab_3)
-((b_1b_2)a)b_3\\
+ (b_2b_1)(b_3a)+(b_2b_1)(ab_3)-((b_2b_1)a)b_3-
b_2((b_1b_3)a)-b_2(a(b_1b_3))+(b_2a)(b_1b_3) \\
= -b_1(b_2(b_3a))-b_1(b_2(ab_3))+b_1((b_2a)b_3)-
b_1((ab_2)b_3)-b_1((b_2a)b_3)+b_1(b_2(ab_3)) \\
+((b_1a)b_2)b_3+(b_2(b_1a))b_3-b_2((b_1a)b_3)+
b_1(b_2(b_3a))+b_1((b_3a)b_2)-(b_1(b_3a))b_2 \\
+ b_1(b_2(ab_3))+b_1((ab_3)b_2)-(b_1(ab_3))b_2-
(b_1(b_2a))b_3-(b_1(ab_2))b_3+((b_1a)b_2)b_3 \\
+ b_2(b_1(b_3a))+b_2((b_3a)b_1)-(b_2(b_3a))b_1+
b_2(b_1(ab_3))+b_2((ab_3)b_1)-(b_2(ab_3))b_1 \\
- (b_2(b_1a))b_3-(b_2(ab_1))b_3+((b_2a)b_1)b_3-
b_2(b_1(b_3a))-b_2(b_1(ab_3))+b_2((b_1a)b_3) \\
- b_2((ab_1)b_3)-b_2((b_1a)b_3)+b_2(b_1(ab_3))+
((b_2a)b_1)b_3+(b_1(b_2a))b_3-b_1((b_2a)b_3).
\end{multline}
The left side of identity B2 gives the following
\begin{multline} \label{B2}
-a(b_1(b_2b_3))+a((b_1b_2)b_3)+a((b_2b_1)b_3)-a(b_2(b_1b_3))\\
= -(ab_1)(b_2b_3)-(b_1a)(b_2b_3)+b_1((ab_2)b_3)-
((b_1a)b_2)b_3-(b_1(b_2a))b_3+b_1((b_2a)b_3)\\
+ b_2((ab_1)b_3)+b_2((b_1a)b_3)-b_2(b_1(ab_3))-
(ab_2)(b_1b_3)-(b_2a)(b_1b_3)+b_2(a(b_1b_3))\\
= -((ab_1)b_2)b_3-(b_2(ab_1))b_3+b_2((ab_1)b_3)-((b_1a)b_2)b_3-(b_2(b_1a))b_3+b_2((b_1a)b_3)\\
+ b_1((ab_2)b_3)+b_1((b_2a)b_3)-b_1(b_2(ab_3))+((ab_1)b_2)b_3+((b_1a)b_2)b_3-(b_1(ab_2))b_3\\
+(b_1(b_2a))b_3+(b_1(ab_2))b_3-((b_1a)b_2)b_3-b_1(b_2(ab_3))-b_1((ab_3)b_2)+(b_1(ab_3))b_2\\
+((ab_2)b_1)b_3+((b_2a)b_1)b_3-(b_2(ab_1))b_3+(b_2(b_1a))b_3+(b_2(ab_1))b_3-((b_2a)b_1)b_3\\
- b_2(b_1(ab_3))-b_2((ab_3)b_1)+(b_2(ab_3))b_1-((ab_2)b_1)b_3-(b_1(ab_2))b_3+b_1((ab_2)b_3)\\
- ((b_2a)b_1)b_3-(b_1(b_2a))b_3+b_1((b_2a)b_3)+b_2((ab_1)b_3)+b_2((b_1a)b_3)-b_2(b_1(ab_3)).
\end{multline}
The identities B3 and B4 give the duals to the expressions
\eqref{B2} and \eqref{B1} respectively.
It is easy to see that all obtained expressions are the combinations
of the elements of the following type
\begin{enumerate}
\item[] $\mathbf{A}1=b_1(b_2(b_3a))+b_1(b_2(ab_3))$,
\item[] $\mathbf{A}2=b_1((ab_2)b_3)+b_1((b_2a)b_3)$,
\item[] $\mathbf{A}3=((b_1a)b_2)b_3+(b_2(b_1a))b_3$,
\item[] $\mathbf{A}4=b_1(b_2(ab_3))+b_2(b_1(ab_3))$,
\item[] $\mathbf{A}5=((b_1a)b_2)b_3+((b_2a)b_1)b_3$,
\item[] $\mathbf{A}6=b_1(b_2(b_3a))+b_1((b_3a)b_2)$,
\item[] $\mathbf{A}7=b_1(b_2(ab_3))+b_1((ab_3)b_2)$,
\item[] $\mathbf{A}8=(b_1(b_2a))b_3+(b_1(ab_2))b_3$,
\item[] $\mathbf{A}9=((ab_1)b_2)b_3+(b_2(ab_1))b_3$,
\item[] $\mathbf{A}10=((ab_3)b_1)b_2+((ab_3)b_2)b_1$,
\item[] $\mathbf{A}11=b_3(b_2(ab_1))+b_3(b_1(ab_2))$,
\end{enumerate}
where $b_1, b_2, b_3 \in \mathfrak{B}(A), a\in A$.
\begin {theo} If for any $b_1, b_2, b_3 \in \mathfrak{B}(A), a\in A$, we have
the equalities $\mathbf{A}i=0$ for $i=1,\dots,11$, then $\mathfrak{B}(A)$ is
a g-alternative algebra.
\end{theo}
\begin{proof} The proof follows directly from the identities \eqref{B1},
\eqref{B2} and the dual identities, which proof that under the
conditions of the theorem we have Axioms~$2_1$ and $2_2$ for the
elements of $\mathfrak{B}(A)$.
\end{proof}
On the other hand the following proposition shows that $\mathfrak{B}(A)$ is
a g-alternative algebra up to the congruence relation $\simeq$.
\begin {Prop} For any $b_1, b_2, b_3 \in \mathfrak{B}(A), a\in A$, we have
$\mathbf{A}i\simeq 0$ for $i=1,\dots,11$.
\end {Prop}
\begin{proof} Direct application of the statements \ref{bar_sac2}, \ref{bar_as2}, \ref{bar_aas2}
and \ref{bar_ap2}.
\end{proof}
\begin{coro}\label{asoci0} If $\asoci(A)=0$, then $\mathfrak{B}(A)$ is a
g-alternative algebra.
\end{coro}
\begin{Lem}\label{asoci10} Let $\{B_i\}_{i\in I}$ be the family of all g-alternative
algebras which have a derived action on $A$. The following
conditions are equivalent:
\begin{enumerate}
\item[(a)] $\asoci(A)=0$;
\item[(b)] $\asoci^1(A)=0$;
\item[(c)] every derived action of $B_i$ on $A$ is anticommutative, for
any $i\in I$ (i.e. $b_ia=-ab_i$, $b_i\in B_i$, $a\in A$) and
$\ann(A)=0$.
\end{enumerate}
\end{Lem}
\begin{proof} The implication (a)$\Rightarrow$(b) is obvious, since
$\asoci^1(A)\subseteq \asoci(A)$.
(b)$\Rightarrow$(c). We have $b_ia+ab_i\in \asoci^1(A)$; since
$\asoci^1(A)=0$,
it follows that every derived action is anticommutative; in particular,
since $A$ has a derived action on itself, $A$ is anticommutative. From this it follows that the right and left
annulators of $A$ coincide and thus $\rann(A)=\lann(A)=\ann(A)$.
As we have noted in Section \ref{properties}, we have an inclusion
$\soci(A)\subseteq \asoci^1(A)$, from which we obtain that
$\soci(A)=0$. Therefore
we obtain that $\ann(A)=\asoci^1(A)=0$.
(c)$\Rightarrow$(a). Since every action of $B_i$ on $A$ is
anticommutative, we have $\soci(A)=0$. Therefore for $x\in
\asoci(A)$ there exists a natural number $n$, such that for any
$a_1,\dots,a_n\in A$ we have $(\dots((xa_1)a_2)\dots a_n)=0$.
Since $\ann(A)=0$, it follows that $(\dots ((xa_1)a_2)\dots
a_{n-1})=0$. Thus applying analogous arguments we obtain that
$xa_1=0$ for any $a_1\in A$ and therefore $x=0$ since $\ann(A)=0$,
which ends the proof.
\end{proof}
\begin{Prop} \label{anti}If $A$ is anticommutative g-alternative algebra over a
field $F$ and $\ann (A)=0$, then $(a_1b)a_2=a_1(ba_2)$, for
any g-alternative algebra $B$ with derived action on $A$ and any
$a_1, a_2\in A, b\in B$.
\end{Prop}
\begin{proof} If $\ch F
\neq 2$, then by Lemma \ref{aas1} (iv) $A=0$, so the equality always
holds. Consider the case $\ch F=2$. Since $A$ is anticommutative,
by Lemma \ref{aas1} (ii), it is antiassociative as well, and
therefore, for any $a\in A$, we have the following equalities
\begin{multline*}
((a_1b)a_2)a=-(a_2(a_1b))a=(a_1(a_2b))a=-(a_1a)(a_2b)=a_2((a_1a)b)=\\-((a_1a)b)a_2=
-(a_1(ab)+a_1(ba)-(a_1b)a)a_2=-(-a(a_1b)+a_1(ba)-(a_1b)a)a_2=\\-((a_1b)a+a_1(ba)-(a_1b)a)a_2=
-(a_1(ba))a_2=a_1((ba)a_2)=-a_1((ba_2)a)=(a_1(ba_2))a, \\
(a_1(ba_2))a=-(a_1a)(ba_2)=(a_1(ba_2))a, \qquad \qquad \qquad \qquad \qquad
\end{multline*}
from which by the condition $\ann (A)=0$ follows the desired
equality.
\end{proof}
Note that the proof of this proposition doesn't follow from Lemma
\ref{aas1} (i) by taking there $x=a_1, y=b, z=a_2$, since in the
proof of $S_1^{\rm as}\sim0$ is involved $\soci(A)$, which
contains an element from $B$.
\begin{Prop}\label{ann0} Let $A$ be a g-alternative algebra with $\ann(A)=0$. The following conditions are equivalent:
\begin{enumerate}
\item[(i)] for any g-alternative algebra $B$ with derived action on $A$
in $\galt$, we have $ba=-ab$ for any $a\in A, b\in B$;
\item[(ii)] $A$ is anticommutative.
\end{enumerate}
\end{Prop}
\begin{proof} Here, as in the previous proposition, we need to
prove only the case, where $\ch F=2$. It is obvious that (i)
$\Rightarrow$ (ii), since A has a derived action on itself.
(ii) $\Rightarrow$ (i). For any
$a',a''\in A$ we have
$a''(a'(ab)+a'(ba))=a''(-a(a'b)+a'(ba))=a''(-a(a'b)+(a'b)a+(ba')a-b(a'a))=
a''(-a(a'b)-a(a'b)+(ba')a-(ba')a-(a'b)a+a'(ba))=-2a''(a(a'b))=0$.
Here we applied anticommutativity of $A$, Axiom~2, Proposition \ref{anti}
and Lemma \ref{aas1} (iii).
\end{proof}
\begin{Prop}\label{antig_ann0} If $A$ is an anticommutative g-alternative algebra and $\ann(A)=0$, then $\mathfrak{B}(A)$ is a g-alternative algebra.
\end{Prop}
\begin{proof} Apply Corollary \ref{asoci0}, Lemma \ref{asoci10} and Proposition \ref{ann0}.
\end{proof}
\section{Sufficient conditions for the existence of an actor in $\galt$}\label{existence_galt}
It is obvious that the action of $\mathfrak{B}(A)$ on $A$ satisfies identities II$_1$ and
II$_2$; but this action in general is not a derived action. We begin
with examples, which show that all other action identities fail in
general.
\begin{ex} \label{ex_prod2} Here we consider the product algebra $A\times A$ of Example \ref{ex_prod} and we show
that the identity II$_3$ is not true in general. We take in
II$_3$:
instead of $a$ the element $(a,a')\in A\times A, b_1=1, b_2=\lambda$ and obtain that in this case II$_3$
is equivalent to the following equality \
$(0,a\lambda)=(0,2a\lambda)-(0,\lambda a)$.
From this, since by assumption $a\lambda=-\lambda a$ and $\ch
F\neq 2$, we obtain that $a\lambda=0$, which is not true in general.
\end{ex}
\begin{ex} Consider the same example of the algebra $A\times A$ as
in Examples \ref{ex_prod} and \ref{ex_prod2}. We take in II$_4$ instead of $a$ the element $(a,a'), b_1=\lambda,
b_2=1$ and obtain \
$(0,\lambda a)=(0,2\lambda a)-(0,a\lambda)$.
As in the previous example, this implies $a\lambda=0$,
which is not true in general.
\end{ex}
\begin{ex} Here we consider the example which shows that the
identity I$_1$ is not always true.
Let $A$ and $R$ be commutative, associative algebras over a
field $F$ with characteristic 2,
and $R$ has a derived action on $A$ in the category of commutative, associative
algebras, i.e. together with the conditions given in Section \ref{def}. We
have $ra=ar$, for any $a\in A$ and $r\in R$. Obviously this will be a derived action
in the category of g-alternative algebras as well. Let
$\Lambda$ be a
g-alternative algebra over the same field $F$, which acts on $A$
in $\galt$, and there exists an element $a' \in A$, such that
$a'\lambda$ is not a zero divisor in $A$. We shall show that
the following equality is not true
\[(r\lambda)(aa')=((r\lambda)a)a'+(a(r\lambda))a'-a((r\lambda)a') \, ,\]
for any $a\in A$. Under the above assumptions on actions and algebras
the computations give the following:
\begin{itemize}
\item $(r\lambda)(aa')=r(\lambda(aa'))+r((aa')\lambda)-(r(aa'))\lambda=
r((\lambda a)a')+r((a\lambda)a')-r(a(\lambda a'))-r((a\lambda)a')+((ra)\lambda)a'$;
\item $((r\lambda)a)a'=(r(\lambda a))a'+(r(a\lambda))a'-((ra)\lambda)a'$;
\item $(a(r\lambda))a'=-(r(a\lambda)a'$;
\item $-a((r\lambda)a')=-a(r(\lambda
a'))-a(r(a'\lambda))+a((ra')\lambda)$.
\end{itemize}
Thus the above equality is equivalent to the following one:
$a((ra')\lambda)=(ar)(a'\lambda)$.
Applying the fact that $A$ is commutative and $\ch F=2$, we have
$a((ra')\lambda)=-(ra')(a\lambda)$; therefore we obtain
$-(ra')(a\lambda)=(ar)(a'\lambda)$,
which can not be true for any $a$, since in this case
$a\lambda=0$ implies $ar=0$, because $a'\lambda$ is not a zero
devisor by assumption.
The cases of the identities I$_2$, I$_3$, I$_4 $ are considered in
analogous ways.
\end{ex}
\begin{theo}\label{asoci0_derived} If $\asoci(A)=0$, then the action of $\mathfrak{B}(A)$ on
$A$ is a derived action.
\end{theo}
\begin{proof} From the definition of the multiplication in $\mathfrak{B}(A)$ it is easy to see that the identities II$_1$ and
II$_2$ of Section \ref{acting}
always hold for the action of $\mathfrak{B}(A)$ on $A$. For the identities
$\mbox{I}_1-\mbox{I}_4$ and II$_3$ and II$_4$ we apply statements \ref{bar_aas1} and \ref{bar_as2},
which show that under the condition of the theorem
$\mathfrak{B}(A)$ has a derived action on $A$. Note that the same can be
proved by the statements \ref{bar sac1}, \ref{bar_ap1},
\ref{bar_sac2} and \ref{bar_ap2}.
\end{proof}
\begin{coro}\label{asoci0_actor} If $\asoci(A)=0$, then $\mathfrak{B}(A)$ is an actor of
$A$.
\end{coro}
\begin{proof} Apply Corollaries \ref{actor_interest}, \ref{asoci0} and Theorem \ref{asoci0_derived}.
\end{proof}
\begin{coro} \label{anti_ann0} If $A$ is anticommutative g-alternative algebra over a field $F$ and
$\ann(A)=0$, then there exists an actor of $A$ and
$\act(A)=\mathfrak{B}(A)$.
\end{coro}
\begin{proof} If $\ch F
\neq 2$, then, by Lemma \ref{aas1} (iv), it follows that $A=0$, and
obviously $\act(A)=\mathfrak{B}(A)=0$. If $\ch F=2$, we apply Lemma
\ref{asoci10}, Propositions \ref{ann0} and \ref{antig_ann0},
Corollary \ref{asoci0_actor} and obtain the result.
\end{proof}
Here we give the construction of an $F$-algebra of
bimultiplications $\bim_{\galt}(A)$ of a g-alternative algebra $A$
over a field $F$. Below is used the notation $f*$ and $*f$ for the
$F$-linear maps $A\rightarrow A$; we will denote by $fa$ (resp.
$af$) the value $(f*)(a)$ (resp. $(*f)(a)$). This kind of notation
(similar to the one of the actions $b*a$ and $a*b$ in a category
of interest) makes simpler to write down the conditions for
bimultiplications; we will see that these conditions are simply
Axioms~$2_1$ and $2_2$ written for the four different ordered
triples. An element of $\bim_{\galt}(A)$ is a pair $f=(f*,*f)$ of
$F$-linear maps from $A$ to $A$, which satisfies the following
conditions
\begin{equation}\label{pair}
\begin{split}
f (a_1 a_2)& = (f a_1) a_2 + (a_1 f)a_2 - a_1(f a_2) \, , \\
(a_1 a_2) f & = a_1 (a_2 f) + a_1 (f a_2) - (a_1 f) a_2 \, ,\\
(f a_1) a_2 & = f (a_1 a_2) + f (a_2 a_1) - (f a_2) a_1 \, , \\
a_1(a_2 f)&=(a_1 a_2)f + (a_2 a_1) f - a_2(a_1 f) \, .
\end{split}
\end{equation}
The product of the elements $f=(f*,*f)$ and $f'=(f'*,*f')$ of
$\bim_{\galt}(A)$ is defined by
\[ ff'=(f*f'*, *f*f')\, , \]
here on the right side $f*f'*$ \ and \ $*f*f'$ are defined by
\begin{align*}
&(f*f'*)(a)=f(f'a)+f(af')-(fa)f' \, ,\\
&(*f*f')(a)=(af)f'+(fa)f'-f(af') \, .
\end{align*}
For the addition we have
\[f+f'=((f*)+f'*,*f+(*f')),\]
where
\begin{align*}
&((f*)+f'*)(a)=fa+f'a \, ,\\
&(*f+(*f'))(a)=af+af'.
\end{align*}
The product of two bimultiplications may not have the properties
\eqref{pair}. Therefore we include in $\bim_{\galt}(A)$ all the new
obtained pairs of $F$-linear maps. Note that different products
can give one and the same pairs of maps, i.e.
$(\varphi*,*\varphi)=(\varphi'*,*\varphi')$ if $\varphi a=\varphi'
a$ and $a \varphi=a \varphi'$, where $\varphi=(\varphi*,*\varphi)$
and $\varphi'=(\varphi'*,*\varphi')$ are certain combinations of
bimultiplications.
It is obvious that $\bim_{\galt}(A)$ is an $F$-algebra and it
is an object of $\galt_G$ in general. In the same way as
Corollary \ref{anti_ann0}, it can be proved
\begin{theo}\label{act_bim}
Let $A$ be an anticommutative g-alternative algebra with
$\ann(A)=0$, then there exists an actor of $A$ and
$\act(A)=\bim_{\galt}(A)$.
\end{theo}
\begin{coro}\label{bim_beta} Under the condition of Theorem \ref{act_bim} we have
$\bim_{\galt}(A)=\mathfrak{B}(A)$.
\end{coro}
\begin{proof} The conditions of Corollary \ref{anti_ann0} are fulfilled, from
which it follows that $\mathfrak{B}(A)$ is an actor of $A$. From Theorem
\ref{act_bim} and the universal property of an actor we obtain the desired
equality.
\end{proof}
\section{The existence of an actor in $\alt$}\label{existence_alt}
As we have noted in Section \ref{def}, by definition of general category of
interest,
we have $\alt_G=\galt_G$. Let $\alt$ be a category of alternative
algebras
over a field $F$ with $\ch F=2$. Then, by Proposition \ref{galt}, $\alt
\subset \galt$. In the categories $\alt$ and $\galt$ Axiom~2 is
the same, according to which the multiplication in the
construction of a universal strict general actor is defined.
Therefore, for any $A\in \alt$, the algebra $\mathfrak{B}_{\alt}(A)$,
constructed for the derived actions in $\alt$, is a subalgebra of
$\mathfrak{B}(A)$, constructed for the derived actions in $\galt$ for the
same alternative algebra $A$. Thus we have the inclusion of
algebras $\mathfrak{B}_{\alt}(A)\subseteq \mathfrak{B}(A)$.
\begin{Prop}\label{ann0_beta} If $A$ is anticommutative
g-alternative algebra
and $\ann(A)=0$, then $\mathfrak{B}(A)$ is also anticommutative and
$\ann(\mathfrak{B}(A))=0$.
\end{Prop}
\begin{proof} If $\ch F
\neq 2$, then, by Lemma \ref{aas1} (iv), it follows that $A=0$, and
obviously $\mathfrak{B}(A)=\ann(\mathfrak{B}(A))=0$. If $\ch F=2$, then by
Proposition \ref{ann0} every derived action on $A$ is
anticommutative; from
this and the condition $\ann(A)=0$, by Lemma \ref{asoci10} it follows that $\asoci(A)=0$.
Applying this fact, from Proposition \ref{bar_sac2}, we obtain that
$\bar{S}_2^{\rm sac}=0$, which gives the equalities $a(b_1b_2)=-a(b_2b_1)$ and
$(b_1b_2)a=-(b_2b_1)a$, for any $a\in A$ and $b_1,b_2 \in \mathfrak{B}(A)$, which
by the construction of $\mathfrak{B}(A)$, proves that $\mathfrak{B}(A)$ is
anticommutative. Now we shall prove that $\ann(\mathfrak{B}(A))=0$.
Suppose $b_1b=0$ for any $b\in \mathfrak{B}(A)$, which means that
$(b_1b)a=0$, for any $a\in A$. By the definition of multiplication in $\mathfrak{B}(A)$ and an action of
$\mathfrak{B}(A)$ on $A$ we have
\[(b_1b)a=b_1(ba)+b_1(ab)-(b_1a)b=0 \, .\]
Since the action of $\mathfrak{B}(A)$ is anticommutative, we
obtain that $(b_1a)b=0$, for any $b\in \mathfrak{B}(A)$, and in particular,
for $b={\bf{\mathfrak{a}}}'$, where $a'$ is any element from $A$. This gives $(b_1a)a'=0$,
for any $a'$, which means that $b_1a$ is an annulator
in $A$, therefore $b_1a=0$, for any $a\in A$, which, by
construction of $\mathfrak{B}(A)$, implies that $b_1=0$.
\end{proof}
Note that in the proof of Proposition \ref{ann0_beta} we could take $b={\bf{\mathfrak{a}}}$
in $b_1b=0$, for any $a\in A$. In this case we should prove that
$(b_1{\bf{\mathfrak{a}}})a'=(b_1a)a'$, for any $a'\in A$. For this we would have
to apply the fact that $\mathfrak{B}(A)$ has a derived action on $A$ in
$\galt$. From this we would obtain that $b_1{\bf{\mathfrak{a}}}$ and $b_1a$ are
equal in $\mathfrak{B}(A)$, i.e. $b_1{\bf{\mathfrak{a}}}=cl(b_1a)$, which will imply that
$(b_1a)a'=0$. From this, since $\ann(A)=0$, it would follow that
$b_1a=0$, and therefore $b_1=0$. As we see this proof is longer
than we have presented.
It is easy to see, and it is noted in \cite {Sch,ZSSS},
that any commutative or anticommutative algebra satisfies the
flexible law E$_1$. Therefore, from this proposition and
Proposition \ref{galt} (i), it follows that, under the conditions of
Proposition \ref{ann0_beta}, $\mathfrak{B}(A)$ is an alternative algebra. The following
corollary proves that $\mathfrak{B}(A)$ is an associative algebra as well.
\begin{coro}\label{beta_alt} If $A$ is anticommutative
g-alternative algebra
with $\ann(A)=0$, then $\mathfrak{B}(A)$ is an associative algebra,
in particular, $\mathfrak{B}(A)\in \alt$.
\end{coro}
\begin{proof} Apply Proposition \ref{ann0_beta} and Lemma \ref{aas1} (iii).
\end{proof}
\begin{theo} \label{act_alt} If $A$ is anticommutative alternative algebra over a
field $F$ with $\ann(A)=0$, then there exists an actor of
$A$ in $\alt$ and
$\act(A)=\mathfrak{B}_{\alt}(A)=\mathfrak{B}(A)$.
\end{theo}
\begin{proof} By Proposition \ref{galt}, for any field $F$, we have $\alt \subseteq
\galt$. If $\ch F \neq 2$, then, by Lemma \ref{aas1} (iv), it follows
that $A=0$, and obviously $\act(A)=\mathfrak{B}(A)=0$. Suppose $\ch F=2$.
Since $A$ is a g-alternative algebra,
we can apply Corollary \ref{beta_alt}, and therefore
$\mathfrak{B}(A)$ is an alternative algebra. By Corollary \ref{anti_ann0} the action of
$\mathfrak{B}(A)$ on $A$
is a derived action in $\galt$.
We shall prove that the action of $\mathfrak{B}(A)$ on $A$ satisfies
the conditions III$_1$ and III$_2$ of Section \ref{acting}, for any $a\in A$
and $b\in \mathfrak{B}(A)$. By Proposition \ref{ann0}, every derived action on $A$ is anticommutative, and by Lemma \ref{asoci10}, under the conditions of the
theorem we obtain $\asoci(A)=0$; therefore applying Lemma
\ref{bar_aas1} and Proposition \ref{bar_as2} we obtain that $\bar{S}_1^{\rm as}=0$ and
$\bar{S}_2^{\rm as}=0$, which imply respectively the equalities III$_1$ and
III$_2$. $\mathfrak{B}(A)$ is an actor of $A$ in $\galt$, therefore by the definition of an actor we obtain, that $\mathfrak{B}(A)$
is an actor of $A$ in $\alt$. Now the result follows from
Theorem \ref{actor_semi}.
\end{proof}
The construction of an $F$-algebra of bimultiplications
$\bim_{\alt}(A)$ of an alternative algebra $A$ over a field $F$ is analogous
to $\bim_{\galt}(A)$, where in addition we require that pairs $f=(f*,*f)$ of $F$-linear
maps from $A$ to $A$ together with conditions \eqref{pair} satisfy the
following two conditions:
\begin{align*}
a(fa)=(af)a \, ,\\
f(af)=(fa)f \, .
\end{align*}
The statements analogous to Theorem \ref{act_bim} and Corollary
\ref{bim_beta} take place for alternative algebras. Obviously,
every commutative associative algebra over a field with
characteristic 2 is anticommutative g-alternative algebra, and
every anticommutative g-alternative algebra $A$ over the same kind
field with $\ann(A)=0$ is associative and commutative (apply Lemma
\ref{aas1} (ii)). Let $\bim(A)$ denote the algebra of bimultiplications
of an associative algebra $A$ \cite{Ho}. It is proved in
\cite{CDL2} that if $A$ is any associative algebra with $\ann(A)=0$, then there
exists an actor of $A$ in the category of associative algebras and
$\act(A)=\bim(A)$. The analogous result we have in the category of
commutative associative algebras, under the same condition an
actor exists and this is the commutative algebra of
multiplications $M(A)$ \cite{LS} (or multipliers \cite{LL}) of
$A$, which is defined as an algebra of $F$-linear maps $f \colon A \to A$
with $f(aa')=f(a)a'$, for any $a,a'\in A$. Let $A$ be an
associative commutative algebra over a field $F$ with
characteristic 2 and $\ann(A)=0$. The equalities $\bar{S}_1^{\rm
as}=0$ and
$\bar{S}_2^{\rm as}=0$ in the
proof of Theorem \ref{act_alt} imply that the action of $\mathfrak{B}(A)$ on $A$ is a
derived action in the category of associative algebras. At the
same time we know that $\mathfrak{B}(A)$ is anticommutative and its action
on $A$ is anticommutative as well, therefore $\mathfrak{B}(A)$ is
commutative and the action on $A$ is commutative too. Therefore,
in the same way as in the proof of Theorem \ref{act_alt}, we conclude that
$\mathfrak{B}(A)$ is an actor of $A$ in the categories of associative and
commutative associative algebras. From the universal property of
an actor we obtain, that if $A$ is commutative associative algebra
over a field $F$ with characteristic 2 and $\ann(A)=0$, then
$\mathfrak{B}(A)=\bim(A)=M(A)$.
\section*{Acknowledgments}
The authors were supported by MICINN (Spain), Grant MTM 2009-14464-C02 (European FEDER support included) and project
Ingenio Mathematica (i-MATH) No. CSD2006-00032 (Consolider Ingenio
2010) and by Xunta de Galicia, Grant PGIDITI06PXIB371128PR. The
second author is grateful to Santiago de Compostela and Vigo
Universities and to Georgian National Science Foundation, Ref. ST06/3-004, for financial supports.
|
1,116,691,497,517 | arxiv | \section{Introduction}\label{intro}}
\IEEEPARstart{F}ederated learning (FL) \cite{DBLP:conf/aistats/McMahanMRHA17,DBLP:journals/tist/YangLCT19,DBLP:conf/aistats/McMahanMRHA17,DBLP:journals/spm/LiSTS20,DBLP:journals/ftml/KairouzMABBBBCC21,li2021personalized} facilitates collaborative training among a set of clients while preserving privacy so that clients can reach a better performance than individual training. Heterogeneity is an inherent problem in federated learning, since clients may have diverse optimization objectives (learning tasks) \cite{DBLP:journals/corr/abs-2106-06843}.
There may exist a cluster structure among clients, a small group of clients have similar data distributions, thus their optimization objectives are consistent, while there is dominant inconsistency among different groups. It is very common in applications such as recommendation systems \cite{sarwar2002recommender,li2003clustering}. Therefore, clustered FL methods \cite{DBLP:conf/nips/GhoshCYR20,DBLP:journals/corr/abs-2005-01026,DBLP:journals/tnn/SattlerMS21,duan2021flexible} are proposed for better personalization by grouping clients into clusters and maintaining a global model in each cluster. The main challenge of clustered heterogeneity is that the latent similarity relationships among clients are unknown. Existing clustered FL researches adopt the conventional server-client communication pattern and estimate clients' cluster identities by iterative \cite{DBLP:conf/nips/GhoshCYR20} or hierarchical \cite{DBLP:journals/tnn/SattlerMS21} methods. However, there are mainly there concerns: (1) Current clustered FL algorithms require the assumption of the number of clusters, it is not effective enough to explore the latent relationships among clients. If the assumed number of clusters is set inappropriately, outliers in a cluster will hinder overall performance. (2) The central server in the current setup bears a potential risk of weakness, the distributed system will break down if it is under malicious attack. (3) If the number of clients is enormous, the central server requires large communication bandwidth \cite{DBLP:conf/nips/LianZZHZL17}. Although the pressure of bandwidth can be relieved by reducing the number of participated clients in each round, it may result in poor convergence in cluster identity estimations.
Consequently, we take advantage of peer-to-peer (P2P) FL where clients communicate with neighbors without a central server and propose a P2P FL algorithm: \textbf{P}ersonalized \textbf{A}daptive \textbf{N}eighbour \textbf{M}atching (PANM), providing a robust and effective method to personalized clustered FL in a decentralized manner. We solve the node clustering problem into a binary classification problem: from the perspective of the client-side, each client only needs to estimate an accessible client is whether in the same cluster as itself or not. Once the neighbor estimation is correct, a clustered communication topology will be inherently established without assuming the number of clusters. Hence, PANM is effective enough to explore the latent relationships among clients.
We note that in previous works, most P2P FL algorithms assume random or fixed communication topologies \cite{DBLP:journals/corr/abs-1901-11173,lalitha2018fully} and they focus on reaching the global consensus by optimization techniques \cite{DBLP:conf/icml/0004KSJ21,DBLP:conf/icml/00010KJS21}. We will show in the experiments that the random and fixed communications will hamper personalization when the objectives of clients are divergent which is realistic in FL scenarios. In addition, there is no global consensus in clustered heterogeneity, but there may exist partial consensus in each cluster. Therefore, our method PANM realizes the construction of adaptive communication topologies in P2P FL, by which clients will achieve partial consensus in the group that shares similar objectives. To the best of our knowledge, this is the first P2P FL algorithm that takes divergent objectives among clients into consideration.
Overall, there are two stages in PANM. In stage one, PANM enables clients to initialize neighbors with high confidence of being same-cluster. In stage two, a heuristic method based on Expectation Maximization (EM) is adopted to enable clients to identify the rest of same-cluster peers and take them as neighbors. The heuristic EM-based method is formulated under the Gaussian Mixture Model assumption of similarities. Our main contributions are as follows.
\begin{itemize}
\item We propose two efficient, effective, and privacy-preserving metrics to evaluate the pair-wise similarity of client objectives in P2P FL. They are based on losses and gradients, respectively.
\item We present a novel P2P FL algorithm: PANM, which enables clients to match neighbors with consistent objectives (same cluster identity), improving local performance.
\item We devise two stages in PANM: confident neighbor initialization and heuristic neighbor matching based on EM. We provide a theoretical analysis of how our method is superior to the P2P FL counterpart.
\item We conduct extensive experiments on a spectrum of Non-IID degrees and network settings, using different datasets. It is shown that PANM outperforms all P2P baselines including Oracle (with prior knowledge of cluster identities). Compared with centralized clustered FL algorithms, PANM is more effective in exploring latent cluster structure and has comparable, even better performance.
\item Additionally, even under low communication budgets, PANM can still achieve superior performance to baselines.
\end{itemize}
The rest of this paper is organized as follows. Section \ref{sect:related} presents related works in clustered FL and P2P FL. Section \ref{sect:method} shows the motivation and the design of PANM, including two metrics for measuring client similarity in P2P FL, the first-stage confident neighbor initialization, and the second-stage heuristic neighbor matching. We also incorporate some theoretical analyses in Section \ref{sect:method}. Results of extensive experiments are shown and analyzed in Section \ref{sect:exp}. We provide further discussion in Section \ref{sect:discussion}. Then we conclude the paper in Section \ref{sect:conclusion}.
\begin{figure*}[t]
\centering
\includegraphics[width=2\columnwidth]{ images/grad1216.png}
\caption{Schematic diagram of optimization paths in centralized FL (a) and P2P FL (b), respectively. In the figure, objective of client \textit{a} is similar with client \textit{b} and dissimilar with client \textit{c}.}
\label{fig1}
\end{figure*}
\section{Related Works} \label{sect:related}
\noindent\textbf{Clustered Federated Learning. } Clustered FL holds the Non-IID assumption that different groups of clients have their own optimization objectives of learning tasks, aggregating models in the same cluster will bring better personalization while aggregating models in inconsistent clusters will cause negative transfers. Current clustered FL researches are mainly under a centralized manner.
Hierarchical clustering methods are used to achieve better personalization. Sattler \textit{et al.} \cite{DBLP:journals/tnn/SattlerMS21} use a hierarchical optimal bi-partitioning algorithm based on cosine similarity of weights or gradients. However, the bi-partitioning method bears high computation costs and requires multiple communication rounds to completely separate all clients with different objectives.
Further, Briggs \textit{et al.} \cite{DBLP:conf/ijcnn/BriggsFA20} design a hierarchical algorithm for a wider range of Non-IID settings and it reduces clustering to a single step to lower computation and communication loads. But it also requires iteratively calculating pair-wise distances between different clusters, which is computationally complex.
In addition, methods derived from K-means are used to recognize clients' cluster identities. FedSEM \cite{DBLP:journals/corr/abs-2005-01026} uses K-means to cluster clients based on clients' $l_{2}$ distances of model weights. Nonetheless, $l_{2}$ distance often suffers from high-dimension, low-sample-size situation which is known as distance concentration phenomenon in high dimension \cite{sarkar2019perfect}. To solve the efficiency problem of cluster estimation, Ghosh \textit{et al.}
\cite{DBLP:conf/nips/GhoshCYR20} propose an iterative algorithm IFCA which incorporates the essence of K-means, IFCA keeps several global models and clients iteratively choose which global model it is prone to contribute to based on local losses of global models. IFCA requires high communication costs, for it needs to broadcast all global models in each round and it is sensitive to the initialization. We also notice that for IFCA, partial participation of clients in each round may result in poor convergence. Besides, Duan \textit{et al.} \cite{DBLP:journals/corr/abs-2010-06870,duan2021flexible} use decomposed cosine similarity of models' updates to group clients and makes both intra-cluster and inter-cluster aggregations and design an efficient newcomer device cold start mechanism. K-means clustering is inherently used in their works.
Note that, all the above-mentioned algorithms in clustered FL are not effective enough in exploring latent relationships among clients. They highly rely on the level of hierarchy or the assumption of the number of clusters, however, these are latent and cannot be obtained as prior knowledge.
\noindent\textbf{Peer-to-peer Federated Learning. } Peer-to-peer federated learning (P2P FL, also known as decentralized FL) alters the centralized topology of conventional FL, it allows clients to communicate with limited neighbors \cite{warnat2021swarm,DBLP:journals/corr/abs-1901-11173}. There is a study comparing decentralized algorithms like gossip learning with centralized federated learning in terms of communication efficiency, it is found that the best gossip variants perform comparably to the best centralized federated learning overall \cite{DBLP:journals/jpdc/HegedusDJ21}. Early works related to P2P FL introduce the P2P FL problem under privacy constraints and provide theoretical guarantees; Lalitha \textit{et al.} \cite{DBLP:journals/corr/abs-1901-11173,lalitha2018fully} use a Bayesian-like approach to let clients collectively learn a model that best fits the observations over the entire network; and Bellet \textit{et al.} \cite{DBLP:conf/aistats/BelletGTT18} make P2P FL differentially private and analyze the trade-off between utility and privacy. They mainly study P2P FL under IID data assumption, but heterogeneity is prevalent in FL practices.
Recent works mostly discuss class imbalance heterogeneity and communication problems. First, to tackle class imbalance, Li \textit{et al.} \cite{DBLP:journals/corr/abs-2012-13063} use mutual knowledge distillation instead of weight averaging, Bellet \textit{et al.} \cite{DBLP:journals/corr/abs-2104-07365} elaborately design a topology from holistic perspective. However, without a central server, the holistic perspective is impractical, it is hard for clients to form such a topology with limited observations. Second, communication of P2P FL can be more efficient by sparsification \cite{DBLP:conf/icdcs/TangSC20}, adaptive partial gradient aggregation \cite{DBLP:journals/caaitrit/JiangH20}, and using max-plus linear system theory to compute throughput \cite{DBLP:conf/nips/MarfoqXNV20}.
Most recently, swarm learning \cite{warnat2021swarm} is brought up as a P2P FL customized for medical researches, utilizing edge computing and blockchain as infrastructures, and it attracts wide attention. It provides strong application practices of P2P FL. While we are formulating this paper, we find a related same-time work (PENS) that has the same motivation as ours but uses different methods \cite{DBLP:journals/corr/abs-2107-08517}. PENS adopts a two-stage strategy. In stage one, in each round, clients choose top $k$ peers as neighbors for aggregation from randomly sampled $l$ neighbor candidates. After stage one, clients select the peers that were chosen as neighbors more than ``\textit{the expected amount of times}'' in stage one as \textit{permanent neighbors}. In stage two, in each round, clients randomly choose $k$ neighbors for aggregation from \textit{permanent neighbors}. It is very possible for PENS to have noisy neighbor estimations, we analyze the superiority of PANM to PENS in Section \ref{sect:method} and \ref{sect:exp}.
\section{Method} \label{sect:method}
\subsection{Notation and Setting}
We formulate it as an empirical risk minimization problem, and the distributed optimization objective is in the sum-structered form $f: \mathbb{R}^{d} \rightarrow \mathbb{R}$, as
\begin{equation} \label{equa1}
f^{\star} \coloneqq
\min_{\mathbf{w}_{1},\cdots,\mathbf{w}_{i} \in \mathbb{R}^{d}}
\left[\frac{1}{n} \sum_{i=1}^{n} f_{i}(\mathbf{w}_{i})\right],
\end{equation}
where there are $n$ clients in the system. The $f_{i}:\mathbb{R}^{d}\rightarrow\mathbb{R}$ is the loss function of client $i\ \left(i\in \{1, \dots, n\}\right)$ on its local dataset, given by: $f_{i}(\mathbf{w})\coloneqq\mathbb{E}_{\xi \sim\mathcal{D}_{i}}\left[F_{i}(\mathbf{w},\xi)\right]$, where $\mathcal{D}_{i}$ denotes the local data distribution of client $i$. In P2P FL setting, each client $i$ maintains its local parameter weights $\mathbf{w}_{i}^{t} \in \mathbb{R}^{d}$, and updates them as
\begin{equation} \label{equa2}
\begin{split}
\mathbf{w}_{i}^{t+1}=\sum_{j \in {\rm N}_{i}^{t'}}w_{i,j}\left(\mathbf{w}_{j}^{t}-\eta\nabla F_{j}\left(\mathbf{w}_{j}^{t}, \xi_{j}\right)\right)\\
+w_{i,i}\left(\mathbf{w}_{i}^{t}-\eta\nabla F_{i}\left(\mathbf{w}_{i}^{t}, \xi_{i}\right)\right),
\end{split}
\end{equation}
where $\xi_{j} \sim\mathcal{D}_{j}$.
${\rm N}_{i}^{t'}$ is the round $t$'s aggregation neighbors of client $i$, randomly sampled from the neighbor list ${\rm N}_{i}^{t}$, ${\rm N}_{i}^{t'}\in {\rm N}_{i}^{t}$. The size of ${\rm N}_{i}^{t'}$ is $k$, while the size of ${\rm N}_{i}^{t}$ is $m$, $k\leqslant m$. $w_{i,j}$ denotes the importance weight of client $j$ to client $i$. In this paper, we set $w_{i,j}=w_{i,i}=\frac{1}{k+1}$. Equation (\ref{equa2}) shows the process of one communication round in P2P FL.
\subsection{Algorithm}
\subsubsection{Metrics for Measuring Client Similarity}
\
\newline
\begin{figure}[t]
\centering
\includegraphics[width=1\columnwidth]{images/ablation_grad.pdf}
\caption{Ablation study of PANMGrad. CIFAR10 with two rotations \{0°,180°\}, 50 clients in each cluster, $l=10,k=5$, trainset size is 400. PANMGrad refers to PANM with metric based on cosines of $\theta_{1}$ and $\theta_{2}$, PANMGrad-theta1 refers to PANM with metric based on $\theta_{1}$, PANMGrad-theta2 refers to PANM with metric based on $\theta_{2}$.}
\label{ablationgrad}
\end{figure}
Metrics for measuring the consistency of optimization objectives are needed to enable clients to select same-cluster peers and filter out outliers. However, due to privacy concerns, in FL, we cannot use data distance measurements like maximum mean discrepancy distance \cite{tzeng2014deep}, since sharing data is forbidden. Loss evaluation is a simple metric, commonly used in the literature \cite{DBLP:conf/nips/GhoshCYR20,DBLP:journals/corr/abs-2107-08517}. By evaluating other clients' models on local data, the model with lower loss value is more possible to have a similar learning task. For P2P FL, we can formulate the similarities based on loss as
\begin{equation} \label{equ6}
{\rm s}_{i, j}=1/F_{i}\left(\mathbf{w}_{j}^{t},\xi_{i} \right),
\end{equation}
where ${\rm s}_{i,j}$ is similarity between client $i$ and $j$, $\xi_{i} \sim \mathcal{D}_{i}$. Since this metric is simple, we adopt this metric in our PANM (named as PANMLoss).
However, calculating loss value is computation-consuming because it requires inferring models on the training dataset. Moreover, local data may not be available for extra computation. Hence, we develop a more efficient metric based on gradients and accumulated weight updates.
In centralized clustered FL, Sattler \textit{et al.} \cite{DBLP:journals/tnn/SattlerMS21} use the cosine similarity of gradients to measure the consistency of optimization objectives, the function can be formulated as
\begin{equation} \label{equ3}
{\rm cos}~ \theta_{i,j}^{1}=\frac{\langle \mathbf{w}_{i}^{t}-\mathbf{w}_{i}^{t-1}, \mathbf{w}_{j}^{t}-\mathbf{w}_{j}^{t-1}\rangle}{\parallel\mathbf{w}_{i}^{t}-\mathbf{w}_{i}^{t-1}\parallel \cdot \parallel\mathbf{w}_{j}^{t}-\mathbf{w}_{j}^{t-1}\parallel},
\end{equation}
where $\theta^{1}$ refers to the angle of vectorized gradients, $\mathbf{w}_{i}^{t}-\mathbf{w}_{i}^{t-1}$ refers to the gradients in the local updates of round $t$. In centralized FL, models are initialized as the same global model at the beginning of local training in each round ($\mathbf{w}_{i}^{t-1}=\mathbf{w}_{j}^{t-1}=\mathbf{w}^{t-1}$), so the cosine function of gradients can effectively imply the consistency, as shown in (a) of Figure \ref{fig1}. Although in P2P FL, without the central server, client model weights diverge since the first round, the measurement of gradients will be noisy, the angle of gradients is shown as $\theta^{1}$ in (b) of Figure \ref{fig1}.
To solve this issue, we notice the accumulated weight updates from the initial model can signify the history optimization directions, and the cosine similarity of the weight updates can imply the consistency of objectives to some extent
\begin{equation} \label{equ4}
\cos\theta_{i,j}^{2}=\frac{\langle \mathbf{w}_{i}^{t}-\mathbf{w}_{0}, \mathbf{w}_{j}^{t}-\mathbf{w}_{0}\rangle}{\parallel\mathbf{w}_{i}^{t}-\mathbf{w}_{0}\parallel \cdot \parallel\mathbf{w}_{j}^{t}-\mathbf{w}_{0}\parallel}.
\end{equation}
The angle of accumulated weight updates is $\theta^{2}$ in (b) of Figure \ref{fig1}.
According to Equation (\ref{equ3}) and (\ref{equ4}), we combine the cosine similarities of $\theta^{1}$ and $\theta^{2}$ to formulate our new metric as
\begin{equation} \label{equ5}
{\rm s}_{i,j}=\alpha \cos\theta_{i,j}^{1}+(1-\alpha)\cos\theta_{i,j}^{2},
\end{equation}
where $\alpha$ is the hyperparameter controlling the weight of two cosine functions, we set $\alpha = 0.5$ in our experiments. Notably, our new metric is robust and effective in the P2P FL setting.
Experiments in Figure \ref{ablationgrad} show that the metric combining two cosine functions in Equation \ref{equ5} can significantly enhance the effectiveness of similarity measurements, compared with using one function alone. We explain that: if using $\cos\theta^{1}$ alone, it will be very noisy since the models are not synchronized. If using $\cos\theta^{2}$ alone, clients which are communicated with in the previous rounds will have similar weights, so they are more prone to have high similarities, but if previous neighbors include outliers, they are also likely to be next-round neighbors. However, these two functions are complementary, the combination can improve performance. For the clients with high $\cos\theta^{1}$ similarity, $\cos\theta^{2}$ will imply the history of accumulated weight updates, if the $\cos\theta^{2}$ similarity is high, the clients are more likely to be same-cluster. For the clients with high $\cos\theta^{2}$ similarity, they have similar weights, and $\cos\theta^{1}$ is more effective when weights are similar (this case is similar to the centralized FL cases, where common model initialization in each round can make $\cos\theta^{1}$ more effective). Thus, the combination of the two functions can more robustly imply the similarities. We adopt the metric in Equation (\ref{equ5}) in PANM, notated as PANMGrad.
Additionally, the similarity computed by Equation (\ref{equ6}) is nearly symmetric: if two clients have distinct objectives, then cross-validation losses on both sides are high, but they are not necessarily equal; while the metric in Equation (\ref{equ5}) is completely symmetric because the inner product operation is symmetric. Symmetric metrics are beneficial to enable clients to reach consensus, which means that our newly proposed metric is superior.
\subsubsection{Confident Neighbor Initialization}
\
\newline
Based on the similarity metrics mentioned in the last subsection, we can devise our P2P FL algorithm PANM. We introduce the first stage of PANM in this subsection.
In the first stage of P2P FL training, clients have to initialize their collaborative neighbors from random sampled peers (${\rm C}_{i}^{t}, |{\rm C}_{i}^{t}|=l$). In PENS \cite{DBLP:journals/corr/abs-2107-08517}, clients select top $k$ peers with maximum similarity as neighbors in each round
\begin{equation} \label{equ7}
\begin{split}
{\rm N}_{i}^{t}&= \mathop{\arg\max}_{\rm N}\sum\limits_{j \in {\rm N}} {\rm s}_{i, j}\\
&s.t.\ {\rm N}\subsetneqq {\rm C}_{i}^{t},|{\rm N}| = k.
\end{split}
\end{equation}
Nevertheless, the probability that all the initialized neighbors are same-cluster is relatively low. Hence, we propose a more confident method: the Confident Neighbor Initialization (CNI) algorithm. In CNI, after the first round, we add the neighbors in the previous round to the candidate list in the current round (as shown in Equation \ref{equ8}), consequently, the confidence of same-cluster neighbors increases over round.
\begin{equation} \label{equ8}
\begin{split}
{\rm N}_{i}^{t}&= \mathop{\arg\max}_{{\rm N}}\sum\limits_{j \in {\rm N}} {\rm s}_{i, j}\\
&s.t.\ {\rm N}\subsetneqq {\rm C}_{i}^{t}\cup {\rm N}_{i}^{t-1},t > 1, |{\rm N}| = k.
\end{split}
\end{equation}
To show the effectiveness of CNI, we provide theoretical analysis, while making the following assumption.
\newtheorem{assumption}{Assumption}
\begin{assumption} \label{assump1}
The metrics in Equation (\ref{equ5}) and (\ref{equ6}) are effective enough, so that for client $i$ ($\forall i \in \{1,\dots,n\}$), we have:
\begin{equation} \label{equ9}
{\rm s}_{i,p} > {\rm s}_{i,q}\quad \forall p \in {\rm SC}_{i}, q \in {\rm DC}_{i}.
\end{equation}
where ${\rm SC}_{i}$ refers to all the clients that are in the same cluster as client $i$, and ${\rm DC}_{i}$ refers to all the clients that are in different clusters with client $i$.
\end{assumption}
Given Assumption \ref{assump1}, we can provide the probability that all the $k$ neighbors are same-cluster in the round $t$ when conducting CNI. We assume there are $n$ clients in the system (including client $i$) and $a$ clients in the same cluster as client $i$ (including client $i$), so we can infer the following theorem.
\newtheorem{thm}{Theorem}
\begin{thm} \label{thm1}
Under Assumption (\ref{assump1}), in the round $t$ when conducting CNI as Equation (\ref{equ8}), for client $i$, the probability that all the $k$ neighbors are same-cluster are ${\rm P}^{t}(k)$, we have
\begin{equation} \label{equ10}
\begin{split}
{\rm P}^{t}(k) &= {\rm G}(k)\ast {\rm P}^{t-1}(k) + {\rm R}(k)\\
&\dots\\
{\rm P}^{2}(k) &= {\rm G}(k)\ast {\rm P}^{1}(k) + {\rm R}(k)\\
{\rm P}^{1}(k) &= {\rm R}(k).
\end{split}
\end{equation}
where {\rm R}(x) and {\rm G}(x) are two functions and the '$\ast$' refers to the discrete convolution computation, defined as
\begin{small}
\begin{equation}
\begin{split}
&{\rm R}(x) = \small{\frac{l!(n-l-1)!}{(n-1)!}\sum\limits_{s=0}^{l-x} \frac{(a-1)!(n-a)!}{s!(l-s)!(n-a-s)!(a-l+s-1)!}}\\
&{\rm G}(x) = \frac{l!(a-1)!(n-a)!(n-l-1)!}{x!(n-1)!(l-x)!(a-x-1)!(n-a-l+x)!}\\
&{\rm G}(x)\ast {\rm P}(x) = \sum\limits^{x-1}_{m=0} {\rm G}(m){\rm P}(x-m).
\nonumber
\end{split}
\end{equation}
\end{small}
If conducting PENS as Equation (\ref{equ7}), the probability is
\begin{equation}
{\rm P}^{t}(k) \equiv {\rm R}(k).
\end{equation}
\end{thm}
Based on Theorem \ref{thm1}, we provide the following corollary.
\newtheorem{corollary}{Corollary}
\begin{corollary} \label{corollary1}
Given Theorem \ref{thm1}, we define a function:\\
${\rm Q}(t) = {\rm P}^{t}(k), t\in \{1,\dots,T\}$.
The function ${\rm Q}(t)$ of CNI is monotone increasing, so we have \\
\begin{equation}
{\rm Q}(t)>{\rm Q}(t-1)>\dots>{\rm Q}(2)>{\rm Q}(1).
\end{equation}
The function ${\rm Q}(t)$ of PENS satisfies
\begin{equation}
{\rm Q}(t)={\rm Q}(t-1)=\dots={\rm Q}(1)={\rm R}(k).
\end{equation}
\end{corollary}
\newtheorem{note}{Note}
\begin{note} \label{note1}
We note that Assumption \ref{assump1} is strong, and we provide this assumption just for theoretical analysis in Theorem \ref{thm1} and Corollary \ref{corollary1}. We will show in Figure \ref{confidfig} that in practice when Assumption \ref{assump1} is not strictly held, results are still consistent and similar to the theoretical findings.
\end{note}
\begin{figure}[t]
\centering
\includegraphics[width=0.8\columnwidth]{images/confidence_fig.pdf}
\caption{Precision of same-cluster neighbors in stage 1. CIFAR10, $n=100, l=10, k=5$, two clusters are formed by rotations \{0°, 180°\}.}
\label{confidfig}
\end{figure}
The Corollary \ref{corollary1} shows that: by CNI, the probability increases over round, while PENS keeps it unchanged at a low value. Intuitively, we calculate the theoretical probabilities and compare them with experimental results in Figure \ref{confidfig}. It is obvious to see that CNI enables PANM to have highly confident neighbors only after several rounds, while the precision of same-cluster neighbors in PENS stays low. However, there are gaps between theory and practice, we explain: at the beginning of distributed training, the models are randomly initialized (not well-trained), therefore the similarity measurements are so noisy that Assumption \ref{assump1} may not be held. After some rounds, when the similarity metrics are effective enough, the results will be closer to the theoretical findings.
Besides the high confidence of neighbors being same-cluster, CNI also facilitates clients to match the clients with global maximum similarities. Experiments in Section \ref{sect:exp} will show it can boost personalization even better than Oracle.
\subsubsection{Heuristic Neighbor Matching}
\
\newline
After confident neighbor initialization, we enable clients to have few neighbors with high precision of being same-cluster. For clustered FL, the recall of clustering is also important, since each client needs to find out the whole community with the same objective. Thus, in the second stage of PANM, we use heuristic neighbor matching to facilitate clients to discover more peers with consistent objectives.
In the second stage of PENS, clients choose peers that are selected more than ``\textit{the expected amount of times}'' in stage one as neighbors. Theorem \ref{thm1} implies that if the setting is difficult, stage-one neighbors of PENS are prone to be noisy, afterward in stage two, the matched neighbors are more likely to include outliers. Besides, PENS requires the hyperparameter ``\textit{the expected amount of times}''; without prior knowledge of cluster information, it is hard to set the hyperparameter to an appropriate value. Therefore, in stage two of PANM, we propose a more effective method based on expectation maximization (EM).
It is obvious that for a client, the same-cluster clients may have high similarities while the ones of the different-cluster are low, so we assume the similarities of the same-cluster obey a consistent distribution while the similarities of the different-cluster obey another distribution. The assumption is as follows.
\begin{assumption} \label{assump2}
For client $i$ ($\forall i \in \{1,\dots,n\}$), the similarities between same-cluster clients and client $i$ obey a Gaussian distribution, parameterized by $\mathcal N(\mu_{0},\sigma_{0}^{2})$, and the similarities between different-cluster clients and client $i$ obey another Gaussian $\mathcal N(\mu_{1},\sigma_{1}^{2})$, as
\begin{equation}
\begin{split}
{\rm s}_{i,p}\sim \mathcal N(\mu_{0},\sigma_{0}^{2}),&\ {\rm s}_{i,q}\sim \mathcal N(\mu_{1},\sigma_{1}^{2}) \\
\forall p \in {\rm SC}_{i},&\ q \in {\rm DC}_{i}.
\end{split}
\nonumber
\end{equation}
We have $\mu_{0} > \mu_{1}$.
\end{assumption}
\begin{figure}[t]
\centering
\includegraphics[width=0.37\columnwidth]{ images/loss_distri.pdf}\hspace{6mm}
\includegraphics[width=0.385\columnwidth]{ images/grad_distri.pdf
\caption{Distributions of similarities. Similarities between client 1 and other clients are shown. CIFAR10, $n=100$, clusters are formed by rotations \{0°, 180°\}.}
\label{lossdistri}
\end{figure}
Assumption \ref{assump2} is quite natural in clustered FL, as in Figure \ref{lossdistri}, the distributions of similarities after stage one of PANM are shown; intuitively, the distributions satisfy our assumption, for there are two distinct Gaussians.
By far, we can formulate the neighbor matching problem into a Gaussian Mixture Model (GMM) problem, a typical solution to GMM problem is EM algorithm. But conventional EM algorithm is not suitable to solve this problem, under the following considerations:
(1) Conventional EM method requires calculating probabilities of all data points in one EM step, but for P2P FL, clients only can communicate with several neighbors in one round; (2) The focus of conventional EM solving GMM problem is to accurately estimate the parameters of Gaussians while our focus is to accurately discriminate cluster identities. Additionally, EM algorithms are sensitive to initialization, better initialization makes it more possible to converge to the global optimum.
To tackle above mentioned matters, we devise our Heuristic Neighbor Matching (HNM) algorithm. In each round, client $i$ randomly samples neighbor candidate list ${\rm C}_{i}^{t} (|{\rm C}_{i}^{t}|=l)$ from non-neighbor clients and also samples a selected neighbor list ${\rm S}_{i}^{t} (|{\rm S}_{i}^{t}|=l$\ if\ $|{\rm N}_{i}^{t}|>l$,\ else\ ${\rm S}_{i}^{t}={\rm N}_{i}^{t})$ from neighbor clients ${\rm N}_{i}^{t}$; client $i$ communicate with these clients and compute similarities $y_{j}={\rm s}_{i,j},\ j \in {\rm M}_{i}^{t}={\rm C}_{i}^{t} \cup {\rm S}_{i}^{t}$. According to Assumption \ref{assump2}, there are two Gaussian distributions in these similarities, the one with higher mean center refers to the same-cluster clients ($\mathcal N(\mu_{0},\sigma_{0}^{2})$), another one refers to the different-cluster ($\mathcal N(\mu_{1},\sigma_{1}^{2})$). Assuming the observed data $y_{j},\ j \in {\rm M}_{i}^{t}$ are generated by the Gaussian Mixture Model:
\begin{equation} \label{equa12}
{\rm Pr}(y|\Theta) = \sum\limits_{r=0}^{1}\beta_{r}\phi(y|\Theta_{r}).
\end{equation}
Here, $\beta_{r}$ refers to the probability that $y$ is generated by distribution $r$, and $\Theta=(\beta_{0},\beta_{1};\Theta_{0},\Theta_{1})$. Our target is using EM algorithm to estimate the distribution identities of $y_{j}$, given by
$$ \gamma_{j,r}=\left\{
\begin{aligned}
1,\ & \text{if}\ j\ \text{belongs\ to\ distribution}\ \mathcal N_{r} \\
0,\ & \text{otherwise}.\\
\end{aligned}
\right.
$$
where $j \in {\rm M}_{i}^{t},\ r \in \{0, 1\}$. Knowing that EM algorithm is sensitive to initialization, with the prior knowledge that most of the clients in ${\rm S}_{i}^{t}$ are same-cluster (it is also possible that it includes outliers), so we can initialize a better parameter as
$$ \gamma_{j,r}^{(1)}=\left\{
\begin{aligned}
1,\ & j\in {\rm S}_{i}^{t}\ \text{and}\ r=1,\ \text{or}\ j\in {\rm C}_{i}^{t}\ \text{and}\ r=0 \\
0,\ & \text{otherwise}.\\
\end{aligned}
\right.
$$
While the latent variable is the distribution parameters: $\Theta_{0}=(\mu_{0},\sigma_{0}), \Theta_{1}=(\mu_{1},\sigma_{1})$, so the complete data is
\begin{equation}
(y_{j}, \Theta_{0}, \Theta_{1}),\ j\in {\rm M}_{i}^{t}.
\nonumber
\end{equation}
Then we formulate the expectation function $Q$, based on the log likelihood function of complete data,
\begin{equation} \label{equa13}
\begin{split}
&Q(\gamma,\gamma^{(a)}) = \mathbb{E}[\log{\rm Pr}(y,\Theta|\gamma)|y,\gamma^{(a)}]\\
&=\sum\limits_{r=0}^{1}\bigg\{
n_{r}\log\mathbb{E}\beta_{r}+\sum\limits_{j\in {\rm M}_{i}^{t}} \gamma_{j,r}[\log(\frac{1}{\sqrt{2\pi}})-\log\mathbb{E}\sigma_{r}\\
&-\frac{1}{2\mathbb{E}\sigma_{r}^{2}}(y_{j}-\mathbb{E}\mu_{r})^{2}]\bigg\},
\end{split}
\end{equation}
where $n_{r}=\sum\limits_{j \in {\rm M}_{i}^{t}}\gamma_{j,r}$.
\textbf{E-step:} Now we need to estimate $\mathbb{E}(\mu_{r},\sigma_{r},\beta_{r})$, notated as $\hat{\mu}_{r},\ \hat{\sigma}_{r},\ \hat{\beta}_{r}$.
\begin{equation}
\begin{split}
\hat{\mu}_{r} = \frac{\sum\limits_{j \in {\rm M}_{i}^{t}} \gamma_{j,r}y_{j}}{n_{r}}, \hat{\beta}_{r} = \frac{n_{r}}{|{\rm M}_{i}^{t}|},\hat{\sigma}_{r}^{2} = \frac{\sum\limits_{j \in {\rm M}_{i}^{t}} \gamma_{j,r}(y_{j}-\hat{\mu}_{r})^{2}}{n_{r}},
\end{split}
\nonumber
\end{equation}
where $ r\in\{0,1\}$.
\textbf{M-step:} Iterative M-step is to find the maximum of the function $Q(\gamma,\gamma^{(a)})$ with respect to $\gamma^{(a)}$, as to set $\gamma^{(a+1)}$ in the next iterative epoch
\begin{equation}
\gamma^{(a+1)} = \mathop{\arg\max}_{\gamma} Q(\gamma,\gamma^{(a)}).
\end{equation}
We use the following function to maximize expectation, since $y_{j}$ more likely belongs to $\mathcal N_{0}$ if $\beta_{0}\phi(y_{j}|\Theta_{0})>\beta_{1}\phi(y_{j}|\Theta_{1})$, vice versa
\begin{equation}
\begin{split}
\gamma_{j,r}^{(a+1)} = &
\mathbbm{1}\Bigg\{ r = \mathop{\arg\max}_{r}
\frac{\hat{\beta}_{r}\phi(y_{j}|\hat{\Theta}_{r})}
{\sum\limits_{c=0}^{1}\hat{\beta}_{c}\phi(y_{j}|\hat{\Theta}_{c})}
\Bigg\}\\
j \in& {\rm M}_{i}^{t}, r \in \{0,1\}.
\end{split}
\end{equation}
Repeat the E-step and M-step until $\gamma^{(a+1)}=\gamma^{(a)}$. Then we obtain the estimated same-cluster neighbors in this round notated as ${\rm H}_{i}^{t}$, where $\gamma_{j,0}=1,\ j \in {\rm H}_{i}^{t},\ {\rm H}_{i}^{t}\subseteqq {\rm M}_{i}^{t}$, then we update our neighbor list
\begin{equation}
{\rm N}_{i}^{t+1} = ({\rm N}_{i}^{t}-{\rm S}_{i}^{t})\cup {\rm H}_{i}^{t}.
\end{equation}
By HNM algorithm, clients can continually update their neighbors, adding new same-cluster clients and removing outliers in neighbor list.
\subsubsection{PANM: Personalized Adaptive Neighbor Matching}
\
\newline
Now we present PANM by combining the above-mentioned algorithms. First, in stage one, client $i (i \in \{1,\dots,n\})$ communicates randomly in the network while conducting confident neighbor initialization for $T_{1}$ rounds. The neighbor list ${\rm N}_{i}^{t}$ in the last round of stage one is set as the initial neighbor list in stage two. Second, in stage two, client $i$ operates gossip communications with neighbors sampled from ${\rm N}_{i}^{t}$ for aggregation and performs heuristic neighbor matching every $\tau$ rounds for updating the neighbor list. The process of PANM is shown in Algorithm \ref{PANM}. To improve readability, we summarize our main notations used in PANM as in Table \ref{tab:notation}.
\begin{table}[t] \small
\caption{Important notations in this paper.}
\centering
\begin{tabular}{c|l}
\toprule
Notation& Meaning\\
\midrule
\textit{i}& Client \textit{i}\\
\textit{n}& Number of all clients\\
\textit{a}& Number of clients in the same cluster as client \textit{i}\\
\textit{k}& Size of aggregation neighbors\\
\textit{l}& Size of neighbor candidate list\\
$\tau$& Round interval of HNM in stage two\\
$\alpha$& Hyperparameter in gradient-based metric\\
${\rm N}_{i}^{t}$& Neighbor list of client \textit{i} in round \textit{t}\\
${\rm C}_{i}^{t}$& Neighbor candidate list of client \textit{i} in round \textit{t}\\
${\rm S}_{i}^{t}$& Selected neighbor in EM-step \\
${\rm M}_{i}^{t}$& Union set of ${\rm C}_{i}^{t}$ and ${\rm S}_{i}^{t}$\\
${\rm H}_{i}^{t}$& Neighbor estimation list in EM-step of client \textit{i}\\
\bottomrule
\end{tabular}
\label{tab:notation
\end{table}
\begin{algorithm}[h]
\caption{PANM: Personalized Adaptive neighbor \\Matching}
\textbf{Input}: $n,k,l,T_{1},T_{2},\eta, e, \tau,\alpha,\mathbf{w}_{0},\mathbf{W}^{1}=\{ \mathbf{w}_{i}^{1}=\mathbf{w}_{0},\\i\in\{1,\dots,n\} \}$;\\
\textbf{Output}: $\mathbf{W}^{T_{1}+T_{2}}$, {\rm N};
\begin{algorithmic}[1]
\STATE Initiate neighbor list: ${\rm N}_{i}$;
\FOR{each round $t=1,\dots, T_{1}+T_{2}$}
\FOR{each client $i, i\in\{1,\dots,n\}$ \textbf{in parallel}}
\STATE Compute $e$ epochs of local training:
\STATE $\mathbf{w}_{i}^{t}\gets\mathbf{w}_{i}^{t}-\eta\nabla F_{i}\left(\mathbf{w}_{i}^{t}, \xi_{i}\right),\xi_{i} \sim\mathcal{D}_{i}$;
\IF {$t\in \{1, \dots, T_{1}\}$}
\STATE ${\rm N}_{i}^{t+1}\gets$ConfiNeighInit(${\rm N}_{i}^{t}$),
\STATE $\mathbf{w}_{i}^{t+1}\gets$ Aggregation(${\rm N}_{i}^{t+1}$,$\mathbf{w}_{i}^{t}$);
\ELSE
\IF {$t \% \tau = 0$}
\STATE ${\rm N}_{i}^{t+1}\gets$ HeurNeighMatch(${\rm N}_{i}^{t}$),
\STATE $\mathbf{w}_{i}^{t+1}\gets$ GossipAggre(${\rm N}_{i}^{t+1}$,$\mathbf{w}_{i}^{t}$);
\ELSE
\STATE ${\rm N}_{i}^{t+1}\gets{\rm N}_{i}^{t}$,
\STATE $\mathbf{w}_{i}^{t+1}\gets$ GossipAggre(${\rm N}_{i}^{t+1}$,$\mathbf{w}_{i}^{t}$);
\ENDIF
\ENDIF
\ENDFOR
\ENDFOR
\end{algorithmic}
\label{PANM}
\end{algorithm}
\section{Experiments and Results}\label{sect:exp}
In this section, we evaluate our methods and compare them with baselines. P2P FL baselines include PENS \cite{DBLP:journals/corr/abs-2107-08517} (state-of-the-art personalized P2P FL algorithm), Random (gossip with random neighbors), Local (without communication), FixTopology (neighbors are randomly sampled at the beginning and fixed during training), Oracle (with prior knowledge of cluster identities, gossip with same-cluster clients). We mention that Oracle is the ideal scenario with ground-truth cluster information which is not realistic in practices, but the following experiments will show our methods can sometimes surpass it.
Centralized FL baselines include IFCA \cite{DBLP:conf/nips/GhoshCYR20} (state-of-the-art centralized clustered FL) and centralized Federated Averaging \cite{DBLP:conf/aistats/McMahanMRHA17}. Our methods include PANMLoss (PANM with metric based on loss), PANMGrad (PANM with metric based on weight updates and gradients).
\subsection{Settings of Datasets}
We use three public benchmark datasets, MNIST \cite{lecun-mnisthandwrittendigit-2010}, CIFAR10 \cite{krizhevsky2009learning}, and FMNIST (Fashion-MNIST) \cite{xiao2017fashion}. To generate clustered Non-IID data, we use rotation transformation and label-swapping, respectively; we note these two Non-IID settings are commonly used in clustered FL (rotation \cite{DBLP:journals/corr/abs-2107-08517,DBLP:conf/nips/GhoshCYR20}, label-swapping \cite{DBLP:journals/tnn/SattlerMS21}). All results are evaluated on each client's local testset, we keep the size of local testset to 100 for all scenarios and present the averaged results among all clients. As detailed below, we describe how we prepare the Non-IID data settings for MNIST, FMNIST, and CIFAR10.
\textbf{Rotation Transformation:} There are two settings for rotation transformation. First is rotation with two clusters (\{0°, 180°\}): clients in cluster 0 keep images without any transformation (0° rotation) while clients in cluster 1 rotate every image in trainset and testset for 180°. Second is rotation with four clusters (\{0°, 90°, 180°, 270°\}): client in cluster 0 keep images without any transformation (0° rotation), clients in cluster 1 rotate every image in trainset and testset for 90°, clients in cluster 2 for 180°, and clients in cluster 3 for 270°. The labels of images keep unchanged and we state that the class distributions in each client are balanced.
\textbf{Swapping Labels:} There are two settings for swapping labels. First is forming two clusters by swapping labels: for clients in cluster 0, images labeled as ``0'' are relabeled as ``1'' and images labeled as ``1'' are relabeled as ``0''; while for clients in cluster 1, images labeled as ``6'' are relabeled as ``7'' and images labeled as ``7'' are relabeled as ``6''. Second is forming four clusters by swapping labels: (1) for clients in cluster 0, images labeled as ``0'' and ``1'' are swapped by labels; (2) for clients in cluster 1, images labeled as ``2'' and ``3'' are swapped; (3) for clients in cluster 2, images labeled as ``4'' and ``5'' are swapped; (4) for clients in cluster 3, images labeled as ``6'' and ``7'' are swapped. Note that the class distributions in each client are balanced.
\subsection{Details of Implementations}
All the experiments are implemented in PyTorch 1.7.1. We have several GPUs for training, including Tesla P40 GPU with 24451MB memory, Quadro RTX 8000 GPU with 48601MB memory, Tesla P100 GPU with 16280MB memory, and Tesla V100 GPU with 16130MB memory. For all experiments, we set the batch size to 128 and the number of local epochs in each round is 3. We adopt the learning rate decay strategy used in the literature. The decay step size is 0.99, which means for each round, the learning rate is set to \textit{the learning rate in the last round} × 0.99. This strategy is adopted in the former work of cluster FL \cite{DBLP:conf/nips/GhoshCYR20}. The initial learning rate is set to 0.08 in the first round. We use the SGD optimizer and set SGD momentum to 0.9. In every setting, we conduct experiments with different random initialization 3 times and averages the results, the mean results and the standard deviations are shown in the tables and figures. We set $T_{1}=100,\ T_{2}=200$ in all experiments.
A three-layer MLP with ReLU activations is adopted as the model for training on MNIST and FMNIST, the first layer is (28*28, 200), the second is (200, 200), and the last is (200, 10). For CIFAR10, a convolution neural network model with ReLU activations which consists of 3 convolutional layers followed by 2 fully connected layers is used. We do not use any image preprocessing transformation, such as flipping and random cropping.
For “\textit{the expected amount of times}” in PENS, we set this hyperparameter to an appropriate value (ceil($T_{1}$×(l+k)/n)) for fair comparison. For centralized FL: IFCA and FedAvg, we set full participation of clients in each round of federated training.
\subsection{Results on Different Datasets}
\begin{figure*}[t]
\centering
\includegraphics[width=1.9\columnwidth]{ images/stage1_matchrecorder.pdf}
\caption{Heatmaps of aggregation records in stage one. CIFAR-10 with two rotations \{0°:\ clients 0-49,\ 180°:\ clients 50-99\}, trainset size is 400, $l=10,k=5$.}
\label{heatmap}
\end{figure*}
Table \ref{dffdataset} shows the accuracies of all methods being compared under different datasets. Although fixed topology and random gossip improve accuracy compared with local training, in contrast to Oracle, the fixed and random topology will impede performance gains. Note that Oracle is with perfect information about cluster identities which is impossible in real FL scenarios, however, our methods surpass Oracle in FMNIST and CIFAR10. We show accuracy and loss curves of results in CIFAR10 in Figure \ref{acclosscifar}, it is demonstrated that PANM (especially PANMLoss) achieves high accuracies in stage one, which means the CNI algorithm is effective, also, PANM converges faster than others. We explain that CNI not only makes sure the neighbors are confident to be same-cluster, it also enables collaboration with neighbors that have maximum similarities among all seen clients, so the performance will be better than randomly sampled from same-cluster peers (Oracle). In Figure \ref{heatmap}, we illustrate the heatmaps of aggregation records between clients in stage one, from which we can see: PENS has noisy aggregations while clients in PANM only aggregates with same-cluster clients; in PANM, clients are prone to communicate with several maximum-similarity clients, the directional communications help PANM to reach better performance. Additionally, we notice that PENS performs better in easy datasets like MNIST, because the objective discrepancy between clusters is not dominant, introducing different-cluster neighbors will not do much harm to local performance.
\begin{table}[!htp] \small
\caption{Results on different datasets. The top two are in bold. For all datasets: trainset size is 200, two rotations \{0°,180°\}, $l=10,k=5$. $n=100$ for CIFAR10 and FMNIST, $n=200$ for MNIST.}
\centering
\begin{tabular}{c|ccc}
\toprule
Methods&MNIST&FMNIST&CIFAR10\\
\midrule
Local&82.57 ±0.28 &76.24 ±0.22 &25.27 ±1.21\\
\midrule
FixTopology&94.71 ±0.09 &85.86 ±0.17 &39.74 ±2.27 \\
Random&95.12 ±0.04 &85.94 ±0.27 &42.96 ±1.42 \\
Oracle&\textbf{95.87 ±0.08} &\textbf{87.01 ±0.26} &\textbf{49.11 ±0.48} \\
PENS&\textbf{96.15 ±0.16} &86.82 ±0.11 &44.78 ±1.12 \\
\midrule
PANMLoss&95.63 ±0.12 &\textbf{87.33 ±0.17} &\textbf{49.19 ±0.79 }\\
PANMGrad&95.65 ±0.09 &86.88 ±0.34 &48.83 ±0.39 \\
\bottomrule
\end{tabular}
\label{dffdataset}
\end{table}
\begin{figure}[t]
\subfloat[]{
\includegraphics[width=1\columnwidth]{ images/swap4_cifar.pdf} }\\
\subfloat[]{
\includegraphics[width=1\columnwidth]{ images/rota2_cifar.pdf} }\\
\subfloat[]{
\includegraphics[width=1\columnwidth]{ images/centralized_cifar.pdf} }
\caption{Accuracy and loss curves. CIFAR10, $n=100,l=10,k=5$, trainset size is 200. (a) 4 clusters with swapping labels; (b) 2 clusters with rotations \{0°,180°\}; (c) comparing with centralized FL methods, 4 clusters with rotations \{0°,90°,180°,270°\}.}
\label{acclosscifar}
\end{figure}
\subsection{Results under Various Heterogeneity}
Table \ref{dffnoniid} shows results under various heterogeneity: randomly swapping out labels and more rotations. For accuracy, it is obvious that our methods are robust in different Non-IID environments. It is notable that in some experiments (CIFAR10 Label-swap(2)(4)), PANM even outperforms Oracle with a large margin. Overall, PANMGrad has similar performances to PANMLoss, but performances vary regarding different settings. This is due to the different perspectives on understanding client similarity, one is based on training loss, another is based on consistency of gradient updates; the two perspectives will take advantage in different scenarios.
\begin{table}[!t] \small
\caption{Results under various heterogeneity. CIFAR10, $n=100,l=10,k=5$, trainset size is 200. Label-swap(2)/(4): two/four clusters with swapping labels, Rotation(4): four clusters with rotation \{0°,90°,180°,270°\}.}
\centering
\begin{tabular}{c|ccc}
\toprule
\ &\multicolumn{3}{c}{Test Accuracy}\\
\midrule
Methods&Label-swap(2)&Label-swap(4)&Rotation(4)\\
\midrule
Local&25.27 ±1.21 &25.27 ±1.21 &25.27 ±1.21\\
\midrule
FixTopology&36.56 ±1.58 &35.08 ±2.70 &31.86 ±0.47 \\
Random&37.32 ±0.26 &37.14 ±1.30 &33.16 ±0.55 \\
Oracle&43.34 ±1.29 &43.32 ±1.06 &\textbf{43.32 ±0.85} \\
PENS&45.72 ±1.34 &\textbf{43.49 ±0.28} &36.64 ±0.58 \\
\midrule
PANMLoss&\textbf{47.12 ±1.30} &\textbf{45.78 ±1.85} &41.43 ±1.83\\
PANMGrad&\textbf{45.84 ±1.92} &42.14 ±1.34 &\textbf{43.99 ±1.26} \\
\bottomrule
\end{tabular}
\begin{tabular}{c|ccc}
\toprule
\ &\multicolumn{3}{c}{Precision of Neighbors}\\
\midrule
Methods&Label-swap(2)&Label-swap(4)&Rotation(4)\\
\midrule
PENS&\textbf{100} &70.83±4.17 &59.75±5.08 \\
PANMLoss&\textbf{100} &\textbf{100} &68.19±28.12\\
PANMGrad&\textbf{100} &83.33±28.87 &\textbf{100} \\
\bottomrule
\end{tabular}
\begin{tabular}{c|ccc}
\toprule
\ &\multicolumn{3}{c}{Recall of Neighbors}\\
\midrule
Methods&Label-swap(2)&Label-swap(4)&Rotation(4)\\
\midrule
PENS&59.18±2.04 &66.67±4.17 &52.78±8.67 \\
PANMLoss&74.15±32.42 &48.61±13.39 &62.50±31.46\\
PANMGrad&\textbf{100} &\textbf{94.44±6.36} &\textbf{98.61±2.41} \\
\bottomrule
\end{tabular}
\label{dffnoniid}
\end{table}
For precision (the fraction of same-cluster clients among neighbors) and recall (the fraction of same-cluster neighbors among all same-cluster clients) in stage two, our methods outperform PENS, which means our EM-based heuristic neighbor matching method is effective to enable clients to match most of the same-cluster clients as neighbors. We also demonstrate the neighbor topologies of PANMLoss and PENS in stage two in Figure \ref{topology}, by PANM, clients evolve to form the four-cluster structure without prior knowledge of cluster identities, whereas PENS constructs a disordered topology.
\begin{figure*}[t]
\centering
\includegraphics[width=2\columnwidth]{ images/NL_topology_rota4.pdf}
\caption{Neighbor topologies in stage two. Four clusters with rotations, CIFAR10, $n=100,l=10,n=5$, trainset size is 200. Each color denotes a cluster.}
\label{topology}
\end{figure*}
\begin{figure}[t]
\centering
\includegraphics[width=1\columnwidth]{ images/linechart.pdf}
\caption{Left: Accuracies when changing trainset size; $n=100,l=10,k=5$, two rotations \{0°,180°\} for all. Right: Accuracies when changing number of clients, $l=10,k=5$, trainset size is 200, two rotations \{0°,180°\} for all.}
\label{figtrainsize}
\end{figure}
\subsection{Impact of Trainset size and Number of Clients}
In the left figure of Figure \ref{figtrainsize}, we compare PANM to the baselines in a setting where we fix the number of clients to 100 while varying the size of the local trainsets. The results show that by increasing the size of the local train set on each client, performance increases for all compared algorithms. We further see that PANM consistently outperforms all baselines (including PENS), and it has comparable performance to Oracle, especially when the trainset size is large. Compared with PENS, PANM surpasses it with a large margin, especially in scenarios where data are sufficient.
In the right figure of Figure \ref{figtrainsize}, results of changing number of clients are shown, in this experiment, we fix the trainset size to 200. Overall, performances increase while adding more clients for participation, and PANM outperforms all baselines in all settings. However, when the number of clients is large, PANM may have inferior performance to Oracle, but it is also superior compared with others.
\begin{table*}[t]
\caption{Impact of $k$ and $l$. CIFAR10 with two rotations \{0°,180°\}, $n=100$, trainset size is 100 for all settings.}
\centering
\begin{tabular}{c|cccccc}
\toprule
\ &\multicolumn{6}{c}{$l, k$}\\
\midrule
Methods&10,5 &10,3 &20,10 &20,5 &30,15 &30,10\\
\midrule
Local&19.93 ±0.36 &19.93 ±0.36 &19.93 ±0.36&19.93 ±0.36 &19.93 ±0.36 &19.93 ±0.36\\
\midrule
FixTopology&36.60 ±2.58 &31.54 ±0.18 &38.31 ±1.45 &34.94 ±0.84 &38.88 ±0.52 &38.77 ±3.80\\
Random&39.19 ±1.04 &38.54 ±2.24 &40.82 ±1.09 &37.34 ±0.80 &42.22 ±2.25 &39.91 ±0.60\\
Oracle&\textbf{43.83 ±0.80} &\textbf{42.28 ±1.25} &\textbf{43.39 ±2.80} &\textbf{43.57 ±0.93} &\textbf{44.79 ±2.36} &\textbf{44.16 ±1.32}\\
PENS&42.42 ±1.58 &\textbf{40.04 ±0.90} &42.57 ±1.74 &41.92 ±1.28 &44.12 ±0.22 &42.80 ±0.98\\
\midrule
PANMLoss&41.75 ±0.24 &39.55 ±1.78 &\textbf{44.02 ±0.59} &\textbf{43.64 ±1.54} &\textbf{46.82 ±0.41} &\textbf{46.19 ±0.84}\\
PANMGrad&\textbf{42.48 ±0.21} &39.53 ±2.99 &38.24 ±4.26 &41.94 ±3.15 &41.16 ±3.60 &42.23 ±1.02\\
\bottomrule
\end{tabular}
\label{lk}
\end{table*}
\subsection{Impact of \textit{l} and \textit{k}} \label{subsec:lk}
$l$ is the number of randomly sampled neighbor candidates, and $k$ is the number of neighbors for aggregation chosen from neighbor candidates or the neighbor list. The choices of $l$ and $k$ depend on communication budgets in the system, larger $l$ and $k$ will bring more communication costs. Besides, the ratio of $l$ and $k$ is also crucial (especially for PENS), it decides the probabilities of neighbors being same-cluster as we have inferred in Theorem \ref{thm1} and Corollary \ref{corollary1}. We conduct experiments by changing $l, k$ as demonstrated in Table \ref{lk}.
We notice that in easy settings (for example, $l=10,k=3$), PENS performs well, but in other settings, PENS has poor results. As we discussed in the paper, if the setting is difficult, PENS is prone to be noisy in neighbor matching.
For oracle, only $k$ matters, because $k$ determines the number of neighbors to collaborate with in each round. As $k$ increases, performances of Oracle also increase. We explain that if the number of neighbors for aggregation increases, it is better for clients to reach partial consensus within clusters, and the model weights are more likely to be similar. It is worth mentioning that PANMLoss is very robust, and it surpasses Oracle in most settings. PANMLoss benefits a lot when $k$ increases.
\begin{table}[!ht] \small
\caption{Comparing with centralized FL. $n=100,l=10,k=5$, trainset size is 200. Best performances in the centralized and decentralized are in bold. Clusters are generated by rotations: 2 for \{0°,180°\}, 4 for \{0°,90°,180°,270°\}.}
\centering
\begin{tabular}{c|ccc}
\toprule
Methods&CIFAR10(4)&FMNIST(4)&FMNIST(2)\\
\midrule
Local&25.27 ±1.21 &76.24 ±0.22 &76.24 ±0.22\\
\midrule
FedAvg&37.03 ±0.74&83.54 ±0.08 &86.86 ±0.16 \\
IFCA(c=2)&40.64 ±2.18 &86.19 ±0.04 &\textbf{88.06 ±0.20} \\
IFCA(c=3)&41.05 ±1.09 &\textbf{86.78 ±0.36} &/ \\
IFCA(c=4)&\textbf{43.65 ±0.77} &86.50 ±0.07 &/ \\
\midrule
Oracle&43.32 ±0.85 &85.45 ±0.38 &87.01 ±0.26 \\
PENS&36.64 ±0.58 &84.68 ±0.27 &86.82 ±0.11 \\
\midrule
PANMLoss&41.43 ±1.83 &\textbf{86.09 ±0.31} &\textbf{87.33 ±0.17} \\
PANMGrad&\textbf{43.99 ±1.26} &85.64 ±0.25 &86.88 ±0.34 \\
\bottomrule
\end{tabular}
\label{ifcatable}
\end{table}
\begin{table}[!ht]
\caption{Performances under low communication budgets, where $n = 100$, the trainset size is 200 and $\Delta = 100\delta$ with $\delta$ being the single model size. We use CIFAR-10 with 4 rotations. For PANM, we set $l = 4,~\tau=10$.}
\centering
\begin{tabular}{c|ccc}
\toprule
Methods &Comm. costs&Max. req. band.&Test acc.\\
\midrule
FedAvg&600$\Delta$ &1$\Delta$ &37.03 ±0.74 \\
IFCA ($c$=2)&900$\Delta$ &2$\Delta$ &40.64 ±2.18 \\
IFCA ($c$=3)&1200$\Delta$ &3$\Delta$ &41.05 ±1.09 \\
IFCA ($c$=4)&1500$\Delta$ &4$\Delta$ &\textbf{43.65 ±0.77} \\
\midrule
PANMLoss ($k$=2)&1118$\Delta$ &0.06$\Delta$ &41.36±0.64 \\
PANMGrad ($k$=2)&1118$\Delta$ &0.06$\Delta$ &42.78±1.68 \\
PANMLoss ($k$=3)&1397$\Delta$ &0.07$\Delta$ &43.30±1.32 \\
PANMGrad ($k$=3)&1397$\Delta$ &0.07$\Delta$ &\textbf{43.34±0.85} \\
\bottomrule
\end{tabular}
\label{tablowcom
\end{table}
\subsection{Comparison with Centralized Clustered FL}
P2P FL takes advantage of bandwidth and reliability as we addressed in Section \ref{intro}; besides, as for clustered FL, P2P solutions are more robust and can exploit the latent cluster structure in a self-evolved manner without assuming the number of clusters. We compare IFCA \cite{DBLP:conf/nips/GhoshCYR20}, the present state-of-the-art centralized clustered FL and centralized Federated Averaging \cite{DBLP:conf/aistats/McMahanMRHA17} with decentralized P2P methods, as shown in Table \ref{ifcatable} and (c) of Figure \ref{acclosscifar}.
In typical centralized FL algorithms, the central server randomly samples $l$ clients for aggregation, but we notice that this will result in bad convergence for IFCA. In our implementations, in a scenario where there are 100 clients with 4 clusters therein, and the central server samples 10 clients in each round (where $l/n=0.1$), IFCA has poor convergence of estimations: all clients are estimated as the same identity. As a result, we have to set full aggregation participation of clients in IFCA, but we remind this will cause large communication burdens and it is unfair to the P2P setting (where we set $n=100, l=10, l/n=0.1$). Even if, in Table \ref{ifcatable}, our P2P method PANM also achieves proportionate performances as a contrast to IFCA. What's more, IFCA requires assumptions on the number of clusters ($c$), we find: if $c$ is set inappropriately, the performance will be poor. According to Table \ref{ifcatable}, in scenario of CIFAR10 with 4 rotations, if set $c=2$, the accuracy of IFCA is 40.64\% while our PANMLoss and PANMGrad reach 41.43\% and 43.99\%. Moreover, from the learning curves in Figure (c) of Figure \ref{acclosscifar}, it is apparent that our P2P methods PANM keeps a more steady and robust learning process while there are some disturbances in the curves of centralized counterparts and our P2P solutions have the same convergence rate with the centralized ones.
We notice that, generally, the overall communication costs of P2P FL are larger than those of centralized FL methods, since every client needs to receive and transmit several models in a communication round. Hence, we implement PANM under low communication budgets so that PANM will have comparable overall communication costs as the centralized, and we conclude the results in Table \ref{tablowcom}. It shows that PANM can have close performances to IFCA when the communication costs are similar but PANM requires much lower bandwidth. We also find that the consequence of inappropriately estimating the number of clusters will be severe for IFCA ($c=4:\ 43.65;\ c=3:\ 41.05$), but PANM is flexible enough to explore the cluster structure under any hyperparameters. The influence of communication budgets is subtle for PANM, therefore it is more robust and effective than the centralized IFCA.
\section{Discussion} \label{sect:discussion}
\subsection{Performance of PANMLoss and PANMGrad}
For simple datasets (e.g. MNIST), we observed that the performance of the two algorithms is comparable. As for CIFAR-10, PANMLoss performs better in rotation (2) and label-swap, while PANMGrad performs better in rotation (4). We explain this by the inter-cluster objective inconsistency (IOI). If IOI is dominant, which means that the learning tasks are diverse, thus the losses are distinct and PANMLoss will be beneficial. We consider the IOI as: label-swap\textgreater rotation, rotation (2)\textgreater rotation (4). However, when IOI is low, the loss-based metric is noisy, the proposed PANMGrad will exploit two cosine functions to provide a more accurate similarity measurement.
Additionally, we notice in Figure \ref{confidfig} that PANMGrad will result in faster convergence of precision of same-cluster neighbors which means the metric based on gradients is more effective in the early stage of training.
\subsection{Applicability of PANM}
For PANMLoss and PANMGrad, the metrics require different computation resources. Generally, PANMGrad is more computationally efficient, since it only requires several inner product calculations on sparse vectors (gradients), while PANMLoss needs inference on local datasets. Hence, under limited computation resources, PANMGrad is more preferable. In addition, as we mentioned before, if IOI is high, PANMLoss is more suggested, otherwise, PANMGrad may be better.
We have mentioned in Subsection \ref{subsec:lk} that we can choose different $l,k$ by the communication budgets and we have shown PANM is still effective under low communication budgets. Furthermore, in our implementations, we assume fully connected communication accessibility, which means that each client is able to communicate with any other client in the system. The fully connected accessibility is not often satisfied in realistic scenarios, but we state that PANM is also applicable under limited accessibility. In PANM, we make lose assumption on accessibility, clients can purify their neighbors within the scope of accessible peers. Thus, PANM is flexible in practice.
\section{Conclusion} \label{sect:conclusion}
In this paper, we study the clustered Non-IID problem in peer-to-peer federated learning and develop PANM that enables clients to match neighbors with similar objectives. We analyze how PANM works, and empirically demonstrate that it significantly improves the performance compared with strong baselines.
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
\bibliographystyle{IEEEtran}
|
1,116,691,497,518 | arxiv | \section*{Acknowledgments}
This research was partially funded by NSF IIS 1910733, DBI 1661375, and IIS-1513616.
We would like to thank Vivek Srikumar for his extremely valuable feedbacks regarding applying {\topoact} to the BERT neural networks.
We also would like to thank Jeff Phillips who forwarded a Twitter message from Chris Olah on March 7, 2019, which inspired this project.
\bibliographystyle{eg-alpha-doi}
\section{Technical Background}
\label{sec:background}
We review technical background on the mapper construction and neural network architecture.
We delay the discussions on activation vectors and feature visualization until~\autoref{sec:methods}.
\para{Mapper graph on point cloud data.}
We give a high-level description of the framework by Singh \etal~\cite{SinghMemoliCarlsson2007} in a point cloud setting.
Given a high-dimensional point cloud $\Xspace \subset \Rspace^d$ equipped with a function $f$ on $\Xspace$, $f: \Xspace \to \Rspace$, the mapper construction provides a topological summary of the data for compact representation and exploration.
It utilizes the topological concept known as the \emph{nerve of a covering}~\cite{Aleksandroff1928}.
An \emph{open cover} of $\Xspace$ is a collection $\Ucal = \{U_i\}_{i \in I}$ of open sets in $\Rspace^d$ with an index set $I$ such that $\Xspace \subset \bigcup_{i \in I} U_{i}$.
Given a cover $\Ucal$ of $\Xspace$, the $1$-dimensional \emph{nerve} of $\Ucal$, denoted as $\Nerve_1(\Ucal)$, is constructed as follows: A finite set $\{i,j\}\subset I$ (i.e., an edge) belongs to $\Nerve_1(\Ucal)$ if and only if the intersection of $U_i$ and $U_j$ is nonempty; if the set $\{i,j\}$ belongs to $\Nerve_1(\Ucal)$, then any of its subsets (i.e., the point $i$ and the point $j$) is also in $\Nerve_1(\Ucal)$. See~\autoref{fig:mapper} for an example. A cover $\Ucal=\{U_1, U_2, U_3, U_4\}$ that contains open rectangles is given for a 2-dimesional point cloud $\Xspace$ in (a). The $1$-dimensional nerve of $\Ucal$, $\Nerve_1(\Ucal)$, is shown in (c). For instance, there is an edge $\{1,2\}$ that belongs to $\Nerve_1(\Ucal)$ since $U_1 \cap U_2 \neq \emptyset$.
\begin{figure}[!ht]
\centering
\includegraphics[width=0.98\columnwidth]{mapper.pdf}
\caption{A simple example of a mapper graph on a point cloud.}
\label{fig:mapper}
\end{figure}
For the mapper construction, we start with a finite cover $\Vcal = \{V_j\}_{j \in J}$ ($J$ being an index set) of the image $f(\Xspace) \subset \Rspace$ of $f$, such that $f(\Xspace) \subseteq \bigcup_j{V_{j}}$, see~\autoref{fig:mapper}(b).
Since $f$ is a scalar function, $V_i$ is an open interval in $\Rspace$.
Let $\Ucal$ denote the cover of $\Xspace$ obtained by considering the clusters of points induced by points in $f^{-1}(V_j)$ for each $j$, see (d).
The $1$-dimensional nerve of $\Ucal$, denoted as $\mapper := \Nerve_1(\Ucal)$, is called the \emph{mapper graph} of $(\Xspace,f)$.
$\mapper$ is a multiscale representation that serves as a topological summary of of $(\Xspace,f)$, \ie, the point cloud $\Xspace$ equipped with a function $f$.
Its construction relies on three parameters: the function $f$, the cover $\Vcal$, and the clustering algorithm.
The function~$f$ plays the role of a \textit{lens}, through which we look at the data, and different lenses provide different insights~\cite{BiasottiGiorgiSpagnuolo2008,SinghMemoliCarlsson2007}.
An interesting open problem for the mapper construction is how to define topological lenses beyond best practices or a rule of thumb~\cite{BiasottiGiorgiSpagnuolo2008,BiasottiMariniMortara2003}.
In practice, functions such as height, distance from the barycenter,
surface curvature, integral geodesic distances, and geodesic distances from a source point have all been used as lenses~\cite{BiasottiGiorgiSpagnuolo2008}.
In this paper, we use the $L_2$ norm of the activation vectors as the lens, although other options are possible (see the supplementary material for a discussion of $L_2$-norm as a lens function).
The cover $\Vcal$ of $f(\Xspace)$ consists of a finite number of open intervals as cover elements, $\Vcal=\{V_j\}_{j \in J}$.
A common strategy is to use uniformly sized overlapping intervals. Let $n$ be the number of intervals and $p$ the amount of overlap between adjacent intervals. Adjusting these parameters increases or decreases the amount of aggregation $\mapper$ provides.
We compute the clustering of the points lying within $f^{-1}(V_i)$ and connect the clusters whenever they have nonempty intersection. A typical algorithm to use is DBSCAN~\cite{EsterKriegelSander1996}, a density-based clustering algorithm; it groups points in high-density regions together and makes points that lie alone in low-density regions outliers. The algorithm requires two input parameters: \emph{minPts} (the number of samples in a neighborhood for a point to be considered as a core point), and \emph{$\epsilon$} (the maximum distance between two samples for one to be considered in the neighborhood of the other).
\autoref{fig:mapper} illustrates a mapper graph construction for a dataset $\Xspace$ sampled from a noisy circle.
The function (lens) used is $f(x) = ||x - p||_2$, where $p$ is the lowest point in the data.
$\Xspace$ is colored by the value of the function.
We divide the range of the $f$ into three intervals $\{V_1, V_2, V_3\}$ with a $30\%$ overlap.
For each interval, we compute the clustering of the points lying within the domain of the lens function restricted to the interval $f^{-1}(V_i)$, and connect the clusters whenever they have a nonempty intersection.
$\mapper$ is the mapper graph whose nodes are colored by the index set, and it preserves the shape of the point cloud data -- a loop.
\para{InceptionV1 architecture.}
We give a high-level overview of InceptionV1~\cite{SzegedyLiuJia2015} (GoogLeNet), the neural network architecture employed in this paper.
However, our framework is not restricted to the specific architecture of a neural network.
InceptionV1 is a CNN that won the ImageNet Large-Scale Visual Recognition Challenge (ILSVRC) for image classification in 2014.
It was trained on ImageNet ILSVRC~\cite{DengDongSocher2009}.
ImageNet consists of over 15 million labeled high-resolution images with roughly $22K$ classes/categories. ILSVRC takes a subset of ImageNet of around $1K$ images in each of $1K$ classes, for a total of 1 million training images, $50K$ validation images, and $100K$ testing images.
The highlights of InceptionV1 architecture include the use of $1\times1$ convolutions, inception modules, and global average pooling.
The $1\times1$ convolution from NIN (networks in networks)~\cite{LinChenYan2014} is used to reduce dimensionality (and computation) prior to expensive convolutions with larger image patches.
A new level of organization is introduced in the form of the \emph{inception module}, which combines different types of convolutions for the same input and stacking all the outputs on top of each other.
InceptionV1 contains nine \emph{inception modules}, each composed of multiple convolution layers.
The demo version of {\topoact} explores the activations of the last layer of each inception module.
The module names such as \emph{mixed3a}, \emph{mixed3b} are shortened as \emph{3a}, \emph{3b}, etc.
This choice is well aligned with previous literature on visual exploration of InceptionV1~\cite{OlahMordvintsevSchubert2017,OlahSatyanarayanJohnson2018,HohmanParkRobinson2020,CarterArmstrongSchubert2019}.
\para{ResNet architecture.}
To demonstrate the generality of topological structures observed across different neural network architectures, we also apply {\topoact} to activation vectors from a ResNet model.
Residual Network or ResNet~\cite{HeZhangRen2016} was one of first neural network architectures that enabled training extremely deep neural networks with up to $1K$ layers.
A neural network $N_D$ of depth $D$ is a subnetwork of any network $N_{D+K}$ of depth $D + k, k >0$.
Theoretically, $N_{D+K}$ should be capable of learning any function that $N_D$ can learn by setting the extra $k$ layers to an identity mapping, and thus perform at least as well as the smaller network.
In practice, however, increasing layers beyond a certain depth leads to a sharp degradation in performance (higher training error and lower classification accuracy on the test set) even when normalization schemes are used for both initialization and intermediate representations.
ResNet overcomes this problem by adding ``shortcut'' connections to a layer that adds the output from layer $k$ to the input of layer $k+i$ where $i$ is usually 2.
In our experiments, we have implemented ResNet-18, a residual network with 18 layers following the layer specifications in~\cite{HeZhangRen2016}. With 200 training epochs, it achieves a classification accuracy of 93.24\% on CIFAR-10 and 91.87\% on CIFAR-100 datasets~\cite{Krizhevsky2009}.
\section{Applying {\topoact} to BERT Neural Network for NLP}
\label{sec:bert}
To demonstrate the utility of {\topoact} with concrete use cases in the wild, we have collaborated with a machine learning (ML) expert in Natural Language Processing (NLP), Dr. Vivek Srikumar from the University of Utah.
We apply {\topoact} to activation vectors obtained from the BERT (Bidirectional Encoder Representations from Transformers) family of models, which is the default representation of text for a variety of NLP tasks.
Although BERT and similar models are undoubtedly helpful in improving predictive accuracy, it is not immediately clear why it helps. This problem arises because BERT embeddings (\ie,~activation vectors) consist of a collection of high-dimensional vectors for every sentence, and this high-dimensional space is not easy to explore.
The highlight of our collaboration is that {\topoact} presents an opportunity in this context by revealing the various syntactic and semantic regularities that each layer of BERT captures, some of which are surprising but interpretable for the ML expert.
In addition, {\topoact} shows potential directions for improving these representations, for instance, targeting a specific downstream NLP task.
For example, in addition to some clear patterns of concepts captured via topological branches, we also see clusters and relationships between them that appear noisy, yet stable.
The existence of these suggests that there may be opportunities for developing topologically aware regularization techniques for training BERT-like models by imposing additional constraints such that the mapper graphs resulting from {\topoact} should exhibit certain structural properties, which is left for future work.
\subsection{BERT and Activation Datasets}
BERT and other transformers-based language models have recently found widespread application as the go-to methods for many tasks in NLP such as sentiment analysis, sentence classification, and domain-specific language modeling \cite{BeltagyLo2019, LeeYoon2020}. Jawahar \etal \cite{JawaharSagot2019} explored and established that contextual embeddings from BERT do indeed encode syntactic structures in earlier layers and compositional structures in later layers.
For our experiments, we use the ``\emph{bert-base-uncased}'' trained network from the \textit{Huggingface's transformers} library \cite{WolfDebut2019} with 12 layers, each with 768 neurons.
We collect activation vectors from BERT on the training set of the Georgetown University Multilayer (GUM) corpus \cite{Zeldes2017}. The data contains 4780 sentences and $81,857$ tokens.
Tokens are individual words, numbers, or punctuation marks that form sentences.
Among the $80K$ tokens, about $11K$ are punctuations.
We collect the activations by passing each sentence through the trained BERT model, collecting per-token activations, and applying {\topoact} to the activations for all 12 layers.
That is, for each of the 12 layers of the BERT neural network, we compute mapper graphs for point clouds of the token's activations in 768 dimensions across various parameter settings, similar to our earlier examples involving InceptionV1 and ResNet-18.
Our experiments applying {\topoact} to BERT activations confirm and expand upon the earlier results in NLP~\cite{BeltagyLo2019, LeeYoon2020, JawaharSagot2019}.
In the following sections, we present some use cases of structures found in the BERT activations through our tool that highlight both local and global structures in both syntactic and semantic regimes.
\subsection{Pronoun Differentiation}
For layer 12 (the last layer) of BERT, we notice a branching structure that highlights the differentiation among pronouns.
Recall that a pronoun is ``a word that can function by itself as a noun phrase and that refers either to the participants in the discourse (\emph{e.g.}, \textbf{I}, \textbf{you}) or to someone or something mentioned elsewhere in the discourse (\emph{e.g.}, \textbf{she}, \textbf{it}, \textbf{this})'', according to Oxford English Dictionary.
\begin{figure}[!ht]
\centering{
\includegraphics[width=.98\linewidth]{me-my-pronouns.pdf}
}
\caption{Pronoun differentiation. Configuration: layer 12, Euclidean norm, 80 intervals, 30\% overlap.}
\label{fig:me-my-pronouns}
\end{figure}
As illustrated in~\autoref{fig:me-my-pronouns}, the chain of nodes with (d) and (e) represents a mixture of the first person singular personal pronoun (i.e, \textbf{me}) and the first person singular personal possessive form (\ie, \textbf{my}). The chain continues along (b) and (c) with the possessive form (\textbf{my}), but the personal pronoun splits into its own branch (a) (\textbf{me}).
Node (a) consists of sentences which employ the word \textbf{me} in phrases such as \emph{``She invited \textbf{me} in for tea''},\emph{``She looks at \textbf{me} and then at Renata''}, \emph{``He soon came back and gave \textbf{me} the good news''}, etc.
Node (b) consists of sentences that employ the word \textbf{my} in phrases such as \emph{``since I was in \textbf{my} twenties''}, \emph{``I wished \textbf{my} da would come home''}.
In particular, it also contains a sentence \emph{``I was riding on \textbf{me} bike and I thought I'd swallowed an insect''}; here the word \textbf{me} is used instead of \textbf{my} likely due to a local dialect.
This indicates that the differentiation between the pronouns goes beyond just the word-level and instead captures semantic differences as well.
This structure is interesting to our ML expert for a couple of reasons. First, we have a cluster of only the first person forms (\textbf{me} and \textbf{my}), and not the second (\textbf{you}) and third person forms (\textbf{he, she, they}). Second, although the pronoun (specifically, the object pronoun \textbf{me}) and the possessive form (\textbf{my}) are related in one sense (i.e., first person singular), they are distinct both in terms of their meanings and grammatical roles.
The branching structure highlights this similarity and their divergence.
\subsection{Contextual Differentiation}
We give two examples involving how branches from {\topoact} capture contextual differentiations.
As shown in~\autoref{fig:water-sea}, the structure of nodes about water highlights the different roles that water may play.
Nodes (a) and (b) reflect oceanic usages (\emph{e.g.},~\textbf{sea}, \textbf{marine}, \textbf{waves}).
Node (d) reflects culinary usages (\emph{``2 cups \textbf{water} (or broth)''}) and starts to connect with other liquids used in cooking (e.g., \emph{``olive \textbf{oil} to taste}'').
And other labeled nodes (\emph{e.g.},~node (e)) reflect meteorological usages (\emph{``the noise of the \textbf{rain}''}), culinary usages (\emph{``rinse...under cold \textbf{water}''}), and oceanic usages.
\begin{figure}[!ht]
\centering{
\includegraphics[width=0.95\linewidth]{water-sea.pdf}
}
\caption{Contextual differentiation. Configuration: layer 9, Euclidean norm, 80 intervals, 30\% overlap.}
\label{fig:water-sea}
\end{figure}
We see similar fine-grained distinctions between \textbf{photographs}, \textbf{photography}, and \textbf{art} in~\autoref{fig:art}.
The structure starts with \textbf{artworks}, \textbf{museum}, and \textbf{art} in node (a), moves on to \textbf{display}, \textbf{exhibition}, and \textbf{exhibits} in node (b), which then gets further refined into \textbf{portrait} and \textbf{painting} in node (c), as well as \textbf{shot} and \textbf{photo} in node (d).
\begin{figure}[!ht]
\centering{
\includegraphics[width=0.98\linewidth]{art.pdf}
}
\caption{Contextual differentiation. Configuration: layer 9, Euclidean distance, 80 intervals, 30\% overlap, Jaccard $=0.01$.}
\label{fig:art}
\end{figure}
The fact that BERT (without any fine-tuning for any task) captures these differences is surprising for our ML expert.
Such an insight could point toward an explanation for how transformer models like BERT and its many variants seem to capture world knowledge and even common-sense knowledge~\cite{BosselutRashkin2019}.
\subsection{Local and Global Syntax}
It is known that the lower layers of BERT characterize more local syntax and non-contextual lexical semantics, whereas the later layers capture global sentential structure and semantics~\cite{TenneyDas2019}.
We see this by comparing the structures we see in layer 3 in (\autoref{fig:local-syntax}) with the ones in layer 9 (\autoref{fig:me-my-pronouns}, \autoref{fig:water-sea}, \autoref{fig:art}).
In lower layers, we observe groupings of words with similar parts of speech that diverge into chains that contain only words with similar meaning, as shown in~\autoref{fig:local-syntax}.
For example, a generic adjective node (c) bifurcates into a set of nodes that describe size (e.g., \textbf{bigger}, \textbf{largest}, \textbf{highest} in node (a)) and a set of nodes that describe goodness (e.g., \textbf{best} and \textbf{better} in node (b)).
\begin{figure}[!ht]
\centering{
\includegraphics[width=0.99\linewidth]{local-syntax.pdf}
}
\caption{Local syntax at an earlier layer. Configuration: layer 3, cosine distance, 80 intervals, 30\% overlap.}
\label{fig:local-syntax}
\end{figure}
The other examples from later layers -- such as layer 9 (\autoref{fig:global-syntax}) -- show that the mapper graph encodes complex relational abstractions that go beyond simply the dictionary meaning of the word.
\subsection{After-When Separation}
The branching structure in~\autoref{fig:global-syntax} represents a surprising and difficult to characterize dichotomy in the usage of words like “after” and “when”. Both these words are used to convey temporal meaning, but, the latter is also used to introduce discourse relationships such as explanations or sometimes even causations. The fact that these two usages are distinctly represented in the BERT embeddings shows that BERT does indeed characterize this subtle difference and, as a result, could serve as a basis for developing future parsers for discourse or rhetorical structure of text.
\begin{figure}[!ht]
\centering{
\includegraphics[width=0.99\linewidth]{after-when.pdf}
}
\caption{After-When separation. Configuration: layer 9, Euclidean distance, 80 intervals, 40\% overlap.}
\label{fig:global-syntax}
\end{figure}
\subsection{Temporal and Locative Prepositions}
The difference between temporal and locative usages of prepositions is well studied, and forms the basis of the Preposition Supersense Project that Dr. Srikumar is part of~\cite{SchneiderSrikumar2015}.
As illustrated in~\autoref{fig:prepositions}, both nodes (a) and (b), and their parent node, represent clusters of prepositions, but the two branches capture distinct meanings rather than mere surface level differences.
To see this, note that both branches include the preposition word \textbf{in}, but the branch (a) represents its temporal usage, whereas (b) represents its locative usage.
This branching structure is reminiscent of the Supersense Hierarchy developed via linguistic analysis~\cite{SchneiderHwang2020a}.
These, and the previous observations, suggest that we can discover linguistic structures from BERT activations using {\topoact}.
Furthermore, {\topoact} also provides investigative directions for why these embeddings have been successful at a wide variety of linguistic tasks.
\begin{figure}[!ht]
\centering{
\includegraphics[width=0.99\linewidth]{prepositions.pdf}
}
\caption{Temporal and locative prepositions. Configuration: layer 9, Euclidean distance, 100 intervals, 30\% overlap.}
\label{fig:prepositions}
\vspace{-4mm}
\end{figure}
\section{Comparison with t-SNE and UMAP}
\label{sec:comparison}
Various dimensionality reduction (DR) techniques have been proposed to analyze and visualize high-dimensional point cloud data~\cite{Cayton2008,LiuMaljovecWang2017}.
Among these, t-SNE~\cite{MaatenHinton2008} and UMAP~\cite{McInnesHealyMelville2018} are most relevant to our proposed work as they have been used previously for exploring neuron activations~\cite{Karpathy2014,NguyenYosinkiClune2016,CarterArmstrongSchubert2019}.
In particular, Carter \etal~\cite{CarterArmstrongSchubert2019} employ both t-SNE and UMAP to project high-dimensional activation vectors to low-dimensions for visual exploration.
In comparison with t-SNE and UMAP, the benefit of {\topoact} is two-fold.
\begin{itemize}
\item {\topoact} provides a global, graph-based representation of the space of activations, which explicitly summarizes the organizational principles (clusters and cluster relations) behind neuron activations. t-SNE and UMAP detect structures visible in the \emph{low-dimensional} embedding, whereas {\topoact} captures complex topological structures -- loops and branches -- in the original \emph{high-dimensional} space.
\item Whereas t-SNE and UMAP focus on preserving proximities within local neighborhoods, {\topoact} explicitly reveals branches and loops that are not necessarily visible via t-SNE/UMAP. These topological structures can be used to guide refined, local structural analysis (\autoref{sec:results}).
\end{itemize}
The kNN (k-nearest neighbor) graph constructed by UMAP can be considered as a topological representation of the high-dimensional data~\cite{McInnesHealyMelville2018}. However, it only approximately preserves the \emph{connectivity} among points within local patches of the manifold, and does not capture structures such as loops or branches.
{\topoact} addresses this important challenge by utilizing the mapper construction.
In addition, several DR techniques have been developed recently~\cite{SilvaMorozovVejdemo-Johansson2009, WangSummaPascucci2011, YanZhaoRosen2018} that explicitly preserve loops and branches; however, none of them give a global summary of all such structures in a single visualization.
In studying the shape of the space of images, Lee \etal~\cite{LeePedersenMumford2003}, Carlsson \etal~\cite{CarlssonIshkhanovDe-Silva2008}, and Xia~\cite{Xia2016} studied the global topological structures of patches from natural images; they used different topological tools (such as persistence homology) and focused on different data problems (such as studying the global structure of natural images from a database) in comparison to {\topoact}.
Finally, a few recent works applied topological techniques in the study of activations.
Gebhart \etal~\cite{GebhartSchraterHylton2019} computed persistent homology over the activation structure of neural networks.
In particular, they characterized the topological structure of the neural network architecture when viewed as a graph with edge weights provided by activations.
Gabella \etal~\cite{GabellaAfamboEbli2019} used both persistent homology and the mapper construction to study the parameter space of neural networks (i.e., weight matrices) during training.
\section{Conclusion}
\label{sec:conclusion}
In this paper, we present {\topoact}, a framework to explore the topology of the activation spaces of neural networks.
We obtain topological summaries of the activation spaces via mapper graphs that capture the organizational principal behind neuron activations.
We apply {\topoact} to trained neural networks such as ResNet and InceptionV1 for image classification, and BERT for contextual word embeddings.
In each case, we present exploration scenarios that provide valuable insights into the image representations or word embeddings learned by different layers of these networks.
This paper is the first step toward understanding the topological structure of the activation spaces in deep neural networks.
\section{Discussion}
\label{sec:discussion}
{\topoact} supports exploratory analysis of numerous interesting topological structures, locally and globally, in the space of activation vectors.
We encourage readers to utilize the live demo of datasets for InceptionV1 and ResNet for such an exploration.
Our approach is not without its limitations.
The exploration scenarios presented here are specific to the choice of input images as well as the choice of activation vectors.
Further analysis is required to determine how stable the results are with respect to these choices.
However, some of these limitations are common to other recent approaches (\emph{e.g.},\cite{CarterArmstrongSchubert2019,HohmanParkRobinson2020}). We offer some topics for discussion and future work.
\para{Generality.}
We focus on CNNs, specifically, InceptionV1 and ResNet-18, in this paper.
However, our approach is not restricted to a particular network architecture.
Mapper graphs could be generated and used as a vehicle for visual exploration whenever neuron activations are present.
{\topoact} can be generalized to explore new datasets coupled with other trained neural network architectures, such as ZFNet~\cite{ZeilerFergus2012}, AlexNet~\cite{KrizhevskySutskeverHinton2012}, and VGGNet~\cite{SimonyanZisserman2015}.
We have showed that our approach is generalizable beyond image classifiers to include textual embedding networks such as BERT.
\para{Parameter tuning.}
Practical and automatic parameter tuning for the mapper construction remains a challenging open problem for the broad TDA community.
Carriere \etal~\cite{CarriereMichelOudot2018} provided the state-of-the-art, albeit theoretical, results on mapper parameter selection under restrictive settings.
Their framework assumed that a point cloud sample taken from the underlying space has a well-behaved, parameterizable probability distribution (formally, an $(a, b)$-standard distribution) and that the sample is sufficiently large, so that the Hausdorff distance between the sample and the underlying space is small.
However, upon careful investigation, these assumptions are not applicable in our setting.
Although we may assume that the activation space is a compact subset of the Euclidean space, we cannot verify that the activation vectors we sample follow the generative model of an $(a, b)$-standard distribution, and that $300K$ vectors form a sufficiently large sample for approximating or possibly reconstructing the underlying space.
On the other hand, the mapper construction comes with ``best practices" in terms of parameter tuning, which rely on a grid search in the parameter space where good parameter combinations are those that produce stable structures. Finding a theoretically sound and yet practical parameter tuning strategy for our mapper graph construction remains open; see the supplementary materials for more discussion on this topic.
For the current version of the {\topoact}, we focus on exploring various mapper graphs with predetermined sets of parameter combinations following the best practices.
Additionally, confirmation bias is a risk when {\topoact} is utilized in practice because users may simply tune the parameters until they see what they want to see in the visualization.
However, confirmation bias cannot be resolved without automatic parameter tuning, which remains an open problem.
\para{Stability.}
Additional theoretical results regarding the stability of mapper construction are available in~\cite{BrownBobrowskiMunch2019} and the references therein.
The investigation is on-going into how stable the mapper graphs are with respect to different sampling techniques.
Under some assumptions on the sampling condition, Brown \etal~\cite{BrownBobrowskiMunch2019} showed that a pair of mapper graphs is close if their underlying point clouds are sampled from the same probability density function concentrated on the underlying topological space.
However, similar to the situation of parameter tuning, the gap between theory and practice is still large.
Filling such a gap is beyond the scope of this paper.
\para{Adversarial attacks.}
An important aspect in understanding the effectiveness of adversarial attacks on neural networks is how an attack alters the intermediate representations, \ie,~the activations.
{\topoact} visualizes these representations from a topological perspective and hence might be useful in analyzing the effect of adversarial attacks at different layers of the network.
\para{Corrective actions during training.}
A branching point (a bifurcation) in the space of activations at a particular layer may indicate the point where the network starts distinguishing a pair of classes.
This knowledge can be useful to inform corrective actions for inputs in the test data that are being misclassified.
For example, if two classes that bifurcate at a particular layer in {\topoact} are still being misclassified as each other, an expert can choose to increase the network width at subsequent layers, or to selectively augment the training data for these classes to encourage better separation.
\section{Exploring the Shape of Activations from InceptionV1}
\label{sec:results}
\begin{figure*}[tb]
\centering
\includegraphics[width=0.9\linewidth]{interface.pdf}
\caption{With {\topoact}, users can interactively explore topological summaries of activations in a neural network for a single layer and across multiple layers. Users can investigate activations at a particular layer under the single-layer exploration mode. The mapper graph panel (a) provides a graph-based topological summary of the activation vectors from $300K$ images across $1K$ classes, where each node of the mapper graph represents a cluster of activation vectors, and each edge encodes the relationships between the clusters. The size of a node is proportional to the number of activations, and the color is mapped to the average $L_2$ norm of activations in the cluster. The edge thickness is proportional to the Jaccard index between two nodes.
The control panel (b) supports the selection of parameters for the mapper construction and visual encoding.
For a chosen cluster in the mapper graph, the data example panel (c) gives textual description of the top three classes within the cluster together with (up to) five image examples from each top class. The feature visualization panel (d) applies feature inversion to generate idealized images, called activation images, for individual activation vectors (obtained from data examples) and for an averaged activation vector within a chosen cluster.}
\label{fig:interface}
\end{figure*}
We present the user interface of {\topoact}, an interactive system used to explore the organizational principles of neuron activations in deep learning image classifiers.
We use InceptionV1 trained on $1$ million ImageNet images across $1K$ classes.
We obtain activation vectors of $300K$ images (300 images per class) via the trained model.
The {\topoact} user interface, \autoref{fig:interface}, contains two exploration modes: single-layer exploration and multilayer exploration.
We present various exploration scenarios using {\topoact} under the single-layer exploration mode that provide valuable insights into learned representations of InceptionV1.
See the supplementary materials for a detailed description on the interface, implementation, and multilayer exploration mode.
For single-layer exploration, the main takeaway from these scenarios is that {\topoact} captures specific topological structures, in particular, branches and loops, in the space of activations that are hard to detect via classic DR techniques; such features offer new perspectives on how a neural network ``sees" the input images.
The topological features identified by {\topoact} can also be used to guide refined, local shape analysis of the space of activations.
\subsection{Discovering Branches from the Space of Activations}
We provide several examples of interesting topological structures that capture relationships between activations during single-layer exploration.
Two main types of topological structures unique to our framework, branches and loops, differentiate {\topoact} from prior work (\emph{e.g.},~\cite{HohmanParkRobinson2020,CarterArmstrongSchubert2019}).
Topologically, branches in a graph represent bifurcations, thus separations, among image classes.
Although we observe variations of similar features along a specific branch, different branches may capture distinct, sometimes unrelated, features.
In order to illustrate the insights gained through user interactions and views between different parts of the system, figures in this section have average activation images (computed from the average of all activations in a node) overlaid on the nodes of the mapper graph.
\para{Leg-face bifurcation.}
Our first example of a bifurcation comes from the layer \emph{4c} of the ImageNet dataset (\emph{overlap-30-epsilon-adaptive}).
\autoref{fig:branch-leg-face} shows two branches emerging from node (d) in the mapper graph; we refer to such a node as the \emph{branching node}.
Node (d) is composed of $381$ activation vectors.
The top three classes within node (d) are \textbf{rugby ball}, \textbf{Indian elephant}, and \textbf{wig}.
Although this clustering of class labels appears to be random, the mapper graph coupled with averaged activation images reveals interesting insights.
As illustrated in~\autoref{fig:branch-leg-face}, the left branch appears to capture the leg of an animal.
The top three classes represented in all the clusters within this branch include various breeds of dogs and horses (a).
The right branch appears to capture features that resemble (distorted) human faces.
Although the class names associated with clusters along the right branch may not suggest a relation to human faces, the data examples associated with these clusters reveal that all the top classes in the right branch contain images with humans, most of which include close-ups of faces (b, c).
Returning to the branching node (d), upon closer inspection, we see that it contains images of \textbf{rugby} players and \textbf{elephants} that include both leg and face features, whereas \textbf{wig} images also include human faces.
Therefore, the activation space bifurcates at the branching node to further differentiate between leg and face features.
\begin{figure}[!ht]
\centering
\includegraphics[width=.99\columnwidth]{leg-face.pdf}
\caption{Leg-face bifurcation. Configuration: layer 4c, Euclidean norm, $70$ intervals, $30\%$ overlap, adaptive $\epsilon$ for DBSCAN.}
\label{fig:branch-leg-face}
\vspace{-2mm}
\end{figure}
We further compare {\topoact} against t-SNE and UMAP projections.
For t-SNE, we use the Multicore-TSNE~\cite{Ulyanov2016} Python library and set the perplexity parameter to be $50$ following the parameter choice used in the \emph{activation atlas}~\cite{CarterArmstrongSchubert2019}. The UMAP projection is performed using its official Python implementation~\cite{McInnesHealySaul2018} with $20$ nearest neighbors and a minimum distance of $0.01$.
As illustrated in \autoref{fig:leg-face-projection}, we select nodes that are involved in the leg-face bifurcation and highlight their corresponding activation vectors in the t-SNE and UMAP projections.
In particular, neither t-SNE nor UMAP reveals a bifurcation as the activation vectors in the projection are scattered over the entire projection.
\begin{figure}[!ht]
\centering
\includegraphics[width=.99\columnwidth]{leg-face-projection.pdf}
\caption{Highlighting activation vectors that belong to the leg-face bifurcation (c) as orange points in the UMAP (a) and t-SNE projection (b).}
\label{fig:leg-face-projection}
\end{figure}
\para{Bird-mammal bifurcation.}
Our second example of a branch comes from the layer \emph{5a} of the ImageNet dataset (\emph{overlap-30-epsilon-fixed}).
\autoref{fig:bird-mammal} shows two branches emerging from the branching node (a), which is composed of $398$ activation vectors.
It contains images of both birds and dogs such as \textbf{oystercatcher} and \textbf{Brittany spaniel}, and the averaged activation image of the branching node appears to be a combination of the left profile of bird faces and right profile of the dog-like faces.
Upon further investigation, the bottom branch that contains nodes (d), (e), and (f) focuses on the features of bird faces: profile views composed of the left eye and beak, with variations of color and textures as we move along the branch.
The clusters in this branch include mainly bird species such as \textbf{bee eater}, \textbf{robin}, and \textbf{lorikeet}.
The variations in the captured features and corresponding data samples can be seen in nodes (d), (e), and (f).
The clusters in the top branch, on the other hand, appear to capture features of mammalian faces: eyes and snouts, with variations in color and texture.
This branch primarily consists of images from classes of mammals, including various dog breeds, \textbf{wolves}, and \textbf{foxes}.
\begin{figure}[!ht]
\centering
\includegraphics[width=.99\columnwidth]{bird-mammal.pdf}
\caption{Bird-mammal bifurcation. Configuration: layer 5a, Euclidean norm, $70$ intervals, $30\%$ overlap, fixed $\epsilon$ for DBSCAN.}
\label{fig:bird-mammal}
\vspace{-4mm}
\end{figure}
\para{Wheel-tread bifurcation.}
Our third example comes from the layer \emph{4c} of the ImageNet dataset (\emph{overlap-30-epsilon-adaptive}).
As illustrated in~\autoref{fig:wheel-tread}, the branching node (a) is a small cluster of size $136$.
All the clusters in this example contain images of various types of automobiles, for example \textbf{minibus}, \textbf{police van}, \textbf{fire engine}, \textbf{limousine}, etc.
The branching node (a) appears to capture what looks like the wheel of a vehicle - a dark round shape with tread-like pattern.
The two branches appear to focus on one of these two features.
Whereas the left branch focuses on the dark round swirling patterns of automobile wheels (b, c, d), the right branch appears to focus more on the tread-like patterns and textures (e, f).
\begin{figure}[!ht]
\centering
\includegraphics[width=.99\columnwidth]{wheel-tread.pdf}
\caption{Wheel-tread bifurcation. Configuration: layer 4c, Euclidean norm, 70 intervals, 30 \% overlap, adaptive $\epsilon$ for DBSAN.}
\label{fig:wheel-tread}
\end{figure}
\subsection{Exploring Loops from the Space of Activations}
Branches capture bifurcations in the types of features across different objects, but some loops seem to capture different aspects of the same underlying object.
\para{Fur-nose-ear-eye loop of mammals.}
Our first example comes from layer \emph{4d} of the ImageNet dataset (\emph{overlap-30-epsilon-fixed}).
\autoref{fig:fur-nose-loop} shows a loop created by six clusters.
The top classes in all six clusters include various dog breeds and a variety of foxes.
All these clusters seem to capture different features (\ie,~body parts) related to these animals.
Based on feature visualization, the leftmost cluster appears to capture the color patterns and the texture of the fur from the body (a).
Going clockwise, the next cluster also captures the color and texture of the fur but from a different body part, possibly the fur and hair surrounding the nose, suggested by the dark spot and the swirling pattern (b).
The next two clusters (c, d) appear to capture animal ears.
The averaged activation image captured by the cluster (e) is not as clearly attributable to a specific part of an animal's body.
As can be observed in (e), this cluster consists of images from a larger variety of animals, from \textbf{foxes} to \textbf{Siamese cats} and \textbf{hogs}.
As a result, the corresponding averaged activation image is a mixture of various colors and slightly different textures.
The last cluster (f) appears to capture the eyes and noses of the animals.
We can observe in (f) that the cluster contains front and side views of dog heads.
\begin{figure}[!ht]
\centering
\includegraphics[width=.99\columnwidth]{fur-nose-loop.pdf}
\caption{Fur-nose-ear-eye loop. Configuration: layer 4d, $70$ intervals, $30\%$ overlap, fixed $\epsilon$ for DBSCAN.}
\label{fig:fur-nose-loop}
\end{figure}
\para{Face-body-leg loop of birds.}
Our next example originates from layer \emph{5a} of the ImageNet dataset (\emph{overlap-30-epsilon-adaptive}).
\autoref{fig:face-body-loop} shows six clusters creating the loop.
The top three classes of all the clusters in the loop consist of bird species, and similar to the previous example, the averaged activation images show us different features (body parts) of the birds captured by these clusters.
Clusters (c, d, e) on the top of the loop appear to capture the left profile views of the bird faces with the left eye and the beak identifiable in the averaged activation images.
These clusters are, in fact, composed of images of birds.
The variation in the color of birds (from red to blue, and to brown) is reflected in the corresponding activation images (b, c, d, e).
Clusters (a, b, f) on the bottom of the loop appear to capture the body and legs along with a feathered texture, although not as clearly as the other three clusters.
As can be seen, cluster (f) also includes images of \textbf{leopard} and \textbf{jaguar} mixed with images of birds (\textbf{partridge} and \textbf{ruffed grouse}) for the representation of texture.
\begin{figure}[!ht]
\centering
\includegraphics[width=.99\columnwidth]{face-body-loop.pdf}
\caption{Face-body-leg loop of birds. Configuration: layer 5a, $70$ intervals, $30\%$ overlap, adaptive $\epsilon$ for DBSCAN.}
\label{fig:face-body-loop}
\end{figure}
\subsection{Studying Global Views of Activation Spaces}
We now explore the global view of an activation space using the single-layer exploration mode.
Instead of focusing on a single type of topological structure such as loops or branches, we investigate the distribution of topological structures.
\begin{figure}[!ht]
\centering
\includegraphics[width=.98\columnwidth]{global-5b.pdf}
\vspace{-2mm}
\caption{A global view of a mapper graph for a fixed layer. Configuration: layer 5b, $70$ intervals, $30\%$ overlap, adaptive $\epsilon$ for DBSCAN.}
\label{fig:global-5b}
\vspace{-4mm}
\end{figure}
As illustrated in \autoref{fig:global-5b}, we investigate the distribution of branches within the largest connected component of a mapper graph, at layer \emph{5b} of the ImageNet dataset (\emph{overlap-30-epsilon-adaptive}).
We pay special attention to branching nodes with high degrees (a, b).
These branching nodes, in some sense, serve as ``anchors'' or ``hubs'' of the underlying space of activations.
We make a few interesting, though speculative, observations.
For each of the two branching nodes in (a) and (b), a mixture of geometric and texture-based images contributes to the representation of the node.
Nodes immediately adjacent to the branching node (a), \ie, those that form branches that merge at node (a), contain geometric objects that are square-shaped (\textbf{envelop}, \textbf{bath towel}), circle-shaped (\textbf{bowl}, \textbf{pasta}), pointy-shaped (\textbf{tie}, \textbf{hammer}), and bottle-shaped (\textbf{beer}).
Nodes immediately adjacent to the branching node (b) have other objects that serve similar purposes, including square-shaped (\textbf{vest}, \textbf{cuirass}), circle-shaped (\textbf{dough}, \textbf{mashed potato}), pointy-shaped (\textbf{ladle}, \textbf{ball point}), and bottle-shaped (\textbf{milk cans}, \textbf{whiskey jug}).
However, (a) and (b) seem to draw these geometric shapes from (almost completely) different classes of images, which may indicate a level of self-similarity within the space of activations that requires further investigation.
\section{{\topoact} User Interface and System Design}
\label{sec:interface}
We provide details regarding the user interface and system design of {\topoact}.
Figure 3 in the main paper illustrates the user interface under single-layer exploration mode.
The control panel includes information regarding the layer of choice (e.g.,~\emph{3a}, \emph{3b}, \emph{4a}), the dataset (across various mapper parameters) under exploration (e.g., \emph{overlap-30-epsilon-fixed}, \emph{overlap-50-epsilon-adaptive}), and a class search box that supports filtering by a set of classes.
It enables projections of the activation vectors using t-SNE and UMAP.
The control panel also contains a check box that superimposes averaged activation images over the graph nodes to provide an alternative overview of the topological summary (see feature visualization panel for details).
It also supports the filtering of graph edges based on the Jaccard index.
\begin{figure}[!ht]
\centering
\includegraphics[width=.8\columnwidth]{class-search.pdf}
\caption{Class search box used to specify a set of classes to be filtered by the mapper graph.}
\label{fig:class-search}
\end{figure}
\para{Class search box with a shopping directory view.}
As illustrated in~\autoref{fig:class-search}, users can type a class name in the search box, which is used to filter the mapper graph.
The search bar uses partial matching to locate a list of possible class names.
Alternatively, users can select a subset of classes from the ``shopping directory'' view in which top classes within the current layer are listed in alphabetical order.
The mapper graph will highlight the clusters that contain any of the user-specified classes among their top three classes.
\begin{figure}[!ht]
\centering
\includegraphics[width=.8\columnwidth]{single-layer-search.pdf}
\caption{A mapper graph highlighting nodes that include classes of large motor vehicles.}
\label{fig:class-search-example}
\end{figure}
When the projection view is enabled, class search will also highlight all activations for that class in the t-SNE/UMAP projection.
As an example, in \autoref{fig:tsne-class-search}, we look at t-SNE projection of activations from layer \emph{5a} of the ImageNet dataset (\emph{overlap-30-epsilon-adaptive}).
Using the shopping directory view, we select several classes of large motor vehicles, for example, \textbf{school bus}, \textbf{tow truck}, \textbf{fire engine}, \textbf{minibus}, \textbf{minivan}, etc.
Each node highlighted in the mapper graph of \autoref{fig:class-search-example} contains at least one of the selected classes among its top three classes.
\begin{figure}[!ht]
\centering
\includegraphics[width=.6\columnwidth]{tsne-class-search-lifeboat.pdf}
\vspace{-2mm}
\caption{Class search in the projection view for the \textbf{lifeboat} class. Users can search one or more classes and visualize them in the t-SNE or UMAP projection.}
\label{fig:tsne-class-search}
\vspace{-4mm}
\end{figure}
\subsection{Single Layer Exploration Mode}
For single layer exploration, the interface is composed of three panels: the mapper graph panel, the data example panel, and the feature visualization panel (see~Figure 3 in the main paper for an illustration).
\para{Mapper graph panel.}
For ImageNet dataset, {\topoact} uses the mapper construction to construct a topological summary from the activation vectors of $300K$ images across $1K$ classes.
Different from dimensionality reduction approaches such as t-SNE~\cite{MaatenHinton2008} and UMAP~\cite{McInnesHealyMelville2018}, {\topoact} computes and captures the shape of the activation space in the original high-dimensional space in the form of a mapper graph and preserves the structural information as much as possible when the mapper graph is drawn on the 2-dimensional plane.
As shown in Figure 3(a) in the main paper, we use a force-directed layout by Dwyer~\cite{Dwyer2009} to visualize the mapper graph.
Each node represents a cluster of ``similar'' activation vectors (in terms of their proximities in Euclidean distance), and each edge encodes the relations between clusters of activation vectors.
Given two clusters of activation vectors $C_u$ and $C_v$, an edge $uv$ connects them if $|C_u \cap C_v| \neq \emptyset$.
Given $C_u$ and $C_v$ connected by an edge $uv$, the edge weight of $uv$ is their Jaccard Index, that is, $J(C_u, C_v) := {|C_u \cap C_v|}/|C_u \cup C_v|$.
Each edge is then visualized by visual encodings (i.e.,~thickness and colormap) that scale proportionally with respect to their weights.
Weights on the edges highlight the strength of relations between clusters.
To explore the mapper graph, users can zoom and pan within the panel.
Hovering over a node in the mapper graph will display simple statistics of the cluster: the number of activation vectors in the cluster and the averaged lens function value.
Clicking on a node will give information on the top three classes (with a membership percentage) within the selected cluster; it will also update the selection for the data example panel and the feature visualization panel, as described below.
\para{Data example panel.}
To make each cluster more interpretable, we combine the original data examples with feature visualization.
For a selected node (cluster) in the mapper graph, we give a textual description of the top three classes in the cluster as well as five data examples from each of the three top classes.
For example, as illustrated in~\autoref{fig:example-view}a, a selected cluster in the mapper graph view for layer \emph{5a} (of \emph{overlap-30-epsilon-adaptive}) contains the three top classes of images: \textbf{fire engine}, \textbf{tow truck}, and \textbf{electric locomotive}.
Its corresponding data example view contains five images sampled from each class to give a concrete depiction of the input images that trigger the activations.
\begin{figure}[!ht]
\centering
\includegraphics[width=.99\columnwidth]{example-view.pdf}
\caption{A data example panel (a) and a feature visualization panel (b) for layer 5a, where (c) contains an averaged activation image for the chosen cluster.}
\label{fig:example-view}
\end{figure}
\para{Feature visualization panel.}
After a user selects a node (cluster) in the mapper graph panel, we display activation images pre-generated for each input image from the data example panel.
These individual activation images are generated by applying feature visualization to individual activation vectors from the $300K$ input images.
The feature visualization displays up to $15$ of such individual activation images, up to $5$ for each of the top classes; see~\autoref{fig:example-view}(b).
Furthermore, we also average the activation vectors that fall within the cluster and run feature inversion on the averaged activation, producing an \emph{averaged activation image} per cluster, as shown in~\autoref{fig:example-view}(c).
Moving across clusters following edges of the mapper graph will help us understand how the averaged activation images vary across clusters.
We obtain a global understanding of not only what the network ``sees'' via these idealized images but also how these idealized images are related to each other in the space of activations.
In addition to the graph view, we can replace each node in the mapper graph by an averaged activation image as a glyph.
This can be perceived as an alternative to the \emph{activation atlas}~\cite{CarterArmstrongSchubert2019} with one crucial difference: the mapper graph captures clusters of activation vectors in their original high-dimensional space and preserves relations between these clusters.
Such a global view provides valuable insights during in-depth explorations.
\para{t-SNE and UMAP projections.}
For comparative purpose, we perform dimensionality reduction on the activation vectors for each layer using t-SNE and UMAP.
The projection is done using all 300K activation vectors onto a 2-dimensional space.
For t-SNE, we use the Multicore-TSNE~\cite{Ulyanov2016} Python library and set perplexity to be $50$ following the parameter choice used in the activation atlas~\cite{CarterArmstrongSchubert2019}. The UMAP projection is performed using its official Python implementation~\cite{McInnesHealySaul2018} with $20$ nearest neighbors and a minimum distance of $0.01$.
t-SNE and UMAP projections are precomputed due to the large number (300K) of activation vectors.
We also provide a linked view between the mapper graph and the t-SNE/UMAP projection. Selecting a node in the mapper graph will highlight its corresponding activation vectors in the t-SNE/UMAP projections.
We provide subsampled versions of these projections (5K, 10k, 50K, 100K, and 300K) to deal with the issue of visual clutter and to accommodate various browser rendering capabilities on a number of devices.
\subsection{Multilayer Exploration Mode}
In the multilayer exploration mode, three adjacent layers are explored side by side; see~\autoref{fig:global-all-layers}(top).
After choosing a particular class or a set of classes using the class search box, {\topoact} highlights nodes (clusters) across all three layers that contain the chosen set of classes among its top three classes.
Other visualization features are inherited from the single layer exploration.
Multilayer exploration helps capture the evolution of classes as images are run through the network and supports structural comparisons of summaries across layers.
Such exploration can be particularly useful when used in conjunction with the class search tool.
As an example of a class search in multilayer mode, we look at layers \emph{4e}, \emph{5a} and \emph{5b} of the \emph{overlap-30-epsilon-adaptive} dataset.
We use the same selection of classes of large motor vehicles used in the earlier example of a class search in single layer mode (\autoref{fig:class-search-example}).
\autoref{fig:class-search-multilayer} shows the class search results, now in the multilayer exploration mode.
\begin{figure}[!ht]
\centering
\includegraphics[width=.98\columnwidth]{multi-layer-multi-class-large-vehicles-4e5a5b-overlap-30-eps.pdf}
\caption{Class search highlights nodes that include classes of large motor vehicles across multiple layers.}
\label{fig:class-search-multilayer}
\vspace{-4mm}
\end{figure}
Under the multilayer exploration mode, we can compare the shape of activation spaces across multiple layers.
As illustrated in~\autoref{fig:global-all-layers}, we show a side-by-side comparison of all layers for the ImageNet dataset (\emph{overlap-30-epsilon-adaptive}).
A further investigation into structural comparisons across layers, such as tracking the evolution of a particular branching node, is nontrivial and left for future work.
\begin{figure}[!ht]
\centering
\includegraphics[width=.99\columnwidth]{global-all-layers.pdf}
\caption{Comparing nine mapper graphs for the ImageNet dataset using multilayer exploration. Configuration: $70$ intervals, $30\%$ overlap, adaptive $\epsilon$ for DBSCAN.}
\label{fig:global-all-layers}
\vspace{-4mm}
\end{figure}
\subsection{System Design}
{\topoact} is open-source via GitHub: \url{https://github.com/tdavislab/TopoAct/}, and web-based with a public demo: \url{https://tdavislab.github.io/TopoAct/}.
It is tested for Google Chrome and Mozilla Firefox.
It is developed using Javascript, HTML, and CSS, along with D3.js and Chart.js.
The $300K$ dataset examples were sampled from ImageNet dataset.
For our mapper graph construction, we used a modified version~\cite{ZhouChalapathiRathore2020} of the open-source KeplerMapper library~\cite{VeenSaul2019} that we optimized to handle the large number of data points that we encountered in our use case.
The construction of mapper graphs across layers was performed on high-performance server machines with $128$, $160$, and $256$ CPU cores, and RAM ranging from $504$ GB to $1024$ GB.
The construction took around 15 minutes for layers with lower dimensional activation vectors (i.e.,~layer \emph{3a} produces 256-dimensional activation vectors) and 25-30 minutes for higher dimensional activation vectors (e.g.,~layer \emph{5b} produces 1024-dimensional activation vectors).
For our choice of $\epsilon$ for the DBSCAN algorithm, we ran PyNNDescent~\cite{PyNNDescent} on a commodity workstation with a 4 core intel i7 (4750HQ) and 8GB of RAM.
Computing $\epsilon$ took on average 5 minutes per layer.
Finally, we used Google Colab~\cite{Bisong2019, GoogleColab} to run our feature visualization with GPUs, either from an Nvidia P100, Nvidia K80, or Nvidia T4 GPU.
Feature visualization of all 300K input images was done via the Lucid library~\cite{Lucid}, which took on average 8 hours.
Feature visualization of average activation vectors took between 2.5 (i.e.,~\emph{3a}) and 6 hours (i.e.,~\emph{5b}) per mapper graph.
\section{Introduction}
\label{sec:introduction}
Deep convolutional neural networks (CNNs) have become ubiquitous in image classification tasks thanks to architectures such as GoogLeNet and ResNet.
Meanwhile, transformer-based models such as BERT are now the state-of-the-art language model for text classification.
However, we do not quite understand how these networks achieve their impressive performance.
One main challenge in deep learning is \emph{interpretability}: How can we make the representations learned by these networks interpretable to humans?
Given a trained deep neural network, we address the interpretability issue by probing neuron activations, \ie,
the combinations of neurons firings, in response to a particular input image.
With a large number of input images for a CNN, we can obtain a global view of what the neurons have learned by studying neuron activations in a particular layer.
We aim to address the following questions: What is the shape of the activation space? What is the organizational principle behind neuron activations? And how are the activations related within a layer and across layers?
We propose to leverage tools from topological data analysis (TDA) to capture global and local patterns of how a trained network reacts to \myupdate{a large number} of input images. In this work:
\begin{itemize}
\item We present {\topoact}, an interactive visual analytics system that uses topological summaries to explore the space of activations in deep learning classifiers for a fixed layer of the network. {\topoact} leverages the mapper construction~\cite{SinghMemoliCarlsson2007} from TDA to capture the overall shape of activation vectors for interactive exploration.
\item We present exploration scenarios where {\topoact} helps us discover valuable, sometimes surprising, insights into learned representations of image classifiers such as InceptionV1~\cite{SzegedyLiuJia2015} and ResNet~\cite{HeZhangRen2016}.
\item We observe structures in the topological summaries, specifically branches and loops, that correspond to evolving activation patterns that help us understand how a neural network reacts to a large group of images. In particular, we find a correlation between semantically meaningful distinctions and topological separations among images from different classes.
\item We further demonstrate the generality and utility of {\topoact} by applying it to activation vectors obtained from text classifiers such as the BERT family of models. Via a collaboration with a machine learning expert, we provide concrete use cases in the wild where {\topoact} reveals syntactic and semantic regularities within layers of BERT that help with hypothesis generation in natural language processing (NLP).
\end{itemize}
Finally, we release an open-source, web-based implementation of the exploration interface on Github: {\url{https://github.com/tdavislab/TopoAct/}}; the current system is also available via a public demo link: {\url{https://tdavislab.github.io/TopoAct/}}.
We expect {\topoact} to benefit the analysis and visualization of neural networks by providing researchers and practitioners the ability to probe black box neural networks from a novel topological perspective.
\begin{itemize}
\item To the best of our knowledge, {\topoact} is the first tool that focuses on exploring complex topological structures -- branches and loops -- within the space of activation vectors. Its exploratory nature helps to inform the global and local organizational principles of activation vectors across different scales.
\item {\topoact} detects if and when activations from different classes become separated, via branches in a fixed layer (see \autoref{sec:results}, and in particular, the deer-horse example in \autoref{sec:CIFAR}), which may be used to inform diagnostic or corrective actions such as selective data augmentation for misclassified inputs or network layer modification for increasing the separation between confounding classes (see \autoref{sec:discussion}).
\end{itemize}
\section{$L_2$ Norm and Adaptive Cover}
\label{sec:l2norm}
In the demo, we used a uniform cover, which caused large variations in cluster sizes.
Although some clusters were composed of only a handful of activation vectors, several clusters had thousands of activation vectors, and large intersections between neighboring clusters.
Finding meaningful relationships across such large clusters is difficult in these cases since the top three classes may not be good representatives of the cluster as a whole.
The branches and loops explored in our examples contain relatively small clusters for which the averaged activation images are more meaningful.
The best way to remedy the large variation in cluster sizes is to use an adaptive cover, in which interval lengths are modified in such a way that each interval contains approximately the same number of points.
Creating adaptive cover elements may be achieved by looking at the distribution of lens function values using histograms.
We now discuss this in more detail.
In general, vectors with a dimension as high as the ones from a neural network (maximum of 1024 dimensions in our case) tend to suffer from the curse of dimensionality, which implies that in very high dimensions, the Euclidean metric or the $L_2$ norm does not exhibit variation - all distances and norms look the same.
\begin{figure}[!h]
\centering
\includegraphics[width=0.95\columnwidth]{overlayed_l2norms.pdf}
\caption{$L_2$ norms of activation vectors across all layers.}
\label{fig:overlayed_l2norms}
\end{figure}
\autoref{fig:overlayed_l2norms} shows the distribution of $L_2$ norms for all layers in the Inception architecture.
Notice that the distribution is bell-shaped with long tails, and the variance of the distribution is reasonably large.
The severity of the curse of dimensionality may be reduced by using an adaptive cover that has more intervals in the denser regions of lens function.
The resulting mapper graphs with such an adaptive cover will contain nodes of comparable sizes.
This is left for future work.
\section{Methods}
\label{sec:methods}
We describe data analytic components of {\topoact}.
First, for a chosen layer of a neural network model (such as InceptionV1), we obtain activation vectors as high-dimensional point clouds for topological data analysis.
Second, we construct mapper graphs from these point clouds to support interactive exploration.
The nodes in the mapper graphs correspond to clusters of activation vectors in high-dimensional space, and the edges capture relationships between these clusters.
Third, for each node (cluster) in the mapper graph, we apply feature visualization to individual activation vectors in the cluster and to the averaged activation vector.
\para{Obtaining activation vectors as point clouds.}
The activation of a neuron is a nonlinear transformation of its input.
To start, we fix a trained model (\ie,~InceptionV1) and a particular layer (~\emph{e.g., 4c}) of interest.
We feed each input image to the network and collect the activations, \ie, the numerical values of how much each neuron has fired with respect to the input, at a chosen layer.
Since InceptionV1 is a CNN, a single neuron does not produce a single activation for an input image, but instead a collection of activations from a number of overlapping spatial patches of the image.
When an entire image is passed through the network, a neuron will be evaluated multiple times, once for each patch of the image.
For example, a neuron within layer \emph{4c} outputs $14 \times 14$ activations per image (for $14 \times14$ patches).
To simplify the construction, in our setting, we randomly sample a single spatial activation from the $14\times 14$ patches, excluding the edges to prevent boundary effects.
For $300K$ images, this gives us $300K$ activation vectors for a given layer.
\autoref{fig:activation-vector} illustrates what we mean by an activation vector.
The dimension of an activation vector depends on the number of neurons in the layer. For instance, layers \emph{3a}, \emph{3b}, and \emph{4a} have $256$, $480$, and $512$ neurons, respectively, producing point clouds of corresponding dimensions.
\begin{figure}[!ht]
\centering
\includegraphics[width=0.90\columnwidth]{activation-vector.pdf}
\caption{Each activation vector is a vector of spatial activations across all channels.}
\label{fig:activation-vector}
\end{figure}
\para{Constructing mapper graphs from activation vectors.}
Given a point cloud of activation vectors, we now compute a mapper graph as its topological summary.
Each node in the mapper graph represents a cluster of activation vectors, and an edge connects two nodes if their corresponding clusters have a nonempty intersection.
Our mapper graphs use the $L_2$-norm of the activation vector as the lens function. We set $n = 70$ cover elements with $p= 20\%, 30\%$, and $50\%$ as the amount of overlap.
We use DBSCAN~\cite{EsterKriegelSander1996} as the clustering algorithm, with minimum points per cluster $\emph{minPts}=5$.
For the $\epsilon$ parameter of the DBSCAN algorithm, which defines core points, we use two variations in our experiments.
In the first variation, we use a fixed $\epsilon = 1,000$ estimated using the distribution of pairwise distances at the middle layer.
In the second variation, we set $\epsilon$ adaptively for each layer, employing the procedure proposed in~\cite{EsterKriegelSander1996}.
Specifically, we generate an approximate kNN graph, sort the distances to the $5$-th nearest neighbor, and select an $\epsilon$ based on the location of a ``elbow'' when these distances are plotted~\cite{EsterKriegelSander1996}.
This way, the $\epsilon$ value is more adaptive to
the activation space of each layer.
Our \emph{adaptive} $\epsilon$ values are $830$ for \emph{3a}, $1070$ for \emph{3b}, $1750$ for \emph{4a}, $1630$ for \emph{4b}, $1330$ for \emph{4c}, $980$ for \emph{4d}, $775$ for \emph{4e}, $790$ for \emph{5a}, and $260$ for \emph{5b}.
These parameter configurations give rise to six datasets currently deployed in our live demo.
Each dataset contains nine mapper graphs (across nine layers of InceptionV1) constructed by a particular set of parameters associated with the mapper construction. The mapper graphs are named according to these parameters.
In particular, each name starts with "overlap-$x$" where $x$ is $20$, $30$, or $50$ to denote $20\%$, $30\%$, or $50\%$ overlap, respectively.
The second half of the name consists of "epsilon-$x$" where $x$ is either "fixed" or "adaptive", indicating whether $\epsilon$ was fixed ($=1000$) or set adaptively for each layer.
For example, \emph{overlap-50-epsilon-fixed} is the dataset containing mapper graphs of nine layers generated using $p= 50\%$ with fixed $\epsilon = 1000$.
\para{Applying feature visualization to activation vectors.}
Activation vectors are high-dimensional abstract vectors.
We employ \emph{feature visualization} to transform them into a more meaningful semantic representation using techniques proposed by Olah \etal~\cite{OlahMordvintsevSchubert2017, OlahSatyanarayanJohnson2018}.
Whereas the neural network transforms an input image into activation vectors, feature visualization goes in the opposite direction.
Given an activation vector $h_{x,y}$ at a spatial position $(x,y)$, feature visualization synthesizes an idealized image that would have produced $h_{x,y}$ via an iterative optimization process.
Normally, this synthesis is achieved by maximizing the dot product $h_{x,y}\cdot v$ of the vector $h_{x,y}$ with the direction $v$.
However, the vector $v$ that maximizes the dot product can have a large orthogonal component.
To counter this, following~\cite{CarterArmstrongSchubert2019}, the dot product is multiplied with a cosine similarity, putting greater emphasis on the angle between vectors.
The optimization process, which is similar to back propagation, begins with a random noise image.
Using gradient descent, this image is iteratively tweaked to maximize the following objective:
${(h_{x,y}\cdot v)^{n+1}}/{(||h_{x,y}|| \,||v||)^n}.$
Subsequently, a \emph{transformation robustness} regularizer~\cite{OlahMordvintsevSchubert2017} is used, which applies small stochastic transformation (jitter, rotate or scale) to the image before applying the optimization step.
Max-pooling can introduce high frequencies in the gradients.
To tackle this problem, the gradient descent is performed in Fourier basis with frequencies scaled to have equal energy, which is equivalent to whitening and de-correlating the data.
Applying feature visualization to all $300K$ activation vectors results in corresponding images that are likely to produce such activations, which we call \emph{activation images}.
Once we obtain a mapper graph, we also apply feature visualization to the averaged activation vector per cluster to obtain an \emph{averaged activation image} for each cluster.
However, feature visualization is not without drawbacks; due to the optimization process and the size of each cluster, it can generate abstract images that remain hard to interpret.
\subsection{Refined Analysis of Topological Structures}
\label{sec:pca}
We can utilize interesting topological structures identified by {\topoact} -- branches and loops -- to obtain topologically meaningful subsets of the activation vectors for further analysis.
We present some examples of {\topoact}-guided principal component analysis (PCA) with the following procedure.
We first identify all nodes that form a branch or a loop within a mapper graph.
We then extract activation vectors (as high-dimensional points) that map to these nodes.
Next, we apply PCA to these points and project them to a 2-dimensional plane.
Consider the leg-face bifurcation from \autoref{fig:branch-leg-face}.~\autoref{fig:pca-leg-face}(a) shows the PCA projection of all points that participate in the bifurcation.
The red points belong to the activation vectors from the ``face'' branch and the blue points belong to the ``leg'' branch.
Similarly, for the wheel-tread bifurcation from \autoref{fig:wheel-tread}, ~\autoref{fig:pca-leg-face}(b) illustrates the PCA projection of its associated points.
For both examples, we could easily observe that points from the two branches lie along two distinct directions.
\begin{figure}[!ht]
\centering
\includegraphics[width=.99\columnwidth]{pca-leg-face-wheel-tread.pdf}
\caption{PCA of the activation vectors that belong to (a) the leg-face bifurcation and (b) the wheel-tread bifurcation.}
\label{fig:pca-leg-face}
\vspace{-4mm}
\end{figure}
For comparative purposes, we apply PCA to all 300K points from layer \emph{4c} and highlight those from the leg-face bifurcation.
As shown in~\autoref{fig:mixed4c-pca}(a), there are no observable branching or clustering structure within this global projection.
To verify that the leg-face bifurcation is not spurious, we construct a minimal bounding box and identify around 86K neighboring points in the space of activations.
We apply PCA to these 86K points and observe that points from the leg-face bifurcation form clusters that are separable from their neighboring points; see ~\autoref{fig:mixed4c-pca}(b), which confirms that the leg-face bifurcation detected by {\topoact} is not spurious.
\begin{figure}[!ht]
\centering
\includegraphics[width=.98\columnwidth]{pca-leg-face-global.pdf}
\caption{PCA applied to the entire set of activation vectors from layer \emph{4c} (a), and to the activation vectors in the neighborhood of leg-face bifurcation (b). We do not see any branching or clustering structure in (a), while (b) reveals the leg-face bifurcation with respect to its neighboring points.}
\label{fig:mixed4c-pca}
\end{figure}
This example demonstrates the advantage of using {\topoact} in combination with classic DR techniques (such as PCA, t-SNE, and UMAP) to perform refined shape analysis of the space of activations.
Although the total number of activation vectors in our dataset is large, the number of significant branches and loops is relatively small, which leads us to hypothesize that a layer is particularly well-trained to identify certain directions in the activation space.
As shown here, {\topoact} can help us identify these directions.
\section{Related Work}
\label{sec:related-work}
We review visual analytics systems for deep learning interpretability as well as various notions of topological summaries.
\para{Visual analytics systems.}
Visual analytics systems have been used to support model explanation, interpretation, debugging, and improvement for deep learning in recent years; see~\cite{HohmanKahngPienta2018} for a survey.
Here we focus on approaches based on neuron activations for interpretability in deep learning.
This line of research attempts to explain the internal operations and the behavior of deep neural networks by visualizing the features learned by hidden units of the network.
Erhan \etal~proposed \emph{activation maximization}~\cite{ErhanDumitruBengio2009}, which uses gradient ascent to find the input image that maximizes the activation of the neuron under investigation.
It has been used to visualize the hidden layers of a deep belief network~\cite{ErhanDumitruBengio2009} and deep auto-encoders~\cite{Le2013}.
Simonyan \etal~\cite{SimonyanVedaldiZisserman2014} used a similar gradient-based approach to obtain salience maps by projecting neuron activations from the fully connected layers of the CNN back on to the input space.
Building on the idea of activation maximization, Zieler \etal~\cite{ZeilerFergus2014} proposed a deconvolutional network that reconstructs the input of convolutional layers from its output.
Yosinski \etal~\cite{YosinskiCluneNguyen2015} introduced the \emph{DeepVis} framework that visualizes the live activations produced on each layer of a CNN as it processes images/videos. Their framework also enabled visualizing features in each layer via regularized optimization.
These methods assume that each neuron specializes in learning one specific type of feature.
However, the same neuron can be activated in response to vastly different types of input images.
Reconstructing a single feature visualization, in such cases, leads to an unintelligible mix of color, scales or parts of objects.
To address this issue, Nguyen \etal~\cite{NguyenYosinkiClune2016} proposed \emph{multifaceted feature visualization}, which synthesizes a visualization of each type of input image that activates a neuron.
Another problem with these visualization approaches is the assumption that neurons operate in isolation.
This problem is addressed by the \emph{model inversion} method proposed by Mahendran \etal~\cite{MahendranVedaldi2015,MahendranVedaldi2016}.
Model inversion looks at the representations learned by the fully connected layers of a CNN, and reconstructs the input from these representations.
Kim \etal~introduced the \emph{TCAV} (Testing with Concept Activation Vectors) framework, which uses directional derivatives of activations to quantify the sensitivity of model predictions to an underlying high-level concept~\cite{KimWattenbergGilmer2018}.
All these techniques can help us understand how a single input or a single class of inputs is ``seen" by the network, but visualizing activations of neurons alone is somewhat limited in explaining the global behavior of the network.
To obtain a global picture of the network,
Karpathy~\cite{Karpathy2014} used t-SNE to arrange input images that have similar CNN codes (i.e.,~fc7 CNN features) nearby in the embedding. Nguyen \etal~\cite{NguyenYosinkiClune2016} projected the training set images that maximally activate a neuron into a low-dimensional space, also via t-SNE.
They clustered the images using k-means in the embedded space, and computed a mean image by averaging the images nearest to the cluster centroid.
Carter \etal~recently proposed the \emph{activation atlas}~\cite{CarterArmstrongSchubert2019}, which combines feature visualization with dimensionality reduction (DR) to visualize averaged activation vectors with respect to millions of input images.
For a fixed layer, the \emph{activation atlas} obtains a high-dimensional activation vector corresponding to each input image.
These high-dimensional vectors are then projected onto low-dimensional space via UMAP~\cite{McInnesHealyMelville2018,McInnesHealySaul2018} or t-SNE~\cite{MaatenHinton2008}.
Finally, feature visualization is applied to averaged activation vectors from small patches of the low-dimensional embedding that allow users to intuitively understand how a particular layer reacts to millions of input images.
Hohman \etal~proposed \emph{SUMMIT}~\cite{HohmanParkRobinson2020}, another framework that summarizes neuron activations of an entire layer of a deep CNN using DR.
In addition to aggregated activations, \emph{SUMMIT} also computes neuron influences to construct an \emph{attribution graph}, which captures relationships between neurons across layers.
\emph{Activation atlas} computes average activation vectors in a low-dimensional embedding, which may introduce errors due to neighborhood distortions.
In comparison, our approach aggregates activation vectors in a different manner.
Using the \emph{mapper} construction, a tool from TDA, we obtain a topological summary of a particular layer by preserving the clusters as well as relationships between the clusters in the original high-dimensional activation space. Our approach preserves more neighborhood structures since the topological summary is obtained within the high-dimensional activation space.
We then study how a particular layer of the neural network reacts to a large number of images through the lens of this topological summary.
\para{Various notions of topological summaries.}
In TDA, various notions of topological summaries have been proposed to understand and characterize the structure of a scalar function $f: \Xspace \to \Rspace$ defined on some topological space $\Xspace$.
Some of these, such as merge trees, contour trees~\cite{CarrSnoeyinkAxen2003}, and Reeb graphs~\cite{Reeb1946}, capture the behavior of the (sub)level sets of a function.
Others, including Morse complexes and the Morse-Smale complexes~\cite{EdelsbrunnerHarerZomorodian2003,EdelsbrunnerHarerNatarajan2003}, focus on the behavior of the gradients of a function.
Fewer topological summaries are applicable for a vector-valued function, including Jacobi sets~\cite{EdelsbrunnerHarer2002,BhatiaWangNorgard2015}, Reeb spaces~\cite{EdelsbrunnerHarerPatel2008,MunchWang2016}, and their discrete variant, the mapper construction~\cite{SinghMemoliCarlsson2007}.
In this paper, we apply the mapper construction to the study of the space of activations to generate topological summaries suitable for interactive visualization.
The mapper construction introduced by~Singh \etal~\cite{SinghMemoliCarlsson2007} has seen widespread applications in data science, including cancer research~\cite{NicolauLevineCarlsson2011,MathewsNadeemLevine2019}, sports analytics~\cite{Alagappan2012}, gene expression analysis~\cite{JeitzinerCarriereRougemont2019}, micro-epidemiology~\cite{Knudson2020}, genomic profiling~\cite{Cho2019}, and neuroscience~\cite{GeniesseSpornsPetri2019,SaggarSpornsGonzalez-Castillo2018}, to name a few; see~\cite{PataniaVaccarinoPetri2017} for an overview.
In visualization, topological approaches such as persistent homology and mapper have recently been applied in graph visualization~\cite{HajijWangRosen2018,HajijWangScheidegger2018,SuhHajijWang2020}.
\section{Applying TopoAct to ResNet Trained on CIFAR}
\label{sec:CIFAR}
To demonstrate the generality of our framework, we provide additional experiments using ResNet trained on the CIFAR-10 and CIFAR-100 datasets~\cite{Krizhevsky2009}.
Both datasets consist of the same set of $60K$ color images of dimension $32 \times 32$, with $50K$ training images and $10K$ test images.
CIFAR-10 has $10$ images classes with $6K$ images per class, and CIFAR-100 has $100$ image classes with $600$ images per class.
The class labels in CIFAR-10 are coarser, such as \textbf{automobiles} and \textbf{mammals}; whereas classes in CIFAR-100 are finer, such as \textbf{bicycle}, \textbf{bus}, \textbf{beaver}, and \textbf{hamster}.
We demonstrate that the insights provided by {\topoact} are not specific to a particular dataset or a particular network architecture.
We give a few exploration scenarios involving branches by applying {\topoact} to ResNet-18 trained on the CIFAR-10 dataset; such examples are similar to those described in~\autoref{sec:results}.
We encourage readers to explore further with our open-source online demo.
\para{Horse-deer bifurcation.}
Our first example is a horse-deer bifurcation from the last layer \emph{4.1.bn2} of the CIFAR-10 dataset, as illustrated in \autoref{fig:horse-deer}.
The left branch that contains nodes (b) and (c) corresponds to images of deer, whereas the right branch with nodes (d), (e) and (f) corresponds to images of horses.
The branching node (a) contains images of both horses and deer.
In addition, none of the earlier layers show such a clear bifurcation between the horse and deer classes. {\topoact} reveals the layer at which the network first begins to differentiate between these two classes.
Such insights would make {\topoact} a useful diagnostic tool for deep learning researchers (see~\autoref{sec:discussion}).
\begin{figure}[!h]
\centering
\includegraphics[width=.99\columnwidth]{horse-deer.pdf}
\vspace{-2mm}
\caption{Horse-deer bifurcation. Configuration: layer {4.1.bn2}, $40$ intervals, $20\%$ overlap.}
\vspace{-2mm}
\label{fig:horse-deer}
\end{figure}
\para{Frog-cat bifurcation.}
Similarly, our second example is a frog-cat bifurcation from the last layer \emph{4.1.bn2} of the CIFAR-10 dataset, as illustrated in \autoref{fig:frog-cat}.
Here, the branching node (a) contains images of frogs and cats.
It then bifurcates into a left branch (with nodes (b) and (c)) that contains only images of cats, and a right branch (with nodes (d), (e), and (f)) that contains only images of frogs.
Even though these are very different types of animals (mammals vs. amphibians), they share similar postures.
\begin{figure}[!h]
\vspace{-2mm}
\centering
\includegraphics[width=.99\columnwidth]{frog-cat.pdf}
\vspace{-2mm}
\caption{Frog-cat bifurcation. Configuration: layer {4.1.bn2}, $100$ intervals, $40\%$ overlap.}
\label{fig:frog-cat}
\vspace{-2mm}
\end{figure}
|
1,116,691,497,519 | arxiv | \section{Introduction}\label{intro}
One of the important task in nuclear and particle physics is
the nonperturbative understanding of
the phase structure of Quantum Chromodynamics (QCD) at
finite temperature ($T$) and real chemical potential $(\mur)$.
However, the confinement-deconfinement transition
has not been fully understood yet.
For example, we cannot find any order parameter of the deconfinement transition
in the case with dynamical quarks
The chiral and confinement-deconfinement transitions are key phenomena
for this purpose, but the confinement-deconfinement transition is not
yet fully understood comparing with the chiral transition.
Although the chiral transition can be described by the
spontaneous breaking of the chiral symmetry, but we can not
find any classical order-parameters of the confinement-deconfinement
transition in the presence of dynamical quarks.
The Polyakov loop, which respect the gauge-invariant holonomy,
is the exact order-parameter of the confinement-deconfinement
transition in the infinite quark mass limit.
However, it is no longer the order parameter in the presence of dynamical quarks.
Although other candidates for the order parameter
of the deconfinement transition have been proposed
~\cite{Bilgici:2008qy,Fischer:2009wc,Kashiwa:2009ki,
Benic:2013zaa,Lo:2013hla,Doi:2015rsa},
they are also not the exact order parameter.
Therefore, we need some extension of ordinary determinations to clearly
discuss and investigate the confinement-deconfinement transition in the
system with dynamical quarks.
Recently, in Refs.~\cite{Kashiwa:2015tna,Kashiwa:2016vrl,Kashiwa:2017yvy},
it is found that
the topological change of QCD thermodynamics at finite $\mui$
can be used to determine the confinement-deconfinement transition,
and based on the non-trivial free-energy degeneracy, the quark number holonomy
which is defined by the contour integral of the quark-number susceptibility of
$\theta=0 \sim 2\pi$ has been proposed as the quantum order-parameter for the
confinement-deconfinement transition.
This argument is based on the analogy of the topological order
discussed in Refs.~\cite{Wen:1989iv} and QCD at $T=0$~\cite{Sato:2007xc}
in the condensed matter physics.
The quark number holonomy counts the gapped points of the quark number density
along $\theta$.
As a results,
it becomes non-zero/zero in the deconfined/confined phase.
In particular,
the quark number density at $\theta=\pi/3$ is important
because it gives the property of the quark number holonomy.
In order to investigate the quark number density at $\theta=\pi/3$,
we use the Dirac-mode expansion \cite{Gongyo:2012vx}.
And we will see the behavior of the quark number density
by removal of the low-lying Dirac modes.
This paper is organized as follows.
In the section 2, we show the heavy quark mass
expansion of the quark number density.
In the section 3, we discuss the Dirac-mode expansion of the quark number density
in both large and small quark mass regime.
Section 4 is devoted to summary and discussions.
\section{large-mass expansion of quark number density}
\label{Sec:HQME}
In this study, we consider the ${\rm SU}(N_{\rm c})$ lattice QCD
on the standard square lattice.
We denote each sites as $x=(x_1,x_2,x_3,x_4) \ (x_\nu=1,2,\cdots,N_\nu)$
and link-variables as $U_\nu(x)$.
We impose the temporal periodic boundary condition for link-variables to
generate configurations in the quenched calculation to manifest the
imaginary-time formalism.
On the lattice, the quark number density is defined as
\begin{align}
n_q
=\frac{1}{V}\sum_x
\Bigl\langle \bar{q}(x) \frac{\partial D}{\partial \mu}q(x) \Bigr\rangle
=\frac{1}{V}\left\langle {\rm Tr}_{\gamma, {\rm c}}
\left[ \frac{\partial D}{\partial \mu} \frac{1}{D+m} \right] \right\rangle,
\label{DefQuarkNumber}
\end{align}
where ${\rm Tr}_{c,\gamma}\equiv \sum_x {\rm tr}_{\gamma, {\rm c}}$ denotes the functional trace and ${\rm tr}_{\gamma, {\rm c}}$ is taken over spinor and color indices.
The Dirac operator $D$ in this article is taken as the Wilson-Dirac operator
with quark mass $m$ and the chemical potential $\mu$ in the lattice unit as
\begin{align}
D
=-\frac{1}{2}\sum_{k=1}^3
\left[P(+k)\hat{U}_k+P(-k)\hat{U}_{-k}\right] -\frac{1}{2}\left[{\mathrm e}^\mu P(+4)\hat{U}_4 +
{\mathrm e}^{-\mu} P(-4)\hat{U}_{-4}\right] +4\cdot \hat{1}
, \label{WilsonDiracOp}
\end{align}
where $\hat{1}$ is the identity matrix.
The link-variable operator $\hat{U}_{\pm\nu}$ is defined by the matrix element
\begin{align}
\langle
x | \hat{U}_{\pm\nu} |x' \rangle=U_{\pm\nu}(x)\delta_{x\pm\hat{\nu},x'},
\label{LinkOp}
\end{align}
with $U_\nu\in {\rm SU}(N_{\rm c})$ and $P(\pm\nu)=1\mp\gamma_\nu$ with
$\nu=1,\cdots,4$.
The chemical potential $\mu$ is the dimensionless on the lattice
and we define $\theta\equiv\mathrm{Im}(\mu)N_\tau$.
We impose the temporal anti-periodicity and spatial periodicity for $D$.
To that end, we add a minus sign to the matrix element of
the temporal link-variable operator $\hat U_{\pm 4}$
at the temporal boundary of $x_4=N_4(=0)$:
\begin{align}
\langle {\bf x}, N_4|\hat U_4| {\bf x}, 1 \rangle
=-U_4({\bf x}, N_4),\ \ \ \ \ \ \
\langle {\bf x}, 1|\hat U_{-4}| {\bf x}, N_4 \rangle
=-U_{-4}({\bf x}, 1)=-U_4^\dagger({\bf x}, N_t).
\label{eq:LVthermal}
\end{align}
In this notation, the Polyakov loop is expressed as
\begin{align}
L\equiv\frac{1}{N_c V}
\sum_x {\rm tr}_c
\Bigl\{\prod_{n=0}^{N_4-1} U_4(x+n\hat{4}) \Bigr\}
=-\frac{1}{N_{\rm c}V}{\rm Tr}_c \{\hat U_4^{N_4}\}.
\label{PolyakovOp}
\end{align}
The minus sign stems from the additional minus on $U_4({\bf s}, N_t)$
in Eq.(\ref{eq:LVthermal}).
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.14]{Fig1.eps}
\caption{
Several paths on the lattice in the case of $N_4=5$.
(a) A temporally closed loop. It is gauge-invariant and $\mu$-dependent.
(b) A spatially closed loop. It is gauge-invariant and $\mu$-independent.
(c) A non-closed loop. It is gauge-variant and thus it vanishes.
}
\label{loops}
\end{center}
\end{figure}
In the heavy quark mass region,
the quark number density (\ref{DefQuarkNumber}) can be expressed by
using the quark mass expansion as
\begin{align}
n_q
=\frac{1}{MV}\left\langle {\rm Tr}_{\gamma, {\rm c}} \left[ \frac{\partial D}{\partial \mu}
\sum_{n=0}^\infty\left(-\frac{\hat{D}}{M}\right)^{n} \right] \right\rangle
\equiv\frac{1}{MV}\sum_{n=0}^\infty \frac{c^{(n)}}{(-M)^n},
\label{QuarkNumberHeavyMass}
\end{align}
where
we define the effective mass $M\equiv m+4$ and the operator $\hat{D}\equiv D-4$.
In the case of the large quark mass,
$c^{(n)}$ with smaller $n$ are dominant.
The $n$-th order contribution $c^{(n)}$ has the terms
\begin{align}
c^{(n)}=c^{(n)}_1+c^{(n)}_2+c^{(n)}_3+\cdots,
\label{n-thContribution}
\end{align}
and $c^{(n)}_i$ is a product of $(n+1)$ link-variables.
The examples of the paths of the products are shown in Fig.~\ref{loops}.
Note that many of them become exactly zero
because of Elitzur's theorem \cite{Elitzur:1975im};
only the gauge-invariant terms corresponding to closed loops are nonzero.
Moreover, spatially closed loops which do not wind the temporal length
are canceled out each other and have no contribution to the quark number density in total in each order $n$.
Noting these important facts, it is confirmed that
the $n$-th order contribution $c^{(n)}$ is constituted of
the gauge-invariant loops with the length $(n+1)$ which winds the temporal direction.
In particular, the nonzero leading term in the expansion (\ref{QuarkNumberHeavyMass})
is the $(n=N_4-1)$-th order term $c^{(N_4-1)}$
which relates to the Polyakov loop ($L$) and its complex conjugate (${\bar L}$).
The leading term, $c^{(N_4-1)}$, to the
quark number density at finite $\theta$ is written as
\begin{align}
c^{(N_4-1)}\sim{\mathrm e}^{i\theta}L-{\mathrm e}^{-i\theta}L^*
=2\sin(\theta+\phi)|L|.
\label{QuarkNumberLeading}
\end{align}
In the heavy quark-mass expansion of the quark number
density~(\ref{QuarkNumberHeavyMass}),
there are higher order terms beyond the leading terms,
which are the Polyakov loop and its conjugate.
By similar analysis in the case of the leading term,
the higher-order terms can be expressed in terms of the quantities
which correspond to loops on the lattice.
For example, the sub-leading terms include a term proportional to the quantity
\begin{align}
c^{(N_4+1)}_1
\equiv
{\rm Tr}_c \{\hat U_4U_1U_4U_{-1}U_4^{N_4-2}\}.
\label{c1}
\end{align}
These terms correspond to the closed paths
which wind the temporal length and bypasses the spatial direction.
As another example, loops winding the temporal direction twice or more
can be contribute to the expansion (\ref{QuarkNumberHeavyMass}).
For example, a loop winding the temporal direction twice
\begin{align}
c^{(2N_4-1)}_1
\equiv
{\rm Tr}_c \{\hat U_4^{2N_4}\}
\label{c2}
\end{align}
is a possible contribution to Eq. (\ref{QuarkNumberHeavyMass})
as the ($n=2N_4-1$)-th order term.
\section{Dirac-mode expansion of the quark number density}
\label{Sec:DME}
In the following,
we consider the quark number density in terms of the Dirac eigenmode.
In large quark mass region,
we analytically investigate it in all order
of the large quark mass expansion (\ref{QuarkNumberHeavyMass}).
In small quark mass region,
we perform the quenched lattice QCD simulation.
The Wilson-Dirac eigenvalues $\lambda_n$ are obtained from the
eigenvalue equation as
\begin{align}
D|n\rangle = \lambda_n|n\rangle, \label{DiracEigenEq}
\end{align}
where $|n\rangle$ is the Wilson-Dirac eigenstate.
Considering the Wilson-Dirac mode expansion of the chiral condensate,
the low-lying eigenmodes of the operators $D$
have dominant contribution to the chiral condensate
known as Banks-Casher relation~\cite{Banks:1979yr, Giusti:2008vb}.
\subsection{large quark mass region}
We start the leading term to express it in terms of the Wilson-Dirac modes.
The leading contribution of the quark number density
in large quark mass region (\ref{QuarkNumberLeading}) is
expressed
by the Polyakov loop and its complex conjugate.
It is already known that the Polyakov loop can be expressed in terms of the
eigenmodes of the naive-Dirac operator which corresponds to the case of $r=0$
\cite{Suganuma:2014wya,Doi:2014zea}
and the Wilson-Dirac operator \cite{Suganuma:2016lnt,Suganuma:2016kva}.
In the following, we derive a different form of the Dirac spectral representation of
the Polyakov loop using the operator $D$ on the square lattice
with the normal non-twisted periodic boundary condition for link-variables,
in both temporal and spatial directions.
We define a key quantity,
\begin{align}
I^{(N_4-1)}={\rm Tr}_{c,\gamma} (D\hat{U}_4^{N_4-1}). \label{I0}
\end{align}
This quantity is defined as the slightly changed quantity from the Polyakov loop
by replacing a temporal link-variable $\hat{U}_4$
to the Wilson-Dirac operator $D$.
This quantity can be calculated as
\begin{align}
I^{(N_4-1)}
=2\mathrm{e}^{-\mu}N_{\rm c}VL. \label{I0_1}
\end{align}
Thus, the quantity $I^{(N_4-1)}$ is proportional to $L$.
Here, other terms vanish because of the Elitzur's theorem
and the trace over the Dirac indecies.
On the other hand,
since $I^{(N_4-1)}$ in Eq. (\ref{I0}) is defined through the functional trace,
it can be expressed in the basis of Dirac eigenmodes as
\begin{align}
I^{(N_4-1)}
=\sum_n\langle n|D\hat{U}_4^{N_4-1}|n\rangle + \mathcal{O}(a)
=\sum_n\Lambda_n \langle n|\hat{U}_4^{N_4-1}| n \rangle + \mathcal{O}(a). \label{I0_2}
\end{align}
The $\mathcal{O}(a)$ term arises because
Wilson-Dirac operator is not normal due to the $\mathcal{O}(a)$ Wilson term and
the completeness of the Wilson-Dirac eigenstates has the $\mathcal{O}(a)$ error:
\begin{align}
\sum_n |n\rangle\langle n|=1+\mathcal{O}(a).
\label{Complete}
\end{align}
However, this error is controllable and can be ignored in close to the continuum limit.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.15]{Fig3.eps}
\caption{
Schematic figures for the functional traces.
They are defined from the gauge-invariant quantities,
Eqs. (\ref{PolyakovOp}), (\ref{c1}) and (\ref{c2}),
by changing a temporal link-variable to the Wilson-Dirac operator.
}
\label{Fig:ECQND}
\end{center}
\end{figure}
Combining Eqs. (\ref{I0_1}) and (\ref{I0_2}),
one can derive the relation between $L$ and the Dirac modes as
\begin{align}
L\simeq\frac{\mathrm{e}^{-\mu}}{2N_{\rm c}V}
\sum_n\Lambda_n\langle n|\hat{U}_4^{N_4-1}| n \rangle,
\label{RelOrig}
\end{align}
with the $\mathcal{O}(a)$ error.
From the formula (\ref{RelOrig}),
it is analytically found that
the low-lying Dirac modes with $|\Lambda_n|\sim0$ have
negligible contribution to the Polyakov loop
because the eigenvalue $\Lambda_n$ plays as the damping factor.
It is also numerically shown that
there is no dominant contribution in the Dirac modes
to the Polyakov loop \cite{Doi:2014zea}.
Thus, one can find the
Dirac spectrum representation of the quark
number density $n_q$ in the leading order
and the low-lying Dirac modes
have little contribution to the quark number density.
The above discussion on the leading term can be applicable to
the Dirac spectrum representation of the higher order terms.
The detailed discussion is shown in our recent paper \cite{Doi:2017dyc}.
In the same way in the case of the leading term,
all the terms in the expansion (\ref{QuarkNumberHeavyMass})
can be expressed in terms of the Wilson-Dirac modes.
Thus, it is analytically found that
the quark number density does not depend on
the density of the low-lying Wilson-Dirac modes in the all-order.
However, this fact is only valid in the sufficiently large quark mass region
since other contributions which can not be expressed by $L$ and
${\bar L}$ can appear in the small $m$ region.
\subsection{Small quark mass region}
Next, we consider the small mass regime.
In this regime, we perform the lattice QCD simulation to investigate the quark
number density.
In this study, we perform the quenched calculation
with the ordinary plaquette action
and then fermionic
observables are evaluated by using the Wilson-Dirac operator (\ref{WilsonDiracOp})
with the imaginary chemical potential $\mu$.
Our calculation is performed in both the confinement phase and
the deconfinement phase.
In the confinement phase,
we consider $6^4$ lattice with $\beta\equiv 6/g^2=5.6$ and $\mu=(0,1745)$
which corresponds to $a\simeq0.25$ fm and $T\simeq133$ MeV.
In the deconfinement phase,
we consider $6^3\times5$ lattice with $\beta=6.0$ and $\mu=(0,2094)$
which corresponds to $a\simeq0.10$ fm and $T\simeq400$ MeV.
Both values of $\mu$ correspond to $\theta\simeq\pi/3$.
In both cases, we set the quark mass as $m=-0.7$ in the lattice unit,
which is equivalent to the hopping parameter $\kappa\equiv1/(2m+8)\simeq0.151515$,
for the calculation of the eigenmodes of the Wilson-Dirac operator
in the small quark mass region \cite{Aoki:1999yr}.
We calculate the quark number density as
\begin{align}
\langle n_q \rangle
=\frac{1}{2V}\left\langle {\rm Tr}_{\gamma, {\rm c}}
\left[
\frac{\partial D}{\partial \mu} \frac{1}{D+m}
-\left(\frac{\partial D}{\partial \mu} \frac{1}{D+m} \right)^\dagger
\right] \right\rangle
\simeq\frac{i}{V}{\rm Im}\left\langle \sum_n
\Bigl\langle n \Bigl| \frac{\partial D}{\partial \mu} \Bigr|n \Bigr\rangle \frac{1}{\Lambda_n+m} \right\rangle.
\label{QuarkNumber_Dirac}
\end{align}
This form trivially takes pure imaginary value up to the
$\mathcal{O}(a)$ error.
Each contribution, $n_q^n$, to the quark number density
of the Dirac mode with $\Lambda_n$
can be defined as
\begin{align}
n_q^n=
\frac{i}{V}{\rm Im} \Bigl \langle n \Bigl| \frac{\partial D}{\partial \mu} \Bigr|n \Bigr\rangle \frac{1}{\Lambda_n+m},
\end{align}
and then the quark number density becomes
\begin{align}
n_q=\sum_n n_q^n.
\end{align}
In Fig.~\ref{Fig:ECQND},
${n}_\mathrm{q}^n (\Lambda)$ is shown in the cases with
$\mu = (0,0.1745)$ and $\mu =(0,0.2094)$.
We here only show results with one particular configuration.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.50]{Fig4_1.eps}
\includegraphics[scale=0.50]{Fig4_2.eps}
\caption{
Each Dirac-mode contribution ${n}_\mathrm{q}^n (\Lambda_n)$
to the quark number density as a function
plotted against the Wilson-Dirac eigenvalue $\Lambda_n$.
Left and right panels show
the results with $\mu =(0,0.1745)$ and $\mu =(0,0.2094)$, respectively.
}
\label{Fig:ECQND}
\end{center}
\end{figure}
To investigate the quark number density in terms of Dirac modes,
we define the infra-red (IR) cutted quark number density with the cutoff
$\Lambda_\mathrm{cut}$ as
\begin{align}
n^\mathrm{cut}_\mathrm{q} (\Lambda_\mathrm{IR})
= \frac{1}{n_\mathrm{q}} \sum_{|\Lambda_n|>\Lambda_{\rm IR}}
{n}_\mathrm{q}^n.
\label{Eq:IR}
\end{align}
In Fig.~\ref{Fig:CQND},
we shown the result at $\mu = (0,0.2094)$,
where the system is in the deconfinement phase.
In general,
we can perform the configuration average in the evaluation of Eq.~(\ref{Eq:IR}).
However, since the averaging well works after summing over all Dirac-modes,
it misses a physical meaning.
Then, we show
$n^\mathrm{cut}_\mathrm{q}$ in one particular configuration.
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.50]{Fig5.eps}
\caption{
The $\Lambda_\mathrm{cut}$-dependence of $n_\mathrm{q}^\mathrm{cut}$
at $\mu =(0,0.2094)$.
}
\label{Fig:CQND}
\end{center}
\end{figure}
Note that the sign of the quark number density
does not changed by removing the low-lying Dirac modes
while the absolute value of the quark number density seems to drastically change.
This tendency has been found in almost all our configurations.
This result means that the absolute value of the quark number density
shares a same property in terms of Dirac modes with the chiral condensate,
while its sign shares the property with the Polyakov loop.
Finally, the topological confinement-deconfinement transition
can be discussed in terms of the Dirac-mode analysis.
The order-parameter of the topological confinement-deconfinement
transition can be expressed~\cite{Kashiwa:2016vrl} as
\begin{align}
\Psi &=
\oint_{0}^{2\pi}
\Bigl\{\mathrm{Im} \Bigl(
\frac{d {\tilde n}_q}{d \theta} \Bigl|_T \Bigr) \Bigr\}
~d\theta,
\label{Eq:psi}
\end{align}
where ${\tilde n}$ is the dimensionless quark number density such as
${\tilde n}_\mathrm{q} = n_\mathrm{q}/T^3$.
It counts gapped points of the quark number density along
$\theta$ direction and thus it becomes zero/non-zero
in the confined/deconfined phase.
The quark number holonomy (\ref{Eq:psi}) can be expressed as
\begin{align}
\Psi &= \pm
2 \nc \lim_{\epsilon \to 0}
\Bigl[ \mathrm{Im}~
{\tilde n}_q ( \theta=\qrw \mp \epsilon )\Bigr],
\label{psi1}
\end{align}
when the RW endpoint which is the endpoint of the RW transition line
becomes the second-order point at
$\theta_\mathrm{RW} =\pi / 3$.
In Eq.~(\ref{psi1}), $\lim_{\epsilon \to 0} n_\mathrm{q} (\qrw \mp
\epsilon)$ characterizes
$\Psi$ and thus $\Psi$ shares the same property about the Dirac modes
with $n_\mathrm{q}$.
The important point here is that
the absolute value of $n_q$ does not have so much meaning even if it is non-zero,
but its sign is important since
the sign flipping at $\theta = (2k-1) \pi/3$
characterizes the gapped points along $\theta$-direction.
From our quenched lattice QCD data, the sign of the quark number density
are insensitive to the low-lying Dirac-modes and this behavior is similar
to the Polyakov loop.
\section{Summary and discussion}
\label{Sec:summary}
In this paper, we have discussed properties of the quark number
density at finite temperature ($T$) and imaginary chemical potential
($\mu_\mathrm{I}$) by using the Dirac-mode expansion.
From the heavy quark mass expansion with the Dirac mode expansion,
we found that low-lying Dirac modes do not dominantly contribute to the
quark number density in all order of the heavy quark mass expansion.
In the small quark mass regime,
we found that the absolute value of the quark number density strongly
depends on low-lying Dirac modes, but its sign does not.
Our result shows that
the quark number holonomy is sensitive to the confinement properties of QCD
and it is the good quantum order parameter
for the confinement-deconfinement transition.
In order to avoid the incompleteness of the Wilson-Dirac modes,
one can consider the hermitian Wilson-Dirac operator $H\equiv\gamma_5 D$.
In fact, the actual calculation is shown in our recent paper \cite{Doi:2017dyc}.
Moreover, the more detailed calculation on the higher order terms in the
large-mass expansion is also included in it.
|
1,116,691,497,520 | arxiv | \section{Introduction}
Lorentz invariance is one of the pillars of the current theories of fundamental physics. It implements the physical equivalence of inertial
observers in absence of gravity, together with the presence of an invariant scale, the speed of light in vacuum. In particular, Lorentz
symmetry is an important ingredient of the Standard Model (SM) of elementary particles. However, the logical consistency of the latter
does not rely on the former, contrary to what happens to unitarity or the absence of gauge anomalies which are necessary requirements
for a quantum theory like the SM. This permits investigating departures from Lorentz invariance in the framework of a quantum field
theory (QFT), and therefore, testing Lorentz invariance as a symmetry of the SM. Although much progress has been made in this direction,
the non-compactness nature of the Lorentz group, the fact that the boost parameter can be arbitrarily large, implies that the unexplored region of this group will always be infinitely great, unlike the rotation group which is compact and can be checked for every angle. Apart from the own interest of testing such a fundamental symmetry, there are motivations to study deviations from Lorentz invariance coming from both the theoretical and the phenomenological sides. On the one hand, some indications coming mostly from theoretical investigations of the quantum-gravity problem have suggested that Lorentz symmetry could be either violated or deformed at high enough energies (see~\cite{BW,ST,LQG,RG,AM,Girelli:2004md,Smolin:2008hd}). On the other hand, some observations might be interpreted to be incompatible with Lorentz invariance, such as a possible violation of the GZK cut-off~\cite{GZK}, anomalies in the propagation of very high energy gamma rays in intergalactic space~\cite{meyer,piran}, or, more recently, the observation of apparent superluminal neutrinos by the OPERA collaboration~\cite{Adam:2011zb} following a proposal~\cite{Ellis:2008fc} to look for signals of Lorentz invariance violation in neutrino propagation.
In this work, we present an analysis of the consequences of such departures from exact Lorentz invariance in the decay of superluminal neutrinos. In particular, we shall focus on charged-lepton pair emission by superluminal neutrinos ($\nu _i \to \nu _i \; e^- \; e^+ $) which is a forbidden process in a Poincar\'e invariant theory because of the charged lepton's masses. The idea that neutrinos can travel faster than light is not recent, and it actually goes back to the eighties (see~\cite{Chodos1984cy} and~\cite{Giannetto:1986rm}). The difficulty in the detection of neutrinos (owing to the weakness of their interactions with matter) entails some of their features remain still unknown, making neutrinos somehow mysterious particles. Since neutrino oscillations suggest that they are not massless particles, it is natural to expect the existence of right-handed neutrinos which are transparent to the electroweak and strong interactions of the SM (they are singlets under $U_Y(1)\times SU_L(2)\times SU_C(3)$ transformations). This special property has been used in the large-extra-dimensions scenario to speculate about the possibility that neutrinos could not be confined to our three-brane, but they are indeed exploring the extra dimensions~\cite{Ark}. Because of this, if the departures from Lorentz invariance had their origin in these additional dimensions, it would be expected that neutrinos were the unique particles capable of directly feeling such deviations from exact Lorentz symmetry (perhaps together with an hypothetical graviton which would also be transparent to the gauge interactions of the SM). The particular quantum numbers of neutrinos in the SM make also possible to write a gauge invariant but Lorentz violating Lagrangian which gives rise to effects of Lorentz invariance violation which are different for neutrinos than for the rest of the SM particles.
Other works have considered neutrinos as the only candidates for Lorentz violating particles in an effective field theory (EFT) framework because in this context, and in presence of interactions, different limiting speeds of different particles at high energies are driven by the renormalization group flow towards a universal speed at low energies (although strong assumptions have to be made in order that the flow be sufficiently fast), the vacuum speed of light (see~\cite{Don}). The weaker the interaction, the weaker the running and therefore the difference of limiting speeds is the greatest when the interactions are the weakest. This suggests that the limiting speed of neutrinos at low energies may be different from the speed of the rest of particles when Lorentz invariance has been broken at high energies.
In this paper, we shall parameterize the deviations from exact Lorentz invariance by a modified dispersion relation for free neutrinos. The consequence of such modification is twofold: on the one hand, there is a new dependence of the velocity on the energy which makes the time of flight different from the one of a special-relativistic theory. On the other hand, the kinematics of particle processes involving neutrinos changes. In fact the dynamics of these processes is also changed because of the different choices of a dynamical matrix element compatible with the modified dispersion relation one can make. Previous works have focused on getting qualitative or quantitative results for the decay width $\nu_i \to \nu_i \, e^- \, e^+ $ or $\nu_i \to \nu_i \, \nu_j \, \bar{\nu_j} $ \cite{Mat,Cohen:2011hx,Carmona:2011zg,Maccione:2011fr,Ciuffoli:2011aa}, without discussing the different alternatives one has for the matrix element, or even without specifying the matrix element they are using in their calculation (however, see~\cite{Bez}). The main \emph{purpose} of the present work is to show how this can affect the results for this process. One could think that a criterion to select a matrix element would be that it could be deduced from a local EFT. However, there exist limitations to this framework. First, one should restrict dispersion relations to analytical expressions so that a momentum power expansion be possible in the EFT. Second, the most likely origin of such deviations from Poincar\'e symmetry arises in attempts to reconcile general relativity and quantum theory, through residual effects (that is, effects that are present when the classical curvature of space-time can be neglected) in the structure of space-time or/and momentum space which modify its classical, Minkowskian nature~\cite{CNC,Kow}. If this were the case, it would not be surprising that this led to non-local effects at low energies since the semi-classical physics of black holes tells us that the fundamental degrees of freedom of gravity cannot be described by a \emph{local} QFT, regardless of whether the latter has a non-trivial fixed point or not, because the densities of states of both theories do not match~\cite{Susskind:1994vu,Shomer}.
Furthermore, doubts on the validity of an EFT description of the low energy limit of quantum gravity come from the difficulties to incorporate the necessary cancellations of contributions to the vacuum energy (cosmological constant problem).
Throughout this work, we shall assume: i) Rotational invariance is preserved. ii) Energy and momentum are conserved in the conventional, additive way. iii) The relevant propagation speed of the neutrinos is the group velocity of their wave packets. iv) Charged leptons are (or can be approximated by) special-relativistic particles. v) Indirect bounds on neutrino masses based on neutrino oscillations and cosmological observations are still valid, in such a way that neutrinos masses can be neglected in the process.
This work has the following structure. In the next section~(\ref{SecPair}) we shall show the general procedure to obtain the decay widths, and we will compute them for four different choices of the matrix element and a general dispersion relation. Section~\ref{SecChoices} is devoted to discuss the properties of the previous matrix elements and their physical interpretation. In Section~\ref{SecEnergy} we shall specify dispersion relations to give definite analytical results for the decay widths and the rates of energy loss due to the pair emission. We shall also provide numerical results for these expressions. Section~\ref{SecTimes} is dedicated to the study of the time of flight of neutrinos in the presence of the aforementioned loss of energy. The following section~(\ref{SecConsistency}) is devoted to analyze the consistency of the superluminal speed interpretation of the observations reported by the OPERA collaboration. Our conclusions and remarks will close this work in Sec.~\ref{SecConcluiding}.
\section{Pair production decay width}
\label{SecPair}
Let us consider the process $\nu(p) \to \nu(p_1) \, e^{-}(p_2) \, e^{+}(p_3)$ induced by the production of a virtual $Z^0$ and subsequent decay into an $e^{-}e^{+}$ pair. This is a kinematically forbidden process in special relativity (SR) which becomes allowed for superluminal neutrinos above a certain energy threshold. We are going to consider a very high energy neutrino so that all three particles in the final state are contained within a small cone around the direction of the momentum of the neutrino in the initial state. The decay width of the process is given by
\begin{equation}
\Gamma =\frac{1}{2E} \left[\prod_{i=1}^3 \int \frac{d^3 {\vec p}_i}{2 E_i (2\pi)^3}\right] (2\pi)^4 \delta(E-\sum_i E_i) \delta^3 ({\vec p}-\sum_i {\vec p}_i) \, \overline{M^2} \,,
\label{Gamma(pp)}
\end{equation}
where $\overline{M^2}$ is the squared amplitude averaged over initial spin states and summed over final spin states,
\begin{equation}
\overline{M^2} = A^{\mu\nu}(p, p_1) p_{2\mu} p_{3\nu}\,,
\label{def(A)}
\end{equation}
and we are using the factorization of the matrix element into a factor depending on the neutrino variables and a factor depending on the electron-positron momenta. The second factor is the standard SR Dirac trace which gives the dependence on the four-momenta $p_2$ and $p_3$. The coefficients $A^{\mu\nu}$ depending on the neutrino momenta $p$, $p_1$ contain all the corrections to SR in the dynamical matrix element.
The decay width can be written as
\begin{equation}
\Gamma = \frac{1}{2E} \int \frac{d^3 \vec{p}_1}{2E_1 (2\pi)^3} A^{\mu\nu}(p, p_1) B_{\mu\nu}(p-p_1)
\label{Gammacov}
\end{equation}
where
\begin{equation}
B_{\mu\nu}(k) = \int \frac{d^3\vec{p}_2}{2E_2 (2\pi)^3} \int \frac{d^3\vec{p}_3}{2E_3 (2\pi)^3} \, p_{2\mu} p_{3\nu} (2\pi)^4 \delta^4(k-p_2-p_3) \,.
\end{equation}
In the approximation where one neglects any modification to SR kinematics for the electron and positron,
\begin{equation}
B_{\mu\nu}(k) = B_1(k^2) \eta_{\mu\nu} k^2 + B_2(k^2) k_\mu k_\nu \,,
\end{equation}
where $k^2 = k_0^2 - {\vec k}^2$, and
\begin{align}
B_1(k^2) &= \frac{1}{96\,\pi} \left(1 - \frac{4m_e^2}{k^2}\right)^{3/2}\,
\theta(k^2 - 4m_e^2)\, \theta(k_0) \nonumber \\
B_2(k^2) &= \frac{1}{96\,\pi} \left(2 + \frac{4m_e^2}{k^2}\right)
\left(1 - \frac{4m_e^2}{k^2}\right)^{1/2} \theta(k^2 - 4m_e^2)\, \theta(k_0) \,,
\end{align}
where $\theta$ is the Heaviside step function. On the other hand, one has
\begin{equation}
k^2 = (p-p_1)^2 = (E-E_1)^2 - (|\vec{p}|-|\vec{p}_1|)^2 - 2 |\vec{p}| |\vec{p}_1| (1-\cos\theta_1)
\label{k2}
\end{equation}
where $\theta_1$ is the angle between the neutrino momenta $\vec{p}$, $\vec{p}_1$. The minimum value of $k^2$ ($4m_e^2$) corresponds to a maximum value of $(1-\cos\theta_1)$
\begin{equation}
(1-\cos\theta_1)_{+} = \frac{(E-E_1)^2 - (|\vec{p}|-|\vec{p}_1|)^2 - 4 m_e^2}{2 |\vec{p}| |\vec{p}_1|}\,.
\end{equation}
If we parameterize the energy-momentum relation for neutrinos as\footnote{We are neglecting neutrino masses in all the discussion.}
\begin{equation}
E = |\vec{p}| \left(1 + \epsilon(|\vec{p}|)\right)
\label{E}
\end{equation}
then we have
\begin{equation}
(E-E_1)^2 - (|\vec{p}|-|\vec{p}_1|)^2 = \left[(E+|\vec{p}|) - (E_1+|\vec{p}_1|)\right] \left[|\vec{p}| \epsilon(|\vec{p}|) - |\vec{p}_1| \epsilon(|\vec{p}_1|)\right]
\end{equation}
and one has a negative result for the maximum value of $(1-\cos\theta_1)$, i.e., pair production is not allowed, in the case of SR kinematics ($\epsilon(|\vec{p}|)=0$). The threshold momentum $|\vec{p}_{th}|$ for pair production verifies:
\begin{equation}
|\vec{p}_{th}|^2\,\epsilon(|\vec{p}_{th}|)=2\, m_e^2 \,
\end{equation}
corresponding to the configuration in which the outgoing neutrino is at rest, and the electron and positron have the same momentum. We shall consider incoming neutrino momenta much greater than this threshold momentum, $|\vec{p}|\gg |\vec{p}_{th}|$, in such a way that one can safely neglect electron and positron masses. In addition, we shall assume that the deviation from SR kinematics is sufficiently small ($|\epsilon(|\vec{p}|)| \ll 1$) so that $|\vec{p}_1| (1-\cos\theta_1)_{+} \ll |\vec{p}|$. In this case, the change in the direction of the neutrino energy flux due to the emission of the $e^{-}e^{+}$ pair is very small (``collinear'' approximation).
We shall also restrict all the discussion to neutrino momenta such that
\begin{equation}
\left[(E+|\vec{p}|) - (E_1+|\vec{p}_1|)\right] \left[|\vec{p}| \epsilon(|\vec{p}|) - |\vec{p}_1| \epsilon(|\vec{p}_1|)\right] \ll M_Z^2 \,.
\end{equation}
In this case we can use the point like approximation for the fermion interaction in the pair production process and then the momentum dependence in the coefficients $A^{\mu\nu}(p, p_1)$, which parametrize all our ignorance on the dynamics, comes from the modified neutrino Dirac trace. Assuming a linear dependence on the components of each momentum in the amplitude then one has
a general form for the angular dependence
\begin{equation}
\begin{split}
A^{\mu\nu}(p, p_1) B_{\mu\nu}(p-p_1) &= \frac{G_F^2 |\vec{p}|^4}{12 \pi} \left[(1-2s_W^2)^2 + (2s_W^2)^2 \right] \, \left[F_0(x_1) + F_1(x_1) (1-\cos\theta_1) + F_2(x_1) (1-\cos\theta_1)^2\right] \\
&\quad\times
\theta \left( (1-\cos\theta_1)_{+} - (1-\cos\theta_1)\right)\,\theta \left(1-x_1+\epsilon(|\vec{p}|) - x_1 \epsilon(x_1|\vec{p}|)\right) \, ,
\label{def(F)}
\end{split}
\end{equation}
when one uses the momentum fraction $x_1 = |\vec{p}_1|/|\vec{p}|$ for the neutrino after pair production,
and where $G_F$ is the Fermi constant and $s_W$ is the sine of the Weinberg angle.
The determination of the coefficients $F_n(x_1)$ of the three angular terms requires a definite dynamical framework.
The angular integral on the neutrino momenta can be made\footnote{The angular integral in $(1-\cos\theta_1)$ goes from $0$ to $(1-\cos\theta_1)_{+} \,$,
except for those outgoing (very close to zero) momenta for which $2<(1-\cos\theta_1)_{+}$, in which case the angular integral goes from $0$ to $2\,$.} and the leading contribution for the decay width is
\begin{equation}
\begin{split}
\Gamma = \frac {G_F^2 |\vec{p}|^5}{192 \, \pi^3} \left[(1-2s_W^2)^2 + (2s_W^2)^2 \right]\,
\int _0 ^1 &\left[F_0(x_1) (1-\cos\theta_1)_{+} + F_1(x_1) \frac{(1-\cos\theta_1)^2_{+}}{2} +
F_2(x_1) \frac{(1-\cos\theta_1)^3_{+}}{3}\right] \\
& \times \theta \left(\epsilon(|\vec{p}|) - x_1 \epsilon(x_1|\vec{p}|)\right) x_1\, dx_1
\label{dGamma/dx1}
\end{split}
\end{equation}
when one consistently retains only the dominant contribution in an expansion in powers of $\epsilon(|\vec{p}|)$.
\subsection{Dynamical matrix element}
Lacking a well defined dynamical framework incorporating superluminal neutrinos, we are going to consider four different simple choices for the dynamical matrix element as a way to illustrate the uncertainties in the evaluation of the pair production process. Naively, one would take the SR matrix element as a first approximation and one would expect that any other choice approaching the SR limit would lead to the same leading order result for the decay width but this is not the case. The reason is that in the SR limit and neglecting masses all four momenta $p$, $p_1$, $p_-$, $p_+$ are light-like and, due to energy-momentum conservation, proportional to each other so that any scalar product (and then the dynamical matrix element) vanishes.
Then the leading contribution to the dynamical matrix element comes from the first non-vanishing correction to the SR limit.
\subsubsection{First example}
An apparently natural choice for the dynamical matrix element corresponds to\footnote{We shall neglect masses in all the matrix elements.}
\begin{equation}
A^{\mu\nu}(p,p_1) = 16 \, G_F^2 \left[(1-2s_W^2)^2 p_1^\mu p^\nu + (2s_W^2)^2 p^\mu p_1^\nu\right]
\label{A1}
\end{equation}
which is the expression of the SR dynamical matrix element but with $p^0=|\vec{p}|(1+\epsilon(|\vec{p}|))$ and $p_1^0=|\vec{p}_1|(1+\epsilon(|\vec{p}_1|))$. In this case one has
\begin{equation}
A^{\mu\nu}(p, p_1) B_{\mu\nu}(k) = \frac{G_F^2}{12 \, \pi} \, \left[(1-2s_W^2)^2 + (2s_W^2)^2 \right] \, \left[2 (p_1\cdot p) k^2 \,+\, 4 (p_1\cdot k) \, (p\cdot k)\right]\theta(k^2)\, \theta(k_0) \, .
\label{AB}
\end{equation}
Using the products
\begin{align}
2 (p_1\cdot p) &= 2 \, \left[|\vec{p}| |\vec{p}_1| \left(\epsilon(|\vec{p}|) + \epsilon(|\vec{p}_1|)\right) + |\vec{p}| |\vec{p}_1| (1-\cos\theta_1)\right] \nonumber \\
k^2 &= 2 \, \left[(|\vec{p}|-|\vec{p}_1|) \left(|\vec{p}| \epsilon(|\vec{p}|) - |\vec{p}_1|
\epsilon(|\vec{p}_1|)\right) - |\vec{p}| |\vec{p}_1| (1-\cos\theta_1)\right] \nonumber \\
2 (p_1\cdot k) &= 2 \, \left[(|\vec{p}|-|\vec{p}_1|) |\vec{p}_1| \epsilon(|\vec{p}_1| +
|\vec{p}_1| \left(|\vec{p}| \epsilon(|\vec{p}|) - |\vec{p}_1| \epsilon(|\vec{p}_1|)\right) + |\vec{p}| |\vec{p}_1| (1-\cos\theta_1)\right] \nonumber \\
2 (p\cdot k) &= 2 \, \left[(|\vec{p}|-|\vec{p}_1|) |\vec{p}| \epsilon(|\vec{p}| +
|\vec{p}| \left(|\vec{p}| \epsilon(|\vec{p}|) - |\vec{p}_1| \epsilon(|\vec{p}_1|)\right) - |\vec{p}| |\vec{p}_1| (1-\cos\theta_1)\right]
\end{align}
we get
\begin{align}
F_0(x_1) &= 4 \left[x_1 (1-x_1)^2 \epsilon(|\vec{p}|) \, \epsilon(x_1|\vec{p}|) + 2 x_1 (1-x_1) \left(\epsilon(|\vec{p}|) + \epsilon(x_1|\vec{p}|)\right) \left(\epsilon(|\vec{p}|) - x_1 \epsilon(x_1|\vec{p}|)\right) + x_1 \left(\epsilon(|\vec{p}|) - x_1 \epsilon(x_1|\vec{p}|)\right)^2\right], \nonumber \\
F_1(x_1) &= 4 \left[- x_1^2 \left(\epsilon(|\vec{p}|) + \epsilon(x_1|\vec{p}|)\right) + 3 x_1 (1-x_1) \left(\epsilon(|\vec{p}|) - x_1 \epsilon(x_1|\vec{p}|)\right)\right], \nonumber \\
F_2(x_1) &= - 8 x_1^2 \,.
\label{Fn}
\end{align}
When the angular integral is done one finds
\begin{equation}
\begin{split}
\Gamma &= \frac {G_F^2 |\vec{p}|^5}{192 \, \pi^3} \left[(1-2s_W^2)^2 + (2s_W^2)^2 \right] \, \int _0^1 dx_1 \,
\theta \left(\epsilon(|\vec{p}|) - x_1 \epsilon(x_1|\vec{p}|)\right) \, \biggl[ 4 x_1 (1-x_1)^3 \epsilon(|\vec{p}|) \epsilon(x_1|\vec{p}|)
\left(\epsilon(|\vec{p}|) - x_1 \epsilon(x_1|\vec{p}|) \right) \\
& \quad + \, 6 x_1 (1-x_1)^2
\left(\epsilon(|\vec{p}|) + \epsilon(x_1|\vec{p}|)\right) \left(\epsilon(|\vec{p}|) - x_1 \epsilon(x_1|\vec{p}|)\right)^2 + \Bigl(4 x_1 (1-x_1)
+ \frac{10}{3} (1-x_1) ^3\Bigr) \left(\epsilon(|\vec{p}|) - x_1 \epsilon(x_1|\vec{p}|)\right)^3\biggl].
\end{split}
\label{Gamma}
\end{equation}
If one is not interested in the angular dependence of the differential decay width, there is a simpler way to get the result in (\ref{Gamma}). Going back to (\ref{AB}), one can use the variable $k^2$ instead of the angle $\theta_1$ to express the products
\begin{equation}
2 (p_1\cdot p) = p^2 + p_1^2 - k^2 {\hskip 1cm}
2 (p_1\cdot k) = p^2 - p_1^2 - k^2 {\hskip 1cm}
\,\, 2 (p\cdot k) = p^2 - p_1^2 + k^2
\end{equation}
and
\begin{equation}
A^{\mu\nu}(p, p_1) B_{\mu\nu}(k) = \frac{G_F^2}{12 \, \pi} \, \left[(1-2s_W^2)^2 + (2s_W^2)^2 \right] \, \left[(p^2 -p_1^2)^2 + (p^2 + p_1^2) k^2 - 2 (k^2)^2\right]\, \theta(k^2)\, \theta(k_0) \, .
\end{equation}
The relation (\ref{k2}) allows replacing the angular integral by an integration over $k^2$, where the upper limit on $k^2$, corresponding to $\theta_1=0$, is given by
\begin{equation}
k_+^2 = 2 |{\vec p}|^2 (1-x_1) \left[\epsilon(|{\vec p}|) - x_1 \epsilon(x_1|{\vec p}|)\right] \,.
\end{equation}
This leads to:
\begin{equation}
\Gamma = \frac {G_F^2}{192 \, \pi^3} \left[(1-2s_W^2)^2 + (2s_W^2)^2 \right] \, \frac{1}{2 |\vec{p}|}\, \int _0 ^1 dx_1 \, \theta \left(k_+^2\right) \left[(p^2 -p_1^2)^2 k_+^2 + \frac{1}{2} (p^2 + p_1^2) (k_+^2)^2 - \frac{2}{3} (k_+^2)^3\right].
\end{equation}
Using
\begin{equation}
p^2 = |\vec{p}|^2 (1 + 2 \epsilon(|\vec{p}|)) {\hskip 2cm}
p_1^2 = |\vec{p}_1|^2 (1 + 2 \epsilon(|\vec{p}_1|))
\end{equation}
one recovers the result (\ref{Gamma}) for the decay width.
The choice (\ref{A1}) for the neutrino dependent factor in the dynamical matrix element cannot always be derived from a quantum field theoretical calculation. In fact, by considering a generic dispersion relation (arbitrary choice of $\epsilon(|\vec{p}|)$) one can have cases where Eq.~(\ref{AB}) and the expression of the decay width take negative values indicating an inconsistency of the \emph{ansatz} for the dynamical matrix element.
\subsubsection{Second example}
One could consider other alternatives to (\ref{A1}) for the matrix element. A very simple choice corresponds to consider a modified neutrino spinor satisfying a modified Dirac equation
\begin{equation}
\left[\gamma^0 E(|\vec{p}|) - \vec{\gamma}\cdot \vec{p}\, (1+\epsilon(|\vec{p}|))\right] \tilde{u}(p) = 0\,.
\label{utilde}
\end{equation}
This modified Dirac equation implies a modified dispersion relation $E(|\vec{p}|)=|\vec{p}| [1+\epsilon(|\vec{p}|)]$. With this modified Dirac neutrino spinors the matrix element can be calculated as in SR and the result for $A^{\mu\nu}$ is now
\begin{equation}
\tilde{A}^{\mu\nu}(p,p_1) = 16 \, G_F^2 \left[(1-2s_W^2)^2 \tilde{p}_1^\mu \tilde{p}^\nu + (2s_W^2)^2 \tilde{p}^\mu \tilde{p}_1^\nu\right]
\label{A2}
\end{equation}
with
\begin{equation}
\tilde{p}^0 = |\vec{p}| [1+\epsilon(|\vec{p}|)] {\hskip 1cm}
\vec{\tilde{p}} = \vec{p}[1+\epsilon(|\vec{p}|)] \,.
\end{equation}
This neutrino factor can be derived from a perturbative field theory calculation by considering the SR vertex for the interaction and a modified free fermion action leading to the simple modification of the Dirac equation (\ref{utilde}).
With the choice (\ref{A2}) for the neutrino factor in the dynamical matrix element one has
\begin{equation}
\begin{split}
\tilde{A}^{\mu\nu}(p, p_1) B_{\mu\nu}(k) &= 16 \, G_F^2 \left[(1-2s_W^2)^2 + (2s_W^2)^2 \right] \, \left[(\tilde{p}_1\cdot \tilde{p}) k^2 B_1(k^2) \,+\, (\tilde{p}_1\cdot k) \, (\tilde{p}\cdot k) B_2(k^2)\right] \\
&= \frac{G_F^2}{12 \pi} \left[(1-2s_W^2)^2 + (2s_W^2)^2 \right] \, \left[2 (\tilde{p}_1\cdot \tilde{p}) k^2 \,+\,4 (\tilde{p}_1\cdot k) \, (\tilde{p}\cdot k)\right]
\, \theta(k^2)\, \theta(k_0) \,.
\end{split}
\end{equation}
In this case we have the products
\begin{align}
2 \tilde{p}_1\cdot \tilde{p} &= 2 |\vec{p}| |\vec{p}_1| (1-\cos\theta_1) \nonumber \\
k^2 &= 2 \, \left[(|\vec{p}|-|\vec{p}_1|) \left(|\vec{p}| \epsilon(|\vec{p}|) - |\vec{p}_1|
\epsilon(|\vec{p}_1|)\right) - |\vec{p}| |\vec{p}_1| (1-\cos\theta_1)\right] \nonumber \\
2 (\tilde{p}_1\cdot k) &= 2 |\vec{p}_1| \left(|\vec{p}| \epsilon(|\vec{p}|) - |\vec{p}_1|
\epsilon(|\vec{p}_1|)\right) + 2 |\vec{p}| |\vec{p}_1| (1-\cos\theta_1) \nonumber \\
2 (\tilde{p}\cdot k) &= 2 |\vec{p}| \left(|\vec{p}| \epsilon(|\vec{p}|) - |\vec{p}_1|
\epsilon(|\vec{p}_1|)\right) - 2 |\vec{p}| |\vec{p}_1| (1-\cos\theta_1)
\end{align}
and
\begin{align}
\tilde{F}_0(x_1) &= 4 x_1 \left(\epsilon(|\vec{p}|) - x_1 \epsilon(x_1|\vec{p}|)\right)^2\,, \nonumber \\
\tilde{F}_1(x_1) &= 8 x_1 (1-x_1) \left(\epsilon(|\vec{p}|) - x_1 \epsilon(x_1|\vec{p}|)\right), \nonumber \\
\tilde{F}_2(x_1) &= - 8 x_1^2 \,.
\label{Ftilden}
\end{align}
The decay width after the angular integration takes a different and simpler form than the decay width result of the first example (\ref{Gamma}):
\begin{equation}
\tilde{\Gamma} = \frac {G_F^2 |\vec{p}|^5}{192 \, \pi^3} \left[(1-2s_W^2)^2 + (2s_W^2)^2 \right] \, \frac{4}{3} \, \int _0^1 dx_1 \, \theta \left(\epsilon(|\vec{p}|) - x_1 \epsilon( x_1|\vec{p}|)\right) \, (1 - x_1^3) \left(\epsilon(|\vec{p}|) - x_1 \epsilon( x_1|\vec{p}|)\right)^3 \,.
\label{Gammatilde}
\end{equation}
Note that the result for the dynamical matrix element and then for the decay width is positive definite independently of the choice of the modified dispersion relation, so that the potential inconsistencies of the use of (\ref{A1}) for the neutrino factor are not present in this second example. This is a direct consequence of a field theoretical derivation of the dynamical matrix element.
\subsubsection{Third example}
One has a third simple choice for the dynamical matrix element with
\begin{equation}
\bar{A}^{\mu\nu}(p,p_1) = 16 \, G_F^2 \left[(1-2s_W^2)^2 \bar{p}_1^\mu \bar{p}^\nu + (2s_W^2)^2 \bar{p}^\mu \bar{p}_1^\nu\right]
\label{A3}
\end{equation}
with $\bar{p}^0 = |\vec{p}|$ and $\vec{\bar{p}} = \vec{p}$ which is just the SR dynamical matrix element.\footnote{Note that in the first choice of the dynamical matrix element one takes into account the modified expression for the energy at the level of the matrix element.} All one has to do is to replace in the first calculation everywhere $p$ by $\bar{p}$. Then one has
\begin{equation}
\begin{split}
\bar{A}^{\mu\nu}(p, p_1) B_{\mu\nu}(k) &= 16 \, G_F^2 \left[(1-2s_W^2)^2 + (2s_W^2)^2 \right] \, \left[(\bar{p}_1\cdot \bar{p}) k^2 B_1(k^2) \,+\, (\bar{p}_1\cdot k) \, (\bar{p}\cdot k) B_2(k^2)\right] \\
&= \frac{G_F^2}{12 \pi} \left[(1-2s_W^2)^2 + (2s_W^2)^2 \right] \, \left[2 (\bar{p}_1\cdot \bar{p}) k^2 \,+\, 4 (\bar{p}_1\cdot k) \, (\bar{p}\cdot k)\right]
\, \theta(k^2)\, \theta(k_0) \,.
\end{split}
\end{equation}
Due to the proportionality $\tilde{p}^\mu = \bar{p}^\mu (1+\epsilon(|\vec{p}|))$ it is obvious that the decay width in this third example coincides with the decay width in the second example up to corrections of order $\epsilon^4$.
\subsubsection{Fourth example}
In the particular case of a modified dispersion relation for the neutrino with a momentum independent choice for $\epsilon$, which corresponds to a momentum independent speed, it is possible to consider a modification of the interaction vertex fixed by gauge invariance from the modified free fermion action.
If we consider a Lagrangian
\begin{equation}
\mathcal{L} = \bar{\nu}_L i \gamma^0 D_0 \nu _L - i\left(1+\frac{\eta_0}{2}\right) \bar{\nu}_L \, \vec{\gamma}\cdot \vec{D}\, \nu _L
\label{Leta0}
\end{equation}
where $D_0$, $\vec{D}$ are covariant derivatives, then one has the simplest generalization of the relativistic Lagrangian for gauge interactions of massless fermions with a constant velocity $v = 1 + \eta_0/2$. In order to derive the dynamical matrix element all one has to do is to replace the Dirac $\gamma$ matrices $\gamma^\mu$ in the neutrino tensor by the modified $\gamma$ matrices $\hat{\gamma}^\mu$ where $\hat{\gamma}^0 = \gamma^0$ and $\vec{\hat{\gamma}} = (1 + \eta_0/2) \vec{\gamma}$. This replacement incorporates the modification in the Dirac equation for the modified Dirac spinors and the modification in the gauge interaction. Using
\begin{equation}
\lbrace\hat{\gamma}^\mu, \hat{\gamma}^\nu\rbrace = 2 \hat{\eta}^{\mu\nu}
\end{equation}
where $\hat{\eta}^{\mu\nu}$ is the modified Minkowski metric $\hat{\eta}^{00}=\eta^{00}=1$, $\hat{\eta}^{ii}=(1+\eta_0)\eta^{ii}=-(1+\eta_0)$ one can show that
\begin{equation}
\mathrm{Tr}\left[\hat{\gamma}^{\mu_1} \hat{\gamma}^{\mu_2} \hat{\gamma}^{\mu_3} \hat{\gamma}^{\mu_4}\right] = 4 \left(\hat{\eta}^{\mu_1\mu_2} \hat{\eta}^{\mu_3\mu_4} - \hat{\eta}^{\mu_1\mu_3} \hat{\eta}^{\mu_2\mu_4} + \hat{\eta}^{\mu_1\mu_4} \hat{\eta}^{\mu_2\mu_3}\right).
\end{equation}
The coefficient of $B_{\mu\nu}(k)$ in the dynamical matrix element can be read in this case from
\begin{equation}
p_{1\rho} p_{\sigma} \left[\hat{\eta}^{\rho\alpha} \hat{\eta}^{\sigma\beta} -
\hat{\eta}^{\rho\sigma} \hat{\eta}^{\alpha\beta} + \hat{\eta}^{\rho\beta} \hat{\eta}^{\sigma\alpha}\right] \left(\delta_\alpha^\mu \delta_\beta^\nu -
\eta_{\alpha\beta} \eta^{\mu\nu} + \delta_\alpha^\nu \delta_\beta^\mu\right)
\label{coefftilde2}
\end{equation}
which is the symmetrized coefficient of $p_{2\mu} p_{3\nu}$ in the product of the two traces. By contracting indices one finds
\begin{equation}
2 \hat{p}^\mu \hat{p}_1^\nu + 2 \hat{p}_1^\mu \hat{p}^\nu + \left(4+3\eta_0\right) (\hat{p}_1\cdot p) \eta^{\mu\nu} - 2 (\hat{p}_1\cdot \hat{p}) \eta^{\mu\nu} - 2 (\hat{p}_1\cdot p) \hat{\eta}^{\mu\nu}\,,
\end{equation}
to be compared with the SR coefficient
\begin{equation}
p_{1\rho} p_{\sigma} \left[\eta^{\rho\alpha} \eta^{\sigma\beta} -
\eta^{\rho\sigma} \eta^{\alpha\beta} + \eta^{\rho\beta} \eta^{\sigma\alpha}\right] \left(\delta_\alpha^\mu \delta_\beta^\nu -
\eta_{\alpha\beta} \eta^{\mu\nu} + \delta_\alpha^\nu \delta_\beta^\mu\right) =
2 p^\mu p_1^\nu + 2 p_1^\mu p^\nu \,.
\label{coeffSR}
\end{equation}
Then the dynamical matrix element will be
\begin{equation}
\begin{split}
\hat{A}^{\mu\nu}(p, p_1) B_{\mu\nu}(k) &= \frac{G_F^2}{12 \pi} \left[(1-2s_W^2)^2 + (2s_W^2)^2 \right] \,\left[-4 (\hat{p}_1\cdot \hat{p}) k^2 + 4 (\hat{p}_1\cdot k) (\hat{p}\cdot k) \right. \\ & \quad + \left. 2 \left(4+3\eta_0\right) (\hat{p}_1\cdot p) k^2 - 2 (\hat{p}_1\cdot p) (\hat{k}\cdot k)\right]\, \theta(k^2)\, \theta(k_0) \, ,
\label{tildetilde}
\end{split}
\end{equation}
with
\begin{alignat}{2}
\hat{p} &= \left(|\vec{p}| (1+\eta_0/2), \vec{p} (1+\eta_0)\right) & \qquad
p &= \left(|\vec{p}| (1+\eta_0/2), \vec{p}\right) \nonumber \\
\hat{p}_1 &= \left(|\vec{p}_1| (1+\eta_0/2), \vec{p}_1 (1+\eta_0)\right) & \qquad
p_1 &= \left(|\vec{p}_1| (1+\eta_0/2), \vec{p}_1\right) \nonumber \\
\hat{k} &= \left((|\vec{p}|-|\vec{p}_1|) (1+\eta_0/2), (\vec{p}-\vec{p}_1) (1+\eta_0)\right) & \qquad
k &= \left((|\vec{p}|-|\vec{p}_1|) (1+\eta_0/2), (\vec{p}-\vec{p}_1)\right) \,.
\end{alignat}
At first order in an expansion in powers of the corrections to SR one has
\begin{align}
2 (\hat{p}_1\cdot \hat{p}) &= 2 |\vec{p}| |\vec{p}_1| (1-\cos\theta_1) -
2 |\vec{p}| |\vec{p}_1| \eta_0 \nonumber \\
k^2 &= - 2 |\vec{p}| |\vec{p}_1| (1-\cos\theta_1) + (|\vec{p}|-|\vec{p}_1|)^2 \eta_0 \nonumber \\
2 (\hat{p}_1\cdot k) &= 2 |\vec{p}| |\vec{p}_1| (1-\cos\theta_1) \nonumber \\
2 (\hat{p}\cdot k) &= - 2 |\vec{p}| |\vec{p}_1| (1-\cos\theta_1) \nonumber \\
2 (\hat{p}_1\cdot p) &= 2 |\vec{p}| |\vec{p}_1| (1-\cos\theta_1) \nonumber \\
(\hat{k}\cdot k) &= - 2 |\vec{p}| |\vec{p}_1| (1-\cos\theta_1)
\end{align}
and then
\begin{align}
\hat{F}_0(x_1) &= 4 x_1 (1-x_1)^2 \eta_0^2\,, \nonumber \\
\hat{F}_1(x_1) &= \left[4 x_1 (1-x_1)^2 - 8 x_1^2\right] \eta_0 \,, \nonumber \\
\hat{F}_2(x_1) &= - 8 x_1^2\,.
\end{align}
After integration on the angle $\theta_1$, we get
\begin{equation}
\hat{\Gamma} = \frac {G_F^2 |\vec{p}|^5}{192 \, \pi^3} \left[(1-2s_W^2)^2 + (2s_W^2)^2 \right] \, \eta_0^3\, \int _0^1 dx_1 \left[x_1 (1-x_1)^4 + \frac{1}{6} (1-x_1)^6\right],
\label{Gammatildetilde}
\end{equation}
which is a third candidate for the pair production decay width, this one limited to the case of a superluminal neutrino with a momentum independent speed $v=1+\eta_0/2$.
\section{Choices of dynamical matrix element and modified dispersion relation from a field theoretical perspective}
\label{SecChoices}
From a field theoretical perspective one should start by considering a generalization of the Lagrangian of the SM containing the neutrino field. From the free part (quadratic in the neutrino field) one could read the generalization of the Dirac equation and the corresponding modified dispersion relation (i.e. the function $\epsilon(|\vec{p}|)$). By also considering the interaction term (product of two neutrino fields and the $Z$-boson field) one could derive the expression for the dynamical matrix element ($\overline{M^2}$).
Since one is considering a generalization of a relativistic gauge theory the most natural way to implement a modified dispersion relation in an extended Lagrangian is to add a new term in the free fermion action that fixes the modification in the dispersion relation and to consider the same dynamical gauge principle (replacing partial by covariant derivatives) that fixes the gauge interaction in the relativistic limit. The fourth example for the dynamical matrix element considered in the previous section is just the simplest choice along these lines with just one spatial derivative in the free part. One could consider generalizations with higher spatial derivative terms that would lead to new interaction terms and new contributions to the dynamical matrix element. A generalization along these lines is restricted to the effective field theory framework (derivative expansion) so that the choice of a modified dispersion relation is very limited and the study of implications of (or constraints on) such a generalization is easier than in other cases. To be precise, the gauge invariance that has been implemented in the Lorentz violating generalization of the SM Lagrangian in Eq.~(\ref{Leta0}) is a $U(1)$ symmetry instead of the complete $SU_L(2)\times U_Y(1)$ gauge invariance of the Lorentz invariant contribution. Should we had considered an $SU_L(2)\times U_Y(1)$ gauge invariant extended Lagrangian, Lorentz violations on different particles would be strongly restricted. In particular since the neutrino field and the left-handed charged lepton field are in a gauge doublet then Lorentz violations in the neutrino sector automatically would have an analogue in the charged lepton sector. The absence of observations of such Lorentz violating effects for electrons implies strong restrictions on posible effects due to Lorentz violations in neutrino physics including the possible energy loss of superluminal neutrinos that we are discussing in detail in this work.
An alternative way to implement a modified dispersion relation is to consider an extension of the relativistic Lagrangian independent of the gauge fields constructed from gauge invariant products of matter fields.\footnote{The consistency of this field theoretical framework in the presence of radiative corrections can be questioned~\cite{Giudice:2011mm} and deserves further study.} This possibility is restricted by the matter field content of the theory. In the case of the SM Lagrangian the fact that the (conjugate of the) doublet scalar field $\tilde{\Phi}$ and the left-handed doublet lepton field $L$
\begin{equation}
\tilde{\Phi} = \left(\begin{array}{c}\phi^{0*} \\
\phi^{-}\end{array}\right) {\hskip 2cm} L = \left(\begin{array}{c}\nu_{L} \\
l_{L}\end{array}\right)
\end{equation}
have the same gauge transformations allow to consider an extended Lagrangian quadratic in the gauge invariant product of these two field doublets
\begin{equation}
\mathcal{L}_{LIV}(\tilde{\Phi}^\dagger L) = \frac{1}{M^2} \left(\bar{L} \tilde{\Phi}\right) i \vec{\gamma} \cdot \vec{\nabla} \, \epsilon(|i\vec{\nabla}|) \left(\tilde{\Phi}^\dagger L\right) \, .
\end{equation}
In the approximation where one neglects the fluctuations in the scalar field this extended Lagrangian reduces to a quadratic Lagrangian in the neutrino left-handed field. This opens the possibility to consider Lorentz violating effects in the neutrino sector with no analogue for other particles so that restrictions from the absence of observations of such effects in other systems do not translate directly into restrictions on possible effects of Lorentz violations in the neutrino sector. This provides us with an example where neutrino physics is a special window to explore departures from SR. As far as one does not require a perturbative treatment of the Lorentz violating interactions in the lepton-scalar sector there is not any restriction on the quadratic extended Lagrangian so that one can consider arbitrary choices for the modified neutrino dispersion relation (arbitrary choice for $\epsilon(|\vec{p}|$)) going beyond a derivative expansion (momentum power expansion). The second example (\ref{A2}) for the matrix element of pair production by superluminal neutrinos can be seen as a result within this framework.
A third alternative to the generalization of a relativistic gauge theory would correspond to assume that the local gauge symmetry is a property of the relativistic limit. In this case one could consider an extended Lagrangian with no restrictions from local gauge invariance. The second example (\ref{A2}) could also be seen as a tree level approximation within this framework which can not be distinguished from the second alternative unless one goes beyond this approximation. An argument in favor of the realization of the second example for the dynamical matrix element within the previous second alternative for a generalized relativistic gauge theory is that it is not clear how the introduction of Lorentz violating terms in a Lagrangian provides a way to escape to the inconsistencies of a gauge non-invariant relativistic field theory with vector fields.
The first example (\ref{A1}) for the dynamical matrix element can not be derived from a field theory perturbative calculation. It therefore requires to consider a generalization of the relativistic gauge theory that goes beyond the field theory framework and then there is no reason to consider restrictions on the choice of modified dispersion relation.
For the second example (\ref{A2}) one can consider a momentum power expansion for the modified dispersion relation ($\epsilon(|\vec{p}|$) if one assumes the validity of the effective field theory framework for the study of Lorentz violating effects or a more general momentum dependence if one assumes that one has to go beyond the effective field theory expansion when one goes beyond the special relativistic limit.
The results for the third example, although can not be derived from a field theory calculation, are equivalent to those of the second example at leading order in the deviations from SR and then do not require any independent discussion until one goes beyond the leading order effects.
Finally in the fourth example for the dynamical matrix element there is a definite modified dispersion relation corresponding to a momentum independent speed (in the massless limit). We could go beyond this case by including gauge invariant higher derivative terms in the Lagrangian (\ref{Leta0}).
\section{Energy loss of superluminal neutrinos}
\label{SecEnergy}
In this section we evaluate the width for pair production, the rate of energy loss, and the energy of a superluminal neutrino after propagation over a given distance for different choices for the dynamical matrix element and the modified dispersion relation. These results are the starting point of an analysis of the observable consequences of having superluminal neutrinos with the uncertainties due to the lack of knowledge of the details of the possible origin of the Lorentz invariance violation in neutrino physics.
\subsection{Decay width}
We have three candidates (\ref{Gamma}), (\ref{Gammatilde}), (\ref{Gammatildetilde}) for the decay width as a functional of the modified dispersion relation corresponding to different choices for the dynamical matrix element of the pair production process.
If we consider the simplest choice for the modified dispersion relation $\epsilon(|\vec{p}|)= \eta_0/2$ (momentum independent speed) with $\eta_0 > 0$ (for negative values of $\eta_0$ the decay width is zero) then the threshold is
$|\vec{p}_{th}|=\sqrt{4\, m_e^2/\eta _0}$ and the decay widths are:
\begin{align}
\Gamma_0 &= \frac{G_F^2 |\vec{p}|^5}{192 \, \pi^3} \left[(1-2s_W^2)^2 + (2s_W^2)^2 \right] \, \xi_0 \left(\frac{\eta_0}{2}\right)^3 \label{Gamma_0} \\
\tilde{\Gamma}_0 &= \frac{G_F^2 |\vec{p}|^5}{192 \, \pi^3} \left[(1-2s_W^2)^2 + (2s_W^2)^2 \right] \, \tilde{\xi}_0 \left(\frac{\eta_0}{2}\right)^3 \label{Gammatilde_0} \\
\hat{\Gamma}_0 &= \frac{G_F^2 |\vec{p}|^5}{192 \, \pi^3} \left[(1-2s_W^2)^2 + (2s_W^2)^2 \right] \, \hat{\xi}_0 \left(\frac{\eta_0}{2}\right)^3 \label{Gammatildetilde_0}
\end{align}
with
\begin{equation}
\xi_0 = \frac{8}{7} {\hskip 2cm}
\tilde{\xi}_0 = \frac{34}{105} {\hskip 2cm}
\hat{\xi}_0 = \frac{16}{35}\,.
\label{gamma0}
\end{equation}
The result for the decay width (\ref{Gamma_0}) corresponding to the first choice for the dynamical matrix element (SR matrix element with the replacement of the SR energy by their modified expression in terms of the momentum) reproduces the result used in Ref.~\cite{Cohen:2011hx} to argue against the consistency of the recent result of OPERA~\cite{Adam:2011zb} for a superluminal neutrino velocity.
The other two results in Eq.~(\ref{gamma0}) also reproduce the two cases studied in Ref.~\cite{Bez}, which in fact can be seen to correspond to the second and fourth matrix elements of the previous section for a momentum independent velocity. Our more direct computation will however allow us to consider other dispersion relations beyond the constant speed case.
We can see from (\ref{gamma0}) that a change in the choice of the dynamical matrix element produces an additional overall factor of $(17/60)$ in the decay width if one takes the dynamical matrix element corresponding to a perturbative field theory calculation with the simplest free action implementing a modified Dirac equation and the SR interaction (second example of the previous section) or an overall factor of $(2/5)$ when one uses an interaction fixed by the dynamical gauge invariance principle (fourth example).
The next choice we consider for the modification in the dispersion relation is
$\epsilon(|\vec{p}|)= |\vec{p}|^n/\Lambda^n$ (with $n$ and $\Lambda$ positive numbers) which corresponds to a
Lorentz violating free term in the Lagrangian with $n$ spatial derivatives when $n$ is a natural number. The energy scale $\Lambda$ is the UV scale that fixes the domain of validity of the effective field theory energy power expansion. In this case, the threshold is given by $|\vec{p}_{th}|^{n+2}= 2\, m_e^2 \, \Lambda ^n$ and the widths are:
\begin{align}
\Gamma_n &= \frac {G_F^2 |\vec{p}|^{5}}{192 \, \pi^3} \left[(1-2s_W^2)^2 + (2s_W^2)^2 \right] \, \left(\frac{|\vec{p}|}{\Lambda}\right)^{3n} \, \xi_n \label{Gamma_n} \\
\tilde{\Gamma}_n &= \frac {G_F^2 |\vec{p}|^{5}}{192 \, \pi^3} \left[(1-2s_W^2)^2 + (2s_W^2)^2 \right] \, \left(\frac{|\vec{p}|}{\Lambda}\right)^{3n} \, \tilde{\xi}_n \label{Gammatilde_n}
\end{align}
with
\begin{align}
\xi_n &= 2 - \frac{12 (n+6)}{(n+3)(n+4)(n+5)} + \frac{12 (n+1)}{(n+2)(2n+3)(2n+5)} - \frac{4(3n+2)}{(3n+4)(3n+5)(3n+7)} \nonumber \\
\tilde{\xi}_n &= 1 - \frac{12}{(n+2)(n+5)} + \frac{6}{(n+3)(2n+3)} - \frac{4}{(3n+4)(3n+7)}\,.
\end{align}
The main difference between (\ref{Gamma_n})-(\ref{Gammatilde_n}) and the results (\ref{Gamma_0})-(\ref{Gammatilde_0}) for the simplest choice for the dispersion relation is the power exponent in the momentum dependence of the decay width. This makes the effect of the production of pairs in the propagation of neutrinos to increase much faster when the energy increases.
As a third choice for the modification of the dispersion relation at high energies ($|\vec{p}|>p_0$), we use $\epsilon(|\vec{p}|) = \lambda^\alpha/|\vec{p}|^\alpha$ with\footnote{If $\alpha>1$ then one has subluminal neutrinos and no pair production.} $0<\alpha\leq1$ and $\lambda$ an additional energy scale required by dimensional arguments. This corresponds to a nonlocal free field theory action trying to illustrate a Lorentz violation in the neutrino sector that goes beyond the effective field theory framework. In this case, the threshold is given by
$|\vec{p}_{th}|^{2-\alpha}= 2\, m_e^2 /\lambda ^{\alpha}$ and the widths (for $|\vec{p}|\gg |\vec{p}_0|$) are:
\begin{align}
\Gamma_{-\alpha} &= \frac {G_F^2 |\vec{p}|^{5}}{192 \, \pi^3} \left[(1-2s_W^2)^2 + (2s_W^2)^2 \right] \, \left(\frac{\lambda}{|\vec{p}|}\right)^{3\alpha} \, \xi_{-\alpha} \label{Gamma-alpha} \\
\tilde{\Gamma}_{-\alpha} &= \frac {G_F^2 |\vec{p}|^{5}}{192 \, \pi^3} \left[(1-2s_W^2)^2 + (2s_W^2)^2 \right] \, \left(\frac{\lambda}{|\vec{p}|}\right)^{3\alpha} \, \tilde{\xi}_{-\alpha}
\label{tildeGamma-alpha}
\end{align}
with
\begin{align}
\xi_{-\alpha} &= 2 - \frac{12 (6-\alpha)}{(3-\alpha)(4-\alpha)(5-\alpha)} + \frac{12 (1-\alpha)}{(2-\alpha)(3-2\alpha)(5-2\alpha)} - \frac{4(2-3\alpha)}{(4-3\alpha)(5-3\alpha)(7-3\alpha)} \nonumber \\
\tilde{\xi}_{-\alpha} &= 1 - \frac{12}{(2-\alpha)(5-\alpha)} + \frac{6}{(3-\alpha)(3-2\alpha)} - \frac{4}{(4-3\alpha)(7-3\alpha)}\,.
\end{align}
In this case the increase of the decay width of pair production with the energy is slower than in the case of a momentum independent velocity of propagation. In addition, note that in the expression of the decay rate there are energy-independent, dimensionless factors (i.e., pure numbers) that depend on the particular form of the dispersion relation and which can differ by one order of magnitude. For example,
\begin{equation}
\dfrac{\tilde \xi_{2}}{\tilde \xi_{-3/4}}\simeq 18.8 \, .
\end{equation}
As a last illustrative example, we shall consider a case in which the dispersion relation is non-analytic and it is given by $\epsilon(|\vec{p}|) = \, e^{-\Lambda/|\vec{p}|}$. In this case, that also goes beyond EFT, the decay width is extremely sensitive to changes in the momentum. When $|\vec{p}|\ll\Lambda$, the dominant contribution to the decay width for the second matrix element is:
\begin{equation}
\tilde{\Gamma} = \frac{G_F^2 |\vec{p}|^5}{192 \pi^3} \, \left[(1-2s_W^2)^2 + (2s_W^2)^2 \right] \, e^{-3\Lambda/|\vec p |} \, ;
\end{equation}
While this expression is small in the region in which is valid, it illustrates the possibility of having a remarkably strong dependence on momentum.
\subsection{Rate of energy loss}
An approximation to the effect of the production of $e^+e^-$ pairs on the propagation of neutrinos can be obtained from the rate of energy loss
\begin{equation}
\frac{d|\vec{p}|}{dt} = - |\vec{p}| \, \int_0^1 dx_1 (1-x_1) \frac{d\Gamma}{dx_1} \,.
\end{equation}
In the case of $\epsilon(|\vec{p}|) = \eta_0/2$ we have
\begin{align}
\frac{d|\vec{p}|}{dt} &= - \frac{G_F^2 |\vec{p}|^6}{192 \, \pi^3} \left[(1-2s_W^2)^2 + (2s_W^2)^2 \right] \, {\xi'_0} \, \left(\frac{\eta_0}{2}\right)^3 \\
\frac{d|\vec{p}|}{dt} &= - \frac{G_F^2 |\vec{p}|^6}{192 \, \pi^3} \left[(1-2s_W^2)^2 + (2s_W^2)^2 \right] \, {\tilde{\xi'}_0} \left(\frac{\eta_0}{2}\right)^3 \\
\frac{d|\vec{p}|}{dt} &= - \frac{G_F^2 |\vec{p}|^6}{192 \, \pi^3} \left[(1-2s_W^2)^2 + (2s_W^2)^2 \right] \, {\hat{\xi'}_0} \left(\frac{\eta_0}{2}\right)^3
\end{align}
with
\begin{equation}
{\xi'_0} = \frac{25}{28} {\hskip 2cm} {\tilde{\xi'}_0} = \frac{11}{42} {\hskip 2cm} {\hat{\xi'}_0} = \frac{5}{14}
\label{xiprimas}
\end{equation}
for the decay width results in (\ref{Gamma}), (\ref{Gammatilde}), and (\ref{Gammatildetilde}), respectively.
If one considers $\epsilon(|\vec{p}|)= |\vec{p}|^n/\Lambda^n$ then one has
\begin{align}
\frac{d|\vec{p}|}{dt} &= - \frac{G_F^2 |\vec{p}|^{6}}{192 \, \pi^3} \left[(1-2s_W^2)^2 + (2s_W^2)^2 \right] \, \left(\frac{|\vec{p}|}{\Lambda}\right)^{3n} \, {\xi'_n} \\
\frac{d|\vec{p}|}{dt} &= - \frac{G_F^2 |\vec{p}|^{6}}{192 \, \pi^3} \left[(1-2s_W^2)^2 + (2s_W^2)^2 \right] \, \left(\frac{|\vec{p}|}{\Lambda}\right)^{3n} \, {\tilde{\xi'}_n}
\label{dpdtn}
\end{align}
with
\begin{equation}
\begin{split}
{\xi'_n} &= \frac{13}{10} - \frac{12 (2n+15)}{(n+3)(n+4)(n+5)(n+6)} + \frac{12(4n+3)}{(2n+3)(2n+4)(2n+5)(2n+6)} \\ & \quad - \frac{2(5n+4)}{(3n+4)(3n+5)(3n+6)} + \frac{10}{3(3n+7)(3n+8)} \\
\end{split}
\end{equation}
\begin{equation}
\begin{split}
{\tilde{\xi'}_n} &= \frac{3}{5} - \frac{24(n+4)}{(n+2)(n+3)(n+5)(n+6)} +
\frac{24(2n+5)}{(2n+3)(2n+4)(2n+6)(2n+7)} \\ & \quad - \frac{8(3n+6)}{(3n+4)(3n+5)(3n+7)(3n+8)}\,.
\end{split}
\end{equation}
With the third choice of modified dispersion relation, $\epsilon(|\vec{p}|) = \lambda^\alpha/|\vec{p}|^\alpha$, the rate of energy loss is given by
\begin{align}
\frac{d|\vec{p}|}{dt} &= - \frac{G_F^2 |\vec{p}|^{6}}{192 \, \pi^3} \left[(1-2s_W^2)^2 + (2s_W^2)^2 \right] \, \left(\frac{\lambda}{|\vec{p}|}\right)^{3\alpha} \, {\xi'_{-\alpha}} \nonumber \\
\frac{d|\vec{p}|}{dt} &= - \frac{G_F^2 |\vec{p}|^{6}}{192 \, \pi^3} \left[(1-2s_W^2)^2 + (2s_W^2)^2 \right] \, \left(\frac{\lambda}{|\vec{p}|}\right)^{3\alpha} \, {\tilde{\xi'}_{-\alpha}}
\label{dpdt-alpha}
\end{align}
with
\begin{equation}
\begin{split}
{\xi'_{-\alpha}} & = \frac{13}{10} - \frac{12 (15-2\alpha)}{(3-\alpha)(4-\alpha)(5-\alpha)(6-\alpha)} + \frac{12(3-4\alpha)}{(3-2\alpha)(4-2\alpha)(5-2\alpha)(6-2\alpha)} \\ & \quad - \frac{2(4-5\alpha)}{(4-3\alpha)(5-3\alpha)(6-3\alpha)} + \frac{10}{3(7-3\alpha)(8-3\alpha)}
\end{split}
\end{equation}
\begin{equation}
\begin{split}
{\tilde{\xi'}_{-\alpha}} &= \frac{3}{5} - \frac{24(4-\alpha)}{(2-\alpha)(3-\alpha)(5-\alpha)(6-\alpha)} +
\frac{24(5-2\alpha)}{(3-2\alpha)(4-2\alpha)(6-2\alpha)(7-2\alpha)} \\ & \quad - \frac{8(6-3\alpha)}{(4-3\alpha)(5-3\alpha)(7-3\alpha)(8-3\alpha)}\,.
\end{split}
\end{equation}
\subsection{Final energy after propagation}
The previous results of the rate of energy loss can be used to get an estimate of the final energy $E_f$ of a neutrino of energy $E_i$ after propagating over a distance $L$.
One has
\begin{equation}
\frac{1}{E_f^5} - \frac{1}{E_i^5} = \frac{1}{E_0^5} {\hskip 2cm}
\frac{1}{E_f^5} - \frac{1}{E_i^5} = \frac{1}{\tilde{E}_0^5} {\hskip 2cm}
\frac{1}{E_f^5} - \frac{1}{E_i^5} = \frac{1}{\hat{E}_0^5}
\end{equation}
with
\begin{equation}
\tilde{E}_0 = \left[\frac{G_F^2 L}{192 \, \pi^3} \left[(1-2s_W^2)^2 + (2s_W^2)^2 \right] \, 5 \, \left(\frac{\eta_0}{2}\right)^3 \, {\tilde{\xi}'_0}\right]^{-1/5}
\label{E0}
\end{equation}
in the case of a momentum independent velocity ($\epsilon(|\vec{p}|)=\eta_0/2$). The result for $E_0$ ($\hat{E}_0$) is obtained from (\ref{E0}) by the replacement of ${\tilde{\xi'}_0}$ by ${\xi'_0}$ (${\hat{\xi'}_0}$).
For a modification of the dispersion relation with $\epsilon(|\vec{p}|)=|\vec{p}|^n/\Lambda^n$ one has
\begin{equation}
\frac{1}{E_f^{5+3n}} - \frac{1}{E_i^{5+3n}} = \frac{1}{E_n^{5+3n}} {\hskip 2cm}
\frac{1}{E_f^{5+3n}} - \frac{1}{E_i^{5+3n}} = \frac{1}{\tilde{E}_n^{5+3n}}
\end{equation}
with
\begin{equation}
{\tilde E}_n = \left[\frac{G_F^2 L}{192 \, \pi^3} \left[(1-2s_W^2)^2 + (2s_W^2)^2 \right] \, (5+3n) \, \frac{1}{\Lambda^{3n}} \, {{\tilde \xi}'_n}\right]^{-1/(5+3n)}
\label{En}
\end{equation}
and a similar result for $E_n$ replacing ${\tilde{\xi'}_n}$ by ${\xi'_n}$.
Finally, in the case $\epsilon(|\vec{p}|) = \lambda^\alpha/|\vec{p}|^\alpha$ one has
\begin{equation}
\frac{1}{E_f^{5-3\alpha}} - \frac{1}{E_i^{5-3\alpha}} = \frac{1}{E_{-\alpha}^{5-3\alpha}} {\hskip 2cm}
\frac{1}{E_f^{5-3\alpha}} - \frac{1}{E_i^{5-3\alpha}} = \frac{1}{\tilde{E}_{-\alpha}^{5-3\alpha}}
\end{equation}
with
\begin{equation}
\tilde{E}_{-\alpha} = \left[\frac{G_F^2 L}{192 \, \pi^3} \left[(1-2s_W^2)^2 + (2s_W^2)^2 \right] \, (5-3\alpha) \, \lambda^{3\alpha} \, {\tilde{\xi}'_{-\alpha}}\right]^{-1/(5-3\alpha)}
\label{E-alpha}
\end{equation}
and $E_{-\alpha}$ with a factor ${\xi'_{-\alpha}}$ instead of ${\tilde{\xi'}_{-\alpha}}$.
\subsection{Some numerical estimates}
We can take the inverse of the decay width of pair production as an estimate of the distance that neutrinos should propagate to have an appreciable loss of energy. From (\ref{Gammatilde_0}) we have
\begin{equation}
\tilde{\Gamma}_0^{-1} = \frac{4.3 \, \times \, 10^{-4}}{\eta_0^3} \, \left(\frac{\text{GeV}}{|\vec{p}|}\right)^5 \, \text{km}
\label{Gamma0-km}
\end{equation}
and similar results for $\Gamma_0^{-1}$ ($\hat{\Gamma}_0^{-1}$) with an additional factor $17/60$ ($17/24$) in the inverse of the decay width.
\begin{comment}
We can change units and rewrite (\ref{Gamma0-km}) as
\begin{equation}
\tilde{\Gamma}_0^{-1} = \frac{1.4 \, \times \, 10^{-32}}{\eta_0^3} \, \left(\frac{\text{TeV}}{|\vec{p}|}\right)^5 \, \text{pc}
\label{Gamma0-pc}
\end{equation}
or
\begin{equation}
\tilde{\Gamma}_0^{-1} = \frac{1.4 \, \times \, 10^{-68}}{\eta_0^3} \, \left(\frac{10^9 \, \text{GeV}}{|\vec{p}|}\right)^5 \, \text{Mpc} \,.
\label{Gamma0-Mpc}
\end{equation}
\end{comment}
In the case of a Lorentz violating free term with $n$ spatial derivatives one has
\begin{equation}
\tilde{\Gamma}_n^{-1} = \frac{1.7 \, \times \, 10^{(57n-5)}}{\tilde{\xi}_n} \, \left(\frac{\Lambda}{10^{19} \, \text{GeV}}\right)^{3n} \, \left(\frac{\text{GeV}}{|\vec{p}|}\right)^{(5+3n)} \, \text{km}
\label{Gamman-km}
\end{equation}
and in the case of a Lorentz violating correction which decreases at large momenta (\ref{tildeGamma-alpha}),
\begin{equation}
\tilde{\Gamma}_{-\alpha}^{-1} = \frac{1.7 \, \times \, 10^{(27\alpha-5)}}{\tilde{\xi}_{-\alpha}} \,
\left(\frac{\text{eV}}{\lambda}\right)^{3\alpha} \, \left(\frac{\text{GeV}}{|\vec{p}|}\right)^{(5-3\alpha)} \, \text{km} \,.
\label{Gamma-alpha-km}
\end{equation}
\begin{comment}
\begin{align}
\tilde{\Gamma}_n^{-1} &= \frac{1.7 \, \times \, 10^{(57n-5)}}{\tilde{\xi}_n} \, \left(\frac{\Lambda}{10^{19} \, \text{GeV}}\right)^{3n} \, \left(\frac{\text{GeV}}{|\vec{p}|}\right)^{(5+3n)} \, \text{km}
\label{Gamman-km} \\
\tilde{\Gamma}_n^{-1} &= \frac{5.7 \, \times \, 10^{(42n-33)}}{\tilde{\xi}_n} \,
\left(\frac{\Lambda}{10^{19} \, \text{GeV}}\right)^{3n} \,
\left(\frac{\text{TeV}}{|\vec{p}|}\right)^{(5+3n)} \, \text{pc}
\label{Gamman-pc} \\
\tilde{\Gamma}_n^{-1} &= \frac{5.7 \, \times \, 10^{(24n-69)}}{\tilde{\xi}_n} \,
\left(\frac{\Lambda}{10^{19} \, \text{GeV}}\right)^{3n} \,
\left(\frac{10^9 \, \text{GeV}}{|\vec{p}|}\right)^{(5+3n)} \, \text{Mpc}
\label{Gamman-Mpc}
\end{align}
and finally in the case of a Lorentz violating correction which decreases at large momenta
(\ref{tildeGamma-alpha}) one has
\begin{align}
\tilde{\Gamma}_{-\alpha}^{-1} &= \frac{1.7 \, \times \, 10^{(27\alpha-5)}}{\tilde{\xi}_{-\alpha}} \,
\left(\frac{\text{eV}}{\lambda}\right)^{3\alpha} \, \left(\frac{\text{GeV}}{|\vec{p}|}\right)^{(5-3\alpha)} \, \text{km}
\label{Gamma-alpha-km} \\
\tilde{\Gamma}_{-\alpha}^{-1} &= \frac{5.7 \, \times \, 10^{(36\alpha-34)}}{\tilde{\xi}_{-\alpha}} \, \left(\frac{\text{eV}}{\lambda}\right)^{3\alpha} \,
\left(\frac{\text{TeV}}{|\vec{p}|}\right)^{(5-3\alpha)} \, \text{pc}
\label{Gamma-alpha-pc} \\
\tilde{\Gamma}_{-\alpha}^{-1} &= \frac{5.7 \, \times \, 10^{(54\alpha-70)}}{\tilde{\xi}_{-\alpha}} \,
\left(\frac{\text{eV}}{\lambda}\right)^{3\alpha} \,
\left(\frac{10^9 \, \text{GeV}}{|\vec{p}|}\right)^{(5-3\alpha)} \, \text{Mpc} \,.
\label{Gamma-alpha-Mpc}
\end{align}
These expressions give us an idea of the sensitivity to an energy-loss due to pair production in the propagation of neutrinos from observations of solar, reactor, accelerator, atmospheric, galactic and extragalactic neutrinos.
\end{comment}
These expressions give us an idea of the sensitivity to an energy-loss due to pair production in the propagation of neutrinos from observations of solar, reactor, accelerator or atmospheric neutrinos.
The results for the energy scales (\ref{E0}), (\ref{En}) and (\ref{E-alpha}),
\begin{align}
\tilde{E}_0 &= \frac{1.6 \, \times \, 10^{-1}}{\eta_0^{3/5}} \, \left(\frac{\text{km}}{L}\right)^{1/5} \, \text{GeV}
\label{E0-km} \\
\tilde{E}_n &= \left(\frac{1.7 \, \times \, 10^{(57n-5)}}{(5+3n){\tilde{\xi}'_n}}\right)^{1/(5+3n)} \, \left(\frac{\Lambda}{10^{19} \, \text{GeV}}\right)^{3n/(5+3n)} \, \left(\frac{\text{km}}{L}\right)^{1/(5+3n)} \, \text{GeV}
\label{En-km} \\
\tilde{E}_{-\alpha}& = \left(\frac{1.7 \, \times \, 10^{(27\alpha-5)}}{(5-3\alpha){\tilde{\xi}'_{-\alpha}}}\right)^{1/(5-3\alpha)} \, \left(\frac{\text{eV}}{\lambda}\right)^{3\alpha/(5-3\alpha)} \, \left(\frac{\text{km}}{L}\right)^{1/(5-3\alpha)} \, \text{GeV}
\label{E-alpha-km}
\end{align}
which allow us to determine the final energy after a propagation over a distance $L$, give us also another estimate of the possible constraints that one can get on (or hints of) Lorentz violating corrections to neutrino physics from the observed high energy neutrino spectrum.
The results in (\ref{E0-km})-(\ref{E-alpha-km}) correspond to our second example (\ref{Gammatilde}) for the dynamical matrix element. For the other examples of matrix elements one has
\begin{equation}
E_0 = \tilde{E}_0 \, \left(\frac{{\tilde{\xi'}_0}}{\xi'_0}\right)^{1/5}
{\hskip 1cm} \hat{E}_0 = \tilde{E}_0 \, \left(\frac{\tilde{\xi}'_0}{{\hat{\xi'}_0}}\right)^{1/5}
{\hskip 1cm} E_n = \tilde{E}_n \, \left(\frac{{\tilde{\xi'}_n}}{\xi'_n}\right)^{1/(5+3n)}
{\hskip 1cm} E_{-\alpha} = \tilde{E}_{-\alpha} \, \left(\frac{{\tilde{\xi'}_{-\alpha}}}{\xi'_{-\alpha}}\right)^{1/(5-3\alpha)}.
\end{equation}
\begin{figure}
\centerline{\includegraphics[scale=0.8]{galactic1.pdf} \hspace{-1cm} \includegraphics[scale=0.8]{galactic2.pdf}}
\caption{Value of the terminal energy $E_n$ (approximate final energy of a neutrino with initial energy higher than $E_n$) for the cases $n=1$ (left) and $n=2$ (right) as a function of the high-energy scale $\Lambda$ written in terms of the Planck energy $E_P$. The curves show the differences with respect to the matrix element used in the calculation and with different distances of propagation, typical of galactic or extragalactic neutrinos.}
\label{fig:galactic}
\end{figure}
For galactic and extragalactic neutrinos, it is convenient to change units and rewrite, e.g., Eq.~(\ref{Gamma0-km}) as
\begin{equation}
\tilde{\Gamma}_0^{-1} = \frac{1.4 \, \times \, 10^{-32}}{\eta_0^3} \, \left(\frac{\text{TeV}}{|\vec{p}|}\right)^5 \, \text{pc}
\label{Gamma0-pc}
\end{equation}
and
\begin{equation}
\tilde{\Gamma}_0^{-1} = \frac{1.4 \, \times \, 10^{-68}}{\eta_0^3} \, \left(\frac{10^9 \, \text{GeV}}{|\vec{p}|}\right)^5 \, \text{Mpc} \,.
\label{Gamma0-Mpc}
\end{equation}
As an illustrative example, Fig.~\ref{fig:galactic} gives the value of $\tilde{E}_n$ for the two most studied cases in the literature of quantum gravity phenomenology, $n=1$ and $n=2$, in the case of galactic (propagation distance of the order of pc) and extragalactic (propagation distance of the order of Mpc) neutrinos. We also show in that figure the differences in the use of the first or second matrix elements in the calculation.
\section{Times of flight of superluminal neutrinos}
\label{SecTimes}
A modification of the neutrino dispersion relation produces an energy loss in the propagation of neutrinos related with the time of flight. Then one can try to look for neutrino observations where one can simultaneously determine the energy loss and time of flight in order to test the consistency of the observations with a given modification of the dispersion relation. The determination of the energy loss is limited by the uncertainties of our knowledge of the sources of very high energy neutrinos. As for the time of flight determination one has together with the uncertainties in the knowledge of the time of emission also the uncertainties due to the limited precision of time measurements.
In the case of a momentum independent superluminal velocity it is straightforward to calculate the time of flight of a neutrino propagating over a distance $L$
\begin{equation}
\Delta t_0 = 3.3 \times 10^3 \, \left(\frac{L}{\text{km}}\right) \, \left(1-\frac{\eta_0}{2}\right) \, \text{ns}
\label{t0-ns}
\end{equation}
or
\begin{equation}
\Delta t_0 = 1.3 \times 10^{14} \, \left(\frac{L}{\text{Mpc}}\right) \, \left(1-\frac{\eta_0}{2}\right) \, \text{s} \,.
\label{t0-s}
\end{equation}
A deviation, with respect to the SR expectation, in the time of flight of neutrinos of the order of $10 \, \text{ns}$ in the propagation over a distance of the order of $10^3 \, \text{km}$ would correspond to $\eta_0 \sim 10^{-5}$. From (\ref{Gamma0-km}) one can see that an inverse decay length of $10^3 \, \text{km}$ for such a value of $\eta_0$ corresponds to a neutrino with $|\vec{p}| \sim 40 \,\text{GeV}$. This provides us with a quantitative comparison of the sensitivity of neutrino times of flight and spectrum observations to a departure from SR kinematics with a momentum independent speed for the neutrinos. Times of flight of extragalactic neutrinos are much more sensitive to deviations from SR with a deviation of the order of a few $\text{s}$ for a neutrino propagating over a $\text{Mpc}$ corresponding to much smaller values of $\eta_0$ ($\eta_0 \sim 10^{-13}$). An inverse decay width of a $\text{Mpc}$ with this value of $\eta_0$ corresponds to a neutrino with $|\vec{p}| \sim \text{TeV}$. An extragalactic neutrino flux extending above these energies excludes then such a deviation from SR.
In the case of more general modifications of the dispersion relation the time of flight of neutrinos will be affected by the energy loss due to the production of $e^+e^-$ pairs. The momentum and then the velocity of the neutrino is changing in the propagation according to (\ref{dpdtn}) or (\ref{dpdt-alpha}). It is very easy to calculate in these two cases the neutrino time of flight over a distance $L$. In the case of $\epsilon(|\vec{p}|)=|\vec{p}|^n/\Lambda^n$ one has
\begin{equation}
\Delta t_n = L - L \, (n+1) \left(\frac{E_f}{\Lambda}\right)^n \, \tau_n
\end{equation}
with
\begin{equation}
\tau_n = \frac{(5+3n)}{(5+2n)} \, \left(\frac{E_n}{E_f}\right)^{(5+3n)} \, \left(1 - \left[1-\left(\frac{E_f}{E_n}\right)^{(5+3n)}\right]^{(5+2n)/(5+3n)} \right)\,.
\end{equation}
In the case of $\epsilon(|\vec{p}|)=\lambda^\alpha/|\vec{p}|^\alpha$ one has
\begin{equation}
\Delta t_{-\alpha} = L - L \, (1-\alpha) \left(\frac{\lambda}{E_f}\right)^\alpha \,
\tau_{-\alpha}
\end{equation}
with
\begin{equation}
\tau_{-\alpha} = \frac{(5-3\alpha)}{(5-2\alpha)} \, \left(\frac{E_{-\alpha}}{E_f}\right)^{(5-3\alpha)} \, \left(1 - \left[1-\left(\frac{E_f}{E_{-\alpha}}\right)^{(5-3\alpha)}\right]^{(5-2\alpha)/(5-3\alpha)} \right)\,.
\end{equation}
In the case of $E_f^5 \cdot L \ll 192 \pi ^3/G_F^2$ one has $E_f \ll E_n$ ($E_f \ll E_{-\alpha}$) and $\tau_n\approx 1$ ($\tau_{-\alpha}\approx 1$), i.e., the time of flight corresponding to a uniform motion with a momentum dependent speed.
\section{Consistency of the superluminal interpretation of OPERA observations}
\label{SecConsistency}
Recently there has been a claim of an observation of neutrinos propagating with superluminal velocities in the CNGS beam from CERN to Gran Sasso~\cite{Adam:2011zb} (OPERA experiment). It is then natural to try to accommodate these observations within the present discussion of the effect of Lorentz violations on the propagation of neutrinos.
Almost all the theoretical discussions related to OPERA consider a dispersion relation with
$\epsilon(|\vec{p}|) =\eta_0/2$. This is due to the absence of a change in the measured times of flight in all the range of energies detected going from $10$ GeV up to $100$ GeV. These results, if confirmed with more statistics, put strong constraints on any momentum dependence of the velocity of propagation of neutrinos at least in the range of energies covered by the OPERA time of flight measurements.
But in fact the OPERA value for $\eta_0=4.7\times 10^{-5}$~\cite{Adam:2011zb} is in obvious conflict\footnote{Assuming electron antineutrinos propagate with the same speed as the muon neutrinos detected by OPERA.} with time of flight limits from SN1987A~\cite{Hirata:1987hu,Bionta:1987qt,Longo:1987ub,Stodolsky:1987vd} as one can see from Eq.~(\ref{t0-s}). This conflict requires to consider a deviation from a momentum independent choice for $\epsilon$ at energies below those explored by OPERA including the range of energies of SN neutrinos (few MeV). There is also a conflict of this large value of $\eta_0$ with high energy neutrino observations owing to its implications on the propagation of high energy atmospheric neutrinos. This requires a strong suppression of the production of $e^+e^-$ pairs at energies well above those explored by OPERA with an appropriate choice of the momentum dependence of $\epsilon$.
This is also needed to avoid incompatibilities with neutrino production owing to the kinematic corrections in the pion decay induced by such large value of $\eta_0$~\cite{Cowsik:2011wv,Bi:2011nd,GonzalezMestres:2011jc}.
The simplest way to try to escape to these contradictions is to use a modified dispersion relation combining a function $\epsilon(|\vec{p}|)$ which increases when going from SN momenta to OPERA momenta, stays almost constant over the energy range of OPERA observations to be consistent with the very mild momentum dependence of the time of flight of neutrinos and then decreases if we go beyond OPERA momenta in order to escape to incompatibilities with the observed high energy neutrino spectra. However, there is still another conflict owing to the effect of the production of $e^+e^-$ pairs in the propagation of neutrinos from CERN to Gran Sasso. According to the previous section, one can see that, for the value of $\eta_0$ required to reproduce OPERA results on time of flights, there is a very drastic correction on the neutrino energies in conflict with observations.
For a momentum independent velocity of propagation in the OPERA energy range, taking Eq.~(\ref{E0-km}) with $\eta_0=4.7\times 10^{-5}$ and $L=731$\,km as the distance traveled by neutrinos in the experiment, we get $\tilde{E}_0=17.0$\,GeV in the case of the second matrix element. In the case of the first matrix element, which is the one considered by Cohen and Glashow in Ref.~\cite{Cohen:2011hx}, we obtain $E_0=13.3$\,GeV (in fact they get a slightly different result, $E_0=12.7$\,GeV, because they use $\eta_0=5.0\times 10^{-5}$, a value which was updated by the OPERA collaboration in November 2011 to $\eta_0=4.7\times 10^{-5}$, which is the one used in our calculation). For the third matrix element, $\hat{E}_0=16.0$\,GeV. This means that together with the rather adhoc choice of the Lorentz violating corrections in the free Lagrangian which would produce the peculiar behavior of the $\epsilon(|\vec{p}|)$ function as described in the previous paragraph, one has to assume a suppression in the dynamical matrix element relevant for the calculation of the energy loss of neutrinos propagating from CERN to Gran Sasso.
From Eq.~(\ref{E0}), we get
\begin{equation}
E_0 = 0.123 \, \left(\frac{L}{\text{km}}\right)^{-1/5} \, \eta_0^{-3/5} \, {\xi'_0}^{-1/5}\, \text{GeV}\,.
\label{bounds}
\end{equation}
Let us consider this relation for a generic matrix element. Since OPERA observes the arrival of neutrinos of energies higher than, let us take, $E_0\geq 60$\,GeV, Eq.~(\ref{bounds}) gives an upper bound for $\xi'_0$:
\begin{equation}
\xi'_0 \leq 4.8 \times 10^{-4}.
\end{equation}
This means that in order to make the OPERA time-of-flight measurement compatible with the non-observation of pair production, one needs a suppression in the dynamical matrix element so that $\xi'\sim \mathcal{O}(10^{-4})$ instead of the $\mathcal{O}(10^{-1})$ values of Eq.~(\ref{xiprimas}) obtained in the three simple examples considered in this work.
One could wonder whether relaxing the OPERA observation of a constant velocity in the 10-100 GeV energy range could make the strong constraints from neutrino decay compatible with the value $\eta_0=4.7\times 10^{-5}$ measured at an energy of $E=17\,$GeV (mean energy in the OPERA experiment).\footnote{Hereafter in this section we will consider the second example of matrix element which is the most favorable for the superluminal interpretation of the OPERA results.}
Studying the terminal energy $\tilde{E}_n$ or $\tilde{E}_{-\alpha}$ (which is always an upper bound of the average final energy $E_f$) from Eqs.~(\ref{En-km}), (\ref{E-alpha-km}), one sees that its maximum value is $\tilde{E}_n=20.1\,$GeV. This is however not enough to explain the arrival of much higher energetic neutrinos, so that one would still need to consider a matrix element with much stronger suppression than those considered in this work. Similarly, analyzing the decay length $\ell$ in units of the distance $L$ between CERN and Gran Sasso, we get from Eqs.~(\ref{Gamman-km}) and~(\ref{Gamma-alpha-km})
\begin{equation}
\dfrac{\ell}{L}\equiv \dfrac{\tilde{\Gamma} _n^{-1} }{L}=r\, \dfrac{(1+n)^3}{\xi _n}\left(\frac{17\, \mbox{GeV}}{E} \right)^{3n}, \, \mbox{when} \; \; \epsilon (|\vec p|)=\left(\frac{|\vec p|}{\Lambda}\right)^n
\label{lengthOPERAn}
\end{equation}
and
\begin{equation}
\dfrac{\ell}{L}\equiv \dfrac{\tilde{\Gamma} _{-\alpha}^{-1} }{L}=r\, \dfrac{(1-\alpha)^3}{\xi _{-\alpha}}\left(\frac{17\, \mbox{GeV}}{E} \right)^{-3\alpha}, \, \mbox{when} \; \; \epsilon (|\vec p|)=\left(\frac{\lambda}{|\vec p|}\right)^{\alpha}
\label{lengthOPERAa}
\end{equation}
where
\begin{equation}
r\equiv \dfrac{1.7\times 10^{-5}}{\left[ v(17\, \mbox{GeV}) -1\right]^3}\left(\frac{\text{GeV}}{E}\right)^5\left(\frac{\text{km}}{L}\right)
\end{equation}
and $v(17\, \mbox{GeV})-1=2.35 \times 10^{-5}$ is the velocity at $17$ GeV required for the superluminal interpretation of the results reported by the OPERA collaboration. Eqs.~(\ref{lengthOPERAn}) and~(\ref{lengthOPERAa}) tell us that the decay length is much shorter than $L$ when the energy is significantly larger than $20$ GeV. This fact can be seen from the dimensionless factor $r$ which is very small for these energies. For example, for $E=50$ GeV, that is roughly a half of the maximum energy detected by OPERA, $r=5.7\times 10^{-3}$ and the maximum value (with respect to $n$ and $\alpha$) of the decay length at this energy is $l=2.8\times 10^{-2}\cdot L$, much shorter than the distance between CERN and Gran Sasso.
\begin{figure}
\centerline{\includegraphics[scale=0.8]{sensitivity.pdf}}\hspace{-1cm}
\caption{Value of the terminal energy $E_0$ (approximate final energy of a neutrino with initial energy higher than $E_0$) for different distances of propagation, as a function of the degree of superluminality in the neutrino speed.}
\label{fig:sensitivity}
\end{figure}
The present analysis gives us an idea of the difficulties one finds to accommodate in a theoretical framework the results announced recently by the OPERA collaboration. In fact, the absence of energy loss in neutrino propagation is a more sensitive way to detect superluminal speeds. Taking our result for the second matrix element in the case of a momentum independent speed of propagation, the observation of the arrival of neutrinos of energies higher than 60\,GeV after a propagation of $L=730\,$km, gives a bound on the possible value of $\eta_0$ through Eq.~(\ref{bounds}),
\begin{equation}
\eta_0 \leq 5.8 \times 10^{-6},
\end{equation}
which, according to Eq.~(\ref{t0-ns}), corresponds to a difference in the time arrival of neutrinos with respect to luminal speeds of $\Delta t_0 \leq 7.0\,$ns. This means that the non-observation of pair production in OPERA is incompatible (under the assumptions we are considering) with its time-of-flight measurement of $\sim 60$\,ns, and that this non-observation is sensitive to tiny time differences, beyond the present precision of OPERA time measurements.
This conclusion is of interest for any experiment trying to repeat the OPERA measurements. Superluminal speeds can be better detected through a deformation in the spectrum (of course, a direct measurement of the time-of-flight is also advisable, since it does not contain any assumption on the pair production process). In order to appreciate this deformation, we need to send neutrinos with energies higher than $E_0$, which depends on the distance of propagation and the degree of superluminality through Eq.~(\ref{bounds}) (we consider the case of a momentum independent speed as an illustrative case). Therefore, an hypothetical experiment should consider both the neutrino energies of the beam and the distance of propagation in the sensitivity to the degree of superluminality. This sensitivity is shown in Fig.~\ref{fig:sensitivity} in terms of $(v-1)$, the difference between neutrino and photon speeds, using our calculations with the second matrix element (the better motivated theoretically). Fig.~\ref{fig:sensitivity} shows that, in order to be sensitive to $(v-1)$ of the order of $10^{-6}$, we should use neutrino beams of energies higher than 110\,GeV if $L\simeq 730\,$km,\footnote{In 2007 the MINOS collaboration reported a superluminal propagation speed $v-1=(5.1\pm 2.9) \times 10^{-5}$ (at $68\%$ C.L.) for a (mostly muon) neutrino beam with an average energy of 3 GeV propagating 735 km between Fermilab and the Soudan mine~\cite{Adamson:2007zzb}. It is expected that the MINOS team will report new results at higher precision in the near future. The OPERA, ICARUS, BOREXINO and LVD experiments at the Gran Sasso underground laboratory will also provide a more precise measurement of the neutrino speed.} while it would suffice to use neutrino beams of energies higher than 70\,GeV if $L\simeq 7000\,$km (the distance between CERN and the Soudan underground mine in USA).
\section{Concluding remarks}
\label{SecConcluiding}
In this work we have investigated the dependence of the charged lepton pair emission by superluminal neutrinos $\nu _i \to \nu _i \; e^- \; e^+ $ on the dispersion relation for neutrinos, and on the dynamical matrix element of the process. General expressions for an arbitrary dispersion relation and for various examples of matrix elements have been obtained. On the one hand, for a given dispersion relation, different choices of the matrix element lead to decay rates which differ by factors of order one. On the other hand, for a given matrix element, different choices of the dispersion relation lead to energy-independent, dimensionless factors (i.e., pure numbers) in the decay rates which can differ by one order of magnitude. These are the main new results of the present investigation. Estimates of the sensitivity of different observations of high energy neutrinos to a possible departure from SR kinematics in the neutrino sector have been presented. The dependence of the results on details of the theory (choice of modified dispersion relation and modified matrix element) is an indication that high energy neutrino physics can be a very good laboratory to explore possible deviations from SR.
Concerning the generality of the analysis, the assumptions on which it is based were already pointed out in the introduction: i) Rotational symmetry is preserved. Nevertheless, some works have built theories or models in which rotational invariance is not exact (see, for instance, Ref.~\cite{Cohen:2006ky}). ii) Energy and momentum are conserved in the conventional, additive way. However, during the last decade, there have been investigations which suggest the possibility of modifying these laws. Most of these explorations come from the quantum space-time realm and they deal with either deformations (see Ref.~\cite{AmelinoCamelia:2010pd} for a review) or violations~(\cite{Mer}) of space-time symmetries. iii) The relevant propagation speed of superluminal particles is the group velocity of their wave packets. Once more, several works have studied other alternatives both in canonical and in non-canonical space-times (\cite{Mignemi:2003ab, Daszkiewicz:2003yr, Ghosh:2007ai}).
While experimental results cannot be confirmed or refuted by theoretical investigations but by new experiments, what this present work shows is that the most likely possibility with respect to the observation reported by the OPERA collaboration is that either the observation of superluminal neutrinos is not confirmed or some of the assumptions indicated in the present work are not valid. In particular, in Doubly Special Relativity scenarios~(\cite{AmelinoCamelia:2010pd}), where the energy-momentum conservation law is modified, forbidden processes in SR are generically forbidden too, as it was firstly pointed out in connection with the OPERA results in Refs.~\cite{Carmona:2011zg,AmelinoCamelia:2011bz}.
Some works have studied the pair production decay width before us. In Refs.~\cite{Cohen:2011hx,Huo:2011ve}, the decay width is computed for the case of constant velocity without specifying, however, what matrix element or relevant Lagrangian (if any) has been used to obtain the result. We reproduce that result for the first example of matrix element which does not come from a $SU(2)$ gauge invariant underlying field theory. The pair production decay width is also computed in Ref.~\cite{Bez} for the momentum independent velocity case with two choices for a four-fermion interaction Lagrangian which are in fact equivalent to our second and fourth matrix elements.
Despite the fact that we have concentrated on the pair production reaction, some of the techniques developed in the present work, and in particular the \textit{collinear} approximation, can be extended to other processes like neutrino splitting $\nu _i \to \nu _i \; \nu _j \; \bar \nu _j$. In the case of a momentum dependent speed for neutrinos, splitting is a relevant mechanism for energy loss in the propagation of superluminal neutrinos.
Other processes contributing to the energy loss of a propagating superluminal neutrino, such as the production of heavier charged leptons and the contribution of virtual $W^\pm$ can also be treated within the same approximation. The results for these processes, as well as an analysis of the uncertainties in the calculation of the production of very high energy superluminal neutrinos with a general dispersion relation will be presented elsewhere~\cite{CCM}.
\section*{Note added}
While this manuscript was being completed, the OPERA collaboration announced the identification of two sources of error in the
determination of the time of flight of neutrinos from CERN to Gran Sasso. One of them was a faulty connection in the optical fiber cable that brings the external GPS signal to the experiment master clock. That this was most probably the origin of the apparent neutrino superluminality was later confirmed (after this paper had been submitted for publication) by the results of muon measurements presented by LVD~\cite{lvd}, which show that the timing between LVD and OPERA became misaligned from the middle of 2008, around the time that OPERA-1 began, and remained stably misaligned by about 73 nanoseconds until the end of 2011, which is when the fiber problem was identified and eliminated. Almost at the same time, the ICARUS collaboration presented new results from the October-November 2011 campaign of measurements, which allowed a very accurate time-of-flight measurement of neutrinos from CERN to LNGS on an event-to-event basis, collecting seven neutrino events which are compatible with luminal speed~\cite{icarus}. Although new measurements are planned during 2012, the new information indicates that most probably the superluminal signal will go away, in agreement with the conclusions of the analysis presented in Section 6.
\section*{Acknowledgments}
This work is supported by CICYT (grant FPA2009-09638) and DGIID-DGA (grant
2010-E24/2).
|
1,116,691,497,521 | arxiv |
\section{Introduction}
\textit{Online} (also known as \textit{simultaneous}) machine translation refers to automatic translation systems which start generating an output hypothesis before the entire input sequence has been consumed \cite{bangalore2012real,sridhar2013segmentation}. Emerging recently as a challenging task, it has been witnessing several works proposed in text-to-text ($T2T$) translation \cite{Ma19acl, Arivazhagan19acl, Ma20iclr,elbayad:hal-02962195}, and in speech-to-text ($S2T$) translation \cite{DBLP:journals/corr/abs-1808-00491,elbayad:hal-02895893,han-etal-2020-end, nguyen2021empirical_icassp}, which attempt to deal with the low latency constraint imposed by the task. Following the \textit{wait-k} policy originally proposed for $T2T$ \cite{Ma19acl} and proven effective when applied to $S2T$ \cite{elbayad:hal-02895893,han-etal-2020-end}, our previous work \cite{nguyen2021empirical_icassp} introduced an adaptive version of \textit{wait-k} which leverages any pre-trained end-to-end offline speech translation model for online speech translation.
However, the model proposed in \cite{nguyen2021empirical_icassp} had a speech encoder based on a \textit{Bi-directional} Long Short-Term Memory (BLSTM) \cite{LSTM} which was not efficient in online mode since re-encoding of the full input was needed each time a new speech block was read.
We show in this work that while replacing BLSTM by \textit{Uni-directional} Long Short-Term Memory (ULSTM) encoding degrades performance in offline mode, it actually improves both efficiency and performance in online mode (this observation was also made for online $T2T$ translation by \cite{elbayad:hal-02962195}). We also investigate how to segment the speech flow in order to alternate optimally between reading (R: encoding input) and writing (W: decoding output) operations. The contributions of this work are the following:
\begin{itemize}
\item Showing that ULSTM speech encoder when using the same (\textit{re-encode}) encoding strategy yields better inference speed and performance in comparison with BLSTM speech encoder,
\item Further improving inference speed and performance of ULSTM speech encoder using a new encoding strategy (ULSTM \textit{Overlap-and-Compensate}),
\item Analyzing the impact of speech flow segmentation on the BLEU/Latency trade-off, comparing three segmentation methods: fixed interval boundaries, oracle word boundaries or randomly set boundaries.
\end{itemize}
\vspace{-10pt}
\section{Background on low latency neural speech translation}
\subsection{Decoding strategies}
Real-life applications require translation systems to start emitting output translation partially before the input sequence is made fully available. Such a low latency constraint has been imposing great challenge to neural sequence-to-sequence translation models, despite their state-of-the-art performance on offline translation tasks. Notable efforts have been going into optimizing quality/latency trade-off of the neural online translation systems, including \cite{cho2016can} who introduces a waiting policy which alternates READ/WRITE operations. Inspired by \cite{cho2016can}, \cite{dalvi2018incremental} designs a static read and write decoding policy, which first reads $S$ input tokens, and alternates between a same number of WRITE and READ operations until the entire source sequence is consumed. In the same spirit, \cite{Ma19acl} proposes a \textit{wait-k} decoding policy which reads $k$ source tokens at the first step, and then alternates single WRITE/READ operations.
Several works on online automatic speech translation got decent results when adapting \textit{wait-k} policy to their task, including \cite{DBLP:journals/corr/abs-1808-00491,elbayad:hal-02895893,han-etal-2020-end, nguyen2021empirical_icassp}. \cite{han-etal-2020-end} made an attempt to build an end-to-end online system which first reads $k$ input frames, then alternates between writing one output token or reading the next $s$ input frames. \cite{nguyen2021empirical_icassp} extends this work, modifying their decoding policy to be able to emit more than one (and maximum $N$) output tokens at a time.
The policy of \cite{nguyen2021empirical_icassp} allows them to exploit any pre-trained offline model in an online decoding mode. However, they only experiment with pre-trained models whose speech encoders use BLSTM layers. \cite{elbayad:hal-02962195} shows that, in online mode, BLSTM models might be an unnecessarily costly choice, and therefore advocates for using ULSTM models instead (for text translation).
In this work, we explore the use of ULSTM models, and make a comparison with their BLSTM counterpart for low latency end-to-end speech translation. We also experiment alternative speech segmentation policies to \cite{nguyen2021empirical_icassp}.
\vspace{-5pt}
\subsection{Evaluation metrics}
Performance of online translation systems is usually illustrated as a trade-off between translation quality and latency. As in offline translation, BLEU remains the most frequently used metrics for measuring translation quality of online systems. Several metrics have been proposed for latency measurement \cite{cho2016can,Ma19acl,cherry2019thinking}, amongst which Average Lagging (AL) proposed by \cite{Ma19acl} is a frequent choice. The original AL metric measures the average rate at which the translation system lags behind an ideal \textit{wait-0} translator. \cite{simuleval2020} argues that this metric has a shortcoming when applied to $S2T$ translation, and proposes an adaptive version which remedies this problem. However, we noticed that this adaptive version is strongly sensitive to the reference's length, which can be arbitrarily long and weakly dependent on the input speech. In some cases, a slight change of the reference length (which might come from a different tokenization method for example) could drastically change the AL value. Furthermore, one should keep in mind that negative values of AL can still occur when the translation system in question gets ahead of the ideal translator (i.e when it predicts output tokens although the already read source frames do not account for them). Despite those shortcomings, we keep using the adaptive AL from \cite{simuleval2020} in this work in order to measure our improvements of results over those of \cite{nguyen2021empirical_icassp}.
\vspace{-10pt}
\section{End-to-end online model}
\textbf{Our previous work \cite{nguyen2021empirical_icassp}} reused an attention-based encoder-decoder architecture described in \cite{nguyen2019ontrac}. The speech encoder stacks two VGG-like CNN blocks \cite{simonyan2014very} before five layers of BLSTM. We stack in each VGG block two 2D-convolution layers, followed by a 2D-maxpooling layer. After these two VGG blocks, the shape $(T \times D)$ of an input speech sequence is transformed to $(T/4 \times D/4)$, with $T$ being the length of the input sequence (number of frames), and $D$ being the features' dimension respectively. The decoder is a stack of two 1024-dimensional LSTM layers, and Bahdanau's attention mechanism~\cite{bahdanau2014neural} is used to bridge the encoder and the decoder. In online mode, the BLSTM speech encoder
must re-encode from the beginning, from left-to-right and from right-to-left, the input speech sequence every time new input frames are read. In terms of decoding strategy, an adaptive version of \textit{wait-k} is proposed in \cite{nguyen2021empirical_icassp}. This deterministic decoding strategy reads at the first reading operation $k$ (\textit{wait} parameter) first acoustic frames of the input speech features sequence. At each reading operation after this, the system continues consuming fixed intervals of $s$ (\textit{stride} parameter) frames (this reading strategy is also referred in this paper as the fixed interval boundaries segmentation method). A writing operation is put after each reading operation, which writes at maximum $N$ (\textit{write} parameter) output tokens.
\textbf{ULSTM Re-encode strategy}
\cite{elbayad:hal-02962195} proves that, for $T2T$ online translation, using a ULSTM encoder gives not only better decoding speed but also better BLEU/AL trade-off. We verify if this idea works for speech as well, comparing BLSTM and ULSTM speech encoders in this work. In order to make this comparison, we retrain an offline model similar to the one presented in \cite{nguyen2021empirical_icassp}, except that the speech encoder is modified to stacked ULSTM layers instead of BLSTM layers after the VGG-like blocks. In this strategy (presented in figure \ref{fig:reencode}) we still re-encode the full speech sequence left-to-right every time we read new input frames, but this \textit{ULSTM-Re-encode} approach frees us from computing the BLSTM's right-to-left re-encoding pass, hence being expected to improve decoding speed.
\textbf{ULSTM Overlap-and-Compensate strategy}
Moving from BLSTM to ULSTM is a first step towards efficiency but re-encoding the full sequence left-to-right each time speech frames are read is still sub-optimal. To avoid this, we tried to feed chunk by chunk of input frames independently
but this solution gave very disappointing results probably because of the quality deterioration of the VGG blocks' output representations due to padding issues near the chunk boundaries (especially in the last several positions of the representations).
Therefore, when dealing with ULSTM speech encoders, we propose an \textit{Overlap-and-Compensate} encoding strategy which allows the encoder to read extra frames from the past in order to compensate some discarded positions in the end of the previous output representation of the VGG-like blocks (figure \ref{fig:overlap_and_compensate}).
\begin{algorithm}[ht]
\SetAlgoLined
\textbf{Input: } sequence $x$; \\
\textbf{Output: } representation $h$; \\
\textbf{Initialization} step $t=1$,
wait parameter $k$, stride parameter $s$, total number of frames read so far $g=k$, $offset=0$,\\
$finish\_read=False$, $h_0=None$,\\
$overlap=round(k/2)$; \textit{\# Overlap half of chunk\_size}\\
\While{$g < |x|$}{
\If{$t>1$}{
$overlap=round(s/2)$;
}
\If{$g>=|x|$}{
$g = |x|$; $overlap=0$;$finish\_read=True$;
}
$x_t=x[offset:g]$; \textit{\# A chunk read at time $t$}\\
$h_t=Encode(x_t, overlap, h_{t-1}, finish\_read)$; \\
$g+=s$; $t+=1$; $offset=g-overlap$;
}
\SetKwFunction{FEncode}{Encode}
\SetKwProg{Fn}{Fuction}{:}{}
\Fn{\FEncode{$x$, $overlap$, $prev\_h$, $finish\_read$}}{
$num\_discard=round(overlap/4)$; \\
$h_{vgg} = VGG(x)$; \\
\If{not $finish\_read$}{
\textit{\# Discard num\_discard positions in the end} \\
$new\_length=|h_{vgg}|-num\_discard$; \\
$h_{vgg}=h_{vgg}[0:new\_length]$;
}
\KwRet $h_{ULSTM} = ULSTM(h_{vgg}, prev\_h)$;
}
\caption{Overlap-and-Compensate encoding strategy}
\label{algorithm:ulstm_overlap}
\end{algorithm}
\begin{figure}[ht]
\centering
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[scale=0.5]{Figures/re-encode.png}
\caption{Re-encode}
\label{fig:reencode}
\end{subfigure}
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[scale=0.5]{Figures/overla-and-compensate.png}
\caption{Overlap-and-compensate}
\label{fig:overlap_and_compensate}
\end{subfigure}
\caption{Different encoding strategies.}
\label{fig:overlap_algorithm}
\end{figure}
Algorithm \ref{algorithm:ulstm_overlap} describes the overlap-and-compensate approach applied to the fixed interval segmentation presented in \cite{nguyen2021empirical_icassp}. It introduces another parameter $overlap$, which decides how many past frames the encoder should read at each encoding step (our \textit{re-encode} strategy corresponds to $overlap=0, offset=0$).
We experiment with $overlap$ corresponding to half of the number of input frames of the current step ($overlap=round(s/2)$).
\vspace{-5pt}
\section{Experimental Setup}
\label{sec:experimental_setup}
\vspace{-5pt}
\textbf{Data} This work focuses on the English-German (EN-DE) language pair. As mentioned in \cite{elbayad:hal-02895893}, the data used to train our models is a combination of MuST-C EN-DE \cite{mustc19}, Europarl EN-DE \cite{europarlst}, and How2 \cite{sanabria18how2} synthetic (i.e. the German translation has been automatically generated by a $T2T$ machine translation system), overall more than 750h of translated speech.
\textbf{Pre-trained models} The offline BLSTM model presented in this work was trained for our participation to IWSLT 2020 \cite{elbayad:hal-02895893}. It scores $21.38$ and $20.54$ BLEU on MuST-C tst-COMMON, and tst-HE, in greedy decoding mode, respectively. We pre-train another offline ULSTM model with exactly the same configuration as the BLSTM model, only replacing BLSTM layers by ULSTM layers. It scores $18.21$ and $17.98$ BLEU on tst-COMMON, and tst-HE, in greedy decoding mode, respectively.
\vspace{-5pt}
\section{Experiments}
\label{sec:experiments}
\subsection{Impact of encoding strategies}
\begin{figure}[ht]
\centering
\begin{subfigure}{.5\textwidth}
\input{Figures/compare_blstm_ulstm_tst-HE.tex}
\caption{MuST-C tst-HE}
\end{subfigure}
\begin{subfigure}{.5\textwidth}
\input{Figures/compare_blstm_ulstm_tst-COMMON.tex}
\caption{MuST-C tst-COMMON}
\end{subfigure}
\caption{Comparing translation models with BLSTM/ULSTM re-encode/ULSTM Overlap encoding strategies, evaluated on MuST-C tst-HE and tst-COMMON.}
\label{fig:compare_blstm_ulstm}
\end{figure}
This subsection compares different models using either BLSTM or ULSTM speech encoders with different encoding strategies (\textit{re-encode} versus \textit{overlap-and-compensate}). We use the same segmentation (arbitrarly fixed interval boundaries) presented in \cite{nguyen2021empirical_icassp}.
Figure \ref{fig:compare_blstm_ulstm} illustrates the BLEU/AL trade-off of BLSTM and ULSTM models with different encoding strategies, evaluated on MuST-C tst-HE and MuST-C tst-COMMON, with different $(k, s, N)$ triplets ($k=[100,200]$, $s=[10,20]$, and $N=[1,2]$). It is noticeable that models with ULSTM speech encoders give consistently better BLEU/AL trade-off than the model with BLSTM speech encoder, on both MuST-C tst-HE and tst-COMMON. Moreover, figure \ref{fig:compare_blstm_ulstm} clearly shows that ULSTM \textit{overlap-and-compensate} strategy outperforms ULSTM \textit{re-encode}, especially in low-latency regimes.
We also investigate the actual time spent decoding each sentence of MuST-C tst-HE using different encoding strategies. In order to do this, we exclusively use the same CPU machine to decode the whole test set using either BLSTM, ULSTM \textit{re-encode} or ULSTM \textit{overlap-and-compensate} encoding strategy. Actual time spent decoding each sentence is captured and averaged over the whole test set. To better illustrate the difference between the encoding strategies, in each latency regime, the time spent of BLSTM is set as a speed unit, and the results of ULSTM \textit{re-encode} and ULSTM \textit{overlap-and-compensate} are reported according to this speed unit.
We observe that amongst all latency regimes, ULSTM models are much faster than the BLSTM model since they only need to encode the input speech in one direction (from left to right). ULSTM \textit{re-encode} is about twice as fast as BLSTM, scoring $0.53$. Remarkably, scoring $0.06$, the ULSTM \textit{overlap-and-compensate} is fastest among all encoding strategies (about $17$ times faster than the BLSTM, and $9$ times faster than ULSTM \textit{re-encode}, respectively). We believe that this huge improvement in terms of computation speed of the ULSTM \textit{overlap-and-compensate} approach is due to the fact that its input chunks are consistently smaller than that of the ULSTM \textit{re-encode} approach.
\vspace{-5pt}
\subsection{Impact of speech input segmentation}
In this section, we investigate the optimal ways to segment the speech flow in order to alternate between reading (R: encoding input) and writing (W: decoding output) operations: fixed interval boundaries (as presented in \cite{nguyen2021empirical_icassp} and in previous experiments of this paper), oracle word boundaries segmentation, and randomly set boundaries segmentation.
\begin{figure}[ht]
\centering
\input{Figures/compare_segmentation_ulstm_overlap_half_tst-HE}
\caption{BLEU/AL trade-off for different speech input segmentation methods, evaluated on MuST-C tst-HE, using ULSTM \textit{overlap-and-compensate} approach.}
\label{fig:compare_segmentations_ulstm_overlap}
\end{figure}
\vspace{-5pt}
\subsubsection{Oracle word boundaries}
\vspace{-5pt}
Questioning whether or not feeding relatively precise word-by-word speech chunks instead of fixed-length chunks \cite{nguyen2021empirical_icassp} would improve the performance, we segment the input audio (phrase level) into words using Montreal Forced Aligner \cite{mcauliffe2017montreal}. Their pre-trained English model (from Librispeech \cite{DBLP:conf/icassp/PanayotovCPK15})\footnote{\url{https://montreal-forced-aligner.readthedocs.io}} is used out of the box.
In terms of decoding strategies, we slightly modify the strategy proposed by \cite{nguyen2021empirical_icassp}:
\begin{itemize}
\item $k$ remains the number of frames the encoder should wait before starting writing, serving as an upper bound. At the first decoding step, the encoder reads the first several chunks of frames (matching the words boundaries) until $total\_number\_of\_frames \geq k$. In this work, we experiment with $k=[0, 50, 100, 150, 200]$.
\item $s$ is the number of source words (chunks of frames) read at each decoding step after the first step. In this work, we keep $s=1$ for all our experiments regarding the oracle word boundaries segmentation.
\item $N$ remains the maximum number of output tokens (characters) written at each decoding step ($N=[1, 2]$).
\end{itemize}
\subsubsection{Randomly set boundaries}
\vspace{-5pt}
The randomly set boundaries segmentation method cuts the audio input into random sized audio chunks. However, to avoid unreasonable fluctuation of the size of each chunk, we set a lower bound (the minimum number of frames) and a higher bound (the maximum number of frames) for each chunk. The number of frames in each chunk is randomly generated within this constraint.
We continuously accumulate these random numbers until their sum exceeds the total number of frames in the input sequence. The number of frames in the last chunk is adjusted so that the sum of frames in all chunks is equal to the input sequence's length. In this work, we experiment with $[low\_boundary, high\_boundary] = [5,10], [5,20], [5,50],[5,100],[10,50],[10,100]$. We experiment with $N=[1,2]$ in this setup.
Algorithm \ref{algorithm:ulstm_overlap} when applied to these two segmentation methods would slightly change: $s=|segment[t]|-|segment[t-1]|$, and $k=|segment[0]|$. Note that as for the oracle word boundaries, $segment[0]$ corresponds to all words read at the first decoding step.
Figure \ref{fig:compare_segmentations_ulstm_overlap} illustrates that the ULSTM \textit{Overlap-and-compensate} encoding strategy performs best with the fixed interval boundaries segmentation. Surprisingly, the oracle word boundaries segmentation does not seem to be beneficial in comparison with the fixed interval boundaries as it almost always takes bigger AL in order to achieve comparable BLEU scores. We suspect that this happens because the average length of each word ($37$ frames) is much bigger than the stride parameter ($s=10$ or $s=20$ frames) that we set for the fixed interval boundaries. Figure \ref{fig:compare_segmentations_ulstm_overlap} also shows that the randomly set boundaries segmentation perform the worst. Their BLEU scores approach $0$ (the red dots at the bottom of figure \ref{fig:compare_segmentations_ulstm_overlap}) when the segment sizes are too small ($[low\_boundary, high\_boundary] = [5, 10]$).
\vspace{-5pt}
\subsection{Highlighting the most difficult utterances for simultaneous decoding}
\cite{elbayad2020online} introduced a metric to measure the lagging difficulty of an utterance: after
estimating source-target ($(x,y)$) alignments (for instance with \textit{fast-align}
\cite{dyer2013simple}), they define a non-decreasing function $z^{align}(t)$, denoting the number of source words needed to translate a target word. This function guarantees that at a given decoding position $t$, $z^{align}(t)$ is larger than or equal to all the source positions aligned with $t$. Lagging difficulty (LD) is then defined as equation (1) below, with $\tau=argmin_t{\{t|z_t=|x|\}}$:
\vspace{-5pt}
\begin{equation}
LD(x, y) = \frac{1}{\tau}\sum_{t=1}^{\tau}z_t^{align}-\frac{|x|}{|y|}(t-1)
\end{equation}
Based on LD, we extract the $100$ most difficult and the $100$ easiest sentences according to the metrics, and report the BLEU/AL trade-off
for these sets of utterances.
Figure \ref{fig:compare_DL} shows that LD metrics could be a good tool for highlighting the most difficult utterances for simultaneous decoding since the AL/BLEU curve for the easiest utterances is clearly above the one for the hardest utterances. This suggest the possibility to build specific challenge sets for end-to-end simultaneous speech translation.
\vspace{-5pt}
\begin{figure}[ht]
\centering
\input{Figures/compare_DL_ulstm_tst-HE}
\caption{BLEU/AL trade-off scored on different subsets of MuST-C tst-HE based on Lagging Difficulty (LD).}
\label{fig:compare_DL}
\end{figure}
\vspace{-20pt}
\section{Conclusions}
This paper advocates for using ULSTM instead of BLSTM speech encoder for online translation systems, as it shows that ULSTM outperforms BLSTM in terms of both inference speed and BLEU/AL trade-off. We further improve inference speed and performance of ULSTM speech encoder by proposing a new encoding strategy called ULSTM overlap-and-compensate. Moreover, this work investigates the impact of segmentation on the BLEU/AL trade-off of the ULSTM overlap-and-compensate strategy, and shows that this encoding method works best with equal sized chunks. We also show that difficulty lagging, an indicator of the complexity of the source sentence, might have a great impact on the performance of the online translation systems.
\vspace{-5pt}
\section{Acknowledgements}
This work was funded by the French Research Agency (ANR) through the ON-TRAC project under contract number ANR-18-CE23-0021, and was performed using HPC resources from GENCI-IDRIS (Grant 20XX-AD011011365).
\bibliographystyle{IEEEtran}
|
1,116,691,497,522 | arxiv | \section{Introduction}
\label{I}
Relevance of structural disorder for the critical behaviour remains
to be an important problem of modern condensed matter physics.
Even a weak disorder may change drastically the behaviour near the
critical point and in this respect may be related to the global
characteristics of a physical system, such as the space dimension,
order parameter symmetry and the origin of interparticle interaction.
In this paper, we are going to discuss some peculiarities of a
paramagnetic-ferromagnetic phase transition in magnets, where the
randomness of structure has the form of substitutional random-site
or random-bond quenched disorder. Solid solutions of magnets with
small concentration of non-magnetic component as well as amorphous
magnets with large relaxation times may serve as an example of
such systems.
Intuitively, it is clear that for a weak enough disorder
the ferromagnetic phase persists in
such systems. Obviously, intuition fails
to predict whether the critical exponents characterizing phase
transition into ferromagnetic state will differ in a disordered
system and in a ``pure" one. The answer here is given by the Harris
criterion \cite{Harris74} which states that the critical exponents
of the disordered system are changed only if the heat capacity
critical exponent of a pure system is positive,
otherwise the critical exponents of a disordered system coincide
with those of a pure one. Returning to $d=3$ dimensional magnets
with $O(m)$ symmetric spontaneous magnetization one is lead to the
conclusion, that here only the critical exponents of uniaxial magnets
described by the $d=3$ Ising model ($m=1$) are the subject of
influence by weak quenched disorder. Indeed, the heat capacity
diverges $\alpha=0.109\pm0.004>0$ \cite{Guida98} for $m=1$, whereas
it does not diverge for the easy-plane and Heisenberg-like magnets:
$\alpha=-0.011\pm0.004$ and $\alpha=-0.122\pm0.010$ for $m=2$ and
$m=3$, respectively \cite{Guida98}.
Note however that the Harris criterion tells about the scaling
behaviour {\em at} the critical point $T_c$. In other words it
predicts (possible) changes in the {\em asymptotic} values of
the critical exponents defined at $T_c$. In real situations one
often deals with the {\em effective} critical exponents governing
scaling when $T_c$ still is not reached \cite{effective}. These are
non-universal. As far as in our study of particular interest
will be the isothermal magnetic susceptibility $\chi_T$ let us define the
corresponding effective exponent by \cite{effective}:
\begin{equation}\label{1}
\gamma_{\rm eff}(\tau)=-\frac{{\rm d}\ln\chi(\tau)}{{\rm d}\ln\tau},
\hspace{2em}\mbox{with} \hspace{2em} \tau=|T-T_c|/T_c.
\end{equation}
In the limit $T\to T_c$ the effective exponent coincides with the
asymptotic one $\gamma_{\rm eff}=\gamma$.
Already in the first experimental studies
of weakly diluted uniaxial (Ising-like) $d=3$ random magnets \cite{note1} the
asymptotic values of critical exponents were found. For the solid solutions,
the exponents do not depend on the concentration of non-magnetic component
and belong to the new universality class \cite{review} as predicted
by the Harris criterion.
We do not know analogous experiments where an influence of
disorder on criticality of easy-plane magnets was examined. However its
irrelevance was experimentally proven \cite{helium} for the superfluid phase
transition in ${\rm He^4}$ which belongs to the same $O(2)$ universality class
as the ferromagnetic phase transition in easy-plane magnets.
As far as the disorder should be irrelevant for the asymptotic
critical behaviour of the Heisenberg magnets, the diluted $d=3$ Heisenberg
magnets should belong to the same $O(3)$ universality class as the pure ones.
Theoretically predicted values of the isothermal magnetic susceptibility,
correlation length, heat capacity, pair correlation function, and the order
parameter {\em asymptotic} critical exponents in this universality class
read \cite{Guida98}:
\begin{eqnarray}\label{2}
&&
\gamma=1.3895\pm0.0050,\,\nu=0.7073\pm0.0035, \, \alpha=-0.122\pm0.009,\\
\nonumber &&
\eta=0.0355\pm0.0025,
\,\beta=0.3662\pm0.0025.
\end{eqnarray}
The experimental picture is more controversial. The bulk of experiments
on critical behaviour of disordered Heisenberg-like magnets performed
up to middle 80-ies is discussed in the comprehensive reviews
\cite{Egami84,Kaul85}.
More recent experiments may be found in
\cite{heis_all,Fahnle83,Kaul88,Kaul94,Babu97,Perumal01}
and references therein. We show typical results of measurements of
the isothermal magnetic susceptibility effective critical exponent
$\gamma_{\rm eff}$ (\ref{1}) in Figs. \ref{fig1}.
As it is seen from the pictures, the behaviour of $\gamma_{\rm eff}$ is
non-monotonic. The exponent differs from its value predicted in the asymptotic
limit (\ref{2}) and is a subject of a wide crossover behaviour. Before
reaching asymptotics $\gamma_{\rm eff}$ possess maximum (except of the fig.
\ref{fig1}.d), the value of the maximum is system dependent: it
differs for different magnets.
It is standard now to rely on the renormalization group (RG) method
\cite{rgbooks} to get a reliable quantitative description of the behaviour
in the vicinity of critical point. Namely in this way the cited above
values (\ref{2}) of the critical exponents of $d=3$
Heisenberg model were obtained. The RG approach appeared
to be a powerful tool to describe asymptotic \cite{review} and
effective \cite{Folk00} critical behaviour of disordered
Ising-like magnets as well. The purpose of the present paper is to
describe the crossover behaviour of disordered Heisenberg-like
magnets in frames of the field-theoretical RG technique. In
particular we want to calculate theoretically the isothermal magnetic
susceptibility effective critical exponent and to explain in this
way the appearance of the peak in its typical experimental
dependencies. The rest of the paper is organized as follows. In
the Section \ref{II} we formulate the model and review main
theoretical results obtained for it so far by means of the RG
technique, effective critical behaviour is analyzed in the Section
\ref{III}, we end by conclusions and outlook in the Section \ref{IV}.
\begin{figure}[htbp]
\centerline{
\epsfxsize=52mm\epsfysize=30mm\epsfbox{f2_22.eps}
\hspace{3.2cm}
\epsfxsize=48mm\epsfysize=30mm\epsfbox{f4_22.eps}}
{\bf a.}\hspace{7.5cm}{\bf b.}
\vspace{0.5cm}\\
\centerline{
\epsfxsize=50mm\epsfysize=30mm\epsfbox{f7_2.eps}
\hspace{3.2cm}
\epsfxsize=48mm\epsfysize=30mm\epsfbox{f6_2.eps}}
\hspace{3cm}{\bf c.}\hspace{7.5cm}{\bf d.}
\vspace{0.9cm}\\
\centerline{
\epsfxsize=52mm\epsfysize=30mm\epsfbox{f5_2.eps}
\hspace{2.8cm}
\epsfxsize=51mm\epsfysize=30mm\epsfbox{f3_2.eps}}
\hspace{3cm}{\bf e.}\hspace{7.5cm}{\bf f.}
\vspace{0.5cm}\\
\centerline{
\epsfxsize=52mm\epsfysize=30mm\epsfbox{f8_2.eps}
\hspace{3.2cm}
\epsfxsize=46mm\epsfysize=30mm\epsfbox{f1_22.eps}}
{\bf g.}\hspace{7.5cm}{\bf h.}
\vspace{0.5cm}
\caption{\label{fig1} Experimentally measured isothermal magnetic susceptibility
effective critical exponent $\gamma_{\rm eff}$ for disordered Heisenberg-like
magnets ($\tau=(T-T_c)/T_c$).
{\bf a.}:
${\rm Fe_{20}Ni_{56}B_{24}}$ (F\"ahnle {\em et al.}, 1983 \cite{Fahnle83});
{\bf b.}:
${\rm Fe_{32}Ni_{36}Cr_{14}P_{12}B_6}$ (Kaul, 1985 \cite{Kaul85});
{\bf c.}:
${\rm Fe_{20}Ni_{60}P_{14}B_6}$,
${\rm Fe_{40}Ni_{40}P_{14}B_6}$ (Kaul, 1985 \cite{Kaul85});
{\bf d.}:
${\rm Fe_{10}Ni_{70}B_{19}Si_1}$ (Kaul, 1988 \cite{Kaul88});
{\bf e.}:
${\rm Fe_{16}Ni_{64}B_{19}Si_1}$ (Kaul {\em et al.}, 1994 \cite{Kaul94});
{\bf f.}:
${\rm Fe_{86}Co_{4}Zr_{10}}$ (Babu {\em et al.}, 1997 \cite{Babu97});
{\bf g.}:
${\rm Fe_{90}Zr_{10}}$ (Babu {\em et al.}, 1997 \cite{Babu97});
{\bf h.}:
${\rm Fe_{90-x}Mn_xZr_{10}}$ (Perumal {\em et al.}, 2001 \cite{Perumal01}).
}
\end{figure}
\section{The model and its RG analysis}
\label{II}
The model of a random quenched magnet we are going
to consider is described by the following Hamiltonian:
\begin{equation} \label{3}
H=-\frac{1}{2}\sum_{{\bf R},{\bf R'}}J(|{\bf R}-{\bf R'}|)
\vec{S}_{{\bf R}} \vec{S}_{{\bf R'}} c_{{\bf R}} c_{{\bf R'}}.
\end{equation}
Here, the sum spans over all sites ${\bf R}$ of $d$-dimensional
hypercubic lattice, $J(|{\bf R}-{\bf R'}|)$ is a short-range
(ferro)magnetic interaction between classical ``spins" $\vec{S}_{{\bf
R}}$ and $\vec{S}_{{\bf R'}}$. We consider the spins $\vec{S}_{{\bf
R}}$ to be $m$-component vectors and the Hamiltonian (\ref{3})
contains their scalar product. Obviously, for the particular case of
Heisenberg spins we will put later $m=3$. The randomness is
introduced into the Hamiltonian (\ref{3}) by the occupation numbers
$c_{\bf R}$ which are equal 1 if the site ${\bf R}$ is occupied by a
spin and $0$ if the site is empty. Considering the case when occupied
sites are distributed without any correlation and fixed in certain
configuration one obtains so-called uncorrelated quenched $m$-vector
model.
In principle, the above information is enough to apply the RG approach
for a study of the critical behaviour of the model (\ref{3}). One should
obtain an effective Hamiltonian corresponding to the model under
consideration and then one analyzes its long-distance properties by
analyzing appropriate RG equations \cite{rgbooks}. But already on this
step there are at least two different possibilities to proceed and
both were exploited for the model (\ref{3}). On one hand, to get the
free energy of the model one can average the logarithm of
configuration-dependent partition function over different possible
configurations of disorder \cite{Brout59}. Then, making use of the
replica trick \cite{Emery75} one arrives to the familiar effective
Hamiltonian \cite{Grinstein76}:
\begin{equation}\label{4}
H_{\rm eff} {=} {-}\!\int\!\! d^dR\!
\left\{\!\frac{1}{2}\sum_{\alpha{=}1}^n\left[{\mu_0}^2|\vec{\phi}^\alpha|^2{+}
|\vec{\nabla}\vec{\phi}^\alpha|^2\right]\!{+}
\frac{u_0}{4!}\sum_{\alpha{=}1}^n|\vec{\phi}^\alpha|^4{+}
\frac{v_0}{4!}\!{\left(\sum_{\alpha{=}1}^n |\vec{\phi}^\alpha|^2\right)\!}^2
\!\right\}
\end{equation}
describing in the replica limit $n\rightarrow 0$
critical properties of the model (\ref{3}).
Here, $\mu_0$ is a bare mass, $u_0>0$ and $v_0\leq0$ are
bare couplings and $\vec{\phi}^{\alpha} \equiv \vec{\phi}^{\alpha}({\bf R})$
is an $\alpha$-replica of
$m$-component vector field. The prevailing amount of RG studies of
the critical behaviour of quenched $m$-vector model was performed on the
base of the effective Hamiltonian (\ref{4}) \cite{review}.
However, one more effective Hamiltonian corresponding to the model
(\ref{3}) is discussed in the literature
\cite{Sobotta78,Sobotta80,Sobotta82,Sobotta85}.
It is obtained exploiting the idea that a quenched disordered system
can be described as an equilibrium system with additional forces of
constraints \cite{Morita64}. In such approach both variables
$\vec{S}_{\bf R}$ and $c_{\bf R}$ are treated equivalently and one
ends up with the effective Hamiltonian which differs from (\ref{4})
and, consequently, leads to different results for the critical
behaviour of the model (\ref{3})
\cite{Sobotta78,Sobotta80,Sobotta82,Sobotta85}.
Whereas the effective Hamiltonian (\ref{4}) was used in the wide
context of general $m$-vector models \cite{review},
the approach of Refs.
\cite{Sobotta78,Sobotta80,Sobotta82,Sobotta85} was mainly used in
explanations of crossover behaviour in Heisenberg-like systems
\cite{west}. Below, we will discuss our results, based on
the effective Hamiltonian (\ref{4}) for $m=3$ and compare them
with those derived in
\cite{Sobotta78,Sobotta80,Sobotta82,Sobotta85}.
As it is well known, the renormalization group (RG) approach makes use of
the scaling symmetry of the system in the asymptotic limit to extract the
universal content and at the same time removes divergencies which occur for
the evaluation of the bare functions in this limit \cite{rgbooks}.
A change in the renormalized couplings $u$, $v$ of the effective Hamiltonian
(\ref{3}) under the RG transformation is described by the flow equations:
\begin{equation} \label{5}
\ell\frac{\rm d}{{\rm d} \ell}u(\ell)=\beta_u\left(u(\ell),v(\ell)\right),\quad
\ell\frac{\rm d}{{\rm d}\ell}v(\ell)=\beta_v\left(u(\ell),v(\ell)\right).
\end{equation}
Here, $\ell$ is the flow parameter related to the distance $\tau$ to the critical
point. The fixed points ($u^*,v^*$) of the system of differential equations
(\ref{5}) are given by:
\begin{equation} \label{6}
\beta_u\left(u^*,v^*\right)=0,\quad
\beta_v\left(u^*,v^*\right)=0.
\end{equation}
A fixed point is said to be stable if the stability matrix
\begin{equation}\label{6a}
B_{ij}\equiv\partial \beta_{u_i}/\partial u_j, \hspace{3em}
i,j=1,2;
\hspace{3em}
u_i=\{u,v\},
\end{equation}
possess in this point eigenvalues $\omega_1,\omega_2$ with positive real parts.
In the limit $\ell\to 0$, $u(\ell)$ and $v(\ell)$ attain the
stable fixed point values $u^*,v^*$. If the stable fixed point
is reachable from the initial conditions (let us recall that for the
effective Hamiltonian (\ref{3}) they read $u> 0,v \leq 0$) it corresponds to
the critical point of the system. The asymptotic critical exponents values
are defined by the fixed point values of the RG
$\gamma$-functions. In particular the isothermal magnetic susceptibility
exponent $\gamma$ is expressed in terms of the RG functions $\gamma_{\phi}$
and ${\bar \gamma}_{\phi^2}$ describing renormalization of the field $\phi$
and of the two-point vertex function with a $\phi^2$ insertion
correspondingly \cite{rgbooks}:
\begin{equation} \label{7}
\gamma^{-1}=1 - \frac{{\bar \gamma}_{\phi^2}}{2-{\gamma}_{\phi}}.
\end{equation}
In Eq. (\ref{7}), the functions $\gamma_{\phi}\equiv \gamma_{\phi}(u,v)$,
${\bar \gamma}_{\phi^2}\equiv{\bar
\gamma}_{\phi^2}(u,v)$ are calculated in the stable fixed
point $u^*,v^*$. In the RG scheme, the effective critical exponents
are calculated in the region,
where couplings $u(\ell),v(\ell)$ have not reached their fixed point values
and depend on $\ell$. In particular for the exponent $\gamma_{\rm eff}$ one
gets: \begin{equation}\label{8}
\gamma^{-1}_{\rm eff}(\tau)=1 - \frac{{\bar \gamma}_{\phi^2}
[u\{\ell(\tau)\},v\{\ell(\tau)\}]}{2-{\gamma}_{\phi}
[u\{\ell(\tau)\},v\{\ell(\tau)\}]}+\dots.
\end{equation}
In (\ref{8}) the part denoted by dots
is proportional to the $\beta$--functions
(\ref{5}) and comes from the change of the amplitude part of the susceptibility.
In the subsequent calculations we will neglect this part, taking
the contribution of the amplitude function to the crossover to be small
\cite{note2}.
For the effective Hamiltonian (\ref{4}), the fixed point structure is
well established \cite{review}.
It is schematically shown in Figs. \ref{fig2}.a, \ref{fig2}.b.
Two qualitatively different scenarios are observed:
for $m>m_c$ the critical behaviour of the disordered magnet is
governed by the fixed point of the pure magnet ($u^*>0$, $v^*=0$),
whereas for $m<m_c$ the new stable fixed point ($u^*>0$, $v^*<0$)
governs the asymptotic critical behaviour of the disordered magnet.
At the marginal dimensionality $m_c$ which separates these two
regimes, the $\alpha$ exponent of the pure magnet equals zero in
agreement with the Harris criterion.
\begin{figure}[htbp]
\begin{center}
{\includegraphic
{max.eps}}
\end{center}
\caption{\label{fig2} Fixed points structure for the effective
Hamiltonian (\ref{4}) at $d=3$ and arbitrary $m$. {\bf a}:
$m>m_c$, {\bf b}: $m<m_c$. Stable fixed points are shown by filled
boxes, unstable ones are shown by filled circles. Only stable
fixed points with coordinates $u^*> 0$, $v^*\leq0$ are reachable
for the model of the quenched magnet (\ref{3}). }
\end{figure}
Best theoretical estimates of
$m_c$ definitely support $m_c<2$: $m_c=1.942\pm0.026$
\cite{Bervillier86}, $m_c=1.912\pm0.004$ \cite{Dudka01}. Consequently,
the fixed point structure of the model of diluted Heisenberg-like magnet
($m=3$) is given by Fig. \ref{fig2}.{\bf a}: the stable reachable
fixed points of the diluted and pure Heisenberg-like magnets do coincide
($u^*\neq 0,v^*=0$),
hence their {\em asymptotic} critical exponents do coincide as well.
However the last statement does not concern the {\em effective} exponents.
These are defined by the running values of the couplings
$u(\ell)\neq 0,v(\ell)\neq 0$ and will be calculated in the next
section.
\section{The RG flows and the effective critical behaviour}
\label{III}
The RG functions of the model (\ref{4}) are known by now in
pretty high orders of the perturbation theory
\cite{review,Pelissetto00}.
For the purpose of present study we will restrict ourselves by the
first approximation where the described crossover phenomena
manifests itself for the Heisenberg-like disordered magnets in
non-trivial way. Within the two loop approximation in the
minimal subtraction RG scheme \cite{Hooft72} the RG-functions read
\cite{Kyriakidis96}:
\begin{eqnarray}
\nonumber
\beta_u(u,v)&=&-u(\varepsilon-\frac{m+8}{6}{u}-2v+
\frac{3m+14}{12}u^2+{\frac {5mn+82}{36}}v^2+\\ \label{9}
&&
\frac{11m+58}{18}uv),
\\ \nonumber
\beta_v(u,v)&=&-v(\varepsilon-\frac{m+2}{3}u-\frac{mn+8}{6}v+
\frac{5(m+2)}{36}u^2 + \\ \label{10}
&&
\frac{3mn+14}{12}v^2+
\frac{11(m+2)}{18}uv),
\\ \label{11}
\gamma_{\phi}(u,v)&=&\frac{m+2}{72}u^2+\frac{mn+2}{72}v^2+
\frac{m+2}{36}uv,
\\ \label{12}
\bar{\gamma}_{\phi^2}(u,v)&=&\frac{m+2}{6}u+\frac{mn+2}{6}v
-\frac{m+2}{12}{u}^{2}-\frac{mn+2}{12}v^2-\frac{m+2}{6}uv.
\end{eqnarray}
Here, $\varepsilon=4-d$ and replica limit $n=0$ is to be taken.
Starting form the expressions (\ref{9})--(\ref{12}) one can
either develop the $\varepsilon$-expansion, or work directly at
$d=3$ putting in (\ref{9}), (\ref{10}) $\varepsilon=1$ and
considering renormalized couplings $u,v$ as the expansion
parameters \cite{Schloms}.
However, such RG perturbation theory series with several couplings
are known to be asymptotic at best \cite{rgbooks}.
One should apply appropriate resummation technique to improve their
convergence to get reliable numerical data on their basis.
We used several different resummation schemes for this purpose.
Here we will give the results obtained by the method which allowed
to analyze the largest region in the parametric $u-v$ space.
The method was proposed in Ref. \cite{Alvarez00} and was successfully
applied to study random $d=3$ Ising model \cite{Pelissetto00}.
Moreover, it was shown that the RG functions of the $d=0$ random Ising
model are Borel-summable by this method \cite{Alvarez00}.
The main idea proposed in Ref. \cite{Alvarez00} is to consider
resummation in variables $u$ and $v$ separately. Taken that the RG
function $f(u,v)$ is given to the order of $p$ loops, one first rewrites
it as a power series in $v$:
\begin{equation} \label{13}
f(u,v)=\sum_{k=0}^pA_k(u)v^k.
\end{equation}
Then each coefficient $A_k(u)$ is considered as power series in $u$ and
resum\-med as a function of a single variable $u$ thus obtaining the
resummed functions $A^{res}_k(u)$. Next one substitutes these
functions into (\ref{13}) and resums the RG function $f$ in single
variable $v$. For the resummation in a single variable one may use
any of familiar methods. Our results are obtained by making use of
the Pad\'e-Borel-Leroy method \cite{Nickel78}.
First, applying the above described resummation procedure to the
$\beta$-functions (\ref{9}), (\ref{10}) we get the pure Heisenberg
fixed point coordinates $u^*=0.8956$, $v^*=0$. The stability matrix
(\ref{6a}) eigenvalues are
positive at this fixed point ($\omega_1=0.577$,
$\omega_2=0.147$) providing its stability. Then for the resummed
values of the asymptotic critical exponents we get \cite{note3}:
\begin{equation}\label{14}
\gamma=1.382,\,\nu=0.701, \, \alpha=-0.104,\,
\eta=0.030,
\,\beta=0.361.
\end{equation}
We do not give the confidence intervals in (\ref{14}), as far as
they can be estimated only by comparison of changes introduced by
different orders of perturbation theory. Note however that the
results (\ref{14}) are in a good agreement with the most accurate
estimates of the exponents in the $O(3)$ universality class (\ref{2}).
This brings about that both the considered here two-loop
approximation as well as the chosen resummation technique give an
adequate description of asymptotic critical phenomena.
Before passing to the effective critical exponents let us first
analyze the corrections to scaling.
For the pure Heisenberg magnet, taking into account the
leading correction to scaling results in the following
formula for the
isothermal susceptibility:
\begin{equation}\label{15}
\chi(\tau)=\Gamma_0\tau^{-\gamma}(1+\Gamma_1\tau^{\Delta}),
\end{equation}
where the correction-to-scaling exponent
is given by $\Delta=\omega\nu$ with
$\omega=\partial\beta_u(u)/\partial u
|_{u=u^*}$ and non-universal critical amplitudes $\Gamma_0, \Gamma_1$.
For the diluted Heisenberg magnet the corresponding formula
includes two leading corrections $\Delta_1$, $\Delta_2$
(see e.g. \cite{Kaul88}):
\begin{equation}\label{16}
\chi_(\tau)=\Gamma'_0\tau^{-\gamma}(1+\Gamma'_1\tau^{\Delta_1}+
\Gamma'_2\tau^{\Delta_2}),
\end{equation}
with critical amplitudes $\Gamma'_0, \Gamma'_1,\Gamma'_2$.
The exponents $\Delta_i$ are expressed in terms of the
stability matrix (\ref{6a}) eigenvalues $\omega_i$ in the pure
Heisenberg fixed point: $\Delta_i=\nu \omega_i$.
At this fixed point, the eigenvalues of the stability matrix
(\ref{6a}) read:
\begin{equation}\label{17}
\omega_1=\frac{\partial\beta_u(u,v)}{\partial u}|_{u^*\neq0,v^*=0},
\hspace{3em}
\omega_2=\frac{\partial \beta_v(u,v)}{\partial
v}|_{u^*\neq0,v^*=0}.
\end{equation}
It is straightforward to see that the value $\omega_1$ coincides with
the exponent $\omega$ of the pure model whereas it may be shown
(see e.g. \cite{Kaul88,Kyriakidis96}) that
the exponent $\omega_2=|\alpha|/\nu$ where $\alpha$ and $\nu$ are
the heat capacity and correlation length critical exponent of the pure
Heisenberg model. On the base of the numerical values of the exponents
(\ref{14}) we get:
\begin{equation}\label{18}
\Delta_1=0.405, \hspace{3em}
\Delta_2=0.104.
\end{equation}
Again, obtained by us in the two-loop approximation numbers
(\ref{18}) can be compared with those in the six-loop
approximation making use of the data (\ref{2}) together with the
value of $\omega$ of pure 3d Heisenberg model
$\omega=0.782\pm0.0013$
\cite{Guida98}. As we have noted above, in order to get the
numerical values of the correction-to-scaling exponents of
diluted Heisenberg model it is no need to consider the RG
functions (\ref{9})--(\ref{12}) in the whole region of couplings
$u,v$: it is enough to know them for the case of the pure model
(i.e. for $u\neq0,v=0$). However, to get the effective exponents
it is necessary to study complete set of the RG functions
(\ref{9})--(\ref{12}) working also in the region where both
couplings $u$ and $v$ differ from zero.
To this end we use the above described
resummation technique in order to restore the convergence of the
RG expansions in couplings $u$, $v$. First we solve the system of
differential equations (\ref{5}) and get the running values of
couplings $u(\ell)$, $v(\ell)$ (\ref{9})--(\ref{12}).
They define the flow in the parametric
space $u,v$ and in the limit $\ell\rightarrow 0$ attain the stable
fixed point value (shown by the filled box in Fig. \ref{fig3}).
\begin{figure}[htbp]
{\includegraphics[width=0.7\textwidth] {flow.eps}} \caption{
\label{fig3} Flows in the parametric space of couplings. The
filled box denotes the stable fixed point $u^*=0.8956,\,v^*=0$.
Curve 1 corresponds to the flow from initial values with $v_0=0$,
curve 2 starts with a small ratio $|v_0/u_0|$ whereas flow 3
corresponds to larger $|v_0/u_0|$.}
\end{figure}
Character of the flow depends on the initial conditions $u_0,v_0$
for solving the system of differential equations (\ref{5}).
For the model (\ref{3}), the coupling $v$ is
proportional to variance of disorder \cite{review}
thus one can use the ratio $|v_0/u_0|$ to define the degree of dilution.
Typical flows which are obtained for different ratios $|v_0/u_0|$
are shown in Fig. \ref{fig3} by curves 1-3. We choose the starting
values in the region with the appropriate signs of couplings
$u>0,\,v<0$ near the origin (in the vicinity of the Gaussian fixed point
$u^*=v^*=0$ shown by the filled circle in the figure). The flow No
1 is obtained for $v_0=0$, it corresponds to the pure Heisenberg
model. The flow No 2 results from the small ratio $|v_0/u_0|$ and
corresponds to the weak disorder whereas the flow No 3 is obtained
for large $|v_0/u_0|$ and corresponds to the stronger dilution.
Obtained running values of coupling constant presented by flows in
Fig. \ref{fig3} allow one to get the effective
critical exponents. Calculating resummed expression for the effective
exponent $\gamma_{\rm eff}$ (\ref{8}) along the flows 1-3 we get the
results shown in the Fig. \ref{fig4}.
\begin{figure}[htbp]
{\includegraphics[width=0.7\textwidth] {expon.eps}} \caption{
\label{fig4} Effective critical exponent $\gamma_{\rm eff}$ versus
logarithm of the flow parameter. The curves correspond to flows
from the Fig. \ref{fig3} denoted by corresponding numbers.}
\end{figure}
Again, the curve 1 corresponds to the effective critical exponent
of the pure Heisenberg model, whereas curves 2 and 3 provide two
possible scenarios for the effective exponents of the disordered
Heisenberg model. Curve 2 corresponds to the weak dilution region:
here, the exponent increases with approach to the
critical point, although the crossover region is larger in
comparison with the pure magnet (compare curves 1 and 2 in Fig.
\ref{fig4}). This may lead to the peculiar situation that the
asymptotic value of the exponent is reached earlier than the
asymptotic values of the coupling.
The effective exponents for the flows
originating from non-zero ratio $|v_0/u_0|$ always attain the value
which are larger than the asymptotic one. But the absolute value of this
``overshooting" for small enough $|v_0/u_0|$ is too small to be
observed experimentally.
An experimental observation of such type of
$\gamma_{\rm eff}$ behaviour of the disordered Heisenberg-like magnet
is provided e.g. by Fig. \ref{fig1}{\bf d}. Different behaviour of
$\gamma_{\rm eff}$ is demonstrated by the curve 3 in Fig. \ref{fig4}.
Here, before reaching the asymptotic region the exponent
possess a distinct peak. Such behaviour is in
agreement with observed experimental data presented by Figs.
\ref{fig1}{\bf a}--\ref{fig1}{\bf c},
\ref{fig1}{\bf e}--\ref{fig1}{\bf h}. The value of
maximum depends on the initial values
for the RG flows. Larger ratio $|v_0/u_0|$ (i.e. stronger disorder)
leads to the larger maximum.
Thus, within unique approach one may explain both scenarios
observed in the diluted Heisenberg-like magnets effective critical exponent
$\gamma_{\rm eff}$ behaviour.
As we have noticed in the section \ref{II}, the crossover behaviour of
random Heisenberg-like magnets was analyzed by means of an alternative
approach in \cite{Sobotta78,Sobotta80,Sobotta82,Sobotta85}. There, the
quenched disordered magnet was described as an equilibrium
one with additional forces of constraints \cite{Morita64}.
This resulted in an effective Hamiltonian which differs from
(\ref{4}). The fixed point structure of this Hamiltonian differs from
those given in Fig. \ref{fig2} and, for different concentrations,
leads to different crossover regimes. In particular, it predicts that
there exists a limiting value of concentration where the critical
behaviour is governed by Fisher-renormalized tricritical exponents
\cite{Sobotta85} which coincide with those of a $d=3$ spherical
model: $\gamma=2$, $\nu=1$, $\alpha=-1$, $\eta=0$, $\beta=1/2$.
There exist two more fixed points which may be stable in the weak
dilution regime. Their stability differs in different orders of
the perturbation theory (compare \cite{Sobotta78} and
\cite{Sobotta82}) but the numerical values of the critical
exponents do not differ essentially at these fixed points. The
maximal possible value of the effective critical exponent
$\gamma_{\rm eff}$ has been estimated as $\gamma_{\rm eff}\simeq 2.6$
\cite{Sobotta82}. However, the distinct
feature of the behaviour of $\gamma_{\rm eff}(\tau)$ obtained in
\cite{Sobotta80} is its monotonic
dependence. Hence, the experimentally observed peaks (see Fig.
\ref{fig1}) can not be explained within such approach.
\section{Conclusions}
\label{IV}
In the present paper we used the field-theoretical RG technique to
study the effective critical behaviour of diluted Heisenberg-like
magnets. The question of particular interest was to explain the
peak in the exponent $\gamma_{\rm eff}$ as function of distance from
$T_c$ observed in some experiments.
Our two-loop calculations refined by
the resummation of the perturbation theory series resulted in
typical behaviour of diluted Heisenberg-like magnets
$\gamma_{\rm eff}$ exponent represented by curves 2 and 3 in Fig.
\ref{fig4}. The exponent can either reach it asymptotic value
without demonstrating distinct maximum or it can first reach the
peak and then cross-over to the asymptotic value from above. The
strength of disorder is a physical reason which discriminates
between these two regimes.
Our calculations are quite general and do not specify any particular
object. In order to fit our curves to certain experiment one should
include into consideration non-universal parameters to specify the
magnetic system. The same concerns the flow parameter $\ell$ which
as we have already noted is related to the distance to the
critical point $\tau$. In principle such calculations may be done.
However we want to emphasize that our analysis shows the reason of
the peak in $\gamma_{\rm eff}(\tau)$ dependence for different disordered
magnets which belong to the $O(3)$ universality class and this reason may
be explained within the traditional RG approach. This concerns not only
the magnetic susceptibility effective critical exponent.
One more example is given by the order parameter effective exponents
$\beta_{\rm eff}$ which has minimum when $\tau$ goes to zero
(see e.g. \cite{Perumal01}). Interpretation of this effect will be the
goal of a separate study
In conclusion we want to note that similar peculiarities of
the effective critical behaviour may be
observed in studies of disordered easy-plane magnets which belong
to the $O(2)$ universality class. Since the heat capacity does not
diverge in such systems, the RG fixed point scenario is given by the
Fig. \ref{fig2}{\bf a} as for the Heisenberg-like disordered magnets.
Up to our knowledge such experiments have not been performed yet and
we hope that our calculations may stimulate them.
M. D. acknowledges the Ernst Mach research fellowship of the
\"Osterreichisher Austauschdienst. This work
was supported in part by \"Osterreichische Nationalbank
Jubil\-\"aums\-fonds through grant No 7694.
|
1,116,691,497,523 | arxiv | \section{Introduction}
A fundamental challenge faced by the particle physics community is to determine the Majorana or Dirac nature of the
neutrino. In order to assess this question, experimentalists are exploring different reactions where the Majorana nature
may be manifested (for a review see e.g. \cite{Zralek:1997sa}). Well known facts concerning this problem are:
\begin{enumerate}
\item a Majorana particle is identical to its own antiparticle and it leads to reactions where the lepton number is not conserved.
The prototypical example of such processes is the neutrinoless double beta decay
\cite{Schechter:1981bd,Deshpande:1984sm,DellOro:2016tmg,Vergados:2016hso} and,
\item massive neutrinos can have helicity $s_{\parallel}\neq-1$ and there are helicity-driven effects yielding a sizeable difference in
the scattering cross sections for Majorana and Dirac neutrinos on electrons \cite{Singh:2006ad,Kayser:1981nw,Barranco:2014cda}.
\end{enumerate}
Most of the experimental effort is focused in the search for neutrinoless double beta decay.
Multiple experiments have been constructed for this purpose like the Heidelberg-Moscow experiment \cite{KlapdorKleingrothaus:2000sn},
IGEX \cite{Aalseth:2002rf}, EXO \cite{Auger:2012ar}, GERDA \cite{Agostini:2017iyd}, Kamland-ZEN \cite{Gando:2012zm} or
CUORE \cite{Arnaboldi:2002du}, among others.
The purpose of this work is to show that the second possibility may lead to observable differences between the Majorana
and Dirac neutrinos. Previous works on this topic have been concentrated in a full conversion of left handed
Majorana neutrino into right handed Majorana anti-neutrino \cite{Barbieri:1991ed,Semikoz:1996up,Pastor:1997pb}. The
non observation of electron anti-neutrinos in solar detectors have set strong limits on the neutrino magnetic moment
\cite{Miranda:2003yh}. Here, we will show that it is not necessary to have a full neutrino-antineutrino conversion in order to obtain a
positive signal of this effect, but only a change in the vector polarization will lead to measurable differences.
We organized the paper as follows: In section \ref{cross_section}, we present the Dirac and Majorana neutrino-electron
elastic scattering cross sections as functions of the polarization vector of the incident neutrino. Next, in
section \ref{change} we study the change of neutrino helicity in the presence of a magnetic field. In
section \ref{sunmagnetic}, assuming a model for the magnetic field of the Sun \cite{Miranda:2000bi}, we study this effect
for solar neutrinos and translate results of the direct measurement of the $^7$Be solar neutrino signal rate performed
by Borexino \cite{Arpesella:2008mt} into a constraint for the neutrino magnetic moment.
Finally, in section \ref{supernova},
we discuss the case of supernova neutrinos where, due to the strong magnetic fields generated in a supernova, in some cases
as strong as $B\sim 10^{15}$ Gauss \cite{Obergaulinger2014,Obergaulinger:2011ic}, even an extremely small neutrino
magnetic moment, as small as $\mu_\nu \sim 10^{-19}\mu_B$, can make a significant change in the polarization
vector, leading to measurable differences between Majorana and Dirac neutrinos in both the number of events and
the energy spectrum.
\section{The $\nu$-e scattering cross section including neutrino polarization }\label{cross_section}
Possible differences in the $\nu$-e scattering cross section between Dirac and Majorana neutrinos have been
previously considered concluding that such differences are proportional to the neutrino masses
\cite{Dass:1984qc,Garavaglia:1983wh}. Because of the smallness of the neutrino masses, in practice, no measurable
difference in the Majorana or Dirac $\nu$-e scattering cross section seems to be possible.
Nevertheless, this conclusion applies to the considered unpolarized cross section and, according to
\cite{Kayser:1981nw,Barranco:2014cda}, when the neutrino polarization is taken into account,
a clear difference between Majorana and Dirac neutrinos in neutrino-electron scattering appears. Defining the incident
neutrino polarization vector in the neutrino rest frame as $s_\nu=\left(0,s_\perp,0,s_\parallel\right)$, we can calculate
the differential cross sections for each case, Dirac and Majorana, in terms of the helicity $s_\parallel$. In
\cite{Kayser:1981nw,Barranco:2014cda} the cross sections were computed in the center of mass frame.
Here we present results in the laboratory frame, which is more suitable for the purposes of this paper.
The differential cross section for $\nu$-$e$ elastic scattering process is given by
\begin{align}\nonumber
&\frac{d^2\sigma^D}{dE^i_\nu dT}=\frac{m_eG^2_F}{4\pi }\bigg\{(g_A+g_V)^2\frac{E^i_{\nu}(E^{i}_\nu-s_\parallel P)}{P^2}\nonumber\\
&+\left((g_A+g_V)^2\frac{(E^i_\nu-T)^2}{P^2}+(g_A^2-g_V^2)\frac{m_eT}{P^2}\right)\left(1-s_\parallel \frac{E^i_\nu}{P}\right)\nonumber\\
&+\left(\frac{s_\parallel}{P}(g_A-g_V)^2(E^i_\nu-T)\left(1+\frac{T}{m_e}\right)+(g_A^2-g_V^2)\left(1-s_\parallel\frac{T}{P}\right)\right)\,,\nonumber\\
\times&\left(\frac{m_\nu}{P}\right)^2\bigg\}\,,\label{CrossDirac}
\end{align}
in the Dirac case, while if the neutrino is a Majorana particle we have
\begin{align}
\nonumber
&\frac{d^2\sigma^M}{dE^i_\nu dT}=\frac{m_e G^2_F}{2\pi P^2}\bigg\{2\left(2g^2_A-g^2_V\right)m^2_\nu\\
& -\frac{4E^i_\nu}{P}g_A g_V s_\parallel T \left(E^i_\nu +\frac{m^2_\nu}{m_e}\right)\left(1-\frac{T}{2E^i_\nu}\right)+\left(g^2_A-g^2_V\right)m_e T\nonumber \\
&+\left(g^2_A+g^2_V\right)\left[2{E^i_\nu}^2\left(1-\frac{T}{E^i_\nu}\right)+T^2\left(1+\frac{m^2_\nu}{m_e T}\right)\right]\bigg\},\label{CrossMajorana}
\end{align}
In last two equations, $P=\left|\vec{P}^i_{\nu}\right|$ is the momentum of the incident neutrino, whereas $T$ represents
the electron recoil energy. From Eqs. \eqref{CrossDirac} and \eqref{CrossMajorana}, with $m_\nu=0$ and $s_\parallel=-1$,
the usual differential cross section for $\nu$-$e$ scattering is recovered.
\begin{figure}
\begin{center}
\includegraphics[width=.45\textwidth]{cross_vs_spin.eps}
\caption{Integrated cross section for the Beryllium line as a function of the
neutrino spin polarization. }\label{Fig1}
\end{center}
\end{figure}
Presently, cosmological data sets a limit on the sum of the neutrino masses $\sum m_\nu < 0.183$ eV
\cite{Giusarma:2016phn,Ade:2015xua}. Furthermore,
terrestrial experiments designed to measure the effect of neutrino masses on the tritium $\beta$-decay spectrum near
its endpoint have set an upper bound on the electron neutrino mass of $m_{\nu_e} < 2.3$ eV at $95\%$ C.L.
\cite{Kraus:2004zw,Lobashev:2001uu}. Thus, neutrino masses are really small, and the only variable able to produce
a difference between a Dirac neutrino and a Majorana neutrino is $s_{||}$. Notice that the leading effects for
$s_{||}\neq -1$ in Eqs. (\ref{CrossDirac}, \ref{CrossMajorana}) are proportional to the electron mass.
In order to illustrate the differences between Dirac and Majorana cross sections driven by $s_{||}$, we calculate the cross
section for a specific polarization
\begin{equation}
\sigma^{M,D}(s_{||})\equiv \int_{T_{min}}^{T_{max}} dT \int_0^{\infty} dE_\nu \lambda(E_\nu) \frac{d^2\sigma^{M,D}}{dE_\nu dT}(E_\nu,T,s_{||})\,,
\end{equation}
where $\lambda(E_\nu)$ is the neutrino spectrum, which depends on the neutrino source under consideration. For
definitiveness, we will use the $^7$Be line of the solar neutrino spectrum. In this case, the spectrum will
be a Dirac delta centred in $E_\nu=0.862$ MeV. For the recoil energy $T$, we will assume a detector with the features
of Borexino, i.e. we consider $T \in [250, 750]$ keV.
In order to exhibit the size of the helicity-driven effects, in Fig. \ref{Fig1} we plot the integrated cross sections for
Dirac and Majorana neutrinos taking
$m_{\nu}=1$ eV in the numeric calculations. We can see three important features from this figure:
\begin{enumerate}
\item a left handed Dirac neutrino has the same cross section as a left handed Majorana neutrino,
\item a right handed Majorana neutrino has the same cross section as a left handed Dirac antineutrino, and
\item a right handed Dirac neutrino has zero cross section, i.e. it is a sterile neutrino.
\end{enumerate}
\begin{figure}
\includegraphics[width=0.45\textwidth]{borexino_boun.eps}
\caption{The function $D(S_{||})$ for the $^7$Be line of the solar spectrum within the Borexino integration intervals.}\label{differenceBorexino}
\end{figure}
Besides these three limits, there are differences in the Majorana and Dirac cross sections for -1 $< s_{\parallel} <$ 1. A way
to quantify this difference is given by means of the function
\begin{equation}
D\left(s_\parallel\right)\equiv\frac{|\sigma^M(s_\parallel)-\sigma^D(s_\parallel)|}{\sigma^D(s_\parallel)}\,,
\label{difference}
\end{equation}
which only depends on the helicity $s_\parallel$. We remark that neutrino-mass-driven effects cancel in the difference
and this function reflects the helicity-driven effects properly. The function $D(s_{||})$ with $\sigma^{M,D}$ integrated
assuming $\lambda(E_\nu)=\delta(E_\nu-0.862 \textrm{MeV})$
and $T \in [0.250,0.750]$ MeV is shown in Fig. \ref{differenceBorexino}. From this plot, we conclude that there is a sizable difference
in the cross section for the scattering of Majorana and Dirac neutrinos with $s_{||}\neq -1$ off electrons.
Neutrinos are born as left handed-particles. Indeed, for instance, in the decay of a pseudo-scalar meson
$P^+ \to \ell^+ + \nu$, the neutrino helicity $s_{||}$ can be computed as $s_{||}=(E-W)|\vec k|/(W E-|\vec k|^2)$
with $W$ and $E$ the energies of the charged lepton $\ell$ and the neutrino, respectively \cite{Barenboim:1996cu}.
Due to the smallness of the neutrino mass, effectively the initial vector polarization can be written as
$s_\nu=\left(0,s_\perp=0,0,s_\parallel=-1\right)$ and, as we have shown before, Dirac or Majorana $\nu$-$e$
cross sections for small neutrino masses and left handed Dirac or Majorana neutrinos are identical.
The challenge we face now is to find physical processes able to change the neutrino helicity $s_\parallel$.
\section{Change of neutrino polarization due to a magnetic field}\label{change}
Fortunately, nature offers a way in which the neutrino helicity may be modified. Indeed, any neutral particle with a
magnetic moment can change its longitudinal part of the polarization vector in the presence of an external magnetic field.
The change in the helicity is given by the Bargmann-Michel-Telegdi equation \cite{Bargmann:1959gz,Semikoz:1992yw}
\begin{equation}
\frac{ds_{||}}{dr}=-2\mu_\nu B_\perp s_{||}\,,\label{spinequation}
\end{equation}
where $B_\perp$ is the component of the external magnetic field perpendicular to the propagation of the neutrino
and $\mu_\nu$ is the effective neutrino magnetic moment.
Neutrinos can have a magnetic moment if they are massive. The effective neutrino magnetic moment is different
for Majorana and Dirac particles. The Majorana neutrino magnetic moment is introduced via the effective
electromagnetic Hamiltonian
$H_{em}^M=-\frac{1}{4}\nu_L^TC^{-1}\lambda\sigma^{\alpha\beta}\nu_LF_{\alpha\beta}+h.c.$ \cite{Schechter:1981hw}.
Here $\lambda=\mu-id$ is an antisymmetric arbitrary complex matrix. On the other hand, the corresponding Dirac
electromagnetic effective Hamiltonian is $H_{em}^D=\frac{1}{2}\bar \nu_R\lambda\sigma^{\alpha\beta}\nu_LF_{\alpha\beta}+h.c.$
and in this case, $\lambda=\mu-id$ is an arbitrary complex hermitian matrix \cite{Grimus:2000tq}.
Experiments are only sensitive to some process-dependent effective neutrino magnetic moment $\mu_\nu$ given
by a superposition of the matrix elements of $\lambda$ (see for instance \cite{Canas:2015yoa}). It is this effective moment
the one that will affect the change of the neutrino helicity $s_{||}$ according to Eq. \eqref{spinequation}, thus we can
forget about the physics behind the specific value of the effective parameter appearing in this equation and treat on
equal footing the evolution of $s_{||}$ for both Dirac and Majorana neutrinos.
Experimental limits on this effective magnetic neutrino moment are shown in Table \ref{expdata}.
\begin{figure}
\begin{center}
\includegraphics[width=.45\textwidth]{evolution_spin.eps}
\caption{Top panel: The magnetic field in the interior of the Sun. Bottom panel: Evolution of the neutrino polarization
$s_{||}$ as it travels in the solar interior.}\label{Fig2}
\end{center}
\end{figure}
The other relevant ingredient to describe the change of $s_{||}$ is the external magnetic field $B_\perp$ and the next
sections are devoted to explore two natural sources of neutrinos produced in an environment with magnetic fields:
the Sun and supernova explosions.
\begin{center}
\begin{table}\label{expdata}
\caption{Stronger limits for different neutrino effective magnetic moments.}
\begin{tabular}{c|c|c}
\hline\hline
Experiment& Limit & Ref. \\
\hline\hline
GEMMA$_{\textrm{Reactor} \bar \nu_e-e^-}$& $\mu_{\nu_e}< 2.9 \times 10^{-11}\mu_B$ & \cite{Beda:2012zz}\\
LSND$_{\textrm{Accelerator} (\nu_\mu,\bar \nu_\mu)-e^-}$ & $\mu_{\nu_\mu}< 6.8 \times 10^{-10}\mu_B$ & \cite{Auerbach:2001wg}\\
Borexino$_{\textrm{Solar} \bar \nu_e-e^-}$& $\mu_{\nu_e}< 5.4 \times 10^{-11}\mu_B$ & \cite{Arpesella:2008mt}\\
\hline\hline
\end{tabular}
\end{table}
\end{center}
\section{Change of the neutrino helicity in the Sun}\label{sunmagnetic}
The question is whether the Sun magnetic field
can produce sizeable changes in the helicity of neutrinos to yield a measurable effect in terrestrial detectors or not.
In order to answer this question, we follow the magnetic profile proposed in \cite{Miranda:2000bi}. It was
obtained by means of full self-consistent and analytical solutions of the magneto-hydrodynamic equations inside the Sun.
The magnetic field inside the Sun was computed as a family of solutions given by
\begin{equation}
B^k_r(r, \theta)=2\hat B^k\cos \theta\left[1 -
\frac{3}{r^2S_{k}}
\left(\frac{\sin(z_{k}r)}{z_{k}r} - \cos(z_{k}r)\right)\right]~,
\nonumber
\end{equation}
\begin{equation}
B^k_{\theta}(r, \theta)=-\hat B^k\sin\theta \left[2 +
\frac{3}{r^2S_{k}}\left(\frac{\sin(z_{k}r)(1-(z_{k}r)^2)}{z_{k}r} -
\cos(z_{k}r)\right)\right]~,\nonumber
\end{equation}
\begin{equation}
B^k_{\phi}(r, \theta)=\hat B^k z_{k}\sin \theta\left[r -
\frac{3}{rS_{k}}\left(\frac{\sin(z_{k}r)}{z_{k}r} -
\cos(z_{k}r)\right)\right]~,
\label{soluciones}
\end{equation}
where $z_{k}$ denote the roots of the spherical Bessel function, $S_{k}\equiv z_k\sin z_k$
and the boundary conditions $B_{\perp}(r=0)=B_{\perp}(r=R_\odot)=0$ are imposed. The polar angle is $\theta$ and the
distance $r$ has been normalized to the solar radius $R_\odot$.
The coefficient $\hat B^k(B_{core})$ is given by
\begin{equation}
\hat B^k = \frac{B_{core}}{2(1 - z_{k}/\sin z_{k})}~.
\end{equation}
There is an upper limit on the magnitude of the Solar magnetic moment in the core. It should be
smaller than $30$ G \cite{boruta}. Furthermore, the solar magnetic field in the convective zone
should be smaller than 100-300 kG \cite{Moreno}.
Latest analysis suggests a lower value \cite{Miranda:2003yh,Friedland:2005xh,Raffelt:2009mm,Friedland:2002pg} hence we consider a conservative maximum
value of the magnetic field in the convective zone of $10$ kG.
Taking advantage of the linearity of the magneto-hydrodynamic equations, any linear combination of $B_K$ is also
a solution, hence the solar magnetic field is computed as $\vec B=\sum_K c_K \vec B_K$. More details in
the method for computing $c_K$ can be found in \cite{Miranda:2000bi}.
Finally, the perpendicular component $B_\perp$, which is relevant to the
neutrino evolution of $s_{||}$, can be computed as $B_{\perp} = \sqrt{B_{\phi}^2 + B_{\theta}^2}$.
In Fig. \ref{Fig2} we show an example of a magnetic field profile for $B_\perp(r)$ in the Sun.
Once we have set the magnetic profile of the Sun, we can solve Eq. \eqref{spinequation} to find $s_{||}(r)$ for different values
of $\mu_\nu$. As an example, in the bottom panel of Fig. \ref{Fig2} we show $s_{||}$ as a function of the
radial coordinate $r$ for the upper bound of the neutrino magnetic moment $\mu_\nu=10^{-11}\mu_B$. From this plot, it is clear that the magnetic field in the Sun may produce a considerable change in the neutrino helicity, thus the effects of this change in observables for terrestrial detectors are worthy of study.
\subsection{On the limit on neutrino magnetic moment}
\begin{figure}
\includegraphics[width=0.45\textwidth]{final_helicity.eps}
\caption{The value of $s_{||}$ evaluated at $r=R_\odot$, i.e. the final value of the neutrino helicity when the neutrino
leaves the Sun for different values of the neutrino magnetic moment $\mu_\nu$.}
\label{Fig4}
\end{figure}
The neutrino magnetic moment $\mu_\nu$ changes the neutrino helicity modifying the expected $\nu$-$e$ elastic
cross section for solar neutrinos colliding with electrons on Earth. Also, it produces an extra term given by the
electromagnetic interaction of the neutrino with the electron given by \cite{Bardin:1970wq,Kyuldjiev:1984kz}
\begin{equation}
\frac{d\sigma^{em}}{dT}=\frac{\pi\alpha^2}{m_e^2}\frac{\mu_\nu^2}{\mu_B^2}\left(\frac{1}{T}-\frac{1}{E_\nu}\right)\,.
\label{CrossMag}
\end{equation}
Furthermore, the inclusion of $\mu_\nu$ may change the neutrino probability oscillation.
Thus, the number of expected events changes accordingly:
\begin{equation}
N^{M,D}_{theo} = n_t t \phi \int dE_\nu dT P(E_\nu, \mu_\nu)\lambda(E_\nu)
\frac{d^2\sigma(E_\nu,\mu_\nu,s_{||})^{M,D}}{dE_\nu dT}, \label{events}
\end{equation}
where $n_t$ is the number of targets, $\phi$ the flux, $t$ the observation time and
$\frac{d^2\sigma(E_\nu,\mu_\nu,s_{||})}{dE_\nu dT}$
includes both s of Eq. \eqref{CrossDirac} or Eq. \eqref{CrossMajorana} and the electromagnetic contribution
given by Eq. \eqref{CrossMag}.
The $^7Be$ line of the solar neutrino spectrum offers a unique way of testing neutrino-electron cross section. Indeed,
in this case $E_\nu=0.862$ MeV, the energy distribution in Eq. \eqref{events} is a Dirac delta
function, the probability is only a function of the $\mu_\nu$ and it is easy to
compute the relative differences in the number of events for Majorana and Dirac neutrinos
\begin{equation}
\frac{|N^M_{theo}-N^D_{theo}|}{N^D_{theo}}=\frac{|\sigma^M(s_\parallel)-\sigma^D(s_\parallel)|}{\sigma^D(s_\parallel)}=D(s_{||}).
\end{equation}
Notice that Eq. \eqref{difference} coincides with the normalized uncertainty in the number of events.
In this concern, the uncertainty in the data of Borexino number of events translates into an uncertainty in the
Dirac-Majorana difference $D\left(s_\parallel\right)$, putting an upper bound on this quantity.
From the number of neutrino events $N=49\pm1.5\textrm{stat}^{+1.5}_{-1.6}\textrm{syst}$ counts/day/100 ton
reported by Borexino \cite{Arpesella:2008mt,Bellini:2011rx} we obtain that the Dirac-Majorana difference should
be less than $4.4\%$. Solving for $D(s_{||})< 0.044$ (see Fig. \ref{differenceBorexino}) we obtain the constraint
$s_{||} < -0.68$ for the helicity of the neutrinos caught in Borexino detectors.
In order to assess possible effects of the helicity change of solar neutrinos in terrestrial detectors, we need the
value of the neutrino helicity when it leaves the Sun, $s_{||}(r=R_\odot)$. This quantity depends on the neutrino magnetic moment.
Assuming that a left handed neutrino was born in the center of the Sun and using the previously described magnetic profile in the Sun,
we calculate $s_{||}(r=R_\odot)$ as a function of the neutrino magnetic moment. Our results are shown in Fig. \ref{Fig4}.
Since most of the magnetic fields between the Earth and the Sun are negligible, neutrinos detected in terrestrial
experiments will have as an upper bound the polarization given by $s_{||}(r=R_\odot)$. The actual polarization can be lower depending
on the site in the Sun where the neutrino is produced, but it is expected that most of them are produced in the core of the Sun.
If the terrestrial detector uses as detection channel the $\nu$-$e$ elastic scattering, for a neutrino helicity different from $s_{||}=-1$, there
will be a difference of neutrino counts due to the different cross sections in Eqs. (\ref{CrossDirac},\ref{CrossMajorana}).
As we mentioned before, the number of Borexino neutrino counts requires $s_{||}$$<$-0.68. The neutrino helicity does
not change when they are travelling from the Sun to the Earth, hence $s_{||}(r=R_\odot,\mu_\nu)<-0.68$. Solving
for $\mu_\nu$ (see Fig. \ref{Fig4}), we finally obtain a new upper bound, $\mu_\nu<1.4 \times 10^{-12}\mu_B$, for
the neutrino magnetic moment. This is an order of magnitude below the best limits in Table \ref{expdata}, even for the
conservative values we are using in the modeling of the sun magnetic field.
\section{Supernova prospects}\label{supernova}
Simulations of stellar core collapse and supernova explosions have been making impressive progress, including the
description of microphysics inputs, i.e. the development of better numerical models that incorporate the important role
of nuclear and weak interaction physics \cite{Janka:2006fh,Janka:2012wk}. The new codes include the neutrino
transport systematically and from those simulations the average $\nu_e$ spectrum emitted by a SN can be
extracted. This spectrum has the quasi-thermal form~\cite{Keil:2002in}:
\begin{equation}
\frac{dN(E)}{dE}=\frac{(1+\alpha)^{1+\alpha}E_{\rm tot}}
{\Gamma(1+\alpha)\bar E^2}
\left(\frac{E}{\bar E}\right)^{\alpha}e^{-(1+\alpha)E/\bar E}\,,\label{SNflux}
\end{equation}
where we use $\bar E=15$~MeV for the average energy, $\alpha=4$ for
the pinching parameter, and $E_{\rm tot}=5\times10^{52}$~erg for the
total amount of energy emitted in $\nu_e$ ~\cite{Keil:2002in}.
\begin{figure}
\includegraphics[width=0.45\textwidth]{neutrino_differenceSN.eps}
\caption{Number of supernova events as a function of the recoil energy for Dirac or Majorana neutrinos.}\label{Fig5}
\end{figure}
In addition to the incorporation of neutrino transport, recently, the influence of magnetic fields on stellar core collapse
and explosion has been explored.
More important, starting with magnetic fields as strong as those predicted by stellar evolution
($B$$\sim$ $10^9$-$10^{10}$ G), due to turbulent flows,
the magnetic field may undergo kinematic amplification of some orders of magnitude
\cite{Obergaulinger2014,Obergaulinger:2011ic}. In some cases,
magnetic field as strong as $10^{15}$ G can emerge for several hundreds of kilometers \cite{Obergaulinger2014}.
In such violent environment, even a tiny neutrino magnetic moment may induce a change in the neutrino helicity
$s_{||}$ and thus may lead to changes in the number of events which will be different for Dirac or Majorana neutrinos.
Indeed, a straightforward calculation using Eq. \eqref{spinequation} for a neutrino with original helicity $s_{||}=-1$
travelling $100$ km in a constant external magnetic field with strength $B_\perp=10^{15}$ G and $\mu_\nu=10^{-19}\mu_B$,
which is the magnetic moment predicted for a Dirac neutrino with a mass $m_\nu=1$ eV \cite{Fujikawa:1980yx},
yields a final helicity of $s_{||}=-0.55$. We can compute now the number of Majorana or Dirac neutrino events as
\begin{equation}
\frac{dN^{D,M}_{events}}{dT}=A \int dE_\nu \frac{dN(E_\nu)}{dE_\nu} \frac{d^2\sigma^{D,M}(E_\nu,T, S_{||})}{dE_\nu dT},
\end{equation}
where $\frac{dN(E_\nu)}{dE_\nu}$ is the average neutrino spectrum, Eq. \eqref{SNflux}, and the differential neutrino cross
sections are given by Eq. \eqref{CrossMajorana} and Eq. \eqref{CrossDirac} for Majorana or Dirac neutrinos, respectively.
The factor $A$ accounts for the number of targets in the detector, the time interval of the detection and the total neutrino flux,
that depends on the distance from the supernova to the Earth. These parameters are specific of the supernova explosion
event thus we are not able to determine the total number of events. Nevertheless, it is possible to perform a qualitative
analysis of the spectrum in arbitrary units (depending on A) to explore the relative impact of the helicity change due to
the magnetic fields in a supernova explosion for Majorana and Dirac neutrinos. In Fig. \ref{Fig5} we plot the distribution
of the number of events as a function of the electron recoil energy $T$ for $s_{||}=-0.55$ exhibiting clear differences for
Dirac and Majorana neutrinos even for the small value $\mu_\nu=10^{-19}\mu_B$ considered in the calculation.
\section{Conclusions}
Massive neutrinos in an environment with magnetic fields can change their helicity $s_{||}$. On the other hand, the
scattering cross section of Dirac and Majorana neutrinos on electrons are different if $s_{||}\neq -1$ . In this work we point
out that nature offer neutrino sources with intrinsic magnetic fields which can provide a sizable change in the neutrino helicity
and the corresponding difference in the cross section between Dirac and Majorana neutrinos scattering off electrons can be
used to study electromagnetic neutrino properties.
Considering a particular model of the Solar magnetic field under conservative assumptions, and the number of events for
solar neutrinos coming from the $^7Be$ line measured by Borexino Collaboration, it is possible to have an improvement
in the current upper limits of the neutrino magnetic moment of at least one order of magnitude. Furthermore, based on
accurate numerical modeling of a supernova explosion, we estimate the change in the neutrino helicity of supernova
neutrinos and show that, even a magnetic moment as small as the predicted by the standard model for Dirac neutrinos,
$\mu_\nu=10^{-19}\mu_B$, produces a sizable difference, both in the total counting and in the spectra, of supernova neutrinos
detected on Earth using $\nu$-$e$ scattering, for Dirac and Majorana neutrinos.
{\bf Aknowledments:} This work was partially supported by Conacyt projects CB-259228, CB-286651, Conacyt SNI and DAIP.
|
1,116,691,497,524 | arxiv | \section{Weak solutions}
In this section, we extend the
weak solution theory in \cite{DG2} to cover the case that the initial population range $\Omega_0 $ is
unbounded. To our knowledge, little is known for problem \eqref{eqfrfisher} with a general unbounded $\Omega_0$.
For future applications, we consider the following more general problem
\begin{equation}\label{eqmain}
\left \{ \begin{array} {ll} \displaystyle\medskip u_t-d \Delta u=g(x,u) \;\; &
\mbox{for $x \in \Omega (t), \; t>0$},\\
\displaystyle\medskip u=0, \;\; u_t=\mu|\nabla_x u|^2 &\mbox{for $x \in \Gamma (t), \; t>0$}, \\
u(0,x)=u_0 (x) \;\; & \mbox{for $x \in \Omega_0$},
\end{array} \right.
\end{equation}
where $\Omega_0$ is a domain in $\mathbb{R}^N$ with Lipschitz boundary $\partial\Omega_0$, the initial function $u_0 (x)$ satisfies
\begin{equation}\label{assumeu0}
u_0 \in C({\overline {\Omega_0}}) \cap H^1_{loc} (\Omega_0)\cap L^{\infty} (\Omega_0),
\;\; u_0>0 \;\; \mbox{in $\Omega_0$}, \;\;\; u_0=0 \;\;
\mbox{on $\partial \Omega_0$},
\end{equation}
and the reaction term $g(x,u)$ is assumed to satisfy
\begin{equation}\label{assumeg}
\left. \begin{array} {ll}
\hbox{(i)} & \smallskip g(x,u) \hbox{ is continuous for } (x,u) \in \mathbb{R}^N \times [0, \infty), \,\,\\
\hbox{(ii)} &\smallskip g(x,u) \hbox{ is locally Lipschitz in } u \hbox{ uniformly for }x \in \mathbb{R}^N,\,\,\\
\hbox{(iii)} &\smallskip g(x,0) \equiv 0,\,\,\\
\hbox{(iv)} & \hbox{there exists } K>0 \hbox{ such that } g(x,u) \leq Ku \hbox{ for all } x \in \mathbb{R}^N \hbox{ and } u \geq 0.
\end{array} \right\}
\end{equation}
To describe the weak formulation of \eqref{eqmain}, it is convenient to start by considering $0<t\leq T$ for some arbitrarily given $T \in (0,
\infty)$. As in \cite{DG2}, the idea is to consider the extended $u$ in the bigger region
$H_T:=[0, T] \times \mathbb{R}^N$ by defining $u(t,x)=0$ for $x \in \mathbb{R}^N \backslash \Omega (t)$, $0\leq t\leq T$, and regard it as a weak solution of an associated equation with certain jumping discontinuity.
Throughout this section, we denote
\begin{equation*
\alpha (w)=\left \{\begin{array}{ll}\displaystyle\medskip w \;\;\;&\mbox{if $w>0$},\\
w-d \mu^{-1} \;\;\; & \mbox{if $w \leq 0$},
\end{array} \right.
\end{equation*}
and
\begin{equation*
{\widetilde u}_0 (x)=\left \{ \begin{array}{ll} \displaystyle\medskip u_0 (x) \;\;\;
& \mbox{for $x \in \Omega_0$},\\
0 \;\;\; &\mbox{for $x \in \mathbb{R}^N \backslash \Omega_0$}.
\end{array} \right.
\end{equation*}
\begin{defi} \label{defweak}
Suppose that $u_0$ satisfies \eqref{assumeu0} and $g$
satisfies \eqref{assumeg}. A nonnegative function $u \in
H^1_{loc} (H_T) \cap L^{\infty}(H_T)$ is called a {\it weak} solution of
\eqref{eqmain} over $H_T$ if
\begin{equation}\label{eqweak}
\displaystyle\medskip \int_0^T \int_{\mathbb{R} ^N} \Big[d \nabla_x u \cdot \nabla_x
\phi-\alpha (u) \phi_t\Big] dx dt-\int_{\mathbb{R} ^N} \alpha ({\widetilde
u}_0) \phi (0,x) dx=\int_0^T \int_{\mathbb{R} ^N} g(x,u) \phi dxdt
\end{equation}
for every function $\phi \in C^{\infty}(H_T)$ such that $\phi$ has compact support\footnote{This means that there exists a ball $B_R$ in $\mathbb{R}^N$ such that
$\phi(t,x)=0$ for $t\in [0, T]$ and $x\in\mathbb{R}^N\setminus B_R$.} and
$\phi=0$ on $\{T\} \times \mathbb{R}^N$.
\end{defi}
Correspondingly, if $u$ satisfies \eqref{eqweak} with ``=" replaced by ``$\geq$ " (resp. ``$\leq$") for every test function $\phi \in C^{\infty}(H_T)$ satisfying $\phi\geq 0$ in $H_T$, $\phi=0$ on $\{T\} \times \mathbb{R}^N$ and $\phi$ has compact support, then we call it a {\it weak supersolution} (resp. {\it weak subsolution}) of \eqref{eqmain} over $H_T$.
Moreover, as in \cite{DG2,Fr1}, for each weak solution (or weak supersolution, or weak subsolution) $u(t,x)$, the function
$\alpha (u(t,x))$ is defined as $u(t,x)$ if $u(t,x)>0$; at points
where $u(t,x)=0$ the function $\alpha (u(t,x))$ is only required to
satisfy $-d \mu^{-1} \leq \alpha (u(t,x)) \leq 0$ and to be such
that it is altogether a measurable function over $H_T$. However, if $w(x)$ is
continuous and positive in $\Omega_0$ and identically zero in $\mathbb{R}^N
\backslash \Omega_0$, then we understand that $\alpha (w)=-d \mu^{-1}$ on $\mathbb{R}^N
\backslash \Omega_0$.
\begin{rem}
Definition \ref{defweak} is an extension of \cite[Definition 2.1]{DG2} for weak solution of problem \eqref{eqmain} with bounded $\Omega_0$ to the
case that $\Omega_0$ is generally unbounded. Indeed, the choice of the test function in \eqref{eqweak} implies that if $\Omega_0$
is bounded and if $u$ is a weak solution to \eqref{eqmain} over $H_T$ by Definition \ref{defweak}, then the restriction of $u$
to $G\times [0,T]$ is a weak solution of \eqref{eqmain} over $G\times [0,T]$ by \cite[Definition 2.1]{DG2}, where $G$ is any given
bounded domain in $\mathbb{R}^N$ such that $\Omega(t)$ stays inside $G$ for $0\leq t\leq T$.
\end{rem}
By a {\it classical solution} of problem \eqref{eqmain} for $0<t\leq T<\infty$, we mean a pair $(u(t,x), \Omega(t))$ such that
$\bigcup_{0\leq t\leq T}\partial\Omega(t)$ is a $C^1$ hypersurface in $\mathbb{R}^{N+1}$, $u$, $\nabla_x u$ are continuous in $\bigcup_{0\leq t\leq T}\overline{\Omega(t)}$, and $u_t$, $u_{x_ix_j}$ are continuous in $\bigcup_{0<t\leq T}\Omega(t)$, and all the equations in \eqref{eqmain} are satisfied in the classical sense.
In this case, there exists $\Phi\in C^1\left(\overline{\bigcup_{0<t\leq T}\Omega(t)}\right)$ such that
$$\Phi(t,x)=0\; \hbox{ on } \Gamma (t),\;\;\; \nabla_x \Phi (x,t)\neq 0 \;\; \mbox{on $\Gamma (t)$}, \;\;\; \Phi
(t,x)<0 \;\;\; \mbox{in $\Omega (t)$},$$
and it follows from
\[
u=0 \mbox{ and } u_t=\mu|\nabla_x u|^2 \mbox{ on } \Gamma(t)
\]
that
\[
\Phi_t=\mu \nabla_x u\cdot \nabla_x \Phi \mbox{ for } x\in \Gamma(t).
\]
\begin{thm} \label{weakclassic}
{\rm (i)} Assume that $(u(t,x), \Omega(t))$ is a classical solution of \eqref{eqmain} for $0<t\leq T$.
Then
$$ w(t,x):=\left \{ \begin{array}{ll} \displaystyle\medskip u(t,x) \;\;\; &\mbox{for $x
\in \Omega (t)$, $0<t \leq T$}, \\
\displaystyle 0 \;\;\; &\mbox{for $x \in \mathbb{R}^N \backslash \Omega (t)$, $0<t\leq T$}
\end{array} \right.$$
is a weak solution of \eqref{eqmain} over $H_T$.
{\rm (ii)} Let $w$ be a weak solution of \eqref{eqmain} on $H_T$. Assume that
there exists a $C^1$ function $\Phi$ in ${H_T}$
satisfying
$$\Omega (t): = \Big\{x \in \mathbb{R}^N: \; w(t,x)>0\Big\}=\Big\{x \in \mathbb{R}^N: \; \Phi
(t,x)<0\Big\}$$
with $\Omega (0)=\Omega_0$, and
$$\nabla_x \Phi \neq 0 \;\;\; \mbox{on\; $\Gamma (t) := \partial
\Omega (t)$}.$$
Setting $u=w$ in $\bigcup_{0<t\leq T} {\overline {\Omega (t)}}$, and
assume that $u$, $\nabla_x u$ are continuous in $\bigcup_{0 \leq
t\leq T} {\overline {\Omega (t)}}$ and that $\nabla_x^2 u$, $u_t$ are
continuous in $\bigcup_{0<t\leq T} \Omega (t)$. Then $(u(t,x),
\Omega(t))$ is a classical solution of \eqref{eqmain} for $t\in (0, T]$.
\end{thm}
\begin{proof} The proof follows that of \cite[Theorem 2.3]{DG2} with some obvious modifications, and we omit the details.
\end{proof}
Next we prove the existence and uniqueness of weak solution to problem \eqref{eqmain}. The strategies of the proof are adapted from the approximation method of \cite{Ka} as used in \cite{DG2, Fr1}, but new techniques are required to handle difficulties arising from the unboundedness of $\Omega_0$.
We begin with the uniqueness part, since the existence proof will use some of the arguments in the uniqueness proof.
\begin{thm}\label{unique}
Suppose that $\mu_1 \geq \mu_2>0$, $u_1$ is a weak supersolution of \eqref{eqmain} over $H_T$ with $\mu=\mu_1$ and $u_2$ is a weak subsolution with $\mu=\mu_2$,
where the initial data $u_0$ and $\Omega_0$ are shared. Then $u_1 \geq u_2$ a.e. in $H_T$. In particular, problem \eqref{eqmain} can have at most one weak solution over $H_T$.
\end{thm}
\begin{proof}
We complete the proof in three steps.
\smallskip
\noindent
{\bf Step 1:} {\it Setup of the approximation method.}
With $u_1$ and $u_2$ as given in the statement of the theorem, we define
$$
\ell (t,x)=\left \{ \begin{array}{ll} \displaystyle\medskip\frac{g(x,u_2 (t,x))-g(x,
u_1 (t,x))}{u_2 (t,x)-u_1 (t,x)} \;\;& \mbox{if $u_2 (t,x) \neq u_1
(t,x)$}, \\
0 \;\;& \mbox{if $u_2 (t,x)=u_1 (t,x)$},
\end{array} \right.
$$
and for $i=1,2$, let $\alpha_i(u)$ denote $\alpha(u)$ with $\mu=\mu_i$, and
$$
e (t,x)=\left \{ \begin{array}{ll} \displaystyle\medskip\frac{u_2 (t,x)-u_1
(t,x)}{\alpha_2 (u_2 (t,x))-\alpha_1 (u_1 (t,x))} \;\;\; &\mbox{if
$u_2 (t,x) \neq u_1 (t,x)$},\\
0 \;\;\; &\mbox{if $u_2 (t,x)=u_1 (t,x)$}.
\end{array} \right.
$$
We then have
\begin{equation}\label{difference}
\int_0^T \int_{\mathbb{R}^N} \big[\alpha_2(u_2)- \alpha_1(u_1)\big]\big(\phi_t+de\Delta \phi +e\ell\phi \big) dxdt \geq d(\mu_2^{-1}- \mu_1^{-1})\int_{\mathbb{R}^N \backslash {\overline {\Omega_0}}} \phi(0,x) dx
\end{equation}
for every nonnegative $\phi \in C^{\infty}(H_T)$ such that $\phi$ has compact support, and
$\phi=0$ on $\{T\} \times \mathbb{R}^N$.
It is easily checked that if we write
$$\alpha_2 (u_2 (t,x))-\alpha_1 (u_1 (t,x))=\bar{\alpha}(t,x)\big[u_2(t,x)-u_1(t,x)\big]$$ when $u_2(t,x)\neq u_1(t,x)$,
then $\bar{\alpha}(t,x) \geq 1 $ for almost all $(t,x)\in H_T$. And hence,
\begin{equation}\label{bde}
0 \leq e(t,x) \leq 1 \;\; \mbox{for almost all $(t,x)
\in H_T$}.
\end{equation}
Moreover, by the condition \eqref{assumeg} on $g$ and the fact that $u_1, u_2 \in
L^\infty (H_T)$, it follows that $\ell \in L^{\infty}(H_T)$. We then approximate $e$ and $\ell$ by smooth functions $e_m\in C^{\infty}(H_T)$ and $\ell_m\in C^{\infty}(H_T)$, respectively, such that for any $R>0$,
\begin{equation}\label{appfns}
\| e - e_m\|_{L^2([0,T]\times B_R)}\to 0,\quad \| \ell - \ell_m\|_{L^2([0,T]\times B_R)}\to 0\,\,\hbox{ as } m\to\infty,
\end{equation}
and that
\begin{equation}\label{appcond}
\inf_{H_T} e_m\geq \frac{1}{m}, \quad \|e_m \|_{L^{\infty}(H_T)}\leq C_1,\quad \|\ell_m \|_{L^{\infty}(H_T)}\leq C_1,
\end{equation}
for some positive constant $C_1$ independent of $m$. Here and in what follows, we use $B_R$ to denote the ball with center the origin and radius $R$.
Now, let $R_0>0$ be fixed, and choose a nonnegative function $f \in C^{\infty}(H_T)$ with $f(t,x)=0$ for all $|x|\geq R_0$ and $0\leq t \leq T$. For any $m\geq 1$ and $R> R_0+1$, we consider the following backward parabolic equation
\begin{equation} \label{backward} \left \{ \begin{array}{ll}
\displaystyle\medskip\frac{\partial \phi}{\partial t}+d e_m \Delta \phi+e_m \ell_m \phi=f \;\;\;&
\mbox{in $[0,T)\times B_R$},\\
\displaystyle\medskip \phi=0 \;\;\; &\mbox{on $\{T\} \times B_R$},\\
\phi=0 \;\; &\mbox{on $[0,T] \times \partial B_R$}.
\end{array} \right.
\end{equation}
This is a nondegenerate problem and it has a unique smooth solution $\phi_m$ (see e.g., \cite{LSU}). Furthermore, the parabolic maximum principle (applied to $\phi_m(T-t,x)$) implies that
$$\phi_m\leq 0 \hbox{ in } [0,T]\times B_R.$$
Next, for $0<\varepsilon\ll 1$, we take a cutoff function $\xi_{\varepsilon}\in C_0^{\infty}(\mathbb{R}^N)$ such that
\begin{equation}\label{modifier1}
0\leq \xi_{\varepsilon} \leq 1 \hbox{ in } \mathbb{R}^N, \quad \xi_{\varepsilon} =1 \hbox{ in } B_{R-2\varepsilon}, \quad \xi_{\varepsilon} =0 \hbox{ in } \mathbb{R}^N \backslash B_{R-\varepsilon}, \
\end{equation}
and that
\begin{equation}\label{modifier2}
\| \nabla \xi_{\varepsilon} \|_{L^{\infty}(\mathbb{R}^N)} \leq C_2\varepsilon^{-1},\quad \|\Delta \xi_{\varepsilon} \|_{L^{\infty}(\mathbb{R}^N)} \leq C_2\varepsilon^{-2}
\end{equation}
for some positive constant $C_2$ independent of $\varepsilon$. Then for any $m>1$ and $0<\varepsilon \ll 1$, define
$$ \psi_m^{\varepsilon} (t,x)=\left \{\begin{array}{ll} \displaystyle\medskip\phi_m(t,x)\xi_{\varepsilon}(x) \;\;\;&\mbox{in $[0,T]\times B_R $},\\
0 \;\;\; & \mbox{in $[0,T]\times (\mathbb{R}^N \backslash B_{R})$}.
\end{array} \right.$$
Clearly, $\psi_m^{\varepsilon}\leq 0$, it belongs to $ C^{\infty}(H_T)$, and vanishes in $(\{T\} \times \mathbb{R}^N)\cup ([0,T]\times (\mathbb{R}^N \backslash B_{R}))$.
We may now take $-\psi_m^{\varepsilon}$ as a text function in \eqref{difference} to obtain, due to $\mu_1\geq \mu_2>0$,
\begin{equation*}
\int_0^T \int_{\mathbb{R}^N} \big[\alpha_2(u_2)- \alpha_1(u_1)\big]\Big((\psi_m^{\varepsilon})_t+de\Delta \psi_m^{\varepsilon} +e\ell\psi_m^{\varepsilon} \Big) dxdt \leq 0.
\end{equation*}
Since $\xi_{\varepsilon}f\equiv f$ in $H_T$, it then follows that
\begin{equation}\label{mainesti}
\left.\begin{array}{ll}
& \displaystyle \medskip \int_0^T \int_{\mathbb{R}^N}\big[\alpha_2(u_2)- \alpha_1(u_1)\big] f dxdt \\
=&\displaystyle \medskip \int_0^T \int_{B_R}\big[\alpha_2(u_2)- \alpha_1(u_1)\big]\Big( \frac{ \partial\phi_m}{\partial t}+d e_m \Delta \phi_m+e_m \ell_m \phi_m\Big) \xi_\varepsilon dx dt\\
\leq & I_m + J_m + K_m,
\end{array}\right.
\end{equation}
where
$$I_m=I_m(\varepsilon):= \int_0^T \int_{B_R}\big[\alpha_2(u_2)- \alpha_1(u_1)\big] d\xi_{\varepsilon}(e_m-e)\Delta\phi_m dxdt, $$
$$J_m=J_m(\varepsilon):= \int_0^T \int_{B_R}\big[\alpha_2(u_2)- \alpha_1(u_1)\big] \xi_{\varepsilon}(e_m\ell_m-e\ell)\phi_m dxdt,$$
and
$$K_m=K_m(\varepsilon):= \int_0^T \int_{B_R}\big[\alpha_2(u_2)- \alpha_1(u_1)\big] \Big(-de\phi_m\Delta \xi_{\varepsilon}-2de\nabla \phi_m\cdot\nabla \psi_{\varepsilon}\Big)dxdt.$$
Our aim is to show, through suitable estimates on $I_m,\; J_m$ and $K_m$, that
\[
\int_0^T \int_{\mathbb{R}^N}\big[\alpha_2(u_2)- \alpha_1(u_1)\big] f dxdt\leq 0,
\]
and the conclusions of the theorem would then follow easily from this inequality.
\smallskip
\noindent
{\bf Step 2:} {\it Estimates of $I_m$, $J_m$ and $K_m$.}
We first consider $I_m$. Let
$$C_3:=\|u_1\|_{L^{\infty}(H_T)}+\|u_2\|_{L^{\infty}(H_T)}+d\mu_1^{-1}+d\mu_2^{-1}.$$
By the H\"{o}lder inequality, we have
\begin{equation*}
\left.\begin{array}{ll}
I_m\!\! &\displaystyle\medskip \leq dC_3 \int_0^T\int_{B_R} |e_m-e||\Delta\phi_m|dxdt\\
&\displaystyle\medskip \leq dC_3 \Big(\int_{0}^T\int_{B_R} \frac{|e_m-e|^2}{|e_m|}dxdt\Big)^{\frac{1}{2}}\Big(\int_{0}^T\int_{B_R} |e_m||\Delta\phi_m|^2 dxdt\Big)^{\frac{1}{2}}\\
&\displaystyle\medskip \leq dC_3 \big\| e_m-e\big\|_{L^2([0,T]\times B_R)}^{\frac{1}{2}} \Big(\int_{0}^T\int_{B_R} \frac{|e_m-e|^2}{|e_m|^2}dxdt\Big)^{\frac{1}{4}}\Big(\int_{0}^T\int_{B_R} |e_m||\Delta\phi_m|^2 dxdt\Big)^{\frac{1}{2}}.
\end{array}\right.
\end{equation*}
It follows from the proof of \cite[Lemma 3.7]{DG2} that there is a positive constant $C_4=C_4(T,f,R)$ such that
$$\Big(\int_{0}^T\int_{B_R} |e_m||\Delta\phi_m|^2 dxdt\Big)^{\frac{1}{2}} \leq C_4.$$
Moreover, by the same analysis as that used in the proof of \cite[Lemma 5]{CH}, we may require that the approximation sequence $e_m$ satisfies
additionally
$$
\Big\| \frac{e}{e_m}\Big\|_{L^2([0,T]\times B_R)} \leq C_5
$$
for some positive constant $C_5=C_5(T,R).$ We thus obtain
\begin{equation}\label{estiim}
I_m\leq C_6 \big\| e_m-e\big\|_{L^2([0,T]\times B_R)}^{\frac{1}{2}},
\end{equation}
with $C_6=d(T|B_R| +C_5^2)^{1/4}C_3C_4$ independent of $m$.
Next, it is easily seen that
\begin{equation*}
\left.\begin{array}{ll}
\medskip J_m\;\leq \;C_3\|\phi_m \|_{L^{\infty}([0,T]\times B_{R})}
&\!\!\!\!\Big(\| e_m\|_{L^2([0,T]\times B_R)} \big\| \ell_m-\ell\big\|_{L^2([0,T]\times B_R)}\\
&\;\;\;\;\;\; + \| \ell \|_{L^2([0,T]\times B_R)} \big\| e_m-e\big\|_{L^2([0,T]\times B_R)} \Big).
\end{array}\right.
\end{equation*}
By the comparison arguments used in \cite[Lemma 3.6]{DG2}, we have
$$\|\phi_m \|_{L^{\infty}([0,T]\times B_{R})} \leq C_7,$$
for some positive constant $C_7=C_7(T,f)$. Hence, by setting $C_8=(T|B_R|)^{1/2}C_1C_3C_7$, we obtain
\begin{equation}\label{estijm}
J_m\leq C_8\Big( \big\| \ell_m-\ell\big\|_{L^2([0,T]\times B_R)} + \big\| e_m-e\big\|_{L^2([0,T]\times B_R)} \Big).
\end{equation}
It remains to estimate $K_m$. Making use of \eqref{bde}, \eqref{modifier1} and \eqref{modifier2}, we have
\begin{equation}\label{estikmfr}
\left.\begin{array}{ll}
K_m &\displaystyle\medskip \leq dC_3 \int_0^T\int_{B_R}\Big( \big|\phi_m\Delta \xi_{\varepsilon}\big| + 2\big| \nabla \phi_m\cdot\nabla \xi_{\varepsilon}\big|\Big)dxdt \\
&\displaystyle\medskip \leq dC_2C_3 \Big(\int_{0}^T\int_{B_{R-\varepsilon}\backslash B_{R-2\varepsilon}} \Big( \frac{|\phi_m|}{\varepsilon^2}+ \frac{2|\nabla \phi_m|}{\varepsilon}\Big)dxdt.
\end{array}\right.
\end{equation}
Since $\phi_m(t,x)=0$ on $[0,T]\times \partial B_R$, it follows by letting $\varepsilon \to 0$ in \eqref{estikmfr} that
\begin{equation}\label{estikmau}
\limsup_{\varepsilon\to 0} K_m(\varepsilon) \leq C_9 R^{N-1} \sup_{0\leq t\leq T,\,x\in\partial B_R} \Big|\frac{\partial \phi_m(t,x)}{\partial \nu_x} \Big|,
\end{equation}
where $C_9=C_9(T,C_2,C_3)$ is a positive constant independent of $m$ and $R$, and
$\nu_x$ is the outward unit normal vector of $B_R$ at $x\in \partial B_R$.
We next estimate $\partial \phi_m(t,x)/\partial \nu_x$ at the boundary $\partial B_R$ by making use of barrier
functions inspired by the proof of \cite[Theorem 2.1]{DV}. Let
$$v(t,x):=-C_{10}\mathrm{e}^{-\beta_1t}(1+|x|^2)^{-\beta_2} \,\hbox{ for } (t,x)\in [0,T]\times B_R, $$
where $C_{10}$, $\beta_1$ and $\beta_2$ are positive constants independent of $R$, to be chosen later.
It is straightforward to verify that
$$v_t=-\beta_1 v\quad \hbox{and} \quad\Delta v \geq 4(\beta_2^2+\beta_2)v \,\,\hbox{ for } (t,x)\in [0,T]\times B_R, $$
and thus,
\begin{equation*}
v_t+d e_m \Delta v+e_m \ell_m v\geq -v \Big (\beta_1-4(\beta_2^2+\beta_2)de_m-e_m\ell_m \Big) \,\hbox{ for } (t,x)\in [0,T]\times B_R.
\end{equation*}
Due to \eqref{appcond} and the fact that $f(t,x)=0$ for all $|x|\geq R_0$ and $0\leq t \leq T$, we may choose
$$\beta_1= 4(\beta_2^2+\beta_2)dC_1+C_1^2+1,$$
and
$$ C_{10}=\mathrm{e}^{\beta_1 T}\big(1+R_0^2\big)^{\beta_2} \max_{(t,x)\in [0,T]\times B_{R_0} } |f(t,x)|,$$
and thus conclude that
\begin{equation*} \left \{ \begin{array}{ll}
\medskip v_t+d e_m \Delta v+e_m \ell_m v \geq f \;\;\;&
\mbox{in $[0,T)\times B_R$},\\
\displaystyle\medskip v<0 \;\;\; &\mbox{on $\{T\} \times B_R$},\\
v < 0 \;\; &\mbox{on $[0,T] \times \partial B_R$}.
\end{array} \right.
\end{equation*}
It then follows from the parabolic maximum principle applied to problem \eqref{backward} that
\begin{equation}\label{phimup}
v(t,x) \leq \phi_m(t,x)\leq 0 \,\hbox{ for } (t,x)\in [0,T)\times B_R.
\end{equation}
We next consider the function
$$v^*(t,x):=\frac{-C_{10}\mathrm{e} ^{-\beta_1 t} \sigma(|x|)}{\sigma(R-1)\big(1+(R-1)^2\big)^{\beta_2}}\,\, \hbox{ for } (t,x)\in [0,T]\times B_R\backslash B_{R-1}, $$
where
$$ \sigma(\rho)=\left \{\begin{array}{ll} \displaystyle\medskip
\rho^{2-N}-R^{2-N}\;\;\;&\mbox{ if } N\geq 3,\\
\log {\big(R/\rho\big)} \;\;\; & \mbox{ if } N=2.
\end{array} \right.$$
The function $v^*(t,x)$ satisfies
$\Delta v^*=0$ and $$v^*_t+e_m \ell_m v^*=-v^*(\beta_1-e_m\ell_m)\geq 0 \mbox{
in $[0,T)\times B_R\backslash B_{R-1}$.}
$$
Since $f(t,x)=0$ for all $0\leq t\leq T$ and $x\in B_R\backslash B_{R-1}$ (note that $R>R_0+1$), it then follows that
$$ v^*_t+d e_m \Delta v^*+e_m \ell_m v^* \geq f \, \hbox{ in $[0,T)\times B_R\backslash B_{R-1}$}.$$
On the other hand, we have, in view of \eqref{phimup},
\begin{equation*} \left \{ \begin{array}{ll}
\displaystyle\medskip v^*<0 \;\;\; &\mbox{on $\{T\} \times B_R\backslash B_{R-1}$},\\
\displaystyle\medskip v^* = 0 \;\; &\mbox{on $[0,T] \times \partial B_R$},\\
v^* =-C_{10}\mathrm{e}^{-\beta_1 t}\big(1+(R-1)^2\big)^{-\beta_2}=v \leq \phi_m \;\; &\mbox{on $[0,T] \times \partial B_{R-1}$}.
\end{array} \right.
\end{equation*}
It then follows from the parabolic maximum principle again that
$$ v^*(t,x) \leq \phi_m(t,x)\leq 0 \,\,\hbox{ for } (t,x)\in [0,T]\times B_R\backslash B_{R-1}.$$
Since both $\phi_m$ and $v^*$ vanish on $\partial B_R$, we thus obtain
$$\frac{\partial v^*(t,x)}{\partial \nu_x} \geq \frac{\partial \phi_m(t,x)}{\partial \nu_x}\geq 0\,\, \hbox{ for } (t,x)\in [0,T] \times \partial B_{R}.$$
This together with the estimate \eqref{estikmau} implies that
\begin{equation*}
\left. \begin{array}{ll}
\displaystyle\medskip
\limsup_{\varepsilon\to 0} K_m(\varepsilon) &\displaystyle\medskip \leq \,\,C_9 R^{N-1} \sup_{0\leq t\leq T,\,x\in\partial B_R} \Big|\frac{\partial v^*(t,x)}{\partial \nu_x} \Big|\\
&\,\,\displaystyle\medskip = \frac{C_9C_{10}R^{N-1}|\sigma'(R)| }{\sigma(R-1)\big(1+(R-1)^2\big)^{\beta_2}}.
\end{array} \right.
\end{equation*}
Thus, if we take $\beta_2=N/2$, then there exists some positive constant $C_{11}=C_{11}(N, C_9, C_{10})$ independent of $R$ and $m$ such that
\begin{equation}\label{estikm}
\limsup_{\varepsilon\to 0} K_m (\varepsilon) \leq C_{11} R^{-1}.
\end{equation}
\smallskip \noindent
{\bf Step 3:} {\it Completion of the proof.}
Combining the estimates in \eqref{estiim}, \eqref{estijm} and \eqref{estikm}, we obtain from \eqref{mainesti} that
\begin{equation*}
\left.\begin{array}{ll}
& \displaystyle \medskip \int_0^T \int_{\mathbb{R}^N}\big[\alpha_2(u_2)- \alpha_1(u_1)\big] f dxdt \\
\leq &\displaystyle \medskip C_6 \big\| e_m-e\big\|_{L^2([0,T]\times B_R)}^{\frac{1}{2}} + C_8\Big( \big\| \ell_m-\ell\big\|_{L^2([0,T]\times B_R)} + \big\| e_m-e\big\|_{L^2([0,T]\times B_R)} \Big) + C_{11}R^{-1}.
\end{array}\right.
\end{equation*}
In view of \eqref{appfns} and the facts that $C_6,\,C_8$ depend on $R$ but not on $m$ and that $C_{11}$ is independent of $m$ and $R$, passing to the limit as $m\to\infty$ followed by letting $R\to\infty$ in the above inequality, we obtain
\begin{equation*}
\int_0^T \int_{\mathbb{R}^N}\big[\alpha_2(u_2)- \alpha_1(u_1)\big] f dxdt \leq 0.
\end{equation*}
Since it holds for all nonnegative smooth function $f(t,x)$ with compact support in $x$, it follows that $\alpha_2(u_2)\leq \alpha_1(u_1)$, and hence in the a.e. sense $u_1(t,x)>0$ whenever $u_2(t,x)>0$, and $u_1\geq u_2$ a.e. in $H_T$. The proof of Theorem \ref{unique} is now complete.
\end{proof}
We now consider the existence of weak solutions of problem \eqref{eqmain} over $H_T$.
\begin{thm}\label{existence}
There exists a unique weak solution $w$ of \eqref{eqmain} over $H_T$.
\end{thm}
The proof of this theorem also follows the approximation arguments used in \cite{DG2,Fr1} and we only provide the details where considerable changes are required. Before giving the proof, we first introduce some notations and approximation functions.
Let $\{\alpha_m (w)\}_{m\in\mathbb{N}}$ be a sequence of smooth functions such that
\begin{equation*
\alpha_m (w) \to \alpha (w) \hbox{ uniformly in any compact subset of } \mathbb{R}^1 \backslash \{0\},
\end{equation*}
and
\begin{equation*
\alpha_m (0) \to -d \mu^{-1}, \quad w-d \mu^{-1} \leq \alpha_m (w) \leq w \hbox{ for all } w \in \mathbb{R}^1.
\end{equation*}
We may choose $\alpha_m (w)$ in such a way that
\begin{equation}\label{restriction}
\alpha_m'(w) \geq 1.
\end{equation}
We now consider the following sequence of approximating problems:
\begin{equation}\label{exappx}
\left \{ \begin{array}{ll}
\displaystyle\medskip\frac{\partial \alpha_m
(w)}{\partial t}-d \Delta w=g(x, w) \;\;\;& \mbox{in $(0,T]\times \mathbb{R}^N$},\\
\displaystyle w (0,x)={\widetilde u}_0 (x) \;\;\; &\mbox{in $\mathbb{R}^N$}.
\end{array} \right.
\end{equation}
For any fixed $m\in\mathbb{N}$, we call a function ${w}$ a {\it bounded supersolution} to problem \eqref{exappx} if
${w}\in C^{1,2}((0,T]\times\mathbb{R}^N)\cap C(H_T)\cap L^{\infty}(H_T)$ and it satisfies
\begin{equation}\label{exappsup}
\left \{ \begin{array}{ll}
\displaystyle\medskip\frac{\partial \alpha_m
({w})}{\partial t}-d \Delta {w}\geq g(x, {w}) \;\;\;& \mbox{in $(0,T]\times \mathbb{R}^N$},\\
\displaystyle {w}(0,x)\geq {\widetilde u}_0 (x) \;\;\; &\mbox{in $\mathbb{R}^N$}.
\end{array} \right.
\end{equation}
Such a $w$ is called a {\it bounded subsolution} to problem \eqref{exappx} if the reversed inequalities in \eqref{exappsup} are satisfied. We have the following comparison result.
\begin{lem}\label{exappcom}
Let $m\in \mathbb{N}$ be fixed.
Assume that $\overline{w}$ and $\underline{w}$ are bounded supersolution and subsolution of problem \eqref{exappx}, respectively. Then $\overline{w}\geq \underline{w}$ in $H_T$.
\end{lem}
\begin{proof}
It is easily verified from \eqref{exappx} that $\overline{w}$ satisfies
\begin{equation}\label{weakapp}
\left.\begin{array}{ll}
\displaystyle\medskip \int_0^T \int_{\mathbb{R} ^N} \Big[d \nabla_x \overline{w} \cdot \nabla_x\phi-\alpha_m (\overline{w}) \phi_t\Big] dx dt
\!\!& \displaystyle\medskip-\,\int_{\mathbb{R} ^N}\alpha_m(\overline{w}(0,x))\phi (0,x) dx\\
\!\!& \displaystyle\medskip \geq\, \int_0^T \int_{\mathbb{R} ^N} g(x,\overline{w}) \phi dx dt
\end{array}\right.
\end{equation}
for every nonnegative function $\phi \in C^{\infty}(H_T)$ such that $\phi$ has compact support and
$\phi=0$ on $\{T\} \times \mathbb{R}^N$. Analogously, $\underline{w}$ satisfies \eqref{weakapp} with the inequality sign reversed. Subtracting these two inequalities, due to \eqref{restriction} and the fact that $\overline{w}(0,x)\geq \underline{w}(0,x)$ in $\mathbb{R}^N$, we obtain
\begin{equation*}
\int_0^T \int_{\mathbb{R}^N} \big[\alpha_m(\underline{w})- \alpha_m(\overline{w})\big]\big(\phi_t+d\widetilde{e}\Delta \phi +\widetilde{e}\widetilde{\ell}\phi \big) dxdt \geq 0,
\end{equation*}
where
$$ \widetilde{\ell} (t,x)=\left \{ \begin{array}{ll} \displaystyle\medskip\frac{g(x,\underline{w} (t,x))-g(x,
\overline{w} (t,x))}{\underline{w} (t,x)-\overline{w}
(t,x) (t,x)} \;\;& \mbox{if $\underline{w}(t,x) \neq \overline{w}
(t,x)$}, \\
0 \;\;& \mbox{if $\underline{w} (t,x)=\overline{w} (t,x)$},
\end{array} \right.$$
and
$$\widetilde{e}(t,x)=\left \{ \begin{array}{ll} \displaystyle\medskip\frac{\underline{w} (t,x)-\overline{w}
(t,x)}{\alpha_m (\underline{w} (t,x))-\alpha_m (\overline{w} (t,x))} \;\;\; &\mbox{if
$\underline{w} (t,x) \neq \overline{w} (t,x)$},\\
0 \;\;\; &\mbox{if $\underline{w} (t,x)=\overline{w} (t,x)$}.
\end{array} \right.$$
Since $\overline{w},\,\underline{w} \in L^{\infty}(H_T)$ and since $\alpha_m'(w) \geq 1$ by \eqref{restriction}, it then follows from the same arguments as those used in the proof of Theorem \ref{unique} that
$$\alpha_m(\underline{w}(t,x))- \alpha_m(\overline{w}(t,x)) \leq 0 \hbox{ for almost all } (t,x)\in H_T.$$ This immediately gives
$$\underline{w}(t,x) \leq \overline{w}(t,x) \hbox{ for all } (t,x)\in H_T, $$
as $\alpha_m(w)$, $\underline{w}(t,x)$ and $\overline{w}(t,x)$ are all continuous functions, and $\alpha_m(w)$ is increasing in $w$. The proof of Lemma \ref{exappcom} is thus complete.
\end{proof}
Clearly, $\underline{w}\equiv 0$ is a subsolution of \eqref{exappx}. On the other hand, by the condition \eqref{assumeg} on $g$ and the property \eqref{restriction} of $\alpha_m$, we can conclude that the function $\overline{w}(t)$ is a bounded supersolution of \eqref{exappx}, where
\begin{equation}\label{deuperw}
\overline{w}(t)=\|\widetilde{u}_0\|_{L^{\infty}(\mathbb{R}^N)} \mathrm{e}^{Kt} \, \hbox{ for } 0\leq t\leq T
\end{equation}
is the unique solution to the problem
$$\frac{d \overline{w}}{d t} =K \overline{w} \,\hbox{ for } 0<t<T,
\quad \overline{w}(0)=\|\widetilde{u}_0\|_{L^{\infty}(\mathbb{R}^N)}.$$
We are ready to show the existence and uniqueness of classical solution to \eqref{exappx}.
\begin{lem}\label{exapplem}
Let $m\in \mathbb{N}$ be fixed and $\overline{w}(t)$ be given as in \eqref{deuperw}. Then problem \eqref{exappx} admits a unique solution $w_m\in C^{1,2}((0,T]\times \mathbb{R}^N)\cap C(H_T)\cap L^\infty(H_T)$. Moreover,
\begin{equation}\label{3.24}
\mbox{$0\leq w_m(t,x) \leq \overline{w}(t)$ for $(t,x)\in H_T$,}
\end{equation}
and for any $R>0$, there exists a positive constant $\widetilde{C}_1=\widetilde{C}_1(T,R)$ independent of $m$ such that
\begin{equation}\label{exh1n}
\| w_m\|_{H^1([0,T]\times B_R)} \leq \widetilde{C}_1\; \hbox{ for all } m \in \mathbb{N}.
\end{equation}
\end{lem}
\begin{proof}
The existence part must be a known result, but we failed to find a proof in the literature. So we include a proof here.
Note that the uniqueness part follows directly from Lemma \ref{exappcom}.
For each $k\in\mathbb{N}$, let $B_{k} \in\mathbb{R}^N$ be the ball with center at $0$ and radius $k$. Consider the following initial boundary value problem
\begin{equation}\label{exapptru}
\left \{ \begin{array}{ll}
\displaystyle\medskip\frac{\partial \alpha_m
(v)}{\partial t}-d \Delta v=g(x, v) \;\;\;& \mbox{in $(0,T]\times B_k$},\\
\displaystyle\medskip v(t,x)=0 \;\;\; &\mbox{on $(0,T]\times \partial B_k$},\\
\displaystyle v (0,x)={\widetilde u}_0 (x) \;\;\; &\mbox{in $B_k$}.
\end{array} \right.
\end{equation}
It is well known that, for each $k\in\mathbb{N}$, problem \eqref{exapptru} admits a unique solution $v_k \in C([0,T]\times {B_k})\cap C^{1,2}((0,T]\times \overline B_k)$ (see e.g., \cite{LSU}). Moreover, it follows
from the comparison result given in \cite[Lemma 3.2]{DG2} that
\begin{equation*
0\leq v_k(t,x) \leq \overline{w}(t) \hbox{ for all } (t,x)\in [0,T]\times B_k.
\end{equation*}
It thus follows that
\begin{equation*
0\leq v_k(t,x) \leq \widetilde{C}_2:=\|\widetilde{u}_0\|_{L^{\infty}(\mathbb{R}^N)} \mathrm{e}^{KT} \hbox{ for all } (t,x)\in [0,T]\times B_k.
\end{equation*}
The comparison argument also gives $v_k(t,x)\leq v_{k+1}(t,x)$ for $t\in [0, T]$ and $x\in B_k$. Thus
\[
v(t,x):=\lim_{k\to\infty} v_k(t,x) \mbox{ exists for every $x\in \mathbb{R}^N$ and $t\in [0, T]$},
\]
and $0\leq v(t,x)\leq \overline w(t)$. By a standard regularity consideration, one sees that $v$ solves \eqref{exappx}, and \eqref{3.24} holds. To mark its dependence on $m$, we will denote $v=w_m$.
Next, we show the estimate \eqref{exh1n}. Clearly, we have
\begin{equation}\label{exbound}
0\leq w_m(t,x)\leq \widetilde{C}_2 \hbox{ for } (t,x)\in H_T.
\end{equation}
For any fixed $R>0$, let $\xi (x)$ be a smooth function such that
$$0 \leq \xi \leq 1 \hbox{ in } \mathbb{R}^N,\; \;\; \xi \equiv 1 \;\; \mbox{in $B_{R}$},
\,\hbox{ and }\, \xi \equiv 0 \;\; \mbox{in $\mathbb{R}^N \backslash B_{2R}$}.$$
If we multiply the differential equation in \eqref{exappx} by $\alpha_m(w_m)\xi^2$, then by following the proof of \cite[Lemma 5.1]{DG2}, we obtain
$$\int_0^T \int_{B_{R}} \big|\nabla_x w_m \big|^2\leq \widetilde{C}_3, $$
for some positive constant $\widetilde{C}_3=\widetilde{C}_3(R,T)$ independent of $m$.
Furthermore, if we multiply the differential equation in \eqref{exappx} by $\frac{\partial w_m}{\partial t} \xi^2$ and estimate the resulting equation over $[0,T]\times B_{2R}$, then the proof of \cite[Lemma 5.2]{DG2} implies that there exists a positive constant $\widetilde{C}_4=\widetilde{C}_4(R,T)$ such that
$$\int_0^T \int_{B_{R}} \Big|\frac{\partial_t w_m}{\partial t} \Big|^2\leq \widetilde{C}_4. $$
Therefore, combining the above, we obtain \eqref{exh1n}.
The proof of Lemma \ref{exapplem} is now complete.
\end{proof}
To complete the proof of Theorem \ref{existence}, we shall need the following estimate.
\begin{lem}\label{lmesgw}
For any $R>0$, there is a positive constant $\widetilde{C}_5$ depending on $R$ and $T$ but
independent of $m$ such that
\begin{equation*}
\int_{B_{R/2}} \Big|\nabla_x w_m (\sigma,x)\Big|^2 dx \leq \widetilde{C}_5,
\; \hbox{ for all } m \in \mathbb{N}, \;\;\sigma \in [0, T].
\end{equation*}
\end{lem}
\begin{proof}
Let $\xi (x)$ be a smooth function such that
$$0 \leq \xi \leq 1 \hbox{ in } \mathbb{R}^N,\; \;\; \xi \equiv 1 \;\; \mbox{in $B_{R/2}$},
\,\hbox{ and }\, \xi \equiv 0 \;\; \mbox{in $\mathbb{R}^N \backslash B_{R}$}.$$
We multiply both sides of the differential equation in
\eqref{exappx} by $\frac{\partial w_m}{\partial t} \xi^2$ and then integrate
the resulting equation over $[0, \sigma] \times B_{R}$. After suitable
integration by parts, we obtain
\begin{equation} \label{3.9}
\left.\begin{array}{lll}
\displaystyle\medskip
\int_0^\sigma \int_{B_{R}} \alpha_m' (w_m)
\Big(\frac{\partial w_m}{\partial t} \Big)^2 \xi^2 dx dt &+& \displaystyle\medskip d
\int_0^\sigma \int_{B_{R}} \nabla_x w_m \cdot \nabla_x \Big(
\frac{\partial w_m}{\partial t} \xi^2 \Big) dx dt \\
\displaystyle & & =\displaystyle \int_0^\sigma \int_{B_{R}} g(x, w_m) \frac{\partial w_m}{\partial t} \xi^2 dxdt.\end{array}\right.
\end{equation}
Moreover, using the assumption \eqref{assumeg} on $g$ and the estimate \eqref{exbound}, we have
\begin{equation*}
\int_0^\sigma \int_{B_{R}} g(x, w_m) \frac{\partial w_m}{\partial t} \xi^2 dxdt \leq \frac{1}{4}\int_0^\sigma \int_{B_{R}}\Big(\frac{\partial w_m}{\partial t}\Big)^2 \xi^2 dxdt \,+\, \widetilde{C}_6(T, R).
\end{equation*}
On the other hand, making use of the estimate \eqref{exh1n}, we deduce that
\begin{equation*}
\left.\begin{array}{ll}
& \displaystyle\medskip d\int_0^\sigma \int_{B_{R}} \nabla_x w_m \cdot \nabla_x \Big(
\frac{\partial w_m}{\partial t} \xi^2 \Big) dx dt\\
=& d\displaystyle\medskip \int_0^\sigma \int_{B_{R}} \xi^2 \nabla_x w_m \cdot \nabla_x
\Big(\frac{\partial w_m}{\partial t} \Big) dx dt+2d\int_0^\sigma
\int_{B_{R}} \xi \frac{\partial w_m}{\partial
t} \nabla_x w_m \cdot \nabla_x \xi dx dt\\
\geq & \displaystyle\medskip \frac{d}{2} \int_{B_{R}} \big|\nabla_x w_m (\sigma, x)\big|^2\xi^2 (x) dx -\frac{d}{2} \int_{B_{R}} \big|\nabla_x {\widetilde u}_0 (x)\big|^2 \xi^2 (x) dx\\
&\displaystyle\medskip \qquad-16d^2\int_0^\sigma \int_{B_{R}} \big(\nabla_x w_m \cdot \nabla_x \xi\big)^2 dx dt-\frac{1}{4}\int_0^\sigma \int_{B_{R}}\Big(\frac{\partial w_m}{\partial t}\Big)^2 \xi^2 dxdt\\
\geq & \displaystyle\medskip \frac{d}{2} \int_{B_{R}} \big|\nabla_x w_m (\sigma, x)\big|^2\xi^2 (x) dx -\frac{d}{2} \int_{B_{R}} \big|\nabla_x {\widetilde u}_0 (x)\big|^2 \xi^2 (x) dx\\
&\displaystyle\qquad-\frac{1}{4}\int_0^\sigma \int_{B_{R}}\Big(\frac{\partial w_m}{\partial t}\Big)^2 \xi^2 dxdt \,-\, \widetilde{C}_7(T, R).
\end{array}\right.
\end{equation*}
Substituting the above estimates into \eqref{3.9}, and recalling that $\alpha_m'(w) \geq 1$ from \eqref{restriction}, we obtain
\begin{equation*}
\left.\begin{array}{ll}
& \displaystyle\medskip \frac{1}{2}\int_0^\sigma \int_{B_{R}} \Big(\frac{\partial w_m}{\partial t} \Big)^2 \xi^2 dx dt + \frac{d}{2}\int_{B_{R}} \big|\nabla_x w_m (\sigma, x)\big|^2\xi^2 (x) dx\\
\leq & \displaystyle \frac{d}{2} \int_{B_{R}} \big|\nabla_x {\widetilde u}_0 (x)\big|^2 \xi^2 (x) dx + \widetilde{C}_6(T, R)+\widetilde{C}_7(T, R).
\end{array}\right.
\end{equation*}
Let
$$\widetilde{C}_5:= d \int_{B_{R}} \big|\nabla_x {\widetilde u}_0 (x)\big|^2 dx +2 \widetilde{C}_6(T, R)+2\widetilde{C}_7(T, R).
$$
It finally follows that
$$\int_{B_{R/2}} \big|\nabla_x w_m (\sigma,x)\big|^2 dx \leq \int_{B_{R}} \big|\nabla_x w_m (\sigma,x)\big|^2 \xi^2(x) dx
\leq \widetilde{C}_5 \;\; \hbox{ for all } m \in \mathbb{N}.$$
This completes the proof.
\end{proof}
We are now ready to give the proof of Theorem \ref{existence}.
\begin{proof} [Proof of Theorem \ref{existence}]
In what follows we will select various subsequences from $\{w_m\}$
and, to avoid inundation by subscripts, we will always denote the
subsequence again by $\{w_m\}$. By the estimate \eqref{exh1n}, and by
Rellich's Lemma and a standard diagonal argument, there is a subsequence of $\{w_m\}$ (denoted by itself) and a
function $w \in H^1_{loc} (H_T)$ such that, for any $R>0$,
\begin{equation*}
w_m \to w \;\; \mbox{weakly in $H^1 ([0,T]\times B_R)$
and strongly in $L^2 ([0,T]\times B_R)$}\, \hbox{ as } m \to \infty.
\end{equation*}
In particular, $w_m \to w$ as $m\to\infty$ and $w \geq 0$ almost everywhere in
$H_T$. Moreover, in view of \eqref{exbound}, we have $0 \leq w \leq \widetilde{C}_2$ in $H_T$, and hence $w\in H^1_{loc} (H_T)\cap L^{\infty}(H_T)$.
Furthermore, Lemma \ref{lmesgw} implies that
\begin{equation*}
\int_{B_{R/2}} |\nabla_x w(t,x)|^2 dx \leq \widetilde{C}_5 \;\;
\mbox{for a.e. $t \in [0,T]$},
\end{equation*}
since the set
$$\Big \{v \in H^1([0,T]\times B_{R/2}): \; \int_{B_{R/2}} \big|\nabla_x
v(t,x)\big|^2 dx \leq \widetilde{C}_5 \;\; \mbox{for a.e. $t \in [0,T]$} \Big \}$$
is closed and convex in $H^1([0,T]\times B_{R/2})$, and such sets are
closed under the weak limit.
With the above preparations, and noting that the test function $\phi$ in Definition \ref{defweak} has compact support, we can follow the same lines as those used in the proof of \cite[Theorem 3.1]{DG2} to verify that $w$ is a weak solution of \eqref{eqmain} over $H_T$.
We do not repeat the details here.
The proof of Theorem \ref{existence} is thereby complete.
\end{proof}
Note that since $T>0$ is arbitrary in Theorem \ref{existence}, the weak solution $u(t,x)$ of \eqref{eqmain} given over $H_T$ can be extended to all $t>0$, and it is unique due to Theorem \ref{unique}.
Next, we present a comparison principle which will be used frequently in the subsequent sections. Suppose that $\mu$ and $\widehat{\mu}$ are two positive constants, $g$ and ${\widehat g}$ both satisfy the assumption \eqref{assumeg}, $\Omega_0$ and ${\widehat \Omega}_0$ are smooth domains in $\mathbb{R}^N$, $u_0$ satisfies \eqref{assumeu0} and ${\widehat u}_0$ satisfies \eqref{assumeu0} with $\Omega_0$ replaced by ${\widehat \Omega}_0$. Let $u$ be a weak subsolution of \eqref{eqmain} corresponding to $(\Omega_0, u_0, g,\mu)$, and ${\widehat u}$ be a weak supersolution corresponding to $({\widehat \Omega}_0, {\widehat u}_0, {\widehat g},\widehat{\mu})$, respectively.
\begin{thm} \label{thmcomp}
Suppose that $\Omega_0 \subset {\widehat \Omega}_0$, $u_0
\leq {\widehat u}_0$, $\mu\leq \widehat{\mu}$ and $g \leq {\widehat g}$. Then $u \leq {\widehat u}$ a.e. in $H_T$.
\end{thm}
\begin{proof}
Let $w$ and $\widehat{w}$ be the weak solutions of \eqref{eqmain} corresponding to $(\Omega_0, u_0, g,\mu)$ and $({\widehat \Omega}_0, {\widehat u}_0, {\widehat g},\mu)$ over $H_T$, respectively. As a direct application of Theorem \ref{unique}, we have
\begin{equation}\label{3.14}
u\leq w, \quad \widehat{u}\geq \widehat{w} \,\,\, a.e. \hbox{ in } H_T.
\end{equation}
On the other hand, let $w_m$ be the solution of problem \eqref{exappx}, and
${\widehat w}_m$ be the solution determined by \eqref{exappx} with
reaction term ${\widehat g}$, and the initial
function is obtained by extending ${\widehat u}_0$ to $\mathbb{R}^N$. By the comparison result Lemma \ref{exappcom}, we
clearly have $w_m \leq {\widehat w}_m$ in $ H_T$. Furthermore, from the proof of Theorem
\ref{existence} and the uniqueness of the weak solution, we know that
$$w_m \to w, \quad \widehat{w}_m \to \widehat{w} \,\,\, a.e. \hbox{ in } H_T \hbox{ as } m\to\infty,$$
and hence $w\leq \widehat{w}$ a.e. in $H_T$. This together with \eqref{3.14} completes the proof.
\end{proof}
The next result is useful for constructing supersolutions and subsolutions to \eqref{eqmain}.
\begin{thm}\label{super-sub}
Assume that
there exists a $C^1$ function $\Phi$ over ${H_T}$
such that, with
$$\Omega (t): = \Big\{x \in \mathbb{R}^N: \; \Phi
(t,x)<0\Big\},
$$
one has
\[
\mbox{
$\Omega (0)=\Omega_0$ and } \; |\nabla_x \Phi| \neq 0 \;\;\; \mbox{on\; $ \partial
\Omega (t)$}.
\]
Suppose that $w(t,x)$ and $\nabla_x w(t,x)$ are continuous in $\bigcup_{0 \leq
t\leq T} {\overline {\Omega (t)}}$, $\nabla_x^2 w(t,x)$, $w_t(t,x)$ are
continuous in $\bigcup_{0<t\leq T} \Omega (t)$. Then $w(t,x)$ (extended by 0 for $x\not\in\Omega(t)$) is a weak supersolution of \eqref{eqmain} for $t\in (0, T]$, provided that
\begin{equation}\label{sup}\left\{
\begin{array}{ll}
w_t-d\Delta w\geq g(x, w),\; w>0 & \mbox{ for } x\in\Omega(t),\; 0<t\leq T,\smallskip
\\
w=0,\; \Phi_t\leq \mu \nabla_x u\cdot\nabla_x\Phi &\mbox{ for } x\in \partial\Omega(t),\; 0<t\leq T,\smallskip
\\
w(0,x)\geq u_0(x) & \mbox{ for } x\in\Omega_0.
\end{array}\right.
\end{equation}
If all the inequalities in \eqref{sup} are reversed, then $w$ is a weak subsolution to \eqref{eqmain}.
\end{thm}
\begin{proof} We only prove the case for weak supersolution, as the proof for the weak subsolution case is similar. We need to show that
\begin{equation}\label{eqweaksuper}
\left.\begin{array}{ll}
\displaystyle\medskip \int_0^T \int_{\mathbb{R} ^N} \Big[d \nabla_x w \cdot \nabla_x
\phi-\alpha (w) \phi_t\Big] dx dt-\int_{\mathbb{R} ^N} &\!\!\!\!\!\!\alpha (\widetilde u_0(x)) \phi (0,x) dx\\
&\displaystyle \geq \int_0^T \int_{\mathbb{R} ^N} g(x,w) \phi dxdt
\end{array}\right.
\end{equation}
for every nonnegative function $\phi \in C^{\infty}(H_T)$ with compact support and satisfying
$\phi=0$ on $\{T\} \times \mathbb{R}^N$.
For each test function $\phi$ as described above, we use
the divergence theorem to calculate the following integral
$$\int _{0}^{T}\int_{\Omega(t)} {\rm div} \Psi \,dxdt \;\mbox{ with }
\Psi(t,x)=(\phi(t,x),0,\cdots,0)\in\mathbb{R}^{N+1},
$$
and obtain, with $S:=\bigcup_{0<t<T}\partial\Omega(t)$,
\begin{equation*}
\left.\begin{array}{ll}
\displaystyle\medskip \int _{0}^{T}\int_{\Omega(t)} \phi_t\; dxdt
\!\! &=\displaystyle\medskip -\int_{S} \phi\; \frac{\Phi_t}{\big|(\Phi_t, \nabla_x \Phi)\big|}d\sigma - \int_{{\Omega}(0)} \phi(0,x)\; dxdt \\
&= \displaystyle\medskip -\int_{0}^T\int_{\partial\Omega(t)} \phi\; \frac{\Phi_t}{\big|\nabla_x \Phi\big|}d S_xdt - \int_{{\Omega}(0)} \phi(0,x) dxdt.
\end{array}\right.
\end{equation*}
Since the unit outward normal of $\Omega(t)$ at $x\in\partial\Omega(t)$ is given by
$\nu=\nu(t,x):=\nabla_x\Phi(t,x)/|\nabla_x\Phi(t,x)|$, by \eqref{sup} we have
\begin{equation*
\frac{\Phi_t}{|\nabla_x\Phi|}\leq \mu \nabla_x w\cdot \nu \mbox{ for } x\in\partial\Omega(t),
\end{equation*}
and hence
\begin{equation}\label{inteul0}
-\frac{d}{\mu}\int _{0}^{T}\int_{{\Omega}(t)} \phi_t\; dxdt \leq d\int_{0}^T\int_{\partial\Omega(t)} \phi\; \nabla_x w\cdot\nu d S_xdt +\frac{d}{\mu} \int_{\Omega(0)} \phi(0,x) dxdt.
\end{equation}
On the other hand, we multiply both sides of the first inequality in \eqref{sup} by $\phi$ and integrate the resulting inequality over $\bigcup_{0<t<T} {\Omega}(t)$, making use of integration by parts and the last inequality in \eqref{sup}, and obtain
\begin{equation*}
\left.\begin{array}{ll}
\displaystyle\medskip \int _{0}^{T}\int_{{\Omega}(t)} \big[\phi_t w-d\nabla_x \phi \nabla_x w\big]dxdt &+\,\,\displaystyle\medskip \int _{0}^{T}\int_{{\Omega}(t)} g(x, w) \phi dxdt\\
&\displaystyle\medskip\leq \,\, -d\int_{0}^T\int_{\partial\Omega(t)} \phi \nabla_xw\cdot\nu d S_xdt
- \int_{{\Omega}(0)}u_0(x)\phi(0,x) dxdt.
\end{array}\right.
\end{equation*}
By this inequality, \eqref{inteul0}, the fact that $w(t,x)=0$ for $x\not\in\Omega(t)$, and the definition of $\alpha$,
we immediately obtain \eqref{eqweaksuper}.
\end{proof}
We now consider the asymptotic behavior of the weak solution to \eqref{eqmain} as $\mu \to \infty$. To emphasize its dependence on $\mu$, we denote by $u_\mu$ the unique weak solution, and denote $\Omega_\mu (t)=\{x: u_\mu (t,x)>0\}$.
\begin{thm} \label{asymu}
Let $u_\mu$ be the unique weak solution to problem \eqref{eqmain}. Then
\begin{equation*}
\lim_{\mu \to \infty} \Omega_\mu (t)=\mathbb{R}^N, \;\; \forall \,t>0,
\end{equation*}
and
\begin{equation*
u_\mu \to U \;\; \mbox{in $C_{loc}^{\frac{1+\nu_0}{2},
1+\nu_0} ((0, \infty) \times \mathbb{R}^N)$ as $\mu \to \infty$},
\end{equation*}
where $\nu_0$ can be any number in $(0,1)$ and $U(t,x)$ is the
unique solution of the Cauchy problem
\begin{equation}\label{cauchyp}
\left \{ \begin{array}{ll}
\displaystyle\medskip U_t-d \Delta U=g(x,U) \;\;&\mbox{in $(0, \infty) \times \mathbb{R}^N$},\\
U(0,x)={\widetilde u}_0 (x) \;\;& \mbox{in $\mathbb{R}^N$}.
\end{array} \right.
\end{equation}
Moreover, $u_\mu (t,x) \leq U(t,x)$ for all $t>0$ and $x \in \Omega_\mu (t)$.
\end{thm}
\begin{proof}
Making use of Theorem \ref{thmcomp}, the proof is almost identical to that of \cite[Theorem 5.4]{DG2}, and we omit the details.
\end{proof}
\begin{rem}
Although our main interest of this paper is for the case $\Omega_0\subset\mathbb{R}^N$ with $N\geq 2$, it is easily seen
from their proofs that all the results in this section remain valid when $N=1$.
\end{rem}
\section{Spreading profile of the Fisher-KPP equation}\label{sec4}
In this section, we study the long-time behavior of the weak solution to problem \eqref{eqfrfisher}, for some special unbounded
$\Omega_0$. More precisely, we assume that there exist $\phi\in (0, \pi)$ and $\xi_1>\xi_2$ such that \eqref{Omega0} holds, namely
\begin{equation}\label{outscone}
\Lambda^\phi+\xi_1e_N \subset {\Omega_0} \subset \Lambda^\phi+\xi_2e_N.
\end{equation}
Our first result on the long-time behavior of \eqref{eqfrfisher} is the following theorem.
\begin{thm}\label{thm6.1}
Let $u(t,x)$ be the unique weak solution of problem \eqref{eqfrfisher} with $\Omega_0$ satisfying \eqref{outscone}.
Denote $\Omega(t)=\big\{x:\, u(t,x)>0\big\}$. Then
\begin{equation}\label{convtorn}
\lim_{t \to \infty} \Omega (t)=\mathbb{R}^N,
\end{equation}
and
\begin{equation}\label{convtoss}
\lim_{t \to \infty} u(t,x)=\frac ab \;\;
\mbox{locally uniformly in $x \in \mathbb{R}^N$}.
\end{equation}
\end{thm}
\begin{proof}
We first prove \eqref{convtorn}. Since $\Omega_0$ satisfies \eqref{outscone}, we can find a ball $B_{r_0}(x_0)\subset \Omega_0$ with radius
\begin{equation}\label{radius}
r_0\geq R^*:= \sqrt{\frac{d}{a}\lambda_1},
\end{equation}
where $\lambda_1$ is the first eigenvalue of the following eigenvalue problem
$$-\Delta \phi=\lambda \phi \hbox{ in } B_{1}(0);\quad \phi=0 \hbox{ on } \partial B_1(0).$$
We then choose a $C^2$ radial function $v_0(r)$ ($r=|x-x_0|$) satisfying
$$v_0(|x-x_0|)\leq u_0(x) \hbox{ for } |x-x_0|<r_0, \,\,\, v_0(r)>0 \hbox{ for } r\in [0,r_0)\,\hbox{ and }\, v_0'(0)=v_0(r_0)=0,$$
and consider the following radially symmetric problem
\begin{equation*
\left \{ \begin{array}{ll}
\displaystyle\medskip v_t-d \Delta v=av-b v^2, \;\;&t>0, \; 0<r<k(t), \\
\displaystyle\medskip v_r (t,0)=0, \;\; v(t,k(t))=0, \;\; &t>0,\\
\displaystyle\medskip k'(t)=-\mu v_r (t, k(t)), \;\; &t>0,\\
\displaystyle k(0)=r_0, \;\;\; v(0,r)=v_0 (r), \;\; & 0 \leq r \leq r_0.
\end{array} \right.
\end{equation*}
It follows from \cite[Theorem 2.1]{DG1} that this problem admits a (unique)
classical solution $(v(t,r), k(t))$ defined for all $t>0$ such that $k'(t)>0$, $ v(t,r)>0$ for $0 \leq r<k(t)$, $t>0$.
Moreover, by \cite[Theorem 2.5]{DG1}, we have
\begin{equation}\label{setbelow}
\lim_{t\to\infty}k(t)=\infty.
\end{equation}
Denote
$$\mathcal{G} (t)=\Big\{x \in \mathbb{R}^N: \; |x-x_0|<k(t)\Big\},$$
and
$$ V(t,x)=v(t,|x-x_0|). $$
We also extend $V(t,\cdot)$ to be zero outside $\mathcal{G} (t)$. Clearly, for any given $T>0$, $(V,\mathcal G)$
is a classical solution of the free boundary problem
\begin{equation}\label{eq6.2}
\left \{ \begin{array} {ll} \displaystyle\medskip u_t-d \Delta u=au-bu^2 \;\; &
\mbox{for $x \in \mathcal{G} (t), \; 0<t\leq T$},\\
\displaystyle\medskip u=0,\; u_t=\mu|\nabla_x u|^2 \;\;&\mbox{for $x \in \partial\mathcal{G}(t), \; 0<t\leq T$}, \\
u(0,x)=v_0 (|x-x_0|) \;\; &\hbox{for } x \in \mathcal{G} (0).
\end{array} \right.
\end{equation}
By Theorems \ref{weakclassic} and \ref{unique}, $V$ is the unique weak solution of $\eqref{eq6.2}$ over $H_T$. It then follows from Theorem \ref{thmcomp} that $u \geq V $ in $H_T$, and hence by the arbitrariness of $T$, we have
$ \mathcal{G} (t)\subset \Omega (t)$ for all $t\geq 0$. This together with \eqref{setbelow} implies \eqref{convtorn}.
It remains to prove \eqref{convtoss}.
We claim that
\begin{equation}\label{auxicom}
{V}(t,x)\leq u(t,x)\leq {U}(t,x) \hbox{ in } [0,\infty)\times \mathbb{R}^N,
\end{equation}
where $U(t,x)$ is the unique solution to the Cauchy problem \eqref{cauchyp}.
Indeed, the first inequality in \eqref{auxicom} follows directly from Theorem \ref{thmcomp}, while the second inequality is a consequence of Theorem \ref{asymu}.
Applying \cite[Theorem 6.2]{DG2} to the equation of ${V}$, we obtain
\begin{equation}\label{stablefb}
\lim_{t \to \infty} {V}(t,x)=\frac ab \;\;\mbox{locally uniformly in $x \in \mathbb{R}^N$}.
\end{equation}
On the other hand, it is well known that
\begin{equation}\label{stablefa}
\lim_{t \to \infty} {U}(t,x)=\frac ab \;\;
\mbox{locally uniformly in $x \in \mathbb{R}^N$}.
\end{equation}
By \eqref{auxicom}, \eqref{stablefb} and \eqref{stablefa}, we immediately obtain \eqref{convtoss}, and the proof of Theorem \ref{thm6.1} is thus complete.
\end{proof}
Next we examine the long-time profile of the free boundary $\partial \Omega(t)$.
For convenience, we first recall the following result from \cite{BDK}.
\begin{prop}\label{semiwave}
Let $d>0$, $a>0$ and $b>0$ be given constants. For any $k\in [0, 2\sqrt{ad})$, the problem
\begin{equation}\label{eqwave}
-dZ''+kZ'=aZ-bZ^2 \hbox{ in } (0,\infty),\quad Z(0)=0
\end{equation}
admits a unique positive solution $Z=Z_k$, and it satisfies $Z_k(r)\to a/b$ as $r\to\infty$. Moreover, $Z'_k(r)>0$ for $r\geq 0$, $Z_{k_1}'(0)>Z_{k_2}'(0)$, $Z_{k_1}(r)>Z_{k_2}(r)$ for $r>0$ and $k_1<k_2$, and for each $\mu>0$, there is a unique $c_*=c_*(\mu,a,b,d)$ such that $\mu Z'_{k}(0)=k$ when $k=c_*$. Furthermore, $c_*(\mu,a,b,d)$ depends continuously on its arguments, is increasing in $\mu$ and $\lim_{\mu\to\infty}c_*(\mu,a,b,d)=2\sqrt{ad}$.
\end{prop}
In our discussion below, since $a$, $b$, $d$ and $\mu$ are always fixed, we use $c_*$ to denote $c_*(\mu,a,b,d)$.
\begin{thm}\label{spreadspeed}
Let $u(t,x)$ be the unique weak solution of problem \eqref{eqfrfisher} with $\Omega_0$ satisfying \eqref{outscone}
for some $\phi\in (\pi/2, \pi)$, and $u_0$ satisfying \eqref{assumeu0} and
\begin{equation}\label{addau0}
\liminf_{d(x,\partial \Omega_0)\to\infty} u_0(x)>0.
\end{equation}
Denote $\Omega(t)=\big\{x:\, u(t,x)>0\big\}$. Then for any $\varepsilon>0$, there exists $T=T(\varepsilon)>0$ such that for all $t>T$
we have
\begin{equation}\label{Omega(t)}
\Lambda^\phi-\left(\frac{c_*}{\sin\phi}-\varepsilon\right)t\,{e}_N\subset \Omega(t)\subset \Lambda^\phi-\left(\frac{c_*}{\sin\phi}+\varepsilon\right)t\,{e}_N.
\end{equation}
Moreover,
\begin{equation}\label{outconebehu}
\lim_{t\to\infty}\left[{\sup}_{x\in \Lambda^\phi-\big(\frac{c_*}{\sin\phi}-\varepsilon\big)t\,{e}_N} \Big|u(t,x)-\frac{a}{b} \Big|\right]=0.
\end{equation}
\end{thm}
We prove this theorem by a series of lemmas. We first show
\begin{equation}\label{Omega(t)-lb}
\Lambda^\phi-\left(\frac{c_*}{\sin\phi}-\varepsilon\right)t\,{e}_N\subset \Omega(t) \mbox{ for all large $t$. }
\end{equation}
By assumption \eqref{addau0}, we can find a one-dimensional function $w_0\in C^2([0,\infty))$ such that
\begin{equation}\label{lowu0}
0<w_0\leq \sigma_0:=\frac{1}{2}\liminf_{d(x,\partial \Omega_0)\to\infty } u_0(x) \hbox{ in } (0,\infty),\quad w_0(0)=0 \quad\hbox{and}\quad w_0'>0 \hbox{ in } [0,\infty).
\end{equation}
Then we consider the following one space dimension free boundary problem
\begin{equation}\label{eqlow}
\left \{ \begin{array}{ll}
\displaystyle\medskip w_t-d w_{yy}=aw-bw^2, \;\;&t>0, \; \rho(t)<y<\infty, \\
\displaystyle\medskip w(t,\rho(t))=0, \;\; &t>0,\\
\displaystyle\medskip \rho'(t)=-\mu w_y (t,\rho(t)), \;\; &t>0,\\
\displaystyle \rho(0)=0, \;\;\; w(0,y)=w_0 (y), \;\; & 0 \leq y <\infty.
\end{array} \right.
\end{equation}
It follows from \cite[Theorem 2.11]{DDL} that, \eqref{eqlow} admits a (unique)
classical solution $(w(t,y), \rho(t))$ defined for all $t>0$ and
$\rho'(t)<0$, $w(t,y)>0$ for $\rho(t) \leq y<\infty$, $t>0$.
Moreover, we have the following comparison result.
\begin{lem}\label{onedimcom}
For any given $\widetilde{T}\in (0,\infty)$, suppose $\widetilde{\rho}\in C^1\big([0,\widetilde{T}]\big)$ and $\widetilde{w}\in C^{1,2}\big(\widetilde{D}_{\widetilde{T}}\big)$ with $\widetilde{D}_{\widetilde{T}}=\big\{(t,y): \, 0\leq t\leq \widetilde{T},\, \widetilde{\rho}(t)\leq y<\infty\big\}$. If
\begin{equation*}\left\{\begin{array}{ll}
\displaystyle\medskip \widetilde{w}_t -d\widetilde{w}_{yy}\geq\widetilde{w}(a-b \widetilde{w}) ,& 0<t\leq \widetilde{T},\,\,\,\widetilde{\rho}(t)<y<\infty,\vspace{3pt}\\
\widetilde{w}(t,\widetilde{\rho}(t))= 0,\,\,\,\, \widetilde{\rho}'(t)\leq -\mu \widetilde{w}_y(t,\widetilde{\rho}(t)),&0<t\leq \widetilde{T},\end{array}\right.
\end{equation*}
and
$$\widetilde{\rho}(0) \leq 0 \quad\hbox{and}\quad w_0(y)\leq \widetilde{w}(0,y)\,\hbox{ in } \, [0,\infty),$$
then the solution $(w,\rho)$ of problem \eqref{eqlow} satisfies
$$ \widetilde{\rho}(t)\leq \rho(t) \,\hbox{ in }\, (0,\widetilde{T}]\,\hbox{ and }\, w(t,y)\leq \widetilde{w}(t,y)\,\hbox{ for } \, 0<t\leq \widetilde{T},\,\, \rho(t)\leq y <\infty.$$
\end{lem}
\begin{proof}
If for any $0\leq t\leq \widetilde{T}$, we extend $w(t,y)$ (resp. $\widetilde{w}(t,y)$) to be zero for $y< \rho(t)$ (resp. $y< \widetilde{\rho}(t)$), then it is easily checked that $w$ (reps. $\widetilde{w}$) is a weak solution (resp. weak supersolution) of the free boundary problem induced from \eqref{eqlow} over $H_{\widetilde{T}}$, and hence the desired comparison result follows from Theorem \ref{thmcomp}. (One could also prove the result directly along the lines of \cite{DL}.)
\end{proof}
We next show the following estimate of $(w,\rho)$.
\begin{lem}\label{eslowra}
For any $\varepsilon>0$, there exists $T_1=T_1(\varepsilon)>0$ such that
\begin{equation}\label{eslowrap1}
\rho(t) \leq -\Big(c_{*}-\frac{2}{3}\varepsilon\Big) t\,\hbox{ for all } t\geq T_1,
\end{equation}
and
\begin{equation}\label{eslowrap2}
\liminf_{t\to\infty}\left[\inf_{y\geq -(c_{*}-\frac 23\varepsilon) t} w(t,y)\right]\geq \frac{a}{b}.
\end{equation}
\end{lem}
\begin{proof}
The proof follows from \cite[Theorem 4.2]{DL} with some modifications. For the sake of completeness,
and also for the convenience of later applications, we include the details below.
We first claim that for any given small $\delta>0$, there exists $t_1=t_1(\delta)>0$ such that
\begin{equation}\label{eqlargt1}
w(t_1,y)\geq \frac{a-\delta}{b+\delta} \hbox{ for all } y\geq 0.
\end{equation}
Indeed, applying the proof of Theorem \ref{thm6.1} to the one-dimensional problem \eqref{eqlow}, we easily
obtain $\lim_{t\to\infty} \rho(t)=-\infty$ and
$$ \lim_{t \to \infty} w(t,y)=\frac{a}{b} \;\;\mbox{locally uniformly for $y \in \mathbb{R}$}.$$
Since $w_0(y)$ is nondecreasing in $y\in[0,\infty)$, it follows from the comparison result stated in Lemma \ref{onedimcom} and the uniqueness of solution to problem \eqref{eqlow} that, for any fixed $t>0$, $w(t,y)$ is nondecreasing in $y\in [\rho(t),\infty)$. We thus obtain
$$ \liminf_{t \to \infty} w(t,y)\geq \frac{a}{b} \;\;\mbox{uniformly for $y \geq 0 $},$$
which clearly implies \eqref{eqlargt1}.
Next, we construct a subsolution to problem \eqref{eqlow}. To do this, we need a few more notations. For any small $\delta>0$, denote $$c_{\delta}:=c_*(\mu,a-\delta,b+\delta,d)$$
and denote by $Z_{\delta}(r)$ the solution of \eqref{eqwave} with $k$, $a$, $b$ replaced by $c_{\delta}$, $a-\delta$, $b+\delta$, respectively. Set
$$
\eta(t)=\eta_\delta(t):=-(1-\delta)^2c_\delta t \,\hbox{ for } \,t>0,
$$
and
$$
\underline{w}(t,y)=\underline{w}_\delta(t,y):=(1-\delta)^2 Z_{\delta}(y-\eta_\delta(t)) \,\hbox{ for } \,\eta_\delta(t)\leq y<\infty,\, t>0.
$$
It is straightforward to verify that
$$\underline{w}(t,\eta(t))=0, \quad \eta'(t)=-\mu \underline{w}_y(t,\eta(t))\,\,\hbox{ for } \, t>0. $$
Moreover, since $Z'_{\delta}>0$ in $[0,\infty)$, direct calculations yield
\begin{equation*}
\left.\begin{array}{ll}
\displaystyle\medskip \underline{w}_t-d\underline{w}_{yy}&=(1-\delta)^{4}c_{\delta}Z_\delta'-d(1-\delta)^{2}Z_\delta''\\
&\displaystyle\medskip\leq (1-\delta)^{2}(Z_\delta'-d Z_\delta'')\\
&\displaystyle\medskip=(1-\delta)^{2}\big[(a-\delta)Z_\delta-(b+\delta) Z^2_\delta\big]\\
&\leq (a-\delta)\underline{w}-(b+\delta)\underline{w}^2
\end{array}\right.
\end{equation*}
for $\eta(t)<y<\infty$, $t>0$. By Proposition \ref{semiwave}, we have
$$ \underline{w}(0,y)=(1-\delta)^2Z_{\delta}({y})<(1-\delta)^2\frac{a-\delta}{b+\delta} \,\,\hbox{ for } 0\leq y <\infty.$$
This together with \eqref{eqlargt1} implies
$$w(t_1,y) \geq \underline{w}(0,y) \,\hbox{ for } 0\leq y<\infty. $$
It then follows from Lemma \ref{onedimcom} that
\begin{equation}\label{comrelow}
\rho(t+t_1) \leq \eta(t), \quad w(t+t_1,y)\geq \underline{w}(t,y) \,\hbox{ for } \, \eta(t)\leq y<\infty,\,t>0.
\end{equation}
Since $\lim_{\delta\to 0} (1-\delta)^2c_{\delta}=c_*:=c_*(\mu,a,b,d)$, for any $\varepsilon>0$, we can find some $\delta_\varepsilon\in (0,\varepsilon)$ such that
\begin{equation}\label{chdelta}
\big|(1-\delta_{\varepsilon})^2c_{\delta_{\varepsilon}}- c_* \big|\leq \varepsilon/2.
\end{equation}
We now fix $\delta=\delta_{\varepsilon}$ in $Z_{\delta}$, $\eta_\delta$ and $t_1(\delta)$.
Then \eqref{comrelow} implies
$$
\rho(t)\leq -(1-\delta_{\varepsilon})^2c_{\delta_{\varepsilon}} (t-t_1) \leq -\Big(c_*-\frac{2}{3}\varepsilon\Big)t-\frac{\varepsilon}{6}t +(1-\delta_{\varepsilon})^2c_{\delta_{\varepsilon}}t_1 \,\hbox{ for } t\geq t_1.
$$
Thus \eqref{eslowrap1} holds with $T_1=6\varepsilon^{-1}(1-\delta_{\varepsilon})^2c_{\delta_{\varepsilon}}t_1$.
It remains to prove \eqref{eslowrap2}. With $\delta=\delta_\varepsilon$ chosen as above, it follows from Proposition \ref{semiwave} that there exists $y_0>0$ sufficiently large such that
\begin{equation}\label{ineqlowinf}
Z_{\delta_{\varepsilon}}(y)\geq \frac{a-2\delta_{\varepsilon}}{b+2\delta_{\varepsilon}} \,\hbox{ for all } \,y\geq y_0.
\end{equation}
On the other hand, by \eqref{chdelta}, we clearly have
$$y-\eta(t-t_1)\geq y+\Big(c_*-\frac{2}{3}\varepsilon\Big)t+\frac{\varepsilon}{6}t-(1-\delta_{\varepsilon})^2c_{\delta_{\varepsilon}}t_1 \,\hbox{ for } t\geq t_1.$$
Thus, if we choose $\widetilde{t}_1=6\varepsilon^{-1}[y_0+(1-\delta_{\varepsilon})^2c_{\delta_{\varepsilon}}t_1]$, then
$$y-\eta(t-t_1)\geq y_0\,\hbox{ for all } y\geq -\Big(c_*-\frac{2}{3}\varepsilon\Big)t,\,t\geq \widetilde{t}_1.$$
This together with \eqref{comrelow} and \eqref{ineqlowinf} implies
$$w(t,y)\geq \underline{w}(t-t_1,y) \geq (1-\delta_{\varepsilon})^2\frac{a-2\delta_{\varepsilon}}{b+2\delta_{\varepsilon}} \,\hbox{ for all} \, y\geq -\Big(c_*-\frac{2}{3}\varepsilon\Big)t,\,t\geq \widetilde{t}_1.$$
Since
$$(1-\delta_{\varepsilon})^2\frac{a-2\delta_{\varepsilon}}{b+2\delta_{\varepsilon}} \to \frac{a}{b} \hbox{ as } {\varepsilon}\to 0, $$
this gives \eqref{eslowrap2}, and the proof of Lemma \ref{eslowra} is now complete.
\end{proof}
\begin{rem}
The conclusions in Lemma 3.5 can be considerably sharpened (though they are not needed in this paper). It is possible to modify the method of \cite{DMZ1} to show that, as $t\to\infty$,
\[
\rho(t)-c_*t\to C\in\mathbb{R}^1, \;\; \sup_{y\in[\rho(t),\infty)}\big|w(t,y)-Z_{c_*}(y-\rho(t))\big|\to 0,
\]
where $Z_{c_*}$ is given in Proposition 3.2.
\end{rem}
\begin{lem}\label{lowbound}
Let $u(t,x)$ and $\Omega(t)$ be given in the statement of Theorem \ref{spreadspeed}. Then for any $\varepsilon>0$, there exists $T_2=T_2(\varepsilon)>0$ such that \eqref{Omega(t)-lb} holds
for all $ t\geq T_2$,
and
\begin{equation}\label{infubound}
\liminf_{t\to\infty}\left[{\inf}_{x\in \Lambda^{\phi}-\big(\frac{c_*}{\sin\phi}-\varepsilon\big)t\,{e}_N } u(t,x)\right]\geq \frac{a}{b}.
\end{equation}
\end{lem}
\begin{proof}
By the assumption \eqref{addau0}, there exists $r_1>0$ sufficiently large such that
$$u_0(x) \geq \sigma_0:= \frac{1}{2}\liminf_{d(\widetilde{x},\partial \Omega_0)\to\infty}u_0(\widetilde{x}) \hbox{ for all } x\in \Omega_0 \hbox{ with } d(x,\partial \Omega_0)\geq r_1. $$
Then, due to the assumption \eqref{outscone}, we have
\begin{equation}\label{initialcom}
u_0(x) \geq \sigma_0 \hbox{ for all } x\in \Lambda^{\phi}+ \Big(\xi_1+\frac{r_1}{\sin\theta}\Big)e_N,
\end{equation}
where $\theta=\pi-\phi$ (and so $\sin\theta=\sin\phi$).
Let $(w,\rho)$ be the unique solution to problem \eqref{eqlow} with initial function $w_0$ satisfying \eqref{lowu0}. For convenience of
notation, we write
\[
z_1:=\Big(\xi_1+\frac{r_1}{\sin\theta}\Big)e_N\quad\hbox{and}\quad \Lambda_{z_1}:= \Lambda^{\phi}+z_1.
\]
For any fixed $z\in \partial \Lambda_{z_1}\backslash \{ z_1\}$, let $\nu_z$ be the inward unit normal vector of $\Lambda_{z_1}$ at $z$, and define
$$
\Omega_{z}(t)
=\Big\{x:\,\, x\cdot \nu_z \geq \rho(t)+r_1+ \xi_1\sin\theta\Big\}$$
($\Omega_{z}(0)$ is illustrated in Figure \ref{fighalfs}), and
$$ w_z(t,x)=w(t,x\cdot\nu_z-r_1- \xi_1\sin\theta).$$
\begin{figure}[h]
\centering
\def9cm{8cm}
\input{halfspace.pdf_tex}
\caption{The domain $\Omega_z(0)$ with given $z\in \partial \Lambda_{z_1}\backslash \{ z_1\}$ }\label{fighalfs}
\end{figure}
We also extend $w_z(t,\cdot)$ to be zero outside $\Omega_z(t)$ for $t\geq 0$. Clearly, $(w_z,\Omega_z)$ is a classical solution of the following problem
\begin{equation*}
\left \{ \begin{array} {ll} \displaystyle\medskip w_t-d \Delta w=w(a-b w) \;\; &
\mbox{for $x \in\Omega(t), \; t>0$},\\
\displaystyle\medskip w=0, \; w_t=\mu|\nabla_x w|^2 \;&\mbox{for $x \in \partial\Omega(t), \; t>0$}, \\
w(0,x)=w_0 (x\cdot\nu_{z}-r_1- \xi_1\sin\theta) \;\; & \mbox{for $x \in \Omega_{z}(0)$},
\end{array} \right.
\end{equation*}
and hence by Theorems \ref{weakclassic} and \ref{unique}, it is the unique weak solution.
By Lemma \ref{eslowra}, for any $\varepsilon>0$, there exists $T_1=T_1(\varepsilon)>0$ such that
\begin{equation*}
\Omega_{z}(t) \supset \Big\{x: \,x\cdot\nu_z\geq -\Big(c_{*}-\frac{2}{3}\varepsilon\Big)t +r_1+ \xi_1\sin\theta\Big\}\,\hbox{ for all } t\geq T_1,
\end{equation*}
and
\begin{equation*}
\liminf_{t\to\infty}\left[{\inf}_{x\cdot \nu_z -r_1- \xi_1\sin\theta\geq -(c_{*}-\frac 23\varepsilon) t} w_z(t,x)\right]\geq \frac{a}{b}.
\end{equation*}
Therefore, if we choose
$$ T_2:=\max\Big\{T_1,\; \frac{3|r_1+ \xi_1\sin\theta|}{\varepsilon}\Big\},$$
then
\begin{equation*}
\Omega_{z}(t) \supset \Big\{x: \,x\cdot\nu_z\geq -(c_{*}-\varepsilon)t \Big\}\,\hbox{ for all } t\geq T_2,
\end{equation*}
and
\begin{equation*}
\liminf_{t\to\infty}\left[{\inf}_{x\cdot \nu_z \geq -(c_{*}-\varepsilon) t} w_z(t,x)\right]\geq \frac{a}{b}.
\end{equation*}
On the other hand, by the choice of $w_0$ in \eqref{lowu0} and the property \eqref{initialcom}, we have
$$\Omega_{z}(0)\subset \Lambda_{z_1} \subset \Omega(0), $$
and
$$w_0 (x\cdot\nu_z-r_1-\xi_1\sin\theta)\leq u_0(x) \,\hbox{ for } x\in \Omega_z(0).$$
Hence we can use Theorem \ref{thmcomp} to compare $u$ and $w_z$ and then obtain
$$w_z(t,x)\leq u(t,x) \,\hbox{ in } [0,\infty)\times\mathbb{R}^N, $$
which clearly implies $\Omega_z(t) \subset \Omega(t)$ for $t\geq 0$.
Thus, we obtain
\begin{equation*}
\Big\{x: \,x\cdot\nu_z\geq -(c_{*}-\varepsilon)t \Big\} \subset \Omega(t) \,\hbox{ for all } t\geq T_2,
\end{equation*}
and
\begin{equation*}
\liminf_{t\to\infty}\left[{\inf}_{x\cdot \nu_z \geq -(c_{*}-\varepsilon) t} u(t,x)\right]\geq \frac{a}{b}.
\end{equation*}
Finally, by the arbitrariness of $z\in \partial \Lambda_{z_1}\backslash \{ z_1\}$, we obtain
\begin{equation*}
\Lambda^\phi-\frac{c_*-\varepsilon}{\sin\phi}t{e}_N\;= \bigcup_{z\in \partial \Lambda_{z_1}\backslash \{ z_1\} }\Big\{x: \,x\cdot\nu_z\geq -(c_{*}-\varepsilon)t \Big\}\; \subset\; \Omega(t) \,\hbox{ for all } t\geq T_2.
\end{equation*}
The desired results then follow if we replace $\varepsilon$ by $\widetilde{\varepsilon}: =\varepsilon/ \sin\phi$.
\end{proof}
Next we prove
\[
\Omega(t)\subset \Lambda^\phi-\left(\frac{c_*}{\sin\phi}+\varepsilon\right)t\,{e}_N \mbox{ for all large $t$}
\]
by constructing a suitable weak supersolution to problem \eqref{eqfrfisher}. We do this with several lemmas.
\begin{lem}\label{lemupu0}
Let $u(t,x)$ and $\Omega(t)$ be given in the statement of Theorem \ref{spreadspeed}. Then for any $\delta>0$, there exist $t_2=t_2(\delta)>0$ and $r_2=r_2(\delta)>0$ such that
\begin{equation}\label{upu0}
u(t,x)\leq \frac{a+\delta}{b-\delta} \,\hbox{ for } t\geq t_2,\,x\in\mathbb{R}^N,
\end{equation}
and
\begin{equation}\label{conez3}
{\Omega(t_2)} \subset \Lambda_{z_2}:=\Lambda^\phi+\Big(\xi_2-\frac{r_2}{\sin\phi}\Big)e_N.
\end{equation}
\end{lem}
\begin{proof}
Let $u^*(t)$ be the unique solution of the problem
$$\frac{d u^*}{dt}=u^*(a-bu^*) \hbox{ for } t>0; \quad u^*(0)=\max\Big\{ \frac{a}{b},\, \|u_0\|_{L^{\infty}(\Omega_0)} \Big\}. $$
Clearly, we have
$$u^*(t)\geq \frac{a}{b} \hbox{ for all } t\geq 0\,\hbox{ and }\,\lim_{t\to\infty} u^*(t)=\frac{a}{b}.$$
Moreover, it follows from Theorem \ref{asymu} and the parabolic comparison principle that
$$u(t,x)\leq U(t,x)\leq u^*(t)\,\hbox{ for } t\geq 0,\, x\in\mathbb{R}^N, $$
where $U(t,x)$ is the unique solution of the Cauchy problem \eqref{cauchyp}. As a consequence, for any $\delta>0$, there exists $t_2=t_2(\delta)>0$ such that
$$u(t,x)\leq u^*(t) \leq \frac{a+\delta}{b-\delta}\,\hbox{ for all } t\geq t_2,\,x\in\mathbb{R}^N, $$
which clearly gives \eqref{upu0}.
It remains to prove \eqref{conez3}.
Let $t_2>0$ be determined as above. It then follows from Proposition \ref{pexsym} in the Appendix
below that there exists $R_0>1$ depending on $t_2$ such that, for any given
radially symmetric function $\widehat{v}_0\in C^2([0,\infty))$ satisfying
\begin{equation}\label{hatv0}
0<\widehat{v}_0(r) < 2\| u_0 \|_{ L^{\infty}(\Omega_0)} \,\hbox{ for } \,r\in (0,\infty),\quad \widehat{v}_0(0)=0,\quad \|\widehat{v}_0\|_{C^1([0,\infty)}\leq 3\| u_0 \|_{ L^{\infty}(\Omega_0)},
\end{equation}
the following free boundary problem
\begin{equation}\label{sec4symubd}
\left \{ \begin{array}{ll}
\displaystyle\medskip \widehat{v}_t-d \Delta \widehat{v}=a\widehat{v}-b\widehat{v}^2, \;\; & 0<t<t_2, \; \widehat{h}(t)<r<\infty,\\
\displaystyle\medskip \widehat{v}(t, \widehat{h}(t))=0, \;\; &0<t<t_2,\\
\displaystyle\medskip \widehat{h}'(t)=-\mu \widehat{v}_r (t, \widehat{h}(t)), \;\; &0<t<t_2,\\
\widehat{h}(0)=R_0, \;\;\; \widehat{v}(0, r)=\widehat{v}_0 (r-R_0), \;\; &R_0 \leq r<\infty,
\end{array} \right.
\end{equation}
admits a unique classical solution $(\widehat{v},\widehat{h})$ defined for $0<t\leq t_2$ with $\widehat{v}(t,r)>0$, $\widehat{h}'(t)<0$ for $0<t\leq t_2,\,\widehat{h}(t) < r<\infty$, and
\begin{equation}\label{freedt2}
\widehat{h}(t_2)\geq R_0/2,
\end{equation}
where due to the radial symmetry, $\Delta \widehat{v}=\widehat{v}_{rr}+\frac{N-1}{r}\widehat{v}_r$.
Set
\[
r_2:=R_0+1 \quad\hbox{and}\quad \Lambda_{z_2}:=\Lambda^\phi+\Big(\xi_2-\frac{r_2}{\sin\phi}\Big)e_N.
\]
Clearly
\[
B_{r_2}(\widehat x_0)\cap (\Lambda^\phi+\xi_2e_N)=\emptyset \;\hbox{ for all } \widehat x_0\in \mathbb{R}^N\setminus \Lambda_{z_2}.
\]
Since $\Omega_0\subset (\Lambda^\phi+\xi_2e_N)$, it is easily seen that $|x-\widehat{x}_0|-R_0>1$
for all $x\in\Omega_0$, $\widehat x_0\in\mathbb{R}^N\setminus \Lambda_{z_2}$. In view of this, we may require that, in addition to the constraint \eqref{hatv0}, $\widehat{v}_0$ also satisfies
\begin{equation}\label{sec5inicom}
\widehat{v}_0(|x-\widehat{x}_0|-R_0) \geq u_0(x) \hbox{ for } x\in\Omega_0,\; \widehat x_0\in\mathbb{R}^N\setminus \Lambda_{z_2}.
\end{equation}
Indeed, this can be ensured by requiring $\widehat{v}_0$ to satisfy $\widehat{v}_0(r) \geq \| u_0 \|_{ L^{\infty}(\Omega_0)}$ for $r\geq 1$.
We now define, for each $\widehat x_0\in \mathbb{R}^N\setminus \Lambda_{z_2}$,
$$\widehat{V}(t,x)=\widehat{v}(t,|x-\widehat{x}_0|)\,\hbox{ for }\,|x-\widehat{x}_0|\geq \widehat{h}(t),$$
and extend it to zero for $|x-\widehat{x}_0|< \widehat{h}(t)$ ($0\leq t\leq t_2$), then it is easily seen that
$\widehat{V}$ is the unique weak solution of the free boundary problem induced from \eqref{sec4symubd} over $[0,t_2]\times \mathbb{R}^N$ with initial function $\widehat{v}_0(|x-\widehat{x}_0|-R_0)$.
Furthermore, due to \eqref{sec5inicom}, we conclude from Theorem \ref{thmcomp} that
$$u(t,x)\leq \widehat{V}(t,x) \,\hbox{ in }\, [0,t_2]\times \mathbb{R}^N.$$
This together with \eqref{freedt2} clearly implies
\begin{equation*}
\Omega(t_2) \subset\Big\{x:\, |x-\widehat{x}_0|\geq \widehat{h}(t_2) \Big\} \subset \Big\{x:\, |x-\widehat{x}_0|\geq R_0/2 \Big\} \;\hbox{ for all } \widehat x_0\in\mathbb{R}^N\setminus \Lambda_{z_2}.
\end{equation*}
It follows that
\[
\Omega(t_2)\subset \bigcap_{\widehat x_0\in\mathbb{R}^N\setminus \Lambda_{z_2}}\Big\{x:\, |x-\widehat{x}_0|\geq R_0/2 \Big\}\subset \Lambda_{z_2}.
\]
The proof of Lemma \ref{lemupu0} is now complete.
\end{proof}
We are now ready to construct a weak supersolution of problem \eqref{eqfrfisher}. For any given small $\delta>0$, denote
$$
c^\delta:=c_*(\mu,a+\delta,b-\delta,d),
$$
and denote by $Z^{\delta}(r)$ the solution of \eqref{eqwave} with $k$, $a$, $b$ replaced by $c^{\delta}$, $a+\delta$, $b-\delta$, respectively. For $R>0$, we define
\[
\xi_R(t):=\xi_2-\frac{R+r_2}{\sin\phi}-\left[(1-\delta)^{-2}\frac{c^\delta}{\sin\phi}\right]\,t \;\hbox{ for } t\geq 0,
\]
with $r_2$ given in Lemma \ref{lemupu0} depending on $\delta>0$, and
\[
\Omega_R(t):=\Lambda_R^\phi+\xi_R(t)e_N \;\hbox{ for } t\geq 0,
\]
with
$$ \Lambda_R^\phi:=\big\{x\in\Lambda^\phi: d(x,\partial\Lambda^\phi)>R\big\}.$$
Then define
\[
\overline u(t,x)= \overline u_R(t,x):=u_R(x-\xi_R(t)e_N) \;\hbox{ for } t\geq0,\;x\in\mathbb{R}^N,
\]
with
\[
u_R(x):=\left\{\begin{array}{ll}
\medskip
(1-\delta)^{-2}Z^\delta(d(x,\partial\Lambda_R^\phi)), & x\in\Lambda_R^\phi,\\
0, & x\not\in \Lambda_R^\phi.
\end{array}
\right.
\]
We are going to show that for suitably chosen $R$, $\overline u_R(t,x)$ is a weak supersolution to the equation satisfied by
$u(t_2+t,x)$; the desired result then easily follows.
\begin{lem}\label{weaksuper}
Let $\overline{u}$ be given as above. Then there exists $R=R(\delta)$ sufficiently large such that $\overline u$
is a weak supersolution of \eqref{eqfrfisher} with $u_0(x)$ replaced by $\overline u(0,x)$.
\end{lem}
\begin{proof}
Let us observe that $\partial\Lambda^\phi_R$ is smooth and it can be decomposed into two parts, a spherical part
\[
\Sigma_R^1:=\partial B_R(0)\cap \Lambda^{\phi-\frac\pi 2},
\]
and part of the surface of the cone $\Lambda^\phi+\frac{R}{\sin\theta} e_N$ (recall $\theta=\pi-\phi$):
\[
\Sigma_R^2:=\Big(\partial\Lambda^\phi+\frac{R}{\sin\theta} e_N\Big)\setminus \Lambda^{\phi-\frac\pi 2}.
\]
Correspondingly, we can decompose $\Lambda_R^\phi$ into two parts:
$$\Lambda_R^\phi: = \Lambda_{R,1}^\phi \cup \Lambda_{R,2}^\phi \;\hbox{ with }\; \Lambda_{R,1}^\phi :=\Lambda_R^\phi \cap \Lambda^{\phi-\frac\pi 2}\; \mbox{ and }\; \Lambda_{R,2}^\phi:=\Lambda_R^\phi
\setminus \Lambda^{\phi-\frac\pi 2}$$
($\Lambda_{R,1}^\phi$ and $\Lambda_{R,2}^\phi$ are illustrated in Figure \ref{figsupersolu}).
\begin{figure}[h]
\centering
\def9cm{9cm}
\input{supersolu.pdf_tex}
\caption{The domains $\Lambda_{R,1}^\phi$ and $\Lambda_{R,2}^\phi$ }\label{figsupersolu}
\end{figure}
In a similar way, for each $t\geq 0$, we can write
\[
\Omega_R(t) =\Omega_R^1(t)\cup \Omega_R^2(t),
\]
with
$$
\Omega_R^1(t):=\Lambda_{R,1}^\phi +\xi_R(t) \; \mbox{ and }\; \Omega_R^2(t):=\Lambda_{R,2}^\phi +\xi_R(t).
$$
Clearly
\[
d(x,\partial\Lambda_R^\phi)=d(x, \Sigma_R^1)=|x|-R \mbox{ if } x\in \Lambda_{R,1}^\phi,
\]
and, by some simple geometrical calculations, for $x=(x', x_N)\in \Lambda_{R,2}^\phi$,
\[
d(x,\partial\Lambda_R^\phi)=d(x,\Sigma_R^2)=|x'|\cos\theta+x_N\sin\theta-R.
\]
It is straightforward to check that $\overline{u}$ and $\nabla_x \overline{u}$ are continuous in $\bigcup_{
t\geq 0} {\overline {{\Omega_R} (t)}}$, that $\nabla_x^2 \overline{u}$, $\overline{u}_t$ are
continuous in $\bigcup_{t>0} \overline{\Omega_R (t)}$.
Next, we show that for $R>0$ sufficiently large,
\begin{equation}\label{checkmaineq}
\overline{u}_t-d\Delta\overline{u}\geq a\overline{u}-b\overline{u}^2 \quad\hbox{for}\quad x\in{\Omega_R}(t),\,\,t>0.
\end{equation}
Denote $z:=x-\xi_R(t)e_N$; direct calculation shows that, for $x\in {\Omega}_R^1(t)$ and $t>0$,
$$
\overline{u}_t= (1-\delta)^{-4} (Z^{\delta})'(z)\frac{z_N}{|z|}\frac{c^{\delta}}{\sin\phi},
$$
and
$$d\Delta\overline{u}=(1-\delta)^{-2} \Big[ d(Z^{\delta})''(z)+\frac{d(N-1)}{|z|}(Z^{\delta})' (z) \Big].$$
Due to $|z|>R$ and $z_N\geq |z|\sin\theta=|z|\sin\phi$ for $x\in {\Omega}_R^1(t)$ (i.e., $z\in \Lambda_{R,1}^\phi$), and $(Z^{\delta})'>0$, we have
\begin{equation*
\left.\begin{array}{ll}
\displaystyle\medskip \overline{u}_t-d\Delta\overline{u}&\displaystyle \geq (1-\delta)^{-2}\Big[ (1-\delta)^{-2}c^{\delta}(Z^{\delta})'-\frac{d(N-1)}{|z|}(Z^{\delta})' - d(Z^{\delta})'' \Big]\\
\displaystyle\medskip&\displaystyle \geq (1-\delta)^{-2}\Big( \Big[(1-\delta)^{-2}c^{\delta}-\frac{d(N-1)}{R}\Big](Z^{\delta})' - d(Z^{\delta})'' \Big)\\
\displaystyle\medskip &\displaystyle \geq (1-\delta)^{-2}\Big( c^{\delta}(Z^{\delta})' +\Big[\delta c^\delta-\frac{d(N-1)}{R}\Big](Z^{\delta})' - d(Z^{\delta})'' \Big).
\end{array}\right.
\end{equation*}
Therefore, if we choose
\begin{equation}\label{chooser}
R\geq \frac{d(N-1)}{\delta c^\delta},
\end{equation}
then for $x\in\Omega_R^1(t)$,
\begin{equation*}
\overline{u}_t-d\Delta\overline{u} \geq (1-\delta)^{-2} \big[ c^{\delta}(Z^{\delta})' - d(Z^{\delta})''\big]
\geq (a+\delta)\overline{u}-(b-\delta)\overline{u}^2\geq a\overline{u}-b\overline{u}^2.
\end{equation*}
For $x\in {\Omega}_R^2(t)$ and $t>0$, it follows from a direct calculation that
$$ \overline{u}_t= (1-\delta)^{-4}c^{\delta} (Z^{\delta})',$$
and
$$d\Delta\overline{u}=(1-\delta)^{-2} \Big[ d(Z^{\delta})''+\frac{d(N-2)\cos\theta (Z^{\delta})' }{|x'|} \Big],$$
where $x':=(x_1,..., x_{N-1})\in \mathbb{R}^{N-1}$.
It is easily checked that for $z=(z', z_N)\in\Lambda_{R,2}^\phi$, we always have $|z'|\geq R\cos\theta$. Thus using $z=x-\xi_R(t)e_N\in\Lambda_{R,2}^\phi$,
we obtain
$$\frac{\cos\theta}{|x'|} \leq \frac{1}{R}.$$
Thus for $R>0$ satisfying \eqref{chooser} and $x\in\Omega_R^2(t)$, we have
\begin{equation*}
\left.\begin{array}{ll}
\displaystyle\medskip \overline{u}_t-d\Delta\overline{u}&\displaystyle \geq (1-\delta)^{-2}\Big[ (1-\delta)^{-2}c^{\delta}(Z^{\delta})'-\frac{d(N-2)}{R}(Z^{\delta})' - d(Z^{\delta})'' \Big]\\
\displaystyle\medskip &\displaystyle \geq (1-\delta)^{-2}\Big[ c^{\delta}(Z^{\delta})' +\Big(\delta c^\delta-\frac{d(N-2)}{R}\Big)(Z^{\delta})' - d(Z^{\delta})'' \Big]\\
\displaystyle\medskip &\displaystyle \geq (1-\delta)^{-2}\Big[ c^{\delta}(Z^{\delta})' - d(Z^{\delta})'' \Big]\\
&\geq (a+\delta)\overline{u}-(b-\delta)\overline{u}^2,
\end{array}\right.
\end{equation*}
and hence,
$$\overline{u}_t-d\Delta\overline{u} \geq a\overline{u}-b\overline{u}^2.$$
We have thus proved that \eqref{checkmaineq} holds for all $R$ satisfying \eqref{chooser}. We henceforth fix such an $R$.
We next define
\[
\Phi(t,x)=R-d(x,\partial\Omega_R(t))=\left\{\begin{array}{ll}
\medskip R-|x-\xi_R(t)e_N|, & x\in\Omega_R^1(t),\smallskip\\
R-\big[|x'|\cos\theta+(x_N-\xi_R(t))\sin\theta\big], & x\in\Omega_R^2(t).
\end{array}
\right.
\]
Clearly, $\Phi$ is smooth, $\Omega_R(t)=\{x: \Phi(t,x)<0\}$ and $|\nabla_x\Phi|\not=0$ for $x\in\partial\Omega_R(t)$. We next show that
\begin{equation}\label{fbdy}
\Phi_t\leq \mu \nabla_x \overline{u}\cdot\nabla_x\Phi \quad \hbox{for} \quad x\in \partial\Omega_R(t),\,\,t>0.
\end{equation}
It is straightforward to calculate that, for $x\in
\partial\Omega_R(t)$ and $t>0$,
\begin{equation*}
\nabla_x \Phi \cdot \nabla_x \overline{u}= -(1-\delta)^{-2}(Z^{\delta})'(0),
\end{equation*}
and
\begin{equation*}
\Phi_t(t,x) =\left\{
\begin{array}{ll}
\displaystyle\medskip -(1-\delta)^{-2}\frac{x_N-\xi_R(t)}{|x-\xi_R(t)e_N|}\frac{c^{\delta}}{\sin\theta} &\displaystyle\medskip \hbox{ if } x\in\Omega_R^1(t), \\
-(1-\delta)^{-2}c^{\delta} & \hbox{ if } x\in\Omega_R^2(t).
\end{array}\right.
\end{equation*}
On the other hand, it is easily seen that for any $z\in \Lambda_{R,1}^\phi$,
$
{z_N}\geq |z| \sin\theta$.
It then follows that
\[
\frac{x_N-\xi_R(t)}{|x-\xi_R(t)e_N|}\geq \sin\theta \;\;\mbox{ for } x\in\Omega_R^1(t),
\]
and hence
$$\Phi_t(t,x) \leq -(1-\delta)^{-2}c^{\delta} \mbox{ for } x\in\Omega_R(t). $$
From this and $\mu(Z^{\delta})'(0)=c^{\delta}$, we deduce \eqref{fbdy}.
We may now apply Theorem \ref{super-sub} to conclude that $\overline u$ is a weak supersolution of \eqref{eqfrfisher} with $u_0(x)$ replaced by
$\overline u(0,x)$.
\end{proof}
\begin{lem}\label{upbound}
Let $u(t,x)$ and $\Omega(t)$ be given in the statement of Theorem \ref{spreadspeed}. Then for any $\varepsilon>0$, there exists $T_3=T_3(\varepsilon)>0$ such that
$$ {\Omega(t)} \subset \Lambda^\phi-\left(\frac{c_*}{\sin\phi}+\varepsilon\right)t\,{e}_N \,\hbox{ for all } t\geq T_3.$$
\end{lem}
\begin{proof}
For any small $\delta>0$, let $t_2=t_2(\delta)$ and $R=R(\delta)$ be given in Lemma \ref{lemupu0} and Lemma \ref{weaksuper}, respectively. By Proposition \ref{semiwave}, there exists $r_3=r_3(\delta)>0$ such that
\begin{equation}\label{waveupu0}
Z^{\delta}(r) \geq (1-\delta)\frac{a+\delta}{b-\delta}\; \hbox{ for all } r\geq r_3.
\end{equation}
We first claim that
\begin{equation}\label{inidomcom}
\Omega(t_2) \subset \Omega_R(0)
\end{equation}
and
\begin{equation}\label{inicom}
u(t_2,x) \leq \overline{u}(0,x+\widetilde r_3 e_N)\quad \hbox{for } \, x\in \Omega(t_2),
\end{equation}
where
\[
\widetilde r_3:=\frac{r_3}{\sin\theta}.
\]
Indeed, it is easily seen from the definition that
\[
\Omega_R(0)\supset \Lambda_{z_2}=\Lambda^\phi+\Big(\xi_2-\frac{r_2}{\sin\phi}\Big)e_N.
\]
Thus, \eqref{inidomcom} is a consequence of $\Omega(t_2)\subset \Lambda_{z_2}$ proved in Lemma \ref{lemupu0}.
We now prove \eqref{inicom}.
For any $x\in \Omega(t_2)$, due to $\Omega(t_2)\subset \Lambda_{z_2}\subset \Omega_R(0)$, we obtain
\[\begin{array}{rl}
\displaystyle d\Big(x+\frac{r_3}{\sin\theta}e_N,\; \partial \Omega_R(0)\Big)\geq& \displaystyle d\Big(x+\frac{r_3}{\sin\theta}e_N,\; \partial \Lambda_{z_2}\Big)\medskip\\
=&\displaystyle d\Big(x,\; \partial \Lambda_{z_2}-\frac{r_3}{\sin\theta}e_N\Big)\medskip\\
\geq &\displaystyle r_3+d(x,\;\partial\Lambda_{z_2})\geq r_3.
\end{array}
\]
Thus, for $x\in\Omega(t_2)$, due to $(Z^{\delta})'(r)>0$ in $(0,\infty)$, we have
\begin{equation*}
\left.\begin{array}{ll}
\displaystyle\medskip \overline{u}(0,x+\widetilde r_3e_N)\!\!\! &\displaystyle\medskip= (1-\delta)^{-2}Z^{\delta}\Big(d\Big(x+\frac{r_3}{\sin\theta}e_N-\xi_{R}(0)e_N,\; \partial\Lambda_R^\phi\Big)\Big)\\
&\displaystyle\medskip= (1-\delta)^{-2}Z^{\delta}\Big(d\Big(x+\frac{r_3}{\sin\theta}e_N,\;\partial\Omega_R(0)\Big)\Big)\\
&\geq (1-\delta)^{-2}Z^\delta(r_3).
\end{array}\right.
\end{equation*}
This together with \eqref{upu0} and \eqref{waveupu0} implies that
$$\overline{u}(0,x+\widetilde r_3 e_N) \geq (1-\delta)^{-1}\frac{a+\delta}{b-\delta}\geq u(t_2,x) \;\hbox{ for } x\in\Omega(t_2),$$
which proves \eqref{inicom}.
By Lemma \ref{weaksuper},
$\overline{u}(t,x+\widetilde r_3 e_N)$ is a weak supersolution of problem \eqref{eqfrfisher} with $u_0$ replaced by $\overline{u}(0,x+\widetilde r_3 e_N)$, and since $u(t+t_2,x)$ is a weak solution of \eqref{eqfrfisher} with $u_0$ replaced by $u(t_2,x)$, it follows from \eqref{inidomcom}, \eqref{inicom} and Theorem \ref{thmcomp} that
$$u(t+t_2,x)\leq \overline{u}(t,x+\widetilde r_3 e_N)\,\hbox{ in } [0,\infty)\times \mathbb{R}^N,$$
and hence,
$$
\Omega(t) \subset \Omega_R(t-t_2)-\widetilde r_3 e_N=\Lambda^\phi_R+\left[ \xi_R(t-t_2)-\widetilde r_3\right] e_N \,\hbox{ for } \,t\geq t_2.
$$
Since $\Lambda_R^\phi\subset \Lambda^\phi$,
we thus obtain
\[
\Omega(t)\subset \Lambda^\phi+\left[ \xi_R(t-t_2)-\widetilde r_3\right] e_N \,\hbox{ for } \,t\geq t_2.
\]
Clearly,
\[
\xi_R(t-t_2)-\widetilde r_3=M-(1-\delta)^{-2}\frac{c^\delta}{\sin\theta}\, t,
\]
with
\[M=M_\delta:=\xi_2-\frac{R+r_2+r_3}{\sin\theta}+(1-\delta)^{-2}\frac{c^\delta}{\sin\theta}\, t_2.
\]
Since
\[
\lim_{\delta\to 0} (1-\delta)^{-2}c^{\delta}=c_*,
\]
for any small $\varepsilon>0$, we can find some $\delta'_\varepsilon\in (0,\varepsilon)$ such that
\begin{equation*}
-(1-\delta'_{\varepsilon})^{-2}\frac{c^{\delta'_{\varepsilon}}}{\sin\theta}\geq - \frac{ c_* }{\sin\theta}-\frac\varepsilon 2.
\end{equation*}
We now fix $\delta=\delta'_{\varepsilon}$ and obtain
\begin{equation*}
\xi_R(t-t_2)-\widetilde r_3 \geq M-\left(\frac{c_*}{\sin\theta}+\frac{\varepsilon}{2}\right)t
= M+\frac{\varepsilon}{2}t -\left(\frac{c_*}{\sin\theta}+{\varepsilon}\right)t \geq -\left(\frac{c_*}{\sin\theta}+{\varepsilon}\right)t
\end{equation*}
for $t\geq T_3$ with
\[
T_3=T_3(\varepsilon):
= \max\Big\{t_2, \, \frac{2}{\varepsilon} |M| \Big\}.
\]
Therefore,
\[
\Omega(t)\subset \Lambda^\phi -\left(\frac{c_*}{\sin\theta}+{\varepsilon}\right)t \,e_N \mbox{ for $t\geq T_3$,}
\]
as desired.
\end{proof}
It is easily seen that \eqref{Omega(t)} in Theorem \ref{spreadspeed} follows from Lemmas \ref{lowbound} and \ref{upbound}, while \eqref{outconebehu} is a direct consequence of \eqref{infubound} and \eqref{upu0}. Thus, Theorem \ref{spreadspeed} is now proved.
Theorem \ref{spreadspeed} implies that if $\Omega_0$ satisfies \eqref{outscone} with $\phi\in (\pi/2, \pi)$, then for all large time, the free boundary $\partial\Omega(t)$ propagates to infinity in the negative $x_N$-direction with speed $c_*/\sin\phi$. Moreover, given any direction
$\nu\in\mathbb S^{N-1}$ pointing outward of $\Lambda^\phi$, if we denote
\[
\psi:=\arccos \left[(-e_N)\cdot\nu\right] \mbox{ (and so $\psi\in (0,\theta)=(0,\pi-\phi)$)},
\]
then the spreading of $\Omega(t)$ in the direction $\nu$ is roughly at speed $c_*/(\sin\phi\cos\psi)$.
In sharp contrast, we will show in
the following theorem, that when $\Omega_0$ satisfies \eqref{outscone} with $\phi\in (0, \pi/2)$, the spreading of $\Omega(t)$
in a set of directions $\nu\in\mathbb S^{N-1}$ pointing outward of $\Lambda^\phi$, including $\nu=-e_N$, is roughly
at the speed $c_*$. (This set of directions $\nu$ is given by $\Sigma_\phi$ below.)
\begin{thm}\label{spreadspeed1}
Let $u(t,x)$ be the unique weak solution of problem \eqref{eqfrfisher} with $\Omega_0$ satisfying \eqref{outscone} for some $\phi
\in (0,\pi/2)$, and $u_0$ satisfying \eqref{assumeu0} and \eqref{addau0}.
Denote $\Omega(t)=\big\{x:\, u(t,x)>0\big\}$. Then for any $\varepsilon>0$, there exists $\widetilde{T}=\widetilde{T}(\varepsilon)>0$ such that
\begin{equation}\label{inconeset}
N\big[\Lambda^\phi, (c_*-\varepsilon)t\big]\, \subset \,{\Omega(t)}\, \subset \, N\big[\Lambda^\phi, (c_*+\varepsilon)t\big] \,\hbox{ for all } t\geq \widetilde{T}.
\end{equation}
Moreover, we have
\begin{equation}\label{inconebehu}
\lim_{t\to\infty}\left[{\sup}_{x \in N[\Lambda^\phi, (c_*-\varepsilon)t]} \Big|u(t,x)-\frac{a}{b} \Big|\right]=0.
\end{equation}
\end{thm}
\begin{proof}
We prove this theorem by two steps.
\smallskip
\noindent
{\bf Step 1:} {\it Proof of the first relation of \eqref{inconeset} and \eqref{inconebehu}.}
Due to the assumptions \eqref{outscone} and \eqref{addau0}, we can find $\xi_0>\xi_1$ such that
$$
u_0(x) \geq \sigma_0:= \frac{1}{2}\liminf_{d(\widetilde{x},\partial \Omega_0)\to\infty,\; \widetilde{x}\in\Omega_0} u_0(\widetilde{x}) \hbox{ for all }
x\in \Lambda^\phi+ \xi_0e_N. $$
Let $R^*>0$ be the positive constant given in \eqref{radius}.
Then we choose a radial function $\widetilde{v}_0\in C^2([0,R^*])$ such that
$$0< \widetilde{v}_0(x)\leq \sigma_0 \hbox{ in } [0,R^*),\quad \widetilde{v}_0'(0)=\widetilde{v}_0(R^*)=0. $$
Next, for any fixed $x_0\in \mathbb{R}^N$ with $B_{R^*}(x_0) \subset \Lambda^\phi+\xi_0e_N$, set $r=|x-x_0|$, and consider the following radially symmetric free boundary problem
\begin{equation}\label{eqsymball}
\left \{ \begin{array}{ll}
\displaystyle\medskip \widetilde{v}_t-d \Delta \widetilde{v}=\widetilde{v} (a-b \widetilde{v}), \;\;&t>0, \; 0<r<\widetilde{k}(t), \\
\displaystyle\medskip \widetilde{v}_r (t,0)=0, \;\; \widetilde{v}(t,\widetilde{k}(t))=0, \;\; &t>0,\\
\displaystyle\medskip \widetilde{k}'(t)=-\mu v_r (t, \widetilde{k}(t)), \;\; &t>0,\\
\displaystyle \widetilde{k}(0)=R^*, \;\;\; \widetilde{v}(0,r)=\widetilde{v}_0 (r), \;\; & 0 \leq r \leq R^*.
\end{array} \right.
\end{equation}
It follows from \cite[Theorems 2.1, 2.5 and Corollary 3.7]{DG1} that problem \eqref{eqsymball} admits a unique classical solution $(\widetilde{v}(t,r),\widetilde{k}(t))$ defined for all $t\geq 0$, and
\begin{equation}\label{speedradial}
\lim_{t\to\infty} \frac{\widetilde{k}(t)}{t}= c_*.
\end{equation}
Furthermore, applying \cite[Theorem 6.4]{DG2} to this problem, we have
\begin{equation}\label{asym-tilde-v}
\lim_{t\to\infty} \max_{r\leq (c_*-\varepsilon/2)t } \left| \widetilde{v}(t,r)-\frac{a}{b} \right|=0
\end{equation}
for every small $\varepsilon>0$.
On the other hand, by our choices of $\widetilde{v}_0$ and $\xi_0$, we have
$$\widetilde{v}_0(|x-x_0|)\leq u_0(x) \quad\hbox{ for all }\, x\in B_{R^*}(x_0). $$
Then extending $\widetilde{v}(t,|x-x_0|)$ to be zero for $|x-x_0|> \widetilde{k}(t)$ and applying the comparison principle Theorem \ref{thmcomp}, we obtain
\begin{equation}\label{comp-tildev-u}
\widetilde{v}(t,|x-x_0|) \leq u(t,x) \quad\hbox{for all }\, t\geq 0, \,x\in\mathbb{R}^N,
\end{equation}
and hence,
$$\Big\{x\in\mathbb{R}^N:\,\,|x-x_0|\leq \widetilde{k}(t) \Big\}\subset \Omega(t) \quad\hbox{for all }\, t\geq 0. $$
This together with \eqref{speedradial} implies that, for any $\varepsilon>0$, there exists $\widetilde{T}_1=\widetilde{T}_1(\varepsilon)>0$ such that
$$\Big\{x\in\mathbb{R}^N:\,\,|x-x_0|\leq \big(c_*-\frac\varepsilon 2\big)t \Big\}\subset \Omega(t) \quad\hbox{for all }\, t\geq \widetilde{T}_1.$$
Note that the above analysis remains valid if $x_0$ replaced by any point $\widetilde{x}_0$ such that $B_{R^*}(\widetilde{x}_0) \subset \Lambda^\phi+\xi_0e_N$, and that the constant $\widetilde{T}_1$ is independent of the choice of such $\widetilde{x}_0$.
It then follows that
$$\bigcup_{ B_{R^*}(x_0) \subset \Lambda^\phi+\xi_0e_N } \Big\{x\in\mathbb{R}^N:\,\,|x-x_0|\leq
\big(c_*-\frac\varepsilon 2\big)t \Big\}\,\subset\, \Omega(t) \quad\hbox{for all }\, t\geq \widetilde{T}_1. $$
Furthermore, it is easily seen that there exists $\widetilde{T}_2=\widetilde{T}_2(\varepsilon) \geq \widetilde{T}_1$ such that, for all $ t\geq \widetilde{T}_2$,
\begin{equation}\label{comp-N-II}
N[\Lambda^\phi, (c_*-\varepsilon)t]\,\subset\, \bigcup_{ B_{R^*}(x_0) \subset \Lambda^\phi+\xi_0e_N } \Big\{x\in\mathbb{R}^N:\,\,|x-x_0|\leq
\big(c_*-\frac\varepsilon 2\big)t \Big\}.
\end{equation}
We thus obtain the first relation of \eqref{inconeset}.
To complete the proof of this step, it remains to show \eqref{inconebehu}. On the one hand, by \eqref{comp-tildev-u} and \eqref{comp-N-II}, we have
$${\inf}_{x \in N[\Lambda^\phi, (c_*-\varepsilon)t]} u(t,x)\, \geq \,\inf \left\{ \min_{|x-x_0|\leq
(c_*-\varepsilon/2)t} \widetilde{v}(t,|x-x_0|):\, {B_{R^*}(x_0) \subset \Lambda^\phi+\xi_0e_N} \right\}$$
for all $t\geq \widetilde{T}_2$. It further follows from \eqref{asym-tilde-v} that
$$\liminf_{t\to\infty}{\inf}_{x \in N[\Lambda^\phi, (c_*-\varepsilon)t]} u(t,x) \,\geq \,\frac{a}{b}.$$
On the other hand, since $u_0$ is bounded, by the arguments used at the beginning of the proof of Lemma \ref{lemupu0}, we have
$$\limsup_{t\to\infty} \sup_{x\in\mathbb{R}^N} u(t,x) \leq \frac{a}{b}.$$
Combining the above, we immediately obtain \eqref{inconebehu}.
\smallskip
\noindent
{\bf Step 2:} {\it Proof of the second relation of \eqref{inconeset}. }
Choose a one-dimensional function $\widetilde{w}_0\in C^2((-\infty,1])\cap L^{\infty}((-\infty,1])$ such that
\begin{equation*}
\widetilde{w}_0(y)\geq \| u_0 \|_{L^{\infty}(\Omega_0)} \hbox{ in } (-\infty,0],\quad \widetilde{w}_0(x)>0 \hbox{ in } (0,1) \quad\hbox{and}\quad \widetilde{w}_0(1)=0.
\end{equation*}
Then we consider the following one-dimensional free boundary problem
\begin{equation}\label{c2eqlow}
\left \{ \begin{array}{ll}
\displaystyle\medskip \widetilde{w}_t-d \widetilde{w}_{yy}=\widetilde{w}(a-b \widetilde{w}), \;\;&t>0, \; -\infty<y<\widetilde{\rho}(t), \\
\displaystyle\medskip \widetilde{w}(t,\widetilde{\rho}(t))=0, \;\; &t>0,\\
\displaystyle\medskip \widetilde{\rho}'(t)=-\mu \widetilde{w}_y (t,\widetilde{\rho}(t)), \;\; &t>0,\\
\displaystyle \widetilde{\rho}(0)=1, \;\;\; \widetilde{w}(0,y)=\widetilde{w}_0 (y), \;\; & -\infty<y\leq 1.
\end{array} \right.
\end{equation}
It follows from \cite[Theorem 2.11]{DDL} that, \eqref{c2eqlow} admits a (unique)
classical solution $(\widetilde{w}(t,y), \widetilde{\rho}(t))$ defined for all $t>0$ and
$\widetilde{\rho}'(t)>0$, $\widetilde{w}(t,y)>0$ for $ -\infty<y<\widetilde{\rho}(t)$, $t>0$.
For any
\[
\nu\in \Sigma_\phi:= \Big\{x\in\mathbb S^{N-1}: \arccos(x\cdot e_N)\in [\phi+\frac\pi 2, \pi]\Big\},
\]
define
$$\widetilde{\Omega}_{\nu}(t)=\Big\{x\in\mathbb{R}^N: x\cdot \nu \leq \widetilde{\rho}(t)\Big\}\quad\hbox{and}\quad
\widetilde{w}_\nu(t,x)=\widetilde{w}(t,x\cdot\nu).$$
Clearly, we have $\Omega_0-\xi_2e_N \subset \widetilde{\Omega}_{\nu}(0)$, and $u_0(\cdot+\xi_2 e_N)\leq \widetilde{w}_{\nu}(0,\cdot)$ in
$\Omega_0-\xi_2e_N$. Then by similar comparison arguments as those used in the proof of Lemma \ref{lowbound}, we obtain
\begin{equation}\label{onedupsolu}
\Omega(t)-\xi_2e_N \subset \widetilde{\Omega}_{\nu}(t) \quad\hbox{for all }\,t\geq 0,\;\nu\in \Sigma_\phi.
\end{equation}
Furthermore, it follows from the proof of \cite[Theorem 4.2]{DL} with similar modifications as those given in the proof of Lemma \ref{eslowra} that, for the given $\varepsilon>0$, there exists $\widetilde{T}_3=\widetilde{T}_3(\varepsilon)>0$ such that
\begin{equation*}
\widetilde{\rho}(t) \leq \big(c_{*}+\frac\varepsilon 2\big) t\,\hbox{ for all } t\geq \widetilde{T}_3.
\end{equation*}
This together with \eqref{onedupsolu} implies
$$
\Omega(t)-\xi_2 e_N \subset \Big\{x\in \mathbb{R}^N:\,\, x\cdot \nu \leq \big(c_{*}+\frac\varepsilon 2\big) t \Big\} \, \hbox{ for all }\, t\geq \widetilde{T}_3,\;\nu\in \Sigma_\phi.$$
Since
\[
N[\Lambda^\phi, \big(c_{*}+\frac\varepsilon 2\big) t]=\bigcup_{\nu\in\Sigma_\phi}\Big\{x\in \mathbb{R}^N:\,\, x\cdot \nu \leq \big(c_{*}+\frac\varepsilon 2\big) t \Big\},
\]
by enlarging $\widetilde T_3$ if necessary (depending on $\xi_2$ and $\varepsilon$), we obtain
\[
\Omega(t)\subset N\big[\Lambda^\phi, (c_*+\varepsilon)t\big] \; \hbox{ for all }\, t\geq \widetilde{T}_3.
\]
which clearly gives the second relation of \eqref{inconeset}.
\end{proof}
\begin{rem}\label{sec6rem2}
The estimates in Theorem \ref{spreadspeed1} can be improved by making use of sharp estimates for the spreading speed for
one space dimension free boundary problems in \cite{DMZ1} and for radially symmetric free boundary problems
in \cite{DMZ2}. We leave the details to the interested reader.
\end{rem}
\section{Appendix}
This appendix is concerned with the existence and uniqueness of classical solutions to an auxiliary radially symmetric problem with initial
range the exterior of a ball. These results have been used to construct the weak supersolution for problem \eqref{eqfrfisher} in the proof of Theorem \ref{spreadspeed}, and here we consider a more general problem which might have other applications.
More precisely, for any given $T>0$, $C_1>0$ and $C_2>0$, we consider the following radially symmetric free boundary problem\begin{equation}\label{symubd}
\left \{ \begin{array}{ll}
\displaystyle\medskip v_t-d \Delta v=\widetilde{g}(r,v), \;\; & 0<t<T, \; h(t)<r<\infty,\\
\displaystyle\medskip v(t, h(t))=0, \;\; &0<t<T,\\
\displaystyle\medskip h'(t)=-\mu v_r (t, h(t)), \;\; &0<t<T,\\
h(0)=R_0, \;\;\; v(0, r)=v_0 (r-R_0), \;\; &R_0 \leq r<\infty,
\end{array} \right.
\end{equation}
where $ \Delta v=v_{rr}+\frac{N-1}{r}v_r$, $R_0>1$ is a constant to be determined by $T,\,C_1,\,C_2$ later, and $v_0$ is a given function in $ C^2([0,\infty))$ satisfying
\begin{equation}\label{symini1}
0<v_0(r) \leq C_1\,\hbox{ for } \,r\in (0,\infty),\quad v_0(0)=0,\quad \|v_0\|_{C^1([0,\infty))}\leq C_2.
\end{equation}
Here we assume that $\widetilde{g}(r,v)$ is a continuous function defined over $\mathbb{R}^+\times \mathbb{R}^+$ satisfying
\begin{equation}\label{tildeg}
\left. \begin{array} {ll}
\hbox{(i)}& \mbox{$\widetilde g(r,v)$ is H\"{o}lder continuous in $r\geq 0$, $v\geq 0$},\\
\hbox{(ii)} &\smallskip \widetilde{g}(r,v) \hbox{ is locally Lipschitz in } v \hbox{ uniformly for }r\geq 0,\,\,\\
\hbox{(iii)} &\smallskip \widetilde{g}(r,0) \equiv 0,\,\,\\
\hbox{(iv)} & \hbox{there exists } K>0 \hbox{ such that } \widetilde{g}(r,v) \leq Kv \hbox{ for all } r \geq 0 \hbox{ and } v \geq 0.
\end{array} \right\}
\end{equation}
\begin{prop}\label{pexsym}
Assume that \eqref{tildeg} is satisfied. For any $T>0$, $C_1>0$ and $C_2>0$, there exists a constant $R_0>1$ depending on $T$, $C_1$ and $C_2$ such that for any $v_0$ satisfying \eqref{symini1},
problem \eqref{symubd} admits a unique solution $(v(t,r), h(t))$ with $h \in C^{1} ([0, T])$, $v \in C^{1,2}(D_T)$, where $D_T=\big\{(t,r): \; t \in [0,T], \; r \in [h(t), \infty)\big\}$. Moreover,
$v(t,r)>0$, $h'(t)<0$ for $0<t\leq T,\,h(t) < r<\infty$, and
\begin{equation}\label{simpty}
h(T)\geq \frac{R_0}{2}.
\end{equation}
\end{prop}
\begin{proof} For given $R_0>1$, following the proof of \cite[Theorem 2.1]{DG1}
we can show that \eqref{symubd} has a unique solution for some small $T>0$. The proof involves the straightening of the free boundary,
and different from \cite{DG1}, the resulting problem here is over an unbounded interval for the new space variable. However, the estimates easily carry over (by using suitable interior estimates, similar to a related situation treated in \cite{DL2}) and so we obtain the local existence and uniqueness all the same. Moreover, all the stated properties in the proposition, except \eqref{simpty}, also hold.
Furthermore, the solution can be extended as long as $h(t)>0$. Let $T_\infty=T_\infty(R_0)$ be the maximal existence time of the solution. If $h(t)\geq R_0/2$ for all $t\in (0, T_\infty)$, then necessarily $T_\infty=\infty$ and thus $h(T)> R_0/2$, and there is nothing left to prove.
Suppose now $h(t_0)<R_0/2$ for some $t_0\in (0, T_\infty)$. Since $h(0)=R_0$, we can find $T_0=T_0(R_0)\in (0, t_0)$ such that
$h(T_0)=R_0/2$. We are going to show that $T_0(R_0)>T$ provided that $R_0$ is sufficiently large, which clearly implies
$h(T)>R_0/2$, as desired. We use an indirect argument and assume that $T_0(R_0)\leq T$ for all $R_0>1$.
By the assumption \eqref{tildeg}, it follows from the parabolic comparison principle that $v(t,r)\leq \bar{v}(t)$ for $r>h(t)$ and $0\leq t\leq T_0$, where $\bar{v}(t)$ is the solution to
$$\frac{d \bar{v}}{dt}=K\bar{v} \hbox{ for } t>0; \quad \bar{v}(0)=\|v_0\|_{L^{\infty}([0,\infty))}.$$
Clearly, $\bar{v}(t)=\|{v}_0\|_{L^{\infty}([0,\infty))} \mathrm{e}^{Kt}$ for $t\geq 0$. Thus, we have
$$v(t,r) \leq C_3:=C_1 \mathrm{e}^{KT} \,\hbox{ for } \; r>h(t),\, 0\leq t\leq T_0.$$
Next we prove that there exists a positive constant $C_4$ independent of $R_0$ such that
\begin{equation}\label{sec5esti1}
-C_4 \leq h'(t)<0 \;\; \hbox{ for } \; t \in (0, T_0].
\end{equation}
This would lead to a contradiction, since it follows that
\[
R_0/2=h(T_0)\geq h(0)-C_4T_0\geq R_0-C_4T>R_0/2 \mbox{ for all large } R_0.
\]
Therefore to complete the proof, it suffices to show \eqref{sec5esti1}.
To this end, for $M>0$ to be determined later, we define
$$\Omega=\Omega_{M}:=\Big\{(t,r): \; 0<t\leq T_0, \;\; h(t)<r<h(t)+M^{-1} \Big\}$$
and construct an auxiliary function
$$w(t,r):=C_3 \big[2M (r-h(t))-M^2 (r-h(t))^2\big]\in (0, C_3)\;\mbox{ for } \; (t,r)\in \Omega.$$
We will show that for some suitable choice of $M>0$, $w(t,r) \geq v(t,r)$ holds over $\Omega$.
Direct calculations give, for $(t,r) \in \Omega$,
$$w_t=2C_3 M \big(-h'(t)\big ) \big[1-M(r-h(t))\big] \geq 0,$$
and
$$-w_r=-2C_3 M \big[1-M (r-h(t))\big]\geq -2C_3M,\quad -w_{rr}=2C_3 M^2. $$
Making use of $1/2\leq R_0/2\leq h(t)$ for $t\in (0, T_0]$, we obtain, for $(t,r)\in\Omega$,
\begin{equation*}
w_t-d \Big(w_{rr}+\frac{N-1}{r} w_r \Big) \geq 2dC_3 M^2-2dC_3M\frac{N-1}{r}\geq 2dC_3\big[M^2-2(N-1)M\big].
\end{equation*}
Thus, if we choose
$$M\geq (N-1)+\sqrt{\frac{K}{2d}+(N-1)^2},$$
where $K$ is given in \eqref{tildeg}, then
\begin{equation*}
w_t-d \Big( w_{rr}+\frac{N-1}{r} w_r \Big) \geq KC_3\geq \widetilde{g}(r,w) \mbox{ for } (t,r)\in\Omega.
\end{equation*}
Let us also note that for $t\in (0,T_0]$,
$$w(t, h(t)+M^{-1})=C_3 \geq v(t, h(t)+M^{-1}), $$ and
$$w(t,h(t))=0=v(t, h(t)).$$
Thus, if our choice of $M$ also ensures
$$v(0,r) \leq w(0, r) \,\hbox{ for }\, r \in [R_0, R_0+M^{-1}],$$
then we can apply the maximum principle to $w-v$ over $\Omega$ to
deduce that $v(t,r) \leq w(t,r)$ for $(t,r) \in \Omega$. It would then follow that
$$v_r (t, h(t)) \leq w_r (t, h(t))=2MC_3,$$
and so
$$h'(t)=-\mu v_r (t,h(t)) \geq -C_4:=-2MC_3 \mu,$$
as we wanted.
To complete the proof, we calculate
$$w_r (0, r)=2C_3M \big[1-M(r-R_0)\big] \geq C_3 M \;\;\; \mbox{for $r \in [R_0, R_0+(2M)^{-1}]$}.$$
Therefore, upon choosing
$$M:=\max \Big\{(N-1)+\sqrt{\frac{K}{2d}+(N-1)^2}, \;\; \frac{2 C_2}{C_3} \Big \},$$
we will have
$$w_r (0, r) \geq 2C_2> v_0'(r)=v_r(0,r) \;\;\; \mbox{for $[R_0, R_0+(2M)^{-1}]$}.$$
Since $w(0,R_0)=v(0,R_0)=0$, the above inequality implies
$$w(0,r) \geq v (0,r) \;\;\; \mbox{for $r \in [R_0, R_0+(2M)^{-1}]$}.$$
For $r \in [R_0+(2M)^{-1}, R_0+M^{-1}]$, we have $w(0,r) \geq C_3/2$ and
$$v(0,r) \leq \|v(0,\cdot) \|_{C^1 ([R_0,R_0+M^{-1}])} M^{-1}\leq C_2 M^{-1}
\leq \frac{C_3}{2},$$
which clearly gives $v (0,r) \leq w(0,r)$. Since $M$ is independent of $R_0$, this completes the proof of \eqref{sec5esti1}.
\end{proof}
|
1,116,691,497,525 | arxiv | \section{INTRODUCTION}
\setcounter{equation}{0}
The electroweak phase transition is at present the
object of extensive investigations \cite{Sin}. If the
phase transition is first order, which is possibly the case
if the mass of the Higgs boson is not too large,
the phase transition occurs via bubble nucleation. Bubble nucleation
can have various consequences for cosmology in the early universe.
The possibility of baryogenesis in bubble walls has been investigated
recently by many authors
(see e.g. \cite{KuRuSha,CoKaNe}), reheating
after the phase transition could be
mediated by bubble nucleation and subsequent coalescence, the creation
of inhomogeneities by bubble formation
could be observable (see e. g. \cite{TuWeWi,LiLeTu,DLHLL} for
representative discussions of the physics of bubble nucleation and
growth).
Bubble nucleation is described usually within the
reaction rate theory formulation
of Langer \cite{La} or, equivalently, the semiclassical
approach to quantum field theory by
Coleman and Callan \cite{Co,CaCo}.
This formulation requires the existence
of a saddle point in configuration space, the minimal bubble,
with one unstable mode, possible zero modes and real frequency
fluctuation modes. The leading term in the tunnelling rate is given
by the negative exponential of the minimal bubble action,
the corrections arise from integrating
out the fluctuations in the Gaussian approximation, leading to
a fluctuation determinant prefactor whose negative logarithm
is the 1-loop effective action. If the leading approximation is
good this prefactor should be of order $1$, substantial prefactors
have however been found in the case of the sphaleron transition,
both from bosonic \cite{CarLi,BaaJu,DyaGoe1} as also fermionic
\cite{DyaGoe2} fluctuations. It is therefore of interest
to investigate how strongly these prefactors modify the
leading order approximation to the bubble nucleation rate.
Here we present an exact computation
of the bosonic fluctuation determinant
of the critical bubble. As the basic action is determined by
the usual Higgs potential with just one minimum at the
classical expectation value, some fluctuation effects
have to be included already at the tree level in order to
allow for minimal bubble solutions. The exact fluctuation
determinant should then reproduce those in order to justify
this modification of the leading order action.
Following the basic work of Coleman and Weinberg \cite{CoWe}
such modified
actions have been proposed by many authors
\cite{DLHLL,KiLi,Sha,AnHa} and used to describe
the bubble nucleation in leading order. To be specific
we use here the one given by Dine et al.\cite{DLHLL}
which was also the basis of a recent approximate computation
of the 1-loop Higgs fluctuations by Kripfganz et al.
\cite{KriLaSch}.
The plan of this paper is as follows:
In the next section we will introduce the model and set up
the basic relations for the bubble nucleation rate.
In section 3 we will discuss the structure of the
fluctuation operator, in particular its
partial wave decomposition.
The computation of its determinant, based on a very useful
theorem, will be described in section 4.
In the final section we will present some
results and conclusions.
\section{Basic relations}
\setcounter{equation}{0}
The three-dimensional high-temperature action
is given, in the formulation by Dine et al. \cite{DLHLL},
by
\begin{eqnarray} \label{htac}
S_{ht} & =&\frac{1}{g_3(T)^2}
\int d^3x \left[\frac{1}{4}F_{ij}F_{ij}+
\frac{1}{2}(D_i\Phi)^\dagger (D_i \Phi)
+ V_{ht}(\bfphi^\dagger\bfphi) \right. \nonumber \\
&&\left.+ \frac{1}{2} A_0 \left(-D_iD_i +\frac{1}{4} \bfphi^\dagger\bfphi
\right )A_0 \right] \; .
\end{eqnarray}
Here the coordinates and fields have been rescaled as \cite{CarMcL}
\begin{equation} \label{scaling1}
\vec x \to \frac{\vec x}{g v(T)}, \;
{\bf \Phi} \to v(T) {\bf \Phi}, \; A \to v(T) A \; .
\end{equation}
The vacuum expectation value $v(T)$ is defined as
\begin{equation} \label{scale1}
v^2(T)=\frac{2D}{\lambda_T} (T_0^2 - T^2)\;.
\end{equation}
$T_0$ is the temperature at which the high-temperature
potential $V_{ht}$ changes its extremum at
${\bf \Phi} = 0$ from a minimum at $T > T_0$ to a maximum at
$T < T_0$. The temperature dependent coupling of the
three-dimensional theory is defined as
\begin{equation} \label{htcoupl}
g_3(T)^2 = \frac{gT}{v(T)} \; .
\end{equation}
In terms of the zero temperature parameters
we have $m_W = g v_0/2$, $m_H = \sqrt{2\lambda} v_0$ with
$v_0 = 246 $ GeV and we use the definitions of Dine et al.
\cite{DLHLL} modified by setting $\Theta_W=0$ and therefore
$m_W=m_Z$ :
\begin{eqnarray} \label{coeffs}
D&=& (3m_W^2+2m_t^2)/8v_0^2 \nonumber \\
E&=& 3 g^3/32 \pi \nonumber \\
B&=& 3 ( 3m_W^4 - 4 m_t^4)/64\pi^2v_0^4 \nonumber\\
T_0^2&=& (m_H^4-8 v_0^2 B)/4D \\
\lambda_T&=& \lambda -3(3m_W^4\ln\frac{m_w^2}{a_BT^2}
-4 m_t^4 \ln\frac{m_t^2}{a_F T^2})/16 \pi^2 v_0^4 \; .
\end{eqnarray}
In terms of these parameters the high-temperature
potential is given by
\begin{equation} \label{htpot1}
V_{ht}(\bfphi^\dagger\bfphi) = \frac{\lambda_T}{4g^2}
\left( (\bfphi^\dagger\bfphi)^2- 2\bfphi^\dagger\bfphi -\frac{4 E}{\lambda_T v(T)}
(\bfphi^\dagger\bfphi)^{3/2} \right) \; .
\end{equation}
The rescaling Eq. (\ref{scaling1}) with the
scale $v(T)$ makes sense only for $T < T_0$. On the other hand
the high-temperature potential has, before rescaling, a
secondary minimum at $|{\bf \Phi}| = \tilde v (T)$ with
\begin{equation}
\tilde v (T) = \frac{3 E T}{2\lambda}+
\sqrt{\left(\frac{3ET}{2\lambda}\right)^2+v^2(T)} \; .
\end{equation}
This minimum is degenerate with the one at ${\bf \Phi} =0$
at a temperature defined implicitly by
\begin{equation}
T_C = T_0/\sqrt{1 - E^2/\lambda_{T_C}} \; .
\end{equation}
$T_C$ marks the onset of bubble formation by
thermal barrier transition.
In the work of Hellmund et al. \cite{HeKriSch} and
Kripfganz et al. \cite{KriLaSch} the vacuum expectation
value of the broken symmetry phase $\tilde v (T)$ is chosen
for the rescaling of the fields, i.e. in Eq. (\ref{scaling1})
$v(T)$ is replaced by $\tilde v(T)$, the high-temperature
coupling constant
Eq.(\ref{htcoupl}) is redefined analoguously and
denoted\footnote{
Our notation differs from the one of Refs. \cite{KriLaSch,HeKriSch}.}
as $\tilde g_3 (T)$.
By this change of scale the high-temperature potential changes
as well; it becomes \footnote{We do not introduce a tilde for
the rescaled fields.} \cite{HeKriSch}
\begin{equation} \label{htpot2}
V_{ht}(\bfphi^\dagger\bfphi) =
\frac{\lambda_T}{4g^2} \left( (\bfphi^\dagger\bfphi)^2-
\epsilon(T) (\bfphi^\dagger\bfphi)^{3/2} + (\frac{3}{2}
\epsilon(T)-2)\bfphi^\dagger\bfphi \right )
\end{equation}
with
\begin{equation} \label{epsdef}
\epsilon(T)=\frac{4}{3}\left ( 1 - \frac{v(T)^2}{\tilde v (T)^2}
\right ) \; .
\end{equation}
The action and its rescaling differ slightly from that of
Hellmund et al. \cite{HeKriSch} and of Kripfganz et al. \cite{KriLaSch}.
In contrast to the former we do not mimic the influence of a Debye
mass by decoupling the longitudinal degrees of freedom.
In contrast to the second one we include only the $\Phi^3$
contribution of
gauge field and would-be Goldstone degrees of freedom as in
Ref. \cite{Sha}. This form of the $\Phi^3$ contribution was
found to yield a good approximation for the exact results
in the case of the sphaleron \cite{CarLi,BaaJu}, at least in the case
$m_H/m_W \ll 1$. We will find, indeed, that this term dominates
the effective action.
The process of bubble nucleation is - within the approach
of Langer \cite{La} and Coleman and Callan
\cite{Co,CaCo}, followed by the work of
Affleck \cite{Af}, Linde \cite{Li} and others
- described by the rate
\begin{equation} \label{rate}
\Gamma/V = \frac{\omega_-}{2 \pi} \left (
\frac{\tilde S}{2\pi}\right )^{3/2}\exp(-\tilde S)~ {\cal J}^{-1/2}
\; . \end{equation}
Here $\tilde S$ is the high-temperature action, Eq. (\ref{htac}),
with the new rescaling, minimized by a classical minimal
bubble configuration (see below), $\cal J$ is the fluctuation
determinant which describes the next-to-leading part of the
semiclassical approach and which will be defined below;
its logarithm is related to the 1-loop
effective action by
\begin{equation}
S^{1-l}_{eff} = \frac{1}{2} \ln {\cal J} \; .
\end{equation}
Finally $\omega_-$ is the absolute value of
the unstable mode frequency.
The classical bubble configuration is described by
a vanishing gauge field and a real spherically symmetric
Higgs field
$\Phi(r) = |{\bf \Phi}| (r)$ which is a solution of the
Euler-Lagrange equation
\begin{equation} \label{Clbub}
-\Phi''(r)-\frac{2}{r}\Phi'(r)+\frac{d V_{ht}}{d\Phi(r)} = 0
\end{equation}
with the boundary conditions
\begin{equation}
\lim_{r\to\infty}
\Phi(r)=0 ~~
{\rm and} ~~ \Phi'(0)=0 \; .
\end{equation}
This differential equation can be
solved numerically e.g. by the shooting method. The solution
will be denoted as $H_0(r)$.
\section{Fluctuation analysis}
In terms of the action $S$ the fluctuation operator is defined
generally as
\begin{equation} \label{flucop}
{\cal M}_{ab} = \frac{\delta^2 S}{\delta \phi_a \delta \phi_b}
|_{\phi=\phi_{bubble}},
\end{equation}
where $\phi_a$ stands for the various gauge and Higgs field
components and $\phi_{bubble}$ is the field configuration
of the minimal bubble.
An analoguous derivative, taken at $\phi = \phi_{vac} \equiv 0$
defines the vacuum fluctuation operator ${\cal M}^0$.
In both configurations the gauge fields vanish, the Higgs field is
given by
\begin{equation} \label{higbub}
{\bf \Phi} = H_0(r) \left( \begin{array}{c} 0 \\ 1 \end{array}
\right)
\end{equation}
in the bubble configuration and vanishes in the vacuum.
The fluctuation determinant $\cal J$ appearing in the rate formula
is defined by
\begin{equation}
{\cal J} =\frac{\det''{\cal M}}{\det {\cal M}^0} \; .
\end{equation}
Here the symbol $\det''$ denotes the determinant with removed
translation zero modes and with the unstable mode frequency
replaced by its absolute value.
The analysis of fluctuations of the minimal bubble
can be related to a similar analysis
performed recently for the electroweak sphaleron
without gauge fixing in Ref. \cite{BaaLa} and in the
't Hooft-Feynman background gauge in Ref. \cite{BaaJu}.
We will use this latter analysis. One can take over the fluctuation
operator with two modifications which represent
at the same time essential simplifications:
\\ \noindent
- the high-temperature effective potential
has to be modified from the one in Eq. (\ref{htpot1}) to
the one in Eq. (\ref{htpot2});
\\ \noindent
- the sphaleron and the broken symmetry vacuum
configurations are replaced by the bubble and
the symmetric vacuum configurations defined above.
Furthermore we use here (see Eq. (\ref{scaling1})) for the coordinates
the scale $(g \tilde v)^{-1}$ instead of the scale
$M^{-1}_W= 2/gv$ used in Ref. \cite{BaaJu}.
The expansion of gauge and Higgs fields reads then \cite{BaaJu}
\begin{eqnarray}
W_\mu^a & = & a_\mu^a \nonumber \\
{\bf \Phi} & = & (H_0 + h + \tau^a \phi^a)\left( \begin{array}{c}
0 \\ 1 \end{array} \right) \; .
\end{eqnarray}
Here the fields denoted with small letters , $ a_\mu^a, h $
and $ \phi_a$ are the fluctuating fields.
Before we discuss fluctuations we have to fix the gauge.
We work here in the 't Hooft-Feynman background gauge.
The gauge conditions read
\begin{equation} \label{backgauge}
{\cal F}_a = \partial_\mu a^\mu_a+ \frac{1}{2}H_0 \phi_a =0 \; .
\end{equation}
The total gauge-fixed action $S_t$ is obtained from the high-temperature
action by adding to it the gauge-fixing action
\begin{equation}
S_{gf} =\frac{1}{\tilde g_3^2(T)}
\int d^3x \frac{1}{2} {\cal F}_a {\cal F}_a
\end{equation}
and the Fadeev-Popov action
\begin{equation}
S_{FP} = \frac{1}{\tilde g_3^2(T)}
\int d^3x \eta^\dagger (-\Delta + \frac{H_0^2}{4}) \eta \; .
\end{equation}
It is the action $S_t= S_{ht}+S_{gf}+S_{FP}$
which has to be used in the definition of the
fluctuation operator (\ref{flucop}).
The Hilbert space of fluctuations decomposes into subspaces
defined by the symmetries of the background field. The
fluctuation operators given below have been derived
from those of Ref. \cite{BaaJu}. This analysis was
based on a $K$ spin basis ($\vec K = \vec J + \vec I$).
Alternatively one might have used here
simply an analysis based on ordinary spin,
i.e. an expansion where
the Higgs field, the Fadeev Popov field
and the time components of the gauge fields are
expanded with respect to spherical harmonics and
the space components of the gauge fields with respect to
vector spherical harmonics $\hat x Y_l^m, r \nabla Y_l^m$ and
$\vec L Y_l^m$.
The electric components of the gauge field and the
isovector (would-be Goldstone) components of the Higgs field
form a coupled $(3 \times 3)$ system. The fluctuation
operator can be written in the
form ${\bf M} = {\bf M}^0 + {\bf V}$.
The free operator ${\bf M}^0$ is diagonal. It consists of
free partial-wave
Klein-Gordon operators
\begin{equation}
{\bf M}^0 =-\frac{d^2}{dr^2}
-\frac{2}{r} \frac{d}{dr} + \frac{l_n(l_n+1)}{r^2} + m_n^2
\end{equation}
with masses $m_n$ given by $(0 ,0, m_H)$ respectively
for the three components and with
centrifugal barriers corresponding to
angular momenta $l_n$ given analoguously by
$(l+1,l-1,l)$. The nonvanishing components of the
potential are
\begin{eqnarray}
V_{11}&=&V_{22}= H_0^2/4 \nonumber \\
V_{33}&=& H_0^2/4 + (\lambda_T/4 g^2)
(4 H_0^2-3 \epsilon H_0) \nonumber \\
V_{13}&=&V_{31}= -\sqrt{\frac{l+1}{2l+1}}\frac{dH_0}{dr} \\
V_{23}&=&V_{32}= \sqrt{\frac{l}{2l+1}}\frac{dH_0}{dr} \; .
\end{eqnarray}
For $l=0$ the second component is absent due to the
vanishing of the vector spherical harmonic $ r \nabla Y_0^0$.
These amplitudes have a triple degeneracy due to
isospin besides the ordinary degeneracy $(2 l +1)$ from spin.
The fluctuation operator for the scalar part of the
Higgs field is given by
\begin{eqnarray} \label{Higgflu}
{\bf M}& =& -\frac{d^2}{dr^2}
-\frac{2}{r} \frac{d}{dr} + \frac{l(l+1)}{r^2}
+ m_H^2 +V_{44}(r) \nonumber \\
V_{44}&=& \frac{\lambda_T}{4 g^2} (12 H_0^2-6\epsilon H_0)
\\
m_H^2 &=& \frac{\lambda_T}{4 g^2} (3\epsilon -4) \; .
\nonumber
\end{eqnarray}
This channel being an isosinglet its degeneracy is just $(2 l +1)$.
The time components of the gauge fields,
the Fadeev Popov fields and the magnetic components of the
vector potentials all satisfy the same equation
\begin{equation} \label{EFP}
{\bf M}_l \psi_5 = \omega^2 \psi_5 \;.
\end{equation}
It consists of a free massless partial wave Klein Gordon
operator
\begin{equation}
{\bf M}^0 =-\frac{d^2}{dr^2}
-\frac{2}{r} \frac{d}{dr} + \frac{l(l+1)}{r^2}
\end{equation}
and a potential
\begin{equation}
V_{55} =\frac{H_0^2}{4}
\end{equation}
which vanishes exponentially as $r \to \infty$.
There is no $l=0$ component of the magnetic vector potential
since the vector spherical harmonic
$\vec L Y_l^m$ vanishes. In the fluctuation determinant
all of these contributions cancel, only the
s-wave Fadeev-Popov contribution survives, due to the lack of
its magnetic counterpart. It is triply degenerate due
to isospin and has to be subtracted.
The partial-wave decomposition of the fluctuation operator
decomposes also its determinant,
\begin{equation}
{\cal J} = \sum (2l+1) {\cal J}_l \; .
\end{equation}
We now need a method for computing numerically the
determinants of the partial wave fluctuation operators.
Such a method
has been developed recently by V. G. Kiselev and the author
\cite{BaaKi} and will be presented briefly in the following
section.
\section{The fluctuation determinant of the \
electroweak bubble}
\setcounter{equation}{0}
A very fast method for computing fluctuation determinants
is based on a theorem on functional determinants;
references to earlier work and an elegant proof are given in
Ref. \cite{CoAS}.
Generalized to a coupled $(n \times n)$ system it can be
stated in the following way:
Let ${\bf f}(\nu,r)$ and ${\bf f}^0(\nu,r)$
denote the $(n \times n)$ matrices formed by
$n$ linearly independent solutions
$f_i^\alpha(\nu,r)$ and $f_i^{\alpha 0}(\nu,r)$ of
\begin{equation}
({\bf M}_{ij}+\nu^2) f_j^\alpha (\nu,r) =0
\end{equation}
and
\begin{equation}
( {\bf M}_{ij}^0 +\nu^2)
f^{\alpha 0}_j (\nu,r) =0 \; ,
\end{equation}
respectively, with regular boundary conditions at $r=0$.
The lower index denotes the $n$ components, the
different solutions are labelled by the greek upper index.
Let these solutions be normalized such that
\begin{equation}
\lim_{r \to 0} {\bf f}(\nu,r)({\bf f}^0(\nu,r))^{-1} = {\bf 1} \;.
\end{equation}
Then the following equality holds:
\begin{equation} \label{Flucdef}
{\cal J}(\nu) \equiv \frac{\det ({\bf M} +\nu^2)}{\det ({\bf M}^0+\nu^2)}
= \lim_{r \to \infty} \frac {\det {\bf f}(\nu,r)}{\det
{\bf f}^0(\nu,r)}
\end{equation}
where the determinants on the left hand side are determinants
in functional space, those on the right
hand side are ordinary determinants of the $n \times n$ matrices
defined above.
If the theorem is applied at $\nu = 0$ it yields the desired ratio of
fluctuation determinants
${\cal J} \equiv {\cal J}(0)$. The consideration of
finite values of $\nu$ is
necessary in the discussion of zero modes.
The theorem has been applied for computing the 1-loop
effective action of a single scalar field on a bubble
background in Ref. \cite{BaaKi} and of a fermion
system on a similar background in Ref. \cite{BaaSoSu} previously.
It was found to yield very precise results, in addition
to providing a very fast computational method.
In the numerical application the solutions $f_n^\alpha$ were written
as \cite{Baa}
\begin{equation}
f_n^\alpha(r) = (\delta_n^\alpha + h_n^\alpha(r))i_{l_n}(\kappa_n r)
\end{equation}
with the boundary condition $h_n^\alpha(r) \to 0$ as $r \to 0$.
The values $l_n$ and $\kappa_n = \sqrt{m_n^2+\nu^2}$ depend on
the channel as specified in the previous section.
This way one generates a set of linearly independent
solutions which near $r=0$ behave like the free solution as required
by the theorem which then takes the form
\begin{equation}
{\cal J}(\nu)
= \lim_{r \to \infty} \ln \det \{\delta_n^\alpha + h_n^\alpha(r)\}
\;.
\end{equation}
The functions $h_n^\alpha(r)$ satisfy the differential
equation \cite{Baa}
\begin{equation}
\frac{d^2}{dr^2}h_n^\alpha(r)+ \left ( \frac{2}{r}+ 2\kappa_n
\frac{i_{l_n}'(\kappa_n r)}{i_{l_n}(\kappa_n r)}\right)
\frac{d}{dr}h_n^\alpha(r)= V_{nn'}(r) \left( \delta_{n'}^\alpha
+ h_{n'}^\alpha(r)\right )\frac{i_{l_{n'}}(\kappa_{n'}r)}
{i_{l_n}(\kappa_n r)} \;
\end{equation}
which can also easily be used for generating the functions
$h_n^\alpha$ order by order in $V$. In particular,
if this differential equation is truncated by leaving out the term
$h_{n'}^\alpha$ on the right hand side, one generates the
first order contribution to $h$ which is the tadpole term.
For more technical details we refer to Refs. \cite{BaaKi,BaaSoSu}.
With the partial wave fluctuation operators
given in the previous section the
application of the theorem to the case of the
electroweak bubble is straightforward.
Some points to be considered are \\
- the subtraction of the divergent
tadpole graphs \\
- double counting of gauge and would-be Goldstone fluctuations \\
- removing the translation zero mode \\
- removing a particular gauge zero mode. \\
We will discuss these briefly. We will add also some
remarks on details of the numerical computation.
\subsection{Tadpole diagrams}
The high-temperature three-dimensional theory has only
linear divergences of the form of tadpole diagrams
which renormalize the mass term of the Higgs field.
They have to be subtracted in the numerical
computation to obtain finite results. This was done in each partial
wave, for which the tadpole contribution may
be computed \cite{BaaKi,BaaSoSu} either by solving a
truncated differential equation or \cite{Baa} as an analytic
expression using the partial wave Green function. After these
contributions have been subtracted, the partial wave
contributions converge as $1/l^2$ and have a finite sum.
Of course this contribution has to be added back,
after having been regularized and renormalized.
Part of these diagrams have already been taken into account
in the renormalization of the four dimensional theory and
in giving the vacuum expectation value (\ref{scale1}) of the Higgs
field a quadratic temperature dependence. Some terms linear in
the temperature survive however and contribute
\cite{CarLi,BaaJu,ArMcL} (after dividing
by the temperature) to the 1-loop effective
action, i.e. the logarithm of the fluctuation determinant.
If the mass of the field in the
loop is $m_i$ and its coupling to the
external field
is described by the potential $V_i$
their contribution to the effective
action is given by $ - m_i/8\pi \int d^3x V_i(r)$.
The fluctuating gauge fields have vanishing mass and do not
contribute. However we receive contributions from the
fluctuating Higgs fields. The mass circulating in the loop
is then $m_H$ which is - including the temperature dependence
and rescaling - given by equation (\ref{Higgflu}).
The potentials are
$V_{33}$ with triple isospin degeneracy and $V_{44}$.
So we have to restitute
the terms
\begin{equation} \label{tad1}
S_{g-tad}^{1-l} =
- \frac{m_H}{2} \int dr r^2 \left(
\frac{3}{4}H_0^2 + 3\frac{\lambda_T}{4g^2} (4 H_0^2-
3 \epsilon H_0) \right )
\end{equation}
for the gauge fields and
\begin{equation} \label{tad2}
S_{h-tad}^{1-l} =
- \frac{m_H}{2} \int dr r^2
6\frac{\lambda_T}{4g^2} (2 H_0^2- \epsilon H_0 )
\end{equation}
for the Higgs field
to the 1-loop effective action.
We should like to remark on two slight
inconsistencies of this procedure.
The first one concerns our choice of the high temperature action.
We have adopted the action of Dine et al. \cite{DLHLL}
since it sets a certain standard and since it has been used also in
Ref. \cite{KriLaSch} to which we want to compare part of our results
(and which shares the inconsistency). In this action
the $T^2$ term does not include the contribution of the
Higgs loop tadpole. This can be seen from the coefficient
D in Eqs. (\ref{coeffs}) which should include a term
$m_H^2$ in addition to $3m_W^2+2 m_t^2$. The contribution was
neglected already in Ref. \cite{AnHa} ``taking the Higgs boson
sufficiently light''. Since the expression is dominated by the
top quark contribution whose fluctuations are not included at
all here this omission may be tolerated at the present level
of accuracy. In a more refined analysis it should and
can easily be remedied.
The second point is the fact that we have
included already 1-loop effects into the tree level action, so
that part of our computation is now at the 2-loop level, without
constituting a complete and systematic 2-loop analysis. This applies
in particular also to the tadpole terms for which this could be
a more severe problem since they are the finite remnants of
divergent graphs. We can appeal here only to an argument - common
in many perturbative calculations - that possible inconsistencies
are of higher order and acceptable at an intermediate level
as they
will be cured in a complete higher order analysis.
\subsection{Double counting of gauge field fluctuations}
As mentioned in section 2 we are working with an action that
contains already the part of the 1-loop effective potential
induced by integrating out the gauge field and would-be
Goldstone boson fluctuations. These are present in the
temperature scale factors and couplings and appear especially
in the high temperature effective potential as the term
proportional to $\Phi^3$. While the $T^2$ contribution to
the vacuum expectation value (\ref{scale1})
comes from the tadpole diagrams and
has been taken into account along with these, the
$\Phi^3$ term is contained in our exact 1-loop effective
action. In order to avoid double counting it, this term has to be
subtracted from our numerical results. The incorporation
of this term into the tree level action was necessary in order
to obtain a first order phase transition and bubble
solution. If this was a good leading order approximation
the gauge field action should be well approximated by this
term. This is indeed the case (see below) but this implies also
that the remaining gauge and would-be Goldstone field contributions
are small differences of large terms, and that they cannot therefore
be expected to be very precise.
\subsection{Translation zero mode and unstable mode}
Translation invariance is broken by the classical solution,
so a zero mode appears. It occurs in the $l=1$ partial
wave of the fluctuation operator of the isoscalar part
of the Higgs field. It is easily removed using the
prescription given in \cite{BaaKi}: one applies the theorem
mentioned above at finite $\nu$ and defines
\begin{equation}
{\cal J}_{l=1,Higgs} = \lim_{\nu \to 0} \lim_{r \to \infty}
\ln \left (\frac{\psi_4(\nu,r)}{\nu^2 i_l(\kappa r)}\right )
\end{equation}
where $\kappa=\sqrt{\nu^2+m_H^2}$. Removing three
eigenvalues $\omega^2 =0 $ gives the fluctuation determinant
${\cal J}$ the dimension $ (energy)^{-6}$. The rate gets then a
dimension $(energy)^3 = 1 /(length)^3$. An additional dimension
$energy=1/time$ comes from the unstable mode prefactor
(see (\ref{rate})).The numerical
computation is based on energy units $ g \tilde v(T)$ which
are given in the Tables below.
The unstable mode makes the determinant of the p-wave
contribution negative. Replacing it by its absolute value
means just to revert the sign of the
determinant before taking the logarithm. We note that,
in contrast to Ref. \cite{KriLaSch} and in analogy to
Refs. \cite{CarLi,BaaJu}, we do not remove the zero mode
from the fluctuation determinant.
\subsection{Gauge zero mode}
Though we have imposed a gauge condition there is one residual
gauge degree of freedom. It is analoguous
to a constant
gauge function for the free theory. Indeed in the latter case
a constant
gauge potential $\Lambda(\vec x)= g_0$ does not contribute to the
vector potential and is therefore not eliminated by the
gauge condition $\partial_\mu a^\mu = 0$. In the case of the
bubble background field there is a similar {\it but nontrivial}
mode which satisfies the background gauge condition and is
therefore not eliminated by it. It manifests itself as a
zero mode
in the electric system for $l=0$.
The form of this mode (and the fact that it is really an
exact zero mode) was found after
extended numerical experiments.
It is given by a gauge
function $g(r)$ which satisfies the same
differential equation as the electric and Fadeev Popov modes,
Eq. (\ref{EFP}), i.e.
\begin{equation} \label{gmode}
g'' +\frac{2}{r} g'-\frac{H_0^2}{4} g =0 \; .
\end{equation}
With regular boundary condition at $r=0$ $g(r)$
becomes constant as $r \to \infty$, in
analogy to the free case.
Then the functions
\begin{eqnarray} \label{gmod}
\psi_1(r)&=& - 2 g'(r) \nonumber \\
\psi_3(r)&=& H_0 g(r)
\end{eqnarray}
satisfy the coupled system for the electric modes at $l=0$
which is given explicitly by
\begin{eqnarray} \label{hdgl}
\psi_1'' + \frac{2}{r} \psi_1' -\frac{2}{r^2} \psi_1
&=& \frac{H_0^2}{4} \psi_1 - H_0' \psi_3 \nonumber \\
\psi_3'' + \frac{2}{r} \psi_3' - m_H^2 \psi_3
&=& \frac{H_0^2}{4} \psi_3 +\frac{\lambda_T}{4g^2}
(4 H_0^2-3\epsilon H_0) \psi_3 - H_0' \psi_1 \; .
\end{eqnarray}
It can be checked easily that this gauge zero mode
satisfies the background gauge condition (\ref{backgauge})
and is therefore
not eliminated by it. Since this zero mode is
not due to a symmetry broken by the
classical solution as the translation mode
it cannot be handled in the usual way.
On the other hand we observe that precisely for
the s-wave the Fadeev-Popov contribution has survived; furthermore
to each Fadeev-Popov mode with finite energy, i. e. a solution
of
\begin{equation}
\psi_5'' +\frac{2}{r}
\psi_5'-\frac{H_0^2}{4} \psi_5 =-\omega_\alpha^2 \psi_5 \;
\end{equation}
there is a solution of the electric s-wave system
constructed exactly as that for the gauge zero mode, i.e.
Eq. (\ref{gmod}) with $g$ replaced by $\psi_5$. So there
is a cancellation of all electric modes of this type
with the corresponding Fadeev-Popov ones, except for the mode
with $\omega_\alpha^2 =0$. There {\it is} of course a solution
of the Fadeev-Popov equation at this energy, but it is
``singular'' at infinity, going to a constant there.
The corresponding mode in the electric system is normalizable,
however, since only its derivative is involved in $\psi_1$ and
its product with the exponentially decreasing function $H_0$ in
$\psi_3$. The cancellation between the s-wave electric
modes (\ref{gmod}) and the Fadeev-Popov ones can be extended
therefore to the zero mode if the boundary condition
at $r \to \infty$ for the latter ones is replaced by
$\psi_5'(r) \to 0$. This can be done in analogy with the
procedure described in the previous section by computing the
fluctuation determinant of the Fadeev-Popov mode at finite
$\nu$ via
\begin{equation} \label{FPbc}
{\cal J}_{l=0,FP}(\nu)=
\ln \lim_{r \to \infty}
\left (\frac{\psi'_5(\nu,r)}{i'_l(\nu r)}\right) \;.
\end{equation}
Then the Fadeev-Popov system at $l=0$ exhibits a zero mode as well,
the limit
\begin{equation}
\lim_{\nu \to 0}
({\cal J}_{l=0,el}(\nu)-{\cal J}_{l=0,FP}(\nu))
\end{equation}
is finite and defines the s-wave part of the fluctuation determinant.
In this way the Fadeev-Popov term cancels
all unwanted longitudinal electric modes for $l=0$ - including
the one with frequency zero. We note that the change of boundary
condition as $r \to \infty$ affects only the s-wave and
only for massless fields.
The definition (\ref{FPbc}) yields results
identical to the usual one
(\ref{Flucdef}) if $l \neq 0$ and/or the fields are
massive.
\subsection{Some numerical details}
The analysis was performed as described in previous publications
\cite{BaaKi,BaaSoSu}. Contributions of angular momenta up to
$l_{max}=30$ were computed numerically, the higher ones were included
by performing a power fit $A l^{-2}+B l^{-3} + C l^{-4}$
through the last $5$ computed contributions
and by adding a corresponding
sum from $l_{max}$ to $\infty$.
This was done already at lower values of $l$, treating
the highest included angular momentum as the actual value of $l_{max}$.
The resulting expressions were found to be independent of
$l$ within typically four
significant digits for $l > 20$.
A more subtle point is the extrapolation to $r=\infty$ implied
in Eq. (\ref{Flucdef}). In the
previous analyses \cite{BaaKi,BaaSoSu} the fields had finite mass and the
approach to $r=\infty$ was exponential. For the massless fields
the Bessel functions $i_l(\kappa r)$ are replaced by
$r^l/(2l+1)!!$ and the functions
$h_1^\alpha$ and $h_2^\alpha$
approach their asymptotic value only as $h_\infty + const./r$.
The extrapolation was performed using this Ansatz.
An exception occurs in the electric p-wave system, where
$h_2^\alpha$ picks up a logarithmic dependence on r due to
the cross term with $h_3^\alpha(r)$
on the r.h.s. of Eq. (\ref{hdgl}) which decreases only
as $1/r^2$. However this logarithmic dependence being
strictly proportional to $\delta^\alpha_3+h^\alpha_3(\infty)$,
i.e. to the third row of the matrix,
it does not contribute in the determinant, as also observed
numerically.
The tadpole contributions were computed in two ways,
once by solving a truncated differential equation
as described in \cite{BaaKi,BaaSoSu} and performing the analoguous
extrapolation, and once as
an integral using the partial wave Green function. In comparing the
two results the extrapolation was found - for the tadpole
contributions - to be reliable to four
significant digits typically.
Judging the accuracy of the results from the stability with
respect to varying extrapolations as $r\to\infty$ and
for large $l$ we would think that the purely numerical
part is accurate to
1 \%. The restituted tadpole contributions are given by
the expressions (\ref{tad1}) and (\ref{tad2}) whose evaluation
implies simple numerical integrals, they
can be considered as exact analytic expressions. This
restitution implies no delicate cancellations.
However even with a precision of 1\% for the numerical results
the final values of the gauge field
contribution have substantial errors since the
numerical part plus the tadpole contribution
is almost cancelled by the analytic
$\Phi^3$ contribution. Unfortunately, in contrast
to the sphaleron computation
\cite{BaaJu}, the cancellation ist not merely one
between two analytic expressions - the tadpole
and $\Phi^3$ contributions - but between the numerical results and
the analytic $\Phi^3$ contribution.
\section{Results and conclusions}
\setcounter{equation}{0}
The numerical results are given in Tables 1 to 4.
Here Table 1 is based on the values Higgs and
gauge boson masses $m_H=m_W=80.2$ GeV,
and a value of the top mass of $m_t= 170$ GeV. For the
vacuum expectation of the Higgs field we used
$v_0 =246$ GeV and for the gauge coupling the value
$g =.6516$. For the computation of Table 2 the values
$m_H=60$ GeV, $m_t=170$ GeV were used. Table 3 corresponds
to values $m_H =60$ GeV and $m_t= 140$ GeV, Table 4
to values $m_H=80.2$ GeV and $m_t= 140$ Gev; these
latter Tables are presented in order to compare with
results obtained in Ref. \cite{KriLaSch}
using the heat kernel expansion.
The values for the temperature chosen correspond to
10 equidistant steps of the quantity $\epsilon(T)$, defined
in Eq. (\ref{epsdef}), between the
onset of bubble nucleation at $\epsilon=2$ and
the critical temperature $T_0$ where bubble nucleation
ends at $\epsilon=4/3$. This choice is equivalent
to the choice of Kripfganz et al. \cite{KriLaSch} who
parametrize this range of temperatures by a variable
$y$ taking values between $0$ and $1$. Since Kripfganz et al. use
a somewhat different effective potential, the relation
between $y$ and $\epsilon$ is not precise, it is
essentially given by $y = 3-2\epsilon $ which we use as a
definition of `our' $y$. At small $y$ the
bubbles are large with thin walls, for $y \simeq 1$ the bubbles
are small and have thick walls.
Tables 1 and 2 are split into a part `a' which contains the essential
parameters for the minimal bubble and, in the last
column, the nucleation rate $R$ {\it without} fluctuation
corrections. The part `b' contains the fluctuation corrections,
i.e. the 1-loop effective action.
The results for $m_t=170$ GeV are given separately for the
isoscalar part of the Higgs field as $S^{1-l}_h$
and for the system of would-be
Goldstone fields and gauge fields (`gauge field contribution'
for short) as
$S^{1-l}_g$,
respectively. We give also separately the parts which were obtained
by the numerical analysis described in section 4, denoted as
$S^{1-l}_{h-num}$ and $S^{1-l}_{g-num}$, respectively.
The difference between
$S^{1-l}_{h}$ and $S^{1-l}_{h-num}$ is the tadpole contibution
$S^{1-l}_{h-tad}$ of Eq. (\ref{tad2}), and analoguously for the
gauge field. Note that the tadpole contributions to the Higgs
field are substantial.
The gauge field contribution $S_g^{1-l}$
contains the $\Phi^3$ part discussed in the previous section.
The numerical value of this term is given in the column
`$\Phi^3$'. This term should be close to the gauge field contribution,
and it is indeed. So the basic action used for computing the
bubble profiles represents a reasonable approximation to the
exact 1-loop effective action. The gauge field contribution
has to be reduced by this term since it would be double-counted
otherwise. The net gauge field contribution is denoted as
$\Delta S^{1-l}_g$ and given in the last column.
The correction to the rate can be simply obtained as a factor
$\exp (-\Delta S_{eff}^{1-l})$ where
$\Delta S_{eff}^{1-l} = S_h^{1-l} + \Delta S_g^{1-l}$.
The dimension
$energy^3$, here in units of $g \tilde v$, is already included
in the minimal bubble rate $R$.
One sees that the fluctuations lead to a substantial
suppression of the nucleation rate.
While the result that the effective action for the
gauge fields is well approximated by the effective
potential, i. e. the $\Phi^3$ term, is very rewarding
a less comfortable feature appears
if one compares the 1-loop effective action with the tree level
action $\tilde S$. If the saddle point approximation which forms
the basis of transition rate formula (\ref{rate}) is justified,
then the 1-loop action should be smaller than the tree level one.
This is not the case. Large 1-loop corrections were found already by
Kripfganz et al. \cite{KriLaSch} when computing the
1-loop effective action for the Higgs-field fluctuation only.
We compare our results to theirs in Tables 3 and 4. Since
these authors define the fluctuation determinant differently
- they remove the unstable mode - we give, besides our
result $S^{1-l}_{Higgs}$, the expression $\ln (A/T^4)$ where
$A$ is the square root ${\cal J}^{-1/2}$ of the
fluctuation determinant with translation {\it and}
unstable modes removed.
The results are close to each other for small $\epsilon$ or
$y \simeq 1$, i .e. for small thick-wall bubbles.
For small $y$, i.e. for large thin-wall bubbles,
our exact results are systematically larger than the
approximate ones of Ref. \cite{KriLaSch}. The question of
finding reliable analytic estimates is certainly an interesting
one, especially the order in which the terms of the
heat kernel expansion are summed. In \cite{BaaSoSu} it was
found that a summation by the number of derivatives
(``derivative expansion'') yields very
precise results if the mass of the fluctuation is much
larger than the inverse size of the background field
configuration. In Ref. \cite{KriLaSch} the terms are
summed with respect to powers of the heat kernel time. The
deviation at small $y$ could be due to the fact that large
bubbles with thin walls have a very substantial derivative
contribution.
It will be interesting to make a more systematic analysis of
various analytical approaches.
The comparison of the Higgs effective action with the
leading minimal bubble action is less favorable than
found in Ref. \cite{KriLaSch}.
This is even more the case if the gauge loops are included, as one
sees from the previous Tables 1 and 2.
In conclusion we state three essential features of our results:
\\ - The 1-loop effective action is substantial, of the order of
and larger than the leading order minimal bubble action. This sheds
some doubt on the applicability of the semiclassical transition
rate theory.
\\ - The sign of the 1-loop effective action is
such that the transition rate is suppressed.
\\ - The 1 -loop ``$\Phi^3$'' contribution which has been
incorporated into the basic effective potential
is reproduced rather well by the 1-loop action. This means that this
term in the effective {\it potential} describes relevant
features of the effective {\it action}.
It will be interesting to pursue this subject further; it could be
of interest to try a selfconsistent extremalization of the sum of
leading and 1-loop action. This would certainly reduce the
total suppression. Furthermore it will be interesting
to see how the inclusion of the fermion determinant
affects the transition rate.
\newpage
\section*{Table Captions}
{\bf Table 1a} Parameters of the minimal bubbles
for
$m_H = 60$ GeV and $m_t = 170$ GeV. The results are given as
a function of temperature in equidistant steps of the
variable $\epsilon$ (Eq. (\ref{epsdef})).
$\tilde v(T)$ is the temperature dependent vacuum expectation
value of Eq. (\ref{scale1}), $\lambda_T$ the
temperature dependent renormalized $\Phi^4$ coupling.
$\tilde S$ is the
minimal bubble action (or energy divided by $T$). $\omega_-^2$
is the square of the frequency of the unstable mode, given in units
of $g \tilde v (T)^2$.
The last column contains the logarithm of the nucleation rate
{\it without} the 1-loop corrections.
\\ \\
{\bf Table 1b}
$S_{h-num}^{1-l}$ is the 1-loop of the isoscalar part of the
Higs field as obtained in the numerical analysis.
$S_{h}^{1-l}$ is the total Higgs part of the 1-loop effective action,
obtained from $S_{h-num}^{1-l}$ by adding the tadpole contribution
$S_{h-tad}^{1-l}$.
$S_{g-num}^{1-l}$ is the 1-loop
gauge and would-be Goldstone field action obtained by the
numerical analysis,
$S_g^{1-l}$ is again obtained by including the tadpole contribution.
The next colum gives the $\Phi^3$ term as included into the
high temperature action.
$\Delta S^{1-l}_g$ is the gauge field action after subtraction of
this $\Phi^3$ contribution. $\Delta S^{1-l}_{eff}$ is the total
effective action after removing the $\Phi^3$ contribution.
\\ \\
{\bf Table 2a} The same as Table 1a for $m_H = m_W = 80.2$ GeV and
$m_t = 170$ GeV.
\\ \\
{\bf Table 2b} The same as Table 1b for $m_H = m_W =80.2$ GeV
and $m_t = 170$ GeV.
\\ \\
{\bf Table 3} Comparison of the Higgs field effective action
with approximate results by Kripf\-ganz et al. for
$m_H =60$ GeV amd $m_t = 140$ GeV. The first entries are as defined
in the previous Tables. The quantity $A$ is the Higgs part
of the fluctuation determinant with removed unstable mode.
Our results are compared to the one of Ref. \cite{KriLaSch},
marked with the subscript $KLS$.
\\ \\
{\bf Table 4} The same as Table 3 for $m_H=80.2$ GeV
and $m_t=140$ GeV.
\newpage
|
1,116,691,497,526 | arxiv | \section{Introduction}
We consider the two-machine flow shop problem with ordered machines in which each job has its smaller processing time
on the first machine and with the objective of determining simultaneously
a minimal common due date $d$ and a minimal number of tardy jobs $n_T$.
More precisely, there is a set of $n$ jobs $J_i$, $i = 1, \ldots, n$ all
of them are available at time zero. Each job $J_i$ must be processed non-preemptively
and sequentially on two machines $M_1$ and $M_2$ with known integer processing times $a_i$ and $b_i$, respectively.
Furthermore it holds $a_i \leq b_i$ for all $i = 1, \ldots, n$. Machines can process at most one job at a time
and the second operation of a job cannot start until the first operation
of that job has been completed. Let $C_i$ and $D_i$ denote the completion
times of job $J_i$ on the machines $M_1$ and $M_2$, respectively. A job $J_i$ is tardy if $D_i > d$, for a given value $d$. The common objective function
is to minimize the maximum completion time $\max(D_i)$ for $i = 1, \ldots, n$, i. e. the makespan of the job sequence.
Using the three-field notation extended to multi-criteria scheduling
problems from \cite{TkBi02}, the general problem can be
denoted as $F2/d_i = d/d, n_T$ and falls into the category of
multi-objective flow shop problems. Therefore, our problem can be denoted as $F2/a_i \leq b_i,
d_i = d/d, n_T$.
For the other related problems (like multi-objective flow shop problems, classical flow shop problems with $m$ machines, proportional flow shop problem, ordered flow shop problem, scheduling problems with job rejections) see \cite{PaKo12} and references therein.
The state of the art Johnson algorithm \cite{Jo54} yields an optimal arrangement of $n$ jobs on two machines with the minimum completion time $C_{max}$, by iteratively selecting a job with the shortest processing time and if that is the first machine -- schedule the job first, otherwise schedule the job as the last. The Johnson sequence for the $F2/a_i \leq b_i, kjobs/C_{max}$ problem for any $k$ jobs is the shortest processing times (SPT) sequence on $M_1$ (where $k$ is any number from 1 to $n$).
The ordinary NP-hardness of the $F2/d_i = d/d, n_T$ problem justifies the search for special cases solvable in polynomial time \cite{TkDeBo07}. One such case is when the problem is fully-ordered, that is when the condition $a_i \leq a_j$ also implies $b_i \leq b_j$, for each $1 \leq i \leq j \leq n$. This problem was analyzed in \cite{DeGuTa00} in the context of the single-objective $F2/d_i = d/n_T$ problem in which the common due date is given. T'kindt et al. \cite{TkDeBo07} surveyed the related literature and developed an exact branch and bound algorithm and also a $O(nD^2)$ pseudo-polynomial dynamic programming algorithm for the $F2/d_i = d/d,n_T$ problem where $D$ is the makespan resulting from applying Johnson's algorithm to the corresponding maximum completion time problem. The equivalence between $F2/d_i = d/d, n_T$ and $F2/kjobs/C_{max}$ problems can be easily demonstrated using their single-machine counterparts \cite{PaKo12}.
The objective of this paper is to show that the problem $F2/a_i \leq b_i, d_i = d/d, n_T$ is solvable in $O(n \log n)$ time. This problem is equivalent to solving the $F2/a_i \leq b_i, kjobs/C_{max}$ problem for every value $k = 1, \ldots, n$ if only $k$ out of $n$ jobs are retained. This is an optimal algorithm, as comparison-based lower bound for sorting is $O(n \log n)$. In this note, we improve the proposed quadratic algorithm by Panwalkar and Koulamas from \cite{PaKo12}, by providing efficient implementation using recently introduced modified binary tree data structure \cite{Il13}.
\section{Optimal algorithm}
\subsection {Data structure}
The binary indexed tree (BIT) or Fenwick tree \cite{Fe94} is an efficient data structure for maintaining the cumulative frequencies that provides efficient methods for calculation and manipulation of the prefix sums. These trees both calculate prefix sums and modify the table in logarithmic time. We will consider the extension of this standard structure to work with minimal/maximal partial summations.
Let $A$ be an array of $n$ elements. The modified binary indexed tree (MBIT) provides the following
basic operations with $O (\log n)$ time complexity (for details see \cite{Il13}):
\begin{enumerate}[($i$)]
\item for given value $x$ and index $i$, add $x$ to the element $A [i]$, $1 \leq i \leq n$.
\item for given interval $[1, i]$, find the sum/min/max of values $A [1], A [2], \ldots, A [i]$, $1 \leq i \leq n$.
\item for given interval $[1, i]$, find the minimum/maximum value among partial sums
$A [1], A [1] + A [2], A [1] + A [2] + A [3], \ldots, A [1] + A [2] + \ldots + A [i]$, $1 \leq i
\leq n$.
\end{enumerate}
The operations can be easily extended to return the index where the extremal value is achieved, by storing an additional index data in each node.
Furthermore, the comparison can be done in such a way that in case of tie -- the maximal index is the leftmost/rightmost one.
\subsection {Pivot job and makespan}
The flow shops have bottleneck machines and the jobs can be numbered in the non-decreasing order of their processing times on any machine yielding the shortest processing time (SPT) sequence. We assume in the sequel that $n$ jobs have been renumbered according to the processing times $a_i$ on machine $M_1$, with ties broken in favor of the shortest $b_i$ values.
The makespan is the total length of the schedule jobs $J$, and this longest path consists of $n + 1$ contiguous processing time elements:
\begin{equation}
\label{eq:makespan}
C_{max} = \max_{1 \leq k \leq n} \left ( \sum_{i = 1}^k a_i + \sum_{j = k}^n b_i \right )= \sum_{i = 1}^n b_i + \max_{1 \leq k \leq n} \left ( \sum_{i = 1}^{k} a_i - b_{i - 1} \right).
\end{equation}
The job $J_i$ at which the critical path changes direction (and machine) is called the pivot job. For a given sequence, there can be several jobs qualifying as pivot jobs and we will identify only the rightmost one among these jobs as the pivot job.
We can define new array $c_i = a_i - b_{i-1}$ with $b_0 = 0$, and also note that the sum of all $b_i$ is constant in each iteration. In order to efficiently find the maximal value of the prefix sums of the array $c$, we can use modified binary indexed tree data structure. Together with the maximal value, we will also store the leftmost index achieving this extremal value in order to determine the pivot job.
If a job $J_i$ is removed from the sequence, then the difference between the old makespan and the new makespan will be called the contribution of job $J_i$ to the current sequence and will be denoted as $\delta_i$. Therefore, we can calculate the contribution of the pivot job in logarithmic time by removing the pivot job, calculating new makespan, and putting back the job $J_i$ back (and reverting all changes to the data structures).
\subsection{Improved algorithm}
The proofs of the following results can be found in \cite{PaKo12}.
\begin{proposition}
\begin{enumerate}[($i$)]
\item For each job $J_i$ on the right of the pivot it holds $\delta_i = b_i$.
\item For each job $J_i$ on the left of the pivot it holds $\delta_i = a_i$, and will not be a candidate for removal as long as the current pivot job and the jobs
to the right remain in the sequence.
\item Removal of the pivot job will make another job on the right the new pivot job (if exists).
\item Removal of any job to the right of the current pivot from the sequence will not change the pivot job and the contributions of any non-pivot jobs.
\end{enumerate}
\end{proposition}
The pseudo-code of improved PK algorithm from \cite{PaKo12} is given in Algorithm \ref{alg:optimal}. The algorithm starts with all jobs sorted as SPT sequence on machine $M_1$. Then, it identifies the job $J_i$ with the maximum contribution $\delta_i$ as the candidate job and removes it from the sequence. Once a job is removed, it is not added to the sequence in subsequent iterations from $1$ to $n$. It should be pointed out that the PK Algorithm emulates the action of the optimal algorithm for the corresponding single-machine problem.
\medskip
In order to speed up the algorithm, we are going to maintain two MBITs for storing the maximal suffix values of $b$ and the maximal prefix partial sums of the array $c$. Note that there is no need for storing the maximums of the array $a$, as the array $a$ is sorted and $a_i \leq b_i$ holds for all $1 \leq i \leq n$. We will also maintain the sum of all $b_i$ in the current sequence and use it in the equation~(\ref{eq:makespan}).
We first construct the data structures $maxB$ and $maxC$ in $O (n \log n)$ time and update them as we remove the jobs from the sequence. The leafs of these tree structures will contain the arrays $b$ and $c$.
\medskip
When the job $J_i$ is removed, we simply set $b_i = 0$ in $maxB$ modified binary indexed tree - and all queries will return correct indices as $b_i > 0$ holds for all existing jobs.
Removal of the job $J_i$ will also involve updating the numbers $c_i$ and is slightly more complicated, as we need to know the jobs to the left and right from $J_i$ in the current sequence. Therefore, we maintain two arrays $left$ and $right$ which will contain the indices of the first remaining jobs from the sequence to the left and right, respectively. More formally, at the beginning it holds $left[i]= i - 1$ and $right[i] = i + 1$ for $1 \leq i \leq n$. When the job $J_i$ is removed, we update the values $c [left[i]]$ and $c [right[i]]$ and make $c [i] = 0$. Furthermore, $right[left[i]] = right [i]$ and $left[right[i]] = left[i]$. In the modified binary tree structure, we will always store the index of the leftmost value which will ensure to always find the existing jobs.
\begin{algorithm}
\label{alg:optimal}
\KwIn{The job sequence $J$ with execution times $a[i]$ and $b[i]$.}
\KwOut{The order of jobs.}
\medskip
Sort the jobs by $a [i]$ and in case of tie by $b [i]$\;
Create MBIT $maxB$ for the maximal suffixes of $b$\;
Create MBIT $maxC$ for the maximal prefix sums of $c [i] = a [i] - b [i - 1]$\;
Create the arrays $left$ and $right$\;
\For{$k = 1$ \KwTo $n$}
{
Calculate the makespan and the pivot job $p$ of the current sequence of jobs\;
Using $maxC$, find the contribution of the pivot job $p$, by removing and putting back the pivot job from the current sequence\;
Using $maxB$, find the job $i$ with the maximal contribution: max of $\delta [p]$, $b [p + 1], \ldots, b [n]$\;
Remove the job $i$ and update the array $c$\;
Update data structures $maxB$ and $maxC$ using the operations, and arrays $left$ and $right$\;
}
\caption{ Calculating the optimal job scheduling. }
\end{algorithm}
Therefore, the preprocessing is taking $O(n \log n)$ time, and each operation in the for loop is $O(\log n)$ time - which makes the total time complexity $O(n \log n)$. Based on the correctness of the algorithm and Proposition 1, we conclude this section with the following proposition.
\begin{proposition}
The described algorithm (always removing the job with the highest contribution) is optimal for the $F2/a_i \leq b_i, kjobs/C_{max}$ and $F2/a_i \leq b_i, d_i = d/d, n_T$ problems with the time and space complexity $O (n\log n)$.
\end{proposition}
The algorithm enumerates the $n + 1$ Pareto optima for each one of these two problems in $O(n \log n)$ time. For the future work, it would be interesting to extend the current approach to other specially-structured flow shop problems with two or more machines and improve the existing flow shop scheduling algorithms using more efficient data structures.
|
1,116,691,497,527 | arxiv | \section{Introduction}
Transmission of a quantum state from one place to another during specific time interval (quantum state transfer) is an important problem in development of quantum communication systems. The simplest model of such transfer is the quantum state transfer in an open 1/2-spin chain. In this case the problem may be formulated as follows. Let the spin chain be placed in the static magnetic field and all spins are directed along the field except the first one which is directed opposite to the field initially at $t=0$ (initial condition). If at some moment of time $t_{tr}>0$ we detect that $k$-th spin is directed opposite to the field then we say that the quantum state
has been transfered from the first node to the $k$-th node of the spin chain. Since the total spin projection must be conserved in this experiment, all other spins will be directed along the magnetic field at $t=t_{tr}$. Only such quantum processes will be considered in this letter. We say that the quantum state is transfered along the spin chain if it is transfered from the first
to the last node of this chain.
Different aspects of the quantum state transfer problem
were studied, for instance, in refs.\cite{FBE,B,CDEL,FR,KF,VGIZ}.
In \cite{CDEL} the possibility of the ideal transfer (i.e. transfer with probability equal to 1) of the initial quantum state along the homogeneous 1/2-spin chains of two- and three- nodes was shown. In order to perform the ideal state transfer along the longer chains, authors suggest to use the inhomogeneous symmetric chains with different coupling constants among the neighboring nodes. But long chains constructed in this way have two
basic disadvantages: (a) coupling constants have particular values for each pair of nodes of the first half of the symmetric chain so that the increase of the length requires recalculation of all coupling constants and (b) spread of coupling constants increases with increase of the length of the chain, which is hard for the practical implementation.
The fact that inhomogeneous spin chains may resolve the problem of the ideal state transfer stimulates the deep study of such chains. Thus, the spin dynamics in alternating spin chains (i.e. chains with two different alternating coupling constants between nearest neighbor nodes) with the XY Hamiltonian at high temperatures was studied in \cite{FR} (odd number of nodes $N$) and in \cite{KF} (even number of nodes $N$). It was demonstrated \cite{KF} that the ideal quantum state transfer along the alternating chain with $N=4$ may be performed for set of different pairs of coupling constants. However, $N=4$ seams to be the maximal length of the alternating chain along which the state may be perfectly transfered, which agrees with \cite{CDEL}. The long-distance entanglement in alternating 1/2-spin chain as well as in homogeneous 1/2-spin chain with small end bonds at zero temperature was studied in \cite{VGIZ}. They found that the maximal entanglement between ends of the long spin chain is possible only in the limit of the exact dimerization.
This letter concerns mainly {\it the high-probability} (rather then ideal) state transfer along the alternating 1/2-spin chains with even number of nodes in an inhomogeneous magnetic field.
The reasoning of the non-ideal state transfer originates from the fact that the ideal state transfer is hardly reachable because of, at least, two following obstacles.
\begin{enumerate}
\item
Nearest neighbor approximation has been used in study of the ideal state transfer in \cite{CDEL,FR,KF}, which is not enough to generate the ideal state transfer in practice where all nodes interact with each other.
\item
Different coupling constants in an inhomogeneous spin chain may not be produced with absolute accuracy.
\end{enumerate}
The Hamiltonian of this system in the approximation of the nearest neighbors interaction may be written in the form
\begin{eqnarray}
{\cal{H}}=\sum_{n=1}^N \omega_n I_n +\sum_{n=1}^{N-1}
D_{n}(I_{n,x}I_{n+1,x} + I_{n,y}I_{n+1,y}),
\end{eqnarray}
where $I_{n,\alpha}$ is the projection operator of the $n$th total spin on the $\alpha$ axis,
$w_n$ is the Larmor spin frequency of the $n$-th node and $D_n$ is a spin-spin coupling constant.
We set
\begin{eqnarray}
w_n=0,\;\;D_n=\left\{\begin{array}{ll}
D_1, & n=1,3,\dots\cr
D_2, & n=2,4,\dots.
\end{array}\right.
\end{eqnarray}
Using Jourdan-Wigner transformation \cite{JW}
\begin{eqnarray}
&&
I^{-}_n=I_{n,x}-i I_{n,y} = (-2)^{n-1}\left(
\prod_{l=1}^{n-1} I_{l,z}
\right) c_n,\\\nonumber
&&
I^{+}_n=I_{n,x}+i I_{n,y} = (-2)^{n-1}\left(
\prod_{l=1}^{n-1} I_{l,z}
\right) c_n^+,\\\nonumber
&&
I_{n,z}=c_n^+c_n-1/2,
\end{eqnarray}
(where $c_n^+$ and $c_n$ are creation and annihilation operators of spin-less fermions) we transform this Hamiltonian to the following one \cite{FR,KF}:
\begin{eqnarray}
&&
{\cal{H}}=\frac{1}{2}{ c}^+ D { c} ,\;\;{c}^+=(c_1^+,\dots,c_N^+),\;\;{ c}=(c_1,\dots,c_N)^t,
\\\nonumber
&&
D=\left(\begin{array}{cccccc}
0 & D_1 &0 & \cdots & 0&0\cr
D_1& 0 & D_2 & \cdots & 0&0\cr
0&D_2& 0 & \cdots & 0&0\cr
\vdots &\vdots &\vdots &\vdots &\vdots &\vdots \cr
0&0&0&\cdots&0&D_j\cr
0&0&0&\cdots&D_j&0
\end{array}\right),\;\;j=\left\{
\begin{array}{ll}
1,& {\mbox{even}}\;\;N\cr
2,& {\mbox{odd}}\;\;N
\end{array}
\right..
\end{eqnarray}
Let
\begin{eqnarray}
|n \rangle=|\underbrace{0\dots 0}_{n-1} 1 \underbrace{0\dots 0}_{N-n} \rangle
\end{eqnarray}
be the state where $n$-th spin is directed opposite to the external magnetic field while all other spins are directed along the field.
It was shown \cite{KF} that
the probability for the system to be initially in the state $|1\rangle$ and
finally in the state $|N\rangle$ is defined by the following expression:
\begin{eqnarray}\label{Probability}
P(t)=\big|\langle N|\exp(-i {\cal{H}} t) |1\rangle\big|^2=\left|
\sum_{j=1}^N u_{Nj} u_{1j} \exp(-it\lambda_j/2)
\right|^2,
\end{eqnarray}
where
$u_{ij}$ are components of the eigenvector $u_j$ corresponding to the eigenvalue $\lambda_j$ of the matrix $D$: $Du_j=\lambda_j u_j$, $u_j=(u_{1j}\dots u_{Nj})^T$.
In the case $N=4$, eq.(\ref{Probability}) yields \cite{KF}:
\begin{eqnarray}\label{P}
P&=&\frac{1}{4}\left| \left(1+\frac{\delta}{\sqrt{\delta^2+4}}\right)\sin\left(\frac{D_1 t}{2} \sqrt{\frac{2+\delta^2 - \delta\sqrt{\delta^2+4}}{2}}\right) -\right.
\\\nonumber
&&
\left. \left(1-\frac{\delta}{\sqrt{\delta^2+4}}\right)\sin\left(\frac{D_1 t}{2} \sqrt{\frac{2+\delta^2 + \delta\sqrt{\delta^2+4}}{2}}\right)
\right|^2,\;\;\delta=\frac{D_2}{D_1}.
\end{eqnarray}
Values $t=\bar t$ and $\delta=\bar \delta$ corresponding to the ideal state transfer are defined by the requirement
\begin{eqnarray}\label{ideal_40}
\sin\frac{\lambda_1 \bar t}{2}=-\sin\frac{\lambda_2 \bar t}{2} =\pm 1,
\end{eqnarray}
which yields
\begin{eqnarray}\label{ideal_4}
&&
D_1\bar t=\frac{2(3+4 k)\pi}{
\sqrt{2(2+\bar\delta^2+\bar\delta\sqrt{4+\bar\delta^2}))}},\;\; \bar\delta=\frac{2|1+2k-2n|}{\sqrt{(3+4k)(1+4n)}},
\\\label{ideal_4b}
&&
D_1\bar t=\frac{2(1+4 k)\pi}{
\sqrt{2(2+\bar\delta^2+\bar\delta\sqrt{4+\bar\delta^2}))}},\;\;
\bar\delta=\frac{2|1-2k+2n|}{\sqrt{(3+4n)(1+4k)}},\\\nonumber
&&
\;\;n,k=0,1,\dots .
\end{eqnarray}
The minimal time interval required for the quantum state transfer: $D_1\bar t_{min}=5.441$ for $\bar\delta=1.155$. It corresponds to the eqs.(\ref{ideal_4}) with $k=n=0$.
In the next section (Sec.\ref{Section}) we study the high-probability (instead of perfect) state transfer along the chains with even (Sec.\ref{Section:N_even}) and odd (Sec.\ref{Section:N_odd}) values of $N$. Conclusions are given in Sec.\ref{Section:Conclusions}.
\section{High-probability state transfer}
\label{Section}
In this section we study the probability of the quantum state
transfer along the chains with different numbers of nodes. It seamed out that
chains with even and odd $N$ exhibit significantly different
properties as follows from Secs.\ref{Section:N_even} and
\ref{Section:N_odd}. We will find out that chains with
even $N$ are preferable for the high-probability state transfer.
\subsection{State transfer along the chain with even $N$.}
\label{Section:N_even}
We use some results of \cite{KF}. Namely, consider the case $\delta=D_2/D_1 > (N+2)/N$ with $w_n=0$, $n=1,\dots,N$.
Then the eigenvalues $\lambda_\nu$ and components of the eigenvectors $u_{k\nu}$ with $1\le \nu \le N$ and $\nu\neq N/2,\nu\neq N/2+1$ are given by the following expressions:
\begin{eqnarray}\label{lambda}
\lambda_\nu &=&\left\{
\begin{array}{ll}
\sqrt{D_1^2 + D_2^2 +2 D_1 D_2 \cos x_\nu},& \nu=1,2,\dots,\frac{N}{2}-1,\cr
-\sqrt{D_1^2 + D_2^2 +2 D_1 D_2 \cos x_\nu},& \nu=\frac{N}{2}+2,\dots,N
\end{array}\right.,\\\nonumber
u_{k\nu}&=& \left\{
\begin{array}{ll}
A_\nu \sin\frac{kx_\nu}{2}, &k=2,4,\dots,N\cr
B_\nu \sin(N-k+1)\frac{x_\nu}{2}, &k=1,3,\dots,N-1
\end{array}
\right. ,\\\nonumber
&&
A_\nu=\sqrt{2}\left(N+1 -\frac{\sin(N+1) x_\nu}{\sin x_\nu}\right)^{-1/2},\;\;B_\nu=A_\nu (-1)^{\nu+1},
\end{eqnarray}
where $x_\nu$ are solutions of the following transcendental equation, $0< x_\nu<\pi$:
\begin{eqnarray}\label{x1}
&&
\delta \sin\frac{N}{2} x_\nu +\sin\left(\frac{N}{2}+1\right) x_\nu=0,\;\;x_{N+1-\nu}=x_\nu,\;\;\nu=1,\dots,\frac{N}{2}-1.
\end{eqnarray}
If $\nu= N/2$ or $\nu= N/2+1$, then
\begin{eqnarray}
\label{lam_N2}
\lambda_{N/2} &=&\sqrt{D_1^2 + D_2^2 -2 D_1 D_2 \cosh y}
,\\\nonumber
\lambda_{N/2+1} &=&-\sqrt{D_1^2 + D_2^2 -2 D_1 D_2 \cosh y},\\\nonumber
\label{u}
u_{k\nu}&=&\left\{
\begin{array}{ll}
A_\nu (-1)^{k/2} \sinh\frac{ky}{2}, &k=2,4,\dots,N\cr
B_\nu (-1)^{(N-k+1)/2} \sinh(N-k+1)\frac{y}{2}, &k=1,3,\dots,N-1
\end{array}
\right.,
\\\nonumber
&&
A_\nu=\sqrt{2}\left(\frac{\sinh(N+1) y}{\sinh y}-N-1\right)^{-1/2},\;\;B_\nu=A_\nu (-1)^{\nu+1},
\end{eqnarray}
where $y$ is a solution of the following transcendental equation, $y>0$:
\begin{eqnarray}\label{y1}
&&
\delta \sinh\frac{N}{2} y -\sinh\left(\frac{N}{2}+1\right) y=0.
\end{eqnarray}
Due to the eqs.(\ref{lambda}-\ref{u}), the
eq.(\ref{Probability}) may be written in the following form:
\begin{eqnarray}\label{Probability2}
P&=&\left|
\sum_{j=1}^{N/2} 2 u_{Nj} u_{1j} \sin(t\lambda_j/2)
\right|^2=\\\nonumber
&&2\left|\sum_{j=1}^{N/2-1} A_j^2(-1)^{j+1}
\sin^2\frac{Nx_j}{2}\sin(t \lambda_j/2)+\right.
\\\nonumber
&&
\left.(-1)^{N/2+1} A_{N/2}^2 \sinh^2\frac{Ny}{2}\sin(t \lambda_{N/2}/2)\right|^2.
\end{eqnarray}
In the numerical simulations below we fix $N$ and vary $\delta$
with the purpose to obtain the maximum of $P$ at some moment of time:
\begin{eqnarray}\label{P_h}
P_h=\max\limits_{\delta,t}P(\delta,t)>0.9.
\end{eqnarray}
The value $0.9$ in the RHS of (\ref{P_h}) is conventional.
Appropriate values of $\delta$ and $t$ will be referred to as
$\delta_h$ and $t_h$ respectively. The state transfer
characterized by the triad $(P_h,\delta_h,t_h)$ will be referred to as
{\it high-probability state transfer}. It is
illustrated in Figs.\ref{Fig:N4_max}-\ref{Fig:N8_max} that this triad is not unique. However, we
are interested in the high-probability state transfer having minimal
$t_h={t_h}_1$. Varying the single parameter $\delta$ we are able to maximize
${P_h}_1$. Values of other ${P_h}_i$, $i>1$, are not important for us.
In the examples below we start with $\delta=2$ and increase $\delta$ obtaining the maximum value of ${P_h}_1$ and appropriate ${t_h}_1$.
To anticipate, the shapes of the graphs $P(t)$ (i.e. superposition of slow and fast oscillations, see
Figs.\ref{Fig:N4_max}-\ref{Fig:N8_max}) together with
eq.(\ref{Probability2}) suggests us to estimate ${t_h}$ in
terms of the minimal of the eigenvalues
$\lambda_{min}={\mbox{min}}(\lambda_1,\dots,\lambda_{N/2})$:
\begin{eqnarray}
\left|\sin \left(t_{h} \lambda_{min}/2\right)\right| \approx 1\;\;\Rightarrow\;\;
{t_h}_1\approx\frac{\pi}{\lambda_{min}}.
\end{eqnarray}
Formulae (\ref{lambda},\ref{lam_N2}) show that $\lambda_{min}=\lambda_{N/2} $, so that
\begin{eqnarray}
{t_h}_1\approx\frac{\pi}{\lambda_{N/2}}.
\end{eqnarray}
Thus, for $N=4$, see Fig.\ref{Fig:N4_max}, we have found ${\delta_h}_1=2.272$, ${P_h}_1=0.999$, $D_1 t_h=8.303$. Eigenvalues are following: $\lambda_1=2.649D_1$, $\lambda_2=0.377D_1$, so that $D_1{t_h}_1\approx\pi\frac{D_1}{\lambda_2} =8.333$.
\begin{figure*}[!htb]
\includegraphics[width=7cm,angle=270]
fig1.EPS}
\caption{Probability
of the state transfer along the four-nodes spin chain with ${\delta_h}_1=2.272$; ${P_h}_1=0.999$ is achieved at $D_1 {t_h}_1=8.303$.
}
\label{Fig:N4_max}
\end{figure*}
For $N=6$, see Fig.\ref{Fig:N6_max}, one has ${\delta_h}_1=2.373$, ${P_h}_1=0.997$, $D_1
{t_h}_1=21.428$. Eigenvalues are following: $\lambda_1=3.060D_1$, $\lambda_2=2.208D_1$, $\lambda_3=0.148D_1$, so that $D_1
{t_h}\approx \pi\frac{D_1 }{\lambda_3} =21.227$.
\begin{figure*}[!htb]
\includegraphics[width=7cm,angle=270]
fig2.EPS}
\caption{
Probability of the state transfer along the six-nodes spin chain with ${\delta_h}_1=2.373$;
${P_h}_1=0.997$ is achieved at $D_1
{t_h}_1=21.428$.}
\label{Fig:N6_max}
\end{figure*}
Similarly, for $N=8$, see Fig.\ref{Fig:N8_max}, one has ${\delta_h}_1=2.557$, ${P_h}_1=0.989$, $D_1
{t_h}_1=58.966$. Eigenvalues are following: $\lambda_1=3.366D_1$, $\lambda_2=2.828D_1$, $\lambda_3=2.070D_1$, $\lambda_4=0.051D_1$. Thus $D_1
{t_h}_1\approx \pi\frac{D_1}{\lambda_4} =61.600$.
\begin{figure*}[!htb]
\includegraphics[width=7cm,angle=270]
fig3.EPS}
\caption{
Probability of the state transfer along the eight-nodes spin chain with ${\delta_h}_1=2.557$; ${P_h}_1=0.989$ is achieved at $D_1
{t_h}_1=58.966$.
}
\label{Fig:N8_max}
\end{figure*}
Finally we remark that, for practical detection of the quantum state transfer, one can select the time interval in the neighbourhood of the main peak ${P_h}_1$ by condition, say, $P>0.8$ everywhere inside of this time interval
and consider that the quantum state has been transfered along the chain if it is detected at the last node during the selected time interval.
\subsubsection{Spin chains with different $N$ and equal $\delta$.}
We demonstrate that the high-probability state transfer is possible along the alternating spin chains having different numbers of nodes $N$ and the same ratio of the coupling constants $\delta$.
We take $N=2 k$, $k=2,\dots,8$ and $\delta=2.380$. Results are collected in the Table 1.
Disadvantage of this state transfer is the fast growth of ${t_h}_1$ with increase of $N$.
\vspace{0.4cm}
\begin{table}
\label{Table1}
\caption{}
\begin{tabular}{|p{1.2cm}|p{1.2cm}|p{1.2cm}|p{1.2cm}|p{1.2cm}|p{1.2cm}|p{1.2cm}|p{1.4cm}|}
\hline
$N$&4&6&8&10&12&14&16\\\hline
$D_1 {t_h}_1$&8.084&21.378&57.654&131.278&265.631&721.119&1403.554\\\hline
${P_h}_1$&0.990 &0.997 &0.957 &0.939&0.949& 0.962&0.901
\\\hline
\end{tabular}
\end{table}
\vspace{0.4cm}
\noindent
\subsubsection{State transfer during the given time interval.}
We also may arrange the high-probability state transfer during the given time interval. For instance, let $N=8$ and suppose that we want to transfer the quantum state from the first node to the last node of the chain at $D_1 {t_h}_1=60$. Function $P(\delta)$ at $t={t_h}_1$ is represented in Fig.\ref{Fig:N8_t}. We see that it has the maximum ${P_h}_1=0.973$ at ${\delta_h}_1=2.510$. Namely the value $\delta={\delta_h}_1$ is required for our purpose.
\begin{figure*}[!htb]
\includegraphics[width=7cm,angle=270]
fig4.EPS}
\caption{
Probability of the state transfer along the
eight-nodes spin chain at the fixed moment $D_1 {t_h}_1=60$; ${P_h}_1=0.973$ is achieved for ${\delta_h}_1=2.510$.
}
\label{Fig:N8_t}
\end{figure*}
\subsubsection{Restrictions of the method: the state transfer to the arbitrary node of the chain}
The possibility to perform the high-probability state transfer between the end nodes of the spin chain suggests us to check whether the high-probability state transfer to {\it the intermediate} nodes of the chain is possible. Expression for the probability of the state transfer to the $k$-th node of the chain, $P_k(t)$, is following:
\begin{eqnarray}
P_k(t)&=&\left|
\sum_{j=1}^{N} u_{kj} u_{1j}e^{-it\lambda_j/2}
\right|^2=\left\{
\begin{array}{ll}
\left|
\sum_{j=1}^{N/2} 2 u_{kj} u_{1j} \sin(t\lambda_j/2)
\right|^2,& {\mbox{even}}\;\;\; k\cr
\left|
\sum_{j=1}^{N/2} 2 u_{kj} u_{1j} \cos(t\lambda_j/2)
\right|^2,& {\mbox{odd}} \;\;\;k.
\end{array}\right. .
\end{eqnarray}
Unfortunately, the answer is negative at least after the numerical simulations of the state transfer along the chains with $N=4,6,8$.
For instance,
graphs of $P_k$, $k=1,2,3,4$ for $N=4$ and $\delta=2.272$ (corresponding to Fig.\ref{Fig:N4_max}) are represented in Fig.\ref{Fig:N432_max}.
\begin{figure*}[!htb]
\includegraphics[width=7cm,angle=270]
fig5.EPS}
\caption{
Comparison of the probabilities $P_k$ for the state transfer to the $k$-th node ($k=2,3,4$) and the probability $P_1$ for the returning in the 1-st node of the chain with $N=4$ and $\delta=2.272$.
}
\label{Fig:N432_max}
\end{figure*}
Remark that the shapes of the functions $P_k(t)$ illustrated in Fig.\ref{Fig:N432_max} may be interpreted as a spin-wave packet moving between the end nodes of the spin chain \cite{FBE}.
\subsection{State transfer along the chain with odd $N$.}
\label{Section:N_odd}
\label{Section:odd_N}
In this subsection we use the basic formulae derived in \cite{FR} where 1/2-spin dynamics of the chain with odd $N$ has been studied. We set $w_n=0$, $n=1,\dots,N$. Then expressions for the eigenvalues $\lambda_\nu$ read:
\begin{eqnarray}\label{lambda_odd}
&&
\lambda_\nu=\left\{\begin{array}{ll}
D_1 \sqrt{\Delta_\nu},&\nu=1,2,\dots,\frac{N-1}{2}\cr
0,&\nu=\frac{N+1}{2}\cr
-D_1 \sqrt{\Delta_\nu},&\nu=\frac{N+3}{2},\frac{N+5}{2},\dots,N.
\end{array}\right.,\\\nonumber
&&
\Delta_\nu=1+2 \delta\cos\frac{2\pi\nu}{N+1}+\delta^2,\\\nonumber
\end{eqnarray}
Expressions for the components of the eigenvectors $u_{j\nu}$ with $1\le \nu\le N$ and $\nu\neq (N+1)/2$ are following:
\begin{eqnarray}\label{u_odd}
&&
u_{j\nu}=\left\{\begin{array}{ll}
\frac{A_\nu D_1}{\lambda_\nu}\left(
\delta \sin\frac{\pi \nu(j-1)}{N+1} +\sin\frac{\pi\nu(j+1)}{N+1}\right)&j=1,3,\dots,N\cr
A_\nu\sin\frac{\pi\nu j}{N+1},& j=2,4,\dots,N-1,
\end{array}\right.,\\\nonumber
&&A_\nu=
\sqrt{\frac{2}{N+1}}.
\end{eqnarray}
In addition,
\begin{eqnarray}\label{uN_odd}
\\\nonumber
&&
u_{j(N+1)/2}=\left\{
\begin{array}{ll}
B(-\delta)^{(N-j)/2},&j=1,3,\dots,N\cr
0,&j=2,4,\dots,N-1.
\end{array}
\right.,\;\;B=\sqrt{\frac{\delta^2-1}{\delta^{N+1}-1}}.
\end{eqnarray}
Eq.(\ref{Probability}) gets the next form:
\begin{eqnarray}\label{P_odd}
&&
P=\left|2\sum_{j=1}^{(N-1)/2} u_{Nj}u_{1j}\cos(\lambda_j t/2)+u_{N(N+1)/2}u_{1(N+1)/2} \right|^2=\\\nonumber
&&
\left|2\sum_{j=1}^{(N-1)/2} A_j^2\frac{D_1^2\delta}{\lambda_j^2}\sin\frac{2\pi j}{N+1}
\sin\frac{\pi j(N-1)}{N+1} \cos(\lambda_j t/2)+B^2(-\delta)^{(N-1)/2} \right|^2.
\end{eqnarray}
Behaviour of the function $P(t)$ is completely different in comparison with the case of even $N$. It was shown \cite{CDEL} that the ideal state transfer is possible for $N=3$ and is impossible for $N>3$ if $\delta=1$. Using numerical simulations we obtain that the high-probability state transfer is possible, in principle, only for $\delta \approx 1$.
For instance, $P(t)$ for $N=5$ and $\delta=1$ is represented in Fig.\ref{Fig:N5_max}. It has a set of maxima. We mark two mostly considerable of them: ${P_h}_1=0.942$ at $D_1{t_h}_1=6.764$ and ${P_h}_2=0.987$ at $D_1{t_h}_2=43.757$.
\begin{figure*}[!htb]
\includegraphics[width=7cm,angle=270]
fig6.EPS}
\caption{
Comparison of probabilities for the state transfer along the two five-spin chains with $\delta=1$ ($\lambda_1=1.732 D_1$, $\lambda_2=D_1$)
and $\delta=2$ ($\lambda_1=2.646 D_1$, $\lambda_2=1.732 D_1$) .
}
\label{Fig:N5_max}
\end{figure*}
One can demonstrate that the amplitude of $P$ decreases with increase of $\delta$. As an example, the functions $P(D_1 t)$ for $N=5$ and two values $\delta=1$ and $2$ are represented in Fig.\ref{Fig:N5_max}.
We emphasise two following features of the high-probability state transfer along the chain with odd $N$.
\begin{enumerate}
\item
The probability is described by an oscillating function similar to the case of even $N$. However, unlike the case of even $N$, the amplitude of $P$ decreases with increase of $\delta$. Because of this fact, we may not effectively use parameter $\delta$ in order to provide the high-probability state transfer during the reasonable time interval $(0,{t_h}_1)$.
\item
The high-probability state transfer ($P_h\sim 0.9$) is observable in the neighbourhood of $\delta = 1$. However, appropriate $t_h$ may be too long for realization in quantum process.
\end{enumerate}
Thus, we conclude that the chains with odd $N$ are less suitable for the high-probability state transfer in comparison with the chains with even $N$.
The obtained result may be justified by the analytical estimation of $P$ for the quantum state transfer along the chain with odd $N$. Using eqs.(\ref{lambda_odd}-\ref{P_odd}) one has
\begin{eqnarray}
&&
P\le (F_1 +F_2)^2,\\\nonumber
&&
F_1=\left|2\sum_{j=1}^{(N-1)/2} A_j^2\frac{D_1^2\delta}{\lambda_j^2}\sin\frac{2\pi j}{N+1}
\sin\frac{\pi j(N-1)}{N+1} \cos(\lambda_j t/2)\right|,\\\nonumber
&&
F_2=\left|B^2(-\delta)^{(N-1)/2}\right|.
\end{eqnarray}
Consider $F_1$ and $F_2$ separately.
For our convenience, we consider the case $\delta\ge 1$ without loss of generality. Using inequality $|\cos(\lambda_j t/2)|\le 1$ and equations (\ref{lambda_odd},\ref{u_odd}) one gets the following chain of inequalities:
\begin{eqnarray}\label{F1}
&&
F_1\le \frac{4}{N+1}\sum_{j=1}^{(N-1)/2} \left|
\frac{D_1^2\delta}{\lambda_j^2} \sin^2\frac{2\pi j}{N+1}
\right|=\\\nonumber
&&
\frac{4}{N+1}\sum_{j=1}^{(N-1)/2}
r_j(\delta)\sin^2\frac{\pi j}{N+1}
\le
\frac{2 \Delta(\delta) }{N+1} \sum_{j=1}^{(N-1)/2}
\left(1-\cos\frac{2\pi j}{N+1}\right)
=
\\\nonumber
&&
\frac{\Delta(\delta) (N-1)}{N+1}.
\end{eqnarray}
Here
\begin{eqnarray}
&&
r_j(\delta)= \frac{\left(2+2\cos\frac{2\pi j}{N+1}\right)}
{\left(\delta+1/\delta+2 \cos\frac{2\pi j}{N+1}\right)},\;\;j=1,\dots,(N-1)/2,\;\;0\le r_j \le 1,\\\label{DELTA}
&&
\Delta(\delta)=\max\Big(r_j(\delta),\;1\le j\le (N-1)/2\Big)=r_1\\\nonumber
&&
=\frac{\left(2+2\cos\frac{2\pi}{N+1}\right)}
{\left(\delta+1/\delta+2 \cos\frac{2\pi}{N+1}\right)}\le 1.
\end{eqnarray}
Deriving (\ref{F1}) we used the inequality $(\delta+1/\delta)\ge 2$ and the identity $\displaystyle \sum_{j=1}^{(N-1)/2}
\cos\frac{2\pi j}{N+1}=0$.
Next,
\begin{eqnarray}\label{F2}
&&
F_2=\delta^{(N-1)/2}\frac{\delta^2-1}{\delta^{N+1}-1}=
\frac{\delta^{(N-1)/2}}{\sum_{k=0}^{(N-1)/2}\delta^{2k}}=
\\\nonumber
&&
\left\{
\begin{array}{ll}
\left(\sum_{k=0}^{(N-3)/4}(1/\delta^{2k+1} +\delta^{2k+1})
\right)^{-1},& {\mbox{odd}}\;\;\;(N-1)/2\cr
\left(\sum_{k=1}^{(N-1)/4}(1/\delta^{2k} +\delta^{2k})
+1\right)^{-1},& {\mbox{even}}\;\;\;(N-1)/2
\end{array}
\right\}\le \frac{2}{N+1}.
\end{eqnarray}
Thus
\begin{eqnarray}\label{P_odd_l}
P\le {\cal{P}}=\left(\frac{\Delta(\delta)(N-1) +2)}{N+1}\right)^2.
\end{eqnarray}
If $\delta=1$, then $\Delta(1)=1$ so that the inequality (\ref{P_odd_l})
yields $P\le 1$, which means that the state transfer may approach
the high-probability state transfer at some moment of time $t_h$. However, $t_h$ may be too long as it was mentioned above.
It follows from the eq.(\ref{P_odd_l}) that ${\cal{P}}$ decreases with
increase of $\delta$. For instance, if $\delta=2$ and $N=5$, then
$\Delta(2)=6/7$ and ${\cal{P}}=361/441 \approx 0.819$ which agrees with Fig.\ref{Fig:N5_max}.
Emphasize that sign $"="$ in the inequalities (\ref{F1}) and (\ref{P_odd_l}) may appear only if the following conditions are valid at some moment of time $t_0$:
\begin{eqnarray}\label{con1}
&&
|\cos(\lambda_j t_0/2)|=1,\;\;\forall \;\;j=1,\dots, (N-1)/2,\\\label{con2}
&&
\delta=1.
\end{eqnarray}
Since the parameter $\delta$ is fixed by the condition (\ref{con2}), one has one parameter $t_0$ in order to satisfy $(N-1)/2$ conditions (\ref{con1}). This is possible if only $N=3$ when the system (\ref{con1}) reduces to the single equation \cite{CDEL}. In the case $N>3$ the sign $"\le"$ must be replaced by the sign $"<"$ in inequalities (\ref{F1}) and (\ref{P_odd_l}).
Similar to the case of even $N$, numerical simulations show that there is no high-probability state transfer to the inner nodes of the chain.
Remark that the high-probability state transfer along the chains with even $N$ may be realized only if $N$ is not too large. In fact, $\lambda_{N/2}\to 0$ and ${t_h}_1 \to \infty$ as
$N\to \infty$, while ${t_h}_1$ may not be too long in quantum
process. This observation removes principal differences between long chains with even and odd $N$ in the quantum state transfer systems.
\section{Conclusions}
\label{Section:Conclusions}
We have demonstrated that the alternating short spin chains with even numbers of nodes $N$ are preferable for the purpose of the quantum state transfer. Although the ideal state transfer for $N>4$ is impossible in the alternating chain, the state transfer along the chain with even $N$
may be performed with high probability. This interesting phenomenon is especially important because the ideal state transfer is hardly achievable in practice.
Authors thank Prof. E.B.Fel'dman for useful discussions.
This work is supported by Russian Foundation for Basic Research through the grant 07-07-00048.
|
1,116,691,497,528 | arxiv | \section{Introduction}
Recently, interest has
increased the interpretation of Core Collapse
Supernovae (CCSNe) data, both spectra and photometry (Richardson et al. 2006;
Elmhamdi et al. 2006; Taubenberger et al. 2009; Maurer et al. 2010;
Elmhamdi et al. 2011).
In particular special
attention has been devoted to the stripped-envelope events (i.e. type Ib-c
hydrogen-deficient SNe). Studying enlarged samples of CCSNe
objects, having good quality observations, can be a potential tool
for assessing the similarities and the diversities within this SNe family,
relating these facts to the physics and possibly to the nature
of their progenitors.
As an example, Elmhamdi et al. 2006 have presented an investigation
of the spectroscopic properties of a selected optical photospheric
spectra of CCSNe,
discussing how hydrogen manifests its presence within this class. The authors
argued for a low mass thin hydrogen layer with very high
ejection velocities above the helium shell to be the most likely
scenario for type Ib SNe.
Although the primary goal of the the cited work was highlighting the hydrogen
traces in CCSNe, an important by-product result concerns the behaviour of
oxygen, in particular the O I 7773 \AA\ line. Based on the
synthetic spectra fits, for this line it seems that at intermediate
photospheric phases, type Ib objects tend to have low
optical depths, while some type Ic SNe, e.g. SN Ic 1987M, are found
to display the deepest O I 7773 \AA\ profile. SNe of type IIb \& II,
at similar
phases, are found to be the objects with the lowest O I 7773 \AA\ optical
depth.
At somewhat later epochs, transient type Ib/c objects display deep
O I 7773 \AA\ troughs. Matheson et al. (2002) arrived at similar conclusions,
indicating the O I 7773 \AA\ line to be stronger in SNe Ic than in SNe Ib.
Interestingly, the deeper and stronger
permitted oxygen line O I 7773 \AA\ in photospheric spectra of SNe Ic and
Ib/c might imply that they are less diluted
by the presence of a helium envelope. We expect indeed the oxygen lines
to be more prominent for a ``naked'' C/O progenitor core. Two further
observational aspects tend to reinforce this belief: first, the
forbidden lines, especially the [O I] 6300,6364 \AA\ doublet, seem to appear
earlier following a SNe sequence ``Ic$-$Ib$-$IIb$-$II''.
The second indication comes from the fact that this nebular emission line
has a velocity width decreasing following the SNe sequence above.
\begin{figure}[htb]
\centerline{\psfig{file=fig1.ps,width=9.3cm,height=10cm}\psfig{file=fig2.ps,width=9.3cm,height=10cm}}
\FigCap{$Left~panel$: a sample of the late time CCSNe spectra. The most
prominent lines are labeled. The corresponding observation date, since
explosion time, are also reported. $Right~panel$: the computed
quasi-bolometric
light curve of SN II 1987A (UBVRI bands); SN IIP 1999em (UBVRI-bands);
SN Ib 1990I (BVRI-bands); SN IIb 1993J (UBVRI bands) and SN Ic
1994I (BVRI bands). The $^{56}$Co to $^{56}$Fe decay slope is also shown
for comparison.}
\end{figure}
In the present work we explore these points and other
issues related to oxygen lines within the CCSNe family,
especially at late nebular phases. We analyze a sample of 26 CCSNe
events, and quantify the [O I] 6300,6364 \AA\ luminosities and
discuss their evolution in time.
The [Ca II] 7291,7324 \AA\ over [O I] 6300,6364 \AA\ flux ratio is also
shown and discussed
as a possible progenitor mass indicator (Fransson $\&$ Chevalier 1989).
Using the the computed luminosities we discuss two methods to translate
these measurements into masses. The estimated oxygen masses are combined
then with the measured nickel masses, determined from
light curves, and compared with yields from three known theoretical CCSNe
models by means of the oxygen-to-iron mass ratio. This is potentially
important since our measurements can be directly correlated
to the progenitor masses from theoretical models.
Worth noting here is the importance of oxygen and iron estimates, from
supernovae explosions, in the chemical enrichment and evolution
of galaxies. In particular, the oxygen abundance is crucial in metal-poor
stars, and is a key issue in modeling the early phases of
the chemical galaxy evolution and as well in constraining the age of globular
clusters (VandenBerg et al. 2000; Melendez et al. 2001).
Oxygen is indeed considered as a major tracer of chemical
evolution, since it is one of primary elements ejected by CCSNe
(i.e. resulting from massive stars). For example the oxygen-to-iron
ratio, when [O/Fe]$>$0 such as in the halo of our Galaxy, is
an indicator of early chemical enrichment by massive stars.
The paper is organized as follows.
In Section 2, we describe the sample and we highlight the main
spectroscopic characteristics and differences within the CCSNe family. Some
constructed quasi-bolometric light curves are also shown and discussed.
The [O I] 6300,6364 \AA\ line luminosity and the
$F_{6300}/F_{6364}$ flux ratio are measured and presented in Section 3.
In Section 4,
we discuss two different methods for estimating oxygen mass in CCSNe.
The methodology of $^{56}$Ni mass estimation is given. We also
evaluate the integrated flux ratio of the forbidden emission lines
[Ca II] 7291,7324 \AA\ and [O I] 6300,6364 \AA.
We conclude with a summary and discussion of our findings in Section 5.
\section{The sample}
The selected sample consists of 26 CCSNe objects$-$13 of them are type
II, one of type IIb and 12 of type Ib-c. Data are gathered mainly
from the literature (i.e. published available data).
Use is made of the ``SUSPECT''\footnote{http://bruford.nhn.ou.edu/$\sim$suspect/
index1.html} and of the
``CfA''\footnote{http://cfa-www.harvard.edu/oir/Research/supernova
/SNarchive.html} Supernova Spectra Archives. Some measurements
are made on late spectra of SN 1996aq, taken from the
Padova-Asiago supernovae database.
A summary of references and descriptive parameters of individual objects
is given in Table 1.
In what follows we adopt the standard reddening laws of
Cardelli et al. (1989).
An example of the analyzed late spectra is shown in Fig.1(left panel),
together with identifications of the most prominent lines. We focus
on the wavelength range 5500-9200 \AA.
At this epoch, when the events are in the radioactive tail phase, and except
for the notable H$\alpha$ emission in type II SNe, the
optical spectra are dominated by emission lines of
[O I] 6300,6364 \AA, [Ca II] 7291,7324 \AA\ and Ca II-IR triplet. The emission
centered at $\lambda \sim$7800 \AA\ is usually attributed to O I 7774 \AA.
These aspects will be highlighted when evaluating the [O I]/[Ca II] line
intensity ratio of the sample objects.
Worth recalling here that the [O I] 6300,6364 \AA\ line
is found to be absent or very weak
in SNe of type IIn\footnote{``n'' stands for
narrow. Type IIn are characterized by a narrow H$\alpha$ emission and
high bolometric light curve with a relatively flat evolution.
In the prototype SN IIn 1988Z indeed there was no sign of the nebular
forbidden lines [O I] 6300,6364 \AA\ and
[Ca II] 7291,7324 \AA. Similar behaviour is observed in SN IIn 1994aj
(Benetti et al. 1998). The well studied events SN IIn 1995N
(Fransson et al. 2002; Pastorello et al. 2005) and SN IIN 1995G
(Chugai $\&$ Danziger 2003; Pastorello et al. 2002) display the
same characterizing peculiarity.
One possibility for this is that their progenitors are not massive,
believed to explode in a very dense CS environment.}.
In Fig.1 (right panel) the constructed ``quasi-bolometric'' light curves of
SNIIP 1999em, SNIIb 1993J, SNIb 1990I and SNIc 1994I are
displayed. The peculiar SNII 1987A is also included for comparison.
The available optical ``U,B,V,R,I'' broad-band photometric data have
been integrated to recover the pseudo bolometric light curves.
Literature references
for the photometry, together with the adopted
parameters are reported in Table 1. We do not include the IR-contribution
since it is available only for SN 1987A, and hence the derived integrated
bolometric light curves represent a lower limit to the ``real'' bolometric
light curves.
The $^{56}$Co to $^{56}$Fe decay slope, $e-$folding time of 111.3 days,
corresponding to the full $\gamma -$ray trapping is also shown.
The difference in the early CCSNe bolometric light curves, i.e.
the peak in type Ib-c
SNe and the plateau in type II SNe, is mainly related to differences
in presupernovae radii and structures. The plateau behaviour in type II
is indicative of massive hydrogen envelopes, although its properties
(duration and luminosity) predict also a dependence on radius, energy
and the ejected amount of $^{56}$Ni (Popov 1993; Elmhamdi et al. 2003).
The lack of such significant hydrogen in the outer layers of type
Ib-c inhibits the presence of a plateau behaviour.
The peak characteristics, luminosity and width, are sensitive to the ejecta
mass, released energy and the $^{56}$Ni mass (Arnett 1982).
The clear faster decline at late phases for type Ib-c SNe is naturally
attributed to the significant $\gamma$-ray escape with decreased
deposition as a result of low mass ejecta in this class of objects, while
owing to the massive hydrogen mantle, type II SNe light curves indicate
that radioactive decay of $^{56}$Co with the consequent trapping of
$\gamma$-rays is the main source of energy powering the light curves at
late times. We note that in type II objects the V-light curve
follows the bolometric light curve
fairly well at late epochs, which simplifies for
example the derivation of the synthesized $^{56}$Ni mass. It is
sufficient then the use of the tail absolute V-light curve of SN 1987A
, for which the ejected $^{56}$Ni mass is accurately known from
observations and detailed
modeling, as a template for $^{56}$Ni mass derivations in other II events
(Elmhamdi et al. 2003b; Hamuy 2003).
The stripped-envelope SNe case is more difficult. The V-band light curve
does not parallel the bolometric one.
It is hence necessary in type Ib-c a bolometric light
curve modeling rather a simple use of the absolute visual bands.
\begin{figure}[htb]
\psfig{file=fig3.ps,width=13cm,height=14cm}\FigCap{The temporal evolution of the [O I] 6300,6364 \AA ~luminosity for
SNe of type II (upper panel) and for SNe Ib-c (lower panel). The $^{56}$Co
to $^{56}$Fe decay slope is also reported.}
\end{figure}
\section{The analysis}
In this section we measure and study the oxygen [O I] 6300,6364 \AA\ line
luminosity for the CCSNe sample. For this purpose
the available spectra were corrected for redshift and reddening effects
and calibrated with photometry when needed.
The recovered integrated
line fluxes, assuming a continuum level, are then converted to luminosities
using adopted distances.
\subsection{Oxygen Luminosity and Mass}
\subsubsection{The luminosity}
Figure 2 highlights the [O I] 6300,6364 \AA\ line luminosity temporal
evolution, starting at 200 days after explosion, for type II SNe
(upper panel) and for type Ib-c SNe (lower panel). For comparison,
the $^{56}$Co to $^{56}$Fe decay slope is also displayed (for an
arbitrary $^{56}$Ni mass; dashed line).
The emission [O I] 6300,6364 \AA\ light curves behave differently
within the CCSNe family. For type II events: the light curves have
a plateau-like maximum at day 200, changing slowly until day $\sim$340.
At this epoch the light curves are already on the exponential decline
phase (Fig.1-right panel).
The line-luminosity dropped then sharply, with a rate of decrease
similar to that of the radioactive decay of $^{56}$Co
(i.e. $e-$folding time of 111.3 days). We note here
that in the case of SN 1987A, it has been argued that dust condensation
affects the line luminosity evolution starting
at day $\sim$530, inducing a further increase in the rate of decrease
(Danziger et al. 1991). At approximately the same time the [O I]
line profile showed a marked shift of the peaks towards
blue wavelengths (Lucy et al. 1989). There are however two exceptions
decaying earlier compared to the rest of the II SNe sample, namely
SN 1970G and SN 1980K. Interestingly these two objects are
respectively type IIP-L and IIL SNe.
For type Ib-c events: starting at the age of 200 days, the light curves
are already on a steep decline, faster to that of the $^{56}$Co radioactive
decay. The ``Chi Square'' fit to SN Ib 1985F luminosity data for example
indicates an
$e-$folding time of about 70 days.
The time at which the deviation to the decline occurs varies
among the CCSNe family, being earlier in Ib-c, followed by IIL and then later
on SNe IIP. Specifically the line luminosity trend is
found to follow the bolometric light curves in the time
range of interest, namely after 200d in type Ib-c and after $\sim 340$d
in type II. In SN 1987A for example, the [O I] luminosity
relative to the bolometric one, i.e. $(L_{[O I]}/L_{bol}$), shows
an almost flat-topped behaviour at the time interval $\sim 340-500$d.
Other effects enter at
later epochs, $>$ 500d, such as dust condensation and the
IR-catastrophe (Danziger et al. 1991; Fransson et al. 1996; Menzies 1991).
On the one hand
this is a direct evidence that the dominant source of ionization and
heating is $\gamma-$rays from the radioactive
decay of $^{56}$Co in the CCSNe variety, with the $\gamma-$rays escaping with
decreased deposition in type Ib-c events, owing to the low
mass nature of their ejecta.
We note also that in both panels the line light curves span more than
one dex in luminosity, which may be related to the variation in the
oxygen yields. Thus the importance of the oxygen mass estimates.
\begin{figure}[htb]
\centerline{\psfig{file=fig4.ps,width=10cm,height=10cm}}
\FigCap{The fit example to the [O I] 6300,6364 \AA ~profile
of SN IIP 1999em at day 390. Shown in thick line is the best fit.
Dashed profiles refer to the decomposition of the best fit into
two Gaussians for the [O I] 6300 \AA\ and [O I] 6364 \AA\ components.}
\end{figure}
\subsubsection{The flux ratio ``$F_{6300}/F_{6364}$'' }
Various investigators have dealt with the doublet flux ratio in
the [O I] 6300,6364 \AA\ line, especially for the well observed
SN 1987A (Spyromilio $\&$ Pinto 1991; McCray 1996).
It is well established that for a large and homologously expanding
atmospheres such as in supernovae
events, the Sobolev approximation holds and consequently simplifies
the radiation transfer and line formation treatments
(Castor 1970; Jeffery $\&$ Branch 1990).
Within this context, a line intensity is given by:
$Flux \propto P_{esc}\times A_{ul}\times N_u $, where $N_u$ is the number
density
of atoms in the upper state, $A_{ul}$ is the Einstein
coefficient, and $P_{esc}$ being the escape probability giving by
$P_{esc}=[1-exp(-\tau)]/\tau$; $\tau$ is the Sobolev optical depth.
For a freely expanding atmosphere the density
$N_u$ goes as $t^{-3}$ and hence the Sobolev optical depth
should behave as $\tau \propto N_u \times t=t^{-2}$.
For the line of interest, i.e. the [O I] 6300,6364 \AA\ doublet, the
two transitions have $A_{6300}=3\times A_{6364}$, and
$\tau_{6300}=3\times \tau_{6364}$; therefore the doublet
flux ratio reads:
\begin{equation}
\frac{F_{6300}}{F_{6364}} = \frac{1-exp(-\tau_{6300})}{1-exp(-\tau_{6364})}
= \frac{1-exp(-\tau_{6300})}{1-exp(-\tau_{6300}/3)}
\end{equation}
A measure of the flux doublet ratio can be then translated into the
optical depth in the 6300 \AA\ line in the SNe ejecta through solving
Eq. 1. In addition we expect an asymptotic value for the ratio to be 1
for the optically thick transitions and to be 3 for the optically
thin case.
Recovering the ratio of the flux doublet is not always an
easy task. Line blending can prevent an adequate intensity measurement.
One possibility is a simple use
of the peak intensities in the components of the doublet
assuming it similar to the ratio of the flux of the doublet components
(Li $\&$ McCray 1992).
Deblending yields to more accurate line ratio determination, especially
in cases with a sever blend. We selected a variety of
objects within our CCSNe sample that allow good estimates of the ratio.
The studied events are: SNe IIP(1988A, 1999em, 2004et), SNe Ib(1985F, 1990I)
and SNe Ic(1998bw, 2002ap). Data for SN 1987A are also included for
comparison. An example is illustrated in Fig.3 for the
observed spectrum of SN IIP 1999em at day 390. Our adopted
methodology consists in fitting the [O I] 6300,6364 \AA\ feature
with a single function formed by 2 Gaussians,
after estimating the continuum level and fixing the wavelength
separation of the doublet, i.e. 64\AA, and with
both Gaussians having the same velocity width. The fit in Fig.3 is
a good one. In some cases
it was not possible to obtain a particularly good overall fit using only two
Gaussians since one is left with either a residual in the blue wing of
the 6300 \AA\ component for narrower lines, or little evidence of the
the 6364 \AA\ component for broader lines. Elmhamdi et al. 2004 have used
similar methodology and
obtained a satisfactory fit in the spectrum of SN Ib 1990I at 254d introducing
a third Gaussian component assuming there is a contribution from
Fe II 6239 \AA\ emission, and allowing its velocity width to be a free
parameter (their Fig. 7).
According to our sample analysis the two approaches, using the deblending
technique and a simple use of the peak fluxes, when possible, agree
within 20$\%$.
\begin{figure}[htb]
\centerline{\psfig{file=fig5.ps,width=11cm,height=12cm}}
\FigCap{$Upper~ panel$: the temporal evolution of the flux ratio
$F_{6300}/F_{6364}$. The overploted curve corresponds to the
LSQ-fit by Eq. 2, indicating a value of $t_{trans} \simeq 426$d (see
text for details).
$Lower~ panel$: the recovered optical depths in the [O I] 6300\AA\ line.
The power law LSQ-fit is also displayed ($\sim t^{-n}$ with $n= 2.089
\pm 0.153$; see text).}
\end{figure}
Figure 4 reports the results of our investigation. The upper panel displays
the temporal evolution of the ratio measurements, while in the lower panel
we plot the corresponding optical depths in the primary component,
computed by solving Eq. 1.
The ratio values demonstrate a temporal trend
from an optically thick phase to an optically thin one. The optical depth
is found to decrease from values as high as $\sim 7$ around day 170
falling to lower values at later phases ($< 0.5$ around 500d). The
continuous line
is the resulted LSQ fit to the data, using a power-law function
(i.e. $\propto t^{-n}$). The best fit gives an index of
n= 2.089 ($\pm 0.153$). This is consistent with the index expected
from the simple expansion assumption discussed above
(i.e. $\tau \propto N_u \times t=t^{-2}$).
Furthermore, we may introduce a time dubbed $t_{trans}$, as the time at which
the line makes the transition to the phase characterized by
$\tau_{6300} = 1$. In fact using $\tau \propto t^{-2}$
and introducing $t_{trans}$, Eq. 1 can be written as:
\begin{equation}
\frac{F_{6300}}{F_{6364}} = \frac{1-exp(-(t_{trans}/t)^2)}{1-exp(-(t_{trans}/\sqrt{3} t)^2)}
\end{equation}
Provided the ratio measurements, we fit the data with Eq.2. The LSQ fit is
shown by the continuous line in the upper panel of Fig. 4. The best fit
indicates a time of $t_{trans} =426.06 \pm 13.24$ d.
The time at which $\tau_{6364} = 1$, is $245.96 \pm 7.64$ d.
This above described analysis demonstrates the consistency within the CCSNe
family on how the components intensity ratio and the optical depths
develop in time.
\subsubsection{The mass}
In this section we adopt two different methods for estimating
the oxygen mass in SNe of type II and SNe of type Ib-c.\\ \\
{$\bullet$ \it Type II SNe:}
We use the recovered [O I] doublet luminosities to determine
the abundance of the oxygen produced through the SN explosion.
At the epoch of about
$1$ year, the luminosity of the [O I] doublet is powered by
the $\gamma$-ray deposition and by
ultraviolet emission arising from the deposition
of $\gamma$-rays in oxygen material. The [O I] doublet luminosity
is related directly to the oxygen mass (Elmhamdi et al. 2003a),
and at a given time one may write:
\begin{equation}
L_t(\mbox{[O I]})=\eta \times L_t(^{56}\mbox{Co}) \times \frac{M_{\rm O}}{M_{\rm ex}}
\end{equation}
where $M_{\rm O}$ is the mass of oxygen,
$M_{\rm ex}$ is the $``$excited" mass in which the bulk of
the radioactive energy is deposited, and $\eta$ is the efficiency
of transformation of the energy deposited in oxygen into
the [O I] doublet radiation. The L$_t(^{56}\mbox{Co})$ refers to
the radioactive decay energy input, given by
$L_t=L_{0}\times$($M_{\rm Ni}$/$M_{\odot}$)e$^{-t/\tau_{\rm Co}}$,
with $L_{0}\simeq 1.32 \times$ 10$^{43}$ ergs s$^{-1}$,
the initial luminosity corresponding to 1 $M_{\odot}$
and $\tau_{\rm Co}=111.3$ d.
Assuming then that all type II events have similar $\eta$ and
$M_{\rm ex}$ at similar phases, we derive rough estimates
of the oxygen mass for the events of the sample adopting [O I] 6300,6364\AA
~light curve of SN 1987A as template. As discussed previously,
the [O I] luminosity is found to have similar decay
rates in SNe II at late phases, following a behaviour similar to
the bolometric light curves. This fact gives some confidence for the
use of SN 1987A [O I] luminosity as a template for the recovery of the
amounts of ejected oxygen. The oxygen mass in SN~1987A is
estimated to be in the range $1.5-2$ $M_{\odot}$
(Fransson et al. 1996; Chugai 1994).
In Table 1, column 7, we report the amounts of the ejected oxygen mass
derived in this manner. The variation range, for each event,
reflects the combination of the variation in the oxygen mass for
SN 1987A (i.e. $1.5-2$ $M_{\odot}$) together with the uncertainties
from the fit procedure.
\\ \\
{$\bullet$ \it Type Ib-c SNe:}
The emission lines which are formed at densities above
their critical ones, given the optical depths are not large, have
the luminosity directly proportional to the mass of the emitting ion.
Such conditions hold for the [O I] 6300,6364 \AA\ doublet line emission
in type Ib-c SNe at nebular phases.
On the one hand, we found moderate optical depths at late phases
(Fig 4, lower panel). On the other hand, the condition of
the high density limit above the critical density for the [O I] line
($\sim7~\times 10^5 ~$cm$^{-3}$) is found to be clearly fulfilled in
the ejecta of type Ib-c SNe (Leibundgut et al. 1991; Elmhamdi et al. 2004).
A possible direct method can be used to check the high density limit
characteristic. Indeed, the density is directly related
to the relative
strengths of the [O I] doublet components (Leibundgut et al. 1991;
Spyromilio $\&$ Pinto. 1991). In both cited works, the variation of the line
doublet ratio as function of the density at a given time is computed, and is
found to be insensitive to the adopted temperature, especially for
the late epochs.
From our computations we take a ratio value of 1.58 at day 250 or 1.8 at day
300 as representatives (i.e. from the best fit in Fig.4-upper panel).
According to Fig. 6 of Leibundgut et al. (1991), the
uncertainty in the temperature leads to the following density
range
$2\times 10^{9}~$cm$^{-3}\leq N_{e} \leq 4\times 10^{9}~$cm$^{-3}$.
In the high density limit, i.e. above the critical density, the mass
of ejected oxygen can be recovered using the
[O I] 6300,6364 \AA\ flux. Uomoto (1986) has shown that the oxygen mass,
in M$_\odot$, is given by:
\begin{equation}
M_{Ox} = 10^{8} \times D^2 \times F(\rm{[O ~I]}) \times \it exp{(2.28/\it T_4)}
\end{equation}
where $D$ is the distance to the supernova (in Mpc), $F$ is the reddening-free
[O I] integrated flux (in ergs s$^{-1}$ cm$^{-2}$) and $T_4$ is the
temperature of the oxygen-emitting gas (in 10$^4$ K).
Worth noting that because of the variation of $F$([O I]) and $T_4$, Eq. 4
implies time-dependence.
Schlegel $\&$ Kirshner (1989), when estimating the ejected
oxygen amounts, have adopted a
constant temperature $T_4$=0.4 at the nebular phase of SNe Ib 1984L and
1985F. The assumption of a
constant temperature at different late phases provides different oxygen
masses as one may expect, since earlier nebular epochs are hotter compared to
latter ones.
Alternatively, the temperature at a given time can be constrained
based on the
[O I] 5577 \AA\ to [O I] 6300,6364 \AA\ flux ratio. Assuming
that the O I lines are formed mainly by collisional excitation, the ratio
is given by the following expression (Houck \& Fransson 1996):
\begin{equation}
\frac{F_{6300}}{F_{5577}}=0.03 \beta_{6300} \times [1+1.44 \it T_3^{-0.034}(\frac{10^8}{N_e})]\times \it exp{(25.83/ \it T_3)}
\end{equation}
\vspace{0.3truecm}\hspace{-0.4truecm}where $T_3$ is the temperature of
the oxygen-emitting gas (in 10$^3$ K), $\beta_{6300}$ is the
[O I] 6300 \AA\ Sobolev escape probability ($\simeq$ 1) and $N_e$ is the
electron density.
\begin{figure}[htb]
\centerline{\psfig{file=fig6.ps,width=11cm,height=11cm}}
\FigCap{The temporal evolution of the [Ca II] 7291,7324 \AA\ over
[O I] 6300,6364 \AA\ integrated flux ratio.
Note the complete
separation of type Ib-c from type II SNe.}
\end{figure}
A problem with this method is the very weak observed [O I] 5577 \AA\ feature
in the late time spectra analyzed here. It is indeed an indication of low
temperatures in the oxygen-emitting zone. However one may estimate an upper
limit on the [O I] 5577 \AA\ flux by integrating over the same velocity
interval of the [O I] 6300,6364 \AA\ profile. It is found that the
temperatures at late phases of interest, $\ge$250d, tend to have values
in the range $3400 - 4200$K.
Results from this described method, through equation 4, are listed in
column 7 in Table 1. For SNe 1994I, 1993J, 1998bw and 2002ap however,
the amounts reported in the table come from the most recent
spectroscopic and photometric modeling (see column 8 for references).
\subsubsection{The [Ca II]/[O I] intensity ratio}
After considering the reddening correction, we have evaluated the
integrated flux ratio of the forbidden emission lines
[Ca II] 7291,7324 \AA\ and [O I] 6300,6364 \AA, for the CCSNe sample objects
having wavelength coverage in these two nebular lines.
Results are displayed in Fig. 5. Two separate
populations are clearly distinguishable. The ratio is found to remain
below unity in type Ib-c, with the exception of SN 1990Ib around day 315
($\sim$1.22).
SNe IIb 1993J, Ic-hypernova 1998bw and IIL 1980K belong to this category
as well. A mean value is measured to be $\sim$ 0.51.
SNe of type II instead concentrate on the top of the figure, with a mean
value of $\sim 3.17$. Additionally, the two classes appears to have
a flat evolution behaviour of the [Ca II]/[O I] ratio.
In a detailed analysis, Fransson $\&$ Chevalier (1987, 1989) have modeled
late emission spectral lines for different supernova progenitors. It is
found that because of the composition and density structures one can
use the relative emission line strengths as a progenitor indicator tool.
In particular the forbidden emission line ratio [Ca II]/[O I] presents
weak dependence on density and temperature of the emitting zones, and
is expected to display an almost constant evolution at late epochs. The ratio
is found to be very sensitive to the core structure and mass. Furthermore
it seems that the ratio tends to increase with decreasing progenitor
mass. The distribution of our measurements in two groups tend to indicate
different progenitor properties, with lower progenitors for type Ib-c, IIb
and IIL classes compared to normal type IIP events. It is worth to note here
however that in type Ib-c SNe there is no hydrogen rich Ca II emitting
zone as is the case for type II objects (de Kool et al. 1998).
\subsubsection{The $^{56}$Ni mass}
As discussed in Sect. 2, the exponential behaviour of the late V-band absolute
light curves of type II SNe is found to be in accordance
with the radioactive decay $^{56}$Co$-$$^{56}$Fe ($M_V \propto
exp(-t/111.3~ $d$)$), which provides a potential
method to recover the amount of the ejected $^{56}$Ni with the use of the
$M_V$ photometry of SN 1987A as a template between $120-400$ d
(Elmhamdi et al. 2004; Hamuy 2003). An amount of $0.075~M_{\odot}$ is
adopted for the $^{56}$Ni mass of SN 1987A (Catchpole et al. 1988,
Suntzeff \& Bouchet 1991). Results from the described methodology are
reported in Tab. 1 (column 6).
There is a significant scatter of the
ejected $^{56}$Ni masses, with an average value of $\approx 0.053~M_{\odot}$.
For type Ib-c events, the broad band light curves do not trace necessary
the bolometrics. The amounts summarized in Tab. 1(column 6) are indeed results
from bolometric light curves and/or spectra modeling (see column 8 for
the corresponding references). An average value of
$\approx 0.18~M_{\odot}$ is computed.
\section{Discussion and conclusions}
We have selected a sample of 26 events within the CCSNe family.
Our main goal was to investigate how the oxygen manifests its presence
at late phases for each SN class, especially the emission
[O I] 6300,6364 \AA\ doublet.
Based on investigating early spectra of CCSNe, the
permitted oxygen line O I 7773 \AA\ seems to get weaker following the
SNe sequence ``Ic$-$Ib$-$IIb$-$II''
(Elmhamdi et al. 2006; Matheson et al. 2001).
For stripped envelope objects, being less diluted by the presence
of an helium envelope, one may expect the oxygen lines
to be more prominent. An observational fact that clearly support
this belief is the earlier appearance of the [O I] 6300,6364 \AA\
following the above order, i.e. ``Ic$-$Ib$-$IIb$-$II''.
The following examples highlight this fact. The
oxygen line emerges at an age of 1-2 months in
type Ic SN 1987M (Filippenko 1997). SN Ic 1994I displayed
evidence for the line at an age
of 50 days, although some hints may even be seen in the $\sim$36 days
spectrum (Clocchiatti et al. 1996b). SN Ic 1997B shows clear evidence
for [O I] 6300,6364 \AA, [Ca II] 7291,7324 \AA\ and also O I 7773 \AA\
on the 2 months spectrum (Gomez $\&$ Lopez 2002).
In SN 1998bw, Ic-hypernova event, the nebular features were already
recognizable in the 43 days spectrum (Patat et al. 2001). While
in SN Ib 1990I it was hinted at the 70 days spectrum (Elmhamdi et al. 2004).
In other type Ib SNe it appears
earlier than in SN 1990I. In SN IIb 1993J, a transition object, the line
was visible in the 62 days
spectrum (Barbon et al. 1995). SN 1996cb, another well
observed IIb event, showed
evidence of the [O I] 6300,6364 \AA\ line around day 80 (Qiu et al. 1999).
In SNe II, however, the line appears later: around day 150 in SN 1987A
(Catchpole et al. 1988) and after day 138 in SN 1992H
(Clocchiatti et al. 1996a). In SN II 1999em it is suggested at a somewhat
earlier phase compared to SNe 1987A and 1992H, namely at day 114.
Furthermore, the width velocities in the [O I] 6300,6364 \AA\ and
in the [Ca II] 7291,7324 \AA\ nebular lines are mainly found to be
higher in SNe Ic than in SNe Ib (Schlegel $\&$ Kirshner 1989;
Matheson et al. 2001).
In the cited papers, the FWHM values were evaluated fitting a
single profile, namely Gaussian, to the total observed profiles.
We mention here that given the line blending in these two lines (doublets),
results should be interpreted with some caution. One should indeed resolve
the lines and compare width velocities of single components rather
than using a single Gaussian fit to the whole feature. It is not
simple however in some cases to get a satisfactory fit, especially
when the blend is sever. According to our analysis, we faced more
difficulties in type Ib and Ic than in type II. This fact is indeed
in itself indicative of higher velocities in stripped envelope objects, while
in SNe II, owing to lower expansion velocities, the
observed [O I] 6300,6364 \AA\ profile
is less complex and the two components are clearly visible.
This can be easily understood if considering the formation of the
line doublets
in cases where the element expansion velocity does not exceed the
the two components separation velocity, i.e. 3000 km s$^{-1}$.
We have checked the width velocities by means of the FWHM of
the single [O I] component at 6300 \AA\ of some of our sample events.
At a phase of $\sim$300d, SNe II 1987A and 1999em
are found to have respectively FWHM $([O~I]~6300~\AA)\sim 2750$
and $2400$ km s$^{-1}$, while for Ib SNe a typical range variation of
$\sim 4000-5000$ km s$^{-1}$ is deduced (e.g. SNe Ib 1985F, 1996N).
SN IIb 1993J, at similar age, has a velocity of $\sim 4600$ km s$^{-1}$,
with a complex [O I] 6300,6364 \AA\ profile due to the presence of
H$\alpha$ at the red wing of the line (Patat et al. 1995).
Higher velocities, $\geq 6000$ km s$^{-1}$, are computed from Ic spectra
(e.g. SNe 1987M, 1994I and 1998bw).
The [O I] 6300,6364 \AA\ nebular line at late epochs is representative
of the expansion velocity. For a given explosion energy, a
greater ejecta mass allows for lower velocities. The found lower
velocities in type II SNe can be attributed then to a large
mass with respect to type Ib-c objects.
We have investigated in details the properties of the
[O I] 6300,6364 \AA\ doublet profile. The measured line luminosities
are found to trace well the bolometric
light curves, behaving differently in type Ib-c than in type II events.
In SNe II, the line light curves show a plateau-like evolution
until $\sim340$ days, following later-on by an exponential decline similar to
the radioactive decay of $^{56}$Co ($e-$folding$\simeq$111.3 days). The
linear type II SNe display an early decline in their [O I] light curves.
type Ib-c light curves display already a decline by an age of 200 days,
steeper than the $^{56}$Co radioactive decay rate.
The $\gamma-$rays from the radioactive decay of $^{56}$Co are hence
the dominant source of ionization and heating in CCSNe variety.
Furthermore, owing to the low mass nature of their ejecta, type Ib-c SNe
are characterized by a decreasing deposition of the $\gamma-$rays escape.
Similar conclusions about the progenitor diversity are argued from
the analysis of the integrated flux ratio of the forbidden emission
lines [Ca II] 7291,7324 \AA\ and [O I] 6300,6364 \AA. The ratio is
found to concentrate in two distinguishable locations, namely around
a value of 3 for type II SNe (mean $\sim3.17$), and below unity
in Ib-c objects (mean $\sim0.51$). This forbidden lines ratio is potentially
sensitive to the progenitor mass star (Fransson $\&$ Chevalier 1987; 1989),
suggesting higher masses in SNe of type II.
The way CCSNe family members transit from optically thick to optically
thin phases is imprinted onto the profile of the [O I] 6300,6364 \AA\ doublet.
We emphasize this point making use of the components flux
ratio $F_{6300}/F_{6364}$,
after deconvolving the two components of the observed line profiles.
Based on a simple description of the physics behind the line formation,
our results indicate a consistency within the CCSNe
on how the components intensity ratio and consequently
the optical depths develop in time. Additionally, the
ratio $F_{6300}/F_{6364}$ can be used to estimate the average density
of the oxygen-emitting zone (Leibundgut et al. 1991;
Spyromilio $\&$ Pinto 1991).
Adopting representative values of the ratio of
1.58 at day 250 or 1.8 at day
300, inferred from our best fit in Fig.4-upper panel,
and according to Fig. 6 of Leibundgut et al. (1991), the
uncertainty in the temperature leads to the following density
range $2\times 10^{9}~$cm$^{-3}\leq N_{e} \leq 4\times 10^{9}~$cm$^{-3}$.
Worth noticing here that based on our approach one may get further
informations about the physical conditions in the oxygen emitting region.
In fact, for an uniform density distribution, one may use the
above derived density range together with the element volume, based
on the [O I] 6300 \AA\ FWHM velocity, to get a rough estimate of the
ejected oxygen mass. This approach is found to provide too large
amounts compared to results reported in Tab. 1 (column 7).
For SN IIP 1999em for example, at 300 days, using the previous derived
velocity of 2400 km s$^{-1}$, we find an ejected oxygen
mass as high as $\sim$40 M$_\odot$, indicating a volume-filling factor
as low as 10$^{-2}$. Similar filling factor is found for SN II 1987A and
SN Ib 1990I. These results suggest
that the oxygen material in CCSNe generally fills its volume in a clumpy
way rather than homogeneously.
We describe and adopt two methods for the oxygen mass estimate in CCSNe.
We provide also estimates of the ejected $^{56}$Ni masses.
It should however be noted that various parameters affect the
recovered values, especially as long as the oxygen mass is concerned,
contributing hence to their uncertainties. Indeed, the luminosity and
temperature measurements are sensitive to the nature and quality of
the spectra (i.e.
severe line blending; multi-component profiles; presence of [O I] 5577 \AA).
According to our test effects, we estimate the uncertainty of the
derived oxygen-mass amounts reported in Tab. 1 to vary within $15-30 \%$.
Furthermore, for cases with clear evidence of O I 7774 \AA\ line, the presence
of ionized oxygen is argued (Mazzali et al. 2010), requiring higher mass of
oxygen than the one needed to produce [O I] 6300,6364 \AA\ alone.
Generally, Nickel masses are found to suffer less uncertainties
compared to oxygen.
The errors in type II SNe, being more homogeneous than SNe Ib-c, do not
exceed $20 \%$ (mainly due to distance and reddening estimates). In SNe Ib-c,
we estimate a variation of $10-30 \%$.''
These estimates
, i.e. oxygen and iron masses, are of importance as they can be
indicative of the core mass and related to the progenitor star nature.
For the purpose of a better investigation, we compute the
[O/Fe]\footnote{$[A/B]=log_{10}(A/B)_{\star}-log_{10}(A/B)_{\odot}$}
yields ratio for each event. A solar value of
$~log_{10}(O/Fe)_{\odot} = 0.82$
dex (Anders $\&$ Grevesse 1989) is adopted. The recovered
amounts are reported in Fig. 6 (horizontal lines) as function of the initial
mass according to the most reliable core collapse SN models in literature
, namely Woosley $\&$ Weaver 1995
(short dot-dashed line), Thielemann, Nomoto $\&$ Hashimoto 1996
(long dashed line) and Nomoto et al. 1997 (long dot-dashed line).
A typical value for type Ia SNe is
as well reported (Nomoto et al. 1984), which shows the nature of type Ia
SNe being iron producers.
Fig. 6 defines two possible concentration regions,
of type II objects from one side and type Ib-c from the other side, and
suggestive of a continuum in the $[O/Fe]$ values.
Indeed, an important issue that can be read out immediately from Fig. 6 is
that the reported results for type Ib-c are found to be located at the
bottom of type II SNe.
These zones may provide constraints on the progenitor masses
in CCSNe family.
Within the core collapse models of type Ib-c SNe, two scenarios are
argued: $first$ a single high mass star ($M_{ms} \geq 35~$ M$_\odot$)
exploding as Wolf-Rayet star after an episode of strong stellar wind, and
$second$ a less massive star ($M_{ms} \sim 13-18~$ M$_\odot$) in a binary
system. Although the progenitor nature of type Ib-c SNe is an open and debated
issue, the position of type Ib-c events in the ``$[O/Fe]~.vs.~M_{ms}$''
plot, according to Fig. 6, might be taken as an observational
support of the intermediate massive stars in binary systems, stripped of
their envelope through binary transfer, as the favoured
progenitors in this class of objects.
Although the uncertainties in the stellar evolution models of massive
stars and in the determination of the oxygen and iron
yields, Fig. 6 provides a methodology to elucidate
the progenitor nature of core collapse SNe and interesting comparisons
can be drawn . However, extended samples and
more reliable determinations, especially for oxygen abundances (late spectra
modeling for example), are clearly
needed to populate the ``$[O/Fe]~.vs.~M_{ms}$'' diagram and hence have a
deeper view on the oxygen to iron ratio and how it changes as function of
the type of SN.
The present quantitative analysis can also provide insights and input
data for the evolution of galaxies and the chemical enrichment
models.
The main goal of the paper was to highlight some late-phases spectra
related properties of CCSNe, providing possible physical
interpretations. As a first step we concentrate our study on 26 selected
events.
For future investigation we aim to enlarge the studied sample including
more recent and up-dated observed CCSNe objects, especially those with
spectra of improved time coverage and spectral resolution
(mostly last decade observations; Elmhamdi et al. In preparation).
\Acknow{We thank the referee for the very helpful and constructive suggestions.
We thank I. John Danziger for his comments on the original
manuscript and for the stimulating discussions.
We are grateful for the use of the ``SUSPECT'' SNe Archive-Oklahoma
University and also of the ``CfA'' Supernova Archive.
We also thank D. K. Sahu for providing published data of SN IIP 2004et,
R. Foley for the discussion about SN Ic 2002ap data and M. Pozzo
for discussing SN IIP 2002hh. This project was supported by King Saud
University , Deanship of Scientific Research, College of Science Research
Center.}
|
1,116,691,497,529 | arxiv | \section{Introduction} \label{intro}
A two-way contingency table gives the joint distribution of two
random variables with a finite number of outcomes. If we denote by
$\{0, \ldots, I-1\}$ and $\{0, \ldots, J-1\}$ the outcomes of
$X_1$ and $X_2$ respectively, the contingency table is represented
by a matrix $P=(p_{ij})$, where $p_{ij}$ is the probability that
$X_1=i$ and $X_2=j$. The table $P$ is also called an $I \times J$
contingency table, in order to emphasize that the variable $X_1$
has $I$ outcomes and the variable $X_2$ has $J$ outcomes.
In the analysis of contingency tables odds ratios, or
cross-product ratios, are major parameters, and their use in the study of $2 \times 2$ tables goes
back to the 1970's. For an explicit discussion on this approach see, e.g.,
\cite{fienberg:80}.
For a $2 \times 2$ table of the form:
\begin{equation} \label{tabella2per2}
\begin{pmatrix}
p_{00} & p_{01} \\
p_{10} & p_{11} \\
\end{pmatrix}
\end{equation}
there is only one cross-product ratio, namely:
\[
r = \frac {p_{00}p_{11}} {p_{01}p_{10}} \, .
\]
In the general $I \times J$ case, there is one cross-product ratio
for each $2 \times 2$ submatrix of the table. Thus, they have the
form
\[
\frac {p_{ij}p_{kh}} {p_{ih}p_{kj}}
\]
for $0 \leq i < k \leq I-1$ and $0 \leq j < h \leq J-1$, see
\cite[Chapter 2]{agresti:02}. In this paper we will consider the
cross-product ratio and other ratios naturally defined.
Odds ratios are used in a wide range of applications, and in
particular in case-control studies in pharmaceutical and medical
research. Following the theory of log-linear models, the
statistical inference for the odds ratios is made under asymptotic
normality, see for example \cite{bishop|fienberg|holland:75}. More
recently, some methods for exact inference have been introduced,
see \cite{agresti:02} and \cite{agresti:01} for details and
further references. For the theory about the Bayesian approach,
see \cite{lindley:64}.
From the point of view of Probability and Mathematical Statistics,
different descriptions of the geometry of the statistical models
for contingency tables are presented in \cite[Chapter
2]{collombier:80}, and in \cite[Section
2.7]{bishop|fienberg|holland:75}, using vector space theory. An
earlier approach to the geometry of contingency tables with fixed
cross-product ratio can be found in \cite{fienberg|gilbert:70}. In
the last few years, the introduction of techniques from
Commutative Algebra gave a new flavor to the geometrical
representation of statistical models, as shown in, e.g.,
\cite[Chapter 6]{pistone|riccomagno|wynn:01},
\cite{pistone|riccomagno|wynn:01ams},
\cite{geiger|heckerman|king|meek:01} and \cite{slavkovic:04}.
In this paper we use Algebraic and Geometric techniques in order
to describe the structure of some models for two-way contingency
tables described through odds ratios.
We first consider the case of $2 \times 2$ contingency tables of
the form (\ref{tabella2per2}) with the constraints $p_{ij} > 0$
for all $i,j=0,1$ and $p_{00}+p_{01}+p_{10}+p_{11}=1$. If we allow
some probabilities to be zero, notice that the ratios are either
zero or undefined. Thus we restrict the analysis to the strictly
positive case.
In a $2 \times 2$ table we consider the three odds ratios:
\[
r_\times = \frac {p_{00}p_{11}} {p_{01}p_{10}} \, ,
\]
\[
r_{||} = \frac {p_{00}p_{10}} {p_{01}p_{11}} \, ,
\]
\[
r_= = \frac {p_{00}p_{01}} {p_{10}p_{11}} \, .
\]
The meaning of the three odds ratios above will be fully explained
in Section \ref{application}.
Let $r_\times = \alpha^2$, $r_{||} = \beta^2$ and $r_==\gamma^2$.
For further use, it is useful to make explicit the following
identities. Considering $r_=$ and $r_{||}$, it is easy to check
that:
\begin{equation} \label{rel1}
\beta \gamma = \frac{p_{00}} {p_{11}} \, ,
\end{equation}
and
\begin{equation} \label{rel2}
\frac \beta \gamma = \frac{p_{01}} {p_{10}} \, .
\end{equation}
In Section \ref{oddsSEC}, we study the geometric properties of
some statistical models for $2 \times 2$ contingency tables. We
consider models obtained by fixing two odds ratios, showing that
the model is represented by a segment in the probability simplex
and studying the behavior of the third ratio. In particular, an
expression for tables with three fixed ratios is derived. We also
recover classical results about models with a fixed odds ratio. In
Section \ref{2x3SEC}, we give a glimpse of the general situation
of $I \times J$ contingency tables. We focus our attention on
$2\times 3$ tables and we present some of the difficulties arising in
the general case. An application to case-control studies is
presented in Section \ref{application}.
\section{Odds Ratios}\label{oddsSEC}
In this section, we use basic geometric techniques to study the
$2\times 2$ tables having two out of the three ratios $r_\times,
r_=$ and $r_{||}$ fixed.
We consider a $2\times 2$ matrix as a point in the real affine
4-space $\mathbb{A}^4$. In particular, with the notation of Equation
(\ref{tabella2per2}), the $p_{ij}$'s are coordinates in
$\mathbb{A}^4$. A $2\times 2$ {\em table} is a matrix in the open
{\em probability simplex}
\[\Delta=\left\lbrace P=(p_{ij})\in\mathbb{A}^4 : \sum p_{ij}=1, p_{ij}> 0 ,i,j=0,1 \right\rbrace \, .
\]
As our goal is to describe odds ratios for tables, we may assume
the ratios to be non-zero positive numbers.
Fixed the first two ratios
\[r_\times=\alpha^2\mbox{\ \ and \ \ } r_{||}=\beta^2\, ,\]
we want to answer the following question:
\begin{quote}
Q1: How can we describe the locus of tables having the assigned
two ratios?
\end{quote}
and also
\begin{quote}
Q2: What are the possible values of the third ratio?
\end{quote}
These questions were posed in the AIM computational algebraic
statistics plenary lecture by Stephen Fienberg in 2003. In this
situation, some interesting comments about treating questions Q1
and Q2 were also made.
Consider the quadratic hypersurfaces of $\mathbb{A}^4$:
\[Q_\alpha: \alpha^2p_{01}p_{10}=p_{00}p_{11}\]
and
\[Q_\beta: \beta^2p_{01}p_{11}=p_{00}p_{10}\, .\]
Notice that a matrix in $Q_\alpha\cap Q_\beta$ is such that
$r_\times=\alpha^2$ and $r_{||}=\beta^2$ as soon as the ratios are
defined. Hence, to answer the first question, it is enough to
study
\[Q_\alpha\cap Q_\beta\setminus Z\, ,\]
where $Z=\left\lbrace P=(p_{ij})\in\mathbb{A}^4 :
p_{00}p_{01}p_{10}p_{11}=0 \right\rbrace$.
We readily see that $Q_\alpha\cap Q_\beta$ contains the
2-dimensional skew linear spaces
\[p_{00}=p_{01}=0 \mbox{ \ \ and \ \ } p_{10}=p_{11}=0\]
and by general facts on quadrics (see \cite[page 301]{Harris}) we
know that there exist two more 2-dimensional skew linear spaces,
$R$ and $S$, such that
\[Q_\alpha\cap Q_\beta=\lbrace p_{00}=p_{01}=0\rbrace\cup\lbrace p_{10}=p_{11}=0\rbrace\cup R\cup S \, . \]
Manipulating equations we notice that a point in $Q_\alpha\cap
Q_\beta\setminus Z$ is such that
\[{p_{00}\over p_{01}}=\alpha^2{p_{10}\over p_{11}}=\beta^2{p_{11}\over p_{10}}\]
and
\[{p_{10}\over p_{11}}=\beta^2{p_{01}\over p_{00}}={1\over\alpha^2}{p_{00}\over p_{01}} \, .\]
Hence, $R$ and $S$ lie in the intersection of the two
3-dimensional spaces
\[(\alpha p_{10}-\beta p_{11})(\alpha p_{10}+\beta p_{11})=0\]
and
\[(\beta p_{01}-{1\over\alpha}p_{00})(\beta p_{01}+{1\over\alpha}p_{00})=0 \, ,\]
where $\alpha$ and $\beta$ are chosen to be positive. Only two out
of the four resulting 2-dimensional linear spaces lie in both
$Q_\alpha$ and $Q_\beta$ and these are $R$ and $S$:
\[R: \alpha p_{10}-\beta p_{11}=\beta p_{01}-{1\over\alpha}p_{00}=0 \, ,\]
\[S: \alpha p_{10}+\beta p_{11}=\beta p_{01}+{1\over\alpha}p_{00}=0 \, ,\]
which have parametric presentations
\[R=\{(\beta u,{1\over\alpha}u,\beta v,\alpha v): u,v\in\mathbb{R}\} \, ,\]
\[S=\{(\beta u,-{1\over\alpha}u,\beta v,-\alpha v): u,v\in\mathbb{R}\} \, .\]
Summing all these facts up, we get
\begin{prop}Fix the ratios $r_\times=\alpha^2$ and $r_{||}=\beta^2$. Then, a matrix has the given ratios if and only if it has the form
\[\left(\begin{array}{cc}
\beta u & {1\over\alpha}u \\
\beta v & \alpha v
\end{array}\right)
\mbox{ or } \left(\begin{array}{cc}
\beta u & -{1\over\alpha}u \\
\beta v & -\alpha v
\end{array}\right)\]
with $u,v$ non-zero real parameters.
\end{prop}
Finally, we have to intersect $R$ and $S$ with the probability
simplex. As we can choose $\alpha$ and $\beta$ to be positive, we
immediately see that $S\cap\Delta=\emptyset$ (there is always a
non-positive coordinate).
To determine $R\cap\Delta$, notice that $R\cap\{\sum p_{ij}=1\}$
is obtained by taking
\[u={1-(\beta+\alpha)v\over{\beta+{1\over\alpha}}}\]
in the parametric presentation of $R$. Hence, we get
\begin{prop}\label{x=PROP}
Fix the ratios $r_\times=\alpha^2$ and $r_{||}=\beta^2$. Then, a
{\em table} has the given ratios if and only if it has the form
\[\left(\begin{array}{cc}
{\beta\over{\beta+{1\over\alpha}}}[1-(\beta+\alpha)v] & {1\over\alpha\beta+1}[1-(\beta+\alpha)v] \\
\beta v & \alpha v
\end{array}\right)\]
where $0< v<{1\over\alpha+\beta}$.
\end{prop}
This answers question Q1: fixed the two ratios, the tables with
those ratios describe a segment in the probability simplex.
\begin{rem}
In \cite[Section 2.7]{bishop|fienberg|holland:75}, a parametric
description of the tables with $r_\times=1$ is written in the form
\begin{equation} \label{par1}
\left(
\begin{array}{cc}
st & s(1-t) \\
(1-s)t & (1-s)(1-t)
\end{array}
\right) \, .
\end{equation}
Let us check that our parametrization contains this as a special
case. In order to do this, we will compute the marginal sums
\begin{equation} \label{par2}
\left(\begin{array}{ccc|c}
\beta[\frac{1}{\beta+1}-v] & \ \ & [\frac{1}{\beta+1}-v] & 1-(\beta+1)v \\
& & & \\
\beta v & & v & (\beta+1)v \\ \hline
\frac{\beta}{\beta+1} & & \frac{1}{\beta+1} & 1 \end{array}\right)
\, .
\end{equation}
Hence, the parametrizations in Equations (\ref{par1}) and
(\ref{par2}) are just the same, simply let
$t=\frac{\beta}{\beta+1}$ and $s=1-(\beta+1)v$.
\end{rem}
\begin{rem}
Suppose to fix $r_\times$ and to ask for a geometric description
of the locus of tables with this ratio. Using Proposition
\ref{x=PROP} we can easily get an answer. For each value of
$r_{||}$ we get a segment of tables, and making $r_{||}$ to vary
this segment describes a portion of a ruled quadratic surface.
Notice that, for $r_\times=1$, this is the result contained in
\cite[Section 2.7]{bishop|fienberg|holland:75}. In particular, we
recall that matrices such that $r_\times$ is fixed form a so called Segre
variety (i.e., in this case, a smooth quadric surface in the
projective three space). For more on this see, e.g.,
\cite{garcia|stillman|sturmfels:05}.
\end{rem}
Answering question Q2 is just a computation, and we see that
\[r_=={1\over \alpha\beta+1}{[1-(\beta+\alpha)v]^2\over v}\, ,\]
where $r_\times=\alpha^2$ and $r_{||}=\beta^2$. Notice that, fixed
$r_\times$ and $r_{||}$, the third ratio can freely vary in
$(0,+\infty)$.
\begin{rem}\label{INVrem} We expressed $r_=$ as a function
$r_=(\alpha,\beta,v)$, and standard computations show that this is
an invertible function of $v$. In particular, we get
\[
v=\frac{1}{\alpha+\beta+\sqrt{(\alpha\beta+1)r_=}} \, .
\]
Thus, given $r_\times=\alpha^2,r_{||}=\beta^2$ and $r_=$, we have
an explicit description of the {\em unique} table with these
ratios (use Proposition \ref{x=PROP}).
\end{rem}
Clearly, completely analogous results hold if we fix the ratios
$r_\times$ and $r_=$.
If we fix the ratios $r_{||}=\beta^2, r_==\gamma^2$ and we argue
as above, we get the following:
\begin{prop} \label{==PROP}
Fix the ratios $r_{||}=\beta^2$ and $r_==\gamma^2$. Then, a {\em
table} has the given ratios if and only if it has the form
\[\left(\begin{array}{cc}
{\beta\over{\beta+{1\over\gamma}}}[1-(\beta+\gamma)v] & \gamma v \\
\beta v & {1\over\beta\gamma+1}[1-(\beta+\gamma)v]
\end{array}\right)\]
where $0< v<{1\over\beta+\gamma}$.
\end{prop}
Again, a trivial computation yields:
\[r_\times=\left({\beta\over \beta\gamma+1}\right)^2{[1-(\beta+\gamma)v]^2\over v^2} \, ,\]
and hence, fixed $r_=$ and $r_{||}$, the third ratio can freely
vary in $(0,+\infty)$, see Remark \ref{INVrem}.
\begin{rem}
In recent literature, there is an increasing attention to the
geometrical structure of probability models for contingency
tables. In particular, in \cite[Chapter 3]{slavkovic:04} the
author presents some results about the geometrical
characterization of probability models for $2 \times 2$
contingency tables in terms of the cross-product ratio and the
conditional distributions. In the same work the connections
between the odds ratios and the classical log-linear and
ANOVA-type representations of the probability models are clearly
stated. We remark that our notation slightly differs from the one
used by A. Slavkovic in her Ph.D. dissertation.
In the same direction, in \cite{luo|wood|jones:04} the graphical
visualization of joint, marginal and conditional distributions on
the probability simplex for $2 \times 2$ contingency tables is
presented.
\end{rem}
\section{The $2\times 3$ case}\label{2x3SEC}
The study of tables with more than two rows and columns would be
of great interest, but the complexity of the problem readily
increases as we show in the $2 \times 3$ case.
Consider the $2\times 3$ matrix
\[
\left(\begin{array}{ccc}p_{00} & p_{01} & p_{02}\\
p_{10} & p_{11} & p_{12}\end{array}\right)
\]
and define odds ratios as above for each $2\times 2$ submatrix. We
complete our previous notation by adding a superscript to denote
the deleted column, e.g.
\[
r_=^{(1)}={p_{00}p_{02}\over
p_{10}p_{12}} \, .
\]
Again, we consider a matrix as a point in a real affine space, in
this case $\mathbb{A}^6$. Notice that the ratios are well defined for
matrices in $\mathbb{A}^6\setminus Z$, where $Z$ denotes the set of
matrices with at least a zero entry.
Relations among the ratios are the cause of the increased
complexity of the higher dimensional cases. For example, as we
will see, two of the ratios can always be freely fixed. But, as
soon as three ratios are considered, constraints come in the
picture.
Easy calculations show that the following relations hold:
\[r_{||}^{(0)}r_{||}^{(2)}=r_{||}^{(1)} \, ,\]
\[r_{\times}^{(0)}r_{\times}^{(2)}=r_{\times}^{(1)} \]
and also
\[r_\times^{(0)}=r_=^{(2)}(r_=^{(1)})^{-1}\, ,\]
\[r_\times^{(1)}=r_=^{(2)}(r_=^{(0)})^{-1}\, ,\]
\[r_\times^{(2)}=r_=^{(1)}(r_=^{(0)})^{-1}\, .\]
These relations, beside producing constraints on the numerical
choice of the ratios, lead to a much more complex geometric
situation. We illustrate this by exhibiting some explicit examples
(worked out with the Computer Algebra systems {\bf Singular} and
{\bf CoCoA}). As references for the software, see \cite{cocoa} and
\cite{singular}.
More precisely, we fix some of the ratios and we describe the
locus of matrices satisfying these relations in
\[\Sigma^\circ=\{P=(p_{ij})\in\mathbb{A}^6 : \sum p_{ij}=1\}\setminus Z \, ,\]
i.e. the space of matrices with non-null entries of sum one. For
the sake of simplicity, we do not consider the positivity
conditions defining the simplex.
In our geometric descriptions, we will slightly abuse terminology,
e.g. we will call a line in $\Sigma^\circ$ a line in $\mathbb{A}^6$
not contained in $Z$; notice that our lines may have some holes
(i.e. the points of intersection with $Z$).
We start by considering the easiest case where two of the ratios
are fixed. Already at this stage, a dichotomy arises and we have
two different situations, as shown in the following examples:
\begin{equation}\label{2eq1}
\ratio{\times}{1}=\ratio{\times}{2}=1 \, ,
\end{equation}
\begin{equation}\label{2eq2}
\ratio{\times}{1}=\ratio{=}{2}=1\mbox{ or
}\ratio{=}{1}=\ratio{||}{2}=1\mbox{ or }
\ratio{||}{1}=\ratio{||}{2}=1\mbox{ or }
\ratio{=}{1}=\ratio{=}{2}=1 \, .
\end{equation}
The locus of matrices in $\Sigma^\circ$ satisfying one of
conditions (\ref{2eq2}) is a 3-dimensional variety of degree 4,
while condition (\ref{2eq1}) describes a 3-dimensional variety of
degree 3. Roughly speaking, the degree (see \cite[page 16]{Harris}
and \cite[page 41]{Shaf}) is a measure of the complexity of the
variety. For a surface in 3-space, for example, the degree bounds
the number of intersections with a line and, in a certain sense,
measures how the surface is folded.
Next, we try to fix three of the ratios, for example:
\begin{equation}\label{3eq1}
\ratio{\times}{0}=\ratio{=}{1}=\ratio{||}{2}=1\mbox{ or }
\ratio{\times}{0}=\ratio{\times}{1}=\ratio{||}{2}=1 \, ,
\end{equation}
\begin{equation}\label{3eq2}
\ratio{\times}{0}=\ratio{\times}{1}=\ratio{=}{2}=1 \, ,
\end{equation}
\begin{equation}\label{3eq3}
\ratio{\times}{0}=4,\ratio{\times}{1}=3,\ratio{=}{2}=2 \, .
\end{equation}
The locus of matrices in $\Sigma^\circ$ satisfying one of
conditions (\ref{3eq1}) is the union of two quadratic surfaces,
while condition (\ref{3eq2}) gives a plane. Moreover, if we
consider the same ratios but we vary their values, as in
(\ref{3eq3}), the locus of matrices is now described by a single
quadratic surface.
Finally, a glimpse of the case of four fixed ratios:
\begin{equation}\label{4eq1}
\ratio{\times}{0}=\ratio{\times}{1}=\ratio{||}{1}=\ratio{||}{2}=1
\, ,
\end{equation}
\begin{equation}\label{4eq2}
\ratio{\times}{0}=\ratio{\times}{1}=\ratio{=}{1}=\ratio{=}{2}=1 \,
,
\end{equation}
In both cases, the locus is described by a curve as expected. But,
condition (\ref{4eq1}) produces the union of four lines, while
condition (\ref{4eq2}) is satisfied by a single line in
$\Sigma^\circ$.
The Computer Algebra systems {\bf Singular} and {\bf CoCoA} were
used to compute primary decompositions (giving the irreducible
components of the loci) and Hilbert functions (giving the
dimension and the degree of the loci).
\section{An application. The case-control
studies}\label{application}
Two-by-two contingency tables are natural models for a large class
of problems known, in medical literature, as case-control studies.
Let us consider a table coming, e.g., from the study of a new
pharmaceutical product, or clinical test, designed for the
detection of a disease. This is an example of a case-control
study.
In a case-control study there are two random variables. The first
variable $X_1$ encodes the presence (level 1) or absence (level 0)
of the disease. The second variable $X_2$ encodes the result of
the clinical test (level 1 if positive, level 0 if negative).
The joint variable $(X_1,X_2)$ has $4$ outcomes, namely:
\[
(0,0),(0,1),(1,0),(1,1)\, .
\]
Its probabilities form a $2 \times 2$ contingency table:
\[
\begin{pmatrix}
p_{00} & p_{01} \\
p_{10} & p_{11} \\
\end{pmatrix} \, .
\]
The probabilities $p_{00}$ and $p_{11}$ are called the probability
of true negative and of true positive, respectively. They
correspond to the cases of correct answer of the clinical test.
The probabilities $p_{10}$ and $p_{01}$ are called the probability
of false positive and of false negative, respectively. They
correspond to the two types of error which can show in a
case-control study. For example, the probability of false negative
is the probability that a diseased subject is incorrectly
classified as not diseased.
A perfect clinical test which correctly classifies all the
subjects would have $p_{01}$ and $p_{10}$ as low as possible,
implying a large value of the odds ratio $r_{\times}$. Therefore,
the odds ratio $r_\times$ measures the validity of the clinical
test. In particular, when $r_{\times}=1$, the random variables are
statistically independent. In our framework this means that, when
$r_\times=1$, the result of the clinical test is independent from
the presence or absence of the disease. Unless one obtains a large
value of $r_{\times}$, the clinical test is judged as non
efficient. The odds ratio $r_{\times}$ is also called Diagnostic
Odds Ratio (DOR) in medical literature.
In such a case-control study, two essential indices are the
specificity and the sensitivity, defined as:
\[
{\rm specificity} = \frac {p_{00}} {p_{00}+p_{01}}
\]
and
\[
{\rm sensitivity} = \frac {p_{11}} {p_{10}+p_{11}} \, .
\]
Specificity is the proportion of true negative among the diseased
subjects, while sensitivity is the proportion of true positive
among the non-diseased subjects.
Straightforward computations show that
\[
r_{\times} = \frac {{\rm specificity} / (1 - {\rm specificity})}
{(1-{\rm sensitivity}) / {\rm sensitivity} } \, .
\]
In view of the definition above, it is easy to show that the
relative magnitude of the sensitivity and specificity is measured
by the odds ratio $r_{||}$. In fact one can show that
\[
\frac {{\rm sensitivity} / (1- {\rm sensitivity})} {{\rm
specificity}/(1 - {\rm specificity})} = \frac 1
{r_{||}}\, .
\]
The ratio above is called Error Odds Ratio (EOR).
In recent literature, the DOR and the EOR are relevant parameters
for the assessment of the validity of a clinical test. They have
received increasing attention in the last few years and a huge
amount of literature has been produced. Hence, we refrain from any
tentative description and refer the interested reader to, for
example, \cite{knottnerus:01}.
The meaning of the third ratio $r_{=}$ is not straightforward as
explained in \cite[Page 21]{bishop|fienberg|holland:75}. However
its statistical meaning can be derived using Equations
(\ref{rel1}) and (\ref{rel2}) shown in Section \ref{intro}.
Finally, we remark that the geometrical structure of the
statistical models for case-control studies is very simple. From
the results in Section \ref{oddsSEC}, one readily sees that the
models are segments or portions of ruled quadratic surfaces.
Moreover, from a Bayesian point of view, Propositions \ref{x=PROP}
and \ref{==PROP} allow to compute the exact range of the free odds
ratio.
\bigskip
\noindent {\bf Acknowledgement.} We wish to thank an anonymous
referee for his/her valuable suggestions and comments for the
improvement of the paper.
\bigskip
\bibliographystyle{alpha}
|
1,116,691,497,530 | arxiv | \section{Introduction}
The holographic principle has a concrete and well-understood realization in anti-deSitter space (AdS) \cite{Maldacena:1997re}. One hopes that the principle extends in some form to (nearly) flat spacetimes like the one we inhabit. The basic fact that the ratio of the boundary to bulk volume goes to a constant at large radius in AdS
and to zero in flat space suggests that flat space holography may differ qualitatively from its AdS counterpart. But exactly how is an outstanding open question.
Two seemingly different approaches to flat space holography are the Banks-Fischler-Shenker-Susskind (BFSS) matrix model \cite{deWit:1988wri,Banks_1997, Susskind:1997cw, Seiberg_1997,Sen:1997we, Polchinski:1999br, Taylor_2001, bigatti1999review, Ydri:2017ncg} and
celestial holography \cite{deBoer:2003vf,He:2015zea,Pasterski:2016qvg,Strominger:2017zoo,Raclariu:2021zjz,Pasterski_2021}. BFSS is a top-down construction equating the momentum-$N$ sector of discrete lightcone quantized (DLCQ) M-theory with a quantum mechanics of $N\times N$ hermitian matrices representing open strings stretching between $N$ D0-branes. Celestial holography is a bottom-up approach applicable to any quantum theory of gravity in flat space, including M-theory, in which the proposed dual field theory lives on the celestial sphere at null infinity. Since the two approaches are applicable to the same theory it is natural to explore their connection.
The starting point of celestial holography (as well as AdS holography) is that both sides of a dual pair must have the same symmetries. Given the bulk description, soft theorems provide an efficient route to finding these symmetries
\cite{Strominger:2017zoo}. So the first question we ask in this paper is `Is the soft graviton theorem realized in BFSS?' We answer this by showing that soft gravitons are matrix subblocks whose rank is held fixed (rather than scaling with $N$ like the hard gravitons) in the large-$N$ limit which recovers the full uncompactified M-theory.\footnote{This is reminiscent of the large-$N$ limit of QCD, where baryons have masses of order $N$ and mesons of order $1$. It would be interesting to see how far this analogy can be pushed.} The soft limit is then nontrivially identified with the M-theory limit. It would be illuminating to derive the soft theorem directly from the matrix model, and would provide a novel test of the latter.
Soft theorems are in general expected to be Ward identities of symmetries. Hence one asks if this expectation holds for the soft theorem in the matrix model. Using the known expression for the BFSS matrix model in a background $U(1)_{RR}$
gauge field \cite{Taylor:1999gq} we show that the soft theorem is the Ward identity of `large' $U(1)_{RR}$ gauge transformations \cite{Strominger:2013lka} which do not die off at past or future timelike infinity.\footnote{This is in accord with the fact that 11D supertranslations with non-zero momentum on the M-theory circle KK reduce to $U(1)_{RR}$ gauge transformations \cite{Marotta:2019cip, Ferko:2021bym}.}
We hope the answers to these basic questions provide a jumping off point for relating these two approaches to flat holography. Many further questions remain unanswered.
We will begin by reviewing the BFSS matrix model in Section \ref{sec: BFSS matrix model}. In Section \ref{sec: soft theorem}, we leverage the soft graviton theorem in M-theory into an analogous one in the matrix model dual. We demonstrate that the soft expansion in the matrix model is a $1/N$ expansion. In Section \ref{sec: RR Gauge Field}, we discuss the interplay between soft theorems and supertranslation symmetry in M-theory arguing that the analog of supertranslation symmetry in the matrix model is a large gauge symmetry of the RR 1-form.
\section{Matrix Model Review}
\label{sec: BFSS matrix model}
In this section we briefly review the relevant features of the BFSS matrix theory \cite{deWit:1988wri,Banks_1997, Susskind:1997cw, Seiberg_1997,Sen:1997we, Polchinski:1999br, Taylor_2001, bigatti1999review, Ydri:2017ncg}, which
conjectures that the compactification of M-theory on a lightlike circle $X^- \sim X^- + 2 \pi R$ with momentum $P^+ = N/R$ is dual to the low-energy dynamics of $N$ D0-branes in 10 dimensions or, equivalently, a certain supersymmetric quantum mechanical theory of $N\times N$ Hermitian matrices.
Readers familiar with BFSS may safely skip this section.
\subsection{BFSS Duality}
Compactification of 11-dimensional M-theory on a spacelike circle gives type IIA string theory \cite{Townsend:1995kk, Witten:1995ex}.
The massless degrees of freedom in M-theory are the 11-dimensional supergraviton multiplet.
The D0-branes in type IIA string theory are identified as the KK-modes of this supergraviton multiplet. The number of units $N$ of momentum around the circle corresponds to the number of D0-branes. The BFSS matrix model concerns a lightlike compactification of M-theory which can be defined as an infinitely large boost of a spacelike one \cite{Seiberg_1997}.
Let us define the lightcone coordinates $X^\pm,$ and lightcone momenta $P^{\pm}$ by
\begin{equation}
X^\pm = \frac{1}{\sqrt{2}}(X^0 \pm X^{10}), \hspace{30pt} P^{\pm} = \frac{1}{\sqrt{2}}(P^0 \pm P^{10}).
\end{equation}
Lightcone quantization is performed on surfaces of constant $X^+$ which plays the role of time, with $P^-$ the Hamiltonian. The lightlike compactification of M-theory is
\begin{equation}
(X^+,X^-) \sim (X^+, X^- + 2\pi R).
\end{equation}
$P^+$ is quantized according to
\begin{equation}\label{qu}
P^+ = N/R.
\end{equation}
BFSS argued that the sector of M-theory with total momentum $P^+ = N/R$ can be described by a rescaled version of the Hamiltonian encoding the low-energy dynamics of $N$ $D0$-branes in type IIA string theory \cite{deWit:1988wri,Banks_1997, Susskind:1997cw}
\begin{equation}
H = \frac{R}{2}\text{Tr}\Bigg[P^I P^I - \frac{1}{2(2\pi l_p^3)^2}[X^I,X^J][X^I,X^J] - \frac{1}{2\pi l_p^3}\Psi^T \Gamma^I[X^I,\Psi]\bigg]
\label{eqn: BFSS Hamiltonian}
\end{equation}
subject to the constraint on physical states
\begin{equation}
f_{ABC} (X^I_B P^I_C - \frac{i}{2} \Psi^\alpha_{B} \Psi^\alpha_{C})|\psi_{\text{phys}}\rangle = 0
\end{equation}
which forces states to be invariant under $U(N)$ transformations.
Here $X^I$ are $N \times N$ Hermitian matrices with the index $I = 1,...,9$ running over the directions transverse to the lightlike compactification. $P^{I}$ are their conjugate momenta. $\Psi^\alpha$ is an $N \times N$ Hermitian matrix-valued spinor of $Spin(9)$ with $\alpha = 1,...,16$ and gamma matrices $\Gamma^I_{\alpha \beta}$. One can decompose these matrices as
\begin{equation}
X^I = X^{I}_A T^A, \hspace{30pt} P^I = P^{I}_A T^A, \hspace{30pt} \Psi^\alpha = \Psi^\alpha_{A} T^A
\end{equation} where $T^A$ are generators of the Lie algebra of $U(N)$ in the adjoint representation normalized so that $\text{Tr}(T^AT^B) = \delta^{AB}$.
We now review some basic properties of this theory \cite{Danielsson:1996uw, Kabat:1996cu, Bachas:1995kx, Sethi_1998, moore2000d, yi1997witten}. The bosonic potential $V \sim \Tr([X^I, X^J]^2)$ is classically at a minimum $V = 0$ when $[X^I,X^J]=0$ for all $I$ and $J$, which implies all matrices can be simultaneously diagonalized. The $N$ eigenvalues are then positions of the $N$ D0-branes. For example, $N$ non-interacting D0-branes travelling along trajectories $x_{i}^I(t)$ with $i = 1,...,N$ are described by the diagonal matrices
\begin{equation}\label{diag_matrix}
X^I(t) = \begin{pmatrix} x^I_1(t) & 0 & \ldots & 0 \\0 & x^I_2(t) & \ldots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \ldots & x^I_N(t)\end{pmatrix}.
\end{equation}
The off-diagonal elements are open strings stretching between the D0-branes.
A clump of $m<N$ coincident D0-branes corresponds to an $m \times m$ sub-block of this matrix for which all of the eigenvalues $x_i^I$ in the sub-block are equal. Quantum mechanically, these clumps are marginally bound states with complicated wave functions. They are dual to the higher KK-momentum supergraviton modes in the M-theory picture.
Widely separated clumps are noninteracting because the strings stretched between them are very massive and forced into their ground states.
\subsection{Scattering }
Consider a set of $n$ gravitons in M-theory with individual momenta $k_{j}^+ = N_j/R,$ with $j = 1,...,n$ and total momentum $P^+_{\text{tot}} = (N_1+\cdots + N_n)/R = N/R.$ Each graviton is dual to a marginally bound clump labeled by the number of D0-branes $N_j$, the transverse momentum $k^I_j$, and the polarization information of the 11D supergraviton multiplet $\epsilon_j$ which in the D0-brane description is encoded by the trace `center of mass' fermions, with the explicit map given in \cite{plefka1998quantum,Plefka_1998,Plefka:1998in,Plefka:1997hm}. The dictionary \cite{Banks_1997, Susskind:1997cw, Seiberg_1997, Becker:2006dvp} between clumps of D0-branes and a collection of M-theory gravitons is:
\begin{table}[H]
\centering
\begin{tabular}{|c c c|}
\hline
\textbf{M-Theory} & ~ & \textbf{BFSS} \\
\hline
$k_{j}^+$ & $\Longleftrightarrow$ & $N_j/R$ \\
$k_j^I$ & $\Longleftrightarrow$ & $k_j^I$ \\
$k_{j}^-$ & $\Longleftrightarrow$ & $R(k_j^I)^2/2N_j$ \\
\hline
\end{tabular}
\caption{Dictionary between momenta of gravitons in M-theory and momenta of D0-brane clumps in the BFSS matrix model. The final relation is determined using the mass-shell condition for M-theory gravitons $0 = -2k_j^+k_j^- + k_j^I k_j^I$}
\label{tab: Kinematics Dictionary}
\end{table}
Widely separated multi-graviton `scattering states' in M-theory with quantum numbers $k_1^\mu,\epsilon_1,...,k_n^\mu,\epsilon_n$ correspond to widely-separated multi-clump states in the matrix model with D0-brane quantum numbers $N_1,k_1^I,\epsilon_1,...,N_n,k_n^I,\epsilon_n$.
In order to formulate the scattering problem in BFSS, one should consider initial and final scattering states corresponding to widely separated wavepacket clumps of D0-branes, evolving past into future using the BFSS Hamiltonian \eqref{eqn: BFSS Hamiltonian} \cite{Becker:1997wh, Becker:1997xw}. The BFSS duality conjecture states that this scattering amplitude matches with the one that one would compute in (lightlike compactified) M-theory, namely
\begin{equation}
\mathcal{A}_{\text{M}}(k_1^\mu,...,k_n^\mu) = \mathcal{A}_{\text{BFSS}}(N_1,k_1^I,...,N_n,k_n^I).
\label{eqn: M-theory and BFSS Amplitude Relation}
\end{equation}
\subsection{The M-theory Limit}
The scattering amplitudes of uncompactified 11-dimensional $M$-theory are obtained by taking the radius large with external momenta held fixed
\begin{equation} \label{mone}R\to \infty, ~~~~~~~k^\mu_j ~~{\rm fixed}.\end{equation}
From the expression for the momenta this is easily seen to be equivalent to
\begin{equation} \label{mtwo}N\sim N_j \sim R \to \infty, ~~~~~~~k^I_j ~~{\rm fixed},\end{equation}
and hence is a variety of large-$N$ limit. In this limit, the discretuum of allowed values of external momentum approach a continuum and any scattering proccess can be studied.
\begin{figure}
\centering
\includegraphics[width = \textwidth]{scattering.png}
\caption{A schematic depiction of equivalent scattering processes viewed through dual lenses of M-theory, D0-brane interactions, and block diagonal matrices in the BFSS model.}
\label{fig:ScatteringProcesses}
\end{figure}
\section{Soft Graviton Theorem in the Matrix Model}
\label{sec: soft theorem}
In this section we show that the soft theorem is realized within BFSS duality and, moreover, that the soft limit is the same as the M-theory limit with external momenta/D0-charges suitably scaled.
\subsection{The Soft Theorem in M-Theory}
Weinberg's soft graviton theorem \cite{Weinberg:1965nx} applies to essentially any gravitational theory in an asymptotically flat spacetime.\footnote{Of course Weinberg considered only four dimensions, but the extension to higher dimensions is straightforward \cite{Kapec_2017,He_2019_1,He_2019,Kapec_2022,Marotta:2019cip}
} In particular it must hold in 11-dimensional M-theory which contains gravitons as part of the low energy effective action.
Consider a generic scattering amplitude $\mathcal{A}_{\text{M}}(k_1^\mu,...,k_n^\mu)$ involving external particles with future-directed momenta $k^\mu_j$. These external particles may be ingoing or outgoing gravitons or some other particles. Momenta are parameterized by a vector $v^I \in \mathbb{R}^d$ and a scale $\omega \in \mathbb{R}_{>0}$ according to
\begin{equation}
k^\mu_j = \omega_j \hat{k}^\mu_j = \frac{\omega_j}{2}(1 + v_j^2, 2 v_j^I, 1 - v_j^2), \hspace{30pt} q^\mu_s = \omega_s \hat{q}^\mu_s = \frac{\omega_s}{2}(1 + v_s^2, 2 v_s^I, 1 - v_s^2) .
\label{eqn: graviton parametrization}
\end{equation}
where we denote the momentum of the soft graviton by $q^\mu_s$. The soft graviton theorem states:
\begin{equation}
\mathcal{A}_\text{M}(q^\mu_s,\epsilon_s^{\mu \nu};k_1^\mu,...,k_n^\mu) = \Bigg[\frac{\kappa}{2} \epsilon^s_{\mu \nu} \sum_{j =1}^n \eta_j \frac{k^\mu_j k^\nu_j}{q_s \cdot k_j} + \mathcal{O}\left(\Big(\frac{\omega_s}{\omega_j}\Big)^0\right) ~ \bigg] \mathcal{A}_\text{M}(k_1^\mu,...,k_n^\mu)
\label{eqn: soft theorem in gravity}
\end{equation}
where $\eta_j = +1 ~ (-1)$ if the $j^{\rm th}$ particle is outgoing (incoming), $\epsilon_s^{\mu \nu}$ is the polarization tensor of the soft graviton, and $\kappa = \sqrt{32\pi G_N}$ \cite{Weinberg:1965nx}.
In the soft limit, the ratios $\omega_s/\omega_j \rightarrow 0$. The coefficient of the leading soft divergence $(\omega_s/\omega_j)^{-1}$ is universal. This leading soft term has corrections which are a power series in $(\omega_s/\omega_j)$.
\subsection{Soft Limit in the Matrix Model}
\label{sec: Soft Limit}
In this subsection we show the soft limit in the BFSS matrix model is the M-theory limit with hard gravitons represented by subblocks whose size grows like $N$ and soft gravitons by subblocks of fixed finite size.
Both soft and hard gravitons are parameterized as in Equation (\ref{eqn: graviton parametrization}) with $q_s^+ = \sqrt{2}\omega_s$ and $k_j^+ = \sqrt{2}\omega_j$. This expression implies
\begin{equation}
\frac{n_s}{R} = \sqrt{2} \omega_s ~ ~ , ~ ~ q_s^I = \omega_s v_s^I \sim \omega_s \hspace{20pt}\text{and}\hspace{20pt} \frac{N_j}{R} = \sqrt{2} \omega_j ~ ~ , ~ ~ k^I_j = \omega_j v_j^I \sim \omega_j
\label{eqn: soft parametrization}
\end{equation}
where we have used the dictionary provided in Table \ref{tab: Kinematics Dictionary} with $N_j$, the block sizes, corresponding to the hard M-theory gravitons, and $n_s$ to the soft ones. Thus, the momentum $k_{j}^+$ for a particular graviton dictates the size of the corresponding block in the matrix model. In the matrix model, the soft limit then reads
\begin{equation}
\textbf{Soft Limit:} ~ ~~~ ~ ~ ~ \frac{n_s}{N_j} = \frac{\omega_s}{\omega_j} \rightarrow 0.
\label{eqn: block ratio}
\end{equation}
Scattering amplitudes with a soft external particle in the BFSS matrix model, thus, correspond to situations where a block of size $n_s$ is dwarfed by the other blocks of size $N_j$. This happens automatically in the M-theory limit \eqref{mtwo} as long as we keep $n_s$ fixed! Hence the soft limit is the same as the M-theory limit, but with a new type of external state constructed from a finite number $n_s$ of D0-branes.
In the M-theory limit, the difference between scattering a graviton with $(N_j,k^I_j)$ versus $(N_j-1,k^I_j)$ with one fewer D0-branes vanishes. This might have led to the naive conclusion that submatrices with sizes or order one don't matter and that the scattering of a single D0-brane $(1, k^I_s)$ vanishes altogether. This is not the case because of the soft pole. Note also the leading term in the scattering amplitude for a bound state with a fixed finite number $n_s$ of D0s differs only by the multiplicative factor $1/n_s$.
We illustrate a $2 \rightarrow 3$ scattering process with soft emission diagrammatically from the M-theory perspective, the D0-brane perspective, and the block diagonal matrix perspective explicitly in Figure \ref{fig:ScatteringProcesses}.
We now write the leading soft graviton theorem of M-theory \eqref{eqn: soft theorem in gravity} in terms of BFSS variables. If we define a convenient basis for graviton polarization tensors
\begin{equation}
\epsilon^{\mu \nu}_{IJ}(v) \equiv \frac{1}{2}\big(\epsilon^\mu_I \epsilon^\nu_J + \epsilon^\nu_I \epsilon^\mu_J\big) - \frac{1}{d}\delta_{IJ}\epsilon^\mu_K \epsilon^{K\nu} \hspace{20pt} \text{with} \hspace{20pt} \epsilon^\mu_J(v) \equiv \partial_J \hat{q}^\mu_s = (v_J,\delta^I_J,-v_J)
\end{equation}
and write the soft graviton polarization as
\begin{equation}
\epsilon_s^{\mu \nu} = e^{IJ} \epsilon_{IJ}^{\mu \nu},
\end{equation}
then after some algebra and using the dictionary \ref{tab: Kinematics Dictionary}, the soft theorem becomes \footnote{There is a small technical subtlety in Equation \eqref{eqn: leading soft theorem 1}. The BFSS matrix model describes M-theory in a sector with momentum $P^+ = N/R$, so all amplitudes must be manifestly momentum conserving in $P^+$ and cannot be off-shell in $P^+$. Equivalently, the number of D0-branes $N$ is always conserved. Therefore, we cannot simply append a small block of size $n_s$ to the matrix, but we must shrink the size of the other blocks slightly. Assuming that the amplitudes are analytic in $N_j$ (in the large $N$ limit, this follows from the analyticity of the M-theory S-matrix) one may perform a first order Taylor expansion to see that the expression will only be corrected at subleading terms.}
\begin{equation}
\mathcal{A}_{\text{BFSS}}(n_s,q^I_s, \epsilon_s,\text{out; in}) = \bigg[-2 \kappa \sum_{j=1}^n \eta_j \frac{N_j}{n_s} \frac{e_{IJ}(v_s-v_j)^I (v_s-v_j)^J}{(v_s-v_j)^2} + \cdots ~ \bigg] \mathcal{A}_{\text{BFSS}}(\text{out; in}).
\label{eqn: leading soft theorem 1}
\end{equation}
Finally, we define the inversion tensor in 9 spatial dimensions as\footnote{This is the same inversion tensor familiar from conformal field theory.}
\begin{equation}
\mathcal{I}^{IJ}(v) = \delta^{IJ} - 2 \frac{v^I v^J}{v^2}.
\label{eqn: polarization basis}
\end{equation}
In terms of this inversion tensor, the leading soft graviton theorem in the BFSS matrix model reads
\begin{equation}
\mathcal{A}_{\text{BFSS}}(n_s,q^I_s,\epsilon_s,\text{out; in}) = \bigg[\kappa \sum_{j=1}^n \eta_j \frac{N_j}{n_s} e_{IJ} \mathcal{I}^{IJ}(v_s-v_j) + \cdots ~ \bigg] \mathcal{A}_{\text{BFSS}}(\text{out; in}).
\label{eqn: leading soft theorem 2}
\end{equation}
Note that the soft pole $\omega_j/\omega_s$ gets recast into the ratio of block sizes $N_j/n_s$, which diverges in the soft limit according to Equation (\ref{eqn: block ratio}).
Sub-leading corrections to this expression are given by an expansion in $n_s/N_j$. Because $n_s \sim \mathcal{O}(1)$ and $N_j \sim \mathcal{O}(N),$ we can identify the subleading terms in the soft expansion on the gravity side with a $1/N$ expansion on the gauge theory side.
It would be illuminating to derive the soft theorem directly from the matrix model. It is not obvious to us even how the factor of $N_j/n_s$ would emerge.
\section{Asymptotic Symmetries in the Matrix Model}
\label{sec: RR Gauge Field}
In this section, we use the soft graviton theorem in 11D to show that the insertion of a single D0-brane in a 10D BFSS scattering amplitude generates a large gauge transformation on the background RR 1-form gauge potential $C_\mu$ in the matrix model.\footnote{The result easily generates to finite bound clumps of D0-branes by dividing by $n_s$. } Since this is a quantum-mechanical model the relevant asymptotic regions are at $t=\pm \infty$. The RR 1-form is of the form $C_\mu = \partial_\mu \theta_{e, v_s}$, for some particular gauge parameter $\theta_{e, v_s}$ given in \eqref{thta} depending on the polarization $e_{IJ}$ and velocity $v_s$ of the soft D0-brane. This is summarized in Equation \eqref{final_eq}, which is the main result of this section. This large $U(1)$ gauge symmetry arises in the KK reduction of the 11D supertranslation symmetry.\footnote{Symmetries associated to 10D supertranslations would have to come from modes independent of the $X^+$ circle and hence involve $n_s=0$. It is not clear to us how to describe these in the matrix model.}
\subsection{Background RR Gauge Potentials}
The standard BFSS matrix model, with the Hamiltonian given by Equation \eqref{eqn: BFSS Hamiltonian}, describes a system of D0-branes living in world where all background fields are turned off. The effect of coupling the D0-branes to external background fields can be incorporated by adding terms to the Lagrangian. In particular the interaction term coupling the D0-branes to the $U(1)_{RR}$ gauge field $C_\mu$ generalizes the usual electromagnetic interaction $Q \int dt \; \dot x^\mu C_\mu(x)$ between a charge $Q$ particle and the gauge field, where $x^\mu$ is the worldline of the particle.
In the matrix model, the precise interaction term was found in \cite{Taylor:1999gq} to be \begin{equation}\label{wati_action}
S_{RR}[C_\mu] = \int dt \sum_{n = 0}^\infty \frac{1}{n!} \big(\partial_{I_1} \cdots \partial_{I_n} C_\mu(t, \vec{0})\big) I^{\mu (I_1 \cdots I_n) }
\end{equation}
where $\mu = 0, \ldots, 9$ and $x^\mu = (t, x^I) = (t, \vec{x})$.
The `multipole moments' of the current $I^\mu$ are defined by
\begin{align} \label{eqn: current}
I^{\mu (I_1 \cdots I_n)}
&= \Tr (\mathrm{Sym}(I^\mu, X^{I_1}, \cdots , X^{I_n} )) +I^{\mu (I_1 \cdots I_n)}_{\text{F}}
\end{align}
where
\begin{equation}
I^\mu = (\mathbb{1}/R,\dot X^I/R).
\end{equation}
Here, $\mathrm{Sym}$ is a symmetrized average over all orderings of the input matrices. $I^{\mu (I_1 \cdots I_n)}_{\text{F}}$ are terms involving at least two fermionic matrices, $\Psi$, which will not be relevant to this paper for reasons discussed in section \ref{sec: asymptotic symmetry}. If one takes the matrices $X^I$ to be diagonal, as in Equation \eqref{diag_matrix}, then the action reduces to the electromagnetic form, as expected.
\subsection{Large $U(1)_{RR}$ Gauge Transformations}
\label{sec: asymptotic symmetry}
If the RR 1-form is pure gauge, then the interaction term \eqref{wati_action} becomes a total derivative. Plugging $C_\mu = \partial_\mu \theta$ into Equation \eqref{wati_action}, one can show that\footnote{To demonstrate this, one must use that the multipole moments satisfy the conservation law
$\partial_t I^{0 (I_1 \cdots I_n)}=I^{I_1 (I_2 \cdots I_n)} + \cdots + I^{I_n (I_1 \cdots I_{n-1})}$.}
\begin{equation}
\begin{split}
S_{RR}[\partial_\mu \theta] &= \frac{1}{R} \int dt ~ \partial_t \left[ \sum_{n = 0}^\infty \frac{1}{n!} (\partial_{I_1} \cdots \partial_{I_n} \theta(t, \vec{0} )) I^{0 (I_1 \cdots I_n)} \right] \\
&= \frac{1}{R} \sum_{n=0}^\infty \frac{1}{n!}(\partial_{I_1} \cdots \partial_{I_n} \theta(t,\vec{0})) I^{0(I_1 \cdots I_n)}\bigg|_{t = -\infty}^{t = +\infty}.
\label{eqn: total derivative}
\end{split}
\end{equation}
Therefore, a pure gauge background field affects amplitudes by a position-dependent phase acting on the initial and final states.
The asymptotic symmetry group of gauge theories is typically defined as the set of `large' gauge transformations which satisfy some set of boundary conditions modulo `small' gauge transformations which vanish at the boundary. For the remainder of this section, we will consider the case where $C_\mu = \partial_\mu \theta$ is pure gauge and given by such a large gauge transformation $\theta$ which is non-vanishing as $t \rightarrow \pm \infty.$ The boundary conditions which $\theta$ must satisfy near past and future timelike infinity \cite{He:2014cra,Campiglia:2015lxa,Kapec:2015ena,Campiglia:2015qka, Strominger:2017zoo} specify that as $t \rightarrow \pm\infty$, the gauge parameter $\theta(t,\vec{x})$ can only depend on the ratio $\vec{x}/t$
\begin{equation}
\theta(t,\vec{x}) \xrightarrow{t \rightarrow \pm \infty} \theta(t, \vec{x}) = \theta(\vec{x}/t)
\end{equation}
implying that these large gauge transformations are parameterized by a single function on $\mathbb{R}^9$.\footnote{This is the limit relevant for nonrelativistic charged massive scattering states of the more general formula for large $U(1)$ gauge transformations. } Outside of the above specification, the gauge parameter is arbitrary.
Now we show that the boundary term \eqref{eqn: total derivative} reduces to a very simple expression on asymptotic scattering states. As functions of $X^I$, asymptotic scattering wavefunctions are sharply peaked in momentum space and non-trivially supported only on matrices of the form
\begin{equation}
X^I = \begin{pmatrix} x^I_1(t) & & \\ & \ddots & \\ & & x^I_N(t) \end{pmatrix} + \Delta X^I, \hspace{0.75 cm} \Delta X^I \sim \mathcal{O}(t^0)
\label{eqn: asy states}
\end{equation}
where $\vec{x}_i(t) = \vec{v}_it + \vec{x}_{i,0}$ tracks the position of the $i^{th}$ D0-brane. Note that $\vec{x}_i(t) = \vec{x}_{j}(t)$ when the $i^{th}$ and $j^{th}$ D0-branes share a bound state. $\Delta X^I$ is a matrix whose values do not grow with time as $t \to \pm \infty$. The entries within the blocks on the diagonal of $\Delta X^I$ correspond to the degrees of freedom of the bound states modulo their center of mass motion. As such, these entries can take values on the order of the spatial size of these bound states.
The off block-diagonal components of $\Delta X^I$ describe strings stretched between distant D0-brane bound states. The mass of the string is proportional to its length, so these strings have mass scaling like $t$. When the string excitations become heavy, the wavefunction for these components gets frozen to the ground state of a quantum (super)harmonic oscillator with frequency $\omega \sim t$. The width of such a wavefunction shrinks as $\sim t^{-1/2}.$ This situation is summarized in Figure \ref{wavepackets}.
\begin{figure}
\centering
\begin{minipage}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{block_t.png}
\caption*{(a)}
\end{minipage}
\begin{minipage}[b]{.65\textwidth}
\centering
\includegraphics[width=\textwidth]{wavepacket.png}
\caption*{(b)}
\end{minipage}
\caption{\label{wavepackets} (a) How different parts of the matrix $X^I$ scale with $t$ for scattering states. (b) The scattering of non-relativistic D0-brane bound state Gaussian wavepackets, which can be taken to have an arbitrarily small angular width. A gauge function $\theta(t,\vec{x})$ which only depends on the ratio $\vec{x}/t$ will take a well-defined value on these wave packets, depending only on their velocity.}
\end{figure}
We may now insert the matrices describing the asymptotic states (Equation \eqref{eqn: asy states}) into the boundary term (Equation \eqref{eqn: total derivative}). We find that the only terms that survive at $t = \pm \infty$ are
\footnote{In this footnote we demonstrate why the fermionic term $I_{\text{F}}^{0 (I_1 \ldots I_n)}$ in Equation \eqref{eqn: current} doesn't contribute to our analysis. First, we notice that the couplings $\partial_{I_1} \cdots \partial_{I_n} \theta(\vec{x}/t)$ contains $n$ spatial derivatives, each pulling down a factor of $1/t$. So, for the term $(\partial_{I_1} \cdots \partial_{I_n} \theta)I^{0(I_1 \cdots I_n)}$ to be non-vanishing at $t = \pm \infty$, $I^{0(I_1 \cdots I_n)} \sim t^n$ at asymptotic times. The bosonic moments, $I^{0(I_1 \cdots I_n)}_{\text{B}} = \text{Tr}(\text{Sym}(X^{I_1} \cdots X^{I_n})),$ scale as $t^n$, since every bosonic matrix has entries growing as $t$. In fact, \textit{only} the entries linear in $t$, which are $v^I t$, survive in this limit. Next, we use the fact that the BFSS action is invariant under $R \mapsto \lambda R, X \mapsto \lambda^{1/3} X, \Psi \mapsto \lambda^{1/2} \Psi, t \mapsto \lambda^{-1/3}t$ \cite{douglas1997d}. This invariance must persist when coupled to a background field if we map $\theta \mapsto \lambda \theta$, which implies that each multipole term $I^{\mu(I_1 \ldots I_n)}$ must have a constant scaling dimension in $\lambda$ for the action to be invariant. Therefore, if two $\Psi$'s are added, three $X$'s must be removed, making the whole term scale three lower powers of $t$ and vanish at the boundary.
}
\begin{align}
S_{RR}[\partial_\mu \theta] & = \sum_{j = 1}^n \eta_j \frac{N_j}{R} \sum_{n=0}^\infty \frac{1}{n!} \big(\partial_{I_1}\cdots \partial_{I_n}\theta(t,0)\big) x_j^{I_1} \cdots x_{j}^{I_n} \bigg|_{t = -\infty}^{t = + \infty} = \sum_{j = 1}^n \eta_j \frac{N_j}{R} \theta(\vec{v}_j)
\end{align}
where we noticed that middle expression is just a Taylor expansion of $\theta(t,\vec{x})$ and used $\theta(t,\vec{x}_j) = \theta(\vec{x}_j/t)$ according to our earlier considerations.
In a quantum amplitude, this addition to the action becomes an overall phase. Therefore, placing BFSS in such a background gauge field modifies the amplitude via
\begin{equation}\label{eq_C_A}
\eval{\mathcal{A}_{\text{BFSS}}(\text{out; in})}_{C_\mu = \partial_\mu \theta} = \exp( i \sum_{j = 1}^n \eta_j \frac{N_j}{R} \theta(\vec{v_j}) ) \eval{\mathcal{A}_{\text{BFSS}}(\text{out; in})}_{C_\mu = 0}.
\end{equation}
From the soft graviton theorem \eqref{eqn: leading soft theorem 2}, if we define the gauge parameter $\theta_{e, v_s}$ by
\begin{equation} \label{thta}
\theta_{e, v_s}(\vec{x}/t) \equiv \frac{e_{IJ} (v_s - x/t)^I (v_s - x/t)^J}{|\vec{v}_s - \vec{x}/t|^2 }
\end{equation}
which depends on the velocity $\vec{v}_s$ and polarization structure $e_{IJ}$ of the soft D0-brane, then by combining \eqref{eq_C_A} and \eqref{eqn: leading soft theorem 1}, we see that
\begin{equation}\label{final_eq}
\lim_{n_s/R \to 0} ~ \frac{n_s}{R} \eval{\mathcal{A}_{\text{BFSS}}(n_s, k^I_s, \epsilon_s, \text{out; in})}_{C_\mu = 0} = - i \eval{\frac{d}{d \varepsilon}}_{\varepsilon = 0} \eval{\mathcal{A}_{\text{BFSS}}(\text{out; in}) }_{C_\mu = \varepsilon \partial_\mu \theta_{e,v_s}}.
\end{equation}
Therefore, the insertion of a single D0-brane in a momentum eigenstate in the amplitude generates the action of a large $U(1)_{RR}$ gauge transformation \eqref{thta} on the asymptotic scattering state.
A general gauge transformation can be generated by an appropriate coherent superposition of a momentum-eigenstate D0-brane.
\section*{Acknowledgements}
We would like to thank Alek Bedroya, Alfredo Guevara, Elizabeth Himwich, Patrick Jefferson, Daniel Kapec, Hong Liu, Juan Maldacena, Shu-Heng Shao, and Nicolas Valdes for stimulating discussions. AT and NM gratefully acknowledge support from NSF GRFP grant DGE1745303.
|
1,116,691,497,531 | arxiv | \section{Introduction}
The letter $G$ will denote a Hausdorff topological group with unit element $e \in G$.
All spaces are assumed to be completely regular and Hausdorff.
By an action of $G$ on a space $X$ we mean a
continuous map $(g, x)\mapsto gx$ of the product $G\times X$ into
$X$ such that $ex=x$ and $(gh)x=g(hx)$, whenever $x\in X$, $g,
h\in G$ and $e$ is the unity of $G$. A space $X$ together with a
fixed action of the group $G$ is called a $G$-space.
If $X$ and $Y$ are $G$-spaces, then a continuous map $f:X\to Y$
is called a $G$-map or an equivariant map, if $f(gx)=gf(x)$ for
every $x\in X$ and $g\in G$. For a point $x\in X$, the subgroup
$G_x=\{g\in G\mid gx=x\}$ of $G$ is called the {\em stabilizer} or
{\em isotropy subgroup} at $x$. Clearly, $G_x\subset G_{f(x)}$
whenever $f$ is a $G$-map and $x\in X$.
If $X$ is a $G$-space, then for a subset $S\subset X$ and a subgroup $H\subset G$, the $H$-hull (or $H$-saturation) of $S$ is defined as follows: $H(S)$= $\{hs \ |\ h\in H,\ s\in S\}$. If $S$ is the one point set
$\{x\}$, then the $H$-hull $H(\{x\})$ usually is denoted by $H(x)$ and called the $H$-orbit of $x$. The orbit
space $X/H$ is always considered in its quotient topology.
A subset $S\subset X$ is called $H$-invariant or, if it coincides with
its $H$-hull, i.e., $S=H(S)$.
A $G$-invariant set is also
called, simply, invariant.
For a closed subgroup $H \subset G$, by $G/H$ we will denote the $G$-space
of cosets $\{xH ~| \ x\in G\}$ under the action induced by left translations, i.e., $g(xH)=(gx)H$ whenever $g\in G$ and $xH\in G/H$.
\medskip
One of the most fundamental facts in the theory of $G$-spaces when $G$ is
a compact Lie group is, the so called, Slice Theorem. In its full generality it
was proved by G.Mostow \cite{mostow} (see also \cite[Corollary 1.7.19]{pal:60}).
\begin{theorem}[Slice Theorem]\label{T:slice}
Let $G$ be a compact Lie group and $X$ a $G$-space. Then for every point $x\in X$, there exists a $G_x$-slice $S\subset X$ such that $x\in S$.
\end{theorem}
We recall the definition of an $H$-slice (cf. \cite[\S 1.7]{pal:60}).
\begin{definition}\label{D:slice} Let $X$ be a $G$-space and $H$ a closed subgroup of $G$. A subset $S\subset X$ is called an $H$-slice in $X$, if:
\begin{enumerate}
\item $ S$ is H-invariant, i.e., $H(S) = S$,
\item $S$ is closed in $G(S)$,
\item if $G \setminus H$, then $gS\cap S = \emptyset$,
\item the saturation $G(S)$ is open in $X$.
\noindent
If in addition $G(S) = X$, then we say that $S$ is a global $H$-slice of $X$.
\end{enumerate}
\end{definition}
To each $H$-slice $S\subset X$, a $G$-map $f_S:G(S)\to G/H$, called the {\it slicing map}, is associated according to the following rule:
$$f_S(gs)=gH\quad \text{for every} \quad g\in G, \ s\in S.$$
Lets check that that $f_S$ is well defined.
Indeed, if $gs=g's'$ for some $g, g'\in G$ and $s,s' \in S$ then
$s =g^{-1}g's' \in S \cap g^{-1}g'S$.
Then item (3) of Definition \ref{D:slice} yields that $g^{-1}g' \in H$ which is equivalent to
$ gH = g'H$, as required.
Thus, the slicing map $f_S$ is well defined.
It is immediate that the slicing map is equivariant, i.e., $f_S(gx) = gf_S(x)$ for all $ x \in G(S)$ and $ g \in G$.
It is also clear that $S=f^{-1}(eH)$, where $eH\in G/H$ stands for the coset of the unit element $e\in G$.
It is a well known fact that the slicing map is continuous whenever the acting group $G$ is compact (see e.g., \cite[Theorem 1.7.7]{pal:60}); this gives the following important external characterization of an $H$-slice.
\begin{theorem}\label{charatercompact} Let $G$ be a compact group, $H$ a closed subgroup of $G$, and $X$ a $G$-space. Then there exists a one-to-one correspondence between all equivariant maps $f:X\to G/H$ and global $H$-slices $S$ in $X$ given by $f\mapsto S_f:=f^{-1}(eH)$. The inverse correspondence is given by $S\mapsto f_S$, above defined.
\end{theorem}
Below, in Theorem \ref{T:slicing}
we generalize Theorem \ref{charatercompact} to the case of proper actions of arbitrary locally compact groups, which are an important generalization of actions of compact groups.
This result is proceeded by Theorem \ref{P:open} which establishes some important properties of small global slices. Then these results are applied in Section \ref{S:orbit} to orbit spaces of proper $G$-spaces. In Section \ref{S:extension} we prove an equivariant extension theorem which is further applied to get an equivariant extension of a continuous map defined on a small cross section. In the final Section \ref{S:BM} the results of Sections \ref{S:slice} and \ref{S:orbit} are applied to give a short proof of the compactness of the Banach-Mazur compacta $BM(n)$, $n\ge 1$.
Recall that the concept of a {\it proper action} of a locally compact group was introduced in 1961 in the seminal work of R. Palais~\cite{pal:61}.
This notion allowed R. Palais to extend a
substantial part of the theory of compact Lie transformation
groups to non-compact ones.
Perhaps, some detailed definitions are in order here.
Let $G$ be a locally compact group and $X$ a $G$-space. Two subsets $U$ and $V$ in $X$ are called thin relative to each other
\cite[Definition 1.1.1]{pal:61}, if the set
$$\langle U,V\rangle=\{g\in G \mid gU\cap V\ne \emptyset\}$$ called {\it the transporter} from $U$ to $V$,
has compact closure in $G$.
A subset $U$ of a $G$-space $X$ is called {\it $G$-small}, or just {\it small}, \ if every point in $X$ has a neighborhood thin relative to $U$.
\begin{definition}[\cite{pal:61}] Let $G$ be a locally compact group.
A $G$-space $X$
is called {\it proper}, if every point in $X$ has a small neighborhood.
\end{definition}
Each orbit in a proper $G$-space is closed, and each stabilizer is compact \cite[Proposition
1.1.4]{pal:61}. It is easy to check that: (1) the product of two $G$-spaces
is proper whenever one of them is so; (2) the inverse image of a proper $G$-space under a $G$-map is
again a proper $G$-space.
Important examples of proper $G$-spaces are the coset spaces
$G/H$ with $H$ a compact subgroup of a locally compact group $G$.
Other interesting examples the reader can find in
\cite{ab:78}, \cite{ant:05}, \cite{kos:65} and \cite{pal:61}.
\
Among other important results, Palais proved in \cite{pal:61}
the following generalization of the Slice Theorem.
\begin{theorem}[Slice Theorem for proper actions]\label{PalaisSlice}
Let $G$ be a Lie group (not necessarily compact) and $X$ a proper $G$-space. Then for every point $x\in X$, there exist an invariant neighborhood $U$ of $x$ and an equivariant map $f:U\to G/G_x$ such that $x\in f^{-1}(eG_x)$.
\end{theorem}
Since the distinguished point $eG_x$ is a global $G_x$-slice for the proper $G$-space $G/G_x$, and since the inverse image of a slice is again a slice (see \cite[Corollary 1.7.8]{pal:60}), we infer that
$S=f^{-1}(eG_x)$ is a $G_x$-slice (in the sense of Definition \ref{D:slice}). Besides, $S$ is a small subset of its saturation $G(S)=U$ since $eG_x$ is a small subset of $G/G_x$.
Therefore, at the first glance, it may seem that Theorem \ref{PalaisSlice} establishes something more than the existence of a slice.
However, this is not the case as it follows from our Theorem \ref{T:slicing} below.
\
\section{Some important properties of small slices}\label{S:slice}
In this section we will prove the following two main results.
\begin{theorem}\label{P:open} Let $G$ be a locally compact group, $X$ a proper $G$-space, $H$ a compact subgroup of $G$, and $S$ a global $H$-slice of
$X$ which is a small subset. Then
\begin{enumerate}
\item the restriction $f:G\times S\to X$ of the action is an open map.
\item the restriction $p:S\to X/G$ of the orbit map $X\to X/G$ is an open map.\end{enumerate}
\end{theorem}
\begin{proof} (1) Let $O$ be an open subset of $G$ and $U$ be an open subset of $S$. It suffices to show that the set $OU=\{gu~|~g\in O, \ u\in U\}$ is open in $X$.
Define $W=\bigcup\limits_{h\in H}(Oh^{-1})\times (hU)$.
Observe that
$$X\setminus OU=f\bigl((G\times S)\setminus W\bigr).$$
Indeed, since $OU=f(W)$ \ and \ $X=f(G\times S)$, the inclusion $X\setminus OU\subset f\bigl((G\times S)\setminus W\bigr)$ follows.
Let us establish the converse inclusion
$f\bigl((G\times S)\setminus W\bigr)\subset X\setminus OU$.
Assume the contrary, that there exists a point $gs\in f\bigr(G\times S)\setminus W\bigr)$ with $(g, s)\in (G\times S)\setminus W$ such that $gs\in OU$. Then $gs=tu$ \ for some \ $(t, u)\in O\times U$. Denote $h=g^{-1}t$. One has
$$s=g^{-1}tu=hu$$
and
$$(g, \ s)=(tt^{-1}g, \ g^{-1}tu)=(th^{-1}, \ hu)\in (O h^{-1})\times (hU).$$
Since both $s$ and $u$ belong to $S$, and $s=hu$, by item (3) of Definition \ref{D:slice}, we conclude that $h\in H$. Consequently, $(O h^{-1})\times (hU)\subset W$, yielding that $(g, s)\in W$, a contradiction. Thus, the equality $X\setminus OU=f\bigl((G\times S)\setminus W\bigr)$ is proved.
Now we observe that $(G\times S)\setminus W$ is closed in $G\times S$, and hence, in $G\times X$.
Since $S$ is a closed small subset of $X$, by \cite[Proposition 1.4(c)]{ab:78}, $f$ is a closed map. This yields that the set
$f\bigl((G\times S)\setminus W\bigr)$ is closed, and hence, $OU$ is open in $X$, as required.
\smallskip
(2) Indeed, let $U$ be an open subset of $S$. By item (1) of this theorem, the saturation $G(U)$ is open in $X$ yielding that the intersection $G(U)\cap S$ is open in $S$. Since
$p^{-1}\big(p(U)\big)=G(U)\cap S$ we conclude that $p(U)$ is open in $X/G$, as required.
\end{proof}
\medskip
This therem is now applied to give an external characterization of a slice in a proper $G$-space.
\begin{theorem}\label{T:slicing}
Let $G$ be a locally compact group, $H$ a compact subgroup of $G$.
Let $X$ be a proper $G$-space
and $S$ a global $H$-slice of $X$ which is a small subset in $X$. Then the slicing map $f_S:X \to G/H$ is continuous and open. If, in addition, $S$ is compact then $f_S$ is also closed.
Conversly, if one has an equivariant map $f:X \to G/H$, then the inverse image $S=f^{-1}(eH)$ is a global $H$-slice which is a small subset of $X$, and $f_S=f$.
\end{theorem}
\begin{proof}
Let $\alpha:G \times S \to X$ be the restriction of the action $G\times X\to X$ and let
$\pi:{G \times S} \to G $ denote the projection.
The quotient map $p:{G } \to G/H, \ p(g)= gH$, is open (and closed since $H$ is compact) and it makes the following diagram commutative:
\[ \xymatrix{G \times S \ar[r]^-\pi \ar[d]^\alpha & G \ar[d]^p
\\
X \ar[r]^f & G/H.
} \]
Since $S$ is a closed small subset of $X$, Theorem \ref{P:open}(1) yields that $\alpha$ is an open map. Since $\pi$ and $p$ are continuous, the equality $f \alpha = p \pi$ implies that $f$ is continuous.
Since the maps $\pi$ and $p$ are open and $\alpha$ is continuous, we infer that $f$ is also open.
If, in addition, $S$ is compact then the map $\pi$ is also closed, which yields that $f$ is closed.
The converse assertion is immediate since the point $eH\in G/H$ is a small global $H$-slice for $G/H$ and an inverse image of a small global $H$-slice is so (see \cite[p.\,10]{pal:61}). The equality $f_S=f$ is
a simple verification.
\end{proof}
\medskip
Since each compact subset of a proper $G$-space is a small subset \cite[p.\,300]{pal:61}, Theorem \ref{T:slicing} has the following immediate corollary.
\begin{corollary}\label{C:slice} Let $G$ be a locally compact group, $H$ a compact subgroup of $G$.
Let $X$ be a proper $G$-space
and $S$ a compact global $H$-slice of $X$. Then the slicing map $f_S:X \to G/H$
is continuous, open and closed.
\end{corollary}
\begin{remark}\label{R:small}
Theorem \ref{T:slicing} is not valid if the $H$-slice $S\subset X$ is not a small subset. Here is a simple counterexample.
Let $G=\mathbb R_{+}$, the multiplicative group of the positive reals and $X=\mathbb R^2\setminus\{0\}$, the Euclidean plane without the origin. Consider the action $G\times X\to X$ defined by means of the ordinary scalar multiplication, i.e., if $\lambda \in G$ and $A=(x, y)\in X$, then $\lambda *A :=\lambda A=(\lambda x, \lambda y)$. It is easy to see that this action is proper. Further, let
$$S:=\big\{(x, \pm \frac{1}{x}) \mid x\in \mathbb R\setminus \{0\}\big\} \cup \big\{(0, \pm 1), (\pm 1, 0)\big\}.$$
Then, clearly, $S$ is a global $H$-slice of $X$ with $H=\{1\}$, the trivial subgroup of $G$ while $S$ is not a small subset of $X$. Also it is easy to see that the corresponding slicing map $f_S:X\to G$ is not continuous.
It is interesting to notice that the unit circle $S=\{A\in X\mid \Vert A\Vert =1\}$ is also a global $H$-slice for $X$. However, in this case the slicing map $f_S:X\to G$ is continuous because $S$, being compact, is a small subset and then Corollary \ref{C:slice} applies. Moreover, in this case $f_S(A)=\Vert A\Vert $, which clearly, is a continuous $G$-map.
\end{remark}
\
\section{Orbit spaces}\label{S:orbit}
Existence of slices facilitates the study of
transformation groups since, for example, it enables the reduction of global
questions about transformation groups to local ones. On the other hand, existence of global slices in proper $G$-spaces enables the reduction of studying the orbit space of a non-compact group action to that of a compact subgroup.
The following result for $G$ a compact group can be found in \cite[Proposition 1.7.6]{pal:60}.
\begin{theorem}\label{Orbitspace}
Let $G$ be a locally compact group, $H$ a compact subgroup of $G$.
Let $X$ be a proper $G$-space
and $S$ a small global $H$-slice of $X$.
Then the inclusion $S\hookrightarrow X$ induces a homeomorphism of the orbit spaces
$S/H$ and $X/G$.
\end{theorem}
\begin{proof}
Let $p:S\to X/G$ be the restriction of the orbit projection $X\to X/G$.
Then, according to Theorem \ref{P:open}, $p$ is continuous and open. Since $p$ is
constant on the $H$-orbits of $S$, it induces a continuous open map $p':S/H\to X/G$.
It remains to show that $p'$ is a bijection. Indeed, if $G(x)\in X/G$ is any point then $x=gs$ for some $g\in G$ and $s\in S$ since $S$ is a global $H$-slice. Then, clearly, $p(s)=G(s)=G(x)$ showing that $p'\big(H(s)\big)=G(x)$. Thus, $p'$ is surjective.
To see that $p'$ is injective, assume that $p'\big(H(s)\big)=p'\big(H(s_1)\big)$ for some $s, s_1\in S$. Then $G(s)=G(s_1)$, yielding that $s=gs_1$ for some $g\in G$. But then, by item (3) of Definition \ref{D:slice}, we get that $g\in H$ which implies that $H(s)=H(gs_1)=H(s_1)$, as desired. This completes the proof that $p':S/H\to X/G$ is a bijection, and hence, a homeomorphism.
\end{proof}
Since every compact subset of a proper $G$-space is small \cite[p.\,300]{pal:61}, Theorem \ref{Orbitspace}
has the following immediate corollary.
\begin{corollary}\label{C:Orbitspace} Let $G$ be a locally compact group, $H$ a compact subgroup of $G$. Let $X$ be a proper $G$-space and $S$ a compact global $H$-slice of $X$.
Then the inclusion $S\hookrightarrow X$ induces a homeomorphism of the orbit spaces
$S/H$ and $X/G$.
\end{corollary}
It turns out that in the presence of compactness of the global $H$-slice $S$, the other assumptions in Theorem \ref{Orbitspace} may essentially be weakened. Namely, the following version of Theorem \ref{Orbitspace} holds true.
\begin{proposition}\label{P:Orbitspace}
Let $G$ be any topological group and $H$ a closed subgroup of $G$.
Let $X$ be a $G$-space
and $S$ a compact global $H$-slice of $X$.
Then the inclusion $S\hookrightarrow X$ induces a homeomorphism of the orbit spaces
$S/H$ and $X/G$ provided that $X/G$ is Hausdorff.
\end{proposition}
\begin{proof}
Let $p:S\to X/G$ and $p':S/H\to X/G$ be as in the proof of Theorem \ref{Orbitspace}.
Since $p'$ is a continuous bijection, it remains to show that it is a closed map. But this
is due to the hypotheses since $S/H$ is compact and $X/G$ is Hausdorff.
\end{proof}
\medskip
\section{Extension to an equivariant map}\label{S:extension}
Let $G$ be a locally compact group and $H\subset G$ a compact subgroup.
If $X$ is a proper $G$-space and $S$ a global $H$-slice of $X$, then it is well known (cf. \cite[Proposition 2.1.3]{pal:61}) that any $H$-equivariant map $f:S\to Y$ uniquely extends to a $G$-equivariant map $F:X\to Y$. This result can be generalized in the following manner (for compact group actions it was proved in \cite[Ch.\,I, Theorem 3.3]{bre:72}).
\begin{theorem}\label{T:section}
Let $G$ be a locally compact group acting properly on the space $X$, and let $Y$ be any $G$-space.
Let $S$ be any closed small subset of $X$, and let $f: S\to Y$ be a continuous map such that whenever
$s$ and $gs$ are both in $S$ (for some $g\in G$), then $f(gs) = gf(s)$. Then
$f$ can be extended uniquely to an equivariant map $F: G(S)\to Y$.
\end{theorem}
\begin{proof} For any $g\in G$ and $s\in S$, put $F(gs) = gf(s)$. To see that $F$ is well
defined let $gs = g's'$. Then $s = (g^{-1}g')s' $ so that $f(s) = f((g^{-1}g')s')
= g^{-1}g'f(s')$, by assumption.
Thus, $gf(s) = g'f(s')$, as desired.
To see that
$F$ is continuous, let $(x_i)$ be a net in $G(S)$ converging to $ x\in G(C)$. Then
$x_i = g_is_i$ and $x=gs$ for some $s, s_i\in S$ and $g, g_i\in G$. Thus,
\begin{equation}\label{1}
g_is_i \rightsquigarrow gs.
\end{equation}
Since $S$ is a small set, we can choose a neighborhood $U$ of $s$ such that the transporter $\langle S, U\rangle$ has compact closure in $G$. Then, by convergence, there exists an index $i_0$ such that
$(g^{-1}g_i)s_i\in U$ whenever $i\ge i_0$. Hence $g^{-1}g_i\in \langle S, U\rangle$ for $i\ge i_0$. Since $\langle S, U\rangle$ has compact closure in $G$, by passing to a subnet we may assume that
$
g^{-1}g_i \rightsquigarrow h
$
for some $h\in G$.
Then $g_i \rightsquigarrow gh$ and
\begin{equation}\label{3}
g_i^{-1} \rightsquigarrow h^{-1}g^{-1}.
\end{equation}
Now, (\ref{1}) and (\ref{3}) imply that $s_i \rightsquigarrow h^{-1}s$. Since $s_i\in S$, by closedeness of $S$ we conclude that $ h^{-1}s\in S$. Then by the hypothesis we have $f(h^{-1}s)=h^{-1}f(s)$ and by continuity of $f$ we get that
$f(s_i)\rightsquigarrow f(h^{-1}s)=h^{-1}f(s).$ Consequently,
$$F(x_i)=F(g_is_i)=g_if(s_i)\rightsquigarrow gh(h^{-1}f(s))=gf(s)=F(gs)=F(x).$$
This proves the continuity of $F$, as desired.
\end{proof}
\medskip
Recall that a continuous map $s:X/G\to X$ is called a cross section for the orbit map $\pi: X\to X/G$ if the composition $\pi s$ is the identity map of $X/G$. It is easy to see that the image $C:=s(X/G)$ is closed in $X$ (since $X$ is Hausdorff). It turns out that if, in addition, $C$ is any small subset of $X$, then it uniquely determines the cross section. Because of this fact, we shall use the term \lq\lq cross section\rq\rq \ for the closed image of a cross section.
More precisely, we have the following result.
\begin{proposition}\label{P:small}
Let $X$ be a proper $G$-space with $G$ a locally compact group and let $\pi :X\to X/G$ be the orbit map. Let $C$ be a small closed
subset of $X$ touching each orbit in exactly one point. Then the map $s: X/G\to X$ defined by $s(\pi(x)) = G(x) \cap C$ is a cross section. Conversely, the image
of any cross section is closed in $ X.$
\end{proposition}
\begin{proof} We need to show that $s$ is continuous. For this let $A\subset C$ be
closed. Since $C$ is a small closed subset, one can apply \cite[Proposition 1.4(c)]{ab:78} according to which the set $s^{-1}(A) = G(A)$ is closed in $X$, as desired.
For the converse, let
$C = s(X/G)$ and let $(x_i)$ be a net in $C$ converging to $x\in X$.
We have $\lim p(x_i)= p(x)$ and $\lim s\big(p(x_i)\big)= s\big(p(x)\big)$. Therefore, $x=\lim x_i =\lim s\big(p(x_i)\big)=s\big(p(x)\big)\in C$ showing that $x\in C$. Thus, $C$ is closed.
\end{proof}
Theorem \ref{T:section} has the following interesting corollary.
\begin{corollary} Let $G$ be a locally compact group, $X$ a proper $G$-space and $Y$ any $G$-space. Assume that $C\subset X$ is a closed small cross section of the orbit map $p:X\to X/G$. Then each continuous map $f: C\to Y$ such that $G_c\subset G_{f(c)}$ for all $c\in C$, has a unique extension to an equivariant map $F: X\to Y$.
\end{corollary}
\begin{proof}
If $c$ and $gc$ belong to $C$ for some $g\in G$, then $gc=c$ since $C$ touchs the orbit $G(c)$ in exactly one point. Thus, $g\in G_c$. Since $G_c\subset G_{f(c)}$, we get that $gf(c)=f(c)=f(gc)$. Thus, the hypotheses of Theorem \ref{T:section} are fulfilled, and hence, its application completes the proof.
\end{proof}
\begin{remark} The example of the proper $G$-space $X$ in Remark \ref{R:small} shows that in Proposition \ref{P:small} one cannot omit the smallness condition on the set $C$.
\end{remark}
\section{An Application}\label{S:BM}
In this section we apply Corollary \ref{C:Orbitspace} to give a short proof of the compactness of the Banach-Mazur compacta $BM(n)$, $n\ge 1$.
As usual, for an integer $n\ge 1$,
$\mathbb R^n$ denotes the $n$-dimensional Euclidean space with the standard norm, and
$GL(n)$ denotes the real full linear group.
We denote by $\mathcal B(n)$ the hyperspace of all compact convex
bodies of $\mathbb R^n$ with odd symmetry about the origin, equipped with the Hausdorff metric
$$d_H(A,B)=\max\left\{ \sup\limits_{b\in B}d(b,A), ~ \sup\limits_{a\in A} d(a, B)\right\},$$
where $d$ is the standard Euclidean metric on $\Bbb R^n$.
We consider the natural action of $GL(n)$ on $\mathcal B(n)$ defined as follows:
$
(g, A)\longmapsto gA; \quad gA=\{ga \mid \ a\in A\}, \ \ \text{for all}
\ \ g\in GL(n), \ \ A\in \mathcal B(n).
$
In \cite{ant:Bull} it was proved that the $GL(n)$-space $\mathcal N(n)$ consisting of all norms $\varphi: \mathbb R^n\to \mathbb R$, endowed with the compact-open topology and the natural action of $GL(n)$, is a proper $GL(n)$-space.
Using the fact that $\mathcal B(n)$ is $GL(n)$-equivariantly homeomorphic to $\mathcal N(n)$ (see \cite[p.\,210]{ant:00}), we infer that $\mathcal B(n)$ is a proper $GL(n)$-space \cite{ant:00}. A direct way of proving the properness of the $GL(n)$-space $\mathcal B(n)$ one can find in \cite[Theorem 3.3]{antjo:13}.
According to a theorem of F.~John \cite{john}, for any $A\in \mathcal B(n)$, there is unique maximal volume ellipsoid $j(A)$ contained in $A$ (respectively, minimal volume ellipsoid $l(A)$ containig $A$). Usually, $j(A)$ is called the John ellipsoid of $A$, and $l(A)$ is called the L\" owner ellipsoid of $A$.
This fact allows to define two {\it special} \ global $O(n)$-slices in $\mathcal B(n)$, where $O(n)$ denotes the orthogonal subgroup of $GL(n)$.
Denote by $J(n)$ the subset of $\mathcal B(n)$ consisting of all bodies $A\in \mathcal B(n)$ for which the ordinary Euclidean unit ball
$\mathbb B^n=\big\{(x_1,\dots, x_n)\in\mathbb R^{n} \ \big | \ \sum_{i=1}^nx_i^2\leq 1\big\}
$
is the maximal volume ellipsoid {\it contained} in $A$. In \cite[Theorem 4]{ant:00} it was proved that $J(n)$
is {\it a global} \ $O(n)$-slice for $\mathcal B(n)$.
Analogously, the subspace $L(n)$ of $\mathcal B(n)$ consisting of all bodies $A\in \mathcal B(n)$ for which $\mathbb B^n$ is the minimal volume ellipsoid {\it containing} $A$, is {\it a global} \ $O(n)$-slice for $\mathcal B(n)$.
Lets give a direct proof that $J(n)$ and $L(n)$ are compact.
\begin{proposition}\label{P:J(n)compact}
$J(n)$ is compact.
\end{proposition}
\begin{proof} It is known that there is a closed ball $D\subset \mathbb R^n$ centered at the origin such that $A\subset D$ for every $A\in J(n)$. Moreover, it was proved by F. John \cite{john} that the radio of $D$ may be taken even $\sqrt n$.
Thus $J(n)$ is a subset of the hyperspace $cc(D)$ of all non-empty compact convex subsets of $D$ endowed with the Hausdorff metric topology.
Since $cc(D)$ is compact (in fact, it is homeomorphic to the Hilbert cube \cite[Theorem 2.2]{Nadler}), it suffices to show that $J(n)$ is closed in $cc(D)$. But this is evident since, if
$(A_k)_{k\in\mathbb N}\subset J(n)$ is a sequence converging to $A\in cc(D)$, then $A$ should contain the unit ball $\mathbb B^n$ since every $A_k$ does. Hence, $A$ has non-empty interior, i.e., it is a convex body, and then, $A\in \mathcal B(n)$. Now we apply \cite[Theorem 4(4)]{ant:00} according to which $J(n)$ is closed in $\mathcal B(n)$. This yields that $A\in J(n)$, and hence, $J(n)$ is closed in $cc(D)$, as desired.
\end{proof}
In a similar way, one can prove the compactness of the global $O(n)$-slice $L(n)$. Here one should take into account that every $A\in L(n)$ contains the closed ball of radio $\frac{1}{\sqrt n}$ centered at the origin of $\mathbb R^n$ (see \cite[p.\,559]{john}).
\smallskip
Thus, $\mathcal B(n)$ is a proper $GL(n)$-space and $J(n)$ is a global $O(n)$-slice of it (see \cite[Theorem 4]{ant:00}). Since by Proposition \ref{P:J(n)compact}, $J(n)$ is compact,
Corollary \ref{C:Orbitspace} immediately yields the following corollary (cf. \cite[Corollary 1]{ant:00}).
\begin{corollary}\label{C:1} The orbit space $\mathcal B(n)/GL(n)$
is homeomorphic to the $O(n)$-orbit space $J(n)/O(n)$.
\end{corollary}
Corollary \ref{C:slice} and Proposition \ref{P:J(n)compact} yield the following corollary.
\begin{corollary} The slcing map $f_{J(n)}:\mathcal B(n)\to GL(n)/O(n)$ corresponding to the global $O(n)$-slice $J(n)$ is continuous, open and closed.
\end{corollary}
The compactness of the orbit space $\mathcal B(n)/GL(n)$ originally was established in \cite{Macbeath}. Corollary \ref{C:1} and Proposition \ref{P:J(n)compact} immediately yield an alternative and short proof of the compactness of $\mathcal B(n)/GL(n)$, known as the Banach-Mazur compactum $BM(n)$ (see \cite{ant:00}).
\begin{corollary} $\mathcal B(n)/GL(n)$ is a compact metrizable space.
\end{corollary}
\
\bibliographystyle{amsplain}
|
1,116,691,497,532 | arxiv | \section{Introduction}
NA60+, a successor of NA60, is a new dimuon experiment that aims to study hard and electromagnetic processes at CERN-SPS energies. It extends and improves the physics program of its predecessor by performing measurements of dileptons and heavy-quark production over the entire SPS energy range and increasing the precision of the NA60 results by using an improved apparatus based on cutting-edge technologies. The NA60+ project will be part of a worldwide experimental program studying the properties of the Quark-Gluon Plasma (QGP) in the region of relatively large baryochemical potentials ($\mu_{\rm B}$).
Figure~\ref{fig:Tetyana}
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.5\linewidth]{hist_rates_detectors_2022_sep_NuPECC_NA60plus.pdf}
\caption{Existing and planned nuclear beams experiments~\cite{Galatyuk:2019lcf}.
}
\label{fig:Tetyana}
\end{center}
\end{figure}
shows current and foreseen experiments in terms of energy coverage and interaction rate for nuclear collisions~\cite{Galatyuk:2019lcf}.
In the SPS colliding energy range, which covers the interval between 6 and 17 GeV per nucleon in the center-of-mass frame, the NA60+ experiment, with a foreseen interaction rate exceeding $10^5$\,s$^{-1}$ has a unique position
reaching counting rates up to two orders of magnitude higher than other experiments.
At these energies, NA60+ can explore the range $230<\mu_{\rm B}<560$\,MeV where the QCD phase diagram should have a first-order phase transition between hadronic matter and QGP which terminates at a second-order critical point. Discovering signals of the first-order phase transition and the location of the critical point represents one of the hottest topics of relativistic heavy-ion physics.
Significant improvement and extension of the physics reach with the NA60+ apparatus will be possible due to a high-intensity beam with at least 10$^7$ Pb ions per spill and state-of-the-art experimental technologies. The setup of the NA60+ experiment is shown in Fig.~\ref{fig:NA60concept}.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.8\linewidth]{setup_na60p_ribs_wb.png}
\caption{GEANT4 rendering of the NA60+ experimental apparatus optimized for low-energy data taking. A C-shaped dipole magnet is shown in blue/yellow. Inside the dipole in light blue are the vertex spectrometer planes. Muon tracking chambers are shown in green, the purple blocks are absorbers, and the toroid magnet is grey.
}
\label{fig:NA60concept}
\end{center}
\end{figure}
It consists of several elements. A target system composed of several sub-targets, possibly including various nuclear species, is foreseen.
Targets are immediately followed by the vertex spectrometer immersed in a 1.5\,T magnetic field of a dipole magnet MEP48, providing a field integral of about 0.5\,Tm. A vertex spectrometer, consisting of 5 (up to 10) identical silicon pixel planes, is positioned at $7<z<38$\,cm starting from the most downstream target. Each plane, featuring a material budget of 0.1\% X$_0$ and intrinsic spatial resolution of $\sim$5 $\mu$m, is formed by four large area monolithic pixel sensors (MAPS) of 15x15\,cm$^2$ each.
An absorber made of BeO with a fixed length of 105\,cm is followed by a graphite absorber that is at least 130\,cm thick, in the version of the set-up optimized for low-energy data taking. The absorber is positioned immediately downstream of the vertex spectrometer, in order to reduce the probability of K and $\pi$ decays into muons and suppress combinatorial background. The choice of the material is dictated by requirements of relatively high density and low Z to minimize multiple scattering of muons. The center of the absorber is made of tungsten, in order to dump the non-interacting ions and forward-emitted fragments. One of the features foreseen in the NA60+ setup is a variable length of the absorber in order to comply with the increase of hadron multiplicity with collision energy.
This can be done by increasing the graphite section up to 335\,cm.
The increase of the absorber thickness also implies that the consequential parts have to be moved downstream, ensuring the rapidity coverage to remain constant with varying collision energy, in the approximate range $0<y<1$.
Muons traversing the absorber lose about 1\,GeV of their energy. They are analyzed in the Muon Spectrometer (MS), which is built out of two stations of pad detectors separated by a toroidal magnet with a 0.5\,T magnetic field. Pad chambers will be based either on Multi-Wire Proportional Chambers (MWPC) or Gas-Electron Multipliers (GEMs) technique. Their goal is to provide approximately 200\,$\mu$m spacial resolution while handling a maximum flux of particles of the order of a few cm$^{-2}$s$^{-1}$.
Spacial and momentum information about muons in the MS is used to match them to tracks in the vertex spectrometer with high efficiency and minimal false-positive rates.
By matching tracks, in coordinate and momentum space, it is possible to accurately measure the muon kinematics, reaching a resolution of less than $10$\,MeV at the $\omega$-meson mass and $\sim 30$\,MeV at the J/$\psi$. Moreover, the high granularity of the vertex detectors allows for accurate measurements of hadronic decays of open charm and strange hadrons.
An additional 180\,cm thick graphite absorber follows the MS. A pair of pad stations, similar in technology to the chambers of the MS, can be used as a trigger and/or as a muon identifier in the experiment.
An R\&D program is in progress for defining the detector aspects. For the MAPS, the studies are advancing in the frame of a collaboration with ALICE, intending to produce, via a stitching technique, sufficient enough surface of detectors with a low-material budget.
For the MWPC, the first prototype has been built, and it is being tested at the detector construction facility at the Weizmann Institute of Science and is planned to be exposed to an SPS test beam. The tracking stations are designed to be built by replicating the developed prototype to obtain the desired geometry. The two upstream muon stations are composed of 12 trapezoid detector modules each, while those downstream of the toroidal magnet and the cylindrical graphite absorber have 36 and 84 modules, respectively. Figure~\ref{fig:wheels} shows the design of the MWPC module (left) and the foreseen arrangement of the modules for the largest stations (right).
For the toroidal magnet, a scaled (1:5) working prototype has been built and tested to assess the feasibility of producing a device with the needed geometry and to provide the necessary information to design the full-scale device.
\begin{figure}[h]
\centering
\includegraphics[width=.43\textwidth]{proto_exploded.png}
\includegraphics[width=.43\textwidth]{3d_wheel.pdf}
\caption{The exploded view of the MWPC detector prototype and the CAD rendering of the largest muon tracking stations (right). }
\label{fig:wheels}
\end{figure}
The NA60+ project, thanks to its rich and specific physics program, will play a unique role by providing for the first time accurate experimental data on dileptons, open charm, and charmonia within the range of SPS energies.
The running plan of the experiment for the first 5-6 years includes monthly periods with a Pb beam, with different incident energy for each year, complemented by a few weeks of proton beam running for calibration purposes and physics studies with $p$-A collisions. This will ensure a fine enough energy scan for the
characterization of the QGP at varying $\mu_{\rm B}$ and the search of signals related to the first-order phase transition.
The proposed new experiment is based on state-of-the-art technologies that would allow the collection of about 20 times larger statistics at each energy, with respect to the former NA60 experiment which only operated at top SPS energy.
Despite the challenging conditions, the experimental layout is well adapted to the physics tasks of the NA60+ experiment.
|
1,116,691,497,533 | arxiv | |
1,116,691,497,534 | arxiv | \section{Introduction}
Let $M$ be an $n$-dimensional, compact, connected, oriented Riemannian
manifold without boundary. The heat kernel is the fundamental solution to
the associated heat equation. That is, it is the unique function $%
K:(0,\infty )\times M\times M$ that satisfies
\begin{eqnarray*}
\left( \frac{\partial }{\partial t}+\Delta _{x}\right) K(t,x,y) &=&0\qquad
\text{ and} \\
\lim_{t\rightarrow 0^{+}}\int_{M}K(t,x,y)\,f(y)\,\mathrm{dvol}(y)
&=&f(x)\qquad \text{for every continuous function }f.
\end{eqnarray*}
This function $K$ can be used to solve the heat equation $\left( \frac{%
\partial }{\partial t}+\Delta _{x}\right) g(t,x)=0$ for any initial
temperature distribution $g(0,x)$. It is well known (\cite{MiP}; see also
\cite{Cha},\cite{Ro}) that for any $x\in M$ and any positive integer $k$,
\begin{equation}
K(t,x,x)=\frac{1}{(4\pi t)^{n/2}}\left( u_{0}(x)+u_{1}(x)t+\dots
+u_{k}(x)t^{k}+O\left( t^{k+1}\right) \right) \text{ as }t\rightarrow 0,
\label{ktxx}
\end{equation}
where $u_{j}(x)$ are smooth functions on $M$ that depend only on geometric
data at the point $x\in M$. In particular, $u_{0}(x)=1$ and $u_{1}(x)=\frac{%
S(x)}{6}$, where $S(x)$ is the scalar curvature of $M$ at $x$. Using the
expansion above, it is possible to prove that the trace of the heat kernel
has a similar asymptotic formula. Let $0=\lambda _{0}<\lambda _{1}\leq
\lambda _{2}\leq \lambda _{3}\dots $ be the eigenvalues of the Laplacian,
counting multiplicities. Then
\begin{eqnarray}
\text{tr}\,\left( e^{-t\Delta }\right) &=&\sum_{m\geq 0}e^{-t\lambda
_{m}}=\int_{M}K(t,x,x)\,\mathrm{dvol}(x) \notag \\
&=&\frac{1}{(4\pi t)^{n/2}}\left( U_{0}+U_{1}t+\dots +U_{k}t^{k}+O\left(
t^{k+1}\right) \right) , \label{trace}
\end{eqnarray}
where $U_{j}=\int_{M}u_{j}(x)\,\mathrm{dvol}(x)$, with $u_{j}(x)$ defined as
above. In particular, $U_{0}=\text{Vol}(M)$. From formula~(\ref{trace}),
Karamata's theorem (see, for example, \cite[pp.~418--423]{Fel}) implies the
Weyl asymptotic formula ([We]; see also \cite[p.~155]{Cha}):
\begin{eqnarray*}
N(\lambda ):= &\#\{\lambda _{m}|\lambda _{m}\leq &\lambda \} \\
&\sim &\frac{\text{Vol}(M)}{(4\pi )^{n/2}\Gamma \left( \frac{n}{2}+1\right) }%
\lambda ^{n/2}
\end{eqnarray*}
as $\lambda \rightarrow \infty $.
The heat kernel has also been studied more generally, such as in the case of
manifolds with boundary or in the case of elliptic operators acting on
sections of a vector bundle over the manifold. Many researchers have studied
this expansion and its generalizations and have worked to compute the
coefficient functions (see \cite{MiP},\cite{Be},\cite{McS},\cite{Gi}),
because the heat kernel is not only used to compute heat flows but is also
used in many areas of geometric and topological analysis. The asymptotic
expansions above (and their generalizations) have been used to study the
spectrum of the Laplacian (see \cite{Be},\cite{BeGM},\cite{Cha},\cite{McS}),
the determinant of the Laplacian (see \cite{OPS1},\cite{Ri1}), conformal
classes of metrics (see \cite{OPS2}), analytic torsion (see \cite{RaS},\cite%
{Che}), modular forms (see \cite{Feg}), index theory (see \cite{ABP},\cite%
{Ro}), stochastic analysis (see \cite{CarZ},\cite{MaSt}), gauge
theory/mathematical physics (see \cite{EFKK}, \cite{Cam},\cite{Bl}), and so
on.
Other researchers have studied generalizations of the heat kernel to orbit
spaces of a group acting on a manifold. In \cite{D2}, the author showed that
if $M$ is a connected $n$--dimensional (not necessarily compact) Riemannian
manifold and $\Gamma $ is a group acting isometrically, effectively, and
properly discontinuously on $M$ with compact quotient $\overline{M}$, the
induced heat operator $e^{-t\overline{\Delta }}$ on the space of functions
on $\overline{M}$ (which is not necessarily a manifold) satisfies
\begin{equation*}
\text{tr}\,\left( e^{-t\overline{\Delta }}\right) =\frac{1}{(4\pi t)^{n/2}}%
\left( \overline{U}_{0}+\overline{U}_{1}t+\ldots +\overline{U}%
_{k}t^{k}+O\left( t^{k+1}\right) \right) ,
\end{equation*}
where $n=\mathrm{dim}(M)$ and $\overline{U}_{0}=\mathrm{Vol}(\overline{M})$.
This is equivalent to calculating the trace of the ordinary heat kernel on $%
M $ restricted to $\Gamma $-invariant functions. In \cite{BrH2}, the
researchers considered a compact, $n$-dimensional Riemannian manifold $M$
along with a compact group $G$ of isometries. Let $E_{\lambda }$ denote the
complex eigenspace of $\Delta $ associated to the eigenvalue $\lambda $, and
let $E_{\lambda }^{G}$ denote the subspace of $E_{\lambda }$ consisting of
eigenfunctions invariant under the induced action of $G$. They show that the
associated equivariant trace for $t>0$ is
\begin{eqnarray}
L(t):= &&\sum_{\lambda \geq 0}e^{-\lambda t}\dim E_{\lambda }^{G} \notag \\
&\sim &(4\pi t)^{-m/2}\left( a_{0}+\sum_{j>k\geq 0}a_{jk}t^{j/2}(\log
t)^{k}\right) \text{ as }t\rightarrow 0, \label{equtrace}
\end{eqnarray}%
where $m=\dim M/G$, $K_{0}$ is less than or equal to the number of different
dimensions of $G$-orbits in $M$, and $a_{0}=$Vol$(M/G)$. The coefficients $%
a_{jk}$ depend only on the metrics on $M$ and $G$ and their derivatives on
the subset $\{(g,x)\,|\,xg=x\}\subset G\times M$. The authors show in
addition that under certain conditions, no logarithmic terms appear in the
asymptotic expansion. Clearly, no logarithmic terms occur if all of the
orbits have the same dimension. Also, if $G$ is connected of rank 1 and acts
effectively on $M$, then no logarithmic terms appear. We remark that the
results of \cite{BrH2} apply to more general situations. If a second order
differential operator has the same principal symbol as the Laplacian, and if
it is geometrically defined and thus commutes with the action of $G$, then
the equivariant trace of the corresponding heat kernel satisfies~(\ref%
{equtrace}). After writing this paper,
I was made aware of the recent work \cite{Dry}, where the authors
compute the asymptotics of the heat kernel on orbifolds, related to the work in
\cite{D2}, \cite{BrH2}, and to Theorem~\ref{orbTheorem} in this paper.
In this paper, we consider a generalization of the trace of the heat kernel
to Riemannian foliations, and we will observe asymptotic behavior similar to
the results for group actions. Suppose that a compact, Riemannian manifold $M
$ is equipped with a \textit{Riemannian foliation} $\mathcal{F}$; that is,
the distance from one leaf of $\mathcal{F}$ to another is locally constant.
For simplicity, we assume that $M$ is connected and oriented and that the
foliation is transversally oriented. In some sense, this is a generalization
of the work in \cite{BrH2} and \cite{D2}, because the orbits of a group
acting by isometries form an example of a Riemannian foliation, if the
orbits all have the same dimension. In \cite{CrP}, the authors explicitly
calculated the heat kernel expansion in this specific case. Of course, the
dimensions of orbits of arbitrary group actions on a manifold are typically
not constant, and the leaf closures of a foliation are generally not orbits
of a group action. In \cite{Ri3}, we showed that many problems in the
analysis of the transverse geometry of Riemannian foliations and that of
group actions are equivalent problems.
A natural question to consider is the following: if we assume that the
temperature is always constant along the leaves of $\left( M,\mathcal{F}%
\right) $, how does heat flow on the manifold? To answer this question, we
must restrict to the space of basic functions $C_{B}^{\infty }(M)$(those
that are constant on the leaves of the foliation) and more generally the
space of basic forms $\Omega _{B}^{\ast }(M)$(smooth forms $\omega $such
that given any vector $X$tangent to the leaves, $i(X)\omega =0$and $%
i(X)d\omega =0$, where $i(X)$denotes the interior product with $X$). The
exterior derivative $d$maps basic forms to basic forms; let $d_{B}$denote $d$%
restricted to $\Omega _{B}^{\ast }(M)$. The relevant Laplacian on forms is
the basic Laplacian $\Delta _{B}=d_{B}\delta _{B}+\delta _{B}d_{B}$, where $%
\delta _{B}$is the adjoint of $d_{B}$on $L^{2}\left( \Omega _{B}^{\ast
}(M)\right) $. The basic heat kernel $K_{B}(t,x,y)$on functions is a
function on $(0,\infty )\times M\times M$that is basic in each $M$factor and
that satisfies
\begin{eqnarray}
\left( \frac{\partial }{\partial t}+\Delta _{B,x}\right) K_{B}(t,x,y) &=&0
\notag \\
\lim_{t\rightarrow 0^{+}}\int_{M}K_{B}(t,x,y)\,f(y)\,\mathrm{dvol}(y) &=&f(x)
\label{basicdef}
\end{eqnarray}
for every continuous basic function $f$. The existence of the basic heat
kernel allows us to answer the question posed at the beginning of this
paragraph. The basic heat kernel on forms is defined in an analogous way.
Many researchers have studied the analytic and geometric properties of the
basic Laplacian and the basic heat kernel (see \cite{A1},\cite{E},%
\cite{EH},\cite{KT},\cite{NRT}, \cite{NTV},\cite{PaRi}). In \cite{E}, the
author proved the existence of the basic heat kernel on functions. The
existence of the basic heat kernel on forms was proved for the case where
the mean curvature form of the foliation is basic in \cite{NRT}. The
existence of the basic heat kernel was proved in general in \cite{PaRi},
where the authors given explicit formulas for the basic Laplacian and basic
heat kernel in terms of the orthogonal projection from $L^{2}$--forms to $%
L^{2}$--basic forms and certain elliptic operators on the space of all forms
on the manifold. A point of difficulty that often arises in this area of
research is that the space of basic forms is not the set of all sections of
any vector bundle, and therefore the usual theory of elliptic operators and
heat kernels does not apply directly to $\Delta _{B}$ and $K_{B}$.
It is natural to try to prove the existence of asymptotic expansions of the
form (\ref{ktxx}) and (\ref{trace}) for the basic heat kernel. We remark
that the basic heat operator is trace class, since the basic Laplacian is
the restriction of an elliptic operator on the space of all functions (see
lower bounds for eigenvalues in \cite{PaRi} and \cite{LRi}). In \cite{Ri2},
it was shown that an analogue of (\ref{ktxx}) exists for the basic heat
kernel. As $t\rightarrow 0$, we have the following asymptotic expansion for
any positive integer $k$:
\begin{equation} \label{kbtxx}
K_{B}(t,x,x)=\frac{1}{(4\pi t)^{{}^{q_{x}/2}}}\left(
a_{0}(x)+a_{1}(x)t+\ldots +a_{k}(x)t^{k}+O\left( t^{k+1}\right) \,\right) ,
\end{equation}
where $q_{x}$ is the codimension of the leaf closure containing $x$ and $%
a_{j}(x)$ are functions depending on the local transverse geometry and
volume of the leaf closure containing $x$. The first two nontrivial
coefficients were computed in \cite{Ri2} and are given in Theorem~\ref%
{oldthm}. In general, the power $q_{x}$ may vary, but its value is minimum
and constant on an open, dense subset of $M$. One might guess that the
asymptotics of the trace of the basic heat operator could be obtained by
integrating the expansion (\ref{kbtxx}), similar to obtaining (\ref{trace})
from (\ref{ktxx}). However, the functions $a_{j}(x)$ for $j\geq 1$ are not
necessarily bounded or even integrable over the dense subset. Example~\ref%
{sphereexample} exhibits this precise behavior. Even if the coefficients $%
a_{j}(x)$ in (\ref{kbtxx}) are integrable, it is not true in general that
these functions can be integrated to obtain the asymptotics of the trace.
Example~\ref{exnonorientable} shows that even if $a_{j}(x)$ is constant,
these coefficients cannot be integrated to yield the trace asymptotics.
Despite these obstacles, we prove that an asymptotic expansion for the trace
of the basic heat operator exists. Let $\overline{q}$ be the minimum
codimension of the leaf closures of $(M,\mathcal{F})$. As $t\rightarrow 0$,
the trace $K_{B}(t)$ of the basic heat kernel on functions satisfies the
following asymptotic expansion for any positive integer $J$:
\begin{equation}
K_{B}(t)=\frac{1}{(4\pi t)^{\overline{q}/2}}\left( a_{0}+\sum_{j>0,~0\leq
k\leq K_{0}}a_{jk}t^{j/2}(\log t)^{k}+O\left( t^{\frac{J+1}{2}}(\log
t)^{K_{0}-1}\right) \right) , \label{tb}
\end{equation}%
where $K_{0}$ is less than or equal to the number of different dimensions of
leaf closures in $\mathcal{F}$, and where
\begin{equation*}
a_{0}=V_{tr}=\int_{M}\frac{1}{\text{Vol}\left( \overline{L_{x}}\right) }\,{%
\mathrm{dvol}}(x).
\end{equation*}%
This is the content of Theorem~\ref{tracebasic}. If the codimension of $%
\mathcal{F}$ is less than $4$, then the logarithmic terms vanish. The idea
of proof is as follows. We rewrite the integral $\displaystyle %
K_{B}(t)=\int_{M}K_{B}(t,x,x)\,\mathrm{dvol}(x)$ in terms of an integral
over $W\times SO(q)$, where $W$ is the \textit{basic manifold}, an $SO(q)$%
-manifold associated to $(M,\mathcal{F})$. Then, we apply the results of
\cite{BrH2}. In Corollary~\ref{Weyl}, we obtain the Weyl asymptotic formula
for the eigenvalues of the basic Laplacian.
In Section~\ref{special}, we derive the first two nontrivial coefficients in
the asymptotic expansion (\ref{tb}) in some special cases, including but not
limited to all possible types of Riemannian foliations of codimension two or
less. In each of these cases, the asymptotic formula contains no logarithmic
terms. We conjecture (Conjecture~\ref{traceconjecture}) that the asymptotic
expansion for the general case has the same features. In Section~\ref{finite}%
, we derive the asymptotics for the case in which all of the leaf closures
have the same dimension, for any codimension. In this case, the leaf closure
space is an orbifold, and en route to the result, we obtain the asymptotics
of the orbifold heat trace, which may be of independent interest. We find
the asymptotics for the transversally orientable, codimension one case in
Section~\ref{codim1}, for the nonorientable codimension one case in Section~%
\ref{nonorientable}, and for codimension two Riemannian foliations in
Section~\ref{codim2}. We remark that the codimension two case yields five
possible types of asymptotic expansions. In Section~\ref{general}, we show
how to simplify the general case by subdividing the basic manifold into
pieces, and this result is used in the calculations of Section~\ref{codim2}.
In Section~\ref{example}, we demonstrate the asymptotic formulas in two
examples of codimension two foliations.
We remark that these asymptotic expansions yield new results concerning the
spectrum of the basic Laplacian. By Corollary~\ref{Weyl}, the eigenvalues of
the basic Laplacian determine the minimum leaf closure codimension and the
transverse volume $V_{tr}$ of the foliation. The results of Section~\ref%
{special} give more specific information in special cases. For example, if
the leaf closure codimension is one, then the spectrum of the basic
Laplacian determines the $L^{2}$ norm of the mean curvature of the leaf
closure foliation. Therefore, the spectrum determines whether or not the
leaf closure foliation is minimal.
In most cases considered in the paper, we assume that the foliations are
transversally oriented for simplicity. In Section~\ref{nonorient}, we
describe the method of obtaining the asymptotics of the basic heat kernel on
Riemannian foliations that are not transversally orientable.
\section{Heat Kernels and Operators on the Basic Manifold}
\label{setup}In this section, we introduce some notation, recall some
results contained in \cite{Ri2}, and then use these results to obtain a
formula for the trace of the basic heat kernel. Let $M$ be an $n$%
-dimensional, closed, connected, oriented Riemannian manifold without
boundary, and let $\mathcal{F}$ be a transversally--oriented, codimension $q$
foliation on $M$ for which the metric is bundle--like. As in the
introduction, we let $\Delta _{B}$ denote the basic Laplacian, and we let $%
K_{B}(t,x,y)$ be the basic heat kernel on functions defined in~(\ref%
{basicdef}).
Let $\widehat M$ be the oriented transverse orthonormal frame bundle of $(M,%
\mathcal{F})$, and let $\pi$ be the natural projection $\pi:\widehat
M\longrightarrow M$. The manifold $\widehat M$ is a principal $SO(q)$-bundle
over $M$. Given $\hat x\in \widehat M$, let $\hat xg$ denote the
well-defined right action of $g\in SO(q)$ applied to $\hat x$. Associated to
$\mathcal{F}$ is the lifted foliation $\widehat{\mathcal{F}}$ on $\widehat M$
. The lifted foliation is transversally parallelizable, and the closures of
the leaves are fibers of a fiber bundle $\rho:\widehat M\longrightarrow W$.
The manifold $W$ is smooth and is called the basic manifold (see \cite[%
pp.~105-108, p.~147ff]{Mo}). Let $\overline{\mathcal{F}}$ denote the
foliation of $\widehat M$ by leaf closures of $\widehat {\mathcal{F}}$.
Endow $\widehat M$ with the metric $g^M+g^{SO(q)}$, where $g^M$ is the
pullback of the metric on $M$, and $g^{SO(q)}$ is the standard, normalized,
biinvariant metric on the fibers. By this, we mean that we use the
transverse Levi--Civita connection (see \cite[p.~80ff]{Mo}) to do the
following. We calculate the inner product of two horizontal vectors in $%
T_{\hat x}\widehat M$ by using $g^M$, and we calculate the inner product of
two vertical vectors using $g^{SO(q)}$. We require that vertical vectors are
orthogonal to horizontal vectors. This metric is bundle--like for both $%
(\widehat M, \widehat {\mathcal{F}})$ and $(\widehat M,\overline {\mathcal{F}%
})$. The transverse metric on $(\widehat M,\overline {\mathcal{F}})$ induces
a well--defined Riemannian metric on $W$. The group $G=SO(q)$ acts by
isometries on $W$ according to $\rho(\hat x)g:=\rho(\hat xg)$ for $g\in
SO(q) $.
The volume form on $\widehat{M}$ can be written as $\mathrm{dvol}_{\overline{%
\mathcal{F}}}\,\rho ^{\ast }\!\mathrm{dvol}_{W}$, where $\mathrm{dvol}_{%
\overline{\mathcal{F}}}$ is the volume form of any leaf closure and $\mathrm{%
dvol}_{W}$ is the volume form on the basic manifold $W$. Let $\phi
:W\rightarrow \mathbb{R}$ be defined by taking $\phi (y)$ to be the volume
of $\rho ^{-1}(y)$. The function $\phi $ is obviously positive and is also
smooth, since $\rho $ is a smooth Riemannian submersion (see a proof of a
similar fact in \cite[Proposition 1.1]{PaRi}). Let $(\ ,\ )$ denote the
pointwise inner product of forms, and let $\left\langle \ ,\ \right\rangle $
denote the $L^{2}$-inner product of forms. Then for all $\alpha ,\gamma \in
\Omega ^{\ast }(W)$,
\begin{eqnarray}
\left\langle \rho ^{\ast }\alpha ,\rho ^{\ast }\gamma \right\rangle _{%
\widehat{M}} &=&\int_{\widehat{M}}\left( \rho ^{\ast }\alpha ,\rho ^{\ast
}\gamma \right) _{\widehat{M}}\,\mathrm{dvol}_{\overline{\mathcal{F}}}\,\rho
^{\ast }\!\mathrm{dvol}_{W} \notag \\
&=&\int_{\widehat{M}}\rho ^{\ast }\left( \alpha ,\gamma \right) _{W}\,%
\mathrm{dvol}_{\overline{\mathcal{F}}}\,\rho ^{\ast }\!\mathrm{dvol}_{W}
\notag \\
&=&\int_{W}\phi \cdot \left( \alpha ,\gamma \right) _{W}\,\mathrm{dvol}%
_{W}=\left\langle \phi \alpha ,\gamma \right\rangle _{W}. \label{metricw}
\end{eqnarray}
We have used the fact that for a bundle--like metric, the pointwise inner
product has the same action on basic forms as the pullback of the pointwise
inner product on the local quotient manifold. For our foliation $(\widehat{M}%
,\overline{\mathcal{F}})$, $W$ is the (global) quotient manifold.
Let $\widehat \Delta_B$ denote the basic Laplacian associated to the lifted
foliation on $\widehat M$, and let $\Delta_{W}$ denote the ordinary
Laplacian on $W$ corresponding to the induced metric on $W$. Note that $\phi$
is invariant under the right action of $SO(q)$ on $W$, so we define the
smooth function $\psi:M\to \mathbb{R}$ by $\pi^{*}\psi=\rho^{*}\phi$. Let $%
\sigma \lrcorner $ denote the adjoint of the wedge product $\sigma \wedge$
for any form $\sigma$.
We define the elliptic operator $\widetilde{\Delta_{W}}:C^\infty (W)\to%
\mathbb{R}$ by
\begin{eqnarray}
\widetilde{\Delta_{W}}&=&\Delta_{W}-\frac 1\phi\, (d\phi )\lrcorner \circ d
\label{twistlap} \\
&=&-g^{ij}\partial_{i}\partial_{j}-b^{j}\partial_{j}\text{ locally on }W,
\notag
\end{eqnarray}
where $g=\left( g_{ij}\right)$ is the metric on $W$ in local coordinates, $%
\left( g^{ij}\right)=g^{-1}$, and $b^{j}=\partial_{i}g^{ij}+g^{ij}%
\partial_{i}\left( \log \left( \phi\sqrt{\det g}\right)\right)$. Let $K_{B}$
and $\widehat {K_{B}}$ denote the basic heat kernels on $M$ and $\widehat M$%
, respectively, and let $\widetilde {K_{W}}$ denote the heat kernel
corresponding to $\widetilde{\Delta_{W}}$ on $W$. Then we have the following
results (see \cite[Theorems 1.1 and 2.4]{Ri2}):
\begin{enumerate}
\item The following equation holds on $C^{\infty }(W)$:
\begin{equation*}
\widehat{\Delta }_{B}\rho ^{*}=\rho ^{*}\widetilde{\Delta _{W}}.
\end{equation*}
\item For every $x,y\in M$, $\hat{x}\in \pi ^{-1}(x)$, and $\hat{y}\in \pi
^{-1}(y)$,
\begin{equation*}
K_{B}(t,x,y)=\int_{G}\widetilde{K_{B}}(t,\hat{x},\hat{y}g)\,\chi (g).
\end{equation*}
\item For every $\hat{x}\in \pi ^{-1}(x)$ and $\hat{y}\in \pi ^{-1}(y)$,
\begin{equation}
\widehat{K_{B}}(t,\hat{x},\hat{y})=\frac{\widetilde{K_{W}}\left( t,\rho (%
\hat{x}),\rho (\hat{y})\right) }{\phi \left( \rho (\hat{y})\right) }.
\label{kerhatkerw}
\end{equation}
\end{enumerate}
We will now write the trace of the basic heat kernel in terms of $\widetilde{%
K_{W}}$. Let $\mathrm{dvol}$, $\widehat{\mathrm{dvol}}$, $\mathrm{dvol}_{W}$%
, and $\chi $ denote the volume forms on $M$, $\widehat{M}$, $W$, and $%
G=SO(q)$, respectively. Then
\begin{eqnarray}
K_{B}(t) &=&\text{trace}\,e^{-t\Delta _{B}} \notag \\
&=&\int_{M}K_{B}(t,x,x)\,\mathrm{dvol} \notag \\
&=&\int_{M}\int_{G}\widehat{K_{B}}\left( t,\hat{x},\hat{x}g\right) \,\chi
(g)\,\mathrm{dvol}(x). \label{trmhat}
\end{eqnarray}%
For any measurable section $s:M\rightarrow \widehat{M}$ that is smooth on an
open, dense subset of $M$, we can describe points of $\widehat{M}$ in terms
of the map $M\times G\rightarrow \widehat{M}$ defined by $(x,h)\mapsto s(x)h$%
. In these \textquotedblleft coordinates,\textquotedblright\ the measure on $%
\widehat{M}$ is $\widehat{\mathrm{dvol}}(x,h)=\mathrm{dvol}(x)\,\chi (h)$.
Therefore, if we change coordinates $\hat{x}\mapsto \hat{x}h$ in~(\ref%
{trmhat}), average over $G$, and use Fubini's Theorem, we get
\begin{eqnarray*}
K_{B}(t) &=&\int_{G}\int_{G}\int_{M}\widehat{K_{B}}\left( t,\hat{x}h,\hat{x}%
hg\right) \,\mathrm{dvol}(x)\,\chi (h)\,\chi (g) \\
&=&\int_{G}\left\langle \widehat{K_{B}}(t,\cdot ,\cdot \,g),1\right\rangle _{%
\widehat{M}}\,\chi (g) \\
&=&\int_{G}\left\langle \frac{\widetilde{K_{W}}(t,\rho (\cdot ),\rho (\cdot
)g)}{\phi (\rho (\cdot ))},1\right\rangle _{\widehat{M}}\,\chi (g)\,\,\,%
\text{by (\ref{kerhatkerw})} \\
&=&\int_{G}\left\langle \widetilde{K_{W}}(t,\cdot ,\cdot \,g),1\right\rangle
_{W}\,\chi (g)\,\,\,\,\,\,\,\,\text{by (\ref{metricw})} \\
&=&\int_{G\times W}\widetilde{K_{W}}(t,w,wg)\,\mathrm{dvol}_{W}(w)\,\chi (g)
\end{eqnarray*}%
We have shown the following:
\begin{proposition}
\label{trbasic1} The trace $K_{B}(t)$ of the basic heat kernel on functions
is given by the formula
\begin{equation*}
K_{B}(t)=\int_{G\times W}\widetilde{K_{W}}(t,w,wg)\,\mathrm{dvol}%
_{W}(w)\,\chi (g).
\end{equation*}
\end{proposition}
\begin{corollary}
The trace of the basic heat kernel on functions on $M$ is the same as the
trace of the heat kernel corresponding to $\widetilde{\Delta _{W}}$
restricted to $SO(q)$--invariant functions on $W$.
\end{corollary}
Therefore, the results of \cite{BrH2} apply, since $\widetilde {\Delta_{W}}$
has the same principal symbol as $\Delta_{W}$ and commutes with the $G$%
--action. In particular, formula~(\ref{equtrace}) holds, where $m=\dim W/G$,
$K_{0}$ is less than or equal to the number of different dimensions of $G$%
-orbits in $W$, and $a_{0}=$Vol$(W/G)$.
The leaf closures of $(M,\mathcal{F})$ with maximal dimension form an open,
dense subset $M_{0}$ of $M$ (see \cite[pp. 157--159]{Mo}). The leaf closures
of the lifted foliation cover the leaf closures of $M$ (see \cite[p.~151ff]%
{Mo}). Given a leaf closure $\overline{L_{x}}$ containing $x\in M$, the
dimension of a leaf closure contained in $\pi^{-1}\left( \overline{L_{x}}%
\right)$ is $\dim \overline{L_{x}}+\dim H_{\hat x}$, where $\hat x\in
\pi^{-1}\left( x\right)$ and $H_{\hat x}$ is the subgroup of $SO(q)$ that
fixes the leaf closure containing $\hat x$. In some sense, the group $%
H_{\hat x}$ measures the holonomy of the leaves contained in the leaf
closure containing $x$ as well as the holonomy of the leaf closure. The
group $H_{\hat x}$ is isomorphic to the structure group corresponding to the
principal bundle $\overline L_{\hat x}\to \overline L_{x}$, where $\overline
L_{\hat x}$ is the leaf closure in $\widehat M$ that contains $\hat x$; the
conjugacy class of $H\subset G$ depends only on the leaf $L_{x}$. Therefore,
on an open, dense subset of $W$, the orbits of $G$ have dimension $\frac {%
q(q-1)}2-m$, where $m$ is the dimension of the principal isotropy groups.
The above discussion also implies that the dimension of $W$ is $\overline q+%
\frac {q(q-1)}2-m$, where $\overline q$ is the codimension of the leaf
closures of $M$ of maximal dimension. As a result, we have that
\begin{equation*}
m=\dim W/G=\dim W-\left(\frac {q(q-1)}2-m\right) =\overline q.
\end{equation*}
Also, the number of different dimensions of $G$-orbits in $W$ is equal to
the number of different dimensions of leaf closures in $(M,\mathcal{F})$.
For $w\in W$, let $wG$ denote the $G$-orbit of $w$. By construction,
\begin{eqnarray*}
\text{Vol}(wG)\cdot \phi (w) &=&\text{Vol}\left( \rho ^{-1}(wG)\right) \\
&=&\text{Vol}\left( \overline{L_{x}}\right) ,
\end{eqnarray*}
where $x$ is chosen so that $x\in \pi \left( \rho ^{-1}(wG)\right) $. Using
the above information, we have
\begin{eqnarray*}
\text{Vol}(W/G) &=&\int_{W}\frac{1}{\text{Vol}(wG)}\,\mathrm{dvol}_{W}(w) \\
&=&\int_{W}\frac{\phi (w)}{\text{Vol}\left( \rho ^{-1}(wG)\right) }\,\mathrm{%
dvol}_{W}(w) \\
&=&\int_{\widehat{M}}\frac{1}{\text{Vol}\left( \rho ^{-1}(\rho (\hat{x}%
)G)\right) }\,\widehat{\mathrm{dvol}}(\hat{x})\qquad \text{by (\ref{metricw})%
} \\
&=&\int_{\widehat{M}}\frac{1}{\text{Vol}\left( \overline{L_{\pi (\hat{x})}}%
\right) }\,\widehat{\mathrm{dvol}}(\hat{x}) \\
&=&\int_{M}\frac{1}{\text{Vol}\left( \overline{L_{x}}\right) }\,{\mathrm{dvol%
}}(x)
\end{eqnarray*}
We remark that the integrals above converge. Proposition 2.1 in \cite{Pa},
which concerns isometric flows, is easily modified to show that the first
integral above converges; the convergence of the other integrals follows.
Using the above discussions and the results of \cite{BrH2}, we have the
following:
\begin{theorem}
\label{tracebasic} Let $\overline{q}$ be the codimension of the leaf
closures of $(M,\mathcal{F})$ with maximal dimension. As $t\rightarrow 0$,
the trace $K_{B}(t)$ of the basic heat kernel on functions satisfies the
following asymptotic expansion for any positive integer $J$:
\begin{equation*}
K_{B}(t)=\frac{1}{(4\pi t)^{\overline{q}/2}}\left( a_{0}+\sum_{j>0,~0\leq
k\leq K_{0}}a_{jk}t^{j/2}(\log t)^{k}+O\left( t^{\frac{J+1}{2}}(\log
t)^{K_{0}-1}\right) \right) ,
\end{equation*}%
where $K_{0}$ is less than or equal to the number of different dimensions of
leaf closures of $(M,\mathcal{F})$, and where
\begin{equation*}
a_{0}=V_{tr}=\int_{M}\frac{1}{\mathrm{Vol}\left( \overline{L_{x}}\right) }\,{%
\mathrm{dvol}}(x).
\end{equation*}%
The coefficients of the logarithmic terms in the expansion vanish if the
codimension of the foliation is less than four.
\end{theorem}
In the last statement of the above proposition, we used the fact that $SO(q)$
is connected, has rank one for $q<4$, and acts effectively on $W$. Also,
note that $a_{jk}$ depends only on the transverse metric, because by the
results in \cite{BrH2} it depends only on infinitesimal metric information
on the set $\{ (g,w)\, |\, wg=w\}$, which is entirely determined by the
transverse geometry of $(M,\mathcal{F})$.
We will call the coefficients $a_{0}$ and $a_{jk}$ the basic heat
invariants; they are functions of the spectrum of the basic Laplacian,
because of the formula (see \cite[Theorem 3.5]{PaRi})
\begin{equation*}
K_{B}(t)=\text{trace}\left( e^{-t\Delta_B}\right) =\sum_{j\geq
0}e^{-t\lambda^{B}_{j}}.
\end{equation*}
Using Karamata's theorem (\cite[pp.~418--423]{Fel}), we also have the
following, which generalizes Weyl's asymptotic formula (\cite{We}):
\begin{corollary}
\label{Weyl} Let $0=\lambda _{0}<\lambda _{1}^{B}\leq \lambda _{2}^{B}\leq
\ldots $ be the eigenvalues of the basic Laplacian on functions, counting
multiplicities. Then the spectral counting function $N_{B}(\lambda )$
satisfies the following asymptotic formula:
\begin{eqnarray*}
N_{B}(\lambda ) &:&=\#\{\lambda _{m}^{B}|\lambda _{m}^{B}\leq \lambda \} \\
&\sim &\frac{V_{\text{tr}}}{(4\pi )^{\overline{q}/2}\Gamma \left( \frac{%
\overline{q}}{2}+1\right) }\lambda ^{\overline{q}/2}
\end{eqnarray*}
as $\lambda \rightarrow \infty $, where $\overline{q}$ and $V_{\text{tr}}$
are defined as above.
\end{corollary}
Observe that we are able to prove Theorem~\ref{tracebasic} and Corollary~\ref%
{Weyl} with very little information about the heat kernel $\widetilde{K_{W}}$%
; we used Proposition~\ref{trbasic1} and the results in \cite{BrH2} alone.
We also remark that although the expansion~(\ref{equtrace}) contains
logarithmic terms, no examples for which these terms are nonzero are known.
In the proof of this expansion \cite{BrH2}, the authors show that $G\times M$
can be decomposed into pieces over which the integral has an expansion with
possibly nonzero logarithmic terms. In the cases for which the authors
proved the nonexistence of logarithmic terms, symmetries cause the sum of
these logarithmic contributions to vanish. We make the following conjectures:
\begin{conjecture}
\label{Gconjecture} Suppose that $\Gamma $ is a compact group that acts
isometrically and effectively on a compact, connected Riemannian manifold $M$%
. Then the coefficients $a_{jk}$ of the equivariant trace formula (\ref%
{equtrace}) satisfy the following:
\begin{itemize}
\item $a_{jk}=0$ for $k>0$.
\item If $M$ is oriented and $\Gamma $ acts by orientation--preserving
isometries, then $a_{j0}=0$ for $j$ odd.
\end{itemize}
\end{conjecture}
\begin{conjecture}
\label{traceconjecture} (Corollary of Conjecture~\ref{Gconjecture}) In
Theorem~\ref{tracebasic}, $a_{jk}=0$ for $k>0$. If in addition $SO\left(
q\right) $ acts by orientation preserving isometries on $W$, then $a_{j0}=0$
for $j$ odd.
\end{conjecture}
We remark that since our foliation is transversally oriented, $SO\left(
q\right) $ acts by orientation preserving isometries precisely when the leaf
closures are all transversally orientable. The $SO\left( q\right) $ action
is not always orientation preserving, as Example~\ref{exnonorientable} shows.
\section{Formulas for the Basic Heat Invariants in Special Cases}
\label{special}We will now explicitly derive the asymptotics of the integral
in Proposition~\ref{trbasic1} in some special cases.
\subsection{Regular Closure\label{finite}}
Suppose that $(M,\mathcal{F})$ has \textit{regular closure}. In other words,
assume that the leaf closures all have the same dimension. Note that this
implies that all of the leaves and leaf closures have finite holonomy. In
this case, the orbits of $SO\left( q\right) $ or $O\left( q\right) $ (and
thus the leaf closures of $(M,\mathcal{F})$) all have the same dimension,
and the space of leaf closures is a Riemannian orbifold. Since the orbits
all have the same dimension, locally defined functions of the metric along
the orbits are smooth (bounded) functions on the basic manifold, and the
volumes of the orbits (and hence the volumes of the leaf closures) are
bounded away from zero. Therefore, the coefficients in the asymptotic
expansion for $K_{B}(t,x,x)$ found in \cite{Ri2} are bounded on the foliated
manifold $M$, but the error term is not necessarily bounded. These local
expressions cannot in general be integrated over $M$ to yield the
asymptotics of the trace of the basic heat operator, as in the method used
to obtain the asymptotic expansion of the trace of the ordinary heat
operator on a manifold, as described in the introduction. Instead, a
calculation of the trace of a second order operator on an orbifold is
required.
Note that the leaf closures of $(M,\mathcal{F})$ themselves form a
Riemannian foliation $(M,\mathcal{F}^{c})$, in which all leaves are compact.
In general the basic Laplacian $\Delta _{B}$ on functions satisfies $\Delta
_{B}=P_{B}\delta d$, where $d$ is the exterior derivative, $\delta $ is the $%
L^{2}$ adjoint of $d$, and $P_{B}$ is the orthogonal projection of $L^{2}$
functions to the $L^{2}$ of basic functions (see \cite{PaRi}). Since the
projection $P_{B}$ on functions is identical for both foliations $(M,%
\mathcal{F})$ and $(M,\mathcal{F}^{c})$, the basic Laplacian on basic
functions of the foliation $(M,\mathcal{F})$ is the same as the basic
Laplacian $\Delta _{B}^{c}$ on basic functions of the foliation $(M,\mathcal{%
F}^{c})$ . Note that the equivalent statement for the basic Laplacian on
forms is false.
Thus, it suffices to solve the problem of calculating the basic heat kernel
asymptotics for the case of closed leaves. Let $p:M\rightarrow N=M\slash
\mathcal{F}^{c}$ be the quotient map, which is a Riemannian submersion away
from the leaf closures with holonomy. Similar to the arguments in Section %
\ref{setup}, the basic Laplacian on functions satisfies
\begin{equation*}
\Delta _{B}\circ p^{\ast }=p^{\ast }\circ \left( \Delta ^{N}-\frac{d\psi }{%
\psi }\lrcorner \circ d\right) ,
\end{equation*}
where $\Delta ^{N}=\delta d$ is the ordinary Laplacian on the orbifold $N$,
and $\psi $ is the function on $N$ defined by $\psi \left( x\right) =\mathrm{%
Vol}\left( p^{-1}\left( x\right) \right) $ if $p^{-1}\left( x\right) $ is a
principal leaf closure and extended to be continuous (and smooth) on $N$.
Note that if $\kappa ^{c}$ is the mean curvature form of $(M,\mathcal{F}%
^{c}) $ and $P_{B}^{1}$ is the $L^{2}$ projection from one-forms to basic
one-forms, then $P_{B}^{1}\kappa ^{c}=-p^{\ast }\left( \frac{d\psi }{\psi }%
\right) $. Since $f$ is basic function on $M$ if and only if $f=p^{\ast }g$
for some function $g$ on $N$, the trace of $K_{B}(t)$ of the basic heat
kernel on functions on $M$ is the trace of $e^{-t\left( \Delta ^{N}-\frac{%
d\psi }{\psi }\lrcorner \circ d\right) }$ on functions on the orbifold $N$.
To calculate this trace, we first collect the following known results.
\begin{lemma}
(See \cite{Cha}, \cite{Gr}, \cite{Ro}) Let $L$ be a second order operator on
functions on a closed Riemannian manifold $N$ of dimension $m$, such that $%
L=\Delta +V+Z,$ where $\Delta $ is the Laplace operator, $V$ is a purely
first-order operator, and $Z$ is a zeroth order operator. Then, the heat
kernel $K_{L}\left( t,x,y\right) $ of $L$, the fundamental solution of the
operator $\frac{\partial }{\partial t}+L$, exists and has the following
properties:
\begin{enumerate}
\item Given $\varepsilon >0$, there exists $c>0$ such that if $r\left(
x,y\right) =\mathrm{dist}\left( x,y\right) >\varepsilon $, then $K_{L}\left(
t,x,y\right) =\mathcal{o}\left( e^{-c/t}\right) $ as $t\rightarrow 0$.
\item If $r\left( x,y\right) =\mathrm{dist}\left( x,y\right) $ is
sufficiently small, then as $t\rightarrow 0$,
\begin{equation*}
K_{L}\left( t,x,y\right) =\frac{e^{-r^{2}\left( x,y\right) /4t}}{\left( 4\pi
t\right) ^{m/2}}\left( c_{0}\left( x,y\right) +c_{1}\left( x,y\right)
t+...+c_{k}\left( x,y\right) +\mathcal{O}\left( t^{k+1}\right) \right)
\end{equation*}
for any $k$, each $c_{j}\left( x,y\right) $ is smooth, and $c_{0}\left(
x,x\right) =1$. The function $c_{j}\left( x,y\right) $ is determined by the
metric and the operators $V$ and $Z$ and their derivatives, evaluated along
the minimal geodesic connecting $x$ and $y$.
\end{enumerate}
\end{lemma}
If the manifold $N$ is instead a Riemannian orbifold, the operators still
can be defined (by their definitions on the local covers using pullbacks),
and a fundamental solution to the heat equation still exists. Note that in
all such cases, the lifted operator $\widetilde{L}=\widetilde{\Delta }+%
\widetilde{V}+\widetilde{Z}$ is equivariant with respect to the local finite
group action. Since the asymptotics of the heat kernel are still determined
locally, we have the following corollary.
\begin{lemma}
Let $L$ be a second order operator on functions on a closed Riemannian
orbifold $N$ of dimension $m$, such that $L=\Delta +V+Z$, where $\Delta $ is
the Laplace operator, $V$ is a purely first order operator, and $Z$ is a
zeroth order operator. Then, the heat kernel $K_{L}\left( t,x,y\right) $ of $%
L$, the fundamental solution of the operator $\frac{\partial }{\partial t}+L$%
, exists and has the following properties:
\begin{enumerate}
\item Given $\varepsilon >0$, there exists $c>0$ such that if $r\left(
x,y\right) =\mathrm{dist}\left( x,y\right) >\varepsilon $, then $K_{L}\left(
t,x,y\right) =\mathcal{O}\left( e^{-c/t}\right) $ as $t\rightarrow 0$.
\item If $r\left( x,y\right) =\mathrm{dist}\left( x,y\right) $ is
sufficiently small, then as $t\rightarrow 0$, and if the minimal geodesic
connecting $x$ and $y$ is away from the singular set of the orbifold, then
\begin{equation}
K_{L}\left( t,x,y\right) =\frac{e^{-r^{2}\left( x,y\right) /4t}}{\left( 4\pi
t\right) ^{m/2}}\left( c_{0}\left( x,y\right) +c_{1}\left( x,y\right)
t+...+c_{k}\left( x,y\right) +\mathcal{O}\left( t^{k+1}\right) \right)
\label{localKernelOrbifold}
\end{equation}
for any $k$, each $c_{j}\left( x,y\right) $ is smooth, and $c_{0}\left(
x,x\right) =1$.
\item Let $z$ be an element of the singular set of $N$, and let $H_{z}$
denote the finite group of isometries such that a neighborhood $U$ of $z$ in
$N$ is isometric to $\widetilde{U}\slash H_{z}$, where $\widetilde{U}$ is
an open set in $\mathbb{R}^{m}$ with the given metric. Let $o\left(
H_{z}\right) $ denote the order of $H_{z}$. Assuming that the neighborhood $U
$ is sufficiently small, there exists $c>0$ such that if $x,y\in U$, then
\begin{equation*}
K_{L}\left( t,x,y\right) =\frac{1}{o\left( H_{z}\right) }\sum_{h\in H_{z}}%
\widetilde{K_{L}}\left( t,x,hy\right) +\mathcal{O}\left( e^{-c/t}\right) ,
\end{equation*}%
where $\widetilde{K_{L}}$ is the heat kernel of the lifted operator $%
\widetilde{\Delta }+\widetilde{V}+\widetilde{Z}$ on $\widetilde{U}$, which
itself satisfies an asymptotic expansion as in (\ref{localKernelOrbifold})
above.
\end{enumerate}
\end{lemma}
The results above are well-known and well-utilized in the cases where $V=Z=0$
(see, for example, \cite{D1}, \cite{D2}), but they are true in the
generality stated.
Next, we establish an estimate and a trigonometric identity.
\begin{lemma}
\label{integralAsymptGamma}Given $a>0$ and $b\in \mathbb{N}$, we have
\begin{eqnarray*}
\int_{0}^{\varepsilon }e^{-\frac{x^{2}a^{2}}{t}}x^{b}dx &=&\int_{0}^{\infty
}e^{-\frac{x^{2}a^{2}}{t}}x^{b}dx-\int_{\varepsilon }^{\infty }e^{-\frac{%
x^{2}a^{2}}{t}}x^{b}dx \\
&=&\frac{\Gamma \left( \frac{b+1}{2}\right) }{2a^{b+1}}t^{\frac{b+1}{2}}+%
\mathcal{O}\left( \left( \frac{\varepsilon ^{2}a^{2}}{t}\right) ^{\frac{b-1}{%
2}}e^{-\frac{\varepsilon ^{2}a^{2}}{t}}\right) ~ \\
&=&\frac{\Gamma \left( \frac{b+1}{2}\right) }{2a^{b+1}}t^{\frac{b+1}{2}}+%
\mathcal{O}\left( \left( \frac{t}{\varepsilon ^{2}a^{2}}\right) ^{N}\right) ~%
\text{as }t\rightarrow 0,
\end{eqnarray*}
for any $N\geq \frac{b+1}{2}$.
\end{lemma}
\begin{proof}
Substituting $u=\frac{x^{2}a^{2}}{t}$, or $x=$ $\frac{\sqrt{ut}}{a}$, we get%
\begin{eqnarray*}
\int_{0}^{\varepsilon }e^{-\frac{x^{2}a^{2}}{t}}x^{b}dx &=&\frac{t^{\frac{b+1%
}{2}}}{2a^{b+1}}\int_{0}^{\varepsilon ^{2}a^{2}/t}e^{-u}u^{\frac{b-1}{2}}du
\\
&=&\frac{t^{\frac{b+1}{2}}}{2a^{b+1}}\int_{0}^{\infty }e^{-u}u^{\frac{b-1}{2}%
}du-\frac{t^{\frac{b+1}{2}}}{2a^{b+1}}\int_{\varepsilon ^{2}a^{2}/t}^{\infty
}e^{-u}u^{\frac{b-1}{2}}du \\
&=&\frac{t^{\frac{b+1}{2}}\Gamma \left( \frac{b+1}{2}\right) }{2a^{b+1}}-%
\frac{t^{\frac{b+1}{2}}}{2a^{b+1}}\Gamma \left( \frac{b+1}{2},\frac{%
\varepsilon ^{2}a^{2}}{t}\right) ,
\end{eqnarray*}%
where $\Gamma \left( A,z\right) $ is the (upper) incomplete Gamma function.
It is known that $\Gamma \left( A,z\right) $ is proportional to $%
e^{-z}z^{A-1}\left( 1+\mathcal{O}\left( \frac{1}{z}\right) \right) $ as $%
\left\vert z\right\vert \rightarrow \infty $, and the formulas above follow.
\end{proof}
\begin{lemma}
\label{trigLemma}For any positive integer $k$,%
\begin{equation*}
\sum_{j=1}^{k-1}\frac{1}{\sin ^{2}\left( \frac{\pi j}{k}\right) }=\frac{%
k^{2}-1}{3}.
\end{equation*}
\end{lemma}
\begin{proof}
Many thanks to George Gilbert. Contact the author for a proof.
\end{proof}
\medskip The goal is evaluate the asymptotics of
\begin{equation*}
K_{B}\left( t\right) =\mathrm{tr}\left( \left. e^{-tP}\right\vert
_{C^{\infty }\left( N\right) }\right) =\int_{N}K_{P}\left( t,x,x\right) ~%
\mathrm{dvol}
\end{equation*}%
as $t\rightarrow 0$, where $P=\Delta ^{N}-\frac{d\psi }{\psi }\lrcorner
\circ d$, but we first proceed with calculating the heat trace of a general
operator $L$ as in (\ref{localKernelOrbifold}) on an orbifold. We now
decompose $N$ as follows. Given an element $z\in N$, let $H_{z}$ denote a
subgroup of the orthogonal group $O\left( \dim N\right) $ such that every
sufficiently small metric ball around $z$ is isometric to a ball in $%
H_{z}\backslash \left( \mathbb{R}^{\dim N},g\right) $, where $g$ is an $H_{z}$%
-invariant metric. The conjugacy class $\left[ H_{z}\right] $ in $O\left(
\dim N\right) $ is called isotropy type of $z$. The stratification of $N$ is
a partition of $N$ into the different isotropy types. The partial order on
these isotropy types is defined as in the general $G$-manifold structure
(see Section \ref{general}). Let $o\left( H_{z}\right) $ denote the order of
$H_{z}$.
As in Section \ref{general}, we decompose $N$ into pieces which include
tubular neighborhoods of parts of the singular strata of the orbifold and
the principal stratum (for which $H_{z}=\left\{ e\right\} $) minus the other
neighborhoods. We may further decompose $N=N_{\varepsilon }\cup \coprod
N_{j}$ as a finite disjoint union, where each $N_{j}\ $is of the form $%
H_{j}\backslash \widetilde{N_{j}}$ with $\widetilde{N_{j}}$ contractible, no
nontrivial element of $H_{j}$ fixing all of $\widetilde{N_{j}}$, and with at
least one point of $\widetilde{N_{j}}$ having isotropy $H_{j}$. We may think
of $N_{j}$ as a tubular neighborhood of an open subset of a singular
stratum, up to sets of measure zero.
Given any isometry $h\in H_{i}\setminus \left\{ e\right\} $, choose a
tubular neighborhood
\[
U_{h,\varepsilon }^{i}=T_{\varepsilon }\left(
H_{j}\backslash \left( \widetilde{N_{i}}\right) ^{h}\right) \cap N_{i}
\]
of
the local submanifold $S_{i}^{h}=\left( H_{j}\backslash \widetilde{N_{i}}%
\right) ^{h}$ of singular points fixed by $h$, and let $\widetilde{%
U_{h,\varepsilon }^{i}}=T_{\varepsilon }\left( \left( \widetilde{N_{i}}%
\right) ^{h}\right) \cap \widetilde{N_{i}}$ denote the local cover. Let $%
\widetilde{U_{e,\varepsilon }^{i}}$ $=\widetilde{N_{i}}$.
\begin{equation*}
N_{\varepsilon }=N\setminus \bigcup\limits_{i}\left( \bigcup\limits_{h\in
H_{i}\setminus \left\{ e\right\} }U_{h,\varepsilon }^{i}\right) .
\end{equation*}%
Then there exists $c>0$ such that
\begin{gather*}
K_{L}\left( t\right) +\mathcal{O}\left( e^{-c/t}\right) \\
=\int_{N_{\varepsilon }}K_{L}\left( t,x,x\right) ~\mathrm{dvol}+\sum_{i}%
\frac{1}{o\left( H_{i}\right) }\sum_{h\in H_{i}}\int_{\widetilde{%
U_{h,\varepsilon }^{i}}}\widetilde{K_{L}}\left( t,x,hx\right) ~\mathrm{dvol}
\\
=\frac{1}{\left( 4\pi t\right) ^{m/2}}\left( \int_{N_{\varepsilon }}%
\mathrm{dvol}+t\int_{N_{\varepsilon }}c_{1}\left( x,x\right) \,\mathrm{dvol}+%
\mathcal{O}\left( t^{2}\right) \right) \\
+\sum_{i}\frac{1}{o\left( H_{i}\right) }\left( \int_{\widetilde{N_{i}}}%
\mathrm{dvol}+t\int_{\widetilde{N_{i}}}c_{1}\left( x,x\right) \,\mathrm{dvol}%
+\mathcal{O}\left( t^{2}\right) \right) \\
+\sum_{i}\frac{1}{o\left( H_{i}\right) \left( 4\pi t\right) ^{m/2}}%
\sum_{h\in H_{i}\setminus \left\{ e\right\} }\int_{\widetilde{%
U_{h,\varepsilon }^{i}}}e^{-r^{2}\left( x,hx\right) /4t}\left( c_{0}\left(
x,hx\right) +\mathcal{O}\left( t^{1}\right) \right) ~\mathrm{dvol}
\end{gather*}%
Note that if $h\neq e$, $S_{i}^{h}$ is a disjoint union of connected
submanifolds $S_{i,j}^{h}$ codimension $d_{h}^{i,j}>0$. Then, following a
calculation in \cite{D1}, we may rewrite the last integral in geodesic
normal coordinates $x$. If $B_{\varepsilon }^{i,j}\left( y\right) $ denotes
the normal exponential ball of radius $\varepsilon $ at $y\in \widetilde{%
S_{i,j}^{h}}$, its volume form $\mathrm{dvol}_{B_{\varepsilon }^{i,j}}$
satisfies $\mathrm{dvol}_{B_{\varepsilon }^{i,j}}=\left( 1+\mathcal{O}\left(
\left\vert x\right\vert ^{2}\right) \right) ~dx$, and the volume form $%
\mathrm{dvol}$ on $\widetilde{U_{h,\varepsilon }^{i}}$ satisfies $\mathrm{%
dvol}=$ $\left( 1+\mathcal{O}\left( \left\vert x\right\vert ^{2}\right)
\right) ~\mathrm{dvol}_{B_{\varepsilon }^{i,j}}~\mathrm{dvol}_{\widetilde{%
S_{i,j}^{h}}}$ for each $j$.
\begin{multline*}
\int_{\widetilde{U_{h,\varepsilon }^{i}}}e^{-r^{2}\left( x,hx\right) \slash
4t}\left( c_{0}\left( x,hx\right) +\mathcal{O}\left( t^{1}\right) \right) ~%
\mathrm{dvol} \\
=\sum_{j}\int_{\widetilde{S_{i,j}^{h}}}\int_{B_{\varepsilon
}^{i,j}}e^{-r^{2}\left( x,hx\right) \slash 4t}\left( 1+\mathcal{O}\left(
r^{2}\right) +\mathcal{O}\left( t\right) \right) ~\mathrm{dvol}%
_{B_{\varepsilon }^{i,j}}~\mathrm{dvol}_{\widetilde{S_{i,j}^{h}}} \\
=\sum_{j}\int_{\widetilde{S_{i,j}^{h}}}\int_{B_{\varepsilon
}^{i,j}}e^{-r^{2}\left( x,hx\right) \slash 4t}\left( 1+\mathcal{O}\left(
\left\vert x\right\vert ^{2}\right) +\mathcal{O}\left( t\right) \right) ~dx~%
\mathrm{dvol}_{\widetilde{S_{i,j}^{h}}} \\
=\sum_{j}\int_{\widetilde{S_{i,j}^{h}}}\int_{\left( I-h\right)
B_{\varepsilon }^{i,j}}e^{-r^{2}\left( u+hx\left( u\right) ,hx\left(
u\right) \right) \slash 4t}\left( 1+\mathcal{O}\left( \left\vert
u\right\vert ^{2}\right) +\mathcal{O}\left( t\right) \right) \cdot\\
\left\vert
\det \left( I-h\right) ^{-1}\right\vert du~\mathrm{dvol}_{\widetilde{%
S_{i,j}^{h}}},
\end{multline*}%
using the change of variables $u=\left( I-h\right) x$. Further (see \cite{D1}%
), there is a change of variables $y\left( u\right) $ such that $u_{j}=y_{j}+%
\mathcal{O}\left( \left\vert y\right\vert ^{3}\right) $ and $r^{2}\left(
u+hx\left( u\right) ,hx\left( u\right) \right) =\sum y_{j}^{2}$, and the
Jacobian for this change of variables is $1+\mathcal{O}\left( \left\vert
y\right\vert ^{2}\right) $. Thus,
\begin{multline*}
\int_{\widetilde{U_{h,\varepsilon }^{i}}}e^{-r^{2}\left( x,hx\right)
\slash 4t}\left( c_{0}\left( x,hx\right) +\mathcal{O}\left( t^{1}\right)
\right) ~\mathrm{dvol} \\
=\left\vert \det \left( I-h\right) ^{-1}\right\vert \sum_{j}\int_{%
\widetilde{S_{i,j}^{h}}}\int_{y\left( \left( I-h\right) B_{\varepsilon
}^{i,j}\right) }e^{-\left\vert y\right\vert ^{2}\slash 4t}\cdot\\
\left( 1+\mathcal{%
O}\left( \left\vert y\right\vert ^{2}\right) +\mathcal{O}\left( t\right)
\right) ~dy~\mathrm{dvol}_{\widetilde{S_{i,j}^{h}}}.
\end{multline*}%
By Lemma \ref{integralAsymptGamma}, we have
\begin{multline*}
\int_{y\left( \left( I-h\right) B_{\varepsilon }^{i,j}\right)
}e^{-\left\vert y\right\vert ^{2}\slash 4t}\left( 1+\mathcal{O}\left(
\left\vert y\right\vert ^{2}\right) +\mathcal{O}\left( t\right) \right) ~dy\\
=\int_{\mathbb{R}^{d_{h}^{i,j}}}e^{-\left\vert y\right\vert ^{2}\slash
4t}dy+\mathcal{O}\left( t^{\left( d_{z}+2\right) \slash 2}\right) \\
=\left( 2\cdot \frac{\Gamma \left( \frac{1}{2}\right) }{2\left( \frac{1}{2}%
\right) }\right) ^{d_{h}^{i,j}}t^{d_{h}^{i,j}\slash 2}+\mathcal{O}\left(
t^{\left( d_{h}^{i,j}+2\right) \slash 2}\right) \\
=\left( 4\pi t\right) ^{d_{h}^{i,j}\slash 2}+\mathcal{O}\left( t^{\left(
d_{h}^{i,j}+2\right) \slash 2}\right) .
\end{multline*}%
Thus,
\begin{multline*}
\int_{\widetilde{U_{h,\varepsilon }^{i}}}e^{-r^{2}\left( x,hx\right) \slash
4t}\left( c_{0}\left( x,hx\right) +\mathcal{O}\left( t^{1}\right) \right) ~%
\mathrm{dvol}\\
=\left\vert \det \left( I-h\right) ^{-1}\right\vert \sum_{j}%
\mathrm{vol}\left( \widetilde{S_{i,j}^{h}}\right) \left( 4\pi t\right)
^{d_{h}^{i,j}\slash 2}+\mathcal{O}\left( t^{\left( d_{h}^{i,j}+2\right)
\slash 2}\right)
\end{multline*}
\vspace{1pt}Hence, letting $\varepsilon $ approach zero and summing up over
the neighborhoods of the singular strata of the orbifold $N$, we have
\begin{gather}
K_{L}\left( t\right) =\frac{1}{\left( 4\pi t\right) ^{m/2}}\left( \int_{N}%
\mathrm{dvol}+t\int_{N}c_{1}\left( x,x\right) \,\mathrm{dvol}\right.\notag\\
+\mathcal{O}%
\left( t^{2}\right) \biggm) \notag \\
+\frac{1}{\left( 4\pi t\right) ^{m/2}}\sum_{i}\frac{1}{o\left(
H_{i}\right) }\sum_{h\in H_{i}\setminus \left\{ e\right\}
}\sum_{j}\left\vert \det \left( I-h\right) ^{-1}\right\vert \mathrm{vol}%
\left( \widetilde{S_{i,j}^{h}}\right) \left( 4\pi t\right)
^{d_{h}^{i,j}\slash 2}\notag\\
+\mathcal{O}\left( t^{\left( d_{h}^{i,j}+2\right)
\slash 2}\right) \notag \\
=\frac{1}{\left( 4\pi t\right) ^{m/2}}\left( \mathrm{vol}\left( N\right)
+t\int_{N}c_{1}\left( x,x\right) \,\mathrm{dvol}\right) \notag\\
+\frac{1}{\left( 4\pi t\right) ^{m/2}}\left( \sum_{i}\frac{1}{o\left(
H_{i}\right) }\sum_{h\in H_{i}\setminus \left\{ e\right\}
}\sum_{d_{h}^{i,j}=1,2}\left\vert \det \left( I-h\right) ^{-1}\right\vert
\mathrm{vol}\left( \widetilde{S_{i,j}^{h}}\right) \left( 4\pi t\right)
^{d_{h}^{i,j}\slash 2}\right) \notag\\
+\mathcal{O}\left( t^{\left( 3-m\right) \slash 2}\right) .
\label{asymptExpOrb1}
\end{gather}%
Note that $\widetilde{S_{i,j}^{h}}$ has codimension 1 precisely when $h$
acts as a reflection, in which case $\left\vert \det \left( I-h\right)
^{-1}\right\vert =\left( 1-\left( -1\right) \right) ^{-1}=\frac{1}{2}$.
Similarly, $\widetilde{S_{i,j}^{h}}$ has codimension 2 exactly when $h$ acts
as a rotation (say by $\theta _{h}=\frac{2\pi }{k}$ for some $k\in \mathbb{Z}%
_{>0}$) in the normal space to $\widetilde{S_{i,j}^{h}}$. In that case,
\begin{multline*}
\left\vert \det \left( I-h\right) ^{-1}\right\vert =\left\vert \det \left(
\begin{array}{cc}
1-\cos \theta _{h} & -\sin \theta _{h} \\
\sin \theta _{h} & 1-\cos \theta _{h}%
\end{array}%
\right) ^{-1}\right\vert\\
=\frac{1}{2-2\cos \left( \theta _{h}\right) }=\frac{%
1}{4\sin ^{2}\left( \frac{\theta _{h}}{2}\right) }.
\end{multline*}%
If this number is summed over all nontrivial elements of a cyclic group
group of rotations by $\left\{ \frac{2\pi }{k},\frac{4\pi }{k},...,\frac{%
2\left( k-1\right) \pi }{k}\right\} $, by Lemma \ref{trigLemma} we have
\begin{equation*}
\sum_{j=1}^{k-1}\frac{1}{4\sin ^{2}\left( \frac{\pi j}{k}\right) }=\frac{%
k^{2}-1}{12}.
\end{equation*}%
In each of these cases, generic points $z$ of $S_{i,j}^{h}$ have isotropy
subgroups $H_{z}$ isomorphic to a cyclic group.
To obtain the asymptotic expansion, we identify two subsets of the singular
part of the orbifold:%
\begin{eqnarray*}
\Sigma _{\mathrm{ref}}N &=&\left\{ z\in N:H_{z}~\text{has order }2\text{ and
is generated by a reflection}\right\} , \\
\Sigma _{k}N &=&\big\{ z\in N:H_{z}\text{ is a cyclic group of order }k \\
&&\qquad \text{ and consists of rotations in a plane}\big\} .
\end{eqnarray*}%
Note that
\begin{eqnarray*}
\mathrm{vol}^{m-1}\left( N_{i}\cap \Sigma _{\mathrm{ref}}N\right) &=&\frac{2%
}{o\left( H_{i}\right) }\sum\limits_{h\text{ reflection}}\mathrm{vol}%
^{m-1}\left( \widetilde{S_{i}^{h}}\right)\\
&=&\sum\limits_{h\text{ reflection}}%
\mathrm{vol}^{m-1}\left( S_{i}^{h}\right) , \\
\mathrm{vol}^{m-2}\left( N_{i}\cap \Sigma _{k}N\right) &=&\frac{k}{o\left(
H_{i}\right) }\sum\limits_{h\text{ rotation of }\frac{2\pi }{k}}\mathrm{vol}%
^{m-2}\left( \widetilde{S_{i}^{h}}\right) \\
=\sum\limits_{h\text{ rotation of }%
\frac{2\pi }{k}}\mathrm{vol}^{m-2}\left( S_{i}^{h}\right)
&=&\sum\limits_{h\text{ rotation by }\frac{2\pi j}{k}}\mathrm{vol}%
^{m-2}\left( S_{i}^{h}\right) \text{ for fixed }j\text{.}
\end{eqnarray*}%
We now combine the results above with (\ref{asymptExpOrb1}) to obtain the
following theorem.
\begin{theorem}\label{orbTheorem}
Let $L$ be a second order operator on functions on a closed Riemannian
orbifold $N$ of dimension $m$, such that $L=\Delta +V+Z$, where $\Delta $ is
the Laplace operator, $V$ is a purely first order operator, and $Z$ is a
zeroth order operator. Then, the trace of the heat operator has the
following asymptotic expansion as $t\rightarrow 0$:%
\begin{eqnarray*}
K_{L}\left( t\right) &=&\frac{1}{\left( 4\pi t\right) ^{m/2}}\Biggm(\mathrm{%
vol}^{m}\left( N\right) +t^{1/2}\frac{\sqrt{\pi }}{2}\mathrm{vol}%
^{m-1}\left( \Sigma _{\mathrm{ref}}N\right) \\
&&+t\left( \int_{N}c_{1}\left( x,x\right) \,\mathrm{dvol}+\frac{\pi \left(
k^{2}-1\right) }{3k}\mathrm{vol}^{m-2}\left( \Sigma _{k}N\right) \right) +%
\mathcal{O}\left( t^{3/2}\right) \Biggm),
\end{eqnarray*}%
where $c_{1}\left( x,x\right) $ is the heat trace coefficient from formula (%
\ref{localKernelOrbifold}) and $\Sigma _{\mathrm{ref}}N$ and $\Sigma _{k}N$
are parts of the singular stratum of the orbifold defined in the paragraph
above.
\end{theorem}
Note that the truth of this theorem is easily checked in the case of a
manifold with boundary or with a manifold quotient by a finite cyclic group
of rotations. Also, note that the coefficient $c_{1}\left( x,x\right) $ may
be computed using standard methods as in \cite{Ro}.
We now wish to apply this result to the foliation case. Here, $N=M\slash
\mathcal{F}^{c}$ is the leaf closure space (a Riemannian orbifold) of a
Riemannian foliation $\left( M,\mathcal{F}\right) $ with regular closure.
The operator of note is
\begin{equation*}
K_{B}\left( t\right) =\mathrm{tr}\left( \left. e^{-tL}\right\vert
_{C^{\infty }\left( N\right) }\right) =\int_{N}K_{L}\left( t,x,x\right) ~%
\mathrm{dvol}
\end{equation*}%
as $t\rightarrow 0$, where $L=\Delta ^{N}-\frac{d\psi }{\psi }\lrcorner
\circ d=\Delta ^{N}+H^{c}$, and $H^{c}$ is the projection of the mean
curvature vector field of the foliation of $M$ by leaf closures to the set
of projectable vector fields, which implies that it descends to a vector
field on $N$. The formula needed is $c_{1}\left( x,x\right) $; we refer to
\cite[formula (3.6)]{Ri2} for a similar calculation, which yields in our case%
\begin{eqnarray*}
c_{1}\left( x,x\right) &=&\frac{S\left( x\right) }{6}+\frac{\Delta ^{N}\psi
\left( x\right) }{2\psi \left( x\right) }+\frac{1}{4}\left\vert H^{c}\left(
x\right) \right\vert ^{2} \\
&=&\frac{S\left( x\right) }{6}+\frac{\Delta ^{N}\psi \left( x\right) }{2\psi
\left( x\right) }+\frac{1}{4}\left\vert \frac{d\psi }{\psi }\right\vert
^{2}\left( x\right) ,
\end{eqnarray*}%
where $S\left( x\right) $ is the scalar curvature at $x\in N$. The theorem
below follows.
\begin{theorem}
Let $\left( M,\mathcal{F}\right) $ be a Riemannian foliation with regular
closure, so that the quotient orbifold $N=M\slash \mathcal{F}^{c}$ by leaf
closures has dimension $m$. If $x\in N$ corresponds to a principal leaf
closure, let $\psi \left( x\right) $ be the volume of the leaf closure, and
extend this function to be smooth on $N$. Let $S$ denote the scalar
curvature of $N$. Further, let $\Sigma _{\mathrm{ref}}N$ be the set of
singular points of $N$ corresponding to true boundary points, and let $%
\Sigma _{k}N$ be the set of singular points of $N$ which have neighborhoods
diffeomorphic to $\mathbb{R}^{n}$ mod a planar cyclic group of rotations of
order $k$. Then the trace of the basic heat kernel on functions satisfies
the following asymptotic formula as $t\rightarrow 0$.
\begin{multline*}
K_{B}\left( t\right) =\frac{1}{\left( 4\pi t\right) ^{m/2}}\Biggm(\mathrm{%
vol}^{m}\left( N\right) +t^{1/2}\frac{\sqrt{\pi }}{2}\mathrm{vol}%
^{m-1}\left( \Sigma _{\mathrm{ref}}N\right) \\
+t\left( \int_{N}\frac{S\,}{6}+\frac{\Delta ^{N}\psi }{2\psi }+\frac{1}{4}%
\left\vert \frac{d\psi }{\psi }\right\vert ^{2}\mathrm{dvol}+\frac{\pi
\left( k^{2}-1\right) }{3k}\mathrm{vol}^{m-2}\left( \Sigma _{k}N\right)
\right) +\mathcal{O}\left( t^{3/2}\right) \Biggm).
\end{multline*}
\end{theorem}
\subsection{Transversally Oriented, Codimension One Riemannian Foliations}
\label{codim1}
Suppose that $\left( M,\mathcal{F}\right) $ is a transversally oriented,
codimension one Riemannian foliation. In this case, the analysis of the
basic manifold is unnecessary, because the basic manifold is isometric to
the space of leaf closures. For such a foliation, either the closure of
every leaf is all of $M$, or the leaves are all compact without holonomy. In
the first case, the basic Laplacian is identically zero, so that the trace
of the basic heat operator satisfies $K_{B}\left( t\right) =1$ for every $t$%
. In the case of compact leaves, the leaves are the fibers of a Riemannian
submersion over a circle. Thus, the basic functions are pullbacks of
functions on the circle, and the basic function $v:M\rightarrow \mathbb{R}$
given by $v(x)=($ the volume of the leaf containing $x)$ is smooth on $M$
and likewise smooth in the circle coordinate. If the circle is parametrized
to have unit speed by the coordinate $s\in \lbrack 0,S)$, then the $L^{2}$
inner products on basic functions and basic one-forms are defined by
\begin{eqnarray*}
\left\langle f,g\right\rangle &=&\int_{0}^{S}f(s)g(s)v(s)\,ds, \\
\left\langle \alpha (s)\,ds,\beta (s)\,ds\right\rangle &=&\int_{0}^{S}\alpha
(s)\,\beta (s)\,v(s)\,ds.
\end{eqnarray*}
Note that $S$ is the transverse volume $V_{tr}$ of $(M,\mathcal{F})$,
defined as in Theorem~\ref{tracebasic}. We then compute that the basic
Laplacian on functions is given by
\begin{equation*}
\Delta _{B}f=-\frac{\partial ^{2}}{\partial s\,^{2}}f-\frac{v^{\prime }}{v}%
\frac{\partial }{\partial s}f.
\end{equation*}
Since the foliation is transversally oriented, we may assume that we have
chosen a unit normal vector field $U=\frac{\partial }{\partial s}$. An
elementary calculation shows that the \textit{total mean curvature} $H(s)$
is given by
\begin{equation}
H(s):=-\ell \int_{L_{s}}\left\langle \mathbf{H}(x),U(x)\right\rangle \,%
\mathrm{dvol}(x)=\frac{v^{\prime }(s)}{v(s)}, \label{totalmean}
\end{equation}
where $\mathbf{H}(x)$ is the mean curvature vector field of the leaf $L_{s}$
corresponding to the coordinate $s$ and $\ell $ is the dimension of each
leaf. Recall that $\mathbf{H}$ is defined as follows. Given $x\in L_{s}$,
let $\{E_{i}\}_{i=1}^{\ell }$ be an orthonormal basis of $T_{x}L$. We define
\begin{equation*}
\mathbf{H}(x)=\frac{1}{\ell }\sum_{i=1}^{\ell }\left( \nabla
_{E_{i}}E_{i}\right) ^{\perp },
\end{equation*}
where $\perp $ denotes the projection onto the normal space. We denote the
\textit{mean curvature} of $L_{s}$ by $h(x)=\Vert \mathbf{H}(x)\Vert $.
In summary, the basic Laplacian on functions $\psi :\left[ 0,V_{tr}\right]
\rightarrow \mathbb{R}$ is given by
\begin{equation}
\Delta _{B}\psi (s)=-\frac{d^{2}}{ds^{2}}\psi (s)-H(s)\frac{d}{ds}\psi (s).
\label{basic1d}
\end{equation}
The trace of the basic heat kernel may now be computed in the standard way
from this operator on the circle. Let an arbitrary point on the circle be
denoted by the coordinate $0$, and let $x$ be any other point within $\frac{%
V_{tr}}{2}$ of $0$. Following the computation in \cite[pp.69--70]{Ro}, we
get the following asymptotic expansion of the basic heat kernel $%
K_{B}(t,x,0) $ as $t\rightarrow 0$:
\begin{equation*}
K_{B}(t,x,0)\sim \frac{e^{-x^{2}/4t}}{\sqrt{4\pi t}}\left(
u_{0}(x)+u_{1}(x)t+u_{2}(x)t^{2}+\ldots \right) ,
\end{equation*}
where $u_{0}(x)=\frac{1}{R_{-1}(x)}$ and $u_{k+1}(x)$ for $k\geq 0$ is given
by
\begin{equation}
R_{k}(x)u_{k+1}(x)=-\int_{0}^{x}\frac{R_{k}(y)}{y}\Delta _{B}u_{k}(y)\,dy,
\label{recurse1}
\end{equation}
where for every $j\geq -1$
\begin{eqnarray*}
R_{j}(x) &=&x^{j+1}\exp \left( \frac{1}{2}\int_{0}^{x}H(t)\,dt\right) \\
&=&x^{j+1}\sqrt{\frac{v(0)}{v(x)}}.
\end{eqnarray*}
Then (\ref{recurse1}) becomes
\begin{equation}
x^{k+1}\sqrt{\frac{v(0)}{v(x)}}u_{k+1}(x)=-\int_{0}^{x}y^{k}\frac{v(0)}{v(x)}%
\Delta _{B}u_{k}(y)\,dy. \label{recurse2}
\end{equation}
Therefore,
\begin{equation*}
u_{0}(x)=\sqrt{\frac{v(x)}{v(0)}},
\end{equation*}
and
\begin{eqnarray*}
u_{1}(x) &=&\frac{1}{x}\sqrt{\frac{v(x)}{v(0)}}\int_{0}^{x}\sqrt{\frac{v(0)}{%
v(y)}}\left( \left( \sqrt{\frac{v(y)}{v(0)}}\right) ^{\prime \prime
}+H(y)\left( \sqrt{\frac{v(y)}{v(0)}}\right) ^{\prime }\right) \,dy \\
&=&\frac{1}{4x}\sqrt{\frac{v(x)}{v(0)}}\int_{0}^{x}\left( \frac{v^{\prime }}{%
v}\right) ^{2}+2\frac{v^{\prime \prime }}{v}\,dy \\
&=&\frac{1}{4x}\sqrt{\frac{v(x)}{v(0)}}\int_{0}^{x}\left( 2H^{\prime
}(y)+3(H(y))^{2}\right) \,dy.
\end{eqnarray*}
Taking the limit as $x\rightarrow 0$, we obtain
\begin{eqnarray*}
u_{0}(0) &=&1 \\
u_{1}(0) &=&\frac{1}{4}\left( 2H^{\prime }(0)+3(H(0))^{2}\right) .
\end{eqnarray*}
By realizing that the coordinate $0$ was labelled arbitrarily and by
integrating the above quantities over the circle, we obtain the following
theorem:
\begin{theorem}
\label{asympcodim1} Let $(M,\mathcal{F})$ be a transversally oriented
Riemannian foliation of codimension one without dense leaves. Then the trace
$K_{B}(t)$ of the basic heat operator has the following asymptotic expansion
as $t\rightarrow 0$. For any nonnegative integer $J$,
\begin{equation*}
K_{B}(t)=\frac{1}{\sqrt{4\pi t}}\left( A_{0}+A_{1}t+\ldots
+A_{J}t^{J}+O\left( t^{J+1}\right) \right) ,
\end{equation*}
where
\begin{eqnarray*}
A_{0}=V_{tr}, \\
A_{1}=\frac{3}{4}\ell ^{2}\left( \Vert h\Vert _{2}\right) ^{2},
\end{eqnarray*}
and the other basic heat invariants may be computed using the recursion
formulas and integrations described above. Here, $V_{tr}$ is the transverse
volume of the foliation, and $\Vert h\Vert _{2}$ is the $L^{2}$ norm of the
mean curvature.
\end{theorem}
\begin{corollary}
Let $(M,\mathcal{F})$ be as in the theorem above. Then the spectrum of the
basic Laplacian on functions determines the $L^{2}$ norm of the mean
curvature. In particular, the foliation is minimal if and only if $A_{1}=0$
in Theorem~\ref{asympcodim1}.
\end{corollary}
\begin{remark}
The above theorem and corollary may be applied in cases of higher
codimension if all of the leaf closures are transversally oriented and have
codimension one.
\end{remark}
\subsection{Codimension One Foliations That Are Not Transversally Orientable}
\label{nonorientable} We now show how the results in the last section need
to be modified if $\left( M,\mathcal{F}\right) $ is not transversally
orientable. We will need the results of this section when we consider the
case of transversally orientable codimension two foliations whose leaf
closures are codimension one and are not necessarily transversally
orientable. If $\left( M,\mathcal{F}\right) $ is a codimension one
Riemannian foliation that is not transversally orientable, it has a double
cover $\left( \widetilde{M},\widetilde{\mathcal{F}}\right) $ that is
transversally orientable. Basic functions on $M$ correspond to basic
functions on $\widetilde{M}$ that are invariant under the orientation
reversing, isometric involution (the deck transformation). Thus the basic
Laplacian is a second order operator on a closed interval with Neumann
boundary conditions instead of a circle. Part of the analysis from the last
section is relevant, so that we obtain the following:
\begin{eqnarray}
\Delta _{B}\psi (s) &=&-\frac{d^{2}}{ds^{2}}\psi (s)-H(s)\frac{d}{ds}\psi (s)
\notag \\
\psi ^{\prime }(0) &=&\psi ^{\prime }\left( V_{tr}\right) =0. \label{BVP}
\end{eqnarray}
The asymptotics of the trace of the associated heat operator is a standard
problem. If $\widetilde{K}\left( t,\widetilde{s}_{1},\widetilde{s}
_{2}\right) $ is the lifted heat kernel to the circle, it corresponds to the
following differential operator on $\left[ -V_{tr},V_{tr}\right] $ $\ $with
periodic boundary conditions:
\begin{equation*}
L\alpha \left( \widetilde{s}\right) =-\frac{d^{2}}{d\widetilde{s}^{2}}\alpha
(\widetilde{s})-\widetilde{H}(\widetilde{s})\frac{d}{d\widetilde{s}}\alpha (%
\widetilde{s}), \label{liftedBVP}
\end{equation*}
where
\begin{equation*}
\widetilde{H}(\widetilde{s})=\left\{
\begin{array}{ll}
H\left( \widetilde{s}\right) & \text{if \ }0<\widetilde{s}<V_{tr} \\
-H\left( -\widetilde{s}\right) & \text{if \ }-V_{tr}<\widetilde{s}<0%
\end{array}
\right. .
\end{equation*}
Note that $\widetilde{H}(\widetilde{s})$ is the logarithmic derivative of
the volume of the leaf on $\left( \widetilde{M},\widetilde{\mathcal{F}}%
\right) $ and thus extends to be a smooth function on the circle. In
particular, this implies that all even derivatives of $\ \widetilde{H}(%
\widetilde{s})$ and $H(\widetilde{s})$\ at $\widetilde{s}=0$ or $V_{tr}$ are
zero. A similar argument shows that the corresponding volume functions $%
v\left( \widetilde{s}\right) $and $\widetilde{v}\left( \widetilde{s}\right) $
have zero odd derivatives at $\widetilde{s}=0$ or $V_{tr}$. The heat kernel $%
K(t,s_{1},s_{2})$ for the original boundary value problem (\ref{BVP})
satisfies
\begin{equation*}
K(t,s,s)=\widetilde{K}\left( t,s,s\right) +\widetilde{K}\left( t,s,-s\right)
,
\end{equation*}
We have that
\begin{equation*}
\widetilde{K}\left( t,\widetilde{s}_{1},\widetilde{s}_{2}\right) \sim \frac{%
e^{-r^{2}/4t}}{\sqrt{4\pi t}}\left( u_{0}\left( \widetilde{s}_{1},\widetilde{%
s}_{2}\right) +u_{1}\left( \widetilde{s}_{1},\widetilde{s}_{2}\right)
t+\ldots \right) ,
\end{equation*}
where $r=$dist$\left( \widetilde{s}_{1},\widetilde{s}_{2}\right) $, and the
functions $u_{j}$ are explicitly computable from the differential equation (%
\ref{liftedBVP}). The trace is computed by the integral
\begin{eqnarray*}
K_{B}(t) &=&\int_{0}^{V_{tr}}K(t,s,s)\,ds \\
&=&\int_{0}^{V_{tr}}\widetilde{K}\left( t,s,s\right) +\widetilde{K}\left(
t,s,-s\right) \,\,ds \\
&=&\frac{1}{2}\int_{-V_{tr}}^{V_{tr}}\widetilde{K}\left( t,s,s\right) +%
\widetilde{K}\left( t,s,-s\right) \,\,ds \\
&\sim &\frac{1}{\sqrt{4\pi t}}\left(
A_{0}+B_{0}t^{1/2}+A_{1}t+B_{1}t^{3/2}+\ldots \right) ,
\end{eqnarray*}
where $A_{j}$ is defined as in the oriented case, and $B_{j}$ depends on the
derivatives of $u_{j}$ evaluated at $(0,0)$ and $\left( V_{tr},V_{tr}\right)
$. The first nontrivial coefficients in the formula are:
\begin{eqnarray*}
A_{0} &=&\frac{1}{2}\int_{-V_{tr}}^{V_{tr}}u_{0}\left( \widetilde{s},%
\widetilde{s}\right) \,d\widetilde{s} \\
B_{0} &=&\frac{\sqrt{\pi }}{2}\left( u_{0}\left( 0,0\right) +u_{0}\left(
V_{tr},V_{tr}\right) \right) \\
A_{1} &=&\frac{1}{2}\int_{-V_{tr}}^{V_{tr}}u_{1}\left( \widetilde{s},%
\widetilde{s}\right) \,d\widetilde{s} \\
B_{1} &=&\frac{\sqrt{\pi }}{2}\left( u_{1}\left( 0,0\right) +u_{1}\left(
V_{tr},V_{tr}\right) \right) \\
&&+\frac{\sqrt{\pi }}{8}\left( \partial _{1}\partial _{1}-2\partial
_{1}\partial _{2}+\partial _{2}\partial _{2}\right) u_{0}\left( 0,0\right) \\
&&+\frac{\sqrt{\pi }}{8}\left( \partial _{1}\partial _{1}-2\partial
_{1}\partial _{2}+\partial _{2}\partial _{2}\right) u_{0}\left(
V_{tr},V_{tr}\right)
\end{eqnarray*}
From the calculations in the transversally oriented case, we have,
\begin{eqnarray*}
u_{0}(s_{1},s_{2}) &=&\sqrt{\frac{v(s_{1})}{v(s_{2})}} \\
u_{1}(s_{1},s_{2}) &=&\frac{1}{4\left( s_{1}-s_{2}\right) }\sqrt{\frac{%
v(s_{1})}{v(s_{2})}}\int_{s_{2}}^{s_{1}}\left( 2\widetilde{H}^{\prime }(y)+3(%
\widetilde{H}(y))^{2}\right) \,dy,
\end{eqnarray*}
which after some calculation implies that
\begin{eqnarray*}
u_{0}\left( \widetilde{s},\widetilde{s}\right) &=&1 \\
u_{1}\left( \widetilde{s},\widetilde{s}\right) &=&\frac{1}{4}\left( 2%
\widetilde{H}^{\prime }(\widetilde{s})+3(\widetilde{H}(\widetilde{s}%
))^{2}\right) \\
\left( \partial _{1}\partial _{1}-2\partial _{1}\partial _{2}+\partial
_{2}\partial _{2}\right) u_{0}\left( 0,0\right) &=&0 \\
\left( \partial _{1}\partial _{1}-2\partial _{1}\partial _{2}+\partial
_{2}\partial _{2}\right) u_{0}\left( V_{tr},V_{tr}\right) &=&0
\end{eqnarray*}
These equations imply that
\begin{eqnarray*}
A_{0} &=&V_{tr} \\
B_{0} &=&\sqrt{\pi } \\
A_{1} &=&\frac{1}{8}\int_{-V_{tr}}^{V_{tr}}\left( 2\widetilde{H}^{\prime }(%
\widetilde{s})+3(\widetilde{H}(\widetilde{s}))^{2}\right) \,d\widetilde{s} \\
&=&\frac{3}{8}\int_{-V_{tr}}^{V_{tr}}(\widetilde{H}(\widetilde{s}))^{2}\,d%
\widetilde{s}=\frac{3}{4}\int_{0}^{V_{tr}}(H(s))^{2}\,ds \\
B_{1} &=&\frac{\sqrt{\pi }}{8}\left( 2\widetilde{H}^{\prime }(0)+3(%
\widetilde{H}(0))^{2}+2\widetilde{H}^{\prime }(V_{tr})+3(\widetilde{H}%
(V_{tr}))^{2}\right) \\
&=&\frac{\sqrt{\pi }}{4}\left( \widetilde{H}^{\prime }(0)+\widetilde{H}%
^{\prime }(V_{tr})\right) =\frac{\sqrt{\pi }}{4}\left( H^{\prime
}(0)+H^{\prime }(V_{tr})\right) .
\end{eqnarray*}
In summary, we have the following theorem.
\begin{theorem}
\label{codim1nonorientable} Let $(M,\mathcal{F})$ be a Riemannian foliation
of codimension one such that the leaves are not dense and the foliation is
not transversally orientable. Then the trace $K_{B}(t)$ of the basic heat
operator has the following asymptotic expansion as $t\rightarrow 0$.
\begin{equation*}
K_{B}(t)=\frac{1}{\sqrt{4\pi t}}\left(
A_{0}+B_{0}t^{1/2}+A_{1}t+B_{1}t^{3/2}+...\right) ,
\end{equation*}%
where
\begin{eqnarray*}
A_{0} &=&V_{tr}, \\
B_{0} &=&\sqrt{\pi } \\
A_{1} &=&\frac{3}{4}\ell ^{2}\left( \Vert h\Vert _{2}\right) ^{2}, \\
B_{1} &=&\frac{\sqrt{\pi }}{4}\left( F(0)+F(V_{tr})\right)
\end{eqnarray*}%
and the other basic heat invariants may be computed using the techniques
described above. Here, $V_{tr}$ is the transverse volume of the foliation, $%
\Vert h\Vert _{2}$ is the $L^{2}$ norm of the mean curvature, and $%
F(0)+F(V_{tr})$ is the sum of the second normal derivatives of the logarithm
of leaf volume, evaluated at the two leaves with $\mathbb{Z}_{2}$ holonomy
(this quantity is independent of the choice of normal at any point of these
leaves).
\end{theorem}
\begin{corollary}
Let $(M,\mathcal{F})$ a codimension one Riemannian foliation. Then the
spectrum of the basic Laplacian on functions determines whether or not the
leaves are dense. If the leaves are not dense, the spectrum also determines
whether or not the foliation is transversally orientable, the $L^{2}$ norm
of the mean curvature, and the average of the second normal derivatives of
the logarithm of leaf volume at the two leaves with $\mathbb{Z}_{2}$
holonomy in the nonorientable case. In particular, the foliation is minimal
if and only if the coefficient $A_{1}=0$.
\end{corollary}
\begin{remark}
The above theorem and corollary may be applied in cases of higher
codimension if all of the leaf closures have codimension one.
\end{remark}
\subsection{The General Case\label{general}}
We now prove some results that will be applied to the codimension two case
in Section~\ref{codim2}. These results are completely general and may be
used to compute the asymptotic expansion in special cases of arbitrary
codimension. We first review results that will be used in our computations.
Recall that $\widetilde{K}(t,w,v):=\widetilde{K_{W}}(t,w,v)$ is the heat
kernel corresponding to the operator $\widetilde{\Delta _{W}}$ defined in~(%
\ref{twistlap}), so that if $\varepsilon >0$ is sufficiently small, dist$%
(w,v)>\varepsilon $ implies that
\begin{equation*}
\widetilde{K}(t,w,v)=O\left( e^{-c/t}\right)
\end{equation*}%
as $t\rightarrow 0$, for some constant $c$ (see, for example, \cite{Gr}). As
a consequence, the asymptotics of the integral in Proposition~\ref{trbasic1}
over $G\times W$ are the same as the asymptotics of the integral over $U$,
where $U$ is any arbitrarily small neighborhood of the compact subset $%
\{(g,w)\in G\times W\,|\,wg=w\}$, up to an error term of the form $O\left(
e^{-c/t}\right) $.
We will now decompose $W$ into pieces and use this decomposition to
partition a neighborhood of $\{(g,w)\in G\times W\,|\,wg=w\}$. Given an
orbit $X$ of $G$ and $w\in X$, $X$ is naturally diffeomorphic to $G/H_{w}$,
where $H_{w}=\{g\in G\,|wg=w\}$ is the (closed) isotropy subgroup. As we
mentioned before, $H_{w}$ is isomorphic to the structure group corresponding
to the principal bundle $\pi :\rho ^{-1}(w)\rightarrow \overline{L}$, where $%
\overline{L}$ is the leaf closure $\pi \left( \rho ^{-1}(w)\right) $ in $M$.
Given a subgroup $H$ of $G$, let $\left[ H\right] $ denote the conjugacy
class of $H$. The isotropy type of the orbit $X$ is defined to be the
conjugacy class $\left[ H_{w}\right] $, which is well--defined independent
of $w\in X$. There are a finite number of isotropy types of orbits in $W$
(see \cite[p.~173]{Bre}). We define the usual partial ordering (see \cite[%
p.~42]{Bre}) on the isotropy types by declaring that
\begin{equation*}
\left[ H\right] \leq \left[ K\right] \iff H\text{ is conjugate to a subgroup
of }K.
\end{equation*}%
Let $\{\left[ H_{i}\right] :i=1,\dots ,r\}$ be the set of isotropy types
occurring in $W$, arranged so that
\begin{equation*}
\left[ H_{i}\right] \leq \left[ H_{j}\right] \Longrightarrow i\geq j
\end{equation*}%
(see \cite[p. 51]{Kaw}). Let $W\left( \left[ H\right] \right) $ denote the
union of orbits of isotropy type $\left[ H\right] $ in $W$. The set $W\left( %
\left[ H_{i}\right] \right) $ is in general a $G$--invariant submanifold of $%
W$ (see \cite[p.~202]{Kaw}). Also, $W\left( \left[ H_{1}\right] \right) $ is
closed, and $W\left( \left[ H_{r}\right] \right) $ is open and dense in $W$ (%
\cite[p.~50,~216]{Kaw}). Thus, $W$ is the disjoint union of the submanifolds
$W\left( \left[ H_{i}\right] \right) $ for $1\leq i\leq r$.
Now, given a proper, $G$-invariant submanifold $S$ of $W$ and $\varepsilon
>0 $, let $T_{\varepsilon }(S)$ denote the union of the images of the
exponential map at $s$ for $s\in S$ restricted to the ball of radius $%
\varepsilon $ in the normal bundle at $S$. It follows that $T_{\varepsilon
}(S)$ is also $G$-invariant. We now decompose $W$ as a disjoint union of
sets $W_{1},\dots ,W_{r}$. If there is only one isotropy type on $W$, then $%
r=1$, and we let $W_{1}=W$. Otherwise, let $W_{1}=T_{\varepsilon }\left(
W\left( \left[ H_{1}\right] \right) \right) $. For $1<j\leq r-1$, let
\begin{equation*}
{\ W_{j}=T_{\varepsilon }\left( W\left( \left[ H_{j}\right] \right) \right)
\setminus \bigcup_{i=1}^{j-1}W_{i}}.
\end{equation*}%
Let
\begin{equation*}
{\ W_{r}=W\setminus \bigcup_{i=1}^{r-1}W_{i}}.
\end{equation*}%
Clearly, $\varepsilon >0$ must be chosen sufficiently small in order for the
following lemma to be valid. We in addition insist that $\varepsilon $ be
chosen sufficiently small so that the asymptotic expansion for $\widetilde{%
K_{W}}(t,x,y)$ is valid if the distance from $x$ to $y$ is less than $%
\varepsilon $. The following facts about this decomposition are contained in
\cite[pp.~203ff]{Kaw}:
\begin{lemma}
\label{decomposition}With $W$, $W_{i}$ defined as above, we have, for every $%
i\in \{1,\ldots ,r\}$:
\begin{enumerate}
\item $\displaystyle W=\bigcup_{i=1}^{r}W_{i}$.
\item $W_{i}$ is a union of $G$-orbits.
\item The closure of $W_{i}$ is a compact $G$-manifold with corners.
\item If $\left[ H_{j}\right] $ is the isotropy type of an orbit in $W_{i}$,
then $j\geq i$.
\item The distance between the submanifold $W\left( \left[ H_{j}\right]
\right) $ and $W_{i}$ for $j<i$ is at least $\varepsilon $.
\end{enumerate}
\end{lemma}
\begin{remark}
The lemma above remains true if at each stage $T_{\varepsilon }\left(
W\left( \left[ H_{j}\right] \right) \right) $ is replaced by any
sufficiently small open neighborhood of $W(H_{j})$ that contains \linebreak $%
T_{\varepsilon }\left( W\left( \left[ H_{j}\right] \right) \right) $, that
is a union of $G$-orbits, and whose closure is a manifold with corners.
\end{remark}
Therefore, by Proposition~\ref{trbasic1}, the trace of the basic heat kernel
is given by
\begin{equation}
K_{B}(t)=\sum_{i=1}^{r}\int_{W_{i}}\int_{G}\widetilde{K}(t,w,wg)\,\chi (g)\,%
\mathrm{dvol}_{W}(w). \label{tracepiece}
\end{equation}
Let $H_{j}$ be the isotropy subgroup of $w\in W\left( \left[ H_{j}\right]
\right) $, and let $\gamma $ be a geodesic orthogonal to $W\left( \left[
H_{j}\right] \right) $ through $w$. This situation occurs exactly when this
geodesic is orthogonal both to the fixed point set $W^{H_{j}}$ of $H_{j}$
and to the orbit $wG$ of $G$ containing $w$. For any $h\in H_{j}$, right
multiplication by $h$ maps geodesics orthogonal to $W^{H_{j}}$ through $w$
to themselves and likewise maps geodesics orthogonal to $wG$ through $w$ to
themselves. Thus, the group $H_{j}$ acts orthogonally on the normal space to
$w\in W\left( \left[ H_{j}\right] \right) $ by the differential of right
multiplication. Observe in addition that there are no fixed points for this
action; that is, there is no element of the normal space that is fixed by
every $h\in H_{j}$. If $G=SO\left( q\right) $ acts by orientation-preserving
isometries, then $H_{j}$ acts on the normal space in the same way. Since $%
H_{j}$ acts without fixed points, the codimension of $W\left( \left[ H_{j}%
\right] \right) $ is at least two in the orientation-preserving case.
On the other hand, if the transformation group $G$ in question is abelian
and acts by orientation-preserving isometries, then the representation
theory of abelian groups implies that the representation space must be
even-dimensional (see \cite[pp.~107--110]{BrotD}). In this case, we would
then conclude that the codimension of each $W\left( \left[ H_{j}\right]
\right) $ is even. We mention this for the following reason. Since $%
\widetilde{\Delta _{W}}$ commutes with the $SO(q)$ action on the basic
manifold $W$, the integrand $\widetilde{K}(t,w,wg)$ in Proposition~\ref%
{trbasic1} is a class function, so that Weyl's integration formula may be
used to rewrite the integral as an integral over a maximal torus $T$ of $%
SO(q)$ (see \cite[pp.~163]{BrotD}). \ Formula (\ref{tracepiece}) becomes
\begin{equation}
K_{B}(t)=\sum_{i=1}^{r}\int_{W_{i}}\int_{T}\widetilde{K}(t,w,wg)\,\eta (g)\,%
\mathrm{dvol}_{W}(w), \label{toruspiece}
\end{equation}%
where $\eta (g)$ is the volume form on $T$ multiplied by a bounded function
of $g\in T$. In the above expression, we may take $W_{i}$ to be those
constructed using $G=T$. Therefore, each $H_{j}$ is a subgroup of the torus,
and the codimension of each $W\left( \left[ H_{i}\right] \right) $ is even
in the orientation-preserving case, as has been explained previously.
As $t\rightarrow 0$, we need only evaluate the asymptotics of the integrals
in (\ref{toruspiece}) on an arbitrarily small neighborhood of the set $%
\{(g,w)\in T\times W\,|\,wg=w\}$. By the construction of $W_{i}$, the
integral over $T$ may be replaced by an integral over a small neighborhood
of $H_{i}$ in $T$. This neighborhood may be described as $N_{\varepsilon
^{\prime }}\left( H_{i}\right) =\left\{ gh\,|\,h\in H_{i}\,,g\in
B_{\varepsilon ^{\prime }}\right\} $, where $B_{\varepsilon ^{\prime }}$ is
a ball of radius $\varepsilon ^{\prime }$ centered at the identity in $\exp
_{e}H_{i}^{\perp }$. Here, $H_{i}^{\perp }$ is the normal space to $%
H_{i}\subset T$ at the identity $e$, and $\exp _{e}$ is the exponential map $%
\exp _{e}:\mathfrak{t}\rightarrow T$. We have
\begin{equation}
K_{B}(t)=\sum_{i=1}^{r}\int_{W_{i}}\int_{B_{\varepsilon ^{\prime }}\times
H_{i}}\widetilde{K}(t,w,wgh)\,\eta ^{\prime }(g,h)\,\mathrm{dvol}%
_{W}(w)+O\left( e^{-c/t}\right) . \label{torus2}
\end{equation}%
We can now make this integral over the torus explicit. A maximal torus of $%
SO(q)$ has dimension $\left[ \frac{q}{2}\right] $, and we define $T$ in the
following way. Let $\Theta =\left( \theta _{1},\ldots ,\theta _{m}\right)
\in (-\pi ,\pi ]^{m}$. If $q=2m$, let
\begin{equation*}
T=\left\{ \left. M(\Theta )\,\right\vert \,\theta _{j}\in (-\pi ,\pi ]\text{
for every }j\right\} ,
\end{equation*}%
where
\begin{equation*}
M(\Theta )=\left(
\begin{array}{ccccc}
\cos \theta _{1} & \sin \theta _{1} & \ldots & 0 & 0 \\
-\sin \theta _{1} & \cos \theta _{1} & \ldots & 0 & 0 \\
\vdots & \vdots & \ddots & \vdots & \vdots \\
0 & 0 & \ldots & \cos \theta _{m} & \sin \theta _{m} \\
0 & 0 & \ldots & -\sin \theta _{m} & \cos \theta _{m}%
\end{array}%
\right)
\end{equation*}%
If $q=2m+1$, $T$ is defined similarly with
\begin{equation*}
M(\Theta )=\left(
\begin{array}{cccccc}
\cos \theta _{1} & \sin \theta _{1} & \ldots & 0 & 0 & 0 \\
-\sin \theta _{1} & \cos \theta _{1} & \ldots & 0 & 0 & 0 \\
\vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\
0 & 0 & \ldots & \cos \theta _{m} & \sin \theta _{m} & 0 \\
0 & 0 & \ldots & -\sin \theta _{m} & \cos \theta _{m} & 0 \\
0 & 0 & \ldots & 0 & 0 & 1%
\end{array}%
\right) .
\end{equation*}%
Then we have the following formulas (\cite[pp. 171, 219--221]{BrotD}) for
the form $\eta $ in formula (\ref{toruspiece}):
\begin{equation*}
\eta (\Theta )=\frac{1}{m!\,2^{m-1}}\prod \left( 1-e^{\pm i\theta _{k}\pm
i\theta _{l}}\right) \,\cdot \frac{d\Theta }{(2\pi )^{m}}
\end{equation*}%
if $q=2m$, and
\begin{equation*}
\eta (\Theta )=\frac{1}{m!\,2^{m}}\prod \left( 1-e^{\pm i\theta _{k}\pm
i\theta _{l}}\right) \prod \left( 1-e^{\pm i\theta _{j}}\right) \,\cdot
\frac{d\Theta }{(2\pi )^{m}}
\end{equation*}%
if $q=2m+1$. Simplifying, we get
\begin{equation*}
\eta (\Theta )=\frac{2^{m^{2}-3m+1}}{m!\,\pi ^{m}}\left( \prod_{1\leq
k<l\leq m}\left( \cos \theta _{k}-\cos \theta _{l}\right) ^{2}\right)
\,d\Theta
\end{equation*}%
if $q=2m$, and
\begin{equation*}
\eta (\Theta )=\frac{2^{m^{2}-2m}}{m!\,\pi ^{m}}\left( \prod_{1\leq k<l\leq
m}\left( \cos \theta _{k}-\cos \theta _{l}\right) ^{2}\right) \,\left(
\prod_{1\leq j\leq m}\left( 1-\cos \theta _{j}\right) \right) \,d\Theta
\end{equation*}%
if $q=2m+1$. Equation (\ref{toruspiece}) becomes
\begin{equation}
K_{B}(t)=\sum_{i=1}^{r}\int_{W_{i}}\int_{\left( -\pi ,\pi \right) ^{m}}%
\widetilde{K}(t,w,w\,M\left( \Theta \right) )\,\,\eta (\Theta )\,\mathrm{dvol%
}_{W}(w), \label{torustrace}
\end{equation}%
using the appropriate choice of $\eta (\Theta )$\ above. The explicit
description of $\,\eta (\Theta )$\ may be used to make equation (\ref{torus2}%
) more explicit as well, but additional calculations and a choice of
coordinates on $\exp _{e}H_{i}^{\perp }$ are necessary.
\subsection{Codimension Two Riemannian Foliations}
\label{codim2}
We now explicitly derive the coefficients in the asymptotic expansion of the
trace of the basic heat operator on transversally oriented Riemannian
foliations of codimension two. We have the following possibilities:
\begin{enumerate}
\item (The trivial case.) The closure of every leaf of $\left( M,\mathcal{F}%
\right) $ is the manifold $M$. In this case, the basic functions are
constants, and the basic heat operator is the identity. Therefore, the trace
$K_{B}\left( t\right) $ of the basic heat operator satisfies $K_{B}\left(
t\right) =1$ for every $t$.
\item Every leaf of $\left( M,\mathcal{F}\right) $ is closed. Then the
results of Section \ref{finite} apply. The basic manifold $W$ is
three-dimensional. \ Given $w\in W$, the $SO\left( 2\right) $ orbit $X$ of $%
w $ is a circle.
\item Each leaf closure of \ $\left( M,\mathcal{F}\right) $ has codimension
one. If \ $\overline{\mathcal{F}}$ denotes the collection of leaf closures
of $\left( M,\mathcal{F}\right) $, $\left( M,\overline{\mathcal{F}}\right) $
is a Riemannian foliation of codimension one. Observe that the basic
functions, $L^{2}$ inner products, and basic Laplacians for $\left( M,%
\mathcal{F}\right) $ and $\left( M,\overline{\mathcal{F}}\right) $\ are the
same, so that we have reduced to the nontrivial codimension one cases. If
the leaf closure foliation is transversally orientable, see Section~\ref%
{codim1}. If the leaf closure foliation is not transversally orientable, see
Section~\ref{nonorientable}.
\item The leaf closures of \ $\left( M,\mathcal{F}\right) $ have minimum
codimension one, but some leaf closures have codimension two. This situation
is the most interesting case that arises. At least one of the orbits has
finite isotropy. Thus, the basic manifold $W$ is a closed two-manifold with
an \ $SO\left( 2\right) $ action whose orbits have two different dimensions.
The circular orbits correspond to the codimension one leaf closures, and the
(isolated) fixed points correspond to the codimension two leaf closures.
Because the group action yields a vector field on $W$ that has index $1$ at
each fixed point, the Euler characteristic of $W$ must be a positive
integer. Therefore, $W$ is a sphere or a projective plane; for simplicity we
consider only the case where $W=S^{2}$. The metric is a function of the
height (orbit) multiplied by the standard metric on the sphere.\ \ The space
of leaf closures of $\left( M,\mathcal{F}\right) $\ is a closed interval.
This case could be considered as a one-dimensional problem (as in Section~%
\ref{codim1}), but the analysis is quite difficult because the mean
curvature of the leaf closures goes to infinity at the leaves with infinite
holonomy. Instead, we use the approach of Section~\ref{general}. The
circular orbits have orbit type $\left( \left\{ e\right\} \right) $, and the
fixed points have orbit type $\left( SO\left( 2\right) \right) $. Since $%
\left( \left\{ e\right\} \right) \leq \left( SO\left( 2\right) \right) $, we
can decompose $W=W_{1}\cup W_{2}$ as in Lemma~\ref{decomposition}, where $%
W_{1}$ is the union of two metric $\varepsilon $-disks centered at the fixed
points and $W_{2}=W\setminus W_{1}$. The maximal torus of $SO\left( 2\right)
$ is itself, and equation (\ref{torustrace}) shows that the trace of the
basic heat operator is
\begin{eqnarray}
K_{B}(t) &=&\int_{W_{1}}\int_{-\pi }^{\pi }\widetilde{K}(t,w,w\,M\left(
\theta \right) )\,\frac{d\theta }{2\pi }\,\,\,\mathrm{dvol}_{W}(w) \notag \\
&+&\int_{W_{2}}\int_{-\pi }^{\pi }\widetilde{K}(t,w,w\,M\left( \theta
\right) )\,\,\frac{d\theta }{2\pi }\,\,\mathrm{dvol}_{W}(w),
\label{sphereparts}
\end{eqnarray}
where $M\left( \theta \right) =\left(
\begin{array}{cc}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta%
\end{array}
\right) \allowbreak $.\newline
We now state a result from \cite{Ri2}:
\end{enumerate}
\begin{theorem}
(in \cite{Ri2}) \label{oldthm} Let $\psi :M\rightarrow \mathbb{R}$ be the
smooth basic function defined by setting $\psi (x)$ equal to the volume of
any leaf closure of $\widehat{M}$ which intersects the fiber $\pi ^{-1}(x)$.
Let $q_{x}$ denote the codimension of the leaf closure $\overline{L}_{x}$
containing $x$ in $M$. Then, as $t\rightarrow 0$, we have the following
asymptotic expansion for any positive integer $k$:
\begin{equation*}
K_{B}(t,x,x)=\frac{1}{(4\pi t)^{{}^{q_{x}/2}}}\left(
a_{0}(x)+a_{1}(x)t+\ldots +a_{k}(x)t^{k}+O\left( t^{k+1}\right) \,\right) ,
\end{equation*}%
where
\begin{equation*}
a_{i}(x)=\sum \frac{2^{l}K_{m_{1}\ldots m_{l}}^{jl}(x)\,I_{m_{1}\ldots
m_{l}}^{Q\,l}}{\mathrm{Vol}\left( \overline{L}_{x}\right) \pi ^{Q/2}}.
\end{equation*}%
The constants $I_{m_{1}\ldots m_{l}}^{Q\,l}$ are defined by
integrals, and the functions
$K_{m_{1}\ldots m_{l}}^{jl}(x)$ are determined
by the methods in \cite{Ri2}. The integer $Q$ is codimension of the
intersection of the leaf closure containing $\hat{x}$ with $\pi ^{-1}(x)$ in
$\pi ^{-1}(x)$. The functions $a_{i}(x)$ are determined by the local
geometry of the foliation at $x\in M$ and by $\psi (x)$. In particular, $%
a_{0}(x)=\frac{1}{\mathrm{Vol}\left( \overline{L}_{x}\right) }$, and
\begin{equation*}
a_{1}(x)=\frac{1}{\mathrm{Vol}\left( \overline{L}_{x}\right) }\Biggm(\frac{%
S^{W}(\rho (\hat{x}))}{6}+\frac{\Delta _{B}\psi }{2\psi }\left( x\right) +%
\frac{3(d\psi ,d\psi )}{4\psi ^{2}}\left( x\right)
\end{equation*}%
\begin{equation*}
+\frac{1}{6}C_{X}^{12}\mathrm{Ric}^{W}\left( \rho (\hat{x})\right) +\frac{1}{%
12}C^{34}C^{12}T\left( \rho (\hat{x})\right) +\frac{1}{6}C^{24}C^{13}T\left(
\rho (\hat{x})\right) -\frac{S^{X}(\rho (\hat{x}))}{3}\Biggm).
\end{equation*}%
The curvature terms are evaluated at any $\hat{x}\in \pi ^{-1}(x)$ and are
independent of that choice. In the above equation, $S^{X}$ and $S^{W}$
denote the scalar curvatures of $X=\rho \left( \pi ^{-1}(x)\right) $ and $W$%
, respectively, $\mathrm{Ric}^{W}$ is the Ricci curvature tensor on $W$, $%
C^{ab}$ denotes tensor contraction in the $a^{\text{th}}$ and $b^{\text{th}}$
slots, and the subscript $X$ means that the contraction is taken over the
tangent space to $X$ at $\rho (\hat{x})$. The $(0,4)$--tensor $T$ on $X$ is
defined by
\begin{eqnarray*}
T(V,W,Y,Z) &=&\left\langle \nabla _{V}^{\perp }W,\nabla _{Y}^{\perp
}Z\right\rangle ^{\perp },\text{ and } \\
T_{kprt} &=&T\left( \partial _{k},\partial _{p},\partial _{r},\partial
_{t}\right) .
\end{eqnarray*}%
The symbol $\perp $ refers to the orthogonal complement of $T_{\rho (\hat{x}%
)}X$.
\end{theorem}
The asymptotics of the second term in (\ref{sphereparts}) can be found by
integrating the asymptotics in Theorem~\ref{oldthm}, because the
coefficients $a_{j}\left( x\right) $ and error estimates are bounded on the
set $\left\{ x\in M\,|\,\rho \left( \pi ^{-1}\left( x\right) \right) \subset
W_{2}\right\} $. \ We note that all of these coefficient functions arise
from such an integral on $W$. Explicitly, for $w\in W_{2}$,
\begin{eqnarray*}
\int_{-\pi }^{\pi }\widetilde{K}(t,w,w\,M\left( \theta \right) )\,\,\frac{%
d\theta }{2\pi } &=&\frac{1}{\sqrt{4\pi t}}\bigm (a_{0}(w)+a_{1}(w)t \\
&+&\ldots +a_{k}(w)t^{k}+O\left( t^{k+1}\right) \,\bigm),
\end{eqnarray*}
where
\begin{equation*}
a_{0}(w)=\frac{\phi \left( w\right) }{\mathrm{Vol}\left( \pi \left( \rho
^{-1}\left( w\right) \right) \right) }=\frac{1}{\mathrm{Vol}\left(
X_{w}\right) },
\end{equation*}
letting $\phi \left( w\right) $ be the volume of the leaf closure $\rho
^{-1}\left( w\right) $ in $\widehat{M}$ and letting $X=X_{w}$ be the orbit
of $w$ in $W$. Observe that we have also used (\ref{metricw}) to convert the
integral over $M$ to an integral over $W$. Similarly, $a_{1}(w)$ is
obtainable from the expression for $a_{1}$ in the theorem. Since the orbits
are one-dimensional and $W$ is two-dimensional,
\begin{eqnarray*}
S^{X}\left( w\right) &=&0\text{ } \\
S^{W}\left( w\right) &=&2K\left( w\right) \text{ } \\
C_{X}^{12}\mathrm{Ric}^{W}(w) &=&\mathrm{Ric}^{W}\left( \sigma _{w},\sigma
_{w}\right) =K\left( w\right) \\
C^{24}C^{13}T(w) &=&C^{34}C^{12}T(w)=\left\Vert \mathbf{H}(w)\right\Vert
^{2}=\kappa \left( w\right) ^{2},
\end{eqnarray*}
where $K\left( w\right) $ is the Gauss curvature of the basic manifold at $w$
and $\kappa \left( w\right) $ is the geodesic curvature of the orbit at $w$.
\ Also, if $x\in \pi \left( \rho ^{-1}\left( w\right) \right) $ and $%
\widehat{x}\in \pi ^{-1}\left( x\right) $,
\begin{eqnarray*}
\frac{\Delta _{B}\psi }{2\psi }\left( x\right) +\frac{3(d\psi ,d\psi )}{%
4\psi ^{2}}\left( x\right) &=&\frac{\pi ^{\ast }\Delta _{B}\psi }{2\pi
^{\ast }\psi }\left( \widehat{x}\right) +\frac{3(\pi ^{\ast }d\psi ,\pi
^{\ast }d\psi )}{4\pi ^{\ast }\psi ^{2}}\left( \widehat{x}\right) \\
&=&\frac{\widehat{\Delta _{B}}\pi ^{\ast }\psi }{2\pi ^{\ast }\psi }\left(
\widehat{x}\right) +\frac{3(d\pi ^{\ast }\psi ,d\pi ^{\ast }\psi )}{4\pi
^{\ast }\psi ^{2}}\left( \widehat{x}\right) \\
&=&\frac{\widehat{\Delta _{B}}\rho ^{\ast }\phi }{2\rho ^{\ast }\phi }\left(
\widehat{x}\right) +\frac{3(d\rho ^{\ast }\phi ,d\rho ^{\ast }\phi )}{4\rho
^{\ast }\phi ^{2}}\left( \widehat{x}\right) \\
&=&\frac{\rho ^{\ast }\widetilde{\Delta _{W}}\phi }{2\rho ^{\ast }\phi }%
\left( \widehat{x}\right) +\frac{3(d\rho ^{\ast }\phi ,d\rho ^{\ast }\phi )}{%
4\rho ^{\ast }\phi ^{2}}\left( \widehat{x}\right) \\
&=&\frac{\Delta _{W}\phi }{2\phi }\left( w\right) +\frac{(d\phi ,d\phi )}{%
4\phi ^{2}}\left( w\right) ,
\end{eqnarray*}
using the definition of $\widetilde{\Delta _{W}}$ (see Section~\ref{setup}).
\ Therefore, we now obtain
\begin{equation*}
a_{1}(w)=\frac{1}{\mathrm{Vol}\left( X_{w}\right) }\left( \frac{1}{2}K\left(
w\right) +\frac{1}{4}\kappa \left( w\right) ^{2}+\frac{\Delta _{W}\phi }{%
2\phi }\left( w\right) +\frac{(d\phi ,d\phi )}{4\phi ^{2}}\left( w\right)
\right) ,
\end{equation*}
and
\begin{eqnarray*}
\int_{W_{2}}\int_{-\pi }^{\pi }\widetilde{K}(t,w,w\,M\left( \theta \right)
)\,\,\frac{d\theta }{2\pi }\,\,\mathrm{dvol}_{W}(w) &=& \\
\frac{1}{\sqrt{4\pi t}}\int_{W_{2}}\bigm(a_{0}(w)+a_{1}(w)t &+&O\left(
t^{2}\right) \,\bigm)\mathrm{dvol}_{W}(w).
\end{eqnarray*}
In the expression above, note that we could take the limit as $\varepsilon
\rightarrow 0$ (that is, as $W_{2}\rightarrow W$) in every term except the
one involving the geodesic curvature $\kappa \left( w\right) $.
\begin{enumerate}
\item Next, we determine the asymptotics of \ the integral over $W_{1}$ in (%
\ref{sphereparts}). Recall that $W_{1}$ is the disjoint union of two metric $%
\varepsilon $-disks $D_{1}$ and $D_{2}$ surrounding the singular orbits. \
Choosing geodesic polar coordinates $\left( r,\gamma \right) $ around one of
the singular orbits, let $C\left( r\right) $ denote the length of the orbit
at radius $r$. Then $C\left( r\right) =2\pi r\left( 1+O\left( r^{2}\right)
\right) $. The metric on the basic manifold is
\begin{equation*}
\left( g_{ij}\left( r,\gamma \right) \right) =\left(
\begin{array}{cc}
1 & 0 \\
0 & \frac{C\left( r\right) ^{2}}{4\pi ^{2}}%
\end{array}
\right) ,
\end{equation*}
where $g_{11}\left( r,\gamma \right) =\left\langle \partial _{r},\partial
_{r}\right\rangle $ and so on. The integral over $D_{1}$ is
\begin{eqnarray*}
\int_{^{D_{1}}}\int_{-\pi }^{\pi } &\ &\widetilde{K}(t,w,w\,M\left( \theta
\right) )\,\,\frac{d\theta }{2\pi }\,\,\mathrm{dvol}_{W}(w) \\
&=&\int_{r=0}^{\varepsilon }\int_{\gamma =-\pi }^{\pi }\int_{-\pi }^{\pi }%
\widetilde{K}(t,\left( r,\gamma \right) ,\left( r,\gamma \right) \,M\left(
\theta \right) )\,\,\frac{d\theta }{2\pi }\,\,d\gamma \frac{C\left( r\right)
}{2\pi }\,dr \\
&=&\int_{r=0}^{\varepsilon }\int_{0}^{\pi }\widetilde{K}(t,\left( r,0\right)
,\left( r,0\right) \,M\left( \theta \right) )\,\,d\theta \,\frac{C\left(
r\right) }{\pi }\,dr,
\end{eqnarray*}
since the integrand is independent of $\gamma $ because of the isometric
action of $M\left( \theta \right) $. The Minakshisundaram-Pleijel expansion
of $\widetilde{K}(t,\left( r,0\right) ,\left( r,0\right) \,M\left( \theta
\right) )$ is of the form
\begin{eqnarray*}
&\ &\widetilde{K}(t,\left( r,0\right) ,\left( r,0\right) \,M\left( \theta
\right) ) \\
&=&\frac{1}{4\pi t}e^{-D^{2}\left( r,\theta \right) /4t}\left( u_{0}\left(
r,\theta \right) +u_{1}\left( r,\theta \right) t+u_{2}\left( r,\theta
\right) t^{2}+O\left( t^{3}\right) \right) ,
\end{eqnarray*}
where $D\left( r,\theta \right) $ is geodesic distance between $\left(
r,0\right) $ and $\left( r,0\right) \,M\left( \theta \right) $. Each $%
u_{j}\left( r,\theta \right) $ is a smooth, bounded function on the $%
\varepsilon $-disk. Next, observe that $D\left( r,0\right) =0$, $D\left(
r,\pm \pi \right) =2r$, and $D^{2}\left( r,\theta \right) =r^{2}\left(
\left( 1-\cos \theta \right) ^{2}+\sin ^{2}\theta \right) +O\left(
r^{3}\right) =\allowbreak 2r^{2}\left( 1-\cos \theta \right) +O\left(
r^{3}\right) $ for small $r$. Using properties of the exponential map, one
could easily show that $D\left( r,\theta \right) $ increases in $\theta $ on
$\left[ 0,\pi \right] $. Hence, after changing coordinates in $\theta $
(affecting the $u_{j}\left( r,\theta \right) $ by $O\left( r\right) $), we
have that
\begin{eqnarray}
&\ &\int_{D_{1}}\int_{-\pi }^{\pi }\widetilde{K}(t,w,w\,M\left( \theta
\right) )\,\,\frac{d\theta }{2\pi }\,\,\mathrm{dvol}_{W}(w) \notag \\
&=&\int_{r=0}^{\varepsilon }\int_{0}^{\pi }\widetilde{K}(t,\left( r,0\right)
,\left( r,0\right) \,M\left( \theta \right) )\,\,d\theta \,\frac{C\left(
r\right) }{\pi }\,dr \notag \\
&=&\frac{1}{4\pi t}\int_{r=0}^{\varepsilon }\int_{0}^{\pi }e^{-\frac{r^{2}}{%
2t}\left( 1-\cos \theta \right) }\left( \overline{u}_{0}\left( r,\theta
\right) +\overline{u}_{1}\left( r,\theta \right) t+O\left( t^{2}\right)
\right) \,d\theta \,\frac{C\left( r\right) }{\pi }\,dr \notag \\
&=&\frac{1}{4\pi t}\int_{r=0}^{\varepsilon }\int_{0}^{\pi }e^{-\frac{r^{2}}{%
2t}\left( 1-\cos \theta \right) }\left( b_{0}\left( r,\theta \right)
+b_{1}\left( r,\theta \right) t+O\left( t^{2}\right) \right) \,d\theta
\,2r\,dr, \label{easympt}
\end{eqnarray}
where by symmetry we may assume that $\left. \frac{\partial ^{j}}{\partial
r^{j}}b_{m}\left( r,\theta \right) \right\vert _{r=0}=0$ for $j$ odd.
We now compute the asymptotics of some integrals that will be used our
calculation. Let
\begin{equation*}
I_{1}\left( t,\varepsilon \right) =\int_{r=0}^{\varepsilon }\int_{0}^{\pi
}e^{-\frac{r^{2}}{2t}\left( 1-\cos \theta \right) }\,d\theta \,2r\,dr.
\end{equation*}
By Tonelli's theorem we change the order of integration, and we then
integrate to get
\begin{equation*}
I_{1}\left( t,\varepsilon \right) =2\varepsilon ^{2}s\int_{0}^{\pi }\frac{%
\left( 1-\,e^{-\frac{1}{2s}\left( 1-\cos \theta \right) }\right) }{\left(
1-\cos \theta \right) }\,\,d\theta ,
\end{equation*}
letting $s=\frac{t}{\varepsilon ^{2}}$. By changing variables by $\cos
\theta =\frac{x-s}{x+s}=1-\frac{2s}{x+s}$, we obtain
\begin{eqnarray*}
I_{1}\left( t,\varepsilon \right) &=&2\varepsilon ^{2}s\int_{0}^{\infty }%
\frac{\left( 1-\,e^{-\frac{1}{x+s}}\right) }{\frac{2s}{x+s}}\,\,\frac{\sqrt{s%
}\,dx}{\sqrt{x}\left( x+s\right) } \\
&=&\varepsilon ^{2}\sqrt{s}\int_{0}^{\infty }\frac{\left( 1-\,e^{-\frac{1}{%
x+s}}\right) }{\sqrt{x}}\,\,dx \\
&=&\varepsilon ^{2}\sqrt{s}\,F\left( s\right) .
\end{eqnarray*}
Using a Lebesgue convergence theorem argument, it can be shown that the
integral $F\left( s\right) $ is smooth in $s$ and can be differentiated
under the integral sign. In fact, it can be shown that
\begin{equation*}
F\left( s\right) =\int_{0}^{\infty }\frac{\left( 1-\,e^{-\frac{1}{x+s}%
}\right) }{\sqrt{x}}\,\,dx=\allowbreak \left( 2\sqrt{\pi }\right) +\left( -%
\frac{1}{2}\sqrt{\pi }\right) s+O\left( s^{2}\right) \allowbreak ,
\end{equation*}
so that
\begin{eqnarray}
I_{1}\left( t,\varepsilon \right) &=&\varepsilon ^{2}\sqrt{\frac{t}{%
\varepsilon ^{2}}}\,F\left( \frac{t}{\varepsilon ^{2}}\right) =\varepsilon
\sqrt{t}\,F\left( \frac{t}{\varepsilon ^{2}}\right) \notag \\
&=&\varepsilon \sqrt{4\pi t}\left( 1-\frac{1}{4}\left( \frac{t}{\varepsilon
^{2}}\right) +O\left( \left( \frac{t}{\varepsilon ^{2}}\right) ^{2}\right)
\right) . \label{I1}
\end{eqnarray}
Next, for any smooth, bounded function $h$ on the $\varepsilon $-disk,
consider the integral
\begin{equation*}
I_{2}\left( t,\varepsilon ,h\right) =\int_{r=0}^{\varepsilon }\int_{0}^{\pi
}e^{-\frac{r^{2}}{2t}\left( 1-\cos \theta \right) }r^{2}h\left( r,\theta
\right) \,d\theta \,2r\,dr.
\end{equation*}
Replacing $r$ with $\varepsilon r$, we obtain
\begin{equation*}
I_{2}\left( t,\varepsilon ,h\right) =\varepsilon
^{4}\int_{r=0}^{1}\int_{0}^{\pi }e^{-\frac{\varepsilon ^{2}r^{2}}{2t}\left(
1-\cos \theta \right) }r^{2}h\left( \varepsilon r,\theta \right) \,d\theta
\,2r\,dr.
\end{equation*}
As before, we let $s=\frac{t}{\varepsilon ^{2}}$, and we substitute $\cos
\theta =\frac{x-s}{x+s}=1-\frac{2s}{x+s}$.
\begin{eqnarray*}
I_{2}\left( t,\varepsilon ,h\right) &=&\varepsilon
^{4}\int_{r=0}^{1}\int_{0}^{\infty }e^{-\frac{r^{2}}{x+s}}r^{2}h\left(
\varepsilon r,\theta \right) \,\frac{\sqrt{s}\,dx}{\sqrt{x}\left( x+s\right)
}\,2r\,dr \\
&=&\varepsilon ^{4}\sqrt{s}\int_{0}^{\infty }\int_{r=0}^{1}e^{-\frac{r^{2}}{%
x+s}}h\left( \varepsilon r,\theta \right) \,\,2r^{3}\,dr\,\frac{\,dx}{\sqrt{x%
}\left( x+s\right) } \\
&=&\varepsilon ^{4}\sqrt{s}\,G\left( s,\varepsilon \right) ,
\end{eqnarray*}
where we have used the fact that the integral converges absolutely and
Fubini's theorem. Again, using a Lebesgue convergence theorem argument, it
can be shown that the integral $G\left( s,\varepsilon \right) $ is smooth in
$s$ and can be differentiated under the integral sign. Moreover, for any
positive integer $m$,
\begin{equation*}
G\left( s,\varepsilon \right) =c_{0}\left( \varepsilon \right) +c_{1}\left(
\varepsilon \right) s+c_{2}\left( \varepsilon \right) s^{2}+...+c_{m}\left(
\varepsilon \right) s^{m}+O\left( s^{m+1}\right) ,
\end{equation*}
where each $c_{j}\left( \varepsilon \right) $ remains bounded as $%
\varepsilon \rightarrow 0$. Therefore,
\begin{eqnarray}
I_{2}\left( t,\varepsilon ,h\right) &=&\varepsilon ^{4}\sqrt{\frac{t}{%
\varepsilon ^{2}}}\,G\left( \frac{t}{\varepsilon ^{2}},\varepsilon \right)
=\varepsilon ^{3}\sqrt{t}\,G\left( \frac{t}{\varepsilon ^{2}},\varepsilon
\right) \notag \\
&=&\varepsilon ^{3}\sqrt{4\pi t}\left( \overline{c}_{0}\left( \varepsilon
\right) +\overline{c}_{1}\left( \varepsilon \right) \frac{t}{\varepsilon ^{2}%
}+O\left( \left( \frac{t}{\varepsilon ^{2}}\right) ^{2}\right) \right) ,
\label{I2}
\end{eqnarray}
where $\overline{c}_{0}\left( \varepsilon \right) $ and $\overline{c}%
_{1}\left( \varepsilon \right) $ are bounded as $\varepsilon \rightarrow 0$.
We will now compute the asymptotics of the integral over $D_{1}$ in equation
(\ref{easympt}). We have that
\begin{eqnarray*}
&\ &\int_{D_{1}}\int_{-\pi }^{\pi }\widetilde{K}(t,w,w\,M\left( \theta
\right) )\,\,\frac{d\theta }{2\pi }\,\,\mathrm{dvol}_{W}(w) \\
&=&\frac{1}{4\pi t}\int_{r=0}^{\varepsilon }\int_{0}^{\pi }e^{-\frac{r^{2}}{%
2t}\left( 1-\cos \theta \right) }\left( b_{0}\left( r,\theta \right)
+b_{1}\left( r,\theta \right) t+O\left( t^{2}\right) \right) \,d\theta
\,2r\,dr \\
&=&\frac{1}{4\pi t}\int_{r=0}^{\varepsilon }\int_{0}^{\pi }e^{-\frac{r^{2}}{%
2t}\left( 1-\cos \theta \right) }\left( b_{0}+O\left( r^{2}\right)
+b_{1}t+O\left( r^{2}t\right) +O\left( t^{2}\right) \right) \,d\theta
\,2r\,dr \\
&=&\frac{1}{4\pi t}\left( I_{1}\left( t,\varepsilon \right) \left(
b_{0}+b_{1}t\right) +I_{2}\left( t,\varepsilon ,h_{0}\right) +I_{2}\left(
t,\varepsilon ,h_{1}\right) t+O\left( I_{2}\left( t,\varepsilon
,h_{2}\right) t^{2}\right) \right)
\end{eqnarray*}
for some appropriate choices of smooth, bounded functions $h_{0}$, $h_{1}$,
and $h_{2}$. Note that we have denoted $b_{j}=\left. b_{j}\left( r,\theta
\right) \right\vert _{r=0}$. Substituting the expressions (\ref{I1}) and (%
\ref{I2}), \ we get
\begin{eqnarray*}
&\ &\int_{D_{1}}\int_{-\pi }^{\pi }\widetilde{K}(t,w,w\,M\left( \theta
\right) )\,\,\frac{d\theta }{2\pi }\,\,\mathrm{dvol}_{W}(w) \\
&=&\frac{1}{\sqrt{4\pi t}}\biggm(\left( \varepsilon b_{0}+O\left(
\varepsilon ^{3}\right) \right) +\bigm(-\frac{b_{0}}{4\varepsilon }%
+\varepsilon b_{1} \\
&\ &+O\left( \varepsilon \right) \bigm)t+O\left( t^{2}\right) \biggm).
\end{eqnarray*}
We observe that by the above construction starting with the asymptotic
expansion of $\widetilde{K}(t,\left( r,0\right) ,\left( r,0\right) \,M\left(
\theta \right) )$, we have $b_{0}=1$. This implies that
\begin{eqnarray*}
&\ &\int_{D_{1}}\int_{-\pi }^{\pi }\widetilde{K}(t,w,w\,M\left( \theta
\right) )\,\,\frac{d\theta }{2\pi }\,\,\mathrm{dvol}_{W}(w) \\
&=&\frac{1}{\sqrt{4\pi t}}\left( O\left( \varepsilon \right) +\left( -\frac{1%
}{4\varepsilon }+O\left( \varepsilon \right) \right) t+O\left( t^{2}\right)
\right) .
\end{eqnarray*}
We have a similar formula for the asymptotics of the integral over the other
$\varepsilon $-disk $D_{2}$. Thus, the integral over $W_{1}$ satisfies
\begin{eqnarray*}
&\ &\int_{W_{1}}\int_{-\pi }^{\pi }\widetilde{K}(t,w,w\,M\left( \theta
\right) )\,\,\frac{d\theta }{2\pi }\,\,\mathrm{dvol}_{W}(w) \\
&=&\frac{1}{\sqrt{4\pi t}}\left( O\left( \varepsilon \right) +\left( -\frac{1%
}{2\varepsilon }+O\left( \varepsilon \right) \right) t+O\left( t^{2}\right)
\right)
\end{eqnarray*}
A simple calculation shows that the $-\frac{1}{2\varepsilon }t$ term exactly
counteracts the blowing up of the term $a_{1}\left( w\right) t$ in (\ref%
{easympt}) as $\varepsilon \rightarrow 0$, as expected. We remark that the
analysis in this section could be extended to find the coefficients of
larger powers of $t$ in the obvious way. Putting these results together, we
have the following theorem.
\begin{theorem}
\label{2d1d}Suppose that $(M,\mathcal{F})$ is a Riemannian foliation of
codimension two, and suppose that the leaf closures of \ $\left( M,\mathcal{F%
}\right) $ have minimum codimension one, but some leaf closures have
codimension two. Then the basic manifold $W$ is a sphere, and the $SO\left(
2\right) $-action has exactly two fixed points $w_{1}$ and $w_{2}$. \ Let $%
W_{\varepsilon }=\left\{ w\in W\,|\,\text{dist}\left( w,w_{j}\right)
>\varepsilon \text{ for }j=1,2\right\} $. As $t\rightarrow 0$, the trace $%
K_{B}(t)$ of the basic heat kernel on functions satisfies the following
asymptotic expansion for any positive integer $J$:
\begin{equation*}
K_{B}(t)=\frac{1}{\sqrt{4\pi t}}\left( A_{0}+A_{1}t+A_{2}t^{2}+\ldots
+A_{J}t^{J}+O\left( t^{{J+1}}\right) \right) ,
\end{equation*}
where
\begin{equation*}
A_{0}=\int_{M}\frac{1}{\mathrm{Vol}\left( \overline{L_{x}}\right) }\,{%
\mathrm{dvol}}(x)
\end{equation*}
and in general $A_{j}=\lim_{\varepsilon \rightarrow 0}\left(
\int_{W_{\varepsilon }}\overline{a}_{j}(w)\,\mathrm{dvol}(w)-f_{j}\left(
\frac{1}{\varepsilon }\right) \right) $ for a specific polynomial $f_{j}$
and where $\overline{a}_{j}(w)=\phi \left( w\right) \,a_{j}\left( \pi \left(
\rho ^{-1}\left( w\right) \right) \right) $ with $a_{j}$ as in Theorem~\ref%
{oldthm}. As before, $\overline{L_{x}}$ denotes the leaf closure containing $%
x\in M$. Specifically,
\begin{equation*}
A_{1}=\lim_{\varepsilon \rightarrow 0}\left( \int_{W_{\varepsilon }}\frac{1}{%
\mathrm{Vol}\left( X_{w}\right) }S(w)\,\mathrm{dvol}(w)-\frac{1}{%
2\varepsilon }\right)
\end{equation*}
for
\begin{equation*}
S(w)=\frac{1}{2}K\left( w\right) +\frac{1}{4}\kappa \left( w\right) ^{2}+%
\frac{\Delta _{W}\phi }{2\phi }\left( w\right) +\frac{(d\phi ,d\phi )}{4\phi
^{2}}\left( w\right) ,
\end{equation*}
where $\phi \left( w\right) $ is the volume of the leaf closure $\rho
^{-1}\left( w\right) $ in $\widehat{M}$ , $X_{w}$ is the orbit of $w$ in $W$%
, $K\left( w\right) $ is the Gauss curvature of the basic manifold at $w$,
and $\kappa \left( w\right) $ is the geodesic curvature of the orbit at $w$.
\end{theorem}
\end{enumerate}
\section{Examples}
\label{example}
In this section we compute two specific examples that demonstrate the
behavior described in the last section. The first example is a transversally
oriented, codimension two Riemannian foliation in which some of the leaf
closures have codimension one and others have codimension two.
\begin{example}
\label{sphereexample} Consider the one-dimensional foliation obtained by
suspending an irrational rotation on the standard unit sphere $S^{2}$. On $%
S^{2}$ we use the cylindrical coordinates $\left( z,\theta \right) $,
related to the standard rectangular coordinates by $x^{\prime }=\sqrt{\left(
1-z^{2}\right) }\cos \theta $, $y^{\prime }=\sqrt{\left( 1-z^{2}\right) }
\sin \theta $, $z^{\prime }=z$. Let $\alpha $ be an irrational multiple of $%
2\pi $, and let the three-manifold $M=S^{2}\times \left[ 0,1\right] /\sim $,
where $\left( z,\theta ,0\right) \sim \left( z,\theta +\alpha ,1\right) $. \
Endow $M$ with the product metric on $T_{z,\theta ,t}M\cong T_{z,\theta
}S^{2}\times T_{t}\mathbb{R}$. Let the foliation $\mathcal{F}$ be defined by
the immersed submanifolds $L_{z,\theta }=\cup _{n\in \mathbb{Z}}\left\{
z\right\} \times \left\{ \theta +\alpha \right\} \times \left[ 0,1\right] $
(not unique in $\theta $). The leaf closures $\overline{L}_{z}$ for $|z|<1$
are two-dimensional, and the closures corresponding to the poles ($z=\pm 1$)
are one-dimensional. Therefore, this foliation satisfies the hypothesis of
Theorem~\ref{2d1d}.
In \cite{Ri2}, we used this example to demonstrate the asymptotic behavior
of \linebreak
$K_{B}\left( t,z,z\right) $ for different values of $z$. We state the
results of some of the computations in \cite{Ri2}. The basic functions are
functions of $\ z$ alone, and the basic Laplacian on functions is $\Delta
_{B}=-\left( 1-z^{2}\right) \,\partial _{z}^{2}+2z\,\partial _{z}$. \ The
volume form on $M$ is $dz\,d\theta \,dt$, and the volume of the leaf closure
at $\ z$ is $\frac{1}{2\pi \sqrt{1-z^{2}}}$ for $|z|<1$. The eigenfunctions
are the Legendre polynomials $P_{n}\left( z\right) $ corresponding to
eigenvalues $m\left( m+1\right) $ for $m\geq 0$. From this information
alone, one may calculate that the trace $K_{B}\left( t\right) $ of the basic
heat operator is
\begin{equation}
K_{B}\left( t\right) =\sum_{m\geq 0}e^{-m\left( m+1\right) t}=\frac{1}{\sqrt{%
4\pi t}}\left( \pi +\frac{\pi }{4}t+O\left( t^{2}\right) \right) .
\label{tracesuspense}
\end{equation}
The basic manifold $W$ corresponding to this foliation is a sphere with
points described by orthogonal coordinates $\left( z,\varphi \right) \in
\lbrack -1,1]\times \left( -\pi ,\pi \right] $. As shown in \cite{Ri2}, the
metric on $W$ is given by $\left\langle \partial _{z},\partial
_{z}\right\rangle =\frac{1}{1-z^{2}},\,\,\left\langle \partial _{\varphi
},\partial _{\varphi }\right\rangle =\frac{4\pi ^{2}\left( 1-z^{2}\right) }{%
4\pi ^{2}\left( 1-z^{2}\right) +z^{2}}$. The following geometric quantities
can be calculated from this data (see \cite{Ri2}):
\begin{eqnarray*}
\mathrm{dvol}_{z,\varphi } &=&\frac{2\pi }{\sqrt{4\pi ^{2}\left(
1-z^{2}\right) +z^{2}}}\,dz\,d\varphi \\
\mathrm{Vol}\left( X_{z}\right) &=&\frac{4\pi ^{2}\sqrt{\left(
1-z^{2}\right) }}{\sqrt{4\pi ^{2}\left( 1-z^{2}\right) +z^{2}}} \\
K\left( z,\varphi \right) &=&2\frac{\left( 4\pi ^{2}-1\right) z^{2}+2\pi ^{2}%
}{\left( 4\pi ^{2}\left( 1-z^{2}\right) +z^{2}\right) ^{2}} \\
\frac{\Delta _{W}\phi }{2\phi }\left( z,\varphi \right) +\frac{(d\phi ,d\phi
)}{4\phi ^{2}}\left( z,\varphi \right) &=&\frac{\left( 4\pi ^{2}-1\right)
\left( \left( -3\pi ^{2}-\frac{3}{4}\right) z^{2}+\left( \pi ^{2}-\frac{1}{4}%
\right) z^{4}+2\pi ^{2}\right) }{\left( 4\pi ^{2}\left( 1-z^{2}\right)
+z^{2}\right) ^{2}} \\
\kappa \left( z,\varphi \right) ^{2} &=&\frac{z^{2}}{\left( 1-z^{2}\right)
\left( 4\pi ^{2}\left( 1-z^{2}\right) +z^{2}\right) ^{2}} \\
\varepsilon &=&\text{dist}\left( z=1,z=1-\varepsilon ^{\prime }\right) =%
\frac{\pi }{2}-\arcsin \left( 1-\varepsilon ^{\prime }\right)
\end{eqnarray*}
Using these computations, we may now compute $A_{0}$ and $A_{1}$ in Theorem~%
\ref{2d1d}.
\begin{equation*}
A_{0}=\int_{M}\frac{1}{\mathrm{Vol}\left( \overline{L_{z}}\right) }\,{%
\mathrm{dvol}}(z,\theta ,t)=\int_{t=0}^{1}\int_{\theta =-\pi }^{\pi
}\int_{z=-1}^{1}\frac{1}{2\pi \sqrt{1-z^{2}}}\,dz\,d\theta \,dt=\pi .
\end{equation*}
Next, we substitute the functions into the formula for $A_{1}$ and simplify:
\begin{eqnarray*}
A_{1} &=&\lim_{\varepsilon \rightarrow 0}\left( \int_{W_{\varepsilon }}\frac{%
1}{\mathrm{Vol}\left( X_{w}\right) }S(w)\,\mathrm{dvol}(w)-\frac{1}{%
2\varepsilon }\right) \\
&=&\lim_{\varepsilon ^{\prime }\rightarrow 0}\left( \int_{\varphi =-\pi
}^{\pi }\int_{z=-1+\varepsilon ^{\prime }}^{1-\varepsilon ^{\prime }}\frac{1%
}{8\pi }\frac{2-z^{2}}{\left( 1-z^{2}\right) ^{3/2}}\,\,dz\,d\varphi -\frac{1%
}{2\left( \frac{\pi }{2}-\arcsin \left( 1-\varepsilon ^{\prime }\right)
\right) }\right) \\
&=&\lim_{\varepsilon ^{\prime }\rightarrow 0}\left( \int_{z=-1+\varepsilon
^{\prime }}^{1-\varepsilon ^{\prime }}\,\,\frac{1}{4}\frac{2-z^{2}}{\left(
1-z^{2}\right) ^{3/2}}\,dz\,-\frac{1}{2\left( \frac{\pi }{2}-\arcsin \left(
1-\varepsilon ^{\prime }\right) \right) }\right) \\
&=&\lim_{\varepsilon ^{\prime }\rightarrow 0}\left( \,\,\allowbreak \left.
\frac{1}{4}\left( \frac{z}{\sqrt{\left( 1-z^{2}\right) }}+\arcsin z\right)
\right\vert _{-1+\varepsilon ^{\prime }}^{1-\varepsilon ^{\prime }}\,-\frac{1%
}{2\left( \frac{\pi }{2}-\arcsin \left( 1-\varepsilon ^{\prime }\right)
\right) }\right) \\
&=&\lim_{\varepsilon ^{\prime }\rightarrow 0}\Bigm(\,\,\allowbreak \frac{1}{2%
}\left( \frac{1-\varepsilon ^{\prime }}{\sqrt{\left( 1-\left( 1-\varepsilon
^{\prime }\right) ^{2}\right) }}+\arcsin \left( 1-\varepsilon ^{\prime
}\right) \right) \\
&\ &-\frac{1}{2\left( \frac{\pi }{2}-\arcsin \left( 1-\varepsilon ^{\prime
}\right) \right) }\Bigm) \\
&=&\lim_{\varepsilon ^{\prime }\rightarrow 0}\left( \frac{\pi }{4}+O\left(
\sqrt{\varepsilon ^{\prime }}\right) \right) =\frac{\pi }{4}.
\end{eqnarray*}
Therefore, Theorem~\ref{2d1d} implies that
\begin{equation*}
K_{B}\left( t\right) =\frac{1}{\sqrt{4\pi t}}\left( \pi +\frac{\pi }{4}%
t+O\left( t^{2}\right) \right) ,
\end{equation*}
which agrees with the direct calculation (\ref{tracesuspense}).
\end{example}
The next example is an example of a codimension two, transversally oriented
Riemannian foliation such that not all of the leaf closures are
transversally orientable.
\begin{example}
\label{exnonorientable} This foliation is the suspension of an irrational
rotation of the flat torus and a $\mathbb{Z}_{2}$-action. Let $X$ be any
closed Riemannian manifold such that $\pi _{1}(X)=\mathbb{Z}\ast \mathbb{Z}$
--- the free group on two generators $\{\alpha ,\beta \}$. We normalize the
volume of $X$ to be 1. Let $\widetilde{X}$ be the universal cover. We define
$M=\widetilde{X}\times S^{1}\times S^{1}\slash \pi _{1}(X)$, where $\pi
_{1}(X)$ acts by deck transformations on $\widetilde{X}$ and by $\alpha
\left( \theta ,\phi \right) =\left( 2\pi -\theta ,2\pi -\phi \right) $ and $%
\beta \left( \theta ,\phi \right) =\left( \theta ,\phi +\sqrt{2}\pi \right) $
on $S^{1}\times S^{1}$. We use the standard product-type metric. The leaves
of $\mathcal{F}$ are defined to be sets of the form $\left\{ (x,\theta ,\phi
)_{\sim }\,|\,x\in \widetilde{X}\right\} $. Note that the foliation is
transversally oriented, but the codimension one leaf closure foliation is
not transversally orientable. The leaf closures are sets of the form
\begin{equation*}
\overline{L}_{\theta }=\left\{ (x,\theta ,\phi )_{\sim }\,|\,x\in \widetilde{%
X},\phi \in \lbrack 0,2\pi ]\right\} \bigcup \left\{ (x,2\pi -\theta ,\phi
)_{\sim }\,|\,x\in \widetilde{X},\phi \in \lbrack 0,2\pi ]\right\}
\end{equation*}
The basic functions and one-forms are:
\begin{eqnarray*}
\Omega _{B}^{0} &=&\left\{ f\left( \theta \right) \right\} \\
\Omega _{B}^{1} &=&\left\{ g_{1}\left( \theta \right) d\theta +g_{2}(\theta
)d\phi \right\} ,
\end{eqnarray*}
where the functions are smooth and satisfy
\begin{eqnarray*}
f\left( 2\pi -\theta \right) &=&f\left( \theta \right) \\
g_{i}\left( 2\pi -\theta \right) &=&-g_{i}\left( \theta \right)
\end{eqnarray*}
From this information, we calculate that $\Delta _{B}f\left( \theta \right)
=-f^{\prime \prime }\left( \theta \right) $. The eigenvalues $\left\{
n\,|\,n\geq 0\right\} $ correspond to the eigenfunctions $\left\{ \cos
n\theta \,|\,n\geq 0\right\} $. Then
\begin{eqnarray*}
K_{B}\left( t,\theta ,\theta \right) &=&\frac{1}{2\pi }+\frac{1}{\pi }%
\sum_{n\geq 1}e^{-n^{2}t}\cos ^{2}\left( n\theta \right) \\
&=&\frac{1}{2\pi }\sum_{n\in \mathbb{Z}}e^{-n^{2}t}\cos ^{2}\left( n\theta
\right) \\
&=&\frac{1}{4\pi }\sum_{n\in \mathbb{Z}}e^{-n^{2}t}+\frac{1}{4\pi }%
\sum_{n\in \mathbb{Z}}e^{-n^{2}t}\cos \left( 2n\theta \right) \\
&=&\frac{1}{4\pi }\mathrm{tr}\left( e^{-t\Delta }\text{ on }L^{2}\left(
S^{1}\right) \right) +\frac{1}{2}K\left( t,0,2\theta \right) ,
\end{eqnarray*}
where $K\left( t,\theta _{1},\theta _{2}\right) $ is the heat kernel on
functions on $S^{1}$. Substituting the expressions for this known kernel and
its trace, we obtain
\begin{multline*}
K_{B}\left( t,\theta ,\theta \right) =\frac{1}{4\pi }\sqrt{\frac{\pi }{t}}%
\sum_{k\in \mathbb{Z}}e^{-k^{2}\pi ^{2}\slash t}+\frac{1}{2}\frac{1}{\sqrt{%
4\pi t}}\sum_{k\in \mathbb{Z}}e^{-\left( 2\theta +2k\pi \right) ^{2}\slash
4t} \\
=\left\{
\begin{array}{ll}
\frac{1}{2\sqrt{\pi t}}+\mathcal{O}\left( e^{-\pi ^{2}/t}\right) & \text{if }%
\theta =k\pi \text{ for some }k\in \mathbb{Z} \\
\frac{1}{4\sqrt{\pi t}}+\mathcal{O}\left( e^{-c\left( \theta \right)
/t}\right) & \text{otherwise, for }c\left( \theta \right) =\min \left\{
\left. \left( \theta +k\pi \right) ^{2}\right\vert ~k\in \mathbb{Z}\right\}%
\end{array}
\right.
\end{multline*}
\newline
The basic manifold $\widehat{W}$ is an $SO(2)$-manifold, defined by $%
\widehat{W}=[0,\pi ]\times S^{1}\slash \sim $, where the circle has length $%
1$ and $\left( \theta =0\text{ or }\pi ,\gamma \right) \sim \left( \theta =0%
\text{ or }\pi ,-\gamma \right) $. This is a Klein bottle, since it is the
connected sum of two projective planes. The group $SO(2)$ acts on $\widehat{W%
}$ via the usual action on $S^{1}$. It is a simple exercise to calculate the
trace of the basic heat operator from the eigenvalues alone:
\begin{equation*}
K_{B}\left( t\right) =\mathrm{tr}\left( e^{-t\Delta _{B}^{0}}\right)
=\sum_{n\geq 0}e^{-n^{2}t}\sim \frac{\sqrt{\pi }}{2}t^{-1/2}+\frac{1}{2}.
\end{equation*}
Note that the existence of the constant term above implies that the
asymptotics of $K_{B}\left( t\right) $ cannot be obtained by integrating the
asymptotics of $K_{B}\left( t,\theta ,\theta \right) $ above.
We now compute the asymptotics of the trace of the basic heat operator using
Theorem~\ref{codim1nonorientable}. The volume of a generic leaf closure is $%
4\pi$, so the transverse volume is $A_{0}=V_{tr}=\frac{\mathrm{Vol}\left( M
\right)}{4\pi}=\pi$. The mean curvature of the leaf closures is identically
zero, so that $A_{1}=B_{1}=0$. Thus, Theorem~\ref{codim1nonorientable}
implies that
\begin{equation*}
K_{B}\left( t \right)=\frac{1}{\sqrt{4\pi t}}\left( \pi +\sqrt{\pi}%
\,t^{1/2}+O\left( t^{2} \right) \right) \sim \frac{\sqrt{\pi }}{2}t^{-1/2}+%
\frac{1}{2} ,
\end{equation*}
as expected.
\end{example}
\section{Riemannian Foliations That Are Not Transversally Orientable}
\label{nonorient}
In most of the cases considered throughout this paper, we have assumed that
the foliation in question is transversally oriented. We now remark that with
a few simple modifications, the results of this paper can be used to find
the asymptotics of the trace of the basic heat operator on a Riemannian
foliation that is not transversally orientable. First of all, the basic
Laplacian on functions is still well-defined on such foliations; the basic
Laplacian can defined using a local orientation and does not depend on the
choice of that orientation. Suppose that $\left( M,\mathcal{F}\right) $ is a
Riemannian foliation on a connected, compact manifold with a bundle--like
metric such that the leaves are not transversally orientable. The foliation
may now be lifted to the (nonoriented) orthonormal transverse frame bundle,
an $O\left( q\right) $ bundle over $M$. The lifted foliation is
transversally parallelizable, and the closures of the leaves of the lifted
foliation fiber over a compact $O\left( q\right) $-manifold $\overline{W}$.
\ The group $O\left( q\right) $ does not act by orientation-preserving
isometries, but otherwise the results of this paper extend by letting the
group $G=O\left( q\right) $. Thus, Theorem~\ref{tracebasic} holds, but some
of the results in Section~\ref{special} would have to be modified to allow
for orientation-reversing holonomy. For example, the powers of $t$ in the
asymptotic expansions of the trace of the basic heat operator would in
general increment by half integers instead of integers. Note that this
phenomenon can occur even in the transversally oriented case, if the leaf
closures are not necessarily transversally oriented (see Example~\ref%
{exnonorientable}).
\begin{acknowledgement}
I thank George Gilbert, Efton Park, Franz Kamber, and Jochen Br\"{u}ning for
helpful conversations.
\end{acknowledgement}
|
1,116,691,497,535 | arxiv | \section{Introduction}
\par Recently, device-to-device (D2D) communication has been envisioned as a promising technology to provide a better user experience. Specifically, proximity D2D users can communicate with each other directly without going through the base station (BS). Taking the advantages of the physical proximity of communicating devices, D2D communication can improve spectrum utilization and energy efficiency, and reduce end-to-end latency. There are mainly two ways for D2D pairs to share the cellular spectrum, namely underlay and overlay D2D \cite{Asadi2014}. In the underlay D2D communication, cellular user (CU) and D2D pairs share the same spectrum, which incurs interference to cellular links. In contrast, overlay D2D communication allows D2D pairs to occupy dedicated spectrum, which could have been assigned to CUs. Nevertheless, in both underlay or overlay D2D, the quality of service (QoS) of CUs will be degraded.
\par Meanwhile, CUs that are far away from BS, often suffer from poor channel quality, so that their QoS requirements are hard to meet. In this context, cooperative relay technology is thought of as a key technology to tackle this problem. Compared to fixed relay stations that incur high expenditure, mobile user relaying is an efficient and flexible solution with low cost.
\par Combining D2D communication and cooperative relay technology, Chen \emph{et al.} \cite{Wei2016TVT} propose a D2D-based cooperative network, where mobile devices serve as relays for CUs. However, the work does not consider any incentive mechanism for mobile devices. In fact, mobile devices, that owned by selfish users, may be unwilling to act as relays for other devices without reward. Inspired by the idea of spectrum \mbox{leasing \cite{Pantisano2012JSAC}}, authors \mbox{in\cite{Wu2017TWC,Cao2015MWC,Chen2015WCSP,Yuan2016PIMRC}} investigate a cooperative D2D system, where the D2D transmitters (DTs) act as relays to assist CUs in exchange for the opportunities to use the licensed spectrum. Thus, the QoS of CUs can be guaranteed and D2D pairs can obtain the transmission opportunities on licensed spectrum. As a result, a win-win situation can be achieved, which motivates CUs and D2D pairs to share the spectrum. However, above works determine the pairing between multiple CUs and multiple D2D pairs at a short timescale (e.g. at LTE scheduling time interval of 1ms), which may incur heavy signaling overhead and thus is not practical in large-scale networks.
\par In this paper, we investigate a cooperative D2D communication system, where CUs and D2D pairs cooperate with each other via spectrum leasing. Unlike previous works, in order to reduce the overhead, we propose a two-timescale resource allocation scheme. In particular, the pairing between multiple CUs and multiple D2D pairs is determined at a long timescale. On the other hand, at a short timescale, a cooperation policy allocates the transmission time for D2D link and cellular link based on the instantaneous channel state information (CSI). Under the proposed scheme, only statistic CSI is required for the pairing problem at the long timescale, while at the short timescale, the BS acquires only the instantaneous CSI between every matched CU-D2D pair to decide the transmission time. As a result, the signal overhead is significantly reduced in comparison to \cite{Wu2017TWC,Cao2015MWC,Chen2015WCSP,Yuan2016PIMRC}.
\par Moreover, we develop a matching game based framework to solve the two-timescale resource allocation problem. Specifically, we investigate the optimal cooperation policy for each D2D pair and its potential CU partner to characterize the long-term payoff of this potential pairing. In general, CUs and D2D pairs may be of self-interest\cite{Song2014}, and thus they can only be paired when they agree to cooperate with each other. The matching game provides an appropriate framework for such pairing problem with two-sided preferences \cite{Bayat2016MSP}. This motivates us to formulate the pairing problem at the long timescale as a one-to-one matching game, based on the long-term payoff of each potential CU-D2D pair. Furthermore, unlike previous one-to-one matching models in D2D networks \cite{Yuan2016PIMRC,Gu2015JSAC,Yuan2019TVT}, we propose to allow the transfer between CUs and D2D pairs as a performance enhancement. Then, an algorithm is proposed, which converges to an \mbox{$\epsilon$-stable} matching.
\par The rest of this paper is organized as follows. In Section II, we introduce the system model. We study the optimal cooperation policy in Section III and investigate the pairing problem in Section IV. Section V gives numerical results. Finally, Section VI concludes this paper.
\par \emph{Notations:} In this paper, $\mathbf{E}\{x\}$ represents the expectation \mbox{of $x$}, and $\mathbf{I}(\cdot)$ denotes the indication function.
\section{System Model}
\par We consider a single cell with a BS denoted by $b$. Because mobile devices are more likely to need help due to their limited power budgets, we focus on uplink resource sharing. There are $M$ CUs on the cell edge suffering from poor channel conditions which could not support their QoS. At the same time, $N$ transmitter-receiver pairs are working in D2D communication mode. There is no dedicated resource allocated for D2D pairs. As a result, D2D pairs serve as relays for CUs in exchange for access to the cellular channels. In the following, we use $\mathcal{M}=\{1,2,\cdots,M\}$ and $\mathcal{N}=\{1,2,\cdots,N\}$ to denote the sets of CUs and D2D pairs, respectively.
\par The time domain is divided into frames of fixed length. Each frame consists of $T_s$ subframe. The channel gain remains constant in each subframe and changes over different subframes. Besides, we assume that the channel gains across different subframes of the same frame are i.i.d. and follow a known distribution. At the channel occupied by CU $m$, the instantaneous channel gains of the cellular link from CU $m$ to BS, the link from CU $m$ to DT $n$, the link from DT $n$ to BS and the D2D link from DT $n$ to D2D receiver (DR) $n$ are represented as $h^m_{mb}, h^m_{mn}, h^m_{nb}, h^m_{nn}$, respectively.
\begin{figure}[!t]
\centering
\includegraphics[width=2.6in]{systemModel.eps}
\captionsetup{font={small}}
\caption{Subframe structure for cooperation.}
\label{systemModel}
\end{figure}
\par We assume that each CU is assisted by at most one D2D pair, since it has been shown that a single relay can achieve the full diversity gain \cite{Kadloor2010TWC}. As depicted in Fig.\ref{systemModel}, the normalized subframe is divided into three phases when D2D pair $n$ cooperates with CU $m$.\footnote{For simplicity, we assume the transmission direction of D2D pair is fixed during the entire frame and DT acts as a relay for CUs. In fact, our proposed scheme can be applied to a more general scenario, where the transmission direction may change and both D2D devices can be selected as a relay.} The first two phases both last $\frac{1-\alpha_{mn}}{2}$ and are used for the relay transmission for CU $m$. Specifically, CU $m$ broadcasts its data with power $P_c$ to the BS and DT $n$ at first. Then, DT $n$ forwards the received data to the BS with power $P_d$. Besides, the first two phases can also be used for the cellular link of CU $m$ when the cellular link has better performance. The last phase lasts $\alpha_{mn}$ and is used for D2D link, where DT $n$ communicates with DR $n$ with power $P_d$. Throughout the paper, we refer to $\alpha_{mn}\in\mathcal{A}\triangleq[0,1]$ as time allocation factor for D2D link.
\par The rate of CU $m$ in the cellular link is
\begin{equation}
r^C_{m}=\ln\left(1+\frac{P_ch^m_{mb}}{N_0}\right),
\end{equation}
where $N_0$ denotes the noise power. In this paper, we take the decode-and-forward with repetition coding as the relay scheme. Thus, when CU $m$ is aided by D2D pair $n$, the rate of CU $m$ during the first two phases is given by
\begin{equation}
r^R_{mn} = \frac{1}{2}\min\left\{\ln\left(\!1\!+\!\frac{P_ch^m_{mn}}{N_0}\!\right),\ln\left(\!1\!+\!\frac{P_ch^m_{mb}}{N_0}\!+\!\frac{P_dh^m_{nb}}{N_0}\!\right)\right\}.
\end{equation}
Since the first two phases can also be used for the cellular link of CU $m$, the achieved rate of CU $m$ during the entire subframe can be represented as
\begin{equation}
R^C_{mn}(\alpha_{mn})=(1-\alpha_{mn})\max\left\{r^C_m,r^R_{mn}\right\}.
\end{equation}
For convenience, we define $r^C_{mn}\triangleq\max\left\{r^C_m,r^R_{mn}\right\}$.
\par At the same time, the rate of D2D pair $n$ during the entire subframe can be given as
\begin{equation}
R^D_{mn}(\alpha_{mn})=\alpha_{mn}\ln\left(1+\frac{P_dh^m_{nn}}{N_0}\right)\triangleq\alpha_{mn}r^D_{mn}.
\end{equation}
\par Thus, we have two variables to determine: pairing between multiple CUs and multiple D2D pairs, and time allocation factor for each CU-D2D pair. To this end, we propose a matching game based framework to determine these two variables at two different timescales. Specifically, based on the instantaneous CSI, the cooperation policy decides the time allocation factor for each CU-D2D pair at each subframe (i.e. at the short timescale). We try to study the optimal cooperation policy to characterize the long-term payoff of each potential CU-D2D pair. Then, based on these long-term payoffs, we use the matching game with transfer to decide the pairing for each frame (i.e. at the long timescale). In other words, the optimal cooperation policy is the bridge between two timescales. In the following two sections, we will study these two subproblems, respectively.
\section{Optimal Cooperation Policy}
\par In this section, we investigate the optimal cooperative policy for each CU-D2D pair. Without loss of generality, we assume that CU $m$ cooperates with D2D pair $n$. Define the state $\mathbf{r}_{mn}\triangleq(r^C_{mn},r^D_{mn})$, which is determined by the instantaneous CSI. The set of all the possible states $\mathbf{r}_{mn}$ is denoted by $\mathcal{R}_{mn}$. The cooperation policy decides the time allocation factor $\alpha_{mn}$ according to the current state $\mathbf{r}_{mn}$. Mathematically, the cooperation policy is a function $\pi:\mathcal{R}_{mn}\rightarrow\mathcal{A}$. Thus, given the state $\mathbf{r}_{mn}$, the rate of D2D pair $n$ and CU $m$ can be represented as $\pi(\mathbf{r}_{mn})r^D_{mn}$ and $\left(1-\pi(\mathbf{r}_{mn})\right)r^C_{mn}$, respectively.
\par he optimal policy aims to maximize the expected rate of the D2D pair while guaranteeing the QoS of the CU. Therefore, the optimization problem is formulated as
\begin{subequations}
\label{equ5}
\begin{alignat}{2}
\max_{\pi}\quad &\mathbf{E}_{\mathbf{r}_{mn}}\left\{\pi(\mathbf{r}_{mn})r^D_{mn}\right\} \label{equ5:a}\\
\text{s.t.}\quad & \mathbf{E}_{\mathbf{r}_{mn}}\left\{\left(1-\pi(\mathbf{r}_{mn})\right)r^C_{mn}\right\}\geq r_{th},\label{equ5:b}
\end{alignat}
\end{subequations}
where $r_{th}$ is the minimum rate requirement for the CU and the constraint (\ref{equ5:b}) is used to guarantee the QoS of the CU. In fact, if $T_s\gg 1$, the objective function (\ref{equ5:a}) and the left-hand side of the constraint (\ref{equ5:b}) are a good approximation of the average rate of D2D pair $n$ and CU $m$ over $T_s$ subframes, respectively. In this section, all the expectations are taken over the random \mbox{variable $\mathbf{r}_{mn}$}. For brevity, we omit the subscript $\mathbf{r}_{mn}$ in the following.
\par Next, we investigate the structure of the optimal cooperation policy in the following theorem.
\begin{theorem}[Structure of Optimal Policy]
\label{thm1}
If the problem (\ref{equ5}) is feasible, the optimal policy $\pi^*$ is given by
\begin{equation}
\pi^*(\mathbf{r}_{mn})=\begin{cases}
0,&\quad\lambda^*r^C_{mn}>r^d_{mn},\\
\alpha^*,&\quad\lambda^*r^C_{mn}=r^d_{mn},\\
1,&\quad\lambda^*r^C_{mn}<r^d_{mn},
\end{cases}
\end{equation}
where
\begin{align}
\lambda^*&=\min\left\{\lambda|\mathbf{E}\{r^C_{mn}\mathbf{I}(\lambda r^C_{mn}\geq r^D_{mn})\}\geq r_{th}\right\},\\
\alpha^* &=\frac{r_{th}-\mathbf{E}\{r^C_{mn}\mathbf{I}(\lambda r^C_{mn}> r^D_{mn})\}}{\mathbf{E}\{r^C_{mn}\mathbf{I}(\lambda r^C_{mn}= r^D_{mn})\}}.
\end{align}
In fact, $\lambda^*$ indicates the minimum threshold which can satisfy the constraint (\ref{equ5:b}), and $\alpha^*$ ensures the equality of (\ref{equ5:b}).
\end{theorem}
\begin{IEEEproof}
We can construct the Lagrangian for the problem (\ref{equ5}) as follows.
\begin{align*}
\mathcal{L}(\pi,\lambda)&=\mathbf{E}\left\{\pi(\mathbf{r}_{mn})r^D_{mn}\right\}+\lambda\left(\mathbf{E}\{(1\!-\!\pi(\mathbf{r}_{mn}))r^C_{mn}\}\!-\!r_{th}\right)\\
&=\mathbf{E}\left\{\pi(\mathbf{r}_{mn})(r^D_{mn}-\lambda r^C_{mn})\right\}+\lambda\mathbf{E}\{r^C_{mn}\}-\lambda r_{th},
\end{align*}
where $\lambda$ is the Lagrange multiplier associated with the constraint (\ref{equ5:b}). For a fixed $\lambda$, it is easy to find out that the following \mbox{policy $\hat{\pi}_{\lambda}$}, which is given in (\ref{solutionOfLagarange}), can maximize the Lagrangian $\mathcal{L}(\pi,\lambda)$.
\begin{equation}
\label{solutionOfLagarange}
\hat{\pi}_{\lambda}(\mathbf{r}_{mn})=\begin{cases}
0,&\quad\lambda r^C_{mn}>r^D_{mn},\\
\alpha^*,&\quad\lambda r^C_{mn}=r^D_{mn},\\
1,&\quad\lambda r^C_{mn}<r^D_{mn}.
\end{cases}
\end{equation}
\par The Lagrange dual function can be given by $g(\lambda) = \max_{\pi}\mathcal{L}(\pi,\lambda)$. Thus, substituting (\ref{solutionOfLagarange}) to $g(\lambda)$, we have
\begin{equation}
\label{dual}
\begin{split}
g(\lambda)
=& \mathcal{L}(\hat{\pi}_{\lambda},\lambda)\\
=& \mathbf{E}\left\{\mathbf{I}(\lambda r^C_{mn}\!<\!r^D_{mn})(r^D_{mn}\!-\!\lambda r^C_{mn})\right\}\!+\!\lambda\mathbf{E}\{r^C_{mn}\}\!-\!\lambda r_{th}.
\end{split}
\end{equation}
\par In the following, we show that $\lambda^*$ minimizes the Lagrange dual function.
\par Assuming $\Delta\lambda>0$. Then, using (\ref{dual}), we have
\begin{align*}
&g(\lambda^*+\Delta\lambda)-g(\lambda^*)\\
&\qquad=\Delta\lambda\mathbf{E}\{r^C_{mn}\}-\Delta\lambda r_{th}-\Delta\lambda\mathbf{E}\left\{r^C_{mn}\mathbf{I}(\lambda^*r^C_{mn}\!<\! r^D_{mn}) \right\}\\
&\qquad\quad-\mathbf{E}\Big\{\!(\lambda^*\!+\!\Delta\lambda)r^C_{mn}\mathbf{I}(\lambda^*r^C_{mn}\!\leq\! r^D_{mn\!}<\!(\Delta\lambda\!+\!\lambda^*)r^C_{mn}\!\Big\}\\
&\qquad\quad+\mathbf{E}\Big\{\!r^D_{mn}\mathbf{I}(\lambda^*r^C_{mn}\!\leq\! r^D_{mn\!}<\!(\Delta\lambda\!+\!\lambda^*)r^C_{mn}\!\Big\}\\
&\qquad\geq\Delta\lambda\mathbf{E}\{r^C_{mn}\}-\Delta\lambda\mathbf{E}\left\{r^C_{mn}\mathbf{I}(\lambda^*r^C_{mn}\!<\! r^D_{mn}) \right\}-\Delta\lambda r_{th}\\
&\qquad=\Delta\lambda\mathbf{E}\left\{r^C_{mn}\mathbf{I}(\lambda^*r^C_{mn}\!\geq\! r^D_{mn}) \right\}-\Delta\lambda r_{th}\\
&\qquad\geq 0,
\end{align*}
where the last inequality is based on the definition of $\lambda^*$.
\par On the other hand, it is easy to verify that $\hat{\pi}_{\lambda^*}(\mathbf{r}_{mn})\leq\hat{\pi}_{\lambda^*-\Delta\lambda}(\mathbf{r}_{mn})$. Consequently, using the definition of $\lambda^*$, we have the following inequalities
\begin{align*}
&g(\lambda^*)-g(\lambda^*-\Delta\lambda)\\
&\qquad\leq -\Delta\lambda\mathbf{E}\left\{\hat{\pi}_{\lambda^*}(\mathbf{r}_{mn})r^C_{mn}\right\}+\Delta\lambda\mathbf{E}\{r^C_{mn}\}-\Delta\lambda r_{th}\\
&\qquad= -\Delta\lambda\mathbf{E}\{r^C_{mn}\mathbf{I}(\lambda^*r^C_{mn}\!<\!r^D_{mn})\}\!+\!\Delta\lambda\mathbf{E}\{r^C_{mn}\}\!-\!\Delta\lambda r_{th}\\
&\qquad= \Delta\lambda\mathbf{E}\{r^C_{mn}\mathbf{I}(\lambda^*r^C_{mn}\geq r^D_{mn})\}-\Delta\lambda r_{th}\\
&\qquad\leq 0.
\end{align*}
\par Thus, we can conclude that $\lambda^*$ is a solution to the dual problem $\min_{\lambda\geq 0}g(\lambda)$. Therefore, we can have
\begin{equation*}
P^*\overset{(a)}\leq g(\lambda^*)=\mathcal{L}(\pi^*,\lambda^*)\overset{(b)}=\mathbf{E}\left\{\pi^*(\mathbf{r}_{mn})r^D_{mn}\right\}\overset{(c)}\leq P^*,
\end{equation*}
where $P^*$ is the optimal value of the problem (\ref{equ5}). The inequality (a) is due to the duality gap. The \mbox{equality (b)} is based on the fact that the policy $\pi^*$ can make the \mbox{constraint (\ref{equ5:b})} hold with equality. Since $\pi^*$ is a feasible solution to the problem (\ref{equ5}), we can obtain the inequality (c).
\par Therefore, we can conclude that $\pi^*$ is an optimal cooperation policy.
\end{IEEEproof}
\par Theorem \ref{thm1} implies that the optimal policy can be a threshold policy, which makes decisions based on the ratio of $r^D_{mn}$ to $r^C_{mn}$. Besides, this theorem also shows that this optimal policy will allocate the entire subframe for D2D transmission (i.e. $\alpha_{mn}=1$) or cellular transmission (i.e. $\alpha_{mn}=0$) in most cases. As a result, such optimal policy enables efficient implementation in practice.
\par Furthermore, note that the term $\mathbf{E}\left\{r^C_{mn}\mathbf{I}(\lambda r^C_{mn}\geq r^D_{mn})\right\}$ increases with increasing $\lambda$. Therefore, we can use binary search to find the threshold $\lambda^*$.
\par At last, we define $v_{mn}\triangleq\mathbf{E}\left\{\pi^*(\mathbf{r}_{mn})r^D_{mn}\right\}$ if the problem (\ref{equ5}) is feasible. In the case of infeasible, we set $v_{mn} = -1$. Thus, $ v_{mn}$ can characterize the long-term payoff of D2D pair $n$ when it cooperates with CU $m$. Besides, if $v_{mn}\geq 0$, we call D2D pair $n$ being \emph{acceptable} to CU $m$. On the contrary, we call D2D pair $n$ being \emph{unacceptable} to CU $m$ when $v_{mn}< 0$.
\section{Matching Game for Pairing Problem}
\par In this section, we study the pairing problem. The assignment is represented as a binary matrix $\mathbf{X}_{M\times N}=[x_{mn}]$, where $x_{mn} = 1$ implies that CU $m$ and D2D pair $n$ are matched. We aim to maximize the long-term sum rate of D2D pairs, which can be formulated as the following problem.
\begin{subequations}
\label{equ11}
\begin{alignat}{3}
\max_{\mathbf{X}}\quad & \sum_{n\in\mathcal{N}}\sum_{m\in\mathcal{M}}x_{mn}v_{mn}&& \label{equ11:a}\\
\text{s.t.}\quad & \sum_{n\in\mathcal{N}}x_{mn}\leq 1, &&\quad\forall m\in\mathcal{M},\label{equ11:b}\\
& \sum_{m\in\mathcal{M}}x_{mn}\leq 1,&&\quad\forall n\in\mathcal{N},\label{equ11:c}\\
& x_{mn}\in\{0,1\},&&\quad\forall m\in\mathcal{M},\forall n\in\mathcal{N}.\label{equ11:d}
\end{alignat}
\end{subequations}
The constraint (\ref{equ11:b}) makes sure that each CU is relayed by at most one DT. Due to the limited battery capacity, each D2D pair can relay at most one CU \cite{Wu2017TWC}, which is represented in the constraint (\ref{equ11:c}). Note that $v_{mn}=-1$ when D2D pair $n$ is unacceptable to CU $m$. Therefore, the CUs will be only matched with acceptable D2D pairs.
\par Originally stemmed from economics \cite{Roth1990Two}, the matching theory provides a framework to tackle the problem of pairing players in two distinct sets, based on each player's individual preference. Since the CUs and D2D pairs are self-interested, we use the matching game to characterize the cooperations between CUs and D2D pairs in the pairing problem. Moreover, Theorem 1 implies that the CU is indifferent over the acceptable D2D pairs while D2D pair may have strict preference over CUs. Therefore, we allow transfer between D2D pairs and CUs to improve the performance. Such model is called matching game with transfer\cite{Bayat2016MSP} and also referred as to assignment game\cite{Roth1990Two}. Specifically, each CU has a price charged to its matched partner. Intuitively, the price of one CU indicates the willingness of D2D pairs to cooperate with that CU.
\begin{definition}
A \emph{one-to-one mapping} $\mu$ is a function from $\mathcal{M}\cup\mathcal{N}$ to $\mathcal{M}\cup\mathcal{N}\cup\{0\}$ such that $\mu(m)=n$ if and only if $\mu(n)=m$, and $\mu(m)\in\mathcal{M}\cup\{0\}$, $\mu(n)\in\mathcal{N}\cup\{0\}$ for $\forall m\in\mathcal{M},\forall n\in\mathcal{N}$.
\end{definition}
\par Note that $\mu(x)=0$ means that the user $x$ is unmatched in $\mu$. The above definition implies that a one-to-one mapping matches a user on one side to the one on the other side unless the user is unmatched. Thus, a mapping $\mu$ can define a feasible solution to the problem (\ref{equ11}). Next, we will introduce the price into the matching model.
\begin{definition}
A \emph{matching} is defined as $\Phi=(\mu,\mathbf{p})$, where $\mu$ is a one-to-one mapping, $\mathbf{p}=(p_1,p_2,\cdots,p_M)$ is the price vector of CUs and $p_m\geq 0,\forall m \in \mathcal{M}$. Moreover, if $\mu(m)=0$, then $p_m=0$.
\end{definition}
\par We denote the utilities of CU $m$ and D2D pair $n$ as $\theta_m$ and $\delta_n$, respectively. Thus, given a matching $\Phi=(\mu,\mathbf{p})$, $\theta_m$ and $\delta_n$ can be represented as
\begin{align}
\theta_m(\Phi)&=p_m,\\
\delta_n(\Phi) &= v_{\mu(n)n}-p_{\mu(n)}.
\end{align}
Here, we let $p_0=0$ and $v_{0n}=0,\forall n\in\mathcal{N}$ for convenience.
\par In the matching theory, the concept of \emph{stability} is important. On the other hand, since the price is usually quantized for exchange between CUs and D2D pairs in practical, we introduce the $\epsilon$-stable matching as follows.
\begin{definition}
Given $\epsilon\geq0$, a matching $\Phi$ is \emph{$\epsilon$-stable}, if and only if the following two conditions are satisfied:
\begin{enumerate}[(1)]
\item $\theta_m(\Phi)\geq 0,\delta_n(\Phi)\geq 0$, for $\forall m\in\mathcal{M},\forall n\in\mathcal{N}$;
\item $\theta_m(\Phi)+\delta_n(\Phi)\geq v_{mn}-\epsilon$, for $\forall m\in\mathcal{M},\forall n\in\mathcal{N}$.
\end{enumerate}
\end{definition}
The condition (1) is called \emph{individual rationality condition} and reflects that a user may remain unmatched if the cooperation is not beneficial. Condition (2) implies that there is no CU-D2D pair $(m,n)$ such that they can form a new matching, where both of them can increase their utilities and one of them can improve its utility by at least $\epsilon$.
\begin{algorithm}[!t]
\caption{Algorithm to Find $\epsilon$-stable Matching}
{\fontsize{8pt}{0.85\baselineskip}\selectfont
\begin{algorithmic}[1]
\renewcommand{\algorithmicrequire}{\textbf{Initialization:}}
\STATE Set $t=1, p_m=\beta_m^t=0,\mu^0(m)=0,\forall m\in\mathcal{M}$;
\renewcommand{\algorithmicrequire}{\textbf{ D2D Pairs' Proposals:}}
\REQUIRE
\STATE Broadcast the price requirement vector $\boldsymbol{\beta}^t=({\beta}^t_1,\beta^t_2,\cdots,\beta^t_M)$;
\FOR{each unmatched D2D pair $n\in\mathcal{N}$}
\STATE Determine its demand $m=D_n(\bm{\beta}^t)$;
\STATE { If $m\neq 0$, D2D pair $n$ proposes to CU $m$ \mbox{($g^t_{mn} = 1$)}. Otherwise, D2D pair $n$ does not proposes ($g^t_{mn} = 0,\forall m\in\mathcal{M}$) and $\mu^t(n)=0$;}
\ENDFOR
\renewcommand{\algorithmicrequire}{\textbf{ CUs' Decision Making:}}
\REQUIRE
\FOR{Each CU $m\in\mathcal{M}$}
\IF{$\sum_{n\in\mathcal{N}}g^t_{mn}\!=\!0$, $\sum_{n\in\mathcal{N}}g^{t-1}_{mn}>0$ and $\mu(m)=0$}
\STATE Set $\mu^t(m)=n^*$, where $n^*=random(\{n|g^{t-1}_{mn}\!=\!1\})$;
\STATE Set $p_m=\beta^{t-1}_m$ and $\beta^{t+1}_m = \beta^{t}_m$;
\STATE Set $g^t_{m^*n^*}=0$, where $m^* = D_{n^*}(\bm{\beta}^t)$;
\ENDIF
\ENDFOR
\FOR {Each CU $m\in\mathcal{M}$}
\IF {$\sum
\limits_{n\in\mathcal{N}}g^t_{mn}\!=\!1$, and $\mu^{t-1}_m\!=\!0$ or $p_m\!<\!\beta^t_m$ are satisfied }
\STATE Set $\mu^{t}(m)=n^*$, where $g^t_{mn^*}=1$;
\STATE Set $p_m=\beta^t_m$ and $\beta^{t+1}_m=\beta^t_m$;
\ELSIF{$\sum_{n\in\mathcal{N}}g^t_{mn}\geq1$}
\STATE Set $\mu^t(m)=0$;
\STATE If $n\!\neq\!0$ and $p_m\!=\!\beta^t_m$ where $n\!=\!\mu^{t-1}(m)$, set $g^t_{mn}\!=\!1$;
\STATE Set $\beta^{t+1}_m=\beta^t_m+\epsilon$;
\ELSE
\STATE Set $\beta^{t+1}_m=\beta^t_m$;
\ENDIF
\ENDFOR
\STATE $t\leftarrow t+1$;
\STATE Go to step 3 until there is no proposal in current loop.
\end{algorithmic}}
\end{algorithm}
\par Algorithm 1 is proposed to find an $\epsilon$-stable matching. In the following, we give a brief description of the algorithm during $t$-th iteration.
\par At first, the price requirement vector $\bm{\beta}^t$ will be broadcast, and $\beta^t_m$ represents the minimum price has to pay if D2D pair wants to propose to CU $m$ at the current iteration. Then, each unmatched D2D pair $n$ selects its favorite CU according to $D_n(\bm{\beta}^t)$, where the demand function is represented as
\begin{equation}
D_n(\bm{\beta}^t)=\begin{cases}
arg\max\limits_ {m\in\mathcal{M}}(v_{mn}-\beta_m^t),&\max\limits _{m\in\mathcal{M}}(v_{mn}-\beta_m^t)\geq0 ,\\
0,&\text{otherwise.}
\end{cases}
\end{equation}
\par In the CUs' decision making stage, the CUs decide if they want to match with the D2D pairs. There are four cases for each CU $m$. The first case (step 8-12) is that CU $m$ is unmatched and receives no proposals after increasing its price requirement, but in the previous iteration, has received proposals from multiple D2D pairs. Then, CU $m$ will select randomly one of those D2D pairs to be matched with and set the price as $p_m=\beta^{t-1}_m$. The second case (step 15-17) is that CU $m$ receives one proposal, and meanwhile, it is unmatched or matched with the price $p_m<\beta^t_m$. In other words, only one D2D pair wants to be matched with CU $m$ with the price $\beta^t_m$. As a result, CU $m$ will be matched with that D2D pair and set the price as $p_m=\beta^t_m$. The third case (step 18-21) is that there are multiple D2D pairs (including the current partner of CU $m$) wanting to be matched with CU $m$ with the price $\beta^t_m$. Then, CU $m$ will increase its price requirement by $\epsilon$ and become unmatched, where $\epsilon> 0$ is the price-step. In the last \mbox{case (step 23)}, CU $m$ will remain the price requirement and do nothing else. The convergence of Algorithm 1 is given in the following theorem.
\begin{theorem}
The Algorithm 1 converges to an $\epsilon$-stable matching.
\end{theorem}
\begin{IEEEproof}
At first, we show that the algorithm converges to a matching. Note that $\beta^t_m$ is non-decreasing. Moreover, it can be found that $\beta^t_m\leq\epsilon+\max_{n\in\mathcal{N}}v_{mn}$. Therefore, the algorithm will converge in finite steps. Use $\Phi=(\mu,\mathbf{p})$ to denote the final result. It is easy to verify that once a CU has received a proposal, the CU will have a partner in $\mu$. Thus, the prices of the CUs unmatched in $\mu$ are zero. Therefore, $(\mu,\mathbf{p})$ is a matching.
\par In the following, we prove that $(\mu,\mathbf{p})$ is $\epsilon$-stable by contradiction.
\par Suppose there exists a CU-D2D pair $(m,n)$ such that $\theta_m+\delta_n< v_{mn} -\epsilon$. Assume $\mu(m)=n'$ and $\mu(n)=m'$. Thus, we can find a price $p'$ such that $p'\geq p_m+\epsilon$ and $v_{mn}-p'>v_{m'n}-p_{m'}$. So, we have
\begin{equation}
\label{equ14}
v_{mn}-p_m>v_{mn}-p'>v_{m'n}-p_{m'}.
\end{equation}
According to the algorithm, (\ref{equ14}) implies that D2D pair $n$ must have proposed to CU $m$. Therefore, there must exist an iteration, denoted by $\tau$-th iteration, where the first case happens for CU $m$. Specifically, CU $m$ receives multiple proposals with $\beta^{\tau-1}_m=p_m$ at $(\tau-1)$-th iteration, and receives no proposal with $\beta^{\tau}_m=p_m+\epsilon$ at $\tau$-th iteration. Furthermore, no D2D pairs propose to CU $m$ afterward. As a result, we have the following inequalities
\begin{equation}
v_{mn}-p'\leq v_{mn}-\beta^\tau_m<v_{m'n}-p_{m'},
\end{equation}
which is inconsistent with (\ref{equ14}).
\par Besides, it is easy to verify $(\mu,\mathbf{p})$ satisfies the individual rationality condition. Therefore, we can conclude that $(\mu,\mathbf{p})$ is $\epsilon$-stable.
\end{IEEEproof}
\section{Simulation Results}
In this section, the performance of the proposed scheme is investigated through simulations. The instantaneous channel gain used in the simulation is $h=\eta L^{-\gamma}$, where $\eta$ is the fast-fading gain with exponential distribution, $\gamma=4$ is the pathloss exponent and $L$ is the distance between transmitter and receiver. For simulation, we consider the scenario where the BS is deployed in the cell center while the radius of the cell is set to 500 m. The CUs are distributed uniformly at the cell edge. Meanwhile, the D2D pairs are uniformly distributed in the area with a distance of 200 m to 400 m from the BS. Other configuration parameters are given in Table.I.
\begin{table}[!t]
\footnotesize
\caption{Configuration Parameters}
\renewcommand{\arraystretch}{1}
\centering
\begin{tabular}{|c|c|}
\hline
\bf{Parameters} & \bf{Value}\\
\hline
Power noise ($N_0$) & -100 dBm\\
\hline
Transmit power of CU ($P_c$) & 20 mW\\
\hline
Transmit power of DT ($P_d$) & 20 mW\\
\hline
Distance of D2D link & Uniformly distributed in $[10, 30]$ m\\
\hline
Minimum rate requirement ($r_{th}$) & 1.8 bps/Hz\\
\hline
Price-step ($\epsilon$) & 1\\
\hline
\end{tabular}
\end{table}
\begin{figure}[!t]
\centering
\captionsetup{font={small}}
\includegraphics[width=2.8in]{AvgUtility}
\caption{The effective average utilities of D2D pairs and CUs versus the number of D2D pairs, where $M=15$.}
\label{AvgUtility}
\end{figure}
\par At first, we investigate the property of our scheme. Fig.\ref{AvgUtility} presents the effective average utilities (EAU) of CUs and D2D pairs versus the number of D2D pairs. The EAU of CUs is defined as $\text{EAU} = \frac{\text{Sum of CUs' utilities}}{\text{Number of matched CUs}}$. The EAU of D2D pairs can be defined in a similar way. It can be observed that with the increasing number of D2D pairs, the EAU of CUs increases while the EAU of D2D pairs decreases. The rationale behind this is that when $N$ is small, there is a strong competition among CUs to acquire the relay service from D2D pairs. Therefore, the prices of CUs are low and each matched D2D pair has high utility. In comparison, when there is a large number of D2D pairs, the available CUs become a scarce resource. As a result, each D2D pair has to pay a higher price for the transmission opportunities on the cellular channels.
\begin{figure}[!t]
\centering
\captionsetup{font={small}}
\includegraphics[width=2.8in]{SumRateD2D}
\caption{The sum rate of D2D pairs with different schemes versus the number of D2D pairs, where $M=15$.}
\label{SumRateD2D}
\end{figure}
\begin{figure}[!t]
\centering
\captionsetup{font={small}}
\includegraphics[width=2.8in]{Outage}
\caption{The outage percentage of CUs with different schemes versus the number of D2D pairs, where $M=15$.}
\label{Outage}
\end{figure}
\par Next, to evaluate the performance gain of the proposed scheme, we compare it with the following schemes: i) the \emph{optimal} scheme provides the optimal solution to the problem (\ref{equ11}); ii) the \emph{matching without transfer} scheme adopts the matching game without transfer to solve the pairing problem (\ref{equ11}), i.e. the prices of CUs are zero; iii) the \emph{random} scheme matches the D2D pairs and CUs randomly. The comparison results are provided in Fig.\ref{SumRateD2D} and Fig.\ref{Outage}.
\par In Fig.\ref{SumRateD2D}, we compare the performance of different schemes in terms of the sum rate of D2D pairs. This figure shows that the proposed scheme achieves near-optimal performance. Besides, owning to allowing transfer between CUs and D2D pairs, the proposed scheme outperforms the matching without transfer scheme, especially in the large $N$ region. It also can be observed that the gain is small in the small $N$ region. This is due to the fact the prices of CUs are close to zero when $N$ is small (which is consistent with the results in Fig.\ref{AvgUtility}). Thus, these two schemes are almost the same in this situation. Moreover, the matching without transfer scheme only matches the CUs with their acceptable partners. Therefore, this scheme can obtain better performance than the random scheme.
\par Fig.\ref{Outage} presents the outage percentage of CUs under different schemes, and the outage refers to the case where the rate requirement of a CU is not satisfied. Compared with the random scheme, the rest three schemes achieve significantly better performance. In particular, these three schemes have a similar outage percentage. The explanation is as follows. The outage never happens if each CU is matched with acceptable D2D pairs. Since all the three schemes only match the CUs with their acceptable D2D pairs, the price has little impact on the outage percentage. Furthermore, when $N\geq20$, the outage percentage of our scheme is close to zero. It implies that our scheme improves the performance of CUs greatly. On the contrary, the outage percentage of the random scheme is larger than 60\%, which indicates that it is essential to have an efficient pairing between CUs and D2D pairs.
\section{Conclusion}
In this paper, we have investigated a cooperative D2D communication system, where D2D pairs and CUs cooperate with each other via spectrum leasing. We have provided a low-overhead design for the system by proposing a two-timescale resource allocation scheme, in which the pairing between CUs and D2D pairs is decided at the long timescale and time allocation factor is determined at the short timescale. Specifically, to characterize the long-term payoff of each potential CU-D2D pair, we investigate the optimal cooperation policy to decide the time allocation factor. Based on these long-term payoffs, we use the matching game with transfer to solve the pairing problem. The simulation results confirm the performance gain of the proposed scheme.
\section*{ACKNOWLEDGMENT}
This work was supported in part by the National Key Research and Development Program of China No.213 and in part by the National Science Foundation of China under Grants 71731004 and Shanghai Municipal Natural Science Foundation under Grants 19ZR1404700.
\bibliographystyle{IEEEtran}
|
1,116,691,497,536 | arxiv | \section{Introduction}
\label{intro}
Absorption of the 21-cm electron spin-flip transition of neutral hydrogen (\HI) is a powerful probe of the reservoir of
star-forming material in the early Universe. At high redshift, 21-cm absorption can provide important insight into star
formation rates and galaxy evolution at a time when chemical abundances were markedly different to the present
day. Furthermore, in combination with other absorbing species, \HI\ 21-cm can provide measurements of the fundamental
constants at large look-back times (\citealt{cdk04} and references therein). In five cases, OH 18-cm absorption has been found
coincident with the \HI\ \citep{cdn99,kc02a,kcdn03,kcl+05}\footnote{For all of these the full-width half maximum (FWHM) of
the OH profile is similar to that of the \HI\ \citep{cdbw07}.} and the hydroxyl radical is of particular interest for
measurement of the constants since it allows highly sensitive measurements from a single absorbing species \citep{dar03}.
However, the detection of either transition at $z > 0.1$ is a rare occurrence. For \HI\ 21-cm, 80 systems have been
detected\footnote{Half of which occur in systems intervening the sight-lines to more distant radio sources, with the
other half associated with a source's host galaxy (see \citealt{cur09a,cw10} and references therein).} and, despite
much searching (\citealt{cwm+10} and references therein), the detection of OH 18-cm is rarer yet with only the
aforementioned five detections to date.\footnote{Three of which are intervening and two associated absorbers.}
\citet{cwm+06,cwc+11} have shown that the molecular fraction in the known molecular absorbers is correlated with the optical--near-infrared
colour, thus indicating that their colours are due to the presence of dust required to shield the molecular gas from
the ambient UV field. Therefore, selecting objects for which an optical redshift is available selects against those with
a high molecular fraction, making the detection of molecular absorbers difficult.
Hence, in an attempt to increase the number of redshifted detections of both species, we have performed full spectral
scans with the Green Bank Telescope (GBT) toward five highly reddened (optical--near-infrared colours of $V - K > 6$)
radio-loud objects (Tanna et al., in prep.). For the reddest ($V - K = 10.26$) of the targets, the $z = 2.64$ quasar
MG\,J0414+0534 (4C\,+05.19), three 21-cm absorption systems have already been detected (see Table~\ref{abs}). In this
letter we report the detection of a further absorber at a redshift of $z = 0.534$, as well as candidate system at $z = 0.339$.
\section{The detection of one (and possibly two) new absorption feature(s)}
\label{oadr}
We have now completed the analysis of the data along the entire redshift space towards MG\,J0414+0534 (see \citealt{cwt+11} for details), upon which we
find a further two possible absorption profiles, near 926 MHz and 1061 MHz.
The absorption feature close to 926 MHz was the least subject to radio frequency interference (RFI), which appears as narrow lines and amplitude fluctuations.
These can be seen in Fig.~\ref{4th-stacked} (top), which essentially constitutes a low-resolution time-lapse series of the individual 146.3 sec exposure scans.
\begin{figure}
\centering \includegraphics[trim=0 0 0 0, angle=0,scale=0.46]{Fig1.ps
\caption{The absorption feature near 926 MHz toward J0414+0534. Top: As seen in individual 146.3 sec exposure
scans of each linear polarisation.
Bottom: Detail showing the averaged spectrum. In this and Fig. \ref{5th-stacked}, the spectrum is shown at
the observed spectral resolution of 3.4 \kms\ and a first order polynomial removed from the bandpass, with the
best fit Voigt profile overlain. The lower
panel shows the standard deviation of each channel from the mean across all individual scans.
}
\label{4th-stacked}
\end{figure}
In the final averaged spectrum (Fig.~\ref{4th-stacked}, bottom), a persistent absorption feature was apparent, which
maintains consistency throughout the observations, with GBT RFI monitoring indicating that this part of the band is
predominantly clear of interference. There remains, however, a two channel wide spike redshifted by $\approx40$ \kms\
from the peak of the profile. The RFI nature of this is confirmed in spectral animations of contiguous 5 sec
integrations, where like the other RFI features in the band, the spike is seen to fluctuate between its minimum and maximum amplitudes
on time-scales of a minute. For the sake of transparency, we do not flag the spike out of final averaged spectrum,
although we blank and interpolate over the affected channels for the measurements described in Sect.~\ref{hi4}.
Given the profile
shape\footnote{For the other possible candidate, OH 18-cm, we expect two features separated by $1.9572/(z+1)$ MHz.} and
its strength, we believe that the feature must arise from 21-cm absorption at $z=0.534$.
Close to 1061 MHz, the dominant RFI was apparent as ``packets'', one of which impinges on one side of a possible absorption
feature. RFI monitoring indicates that these packets (spaced $\sim$1 MHz apart) are due to aircraft radar and much of
the data are heavily affected.
\begin{figure}
\centering \includegraphics[trim=0 0 0 0, angle=0,scale=0.46]{Fig2.ps
\caption{As per Fig. \ref{4th-stacked}, but for the feature near 1061 MHz.}
\label{5th-stacked}
\end{figure}
Again, through the animation of contiguous 5 sec integrations, the packets behave in a similar fashion to the RFI spikes
close to 925 MHz, displaying time dependent fluctuations. Flagging the data most strongly affected leaves
approximately one third of the scans, allowing us to retain some data at all
frequencies, particularly on the blue-shifted side of the line. While the RFI packets are still apparent in the final
averaged spectrum (Fig. \ref{5th-stacked}, bottom), it is clear that the absorption feature at 1061 MHz maintains a
stable, deep profile shape over the course of the observations (even beneath the RFI). Furthermore, the packets vary
regularly in time, flipping between positive, negative and zero fluxes, whereas the absorption profile is consistent and
has an identical appearance in two different IFs (0.9--1.1 GHz and 1.05--1.25 GHz).
The stability of the putative line is confirmed in the standard deviation spectrum, which identifies time
varying features, where the peak is not coincident with the most variable channels (Fig. \ref{5th-stacked}, bottom).
However, given that the near-by RFI is not restricted to one or two channels, this feature requires confirmation from an
independent observation.
\section{Results and discussion}
\label{randd}
\subsection{Absorption by neutral hydrogen}
\label{hi4}
The mean frequency of the first feature (Fig. \ref{4th-stacked}) is $925.738\pm0.009$ MHz, which for \HI\ 21-cm gives,
from the flagged data, a flux averaged mean redshift of $z = 0.53435\pm0.00002$, cf. the peak redshift of $z_{\rm peak}
= 0.53437\pm0.00002$, obtained from the deepest channel in the peak of the profile. There is a decrease in the
bandpass response below 950 MHz, giving a continuum flux of $S = 1.82\pm0.05$ Jy, cf. the 2.3 Jy seen over the rest of the band,
which yields an observed peak optical depth of $\tau_{\rm obs}\equiv\Delta S/S = 0.30\pm0.04$. Unlike the
vast majority of redshifted 21-cm absorption, this peak depth means that the optically thin approximation cannot be
applied (where $\Delta S \lapp0.3\,S$), i.e. $\tau\equiv-\ln\left(1-{\tau_{\rm obs}}/{f_{\rm HI}}\right)\approx
{\tau_{\rm obs}}/{f_{\rm HI}}$. Since, by definition, the covering factor $f_{\rm HI}\leq1$, the peak optical depth is
then $\tau_{_{\rm peak}} \geq 0.36\pm0.06$ and the velocity integrated optical depth $\int\!\tau dv
\geq16\pm 3$ \kms.
This gives a column density of $N_{\rm HI} \geq 2.8\times10^{19}.\,T_{\rm spin}$, where $T_{\rm spin}$ [K] is the mean harmonic spin
temperature of the gas.
The mean frequency of the second feature (Fig. \ref{5th-stacked}) is $1060.975\pm0.004$ MHz, which for \HI\ 21-cm gives a redshift of
$z = 0.338777\pm0.000006$, cf. the peak redshift of $z_{\rm peak} = 0.33882\pm0.00002$.
The peak depth of the line is $\Delta S = 1.059$ Jy and the continuum flux is $S = 2.331\pm0.015$ Jy, giving a peak $\tau_{\rm obs} = 0.45\pm0.01$.
Again, the optically thin approximation is not applicable and so $\tau_{\rm peak} \geq 0.61\pm0.02$, giving
$\int\!\tau dv \geq 37\pm 1$ \kms\ and $N_{\rm HI} \geq 6.7\times10^{19}.\,T_{\rm spin}$.
If genuine, this would be the second deepest redshifted 21-cm feature yet found\footnote{The deepest being $\tau_{_{\rm peak}} \geq0.71$ at $z
= 0.524$ towards AO\,0235+164 \citep{rbb+76}.}, although given our concerns about its authenticity (Sect.~\ref{oadr}) this requires confirmation.
We note there are apparent similarities in the shapes of the two absorption profiles as a function of velocity, although,
in addition to different depths, the widths exhibit a slight difference, with the 926 MHz profile having a full-width
half maximum of FWHM$\,\approx44$ \kms, cf. FWHM\,$\approx61$ \kms\ at 1061 MHz. Since the profile shapes are not consistent in
frequency space, we believe it unlikely that these are due to an instrumental artifact, which lends some weight to the
reality of the 1061 MHz feature. The shape of each line has much broader wings than those produced by a single Gaussian
profile and are best fit by a single Voigt profile,
which we interpret as the effect of pressure broadening convolved with
the velocity dispersion of the gas.
Lastly, in Table \ref{abs} we list the derived column densities together with the other 21-cm absorbers thus far found towards J0414+0534.
\begin{table*}
\centering
\begin{minipage}{150mm}
\caption{The absorbing systems thus far found along the sight-line to J0414+0534. The first column lists the redshift, followed by the feature
with which the absorption is associated. $N_{\rm HI}$ and $N_{\rm OH}$ give the \HI\ 21-cm and OH 18-cm line strengths, respectively,
followed by the normalised OH line strength, in terms of $(f_{\rm HI}/f_{\rm OH}).(T_{\rm ex}/T_{\rm spin})$.
The last column gives the reference for the discovery of the absorption and the line strengths.
\label{abs}}
\begin{tabular}{@{}l l c c c l @{}}
\hline
Redshift & Location & $N_{\rm HI}$ [\scm] & $N_{\rm OH}$ [\scm] & $N_{\rm OH}/N_{\rm HI}$ & Reference\\
\hline
2.63647 & Host galaxy & $7.5\times10^{18}.\,(T_{\rm spin}/f_{\rm HI})$ & $\lapp3\times10^{14}\,(T_{\rm ex}/f_{\rm OH})^{\dagger}$& $\lapp2\times10^{-4}$ & \citet{mcm98} \\%M99\\
0.95974 & Lensing galaxy &$1.6\times10^{18}.\,(T_{\rm spin}/f_{\rm HI})$ & $\leq4.9\times10^{12}.\,(T_{\rm ex}/f_{\rm OH})$& $\leq9\times10^{-6}$& \citet{cdbw07} \\
0.53435 & --- & $\geq2.8\times10^{19}.\,T_{\rm spin}$ & $\leq1.3\times10^{14}.\,(T_{\rm ex}/f_{\rm OH})$& $\leq5\times10^{-6}$& This paper \\%T12\\
0.37895 & Object X & $2.9\times10^{19}.\,(T_{\rm spin}/f_{\rm HI})$ & $\leq3.3\times10^{13}.\,(T_{\rm ex}/f_{\rm OH})$& $\leq2\times10^{-6}$& \citet{cwt+11} \\
0.33878 & --- &$\geq6.7\times10^{19}.\,T_{\rm spin}$ & $\leq5.0\times10^{14}.\,(T_{\rm ex}/f_{\rm OH})$& $\leq8\times10^{-6}$& This paper$^{\ast}$ \\% T12\\
\hline
\end{tabular}
{Notes: $^{\dagger}$OH limit from \citet{cwt+11}, $^{\ast}$if genuine 21-cm absorption (Sect. \ref{oadr}).}
\end{minipage
\end{table*}
Even the weakest (that arising in the lensing galaxy) is likely to qualify as a damped Lyman-$\alpha$ absorption system (DLA), requiring only
$(T_{\rm spin}/f_{\rm HI}) \geq 125$ K to reach the defining $N_{\rm HI}\geq2\times10^{20}$ \scm. That is, all of the absorbers so far
found along the sight-line towards J0414+0534 are likely to be associated with gas-rich galaxies.
\subsection{Optical counterparts}
Confirmation of 21-cm detections can be done through optical imaging and spectroscopy, either by finding absorption
lines in the quasar spectrum at matching redshifts or by identifying the absorbing galaxy. Detecting common optical
absorption lines, such as Mg\,\textsc{ii}, from these systems
is made difficult by the extremely red colour of the quasar's optical spectrum: For $z = 0.5343$ and $z=0.3388$ the
Mg\,\textsc{ii} doublet would appear at 429~nm and 375~nm respectively, where the quasar flux is very low
\citep{htl+92,lejt95}. Furthermore, \citet{mcm98} detect strong 21-cm absorption in the host galaxy and any associated
Lyman-$\alpha$ absorption (centred on 442.6~nm at $z=2.64$) would likely conceal the near-by $z=0.5343$ Mg\,\textsc{ii}
line.
There is evidence from the spectrum of \citet{htl+92} of the $\lambda\lambda5891,5897$ Na\,\textsc{i} doublet at
$z=0.3388$. This identification is made difficult by both the low-resolution of the published spectrum and the presence
of the nearby Ca\,\textsc{ii} H+K line from the lensing galaxy at $z=0.958$. It is, however, suggestive of a galaxy at
the redshift of the 1061 MHz putative detection. The same Na\,\textsc{i} line at $z=0.534$ coincides with strong
sky-line subtraction residuals in the \citeauthor{htl+92} spectrum.
Identifying the corresponding absorbing galaxy is also difficult. No published spectroscopy exists of neighbouring field
galaxies, although \citet{tk99} presented long-slit spectroscopy of the lensing galaxy, taken with the slit placed
across "Object X", the closest field galaxy. \citet{cwt+11} identified two peaks in the low-resolution spectrum at the
wavelengths expected for [O\,\textsc{iii}] at z=0.3789. It is likely, then, that this object does not correspond to
either of the detections presented here. HST images indicate numerous nearby galaxies, but none have redshift
measurements or estimates.\footnote{There are two galaxies, u2fl1\#038 and u2fl1\#044, at relatively large separations
($1.3-1.5$ arcmin), although these are extremely red objects and so likely to be at $z>1$ \citep{yt03}.} Detailed, deep
multi-object and/or integral-field spectroscopy of this field would be required to fully resolve the identification of
the specific absorbing galaxies.
\subsection{The incidence of multiple intervening absorbers along a single sight-line}
\label{sec:inc}
The detection of at least one new absorber now gives at least three intervening systems along the sight-line to
J0414+0534 (Table \ref{abs}). All of these systems are likely to be DLAs (Sect. \ref{hi4}), indicating a similar
sight-line to the $z=3.02$ quasar CTQ\,247, towards which \citet{lmmm01} report the detection of four DLAs. However,
at $z=2.55$, $2.59$ and $2.62$, three of these arise in a single broad feature and the fourth, at $z=1.91$, is inferred
from the metal lines. With a rest-frame equivalent width of $W_{\rm r}^{\lambda1216} = 6.0$~\AA\ for the Lyman-\AL\
line, this is more likely to be a Lyman-limit system than a DLA.
In order to quantify how rare an occurrence the presence of at lease three DLAs along a single sight-line
is, we use the \citet{ppb06} sample of strong (${\rm W}_{\rm r}^{\lambda2796} > 1$~\AA) Mg{\sc \,ii}
absorbers. These are obtained from SDSS DR3 observations of 45\,023 QSO sight-lines, which span $0.35
< z_{\rm MgII} < 2.3$, a similar redshift range to the 21-cm observations towards J0414+0534 at $z=2.64$. However, the
GBT scan below 700 MHz is completely dominated by RFI and so not sensitive to any absorption between the lensing
galaxy and J0414+0534 (Tanna et al., in prep.), restricting the redshifts scanned to $z\lapp1$. The total number of sight-lines probed for a given
redshift value is in excess of 15\,000 for much of this range (see Fig. 2 of \citealt{ppb06}), peaking at 22\,000 --
23\,000 for $0.6<z<0.9$. Of these, there are 2564 unique sight-lines which exhibit ${\rm W}_{\rm r}^{\lambda2796} >
1$~\AA\ absorption at $z\leq1$. Only 78 of these contain two distinct\footnote{Separated by at least 10\,000 \kms (or
$\Delta z\gapp0.03$).} absorbers, with a further four sight-lines containing three absorbers.
Restricting this to DLA strength absorbers (${\rm W}_{\rm r}^{\lambda2796} \gapp 3$~\AA, assuming the ${\rm W}_{\rm
r}^{\lambda2796}$--$N_{\rm HI}$ relation of \citealt{mc09}, see also \citealt{ctp+07}), there are
81 sight-lines which contain a DLA, but none with more than one.
Zero sight-lines with at least two DLA strength absorbers out of $\gapp15\,000$ sight-lines means that,
with at least three distinct DLAs at $z<1$, the sight-line towards J0414+0534 is unprecedented.
Our use of SDSS QSOs \citep{ppb06} selects against reddened sight-lines (see Fig. 1 of \citealt{cwc+11}),
where dust extinction is low, and so extinction may be significant in the sight-line of J0414+0534, given
the very red colour and the high incidence of cold, neutral absorbing gas.
\subsection{Reddening of the quasar light}
\subsubsection{Reddening by dust associated with molecular gas}
\begin{figure*}
\centering \includegraphics[angle=270,scale=0.65]{Fig3.ps
\caption{The OH data at frequencies close to the redshift of the \HI\ absorption, $z_{_{\rm HI}}\pm0.005$.
The left panels show the expected frequencies of OH absorption for the $z=0.534$ absorption
feature and the right panels for the putative $z=0.339$ feature. All but the last panel are part of a 200 MHz
wide scan and so shown at the observed spectral resolution of 3--4 \kms. For 1720 MHz at $z=0.339$, the spectrum
is from a 800 MHz wide scan and shown at the observed resolution of 21 \kms.}
\label{oh-spectra}
\end{figure*}
From \HI\ 21-cm at $z=0.5344$, we expect the 1665 and 1667 MHz OH main lines at 1085.3985 and 1086.67 MHz
and the 1612 and 1720 MHz satellite lines at 1050.75 and 1121.33 MHz, respectively. Likewise, if real,
for \HI\ 21-cm at $z = 0.3388$, we expect the OH 18-cm lines at 1243.94 \& 1245.40 MHz (main) and 1204.22 \& 1285.11 MHz
(satellite). There is, however, no evidence of OH absorption in either case (Fig. \ref{oh-spectra}).
In Table \ref{abs} we list the best OH column density limits for these systems, together with those of the previous
searches. Based on the optical--near-infrared colour of $V-K = 10.26$, we expect molecular fractions of close to unity
\citep{cwc+11} or $N_{\rm OH}/N_{\rm HI}\gapp10^{-4}$ \citep{cwm+06}. This is ruled out for all of the known 21-cm
absorbers, except perhaps that associated with the host which has the weakest limit. Even so, given that only 46\%
of the band is not completely ruined by RFI (Tanna et al., in prep.), the presence of significant columns of molecular gas
elsewhere along this sight-line cannot be ruled out.
\subsubsection{Reddening by dust associated with atomic gas}
Given that there is no evidence of molecular absorption coincident with the redshifts of the 21-cm absorbers, we revisit
the possibility that the reddening is due to dust associated with the intervening atomic gas \citep{cmr+98}. From all of
the published associated $z\geq0.1$ absorption searches, \citet{cw10} found a $3.63\sigma$ correlation between the
21-cm line strength and the $V-K$ colour. Unlike the molecular line strength correlation \citep{cwm+06}, this exhibits
considerate scatter, although it does comprise many more data points and is quite fragile, with the correlation
quickly disappearing as various sub-samples are removed.
Another point of note was that the 21-cm line strength of the associated ($z = 2.64$) absorber was weaker than expected,
based upon the trend defined by the other points (Fig. \ref{red3}).
\begin{figure}
\centering \includegraphics[angle=270,scale=0.77]{Fig4.ps
\caption{The 21-cm line strength versus optical--near-IR colour for the associated $z\geq0.1$ absorbers for which the colours are available \citep{cw10}.
The filled circles are the 21-cm detections and the unfilled circles the non-detections.
The filled star shows the line strength for the $N_{\rm HI} = 7.5\times10^{18}.\,(T_{\rm spin}/f_{\rm HI})$ absorber associated with the host of J0414+0534 and
the hollow stars the total of the line strengths for the four robust detections, and including the putative $z=0.339$ feature (Table \ref{abs}).}
\label{red3}
\end{figure}
Replacing this with the total column density so far observed along this sight-line (the unfilled stars in the figure)\footnote{Setting
the values to $N_{\rm HI} = 2.8$ and $6.7\times10^{19}.\,(T_{\rm spin}/f_{\rm HI})$, in order to maintain consistency with the other measurements.}, moves the line strength
closer to the expected trend and, as before, including the limits via the {\sc asurv} survival analysis package \citep{ifn86},
the total column density increases the
significance of the correlation to $3.66\sigma$ for four absorbers and $3.70\sigma$ including the putative fifth
absorber.
This may be further evidence of dust associated with the dense atomic gas in the intervening systems being the cause of
the extremely red colour of this object, although the individual contributions of these absorbers toward the total
reddening of the quasar light cannot, as yet, be established. Note also, that most of the other sources have only been
searched for 21-cm close to the host redshift (i.e. for associated absorption). Therefore, strictly speaking, these too
represent lower limits, since further absorption systems along their sight-lines have not been ruled out. Furthermore,
as for the molecular gas, substantial atomic absorption may exist in this sight-line at redshifts corresponding to RFI
affected frequencies. In either case, the large number of 21-cm absorbers detected over the useful fraction of the band,
in conjunction with the apparent obscuration of the source, may have far reaching implications for the number of
gas-rich galaxies missed by optical surveys.
\section{Summary}
We have now completed the analysis of a full-band decimetre wave spectral scan towards the very red ($V-K = 10.26$)
quasar MG\,J0414+0534. From the $46$\% of the band not completely ruined by RFI, we have found three (possibly four)
strong intervening absorbers, in addition to the previously detected absorption in the host galaxy at $z=2.64$
\citep{mcm98}. At a total of four (or five), this represents a new record in the number of 21-cm absorbers found along a
single sight-line, the previous being a total of two systems towards PKS\,1830--211 \citep{lrj+96,cdn99}, which has $V-K = 6.25$
(see \citealt{cwm+06}).
The new detections occur at redshifts of $z = 0.3388$ and $z=0.5344$, each being very strong absorbers with column densities of
$N_{\rm HI}\geq6.7\times10^{19}.\,T_{\rm spin}$ and $\geq2.8\times10^{19}.\,T_{\rm spin}$ \scm, respectively.
This qualifies both as damped Lyman-$\alpha$ absorption systems for a paltry $T_{\rm spin} \sim10$~K (an order of
magnitude lower than the lowest yet found, \citealt{ctm+07}). However, given the RFI in the spectrum at 1061 MHz, the
$z = 0.3387$ feature requires confirmation.
Despite the large column densities, no OH absorption was found in either the main or satellite 18-cm lines at these redshifts,
although the very red colour of the background source suggests large molecular fractions somewhere
along the sight-line \citep{cwm+06}. Summing the observed \HI\ column densities does strengthen the atomic gas
abundance/$V-K$ colour correlation \citep{cw10}, although this remains fairly scattered. Therefore, dense molecular gas at an
RFI affected redshift and/or the possibility that J0414+0534 is intrinsically red cannot be ruled out.
However, given that this extremely red sight-line has yielded at least three intervening gas-rich galaxies (which would
likely have remained undiscovered through optical spectroscopy), does have implications for obscured galaxy populations. With the
large field-of-view and instantaneous bandwidths that will be available with the Square Kilometre Array and the
Australian SKA Pathfinder (ASKAP)\footnote{Which will scan \HI\ 21-cm absorption over the $z\lapp1$ redshift range discussed in Sect. \ref{sec:inc},
through the {\em First Large Absorption Survey in \HI} (FLASH).}, large-scale blind surveys of radio sources with faint optical counterparts will soon be possible, allowing
us to quantify the number of such galaxies hidden to optical surveys.
The Centre for All-sky Astrophysics is an Australian Research Council Centre of Excellence, funded by grant CE110001020.
|
1,116,691,497,537 | arxiv | \section{Introduction}
Among the achievements of HERA, one of the major results was the experimental evidence \cite{Derrick:1993xh,Ahmed:1994nw}
that
among the whole set of $\gamma^* p \to X$ deep inelastic scattering events, almost 10\% are diffractive (DDIS), of the form $\gamma^* p \to X Y$ with a rapidity gap between the proton remnants $Y$
and the hadrons $X$
coming from the fragmentation region of the initial virtual photon.
There are two main approaches to theoretically describe
diffraction. The first one involves
a {\em resolved} Pomeron contribution, see Fig.~\ref{ResDirect} (left), while the second one
relies on a {\em direct} Pomeron contribution involving the coupling of a Pomeron with the diffractive state, see Fig.~\ref{ResDirect} (right).
\begin{figure}[h]
\center
\psfrag{q}{\raisebox{-.2cm}{$\gamma^*$}}
\psfrag{l1}{$e^\pm$}
\psfrag{l2}{$e^\pm$}
\psfrag{P}{$\mathbb{P}$}
\psfrag{ld}{}
\psfrag{lu}{}
\psfrag{R}{}
\psfrag{q1}{\raisebox{.2cm}{\ \ jet}}
\psfrag{q2}{\raisebox{-.3cm}{\ \ jet}}
\psfrag{p1}{$p$}
\psfrag{p2}{$Y$}
\includegraphics[scale=.90]{resolved-pomeron-no-arrow.eps}
\qquad
\psfrag{q2}{\raisebox{-.4cm}{\ \ jet}}
\raisebox{.5cm}{\includegraphics[scale=.90]{direct-pomeron-no-arrow.eps}}
\caption{Resolved (left panel) and direct Pomeron (right panel) contributions to two jets production.}
\label{ResDirect}
\end{figure}
For moderate invariant mass $M^2$ of the diffractively produced state $X$, such a state
can be modeled in perturbation theory by a $q \bar{q}$ pair, or by higher Fock states as a $q \bar{q} g$ state for larger values of $M^2$. Based on such a model, with
a two-gluon exchange picture for the Pomeron, a good description of HERA data for diffraction~\cite{Chekanov:2004hy-Chekanov:2005vv-Chekanov:2008fh,
Aktas:2006hx-Aktas:2006hy-Aaron:2010aa-Aaron:2012ad-Aaron:2012hua} could be achieved~\cite{Bartels:1998ea}.
In the direct components considered there, the $q \bar{q} g$ diffractive state has been studied in two particular limits. The first one, valid for very large $Q^2$, corresponds to a collinear approximation in which the transverse momentum of the gluon is assumed to be much smaller than the transverse momentum of the emitter~\cite{Wusthoff:1995hd-Wusthoff:1997fz}.
The second one~\cite{Bartels:1999tn,Bartels:2002ri}, valid for very large $M^2$, is based on the assumption of a strong ordering of longitudinal momenta, encountered in BFKL equation~\cite{Fadin:1975cb-Kuraev:1976ge-Kuraev:1977fs-Balitsky:1978ic}. Both these approaches were combined in order to describe HERA data for DDIS~\cite{Marquet:2007nf}.
It would be natural to extend the HERA studies
to similar hard diffractive events at LHC.
The idea here is to adapt the concept of photoproduction of diffractive jets, which was performed at HERA~\cite{Chekanov:2007rh,Aaron:2010su}, now with a flux of
quasi-real photons in ultraperipheral collisions (UPC)~\cite{Baltz:2007kq-Baur:2001jj}, relying on the notion of equivalent photon approximation. In both cases,
the hard scale is provided by the invariant mass of the tagged jets.
We here report on our computation~\cite{Boussarie:2014lxa} of the $\gamma^* \to q \bar{q} g$ impact factor at tree level with an arbitrary number of $t$-channel gluons described within the Wilson line formalism, also called QCD shockwave approach~\cite{Balitsky:1995ub-Balitsky:1998kc-Balitsky:1998ya-Balitsky:2001re}. As an aside, we rederive the $\gamma^* \to q \bar{q}$ impact factor. In particular, the
$\gamma^* \to q \bar{q} g$ transition is computed without any soft or collinear approximation for the emitted gluon, in contrast with the above mentioned calculations. These results provide a necessary generalization of building blocks for inclusive DDIS (of potential significant phenomenological importance~\cite{Motyka:2012ty}) as well as for two- and three-jet diffractive production.
\section{The shockwave formalism in a nutshell}
Balitsky's shockwave formalism
is very powerful in determining evolution equations and impact factors at next-to-leading order for inclusive processes~\cite{Balitsky:2010ze-Balitsky:2012bs}, at semi-inclusive level for $p_t$-broadening in $pA$ collisions~\cite{Chirilli:2011km-Chirilli:2012jd} or in the evaluation of the triple Pomeron vertex beyond the planar limit~\cite{Chirilli:2010mw}, when compared with usual methods based on summation of contributions of individual Feynman diagrams computed in momentum space. It is an effective way of estimating the effect of multigluon exchange, formulated in coordinate space and thus natural in view of describing saturation~\cite{GolecBiernat:1998js-GolecBiernat:1999qd}.
We introduce the light cone vectors
$n_{1}$ and $n_{2}$%
\begin{equation}
\label{Sudakov-basis}
n_{1}=\left( 1,0,0,1\right) ,\quad n_{2}=\frac{1}{2}\left( 1,0,0,-1\right)
,\quad n_{1}^{+}=n_{2}^{-}=n_{1} \cdot n_{2}=1 \,,
\end{equation}
and the Wilson lines as
\begin{equation}
U_{i}=U_{\vec{z}_{i}}=U\left( \vec{z}_{i},\eta\right) =P \exp\left[{ig\int_{-\infty
}^{+\infty}b_{\eta}^{-}(z_{i}^{+},\vec{z}_{i}) \, dz_{i}^{+}}\right]\,.
\label{WL}%
\end{equation}
The operator $b_{\eta}^{-}$ is the external shock-wave field built from slow gluons
whose momenta are limited by the longitudinal cut-off defined by the rapidity $\eta$
\begin{equation}
b_{\eta}^{-}=\int\frac{d^{4}p}{\left( 2\pi\right) ^{4}}e^{-ip \cdot z}b^{-}\left(
p\right) \theta\left(e^{\eta}-\frac{|p^{+}|}{P^+}\right)\,,\label{cutoff}%
\end{equation}
where $P^+$ is the typical large $+$ momentum of the problem, to be identified with $p_\gamma^+$ later on. We will denote the longitudinal cut-off $\sigma = e^\eta \, P^+ = \alpha \, P^+.$
We use the light cone gauge
$\mathcal{A}\cdot n_{2}=0,$
with $\mathcal{A}$ being the sum of the external field $b$ and the quantum field
$A$%
\begin{equation}
\mathcal{A}^{\mu} = A^{\mu}+b^{\mu},\;\;\;\;\;\;\;\;\;\quad b^{\mu}\left( z\right) =b^{-}(z^{+},\vec{z}\,) \,n_{2}%
^{\mu}=\delta(z^{+})B\left( \vec{z}\,\right) n_{2}^{\mu}\,,\label{b}%
\end{equation}
where
$B(\vec{z})$ is a profile function.
Indeed, let us consider an external gluon field $b^{\mu}$ in its rest frame and boost it along the $+$ direction. One obtains :
\begin{eqnarray}\nonumber
&&b^+ \! \left( x^+,\, x^-, \, \vec{x} \right) \rightarrow \frac{1}{\lambda}b^+ \left( \lambda x^+,\, \frac{1}{\lambda} x^- ,\, \vec{x} \right)\,, \\ \nonumber
&&b^- \! \left( x^+,\, x^-, \, \vec{x} \right) \rightarrow {\lambda} b^- \left( {\lambda x^+},\, {\frac{1}{\lambda} x^-} ,\, \vec{x} \right)\,, \\ \nonumber
&&b^i \, \left( x^+,\, x^-, \, \vec{x} \right) \rightarrow \, \, \, b^i \, \left( \lambda x^+,\, \frac{1}{\lambda} x^- ,\, \vec{x} \right)\,. \\ \nonumber
\end{eqnarray}
Assuming that the field vanishes at infinity, one immediately gets that only its minus component survives the boost in the limit $\lambda \to \infty\,,$ and that it does not depend on $x^-$ and contains $\delta \left( x^+ \right)\,,$ thus justifying the form of $b^\mu$ in Eq.~(\ref{b}).
We use intensively in the following the dipole operator
constructed from the Wilson line (\ref{WL}), namely
$\mathbf{U}_{12}=\frac{1}{N_{c}}\rm{tr}\left( U_{1}U_{2}^{\dagger}\right) -1\,.$
\section{Impact factor for $\gamma\rightarrow q\bar{q}$ transition}
\begin{figure}
\center
\includegraphics[scale=0.65]{lo1__2.eps}
\caption{Diagram contributing to the impact factor for two jet production }
\label{leading}
\end{figure}
At leading order, the diagram contributing to the impact factor for $\gamma\rightarrow q\bar{q}$ transition is shown in Fig. \ref{leading}, in which $z's$ denote the
coordinates of interaction points with the photon and the shock wave.
After projection on the color singlet state and subtraction of the contribution without interaction with the shock wave, the contribution of this diagram can be written in the momentum space as (factorizing out a global QED factor $-i e_q$)
\begin{equation}
M_{0}^{\alpha}=N_c \int d\vec{z}_{1}d\vec{z}_{2}F\left( p_{q},p_{\bar{q}}%
,z_{0},\vec{z}_{1},\vec{z}_{2}\right) ^{\alpha} \mathbf{U}_{12}\,.
\label{M0int}%
\end{equation}
Denoting $Z_{12} = \sqrt{x_{q}x_{\bar{q}}\vec{z}_{12}^{\,\,2}}$, we get for a longitudinally polarized photon
\begin{eqnarray}
\label{FL}
F\left( p_{q},p_{\bar{q}},k,\vec{z}_{1},\vec{z}_{2}\right) ^{\alpha
}\varepsilon_{L\alpha}&=&\theta(p_{q}^{+})\,\theta(p_{\bar{q}}^{+})\frac
{\delta\left( k^{+}-p_{q}^{+}-p_{\bar{q}}^{+}\right) }{(2\pi)^{2}}%
e^{-i\vec{p}_{q}\cdot \vec{z}_{1}-i\vec{p}_{_{\bar{q}}}\cdot\vec{z}_{2}}
\nonumber \\
&\times&
(-2i)\delta_{\lambda_{q},-\lambda_{\bar{q}}}\,x_{q}x_{\bar{q}}%
\,Q\,K_{0}\left(Q \, Z_{12}\right)\,,
\end{eqnarray}
whereas for a transversally polarized photon
\begin{eqnarray}
\label{FT}
F( p_{q},p_{\bar{q}},k,\vec{z}_{1},\vec{z}_{2}) ^{j}%
\varepsilon_{Tj}\!
&=&\theta(p_{q}^{+})\,\theta(p_{\bar{q}}^{+})\frac{\delta(
k^{+}\!\!-\!p_{q}^{+}\!-p_{\bar{q}}^{+}\!) }{(2\pi)^{2}}e^{-i\vec{p}_{q}\cdot\vec
{z}_{1}-i\vec{p}_{_{\bar{q}}}\cdot\vec{z}_{2}}
\nonumber \\
&&\hspace{-2cm}\times
\delta_{\lambda_{q},-\lambda_{\bar{q}}}( x_{q}-x_{\bar{q}%
}+s\lambda_{q}) \frac{\vec{z}_{12} \cdot \vec{\varepsilon}_{T}}{\vec{z}_{12}^{\,\,2}}
Q \,Z_{12} K_{1}(Q\, Z_{12})\,.\!\!\!\!\!
\end{eqnarray}
\section{Impact factor for $\gamma\rightarrow q\bar{q}g$ transition}
\begin{figure}
\center
\includegraphics[scale=0.65]{openProduction1.eps}
\caption{Diagrams contributing to the impact factor for three jet production}
\label{3body}
\end{figure}
In the case of the $q\,\bar q\,g$ Fock final state the contributiong diagrams are shown in Fig.~\ref{3body}.
After projection on the color singlet state and subtraction of the contribution without interaction with the shock wave, the result can be put in the form
\begin{eqnarray}
\nonumber
M^{\alpha} &=& N_c^2 \int d\vec{z}_{1}d\vec{z}_{2}d\vec{z}_{3} \, F_{1}\left( p_{q},p_{\bar{q}}%
,p_{g},z_{0},\vec{z}_{1},\vec{z}_{2},\vec{z}_{3}\right) ^{\alpha} \nonumber \\
&\times& \frac{1}{2}
\left( \mathbf{U}_{32} + \mathbf{U}_{13} - \mathbf{U}_{12} + \mathbf{U}_{32}\,\mathbf{U}_{13} \right)\nonumber
\\
&+& N_c \int d\vec{z}_{1}d\vec{z}_{2} \, F_{2}\left( p_{q},p_{\bar{q}},p_{g},z_{0}%
,\vec{z}_{1},\vec{z}_{2}\right) ^{\alpha}\frac{N_{c}^{2}-1}{2N_{c}} \mathbf{U}_{12}\,.
\label{F2tilde}%
\end{eqnarray}
In this equation, the first two lines and the third one correspond to contributions to the impact factor, respectively, of the diagrams 1 and 2 of Fig.~\ref{3body} and of
the diagrams 3 and 4 of it. The explicit expressions
for the functions $F_i,$
for both longitudinally and transversally polarized photon can be found in
ref.~\cite{Boussarie:2014lxa}.
\section{The 2- and 3-gluon approximation}
We first notice that the dipole operator $\mathbf{U}_{ij}$ involves terms at least of order $g^2$. Hence for only two or three exchanged gluons one can neglect the quadrupole term in the amplitude $M^{\alpha}$ which results in the simpler expression
\begin{eqnarray}
\label{M3gBis}
&& \hspace{-.3cm}M^{\alpha} \overset{\mathrm{g^3}}{=} \frac{1}{2}\int d\vec{z}_{1}d\vec{z}%
_{2} \mathbf{U}_{12} \left[ \left( N_{c}^{2}-1\right)
\tilde{F}_{2}\left( \vec{z}_{1},\vec{z}%
_{2}\right) ^{\alpha} \right.
\nonumber \\
&& \hspace{-.3cm}\left.
+ \int d\vec{z}_{3} \left\{ N_{c}^{2}F_{1}\left(
\vec{z}_{1},\vec{z}_{3},\vec{z}_{2}\right)^{\alpha}
+N_{c}^{2}F_{1}\left( \vec{z}_{3},\vec{z}%
_{2},\vec{z}_{1}\right) ^{\alpha} - F_{1}\left( \vec{z}_{1},\vec{z}_{2},\vec{z}_{3}\right) ^{\alpha} \right\} \right]\,.
%
\end{eqnarray}
Those integrals can be performed analytically
when $\vec{p}_q=\vec{p}_g=\vec{p}_{\bar{q}}=\vec{0}$. They are
otherwise expressible as a simple convergent integral over the interval $[0,1]$.
\section{Towards the next-to-leading-order corrections}
The virtual corrections to the $\gamma^* \to q \bar{q}$ involve two kinds of contributions.
\begin{figure}[h]
\centerline{\includegraphics[scale=0.65]{loop.eps}}
\caption{Diagrams contributing to virtual corrections in which the radiated gluon doesn't cross the shock wave.}
\label{nlo}
\end{figure}
\noindent
The diagrams contributing to virtual corrections in which the radiated gluon does not cross the shock wave are shown in Fig.~\ref{nlo},
and the diagrams in which the radiated gluon interacts with the shock wave are illustrated in the Fig.~\ref{nloSW}.
\begin{figure}[h]
\centerline{\includegraphics[scale=0.75]{nloreal1__2.eps}}
\caption{Diagrams contributing to virtual corrections in which the radiated gluon interacts with the shock wave.}
\label{nloSW}
\end{figure}
One should note that although these virtual corrections only involve one-loop diagrams, the complications arise due to the presence of many different scales. Indeed, our aim is to obtain results in the general kinematics where the virtuality of incoming photon, the $t-$channel momentum transfer and the invariant mass $M^2$ of the diffractive two-jet state are arbitrary. Additionally, this impact factor is a function of the virtuality of $t-$channel exchanged gluons.
We now provide some intermediate results of our computation.
First, we present the matrix element corresponding to diagrams 1, 2, 3 of Fig.~\ref{nlo}~\cite{Boussarie:2015qet}.
We work in
dimensional regularization for the transverse momentum space, i.e. $d=D-2 =2+ 2\,\epsilon\,,$ and introduce the regularization scale $\mu$, and the related dimensionless scale $\tilde{\mu}^2=\mu^2/Q^2\,.$
Denoting $p_{ij} \equiv p_i-p_j\,,$ we introduce $p_\perp=p_{q1\perp}\,,$ $\vec{p}^{\,2}=-p_\perp^2$ and
$w=\vec{p}^{\,2}/Q^2\,.$ For simplicity, we write $x=x_q.$
We get for the case of a longitudinally polarized photon
\begin{eqnarray}
&&\hspace{-1cm}T_{fi}|_{\epsilon_{\alpha}=n_{2\alpha}}=-i g^2\frac{N_{c}^{2}-1}{2N_{c}%
}tr(U(p_{1\bot})U^{\dag}(-p_{2\bot}))\delta(p_{q1\bot}-p_{\gamma\bot}%
+p_{\bar{q}\,2\bot})
\nonumber \\
&\times&\delta(p_{q}^{+}-p_{\gamma}^{+}+p_{\bar{q}}^{+})\theta
(p_{q}^{+})\theta(p_{\bar{q}}^{+})
\nonumber \\
&\times&\frac{\Gamma(1-\epsilon)}{\left(
16\pi^{3}\right) ^{1+\epsilon}}\frac{1}{\sqrt{2p_{\gamma}^{+}}\sqrt
{2p_{q}^{+}}\sqrt{2p_{\bar{q}}^{+}}}
\frac{x(1-x)p_{\gamma}^{+}{}\overline{u}_{p_{q}}\gamma^{+}v_{p_{\bar{q}%
}}}{x(1-x)Q^{2}+\vec{p}{}^{\,\,2}}
\nonumber \\
&\times& \left( \left( 2\ln\left( \frac
{(1-x)x}{\alpha^{2}}\right) -3\right) \left( \ln\left( \frac{\left(
w-x^{2}+x\right) ^{2}}{(1-x)x\tilde{\mu}^{2}}\right) +\frac{1}{\epsilon
}\right) \right. \nonumber \\
&&\left.+\ln^{2}\left( \frac{x}{1-x}\right) -\frac{\pi^{2}}{3}+6\right) \,.
\end{eqnarray}
Expanding the photon momentum in the Sudakov basis (\ref{Sudakov-basis}) as
\begin{equation}
p_\gamma = p_\gamma^+ \, n_1 - \frac{Q^2}{2 p_\gamma^+} \, n_2
\end{equation}
one can explicitly check the electromagnetic gauge invariance for this group of diagrams
since
\begin{equation}
T_{fi}|_{\epsilon_{\alpha}=n_{1\alpha}}=\frac{Q^{2}}{2p_{\gamma}^{+2}}%
T_{fi}|_{\epsilon_{\alpha}=n_{2\alpha}}\,.
\end{equation}
Similarly, for the case of a transversally polarized photon, one gets
\begin{eqnarray}
&& \hspace{-.3cm}T_{fi}|_{transverse}=-i g^2 \frac{N_{c}^{2}-1}{2N_{c}}tr(U(p_{1\bot})U^{\dag
}(-p_{2\bot}))\delta(p_{q1\bot}-p_{\gamma\bot}+p_{\bar{q}2\bot})
\nonumber \\
&&\hspace{-.4cm}\times \,\delta
(p_{q}^{+}-p_{\gamma}^{+}+p_{\bar{q}}^{+})\theta(p_{q}^{+})\theta(p_{\bar{q}%
}^{+})
\nonumber \\
&&\hspace{-.4cm}\times
\frac{\Gamma(1-\epsilon)}{\left( 16\pi^{3}\right) ^{1+\epsilon}%
}\frac{\epsilon_{i}}{\sqrt{2p_{\gamma}^{+}}\sqrt{2p_{q}^{+}}\sqrt{2p_{\bar{q}%
}^{+}}} \frac{-\left( \frac{1}{2}\overline{u}_{p_{q}}[\gamma^{i}\hat{p}_{\bot
}]\gamma^{+}v_{p_{\bar{q}}}+(2x-1)p^{i}\overline{u}_{p_{q}}\gamma
^{+}v_{p_{\bar{q}}}\right) }{2(x(1-x)Q^{2}+\vec{p}{}^{\,\,2})}%
\nonumber \\
&&\hspace{-.4cm} \times\!\!\left[ \! \left( 2\ln\left( \frac{(1-x)x}{\alpha^{2}}\right) -3\right)\!\!
\left( \! \ln\left( \frac{w-x^{2}+x}{\tilde{\mu}^{2}}\right) +\frac
{(1-x)x\ln\left( \frac{(1-x)x}{w-x^{2}+x}\right) }{w}+\frac{1}{\epsilon
}\!\right)
\right. \nonumber \\
&& \hspace{-.4cm}\left.
+\,\ln^{2}\left( \frac{x}{1-x}\right) -\frac{\pi^{2}}{3}+6\right] .
\end{eqnarray}
Second, we present the singular part of diagram 4 of fig.~\ref{nlo} involving final state interaction.
The result for a longitudinally polarized photon reads
\begin{eqnarray}
\label{final-state-long}
&& T_{fi}|_{\epsilon_{\alpha}=n_{2\alpha}}=i\frac{N_{c}^{2}-1}{2N_{c}%
}tr(U(p_{1\bot})U^{\dag}(-p_{2\bot}))
\delta(p_{\gamma\bot}-p_{1q\bot
}-p_{2\bar{q}\bot})
\nonumber\\
&& \times
\delta(p_{\gamma}^{+}-p_{q}^{+}-p_{\bar{q}}^{+})
\frac{\Gamma(1-\varepsilon)}{\left(
16\pi^{3}\right) ^{1+\varepsilon}}
\frac
{1}{\sqrt{2p_{\gamma}^{+}}\sqrt{2p_{q}^{+}}\sqrt{2p_{\bar{q}}^{+}}}\nonumber\\
&& \times\left\{ \frac{(1-x)x\bar{u}_{p_{q}}\gamma^{+}v_{p_{\bar{q}}%
}p_{\gamma}^{+}}{\vec{p}^{\,\,2}+Q^{2}(1-x)x}\left[ \ln^{2}\left(
\frac{(1-x)x}{\alpha^{2}}\right) -\ln^{2}\left( \frac{1-x}{x}\right) \right. \right.\nonumber \\
&&\left. \left. + \, 2\ln\left( \frac{(1-x)x}{\alpha^{2}}\right) \left( \ln\left(
\frac{\left( \vec{p}^{\,\,2}+Q^{2}(1-x)x\right) ^{2}}{Q^{2}(x\vec{p}%
_{\bar{q}}-(1-x)\vec{p}_{q})^{2}}\right) +i\pi\right) \right] +C_{\Vert
}^{fs}\right\} ,\,\,\,\,\,
\end{eqnarray}
while for a transversally polarized photon we obtain
\begin{eqnarray}
\label{final-state-transverse}
&& \hspace{-.2cm} T_{fi}|_{transverse}=i\frac{N_{c}^{2}-1}{2N_{c}}tr(U(p_{1\bot})U^{\dag
}(-p_{2\bot}))
\delta(p_{\gamma\bot}-p_{1q\bot}-p_{2\bar{q}\bot
})\nonumber\\
&&\hspace{-.2cm}\times \, \delta(p_{\gamma}^{+}-p_{q}^{+}-p_{\bar{q}}^{+})
\frac{\Gamma(1-\varepsilon)}{\left( 16\pi^{3}\right)
^{1+\varepsilon}}
\frac{\epsilon_{i}}%
{\sqrt{2p_{\gamma}^{+}}\sqrt{2p_{q}^{+}}\sqrt{2p_{\bar{q}}^{+}}}\nonumber\\
&&\hspace{-.2cm} \times\left\{ -\frac{(2x-1)p_{\bot}^{i}\bar{u}_{p_{q}}\gamma^{+}%
v_{p_{\bar{q}}}+\frac{1}{2}\bar{u}_{p_{q}}\gamma^{+}[\gamma_{\bot}^{i}\hat
{p}_{\bot}]v_{p_{\bar{q}}}}{\left( Q^{2}(1-x)x+\vec{p}^{\,\,2}\right)
}\left[ \frac{1}{2}\ln^{2}\left( \frac{(1-x)x}{\alpha^{2}}\right)
\right. \right. \nonumber \\
&&\hspace{-.2cm} \left. \left. -\frac
{1}{2}\ln^{2}\left( \frac{x}{1-x}\right) +\ln\left( \frac{(1-x)x}%
{\alpha^{2}}\right)
\left( \frac{Q^{2}(1-x)x}{\vec{p}^{\,\,2}}\ln\left(
\frac{Q^{2}(1-x)x}{Q^{2}(1-x)x+\vec{p}^{\,\,2}}\right)
\right.\right.\right.
\nonumber \\
&&\hspace{-.2cm} \left.\left.\left.
+\ln\left(
\frac{(1-x)x\left( Q^{2}(1-x)x+\vec{p}^{\,\,2}\right) }{(x\vec{p}_{\bar{q}%
}-(1-x)\vec{p}_{q})^{2}}\right) +i\pi\right) \right] +C_{\bot}%
^{fs}\right\} \,.
\end{eqnarray}
In eqs.~(\ref{final-state-long},\ref{final-state-transverse}), $C_{\Vert
}^{fs}$ and $C_{\bot}^{fs}$ are finite terms which are
too lengthy to be written here.
\section{Conclusion}
Dijet production in DDIS at HERA was recently analyzed~\cite{Aaron:2011mp}. A precise comparison of
dijet versus triple-jet production, which has not been performed yet at HERA~\cite{Adloff:2000qi}, would be of much interest. Investigations of the azimuthal distribution of dijets in diffractive photoproduction performed by ZEUS~\cite{Guzik:2014iba} show sign of a possible need for a 2-gluon exchange model, which is part of the shock-wave mechanism. Our calculation could be used for phenomenological studies of those experimental results.
Complementary studies could be performed at LHC with UPC events. A full quantitative first principle analysis of this will be possible after completing our program of computing virtual corrections to the $\gamma^* \rightarrow q\bar{q}$ impact factor~\cite{Boussarie:prep}, for which we have provided here intermediate results.
\section*{Acknowledgments}
We thank Ian Balitsky, Cyrille Marquet and St\'ephane Munier for discussions. Andrey~V.~Grabovsky acknowledges support of president scholarship 171.2015.2,
RFBR grant 13-02-01023, Dynasty foundation, Metchnikov grant and University Paris Sud. He
is also
grateful to LPT Orsay for hospitality while part of the
presented work was being done. Renaud Boussarie thanks RFBR for financial support
via grant 15-32-50219.
This work was partially supported by the ANR PARTONS (ANR-12-MONU-0008-01), the COPIN-IN2P3 Agreement and the Th\'eorie-LHC France Initiative. Lech Szymanowski was supported by a grant from the French Ambassy in Poland.
\vskip.2in
|
1,116,691,497,538 | arxiv | \chapter{Audience}
\label{chapter_1}
\large This document is intended to explain the Wireless Backhaul Links Clustering between point to point ethernet transmission links.
With the word clustering the first thing comes to
mind is bundling bunch of physical links to make a single
logical link. While this definition holds true, it is imperative
to know it’s implementation in conjunction with certain key
factors. Upcoming 5G technologies where wireless backhaul
links for point to point wireless Ethernet transmission may
play a vital role in backbone networks, the proliferated needs
of providing higher data rates should be met. With that
demand, challenge of resilience, performance, scalability,
maintainability, and manageability will rise as well. An
optimal solution must be found which can reduce the failover
points, ability to perform upgrade and downgrade with
minimal downtime, and most importantly solution must
overcome all the challenges faced in wireless link
environment. A well tested approach has been presented in
this paper which will accomplish all the mentioned points.
This method will make the clustered wireless backhaul links
more valuable and throughput will increase significantly.
This document targets audience of the class who are taking taken COEN - 332 (Wireless/Mobile Multimedia Networks), as fundamentals to understand this document was explained in the class.
Additionally, lectures coverd layer 2/3 and wireless protocols for control, management and data plane required for this project report.
\tableofcontents{}
\listoffigures
\listoftables
\mainmatter
\chapter{Introduction}
\label{intro}
Wireless Ethernet Transmission is a great way of
extending the backbone network without deploying any
physical network, specially fiber. This technology can be
deployed in any form of topology and network. After FCC
rolled out extra free frequency bands for instance 11GHz
and 5GHz for commercial use, its demand is on rise \cite{goovaerts2016fcc}.
Today Internet Service Providers (ISP), Enterprise
Networking companies are relying on this technology a lot
due to numerous factors such as Cost, Terrains,
maintenance, easy to upgrade, and most importantly easy
to troubleshoot.
With upcoming 5G technologies where connecting 5G
access points to the backbone networks will require to have
a relay link in between to keep the data rate constant for
long links. Wireless Ethernet Transmission technology
will be handy to serve as a middleman in between
backbone networks and 5G access points, where the link is
just too long and data rate most likely will suffer without a
Wireless Ethernet Transmission backhaul link. Another
application of Wireless Transmission Network is campus.
In campus, often it is hard to deploy fiber to extend the
network to other buildings. Since it is in short range, so the
interference wouldn’t matter that much. These backhaul
links comes handy. Fiber gets dropped by the ISP to some places and then the network gets extended by these
backhaul links.
\begin{figure}[t]
\centering
\includegraphics[width=0.6\linewidth]{figures/Algorithm1.png}
\captionsetup{font=footnotesize}
\caption{Primary and Primary-2 selections
}
\label{fig:primaryPrimary2}
\end{figure}
In all these above-mentioned applications, we would
need a higher amount of data rate with reliability and
ability to be up for maximum amount of time. With the
help of current MIMO technologies, specially with IEEE
802.11ac, if we have 2 RF chains, 4 Streams then the max
data rate (Consider optimal RF conditions) is around
1Gbps \cite{Mimosa_B5}. With such high demand of data rate, 1Gbps
will not be sufficient and hence will not be a suitable
solution after all. In 5G the data rates will be way more
than 1Gbps, deploying Wireless Ethernet Transmission
Backhaul links in 5G will not suffice the purpose. An
optimal solution is needed to make data rates higher with
the existing infrastructure. Cluster will bundle multiple
Wireless Ethernet Transmission Backhaul Links and will
become a logically single link to provide higher data rates
needed \cite{802_1X}. There will be a lot more to be explained and
discussed with clustering, few are outlined below:
\begin{itemize}
\item Control Plane messages in unit selections and
assigning duties
\item Load Balancing across links
\item High Availability
\item Flow consistency
\item State replication
\item Configuration replication
\item Management plane, Control plane, Data plane traffic
handling
\end{itemize}
This solution will provide multiple Gbps data rate but also
will provide a scalable, resilient, highly available, easy to
upgrade with minimal disruption solution.
\chapter{Concepts and Explanation}
\large This section will explain the concepts and their relevance in clustering in wireless backhaul links.
\begin{figure}[t]
\centering
\includegraphics[width=0.5\linewidth]{figures/newly_joined_unit_selection_process.pdf}
\captionsetup{font=footnotesize}
\caption{Newly joined unit selection process
}
\label{fig:newlyJoinedUnit}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.7\linewidth]{figures/Physical_topology.pdf}
\captionsetup{font=footnotesize}
\caption{Physical Topology. AP = Access Point, STA = Station
}
\label{fig:physical_topology}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.8\linewidth]{figures/Logical_topology.pdf}
\captionsetup{font=footnotesize}
\caption{Logical Topology. AP = Access Point, STA = Station
}
\label{fig:logical_topology}
\end{figure}
\section{Cluster Formation}
Cluster will be a daemon running on each radio. Which
will bind all the radios wireless links. This will appear one
link for south and north bound devices and will aggregate
the throughput. Cluster will have failover mechanism, load
balancing and an algorithm for communication. Topology
is needed to understand it as shown in figure \ref{fig:physical_topology} and figure \ref{fig:logical_topology} which
shows the difference between physical topology and the
logical topology.
\section{Bootstrap Configuration and Cluster Members}
A unit consists of access-point and station. Each unit
requires minimal bootstrap configuration which includes
cluster name, cluster control link interface, management
cluster link interfaces, data path link interfaces and local
pool of IP address reserved for clustering.
The first unit configured as a bootstrap configuration will
be primary unit and rest become “secondaries”. These are
initial roles; The primary unit assignment will also depend
on the priority set in the bootstrap configuration. Priority
can go from 1-100, where 1 is the highest priority. All
other members are secondary units except one unit with
second highest priority will be primary-2 and secondary
both. Algorithm (figure \ref{fig:primaryPrimary2}) shows the complete scenario. In typical scenario, the very first unit added becomes
the primary, it is because so far this is the only unit present
in the cluster. Apart from bootstrap configuration, all
configuration is on primary only; replication of
configuration on secondary unit starts. When it comes to
physical assets, for instance interfaces, primary’s
configuration is mirrored to secondary unit. For example,
if Ethernet 1 is configured as inside and Ethernet 2 is
configured as outside, then these interfaces on secondary
unit will be inside and outside respectively.
Primary unit selection is based on request message which
consists of priority set by user. As soon as a new unit is
joined in the cluster, an election request will be sent on a
multicast address 224.1.0.10 which is reserved for this
purpose. All units in cluster will listen to this IP and only
primary unit at that moment will reply to the newly joined
unit. If the priority of newly joined unit is greater than the
primary unit then newly joined unit will set itself to
primary (Please see figure \ref{fig:newlyJoinedUnit}). If newly joined unit doesn’t receive any response
back from primary unit then it sets itself and sends a
multicast message to 224.1.0.11 which will force every
unit to set them to secondary. In the case of tie of priorities
(especially the highest priority), different parameters such
as unit name and serial number are used to determine the primary unless it is down or stops responding for some reason, this will trigger a new unit selection
\begin{figure}[t]
\centering
\includegraphics[width=0.8\linewidth]{figures/Spanned_etherchannel_with_lacp.pdf}
\captionsetup{font=footnotesize}
\caption{Spanned EtherChannel with LACP
}
\label{fig:SpannedEtherChannel}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.8\linewidth]{figures/Individual_router_interface_mode.pdf}
\captionsetup{font=footnotesize}
\caption{ Individual router interface mode
}
\label{fig:IndividualRouterInterfaceMode}
\end{figure}
\section{Cluster Interfaces}
All the wireless backhaul links can either be configured
as Spanned EtherChannel or Individual interfaces. All
links has the configured one type only. Spanned
EtherChannel is recommended. One or more links can be
grouped into Spanned EtherChannel that spans all links in
the cluster. Both routed and transparent modes are present
in Spanned EtherChannel. A single IP address is provided
to routed interface if the EtherChannel is configured in
routed mode, on the other hand, transparent mode is
something when we have an IP assigned to BVI (Bridge
virtual Interface) not to the member interface of bridge
group. This will inherit the load balancing as a part of basic
operation.
In Spanned EtherChannel shows in figure \ref{fig:SpannedEtherChannel}, we will
combine all the interfaces into a single logical link using
LACP. This interface will become a bundle of all the
interfaces attached to it. This EtherChannel will spans all
units in the cluster. It is recommended as EtherChannel
aggregates traffic from all the available active interfaces in
the channel. In this mode, all units use the same VIP
(Virtual Internet Protocol address) assigned to the cluster
and the same MAC used in EtherChannel of the cluster.
Another mode is individual routed mode (Figure \ref{fig:IndividualRouterInterfaceMode}) where each link
will maintain its own IP on each data interface. Since this
approach doesn’t use any aggregation link to load balance
and implement any fail over, dynamic routing is needed to
load balance. In individual mode, each radio maintains its
own routing adjacency. The disadvantage of this is slower
convergence and higher processor utilization due to each
unit maintaining its own routing table. In spanned
EtherChannel mode, the primary access point and station
runs dynamic routing. Routing and ARP tables are
synchronized to the secondary units.
A round trip time of 20ms and maintaining a reliable
cluster control link functionality is important. To make
sure that this condition is intact, a ping pong icmp
messages goes from every secondary units to master units.
Cluster control link is the most important link in the cluster
and will have a dedicated interface for cluster control link.
If a cluster control link is down, then the cluster will be
down and it must be brought up by solving the problem
which led it to down and then joining them one by one
manually.
\section{Monitoring units }
To make sure all units are healthy and interfaces are up
and running, a monitoring system is placed. Primary unit
is responsible to monitor every secondary unit by sending
a keepalive message periodically over the CCL (Cluster
Control Link), this monitoring period is configurable. In the event of unit health check failure, it will be removed
from the cluster. Every unit is supposed to send a keepalive message
(UDP multicast packet) (Figure \ref{fig:KeepAliveMessageFormat } to primary unit. Message format:
\begin{itemize}
\item Keepalive message header
\begin{itemize}
\item Type
\item Version
\item Length
\end{itemize}
\item Selection Info Component
\begin{itemize}
\item Unit Priority
\item Serial Number
\item Role (Primary Standby or Secondaries)
\end{itemize}
\item Radio Info Component
\begin{itemize}
\item Mode (Spanned EtherChannel or Individual Router Interface
\item Radio Type (Access Point or Station)
\item SNR
\item Load Balancing Weight
\end{itemize}
\end{itemize}
Here type is the keepalive (CLUSTER\_KEEPALIVE).
Version is 1 and length is the length of the message not
including the header. The message has two more sections
apart from the header. Section info component and Radio
Info component.
Figure \ref{fig:SelectionInfoComponent} shows the message header. Here type is the SELECTION\_INFO\_COMP, length is
the length of the selection info component. Unit priority is
the priority assigned to unit while configuring cluster.
Serial Number of the unit and Role of the unit whether it
is Primary standby or a secondary unit. This section will
let primary unit know about configurations of other units.
Figure \ref{fig:RadioInfoComponent} shows the radio info component message header.
Here the type is RADIO\_INFO\_COMP.
Length is the length of radio info component. It includes
Mode operation whether it is spanned EtherChannel or
Individual router interface. If primary unit finds that
received packet has mismatch of mode of operation, then
it excludes the unit from cluster and sends a unicast
message to forcefully make the unit leave cluster. A high level message flow is depicted in figure \ref{fig:MessageFlowtoPrimaryUnit }.
\section{Monitoring Interfaces and failure status}
As soon as the health monitoring is turned on, all physical
interfaces are monitored, including the main EtherChannel
and redundant interface. An option will be provided to user
to disable the health monitoring per interface. For instance, an EtherChannel is considered to be failed if all the units
in the EtherChannel are failed. If this happens, the
EtherChannel will be removed from the cluster but this
will depend on the minimal port bundling settings. For a
single unit, it will be considered removing from the cluster
if all monitored interfaces fail. The amount of time
removing a unit from the cluster will depend on the type of
interface and whether a unit is established or just joining
the cluster. For any EtherChannel spanned or not, if an
interface is down on an established member, primary unit
removes it after 9 seconds. For first 90 seconds, primary
unit doesn’t monitor any interfaces for the unit which just
joined the cluster. This means a primary unit will not
remove it if the interface state changes during that 90 secs.
In non-EtherChannel, the unit is removed in 500 ms
regardless of the member state. In the case of failure of a
unit, the connection belongs to that unit is seamlessly
transferred to other units and state information for the
traffic flows are shared over the CCL (Cluster Control
Link). In case of primary unit failure itself, other member
with highest priority (Which means unit with the lowest
number priority) takes over. Failed primary unit, after
recovery automatically tries to join the cluster. If rejoining
the cluster of primary unit is failed then all data interfaces
are shut down and only management interface can receive
and send traffic. This management interface remains up
using the IP address the unit received from the cluster IP
pool.
\section{Cluster Rejoining}
The primary factor to join back a unit after being
removed from the cluster is the reason why the unit was
removed. There are three methods how a unit can rejoin
the cluster.
\begin{itemize}
\item Failed cluster control link when initially
joining—After the problem is resolved with the
cluster control link, unit must be manually
rejoined the cluster by re-enabling clustering at
the console port by entering cluster group name,
and then enable.
\item Failed cluster control link after joining the
cluster—Unit automatically tries to rejoin every
5 minutes, indefinitely. This behavior is
configurable.
\item Data Interface Failure - Unit will try maximum of
4 attempts on each 5 minute to rejoin the cluster.
After resolving the issue with the data interface,
a manually enabled clustering is required by
entering cluster group name. This behavior is
configurable.
\end{itemize}
\begin{figure}[t]
\centering
\includegraphics[width=0.8\linewidth]{figures/ClusterMessageHeader.png}
\captionsetup{font=footnotesize}
\caption{Cluster message header.
}
\label{fig:ClusterMessageHeader}
\end{figure}
\begin{table}[h]
\caption{Extra information in connection preservation}
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
\label{SampleTable}
Traffic & State Support & Nodes \\ \hline
Up Time & Yes & \begin{tabular}[c]{@{}c@{}}Keeps track of the \\ system up time\end{tabular} \\ \hline
ARP Table & Yes & \begin{tabular}[c]{@{}c@{}}Individual interface \\ routed mode only\end{tabular} \\ \hline
MAC Address Table & Yes & \begin{tabular}[c]{@{}c@{}}Individual Interface \\ Router Mode Only\end{tabular} \\ \hline
User Identity & Yes & \begin{tabular}[c]{@{}c@{}}Includes AAA (authentication)\\ and radius server related information\end{tabular} \\ \hline
SNMP Engine ID & No & ------------------------------ \\ \hline
VPN Site-Site & No & ------------------------------ \\ \hline
\end{tabular}
\end{center}
\end{table}
\section{Replication of Data path connection state}
There is one owner and one backup owner for every
connection in the cluster. Ownership doesn’t get
transferred to backup owner in case of a failure instead a
provision of TCP/UDP state information gets stored in it
so that the connections can be transferred seamlessly. If for
some reason the owner becomes unavailable, the very first
unit to receive packets from the connection contacts the
backup owner for the relevant connection state and then the backup owner becomes a new owner for this
connection. Obviously, there would be some traffic which
would require information above the TCP or UDP layer.
Replication message is defined below. Cluster message
header is already defined in figure \ref{fig:ClusterMessageHeader}. In the type field of
figure \ref{fig:ReplicationMessageFormat }, type field will be mentioned in
CLUSTER\_REPLICATION, version will be 1 and length
is the length of the packet apart from the cluster message
header.
Often the traffic will be directed by the original unit
where the traffic is flowing. Often UDP/TCP or higher
layer information not needed to be transferred to the
backup unit or in case of transfer to back up unit the traffic
could be from the same origin and destined to the same
destination. Separating the IP layer info (message shown in figure \ref{fig:IPLayerConnectionInfo } from the higher
layer can give flexibility to just check the source and
destination IP and not needed to open the rest of the packet.
This will make the transfer faster.
Upper layer connection info (Message shown in figure \ref{fig:UpperLayerConnectionInfo} will contain vital
information about Upper layer as shown below. Most of
the time, the information doesn’t change and failover will
only have to see the layer-3 packet and forward the upper
layer information. Also, table \ref{SampleTable} shows what extra
information in preserved in connection and its transfer.
\begin{figure}[t]
\centering
\includegraphics[width=0.8\linewidth]{figures/SelectionInfoComponent.png}
\captionsetup{font=footnotesize}
\caption{Selection Info Component
}
\label{fig:SelectionInfoComponent}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.8\linewidth]{figures/RadioInfoComponent.png}
\captionsetup{font=footnotesize}
\caption{Radio Info Component
}
\label{fig:RadioInfoComponent}
\end{figure}
\section{Cluster Management}
To manage the cluster, a separate network must be
created apart from cluster control network. A spanned
EtherChannel or individual interfaces can be used as
management network. Even if spanned EtherChannel is
used for data still for management purposes recommended
approach is individual interface. In it each unit can be
connected directly to individual interfaces (if necessary),
on the other hand a spanned EtherChannel interface only allows to be connected to primary unit via remote
connection.
In case of individual interface, cluster IP address is fixed
which always belongs to the current primary unit. A range
of IP address can be configured for each unit which of
course includes the current primary which can use a local
address from the range. Having a main cluster IP provides
management consistency access to an address; In case of
role change of primary unit to other unit in cluster, this
main IP moves to the new unit which will result in the
seamless transfer of the management of the cluster. For
routing, the local IP is used and it is also useful for
troubleshooting. For an instance, a cluster is managed by
the main cluster IP and this IP is always attached to the
current primary unit but to manage an individual member,
a local IP can do the job. There are outbound management
traffic service like syslog, TFTP etc, each unit (includes
primary unit too) uses this local IP to connect to the server.
In the event of Spanned EtherChannel interface, one main
IP is configured which is attached to the primary unit.
Secondary units will not be allowed to connect directly
using the EtherChannel interface and hence an individual
interface is recommended for configuring the management
interface and user will be easily able to connect to single
unit.
\section{Load Balancing Method}
Spanned EtherChannel have bundled links which load
balances and failovers in case of a link failure. There is
more than one physical links are bundled into the Spanned
EtherChannel (At Max 6), the primary aim is to have
traffic across all the links equally however a IP stickiness
is desired as well. Packets with same destination and
source IP should take the same path every single time for
IP stickiness to be maintained. This will give consistency
and ease of troubleshoot in case of data packet inspection
or loss. In the load balancing algorithm for EtherChannel,
a hash function (Algorithm shown in figure \ref{fig:HashAlgo} calculates a hash value which determines
which link packet will go out. The algorithm for hashing will be “symmetric” which
means the packet from both the directions will have the
same hash, and will be sent to the same unit in spanned
EtherChannel. By default, Source IP and destination IP are
being used and it is recommended too. Another restriction
is to use same type of radio when connecting the units to
the switch so that hashing algorithm applied to all the
packets. In individual routed mode load balancing, each radio
will maintain its own IP address. One method of load
balancing is Policy Based Routing. Traditional Policy
based Routing is based on policy which is applied to
ingress and egress interface based on access-list which will
allow certain type of traffic to be passed. Policy is a map
which allows certain type of traffic to be passed from
certain units. Since it is static, chances are it may not
achieve the optimal load balancing results. Recommended
way of configuring policy is to make sure that forward and
return packets of a connection are directed to the same
physical unit.
\section{Connection Management and Formation}
Roles of connections determine how they are handled in
high availability and normal operation. Connections can
be load balanced among multiple members of the clusters
too. For connection management, we will distribute the
unit roles of three different type:
\begin{itemize}
\item Proprietor
\item Organizer
\item Forwarder
\end{itemize}
from the connection, the director chooses a new owner
from those units from the connection, the director chooses
a new owner from those units.
Function of Organizer is to handles owner lookup requests
which are coming from forwarders and maintain a connection state to serve as a backup if the owner fails.
When a Proprietor receives a new connection, based on our
hashing algorithm of source/destination IP a director is
chosen and a message is sent to the organizer to register
the new connection. If packet arrives at any unit other than
the owner then unit queries the director about which unit
is the owner so that forwarder can forward the packet to
the owner. A connection has only one Organizer. If an
Organizer fails, the owner chooses a new Organizer. Function of “Forwarder” is to forward packets to
Proprietor. If packet received by Forwarder doesn’t own
by it, it goes ahead and queries the organizer for the
proprietor and then it establishes a connection with the
Proprietor so that Forwarder can forward packets received
by it in future. An Organizer can also be a forwarder. Let’s
take an example, if a Forwarder receives a SYN-ACK
packet, it can derive the Proprietor directly from a SYN
cookie in the packet, so it does not need to query the
Organizer. There is a used case when TCP sequence
randomization is not enabled or disabled intentionally,
SYN Cookie is no longer useful and query to Organizer is
required. For short lived flows such as ICMP, DNS etc,
instead of querying, the forwarder immediately sends the
packet to the Organizer, which then sends them to the
Proprietor. A connection can have more than one
forwarder. A good load balancing algorithm is needed
where there are no forwarders and all packets of a
connection are received by the proprietor, our ECMP in
Spanned EtherChannel interface provides this. When a
new connection is directed to a member of the cluster via
load balancing, that unit owns both directions of the
connection. For any connection, if packets are arrived at a
different unit, they will be forwarded to the owner over
CCL (Cluster Control Link). If more optimization is
needed then an external load balancing needs to be in place
for both directions of the flow to arrive at the same unit, it
is redirected back to the original unit.
\begin{figure}[t]
\centering
\includegraphics[width=0.8\linewidth]{figures/MessageFlowToPrimaryUnit.pdf}
\captionsetup{font=footnotesize}
\caption{Message Flow to Primary Unit
}
\label{fig:MessageFlowtoPrimaryUnit }
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.6\linewidth]{figures/KeepAliveMessageFormat.png}
\captionsetup{font=footnotesize}
\caption{Keepalive Message format
}
\label{fig:KeepAliveMessageFormat }
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.7\linewidth]{figures/UpperLayerConnectionInfo.png}
\captionsetup{font=footnotesize}
\caption{Upper Layer Connection Info
}
\label{fig:UpperLayerConnectionInfo}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.8\linewidth]{figures/Algorithm2.png}
\captionsetup{font=footnotesize}
\caption{Hash Algorithm for Load Balancing
}
\label{fig:HashAlgo}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.8\linewidth]{figures/DataPacketFlow.png}
\captionsetup{font=footnotesize}
\caption{Data Packet Flow
}
\label{fig:DataPacketFlow}
\end{figure}
\section{Data Flow}
Figure \ref{fig:DataPacketFlow} shows the data packet flow.
Proprietor role is assigned to unit that initially receives
the connection. The owner maintains the TCP state and
processes packets. A connection has only one owner. If the
original owner fails, then when new units receive packets will become Proprietor.
\begin{itemize}
\item The SYN packet originates from the client and it is
delivered to one Unit (based on the load balancing mechanism). This unit becomes the Proprietor,
proprietor creates the flow, encodes owner
information into a SYN cookie, and forwards the
packet to the server.
\item Since the forwarder doesn’t own the connection, it
uses SYN cookie to decode the owner information,
it creates a forwarding flow to the owner, and
forwards the SYN-ACK to the Proprietor. The
Proprietor sends a state update to the Organizer, and
forwards the SYN-ACK to the client.
\item The Organizer receives the state update from the
Proprietor, creates a flow to the Proprietor, and
records the TCP state information as well as the
Proprietor. The Organizer acts as the backup owner
for the connection.
\item After this any subsequent packets delivered to the
forwarder will be forwarded to the proprietor. If
packets are delivered to any additional unit then a
query to director is needed for the owner and to
establish a flow. Any state change for the flow result
will be a state update from the proprietor to
organizer.
\item Unbalanced flow distributions resulted from load
balancing capabilities of the upstream and
downstream network can be mitigated by redirect
new TCP flows to the other units, this can be
configured while no existing flows will be moved to
the other units.
\end{itemize}
\begin{figure}[t]
\centering
\includegraphics[width=0.5\linewidth]{figures/ReplicationMessageFormat.png}
\captionsetup{font=footnotesize}
\caption{Replication message format
}
\label{fig:ReplicationMessageFormat }
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.8\linewidth]{figures/IPLayerConnectionInfo.png}
\captionsetup{font=footnotesize}
\caption{IP Layer connection info
}
\label{fig:IPLayerConnectionInfo }
\end{figure}
\chapter{Simulation Model}
We simulated wireless backhaul link by using CNET\cite{Simulator} (shown in figure \ref{fig:simulation-model}.
CNET use 5Ghz bandwidth.
CNET provides APIs (predefined) to set up its own simulation model.
We used customized parameters below for simulation.
\begin{itemize}
\item 1472Bytes - MTU Size
\item 1 - 2 Gbps Bandwidth
\item 5GHz - WLAN Frequency
\item 20dBm - WLAN Tx Power
\item 10dBi - WLAN Tx Antenna Gain
\item 10dBi - WLAN Rx Antenna Gain
\end{itemize}
\begin{figure}[t]
\centering
\includegraphics[width=0.65\linewidth]{figures/Simulator.pdf}
\captionsetup{font=footnotesize}
\caption{Simulation model
}
\label{fig:simulation-model}
\end{figure}
These tests are simulated while keeping some
parameters constant.
\begin{itemize}
\item 80Mhz - Channel Width
\item 2 (Number of Streams - 4) - No. of channels
\item 30dBm - Tx Power
\item 36dB - Signal to Noise Ratio (SNR)
\item 0.5\% - Packet Error Rate (PER)
\item (-17) - Error Vector Magnitude (EVM)
\item 8 - MCS Index
\item 256QAM - Modulation Technique
\end{itemize}
PHY rates demonstrated how much ethernet frames can propagate through the wireless backhaul links in optimal conditions.
Additionally, MAC rates are more fair comparison to the TCP throughput than PHY rates.
We already mentioned in previous section about how MAC rates are calculated.
After changing PHY rates, we will note how MAC rates are changing as well.
This will demonstrate the real world scenario about MAC rates, depends on different RF condition.
We used iperf for performing TCP throughput test.
Iperf gives us mechanism to customzie our TCP parameters.
We will set our iperf
parameters as: number of TCP Connection – 50 and TCP Window size - 64K.
PHY Rates shows how much Ethernet frames can cross
the wireless link under optimal condition. However, MAC
rates are dependent on PHY Rates and can give us fair idea
of how TCP throughput varies. Formula for MAC rates:
\begin{equation}
MAC_T = PHY_T * Cycle_T * Efficiency_T
\end{equation}
Where:
Tx MAC Rates = MAC\textsubscript{T} \\
Tx PHY Rates = PHY\textsubscript{T} \\
MAC Duty Cycle = Cycle\textsubscript{T} \\
Tx MAC Efficiency = Efficient\textsubscript{T} \\
PHY\textsubscript{R} $*$ Cycle\textsubscript{R} $*$ Efficiency\textsubscript{R} = MAC\textsubscript{R} \\
Rx MAC Rates = MAC\textsubscript{R} \\
Rx PHY Rates = PHY\textsubscript{R} \\
Rx MAC Duty Cycle = Cycle\textsubscript{R} \\
MAC Efficiency = Efficiency \textsubscript{R} \\
A simple iperf test is performed to collect the TCP
throughput rates. Iperf provides room to define TCP
parameters. For the sake of simplicity, we will set our iperf
parameters as defined below:
\begin{itemize}
\item Number of TCP Connection – 50 and TCP Window size - 64K
\end{itemize}
\begin{figure}[t]
\centering
\includegraphics[width=0.65\linewidth]{figures/MacRatesThroughputWithoutClustering.png}
\captionsetup{font=footnotesize}
\caption{MAC Rates and Throughput without
Clustering
}
\label{fig:MACRatesThroughputWithoutClustering}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.65\linewidth]{figures/ThroughputWithClustering.png}
\captionsetup{font=footnotesize}
\caption{Throughput With Clustering
Clustering
}
\label{fig:ThroughputWithClustering}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.65\linewidth]{figures/ThroughputWithOneUnitRemoved.png}
\captionsetup{font=footnotesize}
\caption{Throughput with one unit removed
}
\label{fig:ThroughputWithOneUnitRemoved}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.65\linewidth]{figures/LatencyFromPrimarytoSecondaries.png}
\captionsetup{font=footnotesize}
\caption{ Latency from primary to secondaries.
}
\label{fig:LatenciesFromPrimaryToSecondaries}
\end{figure}
\chapter{Conclusion}
With CNET APIs infrastructure, and changing duty cycle and efficiency, we could simulate the wireless backhaul links.
Different scenarios were taken into account.
Figures demonstrated that throughput increased.
Additionally, one unit was brought down and we could see the effect on throughput.
In figure \ref{fig:MACRatesThroughputWithoutClustering}, MAC rates and throughput are shown. This
is without clustering and units are working independently
and transmission. These PHY rates are being used in the
rest of the test cases we captured. Individual units give
throughput around 0.8 to 1.3 Mbps. In figure \ref{fig:ThroughputWithClustering}, the throughput is with clustering when all units are working
together and projecting as a single logical unit. Figure \ref{fig:ThroughputWithOneUnitRemoved}
shows throughput when a single unit is removed. Figure \ref{fig:LatenciesFromPrimaryToSecondaries} results shows that throughput got increased
and latency was under 20 ms from primary to secondaries.
When we have one unit removed/failed the throughput got
decreased considerable.
Throughput shown in the result proves that with
clustering we have more resilient, robust, and highly
available solution for providing higher throughput. Even if
we have one unit removed, throughput doesn’t decrease
significantly
\pagebreak
\nocite{*}
\bibliographystyle{unsrt}
|
1,116,691,497,539 | arxiv | \section{Introduction}
The first solar neutrino experiment, the Cl-Ar radiochemical detector \cite{ClAr} built by Ray Davis, Jr. and his colleagues, provided the only direct information about solar neutrinos in the seemingly endless interval between 1968 and 1988. Towards the end of that period the Kamiokande proton-decay detector was being outfitted with new electronics in order to lower the threshold sufficiently to see the neutrinos from $^8$B decay in the sun, which it did in 1989 \cite{Hirata:1989zj}. That period also saw initiation by Vladimir Gavrin's group in the Soviet Union at the newly built Baksan Laboratory of a gallium radiochemical experiment, `SAGE', to see if the $pp$ neutrinos were really there. A similarly motivated effort in the US was abortive, and Till Kirsten launched the Gallex experiment with a mainly European team at Gran Sasso. The ``Standard Solar Model'' (SSM) constructed by Bahcall \cite{Bahc5} predicted that Ga experiments should see a neutrino capture rate of 136 SNU (captures per $10^{36}$ target atoms per second). A rate above 69 SNU was thought to be indicative of errors in the SSM, while a smaller rate would require new neutrino physics. That division, based on the rate in Ga of the $pp$ reaction by itself, today seems naive, and with much merriment Nature produced exactly 69 SNU in the Ga experiments. SAGE reported its first results in 1990 \cite{Gavrin:1990wy,Abdurashitov:1994bc}, and Gallex in 1992 \cite{Anselmann:1993ct}. Every experiment reported rates far below the expectations of the SSM.
The idea that neutrino physics might be responsible for the solar neutrino problem \cite{Gribov:1968kq} was initially greeted with skepticism. Perhaps mixing might take place, but quark mixing angles were known to be small, and why should neutrinos be any different? It was revived by the theoretical discovery of matter enhancement, the Mikheyev-Smirnov-Wolfenstein effect \cite{Mikheyev:1989dy,Wolfenstein:1977ue}, which allowed small vacuum mixing angles to be greatly amplified in matter. (Once again, Nature laughed and gave us nearly maximal mixing {\em and} the MSW effect.)
In 1985 Herb Chen, George Ewan, and collaborators proposed the construction of a large, real-time heavy-water Cherenkov detector to measure the $^8$B rate in both charged-current and neutral-current interactions, thus decoupling the neutrino physics from the astrophysics \cite{Chen:1985na,Aardsma:1987ph}. The Sudbury Neutrino Observatory reported its first results in 2001 \cite{Ahmad:2001an}, comparing the pure CC rate to the elastic scattering (ES) rate seen in the Super-Kamiokande detector \cite{SK}. The conclusion was that indeed the predicted neutrino flux was all there but that 2/3 of the electron neutrinos had converted to mu and tau neutrinos on their way to earth. With steadily increasing precision, the measurements showed also the presence of neutrons liberated by neutral-current interactions with deuterium, again at 3 times the rate that would be expected from the CC data in the absence of neutrino oscillations.
The solar neutrino problem has been resolved, and neutrino flavor change demonstrated in an appearance experiment. Neutrino oscillations and mass explain the observed effects well. The detailed model developed by Bahcall and many other astrophysicists was found to be astonishingly good, predicting the central temperature of the sun to a stunning 1\% accuracy. The Standard Model of particles and fields must now be modified to include massive neutrinos.
What does the future hold for solar neutrinos? Only the radiochemical experiments provide information about the part of the spectrum from 0 to 5 MeV, where $>$ 99\% of the neutrinos reside. To consider the consequences of assuming all is well there, one need only imagine Davis and Bahcall making the same assumption in 1965 and deciding to work on something else. As it turns out, that region is where the transition from matter enhancement to vacuum oscillations takes place. The detailed behavior of the transition is quite sensitive to the presence of nonstandard interactions, and a spectroscopic measurement of neutrinos there would be definitive. Another interesting objective is to realize that, with sufficiently precise data, one could directly test the relationship between neutrino luminosity and radiant energy luminosity. The fact that neutrinos from the center of the sun reach earth in 8 minutes, while photons take some 40,000 years \cite{Bahcall:1989ks}, would yield an eerie look at the future of the sun's life-giving energy.
\section{Solar Neutrino Experiment Results}
Three experiments, SAGE, Super-Kamiokande (SK), and SNO are currently in operation. A fourth, KamLAND, is a terrestrial reactor antineutrino experiment, but it provides information intimately related to the solar neutrino data. Taken together with the earlier results from Cl-Ar, Gallex, and GNO, the data from those experiments provide a remarkably precise picture of the mixing of two neutrinos and of the flux of high-energy neutrinos from the sun.
\subsection{SAGE}
Beginning in January, 1990 with 30 Mg of Ga metal, SAGE \cite{SAGE1} has continuously recorded solar neutrinos via the reaction,
\begin{center}
\begin{tabular}{ll}
$ \nu_x + {^{71}{\rm Ga}} \rightarrow {^{71}{\rm Ge}} + e^- -0.233{\rm \ MeV}$ \hspace{0.5in} & \\
\end{tabular}
\end{center}
In 1995 the target mass was increased to approximately 50 Mg, which, together with numerous improvements in the low-background proportional counters used to detect the decay of 11-day $^{71}$Ge, has resulted in much improved statistical accuracy. Recent work has focussed on the testing and calibration of the experiment with intense artificially produced antineutrino sources of $^{51}$Cr and $^{37}$Ar \cite{SAGE3}. The feasibility of preparing an intense, pure source of $^{37}$Ar via $^{40}$Ca(n,$\alpha$){$^{37}$Ar} in a fast reactor as originally suggested by Haxton \cite{Haxton37} has been convincingly demonstrated. The results, however, are a little surprising (Fig. \ref{fig:SAGE}) and show a consistently lower rate in SAGE and Gallex for both $^{51}$Cr and $^{37}$Ar sources than is expected from the efficiencies and cross sections in use. The SAGE collaboration's conclusion \cite{SAGE2} is, ``The source experiments with Ga should be considered to be a determination of the neutrino capture cross section.'' Since the ground-state cross section is fixed from $\beta$ decay by detailed balance, the correction falls entirely on excited-state cross sections, which already play a relatively minor role in low-energy neutrino capture, and is therefore substantial.
\begin{figure}
\begin{center}
\begin{minipage}[t]{12 cm}
\epsfig{file=SAGESource.eps,scale=0.5}
\end{minipage}
\begin{minipage}[t]{16.5 cm}
\caption{Measured neutrino capture rates in Gallex and SAGE with sources of $^{51}$Cr and $^{37}$Ar \protect\cite{SAGE2}.
\label{fig:SAGE}}
\end{minipage}
\end{center}
\end{figure}
\subsection{Kamiokande and Super-Kamiokande}
Built as proton-decay experiments, SK and its predecessor Kamiokande have produced a remarkable view of the sun through detection of the elastic scattering of $^8$B neutrinos from electrons:
\begin{center}
\begin{tabular}{ll}
$ \nu_x + e^- \rightarrow \nu_x + e^-$ \hspace{0.5in} & (ES)\\
\end{tabular}
\end{center}
SK is very large, with a fiducial mass of 22.5 Gg. The collaboration has recently released a detailed paper \cite{SK2} on the 1496 days of solar neutrino data collected from April 1996 through July 2001. The experiment is at this writing shut down while the number of photomultipliers is restored to the original 11,000 it had before the Nov. 12, 2001 accident. It is anticipated that SK will return to operation with full coverage in the spring of 2006.
\subsection{SNO}
Heavy water (D$_2$O) permits three distinct reactions with solar neutrinos,
\begin{center}
\begin{tabular}{ll}
$\nu_e + d \rightarrow p + p + e^- -1.44{\rm \ MeV}$\hspace{0.5in} & (CC)\\
$ \nu_x + d \rightarrow p + n + \nu_x -2.22{\rm \ MeV} $ & (NC)\\
$ \nu_x + e^- \rightarrow \nu_x + e^-$ & (ES)\\
\end{tabular}
\end{center}
Beginning in November, 1999, the SNO experiment \cite{Boger:1999bb} has run in three configurations, pure D$_2$O \cite{SNO3}, D$_2$O with the addition of 0.195\% by weight of NaCl \cite{SNO4}, and D$_2$O with the deployment of an array of $^3$He-filled proportional counters. SNO has used a fiducial mass of 770 Mg. The last phase is still in progress and will be completed December 31, 2006, at which point the heavy water will be returned to its owners.
\subsection{KamLAND}
Built in the site originally occupied by Kamiokande I, KamLAND is a liquid-scintillator detector sensitive to antineutrinos from reactors in Japan, which fortuitously happen to be situated in a roughly circular pattern about 185 km in radius centered on Kamioka. The signal is very clean and free of most backgrounds thanks to the delayed coincidence between inverse beta decay and neutron capture:
\begin{center}
\begin{tabular}{ll}
$ \overline{\nu_e} + p \rightarrow n + e^+ -1.804{\rm \ MeV}$ \hspace{0.5in} & (Inverse beta)\\
$ n + p \rightarrow d + \gamma +2.223 {\rm \ MeV} $ & (n capture)\\
\end{tabular}
\end{center}
One background, ($\alpha$,n) induced by radon progeny $^{210}$Po, evades the strategy and required separate calculation and subtraction. The fiducial mass of KamLAND was initially chosen to be 408 Mg \cite{KL1}, and in a more recent analysis \cite{KL2} of the complete data set from March, 2002 to January, 2004, 543.7 Mg. The baseline and energy spectrum of the reactor antineutrinos give KamLAND the ability to make a pinpoint determination of the mass splitting $\Delta m_{12}^2$, whereas the solar experiments, both because of matter enhancement and the direct measurement by SNO of the CC/NC ratio, excel at determining the mixing angle $\theta_{12}$.
\subsection{Cl-Ar}
The historic experiment of Davis \cite{ClAr}, which recorded the reaction
\begin{center}
\begin{tabular}{ll}
$ \nu_e + {^{37}{\rm Cl}} \rightarrow {^{37}{\rm Ar}} + e^- -0.814{\rm \ MeV}$ \hspace{0.5in} & \\
\end{tabular}
\end{center}
using 615 Mg of C$_2$Cl$_4$, took its last data in 1998, but the results remain very important in modern analyses. The radiochemical experiments are integral and so do not by themselves give spectroscopic information, but in combination with other experiments their differing thresholds do yield a coarse spectroscopy. The Cl-Ar and Ga results together map out the transition from matter-enhanced to vacuum oscillations.
\subsection{Gallex and GNO}
The Gallex experiment in Gran Sasso ran with 30 Mg of Ga in the form of GaCl$_3$ in the period May, 1991 to January, 1997 \cite{Hampel:1998xg}. That Ga was inherited by the successor experiment GNO \cite{Altmann:2005ix}, which ran successfully from May, 1998 to April, 2003, when it became a casualty of an accidental release of pseudocumene into the environment at Gran Sasso. It is worth emphasizing how valuable it was for the scientific community to see the highly consistent results from Gallex and SAGE, obtained with very different technical approaches by spirited collaborations that would as soon have seen their competition shown to be in error. For many in the larger fields of particle and nuclear physics, this marked a turning point at which the possibility of new neutrino physics had to be taken seriously.
\subsection{Results}
Table \ref{tab:results} gathers in one place the results of solar neutrino experiments, KamLAND, and satellite measurements of the electromagnetic solar irradiance.
\begin{table}
\caption{Results from solar neutrino experiments, KamLAND, and the solar irradiance.}
\medskip
\begin{center}
\begin{tabular}{lrrrlc}
\hline
\hline
Measurement & value & stat. & syst. & units & Ref. \\
\hline
SNO CC/NC & 0.340 & 0.023 & $^{+0.029}_{-0.031}$ & & \cite{SNO4} \\
SNO NC+CC & 4186 & 65 & 244 & Events & \cite{SNO4} \\
SNO ES flux as if $\nu_e$ & 2.35 & 0.22 & 0.15 & $10^{6}$ cm$^{-2}$ s$^{-1}$ & \cite{SNO4} \\
SK ES flux as if $\nu_e$ & 2.35 & 0.02 & 0.08 & $10^{6}$ cm$^{-2}$ s$^{-1}$ & \cite{SK2} \\
Chlorine & 2.56 & 0.16 & 0.14 & SNU & \cite{ClAr} \\
SAGE & 67.2 & 3.7 & $^{+3.6}_{ -3.2}$ & SNU & \cite{SAGE2} \\
Gallex/GNO & 69.3 & \multicolumn{2}{c}{5.5} & SNU & \cite{Altmann:2005ix} \\
KamLAND $P_{ee}$ & 0.658 & 0.044 & 0.047 & & \cite{KL2} \\
Solar Irradiance & 85.31 & & 0.34 & $10^{10}$ MeV cm$^{-2}$ s$^{-1}$ & \cite{BP} \\
\hline
\hline
\end{tabular}
\end{center}
\label{tab:results}
\end{table}
\section{Physics from Solar and Reactor Neutrinos}
With the quite extensive and precise data now available, detailed analyses of neutrino physics and solar astrophysics can be made. Most analyses of neutrino oscillations, until recently, have been in the context of 2-neutrino mixing because the solar and atmospheric physics separate fairly cleanly.
\subsection{Two-mass Mixing}
In the context of two active mass eigenstates, a global analysis of solar and reactor data yields \cite{SNO4} for the joint 2-dimensional 1-$\sigma$ boundary,
\begin{eqnarray*}
\Delta m^2 & = & 8.0^{+0.6}_{-0.4}\times 10^{-5} {\rm \ eV}^2 \\
\theta &= & 33.9^{+2.4}_{-2.2} {\rm \ degrees}
\end{eqnarray*}
For the marginalized 1-$\sigma$ uncertainties, the results are:
\begin{eqnarray*}
\Delta m^2 & = & 8.0^{+0.4}_{-0.3}\times 10^{-5} {\rm \ eV}^2 \\
\theta &= & 33.9^{+1.6}_{-1.6} {\rm \ degrees}
\end{eqnarray*}
Mixing in the solar sector is certainly large, but at the same time maximal mixing is ruled out at more than $5\sigma$. There is residual model dependence in these results arising from the use of SSM fluxes for {\em pp}, {\em pep}, $^7$Be, CNO, and {\em hep} neutrinos. It is, however, quite small as can be seen in Fig. \ref{fig:SKcontour}, from \cite{SK2}. The difference between the innermost and middle contours is from the inclusion of data interpreted via the SSM. The $^8$B flux is allowed to float throughout.
\begin{figure}[tb]
\begin{center}
\begin{minipage}[t]{8 cm}
\epsfig{file=SKcontour.eps,scale=1}
\end{minipage}
\begin{minipage}[t]{16.5 cm}
\caption{The 95\% confidence-level contours for three scenarios: (outermost) KamLAND alone, (middle) KamLAND plus SNO and SK, and (innermost), also including other solar neutrino data and the SSM \protect\cite{SK2}.
\label{fig:SKcontour}}
\end{minipage}
\end{center}
\end{figure}
\subsection{Three-mass Mixing}
The electron neutrino in principle contains components of all 3 mass eigenstates:
\begin{eqnarray*}
U_{e1} & = & \cos\theta_{12}\cos\theta_{13} \\
U_{e2} & = & \sin\theta_{12}\cos\theta_{13} \\
U_{e3} & = & \sin\theta_{13} e^{-i\delta}
\end{eqnarray*}
but the small size of $\theta_{13}$, known to be less than 10 degrees, leads to the convenient and well-known simplification to two-mass mixing with a sacrifice in precision that is for most applications minor. However, solar neutrino and reactor experiments are reaching precisions of a few percent and it is germane to ask,
\begin{itemize}
\item Do solar neutrinos have anything to contribute to the determination of $\theta_{13}$, and,
\item Does inclusion of the third state and the attendant uncertainty in $\theta_{13}$ affect the determinations of the solar mixing parameters?
\end{itemize}
The large mass gap for 13 mixing and the low energies of solar neutrinos means that any effects in the sun from it are independent of energy. The matter resonance would occur at 190 MeV at the center of the sun. On this basis, Fogli {\em et al.} \cite{fogli2} give a relationship between the 3- and 2-neutrino scenarios,
\begin{equation}
P_{ee}^{3\nu}(\delta m^2,\theta_{12},\theta_{13}) = \sin^4\theta_{13} +
\cos^4\theta_{13}\cdot P_{ee}^{2\nu}(\delta m^2,\theta_{12})
\Big|_{N_e\to \cos^2\theta_{13} \,N_e}\ .
\label{e2}
\end{equation}
Qualitatively this consists of an energy-independent conversion of some $\nu_e$ flux into $\nu_2$ and $\nu_3$ via $\theta_{13}$, and a dilution factor of $\cos^2\theta_{13}$ included with the electron density since the beam is no longer prepared as exactly $\nu_e$ when it crosses the 12 matter resonance.
Atmospheric neutrinos give, again in a 2-mass mixing description, $\theta_{23} = 45 \pm 9$ degrees and $\Delta m_{23}^2 = 2.1_{-0.6}^{+1.3} \times 10^{-3}$ eV$^2$ (90\% CL) \cite{SK3}. A global analysis of all available data by Maltoni {\em et al.} \cite{maltoni} in 2003 summarized the situation and drew attention to the role of solar and KamLAND reactor neutrino data in limiting $\theta_{13}$. Figure \ref{fig:maltoni8} is from that work, superimposed with modern limits \cite{SK3} on $\Delta m_{23}^2$. One can see that near the low end of the mass range the tightest limits on $\theta_{13}$ were already coming from solar neutrinos and KamLAND. The relationship between these experiments and $\theta_{13}$ began to be explored even before results were available from KamLAND \cite{concha}. The right-hand panel is from a recent analysis \cite{Valle} including the latest KamLAND data \cite{KL2}.
\begin{figure}[tb]
\begin{center}
\begin{minipage}[]{16.5 cm}
\epsfig{file=Maltoni8.eps,scale=.45}
\epsfig{file=ValleTh13.eps,scale=1.6}
\end{minipage}
\begin{minipage}[t]{16.5 cm}
\caption{(left) Limits on $\theta_{13}$ from Chooz (lines, 90\%, 95\%, 99\%, and 3$\sigma$), and from Chooz+solar+KamLAND (colored regions) \protect\cite{maltoni}. (right) Limits updated for new data \protect\cite{Valle}.
\label{fig:maltoni8}}
\end{minipage}
\end{center}
\end{figure}
Solar and reactor neutrinos allow a separation of the 12 and 13 effects by virtue of matter enhancement, which singles out the 12 component in the solar data. The transition from no matter enhancement at low energies to full matter enhancement above 5 MeV changes $P_{ee}$ from $1- \frac{1}{2}\sin^22\theta_{12}$ to $\sin^2\theta_{12}$. This is illustrated schematically in \cite{roadmap}, from which the left-hand panel of Fig. \ref{fig:matter} is extracted.
\begin{figure}[tb]
\begin{center}
\begin{minipage}[]{16.5 cm}
\epsfig{file=matter.eps,scale=.9}
\epsfig{file=fogliPee.eps,scale=.5}
\end{minipage}
\begin{minipage}[t]{16.5 cm}
\caption{Matter enhancement (MSW) in the LMA region. (left) Schematic, from \protect\cite{roadmap}; (right) Actual from \protect\cite{fogli3}, with $\theta_{13}=0$.
\label{fig:matter}}
\end{minipage}
\end{center}
\end{figure}
The NC and ES measurements then define the total active flux, which, when compared to CC data, gives a ratio that depends on both $\theta_{12}$ and $\theta_{13}$ in the high-energy regime. The final input comes from the total active flux normalization at low energies, which is derived mainly from the luminosity constraint and the standard solar model. A comprehensive global analysis of all available oscillation data as of August, 2005, by the Bari group \cite{fogli3} gives the following results at the 95\% CL:
\begin{eqnarray*}
\sin^2\theta_{13} &=& 0.9_{-0.9}^{+2.3} \times 10^{-2} \\
\Delta m_{12}^2 &=& 7.92(1\pm0.09) \times 10^{-5} {\rm \ eV}^2, \\
\sin^2\theta_{12} &=& 0.314(1^{+0.18}_{-0.18}), \\
\Delta m_{23}^2 &=& 2.4(1^{+0.21}_{-0.26}) \times 10^{-3} {\rm \ eV}^2, \\
\sin^2\theta_{23} &=& 0.44(1^{+0.41}_{-0.22}),
\end{eqnarray*}
The determination of the oscillation parameters can be made in an essentially model-independent way, although in most global analyses, such as \cite{fogli3}, it is customary to include the low-energy solar neutrino data by calculating $P_{ee}$ against the standard solar model \cite{BS05}. As has been emphasized above (Fig. \ref{fig:SKcontour}), however, the low-energy solar neutrino data now play a very minor role in determination of the oscillation parameters, those parameters being fixed by SNO, SK, and KamLAND. This raises the interesting possibility of using that `free energy' in the low-energy data to make new tests of other physics. In particular, a model-independent determination of the total solar neutrino luminosity can now be made and compared to the electromagnetic luminosity.
Global fits can be made numerically, but it is enlightening to look at a set of coupled equations that can be solved straightforwardly to provide best-fit parameters, uncertainties, and correlation coefficients. While not exact, the set of equations makes clear the interrelationships between various kinds of experiment and the parameters that can be determined from them.
The simplifying assumptions include the following:
\begin{itemize}
\item The neutrino phenomenology is the LMA solution with 3 active neutrinos, and KamLAND plus SNO fix the mass to the so-called LMA-I solution. This has the advantage of decoupling $\Delta m_{12}^2 $ from the calculations, since in this part of the LMA space $P_{ee}$ is independent of $\Delta m_{12}^2 $.
\item The possibilities of sterile neutrino admixtures, non-standard interactions, and violation of CPT are neglected.
\item The spectral distortions in both SK and SNO are negligible, connecting the fluxes measured in the experiments (above the 5 MeV threshold) to the total fluxes.
\item The pp, pep, and $^7$Be neutrinos are in the vacuum oscillation region. The `critical energies' in the sun for $^8$B, pp, and $^7$Be are 1.8, 2.2, and 3.3 MeV respectively \cite{roadmap}. See Fig. \ref{fig:matter}.
\item The analysis considers only the solar and KamLAND inputs. Chooz and atmospheric neutrinos also provide constraints on $\theta_{13}$.
\item The CNO flux is set to the SSM \cite{BS05} value, 0.8\%. The flux is included with $\phi_7$, which then is not strictly a ``$^7$Be'' flux.
\end{itemize}
With this framework there are up to 5 unknowns, the 3 (total active) fluxes $\phi_1$, $\phi_7$, and $\phi_8$, and two mixing angles, $\theta_{12}$ and $\theta_{13}$. The mass-squared difference $\Delta m_{12}^2 $ is fixed by the KamLAND reactor oscillation experiment, and $\Delta m_{23}^2 $ by the atmospheric neutrino data. Since the latter is much larger than the former, $\Delta m_{13}^2 \simeq \Delta m_{23}^2 $ for either hierarchy. To extract these 5 unknowns there are 7 equations relating them to experimental observables.
\begin{eqnarray}
P_{ee}^{SNO} & = & \epsilon_{\rm LMA}(\sin^2\theta_{12} \cos^4\theta_{13} + \sin^4\theta_{13}) \\
R_{\rm tot}^{SNO} & = & \phi_8\left[\sigma_{d8}^{\rm N} + \sigma_{d8}^{\rm C} \epsilon_{\rm LMA}(\sin^2\theta_{12} \cos^4\theta_{13} + \sin^4\theta_{13})\right] \\
\Phi_{\rm ES}^{SK, SNO} & = & \eta^{-1} \phi_8 + (1 - \eta^{-1}) \phi_e \nonumber\\
& = & \phi_8 \left[\eta^{-1} + (1-\eta^{-1}) \epsilon_{\rm LMA}(\sin^2\theta_{12}\cos^4\theta_{13} + \sin^4\theta_{13}) \right] \\
R^{\rm Cl} & = & \sigma_{C8} \epsilon_{\rm LMA}\phi_8 (\sin^2\theta_{12} \cos^4\theta_{13} + \sin^4\theta_{13}) \nonumber \\
&& +( \sigma_{C1}\phi_1+ \sigma_{C7}\phi_7)\left[(1-\frac{1}{2}\sin^22\theta_{12})\cos^4\theta_{13} + \sin^4\theta_{13}\right] \\
R^{\rm Ga} & = & \sigma_{G8} \epsilon_{\rm LMA} \phi_8 (\sin^2\theta_{12} \cos^4\theta_{13} + \sin^4\theta_{13}) \nonumber \\
&& +(\sigma_{G1}\phi_1+ \sigma_{C7}\phi_7)\left[(1-\frac{1}{2}\sin^22\theta_{12})\cos^4\theta_{13} + \sin^4\theta_{13}\right] \\
\frac{2I}{Q} & = & 0.980(1-0.088f_{pep})\phi_1+0.939(1-0.003f_{CNO})\phi_7+0.498\phi_8 \\
P_{ee}^{\rm KL} & = & \left[1-\sin^22\theta_{12}\overline{\sin^2\frac{\Delta m_{12}^2 L}{4E}}\right]\cos^4\theta_{13} +\sin^4\theta_{13}
\end{eqnarray}
The first and second equations describe the electron neutrino survival probability (CC/NC ratio) and the total rate of NC+CC interactions in SNO. This particular choice of representation of the NC and CC rates has the advantage of minimizing the correlation between the two equations. The third equation is the equivalent electron neutrino flux measured by SK and SNO, where $\eta = 6.383$ is the cross section ratio for electron neutrinos relative to $\mu$ and $\tau$ neutrinos above 5 MeV. The fourth and fifth equations give the rates in the Cl and Ga detectors in terms of the three (total active) flux components $\phi_1$, $\phi_7$, and $\phi_8$ and the cross sections $\sigma_i$. The $\phi_1$ spectrum includes both the {\em pp} continuum and {\em pep} line features. The sixth equation is the luminosity constraint (see, for example, \cite{spiro,heeger,jnblum}). The seventh equation is the (anti) neutrino survival probability in KamLAND, for which the effective distance argument is averaged over the various reactors that contribute to the signal. From the best-fit parameters for the KL data alone, one finds $$\overline{\sin^2\frac{\Delta m_{12}^2 L}{4E}} = 0.389.$$ A near-unity correction parameter $\epsilon_{\rm LMA}$ is introduced to correct for the small difference between the CC/NC ratio given by the simplified expression $\sin^2\theta_{12} \cos^4\theta_{13} + \sin^4\theta_{13}$ and the value measured and fitted in detailed numerical analyses such as that of Fogli {{\it et al.}} \cite{fogli3}. The value found for $\epsilon_{\rm LMA}$ is 1.10.
In Table \ref{coeffs} the cross sections used are listed. The value of $f_{pep}$ is $f_{pep} = 0.23(2)$\% and $f_{CNO} = 0.8\%$. The cross-section uncertainties in the radiochemical experiments are propagated through the flux equations to be added in quadrature with the experimental uncertainties in the rate.
\begin{table}
\caption{Cross-section coefficients.}
\medskip
\begin{center}
\begin{tabular}{lllr}
\hline
\hline
& & (Effective) & \\
& & Cross Section & Reference \\
& & 10$^{-46}$ cm$^2$ & \\
\hline
SNO & $\sigma_{D8}^{N}$ & 2630 & \cite{SNO4} \\
& $\sigma_{D8}^{C}$ & 8000 & \cite{SNO4} \\
\hline
& $\sigma_{C1}$ & 16 $f_{pep}$ & \protect{\cite{BP,BU}} \\
Cl-Ar & $\sigma_{C7}$ & 2.38(1 + 2.60$f_{CNO}$) & \protect{\cite{Bahc5}} \\
& $\sigma_{C8}$ & 11100 & \protect{\cite{Trinder,Bahc6}} \\
\hline
& $\sigma_{G1}$ & 11.8(1 + 17$f_{pep}$) & \protect{\cite{BU,Hampel}} \\
Gallium & $\sigma_{G7}$ & 76.5(1 + 1.42$f_{CNO}$) & \protect{\cite{Haxton}}
\\ & $\sigma_{G8}$ & 24600 & \protect{\cite{Bahc6}} \\
\hline
\hline
\end{tabular}
\end{center}
\label{coeffs}
\end{table}
The experimentally determined rates and ratios used as input are listed in Table \ref{tab:results}, and the results are summarized in Fig. \ref{fig:lum}. The uncertainties are large and will remain so until a determination of the $^7$Be flux is made, but it is remarkable how a model-independent analysis of the low-energy solar data together with a model-independent determination of neutrino oscillation parameters together produce results very consistent with solar models.
\begin{figure}[]
\begin{center}
\begin{minipage}[]{18 cm}
\epsfig{file=LumC.eps,scale=.45}\epsfig{file=LumU.eps, scale=0.45}
\end{minipage}
\begin{minipage}[t]{16.5 cm}
\caption{Model-independent determination of the low-energy fluxes and the solar luminosity by solar neutrino experiments and KamLAND. The results are expressed as ratios with 1-$\sigma$ uncertainties to the BSB (OP) solar model \protect\cite{BS05} values. The angles $\theta_{12}$ and $\theta_{13}$ were fixed at 34.1$^o$ and 5.4$^o$, respectively \cite{fogli3}. The $^8$B flux depends significantly on the mixing angles, and the value and uncertainty shown are from the numerical fit of Fogli {\it et al.} \ \protect\cite{fogli3}. The left panel shows the results with the luminosity constraint, and the right without it. (Interestingly, there is no indication of a significant problem with the low-energy fluxes in either case.) This procedure sets a general and quite restrictive limit on the contribution of sterile neutrinos to the solar neutrino flux.
\label{fig:lum}}
\end{minipage}
\end{center}
\end{figure}
\section{Nuclear Astrophysics and Solar Neutrinos}
The SSM relies on a large body of painstaking laboratory work in a number of different research areas. In addition, there are laboratory inputs that affect directly the extraction of neutrino oscillation parameters in model-independent analyses. The past few years have seen great progress in improving the accuracy with which important nuclear-physics inputs are known. Much remains to be done, however. For a comprehensive general summary, the reader is referred to the recent paper by Haxton, Parker, and Rolfs \cite{Haxton:2005rw}. A number of additional reactions are included in the discussion below.
\subsection{$^3$He($^3$He,2p)$^4$He and $^{14}$N(p,$\gamma$)$^{15}$O}
The commissioning of the LUNA accelerator in the Gran Sasso Laboratory has made possible two important measurements that would have been overwhelmed at the lowest energies by backgrounds if done on the surface. The $^3$He($^3$He,2p)$^4$He reaction has been measured down to energies completely inclusive of the Gamow peak where stellar burning takes place \cite{LUNA}. The rate-determining step in the CNO cycle is the $^{14}$N(p,$\gamma$)$^{15}$O reaction, and a precise determination of its rate was identified as a priority in the evaluation conducted in 1998 \cite{Adelberger:1998qm}. That was accomplished both at LUNA \cite{Imbriani:2005jz} and at LENA \cite{LENA}, resulting in a recommended S-factor, 1.67 keV b, a factor of 2 smaller than the previously accepted value.
There is insufficient data for a completely model-independent analysis of the CNO flux from the sun, but in 2002, Bahcall, Gonzalez-Garcia, and Pe\~{n}a-Garay \cite{Bahcall:2002jt} carried out an analysis similar to that reported in this paper, a fit to the fluxes as free parameters under the assumption of the LMA solution, but invoking also the luminosity constraint. The higher cross section for CNO neutrino capture as compared to {\em pp} and $^7$Be neutrinos then allows a limit to be set on the CNO flux. Bahcall {\it et al.} \ concluded that the solar CNO luminosity was (at 3$\sigma$) less than 7.3\% of the total luminosity.
Before the new measurements of the $^{14}$N(p,$\gamma$)$^{15}$O cross section, CNO neutrinos constituted 1.6\% of the total flux from the sun in a standard solar model \cite{BP04}. With the revised cross section, the fraction is 0.8\% \cite{BS05}. An analysis \cite{Altmann:2005ix} by the GNO collaboration using Ga data, the luminosity constraint, and the SSM value for the $^7$Be flux, gave as a central value for the CNO flux fraction 0.8\% (!), and a 3$\sigma$-limit of 6.5\%.
\subsection{$^7$Be(p,$\gamma$)$^8$B Reaction}
The cross section of this reaction at the Gamow peak, about 20 keV, is determined by means of theoretical extrapolation from higher energies at which laboratory measurements can be made. The resulting accuracy has thus both a theoretical and an experimental component. Recently, measurements of this cross section with an overall uncertainty of 3.8\% have been reported \cite{Junghans:2003bd}. At this level of precision, this cross section plays a minor role in the overall uncertainty in the SSM prediction of the $^8$B flux. Opacities and the cross section for $^3$He($\alpha,\gamma$)$^7$Be make substantially larger contributions.
\subsection{$^3$He($\alpha,\gamma$)$^7$Be}
Two experimental methods for determining the cross section for this process in the laboratory, direct measurement of capture gammas, and the quantitative assay of the amount of $^7$Be produced, do not agree well, and as a result the rate of this reaction in the sun is uncertain by about 9\%. Several laboratories are undertaking new measurements (LUNA, the University of Washington, and others).
\subsection{Neutrino spectrum from $^8$B decay}
The shape of the neutrino spectrum from $^8$B decay enters directly into determinations of neutrino oscillation parameters, quite independent of any solar models. The shape is not determined in the usual way from weak-interaction theory, because the final state in $^8$Be is unbound and broad, with a beta strength function that calls for experimental measurement. For some time the standard was an analysis \cite{Bahc6} of several experiments undertaken for reasons unrelated to solar neutrino research. A new measurement by Ortiz {\it et al.} \ \cite{Ortiz:2000nf} indicated that the neutrino spectrum at high energies was significantly harder, and that spectrum was used in the analysis of SK and SNO data. However, still more recent experimental work by Winter {\it et al.} \ \cite{Winter:2003ac,Winter:2004kf} gave results in agreement with the earlier spectra and in disagreement with Ortiz {\it et al.}. Results obtained in a new experiment by Bhattacharya {\it et al.} \ \cite{adelb} are in excellent agreement with the results of Winter {\it et al.}.
\subsection{The {\em hep} reaction}
As described in more detail in Haxton {\it et al.} \ \cite{Haxton}, the {\em hep} reaction has a very low intensity and no significant role in energy production in the sun, but the very high endpoint energy (18.77 MeV) makes the reaction potentially observable. Both SNO and SK are in the process of analyzing data to observe or set limits on this process. The s-wave capture is very hindered owing to different principal quantum numbers in the initial and final state wave functions, and about 40\% of the capture proceeds via the p-wave. As a result the beta spectrum does not have a standard allowed shape, but no calculation of the spectral shape has yet been made. The experimental groups need that calculation in order to extract a total rate from the high-energy events above the $^8$B endpoint.
\subsection{Possible ground-state decay of $^8$B}
The decay of $^8$B directly to the ground state of $^8$Be involves a spin change of 2, and so is very hindered. However, the high energy of the resulting neutrinos would place them in the same energy region as the observable part of the {\em hep} reaction, and so it is important to determine if such a branch exists. Preliminary data taken at the University of Washington by Bacrania and Storm \cite{Bacrania} indicates the branch is less than 10$^{-4}$ and thus not a significant contributor.
\section{Non-Standard-Model Scenarios}
A number of `new physics' ideas beyond the now-standard LMA solution with 3 active species have been proposed that could be tested with present and future solar neutrino experimental data. These include sterile neutrino admixtures \cite{Pulido:2005yc,Dev:2005px,deHolanda:2003tx,Bahcall:2002zh}, neutrino magnetic moments and resonant spin-flavor precession \cite{Friedland:2005xh}, and non-standard-model interactions \cite{Friedland:2004pp}.
The analysis above, illustrated in Fig. \ref{fig:lum}, establishes the equality within experimental errors of the electromagnetic luminosity and the active-flavor neutrino luminosity. As such, even at the present level of accuracy, it sets a very general limit on the contribution of sterile neutrinos to the solar neutrino flux, but one that is also considerably more restrictive than those derived with reference to, for example, the SSM prediction of the $^8$B flux. At the 90\% confidence level, sterile neutrinos do not constitute more than roughly 12\% of the flux. Consideration of the potential of future solar neutrino experiments suggests that highly sensitive tests will become possible.
\section{Future Solar Neutrino Experiments}
The SNO experiment will cease data-taking at the end of 2006, but SAGE and SK will continue. They will, it is to be hoped, be joined by several new experiments aimed at precision investigation of the region below 5 MeV in the solar neutrino spectrum. Two experiments, Borexino and KamLAND Solar, are approved and under construction. Three others, LENS, SNO+, and CLEAN, are in an active R\&D phase that will most likely be followed by full proposals. Two experiments, HERON and MOON Solar, have undergone extensive R\&D but are not at present on a trajectory to a full proposal.
\subsection{Borexino}
A liquid-scintillator ES experiment with a 100 Mg fiducial mass, Borexino is under construction in LNGS (Gran Sasso) \cite{Galbiati:2005gs}. An accidental release of pseudocumene scintillator to the environment in 2003 precipitated a suspension of construction while remedies were put in place to prevent a similar occurrence in the future. Most of that work has now been completed and it is expected that permission will be forthcoming to complete the detector as planned.
\subsection{KamLAND Solar}
The KamLAND detector is already in operation as a reactor antineutrino detector. The antineutrino signal provides a delayed coincidence tag (neutron capture) that completely rejects single-event backgrounds. In order to make the detector suitable for detecting low-energy solar neutrinos by elastic scattering, reductions of contained $^{210}$Pb, $^{40}$K, $^{85}$Kr, $^{220}$Rn, $^{232}$Th, and $^{238}$U by factors ranging up to 10$^6$ are required. A distillation plant is under construction. Distillation removes the PPO fluor from the scintillator, and clean replacement material must be found and reintroduced. Another difficulty is cosmogenic activity at the depth of Kamioka, about 2000 mwe. Many such activities are sufficiently short lived that the interval following the passage of a muon can be vetoed, but 20-minute $^{11}$C requires special treatment. It nevertheless leaves a clear window in which the $^7$Be solar neutrino line can be observed.
When KamLAND Solar will be ready for data taking will depend on the performance of the distillation plant and other purification measures. Tests will begin in 2006.
\subsection{SNO+}
While the heavy water from SNO is to be returned to the owners in 2007, the potential use of the cavity, acrylic vessel, and phototube array has not gone unrecognized. SNO+ is a proposal to fill the acrylic vessel with liquid scintillator \cite{Chen:2005yi}. That would result in a detector of similar size and properties as KamLAND, but at the 6010-mwe depth of SNO. Cosmogenic activities would not be a concern, and the reactor antineutrino ``background'' would also be considerably smaller. One potentially thorny problem, the aggressive nature of most scintillator liquids to acrylic, has been resolved with the discovery \cite{Chen:2005yi} that linear alkylbenzenes, common chemical intermediates in detergent manufacturing, make an efficient and non-aggressive scintillator.
\subsection{CLEAN and HERON}
Noble liquids are highly efficient scintillators transparent to their own radiation, and among them He and Ne possess no long-lived isotopes that represent backgrounds to solar neutrinos. Their low boiling points permit fractionation to remove other troublesome contaminants such as $^{85}$Kr. The CLEAN concept \cite{McKinsey:2004rk} involves LNe and a wavelength-shifter on the surface of the container where phototubes are mounted. Event positions can be reconstructed via the luminance distribution at the surface. HERON \cite{Lanou:2005ku} makes use of both scintillation light and electrons liberated by ionizing radiation to reconstruct the energy and position within the fiducial volume, and to reject events that have multiple interaction sites, such as Compton interactions.
\subsection{LENS and MOON}
The LENS experiment, an active, spectroscopic, charged-current experiment, has been under development in various forms for many years \cite{Raghavan:1980pi}. The basic principle is to make use of a neutrino capture to an excited state, the decay of which leads to an isomer. The delayed decay of the isomer provides a tag to reject backgrounds. The most favorable case once again appears to be $^{115}$In, with a 4.76-$\mu$s isomer in the daughter $^{115}$Sn. The Q-value to this state is only -114 keV, but unfortunately, by the standards of solar neutrino rates, In is intensely radioactive, with a half-life of $6 \times 10^{14}$ years. Thus, most R\&D effort has gone into strategies for mitigating this inherent background. The 498-keV beta background and associated bremsstrahlung intrude as chance coincidences. Recent successes in development of an In-loaded scintillator have produced 8\% loading with attenuation scarcely worse than pure pseudocumene, good resolution, and long-term stablility \cite{Raghavan:2005Taup}. Good resolution helps reject the continuous beta background. Another advance has been development of an array of cubical cells to contain the scintillator and guide light predominantly to phototubes on the cell's major axes. In this way good segmentation can be obtained without excessive numbers of channels. A detector of 125-190 Mg fiducial mass, containing 10-15 Mg of In, is currently envisaged.
The MOON concept \cite{Hazama:2005kv,MOON} for solar neutrinos takes advantage of the very large matrix element for $\nu_e$ capture connecting the ground state of $^{100}$Mo to the $1^+$ ground state of $^{100}$Tc, with a Q-value of -168 keV. The subsequent beta decay of 16-s $^{100}$Tc provides a tag to identify the solar neutrino capture and reject backgrounds. The primary technical challenge is the $2\nu\beta\beta$ background from the decay of $10^{18}$-y $^{100}$Mo. In order to reduce the random coincidences to a level below the signal, the detector must be effectively subdivided with a volume resolution corresponding to less than 100 mg of $^{100}$Mo.
\section{Conclusions}
Solar neutrino research has a luminous past and a bright future. It has contributed in a major way to a revolution in fundamental physics that has required the first revision to the Standard Model of particles and fields. Neutrinos are strongly mixed in flavor and have non-zero masses. The next steps for the field are generally agreed upon: precise, spectroscopic measurement of the low-energy fluxes from the sun. A direct confrontation of the luminosity constraint is a matter of considerable importance even beyond the fields of physics and astrophysics. The combination of intensity and remoteness of the solar neutrino source gives it power to test for the presence of non-standard neutrino physics inaccessible to other kinds of experiment. The unexpected can continue to be expected.
|
1,116,691,497,540 | arxiv | \section{Introduction}
\label{Sect:intro}
\subsection{Maldacena holography, fixed-background holography and the Arnsdorf-Smolin puzzle}
The remarkable conjectured `holographic' correspondence \cite{Aharony:1999ti} due originally to Maldacena
\cite{Maldacena:1997re}, between certain superstring theories in the bulk of Anti de Sitter space (AdS) (in suitable
dimensions and times suitable-dimensional spheres and certain other compact spaces
$<$\ref{Note:AdSCFT}$>$)\footnote{The small Roman numerals in angle-brackets refer to the end section, Section
\ref{Sect:notes}, entitled `Notes'.} and certain supersymmetric conformal field theories (CFT) on the AdS conformal
boundary has inspired some authors to study simpler, but more general, types of correspondence between quantum
theories defined on a (fixed) AdS background and associated (again, conformal) quantum theories on the AdS conformal
boundary. In particular, in Rehren's `algebraic holography' \cite{Rehren:1999jn, Rehren:2000tp} a geometrically
natural bijection between bulk wedges and boundary double-cones is seen to induce a bijection between
nets of local *-algebras in the bulk and corresponding nets of local *-algebras on the conformal boundary
$<$\ref{Note:vNvsCstar}$>$ which bijects between bulk sub*-algebras for wedges and boundary sub*-algebras for
double cones, in such a way that the action of the AdS isometry group on the bulk algebra goes over to the action of
the conformal group on the boundary algebra. The nets of local *-algebras involved in this correspondence are of the
sort introduced earlier \cite{Haag} by Haag and others in order to give a general mathematical formulation of quantum
field theory (on a fixed background); the main axioms being those of isotony (i.e.\ that the *-algebra for a
subregion of a given region is a sub*-algebra of the *-algebra for the region) and commutativity at spacelike
separation. And, indeed, it is clear that the Rehren correspondence amounts, at least in some known simple examples
involving bulk `ordinary' linear field theories, to a correspondence between theories which, in both bulk and
boundary, are quantum field theories. In particular, for a bulk theory consisting of the (real) covariant Klein-Gordon
field of mass $m$ satisfying
\begin{equation}
\label{KG}
(g^{ab}\nabla_a\nabla_b+m^2)\phi=0
\end{equation}
on AdS $<$\ref{Note:wrap}$>$ of any spacetime dimension 1+$d$ satisfying vanishing boundary conditions at the
$d$-dimensional conformal boundary, Kay and Larkin \cite{Kay:2007rf} have made use of a classical counterpart to
holography, which they introduced and called `pre-holography', to obtain by second quantization (for integer or half-integer $\Delta$ in Equation (\ref{Delta}) below) concrete examples of Rehren's algebraic holography in which the bulk net of local *-algebras is the usual (C$^*$) algebra for the Klein Gordon theory and the boundary net of local *-algebras
is a certain subalgebra $<$\ref{Note:thinloc}$>$ of the algebra of a real scalar conformal generalized free field theory \cite{Greenberg:1961mr, Duetsch:2002hc} on its conformal boundary with anomalous scaling dimension $\Delta$, where $\Delta$ is related to the mass, $m$, of the bulk field and the bulk spatial dimension ($=$ boundary spacetime dimension) $d$, by
\begin{equation}
\label{Delta}
\Delta =\frac{d}{2} + \frac{1}{2}(d^2+4m^2\ell^2)^{\frac{1}{2}}
\end{equation}
where $\ell$ is the AdS radius (see Equations (\ref{E:a12}) and (\ref{E:104})).
(I.e.\ the algebra on the boundary is determined by having a commutator function equal to $W_b(x,x') - W_b(x',x)$ where $W_b$ is as in Equation (\ref{Wanom}) below.)
Moreover, Bertola et al.\ have proven \cite{Bertola:2000pp} that a Wightman theory (as they define it) on
the bulk of AdS, again in any spacetime dimension, $d+1$, will have, as its boundary limit, a (conformally invariant) Wightman theory on its ($d$-dimensional) conformal boundary $<$\ref{Note:Wightman}$>$ provided the boundary limit of the theory's $n$-point functions is defined so as to correspond to a field mapping:
\begin{equation}
\label{Blim}
\phi^{\mathrm{bulk}} \mapsto \phi^{\mathrm{boundary}}\ \ \hbox{where} \ \ \phi^{\mathrm{boundary}}({x^i})=\lim_{q\rightarrow q_0}
\Xi(q)^{\delta} \phi^{\mathrm{bulk}}(q,x^i)
\end{equation}
where $\Xi(q)$ is a suitable conformal factor $<$\ref{Note:confboundary}$>$ and $\delta$ an appropriate power.
In the case of the covariant Klein Gordon equation for mass $m$ in the bulk with vanishing boundary conditions, this boundary-limit holography links the bulk theory in the geometrically natural global ground state (i.e.\ the ground state with respect to AdS global time) on AdS to the (as made clear in \cite{Kay:2007rf}) same conformal generalized free field theory with anomalous scaling dimension $\Delta$ which figures in Rehren's algebraic holography correspondence and, in this case, the $\delta$ in Equation (\ref{Delta}) above is the same as the $\Delta$ in Equation (\ref{KG}). We remark that, because the relevant quantum field theories in this case are free (respectively generalized-free) the latter boundary theory is fully determined by the boundary limit of the bulk two point function and, when we conformally map the boundary of a Poincar\'e chart to $d$-dimensional Minkowski space, this takes the form,
\begin{equation}
\label{Wanom}
W_b(x,x')=\frac{1}{2\pi^{\frac{d}{2}}}\frac{\Gamma(\Delta)}
{\Gamma(\Delta-d/2+1)}\frac{1} {[-(x^0-x'^0-i\epsilon)^{2}+({\bf x}-{\bf x'})^2]^\Delta}
\end{equation}
which, by the way, specifies exactly what we mean by (the two point function of) a real scalar conformally invariant generalized free field with anomalous scaling dimension $\Delta$.
Clearly there are important differences between the Bertola et al.\ and Rehren versions of holography on the one hand
(below, we shall refer to either/both of these as {\it fixed-background holography}) and the mainstream work on the
Maldacena conjecture (which we shall refer to as {\it Maldacena holography}). In particular, in Maldacena holography,
the bulk theory (i.e.\ a certain superstring theory) involves a dynamical gravitational field rather than a fixed
background. Arnsdorf and Smolin \cite{Arnsdorf:2001qb} have pointed out a seeming paradox, resulting from this
difference. Their point is that, if the quantum field theory on the conformal boundary of AdS$_5$ in the Maldacena work (see Note $<$\ref{Note:AdSCFT}$>$) is both equivalent -- via Rehren's algebraic holography -- with a bulk quantum theory formulable in terms of a net of local *-algebras on a fixed AdS background and also equivalent -- via Maldacena-holography -- with a bulk theory which incorporates quantum gravity, then, it would logically follow that the latter, quantum gravity, theory must be formulable in terms of a net of local *-algebras on a fixed AdS background (with a notion of commutativity at spacelike separation built into its axioms -- see \cite{Rehren:1999jn} $<$\ref{Note:Rehgeo}$>$) -- which would seem to be in contradiction with evidence (based on the bending of light etc.) that Einstein gravity is a background-independent theory.
On the other hand, some of the objections which have been made to fixed-background holography $<$\ref{Note:blog}$>$
appear to be unfounded. These objections have to do with the very different-looking prescriptions for obtaining the bulk-boundary correspondence in Maldacena holography and in fixed-background holography. In a quantum field theoretic version of Maldacena holography, which is believed to capture much of the essence of the Maldacena correspondence in suitable limiting cases, the bulk and boundary theories are thought of as being {\it dual} to one another according to the
schematic prescription (say in a Euclidean formulation) \cite{Witten:1998qj, Gubser:1998bc}
\begin{equation}
\label{dualstring}
<\exp(-\int {\cal O}j)>=\int \exp(-I[\phi(j)])
\end{equation}
where the integral on the right hand side is a path integral and $I(\phi(j))$ represents the action of an appropriate supergravity theory for fields, schematically represented by $\phi(j)$, with boundary limit (defined according to appropriate generalizations of (\ref{Blim}) above) equal to $j$, while, on the left hand side, $\cal O$ is the `dual' field on the boundary of AdS and is integrated over this boundary with $j$.
In a further limiting case which (see also the discussion in \cite{Rehren:2004yu}) involves (say, in the case of the AdS$_5\times S^5$ version of the Maldacena correspondence) a classical supergravity theory in the bulk and a certain large $N$ limit of the corresponding CFT on the conformal boundary, the above schematic formula gets replaced \cite{Witten:1998qj, Gubser:1998bc} by
\begin{equation}
\label{dual}
<\exp(-\int {\cal O}j)>=\exp(-I_{\mathrm{class}}[\phi(j)])
\end{equation}
where $I_{\mathrm{class}}[\phi(j)]$ now represents the (Euclidean) classical action of supergravity for fields (again, schematically represented by $\phi(j)$) with boundary limits (defined according to appropriate generalizations of (\ref{Blim})) equal to $j$. Although, following custom, the correspondences in (\ref{dualstring}) and (\ref{dual}) are referred to as {\it dualities}, we note, in passing, that, as emphasized e.g.\ in \cite{Arnsdorf:2001qb}, which refers to the relation (\ref{dualstring}) as `conformal induction', [see also Note $<$\ref{Note:unfortunate}$>$]
the mapping defined by (\ref{dualstring}) is one way: from bulk to boundary, and, at least not obviously, reversible. We shall return to this point in Section \ref{Sect:discuss}.
In apparent contrast, in, say, the Bertola et al.\ boundary limit holography \cite{Bertola:2000pp}, the relation between bulk and boundary theories is, as we saw above, that the boundary theory is the {\it direct} boundary limit, (\ref{Blim}), of the bulk theory and this, at first sight, appears rather different from the `duality' in e.g.\ the schema (\ref{dual}) $<$\ref{Note:unfortunate}$>$. However, as pointed out by Rehren \cite{Rehren:2004yu} (and as Rehren notes there, as was also realized in some of the string-theory literature itself) mainstream work on Maldacena holography, based on the duality schema (\ref{dual}) e.g.\ by Witten \cite{Witten:1998qj} for a simple model with the bulk covariant Klein Gordon equation (\ref{KG}) (on a fixed bulk background) leads to exactly the same conformal generalized free field theory (specified by (\ref{Wanom})) with anomalous scaling dimension $\Delta$ (\ref{Delta}) on the conformal boundary as that given by Bertola et al.'s boundary-limit prescription. And in fact Duetsch and Rehren \cite{Duetsch:2002wy} (see also \cite{Rehren:2004yu}) have given general arguments, valid for an interacting bulk field theory, as to why the direct boundary limit prescription and the duality prescription based on (\ref{dual}) will give the same boundary theory at least for certain field theories and to all finite orders in perturbation theory -- the reasons being not at all trivial.
So in spite of the differences, there would still seem to be at least a valid and interesting `family resemblance'
between the two types of holography. As we have discussed, the situations considered in Maldacena holography involve
theories with very special properties (involving superstrings in the bulk and very special supersymmetric CFTs on the boundary) and with deep features -- in particular a bulk geometry which participates in the dynamics -- which are not captured in the Bertola et al.\ framework and seemingly also not in the Rehren framework. Nevertheless,
fixed-background holography does appear to capture {\it some} of the more basic features of Maldacena holography while having the virtue of being simple enough to admit of fully mathematically controllable formulations. Even if its study were to lead to the conclusion that a fully quantum field theoretic fixed-background holography must necessarily have certain bad properties, it could still be fruitful to try to understand how Maldacena holography manages to circumvent these bad properties. In particular such a study could perhaps contribute to a better understanding of the relationship between full quantum gravity and quantum field theory in curved spacetime, and indicate how, in general (i.e.\ not just in AdS/CFT situations) quantum gravity (as realized by string theory) manages to resolve some of the difficulties inherent in an approximate semi-classical treatment of quantum gravity based on quantum field theory on fixed backgrounds.
In fact, Rehren has already pointed out in \cite{Rehren:1999jn} that a fully quantum field theoretic (fixed-background) holography will, on his algebraic holography approach, indeed necessarily have (for 1+$d$ dimensional AdS when $d$ is greater than 1) {\it potentially} bad properties in the sense that, while the bulk and boundary theories can both be Wightman theories, they cannot both be Wightman theories which satisfy the time-slice condition $<$\ref{Note:slice}$>$. In the simple scalar field model discussed above, the bulk theory (i.e.\ the covariant Klein Gordon equation) is a Wightman theory satisfying the time-slice axiom but the boundary theory (i.e.\ the conformally invariant generalized free field with anomalous scaling dimension $\Delta$ as in Equation (\ref{Wanom})) doesn't satisfy the time-slice axiom \cite{Duetsch:2002hc} and is also known to certainly have bad properties. In particular, we know, thanks e.g.\ to the general results of \cite{BuchJung86}, that it will have anomalous thermodynamical properties. In general, however, it seems that failing to satisfy the time-slice axiom may not necessarily be bad and there seems to be evidence it could fail (or at least a lack of evidence that it holds) for certain physically realistic theories such as non-abelian gauge theories, cf. \cite{Rehren:1999jn, Rehren:2000tp}. This is relevant to the discussion at the end of Section \ref{Sect:discuss}.
\subsection{Fixed-background holography on Schwarzschild AdS and BTZ}
In the present article, we wish to report on an investigation into the fixed-background version of holography due to Bertola et al.\ (i.e.\ {\it direct-boundary-limit holography}) where, instead of a plain AdS bulk spacetime, one considers an asymptotically AdS black hole spacetime. In particular we shall consider the Bertola et al.\ boundary limit holography $<$\ref{Note:bhalgholo}$>$ for the covariant Klein Gordon equation with (for simplicity) zero mass (and again with vanishing boundary conditions on the conformal boundary) on the (zero angular momentum) BTZ spacetime \cite{Banados:1992gq} which, as is well known, can be regarded as a black hole solution to Einstein's equations with negative cosmological constant in 1+2 dimensions. Mathematically, BTZ is simpler than its counterparts in higher dimensions in that it is not only globally asymptotically AdS but also locally exactly AdS -- arising in fact from AdS by a certain quotient construction \cite{Banados:1992gq} and this construction can be exploited to obtain \cite{Lifschytz:1993eb} the bulk two-point function for the BTZ-analogue of the Hartle-Hawking-Israel state (see below) by the use of an image sum method from the two-point function for the AdS global ground state \cite{Avis:1977yn} (for the case of vanishing boundary conditions on the conformal boundary).
Here, we recall that the Hartle-Hawking-Israel state (HHI state) \cite{Hartle:1976tp, Israel:1976ur} is the unique \cite{Kay:1988mu, Kay:1992gr} quantum state for the covariant Klein Gordon equation on the Kruskal spacetime which is invariant with respect to its one-parameter group of Schwarzschild isometries and for which the two-point function has the Hadamard form \cite{Kay:1988mu} required to have an everywhere finite renormalized stress-energy tensor. Restricted to the exterior Schwarzschild spacetime (i.e.\ say, the right wedge of Kruskal) and regarded as a stationary state with respect to Schwarzschild time-evolution, it is a thermal equilibrium (i.e.\ KMS -- see Section \ref{Sect:prelim}) state at the Hawking temperature. The BTZ analogue of the HHI state obtained by the image sum mentioned above will have appropriately analogous properties, being, again, a Hadamard state and stationary and thermal on the (say) right BTZ wedge at the appropriate Hawking temperature with respect to the (right-BTZ-wedge preserving) BTZ time translations. From now on, we shall also call this an HHI state, relying on the context to make it clear when we are referring to the BTZ analogue. (Similarly we shall refer to the HHI state on Schwarzschild AdS in higher dimensions.)
We shall also consider a simple 1+1 dimensional analogue to this bulk theory in which (cf. again \cite{Lifschytz:1993eb}) what takes the place (see Figure \ref{Fig:ads2}) of the 1+2 dimensional BTZ spacetime is a certain four-wedge
region of the 1+1 dimensional AdS spacetime; here the HHI state is just the restriction to the four-wedge region of the AdS ground state. But of course the one-parameter group of AdS-isometries which gets identified, under the analogy, with `BTZ time-evolution' in (say) the right wedge differs from (global) `AdS-time'. In fact, it stands in relation to global AdS-time in much the same way as Rindler time stands in relation to Minkowski time -- and thus the thermal property can be understood as analogous to the Unruh effect \cite{Unruh:1976db}.
\smallskip
For ease of exposition, we shall actually treat this latter, simpler, 1+1 dimensional model first and in detail (in Section \ref{Sect:1+1}) postponing a concise treatment of the 1+2 dimensional case to Section \ref{Sect:1+2}.
\subsection{The contrasting accounts of the origin of thermality and entropy in the two versions of holography}
Our main purpose will be to compare and contrast what happens in this quantum-field-theory-in-curved-spacetime model with what happens in Maldacena holography when one considers appropriate superstring/supergravity theories on a Schwarzschild-AdS bulk. In particular we shall compare and contrast the different understandings, suggested by each version of holography, of how and why the bulk theory goes over to a thermal theory on the conformal boundary, and we shall compare and contrast the different understandings of the origin of black hole entropy suggested by each version.
As is well-known (see e.g.\ \cite{Maldacena:2001kr}) such a bulk geometry will, when maximally extended, have a conformal boundary which consists of two cylinders (see Figure \ref{Fig:BTZcylinders}) -- each cylinder being the conformal boundary of one of the two spacelike exterior-Schwarzschild-like `wedges' of the maximal extension. One of the notable results \cite{Maldacena:2001kr} of the Lorentzian signature version of Maldacena holography is that the state of the relevant CFT on this disconnected boundary is believed to have a restriction to each single cylinder which is a thermal state at the appropriate Hawking temperature -- the pair of restricted states being mutually entangled in just such a way that the total state on the pair of cylinders is pure. On observing that (using a Euclidean approach and then Wick-rotating) the appropriate replacement for (\ref{dualstring}) formally implies that the bulk and boundary partition functions coincide, and assuming that this remains true in full string-theoretic Maldacena holography, one expects the entropy of the thermal state on each single cylinder to equal the entropy of the bulk which, in turn, one expects (in a suitable limit of weak string coupling and a large ratio $\ell/\ell_s$ where $\ell_s$ is the string length scale and for temperatures above the Witten phase transition \cite{Witten:1998zw} -- see Note $<$\ref{Note:AdSCFT}$>$) to equal
the Hawking value (one quarter of the area of the event horizon $<$\ref{Note:onequarter}$>$) of the Bekenstein-Hawking entropy for the bulk asymptotically AdS black hole. Most remarkably, this is indeed the value of the entropy obtained for the thermal state of the CFT on a single cylinder. (In the case of AdS$_5\times S^5$ with a 1+3 dimensional conformal boundary, this agreement cannot be checked exactly because one can only calcuate the boundary entropy in a different regime -- see Note $<$\ref{Note:AdSCFT}$>$ and the references we cite there. But the agreement is exact in the AdS$_3$ versions with a BTZ bulk and a 1+1 dimensional conformal boundary -- see \cite{Ross:2005sc} and refs. therein.)
One might summarize the situation by saying:
\smallskip
\noindent
\textit{In Maldacena holography, the (bulk) entropy of the black hole (which equals one quarter of the area of its event horizon) is equal to the entropy of the dual field on (say) the right boundary cylinder. In the bulk entropy calculation in Maldacena holography, the entropy is regarded (as it is in Euclidean quantum gravity) as an attribute mainly of the (dynamical) quantum gravitational field.}
\smallskip
We remark, concerning the last sentence of this statement and in preparation for our later discussion (about the Mukohyama-Israel `complementarity' hypothesis -- see below) that, in the case of asymptotically flat black holes, the Hawking value for the entropy (i.e.\ one quarter of the area of the event horizon) is deriveable directly from the bulk partition function if one calculates this within the framework of Euclidean quantum gravity \cite{Gibbons:1976ue, Hawking:1980gf}. This defines it \cite{Gibbons:1976ue} as the path integral of the exponential of minus the appropriate Euclidean quantum gravity action for Einstein gravity with appropriate matter fields and with a suitable boundary geometry periodic in imaginary time with period $\beta=1/{\mathrm{temperature}}$. In fact, Gibbons and Hawking \cite{Gibbons:1976ue} approximate this by the classical action which they point out is entirely the gravitational action of the relevant classical Euclidean black hole solution and they show that this leads to the Hawking value for the entropy. Thus, on the assumption (which Gibbons and Hawking make) that the one and higher-loop terms also contributing to the total action (which represent physically the [gravity and matter] `thermal atmosphere' of the black hole) are small, one re-obtains (approximately) the Hawking value for the entropy. All these developments were shown by Hawking and Page to generalize to asymptotically AdS black holes in \cite{Hawking:1982dh} and indeed, as these authors point out, this case is more satisfactory in that there are stable equilibria involving a black hole surrounded by its thermal atmosphere whereas for Schwarzschild black holes, there is the well-known difficulty of negative specific heat \cite{Gibbons:1976ue, Hawking:1982dh}.
We want to draw particular attention to the importance, in this explanation of the origin of the Hawking value for the entropy, of the assumption that the entropy of the thermal atmosphere is small (or perhaps, alternatively, for some other reason, neglectable). In \cite{Hawking:1980gf} Hawking argues for its smallness by proposing that this term will be well-approximated by the entropy of an equal metrical volume of thermal radiation in flat spacetime and points out that, in relevant situations, this will indeed be small compared to the Hawking value of the black hole entropy. In the work in \cite{Hawking:1982dh} on the thermodynamics of Schwarzschild AdS, it appears to be tacitly assumed that the entropy of the corresponding thermal atmosphere can similarly be neglected, and/or estimated as the (again, small) entropy of thermal radiation at the same temperature in plain AdS for the same cosmological constant (but without a black hole).
As we shall discuss further below, this raises a puzzle (related to the Mukohyama-Israel `complementarity' principle which we discuss below) because, as we shall see, the (Lorentzian) brick wall approach (which we advocate) leads us to conclude that, in addition to the small `volume' piece, the thermal atmosphere entropy also has a piece proportional to the area of the event horizon which is comparable in size to the Hawking value of the black hole entropy. In fact, as we shall see in Section \ref{Sect:brick}, for our scalar Klein Gordon equation on the (1+2 dimensional) BTZ background, the brick wall approach entails that the area piece is substantial while (with the approximation we use) the volume piece vanishes!
Turning next from Maldacena holography back to fixed-background holography, if we assume our bulk theory (i.e.\ the covariant Klein Gordon equation on the maximally extended BTZ spacetime with vanishing boundary conditions on the conformal boundary) to be in the HHI state, then, one of us (LO) \cite{Ortiz:2011mi, lOrt11} has obtained, by taking a direct boundary limit as in (\ref{Blim}), a similar result involving two mutually entangled thermal states (i.e.\ for the conformal generalized free field with anomalous scaling dimension $\Delta$ (\ref{Delta})) on the same pair of boundary cylinders -- again at the relevant Hawking temperature $<$\ref{Note:bhalgholo}$>$.
This result had previously been obtained, with reference to the duality schema (\ref{dual}), for the same covariant Klein Gordon model, by Keski-Vakkuri \cite{KeskiVakkuri:1998nw} (and quoted in the paper by Maldacena \cite{Maldacena:2001kr} to which we referred above). However, the method by which they are obtained in \cite{Ortiz:2011mi, lOrt11}, summarized here in Sections \ref{Sect:1+1} and \ref{Sect:1+2}, may be of interest as an alternative derivation, closer in spirit to the boundary-limit holography work of \cite{Bertola:2000pp}.
One can, in fact, argue on general grounds (cf. \cite{Kay:1985zs}), using the algebraic approach to quantum field theory (see Section \ref{Sect:prelim}) that the direct boundary limit on the pair of cylinders of a bulk theory defined on BTZ would be expected $<$\ref{Note:expect}$>$ to inherit the entanglement and thermal properties of the bulk theory in this way. To argue this, we note first that we would expect that the quantum dynamical system $({\cal A}_{\mathrm{DW}}, \alpha_{\mathrm{DW}}(t))$ consisting of the *-algebra for the BTZ double wedge (i.e.\ in the 1+1 dimensional analogue, the union of the triangles ACF and CED in Figure \ref{Fig:ads2}) together with the one-parameter group of automorphisms induced by the one-parameter group of wedge-preserving BTZ isometries to be equivalent to the quantum dynamical system $({\cal A}_{\mathrm{DC}}, \alpha_{\mathrm{DC}}(t))$ consisting of the *-algebra for the appropriate boundary theory on the `double cylinder' -- i.e.\ the union of the right and left boundary cylinders -- together with the one-parameter group of automorphisms which time-translates towards the future on the right cylinder and toward the past on the left cylinder (this arising from the one-parameter group of conformal isometries -- actually isometries -- on the boundary inherited from the one-parameter group of wedge-preserving isometries in the bulk). Thus, to every bulk state (i.e.\ positive normalized linear functional on ${\cal A}_{\mathrm{DW}}$ -- see Section \ref{Sect:prelim}) -- with given entanglement properties and given thermal properties with respect to $\alpha_{\mathrm{DW}}(t)$, one expects an equivalent boundary state with equivalent entanglement properties and equivalent thermal properties with respect to $\alpha_{\mathrm{DC}}(t)$ $<$\ref{Note:equiv}$>$. Moreover, we expect that suitably smeared boundary limits (in the sense of (\ref{Blim})) of bulk fields will be identifiable with elements of the *-algebra for the appropriate boundary theory on the union of the right and left boundary cylinders in such a way that the action of bulk time-translations goes over to the action of the above-mentioned cylinder time-translations. (Further, we expect all these expectations to generalize from BTZ to Schwarzschild AdS in other dimenions.)
However, a formal calculation now results in an {\it infinite} value for the entropy of each of these (bulk and boundary) thermal states.
This infinite entropy should not have been a surprise for two rather different reasons which we point out below. First, we remark that the bulk and boundary entropies are expected, on general grounds, necessarily to be equal and thus, if one is infinite, then so must the other be. In fact, $<$\ref{Note:Ham}$>$, we expect the Hamiltonian generating the BTZ-analogue of Schwarzschild time-evolution in the BTZ-analogue of the Boulware vacuum representation \cite{Boulware:1974dm} for the field *-algebra in the right BTZ wedge to be unitarily equivalent to the Hamiltonian generating time-evolution in the ground-state representation for the field algebra on the right cylinder. Thus, since the entropy of a thermal state (at any inverse temperature, $\beta$) is given by the standard formulae
$S=-\tr\rho\ln\rho$ where $\rho=e^{-\beta H}/Z$ where $Z=\tr(e^{-\beta H})$, and since the trace is a unitary invariant, if the Hamiltonians are unitarily equivalent, then the entropies must necessarily be equal.
The first reason why we expect this entropy to be infinite, is, thinking of the entropy as the boundary entropy, because of the `bad thermodynamics' of the conformal generalized free field theory with anomalous scaling dimension $\Delta$ (see (\ref{Delta}) and (\ref{Wanom})) on the conformal boundary which we mentioned above.
The second reason why we expect this entropy to be infinite, now thinking of the entropy as bulk entropy, is because we expect the bulk Hamiltonian (and thus also, by the unitary equivalence, the boundary Hamiltonian) to have continuous spectrum whereupon, of course, the trace in the definitions of $Z$ and $\rho$ above will be infinite. We next turn to discuss the reason why we expect the bulk Hamiltonian to have continuous spectrum and explain the relevance of the brick wall model.
The reason for the continuous spectrum is essentially the same as the reason that the spectrum of the `Boulware Hamiltonian' \cite{Boulware:1974dm} on the right Schwarzschild wedge of the Kruskal spacetime (i.e.\ the Hamiltonian generating Schwarzschild time-evolution in the ground-state representation of the *-algebra for the right Schwarzschild wedge) is continuous, even when one encloses the field in a box at some large Schwarzschild radius (the $L$ of Equation (\ref{Sapprox})) and imposes (say) vanishing boundary conditions at the wall of this box. This continuous spectrum, in turn, is what is responsbile for the fact that the entropy of the HHI state for the same system is infinite even when enclosed in a large box. This sort of infinity is in fact well-known in the context of quantum field theory in curved spacetime; it also occurs, e.g.\, for the Minkowski vacuum state restricted to a Rindler wedge (again, even when enclosed in a box at large Rindler spatial coordinate) in Minkowski space. Its origin may regarded as due to the fact that (in the Kruskal example) as far as Schwarzschild wave-modes of our covariant Klein-Gordon equation are concerned, the relevant radial coordinate is not the metrical distance from the horizon $<$\ref{Note:metrical}$>$ but rather the appropriate Regge-Wheeler or `tortoise' coordinate $r^*\in (-\infty, \infty)$, related to the Schwarzschild $r\in (2M,\infty)$ by
\begin{equation}
\label{Schwtortoise}
r^*=r+2M\ln(r/2M-1)
\end{equation}
which, if taken as a sort of `distance' measure places the horizon infinitely far away from any exterior point. (In Minkowski space it is the Rindler coordinate, $x$, related, at Minkowski time $T=0$, to Minkowski $X$ coordinate by $x=\ln X$ -- see e.g.\ \cite{Kay:1985zs}.) Alternatively, it may be regarded as due to the fact that the local temperature (relative to Schwarzschild time or Rindler time -- see Section \ref{Sect:prelim}) tends to infinity as one approaches the horizon -- like inverse metrical distance.
\subsection{'t Hooft's brick wall model and the Mukohyama-Israel puzzle}
Following 't Hooft \cite{'tHooft:1984re} and subsequent authors (see especially the important correction and clarification to the brick wall approach due to Mukohyama and Israel \cite{Mukohyama:1998rf}) it is well known -- for say the Schwarzschild metric -- that, for any linear model field theory, one will obtain a Hamiltonian with discrete spectrum if one encloses the system in a large box and also separates it from the horizon by another wall (called a `brick wall' in \cite{'tHooft:1984re}) just outside the horizon where, say, vanishing boundary conditions are again imposed. If one does this, and takes the system between the brick wall and the box wall to be at the appropriate Hawking temperature, the entropy of the resulting system turns out to be finite. It turns out (in the case of the massless Klein Gordon field) to consist of the sum of a `volume' piece which is close in magnitude to the entropy of the same Klein Gordon field in an equal metrical volume of flat space and an `area piece' which is proportional to the area of the event horizon.
Also following 't Hooft and Mukohyama and Israel, we shall assume that the fields which actually exist in nature can be modelled, for thermodynamic purposes, by a number, $\mathsf{N}$ $<$\ref{Note:MIcorrection}$>$, of `effective (real massless) Klein Gordon fields' in the sense that their combined entropy will be $\mathsf{N}$ times the entropy of a single (real massless) Klein Gordon field.
By adjusting, `by hand', the metrical distance, $\alpha$, from the brick wall to the horizon to be (approximately) the Planck length (i.e.\ by taking $\alpha\approx 1$ in natural units) one then finds that the area piece of the entropy is significantly large. In fact, following the treatment of \cite{Mukohyama:1998rf}, one obtains the Hawking value for the proportionality constant, i.e.\ one quarter, provided one takes ${\mathsf{N}}$ to be of the order of (only) 300 or so. (In fact, for the area piece to take the Hawking value, one requires ${\mathsf{N}}/\alpha^2$ to have the value
$90\pi\simeq 300$. See Note $<$\ref{Note:300}$>$ for details.) This of course does not prove that it is physically correct to take $\alpha\approx 1$ and ${\mathsf{N}} \approx 300$ and, in particular, it does not prove that all the entropy of a black hole arises in this way. But it does suggest that at least a substantial part of it does. And it is then tempting to argue, as 't Hooft and Mukohyama and Israel do, that the entire entropy of a black hole can be seen as arising in this way and that this way of arriving at the entropy should be thought of as a {\it complementary description} of black hole entropy to the description of it as arising from the Gibbons-Hawking Euclidean classical gravitational action $<$\ref{Note:complementarity}$>$ which we discussed above.
Thus with Mukohyama and Israel's proposed complementarity principle, it seems that such a brick-wall modification of a quantum field theory near a horizon is able to mimic what we believe (see Note $<$\ref{Note:onequarter}$>$) would be an exact and correct result for the entropy of a black hole in a full quantum gravitational treatment. Moreover, by the way, if one accepts their argument, then the number 300 may be regarded (see Note $<$\ref{Note:300}$>$) as an order of magnitude prediction for the effective number of matter fields needed in any consistent quantum theory of gravity describing physics in 1+3 dimensions.
The Mukohyama and Israel complementarity principle actually embodies, what is, for us, a second puzzle (our first being that of Arnsdorf and Smolin). After all, as we just remarked, the `volume' piece of the entropy can be estimated as the entropy of thermal radiation at the Hawking temperature in flat space in a box of the same metrical volume. Thus it may, presumably, be identified with the `small' volume piece which Hawking proposed for the entropy of the thermal atmosphere in the Euclidean quantum gravity approach to black hole entropy which we discussed earlier. However, as we anticipated in that discussion, and as we have just seen, with the brick wall approach, the main part of the entropy of the thermal atmosphere is expected to be the piece proportional to the area of the event horizon which, following Mukohyama and Israel \cite{Mukohyama:1998rf} can, as we just saw, plausibly already account for the full Hawking entropy of the black hole. Thus it seems one is forced to conclude, with Mukohyama and Israel, that, for reasons we perhaps do not yet understand, the thermal atmosphere (perhaps both the [small] volume and the [large] area pieces) should not be added to the gravitational entropy of a black hole (which already accounts for the full Hawking value for its entropy) but rather should be regarded as an alternative calculation (or `complementary description') of the same physical quantity. (See Section \ref{Sect:discuss} for further discussion.)
Returning to the case of interest here of Schwarzschild-AdS black holes, a similar brick-wall scenario is applicable (see e.g.\ \cite{hep-th/0011176} -- but see Note $<$\ref{Note:complementarity}$>$) and we note that, in this case, a
large box is no longer needed and gets replaced by imposing vanishing boundary conditions on our quantum field(s) on the conformal boundary (i.e.\ at spacelike infinity). Again one expects the entropy to consist of the sum of a `volume' piece and a piece proportional to horizon area (i.e.\ circumference in the case of BTZ). As we have already remarked above, it turns out \cite{Kim:1996eg} that in the case of 1+2 dimensional BTZ (at least to a good approximation) the entire entropy (see equation (\ref{1plus2bricksecondresult}) in Section \ref{Sect:brick}) is proportional to the circumference of the horizon (i.e.\ the appropriate notion of `horizon area' in 1+2 dimensions) and, strikingly, the `volume piece' vanishes. (Again in line with the Mukohyama-Israel complementarity principle/puzzle, this is in sharp contrast to the assumptions of Hawking and Page in their work \cite{Hawking:1982dh} on the thermodynamics of Schwarzschild AdS, where it is assumed that the only term which goes like horizon area is the term arising from the zero-loop gravitational action.) We shall re-derive this result here, by applying the method and ideas of Mukohyama and Israel \cite{Mukohyama:1998rf} in Section \ref{Sect:brick}. We find, by the way, that, for a brick wall of metrical distance 1 natural unit from the horizon, the entropy will equal the usual Hawking value (one quarter of the circumference) provided the matter fields behave as an effective number of around 34 Klein Gordon fields. (In fact, by the results of Section \ref{Sect:brick}, the entropy will take the Hawking value if ${\mathsf{N}}/\alpha=4\pi^3/(3\zeta(3))\simeq 34$. Cf. the discussion of the 1+3 dimensional Schwarzschild case above, where the corresponding quantity, ${\mathsf{N}}\alpha^2$, needed to take a value around 300.) (As far as we are aware this has not been pointed out before.)
At a computational level, our main new development in the present paper is that we show, in Section \ref{Sect:brick}, in our 1+1 and 1+2 dimensional BTZ models with a single real massless Klein Gordon field $<$\ref{Note:mass}$>$, that the introduction of a suitable brick wall into the bulk alters the state induced from the appropriate bulk thermal state by the direct boundary limit (\ref{Blim}) on our (say) right cylinder boundary (in the case of 1+1 dimensional BTZ, right boundary line) in such a way that the altered boundary state remains a thermal state at the Hawking temperature, but in such a way that the boundary Hamiltonian has a discrete spectrum and the state induced on the boundary has a finite entropy. In fact, in both 1+1 and 1+2 dimensional cases, the finite boundary entropy is equal to the finite brick-wall-modified bulk entropy.
We may summarize much of what we have said above with the statement:
\smallskip
\noindent
\textit{We assume we can mimic a matter atmosphere realistically enough by considering a multiplet of $\mathsf{N}$
(massless) Klein Gordon fields on BTZ. By introducing a suitable brick wall, the entropy of this model thermal atmosphere
becomes finite and is equal to the entropy of the CFT on the conformal boundary obtained from the brick-wall modified bulk
theory with fixed-background (specifically direct-boundary-limit) holography. Moreover, by adjusting the metrical
distance, $\alpha$, from the brick wall to the horizon to be around 1 in natural units (i.e.\ around the `Planck
length' -- see Section \ref{Sect:prelim}), the entropy can be made to equal the Hawking value of one quarter of the
area (i.e.\ circumference) of the event horizon by taking $\mathsf{N}/\alpha$ to be around 34. In the bulk entropy calcuation in the brick wall approach, the entropy is regarded as an attribute of a collection of (mainly) matter fields propagating on a fixed gravitational background.}
\subsection{The connection between the Arnsdorf-Smolin and Mukohyama-Israel puzzles}
A comparison of our two italicized displayed statements above seems to suggest a new and interesting connection between, on the one hand, the puzzling contrast between Maldacena holography and fixed-background holography -- as highlighted e.g.\ by Arnsdorf and Smolin's paradox -- and, on the other hand, the puzzle embodied in the
Mukohyama-Israel complementarity principle, according to which the (entire) entropy of a black hole should describable either as the entropy derived from the zero-loop quantum gravitational partition function, or, in a complementary description, as the entropy of its thermal (mostly matter) atmosphere. In fact our first italicized statements above indicates that Maldacena holography goes together with the first description and our second italicized sentence indicates that fixed-background holography goes together with the second description.
Thus, we have pointed out an apparently interesting and new connection between the two puzzles.
\subsection{Outline of paper and brief preview of conclusions}
After a section (Section \ref{Sect:prelim}) entitled Preliminaries, we recall the relevant facts about direct boundary-limit holography for BTZ, treating the 1+1 dimensional case in Section \ref{Sect:1+1} and the 1+2 dimensional case in Section
\ref{Sect:1+2}. For each case, we construct the HHI state for a single massless Klein Gordon field and show that its direct boundary limit on (say) the right boundary cylinder (= boundary line in the 1+1 case) is a thermal state at the relevant Hawking temperature. Then, in Section \ref{Sect:brick} we introduce the brick wall and show that the brick-wall modified HHI state for a collection of $\mathsf{N}$ massless Klein Gordon fields remains a thermal state on the right boundary cylinder (/line) at the Hawking temperature, but now has a finite entropy given by equation (\ref{1plus2bricksecondresult}) and that this is equal to the entropy of the state induced on the boundary from the brick-wall modified HHI state.
In the first part of our Discussion Section -- Subsection \ref{subsect:summary} of Section \ref{Sect:discuss} -- we summarize our main results. In the second part (Subsections \ref{subsect:working} to \ref{subsect:dual}) we discuss further our two puzzles, i.e.\ those of Arnsdorf and Smolin and of Mukohyama and Israel, in the light of our results and with reference to the AdS$_5$ version of Maldacena holography -- and we discuss what conclusions can be drawn, in the light of our results, regarding these puzzles. We shall work on the assumption that our main results and, in particular, the equality of bulk and boundary entropies (in the presence of suitable brick walls)
will hold for bulk QFT other than the Klein Gordon theory and for Schwarzschild AdS in dimensions other than 1+2. We shall also recall the proposed resolution to the Mukohyama-Israel puzzle due to one of us (BSK) based on the `matter-gravity entanglement hypothesis' of references \cite{Kay:1998vv, Kay:1998cj, Abyaneh:2005tc, Kay:2007rx}. The main tentative conclusions we arrive at, and our arguments for them, may be very briefly summarized as follows: First we shall argue from this latter hypothesis that, in Maldacena AdS/CFT, the algebra of the boundary CFT is isomorphic only to a proper subalgebra of the bulk algebra, albeit (at non-zero temperature) the (GNS) Hilbert spaces of bulk and boundary theories are still the `same' -- the total bulk state being pure, while the boundary state is mixed (thermal). Secondly, we shall argue from the finiteness of its boundary (and hence, on our assumptions, also bulk) entropy at finite temperature, that the Rehren dual of the Maldacena boundary CFT cannot itself be a QFT and must, instead, presumably be something like a string theory. As we say again, at the end of Section \ref{Sect:discuss}, while these tentative conclusions still do not resolve the Arndorf-Smolin puzzle, they seem to go at least some of the way towards a possible resolution.
\section{Preliminaries}
\label{Sect:prelim}
We adopt signature (+,-,-,-). Concerning units: In the main part of the paper, we omit all factors of $c$, $\hbar$, $G$, and $k$ (Boltzmann's constant) from our formulae and so regard all physical quantities as pure numbers. In 1+3 dimensions, where theory can be compared with experiment, this of course amounts to using Planck units in which lengths are multiples of the Planck length ($\approx 10^{-33}$ cm), times are multiples of the Planck time ($\approx 10^{-43}$ sec) and masses are multiples of the Planck mass ($\approx 10^{-5}$ g). (In Note $<$\ref{Note:AdSCFT}$>$ and Section \ref{Sect:discuss} we shall restore $G$.) We use the symbol `$\approx$' to denote `of the order of' and `$\simeq$' to denote approximate equality.
The appropriate mathematical framework for quantum theory in curved spacetime is the algebraic approach to quantum field theory -- see \cite{Haag:1967sg} for the general framework and \cite{Kay:2006jn} and \cite{Kay:1988mu} for the specific application to linear quantum field theories in curved spacetime such as that of our Klein Gordon equation (\ref{KG}). We briefly recall some of the salient points although many of the details will only be needed in the Notes section (Section \ref{Sect:notes}). Observables are regarded as self-adjoint elements of a *-algebra, for which we shall typically use the symbol $\cal A$, and a state, $\omega$, on $\cal A$ is a positive normalized linear functional and, in the case of a quantum field theory, is specified in practice by listing all of its (smeared) $n$-point functions.
The GNS representation of a *-algebra, $\cal A$, for a choice of state, $\omega$, is a representation, $\rho$, of $\cal A$ as operators on a Hilbert space $\cal H$, together with a vector, $\Omega\in {\cal H}$ such that $\Omega$ is {\it cyclic} for $\rho$ (meaning the set $\rho({\cal A})\Omega$ is dense in $\cal H$) and satisfies
$\omega(A)=\langle\Omega|\rho(A)\Omega\rangle \ \forall A\in {\cal A}$. Moreover, if we have an algebraic quantum dynamical system $({\cal A}, \alpha(t))$ consisting of a *-algebra, ${\cal A}$, together with a one-parameter group, $\alpha(t)$, of `time-translation' automorphisms, $\alpha(t)$, then there will be a one-parameter unitary group $U(t)$ (which we shall assume to be strongly continuous and which will, hence, arise in the form $U(t)=e^{-iHt}$) for a self-adjoint `Hamiltonian generator', $H$, on the GNS Hilbert space $\cal H$ which {\it implements} $\alpha(t)$ in the sense that $\rho(\alpha(t)(A))=U(-t)\rho(A)U(t) \ \forall A\in {\cal H}$. It is well-known and easy to show that $U(t)$ is fixed uniquely if we demand $U(t)\Omega=\Omega$ (i.e.\ $H\Omega=0$) for all $t\in {\cal H}$. When such a condition holds for each of two equivalent quantum dynamical systems
$({\cal A}_1, \alpha_1(t))$ and $({\cal A}_2, \alpha_2(t))$, then one easily sees that the Hamiltonian generators, $H_1$ and $H_2$, say, must then be equivalent in the sense that there exists an isomorphim $V: {\cal H}_1\rightarrow {\cal H}_2$ such that
\begin{equation}
\label{equivHam}
VH_1=H_2V.
\end{equation}
Whenever we refer to a `ground state' or to a `thermal state' at some given `inverse temperature', there is always, at least implicitly, a notion of `time' being assumed and so these notions are defined formally relative to a given quantum dynamical system $({\cal A}, \alpha(t))$. A state, $\omega$ on the $\cal A$ of such a dynamical system is said to be
a ground state if its Hamiltonian generator, defined, and made unique as in the above paragraph, is a positive operator.
Ground states are characterised by having two-point functions which are translationally invariant and, regarded as functions of $t=t_1-t_2$, say, are the boundary limits of holomorphic functions in the lower half $t$ plane (and an appropriate statement for other $n$-point functions). The two-point function of a thermal state, at inverse temperature, $\beta$, will satisfy the KMS condition (see again \cite{Haag} for the general theory and especially \cite{Haag:1967sg}; see also \cite{Kay:2006jn} and \cite{Kay:1988mu} for the case of linear field theories in curved spacetimes and see also \cite{saFullsnRui87}) which means that it is, again, translationally invariant in time and extends to a function, $G$ of $t=t_1-t_2$ which is holomorphic in the strip, $-\beta <\mathrm{Im} t < 0$ such that the boundary value, $G(-t)$ equals the boundary value $G(t-i\beta)$.
We note that, if $H$ is a Hamiltonian on a Hilbert space $\cal H$ and if $Z={\rm tr}(e^{-\beta H})$ is finite, then the state, $\omega_\beta$, defined on operators, $A$, on $\cal H$ by $\omega_\beta(A)=
{\rm tr}(e^{-\beta H} A)/Z$ will be a KMS state at inverse temperature $\beta$ and its GNS Hilbert space will, in a certain sense explained in \cite{Kay:1985yw} (cf. also the `thermo-field dynamics' of Takahashi and Umezawa \cite{TakUme, Takahashi:1996zn}) be larger than $\cal H$ (in fact ${\cal H}\otimes {\cal H}$) and may be thought of as carrying a representation of a pure state (the `purification' of $\omega_\beta$) on a larger system whose reduced density operator on $\cal H$ is equal to $e^{-\beta H}/Z$ . This remark will be relevant in Note $<$\ref{Note:double}$>$ and in Subsection \ref{subsect:matgrav} of Section \ref{Sect:discuss}.
In a spherically symmetric spacetime of dimension 1+$d$ ($d>1$) with a metric of form
\begin{equation}
\label{generalsphmetric}
ds^2=f(r)dt^2-f(r)^{-1}dr^2-r^2ds_{{\mathrm S}^{d-1}}^2,
\end{equation}
where $ds_{{\mathrm S}^{d-1}}^2$ represents the metric on a sphere of dimension $d-1$, then, for a given temperature, $\cal T$, with respect to $t$, one can also refer to a `local temperature' (see (\ref{Tlocal})) equal to $f^{-1/2}{\cal T}$. (This also applies in $d=1$ with the last term in the above equation absent.) In the case of the Schwarzschild metric, $\cal T$ then coincides with the temperature at infinity since the relevant $f$ tends to $1$ at infinity. In the case of the Schwarzschild-AdS metric (such as our BTZ metrics, given by (\ref{E:bb3}) and (\ref{E:16}) in 1+1 dimensions and (\ref{E:109}) and (\ref{E:110}) in 1+2 dimensions) the local temperature is infinity on the horizon and zero at infinity. However, because Schwarzschild-AdS time (in 1+1 and 1+2 dimensions, what we call here BTZ-time) in the bulk goes over to `cylinder-time' on the conformal boundary, temperature with respect to cylinder time on the conformal boundary will coincide with temperature with respect to BTZ-time in the bulk.
Finally, we note that, for metrics of the above spherically symmetric form,
if a horizon is located at $r_+$, indicated by the vanishing of $f(r_+)$, then its surface gravity, $\kappa$ is equal to $f'(r_+)/2$.
\section{The boundary limit of the massless Klein Gordon QFT on 1+1 dimensional AdS and 1+1 dimensional BTZ}
\label{Sect:1+1}
We first briefly recall some well-known facts about coordinate systems for 1+1 dimensional AdS (we'll just call it AdS for the rest of this section) and then, following Lifschytz and Ortiz \cite{Lifschytz:1993eb}, define what we mean by 1+1 dimensional BTZ.
1+1 dimensional AdS with radius $\ell$ can be defined to be the surface
\begin{equation}
\label{E:a12}
u^{2}+v^{2}-x^{2}=\ell^{2}
\end{equation}
embedded in $\mathbb{R}^{3}$ with metric
\begin{equation}
\label{E:2}
ds^{2}=du^{2}+dv^{2}-dx^{2}.
\end{equation}
The metric is given by the pullback of
(\ref{E:2}) to (\ref{E:a12}) under the inclusion map. We are interested in three parametrizations of (\ref{E:a12}). The first is with
\begin{equation}
\label{E:a13}
u=\ell\sec\rho\sin\lambda
\hspace{0.7cm}v=\ell\sec\rho\cos\lambda\hspace{0.7cm}x=\ell\tan\rho,
\end{equation}
where $\lambda\in[0, 2\pi)$ and $\rho\in(-\pi/2, \pi/2)$ which gives the metric in global coordinates:
\begin{equation}
\label{E:a14}
ds^{2}=\ell^{2}\sec^{2}\rho\left(d\lambda^{2}-d\rho^{2}\right).
\end{equation}
The second is
\begin{equation}
\label{E:77}
u=\frac{\ell T}{z}\quad
v=\frac{1}{2z}\left(z^{2}+\ell^{2}-T^{2}\right)\quad
x=\frac{1}{2z}\left(-z^{2}+\ell^{2}+T^{2}\right),
\end{equation}
where $(T,z)\in(-\infty,\infty)\times(0,\infty)$
which gives the metric in Poincar\'e coordinates:
\begin{equation}
\label{E:79}
ds^{2}=\frac{\ell^{2}}{z^2}\left(dT^{2}-dz^{2}\right).
\end{equation}
A third possibility (cf. \cite{Lifschytz:1993eb}) is to choose a positive number, $M$ (which will get interpreted as the `BTZ mass') and to parametrize (\ref{E:a12}) with
\[
v=\ell\left(\frac{r^{2}-r_{+}^{2}}{r_{+}^{2}}\right)^{1/2}\sinh\kappa
t\hspace{0.5cm}x=\ell\left(\frac{r^{2}-r_{+}^{2}}{r_{+}^{2}}\right)^{1/2}\cosh\kappa t
\]
\begin{equation}
\label{E:bb2}
u=-\ell\frac{r}{r_{+}},
\end{equation}
where $r_+=\ell\sqrt{M}$ and $\kappa=\frac{r_{+}}{\ell^{2}}$ $(=\sqrt M/\ell)$. This gives the
metric as
\begin{equation}
\label{E:bb3}
ds^{2}=f(r)dt^{2}-f(r)^{-1}dr^{2},
\end{equation}
($t\in (-\infty,\infty), r=(r_+,\infty)$) where
\begin{equation}
\label{E:16}
f(r)=-M+\frac{r^{2}}{\ell^{2}}.
\end{equation}
We call $(t,r)$ BTZ coordinates. We note that
$\kappa=f'(r_+)/2$ and so (see Section \ref{Sect:prelim}) it is equal to the surface gravity of the BTZ horizon at $r=r_+$
The charts for the three coordinate systems cover different regions of AdS spacetime as shown in
Figure \ref{Fig:ads2}.
\begin{figure}
\centering
\includegraphics[scale=0.4]{ads2dmodified}
\caption{\label{Fig:ads2} Regions in AdS spacetime in 1+1 dimensions covered by Poincar\'e and BTZ
coordinates. The region ABD is covered by Poincar\'e coordinates whereas the region CED (the `right exterior region' is covered by BTZ coordinates. Global coordinates cover all the manifold. The 1+1 dimensional BTZ spacetime consists of the `four-wedge' region AEDF bounded by the line AE (not drawn).}
\end{figure}
In order to draw this diagram we have conformally mapped AdS to the strip $-\pi/2<\rho<\pi/2$ in Minkowski spacetime
(see Note $<$\ref{Note:confboundary}$>$) and attached a (disconnected) boundary at $\rho=-\pi/2$ and $\rho=\pi/2$. We also assume from now on (see Note $<$\ref{Note:wrap}$>$) that we are on the covering space i.e.\ we let $\lambda$ now range over $\mathbb{R}$. Our BTZ coordinates range over $t\in (-\infty, \infty)$, $r\in (r_+, \infty)$ and cover the right-wedge region CDE which corresponds to the (right) exterior BTZ spacetime. The full (1+1 dimensional) BTZ spacetime consists of the square (or `four-wedge') region AEDF.
In the massless case and for the two-dimensional AdS metric (\ref{E:a14}), our (classical, real) Klein Gordon equation (\ref{KG}) amounts to the wave equation
\begin{equation}
\label{2dwave}
\frac{\partial^2\phi}{\partial\lambda^2}-\frac{\partial^2\phi}{\partial\rho^2}=0
\end{equation}
and, in consequence $<$\ref{Note:cnfmlwt}$>$, the massless quantum two-point function $<$\ref{Note:2ptfn}$>$,
\[
{\cal G}_{\mathrm{global}}(\lambda_1,\rho_1; \lambda_2,\rho_2)=\omega_{\mathrm{globalground}}(\phi(\lambda_1,\rho_1)\phi(\lambda_2,\rho_2)),
\]
in the global AdS ground state is the same as that for the ground state of (\ref{2dwave}) with respect to Minkowski-time on our strip of 1+1 dimensional Minkowski space with vanishing boundary conditions at $\rho=\pm \pi/2$, which, working for convenience with $y$, defined by $y=\pi/2-\rho$, is easily calculated to be
\begin{equation}
\label{2dmasslessG}
{\cal G}_{\mathrm{global}}(\lambda_1,y_1; \lambda_2,y_2)=\frac{1}{\pi}\sum_{n=1}^\infty \frac{e^{-in(\lambda_1-\lambda_2)}}{n}
\sin n y_1\sin n y_2.
\end{equation}
From this, one finds that the direct boundary limit $G_{\mathrm{global}}$ -- obtained from ${\cal G}_{\mathrm{global}}$, according to (\ref{Blim}) applied to (\ref{E:a14}), by taking the limit as $\rho_1$ and $\rho_2$ tend towards say, the right boundary, $\rho=\pi/2$, i.e as $y_1$ and $y_2$ tend to $y=0$ -- of $(\sin y_1\sin y_2)^{-1}{\cal G}_{\mathrm{global}}(\lambda_1,y_1; \lambda_2,y_2)$
is easily seen to be
\begin{equation}
\label{E:71}
G_{\mathrm{global}}\left(\lambda_{1},\lambda_{2}\right)=\frac{1}{2\pi}\frac{1}{\cos\left(\lambda_{1}-\lambda_{2}-i\epsilon\right)-1}.
\end{equation}
(See \cite{pLark07} for details but note that, there, the boundary limit is taken at the left boundary!)
We can also compute the boundary limit of the same two point function (i.e.\ for the global AdS ground state) -- this time coordinatizing the boundary by Poincar\'e time $T$ (which is the `Minkowski' time coordinate in the interpretation of the relevant region, i.e.\ the line DB in Figure \ref{Fig:ads2}, of the boundary as 1+0-dimensional Minkowski space). One way to do this is to rely on the proof, \cite{Spradlin:1999bn}, by Spradlin and Strominger that, in 1+1 dimensions, the AdS ground state and the Poincar\'e ground state coincide. The two-point function in the AdS ground state, restricted to a Poincar\'e chart, can then (see again $<$\ref{Note:cnfmlwt}$>$) in view of (\ref{E:79}), be identified with the
two point function
\begin{equation}
\label{Poincaretwopointfn}
{\cal G}_{\mathrm{Poincare}}(T_1,z_1; T_2,z_2)=\frac{1}{\pi}\int_0^\infty \frac{e^{-ik(T_1-T_2)}}{k}
\sin kz_1\sin kz_2\,dk
\end{equation}
for the ground state on the right half of 1+1 dimensional Minkowski space with vanishing boundary conditions at $z=0$.
Taking the direct boundary limit -- obtained, according to (\ref{Blim}) applied to (\ref{E:79}) -- by taking the limit as
$z_1$ and $z_2$ tend towards the right-boundary at $z=0$ -- of $(z_1z_2)^{-1}{\cal G}_{\mathrm{Poincare}}(T_1, z_1; T_2, z_2)$, one obtains
\begin{equation}
\label{E:92}
G_{\mathrm{Poincare}}(T_{1},T_{2})=-\frac{1}{\pi}\frac{1}{\left(T_{1}-T_{2}-i\epsilon\right)^{2}}
\end{equation}
which agrees, as it of course must, with (\ref{Wanom}) for the case $d=1$, $\Delta=1$.
One may alternatively obtain (\ref{E:92}) from (\ref{E:71}) by noticing first that (by comparing $u/v$ in the limit as
$x$ tends to $\infty$ in (\ref{E:a13}) with $u/v$ in the limit as $x$ tends to $\infty$ in (\ref{E:77})) $T$ coordinates are related to $\lambda$ coordinates on the boundary by
\begin{equation}
\label{E:81}
\frac{T}{\ell}=\tan\left(\frac{\lambda}{2}\right).
\end{equation}
From this expression, it follows that, if we define the metrics on the boundary, $g=d\lambda^2$ and $\tilde g=dT^2$, then $g$ and $\tilde g$ are related by
\begin{equation}
\label{E:84}
\tilde g=\Omega^{2}g
\end{equation}
where
\begin{equation}
\label{gpOmega}
\Omega=\frac{\ell^{2}+T^{2}}{2\ell},
\end{equation}
whereupon, by the formalism of Note $<$\ref{Note:cnfmlwt}$>$, we must have
\[
G_{\mathrm{Poincare}}(T_1, T_2)=\Omega(T_1)^{-\Delta}\Omega(T_2)^{-\Delta}G_{\mathrm{global}}(\lambda_1(T_1),\lambda_2(T_2))
\]
where (by (\ref{Delta}) for $d=1$ and $m=0$) $\Delta=1$. With $\Omega$ as in (\ref{gpOmega}) and the relationship (\ref{E:81}) between $T$ and $\lambda$, this easily reproduces our formula (\ref{E:92}) for $G_{\mathrm{Poincare}}$ given the formula, (\ref{E:71}) for $G_{\mathrm{global}}$.
It is easy to see that the two-point function, $G_{\mathrm{Poincare}}(T_1, T_2)$ of (\ref{E:92}) satisfies the
ground-state condition with respect to $T$-translations. In fact, apart from a 1/4 factor (see Note $<$\ref{Note:1plus1Mink}$>$) it is equal to the twice-differentiated ground state two point function on a null line for the massless Klein Gordon equation in 1+1 dimensional Minkowski space -- where we identify $T$ here with, say the null-coordinate $U=T+X$ on the latter.
Next, we wish to find the boundary limit, $G_{\mathrm{BTZ}}(t_1, t_2)$, of the bulk two-point function for the global ground state of AdS, when expressed in terms of BTZ time, where we now think of the global ground state (see Section
\ref{Sect:intro}) as the appropriate version of the HHI state when the 4-wedge region of AdS is interpreted as 1+1 dimensional BTZ. To find this, notice first that
(by comparing $v/x$ in the limit as
$\rho$ tends to $\pi/2$ in (\ref{E:a13}) with $v/x$ in the limit as $z$ tends to $0$ in (\ref{E:bb2})) $t$ coordinates are related to $\lambda$ coordinates on the boundary by
\begin{equation}\label{E:93a}
\tanh\kappa t=\cos\lambda.
\end{equation}
From this expression, it follows that, if we define the metrics on the boundary, $g=d\lambda^2$ and $\tilde g=dt^2$, then $g$ and $\tilde g$ are related by an equation of form (\ref{E:84}) where now
\begin{equation}
\label{E:98}
\Omega(t)=\frac{\cosh\kappa t}{\kappa},
\end{equation}
whereupon, again by the formalism of Note $<$\ref{Note:cnfmlwt}$>$, we must have
\[
G_{\mathrm{BTZ}}(t_1, t_2)=\Omega(t_1)^{-\Delta}\Omega(t_2)^{-\Delta}G_{\mathrm{global}}(\lambda_1(t_1),\lambda_2(t_2))
\]
where (again by (\ref{Delta}) for $d=1$ and $m=0$) $\Delta=1$. With $\Omega$ as in (\ref{E:98}) and the relationship (\ref{E:93a}) between $t$ and $\lambda$, this easily gives
\begin{equation}\label{E:100}
G_{\mathrm{BTZ}}(t_{1},t_{2})=-\frac{1}{4\pi}\frac{\kappa^{2}}{\sinh^{2}\left(\kappa\frac{t_{1}-t_{2}-i\epsilon}{2}\right)},
\end{equation}
which, as one may easily check, is a KMS state at inverse temperature $2\pi/\kappa$. In fact (again up to a factor of $1/4$) the two point function (\ref{E:100}) is equal to the twice-differentiated thermal two point function at the same temperature on a null line for the massless Klein Gordon equation in 1+1 dimensional Minkowski space -- where we identify $t$ here with, say, the null-coordinate $u=t+x$ on the latter. (Below we shall think of $(t, x)$ as `Rindler' coordinates on a right wedge of a different copy of Minkowski space and $u$ as its restriction to a null half-line.)
It is interesting to notice, in fact, that the relation between the (Poincar\'e) $T$ coordinate and the (BTZ) $t$ coordinate on the boundary (DE in Figure \ref{Fig:ads2}) is (e.g.\ from (\ref{E:81}) and (\ref{E:93a}))
\begin{equation}
\label{Ttrelation}
-T=\ell e^{-\kappa t}.
\end{equation}
and we could, of course, alternatively, have obtained (\ref{E:100})
directly from (\ref{E:92}) and (\ref{Ttrelation}) $<$\ref{Note:TtGstory}$>$.
In fact, the situation for the two-point functions (\ref{E:92}) and (\ref{E:100}) on the (timelike) boundary line DEB in Figure \ref{Fig:ads2} is closely similar to the situation on a full null-line -- say the line coordinatized by $U=T+X$ in 1+1 dimensional Minkowski space $<$\ref{Note:1plus1Mink}$>$, the only difference being the factor of $1/4$ and the need to twice-differentiate as we noted above in the case of each two-point function. Aside from this difference, this close similarity, or analogy, is realized by identifying our `Poincar\'e' coordinate, $T$, with $U$ and (cf. (\ref{Ttrelation})) our `BTZ' coordinate, $t$ (which coordinatizes the boundary half-line DE) with, $u$, related to $U$ by (cf. (\ref{Ttrelation})) $-U=\ell e^{\kappa u}$ which coordinatizes, say, the `left' half of the chosen null line in Minkowski space. On the Minkowski side of this analogy, $u$ translations correspond to (the restriction to the null line of) Lorentz boosts and the thermality of $G_{\mathrm{BTZ}}$ amounts to the Unruh effect. Here we mean, by the Unruh effect (see \cite{Unruh:1976db} and also \cite{Kay:1985zs}) the thermal nature of the usual ground state in Minkowski space when restricted to, say, the right Minkowski-wedge and regarded with respect to the `time-evolution' given by the one-parameter family of wedge-preserving Lorentz boosts (scaled by $\kappa$ to make our analogy closer) -- except we are considering here, instead of the full wedge algebra, the algebra for one of the two null half-lines which bounds the right wedge (say the one to the future) -- this being, by the way, for the massless scalar field in 1+1 dimensions, equivalent in an obvious way to the restriction to left-moving modes. Moreover, (cf. \cite{Kay:1985zs}) just as the Minkowski ground state is a `double KMS state' in the sense of \cite{Kay:1985yw} (see also Note $<$\ref{Note:equiv}$>$) on the union of the right and left wedge of Minkowski space -- and, more relevantly, by restriction, on the full null-line we are considering, it is equally a `double KMS state' with respect to the union of the right half-line ($U>0$) and the left half line ($U<0$) (where `time-evolution' is given by $u$-translations where $u$ is defined by $U=\ell e^\kappa u$ on the right half line and by $-U=\ell e^{-\kappa u}$) on the left half-line. In particular, and adopting `heuristic' language, it is thermal on each half-line ($U>0$ and $U<0$) but `entangled' in just such a way as to be a pure state (in fact a ground state for $U$-translations) on the full $U$ line. And all this translates, pursuing our analogy in reverse, to the fact that the AdS ground state on our bulk AdS, restricted to the right (now timelike) boundary line DEB is a ground state with respect to $T$-translations on the full line DEB in Figure \ref{Fig:ads2} but it is a `double KMS state' with respect to $t$-translations (i.e.\ `BTZ time-translations') on the half-line $T>0$ which coordinatizes EB (with $t$ now defined by $T=\ell e^{\kappa t}$) and with respect to our BTZ $t$-translations on the half-line $T<0$ which coordinatizes DE (with $t$ as in (\ref{Ttrelation})). Finally, we remark that, because of the way solutions to our bulk field theory (i.e.\ Equation (\ref{2dwave})) propagate, the situation on the half line EB (see again Figure \ref{Fig:ads2}) is essentially the same as the situation on the half-line FA and so we may alternatively think in terms of the boundary restriction (now to the left boundary-half-line, FA, and the right boundary-half-line, DE) of the global AdS ground state as being a double KMS state with respect to our $t$-translations on DE and the appropriate notion of $t$-translations on FA. This latter situation is, as we shall see in Section \ref{Sect:1+2}, analogous to what happens in the case of bulk 1+2 dimensional BTZ, for the two boundary cylinders there.
\section{The boundary limit of the Klein Gordon QFT on 1+2 dimensional AdS and BTZ}
\label{Sect:1+2}
1+2 dimensional AdS with radius $\ell$, related to the cosmological constant, $\Lambda$, by $\Lambda=-1/\ell^2$ (we'll just call it AdS from now on in the rest of this section) can be defined to be the surface
\begin{equation}
\label{E:104}
u^{2}+v^{2}-x^{2}-y^{2}=\ell^{2}
\end{equation}
embedded in $\mathbb{R}^{4}$ with metric
\begin{equation}\label{E:1}
ds^{2}=du^{2}+dv^{2}-dx^{2}-dy^{2}.
\end{equation}
We are interested again in three parameterizations of
(\ref{E:104}) leading to global, Poincar\'{e} and BTZ coordinates. Global
coordinates $(\lambda,\rho,\theta)$ can be defined by
\begin{eqnarray}\label{E:105}
v=\ell\sec\rho\cos\lambda\hspace{1cm}u=\ell\sec\rho\sin\lambda\nonumber\\
x=\ell\tan\rho\cos\theta\hspace{1cm}y=\ell\tan\rho\sin\theta,
\end{eqnarray}
where
$(\lambda,\rho,\theta)\in[-\pi,\pi)\times[0,\pi/2)\times[-\pi,\pi)$.
$-\pi<\lambda\leq\pi$, $0\leq\rho<\pi/2$ and $-\pi\leq\theta<\pi$.
In these coordinates the metric is
\begin{equation}
\label{E:106aa}
ds^{2}=\ell^{2}\sec^{2}\rho\left(d\lambda^{2}-d\rho^{2}-\sin^{2}\rho
d\theta^{2}\right).
\end{equation}
Poincar\'{e} coordinates $(T,k,z)$ are given by
\begin{eqnarray}\label{E:107}
v=\frac{1}{2z}\left(z^{2}+\ell^{2}+k^{2}-T^{2}\right)\hspace{1cm}u=\frac{\ell T}{z}\nonumber\\
x=\frac{1}{2z}\left(\ell^{2}-z^{2}+T^{2}-k^{2}\right)\hspace{1cm}y=\frac{\ell k}{z}.
\end{eqnarray}
In these coordinates the metric is
\begin{equation}
\label{E:108}
ds^{2}=\frac{\ell^{2}}{z^{2}}\left(dT^{2}-dk^{2}-dz^{2}\right)
\end{equation}
where
$(T,k,z)\in(-\infty,\infty)\times(-\infty,\infty)\times(0,\infty)$.
From (\ref{E:108}) we see that the boundary, at $z=0$, of the region of AdS
covered by Poincar\'{e} coordinates is conformal to 1+1 dimensional
Minkowski space.
The third parametrization, again, for a given choice of $M\in \mathbb{R}^+$ (which will again become the `BTZ mass') is given by
\begin{eqnarray}\label{E:112}
u=\ell\sqrt{\frac{r^{2}-r_{+}^{2}}{r_{+}^{2}}}\sinh\kappa t\hspace{1cm}v=\ell\frac{r}{r_{+}}\cosh \ell\kappa\varphi\nonumber\\
y=\ell\sqrt{\frac{r^{2}-r_{+}^{2}}{r_{+}^{2}}}\cosh\kappa
t\hspace{1cm}x=\ell\frac{r}{r_{+}}\sinh \ell\kappa\varphi,
\end{eqnarray}
where $r_+=\ell\sqrt{M}$ and $\kappa=r_{+}/\ell^{2}$ ($=\sqrt{M}/\ell$) and
$(t,r,\varphi)\in(-\infty,\infty)\times(r_+,\infty)\times(-\infty,\infty)$.
The metric in this case is
\begin{equation}
\label{E:109}
ds^{2}=f(r)dt^{2}-f(r)^{-1}dr^{2}-r^{2}d\varphi^{2},
\end{equation}
where
\begin{equation}
\label{E:110}
f(r)=\frac{r^{2}}{\ell^{2}}-M.
\end{equation}
The resulting coordinate patch, together with the metric (\ref{E:109}) becomes one of the exterior (i.e.\ $r > r_+$) regions of the BTZ black hole if we make $\varphi$ periodic (i.e.\ if we quotient by the equivalence relation $\varphi\sim \varphi+2\pi$) \cite{Banados:1992gq}. The maximally extended BTZ spacetime has two exterior regions and it turns out \cite{Ortiz:2011mi, lOrt11}
that the intersection with the conformal boundary of AdS of the corresponding regions {\it before} we make $\varphi$ periodic are the subregions of the intersection of the Poincar\'e chart with the boundary as depicted in Figure \ref{Fig:btzpoincare}. One of us (LO, see \cite{lOrt11, Ortiz:2011mi}) has shown that if one quantizes the Klein Gordon equation, (\ref{KG}), for an arbitrary mass on AdS with vanishing boundary conditions on the conformal boundary, then the boundary limit of the two point function in either the global ground state or the Poincar\'e ground state $<$\ref{Note:PoincareGlobalEquiv}$>$ is, when expressed in Poincar\'e coordinates, given by
\begin{figure}
\centering
\includegraphics[trim=0cm 4cm 0cm 0cm, clip=true, width= 8.2cm, height=9cm]{btzpoincare}
\caption{\label{Fig:btzpoincare}A Penrose diagram showing how the intersection of the covering space of BTZ with the AdS boundary is related to the intersection of our Poincar\'{e} chart with the AdS boundary. The big diamond is the intersection of our Poincar\'{e} chart with the AdS boundary, whereas the small diamonds, A and B, are, respectively, the intersections with the boundary of the covering spaces of the two exterior BTZ regions.}
\end{figure}
\begin{equation}
\label{E:140ll}
G_{\mathrm{Poincare}}(T_1, k_1; T_2, k_2)=\frac{1}{2\pi}\frac{1}{((k_1-k_2)^2-(T_1-T_2-i\epsilon)^2)^\Delta}
\end{equation}
where $\Delta$ is given by (\ref{Delta}) with $d=2$.
From (\ref{E:107}) and (\ref{E:112}) we find that, say in the right diamond region labelled B in Figure (\ref{Fig:btzpoincare}),
the Poincar\'e $(T,k)$ coordinates are related to our BTZ $(t,\varphi)$ coordinates by
\begin{equation}\label{E:116}
T=\ell e^{-\kappa\ell\varphi}\sinh\left(\kappa
t\right)\hspace{1cm} k=\ell e^{-\kappa\ell\varphi}\cosh\left(\kappa t\right).
\end{equation}
Defining the metrics, $g=dt^2-\ell^2d\varphi^2$ and $\tilde g=dT^2-dk^2$, we obtain
\begin{equation}\label{E:117}
\tilde g=\Omega(\varphi)^{2}g,
\end{equation}
where
\begin{equation}\label{E:140rr}
\Omega(\varphi)=\frac{e^{\kappa\ell\varphi}}{\ell\kappa}.
\end{equation}
By a similar analysis to that in the previous section, we have, by (\ref{confstate}) in Note $<$\ref{Note:cnfmlwt}$>$,
\begin{equation}
\label{E:140tt}
\fl G_{\mathrm{preBTZ}}(t_1, \varphi_1'; t_2, \varphi_2')=\frac{1}{2\pi
2^\Delta}\frac{\kappa^{2\Delta}}{(\cosh\kappa\ell(\varphi_1' - \varphi_2')-\cosh\kappa(t_1-t_2-i\epsilon))^\Delta}.
\end{equation}
We have called this two point function, for the global ground state of AdS in BTZ coordinates (before making $\varphi$ periodic) `$G_{\mathrm{preBTZ}}$'.
Now, as discussed in the introduction, we expect the HHI state on BTZ to be the natural state induced on BTZ by the AdS ground state on AdS when one quotients by the equivalence relation $\varphi\sim \varphi+2\pi$. Moreover it is clear that the quotient of the boundary by $\varphi\sim \varphi+2\pi$, consists of two disconnected cylinders, each of radius $\ell$ (see Figure (\ref{Fig:BTZcylinders})), one being the boundary of the right exterior BTZ region and one the boundary of the left exterior BTZ region. In consequence, the two-point function in the HHI state on, say the right exterior region of true BTZ will be defined on the right cylinder and will be given by the image sum $<$\ref{Note:image}$>$
\begin{equation}
\label{E:140vv}
\fl G_{\mathrm{BTZ}}(t_1, \varphi_1'; t_2, \varphi_2')=\sum_{n\in{\mathbb Z}}\frac{1}{2\pi
2^\Delta}\frac{\kappa^{2\Delta}}{(\cosh\kappa\ell(\varphi_1 - \varphi_2+2\pi n)-\cosh\kappa(t_1-t_2-i\epsilon))^\Delta}.
\end{equation}
\medskip
\begin{figure}
\includegraphics[scale = 1.2, trim = 0 0 580 650]{btzboundarycylinders}
\caption{\label{Fig:BTZcylinders}\footnotesize{Schematic picture of BTZ spacetime with its two boundary cylinders}}
\end{figure}
\noindent
(\ref{E:140vv}) has been obtained before \cite{KeskiVakkuri:1998nw, Maldacena:2001kr}. (See Section \ref{Sect:intro}.)
$G_{\mathrm{BTZ}}$ clearly inherits the KMS property from $G_{\mathrm{preBTZ}}$ -- being, in fact, the two-point function on our cylinder of radius $\ell$ at inverse temperature $2\pi/\kappa$, i.e.\ the inverse Hawking temperature of the BTZ black hole. In fact, as already discussed by Maldacena in \cite{ Maldacena:2001kr}, the boundary limit of the HHI state on the union of the left cylinder and the right cylinder is expected to be, what in our language (see Note $<$\ref{Note:equiv}$>$ and \cite{Kay:1985yw, Kay:1985zs}) we would call a double-KMS state -- i.e.\ KMS with respect to the appropriate $t$ (going to the future on the right cylinder and the past on the left cylinder) when restricted to either of the cylinders but entangled in just such a way as to be pure. (See \cite{ Maldacena:2001kr} for the two point function with one point in the right cylinder and one in the left and cf. similar formulae in \cite{Kay:1985yw} and \cite{Kay:1985zs})
(A similar remark could of course have been made about $G_{\mathrm{preBTZ}}$.)
\section{The brick wall modification of the HHI state for the Klein Gordon equation on BTZ and its boundary limit in $d=1$ and $d=2$}
\label{Sect:brick}
In Sections \ref{Sect:1+1} and \ref{Sect:1+2} we have seen that, in 1+1 dimensional and 1+2 dimensional BTZ, the appropriate HHI state -- i.e.\ the global ($=$ Poincar\'e) AdS ground state in the 1+1 dimensional case and the state (with two point function defined by the appropriate image sum) inherited from the global AdS ground state (see Note $<$\ref{Note:PoincareGlobalEquiv}$>$) under the BTZ quotient construction in the 1+2 dimensional case -- goes over to a state on the (say, right) boundary line (in 1+1) or cylinder (in 1+2) which is a KMS state at the appropriate Hawking temperature. Moreover, although we have not derived this explicitly here, it is straightforward to check that each of these bulk HHI states is a KMS state at the Hawking temperature with respect to BTZ time translations in, say, the right BTZ wedge (and is a `double KMS state' on the bulk double wedge -- cf. e.g.\ the discussion after (\ref{Ttrelation}) and Note $<$\ref{Note:equiv}$>$). So it is fair to say that, in boundary-limit holography for BTZ, the thermal nature of the HHI state in the bulk is faithfully reflected by the thermal nature of its boundary limit on the (say) right conformal boundary. However, in line with our discussion in the introduction, in both cases, both bulk and boundary state have infinite entropy -- corresponding to the fact that, if we attempt, formally, to express these states as Gibbs states, with, say, in the case of
1+1 dimensional bulk BTZ, two-point function defined by
\[
{\cal G}_{\mathrm{BTZ}}(t_1,r_1; t_2,r_2)=\tr(e^{-\beta H}\phi(t_1,r_1)\phi(t_2,r_2))/Z
\]
etc., where the fields are represented in the appropriate ground-state (or `Boulware') representation, $Z=\tr(e^{-\beta H})$, $\beta=2\pi/\kappa$, and $H$ is the appropriate Hamiltonian, then $H$ will have continuous spectrum and so the trace in the standard formula $S=-\tr\rho\ln\rho$ where $\rho=e^{-\beta H}/Z$ (as well as the trace in the definition, $Z=\tr(e^{-\beta H})$, of $Z$ itself) will not really exist -- or be `infinite'. (And similarly on the boundary and similarly for bulk and boundary in the 1+2 dimensional case).
To remedy this, we will adopt the brick-wall approach of 't Hooft \cite{'tHooft:1984re} and of Mukohyama and Israel
\cite{Mukohyama:1998rf} and others, as discussed in the introduction, and introduce a suitable `brick wall' in the bulk so as to make the bulk entropy finite. We shall then take the boundary limit of the brick-wall modified thermal two-point function at the Hawking temperature and, in the case of our 1+1 dimensional BTZ with our zero-mass Klein Gordon equation, find the explicit form of the boundary-limit two-point function and show that this is both thermal at the Hawking temperature (for an appropriately, brick-wall modified boundary Hamiltonian) and also has the same finite entropy as the bulk brick-wall modified state. We shall calculate the bulk entropy in the 1+1 dimensional brick wall model by the Mukohyama-Israel approximation method \cite{Mukohyama:1998rf} and show that, for this case, this method reproduces the exact result.
In the 1+2 dimensional BTZ case, we shall re-obtain, by the same Mukohyama-Israel approximation method, the result of Kim et al.\ \cite{Kim:1996eg} which is that (at least to a good approximation) the brick-wall modified bulk entropy is proportional to the `area' (i.e circumference) of the event horizon. We shall also obtain in-principle exact sums (albeit difficult to perform explicitly) for the bulk brick-wall modified two-point function and its boundary limit and show, without summing explicitly, that the bulk and boundary entropies must be exactly equal.
\subsection{The brick wall model and its boundary limit for 1+1 dimensional BTZ}
We locate our brick wall, in BTZ coordinates, at $r=r_+ + \epsilon$ where $\epsilon$ is suitably small. This is a natural choice since it respects BTZ time-translational invariance. In Figure \ref{Fig:ads2}, the brick wall is a smooth curve joining D to E and passing just to the right of the piecewise-linear curve DCE. Our aim is to quantize our field to the right of this brick wall, with vanishing boundary conditions on the brick wall as well as on the conformal boundary. The Hamiltonian will, as expected, turn out to have discrete spectrum. So we will be able to construct a thermal state (with inverse Hawking temperature, $\beta=2\pi/\kappa$) as a Gibbs state in the ground-state representation.
To do all this, it is convenient to work in $(t,r^*)$ coordinates, where the tortoise-like coordinate (cf. (\ref{Schwtortoise})) $r^*$, is related to the BTZ $r$ coordinate of (\ref{E:bb3}) and (\ref{E:16}) by
\begin{equation}
\label{defrstar}
\frac{dr^*}{dr}=f(r)^{-1}=\left(\frac{r^2}{\ell^2}-M\right)^{-1}=\frac{\ell^2}{(r+r_+)(r-r_+)},
\end{equation}
with solution
\begin{equation}
\label{BTZtortoise}
r^{*}=\frac{\ell^{2}}{2r_{+}}\textrm{ln}\frac{r-r_{+}}{r+r_{+}}.
\end{equation}
The metric is then given by
\begin{equation}
\label{E:141b}
ds^{2}=f(r(r^*))\left(dt^2-{dr^*}^2\right).
\end{equation}
On the full exterior BTZ region, $r^*$ ranges from $-\infty$ at the BTZ horizon to $0$ at the AdS boundary (where $r=\infty$). However, the brick wall will lie at the finite value $r^*=-B$, where $B=(\ell^2/(2r_{+}))\ln(2r_+/\epsilon+1)$. So, the classical theory which we wish to quantize is the massless covariant Klein-Gordon equation, which, in $(t,r^*)$ coordinates, becomes
\begin{equation}
\label{E:142b}
\left(\frac{\partial^{2}}{\partial
t^{2}}-\frac{\partial^{2}}{\partial
{r^{*}}^{2}}\right)\phi(t,r^{*})=0,
\end{equation}
with the boundary conditions
\begin{equation}
\label{E:143b}
\phi(t,-B)=0=\phi(t,0), \ \ \forall t\in {\mathbb R}.
\end{equation}
By essentially the same arguments which we gave for the form of ${\cal G}_{\mathrm{global}}$ in Equation (\ref{2dmasslessG}), the two-point function for the ground state for this theory then takes the form
\[
{\cal G}_{\mathrm{BTZ1+1brick}}(t_1,r^*_1; t_2, r^*_2) = \hbox{`}\langle 0|\hat{\phi}(t_1,r^*_1)\hat{\phi}(t_2, r^*_2)|0\rangle\hbox{'}
\]
\begin{equation}
\label{E:148b}
=\sum_n\frac{1}{B\omega_n}e^{-i\omega_n(t_1-t_2)}\sin(\omega_n
r^*_1)\sin(\omega_n r^*_2)
\end{equation}
where $\omega_n$ stands for $n\pi/B$, $n=1,2,3,\dots$. In other words, it is the same as the two-point function of a (real) massless Klein Gordon field in a strip, $t \in {\mathbb R}, r^*\in (-B, 0)$ of 1+1 dimensional Minkowski space with vanishing boundary conditions at its edges. In fact, as may readily be checked, (\ref{E:148b}) may be derived by the usual creation and annihilation operator method, for mode functions adapted to these boundary conditions.
To get a suitable boundary limit of ${\cal G}_{\mathrm{BTZ1+1brick}}$, we expect, in accordance with (\ref{Blim}), that the right thing to do will be to multiply (\ref{E:148b}) by $\ell^{-2}f(r(r_1^*))^{1/2}f(r(r_2^*))^{1/2}$ and then to take the limit as $r_1^*$ and $r_2^*$ both tend to zero. To this end, we notice first that $f(r)^{1/2}\sim r/\ell$. Also, recalling that $r(r^*)$ is given by Equation (\ref{BTZtortoise}) one easily sees that the inverse function, $r(r^*)$ is
\begin{equation}
\label{BTZesiotrot}
r(r^*)=-r_+\coth\left(\frac{r_+r^*}{\ell^2}\right)
\end{equation}
from which we see that when $r$ is large (i.e.\ $r^*$ is close to $0$) we have $r\sim -\ell^2/r^*$ (we can actually also see this directly from (\ref{BTZtortoise})) and hence
$f(r(r^*))\sim r(r^*)/\ell\sim -\ell/r^*$. In fact, one easily sees, with these considerations, that the above `right thing to do' amounts to taking the limit as $r_1^*$ and $r_2^*$ tend to zero of $1/(r_1^*r_2^*)$ times ${\cal G}_{\mathrm{BTZ1+1brick}}(t_1,r^*_1; t_2, r^*_2)$ which is easily seen to be
\begin{equation}\label{E:149b}
G_{\mathrm{BTZ1+1brick}}(t,t')=\sum_n\frac{\omega_n}{B}e^{-i\omega_n(t-t')}.
\end{equation}
As one can see from the discreteness of the sum in (\ref{E:148b}) (or from the discreteness of the modes in its derivation with creation and annihilation operators) the `one-particle Hamiltonian', $h$, whose second quantization $d\Gamma(h)=
0 \oplus h \oplus (h\otimes 1 + 1\otimes h) \oplus \dots$ (see e.g.\ \cite{RSVol2}) generates bulk BTZ time-evolution in the region to the right of our brick wall, has discrete spectrum, with, in fact, eigenvalues
$\omega_n=n\pi/B$, $n=1,2,3\dots$ (and multiplicity 1). Thus $d\Gamma(h)$ will have discrete spectrum too. And, from (\ref{E:149b}), we may conclude that, also, the one-particle Hamiltonian generating time-evolution on the BTZ boundary for the (conformal, generalized-free, with scaling dimension $\Delta=1$)) boundary restriction of this bulk theory has the same spectrum, as will also {\it its} second quantization. In fact it will clearly be unitarily equivalent to $h$, so we will also call it $h$ and its second quantization $d\Gamma(h)$.
Because the spectra are discrete -- more precisely, because
\begin{equation}
\label{partition}
Z=\tr\left(e^{-\beta d\Gamma(h)}\right)
\end{equation}
is finite -- we may thus compute the bulk and boundary thermal two-point functions
\[
{\cal G}_{\mathrm{BTZ1+1brick}}^\beta(t_1,r^*_1; t_2, r^*_2) = Z^{-1}\tr(e^{-d\Gamma(h)}\hat{\phi}(t_1,r^*_1)\hat{\phi}(t_2, r^*_2))
\]
and
\[
G_{\mathrm{BTZ1+1brick}}^\beta(t_1, t_2) = Z^{-1}\tr(e^{-d\Gamma(h)}\hat{\phi}(t_1)\hat{\phi}(t_2))
\]
at the appropriate inverse Hawking temperature $\beta=2\pi/\kappa$. E.g.\ by using creation and annihilation operator methods again, these are easily calculated to be
\begin{equation}
\label{E:150b}
\fl {\cal G}_{\mathrm{BTZ1+1brick}}^\beta(t_1,r^*_1; t_2, r^*_2)=\sum_n\frac{\sin(\omega_n
r_1^{*})\sin(\omega_n {r_2^{*}})}{B\omega_n}
\left(\frac{e^{-i\omega_n(t-t_1)}+e^{i\omega_n(t-t_1)}e^{-\beta\omega_n}}{1-e^{-\beta\omega_n}}\right)
\end{equation}
and
\begin{equation}
\label{E:152b}
G_{\mathrm{BTZ1+1brick}}^\beta(t_1, t_2) =
\sum_{n}\frac{\omega_n}{B}\left(\frac{e^{-i\omega_n(t_1-t_2)}+e^{i\omega_n(t_1-t_2)}e^{-\beta\omega_n}}{1-e^{-\beta\omega_n}}\right)
\end{equation}
both of which are easily seen to satisfy the KMS condition (see Section \ref{Sect:prelim}) for this $\beta$. We also notice that, just as we obtained
$G_{\mathrm{BTZ1+1brick}}$ (\ref{E:149b}) from ${\cal G}_{\mathrm{BTZ1+1brick}}$ (\ref{E:148b}), we could have obtained $G_{\mathrm{BTZ1+1brick}}^\beta$ (\ref{E:152b}) as the limit as $r_1^*$ and $r_2^*$ tend to zero of $1/(r_1^*r_2^*)$ times ${\cal G}_{\mathrm{BTZ1+1brick}}^\beta$ (\ref{E:150b}).
Recalling that $\omega_n$ stands for $n\pi/B$, we notice that (\ref{E:152b}) can be rewritten as
\begin{equation}
\label{Gbrickbetan}
\fl G_{\mathrm{BTZ1+1brick}}^\beta(t_1, t_2) =
\sum_{n=1}^\infty\frac{\pi n}{B^2}\left(\frac{e^{-i(n\pi/B)(t_1-t_2)}+e^{i(n\pi/B)(t_1-t_2)}e^{-\beta n\pi/B}}{1-e^{-\beta n\pi/B}}\right).
\end{equation}
In the limit as the brick wall is removed, $B$ tends to infinity and we expect this to be well-approximated by replacing the sum over $n$ by an integral over, say, $y$. After making the substitution $x=y/B$ we will thus have
\begin{equation}
\label{Gbrickbetax}
\int_0^\infty \pi x\left(\frac{e^{-i(\pi x)(t_1-t_2)}+e^{i(\pi x)(t_1-t_2)}e^{-\beta\pi x}}{1-e^{-\beta\pi x}}\right)dx
\end{equation}
which $<$\ref{Note:Gradshteyn}$>$ is equal to
\[
-\frac{\pi}{\beta^2}\frac{1}{\sinh^2\left(\frac{\pi(t-t'-i\epsilon)}{\beta}\right)}
\]
which, recalling that $\kappa=2\pi/\beta$, agrees with (\ref{E:100}). This confirms that the brick wall does what we anticipated in Section \ref{Sect:intro} that it would do in this case:
\emph{The boundary limit of the thermal state at the Hawking temperature built on the modification, due to the presence of the brick wall, to the boundary limit of the BTZ ground state (i.e.\ the `Boulware state') tends, as the brick wall is removed, to the boundary limit of the BTZ HHI state.}
Although we have not shown it expicitly here, the corresponding (and more straightforward) statement also holds for the bulk theory: The bulk thermal state at the Hawking temperature built on the brick-wall modified BTZ ground state tends, as the brick wall is removed, to the BTZ HHI state (restricted to the right BTZ wedge).
Moreover, we saw above that (again in line with what we anticipated in Section \ref{Sect:intro} -- see Note $<$\ref{Note:Ham}$>$) in the presence of the brick wall, the Hamiltonians for the bulk and boundary theory are unitarily equivalent mathematical operators. But now, since they have discrete spectrum (and a finite partition function (\ref{partition})) they will, in particular, have finite (and equal!) von Neumann entropies, given by $S=-\tr(\rho\ln\rho)$ for $\rho=e^{-\beta H}/Z$, $Z=\tr(e^{-\beta H})$, $H$ being what we called above $d\Gamma(h)$. We next calculate the value of this entropy.
In this simple massless 1+1 dimensional BTZ case, we can in fact calculate the entropy exactly: In fact,
thinking of it as bulk entropy, in view of Equations (\ref{E:142b}) and (\ref{E:143b}) etc., for given $\beta=1/{\cal T}$, it is just the usual entropy for a special relativistic real free scalar field in a `one-dimensional box' (i.e.\ line) of length $B$ in thermal equilibrium at temperature, $\cal T$, which is given (to a very good approximation $<$\ref{Note:scalarentropy}$>$) by the (standard) formula:
\begin{equation}
\label{1dentropy}
S=\frac{\pi B{\cal T}}{3}.
\end{equation}
Taking $\cal T$ here to be the appropriate Hawking temperature, which we take to be $\kappa/(2\pi) = r_+/(2\pi \ell^2)$ ($= M^{1/2}/(2\pi \ell)$), this is
\begin{equation}
\label{1dentropywithTH}
S=\frac{r_+ B}{6\ell^2}.
\end{equation}
It is also instructive to apply the approximation method of Mukohyama and Israel \cite{Mukohyama:1998rf} to this case.
For any metric with the spherically symmetric `Schwarzschild-like' form (\ref{generalsphmetric}) of any spacetime dimension this method amounts to assuming that the total entropy, $S$, of a thermal state at temperature $\cal T$ in a region, $r_+ + \epsilon < r < L$, say ($r_+$ the value of $r$ at the relevant horizon) should be well approximated by the integral
\begin{equation}
\label{Sapprox}
S=\int_{r_+ +\epsilon}^L s({\cal T}_{\mathrm{loc}}(r)) A(r)\frac{dr}{f(r)^{1/2}}
\end{equation}
where, for any temperature, $\cal T$, $s({\cal T})$ is the entropy density of a free real scalar field in Minkowski space of the same dimension and the local temperature, ${\cal T}_{\mathrm{loc}}(r)$ is given by
\begin{equation}
\label{Tlocal}
{\cal T}_{\mathrm{loc}}(r)={\cal T}f^{-1/2}(r).
\end{equation}
$A(r)$ in (\ref{Sapprox}) denotes the area of the $d$-1 sphere of radius $r$ ($=2\pi r$ in 1+2 dimensions, $4\pi r^2 $ in 1+3 dimensions, etc.; in 1+1 dimensions, we replace $A(r)$ by 1) so that $A(r)dr/f(r)^{1/2}$ represents the proper volume element.
Applying the formula (\ref{Sapprox}) to find the total entropy outside the brick wall in 1+1 dimensional BTZ, we take $L=\infty$. Also we have, by (\ref{E:16}), $f(r)=(r^2/\ell^2-M)^{-1}$ while the entropy density $s({\cal T})$ (by Note $<$\ref{Note:scalarentropy}$>$) is $\pi {\cal T}/3$. So (\ref{Sapprox}) becomes
\[
S=\frac{\pi {\cal T}}{3}\int_\epsilon^\infty \left(\frac{r^2}{\ell^2}-M\right)^{-1}dr
\]
which, in view of (\ref{defrstar}), is just $\pi {\cal T}/3 \int_{-B}^0 dr^* = \pi B{\cal T}/3$ which agrees with (\ref{1dentropy}) exactly! (And hence will also lead to the formula (\ref{1dentropywithTH})). Our main reason for deriving (\ref{1dentropy}), (\ref{1dentropywithTH}) is as a preliminary to deriving the corresponding formulae for 1+2 dimensional BTZ to which we next turn. See also Note $<$\ref{Note:300}$>$.
\subsection{The brick wall model and its boundary limit for 1+2 dimensional BTZ}
All of the above generalizes to 1+2 dimensional BTZ. The (zero-mass) covariant Klein-Gordon equation (\ref{KG})
becomes
\begin{equation}
\label{BTZ1+2KG}
\frac{\partial^2\phi}{\partial t^2}-\frac{1}{r}\left(\frac{r^2}{l^2}-M\right)\frac{\partial}{\partial r}\left(r\left(\frac{r^2}{l^2}-M\right)\frac{\partial\phi}{\partial r}\right)-\left(\frac{r^2}{l^2}-M\right)\frac{\partial^2\phi}{\partial\varphi^2}=0
\end{equation}
and may be solved, subject to our vanishing boundary conditions at $r=r_+ +\epsilon$ and $r=\infty$, by the usual method of separation of variables. We seek (complex-valued) solutions of form
\[
\phi(t,r,\varphi)=e^{-i\omega t}e^{in\varphi}f_n(r)
\]
where $f_n(r)$ satisfies the appropriate radial equation
\begin{equation}
\label{radialeq}
\fl -\frac{1}{r}\left(\frac{r^2}{l^2}-M\right)\frac{d}{dr}\left(r\left(\frac{r^2}{l^2}-M\right)\frac{df_n(r)}{dr}\right)
+n\left(\frac{r^2}{l^2}-M\right)f_n(r)=\omega^2f_n(r).
\end{equation}
We remark that (\ref{radialeq}) may be written, alternatively,
in the form of the Schr\"odinger-like equation with positive potential:
\[
\left(-\frac{d^2}{{dr^*}^2}+\left(\frac{r^2}{l^2}-M\right)\left(n^2+\frac{3}{4l^2}+\frac{M}{4r^2}\right)\right)r^{1/2}f_n(r^*)=\omega^2r^{1/2}f_n(r^*)
\]
where $r^*$ is related to $r$ in the same way as in (\ref{BTZtortoise}) and (\ref{BTZesiotrot}).
For each $n$, this will clearly have solutions which satisfy our boundary conditions only for certain discrete values of $\omega$, which we shall call $\omega_n^m$. We also recall \cite{lOrt11} that, for any $\omega$, Equation (\ref{radialeq}) has (up to a multiplicative constant) the unique solution which vanishes at $r=\infty$, $f_n(r)=R_{\omega,n}(r)$, where
\begin{equation}
\label{hyper}
\fl R_{\omega,n}(r)=M^{\alpha+\beta}\left(\frac{r^2}{r_+^2}-1\right)^\alpha \left(\frac{r^2}{r_+^2}\right)^{\beta-\alpha}
F\left(\alpha+\beta+1,\alpha-\beta+1;2;\frac{r_+^2}{r^2}\right),
\end{equation}
where $\alpha=ir_+\omega/(2M)$ and $\beta=ir_+n/(2M\ell)$.
(cf. \cite{Kenmoku:2008qx} who give an apparently different expression).
The discrete allowed values, $\omega_n^m$, of $\omega$ may then, in principle, be found by imposing the boundary condition $R_{\omega,n}(r)=0$ at $r=r_+ +\epsilon$ and we shall, from now on, let $f_{nm}(r)$ stand for
$R_{\omega_n^m,n}(r)$ (defined, as in (\ref{hyper}), for $\alpha=ir_+\omega_n^m/(2M)$ and $\beta$ as before).
We will then have (see again \cite{lOrt11}) that the functions
\begin{equation}
\label{E:157b}
F_{nm}(t,\varphi, r)=\left(\frac{A}{\omega_n^m}\right)^{1/2}e^{-i\omega t}e^{in\phi}f_{n\omega}(r)
\end{equation}
will be a complete set of positive frequency modes for a suitable constant, $A$.
By creation and annihilation operator methods, one may then show (in analogy with equation (\ref{E:148b})), that the ground state two-point function will take the form
\[
{\cal G}_{\mathrm{BTZ1+2brick}}(t_1, \varphi_1, r_1; t_2, \varphi_2, r_2) = \hbox{`}\langle 0|\hat{\phi}(t_1, \varphi_1, r_1)\hat{\phi}(t_2, \varphi_2, r_2)|0\rangle\hbox{'}
\]
\begin{equation}
\label{E:160b}
=\sum_{nm}\frac{A}{\omega_n^m}
e^{-i\omega_n^m(t_1-t_2)}e^{in(\varphi_1-\varphi_2)}f_{nm}(r_1)f_{nm}(r_2)^*.
\end{equation}
The boundary limit of (\ref{E:160b}) will be defined (combining (\ref{Blim}) with Equation (\ref{1+2conf}) in Note $<$\ref{Note:confboundary}$>$ -- cf. the passage from (\ref{E:148b}) to (\ref{E:149b}))
as the limit as $r_1^*$ and $r_2^*$ tend to zero of
$1/(r_1^*r_2^*)$ times -- equivalently the limit as $r_1$ and $r_2$ tend to infinity of $(r_1/r_+)^2(r_2/r_+)^2$ times -- ${\cal G}_{\mathrm{BTZ1+2brick}}(t_1,\varphi_1, r_1; t_2, \varphi_2, r_2)$, which is easily seen to be
\begin{equation}
\label{E:161b}
G_{\mathrm{BTZ1+2brick}}(t_1, \varphi_1; t_2, \varphi_2)=\sum_{nm}\frac{A}{\omega_n^m}e^{-i\omega_n^m(t_1-t_2)}e^{in(\varphi_1-\varphi_2)}
\end{equation}
(in analogy with equation (\ref{E:148b})).
Similarly, in analogy with equations (\ref{E:150b}) and (\ref{E:152b}), we'll have
\[
{\cal G}_{\mathrm{BTZ1+2brick}}^\beta(t_1, \varphi_1, r_1; t_2, \varphi_2, r_2)=\sum_{nm}\frac{A}{\omega_n^m}
e^{in(\varphi_1-\varphi_2)}f_{nm}(r_1)f_{nm}(r_2)^* \ \ \times
\]
\begin{equation}
\label{1+2betacal}
\left(\frac{e^{-i\omega_n^m(t_1-t_2)}+e^{i\omega_n^m(t_1-t_2)}e^{-\beta\omega_n^m}}{1-e^{-\beta\omega_n^m}}\right)
\end{equation}
and
\begin{equation}
\label{1+2beta}
\fl G_{\mathrm{BTZ1+2brick}}^\beta(t_1, \varphi_1; t_2, \varphi_2) =
\sum_{nm}\frac{A}{\omega_n^m}e^{in(\varphi_1-\varphi_2)}\left(\frac{e^{-i\omega_n^m(t_1-t_2)}+e^{i\omega_n^m(t_1-t_2)}e^{-\beta\omega_n^m}}{1-e^{-\beta\omega_n^m}}\right).
\end{equation}
both of which are, again, easily seen to satisfy the KMS condition (see Section \ref{Sect:prelim}). Further,
just as we showed above that the limit in which the brick wall is removed of $G_{\mathrm{BTZ1+1brick}}^\beta$
(\ref{E:152b}), for $\beta$ equal to the inverse Hawking temperature, equals the boundary limit, (\ref{E:100}), of the two-point function of the HHI state on 1+1 dimensional BTZ, so we expect that the limit in which
the brick wall is removed of $G_{\mathrm{BTZ1+2brick}}^\beta$ will be equal to the boundary limit, (\ref{E:140vv}), of the two-point function of the HHI state on 1+2 dimensional BTZ. However we have not shown this explicitly and, to do so, one would need to have more mathematical control on the values of the $\omega_n^m$.
What is important for us is that it is clear from the above that (even in the absence of any mathematical control on the values of the $\omega_n^m$) similarly to the 1+1 dimensional case, the one particle Hamiltonians
generating time evolution in both bulk and boundary in the ground-state representations corresponding to each of the brick-wall-modified two-point functions, ${\cal G}_{\mathrm{BTZ1+2brick}}$ (\ref{E:160b}) and $G_{\mathrm{BTZ1+2brick}}$ (\ref{E:161b}) are both unitarily equivalent to one another (both being the second quantizations, $d\Gamma(h)$, of the one-particle Hamiltonian, $h$, with non-degenerate spectrum now consisting of the $\omega_n^m$ (taken with appropriate multiplicity if/when any of these turn out to be degenerate).
So, just as in the 1+1 dimensional case, we expect that, in the presence of the brick wall, both bulk and boundary will have finite $<$\ref{Note:omeganmcondition}$>$ (and equal!) von Neumann entropies, given by $S=-\tr(\rho\ln\rho)$ for $\rho=e^{-\beta H}/Z$, $Z=\tr(e^{-\beta H})$, $H$ now meaning $d\Gamma(h)$ with $h$ (with spectrum consisting of the $\omega_n^m$) as in the previous paragraph.
It turns out to be difficult to compute this entropy exactly, because of the difficulty of obtaining sufficient mathematical control over the $\omega_n^m$ although an attempt to do so was made by Ichinose and Satoh \cite{Ichinose:1994rg}. Instead, we have applied the approximation method of Mukohyama and Israel, which we also used above in the 1+1 dimensional case, to obtain Equations (\ref{1plus2brickfirstresult}) and (\ref{1plus2bricksecondresult}) below. Essentially the same results as these (together with generalizations to rotating BTZ) have been obtained before, by Kim et al.\ \cite{Kim:1996eg} who also comment on the work of Ichinose and Satoh and explain how it can be reconciled with their ($=$ our) result. We give the details below since our method (i.e.\ the application of the method of Mukohyama and Israel \cite{Mukohyama:1998rf}), is particularly short and direct and also since, by basing our discussion on the approach of Mukohyama and Israel \cite{Mukohyama:1998rf} we are able to draw a conclusion (about the effective number of Klein Gordon fields needed in a consistent quantum theory of gravity) which does not appear to have been drawn before in a 1+2 dimensional (BTZ) context.
Our starting point is again Equation (\ref{Sapprox}). Applying this formula to find the total entropy outside the brick wall in 1+2 dimensional BTZ, we again take $L=\infty$ and again have, by (\ref{Tlocal}), ${\cal T}_{\mathrm{loc}}(r)={\cal T}f^{-1/2}(r)$ where, by (\ref{E:110}), $f(r)$ is again equal to $(r^2/\ell^2-M)^{-1}$. The entropy density, $s({\cal T})$ (by Note $<$\ref{Note:scalarentropy}$>$) will now be given by
\begin{equation}
\label{2dimentropydens}
s({\cal T})=3\zeta(3){\cal T}^2/(2\pi)
\end{equation}
and $A(r)$ is $2\pi r$. So (\ref{Sapprox}) becomes
\[
S=3\zeta(3){\cal T}^2\int_{r_+ + \epsilon}^\infty r\left(\frac{r^2}{\ell^2}-M\right)^{-3/2}dr
\]
\[
=-\frac{3\zeta(3){\cal T}^2\ell^3}{2}\left((r_+ +\epsilon)^2-r_+^2\right)^{-1/2}
\]
where, for the last line, we recall that $M=r_+^2/\ell^2$.
For small $\epsilon$, this will be well-approximated by
\begin{equation}
\label{1plus2brickfirstresult}
\frac{3\zeta(3)}{2}{\cal T}^2\ell^3(2\epsilon r_+)^{-1/2}.
\end{equation}
We next set $\cal T$ equal to the Hawking temperature,
\begin{equation}
\label{THawking}
{\cal T}=\kappa/(2\pi) = r_+/(2\pi \ell^2),
\end{equation}
and, following Mukohyama and Israel, express our result in terms of the metrical distance, $\alpha$, of the brick wall from the horizon, related to $\epsilon$ by
\[
\alpha=\int_{r_+}^{r_++\epsilon}f(r)^{-1/2}dr=\ell\int_{r_+}^{r_++\epsilon}\frac{dr}{(r^2-r_+^2)^{1/2}}\simeq 2\ell\left(\frac{\epsilon}{2r_+}\right)^{1/2},
\]
where for the last approximate equality, we have approximated $(r^2-r_+^2)$ by $2r_+(r-r_+)$. Equivalently
\begin{equation}
\label{epsilonalpharelation}
\epsilon\simeq\frac{r_+}{2\ell^2}\alpha^2.
\end{equation}
With (\ref{THawking}) and (\ref{epsilonalpharelation}), (\ref{1plus2brickfirstresult}) becomes $3\zeta(3)r_+/(8\pi^2 \alpha)$. The entropy, $S_{\mathsf{N}}$, of a collection of matter fields equivalent (see Section \ref{Sect:intro}) to $\mathsf{N}$ effective Klein Gordon fields may then be written in the suggestive form $<$\ref{Note:300}$>$
\begin{equation}
\label{1plus2bricksecondresult}
S_{\mathsf{N}}={\mathsf{N}}\left(\frac{3\zeta(3)}{4\pi^3}\right)\frac{1}{\alpha}\left(\frac{2\pi r_+}{4}\right).
\end{equation}
We see from (\ref{1plus2bricksecondresult}) that the entropy is proportional to the appropriate notion of `horizon area' for 1+2 dimensions -- i.e.\ circumference, $2\pi r_+$. Moreover, following the line of argument of Mukohyama and Israel \cite{Mukohyama:1998rf}, if the metrical distance, $\alpha$, from the brick wall to the horizon is taken to be $\approx$ 1 in natural units (see Section \ref{Sect:prelim}) then, for $\mathsf{N}$ around $4\pi^3/(3\zeta(3)) \approx 34$, the total matter entropy will take the Hawking value of $2\pi r_+/4$. This suggests that a consistent theory of quantum gravity (with cosmological constant) in 1+2 dimensions will require a collection of matter fields equivalent to a number, $\mathsf{N}$, within an order of magnitude of 34 or so of (massless, real) effective Klein Gordon fields.
We remark that, with a brick wall in the (1+3 dimensional) Schwarzschild metric, as discussed in \cite{'tHooft:1984re} and
\cite{Mukohyama:1998rf}, and for finite $L$ in (\ref{Sapprox}), the entropy arises as a sum of a (large) `area piece', due to a small region near the brick wall and proportional to horizon area, and a (typically smaller) `volume piece' associated with the region far away from the horizon. In contrast, in the case of BTZ, even though $L$ is now taken to be $\infty$, we see from Equation (\ref{1plus2bricksecondresult}) that -- at least in our ($=$ Kim et al.'s \cite{Kim:1996eg}) approximation -- our entire entropy consists of an `area piece' and the `volume piece' vanishes!
\section{Discussion}
\label{Sect:discuss}
\subsection{Summary of Results}
\label{subsect:summary}
We have provided general evidence that, at least for our simple massless Klein Gordon equation model, there seem to be well-defined versions of fixed-background holography, not only for plain AdS, but also for certain asymptotically AdS spacetimes, such as, in $1+ d$ dimensions, the Schwarzschild-AdS black hole spacetime. In particular, specializing to 1+2 dimensions (BTZ) we have confirmed that the boundary theory is (locally) the same (i.e.\ that for the conformal generalized free field with anomalous dimension $\Delta=1$) as that of plain 1+2 dimensional AdS and also that, as expected, the BTZ counterpart to the HHI state has, as its boundary limit (on, say, the right boundary cylinder) a thermal state of that boundary theory, at the appropriate Hawking temperature. Moreover, and this was our main new result at a technical level, we have shown that, while these states of this theory have infinite entropy, in both bulk and boundary, if one imposes a suitable brick-wall cutoff in the bulk, labelled by the metrical distance, $\alpha$, from the brick wall to the horizon, then both bulk and boundary theories will have equal, finite, entropies, proportional to the `area' (i.e.\ circumference) of the event horizon. Moreover, by the way, we have pointed out, following 't Hooft and Mukohyama and Israel, that, if one assumes that a suitable number, $\mathsf{N}$, of our Klein Gordon fields will serve to model the thermodynamic behaviour of the basic (matter and gravity) fields which make up a consistent theory of quantum gravity (in 1+2 dimensions, with cosmological constant) then, in order for the entropy of both bulk and boundary to equal the, presumed physically correct, Hawking value of one quarter of the `area' of the event horizon (i.e.\ $\pi r_+/2$), ${\mathsf{N}}/\alpha$ needs to be taken to be approximately 34. (So, in particular, if $\alpha$ is around 1, then $\mathsf{N}$ needs to be around 34.)
Below, we shall attempt to draw some tentative conclusions from our results regarding the relationship between
fixed-background holography (in particular, Rehren's algebraic holography) and mainstream Maldacena holography.
In the course of our discussion, we shall recall the `matter-gravity entanglement hypothesis' of one of us (BSK) and the way in which this claims to offer a resolution to the Mukohyama-Israel `complementarity' puzzle. Some (but not all) of our tentative conclusions assume this hypothesis and this resolution to be correct. In particular, we point out that this resolution seems to suggest a radically different from usual understanding of the nature of Maldacena holography (namely as a one-way map from bulk to boundary rather than a bijection). In the sequel, we shall find it convenient (despite our earlier terminological conventions in Section \ref{Sect:intro} and Note $<$\ref{Note:unfortunate}$>$) to adopt Arnsdorf and Smolin's terminology `Rehren dual' to denote the bulk theory which corresponds to a given boundary CFT according to Rehren's algebraic holography. One of our tentative conclusions which doesn't depend on our `matter-gravity entanglement hypothesis' (although which survives in an interestingly modified form with it) concerns the nature of the Rehren dual of the boundary CFT which arise in Maldacena's AdS/CFT correspondence. Finally, we shall also comment further on the Arnsdorf-Smolin puzzle in the light of our tentative conclusions $<$\ref{Note:Smolinprovisos}$>$.
We begin, in the next subsection, by gathering together some, we think reasonable, extrapolations from what we have shown:
\subsection{Tentative generalizations and working assumptions}
\label{subsect:working}
We adopt a number of working assumptions throughout the sequel: First: that the result which we found in the 1+2 dimensional case holds for Schwarzschild AdS (with arbitrary mass $M$ and radius $\ell$) in arbitrary dimension 1+$d$ and for other quantum theories (not just the Klein-Gordon field) defined in terms of nets of local *-algebras on the fixed Schwarzschild-AdS background. In particular, we shall assume that, for such quantum theories on such bulk spacetimes, the boundary theory, defined by a suitable generalization of (\ref{Blim}) on the appropriate 1+($d$-1) dimensional (say right) boundary cylinder will be locally the same boundary conformal quantum theory that one obtains on the boundary for the same bulk quantum theory on AdS. Moreover, we shall assume that the HHI state will go over to a thermal state at the appropriate Hawking temperature on the boundary theory. We shall assume further that, if necessary with a suitable brick wall in the bulk to make it finite, the entropy of the induced state on the boundary will always equal the bulk entropy.
Turning to consider the reverse direction of boundary to bulk: Second: It also seems reasonable to expect (see $<$\ref{Note:expect}$>$) that, given our conformal generalized free field with anomalous dimension given by (\ref{Delta}) for some choice of $d$ and $m$, defined on a 1+($d$-1) dimensional cylinder of radius $\ell$, then the bulk quantum theory corresponding to the covariant Klein Gordon equation (\ref{KG}) for mass $M$ on the right Schwarzschild wedge of the 1+$d$ dimensional Schwarzschild-AdS spacetime of mass $M$ and radius $\ell$ will be related to it by boundary limit holography. (For the prospects for it also being related according to a suitable extension of Rehren's algebraic holography see Note $<$\ref{Note:bhalgholo}$>$). Furthermore, we would expect a thermal state for any temperature, $\cal T$, on the former CFT to go over to a thermal state at the same temperature on each of the latter, bulk right Schwarzschild wedges (as well, of course, as on all of plain AdS). However, in view of the uniqueness results in \cite{Kay:1988mu} and \cite{Kay:1992gr}, for given $M$ and $\ell$, we would only expect one of these bulk states to extend in a non-singular way from the right Schwarzschild wedge to the full *-algebra for the full (maximally extended) Schwarzschild-AdS bulk -- namely the one for which $\cal T$ coincides with the Hawking temperature. (And when it does extend, the extended state will be the HHI state.)
We shall also assume that this second working assumption generalizes to an arbitrary boundary CFT. i.e.\ we shall assume that, given an arbitrary CFT whose Rehren dual is a given quantum theory in the bulk of plain AdS, then, if defined on a 1+ ($d$-1) dimensional cylinder of radius $\ell$, it will be related by boundary limit holography to the \textit{same} $<$\ref{Note:same}$>$ bulk theory defined on the right Schwarzschild wedge of the 1+$d$ dimensional Schwarzschild AdS spacetime of arbitrary mass, $M$, and radius $\ell$. A thermal state at temperature, $\cal T$, of the CFT on the boundary cylinder could then be seen as arising from the boundary limit of a thermal state at the same temperature of the same bulk theory. But we would only expect such a state to arise from a non-singular state on the *-algebra when the same theory is defined on the maximally extended Schwarzschild AdS with mass, $M$, such that $\cal T$ is the Hawking temperature for that particular $M$ and $\ell$. (And when it does extend, the extended state would again be the HHI state.)
With these working assumptions, we next remark that, in fixed background holography, the very fact that the same boundary theory (say our conformal generalized free field with anomalous dimension $\Delta$ on a 1+($d$-1) dimensional cylinder of radius $\ell$) is related to more than one bulk theory (i.e.\ the Klein Gordon equation on 1+$d$ dimensional plain AdS and also on the right Schwarzschild wedge of Schwarzschild AdS for each value of $M$) is a feature that is not shared with (presumably background independent!) mainstream Maldacena AdS/CFT. In the latter theory, we expect, of course, that the different bulk geometries themselves arise from classical approximations to different states of the same bulk theory (in particular a ground state and a thermal state at the relevant Hawking temperature $\sqrt{M}/(2\pi\ell)$ for each $M$). This contrast is well-illustrated by the Maldacena scenario involving an AdS$_5\times \mathbb{S}^5$ bulk, where \cite{Witten:1998zw} at low temperatures, a thermal state on a cylinder of the boundary CFT is believed to correspond to a state of the bulk theory $<$\ref{Note:double}$>$ which, in the large $N$/weak coupling limit consists of the relevant (Kaluza Klein reduced) supergravity theory on a bulk spacetime which is locally plain AdS$_5$, while at high temperatures (i.e.\ above the relevant phase transition \cite{Witten:1998zw}) and in the same limit, it consists of the same supergravity theory, but now on a five dimensional Schwarzschild-AdS background with an appropriate mass, $M$; whereas, on the other hand, one can, presumably, ask about the fixed-background holography counterpart to thermal states of the boundary CFT on the given radius-$\ell$ boundary cylinder at any temperature for bulk Schwarzschild-AdS backgrounds of any radius $\ell$ and any mass $M$ (or for plain AdS) -- and these will constitue a one-parameter family (i.e.\ parametrized by $M$) of \textit{different} fixed-background holographic relationships.
\subsection{The connection between the Mukohyama-Israel puzzle and the Arnsdorf-Smolin puzzle}
\label{subsect:MI/ASconnection}
For definiteness, and also to make closer contact with \cite{Arnsdorf:2001qb}, let us take as our examplar of a quantum gravity theory the bulk (string) theory involved in the Maldacena AdS$_5$ correspondence (see Note $<$\ref{Note:AdSCFT}$>$). It is reasonable and usual to suppose that a state of this theory which physically represents a (stable) quantum black hole in equilibrium with its thermal atmosphere will have an approximate description in terms of a classical 5-dimensional Schwarzschild AdS spacetime together with a thermal atmosphere consisting of the fields which result from a Kaluza Klein reduction of the relevant supergravity theory on the 5 dimensional Schwarzschild AdS times $\mathbb{S}^5$ -- defined on (say) the right Schwarzschild wedge $<$\ref{Note:double}$>$. We recall from Section \ref{Sect:intro} that the Mukohyama-Israel puzzle is the puzzle as to how it comes about that the entropy of this state can be calculated -- see again Note $<$\ref{Note:AdSCFT}$>$ -- in two seemingly quite different ways: On the one hand, it can be calculated as the gravitational entropy, i.e.\ one quarter of the area of the event horizon (divided by the appropriate string theory formula for Newton's constant) of the classical black hole background, while, on the other hand, it can presumably also be calculated as the entropy of the thermal atmosphere -- the latter calculation presumably requiring a suitable brick wall cutoff in order to compensate for a presumed flaw in the description of the thermal atmosphere in terms of quantum fields on a fixed curved spacetime (which will, again presumably, get repaired in a more correct quantum gravitational description -- [but] see Subsection \ref{subsect:dual} below).
The connection with the Arnsdorf-Smolin puzzle (see \cite{Arnsdorf:2001qb} and Section \ref{Sect:intro}) is that there are now two seemingly quite different ways of arguing that the entropy of the quantum black hole state is equal to the entropy of the boundary CFT which arises in the AdS$_5$ version of AdS/CFT. Firstly, thinking of the entropy of the state as given by the former gravitational entropy calculation, we can argue that this must be equal to the entropy of the boundary CFT using Maldacena holography. On the other hand, if we assume that the (brick wall modified) thermal atmosphere is related to the same boundary CFT by fixed background holography, then, with our working assumptions, thinking of the entropy of our state as the entropy of this thermal atmosphere, we can argue that this must be equal to the entropy of the boundary CFT but, this time, the argument for the equality is based on fixed background holography.
Admittedly, the above remarks, as they stand, amount to just an interesting connection between our two puzzles rather than a resolution of either of them. However, this connection would seem to suggest one interesting conclusion: Namely, that the bulk theory which is related to our boundary CFT by boundary limit holography must at least be `something like' our thermal atmosphere -- i.e.\ the (Kaluza Klein reduction of) our supergravity theory on the right Schwarzschild wedge. And, with our (generalization of our) second working assumption, this then implies that, in {\it plain} AdS, the Rehren dual to our boundary CFT must be `something like' the same supergravity theory. This would seem to support one of the tentative possible resolutions of their paradox raised by Arnsdorf and Smolin themselves (see the paragraph labelled `M1' on page 7 of \cite{Arnsdorf:2001qb} where they write that, ``in the [large $N$ and] $g_s\rightarrow 0$ limit, Maldacena holography and the Rehren duality could coincide''). See however Subsection \ref{subsect:dual} below where we will offer a clarification of this and, in particular, argue for a certain clarification of the phrase `something like' here.
\subsection{The matter-gravity entanglement hypothesis and its implications for the Mukohyama-Israel puzzle and for Maldacena holography}
\label{subsect:matgrav}
In a series of articles \cite{Kay:1998vv, Kay:1998cj, Abyaneh:2005tc, Kay:2007rx} dating back to 1998 (see especially End Notes (i)-(v), in \cite{Kay:2007rx}) one of us (BSK) proposed a physical interpretation of quantum gravity which, while seemingly consistent with conservative principles -- it assumes quantum gravity is a conventional quantum theory with a unitary time evolution -- entails a radically different from usual picture of quantum black holes. The main idea is to posit that any closed quantum gravitational system (be it one that contains a black hole or otherwise) is described by an (ever pure and unitarily evolving) pure state (here, we take `pure state' to be synonymous with `vector state'), $\Psi_{\mathrm{total}}$ on a total Hilbert space, ${\cal H}_{\mathrm{total}}$ which is a tensor product of a gravity Hilbert space, ${\cal H}_{\mathrm{gravity}}$, and a matter Hilbert space ${\cal H}_{\mathrm{matter}}$. However, rather than identifying its physical entropy with the von Neumann entropy of the total state (which would of course be ever zero for a total pure state) it is proposed to identify it with the total state's matter-gravity entanglement entropy. This can perfectly well be non-zero, and increasing with time, even though the total state is assumed to be pure and to evolve unitarily. It thereby seems to offer a simple and attractive solution to the information loss puzzle \cite{Hawking:1976ra}.
As explained in End Notes (i) and (iii) in \cite{Kay:2007rx} this `matter-gravity entanglement hypothesis' also seems to offer a natural resolution to, what we have called in the present paper, the Mukohyama-Israel puzzle. For, in a vector state, the entanglement entropy of a bipartite system must, by an easy
theorem, be both the von Neumann entropy of the reduced density operator on the first system and equally the von Neumann entropy of the reduced density operator on the second system. But, applying this theorem to our pure entangled state of matter-gravity, our physical entropy must be the von Neumann entropy of the reduced density operator of the gravitational field, which is plausibly the same thing as the gravitational entropy (i.e.\ one quarter of the area of the event horizon); but it must also be the von Neumann entropy of the reduced density operator of the matter field which is plausibly (close to -- see below) the same thing as the state of the (perhaps brick wall modified) thermal atmosphere in a quantum field theory in curved spacetime description.
As far as the operators of the quantum gravity theory are concerned, these presumably are described by a total *-algebra which is a tensor product of a gravity *-algebra, ${\cal A}_{\mathrm{gravity}}$, and a matter *-algebra,
${\cal A}_{\mathrm{matter}}$ (the full tensor product being represented on ${\cal H}_{\mathrm{total}}$). Combining all this with the conclusions of the previous subsection, we seem to be led to the following view as to the nature of Maldacena holography in which, rather than being a bijection between the total bulk theory and the boundary CFT:
\medskip
\noindent
\textit{Maldacena holography (say in its AdS$_5$ version) is a one-way mapping from bulk to boundary in which the boundary CFT is identified with (and arises as a suitable boundary limit of) bulk operators of form $I \otimes {\cal A}_{\mathrm{matter}}$ ($I$ here denoting the identity operator on the gravity Hilbert space).}
\smallskip
\noindent
\textit{Moreover, the proper understanding of the well-established (see Section \ref{Sect:intro} and Note $<$\ref{Note:AdSCFT}$>$) equality between the bulk gravitational entropy and the boundary CFT entropy in the case of a thermal state on the 1+3 dimensional boundary cylinder (at a temperature above the Witten phase transition temperature) is that the total bulk state (as usual, approximately described by the relevant classical Schwarzschild-AdS background with the relevant Kaluza Klein-reduced supergravity fields/strings propagating on it) is a {\it pure} total state, $\Psi_{\mathrm{total}}$, of matter-gravity, entangled in just such a way that the reduced density operators (i.e.\ of
$|\Psi_{\mathrm{total}}\rangle\langle\Psi_{\mathrm{total}}|$) on each of ${\cal H}_{\mathrm{matter}}$ and
${\cal H}_{\mathrm{gravity}}$ are each separately (approximately) thermal at the relevant Hawking temperature. When we say that the entropies of bulk and boundary are equal, what is really happening is that the von Neumann entropy of the (mixed) boundary thermal density operator (equals the von-Neumann entropy of the reduced density operator on ${\cal H}_{\mathrm{matter}}$ which) equals the matter-gravity entanglement entropy of the (pure!) bulk state.}
\medskip
These italicised statements seem quite heretical although we are unaware of any decisive evidence against them. We note, in this regard, that Arnsdorf and Smolin have argued in \cite{Arnsdorf:2001qb} that it may well be consistent with what is known (was known as of mid 2001) about Maldacena holography that it might just be a one-way map from bulk to boundary, rather than a bijection (see also a similar point by Giddings, reiterated in his recent paper \cite{arXiv:1105.6359}); and such a result, would, as they themselves point out, go at least part of the way towards resolving their paradox. In any case, even if there may turn out to be evidence against it, such evidence should, we feel, be weighed against the virtue of the above view that it would seem to fit better with the hope that string theory can be part of the solution to the information loss puzzle than the conventional view about the interpretation of string theory (where one adopts the conventional picture of black hole equilibrium states as non-pure states -- a picture which seems to be at odds with any such resolution).
Of course, for these italicized statements to be less than vague, we would need to specify what might be meant by the `matter-gravity split' in a string-theory context. We have to admit that we presently do not have a clear answer to this but would make the following remarks: First (cf. especially Note (ii) in \cite{Kay:2007rx}) the splitting between matter and gravity is only expected to be an approximate notion which only makes sense (well) below the Planck energy. Secondly, in a more physically realistic string theory (where, say there are compact `extra' dimensions which are small) then we can, of course, identify, at low energies, in a field-theoretic approximation involving a Kaluza-Klein reduced supergravity theory etc., what are photons, electrons etc. (i.e.\ matter) and what are gravitons (gravity) etc. Thirdly, as Susskind \cite{Susskind:1994vu} has argued, we expect a black hole to correspond in the weak string-coupling limit to states of very long strings so it seems plausible that such states of very long strings should get lumped together as part of the gravitational field. Finally, it is tempting to wonder if one can, naively, make the identification: \textit{open strings} $\leftrightarrow$ \textit{matter}; \textit{closed strings} $\leftrightarrow$ \textit{gravity}. But it is then not clear how this would fit with our above remark about long strings. We should also remark that there may be more than one way of realizing a splitting between closed and open strings in the bulk. In fact, quite aside from the fact that such a splitting can of course anyway only make sense in an approximate way at weak string coupling, there may also possibly be alternative dual (now really in the sense of equivalent!) descriptions of the bulk theory in which the parts of the theory which are identified as closed-string and open-string sectors are different. For example, IY Park \cite{Park:1999xz} has argued for a dual (i.e. equivalent) description of the bulk theory in which, rather than a Type IIB theory, it is a theory with open strings (some with fully Neumann boundary condistions) as well as closed strings and D-branes.
As a final remark on this issue: it may be that what we need to do is turn things around and simply {\it define} ${\cal A}_{\mathrm{matter}}$ to be the Rehren dual (see Subsection \ref{subsect:dual}) of the *-algebra of the Maldacena boundary CFT.
We next make a couple of further comments on the scenario proposed in our italicised statements above. First, another way of expressing things is that the pure vector state, $\Psi_{\mathrm{total}}$, in the total Hilbert space ${\cal H}_{\mathrm{total}}$ which describes the bulk matter-gravity equilibrium {\it purifies} the thermal density operator on ${\cal H}_{\mathrm{matter}}$ (cf. \cite{Kay:1985yw,Kay:1985yx,Kay:1985zs, Kay:1988mu}). Secondly, on our scenario, an arbitrary element of ${\cal H}_{\mathrm{total}}$ will be approximated as closely as one likes by acting on $\Psi_{\mathrm{total}}$ with operators of the form $I \otimes {\cal A}_{\mathrm{matter}}$ -- which is identified with the *-algebra of operators of the boundary CFT. So in this sense, it would still be correct to say (cf. \cite{Maldacena:1997re}) that \textit{the bulk (string theory) Hilbert space is the same as the boundary CFT Hilbert space}. Cf. the analogy with the `right-wedge Reeh Schlieder property' which we point out in Note $<$\ref{Note:double}$>$. Another way of saying this is that the total Hilbert space can be equated with the GNS Hilbert space (see Section \ref{Sect:prelim}) for the (algebraic) state ${\rm trace}(\rho(\cdot))$ on ${\cal A}_{\mathrm{matter}}$. In other terminology, it is the Hilbert space in the sense of thermofield dynamics \cite{TakUme, Takahashi:1996zn}. However, at zero temperature (i.e.\ for the ground state on the boundary CFT -- now on Minkowski space) when the bulk is plain AdS this will no longer be the case. In fact, in line with our `matter-gravity entanglement hypothesis' and the expectations above, we would expect the total ground state to be (approximately) a tensor product (i.e.\ an unentangled state) of a gravity ground state and a matter ground state and therefore its entropy (as we define it) will be zero -- which is of course in line with the conventional expectation.
Lastly, in the paper \cite{arXiv:1105.6359} by Giddings which we cited in a parenthetical remark above, it is speculated that the bulk-to-boundary mapping may be such that the boundary theory only captures an ``appropriately coarse-grained'' description of bulk physics. We remark that our proposal in our italicized statement above might perhaps be regarded as a concrete implementation of this idea. After all, the process of taking a partial trace (in our proposal, over gravity so as to obtain the reduced density operator of matter in the bulk which will then be equivalent to the thermal state of the boundary CFT) can be regarded as a particular sort of quantum counterpart to classical coarse-graining -- a sort which in our context (in particular, because we suppose the bulk to be in a pure state) is, as we indicated above, consistent with the resolution to the information loss puzzle proposed in \cite{Kay:1998vv, Kay:1998cj, Abyaneh:2005tc, Kay:2007rx}.
\subsection{More about the Rehren dual of the boundary CFT of Maldacena AdS$_5$ holography}
\label{subsect:dual}
Finally, we return to our tentative conclusion, in Subsection \ref{subsect:MI/ASconnection}, that the Rehren dual of the Maldacena boundary CFT must at least be `something like' our thermal atmosphere -- i.e.\ our (Kaluza Klein reduction of) our supergravity theory on our right Schwarzschild wedge. First of all, we recall that Rehren pointed out in \cite{Rehren:2004yu} that the large $N$ limit of the Maldacena CFT is not itself a quantum field theory -- and it is of course strictly only this large $N$ limit which (at fixed 't Hooft coupling) is believed to correspond to classical supergravity. So the question we ask ourselves is: What is the Rehren dual of the Maldacena CFT at large but finite $N$? What we would seem to learn from our results (with the extrapolations and working assumptions we made in Subsection \ref{subsect:working} ) is that \textit{this cannot be an ordinary quantum field theory}! After all, if it were an ordinary quantum field theory, it would presumably have a formally infinite entropy requiring a brick wall to be made finite. And yet the Maldacena CFT (unlike e.g.\ the conformal generalized free field whose Rehren dual is the Klein Gordon equation) \textit{is} presumably an ordinary quantum field theory and therefore is expected to have a finite entropy density and hence a finite total entropy when defined on a cylinder, {\it without} the need for any brick wall in the bulk! As to what its Rehren dual could be, it would seem reasonable to suggest that the phrase `something like our thermal atmosphere' should, in the light of this, be (partially) clarified in the following way:
\medskip
\noindent
\textit{Whatever the Rehren dual of the Maldacena CFT (i.e.\ of ${\cal N}=4$ supersymmetric Yang Mills theory) may be, for fixed $N$ and at finite Yang-Mills coupling constant $g_{\mathrm{YM}}$, in the limit $g_{\mathrm{YM}}\rightarrow 0$, it is something like the (Kaluza Klein reduction of) the bulk string theory of AdS$_5 \times {\mathbb S}^5$ in the limit of zero string coupling $g_s$ (but at finite string length $\ell_s$).}
\medskip
Part of our rationale for saying this is that, at zero $g_{\mathrm{YM}}$ (which corresponds -- see Note $<$\ref{Note:AdSCFT}$>$ -- to zero
$g_{s}$) the string theory will be a theory which makes sense on a fixed background. Another part is that we suppose that (such a limit of) a string theory will have a better chance than a quantum field theory of not needing a brick wall in order to have a finite entropy.
This (still partly vague) conclusion did not depend on the `matter-gravity entanglement hypothesis' of \cite{Kay:1998vv, Kay:1998cj, Abyaneh:2005tc, Kay:2007rx}. However, on that hypothesis, it would be natural to add the further clarification that it is in fact `something like the {\it matter sector} of the (Kaluza Klein reduction of) the bulk string theory'. (If/when that can be given a clear meaning.)
In support of either of these tentative conclusions (i.e.\ with or without the above further clarification) we recall first that Rehren himself has suggested (in \cite{Rehren:2000tp}) that the Rehren dual of the Maldacena CFT may well involve strings. Also there is work (e.g.\ by Dimock \cite{Dimock:2000yv, Dimock:2001dy}) suggesting that it will be possible to incorporate (at least free, open) string field theory into the framework of nets of local *-algebras.
All these tentative conclusions are, of course, more or less speculative. But, assuming they are on the right track, they would seem to go at least some of the way towards a possible resolution of the Arnsdorf-Smolin puzzle. The puzzle will probably only be fully resolved when we have a satisfactory background-independent formulation of string theory, and if and when we have a clearer understanding of the nature of the Rehren dual of the Maldacena boundary CFT. Such a clearer understanding of the Rehren dual would, on our matter-gravity entanglement hypothesis, as we remarked above, also presumably clarify how the matter-gravity split should be defined in the bulk string theory.
\section{Notes}
\label{Sect:notes}
\begin{enumerate}
\item
\label{Note:AdSCFT}
The principal examples of AdS/CFT involve AdS$_5\times {\mathbb S}^5$ and AdS$_3\times {\mathbb S}^3\times X$ bulks where $X$ denotes either $\mathbb{T}^4$ or $K3$. See \cite{Aharony:1999ti} or \cite{Ross:2005sc}. The latter review by Ross provides a useful bridge between work on quantum field theory in curved spacetime and on Euclidean quantum gravity on the one hand and work on AdS/CFT on the other.
\smallskip
We briefly recall, in the remainder of this note, more details about the AdS$_5\times {\mathbb S}^5$ example since we shall refer to this extensively in Section \ref{Sect:discuss}. We shall also include an outline of the calculation of the entropy of a thermal state of its boundary CFT, when defined on a cylinder, in the regime in which the bulk is well described by a classical Schwarzschild AdS gravitational field, since this result is central to much of our discussion and appears difficult to find in a single place. (The account below is pieced together from \cite{Maldacena:1997re, Gubser:1996de, Witten:1998zw, Klebanov:2000me, Ross:2005sc}. See also e.g.\ \cite{Ortin} for the definition of Newton's constant in string theory. )
\smallskip
In the AdS$_5\times {\mathbb S}^5$ example, the bulk theory is Type IIB string theory (with string length $\ell_s$ and string coupling constant $g_s$) with 5 units of Ramond Ramond flux on AdS$_5\times {\mathbb S}^5$ . Both the AdS$_5$ and the ${\mathbb S}^5$ have the same radius $\ell$. The boundary theory is ${\cal N}=4$ supersymmetric Yang Mills theory for the gauge group $SU(N)$ with coupling constant $g_{YM}$. The `dictionary' that relates the two theories includes the entries (now only setting $\hbar$ and $c$ equal to 1): $g_{\mathrm{YM}}^2=4\pi g_s$; $g_{\mathrm{YM}}^2N$ ($=$ the `'t Hooft coupling') $=(\ell/\ell_s)^4$. Also, Newton's constant, $G_{10}$, in 10 dimensions is related to $g_s$ and $\ell_s$ by $G_{10}=8\pi^6g_s^2\ell_s^8$ while, we remark that, in a Kaluza Klein reduced picture, the 5-dimensional Newton's constant, $G_5$, is equal to $G_{10}$ divided by the `area',
$\pi^3\ell^5$, of the 5-sphere. (Below, we will also need that the `area' of a 3-sphere of radius $R$ is $2\pi^2R^3$.)
When $N$ is very large, $g_{\mathrm{YM}}$ is small, but the 't Hooft coupling, $g_{\mathrm{YM}}$, is large, then the bulk theory is believed to be well-approximated by classical supergravity.
\smallskip
The entropy of the supersymmetric Yang Mills theory at temperature $\cal T$ on a 1+3 dimensional cylinder of radius $\ell$ (and hence spatial volume, say $V$, equal to $2\pi^2\ell^3$) in this regime is calculated, according to AdS/CFT (assuming, for the given $\ell$, that $\cal T$ is above the Hawking-Page-like \cite{Hawking:1982dh} phase transition discussed in \cite{Witten:1998zw}) by simply equating it with the Hawking entropy of the 10-dimensional black hole, consisting of a 5-dimensional classical Schwarzschild AdS of asymptotic radius $\ell$ times the 5-sphere of the same radius $\ell$, for which the Hawking temperature is $\cal T$. There are actually two such classical metrics, as one can see as follows. First, we notice that the metric takes \cite{Witten:1998zw} the form (\ref{generalsphmetric}) for $d=4$ with $f(r)= 1-\mu/r^2+r^2/\ell^2$ (where, by the way, $\mu$ is related \cite{Witten:1998zw} to the black hole mass, $M$, by $\mu=8\pi G_5/(3\pi^2)$) and one easily sees that, for each $\mu$, there is a unique value of $r$ for which $f(r)=0$, namely $r_+$ ($\simeq \sqrt\mu$). Secondly, the surface gravity, $\kappa$ (see Section \ref{Sect:prelim}) is $f'(r_+)/2$, which (using $f(r)=0$ again) is easily seen to be $2r_+/\ell^2 + 1/r_+$. Equating $\cal T$ with the Hawking temperature, $\kappa/(2\pi)$, we find that $r_+$ must be a root of the quadratic equation $2r_+^2-(2\pi\ell^2{\cal T})r_++\ell^2=0$. Just as in the $1+3$ dimensional case \cite{Hawking:1982dh}, there are two of these: one `small', representing an unstable equilibrium, the other `large', with $r_+\simeq \pi\ell^2{\cal T}$ representing a stable equilibrium. Choosing the latter, the entropy is given by $S=$ `Area$/(4G)$' $=(2\pi^2r_+^3)(\pi^3\ell^5)/(4G_{10})$ (or alternatively, but equivalently, by $2\pi^2r_+^3/(4G_5)$) which we can write as $(\pi^3/4)(V\ell^3{\cal T}^3/G_{10})$ which, using the dictionary entries and the formula for $G_{10}$ and the definition of $V$ above, is equal to $\pi^2N^2V{\cal T}^3/2$. This is to be compared with the value $2\pi^2N^2V{\cal T}^3/3$ for the boundary CFT at fixed $N$ and weak Yang Mills coupling -- see \cite{Klebanov:2000me} both for the derivation of this latter formula and for a discussion of the physical significance of the factor of $3/4$ discrepancy. As explained there, one takes these calculations as (part of the) evidence that, for any $N$, and any value of the Yang Mills coupling, the boundary entropy equals the bulk entropy.
\item
\label{Note:thinloc} The *-algebra isomorphism in \cite{Kay:2007rf} involves a certain `thinning-out' of test functions -- see \cite{Kay:2007rf} and Note $<$\ref{Note:vNvsCstar}$>$. There is also a significant caveat about the nature of the localization of the smeared boundary fields in the isomorphism, explained in Note [14] of \cite{Kay:2007rf}
\item
\label{Note:vNvsCstar}
Rehren had in mind von Neumann algebras. Kay and Larkin's construction \cite{Kay:2007rf} referred to below will (see the place in \cite{Kay:2007rf} where reference is made to \cite{Kay:2006jn}) be in terms, say, of $C^*$ algebras. However one can easily see that it can be extended, once one chooses to represent fields in the global AdS vacuum representation, to a von Neumann algebra correspondence and, indeed, it seems conceivable that the nicety Kay and Larkin found in \cite{Kay:2007rf} about the need to `thin-out' the test functions on the boundary in order to obtain a bulk-boundary isomorphism will go away when such an extension is made.
\item
\label{Note:wrap}
We should point out that the algebraic holography work of Rehren concerns the ${\mathbb Z}_2$ quotient of true AdS (i.e.\ with closed timelike curves) while the Kay-Larkin work is done on the covering space and no ${\mathbb Z}_2$ quotient is taken. We shall assume throughout this paper that, unless it is clear otherwise from the context, when we refer to AdS, we mean the covering space (what is sometimes called CAdS) of true AdS.
\item
\label{Note:Wightman}
We remark that to have something which deserves to be called a Wightman theory in the bulk is roughly equivalent to a specification of a bulk field algebra together with the choice of the ground state for the time translations of global coordinates. The Wightman functions will then be interpretable as expectation values of products of fields in that state. Also, when we refer to a Wightman theory on the conformal boundary, what we really mean is that the theory will be a Wightman theory on $d$ dimensional Minkowski space when that is identified in the usual way (see \cite{Kay:2007rf}) with the boundary of a Poincar\'e chart.
\item
\label{Note:confboundary}
In Equation (\ref{Blim}), we assume $(q,x^i)$, $x^i\in {\mathbb R}^d$ denotes any system of coordinates such that the
metric of AdS takes the form
\[
ds_{\mathrm{AdS}}^2=\ell^2\Xi(q)^2(dq^2+ds_{\mathrm{boundary}}(x^i)^2)
\]
where $\ell$ is the AdS radius (see (\ref{E:a12}) and (\ref{E:104})) and $\Xi(q)$ is positive and tends to infinity as $q$ tends to $q_0$, which is the location of the conformal boundary.
\smallskip
In Equation (\ref{E:a14}), which relates to 1+1 dimensional AdS in global coordinates, $q$ is identified with $\rho$, $x^i$ with $\lambda$, $\Xi(q)$ with $\sec(\rho)$, and $q_0$ with $\rho=\pm \pi/2$ etc. Similarly for Equations (\ref{E:79}) (Poincar\'e coordinates), (\ref{E:106aa}) (global coordinates on 1+2 dimensional BTZ) and (\ref{E:108}) (Poincar\'e coordinates). In the case of BTZ coordinates in 1+1 and 1+2 dimensions, we remark that Equation (\ref{E:bb3}) can be rewritten, using our tortoise-like coordinate (see (\ref{BTZtortoise}) and (\ref{BTZesiotrot})) in the above form as (\ref{E:141b})
($ds^{2}=f(r(r^*))\left(dt^2-{dr^*}^2\right)$) while equation (\ref{E:108}) can be rewritten as
\begin{equation}
\label{1+2conf}
ds^2=f(r(r^*))\left(dt^2-{dr^*}^2-\frac{r(r^*)^2}{f(r(r^*))}d\varphi^2\right)
\end{equation}
where $f(r)$ is given by (\ref{E:16}).
\item
\label{Note:Rehgeo}
We note, by the way, that the notion of commutativity at spacelike separation in \cite{Rehren:1999jn} is different from the usual one and corresponds to the non-existence of timelike geodesics connecting the two regions in question rather than the, more usual, non-existence of timelike curves. However, if one reformulates Rehren's algebraic holography in terms of the covering space (CAdS) of AdS (as is done implicitly in \cite{Kay:2007rf} -- see Note $<$\ref{Note:wrap}$>$ and/but Note $<$\ref{Note:vNvsCstar}$>$) then the relevant notion can be taken to be non-existence of timelike curves connecting the two regions.
\item
\label{Note:blog}
The objections to algebraic holography referred in the main text include a number of articles in the blogosphere; see in particular \hfil\break
http://golem.ph.utexas.edu/~distler/blog/archives/000987.html
\item
\label{Note:slice}
The time-slice condition on a quantum field theory on a given spacetime with a choice of time-coordinate is that the
*-algebra for the region of the spacetime between two times is equal to the *-algebra of the full spacetime.
\item
\label{Note:unfortunate}
The terminology issue is possibly confused further because, in a number of references (including \cite{Arnsdorf:2001qb} as well as the paper, \cite{Kay:2007rf}, co-authored by one of us) the Rehren algebraic holography correspondence between quantum theories in the bulk and on the boundary is also referred to as a `duality' -- the word being used, in these references, as if synonmyous with `isomorphism'. We shall also use the word `dual' ourselves in this sense in Section \ref{Sect:discuss} when we talk about the `Rehren dual' of the Maldacena boundary CFT.
\item
\label{Note:bhalgholo}
In addition to direct boundary-limit holography, we also expect the algebraic holography of Rehren to extend from plain AdS to Schwarzschild AdS (and in particular, BTZ) in a suitable way:
\smallskip
If, for a given double cone on, (say) the right Schwarzschild-AdS boundary cylinder, we define (cf. the equivalent definition given by \cite{Arnsdorf:2001qb} to Rehren's definition of `wedge' for plain AdS) the bulk wedge which corresponds to it to be the `causal completion' in the bulk of the given boundary double cone (i.e.\ the set of all points in the bulk which can be connected by both past and future directed causal curves to points in the given double cone) then it is clear that spacelike related boundary double cones will correspond to spacelike related bulk wedges. So at least an important part of the main `geometric Lemma' of \cite{Rehren:1999jn} will generalize to the case of Schwarzschild AdS and still give us a bijection between the net of local *-algebras for the right cylinder boundary and the net of local *-algebras for the exterior right wedge region of the BTZ bulk which maps the sub*-algebras for double cones to the sub*-algebras for wedges. However, some of the properties of the bijection in the case of plain AdS may not generalize because e.g.\ of the lower degree of symmetry. In particular, it seems that all the bulk wedges which correspond to boundary cones `point in the same direction' so one cannot take intersections of them to obtain bulk double cones.
\smallskip
We would also expect there to be a suitable generalization of the pre-holography work of \cite{Kay:2007rf} showing that, subject to the caveats mentioned in Note $<$\ref{Note:thinloc}$>$,
in the case of the bulk (real) scalar Klein Gordon field, this bijection will be between the net of local *-algebras for the
bulk covariant Klein Gordon equation on the bulk right wedge of Schwarzschild AdS and the net of local *-algebras for the
conformally invariant generalized free field of (\ref{Wanom}) on the right cylinder.
\smallskip
Further we expect this bijection to extend (both abstractly and in the concrete case of the Klein Gordon bulk field) in a suitable way to a bijection from the net of local *-algebras for the full bulk Schwarzschild AdS and the net of local *-algebras for the union of the right and left boundary cylinder.
\item
\label{Note:onequarter}
Given that we raise a number of puzzles and apparent paradoxes in our introduction which relate to black hole entropy, it is worth saying explicitly that one thing we shall not question is the belief that the entropy of a black hole has (at least approximately) the Hawking value of $1/4$ of the area of its event horizon. The strongest reason for believing this for a physical black hole is the original thermodynamic argument from the Hawking radiation formula as given in \cite{Hawking:1974sw}. But it is also believed to hold for AdS black holes (see e.g.\ \cite{Hawking:1982dh}) and in dimensions other than 4 and we shall not question these beliefs either. For the 1+2 dimensional BTZ black hole, it also holds (see e.g.\ \cite{Ross:2005sc}) with `area', of course, understoood to mean circumference -- i.e.\ (in the notation of Section \ref{Sect:1+1} and \ref{Sect:1+2}) $2\pi r_+$. (But we do not have anything new to say here as to why the relevant coefficient of area takes the value $1/4$.)
\item
\label{Note:expect}
Our remarks which contain the words `expectations' and `arguments' in the main text and in the notes (especially Notes $<$\ref{Note:bhalgholo}$>$, $<$\ref{Note:equiv}$>$, and $<$\ref{Note:Ham}$>$) should be interpreted not as mathematical proofs but rather as conjectures or sketches which still need to be filled in; they may turn out to require provisos and extra conditions etc. For example, in view of the results in \cite{Kay:2007rf}, it seems possible that
(and it would be interesting to investigate whether) the statement in the main text to the effect that the direct boundary limit on bulk fields would be expected to inherit the entanglement and thermal properties of the bulk theory might (in the case in which the bulk field theory is (\ref{KG})) turn out to hold only when the mass, $m$ is such that $\Delta$ in (\ref{Delta}) is an integer or half-integer.
\smallskip
The results that we will obtain in Sections (\ref{Sect:1+1}) and (\ref{Sect:1+2}) will show that our expectations are all actually fulfilled at least for the zero mass Klein Gordon fields which we study there.
\item
\label{Note:equiv} We expect that the *-algebra ${\cal A}_{\mathrm{DW}}$ (which will, up to possible small technicalities, be the same as the *-algebra for full BTZ) will arise in the form of a tensor product of *-algebras, ${\cal A}_{\mathrm{LW}}$ for the left wedge (i.e.\ the triangle ACF in Figure \ref{Fig:ads2}) and ${\cal A}_{\mathrm{RW}}$ for the right wedge (the triangle ECD in Figure \ref{Fig:ads2}) in such a way that the pair $({\cal A}_{\mathrm{DW}}, \alpha_{\mathrm{DW}}(t))$, together with the appropriate wedge-reflection involutary antiautomorphism, $\iota_{\mathrm{W}}$ (see \cite{Kay:1985yw} and \cite{Kay:1985zs}) will be what is called in \cite{Kay:1985yw} a `double dynamical system' and similarly we expect that the *-algebra ${\cal A}_{\mathrm{DC}}$ will arise in the form of the tensor product of *-algebras, ${\cal A}_{\mathrm{LC}}$ for the left cylinder (see Figure \ref{Fig:BTZcylinders}) and ${\cal A}_{\mathrm{RC}}$ for the right cylinder, in such a way that the pair $({\cal A}_{\mathrm{DC}}, \alpha_{\mathrm{DC}}(t))$, together with an appropriate involutary antiautomorphism, $\iota_{\mathrm{C}}$, which maps between ${\cal A}_{\mathrm{LC}}$ and ${\cal A}_{\mathrm{RC}}$ and reverses the sense of time will also be a `double dynamical system' (and we expect, further, that these, in turn, will each arise by second quantization from appropriate `double linear dynamical systems' for the appropriate underlying classical theories as in \cite{Kay:1985yx}). With this terminology, the expectation, stated in the main text, that ``the direct boundary limit on the pair of cylinders of a bulk theory defined on BTZ would be expected to inherit the entanglement and thermal properties of the bulk theory'' can then be formulated and argued for more precisely by saying, first, that we expect that the direct boundary limit will induce a natural isomorphism between the double dynamical systems $({\cal A}_{\mathrm{DW}}, \alpha_{\mathrm{DW}}(t), \iota_{\mathrm{W}})$ and $({\cal A}_{\mathrm{DC}}, \alpha_{\mathrm{DC}}(t), \iota_{\mathrm{C}})$ and, second, that the HHI state on the bulk spacetime will be a `double KMS state', in the sense of \cite{Kay:1985yw}, at the Hawking temperature, on $({\cal A}_{\mathrm{DW}}, \alpha_{\mathrm{DW}}(t), \iota_{\mathrm{W}})$. Hence, by the natural isomorphism just mentioned, the state induced on the boundary by taking the direct boundary limit of fields, will be a double KMS state at the Hawking temperature on the (isomorphic) double dynamical system $({\cal A}_{\mathrm{DW}}, \alpha_{\mathrm{DW}}(t), \iota_{\mathrm{W}})$. (We note that in the case of 1+1 dimensional BTZ, the right and left cylinders of course reduce to lines.)
\item
\label{Note:Ham}
The quantum dynamical system (see Section \ref{Sect:prelim}) consisting of the bulk (total) *-algebra, ${\cal A}_{\mathrm{RW}}$, for, say, the bulk quantum field theory of (\ref{KG}), on, say the right BTZ wedge, together with the automorphism group, $\alpha_{\mathrm{RW}}(t)$, corresponding to the one-parameter subgroup, of right-BTZ-wedge preserving AdS isometries (see the next remark) is expected to be equivalent to the quantum dynamical system consisting of the *-algebra, ${\cal A}_{\mathrm{RC}}$, for the conformal generalized free field with anomalous scaling dimension $\Delta$ on the right cylinder together with the one-parameter group of automorphisms, $\alpha_{\mathrm{RW}}(t)$, which time-translates towards the future in the right cylinder.
\smallskip
We remark that $\alpha_{\mathrm{RW}}(t)$ will, of course, just be the restriction from ${\cal A}_{\mathrm{DW}}$
to ${\cal A}_{\mathrm{RW}}$ of the ${\cal A}_{\mathrm{DW}}$ mentioned in Note $<$\ref{Note:equiv}$>$
and in the paragraph in the main text to which that note refers (and similarly for ${\cal A}_{\mathrm{DC}}$
and ${\cal A}_{\mathrm{RC}}$).
\smallskip
It follows (see Section \ref{Sect:prelim}) that the Hamiltonian generating dynamics in the GNS representation of the ground state on $({\cal A}_{\mathrm{RW}}, \alpha_{\mathrm{RW}}(t))$ (i.e.\ of the BTZ analogue of the Boulware state \cite{Boulware:1974dm}) will be unitarily equivalent to the Hamiltonian generating dynamics in the GNS representation of the ground state for $({\cal A}_{\mathrm{RC}}, \alpha_{\mathrm{RW}}(t))$.
\item
\label{Note:metrical}
When we refer, in the main text, to the metrical distance from the horizon, we mean the metrical distance from the horizon within a surface of constant Schwarzschild time.
\item
\label{Note:MIcorrection}
Our symbol $\mathsf{N}$ for what might be termed the `effective number of (real) Klein Gordon fields in nature'
corresponds to Mukohyama and Israel's $\cal N$ \cite{Mukohyama:1998rf} . It plays a similar role to the quantity denoted by $Z$ in
\cite{'tHooft:1984re}. Actually our $\mathsf{N}$ is defined so as to be $90/\pi^4$ ($\approx 0.92$) times the $\mathsf{N}$ defined in \cite{Mukohyama:1998rf}. This (unimportant) discrepancy is slightly
complicated by the fact that there appears to be an (unimportant) error in equation (3.14) of \cite{Mukohyama:1998rf}
which seemingly only holds if one takes $\mathsf{N}$ there to be as we have defined it here (and not as it was defined earlier in \cite{Mukohyama:1998rf}).
\item
\label{Note:complementarity}
The Mukohyama-Israel complementarity argument is attractive insofar as the alternatives seem unattractive: After all, if one were to take $\alpha$ to lead to a different fraction (say $f$) than one quarter of the area of the event horizon, then there would only seem to be two reasonable possibilities: Either this is to be equated with the total entropy of the black hole, or it should be added to the entropy arising from the Gibbons-Hawking Euclidean classical gravitational action. In either case, though, this would result in a fraction of the area of the event horizon different from one quarter -- namely $f$ on the first view and $1/4 + f$ on the second view.
\smallskip
We note that there is a more sophisticated variant of the view that the area part of the thermal atmosphere entropy and the entropy derived from the classical gravitational action should be added together, according to which
(see \cite{Susskind:1994sm}) the overall coefficient of ``one quarter area'' (divergent as $\alpha\rightarrow 0$ in our formulae (\ref{1plus2bricksecondresult}) and (\ref{originalbrick})) is interpreted as an inverse renormalized Newton's constant. (See e.g. \cite{hep-th/0011176}.) However there seem to be difficulties with this view too. In particular (cf. \cite{Barbon:1995im}) it seems to rely on a viewpoint, according to which the distance from the brick wall to the horizon (called $\alpha$ here and $\epsilon$ in \cite{Susskind:1994sm} and \cite{Barbon:1995im}) is regarded as a comparable quantity to the invariant-geodesic distance between a pair of close-by events -- used, in point-splitting regularization, to regulate the ultra-violet divergences in products of matter fields at a single point. However, it is not clear that these are comparable quantities and, in fact, as pointed out in \cite{Barbon:1995im}, an attempted comparison would lead to frame-dependent results.
\item
\label{Note:mass}
For 1+1 dimensional BTZ, we confine our discussion, for simplicity, to the massless case.
\item
\label{Note:cnfmlwt}
In general (see e.g.\ \cite{Wald} as well as \cite{pdFranpMatdSene97}), a conformally invariant scalar field, $\phi$, on a manifold, $M$, equipped with the spacetime metric, $g$, transforms to the field, $\tilde\phi$, on the same manifold equipped with the metric
$\tilde g=\Omega^2 g$ according to
\[
\tilde\phi=\Omega^{-\Delta}\phi
\]
where $\Delta$ is the scaling dimension of $\phi$. Related to this, if $G(x_1, x_2)=\omega(\phi(x_1)\phi(x_2))$ is the expectation value of a product of fields (i.e.\ a `two-point function') in some state, $\omega$, on $(M, g_{ab})$, then
\begin{equation}
\label{confstate}
\tilde G(x_1,x_2)=\Omega(x_1)^{-\Delta}\Omega(x_2)^{-\Delta}G(x_1,x_2)
\end{equation}
will define the corresponding two-point function -- i.e.\ the expectation value, $\tilde\omega(\tilde\phi(x_1)\tilde\phi(x_2))$, of the product of the conformally transformed fields in the appropriately `conformally transformed' state $\tilde\omega$. We note that equation (\ref{confstate}) (and its counterparts for higher $n$-point functions) actually tells us what we mean here by the `conformally transformed state'.
\smallskip
For the massless Klein Gordon equation on the bulk of AdS$_2$, we rely on the fact that that (just in 1+1 dimensions -- in any other spacetime dimension, $n$, one would need, \cite{Wald}, to include a conformal coupling term $\xi R\phi$ on the left hand side of
(\ref{KG}) where $\xi=(n-2)/(4(n-1))$) the massless Klein Gordon equation is conformally invariant with (non-anomalous) scaling dimension $\Delta=0$. (For the $n$ dimensional conformally coupled version of (\ref{KG}) indicated above, $\Delta$ is related to $n$ by the `non-anomalous' relation $\Delta=n/2-1$.) It is thanks to all this that the global two-point function on 1+1 dimensional AdS is, in the massless case, equal to the two-point function (\ref{2dmasslessG}) for the ground state on our strip of Minkowski space, and similarly that the Poincar\'e two-point function is equal to the two point function (\ref{Poincaretwopointfn}) for the ground state on the right half of Minkowski space.
\item
\label{Note:2ptfn}
Here, we use the terminology and notation of algebraic quantum field theory \cite{Haag, Kay:2006jn} where our ground state $\omega_{\mathrm{globalground}}$ is a state in the sense of a `positive normalized linear functional' (say, on the algebra of smeared fields). We might write our two-point function in `physicist's notation' as
$\langle O_{\mathrm{globalground}}|(\phi(\lambda_1,\rho_1)\phi(\lambda_2,\rho_2)|O_{\mathrm{globalground}}\rangle$.
\item
\label{Note:TtGstory}
To get (\ref{E:100}) directly from (\ref{E:92}), we can define $g=dT^2$ and $\tilde g=dt^2$. Then we easily have, by (\ref{Ttrelation}), $\tilde g=\Omega^2g$ with $\Omega=1/(\kappa T)=e^{\kappa t}/(\kappa \ell)$, whereupon by (\ref{confstate}) in Note $<$\ref{Note:cnfmlwt}$>$, $G_{\mathrm{BTZ}}$ is easily seen to be given by (\ref{E:100}).
\item
\label{Note:1plus1Mink}
As is well-known, or easy to calculate, the two point function, ${\cal G}_{\mathrm{Minkowski}}$, for the ground state with respect to Minkowski time translations of the massless Klein Gordon equation on 1+1 dimensional Minkowski space can be written
\[
\fl {\cal G}_{\mathrm{Minkowski}}(T_1, T_2; X_1, X_2)=-\frac{1}{4\pi}\ln((T_1-T_T)^2-(X_1-X_2)^2-2i\epsilon(T_1-T_2)) + C
\]
where $C$ is an `ill-defined' constant which doesn't matter for us because it will go away when we take derivatives.
Thus, if we introduce the double-null coordinates, $U=T+X$ and $V=T-X$, it can be written (now ignoring the constant) as the sum of
\begin{equation}
\label{left}
G_{\mathrm{MinkowskiLeft}}(U_1, U_2)=-\frac{1}{4\pi}\ln(U_1-U_2-i\epsilon)
\end{equation}
and
\[
G_{\mathrm{MinkowskiRight}}(V_1, V_2)=-\frac{1}{4\pi}\ln(V_1-V_2-i\epsilon),
\]
which can be thought of as the restrictions of ${\cal G}_{\mathrm{Minkowski}}$ to the null lines $T=X$ and $T=-X$
(or alternatively as the restrictions to the sectors consisting of left-moving and right-moving modes).
\smallskip
In the main text, we point out that, if we replace $T$ by $U$, the two-point function (\ref{E:92}) $G_{\mathrm{Poincare}}$ can be identified with the the double derivative, ${\partial^2/\partial U_1\partial U_2}$ of $G_{\mathrm{MinkowskiLeft}}$ up to a factor of 4. Moreover, if we introduce Rindler coordinates on the right Rindler-wedge of our 1+1 dimensional Minkowski space via $U=\ell e^{\kappa u}$, $-V=\ell e^{-\kappa v}$ (and then introduce $t$ and $x$ according to $u=t+x$ and $v=t-x$) for $U$ positive and $V$ negative (and similarly we introduce Rindler coordinates on the left wedge by $-U=\ell e^{-\kappa u}$, $V=\ell e^{\kappa v}$ for $U$ negative and $V$ positive and then again define $t$ and $x$ according to $u=t+x$ and $v=t-x$) then we notice that the analogy extends to relate the fact that the 1+1 massless Minkowski ground state (say restricted to the null line $T=X$) is a `double KMS state' in the sense of \cite{Kay:1985yw, Kay:1985zs} and Note $<$\ref{Note:equiv}$>$ with respect to Lorentz boosts to the fact that the global 1+1 dimensional AdS ground state, restricted to the right boundary line, is a double KMS state (now with respect to the restriction to the right AdS boundary line of BTZ time translations -- i.e.\ what we call $t$-translations in the main text -- on the region $T<0$ and the corresponding $t$-translations on the region $T>0$). In particular, it relates the fact that $G_{\mathrm{MinkowskiLeft}}$ is a KMS state (at inverse temperature $2\pi/\kappa$) with respect to Lorentz boosts (which act as $u$-translations). In fact, we easily have (cf. (\ref{E:100})) that
${\partial^2/\partial u_1\partial u_2}$ of $G_{\mathrm{MinkowskiLeft}}(e^{\kappa u_1}, e^{\kappa u_2})$ is
\[
=-\frac{1}{16\pi}\frac{\kappa^{2}}{\sinh^{2}\left(\kappa\frac{u_{1}-u_{2}-i\epsilon}{2}\right)},
\]
which is easily seen to satisfy the KMS condition at inverse temperature $2\pi/\kappa$.
(Above, we included factors of $\ell$ and $\kappa$ to make the analogy closer where $\ell$ is identified with the AdS radius and $\kappa$ is identified with the BTZ surface gravity of the main text.)
\item
\label{Note:PoincareGlobalEquiv}
Unlike in the 1+1 dimensional case, which was settled by Spradlin and Strominger in \cite{Spradlin:1999bn}
it appears an open question as to whether the global and Poincar\'e ground states coincide in 1+2 dimensional AdS. (It seems very likely that they do though!) In \cite{Ortiz:2011mi, lOrt11}, one of us (LO) shows, in any case, that both ground states have the same boundary limit for their bulk two-point functions (as given by (\ref{E:140ll})).
\item
\label{Note:image}
As pointed out in \cite{Kay:2006jn} (see the Section there entitled `Warnings') in general, a two-point function defined by an image sum does not necessarily satisfy the necessary positivity properties to be the expectation value of a product of quantum fields. However, one can see that this is not a problem in the case of our $G_{\mathrm{BTZ}}$ as defined by (\ref{E:140vv}).
\item
\label{Note:Gradshteyn}
The passage from (\ref{Gbrickbetan}) to (\ref{Gbrickbetax}) can be made on noticing that (see e.g.\ formula {\bf 3.41} 31, page 327 of Gradshteyn \cite{Gradshteyn})
\[
\int_0^\infty\frac{e^{-qx}+e^{q-p}x}{1-e^{-px}}xdx=\left(\frac{\pi}{p}{\rm cosec}{\frac{q \pi}{p}}\right)^2 \quad 0<q<p
\]
with some care over `epsilons'.
\item
\label{Note:scalarentropy}
To derive equations (\ref{1dentropy}) and (\ref{2dimentropydens}), we start with $S=-\tr(\rho\ln\rho)$, which, since $\rho=e^{-\beta H}/Z$, is given by the (standard, statistical mechanical) formula
\begin{equation}
\label{SfromZ}
S=\left(1-\beta\frac{\partial}{\partial\beta}\right)\ln Z
\end{equation}
where $Z=\tr(e^{-\beta d\Gamma(h)})$. To calculate $Z$ for a one-dimensional box of side $L$ say, we may think of our real scalar quantum field as a collection of quantum harmonic oscillators with angular frequencies $\omega_m=\pi m/L$.
Each such single oscillator will obviously have the partition function (ignoring zero-point energy), $Z_m= \sum_{n=0}^\infty e^{-\beta n\omega_m}=(1-e^{-\beta\omega_m})^{-1}$. So the collection of oscillators will have partition function given by
\begin{equation}
\label{1dlogZ}
\ln Z=\sum_{m=1}^\infty \ln\frac{1}{1-e^{1-\beta\pi m/L}}.
\end{equation}
We approximate the sum by the integral
\[
\frac{L{\cal T}}{\pi}\int_0^\infty\ln\left(\frac{1}{1-e^{-x}}\right)dx
\]
(${\cal T}=1/\beta$)
\[
=\frac{L{\cal T}}{\pi}\frac{\pi^2}{6}=\frac{\pi L{\cal T}}{6}.
\]
By (\ref{SfromZ}) we then have $S$ $(=Z-\beta\partial Z/\partial\beta)=Z+{\cal T}dZ/d{\cal T}=\pi L{\cal T}/3$.
\smallskip
For a two-dimensional, say square, box of side $L$, the calculation is similar, except we now have modes labelled by two integers, say $m$ and $p$ with angular frequencies $\omega_{mp}=\pi (m^2+p^2)^{1/2}/L$. In consequence,
(\ref{1dlogZ}) gets replaced by
\begin{equation}
\label{2dlogZ}
\ln Z=\sum_{m=1}^\infty\sum_{p=1}^\infty \ln\frac{1}{1-e^{1-\beta(\pi/L)(m^2+p^2)^{1/2}}}
\end{equation}
which we can approximate as an integral over the positive quadrant of ${\mathbb R}^2$, which, when we convert to polar coordinates, becomes
\[
\frac{L^2{\cal T}^2}{\pi^2}\frac{2\pi}{4}\int_0^\infty x\ln\left(\frac{1}{1-e^{-x}}\right)dx
\]
\[
=\frac{2}{4\pi}\zeta(3)L^2{\cal T}^2
\]
whereupon $S=Z+{\cal T}dZ/d{\cal T}=3\zeta(3)L^2{\cal T}^2/(2\pi)$ where $\zeta(3)$ is the Riemann zeta function of 3 (approximately $1.202$).
\smallskip
The obvious 3-dimensional counterpart to these calculations gives the well-known results, for the partition function and entropy of a real scalar field: $\ln Z=L^3{\cal T}^3\pi^2/90$, $S=2\pi^2{\cal T}^3/45$.
\item
\label{Note:omeganmcondition}
In order for the brick-wall modified $\rho$ and $Z$ to exist in the 1+2 dimensional case, we require, of course, that
$Z=\tr(e^{-\beta H})$ where $H$ is the second quantization, $d\Gamma(h)$ of the one-particle Hamiltonian, $h$, whose spectrum consists of the $\omega_n^m$. We haven't actually checked this since we haven't obtained mathematical control on the $\omega_n^m$. However, we expect it to be finite.
\item
\label{Note:300}
The counterpart to our Equation (\ref{1plus2bricksecondresult}) for a collection of $\mathsf{N}$ Klein-Gordon fields in a finite box, outside a suitable brick wall in the (1+3 dimensional) Schwarzschild spacetime is Mukohyama and Israel's \cite{Mukohyama:1998rf} Equation (3.14) (see also Note $<$\ref{Note:MIcorrection}$>$) for the area piece of their entropy, which can be written in the form
\begin{equation}
\label{originalbrick}
S_{\mathsf{N}} = \left(\frac{\mathsf{N}}{90\pi}\right)\frac{1}{\alpha^2}\left(\frac{4\pi r^2}{4}\right)
\end{equation}
where $r$ is the Schwarzschild radius. In \cite{Mukohyama:1998rf} it was concluded from this that ``$\alpha$ is very near the Planck length if the effective number, $\mathsf{N}$, of basic quantum fields in nature is on the order of 300'' ($\approx 90\pi$). One might equally conclude, though, that, if quantum gravitational effects are correctly taken into account by setting $\alpha$ (the metrical distance from the brick wall to the horizon) approximately equal to the Planck length, then the number of basic quantum fields in nature (or perhaps we should say the number of basic quantum fields in a consistent theory of quantum gravity) must be of the order of 300. We draw a corresponding conclusion in the main text from our Equation (\ref{1plus2bricksecondresult}).
\smallskip
There doesn't appear to be a physically meaningful corresponding conclusion for the case of 1+1 dimensional BTZ. In this case, it is easy to see that the metrical distance, $\alpha$, from the brick wall to the horizon is related to $B$ (see after Equation (\ref{E:141b})) by the approximate formula
\begin{equation}
\label{1plus1brickdist}
\ln\left(\frac{\alpha}{2\ell}\right)\simeq -\frac{r_+B}{\ell}.
\end{equation}
So for $\mathsf{N}$ (real, massless) Klein Gordon fields, we would have, by (\ref{1dentropywithTH}) and (\ref{1plus1brickdist}), the 1+1 dimensional counterpart to Equation (\ref{1plus2bricksecondresult})
\[
S_{\mathsf{N}}=\left(\frac{2{\mathsf{N}}}{3}\ln\left(\frac{2\ell}{\alpha}\right)\right)\left(\frac{1}{4}\right).
\]
If one considers the Hawking value of the entropy to be $1/4$ in this case, then we observe that, for this to take the Hawking value, we would need to set ${\mathsf{N}}=(3/2)/(\ln(2\ell/\alpha))$ which, for $\alpha=1$ and $\ell$ much larger than 1 in natural units, would be less than 1!
\item
\label{Note:Smolinprovisos}
In their paper, \cite{Arnsdorf:2001qb}, Arnsdorf and Smolin consider the possibilities that the resolution to their puzzle might be either that string theory is not a theory of quantum gravity or that the Maldacena dual (i.e. ${\cal N}=4$ supersymmetric Yang Mills theory) in the AdS$_5$ version of holography does not exist as a quantum field theory in Minkowski space satisfying the basic axioms of nets of local *-algebras. While we of course can not discount either of these possibilities, we shall aim in Section \ref{Sect:discuss} to arrive at at least a partial resolution to the Arnsdorf Smolin puzzle without contemplating either of these possibilities.
\item
\label{Note:same}
For the subtleties involved in determining what is meant by the `same quantum field theory on two different curved spacetime backgrounds' see \cite{Fewster:2011pe, Fewster:2011pn}.
\item
\label{Note:double}
When we ask about the state of quantum gravity (as described by string theory) corresponding to a thermal state defined on a boundary cylinder, then the picture (see Figure \ref{Fig:BTZcylinders}) of the full maximally extended Schwarzschild AdS spacetime with its two boundary cylinders is probably misleading. This picture is meaningful in the context of quantum field theory on a fixed curved spacetime. But, already in Euclidean quantum gravity (see \cite{Gibbons:1976ue}, \cite{Hawking:1980gf}), it seems not to be meaningful, for a given cylinder, to regard it as a `right cylinder' and to assume the existence of a left wedge and a left cylinder. (Note that this point is unrelated to the `single exterior black holes' considered by Louko and Marolf \cite{Louko:1998hc} and by Maldacena \cite{Maldacena:2001kr}.) With our `matter-gravity entanglement hypothesis' (discussed in Subsection \ref{subsect:matgrav}) one might say, indeed, that the role which would be played by the left wedge is, in a sense, taken over by the quantum gravitational field. In particular, our expectation (see Subsection \ref{subsect:matgrav}) that the vector state, $\Psi_{\mathrm{total}}$ in ${\cal H}_{\mathrm{total}}$ which represents the equilibrium involving our AdS black hole and its thermal atmosphere `purifies' our thermal density operator on ${\cal H}_{\mathrm{matter}}$ is analogous to the way in which the double-wedge Hilbert space purifies the thermal density operator for the right wedge in a quantum field theory in curved spacetime context. Similarly our expectation, in Subsection \ref{subsect:matgrav}, that an arbitrary vector in ${\cal H}_{\mathrm{total}}$ can be arbitrarily closely approximated by vectors obtained by acting with elements of $I\otimes {\cal A}_{\mathrm{matter}}$ on $\Psi_{\mathrm{total}}$ is analogous to the `Reeh Schlieder property' of the right wedge algebra in the fixed background context. (Cf. \cite{Kay:1985zs}).
\end{enumerate}
\ack
LO thanks the Mexican National Council for Science and Technology (CONACYT) for funding his research studentship in York. BSK is grateful to Michael Kay for helpful comments and suggestions.
\section*{References}
|
1,116,691,497,541 | arxiv | \section{Introduction}
The classification of associative algebras was instituted by Benjamin Peirce
in the 1870's \cite{pie}, who gave a partial classification of the complex
associative algebras of dimension up to 6, although in some sense, one can
deduce the complete classification from his results, with some additional
work. The classification method relied on the following remarkable fact:
\begin{thm}
Every finite dimensional algebra which is not nilpotent contains a nontrivial
idempotent element.
\end{thm}
A nilpotent algebra $A$ is one which satisfies $A^n=0$ for some $n$, while an
idempotent element $a$ satisfies $a^2=a$. This observation of Peirce eventually
leads to two important theorems in the classification of finite dimensional associative
algebras. Recall that an algebra is said to be simple if it has no nontrivial
proper ideals, and it is not the trivial 1-dimensional nilpotent algebra over \mbox{$\mathbb K$}\, which is given
by the trivial product.
\begin{thm}[Fundamental Theorem of Finite Dimensional Associative Algebras]
Suppose that $A$ is a finite dimensional algebra over a field \mbox{$\mathbb K$}. Then $A$ has
a maximal nilpotent ideal $N$, called its radical. If $A$ is not nilpotent, then $A/N$
is a semisimple
algebra, that is, a direct sum of simple algebras.
\end{thm}
In fact, in the literature, the definition of a semisimple algebra is often given as
one whose radical is trivial, and then it is a theorem that semisimple algebras are
direct sums of simple algebras. Moreover, when $A/N$ satisfies a property called separability
over \mbox{$\mathbb K$}, then $A$ is a semidirect product of its radical and a semisimple algebra.
Over the complex numbers, every semisimple algebra is separable. To apply this theorem
to construct algebras by extension, one uses the following characterization of simple algebras.
\begin{thm}
[Wedderburn] If $A$ is a finite dimensional algebra over \mbox{$\mathbb K$}, then $A$ is simple iff
$A$ is isomorphic
to a tensor product $M\otimes D$, where $M=\mathfrak{gl}(n,\mbox{$\mathbb K$})$ and
$D$ is a division algebra over \mbox{$\mathbb K$}.
\end{thm}
One can also say that $A$ is a matrix algebra with coefficients in a division algebra over \mbox{$\mathbb K$}.
An associative division algebra is a
unital associative algebra where every nonzero element has a multiplicative
inverse. (One has to modify this definition in the case of graded algebras, but we will
not address this issue in this paper.) Over the complex numbers, the only division algebra
is $\mbox{$\mathbb C$}$ itself, so Wedderburn's theorem says that the only simple algebras are the
matrix algebras. In particular, there is exactly one simple 4-dimensional complex associative
algebra, $\mathfrak{gl}(2,\mbox{$\mathbb C$})$, while there is one additional semisimple algebra,
the direct sum of 4 copies of $\mbox{$\mathbb C$}$.
According to our investigations, there are two basic prior approaches
to the classification. The first is the old paper by Peirce \cite{pie}
which attempts to classify all the nilpotent algebras, including
nonassociative ones. There are some evident mistakes in that paper, for
example, it gives a classification of the commutative nilpotent
associative algebras which contains nonassociative algebras as well.
The second approach \cite{mas} classifies the unital algebras only. It turns out
that classification of unital algebras is not sufficient.
Let us consider the unital algebra of one higher dimension which is
obtained by adjoining a multiplicative identity as the unital
enlargement of the algebra. Two nonisomorphic non-nilpotent algebras can
have isomorphic unital enlargements, so they cannot be recovered so
easily. Nevertheless, let us suppose that there were some efficient
method of constructing all unital algebras of arbitrary dimension, and
to determine their maximal nilpotent ideals. In that case, we could
recover all nilpotent algebras of dimension $n$ from their
enlargements. Moreover, to recover all algebras of dimension $n$, one
would only have to consider extensions of nilpotent algebras of
dimension $k$ by semisimple algebras of dimension $n-k$, where $0\leq k
\leq n$. Our method turns out to be efficient in
constructing extensions of nilpotent algebras by semisimple ones.
Thus, even if the construction of unital algebras could be carried out
simply, which is by no means obvious from the literature, one would
still need our methodology to construct most of the algebras. So the
role of our paper is to explore the construction method which leads to
the description of all algebras.
The main goal of this paper is to give a complete description of the
moduli space of nonnilpotent 4-dimensional associative algebras,
including a computation of the miniversal deformation of every element.
We get the description with the help of extensions, which is the
novelty of our approach. The nilpotent cases will be classified in
another paper.
We also give a canonical stratification
of the moduli space into projective orbifolds of a very simple type,
so that the strata are connected only by deformations
factoring through jump deformations, and the elements of a particular
stratum are given by neighborhoods determined by smooth
deformations.
The authors thank the referees for their useful comments.
\section{Construction of algebras by extensions}
In \cite{fp11}, the theory of extensions of an algebra $W$ by an algebra $M$ is
described. Consider the exact sequence
$$
0\rightarrow M\rightarrow V\rightarrow W\rightarrow 0
$$
of associative \mbox{$\mathbb K$}-algebras, so that $V=M\oplus W$ as a \mbox{$\mathbb K$}-vector space, $M$ is an
ideal in the algebra $V$, and $W=V/M$ is the quotient algebra. Suppose that
$\delta\in C^2(W)$ and $\mu\in C^2(M)$ represent the algebra structures on
$W$ and $M$ respectively. We can view $\mu$ and $\delta$ as elements of $C^2(V)$.
Let $T^{k,l}$ be the subspace of $T^{k+l}(V)$ given recursively
by
\begin{align*}
T^{0,0}&=\mbox{$\mathbb K$}\\
T^{k,l}&=M\otimes T^{k-1,l}\oplus V\otimes T^{k,l-1}
\end{align*}
Let
$C^{k,l}=\mbox{\rm Hom}(T^{k,l},M)\subseteq C^{k+l}(V)$.
If we denote the algebra structure on $V$ by $d$, we have
$$
d=\delta+\mu+\lambda+\psi,
$$
where $\lambda\in C^{1,1}$ and $\psi\in C^{0,2}$. Note that in this notation,
$\mu\in C^{2,0}$. Then the condition that $d$ is associative: $[d,d]=0$ gives the
following relations:
\begin{align*}
[\delta,\lambda]+\tfrac 12[\lambda,\lambda]+[\mu,\psi]&=0,
\quad\text{The Maurer-Cartan equation}\\
[\mu,\lambda]&=0,\quad\text{The compatibility condition}\\
[\delta+\lambda,\psi]&=0,\quad\text{The cocycle condition}
\end{align*}
Since $\mu$ is an algebra structure, $[\mu,\mu]=0$. Then if we define
$D_\mu$ by $D_\mu(\varphi)=[\mu,\varphi]$, then $D^2_\mu=0$.
Thus $D_\mu$ is a differential on $C(V)$.
Moreover $D_\mu:C^{k,l}\rightarrow C^{k+1,l}$. Let
\begin{align*}
Z_\mu^{k,l}&=\ker(D_\mu:C^{k,l}\rightarrow C^{k+1,l}),\quad\text{the $(k,l)$-cocycles}\\
B_\mu^{k,l}&=\operatorname{Im}(D_\mu:C^{k-1,l}\rightarrow C^{k,l}),\quad\text{the $(k,l)$-coboundaries}\\
H_\mu^{k,l}&=Z_\mu^{k,l}/B_\mu^{k,l},\quad\text{the $D_u$ $(k,l)$-cohomology}
\end{align*}
Then the compatibility condition means that $\lambda\in Z^{1,1}$.
If we define $D_{\delta+\lambda}(\varphi)=[\delta+\lambda,\varphi]$, then it is not
true that $D^2_{\delta+\lambda}=0$, but
$D_{\delta+\lambda}D_\mu=-D_{\mu}D_{\delta+\lambda}$, so that $D_{\delta+\lambda}$ descends
to a map $D_{\delta+\lambda}:H^{k,l}_\mu\rightarrow H^{k,l+1}_\mu$, whose square is zero, giving
rise to the $D_{\delta+\lambda}$-cohomology $H^{k,l}_{\mu,\delta+\lambda}$.
Let the pair $(\lambda,\psi)$ give rise to a codifferential $d$, and $(\lambda,\psi')$
give rise to another codifferential $d'$. Then if we express $\psi'=\psi+\tau$, it is
easy to see that $[\mu,\tau]=0$, and $[\delta+\lambda,\tau]=0$, so that the image $\bar\tau$
of $\tau$ in $H^{0,2}_\mu$ is a $D_{\delta+\lambda}$-cocycle, and thus $\tau$ determines
an element $\{\bar\tau\}\in H^{0,2}_{\mu,\delta+\lambda}$.
If $\beta\in C^{0,1}$, then $g=\exp(\beta):\T(V)\rightarrow\T(V)$ is given by
$g(m,w)=(m+\beta(w),w)$. Furthermore $g^*=\exp(-\operatorname{ad}_{\beta}):C(V)\rightarrow C(V)$ satisfies
$g^*(d)=d'$, where $d'=\delta+\mu+\lambda'+\psi'$ with
$\lambda'=\lambda+[\mu,\beta]$ and $\psi'=\psi+[\delta+\lambda+\tfrac12[\mu,\beta],\beta]$.
In this case, we say that $d$ and $d'$ are equivalent extensions in the restricted sense.
Such equivalent extensions are also equivalent as codifferentials on $\T(V)$.
Note that $\lambda$
and $\lambda'$ differ by a $D_\mu$-coboundary, so $\bar\lambda=\bar\lambda'$ in
$H^{1,1}_\mu$. If $\lambda$ satisfies the MC equation for some $\psi$, then
any element $\lambda'$ in $\bar\lambda$ also gives a solution of the MC equation,
for the $\psi'$ given above. The cohomology classes of those $\lambda$ for which
a solution of the MC equation exists determine distinct restricted equivalence classes
of extensions.
Let $G_{M,W}=\mbox{\bf GL}(M)\times\mbox{\bf GL}(W)\subseteq\mbox{\bf GL}(V)$. If $g\in G_{M,W}$ then $g^*:C^{k,l}\rightarrow
C^{k,l}$, and $g^*:C^k(W)\rightarrow C^k(W)$, so $\delta'=g^*(\delta)$ and $\mu'=g^*(\mu)$
are codifferentials on $\mathcal T(M)$ and $\mbox{$\T(W)$}$ respectively.
The group $G_{\delta,\mu}$ is the
subgroup of $G_{M,W}$ consisting of those elements $g$ such that $g^*(\delta)=\delta$
and $g^*(\mu)=\mu$. Then $G_{\delta,\mu}$ acts on
the restricted equivalence classes of extensions, giving the equivalence classes
of general extensions. Also $G_{\delta,\mu}$
acts on $H^{k,l}_\mu$, and induces an action on the classes $\bar\lambda$ of $\lambda$
giving a solution to the MC equation.
Next, consider the group $G_{\delta,\mu,\lambda}$ consisting
of the automorphisms $h$ of $V$ of the form $h=g\exp(\beta)$, where
$g\in G_{\delta,\mu}$, $\beta\in C^{0,1}$ and $\lambda=g^*(\lambda)+[\mu,\beta]$.
If $d=\delta+\mu+\lambda+\psi+\tau$, then $h^*(d)=\delta+\mu+\lambda+\psi+\tau'$ where
\begin{equation*}
\tau'=g^*(\psi)-\psi+[\delta+\lambda-\tfrac12[\mu,\beta],\beta]+g^*(\tau).
\end{equation*}
Thus the group $G_{\delta,\mu,\lambda}$ induces an action on $H^{0,2}_{\mu,\delta+\lambda}$
given by $\{\bar\tau\}\rightarrow\{\overline{\tau'}\}$.
The general group of equivalences of extensions of the algebra structure $\delta$ on $W$
by the algebra structure $\mu$ on $M$ is given by the group of automorphisms of $V$ of
the form $h=\exp(\beta)g$, where $\beta\in C^{0,1}$ and $g\in G_{\delta,\mu}$. Then we
have the following classification of such extensions up to equivalence.
\begin{thm}[\cite{fp11}]
The equivalence classes of extensions of $\delta$ on $W$ by $\mu$ on $M$ is classified
by the following:
\begin{enumerate}
\item Equivalence classes of $\bar\lambda\in H^{1,1}_\mu$ which satisfy the MC equation
\begin{equation*}
[\delta,\lambda]+\tfrac12[\lambda,\lambda]+[\mu,\psi]=0
\end{equation*}
for some $\psi\in C^{0,2}$, under the action of the group $G_{\delta,\mu}$.
\item Equivalence classes of $\{\bar\tau\}\in H^{0,2}_{\mu,\delta+\lambda}$ under the
action of the group $G_{\delta,\mu,\lambda}$.
\end{enumerate}
\end{thm}
Equivalent extensions will give rise to equivalent algebras on $V$, but it may
happen that two algebras arising from nonequivalent extensions are equivalent.
This is because the group of equivalences of extensions is the group of invertible
block upper
triangular matrices on the space $V=M\oplus W$, whereas the the equivalence
classes of algebras on $V$ are given by the group of all invertible
matrices, which is larger.
The fundamental theorem of finite dimensional algebras allows us to restrict our
consideration of extensions to two cases. First, we can consider those extensions
where $\delta$ is a semisimple algebra structure on $W$, and $\mu$ is a nilpotent
algebra structure on $M$. In this case, because we are working over $\mbox{$\mathbb C$}$, we can
also assume that $\psi=\tau=0$. Thus the classification of the extension reduces
to considering equivalence classes of $\lambda$.
Secondly, we can consider extensions
of the trivial algebra structure $\delta=0$ on a 1-dimensional space $W$ by
a nilpotent algebra $\mu$. This
is because a nilpotent algebra has a codimension 1 ideal $M$, and the restriction
of the algebra structure to $M$ is nilpotent. However, in this case, we cannot assume
that $\psi$ or $\tau$ vanish,
so we need to use the classification theorem above to determine the
equivalence classes of extensions. In many cases, in solving the MC equation for
a particular $\lambda$, if there is any $\psi$ yielding a solution, then $\psi=0$
also gives a solution, so the action of $G_{\delta,\mu,\lambda}$ on $H^{0,2}_\mu$
takes on a simpler form than the general action we described above.
In addition to the complexity which arises because we cannot take the cocycle term
$\psi$ in the extension to be zero, there is another issue that complicates the construction
of the extensions. If an algebra is not nilpotent, then it has a maximal nilpotent ideal
which is unique, and it can be constructed as an extension of a semisimple algebra by this
unique ideal. Both the semisimple and nilpotent parts in this construction are completely
determined by the algebra. Therefore, a classification of extensions up to equivalence of
extensions will be sufficient to classify the algebras. This means that the equivalence classes
of the module structure $\lambda$ determine the algebras up to isomorphism.
For nilpotent algebras, we don't have this assurance. The same algebra structure may arise by
extensions of the trivial algebra structure on a 1-dimensional space by two different nilpotent
algebra structures on the same $n-1$-dimensional space.
In addition, the deformation theory of the nilpotent algebras is far more involved than the deformation
theory of the nonnilpotent algebras. Thus, we decided to discuss the nilpotent 4-dimensional complex
algebras in a separate paper. In this paper, we only look at extensions of semisimple algebras by
nilpotent algebras, which is precisely what is necessary to classify
all non-nilpotent algebras.
\section{Associative algebra structures on a 4-dimensional vector space}
Denote the basis elements of a 4-dimensional associative algebra by
$f_1,f_2,f_3, f_4$ and let $\psi_k^{ij}$ denote the product
$f_if_j=f_k$.
We will
recall the classification of algebras on a 2-dimensional space given in \cite{bdhoppsw2},
and the classification of algebras on a 3-dimensional space given in \cite{fpp1}.
\begin{table}[h]
\begin{center}
\begin{tabular}{lccccc}
Codifferential&$H^0$&$H^2$&$H^1$&$H^3$&$H^4$\\ \hline \\
$d_1=\psi_1^{11}+\psi_2^{22}$&2&0&0&0&0\\
$d_2=\psi_2^{22}+\psi^{12}_1$&0&0&0&0&0\\
$d_3=\psi_2^{22}+\psi_1^{21}$&0&0&0&0&0\\
$d_4=\psi_2^{22}+\psi_1^{12}+\psi_1^{21}$&2&1&1&1&1\\
$d_5=\psi_2^{22}$&2&1&1&1&1\\
$d_6=\psi_1^{22}$&2&2&2&2&2\\\\ \hline
\end{tabular}
\end{center}
\label{Table 1}
\caption{Two dimensional complex associative algebras and their
cohomology}
\end{table}
\begin{table}[h]
\begin{center}
\begin{tabular}{lccccc}
Codifferential&$H^0$&$H^2$&$H^1$&$H^3$&$H^4$\\ \hline \\
$d_1=\psa{33}3+\psa{22}2+\psa{11}1$&3&0&0&0&0\\
$d_{2}=\psi_2^{22}+\psi_3^{33}+\psi_1^{21}+\psi_1^{13}$&1&0&0&0&0\\
$d_{3}=\psi_2^{22}+\psi_3^{33}+\psi_1^{12}$&1&0&0&0&0\\
$d_{4}=\psi_2^{22}+\psi_3^{33}+\psi_1^{21}$&1&0&0&0&0\\
$d_{5}=\psi_2^{22}+\psi_3^{33}+\psi_1^{21}+\psi_1^{12}$&3&1&1&1&1\\
$d_{6}=\psi_2^{22}+\psi_3^{33}$&3&1&1&1&1\\
$d_{7}=\psi_3^{33}+\psi_1^{22}+\psi_1^{31}+\psi_2^{32}+\psi_1^{13}+\psi_2^{23}$&3&2&2&2&2\\
$d_{8}=\psi_3^{33}+\psi_1^{22}$&3&2&2&2&2\\
$d_{9}=\psi_3^{33}+\psi_1^{31}+\psi_2^{32}$&0&3&0&0&0\\
$d_{10}=\psi_3^{33}+\psi_1^{13}+\psi_2^{23}$&0&3&0&0&0\\
$d_{11}=\psi_3^{33}+\psi_1^{31}+\psi_2^{23}$&0&1&0&1&0\\
$d_{12}=\psi_3^{33}+\psi_1^{13}+\psi_2^{32}+\psi_2^{23}$&1&1&1&1&1\\
$d_{13}=\psi_3^{33}+\psi_1^{31}+\psi_2^{32}+\psi_2^{23}$&1&1&1&1&1\\
$d_{14}=\psi_3^{33}+\psi_2^{32}$&1&1&2&2&2\\
$d_{15}=\psi_3^{33}+\psi_2^{23}$&1&1&2&2&2\\
$d_{16}=\psi_3^{33}+\psi_2^{32}+\psi_2^{23}$&3&2&2&2&2\\
$d_{17}=\psi_3^{33}+\psi_1^{31}+\psi_1^{13}+\psi_2^{32}+\psi_2^{23}$&3&4&6&12&24\\
$d_{18}=\psi_3^{33}$&3&4&8&16&32\\
$d_{19}=\psi_2^{13}+\psi_2^{31}+\psi_1^{33}$&3&3&3&3&3\\
$d_{20}(0:0)=\psi_2^{33}$&3&5&9&17&33\\
$d_{20}(1:0)=\psi_2^{13}+\psi_2^{33}$&1&2&5&8&11\\
$d_{20}(1:1)=\psi_2^{13}+\psi_2^{31}+\psi_2^{33}$&3&4&5&7&8\\
$d_{20}(1:-1)=\psi_2^{13}-\psi_2^{31}+\psi_2^{33}$&1&2&3&4&5\\
$d_{20}(p:q)=\psi_2^{13}p+\psi_2^{31}q+\psi_2^{33}$&1&2&3&3&4\\
$d_{21}=\psi_2^{13}-\psi_2^{31}$&1&4&5&8&9\\\\ \hline
\end{tabular}
\end{center}
\label{coho3 table}
\caption{Three dimensional complex associative algebras and their
cohomology}
\end{table}
Actually, we only need to know the nilpotent algebras from lower dimensions as well
as the semisimple algebras. In dimension 1, there is one nontrivial algebra structure
$d_1=\psa{11}1$, which is just complex numbers $\mbox{$\mathbb C$}$.
Thus, in dimension 2, the algebra $d_1=\psi_1^{11}+\psi_2^{22}$ is semisimple, while
the algebra $d_6=\psi_1^{22}$ is nilpotent. These are the only algebras of dimension 2 (other than
the trivial algebra) which play a role in the construction of 4-dimensional algebras by
extensions. The algebra $d_1$ is just the direct sum $\mbox{$\mathbb C$}^2=\mbox{$\mathbb C$}\oplus\mbox{$\mathbb C$}$.
In the case of 3-dimensional algebras, only $d_1=\psi_1^{11}+\psi_2^{22}+\psi_3^{33}$ is semisimple, and only the
algebras $d_{19}=\psi_2^{13}+\psi_2^{31}+\psi_1^{33}$, $d_{20}(p:q)=\psi_2^{13}p+\psi_2^{31}q+\psi_2^{33}$, and $d_{21}=\psi_2^{13}-\psi_2^{31}$ are nilpotent. The algebra $d_1$ is
just the direct sum of three copies of $\mbox{$\mathbb C$}$.
Note that $d_{20}(p:q)$ is
a family of algebras parameterized by the projective orbifold $\mathbb P^1/\Sigma_2$. By this
we mean that the algebras $d_{20}(p:q)$ and $d_{20}(tp:tq)$ are isomorphic if $t\ne0$,
which gives the projective parameterization,
and that the algebras $d_{20}(p:q)$ and $d_{20}(q:p)$ are also isomorphic, which gives
the action of the group $\Sigma_2$ on $\mathbb P^1$.
In constructing the elements of the moduli space by extensions, we need to consider
three possibilities, extensions of the semisimple algebra structure on a 3-dimensional space
$W$ by the
trivial algebra structure on a 1-dimensional space $M$,
extensions of the semisimple algebra structure on a 2-dimensional space by a nilpotent
algebra on a 2-dimensional space, and extensions of either the simple or the trivial
1-dimensional algebras by a nilpotent 3-dimensional algebra.
\medskip
Consider the general setup,
where an $n$-dimensional space
$W=\langle f_{m+1},\dots f_{m+n}\rangle$ is extended
by an $m$-dimensional space $M=\langle f_1,\cdots, f_m\rangle$. Then the module structure is
of the form
$$
\lambda=\psa{kj}i(L_k)^i_j+\psa{jk}i(R_k)^i_j,\quad i,j=1,\dots m, k=m+1\dots m+n,
$$
and we can consider $L_k$ and $R_k$ to be $m\times m$ matrices. Then we can express
the bracket $\tfrac12[\lambda,\lambda]$, which appears in the MC equation in terms
of matrix multiplication.
\begin{align}
\tfrac12[\lambda,\lambda]=
\psa{jkl}i(R_lR_k)^i_j+\psa{kjl}i(L_kR_l-R_kL_l)^i_j-\psa{klj}i(L_kL_l)^i_j,
\label{lambda-lambda}
\end{align}
where $i,j=1,\dots m$, and $k,l=m+1,\dots m+n$.
Next, suppose that $\delta=\psa{m,m}m+\cdots+ \psa{m+n,m+n}{m+n}$ is the semisimple
algebra structure $\mbox{$\mathbb C$}^n$ on $W$. Then we can also express $[\delta,\lambda]$ in
terms of matrix multiplication.
\begin{equation}
[\delta,\lambda]=\psa{kkj}i(L_k)^i_j-\psa{jkk}i(R_k)^i_j.
\label{delta-lambda}
\end{equation}
Since $\delta$ is semisimple, one can ignore the cocycle $\psi$ in constructing an extension,
so the MC equation is completely determined by the equations (\ref{delta-lambda}) and
(\ref{lambda-lambda}), so we obtain the conditions.
Therefore, the MC equation holds precisely when
\begin{align*}
L_k^2=L_k,\quad R_k^2=R_k,\quad L_kL_l=R_kR_l=0\text{ if $k\ne l$},\quad L_kR_l=R_lL_k.
\end{align*}
As a consequence, both $L_k$ and
$R_k$ must be commuting nondefective matrices whose eigenvalues are either 0 or 1, which
limits the possibilities.
Moreover, it can be shown that $G_{\delta}$, the group of automorphisms of $W$ preserving
$\delta$ is just the group of permutation matrices. Thus if $G=\operatorname {diag}(G_1,G_2)$ is
a block diagonal element of $G_{\delta,\mu}$, the matrix $G_2$ is a permutation matrix.
The action of $G$ on $\lambda$ is given by simultaneous conjugation of the matrices
$L_k$ and $R_k$ by $G_1$, and a simultaneous permutation of the $k$-indices determined
by the permutation associated to $G_2$.
When $\mu$ is zero, this is the entire story. When $\mu\ne0$, the matrices $G_1$ are
required to preserve $\mu$, and the compatibility condition $[\mu,\lambda]$ also complicates
the picture.
It is important to note that given an $m$ and a nilpotent element $\mu$ on an $m$-dimensional
space $M$, there is an $n$ beyond which the extensions of the semisimple codifferential on an
$N$ dimensional space with $N$ greater than $n$ are simply direct sums of the extensions
of the $n$-dimensional semisimple algebra $\mbox{$\mathbb C$}^n$ and the semisimple algebra $\mbox{$\mathbb C$}^{N-n}$.
We say that the extension theory becomes stable at $n$. Moreover, the deformation picture
stabilizes as well.
In higher dimensions, there are semisimple algebras which are not of the form $\mbox{$\mathbb C$}^n$. Also,
as $m$ increases, the complexity of the nontrivial nilpotent elements $\mu$ increases as
well. In dimension 4, there is a simple algebra, $\mbox{$\mathfrak{gl}$}(2,\mbox{$\mathbb C$})$, represented by the
codifferential $d_1$, and a semisimple algebra $\mbox{$\mathbb C$}^4$, represented by the algebra $d_2$.
All other 4-dimensional nonnilpotent algebras are extensions of a semisimple algebra of
the type $\mbox{$\mathbb C$}^n$, for $n=1,2,3$.
\section{Extensions of the 3-dimensional semisimple algebra $\mbox{$\mathbb C$}^3$
by the 1-dimensional trivial
algebra $\mbox{$\mathbb C$}_0$}
Let $W=\langle f_2, f_3,f_4\rangle$ and $M=\langle f_1\rangle$.
The matrices $L_k$ and $R_k$ determining $\lambda$ are $1\times1$ matrices, in
other words, just numbers; in fact, they are either 0 or 1. By applying a permutation
to the indices $2,3,4$, we can assume that either all the $L_k$ vanish, or $L_2=1$
and both $L_3$ and $L_4$ vanish. In the first case, either $R_2=1$ or $R_2=0$ and
$R_3=R_4=0$. In the second case, we can either have $R_2=1$ and $R_3=0$, $R_2=0$ and
$R_3=1$, or both $R_2$ and $R_3$ vanish. In all three cases, $R_4=0$.
Note that in all of these solutions, we can assume that $L_4=R_4=0$. For extensions
by a 1-dimensional space $M$, the extension picture stabilizes at $n=2$, and we are
looking at $n=3$. Thus the five solutions for $\lambda$ here, which
give the codifferentials $d_3,\cdots, d_7$ correspond to the five 3-dimensional codifferentials
$d_2,\cdots, d_6$.
\section{Extensions of the 2-dimensional semisimple algebra $\mbox{$\mathbb C$}^2$
by a 2-dimensional nilpotent algebra}
Let $W=\langle f_3,f_4\rangle$ and $M=\langle f_1,f_2\rangle$.
There are two choices of
$\mu$ in this case, depending on whether the algebra structure on $M$ is the
trivial or nontrivial nilpotent
structure. Although we cannot calculate $G_{\delta,\mu}$ without knowing $\mu$, we can say
that the matrix $G_2$ in the expression above for an element of $G_{M,W}$ must be one
of the two permutation matrices.
\subsection{Extensions by the nontrivial nilpotent algebra}
In this case $\mu=\psa{22}1$. In order for $[\mu,\lambda]=0$, using
equations (\ref{delta-lambda}) and (\ref{lambda-lambda}), we must have
$$ (L_k)^2_1=(R_k)^2_1=0, (L_k)^1_1=(L_k)^2_2=(R_k)^1_1=(R_k)^2_2,
$$
for all $k$. It follows that $L_k$ and $R_k$ are upper triangular matrices with
the same values on the diagonal, and since they are also nondefective matrices, they
must be diagonal, and therefore are either both equal to the identity or both the zero
matrix. It follows that by applying a permutation, we obtain either the solution
$\lambda=0$ or $L_3=R_4=I$, and $L_k=R_k=0$ for $k>3$. In fact, we see that this
case stabilizes when $n=1$, and we are looking at the case $n=2$. Thus the two solutions
$d_8$ and $d_9$ correspond to the three dimensional algebras $d_7$ and $d_8$. In fact,
$d_7$ arises by the following consideration. Given an algebra on an $n$-dimensional space,
there is an easy way to extend it to a unital algebra on an $n+1$-dimensional space, by taking any
vector not in the original space and making it play the role of the identity. This is how $d_7$ arises.
The algebra $d_8$ also arises in a natural way as the direct sum of the algebra structures $\delta$ and $\mu$.
For this $\mu$, these are the only such structures which arise, and this is somewhat typical.
\subsection{Extensions by the trivial nilpotent algebra}
In this case, $L_k$ and $R_k$ are $2\times2$ matrices.
The nontrivial permutation has the effect of
interchanging $L_3$ and $L_4$ as well as $R_3$ and $R_4$. The matrix $G_1$ acts on
all four of the matrices by simultaneously conjugating them.
By permuting if necessary, one can assume that $L_3$ is either a nonzero matrix,
or both $L_3$ and $L_4$ vanish. Moreover, by conjugation in case $L_3$ is not
the identity or the zero matrix, we have $L_3=\operatorname {diag}(1,0)$, which we will denote by
$T$. Now if $L_3=I$, then since $L_3L_4=0$, we must have $L_4=0$, but if $L_3=T$,
then the condition $L_3L_4=0$ forces to either be
$B=\operatorname {diag}(0,1)$ or 0. A similar analysis applies to the $R$ matrices.
Let us consider a case by case analysis.
If $L_3=I$ then $L_4=0$. Since $I$ is invariant
under conjugation, we can still apply a conjugation to put $R_3$ in the form
$I$, $T$ or $0$. If $R_3=I$, then $R_3=0$. If $R_3=T$, then either $R_4=B$ or
$R_4=0$. If $R_3=0$, then $R_4$ may equal $I$, $T$ or $0$. This gives six
solutions.
Next, assume $L_3=T$ and $L_4=B$.
Since we have used up the conjugation in putting $L_3$ and $L_4$ in
diagonal form, we can only use the fact that since $R_3$ and $R_4$
commute with $L_3$ and $L_4$, they can be simultaneously diagonalized, so we may
assume they are diagonal. Thus $R_3$ is either $I$, $T$ $B$ or $0$.
If $R_3=I$ then $R_4=0$. If $R_3=T$ then
$R_4=B$ or $R_4=0$. If $R_3=D$ then $R_4=T$ or $R_4=0$. If $R_3=0$, then $R_4$ is
either $I$, $T$, $B$ or $0$. This gives 9 possibilities, but there is one more thing
which we have to be careful of. A certain conjugation interchanges $T$ and $D$, so that if we
first apply the nontrivial permutation and then the conjugation which interchanges $T$ and $B$,
we find that the $L_3=T$, $L_4=B$, $R_3=I$ and $R_4=0$ is the same as if $R_3=0$ and
$R_4=I$. Similarly, $R_3=T$ and $R_4=0$ transforms to $R_3=0$ and $R_4=B$. Finally,
$R_3=B$ and $R_4=0$ transforms to $R_3=0$ and $R_4=B$. Thus instead of 9 cases, we
only obtain 6.
If $L_3=T$ and $L_4=0$, then we obtain the same 9 cases for $R_3$ and $R_4$ as when
$L_3=T$ and $L_4=B$, except this time, there are no hidden symmetries, so we get
exactly 9 cases.
Finally, when $L_3=L_4=0$, if $R_3=I$ then $R_4=0$, while if $R_3=T$, then $R_4=B$
or $R_4=0$, and if $R_3=0$, then $R_4=0$, giving 4 more cases.
This gives a total of 25 nonequivalent extensions, and they are also all nonequivalent as
algebras, corresponding to $d_{10},\dots d_{34}$. In this case, $n=2$ is not the
stable case. It is not hard to see that $n=4$ gives the stable case, corresponding to
the 6-dimensional moduli space.
\section{Extensions of the 1-dimensional simple algebra $\mbox{$\mathbb C$}$
by a 3-dimensional nilpotent algebra}
Here $M=\langle f_1,f_2,f_3\rangle$ and $W=\langle f_4\rangle$, and $L_4$
and $R_4$ are $3\times 3$ matrices, which for simplicity, we will just denote by $L$ and $R$.
Elements in $C^{0,1}$ are of the form Let $\beta=\pha{4}1b_1+\pha{4}2b_2+\pha43b_3$.
\subsection{Extensions by the nilpotent algebra $\mu=\psa{13}2+\psa{31}2+\psa{33}1$}
In order for $[\mu,\lambda]=0$, using equations (\ref{delta-lambda})
and (\ref{lambda-lambda}), we must have
$$L=\left[\begin{matrix}L^1_1&0&L^1_3\L^1_3&L^1_1&L^2_3\\0&0&L^1_1\end{matrix}
\right],
R=\left[\begin{matrix}L^1_1&0&L^1_3\L^1_3&L^1_1&R^2_3\\0&0&L^1_1\end{matrix}
\right].$$
Taking into account the MC equation, we obtain that $L$ and $R$ must be diagonal matrices,
so we only get two solutions, depending on whether $L$ and $R$ both vanish or are both
equal to the identity matrix. Note that this is the stable case. It corresponds to the
codifferentials $d_{35}$ and $d_{36}$. Notice that we obtain one unital algebra $d_{35}$ and one algebra
which is a direct sum, $d_{36}$.
\subsection{Extensions by the nilpotent algebra $\mu=\psa{13}2p+\psa{31}2q+\psa{33}2$}
Here the situation depends on the projective coordinate $(p:q)$, which is parameterized by $\mathbb P^1/\Sigma_2$.
There are special cases when $p=q=0$ or $p=1$ and $q=0$. These three cases arise
from the compatibility condition $[\mu,\lambda]=0$, which generically has one solution,
but has additional solutions when either $p$ or $q$ vanishes, and when both $p$ and $q$ vanish.
\subsubsection{The generic case}
In this case, in order for $[\mu,\lambda]=0$, we must have
$$L=\left[\begin{matrix}L^1_1&0&0\L^2_1&L^1_1&L^2_3\\0&0&L^1_1\end{matrix}
\right],
R=\left[\begin{matrix}L^1_1&0&0\\R^2_1&L^1_1&R^2_3\\0&0&L^1_1\end{matrix}
\right].$$
Taking into account the MC equation, we obtain that $L$ and $R$ must be diagonal matrices,
so we only get two solutions, depending on whether $L$ and $R$ both vanish or are both
equal to the identity matrix. Note that this is the stable case. It corresponds to the
algebras $d_{37}(p:q)$ and $d_{38}(p:q)$. Note that both of these are families
parameterized by $\mathbb P^1/\Sigma_2$.
\subsubsection{The case $p=1$, $q=0$}
In this case, in order for $[\mu,\lambda]=0$, we must have
\begin{equation*}
L=\left[ \begin {array}{ccc}
L^1_1&0&L^1_1-L^3_3\\\noalign{\medskip}
L^2_1&L^1_1&L^2_3\\\noalign{\medskip}
0&0&L^3_3
\end {array} \right] , R=\left[\begin {array}{ccc}
L^3_3&0&L^3_3-R^3_3\\\noalign{\medskip}
R^2_1&R^3_3&R^2_3\\\noalign{\medskip}
0&0&R^3_3
\end {array} \right].
\end{equation*}
Since $[\mu,\beta]=\psa{43}2(b_1+b_3)+\psa{14}2b_3+\psa{34}2b_3$, we can further
assume that $L^2_3=0$. Moreover, the eigenvalues of $L$ are $L^1_1$ and $L^3_3$,
while those of $R$ are $L^3_3$ and $R^3_3$, and these numbers must be either 0
or 1,
yielding 8 possibilities. In fact, each one of these 8 choices corresponds to a
solution
solution of the MC equation. Let
\begin{equation*}
T_1=\left[\begin{matrix}
1&0&1\\
0&1&0\\
0&0&0
\end{matrix}\right],
T_2=\left[\begin{matrix}
0&0&-1\\
0&0&0\\
0&0&1
\end{matrix}\right],
B_1=\left[\begin{matrix}
1&0&1\\
0&0&0\\
0&0&0
\end{matrix}\right],
B_2=\left[\begin{matrix}
0&0&-1\\
0&1&0\\
0&0&1
\end{matrix}\right].
\end{equation*}
Then the 8 solutions are $L=I,R=I$;\enspace $L=0,R=0$;\enspace
$L=I,R=B_1$;\enspace
$L=T_1,R=B_2$;\enspace
$L=T_2,R=B_2$;\enspace $L=0,R=B_2$;\enspace $L=T_2,R=B_1$ and $L=T_1,R=0$, corresponding to
the codifferentials
$d_{37(1:0)}$,\enspace $d_{38(1: 0)}$,\enspace $d_{39},\cdots, d_{44}$.
This is not the stable case, because given an $L,R$ pair above,
there is another such pair, which satisfies the requirements that the products of the
$L$ matrices vanish, the products of the $R$ matrices vanish, and the $L$ and $R$ matrices commute.
In fact, it is not hard to see that $n=2$ gives the stable case, which will occur for
5-dimensional algebras.
\subsubsection{The case $p=0$, $q=0$}
In this case, in order for $[\mu,\lambda]=0$, we must have
\begin{equation*}
L=\left[ \begin {array}{ccc}
L^1_1&0&L^1_3\\\noalign{\medskip}
L^2_1&L^3_3&L^2_3\\\noalign{\medskip}
0&0&L^3_3
\end {array} \right] , R=\left[\begin {array}{ccc}
R^1_1&0&R^1_3\\\noalign{\medskip}
R^2_1&L^3_3&R^2_3\\\noalign{\medskip}
0&0&L^3_3
\end {array} \right].
\end{equation*}
Since $[\mu,\beta]=\psa{43}2b_3+\psa{34}2b_3$, we can further
assume that $L^2_3=0$. Moreover, the eigenvalues of $L_4$ are $L^1_1$ and $L^3_3$,
while those of $R_4$ are $L^3_3$ and $R^3_3$, and these numbers must be either 0 or 1,
yielding 8 possibilities. In fact, each one of these 8 choices corresponds to a unique
solution of the MC equation, which has some parameters. However, at this point we still
have not taken into account the action of $G_{\delta,\mu}$. An element in
$G_\mu$ is a matrix of the form $G=
\left[ \begin {smallmatrix}
g^1_1&0&g^1_3\\\noalign{\medskip}
g^2_1&(g^3_3)^2&g^2_3\\\noalign{\medskip}
0&0&g^3_3
\end {smallmatrix} \right]$,
and it acts on $\lambda$ by conjugating $L$ and $R$ simultaneously. This action is sufficient
to eliminate the parameters in the solutions for $L$ and $R$. Let $T=\operatorname {diag}(1,0,0)$ and
$B=\operatorname {diag}(0,1,1)$.
The 8 solutions are $L=I,R=I$, $L=0,R=0$, $L=I,R=B$, $L=T,R=T$,
$L=B,R=I$, $L=0,R=T$, $L=B,R=B$ and $L=T,R=0$, corresponding to
the codifferentials $d_{37(0: 0)}$, $d_{38(0: 0)}$,
$d_{45},\cdots, d_{50}$.
This is not the stable case, because given an $L,R$ pair above,
there is another such pair, which satisfies the requirements that the products of the
$L$-s vanish, the products of the $R$-s vanish, and the $L$ and $R$ matrices commute.
In fact, it is not hard to see that $n=2$ gives the stable case, which will occur for
5-dimensional algebras.
\subsection{Extensions by the nilpotent algebra $\mu=\psa{13}2-\psa{31}2$}
In order for $[\mu,\lambda]=0$, we must have
$$L=\left[ \begin {matrix}
L^3_3&0&0\\\noalign{\medskip}
L^2_1&L^3_3&L^2_3\\\noalign{\medskip}
0&0&L^3_3
\end {matrix} \right] , \left[ \begin {matrix}
L^3_3&0&0\\\noalign{\medskip}
R^2_1&L^3_3&R^2_3\\\noalign{\medskip}
0&0&L^3_3
\end {matrix} \right]$$
Taking into account the MC equation, we obtain that $L$ and $R$ must be diagonal matrices,
so we only get two solutions, depending on whether $L$ and $R$ both vanish or are both
equal to the identity matrix. Note that this is the stable case. It corresponds to the
codifferentials $d_{51}$, which is the unital extension, and $d_{52}$, which is the direct sum extension.
\subsection{Extensions by the trivial nilpotent algebra}
Since $\mu=0$, we don't get any restrictions on $\lambda$ from the compatibility
condition, but since $G_{\mu}=\mbox{\bf GL}(3,\mbox{$\mathbb C$})$, we can assume that $L$ is in Jordan normal
form, and since $L$ is nondefective, this implies that $L$ is diagonal. From
this it follows that $L$ can only be one of $I$, $T_1=\operatorname {diag}(1,1,0)$ $T_2=\operatorname {diag}(1,0,0)$
or 0.
When $L=I$ or $L=0$, it is invariant under conjugation, so we may conjugate $R$ to be one
of the same 4 matrices $I$, $T_1$, $T_2$ or 0. When $L=T_1$, $R$ can also be conjugated
to make it diagonal, and we obtain that $R$ is one of the six matrices $I$, $0$, $T_1$, $T_2$,
$B_1=\operatorname {diag}(1,0,1)$ or $B_2=\operatorname {diag}(0,0,1)$. When $L=T_2$, $R$ can again be conjugated to make
it diagonal, and it is one of the six matrices
$I$, $0$, $T_1$, $T_2$, $B_3=\operatorname {diag}(0,1,1)$,or $B_4=\operatorname {diag}(0,1,0)$. This gives the 20
codifferentials $d_{53},\cdots, d_{72}$.
This is not the stable case, and it is not hard to see that the stable case occurs
when $\dim(W)=6$.
\begin{table}[ht]
\begin{center}
\begin{tabular}{ccllrc}
M&W&$\delta$&$\mu$&N&Range\\\hline \\
1&$3$&$\psa{44}4+\psa{33}3+\psa{22}2$&$\psa{11}1$&1&$d_2$\\
1&$3$&$\psa{44}4+\psa{33}3+\psa{22}2$&$0$&5&$d_3,\cdots, d_7$\\
2&$2$&$\psa{44}4+\psa{33}3$&$\psa{22}1$&2&$d_8,d_9$\\
2&$2$&$\psa{44}4+\psa{33}3$&$0$&25&$d_{10},\cdots, d_{34}$\\
3&$1$&$\psa{44}4$&$\psa{31}2+\psa{13}2+\psa{33}1$&2&$d_{35},d_{36}$\\
3&$1$&$\psa{44}4$&$\psa{31}2q+\psa{13}2p+\psa{33}2$&2&$d_{37}(p:q),d_{38}(p:q)$\\
3&$1$&$\psa{44}4$&$\psa{13}2+\psa{33}2$&6&$d_{39},\cdots, d_{44}$\\
3&$1$&$\psa{44}4$&$\psa{33}2$&6&$d_{45},\cdots, d_{50}$\\
3&$1$&$\psa{44}4$&$\psa{13}2-\psa{31}2$&2&$d_{51},\cdots, d_{52}$\\
3&$1$&$\psa{44}4$&$0$&20&$d_{53},\cdots, d_{72}$\\
\\ \hline
\end{tabular}
\end{center}
\label{coho04 table}
\caption{Table of Extensions of $\delta$ on $W$ by $\mu$ on $M$}
\end{table}
Note that the simple algebra $d_1$ does not appear in the table above, because it does not arise as an extension.
\section{Hochschild Cohomology and Deformations}
Suppose that $V$ is a vector space, defined over a field $\mbox{$\mathbb K$}$
whose characteristic is not 2 or 3, equipped with an associative multiplication
structure $m:V\otimes V\rightarrow V$. The associativity relation can be given in the form
\begin{equation*}
m\circ(m\otimes 1)=m\circ(1\otimes m).
\end{equation*}
The notion of isomorphism or \emph{equivalence} of associative algebra structures is given as
follows. If $g$ is a linear automorphism of $V$, then define
\begin{equation*}
g^*(m)=g^{-1}\circ m\circ (g\otimes g).
\end{equation*}
Two algebra structures $m$ and $m'$ are equivalent if there is an automorphism
$g$ such that $m'=g^*(m)$. The set of equivalence classes of algebra structures
on $V$ is called the \emph{moduli space} of associative algebras on $V$.
\emph{Hochschild cohomology} was introduced in \cite{hoch}, and was used by Gerstenhaber in \cite{gers} to
classify infinitesimal deformations of associative algebras.
We define the Hochschild coboundary operator $D$ on $\mbox{\rm Hom}(\T(V),V)$ by
\begin{align*}
D(\varphi)(a_0,\cdots, a_n)=&a_0\varphi(a_1,\cdots, a_n)+\s{n+1}\varphi(a_0
,\cdots, a_{n-1})a_n\\&+\sum_{i=0}^{n-1}\s{i+1}\varphi(a_0,\cdots, a_{i-1},a_ia_{i+1},a_{i+2},\cdots, a_n)
.
\end{align*}
We wish to transform this classical viewpoint into the more modern viewpoint of
associative algebras as being given by codifferentials on a certain coalgebra.
To do this, we first introduce the \emph{parity reversion} $\Pi V$ of a
\mbox{$\Z_2$}-graded vector space $V$. If $V=V_e\oplus V_o$ is the decomposition of $V$
into its even and odd parts, then $W=\Pi V$ is the \mbox{$\Z_2$}-graded vector space
given by $W_e=V_o$ and $W_o=V_e$. In other words, $W$ is just the space $V$
with the parity of elements reversed.
Given an ordinary associative algebra,
we can view the underlying space $V$ as being \mbox{$\Z_2$}-graded, with $V=V_e$.
Then its parity reversion $W$ is again the same space, but now all elements
are considered to be odd. One can avoid this gyration for ordinary spaces,
by introducing a grading by exterior degree on the tensor coalgebra of $V$,
but the idea of parity reversion works equally well when the algebra is \mbox{$\Z_2$}-graded,
whereas the method of grading by exterior degree does not.
Denote the tensor (co)-algebra of $W$ by $\mbox{$\T(W)$}=\bigoplus_{k=0}^\infty W^k$,
where $W^k$ is the $k$-th tensor power of $W$ and $W^0=\mbox{$\mathbb K$}$. For brevity, the
element in $W^k$ given by the tensor product of the elements $w_i$ in $W$ will
be denoted by $w_1\cdots w_k$. The coalgebra structure on $\mbox{$\T(W)$}$ is given by
\begin{equation*}
\Delta(w_1\cdots w_n)=\sum_{i=0}^n w_1\cdots w_i\otimes w_{i+1}\cdots w_n.
\end{equation*}
Define $d:W^2\rightarrow W$ by $d=\pi\circ m\circ (\pi^{-1}\otimes\pi^{-1})$, where
$\pi:V\rightarrow W$ is the identity map, which is odd, because it reverses the parity
of elements. Note that $d$ is an odd map. The space $C(W)=\mbox{\rm Hom}(\mbox{$\T(W)$},W)$ is
naturally identifiable with the space of coderivations of $\mbox{$\T(W)$}$. In fact, if
$\varphi\in C^k(W)=\mbox{\rm Hom}(W^k,W)$, then $\varphi$ is extended to a coderivation of $\mbox{$\T(W)$}$
by
\begin{equation*}
\varphi(w_1\cdots w_n)=
\sum_{i=0}^{n-k}\s{(w_1+\cdots+ w_i)\varphi}w_1\cdots
w_i\varphi(w_{i+1}\cdots w_{i+k})w_{i+k+1}\cdots w_n.
\end{equation*}
The space of coderivations of $\mbox{$\T(W)$}$ is equipped with a \mbox{$\Z_2$}-graded Lie algebra
structure given by
\begin{equation*}
[\varphi,\psi]=\varphi\circ\psi-\s{\varphi\psi}\psi\circ\varphi.
\end{equation*}
The reason that it is more convenient to work with the structure $d$ on $W$
rather than $m$ on $V$ is that the condition of associativity for $m$
translates into the codifferential property $[d,d]=0$. Moreover, the
Hochschild coboundary operation translates into the coboundary operator $D$ on
$C(W)$, given by
\begin{equation*}
D(\varphi)=[d,\varphi].
\end{equation*}
This point of view on Hochschild cohomology first appeared in \cite{sta4}. The
fact that the space of Hochschild cochains is equipped with a graded Lie
algebra structure was noticed much earlier \cite{gers,gers1,gers2,gers3,gers4}.
For notational purposes, we introduce a basis of $C^n(W)$ as follows. Suppose
that $W=\langle w_1,\cdots, w_m\rangle$. Then if $I=(i_1,\cdots, i_n)$ is a
\emph{multi-index}, where $1\le i_k\le m$, denote $w_I=w_{i_1}\cdots w_{i_n}$.
Define $\varphi^{I}_i\in C^n(W)$ by
\begin{equation*}
\varphi^I_i(w_J)=\delta^I_Jw_i,
\end{equation*}
where $\delta^I_J$ is the Kronecker delta symbol. In order to emphasize the
parity of the element, we will denote $\varphi^I_i$ by $\psi^I_i$ when it is an odd
coderivation.
For a multi-index $I=(i_1,\cdots, i_k)$, denote its \emph{length} by $\ell(I)=k$. Then
since $W$ is a completely odd space,
the parity of $\varphi^I_i$ is given by $|\varphi^I_i|=k+1\pmod2$. If
$K$ and $L$ are multi-indices, then denote $KL=(k_1,\cdots, k_{\ell(K)},l_l,\cdots,
l_{\ell(L)})$. Then
\begin{align*}
(\varphi^I_i\circ\varphi^J_j)(w_K)&=
\sum_{K_1K_2K_3=K}\s{w_{K_1}\varphi^J_j} \varphi^I_i(w_{K_1},\varphi^J_j(w_{K_2}), w_{K_3})
\\&=
\sum_{K_1K_2K_3=K}\s{w_{K_1}\varphi^J_j}\delta^I_{K_1jK_3}\delta^J_{K_2}w_i,
\end{align*}
from which it follows that
\begin{equation}\label{braform}
\varphi^I_i\circ\varphi^J_j=\sum_{k=1}^{\ell(I)}\s{(w_{i_1}+\cdots+ w_{i_{k-1}})\varphi^J_j}
\delta^k_j
\varphi^{(I,J,k)}_i,
\end{equation}
where $(I,J,k)$ is given by inserting $J$ into $I$ in place of the $k$-th
element of $I$; \hbox{\it i.e.}, $(I,J,k)=(i_1,\cdots, i_{k-1},j_1,\cdots, j_{\ell(J)},i_{k+1},\cdots,
i_{\ell(I)})$.
Let us explain the notion of an infinitesimal deformation in terms of the
language of coderivations. We say that
\begin{equation*}
d_t=d+t\psi
\end{equation*}
is an infinitesimal deformation of the codifferential $d$
precisely when $[d_t,d_t]=0 \mod t^2$.
This condition immediately reduces to the cocycle condition $D(\psi)=0$. Note
that we require $d_t$ to be odd, so that $\psi$ must be an odd coderivation.
One can introduce a more general idea of parameters, allowing both even and odd
parameters, in which case even coderivations play an equal role, but we will
not adopt that point of view in this paper.
For associative algebras, we require that $d$ and $\psi$ lie in $C^2(W)$.
Since in this paper, our algebras are ordinary algebras, so that the
parity of an element in $C^n(W)$ is $n+1$,
elements of $C^2(W)$ are automatically odd.
We need the notion of a versal deformation, in order to
understand how the moduli space is glued together.
To explain versal deformations we introduce
the notion of a deformation with a local base. For details see
\cite{fi,fi2}.
A local base $A$ is a \mbox{$\Z_2$}-graded commutative, unital
$\mbox{$\mathbb K$}$-algebra with an augmentation $\epsilon:A\rightarrow\mbox{$\mathbb K$}$,
whose kernel $\mbox{$\mathfrak m$}$ is the unique maximal ideal in $A$,
so that $A$ is a local ring. It follows that $A$ has a
unique decomposition $A=\mbox{$\mathbb K$}\oplus\mbox{$\mathfrak m$}$ and $\epsilon$ is
just the projection onto the first factor. Let $W_A=W\otimes A$
equipped with the usual structure of a right $A$-module.
Let $T_A(W_A)$ be the tensor algebra of $W_A$ over $A$,
that is $T_A(W_A)=\bigoplus_{k=0}^\infty T^k_A(W_A)$ where $T^0_A(W_A)=A$ and
$T^{k+1}_A(W_A)=T^k(W_A)_A\otimes_A W_A$. It is not difficult to show
that $T^k_A(W_A)=T^k(W)\otimes A$ in a natural manner, and thus $T_A(W_A)=T(W)\otimes A$.
Any $A$-linear map $f:T_A(W)\rightarrow T_A(W)$ is induced by its
restriction to $T(W)\otimes \mbox{$\mathbb K$}=T(W)$ so we can view an $A$-linear coderivation
$\delta_A$ on $T_A(W_A)$ as a map $\delta_A:T(W)\rightarrow T(W)\otimes A$.
A morphism $f:A\rightarrow B$ induces
a map $$f_*:\operatorname{Coder}_A(T_A(W_A))\rightarrow \operatorname{Coder}_B(T_B(W_B))$$ given by
$f_*(\delta_A)=(1\otimes f)\delta_A$, moreover if $\delta_A$ is a codifferential
then so is $f_*(A)$.
A codifferential $d_A$ on $T_A(W_A)$ is said to be a deformation
of the codifferential $d$ on $T(W)$ if $\epsilon_*(d_A)=d$.
If $d_A$ is a deformation of $d$ with base $A$ then we can express
\begin{equation*}
d_A=d+\varphi
\end{equation*}
where $\varphi:T(W)\rightarrow T(W)\otimes\mbox{$\mathfrak m$}$. The condition for $d_A$ to be a codifferential is
the Maurer-Cartan equation,
\begin{equation*}
D(\varphi)+\frac12[\varphi,\varphi]=0
\end{equation*}
If $\mbox{$\mathfrak m$}^2=0$ we say that $A$ is an infinitesimal algebra and a deformation
with base $A$ is called infinitesimal.
A typical example of an infinitesimal base is $\mbox{$\mathbb K$}[t]/(t^2)$;
moreover, the classical notion of an infinitesimal deformation:
$d_t=d+t\varphi$
is precisely an infinitesimal deformation with base $\mbox{$\mathbb K$}[t]/(t^2)$.
A local algebra $A$ is complete if
\begin{equation*}
A=\displaystyle{\invlm}_kA/\mbox{$\mathfrak m$}^k
\end{equation*}
A complete, local augmented $\mbox{$\mathbb K$}$-algebra is called formal
and a deformation with a formal base is called a formal deformation,
see \cite{fi2}.
An infinitesimal base is automatically formal, so every infinitesimal deformation
is a formal deformation.
An example of a formal base is $A=\mbox{$\mathbb K$}[[t]]$ and a
deformation of $d$ with base $A$ can be expressed in the form
$$d_t=d+t\psi_1+t^2\psi_2+\dots$$
This is the classical notion of a formal deformation.
It is easy to see that the condition for $d_t$ to be a formal deformation reduces to
\begin{align*}
D(\psi_{n+1})=-\frac12\sum_{k=1}^{n}[\psi_k,\psi_{n+1-k}],\quad n=0,\dots
\end{align*}
An automorphism of $W_A$ over $A$ is an $A$-linear isomorphism $g_A:W_A\rightarrow W_A$ making
the diagram below commute:
\begin{figure}[h!]
$$\xymatrix{
W_A \ar[r]^{g_A} \ar[d]^{\epsilon_*} & W_A \ar[d]^{\epsilon_*} \\
W \ar[r]^I & W}$$
\end{figure}
The map $g_A$ is induced by its restriction to $T(W)\otimes\mbox{$\mathbb K$}$ so we can view $g_A$ as a map
$$g_A:T(W)\rightarrow T(W)\otimes A$$
so we ca express $g_A$ in the form
$$g_A=I+\lambda$$
where $\lambda:T(W)\rightarrow T(W)\otimes\mbox{$\mathfrak m$}$. If $A$ is infinitesimal then $g_A^{-1}=I-\lambda$.
Two deformations $d_A$ and $d_A'$ are said to be equivalent over $A$ if
there is an automorphism $g_A$ of $W_A$ over $A$ such that $g_A^*(d_A)=d_A'$.
In this case we write $d'_A\sim d_A$.
An infinitesimal deformation $d_A$ with base $A$ is called
universal if whenever $d_B$ is an infinitesimal deformation with
base $B$, there is a unique morphism $f:A\rightarrow B$ such that $f_*(d_A)\sim d_B$.
\begin{thm}[\cite{ff2}]
If $\dim H^2_{odd}(d)<\infty$ then there is a universal
infinitesimal deformation $\mbox{$d^\text{inf}$}$ of $d$, given by
$$\mbox{$d^\text{inf}$}=d+\delta^it_i$$
where $H^2_{odd}(d)=\langle\bar{\delta^i}\rangle$ and $A=\mbox{$\mathbb K$}[t_i]/(t_it_j)$ is
the base of deformation.
\end{thm}
In the theorem above $\bar\delta^i$ is the cohomology class determined by the cocycle $\delta^i$.
A formal deformation $d_A$ with base $A$ is called versal if given
any formal deformation of $d_B$ with base $B$ there is a morphism
$f:A\rightarrow B$ such that $f_*(d_A)\sim d_B$.
Notice that the difference between the versal and the universal
property of infinitesimal deformations is that $f$ need
not be unique. A versal deformation is called
\emph{miniversal} if $f$ is unique whenever $B$ is
infinitesimal. The basic result about versal deformations is:
\begin{thm}[\cite{fi,fi2,fp1}]
If $\dim H^2_{odd}(d)<\infty$ then a miniversal deformation of $d$ exists.
\end{thm}
The following result can be used in some special cases to compute the versal deformations.
\begin{thm}
Suppose $H^2_{odd}(d)=\langle\bar{\delta^i}\rangle$ and
$[\delta^i,\delta^j]=0$ for all $i,j$ then the infinitesimal deformation
$$\mbox{$d^\text{inf}$}=d+\delta^it_i$$
is miniversal, with base $A=\mbox{$\mathbb K$}[[t_i]].$
\end{thm}
The construction of the moduli space as a geometric object
is based on the idea that codifferentials
which can be obtained by deformations with small parameters
are ``close'' to each other. From the small deformations,
we can construct 1-parameter families or even multi-parameter families,
which are defined for small values of the
parameters, except possibly when the parameters vanish.
If $d_t$ is a one parameter family of deformations, then two things can occur.
First, it may happen that
$d_t$ is equivalent to a certain codifferential $d'$ for every small value of $t$ except zero.
Then we say that $d_t$ is a jump deformation from $d$ to $d'$. It will
never occur that $d'$ is equivalent to $d$, so there are no
jump deformations from a codifferential to itself.
Otherwise, the codifferentials $d_t$ will all be nonequivalent if $t$ is small enough.
In this case, we
say that $d_t$ is a smooth deformation. (In detail, see \cite{fp10}.)
In \cite{fp10}, it was proved for Lie algebras that given three
codifferentials $d$, $d'$ and $d''$,
if there are jump deformations from $d$ to $d'$ and from $d'$ to $d''$,
then there is a jump deformation from
$d$ to $d''$. The proof of the corresponding statement for associative algebras
is essentially the same.
Similarly, if there is a jump deformation from $d$ to $d'$, and a family of smooth deformations
$d'_t$, then there is a family $d_t$ of smooth deformations of $d$,
such that every deformation in the image of
$d'_t$ lies in the image of $d_t$, for sufficiently small values of $t$.
In this case, we say that the
smooth deformation of $d$ factors through the jump deformation to $d'$.
In the examples of complex moduli spaces of Lie and associative algebras which we have studied,
it turns out that there is a natural
stratification of the moduli space of $n$-dimensional algebras by orbifolds,
where the codifferentials
on a given strata are connected by smooth deformations
which don't factor through jump deformations.
These smooth deformations determine the local neighborhood structure.
The strata are connected by jump deformations, in the sense that
any smooth deformation from a codifferential on one strata
to another strata factors through a jump deformation.
Moreover, all of the strata are given by projective
orbifolds.
In fact, in all the complex examples we have studied,
the orbifolds either are single points,
or $\mbox{$\mathbb C$}\mathbb P^n$ quotiented out by either $\Sigma_{n+1}$ or a subgroup, acting on
$\mbox{$\mathbb C$}\mathbb P^n$ by permuting the coordinates.
\section{Deformations of the elements in our moduli space}
We have ordered the codifferentials so that
a codifferential only deforms to a codifferential earlier on the list.
Partially, this was accomplished the ordering of the different choices of $M$ and $W$.
That such an ordering is possible is due to
the fact that jumps between families have a natural
ordering by descent.
The radical of an algebra $A$ is the same as the radical of its opposite algebra,
ideals in an algebra are the same as the ideals in its opposite algebra $A^\circ$,
and the quotient of the opposite algebra by an ideal is naturally isomorphic to
the quotient of the opposite algebra by the same ideal, it follows that the semisimple quotient
of an algebra is the same as its opposite algebra.
Also the center
of an algebra coincides the center of its opposite algebra.
Moreover, if an algebra $A$ deforms to an
algebra $B$, then its opposite algebra $A^\circ$ deforms to $B^\circ$.
A commutative algebra is isomorphic to its opposite algebra, but an
algebra may be isomorphic to its opposite algebra without being equal to it.
For example, a matrix algebra is always
isomorphic to its opposite algebra, and
the simple $1|1$-dimensional algebra is isomorphic to its opposite,
but neither of these algebras is commutative.
We shall summarize most of the relevant information about the algebras in tables below.
Since there are too many codifferentials to list in a single table,
we will split them up into several tables. In one set of tables,
we will give the codifferential which represents the algebra, as well as
information about the cohomology spaces
$H^0$ through $H^3$. In another set of tables, we will note which algebras are pairs of opposite
algebras, give a basis for the center of the algebra, and indicate which algebras it deforms
to. It would take up too much space to give the versal deformations for each of these algebras,
but all of them were computed using the constructive method we have outlined above.
\begin{table}[h]
\begin{center}
\begin{tabular}{lcccc}
Codifferential&$H^0$&$H^1$&$H^2$&$H^3$\\ \hline \\
$d_{1}=\psa{11}1+\psa{12}2+\psa{23}1+\psa{24}2+\psa{32}4+\psa{31}3+\psa{43}3+\psa{44}4$&$1$&$0$&$0$&$0$\\
$d_{2}=\psa{33}3+\psa{44}4+\psa{22}2+\psa{11}1$&$4$&$0$&$0$&$0$\\
$d_{3}=\psa{33}3+\psa{44}4+\psa{22}2+\psa{21}1+\psa{13}1$&$2$&$0$&$0$&$0$\\
$d_{4}=\psa{33}3+\psa{44}4+\psa{22}2+\psa{12}1$&$2$&$0$&$0$&$0$\\
$d_{5}=\psa{33}3+\psa{44}4+\psa{22}2+\psa{21}1$&$2$&$0$&$0$&$0$\\
$d_{6}=\psa{33}3+\psa{44}4+\psa{22}2+\psa{21}1+\psa{12}1$&$4$&$1$&$1$&$1$\\
$d_{7}=\psa{33}3+\psa{44}4+\psa{22}2$&$4$&$1$&$1$&$1$\\
$d_{8}=\psa{33}3+\psa{44}4+\psa{22}1+\psa{31}1+\psa{32}2+\psa{13}1+\psa{23}2$&$4$&$2$&$2$&$2$\\
$d_{9}=\psa{33}3+\psa{44}4+\psa{22}1$&$4$&$2$&$2$&$2$\\
\end{tabular}
\end{center}
\label{d1-d9}
\caption{The cohomology of the algebras $d_1\dots d_9$}
\end{table}
\subsection{The algebras $d_1\dots d_9$}
The algebra $d_1$ represents the matrix algebra $\mbox{$\mathfrak{gl}$}(2,\mbox{$\mathbb C$})$. As such, it is simple, and so has no ideals,
no deformations, and its center consists of the multiples of the identity, so has dimension 1. Thus
$\dim H^0=1$ and $\dim H^n=0$ otherwise.
The algebra $d_2$ is the semisimple algebra which is the direct sum of four copies of $\mbox{$\mathbb C$}$. Being semisimple,
it is also cohomologically rigid, but it is commutative, so $\dim H^0=4$.
The algebras $d_3$ $d_4$ and $d_5$ are all rigid, with center of dimension 2. The algebras $d_4$ and $d_5$ are opposite
algebras.
The algebra $d_6$ is the direct sum of $C^2$ with the algebra given by adjoining an identity to turn the 1-dimensional trivial algebra into a 2-dimensional unital algebra. It is unital, commutative, is not rigid, and in fact has a jump deformation to $d_2$.
The algebra $d_7$ is the direct sum of the trivial 1-dimensional algebra (which we denote as $\mbox{$\mathbb C$}_0$) with the semisimple
3-dimensional algebra $\mbox{$\mathbb C$}^3$. It is commutative but not unital, and also has a jump deformation to $d_2$.
The algebra $d_8$ arises by as a direct sum of $\mbox{$\mathbb C$}$ and the algebra which arises from adjoining an identity to the 2-dimensional nontrivial nilpotent algebra. It is unital and commutative. It has jump deformations to $d_2$ and $d_6$.
The algebra $d_9$ is the direct sum of the nontrivial 2-dimensional nilpotent algebra and $\mbox{$\mathbb C$}^2$. It is not unital, but is commutative. It has deformations to $d_2$, $d_6$ and $d_7$.
\begin{table}[h,t]
\begin{center}
\begin{tabular}{lcccc}
Codifferential&$H^0$&$H^1$&$H^2$&$H^3$\\ \hline \\
$d_{10}=\psa{33}3+\psa{44}4+\psa{31}1+\psa{32}2+\psa{14}1$&$0$&$0$&$0$&$0$\\
$d_{11}=\psa{33}3+\psa{44}4+\psa{41}1+\psa{13}1+\psa{23}2$&$0$&$0$&$0$&$0$\\
$d_{12}=\psa{33}3+\psa{44}4+\psa{31}1+\psa{42}2$&$0$&$0$&$0$&$0$\\
$d_{13}=\psa{33}3+\psa{44}4+\psa{13}1+\psa{24}2$&$0$&$0$&$0$&$0$\\
$d_{14}=\psa{33}3+\psa{44}4+\psa{31}1+\psa{24}2$&$0$&$0$&$0$&$0$\\
$d_{15}=\psa{33}3+\psa{44}4+\psa{31}1+\psa{14}1+\psa{42}2$&$0$&$0$&$0$&$0$\\
$d_{16}=\psa{33}3+\psa{44}4+\psa{13}1+\psa{41}1+\psa{24}2$&$0$&$0$&$0$&$0$\\
$d_{17}=\psa{33}3+\psa{44}4+\psa{31}1+\psa{32}2+\psa{14}1+\psa{24}2$&$1$&$3$&$0$&$0$\\
$d_{18}=\psa{33}3+\psa{44}4+\psa{31}1+\psa{32}2$&$1$&$3$&$0$&$0$\\
$d_{19}=\psa{33}3+\psa{44}4+\psa{13}1+\psa{23}2$&$1$&$3$&$0$&$0$\\
$d_{20}=\psa{33}3+\psa{44}4+\psa{23}2+\psa{31}1$&$1$&$1$&$0$&$1$\\
$d_{21}=\psa{33}3+\psa{44}4+\psa{13}1+\psa{32}2+\psa{41}1+\psa{24}2$&$1$&$1$&$1$&$1$\\
$d_{22}=\psa{33}3+\psa{44}4+\psa{32}2+\psa{23}2+\psa{31}1$&$2$&$1$&$1$&$1$\\
$d_{23}=\psa{33}3+\psa{44}4+\psa{32}2+\psa{23}2+\psa{13}1$&$2$&$1$&$1$&$1$\\
$d_{24}=\psa{33}3+\psa{44}4+\psa{31}1+\psa{14}1$&$2$&$1$&$1$&$1$\\
$d_{25}=\psa{33}3+\psa{44}4+\psa{31}1+\psa{42}2+\psa{24}2$&$2$&$1$&$1$&$1$\\
$d_{26}=\psa{33}3+\psa{44}4+\psa{13}1+\psa{42}2+\psa{24}2$&$2$&$1$&$1$&$1$\\
$d_{27}=\psa{33}3+\psa{44}4+\psa{13}1+\psa{41}1+\psa{42}2+\psa{24}2$&$2$&$1$&$1$&$1$\\
$d_{28}=\psa{33}3+\psa{44}4+\psa{31}1+\psa{14}1+\psa{42}2+\psa{24}2$&$2$&$1$&$1$&$1$\\
$d_{29}=\psa{33}3+\psa{44}4+\psa{32}2$&$2$&$1$&$2$&$2$\\
$d_{30}=\psa{33}3+\psa{44}4+\psa{23}2$&$2$&$1$&$2$&$2$\\
$d_{31}=\psa{33}3+\psa{44}4+\psa{32}2+\psa{23}2$&$4$&$2$&$2$&$2$\\
$d_{32}=\psa{33}3+\psa{44}4+\psa{31}1+\psa{13}1+\psa{42}2+\psa{24}2$&$4$&$2$&$2$&$2$\\
$d_{33}=\psa{33}3+\psa{44}4+\psa{31}1+\psa{13}1+\psa{32}2+\psa{23}2$&$4$&$4$&$6$&$12$\\
$d_{34}=\psa{33}3+\psa{44}4$&$4$&$4$&$8$&$16$\\
\end{tabular}
\end{center}
\label{d10-d34}
\caption{The cohomology of the algebras $d_{10}\dots d_{34}$}
\end{table}
\subsection{The algebras $d_{10}\dots d_{34}$}
The algebras $d_{10}\dots d_{20}$ are all nonunital, noncommutative, and rigid. The pairs of opposite algebras
are $d_{10}$ and $d_{11}$, $d_{12}$ and $d_{13}$, $d_{15}$ and $d_{16}$, and $d_{18}$ and $d_{19}$. The algebras
$d_{14}$, $d_{17}$ and $d_{20}$ are all isomorphic to their opposite algebras.
The algebra $d_{21}$ is unital, but not commutative, and it has a jump deformation to $d_1$.
The algebras $d_{22}$ and $d_{23}$ are nonunital, noncommutative opposite algebras, with $d_{22}$ having a jump
deformation to $d_{5}$, while $d_{23}$ jumps to its opposite algebra $d_{4}$. Similarly $d_{25}$ and $d_{26}$ are also
nonunital, noncommutative opposite algebras which jump to the same two elements in the same order.
The algebra
$d_{24}$ is a nonunital, noncommutative algebra which is isomorphic to its opposite algebra, and it jumps to $d_3$.
The algebras $d_{27}$ and $d_{28}$ are unital, noncommutative opposite algebras both of which have jump deformations
to $d_3$.
The algebras $d_{29}$ and $d_{30}$ are nonunital, nonunital opposite algebras, with $d_{29}$ jumping to $d_3$ and $d_5$,
while $d_{30}$ jumps to $d_3$ and $d_4$.
The algebra $d_{31}$ is nonunital but is commutative, and it has jump deformations to $d_2$, $d_6$ and $d_7$, all of which
are commutative. Note that a commutative algebra may deform to a noncommutative algebra, but the converse is impossible.
The algebra $d_{32}$ is both unital and commutative and jumps to $d_2$ and $d_6$. Note that a unital algebra can only deform
to another unital algebra, and both $d_2$ and $d_6$ are unital.
The algebra $d_{33}$ which arises by first taking the trivial 2-dimensional algebra, adding a multiplicative identity to make
it unital, and then taking a direct sum with $\mbox{$\mathbb C$}$, is both unital and commutative. It deforms to $d_2$, $d_3$, $d_6$ and $d_8$.
Note that $d_3$ is not commutative, illustrating the fact that a commutative algebra can deform to a noncommutative algebra.
The final algebra in this group, $d_{34}$, is the direct sum of the trivial 2-dimensional algebra $\mbox{$\mathbb C$}_0^2$ with $\mbox{$\mathbb C$}^2$, so it
is not unital, but is commutative. This algebra has a lot of deformations, with jump deformations to $d_2$, $d_4$, $d_5$, $d_6$,
$d_7$ and $d_9$. Note that even though $d_{34}$ is isomorphic to its opposite, it has jump deformations to $d_4$ and $d_5$, which
are not their own opposites. However, they are opposite algebras, illustrating the fact that if an algebra which is isomorphic
to its opposite deforms to another algebra, it also deforms to the opposite of that algebra.
\begin{table}[h,t]
\begin{center}
\begin{tabular}{lcccc}
Codifferential&$H^0$&$H^1$&$H^2$&$H^3$\\ \hline \\
$d_{35}=\psa{44}4+\psa{31}2+\psa{13}2+\psa{33}1+\psa{41}1+\psa{42}2+\psa{43}3+\psa{14}1+\psa{24}2+\psa{34}3$&$4$&$3$&$3$&$3$\\
$d_{36}=\psa{44}4+\psa{31}2+\psa{13}2+\psa{33}1$&$4$&$3$&$3$&$3$\\
$d_{37}(p:q)=\psa{44}4+q\psa{31}2+p\psa{13}2+\psa{33}2+\psa{41}1+\psa{42}2+\psa{43}3+\psa{14}1+\psa{24}2+\psa{34}3$&$2$&$2$&$1$&$0$\\
$d_{37}(1:1)=\psa{44}4+\psa{31}2+\psa{13}2+\psa{33}2+\psa{41}1+\psa{42}2+\psa{43}3+\psa{14}1+\psa{24}2+\psa{34}3$&$4$&$4$&$5$&$6$\\
$d_{37}(1:-1)=\psa{44}4-\psa{31}2+\psa{13}2+\psa{33}2+\psa{41}1+\psa{42}2+\psa{43}3+\psa{14}1+\psa{24}2+\psa{34}3$&$2$&$2$&$1$&$1$\\
$d_{37}(1:0)=\psa{44}4+\psa{13}2+\psa{33}2+\psa{41}1+\psa{42}2+\psa{43}3+\psa{14}1+\psa{24}2+\psa{34}3$&$2$&$2$&$3$&$5$\\
$d_{37}(0:1)=\psa{44}4+\psa{31}2+\psa{33}2+\psa{41}1+\psa{42}2+\psa{43}3+\psa{14}1+\psa{24}2+\psa{34}3$&$2$&$2$&$3$&$5$\\
$d_{37}(0:0)=\psa{44}4+\psa{33}2+\psa{41}1+\psa{42}2+\psa{43}3+\psa{14}1+\psa{24}2+\psa{34}3$&$4$&$5$&$7$&$13$\\
$d_{38}(p:q)=\psa{44}4+q\psa{31}2+p\psa{13}2+\psa{33}2$&$2$&$2$&$3$&$3$\\
$d_{38}(1:1)=\psa{44}4+\psa{31}2+\psa{13}2+\psa{33}2$&$4$&$4$&$5$&$7$\\
$d_{38}(1:-1)=\psa{44}4-\psa{31}2+\psa{13}2+\psa{33}2$&$2$&$2$&$3$&$4$\\
$d_{38}(1:0)=\psa{44}4+\psa{13}2+\psa{33}2$&$2$&$2$&$5$&$8$\\
$d_{38}(0:1)=\psa{44}4+\psa{31}2+\psa{33}2$&$2$&$2$&$5$&$8$\\
$d_{38}(0:0)=\psa{44}4+\psa{33}2$&$4$&$5$&$9$&$17$\\
$d_{39}=\psa{44}4+\psa{13}2+\psa{33}2+\psa{41}1+\psa{42}2+\psa{43}1-\psa{34}1+\psa{24}2+\psa{34}3$&$1$&$1$&$0$&$0$\\
$d_{40}=\psa{44}4+\psa{13}2+\psa{33}2-\psa{43}1+\psa{43}3+\psa{14}1+\psa{34}1$&$1$&$1$&$1$&$0$\\
$d_{41}=\psa{44}4+\psa{13}2+\psa{33}2+\psa{41}1+\psa{42}2+\psa{43}1$&$0$&$1$&$1$&$1$\\
$d_{42}=\psa{44}4+\psa{13}2+\psa{33}2-\psa{34}1+\psa{24}2+\psa{34}3$&$0$&$1$&$1$&$1$\\
$d_{43}=\psa{44}4+\psa{13}2+\psa{33}2-\psa{43}1+\psa{43}3+\psa{14}1+\psa{24}2+\psa{34}3$&$0$&$1$&$1$&$1$\\
$d_{44}=\psa{44}4+\psa{13}2+\psa{33}2+\psa{41}1+\psa{42}2+\psa{43}3+\psa{14}1+\psa{34}1$&$0$&$1$&$1$&$1$\\
$d_{45}=\psa{44}4+\psa{33}2+\psa{41}1+\psa{24}2+\psa{42}2+\psa{34}3+\psa{43}3$&$2$&$2$&$2$&$2$\\
$d_{46}=\psa{44}4+\psa{33}2+\psa{14}1+\psa{24}2+\psa{42}2+\psa{34}3+\psa{43}3$&$2$&$2$&$2$&$2$\\
$d_{47}=\psa{44}4+\psa{33}2+\psa{14}1$&$2$&$2$&$3$&$3$\\
$d_{48}=\psa{44}4+\psa{33}2+\psa{41}1$&$2$&$2$&$3$&$3$\\
$d_{49}=\psa{44}4+\psa{33}2+\psa{14}1+\psa{41}1$&$4$&$3$&$3$&$3$\\
$d_{50}=\psa{44}4+\psa{33}2+\psa{24}2+\psa{42}2+\psa{34}3+\psa{43}3$&$4$&$3$&$3$&$3$\\
$d_{51}=\psa{44}4-\psa{31}2+\psa{13}2+\psa{41}1+\psa{42}2+\psa{43}3+\psa{14}1+\psa{24}2+\psa{34}3$&$2$&$4$&$6$&$8$\\
$d_{52}=\psa{44}4-\psa{31}2+\psa{13}2$&$2$&$4$&$5$&$8$\\
\end{tabular}
\end{center}
\label{d35-d52}
\caption{The cohomology of the algebras $d_{35}\dots d_{52}$}
\end{table}
\subsection{The algebras $d_{35}\dots d_{52}$}
The algebra $d_{35}$ which arises by adjoining an identity to the 3-dimensional nilpotent algebra
$d_{19}=\psa{31}2+\psa{13}2+\psa{33}1$ is both unital and commutative. It has jump deformations to
$d_{2}$, $d_6$, $d_8$ and $d_{32}$.
The algebra $d_{36}$ which is the direct sum of $\mbox{$\mathbb C$}$ and the 3-dimensional nilpotent algebra $d_{19}$ above, is nonunital
but commutative, and it deforms to $d_2$ $d_6$, $d_7$, $d_8$, $d_9$, and $d_{31}$.
The family of algebras $d_{37}(p:q)$ is parameterized by $\mathbb P^1/\Sigma_2$, which means that it is a projective family,
in the sense that $d_{37}(p:q)\sim d_{37}(up:uq)$ when $u\in\mbox{$\mathbb C$}^*$, and is invariant under the action of $\Sigma_2$ by
interchanging coordinates, in the sense that $d_{37}(p:q)\sim d_{37}(q:p)$.
We remark that there is an element
$d_{37}(0:0)$ corresponding to what is called the generic point in $\mathbb P^1$. This point is usually omitted in the definition of $\mathbb P^1$, because including this generic point makes $\mathbb P^1$ a non-Hausdorff space. In fact, this non-Hausdorff behavior is reflected in
the deformations of the point $d_{37}(0:0)$, so the inclusion of the corresponding codifferential in the family here is quite natural.
With the families, there is a generic deformation pattern, and then there are some special values of the parameter $(p:q)$ for which the deformation pattern is not generic in the sense that there are additional deformations. Generically, this family consists
of unital but not commutative algebras. Because the opposite algebra to $d_{37}(p:q)$ is $d_{37}(q:p)$ which is isomorphic to the
original algebra, all of the elements of this family are isomorphic to their own opposite algebras.
Generically, an element in
this algebra deforms in a smooth way to other elements in the family, and these are the only deformations. We say that the
deformations are along the family. In fact, in every family of codifferentials, there are always smooth deformations along the family. In this case, these are the only deformations which occur generically.
The element $d_{37}(1:0)$ has additional jump deformations to $d_3$, $d_{27}$ and $d_{28}$. The element $d_{1:1}$ is commutative,
and has
additional jump deformations to $d_2$, $d_6$ $d_8$, $d_{32}$ and $d_{35}$, all of which are unital, commutative algebras.
If an element in a family has a deformation to an algebra, then the generic element in the family will also deform to it.
Moreover, the generic element always has jump deformations to all other
elements in the family, so $d_{37}(0:0)$ has
jump deformations to $d_{37}(p:q)$ for all $(p:q)$ except $(0:0)$.
Thus we automatically know that $d_{37}(0:0)$ has jump deformations the elements to which $d_{37}(1:0)$ and $d_{37}(1:1)$ deform.
In addition, there is a jump deformation from $d_{37}(0:0)$ to $d_{33}$. We also note that $d_{37}(0:0)$ is commutative.
The family $d_{38}(p:q)$ is also parameterized projectively by $\mathbb P^1/\Sigma_2$. Generically, the elements of the family are not commutative, and the only deformations are smooth deformations along the family.
The algebra $d_{38}(1:0)$ also has jump deformations to $d_3$, $d_4$, $d_5$, $d_{22}$, $d_{23}$, $d_{29}$, and $d_{30}$.
The algebra $d_{38}(1:1)$ is commutative and has additional jump deformations to $d_2$, $d_6$, $d_7$, $d_8$, $d_9$, $d_{31}$, and $d_{36}$. Finally, the generic element, which is commutative, in addition to all deformations above, and jump deformations to every
other element of the family, also has jumps to $d_3$, $d_{33}$ and $d_{34}$.
The algebras $d_{39}\dots d_{48}$ are neither unital nor commutative. The algebras $d_{39}$ are isomorphic to their opposite algebras. The algebra $d_{39}$ is rigid, while the algebra $d_{40}$ has a jump deformation to $d_1$. The algebras $d_{41}$ and $d_{42}$ are opposite algebras, with $d_{41}$ having a jump deformation to $d_{10}$, while $d_{42}$ deforms to the opposite algebra $d_{11}$.
The algebras $d_{43}$ and $d_{44}$ are opposites, with $d_{43}$ jumping to $d_{13}$ and $d_{44}$ jumping to its opposite $d_{12}$,
The algebras $d_{45}$ and $d_{46}$ are opposites, with $d_{45}$ jumping to $d_5$, $d_{22}$ and $d_{25}$, while $d_{46}$ jumps to
their opposite algebras $d_{4}$, $d_{23}$ and $d{26}$. The algebras $d_{47}$ and $d_{48}$ are opposite algebras, with
$d_{47}$ having jump deformations to $d_3$, $d_4$, $d_{24}$, $d_{26}$, $d_{27}$, and $d_{30}$, while $d_{48}$ jumps to
$d_3$, $d_5$, $d_{24}$, $d_{25}$, $d_{28}$, and $d_{29}$.
The algebras $d_{49}$ and $d_{50}$ are nonunital but are commutative, with $d_{49}$ jumping to $d_2$, $d_6$, $d_7$, $d_9$, $d_{31}$, and $d_{32}$, while $d_{50}$ jumps to $d_2$, $d_6$, $d_7$, $d_8$, and $d_{31}$.
The algebra $d_{51}$ is unital, with jump deformations to $d_1$, $d_{21}$, $d_{37}(1:-1)$, and deforms smoothly near $d_{37}(1:-1)$. This type of smooth deformation is said to factor through the jump deformation to $d_{37}(1:-1)$.
The algebra $d_{52}$ is neither unital nor commutative, but is isomorphic to its own opposite algebra. It has jump deformations to
$d_{20}$ and $d_{38}(1: -1)$, as well as deforming smoothly in a neighborhood of $d_{38}(1: -1)$.
\begin{table}[h,t]
\begin{center}
\begin{tabular}{lcccc}
Codifferential&$H^0$&$H^1$&$H^2$&$H^3$\\ \hline \\
$d_{53}=\psa{44}4+\psa{14}1+\psa{24}2+\psa{43}3$&$0$&$4$&$0$&$8$\\
$d_{54}=\psa{44}4+\psa{41}1+\psa{42}2+\psa{34}3$&$0$&$4$&$0$&$8$\\
$d_{55}=\psa{44}4+\psa{41}1+\psa{42}2+\psa{43}3$&$0$&$8$&$0$&$0$\\
$d_{56}=\psa{44}4+\psa{14}1+\psa{24}2+\psa{34}3$&$0$&$8$&$0$&$0$\\
$d_{57}=\psa{44}4+\psa{14}1+\psa{41}1+\psa{24}2+\psa{43}3$&$1$&$2$&$3$&$5$\\
$d_{58}=\psa{44}4+\psa{14}1+\psa{41}1+\psa{42}2$&$2$&$2$&$3$&$4$\\
$d_{59}=\psa{44}4+\psa{14}1+\psa{41}1+\psa{24}2$&$2$&$2$&$3$&$4$\\
$d_{60}=\psa{44}4+\psa{41}1+\psa{34}3$&$1$&$2$&$4$&$6$\\
$d_{61}=\psa{44}4+\psa{14}1+\psa{41}1+\psa{42}2+\psa{43}3$&$1$&$4$&$4$&$4$\\
$d_{62}=\psa{44}4+\psa{14}1+\psa{41}1+\psa{24}2+\psa{34}3$&$1$&$4$&$4$&$4$\\
$d_{63}=\psa{44}4+\psa{41}1+\psa{42}2$&$1$&$4$&$5$&$5$\\
$d_{64}=\psa{44}4+\psa{14}1+\psa{24}2$&$1$&$4$&$5$&$5$\\
$d_{65}=\psa{44}4+\psa{14}1+\psa{41}1+\psa{24}2+\psa{42}2$&$4$&$5$&$7$&$13$\\
$d_{66}=\psa{44}4+\psa{14}1+\psa{41}1+\psa{24}2+\psa{42}2+\psa{43}3$&$2$&$4$&$8$&$16$\\
$d_{67}=\psa{44}4+\psa{14}1+\psa{41}1+\psa{24}2+\psa{42}2+\psa{34}3$&$2$&$4$&$8$&$16$\\
$d_{68}=\psa{44}4+\psa{14}1+\psa{41}1$&$4$&$5$&$9$&$17$\\
$d_{69}=\psa{44}4+\psa{41}1$&$2$&$4$&$10$&$20$\\
$d_{70}=\psa{44}4+\psa{14}1$&$2$&$4$&$10$&$20$\\
$d_{71}=\psa{44}4+\psa{14}1+\psa{41}1+\psa{24}2+\psa{42}2+\psa{34}3+\psa{43}3$&$4$&$9$&$24$&$72$\\
$d_{72}=\psa{44}4$&$4$&$9$&$27$&$81$\\
\end{tabular}
\end{center}
\label{d53-d72}
\caption{The cohomology of the algebras $d_{53}\dots d_{72}$}
\end{table}
\subsection{The algebras $d_{53}\dots d_{72}$}
The algebras $d_{53}\dots d_{68}$ are neither unital nor commutative. The algebras $d_{53}$ and $d_{54}$, and $d_{55}$ and $d_{56}$,
are pairs of opposite algebras, and they are all rigid.
The algebra $d_{57}$ is isomorphic to its opposite algebra, and jumps to $d_{14}$, $d_{20}$ and $d_{39}$.
the algebras $d_{58}$ and $d_{59}$ are opposites, with the former jumping to $d_3$, $d_5$, $d_{22}$, $d_{27}$, and $d_{29}$,
while the latter jumps to $d_3$, $d_4$, $d_{23}$, $d_{28}$, and $d_{30}$.
The algebra $d_{60}$ is isomorphic to its opposite algebra, and jumps to $d_1$, $d_{15}$, $d_{16}$, $d_{20}$, $d_{21}$, and $d_{40}$.
The algebras $d_{61}$ and $d_{62}$ are opposites, with $d_{61}$ jumping to $d_{12}$, $d_{18}$ and $d_{44}$, while
$d_{62}$ jumps to $d_{13}$ $d_{19}$ and $d_{43}$. The algebras $d_{63}$ and $d_{64}$ are opposite algebras, with
$d_{63}$ having jump deformations to $d_{10}$, $d_{17}$, $d_{18}$, and $d_{41}$, while $d_{64}$ jumps to
$d_{11}$, $d_{17}$, $d_{19}$, and $d_{42}$.
The algebra $d_{65}$ is its own opposite algebra, and it jumps to $d_2$, $d_3$, $d_6$, $d_7$, $d_8$, $d_{24}$, $d_{31}$, $d_{33}$,
and $d_{50}$. The algebras $d_{66}$ and $d_{67}$ are opposite algebras, with $d_{66}$ having jump deformations to
$d_5$, $d_{10}$, $d_{15}$, $d_{22}$, $d_{25}$, and $d_{45}$, and $d_{67}$ jumping to
$d_4$, $d_{11}$, $d_{16}$, $d_{23}$, $d_{26}$, and $d_{46}$.
The algebra $d_{68}$ is not unital, but is commutative, and it jumps to $d_2$, $d_4$, $d_5$, $d_6$, $d_7$,
$d_9$, $d_{25}$, $d_{26}$, $d_{31}$, $d_{32}$, $d_{34}$ and $d_{49}$. The opposite algebras $d_{69}$ and $d_{70}$ are neither unital
nor commutative, with the former deforming to $d_3$, $d_5$, $d_{11}$, $d_{14}$, $d_{15}$, $d_{24}$, $d_{25}$, $d_{28}$,
$d_{29}$, $d_{48}$, and the latter to $d_3$, $d_4$, $d_{10}$, $d_{13}$, $d_{14}$, $d_{16}$, $d_{24}$, $d_{26}$, $d_{27}$, $d_{30}$, and $d_{47}$.
The algebra $d_{71}$ is the algebra arising by adjoining an
identity to the trivial 3-dimensional algebra $\mbox{$\mathbb C$}^3_0$, so it is unital and commutative, and it has deformations to every unital
algebra, that is, to $d_1$, $d_2$, $d_3$, $d_6$, $d_8$, $d_{17}$, $d_{21}$,
$d_{27}$, $d_{28}$, $d_{32}$, $d_{33}$, $d_{35}$, every element of the family $d_{37}(p:q)$, and $d_{51}$.
Finally, the algebra $d_{72}$ is the direct sum of $\mbox{$\mathbb C$}$ and $\mbox{$\mathbb C$}^3_0$, so it is nonunital and commutative. It has jump deformations to
$d_2\dots d_9$, $d_{18}$, $d_{19}$, $d_{20}$, $d_{22}$, $d_{23}$, $d_{29}$, $d_{30}$, $d_{31}$, $d_{33}$,
$d_{34}$, $d_{36}$, all members of the family $d_{38}(p:q)$, and $d_{52}$.
\begin{table}[h,t]
\begin{center}
\begin{tabular}{lll}
Codifferential&Gabriel Number&Structure\\ \hline \\
$d_{1}$&10& $\mbox{$\mathfrak{gl}$}(2,\mbox{$\mathbb C$})$\\
$d_2$&1&$\mbox{$\mathbb C$}^4$\\
$d_3$&13&$\mbox{$\mathbb C$}\oplus\{\left[\begin{smallmatrix}a&b\\0&c\end{smallmatrix}\right]|a,b,c\in\mbox{$\mathbb C$}\}$\\
$d_6$&2&$\mbox{$\mathbb C$}^2\oplus\mbox{$\mathbb C$}[x]/(x^2)$\\
$d_8$&4&$\mbox{$\mathbb C$}\oplus\mbox{$\mathbb C$}[x]/(x^3)$\\
$d_{17}$&17&$\left\{\left[\begin{smallmatrix}a&0&0\\0&a&0\\c&d&b\end{smallmatrix}\right]|a,b,c,d\in\mbox{$\mathbb C$}\right\}$\\
$d_{21}$&11&$\left\{\left[\begin{smallmatrix}a&0&0&0\\0&a&0&d\\c&0&b&0\\0&0&0&b\end{smallmatrix}\right]|a,b,c,d\in\mbox{$\mathbb C$}\right\}$\\
$d_{27}$&15&$\left\{\left[\begin{smallmatrix}a&c&d\\0&a&0\\0&0&b\end{smallmatrix}\right]|a,b,c,d\in\mbox{$\mathbb C$}\right\}$\\
$d_{28}$&14&$\left\{\left[\begin{smallmatrix}a&0&0\\c&a&0\d&0&b\end{smallmatrix}\right]|a,b,c,d\in\mbox{$\mathbb C$}\right\}$\\
$d_{32}$&3&$\mbox{$\mathbb C$}[x]/(x^2)\oplus\mbox{$\mathbb C$}[y]/(y^2)$\\
$d_{33}$&6&$\mbox{$\mathbb C$}\oplus\mbox{$\mathbb C$}[x,y]/(x^2,xy,y^2)$\\
$d_{35}$&5&$\mbox{$\mathbb C$}[x]/(x^4)$\\
$d_{37}(p:q)$&$18(u)$&$\mbox{$\mathbb C$}[x,y]/(x^2,y^2,yx,-uxy),\quad u\ne-1$\\
$d_{37}(1:-1)$&19&$\mbox{$\mathbb C$}[x,y]/(y^2,x^2+yx,xy+yx)$\\
$d_{37}(1:1)$&7&$\mbox{$\mathbb C$}[x,y]/(x^2,y^2)$\\
$d_{37}(1:0)$&16&$\mbox{$\mathbb C$}[x,y]/(x^2,y^2,yx)$\\
$d_{37}(0:0)$&8&$\mbox{$\mathbb C$}[x,y]/(x^3,xy,y^2)$\\
$d_{51}$&12&$\bigwedge\mbox{$\mathbb C$}^2$\\
$d_{71}$&9&$\mbox{$\mathbb C$}[x,y,z]/(x,y,z)^2$\\
\end{tabular}
\end{center}
\label{Structure of Unital 4-dimensional algebras}
\caption{The structure of 4-dimensional unital algebras}
\end{table}
\section{Unital Algebras}
There are 15 unital algebras, including all of the elements in the family $d_{37}(p:q)$.
According to \cite{mas}, unital complex 4-dimensional associative algebras were classified by P. Gabriel \cite{gab}, and
this classification is in agreement with the unital algebras we determined by our methods. Note that no nilpotent algebra
can be unital, so the classification of the nonnilpotent algebras given here is sufficient to determine all of the unital
algebras.
\begin{table}[h,t]
\begin{center}
\begin{tabular}{ll}
Codifferential&Structure\\ \hline \\
$d_{2}$& $\mbox{$\mathbb C$}^4$\\
$d_{6}$& $\mbox{$\mathbb C$}^2\oplus\mbox{$\mathbb C$}[x]/(x^2)$\\
$d_7$&$\mbox{$\mathbb C$}^3\oplus\mbox{$\mathbb C$}_0$\\
$d_8$&$\mbox{$\mathbb C$}\oplus\mbox{$\mathbb C$}[x]/(x^3)$\\
$d_9$&$\mbox{$\mathbb C$}^2\oplus x\mbox{$\mathbb C$}[x]/(x^3)$\\
$d_{31}$&$\mbox{$\mathbb C$}\oplus\mbox{$\mathbb C$}_0\oplus\mbox{$\mathbb C$}[x]/(x^2)$\\
$d_{32}$&$\mbox{$\mathbb C$}[x]/(x^2)\oplus\mbox{$\mathbb C$}[y]/(y^2)$\\
$d_{33}$&$\mbox{$\mathbb C$}\oplus\mbox{$\mathbb C$}[x,y]/(x^2,xy,y^2)$\\
$d_{34}$&$\mbox{$\mathbb C$}^2\oplus\mbox{$\mathbb C$}^2_0$\\
$d_{35}$&$\mbox{$\mathbb C$}[x]/(x^4)$\\
$d_{36}$&$\mbox{$\mathbb C$}\oplus x\mbox{$\mathbb C$}[x]/(x^4)$\\
$d_{37}(1:1)$&$\mbox{$\mathbb C$}[x,y]/(x^2,y^2)$\\
$d_{37}(0:0)$&$\mbox{$\mathbb C$}[x,y]/(x^3,xy,y^2)$\\
$d_{38}(1:1)$&$\mbox{$\mathbb C$}\oplus(x,y)\le\mbox{$\mathbb C$}\oplus\mbox{$\mathbb C$}[x,y]/(x^2-xy,y^2)$\\
$d_{38}(0:0)$&$\mbox{$\mathbb C$}\oplus\mbox{$\mathbb C$}_0\oplus x\mbox{$\mathbb C$}[x]/(x^3)$\\
$d_{49}$&$\mbox{$\mathbb C$}[x]/(x^2)\oplus y\mbox{$\mathbb C$}[y]/(y^3)$\\
$d_{50}$&$\mbox{$\mathbb C$}_0\oplus\mbox{$\mathbb C$}[x]/(x^3)$\\
$d_{68}$&$\mbox{$\mathbb C$}_0^2\oplus C[x]/(x^2)$\\
$d_{71}$&$\mbox{$\mathbb C$}[x,y,z]/(x,y,z)^2$\\
$d_{72}$&$\mbox{$\mathbb C$}\oplus\mbox{$\mathbb C$}_0^3$\\
\end{tabular}
\end{center}
\label{Commutative Nonnilpotent}
\caption{The structure of nonnilpotent 4-dimensional commutative algebras}
\end{table}
\bigskip
\begin{figure}[htp]%
\begin{center}
\includegraphics[scale=0.42]{unital.eps
\caption{Deformations between unital algebras}%
\label{unital}%
\end{center}
\end{figure}
\section{Commutative Algebras}
There are 20 distinct nonnilpotent commutative algebras, of which 9 are unital. Every commutative algebra is a direct sum of
algebras which are ideals in quotients of polynomial algebras. Every finite dimensional unital commutative algebra is a quotient of a polynomial algebra, while every
finite dimensional nonunital algebra is an ideal in such an algebra. The
algebra $\mbox{$\mathbb C$}$ is representable as $\mbox{$\mathbb C$}[x]/(x)$, while the trivial algebra $\mbox{$\mathbb C$}_0$ is representable as the ideal $x\mbox{$\mathbb C$}[x]/(x^2)$.
In Table 7, the ideal $(x,y)$ in $\mbox{$\mathbb C$}[x,y]/(x^2-xy,y^2)$ has dimension 3 as a vector space over $\mbox{$\mathbb C$}$, and the algebra
$d_{38}(1:1)$ is expressed as a direct sum of $\mbox{$\mathbb C$}$ and that ideal, which gives a 4-dimensional algebra.
For completeness here, in Table 8, we give the nilpotent commutative algebras as well. The codifferential number
given relates to the dsscription of codifferentials which will appear in a sequel. These algebras were
classified by Hazlett \cite{hazl}, and also given in \cite{mazz}. There are 8 nontrivial commutative algebras.
We note that commutative algebras may deform into noncommutative algebras, but noncommutative algebras never deform into
a commutative algebras. The fact that commutative algebras have noncommutative deformations plays an important role in physics,
and deformation quantization describes a certain type of deformation of a commutative algebra into a noncommutative one.
\begin{table}[h,t]
\begin{center}
\begin{tabular}{ll}
Codifferential&Structure\\ \hline \\
$d_{74}$&$x\mbox{$\mathbb C$}[x]/(x^5)$\\
$d_{75}(1:1)$&$(x,y)\le \mbox{$\mathbb C$}[x,y]/(x^2-y^2,yx^2,xy^2) $\\
$d_{75}(0:0)=d_{86}(1:1)$&$\mbox{$\mathbb C$}_0\oplus(x,y)\le\mbox{$\mathbb C$}_0\oplus\mbox{$\mathbb C$}[x,y]/(x^2-y^2,xy)$\\
$d_{76}$&$(x,y)\le\mbox{$\mathbb C$}[x,y]/(y^3-x^2,xy)$\\
$d_{79}(1:1)$&$(x,y)\le\mbox{$\mathbb C$}[x,y]/(y^2,x^2y,x^3)$\\
$d_{79}(0:0)=d_{86}(0:0)$&$\mbox{$\mathbb C$}_0^2\oplus x\mbox{$\mathbb C$}[x]/(x^3)$\\
$d_{83}$&$\mbox{$\mathbb C$}_0\oplus x\mbox{$\mathbb C$}[x]/(x^4)$\\
$d_{85}$&$(x,y,z)\le\mbox{$\mathbb C$}[x,y,z]/(x^2-y^2,y^2-yz,xy,xz,z^2)$\\
$d_0$&$\mbox{$\mathbb C$}_0^4$\\
\end{tabular}
\end{center}
\label{Commutative Nilpotent}
\caption{Nilpotent 4-dimensional commutative algebras}
\end{table}
\bigskip
\begin{figure}[htp]%
\begin{center}
\includegraphics[scale=0.45]{commutative.eps
\caption{Deformations between nonnilpotent commutative algebras}%
\label{commutative}%
\end{center}
\end{figure}
\section{Levels of algebras}
It would be difficult to construct a picture showing the jump
deformations for all 72 families of nonnilpotent complex 4-dimensional
algebras, as we did for the unital and commutative algebras, because
there are too many of them. Instead, we give a table showing the
levels of each algebra. To define the level, we say that a rigid
algebra has level 1, an algebra which has only jump deformations to an
algebra on level one has level two and so on. To be on level $k+1$, an
algebra must have a jump deformation to an algebra on level $k$, but no
jump deformations to algebras on a level higher than $k$. For families,
if one algebra in the family has a jump to an element on level $k$,
then we place the the entire family on at least level $k+1$. Thus, even
though generically, elements of the family $d_{37}(p:q)$ deform only to
members of the same family, there is an element in the family which has
a jump to an element on level 4. For the generic element in a family,
we consider it to be on a higher level than the other elements because
it has jump deformations to the other elements in its family.
\begin{table}[h,t]
\begin{center}
\begin{tabular}{ll}
Level&Codifferentials\\ \hline \\
$1$&1,2,3,4,5,10,11,12,13,14,15,16,17,18,19,20,39,53,54,55,56\\
$2$&6,7,21,22,23,24,25,26,27,28,29,30,40,41,42,43,44,57\\
$3$&8,9,31,32,45,46,47,48,58,59,60,61,62,63,64\\
$4$&33,34,35,36,49,50,66,67,69,70\\
$5$&65,68,$37(p:q)$, $38(p:q)$\\
$6$&$37(0:0)$,$38(0:0)
$ 51,52\\
$7$&71,72\\
\\ \hline
\end{tabular}
\end{center}
\label{Levels}
\caption{The levels of the algebras}
\end{table}
\bibliographystyle{amsplain}
|
1,116,691,497,542 | arxiv | \section{Details for Section \ref{sec:dec}}
In this section, we give details of lemmas from
Section \ref{sec:dec}.
\subsection{Equivalence of $\textsf{PS 2.0-rlx}$ and ${\mathsf{LoHoW}}$}
To prove Theorem \ref{thm:eqv},
we show the following: Given a program $\prog$,
starting from the initial machine state $\mathcal{MS}_{\mathsf{init}}
=((J_{\mathsf{init}}, R_{\mathsf{init}}), \mathsf{V}_{\mathsf{init}}, \mathsf{PS}_{\mathsf{init}},
M_{\mathsf{init}}, G_{\mathsf{init}})$
in $\textsf{PS 2.0-rlx}$, we can reach in $\textsf{PS 2.0-rlx}$ the machine state
$\mathcal{MS}_n=((J_n, R_n), \mathsf{V}_{n}, \mathsf{PS}_{n},
M_{n}, G_{n})$ with $\mathsf{PS}_{n}(p)=\emptyset$ for all $p \in \procset$ iff,
starting from an initial ${\mathsf{LoHoW}}$ two phases state $\mathsf{St}_{\mathsf{init}}=({\tt{std}}, p, {\mathfrak{st}}_{init}, {\mathfrak{st}}_{init})$,
we reach the state
$({\tt{std}}, -, ((J_n, R_n),{\mathsf{\bf HW}}_n), -)$, such that ${\mathsf{\bf HW}}_n(x)$ does not contain any memory type of the form $({\tt{prm}},-,p,-)$ or $({\tt{prm}},-,p,-,-)$ for all $x \in \varset$.
The equivalence of the runs follows from the fact that
the sequence of instructions followed in each phase ${\tt{std}}$ and ${\tt{cert}}$ are same in
both $\textsf{PS 2.0-rlx}$ and ${\mathsf{LoHoW}}$
; ${\mathsf{LoHoW}}$ allows lossy transitions which does not affect reachability. Moreover, the ${\mathsf{LoHoW}}$ run satisfies the following invariants.
\noindent{\bf{Invariants for ${\mathsf{\bf HW}}$}}.
The following invariants hold good for ${\mathsf{\bf HW}}(x)$ for all $x \in \varset$.
We then say that ${\mathsf{\bf HW}}(x)$ is faithful to the sub memory $M(x)$ and the view mapping.
\begin{itemize}
\item[(\textbf{Inv1})] For all $x \in \varset$,
${\mathsf{\bf HW}}(x)$ is {\em well-formed} : for each process $\proc \in \procset$, there is a unique
position $i$ in ${\mathsf{\bf HW}}(x)$ having $p$ in its pointer set;
\item[(\textbf{Inv2})] For all $i >
{\mathsf{ptr}}(p,{\mathsf{\bf HW}}(x))$, we have ${\mathsf{\bf HW}}(x)[i] {\notin} \{({\mathsf{msg}}, -, p, -), ({\mathsf{msg}}, -, p, -, -)\}$.
This says that memory types at positions greater than the pointer of $p$
cannot correspond to messages added by $p$ to $M(x)$.
\end{itemize}
\begin{lemma}
The higher order words ${\mathsf{\bf HW}}(x)$ for all $x \in \varset$ appearing in the states of
a ${\mathsf{LoHoW}}$ run satisfy invariants \textbf{Inv1} and
\textbf{Inv2}.
\label{lem:inv}
\end{lemma}
Lemma \ref{lem:inv} can be proved by inducting on the length of a ${\mathsf{LoHoW}}$ run, starting
from the initial states, using the following.
\begin{itemize}
\item For each memory type $({\mathsf{msg}}, v, p, S, -)$ or
$({\mathsf{msg}}, v, p, S)$
in ${\mathsf{\bf HW}}(x)$, there is a message in $M(x)$ which
was added by process $p$, having value $v$.
Similarly, for each memory type $({\tt{prm}}, v, p, S, -)$ or
$({\tt{prm}}, v, p, S)$
in ${\mathsf{\bf HW}}(x)$, there is a promise in $M(x)$ which
was added by process $p$, having value $v$.
\item
The order between memory types in ${\mathsf{\bf HW}}(x)$
and the corresponding
messages in $M(x)$ are the same. That is, for $i < j$, the messages or promises $m, m' \in M(x)$
corresponding to ${\mathsf{\bf HW}}(x)[i]$ and ${\mathsf{\bf HW}}(x)[j]$ are such that
$m.{\textcolor{cobalt}{\tt{to}}} < m'.{\textcolor{cobalt}{\tt{to}}}$.
\item the elements in the pointer set of
a memory type $m$ in ${\mathsf{\bf HW}}(x)$ are exactly the set of processes whose local view
is the ${\textcolor{cobalt}{\tt{to}}}$ stamp of the element of $M(x)$ corresponding to $m$.
\end{itemize}
The base case is easy : the initial two-phases ${\mathsf{LoHoW}}$ state has the same local process states
as the initial $\textsf{PS 2.0}\xspace$ machine state; moreover, the invariants trivially hold, since all process
pointers are at the same position.
For the inductive hypothesis, assume that both invariants
hold in a ${\mathsf{LoHoW}}$ run after $i$ steps. To show that they continue
to hold good after $i+1$ steps, we have to show that for all
${\mathsf{LoHoW}}$ transitions that can be taken after $i$ steps, they are preserved. Assume that the two phases ${\mathsf{LoHoW}}$ state
at the end of $i$ steps is $({\tt{std}}, p, {\mathfrak{st}}, {\mathfrak{st}}')$. The proof for the case when
we have a state $({\tt{cert}}, p, {\mathfrak{st}}, {\mathfrak{st}}')$ after $i$ steps of the ${\mathsf{LoHoW}}$ run is similar.
\begin{itemize}
\item Assume that we have the transition $\xrightarrow[p]{\rd(x,v)}$. Then ${\mathsf{ptr}}(p, {\mathsf{\bf HW}}(x))$
is updated in the resultant state, and so are $(J,R)$, Clearly, the higher order word in the resultant state satisfies both invariants
since the starting state does.
\item Assume that we have the transition
$\xrightarrow[p]{\wt(x,v)}$. Then
we remove $p$ from the pointer set at position $i={\mathsf{ptr}}(p, {\mathsf{\bf HW}}(x))$. A new simple word is added at a position $>i$, or a memory type $({\mathsf{msg}}, v, p, \{p\})$ is added at a position $j>i$, right next to a $\#$, by moving
the memory type at $j$ to position $j-2$.
In either case, the
resultant higher order word satisfies both invariants, since
the starting state does.
\item The update rule $\xrightarrow[p]{{\tt{U}}(x, v_r, v_w)}$
combines the above two cases, by first performing a read and then atomically the write. From the above two cases, the invariants can be seen to hold good in the higher order words in the state obtained after the transition.
\item Consider the Promise rule. In this case, we do not remove $p$ from its pointer set, and only
add the memory type $({\tt{prm}}, v, p, \{\})$ ahead of ${\mathsf{ptr}}(p, {\mathsf{\bf HW}}(x))$. Note that
\textbf{Inv2} only requires that there are no memory types of the form $({\mathsf{msg}}, v, p, S)$
or $({\mathsf{msg}}, v, p, S,-)$ ahead of ${\mathsf{ptr}}(p, {\mathsf{\bf HW}}(x))$. Clearly, both invariants continue to hold.
\item Consider a fulfil rule obtained as a write.
In this case, $p$ is deleted from the position ${\mathsf{ptr}}(p, {\mathsf{\bf HW}}(x))$; and the memory type $({\tt{prm}}, v, p, S)$ (or $({\tt{prm}}, v, p, S.-)$) is replaced with $({\mathsf{msg}}, v, p, S \cup \{p\})$ (or $({\mathsf{msg}}, v, p, S \cup \{p\})$). It is easy to see both invariants holding good.
\item Consider the reservation rule. This
does not affect the invariants since we only tag the last component of a memory type with the process making the reservation.
\item Consider the SC fence rule. If ${\mathsf{ptr}}(p, {\mathsf{\bf HW}}(x)) > {\mathsf{ptr}}(g, {\mathsf{\bf HW}}(x))$, then,
in the resultant word, $p$ is moved to ${\mathsf{ptr}}(g, {\mathsf{\bf HW}}(x))$. The case when ${\mathsf{ptr}}(p, {\mathsf{\bf HW}}(x)) < {\mathsf{ptr}}(g, {\mathsf{\bf HW}}(x))$,
is handled by moving $g$ to ${\mathsf{ptr}}(p, {\mathsf{\bf HW}}(x))$. Since this is the only change in the resultant higher
order words, clearly, both invariants hold good.
\end{itemize}
Notice that the arguments above hold good for both modes $a \in \{{\tt{std}}, {\tt{cert}}\}$.
To prove Theorem \ref{thm:eqv},
we show the following: Given a program $\prog$,
starting from the initial machine state $\mathcal{MS}_{\mathsf{init}}
=((J_{\mathsf{init}}, R_{\mathsf{init}}), \mathsf{V}_{\mathsf{init}}, \mathsf{PS}_{\mathsf{init}},
M_{\mathsf{init}}, G_{\mathsf{init}})$
in $\textsf{PS 2.0-rlx}$, we can reach the machine state
$\mathcal{MS}_n$=$((J_n, R_n), \mathsf{V}_{n}, \mathsf{PS}_{n},
M_{n}, G_{n})$ with $\mathsf{PS}_{n}(p)=\emptyset$ for all $p \in \procset$ iff,
starting from an initial ${\mathsf{LoHoW}}$ two phases state $\mathsf{St}_{\mathsf{init}}=({\tt{std}}, p, {\mathfrak{st}}_{init}, {\mathfrak{st}}_{init})$,
we reach the state
$({\tt{std}}, -, ((J_n, R_n),{\mathsf{\bf HW}}_n), -)$, such that ${\mathsf{\bf HW}}_n(x)$ does not contain any memory type of the form $({\tt{prm}},-,p,-)$ or $({\tt{prm}},-,p,-,-)$ for all $x \in \varset$.
The equivalence of the runs follows from the fact that
the sequence of instructions followed in each phase ${\tt{std}}$ and ${\tt{cert}}$ are same in
both $\textsf{PS 2.0-rlx}$ and ${\mathsf{LoHoW}}$
; ${\mathsf{LoHoW}}$ allows lossy transitions which does not affect reachability. Moreover, the ${\mathsf{LoHoW}}$ run satisfies the following invariants.
\noindent{\bf{Invariants for ${\mathsf{\bf HW}}$}}.
The following invariants hold good for ${\mathsf{\bf HW}}(x)$ for all $x \in \varset$.
We then say that ${\mathsf{\bf HW}}(x)$ is faithful to the sub memory $M(x)$ and the view mapping.
\begin{itemize}
\item[(\textbf{Inv1})] For all $x \in \varset$,
${\mathsf{\bf HW}}(x)$ is {\em well-formed} : for each process $\proc \in \procset$, there is a unique
position $i$ in ${\mathsf{\bf HW}}(x)$ having $p$ in its pointer set;
\item[(\textbf{Inv2})] For all $i >
{\mathsf{ptr}}(p,{\mathsf{\bf HW}}(x))$, we have ${\mathsf{\bf HW}}(x)[i] {\notin} \{({\mathsf{msg}}, -, p, -), ({\mathsf{msg}}, -, p, -, -)\}$.
This says that memory types at positions greater than the pointer of $p$
cannot correspond to messages added by $p$ to $M(x)$.
\end{itemize}
\subsection*{All ${\mathsf{\bf HW}}(x)$ respect Invariants \textbf{Inv1} and
\textbf{Inv2}}
\begin{lemma}
The higher order words ${\mathsf{\bf HW}}(x)$ for all $x \in \varset$ appearing in the states of
a ${\mathsf{LoHoW}}$ run satisfy invariants \textbf{Inv1} and
\textbf{Inv2}.
\label{lem:inv}
\end{lemma}
Lemma \ref{lem:inv} can be proved by inducting on the length of a ${\mathsf{LoHoW}}$ run, starting
from the initial states, using the following.
\begin{itemize}
\item For each memory type $({\mathsf{msg}}, v, p, S, -)$ or
$({\mathsf{msg}}, v, p, S)$
in ${\mathsf{\bf HW}}(x)$, there is a message in $M(x)$ which
was added by process $p$, having value $v$.
Similarly, for each memory type $({\tt{prm}}, v, p, S, -)$ or
$({\tt{prm}}, v, p, S)$
in ${\mathsf{\bf HW}}(x)$, there is a promise in $M(x)$ which
was added by process $p$, having value $v$.
\item
The order between memory types in ${\mathsf{\bf HW}}(x)$
and the corresponding
messages in $M(x)$ are the same. That is, for $i < j$, the messages or promises $m, m' \in M(x)$
corresponding to ${\mathsf{\bf HW}}(x)[i]$ and ${\mathsf{\bf HW}}(x)[j]$ are such that
$m.{\textcolor{cobalt}{\tt{to}}} < m'.{\textcolor{cobalt}{\tt{to}}}$.
\item the elements in the pointer set of
a memory type $m$ in ${\mathsf{\bf HW}}(x)$ are exactly the set of processes whose local view
is the ${\textcolor{cobalt}{\tt{to}}}$ stamp of the element of $M(x)$ corresponding to $m$.
\end{itemize}
The base case is easy : the initial two-phases ${\mathsf{LoHoW}}$ state has the same local process states
as the initial $\textsf{PS 2.0}\xspace$ machine state; moreover, the invariants trivially hold, since all process
pointers are at the same position.
For the inductive hypothesis, assume that both invariants
hold in a ${\mathsf{LoHoW}}$ run after $i$ steps. To show that they continue
to hold good after $i+1$ steps, we have to show that for all
${\mathsf{LoHoW}}$ transitions that can be taken after $i$ steps, they are preserved. Assume that the two phases ${\mathsf{LoHoW}}$ state
at the end of $i$ steps is $({\tt{std}}, p, {\mathfrak{st}}, {\mathfrak{st}}')$. The proof for the case when
we have a state $({\tt{cert}}, p, {\mathfrak{st}}, {\mathfrak{st}}')$ after $i$ steps of the ${\mathsf{LoHoW}}$ run is similar.
\begin{itemize}
\item Assume that we have the transition $\xrightarrow[p]{\rd(x,v)}$. Then ${\mathsf{ptr}}(p, {\mathsf{\bf HW}}(x))$
is updated in the resultant state, and so are $(J,R)$, Clearly, the higher order word in the resultant state satisfies both invariants
since the starting state does.
\item Assume that we have the transition
$\xrightarrow[p]{\wt(x,v)}$. Then
we remove $p$ from the pointer set at position $i={\mathsf{ptr}}(p, {\mathsf{\bf HW}}(x))$. A new simple word is added at a position $>i$, or a memory type $({\mathsf{msg}}, v, p, \{p\})$ is added at a position $j>i$, right next to a $\#$, by moving
the memory type at $j$ to position $j-2$.
In either case, the
resultant higher order word satisfies both invariants, since
the starting state does.
\item The update rule $\xrightarrow[p]{{\tt{U}}(x, v_r, v_w)}$
combines the above two cases, by first performing a read and then atomically the write. From the above two cases, the invariants can be seen to hold good in the higher order words in the state obtained after the transition.
\item Consider the Promise rule. In this case, we do not remove $p$ from its pointer set, and only
add the memory type $({\tt{prm}}, v, p, \{\})$ ahead of ${\mathsf{ptr}}(p, {\mathsf{\bf HW}}(x))$. Note that
\textbf{Inv2} only requires that there are no memory types of the form $({\mathsf{msg}}, v, p, S)$
or $({\mathsf{msg}}, v, p, S,-)$ ahead of ${\mathsf{ptr}}(p, {\mathsf{\bf HW}}(x))$. Clearly, both invariants continue to hold.
\item Consider a fulfil rule obtained as a write.
In this case, $p$ is deleted from the position ${\mathsf{ptr}}(p, {\mathsf{\bf HW}}(x))$; and the memory type $({\tt{prm}}, v, p, S)$ (or $({\tt{prm}}, v, p, S.-)$) is replaced with $({\mathsf{msg}}, v, p, S \cup \{p\})$ (or $({\mathsf{msg}}, v, p, S \cup \{p\})$). It is easy to see both invariants holding good.
\item Consider the reservation rule. This
does not affect the invariants since we only tag the last component of a memory type with the process making the reservation.
\item Consider the SC fence rule. If ${\mathsf{ptr}}(p, {\mathsf{\bf HW}}(x)) > {\mathsf{ptr}}(g, {\mathsf{\bf HW}}(x))$, then,
in the resultant word, $p$ is moved to ${\mathsf{ptr}}(g, {\mathsf{\bf HW}}(x))$. The case when ${\mathsf{ptr}}(p, {\mathsf{\bf HW}}(x)) < {\mathsf{ptr}}(g, {\mathsf{\bf HW}}(x))$,
is handled by moving $g$ to ${\mathsf{ptr}}(p, {\mathsf{\bf HW}}(x))$. Since this is the only change in the resultant higher
order words, clearly, both invariants hold good.
\end{itemize}
Notice that the arguments above hold good in both modes $a \in \{{\tt{std}}, {\tt{cert}}\}$.
\subsection*{Proof of Theorem \ref{thm:eqv}}
To show the equivalence of $\textsf{PS 2.0-rlx}$ and ${\mathsf{LoHoW}}$ we show that the transitions
in each phase of $\textsf{PS 2.0-rlx}$ (standard, certification) is handled
in ${\mathsf{LoHoW}}$ by an appropriate state $({\tt{std}}, -,-, -)$
or $({\tt{cert}}, -, -, -)$, and conversely. The first direction we consider is from $\textsf{PS 2.0-rlx}$ to ${\mathsf{LoHoW}}$.
To see the proof, we consider the four kinds of transitions between phases.
\begin{itemize}
\item Switching from \emph{certification} phase
to the \emph{standard} phase is possible in $\textsf{PS 2.0-rlx}$ only when the
promise set of the process in the certification phase
becomes empty.
Any process can non deterministically
begin the standard phase when the certification
of one process ends successfully.
These conditions are the simulated in ${\mathsf{LoHoW}}$ by
allowing a transition from a two phases state $({\tt{cert}}, p, ((J,R), {\mathsf{\bf HW}}), ((J',R'), {\mathsf{\bf HW}}'))$ to
$({\tt{std}}, q, ((J,R), {\mathsf{\bf HW}}), ((J',R'), {\mathsf{\bf HW}}'))$ only when there are no memory types
$({\tt{prm}}, -, p, -)$ in ${\mathsf{\bf HW}}'$.
\item The switch from \emph{standard} phase to \emph{certification} phase happens in $\textsf{PS 2.0-rlx}$
from a capped memory. This is simulated in ${\mathsf{LoHoW}}$ as follows. When entering the certification phase, ${\mathsf{LoHoW}}$ duplicates the higher
order words.
When the last memory type in any ${\mathsf{\bf HW}}(x)$ is not tagged by the reservation of a process $q \neq p$, the duplicated higher order word
accounts for the capped memory, since we do not allow insertions in between during certification.
When the last memory type in ${\mathsf{\bf HW}}(x)$ is tagged by a reservation
of process $q \neq p$, then we add a new simple word $\# ({\mathsf{msg}}, -, q, \{\})$ at the end of the duplicated higher order
word. This respects the semantics of reservation by a process $q \neq p$.
Thus, the capped memory during certification of $\textsf{PS 2.0-rlx}$ is simulated in ${\mathsf{LoHoW}}$ by disallowing insertions inside a higher order word, and making explicit the reservations of a process.
\item Once we are in a phase an continue in that phase,
the proof in both directions
is done by showing that each instruction simulated
in $\textsf{PS 2.0-rlx}$ can be simulated by the corresponding rule
in ${\mathsf{LoHoW}}$ preserving the invariants, and conversely.
\end{itemize}
The first direction from $\textsf{PS 2.0-rlx}$ to ${\mathsf{LoHoW}}$ is done as follows. For each transition by a process $p$ on an instruction in $\textsf{PS 2.0-rlx}$, we show that we can simulate the same instruction in ${\mathsf{LoHoW}}$.
\begin{enumerate}
\item Consider the read $\rd(x,v)$ rule in $\textsf{PS 2.0-rlx}$.
In ${\mathsf{LoHoW}}$, the read rule updates ${\mathsf{ptr}}(p,{\mathsf{\bf HW}}(x))$ in such a way that
${\mathsf{\bf HW}}(x)$ is faithful to $M(x)$ and the view $\mathsf{V}$. In case the read operation
in $\textsf{PS 2.0-rlx}$ uses a message whose ${\textcolor{cobalt}{\tt{to}}}$ time stamp is not the local view of any process, the corresponding memory type may or may not be
present in ${\mathsf{\bf HW}}(x)$ due to lossiness. Considering the case when this memory type is not lost, it is used exactly in the same manner as the respective message in $\textsf{PS 2.0-rlx}$. {\bf{Rule 1}} from Figure \ref{ps-program_sem} handles this. \\
\item Consider the $\wt(x,v)$ rule in $\textsf{PS 2.0-rlx}$. In ${\mathsf{LoHoW}}$, the write rule either appends memory types or adds simple words
to ${\mathsf{\bf HW}}(x)$ in the \emph{standard} phase, and appends
the memory type at the end of ${\mathsf{\bf HW}}_x$ in a \emph{certification} phase
due to the capping of memory.
${\mathsf{\bf HW}}(x)$ is faithful to $M(x)$ and $\mathsf{V}$ in these simulations. Mapping memory types in ${\mathsf{\bf HW}}(x)$ to $M(x)$,
the relative ordering of the new memory type which gets added with respect to existing memory types in ${\mathsf{\bf HW}}(x)$ is exactly same as the order
the newly added message has, with respect to others in $M(x)$ in either phase.
{\bf{Rules 3,4}} in
Figure \ref{ps-program_sem} handles this.
A $\wt(x,v)$ rule can be done in $\textsf{PS 2.0-rlx}$ during
a \emph{certification} phase by splitting a promise, or
in \emph{standard} phase for the fulfilment of a promise. These cases are handled respectively in ${\mathsf{LoHoW}}$ by (1) inserting a new message immediately preceding a promise in ${\mathsf{\bf HW}}_x$, and (2)replacing
a promise memory type $({\tt{prm}}, -, -, -)$ with a message memory type
$(\msg, -, -, -)$ and updating the pointer of $p$ in each case.
{\bf{Rule 2}} in Figure \ref{ps-program_sem} handles these cases.\\
\item Consider the ${\tt{U}}(x,v_r, v_w)$ rule in $\textsf{PS 2.0-rlx}$. In ${\mathsf{LoHoW}}$,
the RMW rule appends memory types to
a simple word. The memory type corresponding to the
message $m$ in $M(x)$ on which RMW is done, if available in ${\mathsf{\bf HW}}(x)$, will be the rightmost in a simple word (right to a $\#$) in the
\emph{standard} phase, ahead of ${\mathsf{ptr}}(p, {\mathsf{\bf HW}}_x)$, while
in the \emph{certification} phase, this will be the rightmost
symbol in ${\mathsf{\bf HW}}_x$ due to the implementation capped memory.
The memory type which is appended to $\#$ after moving
$m$ to the left of $\#$, corresponds to the new addition, right adjacent
to $m$ in $M(x)$. The append
operation captures the adjacency of the new message added
to $M(x)$ with respect to the one on which RMW is performed.
This results in ${\mathsf{\bf HW}}(x)$ being faithful to $M(x)$ and view $\mathsf{V}$.
{\bf{Rules 8, 9}} in Figure \ref{ps-program_sem} handle these cases.
An ${\tt{U}}(x,v_r, v_w)$ rule can be done in $\textsf{PS 2.0-rlx}$ during
a \emph{certification} phase by splitting a promise, or
in \emph{standard} phase for the fulfilment of a promise. These cases are handled respectively in ${\mathsf{LoHoW}}$ by (1) inserting a new message immediately preceding a promise in ${\mathsf{\bf HW}}_x$, and (2)replacing
a promise memory type $({\tt{prm}}, -, -, -)$ with a message memory type
$(\msg, -, -, -)$ and updating the pointer of $p$ in each case.
{\bf{Rule 10}} in Figure \ref{ps-program_sem} handles these cases.\\
\item Next consider the promise rule in $\textsf{PS 2.0-rlx}$ by a process $p$. Promises take place only in the \emph{standard} phase.
The simulation in ${\mathsf{LoHoW}}$ is similar to the write rule. A new memory type $({\tt{prm}}, v,p, \{\})$ is added to ${\mathsf{\bf HW}}(x)$
at a position $>{\mathsf{ptr}}(p, {\mathsf{\bf HW}}(x))$ with
an empty pointer set. This corresponds to the fact that
the process $p$ which makes the promise has its local view smaller than the ${\textcolor{cobalt}{\tt{to}}}$ time stamp of the promise. Promise memory types
are not lost from ${\mathsf{\bf HW}}(x)$. {\bf{Rule 12}} in Figure \ref{ps-program_sem} handles this.
Notice that When the promise is fulfilled, $p$ is added to the pointer set
of $({\tt{prm}}, v, p,S)$ and the ${\tt{prm}}$ memory type
is replaced with the ${\mathsf{msg}}$ memory type. This corresponds to removing a promise
from the promise set of $P$. As already explained above, {\bf{rules 2, 10}}
in Figure \ref{ps-program_sem} handle this.
Thus, ${\mathsf{\bf HW}}(x)$ is faithful
also to the promise set. If there is a promise which cannot be fulfilled in $\textsf{PS 2.0-rlx}$, the corresponding promise memory type
will stay in ${\mathsf{\bf HW}}(x)$, disallowing to reach a
state $({\tt{std}}, -, -, -)$ in ${\mathsf{LoHoW}}$. \\
\item Let us now look at reservations in $\textsf{PS 2.0-rlx}$. These are done
in the \emph{standard} phase.
The reserve rule done by a process $p$
reserves a timestamp interval
adjacent to an existing message $m$ in $M(x)$.
To simulate this in ${\mathsf{LoHoW}}$, if the memory type corresponding to $m$ is available
in ${\mathsf{\bf HW}}(x)$, then it will be the rightmost in a simple word
of ${\mathsf{\bf HW}}(x)$. The reservation is done by tagging this memory type as a reservation by $p$, thereby blocking this memory type
from participating in any RMW. {\bf{Rule 6}} in Figure \ref{ps-program_sem} handles this.
Similar to splitting promise intervals in a \emph{certification} phase
in $\textsf{PS 2.0-rlx}$, reservation intervals are also allowed to be split
in $\textsf{PS 2.0-rlx}$ during certification.This can happen as part of a write
or an update in $\textsf{PS 2.0-rlx}$.
To simulate this in ${\mathsf{LoHoW}}$, we allow a process $p$ to make use of its reservation.
\noindent
$\bullet$ {\it Splitting a reservation.}
$\mathfrak{ch} \underset{j}{\stackrel{SR}{\hookleftarrow}} m$ is defined only if
$\mathfrak{ch}[j]$ is of the form $(r',v',q,S,p)$. Let $\mathfrak{ch}'$ be the higher order word defined as $\mathsf{del}(\mathfrak{ch},p)$. Then, the extended higher order $\mathfrak{ch} \underset{j}{\stackrel{SR}{\hookleftarrow}} m$ is defined as $\mathfrak{ch}'[1,j-2] \cdot (r',v',q,S)\cdot \# (r,v,p,\{p\},p) \cdot \mathfrak{ch}'[j+1,|\mathfrak{ch}|]$. Observe that the new message $ (r,v,p,\{p\},p)$ is added to the right of the position $j$ which corresponds to the slot that has been reserved by $p$. This special splitting rule will be used during the certification phase. This will allow the process $p$ to use the reserved slots. Recall that it is not allowed to add memory types in the middle of the higher order words (other than the reserved ones) during the certification phase.
This is achieved by removing $p$ from its pointer set and replacing $\#(r', v', q, S, p)$
in ${\mathsf{\bf HW}}_x$ with $(r',v',q,S)\#(r,v,p, \{p\},p)$. {\bf{Rules 5, 11}}
in Figure \ref{ps-program_sem} handle these. \\
\item Cancelling a reservation in $\textsf{PS 2.0-rlx}$ frees up the reserved timestamp interval in $M(x)$.
To simulate this in ${\mathsf{LoHoW}}$, if the corresponding tagged memory type is available in ${\mathsf{\bf HW}}(x)$, then it is unblocked from doing RMW by removing the reserve tag of $p$ from it. {\bf{Rule 7}}
in Figure \ref{ps-program_sem} handles this. \\
\item Finally, SC fence rules in $\textsf{PS 2.0-rlx}$ updates the views
of the performing process to the most recent one.
To simulate this in ${\mathsf{LoHoW}}$, a dummy process $g$ simulating the global view is added. We
update the pointer sets of $p$ (or $g$) depending on
which one is ahead. {\bf{Rule 13}} in Figure \ref{ps-program_sem}
handles this.
\end{enumerate}
Thus, for every run that reaches a consistent state in $\textsf{PS 2.0-rlx}$ with local process states $(J,R)$, there is a run in ${\mathsf{LoHoW}}$ that reaches
a two phases state $({\tt{std}}, -, ((J,R),{\mathfrak{st}}), -)$ following the same sequence of instructions. Note that {\bf{rules 1- 13}} in Figure \ref{ps-program_sem} are mutually non interfering since they apply
to distinct rules and phases. Thus, for each rule in $\textsf{PS 2.0-rlx}$ we have a unique rule in ${\mathsf{LoHoW}}$ from Figure \ref{ps-program_sem} which simulates that while the $\textsf{PS 2.0-rlx}$ is any of the phases, \emph{standard} or
\emph{certification}.
The converse argument from ${\mathsf{LoHoW}}$ to $\textsf{PS 2.0-rlx}$ is similar.
The crucial argument is the memory types in each ${\mathsf{\bf HW}}(x)$ form a
subset of $M(x)$, which has all the ``necessary'' messages (promises, non empty memory types in non redundant simple words).
Lossiness of empty memory types/redundant simple words in ${\mathsf{\bf HW}}(x)$ can be interpreted as messages
which are skipped over, or which have already been used in
$M(x)$. It is easy to see that any sequence of transitions of instructions
in ${\mathsf{LoHoW}}$ can be simulated by exactly the same instruction sequence
in $\textsf{PS 2.0-rlx}$.
\subsection{Proof of Lemma \ref{computing-pre}}
\label{app:pre}
Recall that $\tt{minpre}(c)$ is defined as $\mathtt{min}(\mathtt{Pre}(\upclos{\{c\}}) \cup \upclos{\{c\}})$.
In the following, we show the set $\tt{minpre}(c)$ is effectively computable for any two-phases K-${\mathsf{LoHoW}}$ state $c$.
To do that, we will use a transducer based approach. Lemma \ref{computing-pre} is an immediate consequence of Lemma \ref{min-reg}, Lemma \ref{reg-up}, Lemma \ref{product-trans}, and Lemma
\ref{trans-qs}.
Lemma \ref{reg-up} shows the regularity of $\upclos{\{c\}}$,
Lemma \ref{trans-qs} and \ref{product-trans} show the regularity of
$\mathtt{Pre}(\upclos{\{c\}})$, while Lemma \ref{min-reg} shows the effective computability of $\mathtt{min}(\mathtt{Pre}(\upclos{\{c\}}) \cup \upclos{\{c\}})$.
\smallskip
\noindent
{\bf Finite-state automata}. A finite state automaton $A$ is a tuple $A=(\Sigma_1,P,I,E,F)$, where $\Sigma_1$ is the finite input alphabet, $P$ is a finite set of states,
$I,F\subseteq P$ are subsets of initial and final states, and $E\subseteq P\times\Sigma_1\times P$
is a finite set of transition rules. A word $u=a_1\dots a_n$ is accepted by $A$ if there is a run $p_0 \act{a_1} p_1 \act{a_2} \dots p_{n-1} \act{a_n}p_n$ such that $p_0\in I$, $p_n\in F$ and $(p_{i-1},a_i,p_i)\in E$. We use $L(A)$ to denote the set of words accepted by $A$.
\smallskip
\noindent
{\bf Regular set of two-phases K-${\mathsf{LoHoW}}$-states}
We use an encoding of two-phases K-${\mathsf{LoHoW}}$ states as words over a finite alphabet, and
use this encoding to define a regular set of two-phases K-${\mathsf{LoHoW}}$ states.
Let ${\mathfrak{st}}$ denote $((J,R), {\mathsf{\bf HW}}))$. Consider a two-phases K-${\mathsf{LoHoW}}$ state $c=({\tt{std}}, p, {\mathfrak{st}}, {\mathfrak{st}}')$ or
$({\tt{cert}}, p, {\mathfrak{st}}, {\mathfrak{st}}')$.
Recall that $(J, R)$ gives the local instruction
labels of all processes and the local register values. Assuming
we have locations $x_1, \dots, x_m$, ${\mathsf{\bf HW}}=(\mathfrak{ch}_{x_i})_{1 \leq i \leq m}$.
The state $c$ is encoded
by the word $w={\tt{std}} \$ p \$ J \$R \$_0 \mathfrak{ch}_{x_1}\$_1\dots \mathfrak{ch}_{x_m}\$_m \ddagger J'\$'_0 \mathfrak{ch}'_{x_1} \$' \mathfrak{ch}'_{x_2} \dots
\mathfrak{ch}'_{x_m}\$'_m$ or
${\tt{cert}} \$ p \$ J \$R \$_0 \mathfrak{ch}_{x_1}\$_1\dots \mathfrak{ch}_{x_m}\$_m\ddagger J'\$'_0 \mathfrak{ch}'_{x_1} \$' \mathfrak{ch}'_{x_2} \dots
\mathfrak{ch}'_{x_m}\$'_m$
where
$J$ defines the local state of each process, and
the $\ddagger, \$_i, \$'_i$'s act as delimiters between the contents of the higher order words.
$w$ is denoted $Enc(c)$.
$w$ is a correct encoding, if, on ``decoding'' $w$, we obtain a
unique $decode(w)=({\tt{std}}, p, {\mathfrak{st}}, {\mathfrak{st}}')$ or
$({\tt{cert}}, p, {\mathfrak{st}}, {\mathfrak{st}}')$
where, each $\mathfrak{ch}_x \in (\Sigma^* \# (\Sigma \cup \Gamma))^+$
appearing in ${\mathfrak{st}}$
satisfies the invariants $({\bf{Inv1}})$ and $({\bf{Inv2}})$.
Given a set $R$ of two-phases K-${\mathsf{LoHoW}}$ states,
let $Enc(R)$ represent the set of its word encodings.
We say that a set $R$ of two-phases K-${\mathsf{LoHoW}}$ states is regular if and only if there is a finite state automaton that accepts $Enc(R)$.
\begin{lemma}
Given a regular set $R$ of two-phases K-${\mathsf{LoHoW}}$ states, we can effectively compute $\min{(R)}$.
\label{min-reg}
\end{lemma}
\begin{proof}
Let $A=(\Sigma_1,P,I,E,F)$ be the finite state automaton that accepts $Enc(R)$. The main idea
to effectively compute $\min{(R)}$
is to bound the size of the words accepted by $A$ that encode minimal two-phases K-${\mathsf{LoHoW}}$ states. Observe that the cycles in $A$ can be only labeled by the empty memory type. Otherwise there will be a violation of invariant $({\bf{Inv1}})$. Now consider a word $w$ accepted by $A$. We will first construct another word $w'$ from $w$ such $ decode(w') \sqsubseteq decode(w)$ and the number of $\#e$ where
$e$ is an empty memory type from the subset
$({\mathsf{msg}}, -, \procset, \{\})$ of $\Sigma$
or $({\mathsf{msg}}, -, \procset, \{\},-)$ of $\Gamma$
occurring in $w'$ is polynomially bounded by the size of $A$.
In the following, for convenience, we use macro transitions
on $\#a$ rather than two separate transitions
on $\#$ followed by a transition for $a$.
Let us assume that $w$ is accepted by $A$ using the following run $p_0 \act{\#a_1} p_1 \act{\#a_2} \dots p_{k-1} \act{\#a_k}p_k$.
Let $i_1 <i_2 < \cdots< i_b$ be the maximal sequence of indices such that $a_{i_j}$ is an empty memory type
$\in ({\mathsf{msg}}, -, \procset, \{\})$ or $({\mathsf{msg}}, -, \procset, \{\},-)$.
Now if $b > |P| \cdot |\Sigma_1|$, then there are two indices $i_j$ and $i_\ell$ such that $i_j <i_\ell$, $a_{i_j}=a_{i_{\ell}}$ and $p_{i_{j}-1}=p_{i_{\ell}-1}$. Furthermore, all the symbols occurring between $i_j$ and $i_\ell$ are empty memory types
(from $({\bf{Inv1}})$). This means that $p_0 \act{\#a_1} p_1 \act{\#a_2} \dots p_{i_{j}-1} \act{\#a_{i_j}} p_{i_{\ell}} \cdots p_{k-1} \act{\#a_k}p_k$ is an accepting run of $A$ (accepting the word $w_1$). Furthermore, $ decode(w_1) \sqsubseteq decode(w)$. We can now proceed iteratively on $w_1$ in order to obtain the word $w'$ that is accepted by $A$, $ decode(w') \sqsubseteq decode(w)$, s.t. the number of $\#e$, with $e$ an empty memory type
from $\Sigma \cup \Gamma$
occurring in $w'$ is bounded by $|P| \cdot |\Sigma_1|$. Observe that the
number of $\#b$ where $b$ is a non empty memory type from $\Sigma \cup \Gamma$
occurring in $w'$ is also bounded by $|\procset|$+K+1 : these are
either K promise memory types $({\tt{prm}}, -, -, -)$ or
$({\tt{prm}}, -, -, -,-)$
or those of the form
$({\mathsf{msg}}, -, -, S)$ or
$({\mathsf{msg}}, -, -, S,-)$
where $S \neq \emptyset$). For the latter, we have a bound of $|\procset|+1$. This
comes from ${\bf{Inv1}}$ since each process in $\procset \cup \{g\}$ appears in a unique pointer set.
Thus, the number of $\#e$ where $e \in \Sigma \cup \Gamma$
occurring in $w'$ is polynomially bounded by the size of $A$.
Now from $w'$ we will construct another word $w''$ accepted by $A$ and such that $ decode(w'') \sqsubseteq decode(w')$ and $|w''|$ is polynomially bounded by the size of $A$. Let $\rho:= g_0 \act{\#b_1} g_1 \act{\#b_2} \dots g_{t-1} \act{\#b_t}g_t$ be the run of $A$ accepting $w'$. Let $i_1 <i_2 < \cdots< i_r$ be the maximal sequence of indices such that $b_{i_j} \in \Sigma \cup \Gamma$.
Observe that $r$ is polynomially bounded by the size of $A$ as we have shown previously. Assume $i_0=1$ and $i_{r+1}=t$. Now we can iteratively remove any cycle between two indices $i_{f}$ and $i_{f+1}$ in $\rho$ that is only labeled by empty memory types from
$\Sigma$
to obtain $w''$ satisfying the previous conditions.
\end{proof}
\begin{lemma}
Given a regular set $R$ of K-${\mathsf{LoHoW}}$ states, the set $R \uparrow$ is also regular.
\label{reg-up}
\end{lemma}
\begin{proof}
Let $A=(\Sigma_1,P,I,E,F)$ be the finite state automaton that accepts $Enc(R)$. To construct a finite state automaton $A'$ that accept $Enc(R\uparrow)$, we proceed as follows: The automaton $A'$ is constructed by replacing each macro transition $(p,ba,p') \in E$ labeled by the letter $a \in \Sigma$, $b \neq \#$
by the following macro-transition $(p, e^* \cdot ba \cdot e^* ,p')$ in $A'$, where $e$ is over the empty memory types of $\Sigma$.
Furthermore, any macro transition $(p,\#a,p') \in E$ labeled by the letter $a \in \Sigma \cup \Gamma$
is replaced in $A'$ by the macro-transition $(p, \#a \cdot (w \#b)^*,
p')$, where $w \in \Sigma^*$ is over the empty memory types of $\Sigma$ and $b \in \Sigma \cup \Gamma$ is an empty memory type in $\Sigma \cup \Gamma$. We can also have a loop on
empty memory types of $\Sigma$ on the initial state.
Observe that any macro-transition can be easily translated to a sequence of simple transitions by using extra-intermediary states.
\end{proof}
\smallskip
\noindent{\bf {Rational Transducers}}. A rational transducer $T$ is a non-deterministic finite state automaton which outputs words on each transition.
Formally, a \emph{rational transducer} is a tuple $T=(\Sigma_1,\Sigma_2,Q,I,E,\eta,F)$, where $\Sigma_1,\Sigma_2$ are finite input and output alphabets, $Q$ is a finite set of states,
$I,F\subseteq Q$ are subsets of initial and final states, $E\subseteq Q\times\Sigma_1\times Q$
is a finite set of transition rules, and $\eta:E \rightarrow 2^{\Sigma_2^*}$ is a
function
specifying a regular
language of partial outputs for each
transition rule (i.e., $\eta(e)$ is a regular language for all $e \in E$).
The relation defined by $T$ contains
pairs $(u,v)$ of input and output words,
where $u=a_1\dots a_n$ and $v=v_1\dots v_n$,
for which there is a run
$q_0 \act{a_1 \:|\: v_1} q_1 \act{a_2 \:|\: v_2} \dots q_{n-1} \act{a_n \:|\: v_{n}}q_n$
such that
$q_0\in I$, $q_n\in F$, $(q_{i-1},a_i,q_i)\in E$,
$v_i\in \eta(q_{i-1},a_i,q_i)$.
The set of pairs $(u,v)$ defined by $T$ is denoted $L(T)$.
\begin{lemma}
Given a regular language $R$ (described by a finite-state automaton), we can easily compute a finite state automaton $A$ such that $L(A)= \{u \,|\, (u,v) \in L(T) \,\wedge\, v \in R\}$.
\label{product-trans}
\end{lemma}
\begin{proof}
Trivial.
\end{proof}
\begin{lemma}
It is possible to construct a transducer $T$ that accepts any pair $(Enc(s),Enc(s'))$, with $s$ and $s'$ are two two-phases K-${\mathsf{LoHoW}}$-states, such that $s'$ is reachable from $s$ in one step.
\label{trans-qs}
\end{lemma}
\begin{proof}
Observe that the class of rational transducers are closed under union and therefore it is sufficient to construct the transducer $T$ for each transition rule. Furthermore, we always assume that the input and output tape of the transducer $T$ satisfy the two invariants $({\bf{Inv1}})$ and $({\bf{Inv2}})$ (these can be easily specified as a regular language). The proof is about simulating the rules in the transition system in ${\mathsf{LoHoW}}$ as defined in Section \ref{sec:formal}. We reproduce the rules for easy reference.
\subsection*{The global transition rules in ${\mathsf{LoHoW}}$}
Given $\mathsf{St}=(\pi, p, {\mathfrak{st}}_{{\tt{std}}}, {\mathfrak{st}}_{{\tt{cert}}})$ and $\mathsf{St}'=(\pi', p', {\mathfrak{st}}'_{{\tt{std}}}, {\mathfrak{st}}'_{{\tt{cert}}})$, we have $\mathsf{St} \rightarrow \mathsf{St}'$ iff one of the following cases hold:
\begin{itemize}
\item[(a)] {\bf During the standard phase.} $\pi=\pi'={\tt{std}}$, $p=p'$, ${\mathfrak{st}}_{{\tt{cert}}}={\mathfrak{st}}'_{{\tt{cert}}}$ and ${\mathfrak{st}}_{{\tt{std}}} \xrightarrow[p]{{\tt{std}}}{\mathfrak{st}}'_{{\tt{std}}}$. This corresponds to a simulation of a standard step of the process $p$.
\item[(b)] {\bf During the certification phase.} $\pi=\pi'={\tt{cert}}$, $p=p'$, ${\mathfrak{st}}_{{\tt{std}}}={\mathfrak{st}}'_{{\tt{std}}}$ and ${\mathfrak{st}}_{{\tt{cert}}} \xrightarrow[p]{{\tt{cert}}}{\mathfrak{st}}'_{{\tt{cert}}}$. This corresponds to a simulation of a certification step of the process $p$.
\item[(c)] {\bf From the standard phase to the certification phase.} $\pi={\tt{std}}$, $\pi'={\tt{cert}}$, $p=p'$, ${\mathfrak{st}}_{{\tt{std}}}={\mathfrak{st}}'_{{\tt{std}}}= (({\sf J}, {\sf R}), {\mathsf{\bf HW}})$, and ${\mathfrak{st}}'_{{\tt{cert}}}$ is of the form $(({\sf J}, {\sf R}), {\mathsf{\bf HW}}')$ where for every $x \in \varset$, ${\mathsf{\bf HW}}'(x)={\mathsf{\bf HW}}(x) \# ({\mathsf{msg}},v,q,\{\})$ if ${\mathsf{\bf HW}}(x)$ is of the form $w \cdot \# (-,v,-,-,q)$ with $q \neq p$, and ${\mathsf{\bf HW}}'(x)={\mathsf{\bf HW}}(x)$ otherwise. This corresponds to the copying of the standard ${\mathsf{LoHoW}}$ state to the certification ${\mathsf{LoHoW}}$ state in order to check if the set of promises made by the process $p$ can be fulfilled. The higher order word ${\mathsf{\bf HW}}'(x)$ (at the beginning of the certification phase) is almost the same as ${\mathsf{\bf HW}}(x)$ (at the end of the standard phase) except when the rightmost memory type $(-,v,-,-,q)$ of ${\mathsf{\bf HW}}(x)$ is tagged by a reservation of a process $q \neq p$. In that case, we append the memory type $ ({\mathsf{msg}},v,q,\{\})$ at the end of ${\mathsf{\bf HW}}(x)$ to obtain ${\mathsf{\bf HW}}'(x)$. Note that this is in accordance
to the definition of capping memory before going into certification: to cite, (item 2 in capped memory of \cite{promising2}), a cap message
is added for each location unless it is a reservation made by the process going in for certification.
\subsection*{Copying ${\mathsf{\bf HW}}$ to ${\mathsf{\bf HW}}'$ symbol by symbol}
We can implement copying of ${\mathsf{\bf HW}}$ to ${\mathsf{\bf HW}}'$ by copying symbol by symbol
as follows. Consider any ${\mathsf{\bf HW}}(x)$. Let ${\mathsf{\bf HW}}=(a_x W_x)_{x \in \varset}$ where ${\mathsf{\bf HW}}(x)=a_xW_x \in (\Sigma^* \# (\Sigma \cup \Gamma))^*$, $|a_x|=1$. Define the function $\mathsf{copy}$ on the two phases ${\mathsf{LoHoW}}$ state $({\tt{std}}, p, ((J,R), (a_xW_x)_{x \in \varset}), -)$, and then recursively to subsequent states
until we end up in $({\tt{cert}}, p, ((J,R),{\mathsf{\bf HW}}), ((J,R), {\mathsf{\bf HW}}))$.
The $\mathsf{copy}$ function is defined recursively as follows.
\begin{itemize}
\item[(Base)] $\mathsf{copy}({\tt{std}}, p, ((J,R), (a_xW_x)_{x \in \varset}), -)=
(cc, p, ((J,R), (\overline{a}_x W_x)_{x \in \varset}), ((J,R),(a_x)_{x \in \varset}))$. This is copying the first
symbol of each ${\mathsf{\bf HW}}(x)$. $cc$ is an intermediate phase used only in copying. Notice that the over lined symbol shows the progress of copying, one symbol each time.
\item[(Inter)] Next, we copy subsequent symbols. $\mathsf{copy}(cc, p, ((J,R), (\overline{\alpha} a_x U_x)_{x \in \varset}), ((J, R), (W_x)_{x \in \varset}))$ is defined as
$(cc, p, ((J,R), (\overline{\alpha a}_x U_x)_{x \in \varset}), ((J, R), (W_xa_x)_{x \in \varset}))$.
\item[(Last)] Finally, when all higher order words have been copied, we move from $cc$ to ${\tt{cert}}$. When a higher word has been completely copied, it has the form $\overline{\alpha}$, where $\alpha \in (\Sigma^* \# \Gamma)^+$. Then we define
$\mathsf{copy}(cc, p, ((J,R), (\overline{\alpha}_x)_{x \in \varset}), ((J, R), (W_x)_{x \in \varset}))$ as $({\tt{cert}}, p, ((J,R), (\alpha_x)_{x \in \varset}), ((J, R), (W_x)_{x \in \varset}))$, by removing the overline, and having the phase ${\tt{cert}}$.
\end{itemize}
If the last symbol $a_x$ in ${\mathsf{\bf HW}}(x)$ is of the form $(-,v,-,-,q)$, for $q \neq p$, then $\mathsf{copy}$ appends
$a_x\#({\mathsf{msg}}, v, q, \{\})$ instead of just $a_x$ in (Inter).
\item[(d)] {\bf From the certification phase to standard phase.} $\pi={\tt{cert}}$, $\pi'={\tt{std}}$, ${\mathfrak{st}}_{{\tt{std}}}={\mathfrak{st}}'_{{\tt{std}}}$, ${\mathfrak{st}}_{{\tt{cert}}}={\mathfrak{st}}'_{{\tt{cert}}}$, and ${\mathfrak{st}}_{{\tt{cert}}}$ is of the form $(({\sf J}, {\sf R}), {\mathsf{\bf HW}})$ with ${\mathsf{\bf HW}}(x)$ does not contain any memory type of the form $({\tt{prm}},-,p,-)$/$({\tt{prm}},-,p,-,-)$ for all $x \in \varset$ (i.e., all promises made by $p$ are fulfilled).
\end{itemize}
\paragraph{Description of the Transducer}
We consider 4 cases based on the 4 cases we have
in the transition rules (a)-(d) as above.
\begin{enumerate}
\item We first consider the case when $s$ and $s'$ have the same phase (${\tt{std}}$ or ${\tt{cert}}$). If the location involved in the instruction is $x_i$, then
the transducer copies all $\mathfrak{ch}_{x_j}$, $j \neq i$ as is.
For $\mathfrak{ch}_{x_i}$, if the phase we have
in $Enc(s)$ is ${\tt{std}}$, then the transducer copies $\ddagger$
as well as all symbols after that in the output, while
if the phase we have in $Enc(s)$ is ${\tt{cert}}$, the the transducer copies $\ddagger$
as well as all symbols before that in the output. This is common
to all items below and we will not mention it separately.
\begin{enumerate}
\item Consider a ${\tt{Read}}$ instruction of the form $\lambda: \$r=x_i$ of the process $p$. Then the transducer will first guess the value $v$ that will be read and update the local states of the processes (as an output). The only change that the transducer will do concerns the $i$-th higher order word $\mathfrak{ch}_{x_i}$.
For each symbol that the transducer reads on the input tape of $\mathfrak{ch}_{x_i}$ before $\ddagger$, it outputs the same symbol. Once the symbol pointed by the process $p$ is read on the input tape, the transducer will check the value of each symbol read on the input tape and if it corresponds to $v$, the transducer will non-deterministically add $p$ to its pointer set, otherwise it will output the same read symbol (while removing $p$ from its pointer set, which has bee read, if needed).
\item Consider a ${\tt{Write}}$ instruction of the form $\lambda: x_i= \$r$ of the process $p$. Then the transducer will first update the local states of the processes (as an output). The only change that the transducer will do concerns the $i$-th higher order word $\mathfrak{ch}_{x_i}$.
For each symbol that the transducer reads on the input tape of $\mathfrak{ch}_{x_i}$, it outputs the same symbol. Once the symbol having $p$ in its pointer set is read on the input tape, the transducer will output the same read symbol (while removing $p$ from the pointer set). When the transducer reads a symbol after $\#$, it can decide to output the new message corresponding to the write instruction and after that, go on by outputting any read symbol.
\item The case of RMW is very similar to the case of a ${\tt{Write}}$ instruction of the process $p$.
\item The case of a promise rule is similar to the ${\tt{Write}}$.
The main difference is that when the transducer reads the symbol pointed by the process $p$ on the input tape, the transducer will output the same read symbol (without removing $p$ from the pointer set). When the transducer reads a symbol right after $\#$,
it can decide to output the new promise message, such that the pointer set is empty. After that, it goes on by outputting any read symbol.
\item The case of a reservation rule is similar to RMW.
\item The case of a cancel rule by a process $p$ is as follows.
The transducer reads on symbols and outputs the same, till it finds the symbol $(-, -, -, -, p)$.
On reading this, it outputs $\epsilon$. After that, it goes on by outputting any read symbol.
\item The case of a fulfil rule is as follows. The transducer
outputs what it reads till it finds a symbol having $p$ in its pointer set.
It outputs the same symbol removing $p$ from the pointer set. Then it continues outputting the read symbol till it reads a symbol
$({\tt{prm}}, v, p, S)$. It outputs $({\mathsf{msg}}, v, p, S \cup \{p\})$ by adding $p$
to the pointer set. After that, it goes on by outputting any read symbol.
\item Consider a ${\mathsf{SC\text{-}fence}}$ instruction. In this case the transducer will output any read symbol except the ones that have $g$ or ${p}$ in its pointer set. If $p$ and $g$ are in the same pointer set, then the transducer will continue outputting any read symbol. If the transducer reads the first encountered symbol that contains only $p$ or $g$ in its pointer set, then the transducer will output the same symbol without the pointer set containing either $g$ or $p$. Once the transducer reads the second encountered symbol whose pointer set contains only $p$ or $g$ then the transducer will output the same symbol with the pointer set containing both $g$ and $p$. This is done for each $\mathfrak{ch}_{x_i}$.
\end{enumerate}
\item If the phase in $Enc(s)$ is ${\tt{cert}}$ and that of $Enc(s')$ is ${\tt{std}}$,
then the transducer simply replaces ${\tt{cert}}$ by ${\tt{std}}$, and
the process $p$ by any process $q$, and copies the rest as is in the output.
\item If the phase in $Enc(s)$ is ${\tt{std}}$ and that of $Enc(s')$ is ${\tt{cert}}$,
then the transducer implements the $\mathsf{copy}$ function described
above. Each $\mathsf{copy}$ is implemented by a transducer, and
the final result is obtained by composing all these transducers.
Note that rational transducers are closed under composition,
so it is possible to obtain one rational transducer
that achieves the effect of all the $\mathsf{copy}$ functions, starting
with the ${\tt{std}}$ phase and ending in the ${\tt{cert}}$ phase.
Note that this is easily done, since
in each step, the transducer progressively marks a symbol before $\ddagger$ with overline, and copies the same at the end.
\end{enumerate}
\end{proof}
\subsection{Proof of Lemma \ref{lem:mon}}
\label{app:mono}
Consider K-${\mathsf{LoHoW}}$ states $c_1, c_2$ s.t. $c_1 \rightarrow c_2$,
and let $c_3$ be a state s.t. $c_1 \sqsubseteq c_3$.
We make a case analysis based on the transition chosen.
Let $c_1=({\tt{std}}, p, ((J_1,R_1), {\mathsf{\bf HW}}_1), ((J_2,R_2), {\mathsf{\bf HW}}_2))$,
$c_2=(\pi, q, ((J_3,R_3), {\mathsf{\bf HW}}_3), ((J_4,R_4), {\mathsf{\bf HW}}_4))$,
$c_3=({\tt{std}}, p, ((J_1,R_1), {\mathsf{\bf HW}}_5), ((J_2,R_2), {\mathsf{\bf HW}}_6))$, and
$c_4=(\pi, q, ((J_7,R_7), {\mathsf{\bf HW}}_7), (J_8, R_8), {\mathsf{\bf HW}}_8))$.
The case when $c_1=({\tt{cert}}, p, -, -)$ is similar to the case we discuss here.
\begin{enumerate}
\item Consider the transition $c_1 \xrightarrow[\proc]{{\lambda: \$r=x}} c_2$
by a read instruction $\$r=x$ in process $p$.
Then $\exists k \leq j$, $k={\mathsf{ptr}}(p,{\mathsf{\bf HW}}_1(x))$, and
the memory type at ${\mathsf{\bf HW}}_1(x)[j]$ has the form $(-, v, -, S)$, $v=R(\$r)$.
${\mathsf{\bf HW}}_3(x)$ is obtained by updating ${\mathsf{ptr}}(p,{\mathsf{\bf HW}}_1(x))$ to $j$,
so that $p$ is in the pointer set $S$. Since
$c_1 \sqsubseteq c_3$, there is an increasing function
$f$ from the positions of ${\mathsf{\bf HW}}_1(x)$
to that of ${\mathsf{\bf HW}}_5(x)$ such that
$f(k) \leq f(j)$, ${\mathsf{ptr}}(p,{\mathsf{\bf HW}}_5(x))=f(k)$ in ${\mathsf{\bf HW}}_5(x)$ and
the memory type at $f(j)$ has the form
$(-,v,-,S'')$, $v=R(\$r)$. Indeed, one can update ${\mathsf{ptr}}(p,{\mathsf{\bf HW}}_5(x))$
to $f(j)$, obtaining a state $c_4$ from $c_3$. The local process
states of $c_4$ is same as that of $c_2$.
All higher order words ${\mathsf{\bf HW}}_5(y)$, $y \neq x $ of $c_3$ remain unchanged in $c_4$ (and
all higher order words ${\mathsf{\bf HW}}_1(y)$, $y \neq x $ of $c_1$ remain unchanged in $c_2$), hence
the $\sqsubseteq$ relation holds for these higher order words in $c_2, c_4$.
The same function $f$ between positions of ${\mathsf{\bf HW}}_1(x)$ and
${\mathsf{\bf HW}}_5(x)$ can be used on positions of ${\mathsf{\bf HW}}_3(x)$
of $c_2$ and ${\mathsf{\bf HW}}_7(x)$ of $c_4$ to see that $c_2 \sqsubseteq c_4$ and $c_3 \xrightarrow[\proc]{\lambda: \$r=x} c_4$.
\item Consider the transition $c_1 \xrightarrow[\proc]{{\lambda: x = \$r}} c_2$. Then, there is a position $k$ in ${\mathsf{\bf HW}}_1(x)$ such that
$k={\mathsf{ptr}}(p,{\mathsf{\bf HW}}_1(x))$. Let the memory type at ${\mathsf{\bf HW}}(x)[k]$ be $(-, v_1, -, S_1 \cup \{p\})$.
After the transition, we obtain
${\mathsf{\bf HW}}_3(x)$ such that ${\mathsf{ptr}}(p,{\mathsf{\bf HW}}_3(x)) = j-1 > k$. There are 2 possibilities.
\begin{itemize}
\item[(a)] $j-1, j$ form the positions of the 2 symbols $\#, ({\mathsf{msg}}, R(\$r), p, \{p\})$ in the newly added
simple word in ${\mathsf{\bf HW}}_1(x)$. Figure \ref{wt-order} depicts this case.
Notice that in ${\mathsf{\bf HW}}_1(x)$, $k={\mathsf{ptr}}(p, {\mathsf{\bf HW}}_1(x))$, and positions $j-3, j-2$ represent
the last two positions of a simple word. The new simple word
is added right after this in ${\mathsf{\bf HW}}_3(x)$, at positions $j-1, j$.
\begin{figure}[h]
\includegraphics[scale=.13]{app-proof1.pdf}
\caption{The higher order words in $c_1, c_2, c_3, c_4$ in case(a). The two pink positions correspond to the newly added simple word.
The positions $j-3, j-2$ have $\#$ and $a \in \Sigma \cup \Gamma$
denoting the end of a simple word in ${\mathsf{\bf HW}}_1(x)$, so that a new simple word can be inserted right after. The position $k$
in ${\mathsf{\bf HW}}_1(x)$ is ${\mathsf{ptr}}(p, {\mathsf{\bf HW}}_1(x))$.
${\mathsf{\bf HW}}_1(x) \sqsubseteq {\mathsf{\bf HW}}_5(x)$ witnessed by the increasing function $f$.
${\mathsf{\bf HW}}_3(x), {\mathsf{\bf HW}}_7(x) $ respectively are obtained from ${\mathsf{\bf HW}}_1(x), {\mathsf{\bf HW}}_5(x)$ by the $\wt(x,v)$ transition.
}
\label{wt-order}
\end{figure}
Since $c_1 \sqsubseteq c_3$, let $f$ be an
increasing function from the positions of ${\mathsf{\bf HW}}_1(x)$ to those
of ${\mathsf{\bf HW}}_5(x)$. ${\mathsf{\bf HW}}_7(x)$ is obtained from ${\mathsf{\bf HW}}_5(x)$ by inserting the new simple word right after position $f(j-2)$, at positions $f(j-2)+1, f(j-2)+2$. The position $f(j-1)$
in ${\mathsf{\bf HW}}_5(x)$ is shifted to the right by two positions in ${\mathsf{\bf HW}}_7(x)$. Thus, we can define an increasing function from positions
of ${\mathsf{\bf HW}}_3(x)$ and ${\mathsf{\bf HW}}_7(x)$ as follows.
\begin{itemize}
\item For $i \in \{1, \dots, j-2\}$, $g(i)=f(i)$,
\item $g(j-1)=f(j-2)+1, g(j)=f(j-2)+2$, (note that $g(j-1),g(j)$ are the two new positions in ${\mathsf{\bf HW}}_7(x)$ corresponding to the new positions $j-1, j$ in ${\mathsf{\bf HW}}_3(x)$),
\item For $i \in \{j+1, \dots, n+2\}$, $g(i)=f(i-2)+2$
\end{itemize}
It is easy to see that $g$ is an increasing function between
the positions of ${\mathsf{\bf HW}}_3(x)$ and ${\mathsf{\bf HW}}_7(x)$ : we know that $f(j-2) < f(j-1)$. Hence, $g(j) =f(j-2)+2 < f(j-1)+2 = g(j+1)$.
This also gives ${\mathsf{\bf HW}}_3(x) \sqsubseteq {\mathsf{\bf HW}}_7(x)$.
\item[(b)] $j-1$ is the position obtained by appending to a simple word in ${\mathsf{\bf HW}}_1(x)$.
\begin{figure}[h]
\includegraphics[scale=.13]{app-proof2.pdf}
\caption{The higher order words in $c_1, c_2, c_3, c_4$ in case(b). The pink position in ${\mathsf{\bf HW}}_3(x)$ corresponds to the newly added memory type, right after $\#$ at position $j-3$ in ${\mathsf{\bf HW}}_1(x)$.
$a \in \Sigma$ at position $j-2$ in ${\mathsf{\bf HW}}_1(x)$ is shifted to the
left of $\#$ in ${\mathsf{\bf HW}}_3(x)$. The position $k$
in ${\mathsf{\bf HW}}_1(x)$ is ${\mathsf{ptr}}(p, {\mathsf{\bf HW}}_1(x))$.
${\mathsf{\bf HW}}_1(x) \sqsubseteq {\mathsf{\bf HW}}_5(x)$ witnessed by the increasing function $f$.
${\mathsf{\bf HW}}_3(x), {\mathsf{\bf HW}}_7(x) $ respectively are obtained from ${\mathsf{\bf HW}}_1(x), {\mathsf{\bf HW}}_5(x)$ by the $\wt(x,v)$ transition.
}
\label{wt-orderb}
\end{figure}
\end{itemize}
Figure \ref{wt-orderb} illustrates this case. ${\mathsf{\bf HW}}_1(x) \sqsubseteq {\mathsf{\bf HW}}_5(x)$ is witnessed
by the increasing function $f$. The new memory type is added at position $f(j-2)+1$ (right next to $\#$), and all subsequent symbols
are shifted right by one position.
It is easy to see that ${\mathsf{\bf HW}}_7(x)$ is obtained from ${\mathsf{\bf HW}}_5(x)$
by the $\wt(x,v)$ transition. The increasing function $g$ from the
positions of ${\mathsf{\bf HW}}_3(x)$ to that of ${\mathsf{\bf HW}}_7(x)$ is defined as follows.
\begin{itemize}
\item For $i \in \{1, \dots, j-2\}, g(i)=f(i)$,
\item $g(j-1)=f(j-2)+1$,
\item For $i \in \{j, \dots, n+1\}$, $g(i)=f(i-1)+1$
\end{itemize}
Notice that $g$ is an increasing function: $g(j-2)=f(j-2) < f(j-2)+1=g(j-1)$,
$g(j)=f(j-1)+1 > f(j-2)+1=g(j-1)$, and the same relationship holds for subsequent indices.
\item The case of $c_1 \xrightarrow[\proc]{\lambda: {\mathbf{CAS}}(x_i, \$r_1, \$r_2)} c_2$ is similar to the write.
\item The case of a promise rule is exactly same as the write rule, as
far as monotonicity is concerned.
\item The case of promise fulfilment is trivial for monotonicity,
since we only shift the pointer of $p$, and update ${\tt{prm}}$ to ${\mathsf{msg}}$
in the memory type.
\item The case of reservation follows exactly like case (b)
of the write rule.
\item The case of cancellation is trivial for monotonicity
since the operation does not change the length of the word.
\item The case of $\xrightarrow[\proc]{\mathsf{SC\text{-}fence}}$ is trivial
by using the observation that
the relative ordering of the pointers $p$ and $g$ are same
in ${\mathsf{\bf HW}}_1(x)$ and ${\mathsf{\bf HW}}_5(x)$. ${\mathsf{\bf HW}}_3$ and ${\mathsf{\bf HW}}_7$ are obtained respectively
by moving the pointers of $p, g$ to the rightmost one (whichever it is). So the same
increasing function that was used for ${\mathsf{\bf HW}}_1 \sqsubseteq {\mathsf{\bf HW}}_5$ will work for
${\mathsf{\bf HW}}_3 \sqsubseteq {\mathsf{\bf HW}}_7$.
\end{enumerate}
\newpage
\section{Source to Source Translation and Proof of Correctness}
\subsection{Intuition for the Translation}
\paragraph{$2K$ Timestamps} We bound the number of essential events by $K$. Why do $2K$ timestamps suffice?. Intuitively timestamps are used to determine relative order between the events. We track timestamps of the view-switching messages (messages read by other processes), promises and reservations. For each view-switch there are two timestamps of consequence. The timestamp of the reading process before the read and the timestamp of the message to be read. Hence for each view switch, the comparison operation requires us to maintain two timestamps. For a promises (reservation) we maintain the timestamp of the promise (reservation). We do not explicitly store timestamps of messages that will not view switch. These messages however may be read by the same process that generated them. We keep track of whether the latest write can be read by the same process by using some thread-local state.
\paragraph{$K+n$ Contexts} It suffices to have $K+n$ contexts since we can run the processes in the order in which they generate view-switching messages. In each context, the process only depends on the essential messages generated in previous contexts. If this were not the case we would get a deadlock. We require $n$ additional contexts to initiallize each process.
\subsection{Glossary of Global and Local Variables used in the SC Program}
\label{app:ctc}
We first give a glossary of all the variables used in the code. The list contains variables global to all processes or local to a process. A small description of their role is also mentioned, which serve as invariants.
\newcommand{\appvar}[1]{\lstinline[style=customc,basicstyle=\normalsize\ttfamily]{#1}}
\begin{enumerate}
\item \appvar{numEE} : a global variable, initialized to 0, keeps track of the number of essential events (promises, reservations and view switches) so far. Each time an essenial event occurs, \appvar{numEE} is incremented.
\item \appvar{numContexts} : a global variable, initialized to 0, keeps track of the number of context switches so far. This is used in the translation to SC.
\item \appvar{view[x].v} : a local variable, stores the value of $x \in \varset$ in the local view of the process
\item \appvar{view[x].t} : local variable, stores the time stamp $\in \mathsf{Time}$ of $x \in \varset$ in the local view of the process.
\item \appvar{view[x].l} : local variable, boolean, which is set to true when \appvar{view[x].t} is a valid timestamp, and can be used in comparisons with timestamps of other messages.
\item \appvar{view[x].f} : local variable, boolean. A true value indicates that \appvar{view[x].v} is recent, and can be used for reading locally.
\item \appvar{view[x].u} : local variable, boolean. A true value indicates that the sequence of events starting from the one that resulted in the timestamp \appvar{view[x].t} till the most recent, form a chain of ${\mathbf{CAS}}$ operations on $x$. Whenever a write is published, \appvar{view[x].u} is set to true. \appvar{view[x].u} is set to false on an unpublished write. On a sequence of ${\mathbf{CAS}}$ operations, \appvar{view[x].u} is left unchanged.
\item \appvar{checkMode} : local variable, boolean. Set to true when the process is in certification phase, which means the process is making and certifying promises.
\item \appvar{liveChain[x]} : local variable, for each $x \in \varset$, boolean. Can be true only when \appvar{checkMode} is true. A true value represents that the last write done while the process is in certification phase is not a published promise message.
\item \appvar{extView[x]} : local variable, for each $x \in \varset$, boolean. A true value represents that the local value \appvar{view[x].v} of the process comes from a message generated external to the certification phase.
\item \appvar{blockPromise[x]} : a global boolean array, which for each $x \in \varset$ stores whether promises should be blocked on variable $x$. This is used in the case of ${\tt{ra}}$ writes when we cannot have promises on the same variable later (refer to \textsf{PS 2.0}\xspace, ${\tt{ra}}$ accesses).
\item \appvar{avail[x][t]} : for each $x \in \varset$, a global boolean array of length $2K+1$ corresponding to the $2K+1$ time stamps, checks availability of a time stamp on a fresh write.
\item \appvar{usedReservations[x][t]} : denotes whether the reservation on variable $x$ with timestamp $t$ has been used by the process during the certification check. If this not true, the reservation will be cancelled.
\item \appvar{reserv[x][t]} : denotes whether the reservation following timestamp t on variable $x$ has been claimed, and if so which process has claimed it.
\item \appvar{upd[x][t]} : for each $x \in \varset$, a global boolean array of length
$2K+1$ corresponding to the $2K+1$ time stamps, checks whether a certain timestamp has been used to read in a ${\mathbf{CAS}}$.
\item \appvar{globalTimeMap[x]}
: global variable, for each $x \in \varset$, stores
a time stamp $\in \mathsf{Time}$. This is used for simulating SC Fences where this functions as the $G$ timemap from \textsf{PS 2.0}\xspace.
\item \appvar{messageStore} : This is an array of messages, where each message is of type $\mathsf{Message}$ as described in the main paper. The length of the array is $K$, the bound on the number of promises + view switches.
\item \appvar{messagesUsed} : a number from 0 to $K$ which keeps track of the number of populated messages in $\mathit{messageStore}$.
\item \appvar{messageNum} : a number from 0 to $K$ which chooses a number from the available free cells in \appvar{messageStore}.
\end{enumerate}
In addition, the \appvar{message} object stores the following data:
\begin{enumerate}
\item \appvar{mess.var} is the shared variable on which the message has been generated
\item \appvar{mess.t[x]} stores for each $x \in \varset$ the timestamp of $x$ in the view object stored in the message
\item \appvar{mess.l[x]} stores for each variable $x \in \varset$, a boolean signifying whether the corresponding timestamp stored in \appvar{mess.t[x]} was one of the exact timestamps $\in \{0 ... K\}$ or an abstract timestamp.
\item \appvar{mess.val} stores the value of the message
\item \appvar{mess.flag} stores the promise state of the message, that is whether (1) it is has been fulfilled/is not a promise (2) if it is a promise then the process that it belongs to. \appvar{mess.flag} takes values from \appvar{0, -1, PIDs}. If it is a simple message (not a promise), \appvar{mess.flag = 0}. If it is a promise, \appvar{mess.flag} is set to the \appvar{PID} of the process which has made the promise. \appvar{mess.flag} is set to \appvar{-1} when the process has temporarily certified it in the current certification phase but will be reset tp \appvar{PID} after exiting the certification phase.
\end{enumerate}
Next we discuss the context switching modules.
\subsection{Context Switching Modules}
\paragraph{$\mathsf{CSI}$ Context-Switch-In}
The $\mathsf{CSI}$ module switches the process into context by setting \appvar{active} to true and incrementing \appvar{numContexts}. Finally we check \appvar{numContexts} does not exceed the context switch bound.
\lstinputlisting[caption=$\mathsf{CSI}$,style=customc,mathescape]{ps-csi.c}
\paragraph{$\mathsf{CSO}$ Context-Switch-Out}
The $\mathsf{CSO}$ module has two functions- (1) moving the process from normal to check mode and (2) switching the process out of context. When a process enters the CSO block, with \appvar{checkMode} set to false, it enters the `if' branch on line 2, sets \appvar{checkMode} to true and saves the return label (of the current instruction pointer) in \appvar{retAddr} and saves the process state before entering check mode (lines 9-10). This ensures that the process returns to the current instruction after the consistency check. Now after the consistency check phase the process switches out of context. At this point, \appvar{checkMode} is \appvar{true}, and hence the process enters the `else' branch on line 13. Consequently, we check whether there are no outstanding uncertified promises for the process (line 15). All the promises that have been certified are reset to belong to the process by setting \appvar{mess.flag} to the PID (lines 16-18). Then it is checked that there are no uncertified splitting insertions, by ensuring that \appvar{liveChain[x]} is not true (lines 20-22). Finally we check for unused reservations during ceritification and cancel them (lines 23-30). Once these checks for cnsistent configuration are complete, we reload the saved state from before the consistency check phase and reload the return address from \appvar{retAddr}. Then we move control to the instruction label in \appvar{retAddr}. After returning control to \appvar{label}, we set \appvar{checkMode} and \appvar{active} to false and exit context.
\lstinputlisting[caption=$\mathsf{CSO}$,multicols=2,style=customc,mathescape]{ps-cso.c}
\paragraph{\appvar{loadState} and \appvar{saveState} subroutines}
The \appvar{saveState} subroutine copies the local state of the calling process and the global state into a what we refer to as `copy' variables. We note that it does not however copy \appvar{numEE}, \appvar{reserv[x][t]} and contents of \appvar{messageStore}. The reason for this being, the promises the process makes in check mode are retained even after exiting check mode is made false. Hence the increments made to \appvar{numEE} and the messages added to \appvar{messageStore} should be maintained even after exiting check mode. This is even true for reservations, which are marked in \appvar{reserv[x][t]}, which are maintained evef after the process exits check mode.
Analogously in \appvar{loadState}, we load the contents of the (saved) `copy variables' into their original counterparts. Another subtle point to be noted is that when the process publishes a message (as a promise) when \appvar{checkMode} is true, we also update the `copy' variables corresponding to \appvar{avail[x][t]}. This is done so that when the process returns to normal mode, the changes are reflected in their original counterparts (which is essential since promise messages are maintained beyond the time \appvar{checkMode} is false and hence their timestamps must be unavailable).
\subsection{Reads}
We provide the translation codes for reads of both access types, ${\tt{rlx}}$ and ${\tt{ra}}$. We will first explain with respect to ${\tt{rlx}}$ access reads.
\paragraph{${\tt{rlx}}$ reads}
The read can be one of two types, view switching, in which a message from \appvar{messageStore} is acquired or a non view-switching (local) read. We guess non-deterministically, one amongst these.
In case of a local read (line 2), the process checks that the local value is usable (line 3) by checking \appvar{view[x].f} which denotes whether \appvar{view[x].v} is a valid value which can be read. It then loads its local value \appvar{view[x].v} into $\reg$. The local value may become unusable if the process crosses an SC-fence which increases its \appvar{view[x].t} (see $\mathsf{SC\text{-}fence}$).
In the case of a view-switching read (line 6), we check that we have not reached the essential-event bound $K$ (line 7). We ensure that \appvar{liveChain[x]} is false before the read in order to forbid additive insertions when checking consistency. Recall from the \appvar{liveChain} invariant that \appvar{liveChain[x]} is true only when the process is in certification mode and the last write on $x$ was neither published as a promise message nor was it certified with a reservation. Reading a message from the memory when $\mathit{liveChain[x]}$ is true implies additive insertion during certification, as illustrated by the following example.
\textit{liveChain}
Assume the process is in the promise certification mode, with $\mathit{view[x].t}$ set to $t_1$, and let the first write use a timestamp $t_2 > t_1$ with the message not published as promise, with $\mathit{liveChain}[x]$ as true. Now the instruction a:=x uses a message in the memory with a timestamp $t_3 \geq t_2$.
\setlength{\columnsep}{7pt}
\setlength{\intextsep}{7pt}
\begin{wrapfigure}{r}{3cm}
\fcolorbox{black}{yellow!20}{
\begin{tabular}{ll}
x:=1; & // $t_2$ \\
a:=x; & // $t_3$ \\
x:=2; & // $t_3+1$\\
\end{tabular}}
\end{wrapfigure}
If the next write certifies a promise message, the interval in the message will be $t_3+1$, since \appvar{liveChain[x]} is true. This results in two writes during the certification, with non-adjacent timestamps $t_2, t_3+1$, with \textit{only} the latter being promised. This behaviour is forbidded in \textsf{PS 2.0}\xspace due to capped memories. Notice that if the earlier write also resulted in a promise message then we do not have additive insertion (since both are promised) and the read with timestamp $t_2$ is allowed since \appvar{liveChain[x]} is false.
Finally a new message is fetched from \appvar{messageStore} with a larger timestamp that the one in the current view (lines 8-11), the process view is updated to include that new message. Whenever a process makes a global read during check mode, it must reads from a message which has been created outside its current certification phase. Hence, \appvar{extView[x]} will be set to true (see \appvar{extView} invariant in the glossary).
\lstinputlisting[caption=$\texttt{read}_{\tt{rlx}}$,multicols=2,style=customc]{ps-read-rlx.c}
\paragraph{${\tt{ra}}$ reads}. This case is almost similar to the earlier and hence only state the point of difference. The main difference is that due to ${\tt{ra}}$ access, we merge (take the join of) all the timestamps rather than just $x$ as we did for ${\tt{rlx}}$.
\lstinputlisting[caption=$\texttt{read}_{\tt{ra}}$,multicols=2,style=customc]{ps-read-ra.c}
\subsection{Writes}
We now provied the translation of a write instruction $x=\reg$ of process. Once again we simulate two access modes, ${\tt{rlx}}$ and ${\tt{ra}}$. we first describe the relaxed mode and then discuss the changes for the ${\tt{ra}}$ mode.
\paragraph{${\tt{rlx}}$ writes} \textit{When in normal mode} \newline
Let us first consider execution in the normal phase (i.e., when \appvar{checkMode} is false). The value of $\mathit{val}(\reg)$ is recorded in the local view, \appvar{view[x].v} and \appvar{view[x].f} is set to true meaning that the value in \appvar{view[x].v} is a valid value and can be read from.
Then, we non-deterministically choose one of three possibilities for the write:
it either (i) is not assigned a fresh timestamp,
(ii) is assigned a fresh timestamp,
(iii) fulfils some outstanding promise. These nondeterministic branches are given on lines 5, 24 and 60 of the code.
\lstinputlisting[caption=$\texttt{write}_{\tt{rlx}}$,multicols=2,style=customc]{ps-write-rlx.c}
In case (i), no message is created, and \appvar{view[x].l} is set to false, signifying that the timestamp recorded in the view does not correspond to the most recent write to $x$ and should therefore not be used in the comparisons. The `if' branch on line 7 is not taken \appvar{checkMode} is false.
In case (ii), since in this case, the timestamp in the view is by definition valid, we set \appvar{view[x].l} to true (line 25). Since the write is relaxed, the message generated will only store the timestamp on the variable written to (i.e. $x$) and 0 for all other variables (line 27-30). Now we allocate a new timestamp to the write. Since we are in normal mode, \appvar{liveChain[x]} is false (see \appvar{liveChain} invariant in glossary). Thus we choose a timestamp nondeterministically (line 36) and store it into \appvar{view[x].t}. We use the \appvar{avail[x][.]} array to ensure that allocated timestamps are unique: (1) we check that the selected timestamp is available (i.e., not allocated) on line 40, and remove it from the array of available stamps (line 41).
Now this message can either be published (for cnsumption by another thread) or not. In the former case, the appropriate message is constructed with \appvar{newView}, \appvar{newViewL}. Note that the last component of the message stores the flag \appvar{mess.flag}. This flag is set to false since the message is not a promise (see \appvar{mess.flag} invariant in glossary). In the latter case non of this is done (`else' branch on line 55). The \appvar{assume(!checkMode)} is satisfied.
In case (iii) Finally, if the process decides to fulfill a promise, a message is fetched from \appvar{messageStore} and checked to be an unfulfilled promise by the current process (checking \appvar{flag == p} on line 68), and \appvar{mess.flag} is set to $0$ and message reinserted into \appvar{messageStore}. Additionally we set \appvar{extView[x]} to true maintaining the \appvar{extView} invariant.
\paragraph{${\tt{rlx}}$ writes} \textit{When in check mode} \newline
Let us now consider a write executing in the certification phase (i.e., when \appvar{checkMode} is true). We will only highlight differences between the normal and certification phase writes.
In case (i), that is when a fresh timestamp is not assigned, the write is certified either by deferring certification to a promise by using splitting insertion (line 9) or by the a presence of a reservation (line 15). In the case where, \appvar{liveChain[x]} is already true (line 7), certification for the current sequence of writes is already deferredand hence we do none of the two. While certification by either of splitting/reservation we nondeterministically choose an timestamp \appvar{t} after which the current write occurs (line 12). We note that this is not the timestamp of the write itself, but specifies between which two timestamps from $\mathsf{Time}$ the write occurs. If we rely on splitting insertion (line 9), we set \appvar{liveChain[x]} to true, and In case of certification by reservation we reserve an interval adjacent to the timestamp \appvar{t} (line 19) after ensuring that it is available (line 18). Finally since this reservation has been used in some certification, we mark this fact (line 20).
In cases (ii), the write is assigned a timestamp from $\mathsf{Time}$ and hence consequently published as a promise. We allocate a fresh timestamp and store it into \appvar{view[x].t}. The most important point to note is that we maintain and use the \appvar{liveChain} invariant whenever a fresh timestamp is assigned.
Indeed, if \appvar{liveChain} is true, the process must assign consecutive timestamps, otherwise it can non-deterministically choose any timestamp greater than \appvar{view[x].t} (line 32-37). Additionally, when generating a message, the \appvar{mess.flag} is set to \appvar{-1} denoting that the message is promise but has been certified and publish the message. We also increment \appvar{numEE} (line 48) as a promise is an essential event.
In case (iii) we fulfill an older promise, and thus first retrieve an uncertified promise belonging to the current process (\appvar{mess.flag == PID}) from \appvar{messageStore} (line 68). The main difference with the normal mode is that we set \appvar{mess.flag} to \appvar{-1} signifying that the promise is (temporarily) certified but not fulfilled. We set the \appvar{extView[x]} to false signifying that the processes' view has come from \appvar{checkMode} and hence is not external.
\lstinputlisting[caption=$\texttt{write}_{\tt{ra}}$,multicols=2,style=customc]{ps-write-ra.c}
\paragraph{${\tt{ra}}$ writes}
The ${\tt{ra}}$ writes have some minor differences w.r.t ${\tt{rlx}}$. Firstly, the timestamps for all variables \appvar{view[x][t]} are added to the published messages, (lines 27-30). Next we set \appvar{blockPromise[x]} to true signifying that henceforth there cannot be any promises on $x$ (refer to \textsf{PS 2.0}\xspace, ${\tt{ra}}$ accesses). This also implies that cases (ii) and (iii) (generating new promises and certifying earlier promises) is not possible for ${\tt{ra}}$ writes as enforced on (line 23). Note that \appvar{blockPromise[x]} is also assumed to be false in ${\tt{rlx}}$ writes when either generating new promises (ii) or certifying earlier ones (iii).
\subsection{${\mathbf{CAS}}$ operations}
We only provide code for the ${\mathbf{CAS}}({\tt{rlx}},{\tt{rlx}})$ variant since the others are implemented similarly, carrying over the access dependent changes from the corresponding \texttt{read} and \texttt{write} codes. ${\mathbf{CAS}}$ is bootstrapping a $\texttt{read}$ and $\texttt{write}$, additioanlly enforcing that the timestamps are consecutive.
\lstinputlisting[caption=${\mathbf{CAS}}$,multicols=2,style=customc]{ps-write-rlx.c}
\subsection{Fences}
\paragraph{$\mathsf{SC\text{-}fence}$} The $\mathsf{SC\text{-}fence}$ command essentially merges the thread local view with the globally stored view in \appvar{globalTimeMap}. For each shared variable $x$ we do the following. On line 3 we check whether the globally stored view \appvar{globalTimeMap[x]} is greater than the process local view. if that is the case, we increase the process-local view \appvar{view[x].t} to the globally stored view. Additionally, we set \appvar{view[x].f} to false since, the value in \appvar{view[x].val} is no more valid (cannot be read from again, since the process timestamp has increased). In the order case, (line 8), we raise the \appvar{globalTimeMap[x]} either to \appvar{view[x].t} (if it is valid, checked by line 9) or to the next higher timestamp, \appvar{view[x].t + 1}.
\lstinputlisting[caption=$\mathsf{SC\text{-}fence}$,multicols=2,style=customc,mathescape]{ps-fence.c}
\input{trans-corr.tex}
\newpage
\section{Complete Experimental Results}
We report the results of experiments we have performed with $\mathsf{PS2SC}$. We have two objectives: (1) studying the performance of $\mathsf{PS2SC}$~ on benchmarks which are unsafe only with promises and (2) comparing $\mathsf{PS2SC}$~ with other model checkers when operating in the promise free mode.
In the first case, we show that $\mathsf{PS2SC}$~ is able to uncover bugs in examples with low interaction with the shared memory. When this interaction increases, however, $\mathsf{PS2SC}$~ performs poorly, owing to the huge non-determinism required by \textsf{PS 2.0}\xspace. However, with partial promises, $\mathsf{PS2SC}$~ is once again able to uncover bugs in reasonable amounts of time.
In the second case, our observations highlight the ability to detect hard to find bugs with small $K$ for unsafe benchmarks, and scalability by altering $K$ as discussed earlier in case of safe benchmarks. We compare $\mathsf{PS2SC}$ with three state-of-the-art stateless model checking tools, $\cdsc$ \cite{cdsc}, $\genmc$ \cite{genmc} and $\rcmc$ \cite{rcmc} that support the promise-free subset of the \textsf{PS 2.0}\xspace semantics.
We now report results of all the experiments we have performed with $\mathsf{PS2SC}$. In the tables that follow we
provide the value of $K$ used (for our tool only). We also specify the value of $L$ used (for all tools).
We do not consider compilation time for any tool while reporting the results. For our tool, the time reported is the time taken by the CBMC backend for analysis. The timeout used is 1 hour for all benchmarks. All experiments are conducted on a machine equipped with a 3.00 GHz Intel Core i5-3330 CPU and 8GB RAM running a Ubuntu 16 64-bit operating system. We denote timeout by `TO', and memory limit exceeded `MLE'.
\subsection{Experimenting with Promises}
In this section we experiment with $\mathsf{PS2SC}$~ in the \textit{promise-enabled} mode.
\paragraph{Litmus Tests}
We first test the tool on a number of litmus tests obtained from various sources. This has two objectives: (a) to perform sanity checks on the correctness of the tool (b) to gain an understanding of the causes of performance bottlenecks when handling promises. The results of these tests are summarized in Table \ref{tab:prom0} below. We tested $\mathsf{PS2SC}$~ on many litmus tests from \cite{promising,promising2,weakestmo,Svendsen:2018}. In these $\mathsf{PS2SC}$~ terminated with the correct result within one minute, with the value of $K$
used for the unsafe trace being atmost 5. We also tested $\mathsf{PS2SC}$~ on the Java Causality Tests of Pugh \cite{jmm}, which were also experimented on in \citet{mrder}. In these too we were able to verify most examples within one minute. However, $\mathsf{PS2SC}$~ timed out (TO = 30 mins) on two tests.
\begin{table}[!htb]
\small
\begin{minipage}{0.5\textwidth}\centering
\begin{tabular}{cccc}
\hline
\textbf{testcase} & $K$ & \textbf{$\mathsf{PS2SC}$}
\\
\hline\hline
ARM\_weak & 4 & 0.765s \\
Upd-Stuck & 4 & 1.252s \\
split & 4 & 25.737s \\ \hline
LB & 3 & 1.469s \\
LBd & 3 & 1.481s \\
LBfd & 3 & 1.512s \\
LBcu & 4 & 5.253s \\
LB2cu & 4 & 5.748s \\ \hline
CYC & 5 & 1.967s \\
Coh-CYC & 5 & 42.67s
\end{tabular}
\end{minipage}\begin{minipage}{0.5\textwidth}\centering
\begin{tabular}{cccc}
\hline
\textbf{testcase} & Testcase-Safety & $K$ & \textbf{$\mathsf{PS2SC}$}
\\
\hline\hline
Pugh2 & Unsafe & 3 & 13.725s \\
Pugh3 & Unsafe & 3 & 12.920s \\
Pugh6 & Unsafe & 3 & 0.360s \\
Pugh8 & Unsafe & 3 & 1.67s \\ \hline
Pugh4 & Safe & 5 & 3.244s \\
Pugh5 & Safe & 5 & 4.811s \\
Pugh10 & Safe & 5 & 3.868s \\
Pugh13 & Safe & 5 & 3.345s \\ \hline
Pugh14 & - & 3 & TO \\
Pugh15 & - & 3 & TO \\
\end{tabular}
\end{minipage}
\caption{Performance of $\mathsf{PS2SC}$~ on \textsf{PS 2.0}\xspace~ idioms}
\label{tab:prom0}
\end{table}
\vspace{-0.6cm}
\paragraph{Modular Promises}
In this section we ask whether the source-to-source translation technique can effectively scale while handling promises for \textsf{PS 2.0}\xspace. In conclusion, we note that our approach performs well on programs requiring limited global memory interaction. When this interaction increases $\mathsf{PS2SC}$~ times out, owing to the huge non-determinism of \textsf{PS 2.0}\xspace. However, the modular approach of partial-promises enables us to recover effective verification.
\begin{table}[h]
\small
\begin{tabular}{cccc}
\hline
\textbf{testcase} & $K$ & \textbf{$\mathsf{PS2SC}$}[1p]
\\
\hline\hline
fib\_global\_2 & 4 & 55.972s \\
fib\_global\_3 & 4 & 2m4s \\
fib\_global\_4 & 4 & 4m20s \\ \hline
exp\_global\_1 & 4 & 19m37s \\
exp\_global\_2 & 4 & 41m12s \\ \hline
tri\_global\_2 & 4 & 52.973s \\
tri\_global\_3 & 4 & 1m57s \\
tri\_global\_4 & 4 & 3m58s \\ \hline
\end{tabular}
\caption{Performance of $\mathsf{PS2SC}$~ on cases with global update}
\end{table}
\vspace{-0.6cm}
\subsection{Comparing Performance with Other Tools}
\begin{table}[h!]
\small
\begin{tabular}{ccccccc}
\hline
\textbf{benchmark} & $L$ & $K$ & \textbf{$\mathsf{PS2SC}$} & \textbf{CDSChecker} & \textbf{GenMC} & \textbf{RCMC} \\ \hline\hline
exponential\_5\_unsafe & 10 & 10 & 1.312s & 0.900s & 0.135s & 6.692s \\
exponential\_10\_unsafe & 10 & 10 & 1.854s & 1.921s & 0.367s & 3m41s \\
exponential\_25\_unsafe & 25 & 10 & 3.532s & 7.239s & 3.736s & TO \\
exponential\_50\_unsafe & 50 & 10 & 6.128s & 36.361s & 39.920s & TO \\
exponential\_70\_unsafe & 10 & 10 & 9.509s & 1m33s & 2m29s & TO \\
\hline
fibonacci\_2\_unsafe & 2 & 20 & 2.746s & 2.332s & 0.084s & 0.086s \\
fibonacci\_3\_unsafe & 3 & 20 & 9.392s & 46m8s & 0.462s & 0.544s \\
fibonacci\_4\_unsafe & 4 & 20 & 34.019s & TO & 12.437s & 18.953s \\ \hline
fibonacci\_2\_safe & 2 & 20 & 6.454s & 8.900s & 0.096s & 0.162s \\
fibonacci\_3\_safe & 3 & 20 & 30.936s & TO & 0.910s & 3.884s \\
fibonacci\_4\_safe & 4 & 20 & 2m16s & TO & 1.140s & 2m36s \\ \hline
\end{tabular}
\caption{Comparison of performance on a set of parameterized benchmarks}
\vspace{-0.6cm}
\end{table}
\begin{table}[h]
\vspace{0.6cm}
\small
\begin{tabular}{ccccccc}
\hline
\textbf{benchmark} & $L$ & $K$ & \textbf{$\mathsf{PS2SC}$} & \textbf{CDSChecker} & \textbf{GenMC} & \textbf{RCMC} \\ \hline\hline
hehner2\_unsafe & 4 & 5 & 7.207s & 0.033s & 0.094s & 0.087s \\
hehner3\_unsafe & 4 & 5 & 28.345s & 0.036s & 2m53s & 1m13s \\ \hline
linuxlocks2\_unsafe & 2 & 4 & 0.547s & 0.032s & 0.073s & 0.078s \\
linuxlocks3\_unsafe & 2 & 4 & 1.031s & 0.031s & 0.083s & 0.081s \\ \hline
queue\_2\_safe & 4 & 4 & 0.180s & 0.031s & 0.082s & 0.085s \\
queue\_3\_safe & 4 & 4 & 0.347s & 0.037s & 0.090s & 0.092s \\ \hline
\end{tabular}
\caption{Comparison of performance on concurrent data structures based benchmarks}
\vspace{-0.6cm}
\end{table}
\begin{table}[h]
\small
\begin{tabular}{ccccccc}
\hline
\textbf{benchmark} & $L$ & $K$ & $\mathsf{PS2SC}$ & \textbf{CDSChecker} & \textbf{GenMC}& \textbf{RCMC} \\ \hline\hline
readerwriter\_7 & 0 & 5 & 0.719s & 0.005s & 0.057s & 0.690s \\
readerwriter\_8 & 0 & 5 & 0.839s & 0.006s & 0.056s & 7.425s \\
readerwriter\_9 & 0 & 5 & 1.068s & 0.007s & 0.053s & 1m17s \\
readerwriter\_10 & 0 & 5 & 1.393s
& 0.007s & 0.056s & 14m49s \\ \hline
redundant\_co\_10 & 10 & 5 & 0.470s & 0.114s & 0.087s & 38m12s \\
redundant\_co\_20 & 20 & 5 & 1.031s & 0.548s & 0.218s & TO \\
redundant\_co\_50 & 50 & 5 & 3.219s & 8.965s & 4.143s & TO \\
redundant\_co\_70 & 70 & 5 & 6.093s & 13.843s & 18.185s & TO \\\hline
\end{tabular}
\caption{Evaluation using two synthetic safe benchmarks. We note that the value of $K$ is chosen to be large enough to consider all executions.}
\end{table}
\begin{table}[h]
\small
\begin{tabular}{ccccccc}
\hline
\textbf{benchmark} & $L$ & $K$ & \textbf{$\mathsf{PS2SC}$} & \textbf{CDSChecker} & \textbf{GenMC} & \textbf{RCMC} \\ \hline\hline
peterson1U(4) & 1 & 6 & 1.408s & 0.039s & TO & 9.129s \\
peterson1U(6) & 1 & 6 & 7.286s & 0.010s & TO & TO \\
peterson1U(8) & 1 & 6 & 47.786s & TO & TO & TO \\
peterson1U(10) & 1 & 6 & 4m19s & TO & TO & TO \\
\hline
szymanski1U(4) & 1 & 2 & 1.015s & 0.043s & MLE & TO \\
szymanski1U(6) & 1 & 2 & 2.771s & TO & MLE & TO \\
szymanski1U(8) & 1 & 2 & 6.176s & TO & TO & TO \\
szymanski1U(10) & 1 & 2 & 12.203s & TO & TO & TO \\
\hline
\end{tabular}
\caption{Comparison of performance on mutual exclusion benchmarks with a single unfenced process}
\vspace{-0.5cm}
\end{table}
\begin{table}[h]
\vspace{0.5cm}
\small
\begin{tabular}{ccccccc}
\hline
\textbf{benchmark} & $L$ & $K$ & \textbf{$\mathsf{PS2SC}$} & \textbf{CDSChecker} & \textbf{GenMC} & \textbf{RCMC} \\ \hline\hline
peterson1C(3) & 1 & 2 & 0.487s & 0.053s & 0.083s & 0.087s \\
peterson1C(4) & 1 & 2 & 1.193s & 3.500s & TO & 3.360s \\
peterson1C(5) & 1 & 2 & 2.713s & TO & TO & TO \\
peterson1C(6) & 1 & 2 & 6.045s & TO & TO & TO \\
peterson1C(7) & 1 & 2 & 11.008s & TO & TO & TO \\ \hline
peterson2C(3) & 1 & 2 & 0.481s & 0.032s & 0.099s & 0.091s \\
peterson2C(4) & 1 & 2 & 1.241s & 0.037s & TO & 9.162s \\
peterson2C(5) & 1 & 2 & 2.801s & 1m47s & TO & TO \\
peterson2C(6) & 1 & 2 & 6.528s & TO & TO & TO \\
peterson2C(7) & 1 & 2 & 11.030s & TO & TO & TO \\ \hline
\end{tabular}
\caption{Comparison of performance on completely fenced peterson mutual exclusion benchmarks with a bug introduced in the critical section of a single process}
\end{table}
\vspace{-0.5cm}
\begin{table}[h]
\small
\begin{tabular}{ccccccc}
\hline
\textbf{benchmark} & $L$ & $K$ & $\mathsf{PS2SC}$ & \textbf{CDSChecker} & \textbf{GenMC} & \textbf{RCMC} \\ \hline\hline
peterson(3) & 1 & 2 & 0.878s & TO & 9.665s & 26.208s \\
peterson(2) & 1 & 2 & 0.321s & 0.325s & 0.087s & 0.068s \\ \hline
peterson(3) & 2 & 4 & 1.695s & TO & MLE & TO \\
peterson(2) & 2 & 4 & 0.539s & 15m22s & 0.039s & 0.428s \\ \hline
peterson(3) & 4 & 4 & 15.900s & TO & MLE & TO \\
peterson(2) & 4 & 4 & 3.412s & TO & TO & TO \\ \hline
\end{tabular}
\caption{Evaluation using safe mutual exclusion protocols}
\vspace{-0.5cm}
\end{table}
\vfill
\section{Introduction}
\label{sec:intro}
An important long-standing open problem in PL research has been to define a weak memory model that
captures the semantics of concurrent memory accesses in languages like Java and C/C++.
A model is considered good if it can be implemented efficiently
(i.e., if it supports all usual compiler optimizations
and its accesses are compiled to plain x86/ARM/Power/RISCV accesses),
and is easy to reason about.
After many attempts at solving this problem (e.g., \cite{jmm,zhang-feng:2013,bubbly,crary-sullivan:2015,rc11,jeffrey-riely:2019,batty:2011}),
a breakthrough was achieved by Kang et al.~\cite{promising}, who introduced the \emph{promising semantics}.
This was the first model that supported basic invariant reasoning, the DRF guarantee,
and even a non-trivial program logic~\cite{Svendsen:2018}.
In the promising semantics, the memory is modeled as a set of timestamped messages, each corresponding to a write made by the program.
Each process/thread records its own view of the memory---i.e., the latest timestamp for each memory location that it is aware of. A message has the form $\msgnew{x}{v}{f}{t}{V}$ where $x$ is a location, $v$ a value to be stored for $x$, $(f,t]$ is the timestamp
interval corresponding to the write and $V$ is the local view
of the process who made the write to $x$.
When reading from memory, a process can either return the value stored at the timestamp in its view
or advance its view to some larger timestamp and read from that message.
When a process $p$ writes to memory location $x$, a new message with a timestamp larger than $p$'s view of $x$ is created,
and $p$'s view is advanced to include the new message.
In addition, in order to allow load-store reorderings, a process is allowed to \emph{promise} a certain write in the future. A promise is also added as a message in the memory, except that the local view of the process is not updated using the timestamp interval in the message.
This is done only when the promise is eventually fulfilled.
A {\em consistency} check is used to ensure that every promised message can be {\em certified} (i.e., made fulfillable) by executing that process
on its own. Furthermore, this should hold from any future memory (i.e., from any extension of the memory with additional messages). The quantification prevents deadlocks (i.e., processes from making promises they are not able to fulfil).
The promising semantics generally allows program executions to contain unboundedly many concurrent promised messages,
provided that all of them can be certified. As one can immediately see, this is a fairly complex model,
and beyond its support for some basic reasoning patterns,
it is not at all obvious whether it is easy to reason about concurrent programs running under this model.
Furthermore, the unbounded number of future memories, that need to be checked, makes the verification of even simple programs practically infeasible. Moreover, a number of transformations based on global value range analysis as well as register promotion were not supported in ~\cite{promising}.
To address the above concerns, a new version of the promising semantics $\textsf{PS 2.0}\xspace$ \cite{promising2} has been
proposed, by redesigning key components of the promising semantics \cite{promising}. Mainly, $\textsf{PS 2.0}\xspace$ supports register promotion and global value range analysis, while capturing all features (thread local optimizations, DRF guarantees, hardware mappings) of the promising semantics of \cite{promising}. $\textsf{PS 2.0}\xspace$ simplifies also the consistency check and instead of checking the promise fulfilment from all future memories, $\textsf{PS 2.0}\xspace$
checks for promise fulfilment only from a specially crafted extension
of the current memory called capped memory. $\textsf{PS 2.0}\xspace$ also introduces
the notion of reservations, which allows a process to secure an timestamp interval in order to perform a future atomic read-modify-write instruction. The reservation
blocks any other message from using that timestamp interval.
Reservations allows register promotions.
The wide umbrella of features of $\textsf{PS 2.0}\xspace$ allowing two memory access modes,
relaxed (${\tt{rlx}}$) and release-acquire (${\tt{ra}}$) along with promises, reservations and subsequent certification make $\textsf{PS 2.0}\xspace$ a very complex
model. While the $\textsf{PS 2.0}\xspace$ semantics is a breakthrough
contribution, a natural and fundamental question is to investigate the verification of concurrent programs under $\textsf{PS 2.0}\xspace$. For that, investigating the decidability
of verification problems as well as defining efficient analysis techniques are two extremely important problems.
One of the problems addressed in this paper
is to ivestigate the decidability of the reachability problem for $\textsf{PS 2.0}\xspace$.
Let $\textsf{PS 2.0}\xspace$-${\tt{rlx}}$ and $\textsf{PS 2.0}\xspace$-${\tt{ra}}$
represent respectively, the fragment of $\textsf{PS 2.0}\xspace$ allowing only relaxed
(${\tt{rlx}}$) and release-acquire (${\tt{ra}}$) memory accesses.
The reachability with only ${\tt{ra}}$ accesses has been shown to be undecidable \cite{pldi2019}, even without
the features of promises and reservations. That leaves only the fragment $\textsf{PS 2.0-rlx}$ of $\textsf{PS 2.0}\xspace$
for investigation. We show that if unbounded number of promises is allowed, the reachability problem is
undecidable in $\textsf{PS 2.0-rlx}$, while it becomes decidable if we bound
the number of promises at any time (however, the total number of promises made with a run can be unbounded). Our undecidability is obtained with just 2 threads,
with an execution where the number of context switches between the two processes is three, where a context is a computation segment
in which one process is active.
The proof of decidability is done
by proposing a new memory model with higher order words ${\mathsf{LoHoW}}$, and showing the equivalence
of $\textsf{PS 2.0-rlx}$ and ${\mathsf{LoHoW}}$. Under the bounded promises assumption,,
we use the decidability of the coverability problem of well structured transition systems (WSTS) \cite{wsts2,wsts1} to
show that the reachability problem for ${\mathsf{LoHoW}}$ with bounded number of promises is decidable.
Given this high complexity for $\textsf{PS 2.0-rlx}$ with bounded number of promises and the undecidability result for $\textsf{PS 2.0}\xspace$-${\tt{ra}}$ \cite{pldi2019}, we consider a bounded version of the reachability problem. To this end, we propose a
parametric under-approximation in the spirit of context bounding \cite{cb2,DBLP:conf/cav/TorreMP09,DBLP:journals/fmsd/LalR09,demsky,MQ07,DBLP:conf/tacas/QadeerR05,pldi2019,cb3}.
The bounding concept chosen for concurrent programs depends on aspects related to the
interactions between the processes. In the case of SC programs, context bounding
has been shown experimentally to have extensive behaviour coverage for bug detection \cite{MQ07,DBLP:conf/tacas/QadeerR05}.
A context in the SC setting is a computation segment where only one process is active.
The concept of context bounding has been extended for weak memory models. For instance, in TSO,
the notion of context is extended to one where all updates to the main memory are done only from the buffer
of the active thread \cite{cb2}.
In the case
of RA \cite{pldi2019}, context bounding was extended to view bounding,
using the notion of view-switching messages.
Since $\textsf{PS 2.0}\xspace$ subsumes RA, we propose a
bounding notion that extends the view bounding
proposed in \cite{pldi2019}. Using this new bounding notion, we propose a source to source translation
from programs under $\textsf{PS 2.0}\xspace$ to context-bounded executions
of the transformed program in SC.
The main challenge in the code-to-code translation of \cite{pldi2019} was to keep track of the causality between different variables. In our case, the challenge is fundamentally different and is to provide a procedure that
(i) handles different memory accesses ${\tt{rlx}}$ and ${\tt{ra}}$,
(ii) guesses the promises and reservations in a non-deterministically manner, and (iii) verify that each promise so guessed is fulfilled using the capped memory.
This reduction is implemented in a tool, called $\mathsf{PS2SC}$.
Our experimental results demonstrate the effectiveness of our approach.
We exhibit cases where hard-to-find bugs are detectable using a small view-bound $K$.
Our tool displays resilience to trivial changes in the position of bugs and the order of processes.
\smallskip
\noindent
{\textbf{Related Work.}}
The decidability of the verification problems for programs running under weak memory models has been addressed for TSO \cite{ABBM10}, $\textsf{PS 2.0}\xspace$-${\tt{ra}}$ \cite{pldi2019}, Power \cite{Power_Netys20}, and for a subclass of $\textsf{PS 2.0}\xspace$-${\tt{ra}}$ \cite{DBLP:conf/pldi/LahavB20}. To the best of our knoweldge, this the first time that this problem is investigated for $\textsf{PS 2.0-rlx}$ and $\mathsf{PS2SC}${} is the first tool for automated verification of programs under $\textsf{PS 2.0}\xspace$, which also works
for the promising semantics \cite{promising}.
Most of the existing work concerns the development of
stateless model checking (SMC), coupled with (dynamic) partial order
reduction techniques (e.g., \cite{phong,rcmc,genmc,demsky,cdsc}) and do not handle promises. Context-bounding has been proposed in \cite{DBLP:conf/tacas/QadeerR05} for programs running under SC. This work has been extended in different directions and has led to efficient and scalable techniques for the analysis of concurrent programs (see e.g., \cite{MQ07,DBLP:journals/fmsd/LalR09,DBLP:conf/cav/TorreMP09,madhu3,DBLP:conf/popl/EmmiQR11,DBLP:conf/cav/TorreMP10}). In the context of weak memory models, context-bounded analysis has been only proposed to programs running under TSO/PSO in \cite{cb2,DBLP:conf/sefm/TomascoN0TP17} and under POWER in \cite{cb3}.
\section{Preliminaries}
\label{sec:prels}
In this section, we introduce the simple programming language and the notation
that will be used throughout. Then, we review $\textsf{PS 2.0}\xspace$ definition, and present
the model following \cite{promising2}.
\subsection{Notations}
Given two natural numbers $i, j \in \mathbb{N}$ s.t. $i \leq j$, we use $[i,j]$ to denote the set $\{k \,|\, i \leq k \leq j\}$.
Let $A$ and $B$ be two sets. We use $f: A \rightarrow B$ to denote that $f$ is a function from $A$ to $B$. We define $f[a \mapsto b]$ to be the function $f'$ such that $f'(a)=b$ and $f'(a')=f(a')$ for all $a' \neq a$.
For a binary relation $R$, we use $\rtstep R$ to denote its reflexive and transitive closure.
Given an alphabet $\Sigma$, we use $\Sigma^*$ (resp.\ $\Sigma^+$) to denote the set of possibly empty (resp.\ non-empty) finite words over $\Sigma$.
Let $w= a_1 a_2 \cdots a_n$ be a word over $\Sigma$, we use $|w|$ to denote the length of $w$.
Given an index $i$ in $[1,|w|]$, we use $w[i]$ to denote the $i^{\text{th}}$ letter of $w$.
Given two indices $i$ and $j$ s.t. $1\leq i \leq j \leq |w|$, we use $w[i,j]$ to denote the word $a_i a_{i+1} \cdots a_j$. Sometimes, we consider a word as a function from $[1,|w|]$ to $\Sigma$.
\subsection{Program Syntax}
The simple programming language we use
is described in Figure \ref{program_syntax}.
A program $\prog$ consists of a set $\varset$ of (global) variables or memory locations,
and the definition of a set $\procset$ of processes.
Each process $\proc$ declares a set $\regsetof\proc$ of (local) {\it registers} followed by a sequence of labeled instructions. We assume that these sets of registers are disjoint and we use $\regset:=\cup_\proc\regsetof\proc$ to denote their union. We assume also a (potentially unbounded) data domain $\mathsf{Val}$ from which the registers and locations take values.
All locations and registers are assumed to be initialized with the
special value $0 \in \mathsf{Val}$ (if not mentioned otherwise).
An instruction $\instr$ is of the form $\lbl : \stmt$ where
$\lbl$ is a unique label and $\stmt$ is a statement. We use
$\mathbb{L}_{\proc}$ to denote the set of all labels
of the process $\proc$, and $\mathbb{L}=\bigcup_{\proc \in \procset}\mathbb{L}_{\proc}$ the set of all labels of all processes.
We assume that the execution of the process $\proc$ starts always with a unique initial instruction labeled by
$\lambda_{\rm init}^{\proc}$.
A write instruction is of the form $\xvar^o=\reg$ assigns the value of register $\reg$ to the location $\xvar$, and
$o$ denotes the access mode. If $o=\mathsf{rlx}$, the write is a \emph{relaxed} write, while
if $o=\mathsf{ra}$, it is a \emph{release} write.
A read instruction $\reg=\xvar^o$ reads the value of the location $\xvar$ into the local register $\reg$. Again,
if the access mode $o=\mathsf{rlx}$, it is a \emph{relaxed} read, and
if $o=\mathsf{ra}$, it is an \emph{acquire} read. Atomic updates or $\mathsf{RMW}$
instructions are either
compare-and-swap (${\mathbf{CAS}}^{o_r,o_w}$) or ${\mathbf{FADD}}^{o_r, o_w}$. Both have a pair of accesses ($o_r,o_w \in \{ \mathsf{rel}, \mathsf{acq},
\mathsf{rlx}\}$) to the same location -- a read followed by a write. Following \cite{promising2}, ${\mathbf{FADD}}(x,v)$ stores the value of $x$ into a register $\reg$, and adds $v$
to $x$, while ${\mathbf{CAS}}(x, v_1, v_2)$ compares an expected value $v_1$ to the value in $x$, and
if the values are same, sets the value of $x$ to $v_2$. The old value of $x$
is then stored in $\reg$.
\tikzset{background rectangle/.style={draw=black,rounded corners,fill = black!2}}
\setlength\intextsep{0pt}
\begin{wrapfigure}[12]{r}{5.5cm}
\begin{tikzpicture}[codeblock/.style={line width=0.3pt, inner xsep=0pt, inner ysep=0pt}, show background rectangle]
\node[codeblock] (init) at (current bounding box.north east) {
\footnotesize
{
$
\begin{array}{l}
~~~~\prog ::= \keyword{var}~ x^*(\keyword{proc} ~p|| \dots || \keyword{proc} ~p) \\
~~~~ \keyword{proc}~ p::=\regset(p)~ \instr^*\\
~~~\instr::=\lambda:\stmt\\
~\stmt \in \mathsf{St}::= \\
~~\;\;\mathsf{skip} ~~\;\;|s;s ~~\;\;|\keyworr{assume}(\xvar=e)\\
~~\;\;|\kwdo ~s^* ~\kwwhile~ e ~~\;\;|\kwwhile ~e ~\kwdo~ s^* \kwdone \\
~~\;\;|\kwif~ e ~ \kwthen~ s~ \kwelse ~s ~\\
~~\;\;|\reg~:=~ e ~~\;\;|\reg~:=~ x^o ~~\;\;|x^o~:=~ \reg \\
~~\;\;|\reg~:=~\mathbf{FADD}^{o,o}(x,v) \\~~\;\;|\reg~:=~\mathbf{CAS}^{o,o}(x,v,v)~~\;\;|\mathsf{SC\text{-}fence}\\
o \in \mathsf{Mode} ::= \mathsf{rlx}|\mathsf{ra}
\end{array}
$
}
};
\end{tikzpicture}
\caption{\footnotesize Syntax of concurrent programs.}
\label{program_syntax}
\end{wrapfigure}
A {\em local} assignment instruction $\reg=e$ assigns to the register $\reg$ the value of $e$, where $e$ is an expression over a set of operators,
constants as well as the contents of the registers of the current process, but not referring to the set of locations.
The fence instruction $\keyword\mathsf{SC\text{-}fence}$ is used to enforce sequential consistency if it is placed between two memory access operations.
Finally, the conditional, assume and iterative instructions
have the standard semantics.
For simplicity, we will write $\keyworr{assume}(\xvar=e)$ instead of
${\reg=\xvar}; \keyworr{assume}(\reg=e)$.
This notation is extended in the straightforward manner to conditional statements.
\subsection{The Promising Semantics}
\label{sec:ps}
In this section, we recall the promising semantics \cite{promising2}. We present here $\textsf{PS 2.0}\xspace$ with three
memory accesses, \emph{relaxed} (this is the default mode),
\emph{release writes} (${\tt{rel}}$) and \emph{acquire reads}
(${\tt{acq}}$). Read-modify-writes (RMW) instructions have two access modes - one for read
and one for write. We keep aside
the release and acquire fences (and subsequent
access modes) which are part of $\textsf{PS 2.0}\xspace$, since
they do not affect the results of this paper.
\smallskip\noindent{\bf Timestamps.}
$\textsf{PS 2.0}\xspace$ uses timestamps to maintain a total order over all the writes to the same variable.
We assume an infinite set of timestamps $\mathsf{Time}$, densely totally ordered by $\leq$,
with $0$ being the minimum element. A \emph{view} is a timestamp function $V : \varset \rightarrow \mathsf{Time}$ records the largest known timestamp for each location.
Let $\mathbb{T}$ be the set containing all the timestamp functions, along with the special
symbol $\bot$.
Let $V_{\rm init}$ represent the initial view where all locations are mapped to $0$. Given two views $V$ and $V'$, we use $V \leq V'$ to denote that $V(x) \leq V'(x)$ for $x \in \varset$. The merge operation $\sqcup$ between the two views $V$ and $V'$ returns the pointwise maximum of $V$ and $V'$,
i.e., $(V \sqcup V')(y)$ is the maximum of $V(y)$ and $V'(y)$. Let $\mathcal{I}$ denote the set of all intervals over $\mathsf{Time}$.
The timestamp intervals in $\mathcal{I}$ have the form $(f,t]$ where either $f=t=0$ or $f < t$, with $f, t \in \mathsf{Time}$. Given an interval $I=(f,t] \in \mathcal{I}$, $I.{\textcolor{cobalt}{\tt{frm}}}$ and $I.{\textcolor{cobalt}{\tt{to}}}$ denote $f, t$ respectively.
\smallskip\noindent{\bf Memory.}
In $\textsf{PS 2.0}\xspace$, the memory is modelled as a set of concrete \emph{messages} (which we just call messages), and \emph{reservations}.
Each message represents the effect of a write or a RMW operation and each reservation is a timestamp interval reserved for future use.
In more detail, a message $m$ is a tuple $\msgnew{\xvar}{v}{f}{t}{V}$ where $\xvar \in \varset$, $v \in \mathsf{Val}$, $(f, t] \in \mathcal{I}$ and $V\in \mathbb{T}$. A reservation $r$ is a tuple $\reserv{x}{f}{t}$. Note that a reservation, unlike a message, does not commit to any particular value, but only specifies the interval which is reserved.
We use $m.{\mathsf{loc}}$ ($r.{\mathsf{loc}}$), $m.\tt{{val}}$, $m.{\textcolor{cobalt}{\tt{to}}}$ ($r.{\textcolor{cobalt}{\tt{to}}}$), $m.{\textcolor{cobalt}{\tt{frm}}}$ ($r.{\textcolor{cobalt}{\tt{frm}}}$) and $m.{\mathsf{View}}$ to denote respectively $x$, $v$, $t$, $f$ and $V$.
Two elements (either messages or reservations) are said to be \emph{disjoint} ($m_1 \# m_2$) if they concern different variables ($m_1.{\mathsf{loc}}\neq m_2.{\mathsf{loc}}$) or their intervals do not overlap ($m_1.{\textcolor{cobalt}{\tt{to}}} < m_2.{\textcolor{cobalt}{\tt{frm}}} \lor m_1.{\textcolor{cobalt}{\tt{frm}}} > m_2.{\textcolor{cobalt}{\tt{to}}}$).
Two sets of elements $M, M'$ are disjoint, denoted $M \# M'$, if $m \# m'$ for every $m \in M, m' \in M'$.
Two elements $m_1, m_2$ are \emph{adjacent} denoted ${\mathsf{Adj}}(m_1,m_2)$ if $m_1.{\mathsf{loc}}=m_2.{\mathsf{loc}}$ and $m_1.{\textcolor{cobalt}{\tt{to}}}=m_2.{\textcolor{cobalt}{\tt{frm}}}$. A memory $M$ is a set of pairwise disjoint messages and reservations. Let $\widetilde{M}$ be the subset of $M$ containing
only messages (no reservations). For a location $x$, let $M(x)$ be
$\{m \in M \mid m.{\mathsf{loc}}=x\}$. Given a view $V$ and a memory $M$, we say $V \in M$ if $V(x)=m.{\textcolor{cobalt}{\tt{to}}}$ for some
message $m \in \widetilde{M}$ for every $x \in \varset$. Let ${\mathbb M}$ denote the set of all memories.
\smallskip
\noindent
{\it Insertion into Memory.} Following \cite{promising2},
a memory $M$ can be extended with a \emph{message} (due to the execution of a write/RMW instruction) or a \emph{reservation} $m$ with $m.{\mathsf{loc}} = x$, $m.{\textcolor{cobalt}{\tt{frm}}} = f$ and $m.{\textcolor{cobalt}{\tt{to}}} = t$ in a number of ways:
\smallskip
\noindent
{[Additive insertion]}
$M \stackrel{A}{\hookleftarrow} m$ is defined only if (1) $M\# \{m\}$; (2) if $m$ is a message, then no message $m' \in M$ has $m'.{\mathsf{loc}} = x$ and $m'.{\textcolor{cobalt}{\tt{frm}}} = t$; and (3) if $m$ is a reservation, then there exists a message $m'\in \widetilde{M}$ with $m'.{\mathsf{loc}} = x$ and $m'.{\textcolor{cobalt}{\tt{to}}} = f$. The extended memory $M \stackrel{A}{\hookleftarrow} m$ is then $M \cup \{m\}$.
\noindent
{[Splitting insertion]}
$M \stackrel{S}{\hookleftarrow} m$ is defined if $m$ is a message, and, if there exists a message $m'=(x, v', (f,t'],V)$ with $t < t'$ in $M$. Then $M$ is updated to $M \stackrel{S}{\hookleftarrow} m = (M\backslash\{m'\} \cup \{m, (x,v', (t,t'],V)\})$.
\noindent
[Lowering Insertion]
$M \stackrel{L}{\hookleftarrow} m$ is only defined if there exists $m'$ in $M$ that is identical to $m=\msgnew x v f t V$ except for $m.{\mathsf{View}} \leq m'.{\mathsf{View}}$. Then, $M$ is updated to $M \stackrel{L}{\hookleftarrow} m = M\backslash \{m'\} \cup \{m\}$.
\smallskip
\noindent
[Cancellation]
$M \stackrel{C}{\hookleftarrow} m$ is defined if $m$ is a reservation in $M$.
Then $M$ is updated as $M\setminus \{m\}$.
\smallskip\noindent{\bf Transition System of a Process.}
Given a process $\proc \in \procset$, a state $\sigma$ of $\proc$ is defined as a pair $(\lambda,R)$ where $\lambda \in \mathbb{L}$ is the label of the next instruction to be executed by $\proc$ and $R : \regset \rightarrow \mathsf{Val}$ maps each register of $\proc$ to its current value. (Observe that we use the set of all labels $\mathbb{L}$ (resp. registers $\regset$) instead of $\mathbb{L}_{\proc}$ (resp. $\regsetof\proc$) in the definition of $\sigma$ just for the sake of simplicity.) Transitions between the states of $\proc$ are of the form $ (\lambda,R)\xRightarrow[p]{t} (\lambda',R')$ with $t \in \{\epsilon, \rd(o,x,v), \wt(o,x,v), {\tt{U}}(o_r, o_w, x, v_r, v_w), \mathsf{SC\text{-}fence}\,|\, x \in \varset, v \in \mathsf{Val}, o\in \{{\tt{rlx}},{\tt{ra}}\} \}$. A transition of the form $ (\lambda,R)\xRightarrow[p]{\rd(o,x,v)} (\lambda',R')$ denotes the execution of a read instruction of the form $\reg=\xvar^o$ labeled by $\lambda$ where $(1)$ $\lambda'$ is the label
of the next instructions that can be executed after the execution of the instruction labelled by $\lambda$, and $(2)$ $R'$ is the mapping that results from the replacement of the value of the register $\reg$ in $R$ by $v$. The transition relation $ (\lambda,R)\xRightarrow[p]{t} (\lambda',R')$ is defined in similar manner for the other cases of $t$ where ${\mathsf{wrt}}(o,x,v)$ stands for a write instruction that writes the value $v$ to $x$, ${\tt{U}}(o_r, o_w, x, v_r, v_w)$ stands for a RMW that reads the value $v_r$ from $x$ and write $v_w$ to it, $\mathsf{SC\text{-}fence}$ stands for a $\mathsf{SC\text{-}fence}$ instruction, and $\epsilon$ stands for the execution of the other local instructions. Observe that $o, o_r, o_w$ are the access modes which can be ${\tt{rlx}}$
or ${\tt{ra}}$. We use ${\tt{ra}}$ for both release and acquire.
Finally, we use $ (\lambda,R)\xrightarrow[p]{t} (\lambda',R')$ with $t \in \{\rd(o,x,v), \wt(o,x,v), {\tt{U}}(o_r, o_w, x, v_r, v_w), \mathsf{SC\text{-}fence}\,|\, x \in \varset, v \in \mathsf{Val}, o\in \{{\tt{rlx}},{\tt{ra}}\} \}$ to denote that $ (\lambda,R)\xRightarrow[p]{\epsilon} \sigma_1 \xRightarrow[p]{\epsilon} \cdots \xRightarrow[p]{\epsilon} \sigma_n \xRightarrow[p]{t} \sigma_{n+1} \xRightarrow[p]{\epsilon} \cdots \xRightarrow[p]{\epsilon} (\lambda',R')$.
\smallskip\noindent{\bf Machine States.}
A machine state $\mathcal{MS}$ is a tuple $(({\sf J}, {\sf R}), \sf {VS}, {\sf PS}, M, G)$, where
${\sf J} : \procset \mapsto \mathbb{L}$ maps each process $p$ to the label of the next instruction to be executed,
${\sf R} : \regset \rightarrow \mathsf{Val}$ maps each register to its current value,
${\sf VS} = \procset \rightarrow \mathbb{T}$ is the process view map, which maps each process to a view,
$M$ is a memory and $PS: \procset \mapsto {\mathbb M}$ maps each process to a set of messages (called \emph{promise} set), and
$G \in \mathbb{T}$ is the global view (that will be used by SC fences).
We use $\confset$ to denote the set of all machine states.
Given a machine state $\mathcal{MS}=(({\sf J}, {\sf R}), \sf {VS}, {\sf PS}, M, G)$ and a process $p$, let $\mathcal{MS}{\downarrow}p$ denote the projection, $(\sigma, \sf {VS}(p), {\sf PS}(p), M, G)$ with $\sigma=({\sf J}(p), {\sf R}(p))$, of the machine state to the process $p$.
We call $\mathcal{MS} {\downarrow} p$ the process configuration.
We use $\confset_p$ to denote the set of all process configurations.
The initial machine state $\mathcal{MS}_{\rm init}=(({\sf J}_{\rm init}, {\sf R}_{\rm init}), {\sf VS}_{\rm init}, {\sf PS}_{\rm init}, M_{\rm init},G_{\rm init})$ is one where:
(1) ${\sf J}_{\rm init}(p)$ is the label of the initial instruction of $\proc$; (2) ${\sf R}_{\rm init}(\reg)=0$ for every $\reg \in \regset$;
(3) for each $p$, we have ${\sf VS}(p) = V_{\rm init}$ as the initial view (that maps each location to the timestamp 0);
(4) for each process $p$, the set of promises ${\sf PS}_{\rm init}(p)$ is empty;
(5) the initial memory $M_{\rm init}$ contains exactly one initial message $(x,0, (0, 0], V_{\rm init})$ for each location $x$;
and (6) the initial global view maps each location to $0$.
\input{fig-inf-rules}
\smallskip\noindent{\bf Transition Relation.}
We first describe the transition $(\sigma, V, P, M, G) \xrightarrow[p]{} (\sigma', V', P', M', G')$ between process configurations in $\confset_p$ %
from which we induce the transition relation between machine states.
\smallskip
\noindent{\it Process Relation.}
The formal definition of $\xrightarrow[p]{}$ is in Figure \ref{program_sem}. Below, we explain these inference rules.
\noindent{\bf {Read}}. A process $p$ can read from $M$ by observing a
message $m=(x,v, (f,t], K)$ if $V(x) \leq t$ (i.e., $\proc$ must not be aware of a later message for $\xvar$).
In case of a relaxed read $\rd({\tt{rlx}}, x, v)$,
the process view of $x$ is updated to $t$, while
for an acquire read $\rd({\tt{ra}}, x, v)$, the process view is updated
to $V[x \mapsto t] \sqcup K$. The global memory $M$, the set of promises $P$, and the global view $G$ remain the same.
\noindent{\bf {Write}}. A process can add a fresh message
to the memory ($\mathsf{MEMORY:NEW}$) or fulfil an outstanding promise
($\mathsf{MEMORY : FULFILL}$). The execution of a write ($\wt({\tt{rlx}}, x, v)$) results in a message $m$ with location
$x$ along with a timestamp interval $(-, t]$. Then, the process view of location $x$ is updated to $t$.
In case of a release write ($\wt({\tt{ra}}, x, v)$) the updated process view is also attached
to $m$, and ensures that the process does not have an outstanding promise
on location $x$. ($\mathsf{MEMORY : FULFILL}$) allows
to split a promise interval or lower its view before fulfilment.
\noindent{\bf {Update}}. When a process performs a RMW, it first reads a message
$m=(x, v, (f,t], K)$ and then writes an update message
with ${\textcolor{cobalt}{\tt{frm}}}$
timestamp equal to $t$; that is, a message
of the form $m'=(x, v', (t, t'], K')$. This forbids any other write
to be placed between $m$ and $m'$. The access modes of the reads and writes in the update
follow what has been described for the read and write above.
\noindent{\bf Promise, Reservation and Cancellation.}
A process can non-deterministically \emph{promise} future writes which
are not release writes.
This is done by adding a message $m$ to the memory $M$
s.t. $m \#M$ and to the set of promises $P$. Later, a relaxed write instruction can fulfil an existing promise. Recall that the execution of a release write requires that the set of promises to be empty and thus it can not be used to fulfil a promise.
In the reserve step, the process reserves a timestamp interval to be used
for a later RMW instruction reading from a certain message without fixing the value it will write.
A reservation is added both to the memory and the promise set. The process can drop the reservation from both sets using the cancel step in non-deterministic manner.
\noindent{\bf SC fences.}The process view $V$ is merged with the global view $G$, resulting
in $V \sqcup G$ as the updated process view and global view.
\smallskip
\noindent
{\it Machine Relation.}
We are ready now to define the induced transition relation between machine states.
For machine states $\mathcal{MS}=((J, R), VS, PS, M, G)$ and $ \mathcal{MS}'=((J', R'), VS', PS', M', G')$, we write
$\mathcal{MS} \xrightarrow[p]{} \mathcal{MS}' $ iff $(1)$ ${\mathcal{MS} {\downarrow} p} \xrightarrow[p]{} {\mathcal{MS} {\downarrow} p}$ and $(J(p'),VS(p'), PS(p')) = (J'(p'),VS'(p'), PS'(p'))$ for all $p' \neq p$.
\smallskip
\noindent{\bf Consistency.}
According to Lee et al. \cite{promising2}, there is one final requirement on machine states called \emph{consistency},
which roughly states that, from every encountered machine state encountered,
all the messages promised by a process $\proc$ can be {\em certified} (i.e., made fulfillable) by executing $\proc$ on its own from a certain
future memory (called capped memory), i.e., extension of the memory with additional
reservation. Before defining consistency, we need to introduce capped memory.
\smallskip
\noindent
\emph{Cap View, Cap Message and Capped Memory.} The last element of a memory $M$ with respect to a location $x$, denoted by $\overline{m}_{M, x}$, is
an element from $M(x)$ with the highest timestamp among all elements of $M(x)$ and is defined as
$\overline{m}_{M,x} = \max_{m\in M(x)} m.{\textcolor{cobalt}{\tt{to}}}$.
The \emph{cap view} of a memory $M$, denoted by $\widehat{V}_M$, is the view which assigns to each
location $x$, the ${\textcolor{cobalt}{\tt{to}}}$ timestamp in the message $\overline{m}_{\widetilde{M},x}$. That is,
$\widehat{V}_M = \lambda x. \overline{m}_{\widetilde{M}, x}.{\textcolor{cobalt}{\tt{to}}}$. Recall that $\widetilde{M}$ denote the subset of $M$ containing
only messages (no reservations).
The \emph{cap message} of a memory $M$ with respect to a location $x$,
is given by the message
$\widehat{m}_{M,x} = (x, \overline{m}_{\widetilde{M}, x}.{\tt{val}}, (\overline{m}_{M, x}.{\textcolor{cobalt}{\tt{to}}}, \overline{m}_{M, x}.{\textcolor{cobalt}{\tt{to}}} + 1], \widehat{V}_{M})$.
Then, the capped memory of a memory $M$, wrt. a set of promises $P$, denoted by $\widehat{M}_P$, is an extension of $M$, defined as: $(1)$ for every $m_1,m_2 \in M$ with $m_1.{\mathsf{loc}} = m_2.{\mathsf{loc}},~ m_1.{\textcolor{cobalt}{\tt{to}}} <
m_2.{\textcolor{cobalt}{\tt{to}}}$, and there is no message $m' \in M(m_1.{\mathsf{loc}})$ such
that $m_1.{\textcolor{cobalt}{\tt{to}}} < m'.{\textcolor{cobalt}{\tt{to}}} < m_2.{\textcolor{cobalt}{\tt{to}}}$, we include a reservation
$(m_1.{\mathsf{loc}}, (m_1.{\textcolor{cobalt}{\tt{to}}}, m_2.{\textcolor{cobalt}{\tt{frm}}}])$ in $\widehat{M}_P$, and $(2)$ we include a cap message $\widehat{m}_{M,x}$ in $\widehat{M}_P$ for every variable $x$ unless $\overline{m}_{M,x}$ is a reservation in $P$.
\smallskip
\noindent
\emph{Consistency of machine states.}
A machine state $\mathcal{MS}=((J, R), VS, PS, M, G)$ is \emph{consistent}
if every process $p\in\procset$ can certify/fulfil all its promises from the capped memory $\widehat{M}_{PS(p)}$, i.e.,
$ ((J, R), VS, PS, \widehat{M}_{PS(p)}, G) \rtstep{\xrightarrow[p]{}} ((J', R'), VS', \emptyset, M', G')$.
\medskip
\noindent
\textbf{The Reachability Problem in $\textsf{PS 2.0}\xspace$.}
A run of $\prog$ is a sequence of the form:
$\mathcal{MS}_{0} \rtstep{\xrightarrow[p_{i_1}]{}}
\mathcal{MS}_{1} \rtstep{\xrightarrow[p_{i_2}]{}}
\mathcal{MS}_{2} \rtstep{\xrightarrow[p_{i_3}]{}}
\ldots
\xrightarrow[p_{i_n}]{*}
\mathcal{MS}_{n}$
where $\mathcal{MS}_0=\mathcal{MS}_{\rm init}$ is the initial machine state and $\mathcal{MS}_1,\ldots,\mathcal{MS}_n$ are consistent machine states.
In this case, the machine states $\mathcal{MS}_0,\ldots,\mathcal{MS}_n$ are said to be reachable from $\mathcal{MS}_{\rm init}$.
Given an instruction label function $J: \procset \rightarrow \mathbb{L}$ that maps each process $\proc \in \procset$ to an instruction label in $ \mathbb{L}_{\proc}$,
the \emph{reachability} problem asks
whether there exists a machine state of the form $((J,R),V,P,M,G)$ that is reachable from $\mathcal{MS}_{\rm init}$.
In the case of a positive answer to this problem, we say that $J$ is reachable in $\prog$ in $\textsf{PS 2.0}\xspace$.
\subsection{Examples}
\tikzset{background rectangle/.style={draw=black,sharp corners,fill = yellow!5}}
\begin{wrapfigure}{r}{0.5\textwidth}
\begin{tikzpicture}[codeblock/.style={line width=0.2pt, inner xsep=0pt, inner ysep=0pt}, show background rectangle]
\node[codeblock] (init) at (current bounding box.north east) {
\footnotesize
\begin{tabular}[h]{l||l}
\begin{lstlisting}[style=examples,tabsize=4]
$r1=x
if($r1 != 2){
z=1
$r1=z
assume($r1=3)
z=2
}
else{
z=2 //
}
\end{lstlisting}
&
\begin{lstlisting}[style=examples,tabsize=4]
z=3
$r2=z
assume($r2=2)
x=2
\end{lstlisting}
\end{tabular}
};
\end{tikzpicture}
\caption{\footnotesize The annotated behaviour is not reachable.}
\label{eg1}
\end{wrapfigure}
In the following, we describe some examples to demonstrate $\textsf{PS 2.0}\xspace$. For readability, instead of referring to
reachable instruction labels, we consider possible
program outcomes represented using the program comment annotation ``\textcolor{sangria}{//}''.
All writes and reads are relaxed in both examples below.
\begin{example}
The annotated program outcome in Figure \ref{eg1} is not allowed by $\textsf{PS 2.0}\xspace$.
\noindent We list the execution steps of $\textsf{PS 2.0}\xspace$ showing that the annotated behaviour is not possible. We give a proof by contradiction.
Assume that the annotated behaviour is possible. The only way for this is that the first process $p_1$ (whose code on the left side) to execute the $\kwelse$ branch. For this, it needs to read 2 from x.
This can be provided only by the second process using the write x=2. For this to happen,
$p_2$ first executes the write z=3 by adding a message (z, 3, $(r,s], \bot)$ to the memory. Next,
$p_2$ has to read a message
of the form (z, 2, $(f,t], \bot)$ which can only be generated
by $p_1$ as a promise.
\vspace{.2cm}
Note that $p_1$
can promise the write $z=2$ in its $\kwif \dots \kwthen$ branch. To certify this promise, $p_1$ starts from
the capped memory, and first executes the write z=1 in the $\kwif \dots \kwthen$ branch. To do this,
it can split the promise interval $(f,t]$ and add a message (z, 1, $(f, t'], \bot)$ while modifying
(z, 2, $(f,t], \bot)$ in the memory to (z, 2, $(t',t], \bot)$. Note that since
we work from the capped memory, there are no available intervals
in $[0, max(t,s)]$, and the only way to add a message for the write z=1 of $p_1$,
in such a way that $p_1$ can read the 3 written by $p_2$, and also to fulfil its promise,
is to split the promise interval. Next, $p_1$ reads (z, 3, $(r,s], \bot)$ to
go past the $\keyworr{assume}$(z=3) statement. This imposes $f<t' \leq r < s$. However,
since $p_2$
wrote 3 to z before reading the promise (z, 2, $(t',t], \bot)$, we also need $r < s \leq f <t'$ which contradicts
$f < r$. Hence, the annotated behaviour is not reachable, since $p_1$ fails the certification.
\end{example}
\begin{example}
In Figure \ref{eg2}, we present an example having a run realising the program outcome
which has unboundedly many reservations and subsequent cancellations.
\tikzset{background rectangle/.style={draw=black,sharp corners,fill = yellow!5}}
\begin{figure}[h]
\begin{tikzpicture}[codeblock/.style={line width=0.3pt, inner xsep=0pt, inner ysep=0pt}, show background rectangle, sharp corners]
\node[codeblock] (init) at (current bounding box.north east) {
\begin{tabular}{c||c||c}
\begin{lstlisting}[style=examples,tabsize=2]
$r1=z
if($r1 = 2)
{
x=2 //
}
else{
do{
$r4 = FADD(y,1)
}while (w=0)
x=2
}
\end{lstlisting}
&
\begin{lstlisting}[style=examples,tabsize=2]
w=1
$r2=x
assume($r2 == 2)
z=2
\end{lstlisting}
&
\begin{lstlisting}[style=examples,tabsize=2]
do
y=$r3
while(w=0)
\end{lstlisting}
\end{tabular}
};
\end{tikzpicture}
\caption{\footnotesize The annotated behaviour is reachable.}
\label{eg2}
\end{figure}
\noindent We list the execution steps of $\textsf{PS 2.0}\xspace$ leading to the annotated behaviour. Items prefixed with ``C'' represent certification steps.
\begin{itemize}
\item[(1)] Process 2 writes 1 to $w$.
\item[(2)]
Process 3 writes arbitrarily many messages $(y, 0, (f_1, t_1], \bot), (y, 0, (f_2, t_2], \bot) \dots (y, 0, (f_k, t_k], \bot)$ such that $t_1 < f_2 < t_2 < f_3 \dots < f_k<t_k$, until it reads the value 1 from $w$. The number of messages written depends on the number of iterations of $\kwwhile$.
\item[(3)] Process 1 promises $(x, 2, (f, t], \bot)$ corresponding to the write $x=2$ in the $\kwelse$ branch.
\item[(4)] Process 1 makes arbitrarily many reservations $(y, (t_1, t'_1]),
(y, (t_2, t'_2]), \dots, (y, (t_{k-1}, t'_{k-1}])
$ such that $t'_1< f_2< t'_2 < f_3 \dots t'_{k-1} < f_k<t_k$ and $(y, (t_k, t_{k+1}])$.
\item[(C1)] Starting from the capped memory, process 1
cancels the reservations one by one, while executing the
${\mathbf{FADD}}$ instructions, thereby adding messages
$(y, 1, (t_i, t'_i], \bot)$ to the memory.
\item[(C2)] Process 1 fulfils its promise.
\item[(5)] Process 2 reads the message $(x, 2, (f,t], \bot)$
and adds the message $(z, 2, (f'', t''], \bot)$ for the write $z=2$.
\item[(6)]Process 1 reads $(z, 2, (f'', t''], \bot)$
and fulfils $(x, 2, (f, t], \bot)$ reaching the program outcome.
\end{itemize}
\end{example}
\section{Undecidability of Consistent Reachability in $\textsf{PS 2.0}\xspace$}
\label{sec:undec}
In this section, we show that reachability is undecidable
for $\textsf{PS 2.0}\xspace$ even for finite-state programs. The proof is by a reduction from Post's Correspondence Problem (PCP) \cite{post}. Our proof works with the fragment of $\textsf{PS 2.0}\xspace$ having only relaxed (${\tt{rlx}}$) memory accesses and crucially uses unboundedly many promises to ensure that a process cannot skip any writes made by another process.
It also works even when we restrict our analysis to executions that can be split into a bounded number of contexts,
where within each context, only one process is active. We need just 3 context switches.
Our undecidability result is also \emph{tight} in the sense that
the reachability problem becomes decidable when we restrict ourselves to machine states where the number of promises is bounded. Given our proof (Theorem \ref{thm:undec}) where undecidability is obtained
with the ${\tt{rlx}}$ fragment of $\textsf{PS 2.0}\xspace$, a natural question is the
decidability status of the ${\tt{ra}}$ fragment of $\textsf{PS 2.0}\xspace$.
This is known to be undecidable from \cite{pldi2019} even in the absence of promises. Let us call the fragment of $\textsf{PS 2.0}\xspace$ with only ${\tt{rlx}}$ memory accesses $\textsf{PS 2.0-rlx}$.
\begin{theorem}
\label{undecidability}
The reachability problem for concurrent programs over a finite data domain is undecidable under $\textsf{PS 2.0}\xspace$.
In fact, the undecidability still holds for the $\textsf{PS 2.0-rlx}$ fragment.
\label{thm:undec}
\end{theorem}
{
\begin{figure*}[h]
\centering
\colorbox{black!5}{\footnotesize
\begin{tabular}{|@{}c@{}|@{}c@{}|@{}c@{}|@{}c@{}|}
\hline\hline
Process $\proc_1$ & Process $\proc_2$& $\keyword{Module^{\proc_1}_{v_i}}$ & $\keyword{Module^{\proc_2}_{u_i}}$ \\
\hline\hline
$\begin{array}[t]{l}
\lcomme ~\textcolor{blush}{generation~ mode}~ \rcomme
\\ \kwif\ \mathit{validate} = 0\ \kwthen
\\\quad \kwwhile\ \mathit{term} = 0\ \kwdo
\\\qquad \mathit{index} = 1
\\\qquad \keyword{Module^{\proc_1}_{u_1}}
\\\qquad \mathit{index} = \#
\\\qquad \ldots
\\\qquad \mathit{index} = n
\\\qquad \keyword{Module^{\proc_1}_{u_n}}
\\\qquad \mathit{index} = \#
\\\quad \kwdone
\\\quad \mathit{index} = \S
\\\lcomme ~\textcolor{blush}{validation~ mode}~ \rcomme
\\\kwelse
\\\quad \reg' = \mathit{index}'
\\\quad \keyworr{assume}(\reg' \in [1,n])
\\\quad \kwwhile\ \reg' \neq \S \ \kwdo
\\\qquad \kwif\ \reg' = 1\ \kwthen
\\\qquad \quad\keyword{Module^{\proc_1}_{v_1}}
\\\qquad \kwelse\ \kwif\ \reg' = 2\ \kwthen
\\\qquad \quad\keyword{Module^{\proc_1}_{v_2}}
\\\qquad \ldots
\\\qquad \kwelse\ \kwif\ \reg' = n\ \kwthen
\\\qquad \quad\keyword{Module^{\proc_1}_{v_n}}
\\\qquad \kwendif
\\\qquad \keyworr{assume}(\mathit{index}' = \#)
\\\qquad \reg' = \mathit{index}'
\\\qquad \keyworr{assume}(\mathit{index}' \neq \#)
\\\quad \kwdone
\\\quad \mathit{index} = \S
\\\quad \textcolor{red}{\keyworr{assume}(\mathit{true})}\pouto
\\\kwendif
\end{array}$
&
$\begin{array}[t]{l}
\mathit{term} = 1;
\\\reg = \mathit{index};
\\\keyworr{assume}(\reg \in [1,n])
\\\kwwhile\ \reg \neq \S \ \kwdo
\\\quad \kwif\ \reg = 1\ \kwthen
\\\quad \quad\keyword{Module^{\proc_2}_{u_1}}
\\\quad \kwelse\ \kwif\ \reg = 2\ \kwthen
\\\quad \quad\keyword{Module^{\proc_2}_{u_2}}
\\\quad \ldots
\\\quad \kwelse\ \kwif\ \reg = n\ \kwthen
\\\quad \quad\keyword{Module^{\proc_2}_{u_n}}
\\\quad \kwendif
\\\quad \keyworr{assume}(\mathit{index} = \#)
\\\quad \reg = \mathit{index}
\\\quad \keyworr{assume}(\mathit{\reg} \neq \#)
\\\kwdone
\\\mathit{validate} = 1
\\\mathit{index}' =\S
\\\textcolor{red}{\keyworr{assume}(\mathit{true})}\poutt
\end{array}$
&
$\begin{array}[t]{l}
\keyworr{assume}(y = v_i [1])
\\\keyworr{assume}(y = \#)
\\\keyworr{assume}(y = v_i [2])
\\\ldots
\\\keyworr{assume}(y = v_i [|v_i|])
\\\keyworr{assume}(y = \#)
\\x =v_i[1]
\\x = \#
\\x =v_i[2]
\\\ldots
\\x = v_i [|v_i|]
\\\mathit{index} = i
\\\mathit{index} = \#
\\~
\\\hline\hline
\multicolumn{1}{c}{ \keyword{Module^{\proc_1}_{u_i}} }
\\\hline\hline
x = u_i [1]
\\x = \#
\\x = u_i [2]
\\\ldots
\\x = u_i [|u_i|]
\\x = \#
\end{array}$
&
$\begin{array}[t]{l}
\keyworr{assume}(x = u_i [1])
\\\keyworr{assume}(x = \#)
\\\keyworr{assume}(x = u_i [2])
\\\ldots
\\\keyworr{assume}(x = u_i [|u_i|])
\\\keyworr{assume}(x = \#)
\\y =u_i[1]
\\y = \#
\\y =u_i[2]
\\\ldots
\\y = u_i [|u_i|]
\\\mathit{index}' = i
\\\mathit{index}' = \#
\end{array}$
\\\hline\hline
\end{tabular}}
\caption{Simulation of the PCP problem using two processes.}
\label{tab:prog:unde}
\end{figure*}
The rest of this section is devoted to the proof of Theorem \ref{thm:undec}. The
undecidability is obtained by a reduction from Post's Correspondence Problem (PCP) \cite{post}.
A PCP instance consists of two sequences $u_1, \ldots, u_n$ and $v_1, \ldots, v_n$ of non-empty words
over some alphabet $\Sigma$. Checking whether there exists a sequence of indices $j_1, \dots, j_k \in \{1, \dots, n\}$
s.t. $u_{j_1} \dots u_{j_k}=v_{j_1} \dots v_{j_k}$ is undecidable.
We construct a concurrent program with two processes $p_1$ and $p_2$ (see Figure \ref{tab:prog:unde}),
six memory locations $\varset=\{x, y, \mathit{validate}, \mathit{index}, \mathit{index}',\mathit{term}\}$,
and two registers $\{\reg,\reg'\}$.
The finite data domain of $\prog$ is defined as $\mathsf{Val}=\Sigma \cup \{0,1, \dots, n\} \cup \{\S,\#\}$,
where $\S$ and $\#$ are two special symbols (not in $\Sigma \cup \{0,1,\dots,n\}$).
All the locations and registers are initialized to zero. We show that reaching the instructions annotated by
$\pouto$ and $\poutt$ in $p_1, p_2$ is possible iff the PCP instance has a solution. We give below an overview of the execution steps leading to the annotated instructions.
\begin{itemize}
\item[(1)]To begin, process $p_2$ writes 1 to the location $term$.
\item[(2)]Process $p_1$
promises to write letters of $u_i$ (one by one) to location $x$, and the respective indices $i$ to the location $index$.
The number of made promises is arbitrary, since it depends on the length of the PCP solution. Observe that the sequence of promises made to the variable $index$ corresponds to the guessed solution of the PCP problem.
\item[(C1)] Using the $\kwif$ branch, $p_1$ certifies its promise before switching out of context. Note that fulfilment of promises is yet to be done.
\item[(3)] Process $p_2$ reads from the sequences of promises written to $x$ and $index$ and copies them (one by one) to variables $y$ and $index'$ respectively, and reaches $\poutt$.
\item[(4)] The $\kwelse$ branch in $p_1$ is enabled at this point, where $p_1$ reads the sequence of indices from $index'$, and
each time it reads an index $i$ from $index'$, it checks that it can read the sequence of letters of $v_i$ from $y$.
\item[(C1)] $p_1$ copies (one by one) the sequence of observed values from $y$ and $index'$ back to $x$ and $index$ respectively. To fulfil the promises,
it is crucial that the sequence of read values from $index'$ (resp. $y$) is the same as the sequence of written values to $index$ (resp. $x$).
Since $y$ holds a sequence $v_{i_1}\dots v_{i_k}$, the promises are fulfilled iff this sequence
is same as the promised sequence $u_{i_1} \dots u_{i_k}$. This happens only when $i_1, \dots, i_k$ is a PCP solution.
\item[(5)] At the end of promise fulfilment, $p_1$ reaches $\pouto$.
\end{itemize}
Let us now give more details about the code of the two processes given in Figure~\ref{tab:prog:unde}.
Depending on the value of the $\mathit{validate}$ flag read,
process $p_1$ can run in generation mode ($\kwthen$ branch) or validation mode ($\kwelse$ branch).
In generation mode,
$p_1$ writes in sequential manner the sequence of indices (alternated with the special symbol $\#$) of a potential solution of the PCP problem to the location $\mathit{index}$ and writes, letter by letter, the sequence of letters of the word $u_i$ to location $x$ each time $p_1$ sets the location $\mathit{index}$ to $i$ (using the $\keyword{Module^{\proc_1}_{u_i}}$ procedure).
In validation mode, $p_1$ reads from locations $\mathit{index}'$ and $y$
and writes back what it has read, to the locations $\mathit{index}$ and $x$, respectively (using the $\keyword{Module^{\proc_1}_{v_i}}$).
The second process proceeds in a similar manner as the $\kwelse$ branch of the first process:
It reads from locations $\mathit{index}$ and $x$ and writes the values read to $\mathit{index}'$ and $y$, respectively (using the $\keyword{Module^{\proc_2}_{u_i}}$).
We will show that a solution of the PCP problem exists iff we can reach the annotations $\pouto, \poutt$ respectively in processes $p_1, p_2$.
Assume that a solution of the PCP problem exists.
This means that there is a sequence of indices $i_1,i_2,\ldots,i_k$ such that $v_{i_1} v_{i_2} \cdots v_{i_k}= u_{i_1} u_{i_2} \cdots u_{i_k}$.
Let $w= u_{i_1} u_{i_2} \cdots u_{i_k}$.
Let us show that the pair of annotations $\pouto, \poutt$ are reachable in $\prog$.
For that aim, consider the following run of the program $\prog$:
$p_2$ starts first by setting the location $\mathit{term}$ to $1$.
Then, $p_1$ will use the $\kwthen$ branch of its conditional statement and make the two following sequences of promises
$(\mathit{index},i_1,(1,2]), (\mathit{index},i_2,(2,3]), \ldots, (\mathit{index},i_k,(k,k+1])$ and
$(x,w[1],(1,2]), (x,w[2],(2,3]), \ldots, (x,w[|w|],(|w|,|w|+1])$.
Observe that $p_1$ can certify such sequences of promises by iterating its iterative statement in the $\kwthen$ branch of its alternative statements.
Once these promises are performed, $p_2$ reads these two sequences and writes them back to the locations $\mathit{index}'$ and $y$, respectively.
$p_2$ then sets the location $validate$ to $1$.
Now $p_1$ can resume its execution by reading the location $validate$ written by the second process
and enter its $\kwelse$ branch of its alternative statement.
Then, $p_1$ will iteratively read the values written by $p_2$ on the location $\mathit{index}'$ and $y$ and write them back to the locations $\mathit{index}$ and $x$, respectively.
By doing this $p_1$ fulfils also the sequence of promises that has been issued.
Now assume that we can reach the pair of annotations
$\pouto, \poutt$.
In order for $p_1$ to reach $\pouto$, it must execute the $\kwelse$ branch of its conditional statement.
Let us assume it does so.
Then, $p_1$ will read the sequence of indices $i_1,i_2,\ldots,i_k$ written by the process $p_2$ on the location $\mathit{index}'$.
Let us assume that the process $p_2$ writes the sequence of indices $j_1,j_2,\ldots, j_m$ on the location $\mathit{index}'$ (by reading the sequence of promises made by $p_1$).
Each time that the process $p_1$ reads an index from the location $\mathit{index}'$, it writes it back on the location $\mathit{index}$.
The process $p_1$ (resp.\ $p_2$) alternates between writing/reading an index in $\{1,\dots,n\}$ and the special symbol $\#$ in order to make sure that each written index is at most read once.
In similar manner, the process $p_2$ reads the sequence of indices $j_1,j_2, \ldots, j_m$ written by the process $p_1$ on the location $\mathit{index}$ and it writes it back on the locations $\mathit{index}'$.
This implies that the sequence $j_1,j_2, \ldots, j_m$ is a subsequence of $i_1,i_2, \ldots,i_k$ (since the process $p_2$ can miss reading some written indices by the process $p_1$) and also that the sequence $i_1,i_2, \ldots,i_k$ is a subsequence of $j_1,j_2, \ldots, j_m$ (since $p_1$ can miss reading some written index by the process $p_2$).
Thus, we have that the sequences $i_1,i_2, \ldots,i_k$ and $j_1,j_2, \ldots, j_m$ are the same.
Every time the process $p_1$ (resp.\ $p_2$) reads an index $i$ from the location $\mathit{index}'$ (resp.\ $\mathit{index}$),
it (1) tries to read in sequential manner the sequence of letters appearing in $v_i$ (resp.\ $u_i$) (alternated with the special symbol $\#$) from the location $y$ (resp.\ $x$),
and (2) writes the same sequence of letters to the location $x$ (resp.\ $y$).
Using a similar argument as in the case of indices, we can deduce that if $p_1$ (resp.\ $p_2$) writes the words $v_{i_1} v_{i_2} \cdots v_{i_k}$ (resp.\ $u_{j_1} u_{j_2} \cdots u_{j_m}$), letter by letter (with an alternation with the symbol$\#$), to the location $x$ (resp.\ $y$), then $v_{i_1} v_{i_2} \cdots v_{i_k}$ (resp.\ $u_{j_1} u_{j_2} \cdots u_{j_m}$) is a subsequence of $u_{j_1} u_{j_2} \cdots u_{j_m}$ (resp.\ $v_{i_1} v_{i_2} \cdots v_{i_k}$).
Thus, if the pair of annotations $\pouto, \poutt$ are reachable then there exist two sequences $i_1,i_2, \ldots,i_k$ and $j_1,j_2, \ldots, j_m$, written, respectively, by $p_1$ and $p_2$ such that $i_1,i_2, \ldots,i_k$ is equal to $j_1,j_2, \ldots, j_m$, and
$v_{i_1} v_{i_2} \cdots v_{i_k}$ is equal to $u_{j_1} u_{j_2} \cdots u_{j_m}$.
Observe that sequence of indices $i_1,i_2,\ldots,i_k$ is non-empty due to the assume statement $\keyworr{assume}(\reg' \in [1,n] )$.
\section{Decidable Fragments of $\textsf{PS 2.0}\xspace$}
\label{sec:dec}
Since keeping ${\tt{ra}}$ memory accesses renders the reachability problem undecidable \cite{pldi2019} and so does having unboundedly many promises when having ${\tt{rlx}}$ memory accesses (Theorem \ref{thm:undec}), we address in this section the decidability problem for $\textsf{PS 2.0-rlx}$ with a bounded number of promises in any reachable configuration. Observe that bounding the number of promises in any reachable machine state does not imply that the total number of promises made during that run is bounded.
Let $\textsf{bdPS 2.0-rlx}$ represent the restriction of $\textsf{PS 2.0-rlx}$ to boundedly many promises where the number of promises in each reachable machine state is smaller or equal to a given constant. In the following, we show the decidability of the reachability problem for
$\textsf{bdPS 2.0-rlx}$. For establishing this result, we
introduce an alternate memory model for concurrent programs which we call ${\mathsf{LoHoW}}$ (for
``lossy higher order words''). We present the operational semantics of ${\mathsf{LoHoW}}$, and show that
$\textsf{PS 2.0-rlx}$ is operationally equivalent to ${\mathsf{LoHoW}}$.
Then, under the bounded promise assumption, we show how ${\mathsf{LoHoW}}$ is used to decide the reachability problem for $\textsf{bdPS 2.0-rlx}$.
\subsection{Introduction to ${\mathsf{LoHoW}}$}
Given an alphabet $A$, a simple word over $A$ is an element of $A^*$, while
a higher order word is an element of $(A^*)^*$ (i.e., word of words). A \emph{state} of ${\mathsf{LoHoW}}$ maintains a collection of higher order words, one per location, along with the states of all processes.
The higher order word $\mathfrak{ch}_x$ corresponding to the location $x$ is
a word of simple words, representing the sub memory $M(x)$
in $\textsf{PS 2.0-rlx}$. Each simple word in $\mathfrak{ch}_x$
is an ordered sequence of ``memory types'', that is,
messages or promises in the memory corresponding to $x$,
maintained in the order of their ${\textcolor{cobalt}{\tt{to}}}$ timestamps in the memory.
Unlike $\textsf{PS 2.0-rlx}$, the ${\mathsf{LoHoW}}$ does not store
timestamps in the messages and promises; instead, it takes advantage of the word order which induces a natural ordering amongst these without explicit use of timestamps. The key information to encode in each memory type occurring in $\mathfrak{ch}_x$ is:
(1) whether it is a message (${\mathsf{msg}}$) or a promise (${\tt{prm}}$),
(2) which process ($p$)
added it to the memory, and the value ($\tt{{val}}$) it holds, (3) the set $S$ (called pointer set) of processes that are aware of this message/promise (processes which point to this message/promise), and (4) whether
the time interval to the right has been reserved by some process.
\smallskip
\noindent{\bf {Memory Types}.} A \emph{memory type} is an element of
$\Sigma=\{{\mathsf{msg}}, {\tt{prm}}\} \times \mathsf{Val} \times \procset \times 2^{\procset}$ $\cup \Gamma= \{{\mathsf{msg}}, {\tt{prm}}\} \times \mathsf{Val} \times \procset \times 2^{\procset} \times \procset$.
The first component represents a message (${\mathsf{msg}}$) or a promise (${\tt{prm}}$) in the memory $M$ of $\textsf{PS 2.0-rlx}$, the second component the value in the message/promise, the third component is the process
which adds the message/promise to the memory and the fourth component is a
\emph{pointer set}, which contains all processes whose local view agree with the ${\textcolor{cobalt}{\tt{to}}}$ time stamp of the message/promise. In the case
of $\Gamma$, we have a fifth component which holds the id of the process that has
reserved the time slot to the right of this message/promise.
For a memory type $m=(r,v,p,S)$ (or $m=(r,v,p,S,q)$), we use $m.value$ to denote $v$.
For a memory type $m=(r,v,p,S)$ (resp. $m=(r,v,p,S,q)$) and a process $h \in \procset$, we use $add(m,h)$ to denote the memory type $m=(r,v,p,S \cup \{h\})$ (resp. $m=(r,v,p,S \cup \{h\},q)$). We use also $delete(m,h)$ to denote the memory type $m=(r,v,p,S \setminus \{h\})$ (resp. $m=(r,v,p,S \setminus \{h\},q)$). This corresponds to the addition/deletion of the process $h$ to/from the set of pointers of the memory type $m$.
\smallskip
\noindent{\bf{Simple Words}.} A simple word is a word $\in \Sigma^* \# (\Sigma \cup \Gamma)$,
and each $\mathfrak{ch}_x$ is a word
$\in (\Sigma^* \# (\Sigma \cup \Gamma))^+$. $\#$ is a special symbol not in $\Sigma \cup \Gamma$, which separates the last symbol from the rest
of the simple word. Consecutive symbols of $\Sigma$
in a simple word represent adjacent messages/promises
in the memory of $\textsf{PS 2.0-rlx}$, and are hence unavailable
for a RMW. The special symbol $\#$ segregates these
from the last symbol of $\Sigma \cup \Gamma$ in a simple word. $\#$ does
not correspond to any element from the memory; its job is simply
to demarcate the messages/promises which are not available for RMW
from the last symbol of the simple word.
If the last symbol in a simple word is
in $\Sigma$, then it is available for a RMW; if the last symbol is in $\Gamma$, then it is not available for a RMW since the next message adjacent to this symbol is a reservation. The last symbol from $\Sigma \cup \Gamma$ in a simple word $\Sigma^* \# (\Sigma \cup \Gamma)$ thus represents a message/promise (combined with or not a reservation) in the memory which is adjacent
to the messages represented by the symbols immediately preceding $\#$ (if any).
\vspace{0.3cm}
\begin{figure}[h]
\includegraphics[scale=.25]{ptr.pdf}
\caption{A higher order word $\mathfrak{ch}$.}
\label{fig:ch1}
\end{figure}
\vspace{0.3cm}
\noindent{\bf{Higher order words}}.
A \emph{higher order word} is a sequence of simple words. Figure \ref{fig:ch1} depicts a higher order word with four simple words. We use a left to right order
in both simple words and higher order words. Furthermore, we extend in the straightforward manner the classical word indexation strategy to higher order words. For example, the symbol at the third position of the higher order word $\mathfrak{ch}$ given in Figure \ref{fig:ch1} is $\mathfrak{ch}[3]=({\mathsf{msg}},2,p,\{p,q\})$. A higher order word $\mathfrak{ch}$ is {\em well-formed} iff for every $\proc \in \procset$, there is a unique
position $i$ in $\mathfrak{ch}$ having $p$ in its pointer set; that is, $\mathfrak{ch}[i]$ is of the form $(-, -, -, S) \in \Sigma$ or
$(-, -, -, S, -) \in \Gamma$
s.t. $p \in S$. Observe that the higher order word given in Figure \ref{fig:ch1} is well-formed. We will use ${\mathsf{ptr}}(p,\mathfrak{ch})$ to denote the unique position $i$ in $\mathfrak{ch}$ having $p$ in its pointer set. Next, we assume that all the manipulated higher order words are well-formed.
As already mentioned, for each $x \in \varset$, we have a
higher order word $\mathfrak{ch}_x$. The higher order word $\mathfrak{ch}_x$
represents the entire space $[0, \infty)$ of available timestamps. Each simple word
in $\mathfrak{ch}_x$ represents a timestamp interval $(f, t]$, with consecutive
simple words representing disjoint timestamp intervals (while preserving order). The memory types
in each simple word take up \emph{adjacent} timestamp intervals, spanning
the timestamp interval of the simple word. This adjacency of timestamp intervals
within simple words is mainly used in RMW steps and reservations.
The memory type in $\Sigma$ occurring at the end of a simple word denotes a message/promise which is available for a RMW operation. The memory type in $\Gamma$ occurring at the end of a simple word denotes a message/promise followed by a reservation and therefore it is not available for a RMW operation.
The memory types at positions other than the rightmost
in a simple word, represent messages/promises which are not available for RMW. Figure \ref{fig:ch2} presents a mapping from a memory of $\textsf{PS 2.0-rlx}$ to a collection of higher order words (one per location) in ${\mathsf{LoHoW}}$.
Given a higher order word $\mathfrak{ch}$, a position $i \in \{1,\ldots, |\mathfrak{ch}|\}$, and $p \in \procset$ , we use $add(\mathfrak{ch},p,i)$ (resp. $delete(\mathfrak{ch},p)$) to denote the higher order word $\mathfrak{ch}[1,i-1] \cdot add(\mathfrak{ch}[i],p) \cdot \mathfrak{ch}[i+1,|\mathfrak{ch}|]$ (resp. $\mathfrak{ch}[1,i-1] \cdot delete(\mathfrak{ch}[{\mathsf{ptr}}(p,\mathfrak{ch})],p) \cdot \mathfrak{ch}[i+1,|\mathfrak{ch}|]$). This corresponds to the addition/deletion of $p$ to/from the set of pointers of $\mathfrak{ch}[i]$/$\mathfrak{ch}[{\mathsf{ptr}}(p,\mathfrak{ch})]$. We use $move(\mathfrak{ch},p,i)$ to denote $add(delete(\mathfrak{ch},p),p,i)$.
\vspace{0.3cm}
\begin{figure}[h]
\centering
\newtcbox{\colorboxouline}[1][]{boxsep=0.5pt,left=0.1pt,right=0.1pt,top=1pt,bottom=1pt,colframe=magenta,colback=white,boxrule=0pt,toprule=1pt,bottomrule=1pt,sharp corners,#1}
\input{newstyle.tikzstyles}
\scalebox{0.6}{
\input{memoryps2}
}
\caption{A mapping from memories $M(x), M(y)$
to higher order words $\mathfrak{ch}_x, \mathfrak{ch}_y$, respectively.}
\label{fig:ch2}
\vspace{0.2cm}
\end{figure}
\smallskip
\noindent{\bf {Initializing higher order words}.} For each location $x \in \varset$, the initial higher order word $\mathfrak{ch}^{\rm init}_x$ is defined as \includegraphics[scale=.30]{initmsg.pdf}, where $\procset$ is the set of all processes
and $p_1$ is some process in $\procset$. The set of all higher order words $\mathfrak{ch}_x^{\rm init}$ for all locations $x$
represents the initial memory of $\textsf{PS 2.0-rlx}$ where all locations have value 0, and all processes
are aware of the initial message.
\smallskip
\noindent{\bf Simulating Reads, Writes, RMWs in ${\mathsf{LoHoW}}$.}
In the following, we informally describe how to handle $\textsf{PS 2.0-rlx}$ instructions
in ${\mathsf{LoHoW}}$. Since we only have
the ${\tt{rlx}}$ access mode, we denote
Reads, Writes and RMWs as $\wt(x,v)$, $\rd(x,v)$ and ${\tt{U}}(x, v_r, v_w)$, dropping the access modes.
\paragraph{Reads} A $\rd(x,v)$ step by a process $p$ (reading $v$ from $x$) is handled as follows in ${\mathsf{LoHoW}}$.
There exists an index $j \geq {\mathsf{ptr}}(p,\mathfrak{ch}_x)$ in $\mathfrak{ch}_x$ such that $\mathfrak{ch}_x[j]$ is of the form $(-, v, -, S')$ or $(-, v, -, S',-)$. This corresponds to the existence of a memory type holding the value $v$ in $\mathfrak{ch}_x$ and this symbol is on the right of the current view/pointer of the process $p$.
Add $p$ to the set of pointers $S'$ and remove it from its previous position.
\paragraph{Writes} A $\wt(x,v)$ step by a process $p$ (writing the value $v$ to the location $x$) in $\textsf{PS 2.0-rlx}$ is done by adding a new message with a timestamp
higher than the local view of $p$ for $x$: the timestamp interval of this new message can be adjacent
to the timestamp of the local view of $p$, or much ahead. These two possibilities
are captured in ${\mathsf{LoHoW}}$ as follows.
(1) Add the simple word \includegraphics[scale=.22]{wtmsg.pdf} to $\mathfrak{ch}_x$ to the right of ${\mathsf{ptr}}(p,\mathfrak{ch}_x)$, or
(2) there is a symbol $\alpha \in \Sigma$ and two words $w$ and $w'$ such that $\mathfrak{ch}_x=w \cdot \# \cdot\alpha\cdot w'$. Then, update the higher order word $\mathfrak{ch}_x$ to $ w\cdot \alpha\cdot \# \cdot({\mathsf{msg}}, v, p, \{p\})\cdot w'$.
Finally, remove $p$ from its previous pointer set.
\paragraph{(RMW)} Capturing RMWs is similar to the execution of a read followed by a write.
In $\textsf{PS 2.0-rlx}$, a process $p$ performing RMW reads from a message with a timestamp interval
$(,t]$ and adds a message to the memory with timestamp interval $(t,-]$. This is handled as follows in ${\mathsf{LoHoW}}$, and shows
the need for the higher order words. Consider a ${\tt{U}}(x, v_r, v_w)$ step by $p$. Then,
there is a simple word \includegraphics[scale=.22]{rmw.pdf} in $\mathfrak{ch}_x$ having
$(-, v_r, -, S)$ as the last memory type in it, and the position of the memory type $(-, v_r, -, S)$ is on the right of the current pointer of $p$ in $\mathfrak{ch}_x$.
$p$ is removed from its pointer set,
$\#(-, v_r, -, S)$ is replaced with $(-, v_r, -, S\backslash\{p\})\#$ and
$(-, v_w, p, \{p\})$ is appended, resulting in extending
\includegraphics[scale=.22]{rmw.pdf} to
\includegraphics[scale=.22]{rmw1.pdf}.
\begin{example}
We illustrate the read, write and RMW in ${\mathsf{LoHoW}}$ on an example. Figure \ref{fig:sim} depicts a run in $\textsf{PS 2.0-rlx}$ and the corresponding run in ${\mathsf{LoHoW}}$. The run of $\textsf{PS 2.0-rlx}$ shows how the memory evolves, and the corresponding run
in ${\mathsf{LoHoW}}$ faithfully simulates this using higher order words $\mathfrak{ch}_x$ and $\mathfrak{ch}_y$.
\begin{center}
\begin{tabular}[t]{c||c}
\begin{lstlisting}[style=examples,tabsize=3]
x:=1
y:=2
x:=3
\end{lstlisting}
&
\begin{lstlisting}[style=examples,tabsize=3]
x:=5
$r1:=x //3
$r2:= FADD(y,1) //2
\end{lstlisting}
\end{tabular}
\end{center}
\begin{figure}[ht]
\includegraphics[scale=.14]{sim2.pdf}
\caption{Below, a run in $\textsf{PS 2.0}\xspace$ showing the changes to memory, and above, the corresponding run in ${\mathsf{LoHoW}}$. Observe that $init$ stands for the initial memory.}
\label{fig:sim}
\end{figure}
\end{example}
\noindent{\bf Promises in ${\mathsf{LoHoW}}$.}
Next, we discuss how to handle promises.
\paragraph{Promises} Handling promises made by a process $p$ in $\textsf{PS 2.0-rlx}$ is similar to handling $\wt(x,v)$: we add the simple word \includegraphics[scale=.22]{prm.pdf} in $\mathfrak{ch}_x$
to the right of the position ${\mathsf{ptr}}(p, \mathfrak{ch}_x)$, or
append $({\tt{prm}}, v, p, \{\})$ at the end of a simple word with a position larger than ${\mathsf{ptr}}(p,\mathfrak{ch}_x)$.
Other than tagging the symbol as a promise (${\tt{prm}}$), the pointer set is empty.
\smallskip
\noindent{\bf Reservations and Cancellations in ${\mathsf{LoHoW}}$.} Next, we come to one of the new features of $\textsf{PS 2.0}\xspace$ over the first version, namely, reservations and cancellations. In $\textsf{PS 2.0-rlx}$, a process $p$ makes a reservation by adding the pair $(x, (f,t])$
to the memory, given that there is a message/promise in the memory with timestamp interval $(-,f]$. In ${\mathsf{LoHoW}}$ this is captured by ``tagging''
the rightmost memory type (message/promise) in a simple word with the name of the process that makes the reservation. This requires us to consider the memory types from $\Gamma=\{{\mathsf{msg}}, {\tt{prm}}\} \times \mathsf{Val} \times \procset \times 2^{\procset}\times \procset$
where the last component stores the process which made the reservation.
Such a memory type always appears at the end of a simple word, and represents that the next timestamp interval adjacent to it has been reserved. Observe that we can not add new memory types to the right of a memory type of the form
$({\mathsf{msg}}, v, p, S, q)$.
Thus, reservations are handled as follows.
\begin{enumerate}
\item[(Res)]
Assume the rightmost symbol
in a simple word as $({\mathsf{msg}}, v, p, S)$. To capture the
reservation by $q$, $({\mathsf{msg}}, v, p, S)$ is replaced with $({\mathsf{msg}}, v, p, S,q)$.
\item[(Can)] A cancellation is done by removing the last component
$q$ from $({\mathsf{msg}}, v, p, S,q)$ resulting
in $({\mathsf{msg}}, v, p, S)$.
\end{enumerate}
\noindent{\bf Empty Memory Types, Redundant simple words.}
When a process $p$
reads from a message, the pointer of $p$ is updated, and moves forward. As a result, we may have memory types of the form $({\mathsf{msg}}, v, p, \{\})$ as well as $({\mathsf{msg}}, v, p, \{\},q)$
representing those messages in the memory whose pointer set is empty.
Call such symbols of $\Sigma \cup \Gamma$ \emph{empty memory types}. It is then possible to lose an \emph{empty memory type} of $\Sigma$ from a simple word if it is not at the rightmost position. This will not have any consequence with respect to the reachability problem, since processes can non-deterministically skip reading some messages in the memory. Likewise, a simple word of the form $w \# m \in \Sigma^* \# (\Sigma \cup \Gamma)$ where all symbols
in $w$ are empty memory types from $\Sigma$ and $m$ is an empty memory type from $\Sigma \cup \Gamma$
can be lost entirely. Such simple words are called \emph{redundant simple words}. Given this, what cannot be lost from $\mathfrak{ch}_x$? The following:
\begin{itemize}[leftmargin=*]
\item memory types $({\tt{prm}}, -,-,-)$ or $({\tt{prm}}, -,-,-,-)$
representing promises. This is due to the fact promises should be fulfilled and therefore can not be lost.
\item non \emph{empty memory types}: the pointer set of these contain at least one process. Since losing any of these memory types will result in losing the pointer/view of at least one of the processes.
\item Only rightmost memory type (right next to $\#$) in a simple word. Losing only this memory type will result in a non well-defined higher order word.
\end{itemize}
\smallskip
\noindent{\bf Certification and Fulfilment.}
In $\textsf{PS 2.0-rlx}$, certification, for a process $\proc$, happens from the capped memory, where
intermediate time slots (other than reserved ones) are blocked, and any new message can be added
only at the maximal timestamp.
This is handled in ${\mathsf{LoHoW}}$ by one of the following:
\begin{itemize}[leftmargin=*]
\item addition
of new memory types is only allowed only at the right end of any $\mathfrak{ch}_x$,
\item If the rightmost memory type $m$ in $\mathfrak{ch}_x$ is of the form $(-, v, -, -, q)$ with $q \neq p$ (i.e., tagged by a reservation for $q$), then a simple word $\#({\mathsf{msg}}, v, q,\{\})$ is appended at the end of $\mathfrak{ch}_x$.
\end{itemize}
Memory is altered in $\textsf{PS 2.0-rlx}$ during certification phase to check for promise fulfilment,
and at the end of the certification phase,
we resume from the memory
which was there before. To capture this in ${\mathsf{LoHoW}}$, we work on a duplicate of $(\mathfrak{ch}_x)_{x \in \varset}$
in the certification phase. Notice that the duplication
allows losing some of empty memory types and redundant simple words non deterministically (as described in the previous paragraph).
This copy of $\mathfrak{ch}_x$ is then modified during certification, and
is discarded once we finish the certification phase.
The fulfilment of a promise by $p$ using the rule $\stackrel{L}{\hookleftarrow}$ (see rule $\mathsf{(MEMORY: FULFILL)}$ in Figure \ref{program_sem}) will be handled in a similar manner as using the rule $\stackrel{A}{\hookleftarrow}$ (since we are only dealing with the fragment of $\textsf{PS 2.0}\xspace$ restricted to ${\tt{rlx}}$). This will result in replacing a memory type of the form $({\tt{prm}}, v, p, S)$ (resp. $({\tt{prm}}, v, p, S,q)$) by $({\mathsf{msg}}, v, p, S)$ (resp. $({\mathsf{msg}}, v, p, S,q)$) if this memory type is in a position which is on the right of the current pointer of the process $p$. Then, the process $p$ is added to the pointer set $S$ while removing it from the previous pointer set it belongs to.
The fulfilment of a promise by a process $p$ in $\textsf{PS 2.0}\xspace$ using the rule $\stackrel{S}{\hookleftarrow}$ (see rule $\mathsf{(MEMORY: FULFILL)}$ in Figure \ref{program_sem}) results in splitting the intervals of the promise, when adding a new message $\msgnew{\xvar}{v'}{f}{t}{\bot}$ to the memory.
To capture this, we allow insertion
of a memory type right before the promise whose interval
is split. This will result in replacing a memory type of the form $({\tt{prm}}, v, p, S)$ (resp. $\#({\tt{prm}}, v, p, S,q)$) by $({\mathsf{msg}}, v', p, \{p\}) ({\tt{prm}}, v, p, S)$ (resp. $({\mathsf{msg}}, v', p, \{p\}) \# ({\tt{prm}}, v, p, S,q)$) if this memory type is in a position which is on the right of the current pointer of the process $p$. Then, the process $p$ is removed from the previous pointer set it belongs to. We may also need to update the position of the separator $\#$ so that it is just before the last symbol of a simple word.
\smallskip
\noindent{\bf SC fences}. SC-fences are handled by adding a dummy process $g$
to $\procset$. Whenever a process $p$ performs a SC fence,
$g, p$ are added to the same pointer set, by moving
$g$ ($p$) to the pointer set of $p$ ($g$) depending on which is
more to the right.
\begin{example}
Figure \ref{fig:sim3} illustrates a run in ${\mathsf{LoHoW}}$ on a program where promises are necessary to reach the annotated part $\textcolor{sangria}{//}$.
To reach the annotated part in P1, the execution proceeds as follows. C1, C2 represent
two certification phases.
\begin{enumerate}
\item[(1)] P1 promises the write of 42 to $x$, by a message $(x, 42, (f,t], \bot)$.
\item[(C1)] To certify, P1 begins from the capped memory,
and enters the else branch. It begins a duplicate of the higher order words, and works on them in this phase.
\begin{itemize}
\item Since all positions in $(0,t]$ are blocked,
P1 splits the interval $(f,t]$ to write 41 to
$x$, and modifies the memory to $(x, 42, (t',t], \bot)$,
$(x, 41, (f,t'], \bot)$.
\item P1 fulfils its promise
\end{itemize}
\item[(2)] P2 reads 42 from $x$ and writes 42 to $z$
\item[(3)] P1 reads 42 from $z$
\item[(4)] P1 fulfils its promise, and reaches the annotated part.
\end{enumerate}
\begin{figure}[ht]
\includegraphics[scale=0.13]{sim3.pdf}
\caption{Run in ${\mathsf{LoHoW}}$. The certification phase works on the duplicates of $\mathfrak{ch}_x, \mathfrak{ch}_z$ denoted in yellow.}
\label{fig:sim3}
\end{figure}
\label{eg:sim3}
\end{example}
\subsection{Formal Model of ${\mathsf{LoHoW}}$}
\label{sec:formal}
In the following, we formally define ${\mathsf{LoHoW}}$ and state the equivalence of the reachability problem in $\textsf{PS 2.0-rlx}$
and ${\mathsf{LoHoW}}$.
\smallskip
\noindent{\bf Insertion into higher order words.}
A higher order word $\mathfrak{ch}$ can be extended in position $1 \leq j\leq |\mathfrak{ch}|$ with a memory type $m$ of the form $(r, v, p, \{p\})$ in a number of ways:
\smallskip
\noindent
$\bullet$ {\it Insertion as a new simple word.}
$\mathfrak{ch} \underset{j}{\stackrel{N}{\hookleftarrow}} m$ is defined only if
$\mathfrak{ch}[j-1]= \#$ (i.e., the position $j$ is the end of a simple word). Let $\mathfrak{ch}'$ be the higher order word defined as $delete(\mathfrak{ch},p)$ (i.e., removing $p$ from its previous set of pointers). Then, the extended higher order $\mathfrak{ch} \underset{j}{\stackrel{N}{\hookleftarrow}} m$ is defined as $\mathfrak{ch}'[1,j] \cdot \# m \cdot \mathfrak{ch}'[j+1,|\mathfrak{ch}|]$ (i.e., inserting the new simple word just after the position $j$).
\smallskip
\noindent
$\bullet$ {\it Insertion at the end of a simple word.}
$\mathfrak{ch} \underset{j}{\stackrel{E}{\hookleftarrow}} m$ is defined only if
$\mathfrak{ch}[j-1]= \#$ (i.e., the position $j$ is the end of a simple word) and $\mathfrak{ch}[j] \in \Sigma$ (i.e., the last memory type in the simple word should be free from reservations). Let $\mathfrak{ch}'$ be the higher order word defined as $delete(\mathfrak{ch},p)$. Then, the extended higher order $\mathfrak{ch} \underset{j}{\stackrel{E}{\hookleftarrow}} m$ is defined as $w_1 \cdot m' \cdot \# m \cdot w_2$ with $\mathfrak{ch}'= w_1 \cdot \# m' \cdot w_2 $, and $m' \in \Sigma$, and $|w_1 \cdot \# m'|= j$ (i.e., inserting the new memory type just after the position $j$).
\smallskip
\noindent
$\bullet$ {\it Splitting a promise.}
$\mathfrak{ch} \underset{j}{\stackrel{SP}{\hookleftarrow}} m$ is defined only if
$\mathfrak{ch}[j]$ is of the form $({\tt{prm}},-,p,-)$ or $({\tt{prm}},-,p,-,-)$ (i.e., the memory type at position $j$ is a promise). Let $\mathfrak{ch}'$ be the higher order word defined as $delete(\mathfrak{ch},p)$. Then, the extended higher order $\mathfrak{ch} \underset{j}{\stackrel{SP}{\hookleftarrow}} m$ is defined as $(1)$ $\mathfrak{ch}'[1,j-2] \cdot m \cdot \# m' \cdot \mathfrak{ch}'[j+1,|\mathfrak{ch}|]$ if $\mathfrak{ch}'[j]=m'$ and $\mathfrak{ch}'[j-1]=\#$, or $(2)$ $\mathfrak{ch}'[1,j-1] \cdot m \cdot m' \cdot \mathfrak{ch}'[j+1,|\mathfrak{ch}|]$ if $\mathfrak{ch}'[j]=m'$ and $\mathfrak{ch}'[j-1] \neq \#$. Observe that in both cases we are inserting
the new memory type $m$ just before the position $j$.
\smallskip
\noindent
$\bullet$ {\it Fulfilment of a promise.}
$\mathfrak{ch} \underset{j}{\stackrel{FP}{\hookleftarrow}} m$ is defined only if
$\mathfrak{ch}[j]$ is of the form $({\tt{prm}},v,p,S)$ or $({\tt{prm}},v,p,S,q)$. Let $\mathfrak{ch}'$ be the higher order word defined as $delete(\mathfrak{ch},p)$. Then, the extended higher order $\mathfrak{ch} \underset{j}{\stackrel{FP}{\hookleftarrow}} m$ is defined as $\mathfrak{ch}'[1,j-1] \cdot m' \cdot \mathfrak{ch}'[j+1,|\mathfrak{ch}'|]$ with $m'=({\mathsf{msg}},v,p,S\cup \{p\})$ if $\mathfrak{ch}[j]=({\tt{prm}},v,p,S)$ and $m'=({\mathsf{msg}},v,p,S\cup \{p\},q)$ if $\mathfrak{ch}[j]=({\tt{prm}},v,p,S,q)$.
\smallskip
\noindent
$\bullet$ {\it Splitting a reservation.}
$\mathfrak{ch} \underset{j}{\stackrel{SR}{\hookleftarrow}} m$ is defined only if
$\mathfrak{ch}[j]$ is of the form $(r',v',q,S,p)$. Let $\mathfrak{ch}'$ be the higher order word defined as $delete(\mathfrak{ch},p)$. Then, the extended higher order $\mathfrak{ch} \underset{j}{\stackrel{SR}{\hookleftarrow}} m$ is defined as $\mathfrak{ch}'[1,j-2] \cdot (r',v',q,S)\cdot \# (r,v,p,\{p\},p) \cdot \mathfrak{ch}'[j+1,|\mathfrak{ch}|]$. Observe that the new message $ (r,v,p,\{p\},p)$ is added to the right of the position $j$ which corresponds to the slot that has been reserved by $p$. This special splitting rule will be used during the certification phase. This will allow the process $p$ to use the reserved slots. Recall that it is not allowed to add memory types in the middle of the higher order words (other than the reserved ones) during the certification phase.
\smallskip
\noindent{\bf Making/Canceling a reservation.}
A higher order word $\mathfrak{ch}$ can also be modified through making/cancelling a reservation at a position $1 \leq j\leq |\mathfrak{ch}|$ by a process $p$. Thus, we define the operation ${Make}(\mathfrak{ch},p,j)$ (resp. ${Cancel}(\mathfrak{ch},p,j)$) that reserves (resp. cancels) a time slot at the position $j$. ${Make}(\mathfrak{ch},p,j)$ (resp. ${Cancel}(\mathfrak{ch},p,j)$) is only defined if $\mathfrak{ch}[j]$ is of the form $(r,v,q,S)$ (resp. $(r,v,q,S,p)$) and $\mathfrak{ch}[j-1]=\#$. Then, the extended higher order ${Make}(\mathfrak{ch},p,j)$ (resp. ${Cancel}(\mathfrak{ch},p,j)$) is defined as $\mathfrak{ch}[1,j-1] \cdot (r,v,q,S,p)\cdot \mathfrak{ch}[j+1,|\mathfrak{ch}|]$ (resp. $\mathfrak{ch}[1,j-1] \cdot (r,v,q,S)\cdot \mathfrak{ch}[j+1,|\mathfrak{ch}|]$).
\smallskip
\noindent{\bf{Process configuration in ${\mathsf{LoHoW}}$}.} A configuration of $\proc \in \procset$ in ${\mathsf{LoHoW}}$ consists of a pair $(\sigma, {{\mathsf{\bf HW}}})$ where $(1)$ $\sigma$ is the process state maintaining the
instruction label and the register values (see Subsection \ref{sec:ps}), and ${{\mathsf{\bf HW}}}$ is a mapping from the set of locations to higher order words.
The transition relations $\xrightarrow[\proc]{{\tt{std}}}$ and $\xrightarrow[\proc]{{\tt{cert}}}$ between process configuration is given in Figure \ref{ps-program_sem}. The transition relation $\xrightarrow[\proc]{{\tt{cert}}}$ is used only in the certification phase while $\xrightarrow[\proc]{{\tt{std}}}$ is used to simulate the standard phase of $\textsf{PS 2.0-rlx}$.
A read operation in both phases (standard and certification) is handled by reading a value from a memory type which is on the right of the current pointer of $\proc$. A write operation, in the standard phase, can result in the insertion, on the right of the current pointer of $p$, of a new memory type at the end of a simple word or as a new simple word. The memory type resulting from a write in the certification phase is only allowed to be inserted at the end of the higher order word or at the reserved slots (using the rule splitting a reservation). Write can also be used to fulfil a promise or to split a promise (i.e., partial fulfilment) during the both phases. Making/canceling a reservation will result in tagging/untagging a memory type at the end of a simple word on the right of the current pointer of $p$. The case of RMW is similar to a read followed by a write operations (whose resulting memory type should be inserted to the right of the read memory type). Finally, a promise can only be made during the standard phase and the resulting memory type will be inserted at the end of a simple word or as a new word on the right of the current pointer of $p$.
\input{fig-inf-rules-queue.tex}
\smallskip
\noindent{\bf{Losses in ${\mathsf{LoHoW}}$.}}
Let $\mathfrak{ch}$ and $\mathfrak{ch}'$ be two higher order words in $(\Sigma^* \# (\Sigma \cup \Gamma))^+$. Let us assume that $\mathfrak{ch}= u_1 \#a_1 u_2 \# a_2 \dots u_k \# a_k$
and $\mathfrak{ch}'= v_1 \# b_1 v_2 \# b_2 \dots v_m \# b_m$, with
$u_i, v_i \in \Sigma^*$ and $a_i, b_j \in \Sigma \cup \Gamma$. We extend the subword relation $\sqsubseteq$ to higher order word as follows:
$\mathfrak{ch} \sqsubseteq \mathfrak{ch}'$ iff there is a strictly increasing function
$f: \{1, \dots, k\} \rightarrow \{1, \dots, m\}$ s.t.
$(1)$ $u_i \sqsubseteq v_{f(i)}$ for all $1 \leq i \leq k$, $(2)$
$a_i=b_{f(i)}$, and $(3)$ we have the same number of memory types of the form $({\tt{prm}},-,-,-)$ or $({\tt{prm}},-,-,-,-)$ in $\mathfrak{ch}$ and $\mathfrak{ch}'$.
The relation $\sqsubseteq$ corresponds to the loss of some special empty memory types and redundant simple words (as explained earlier).
The relation $\sqsubseteq$ is extended to mapping from locations to higher order words as follows: ${{\mathsf{\bf HW}}} \sqsubseteq {{\mathsf{\bf HW}}'}$ iff ${{\mathsf{\bf HW}}}(x) \sqsubseteq {{\mathsf{\bf HW}}'}(x)$ for all $x \in \varset$.
\smallskip
\noindent{\bf{${\mathsf{LoHoW}}$ states}.}
A ${\mathsf{LoHoW}}$ state ${\mathfrak{st}}$ is a tuple $(({\sf J}, {\sf R}), {\mathsf{\bf HW}})$
where
${\sf J} : \procset \mapsto \mathbb{L}$ maps each process $p$ to the label of the next instruction to be executed,
${\sf R} : \regset \rightarrow \mathsf{Val}$ maps each register to its current value, and ${\mathsf{\bf HW}}$ is a mapping from locations to higher order words. The initial ${\mathsf{LoHoW}}$ state ${\mathfrak{st}}_{\rm init}$ is defined as
$(({\sf J}_{\rm init}, {\sf R}_{\rm init}), {\mathsf{\bf HW}}_{\rm init})$ where:
(1) ${\sf J}_{\rm init}(p)$ is the label of the initial instruction of $\proc$; (2) ${\sf R}_{\rm init}(\reg)=0$ for every register $\reg \in \regset$; and $(3)$ ${\mathsf{\bf HW}}_{\rm init}(x)=\mathfrak{ch}^{\rm init}_x$ for all $x \in \varset$.
Now we are ready to define the induced transition relation between ${\mathsf{LoHoW}}$ states. For two ${\mathsf{LoHoW}}$ states ${\mathfrak{st}}=(({\sf J}, {\sf R}), {\mathsf{\bf HW}})$ and ${\mathfrak{st}}'=(({\sf J}', {\sf R}'), {\mathsf{\bf HW}}')$ and $a \in \{{\tt{std}},{\tt{cert}}\}$, we write
${\mathfrak{st}} \xrightarrow[p]{a} {\mathfrak{st}}' $ iff one of the following cases holds: $(1)$ $(({\sf J}(p), {\sf R}), {\mathsf{\bf HW}}) \xrightarrow[p]{a} (({\sf J}'(p), {\sf R}'), {\mathsf{\bf HW}}')$ and ${\sf J}(p')={\sf J}'(p')$ for all $p' \neq p$, or $(2)$ $({\sf J}, {\sf R})=({\sf J}', {\sf R}')$ and ${{\mathsf{\bf HW}}} \sqsubseteq {{\mathsf{\bf HW}}'}$.
\smallskip
\noindent{\bf{Two phases ${\mathsf{LoHoW}}$ states}}. A two-phases state of ${\mathsf{LoHoW}}$ is $\mathsf{St}=(\pi, p, {\mathfrak{st}}_{{\tt{std}}}, {\mathfrak{st}}_{{\tt{cert}}})$
where $\pi \in \{{\tt{cert}}, {\tt{std}}\}$ is a flag describing whether
the ${\mathsf{LoHoW}}$ is in ``standard'' phase or ``certification'' phase, $p$ is the process which evolves in one of these phases,
while ${\mathfrak{st}}_{{\tt{std}}}$, ${\mathfrak{st}}_{{\tt{cert}}}$ are two ${\mathsf{LoHoW}}$ states (one for each phase).
When the ${\mathsf{LoHoW}}$ is in the standard phase, then ${\mathfrak{st}}_{{\tt{std}}}$
evolves, and when the ${\mathsf{LoHoW}}$ is in certification phase,
${\mathfrak{st}}_{{\tt{cert}}}$ evolves. A two-phases ${\mathsf{LoHoW}}$ state is said to be initial if it is of the form $({\tt{std}}, p, {\mathfrak{st}}_{\rm init}, {\mathfrak{st}}_{\rm init})$, where
$p \in \procset$ is any process. The transition relation $\rightarrow$
between two-phases ${\mathsf{LoHoW}}$ states is defined as follows: Given $\mathsf{St}=(\pi, p, {\mathfrak{st}}_{{\tt{std}}}, {\mathfrak{st}}_{{\tt{cert}}})$ and $\mathsf{St}'=(\pi', p', {\mathfrak{st}}'_{{\tt{std}}}, {\mathfrak{st}}'_{{\tt{cert}}})$, we have $\mathsf{St} \rightarrow \mathsf{St}'$ iff one of the following cases hold:
\begin{itemize}[leftmargin=*]
\item {\bf During the standard phase.} $\pi=\pi'={\tt{std}}$, $p=p'$, ${\mathfrak{st}}_{{\tt{cert}}}={\mathfrak{st}}'_{{\tt{cert}}}$ and ${\mathfrak{st}}_{{\tt{std}}} \xrightarrow[p]{{\tt{std}}}{\mathfrak{st}}'_{{\tt{std}}}$. This corresponds to a simulation of a standard step of the process $p$.
\item {\bf During the certification phase.} $\pi=\pi'={\tt{cert}}$, $p=p'$, ${\mathfrak{st}}_{{\tt{std}}}={\mathfrak{st}}'_{{\tt{std}}}$ and ${\mathfrak{st}}_{{\tt{cert}}} \xrightarrow[p]{{\tt{cert}}}{\mathfrak{st}}'_{{\tt{cert}}}$. This corresponds to a simulation of a certification step of the process $p$.
\item {\bf From the standard phase to the certification phase.} $\pi={\tt{std}}$, $\pi'={\tt{cert}}$, $p=p'$, ${\mathfrak{st}}_{{\tt{std}}}={\mathfrak{st}}'_{{\tt{std}}}= (({\sf J}, {\sf R}), {\mathsf{\bf HW}})$, and ${\mathfrak{st}}'_{{\tt{cert}}}$ is of the form $(({\sf J}, {\sf R}), {\mathsf{\bf HW}}')$ where for every $x \in \varset$, ${\mathsf{\bf HW}}'(x)={\mathsf{\bf HW}}(x) \# ({\mathsf{msg}},v,q,\{\})$ if ${\mathsf{\bf HW}}(x)$ is of the form $w \cdot \# (-,v,-,-,q)$ with $q \neq p$, and ${\mathsf{\bf HW}}'(x)={\mathsf{\bf HW}}(x)$ otherwise. This corresponds to the copying of the standard ${\mathsf{LoHoW}}$ state to the certification ${\mathsf{LoHoW}}$ state in order to check if the set of promises made by the process $p$ can be fulfilled. The higher order word ${\mathsf{\bf HW}}'(x)$ (at the beginning of the certification phase) is almost the same as ${\mathsf{\bf HW}}(x)$ (at the end of the standard phase) except when the rightmost memory type $(-,v,-,-,q)$ of ${\mathsf{\bf HW}}(x)$ is tagged by a reservation of a process $q \neq p$. In that case, we append the memory type $ ({\mathsf{msg}},v,q,\{\})$ at the end of ${\mathsf{\bf HW}}(x)$ to obtain ${\mathsf{\bf HW}}'(x)$. Note that this is in accordance
to the definition of capping memory before going into certification: to cite, (item 2 in capped memory of \cite{promising2}), a cap message
is added for each location unless it is a reservation made by the process going in for certification.
It is easy to see that this transition rule can be implemented by a sequence of transitions
which copies one symbol at a time, from ${\mathsf{\bf HW}}$ to ${\mathsf{\bf HW}}'$.
\item {\bf From the certification phase to standard phase.} $\pi={\tt{cert}}$, $\pi'={\tt{std}}$, ${\mathfrak{st}}_{{\tt{std}}}={\mathfrak{st}}'_{{\tt{std}}}$, ${\mathfrak{st}}_{{\tt{cert}}}={\mathfrak{st}}'_{{\tt{cert}}}$, and ${\mathfrak{st}}_{{\tt{cert}}}$ is of the form $(({\sf J}, {\sf R}), {\mathsf{\bf HW}})$ with ${\mathsf{\bf HW}}(x)$ does not contain any memory type of the form $({\tt{prm}},-,p,-)$/$({\tt{prm}},-,p,-,-)$ for all $x \in \varset$ (i.e., all promises made by $p$ are fulfilled).
\end{itemize}
\smallskip
\noindent{\bf {The Reachability Problem in ${\mathsf{LoHoW}}$}}.
Given an instruction label function $J: \procset \rightarrow \mathbb{L}$ that maps each $\proc \in \procset$ to a label in $ \mathbb{L}_{\proc}$,
the \emph{reachability} problem in ${\mathsf{LoHoW}}$ asks
whether there exists a two phases ${\mathsf{LoHoW}}$ state $\mathsf{St}$ of the form $({\tt{std}}, -, ((J,R), {\mathsf{\bf HW}}), ((J',R'), {\mathsf{\bf HW}}'))$ s.t. $(1)$ ${\mathsf{\bf HW}}(x)$ and ${\mathsf{\bf HW}}'(x) $ do not contain any memory type of the form $({\tt{prm}},-,p,-)$/$({\tt{prm}},-,p,-,-)$ for all $x \in \varset$, and $(2)$ $\mathsf{St}$ is reachable in ${\mathsf{LoHoW}}$ (i.e., $\mathsf{St}_0 \rtstep{\xrightarrow[]{}} \mathsf{St}'$
where $\mathsf{St}_0$ is an initial two-phases ${\mathsf{LoHoW}}$ states).
In the case of a positive answer to this problem, we say that $J$ is reachable in $\prog$ in ${\mathsf{LoHoW}}$.
\begin{theorem}
\label{equivalence}
An instruction label function $J$ is reachable in a program $\prog$ in ${\mathsf{LoHoW}}$ iff $J$ is reachable in $\prog$ in $\textsf{PS 2.0-rlx}$.
\label{thm:eqv}
\end{theorem}
\subsection{Decidability of ${\mathsf{LoHoW}}$ with Bounded Promises}
\label{sec:decproof}
The equivalence of the reachability
in ${\mathsf{LoHoW}}$ and $\textsf{PS 2.0-rlx}$, coupled with Theorem \ref{thm:undec} shows that reachability is undecidable in ${\mathsf{LoHoW}}$.
To recover decidability, we look at ${\mathsf{LoHoW}}$ with only bounded number of the promise memory type in any higher order word. Let K-${\mathsf{LoHoW}}$ denote ${\mathsf{LoHoW}}$ with a number of promises bounded by $K$. (Observe that K-${\mathsf{LoHoW}}$ corresponds to $\textsf{bdPS 2.0-rlx}$.)
\begin{theorem}
The reachability problem is decidable for K-${\mathsf{LoHoW}}$.
\label{decidability-qs} \end{theorem}
As a corollary of Theorem \ref{decidability-qs}, the decidability
of reachability follows for $\textsf{bdPS 2.0-rlx}$.
The proof makes use of the framework of \emph{Well-Structured Transition Systems} (WSTS) \cite{wsts2,wsts1}, and follows from lemmas \ref{lem:wqo} to \ref{computing-pre}.
\smallskip
\noindent{\bf Well-Structured Transition Systems (WSTS).}
We recall the main ingredients of WSTS.
For more details, the reader is referred to \citet{wsts1,wsts2}.
\smallskip
\noindent{\it Well-quasi Orders.}
Given a (possibly infinite set) $C$,
a quasi-order on $C$ is a reflexive and transitive relation ${\preceq} \subseteq C \times C$.
An infinite sequence $c_1, c_2, \dots$ in $C$ is said to be saturating if there exists
indices $i < j$ s.t. $c_i \preceq c_j$. A quasi-order $\preceq$ is said to be a well-quasi order (wqo)
on $C$ if every infinite sequence in $C$ is saturating. Given a quasi-order $\preceq$ on $C$,
the \emph{embedding order} $\sqsubseteq$ on $C^*$ (i.e., the set of finite words over $C$) is defined as
$a_1a_2 \dots a_m \sqsubseteq b_1 b_2 \dots b_n$ if there exists a strictly increasing
function $g: \{1,2,\dots,m\} \rightarrow \{1, 2, \dots, n\}$ s.t. for all $1 \leq i \leq m$,
$a_i \preceq b_{g(i)}$. It is well-known that if
$\preceq$ is a wqo on $C$, then the embedding order $\sqsubseteq$ is also a wqo on $C^*$ \cite{higman}.
\smallskip\noindent{\it Upward Closure.}
Given a wqo $\preceq$ on a set $C$, a set $U \subseteq C$ is upward closed if for every $a \in U$ and $b \in C$, with $a \preceq b$,
we have $b \in U$. The upward closure of a set $U \subseteq C$ is
$\upclos{U} =\{b \in C \mid \exists a \in U, a \preceq b\}$.
It is known that every upward closed set $U$ can be characterized by a finite \emph{minor}.
A minor $M \subseteq U$ is s.t.\
(i) for each $a \in U$, there is a $b \in M$ s.t.\ $b \preceq a$, and
(ii) for all $a, b \in M$ s.t. $a \preceq b$, we have $a=b$.
For an upward closed set $U$, let $\mathsf{min}$ be the function that returns the minor of $U$.
\smallskip\noindent{\it Well-Structured Transition Systems (WSTS)}.
Let $\mathcal{T}$ be a transition system with (possibly infinite) set of states $C$, initial states $C_{\mathsf{init}}$ and transition relation $\rightsquigarrow \subseteq C \times C$.
Let $\preceq$ be a well-quasi ordering on $C$.
We define the set of predecessors of a subset $U \subseteq C$ of states as ${\mathtt{Pre}}(U)=\{c \in C \mid \exists c' \in U.\; c \rightsquigarrow c'\}$.
For a state $c$, we denote the set $\mathtt{min}(\mathtt{Pre}(\upclos{\{c\}}) \cup \upclos{\{c\}})$ as $\mathtt{minpre}(c)$.
$\mathcal{T}$ is called well-structured if $\rightsquigarrow$ is \emph{monotonic} w.r.t.\ $\preceq$ : that is, given $c_1, c_2$ and $c_3$ in $C$, if $c_1 \rightsquigarrow c_2$ and $c_1 \preceq c_3$, then there exists a state $c_4$ s.t. $c_3 \stackrel{*}{\rightsquigarrow} c_4$ and $c_2 \preceq c_4$.
Given a finite set of states $C_{\tt{target}} \subseteq C$,
the \emph{coverability} problem asks if there is a state $c' \in \upclos{C_{\tt{target}}}$ reachable in $\mathcal{T}$.
The following conditions are sufficient for the decidability of this problem:
(i) for every two states $c_1, c_2 \in C$, it is decidable if $c_1 \preceq c_2$,
(ii) for every $c \in C$, we can check if $\upclos{\{c\}} \cap C_{\mathsf{init}} \neq \emptyset$, and
(iii) for each $c \in C$, the set $\tt{minpre}(c)$ is finite and computable.
The algorithm for checking WSTS coverability is based on a backward analysis.
The sequence $(U_i)_{i \geq 0}$ with
$U_0= \min(C_{\tt{target}})$ and
$U_{i+1}=\min({\tt{Pre}}(\upclos{U_i}) \cup \upclos{U_i})$
reaches a fixpoint and is computable \cite{wsts2,wsts1}.
\smallskip
\noindent{\bf{${\mathsf{LoHoW}}$ with bounded promises is a WSTS}.}
We will show that the K-${\mathsf{LoHoW}}$ transition system
is a well-structured transition system.
Let $C$ denote the set of two-phases K-${\mathsf{LoHoW}}$ states of $\prog$. Given an instruction label function $J: \procset \rightarrow \mathbb{L}$, let $C_{\tt{target}}$ be a finite subset of $C$ of the form $({\tt{std}}, -, ((J,R), {\mathsf{\bf HW}}), ((J',R'), {\mathsf{\bf HW}}'))$ such that for every $x \in \varset$, we have: $(1)$ ${\mathsf{\bf HW}}(x)$ and ${\mathsf{\bf HW}}'(x)$ do not contain any memory type of the form $({\tt{prm}},-,p,-)$/$({\tt{prm}},-,p,-,-)$, and $(2)$ $|{\mathsf{\bf HW}}(x)|, |{\mathsf{\bf HW}}'(x)| \leq |\procset|$. We define the well-quasi ordering $\sqsubseteq$ on $C$ in a way that the upward closure of $C_{\tt{target}}$ consists of all two-phases K-${\mathsf{LoHoW}}$ states of the form $({\tt{std}}, -, ((J,R), {\mathsf{\bf HW}}), ((J',R'), {\mathsf{\bf HW}}'))$ such that for every $x \in \varset$, ${\mathsf{\bf HW}}(x)$ and ${\mathsf{\bf HW}}'(x)$ do not contain any memory type of the form $({\tt{prm}},-,p,-)$/$({\tt{prm}},-,p,-,-)$.
Then, the coverability of $C_{\tt{target}}$
is equivalent to the reachability of $J$ in K-${\mathsf{LoHoW}}$.
In the following, we define the well-quasi ordering $\sqsubseteq$ on on $C$ (Lemma \ref{lem:wqo}). Then, we show the monotonicity of the K-${\mathsf{LoHoW}}$ transition relation $\rightarrow$ w.r.t.\ $\sqsubseteq$ (Lemma \ref{lem:mon}). Finally, we show how to compute the set of predecessors of a given two-phases $K$-{${\mathsf{LoHoW}}$} state (Lemma \ref{computing-pre}). Observe that the first and second sufficient conditions for the decidability of the coverability problem, namely comparing two states and checking whether an upward closure set contains the initial state, are trivial (the second condition can be reduced whether a minimal state is equal to the initial state).
The ordering $\sqsubseteq$ defined on mapping from locations to higher order words
can be extended to two phases K-${\mathsf{LoHoW}}$ states by component wise extension:
$(\pi, p, ((J_1,R_1), {\mathsf{\bf HW}}_1), ((J_2,R_2), {\mathsf{\bf HW}}_2)) \sqsubseteq (\pi', p', ((J'_1,R'_1), {\mathsf{\bf HW}}'_1), ((J'_2,R'_2), {\mathsf{\bf HW}}'_2))$ holds iff
$\pi'=\pi$, $p'=p$, $(J_1,R_1)=(J'_1,R'_1)$, $(J_2,R_2)=(J'_2,R'_2)$, ${\mathsf{\bf HW}}_1 \sqsubseteq {\mathsf{\bf HW}}'_1$, and ${\mathsf{\bf HW}}_2 \sqsubseteq {\mathsf{\bf HW}}'_2$. Since the embedded ordering $\sqsubseteq$ is a wqo on higher order words when the number of promises is bounded \cite{higman}, we obtain the following lemma.
\begin{lemma}
The relation $\sqsubseteq$
is a well-quasi ordering on the two phases K-${\mathsf{LoHoW}}$ states.
\label{lem:wqo}
\end{lemma}
Consider now a two-phases K-${\mathsf{LoHoW}}$ state $\mathsf{St}$ of the form $({\tt{std}}, -, ((J,R), {\mathsf{\bf HW}}), ((J',R'), {\mathsf{\bf HW}}'))$ such that for every $x \in \varset$, ${\mathsf{\bf HW}}(x)$ and ${\mathsf{\bf HW}}'(x)$ do not contain any memory type of the form $({\tt{prm}},-,p,-)$/$({\tt{prm}},-,p,-,-)$, then it is easy to see that $\mathsf{St} \in \upclos{C_{\tt{target}}}$.
This implies that:
\begin{lemma}
The coverability of $C_{\tt{target}}$ is equivalent
to the reachability of $J$ in K-${\mathsf{LoHoW}}$.
\label{lem:cover-equivalence}
\end{lemma}
\noindent{\bf{Monotonicity}}.
The following lemma shows the monotonicity of the K-${\mathsf{LoHoW}}$ transition relation $\rightarrow$ w.r.t.\ $\sqsubseteq$.
This allows the backward algorithm for coverability to work with only upward closed sets,
since the set of predecessors of an upward closed set is also upward closed \cite{wsts2,wsts1}.
\begin{lemma}
The transition relation $\rightarrow$ is monotonic w.r.t.\ $\sqsubseteq$.
\label{lem:mon}
\end{lemma}
\noindent
{\bf Computing the set of predecessors.}
The last sufficient condition for the decidability of the coverability problem in $K$-${\mathsf{LoHoW}}$ is stated by the following lemma
\begin{lemma}
\label{computing-pre}
For each two-phases K-${\mathsf{LoHoW}}$ state $c$, the set $\tt{minpre}(c)$ is effectively computable.
\end{lemma}
Next, we state that the reachability problem for K-${\mathsf{LoHoW}}$ (even for $K=0$) is highly non-trivial (i.e., non-primitive recursive). The proof is done by reduction from the reachability problem for lossy channel systems, in a similar to the case of TSO \cite{ABBM10} where we insert $\mathsf{SC\text{-}fence}$ instructions everywhere in the process that simulates the lossy channel process (in order to ensure that no promises can be made by that process).
The proof is done by reduction from the reachability problem for lossy channel systems (LCS). We construct a
concurrent program with 2 processes, the first process $p_1$ keeps track of the finite state control of the LCS, while the
second process $p_2$ simulates the lossy channel. Two shared variables $x_c, y_c$ are used to simulate the lossy channel $c$.
$p_1$ writes to $x_c$ on each transition that writes to $c$ in the LCS. $p_2$ reads from $x_c$ and writes to $y_c$.
A read from the channel $c$ in the LCS is simulated by $p_1$ reading from $y_c$, thereby simulating the lossiness of $c$
($p_2$ can skip some messages of $x_c$, and $p_1$ can also skip some messages of $y_c$). Every two instructions
of $p_1, p_2$ have a $\mathsf{SC\text{-}fence}$ to ensure no promises can be made (and fulfilled).
\begin{theorem}
The reachability problem for K-${\mathsf{LoHoW}}$ is non-primitive recursive.
\label{thm:npr}
\end{theorem}
\section{Source to Source Translation}
\label{sec:c2c}
We consider a
parametric under-approximation in the spirit of context bounding \cite{cb2}, \cite{DBLP:conf/cav/TorreMP09}, \cite{DBLP:journals/fmsd/LalR09}, \cite{demsky}, \cite{MQ07}, \cite{DBLP:conf/tacas/QadeerR05}, \cite{pldi2019}, \cite{cb3}. The bounding concept chosen for concurrent programs depends on aspects related to the
interactions between the processes. In the case of SC programs, context bounding has been shown experimentally to have extensive behaviour coverage for bug detection \cite{MQ07}, \cite{DBLP:conf/tacas/QadeerR05}.
A context in the SC setting is a computation segment where only one process is active. The concept of context bounding has been extended for weak memory models. For instance, in TSO, the notion of context is extended to one where all updates to the main memory are done only from the buffer of the active thread \cite{cb2}.
In the case of POWER \cite{cb3}, context was extended to consider propagation actions performed by the active process. In the case of $\textsf{PS 2.0}\xspace$-${\tt{ra}}$ without promises and reservations \cite{pldi2019}, context bounding was extended to view bounding, using the notion of view switching messages. The notion of bounding appropriate for a model depends on its underlying complexity. From a theoretical point of view, we have already seen that $\textsf{PS 2.0}\xspace$ is very complex, and bounding contexts is not sufficient. Our bounding notion for $\textsf{PS 2.0}\xspace$ is based on its various features which includes relaxed as well as RA memory accesses, promises and certification. Since $\textsf{PS 2.0}\xspace$ subsumes RA, we recall
the bounding notion used in RA first, using \emph{view altering} messages.
\noindent{\textit{View Altering Reads}}. A read from the memory
is view altering if it changes the view of the process reading it.The message which is reads from in turn is called a view altering message. The under approximate analysis
for RA \cite{pldi2019} considered view bounded runs, where the number of view altering reads is bounded.
\noindent{\textit{Essential Events}}. An essential event in a run $\rho$ of a concurrent program
under $\textsf{PS 2.0}\xspace$ is either a promise, a reservation or a
view altering read by some process in the run.
\noindent{\textit{Bounded Context}}. A context is an uninterrupted sequence of actions by a single process. In a run having $K$ contexts,
the execution switches from one process to another $K-1$ times. A $K$ bounded context run is one where the number of context switches are bounded by $K \in \mathbb{N}$. The $K$ bounded context reachability problem in SC checks for the existence
of a $K$ bounded context run reaching some chosen instruction. A SC program is called a $K$ bounded context program if all
its runs are $K$ bounded context.
Now we define the notion of bounding for $\textsf{PS 2.0}\xspace$.
\noindent{\bf {The Bounded Consistent Reachability Problem}}.
Consider a run $\rho$ of a concurrent program under $\textsf{PS 2.0}\xspace$, $\mathcal{MS}_{0} \rtstep{\xrightarrow[p_{i_1}]{}}
\mathcal{MS}_{1} \rtstep{\xrightarrow[p_{i_2}]{}}
\mathcal{MS}_{2} \rtstep{\xrightarrow[p_{i_3}]{}}
\ldots
\rtstep{\xrightarrow[p_{i_n}]{}}
\mathcal{MS}_{n}$.
A run $\rho$ of a concurrent program $\prog$ under $\textsf{PS 2.0}\xspace$ is called \emph{$K$ bounded} iff the
number of essential events in $\rho$ is $\leq K$.
The $K$ bounded reachability problem for $\textsf{PS 2.0}\xspace$ checks for the existence
of a run $\rho$ of $\prog$ which
is $K$-bounded. Assuming $\prog$ has $n$ processes,
we propose an algorithm that reduces the $K$ bounded reachability problem
to a $K+n$ bounded context reachability problem under SC.
\noindent{\bf{Translation Overview}}.
Let $\prog$ be a concurrent program under $\textsf{PS 2.0}\xspace$ with set of processes $\procset$ and locations $\varset$.
Our algorithm relies on a source to source translation of $\prog$ to a bounded context SC program $\sem{\prog}$, as shown in Figure \ref{transl} and operates on the same data domain. The translation
adds a new process (\textsc{Main}) that initializes the global variables of $\sem{\prog }$.
The translation of a process $\proc\in\procset$ adds local variables, which are initialized by the function $\textsc{InitProc}$.
\begin{figure}
\small
\tikzset{background rectangle/.style={fill=black!5,rounded corners,draw=black}}
\resizebox{4\textwidth/5}{!}{
\begin{tikzpicture}[codeblock/.style={line width=0.5pt, inner xsep=0pt, inner ysep=0pt}, show background rectangle]
\node[codeblock] (init) at (current bounding box.north west) {
{
$\arraycolsep=0.7pt\def1.4{1.4}
\begin{array}{rl}
\sem{Prog} &\coloneqq (\langle \text{global vars}\rangle; \langle \textsc{Main} \rangle ; (\sem{\texttt{proc } p \texttt{ reg } \reg^* i^*})^* \\
\sem{ \texttt{proc } p \texttt{ reg } \reg^*\; i^* } &\coloneqq \texttt{proc } p \texttt{ reg } \reg^*
\langle \text{local vars} \rangle \langle\textsc{InitProc}\rangle
\langle\textsc{CSO}\rangle^{p, \lambda_0} (\sem{ i}^p)^* \\
\sem{ \lambda\space:\space i }^p &\coloneqq
\lambda\space: \langle\textsc{CSI}\rangle; \sem{s}^p; \langle\textsc{CSO}\rangle^{p, \lambda} \\
\sem{ \kwif\ \mathit{exp} \ \kwthen\ i^* \ \kwelse\ i^* }^p &\coloneqq \kwif\ \mathit{exp} \ \kwthen\ (\sem{ i}^p)^* \ \kwelse (\sem{ i}^p)^*\\
\sem{ \kwwhile\ \mathit{exp} \ \kwdo\ i^* }^p &\coloneqq \kwwhile\ \mathit{exp}\ \kwdo\ (\sem{ i}^p)^* \\
\sem{ \keyworr{assume}(\mathit{exp})}^p &\coloneqq \keyworr{assume}(\mathit{exp}) \\
\sem{ \reg = \mathit{exp}}^p &\coloneqq \reg = \mathit{exp} \\
\sem{ x = \reg }^{p}_{o \in \{{\tt{rlx}}, {\tt{ra}}\}} &\coloneqq \text{ see write Pseudocode }\\
\sem{ \reg = x }^{p}_{o \in \{{\tt{rlx}}, {\tt{ra}}\}} &\coloneqq \text{ see read Pseudocode } \\
\end{array}$
}
};
\end{tikzpicture}
}
\caption{Source-to-source translation map}
\label{transl}
\end{figure}
This is followed by the code block
$\langle CSO \rangle^{p, \lambda_0}$ (Context Switch Out)
that optionally enables the process to switch out of context. For each instruction $i$ appearing in the code of $p$, the map $\sem{i}^p$
transforms it into a sequence of instructions as follows : the code block
$\langle CSI \rangle$ (Context Switch In) checks if the process is active in the current context; then it transforms each statement $s$ of instruction $i$
into a sequence of instructions following the map $\sem{s}^p$, and finally executes the code block $\langle CSO \rangle^{p, \lambda}$. $\langle CSO \rangle^{p, \lambda}$ facilitates two things: when the process is at an instruction label $\lambda$,
(1)
allows $p$ to make promises/reservations after $\lambda$, s.t. the control is back at $\lambda$ after certification;
(2) it ensures that the machine state is consistent when $p$ switches out of context.
Translation of $\keyworr{assume}$, $\kwif$ and $\kwwhile$ statements
keep the same statement. Translation of read and write statements are described later. Translation of RMW statements are omitted for ease of presentation.
\input{timeline}
The set of promises a process makes has to be constrained with respect to the set of promises that it can certify, since processes can generate arbitrarily many promises/reservations, while, in reality only a few of them will be certifiable.
To address this, in the translation, processes run in two modes :
a `normal' mode and a `check' (\textit{consistency check}) mode. In the normal mode, a process does not make any promises or reservations. In the check mode, the process may make promises and reservations and
subsequently certify them before switching out of context. In any context, a process first enters the normal mode, and then, before exiting the context it enters the check mode. The check mode is used by the process to (1) make new promises/reservations and (2) certify consistency of the machine state.
We also add an optional parameter, called \textit{certification depth} (\texttt{certDepth}), which constrains the number of steps a process may take in the check mode to certify its promises.
Figure \ref{fig:scrun} shows the structure of a translated run under SC.
To reduce the $\textsf{PS 2.0}\xspace$ run into a bounded context SC run, we use the bound on the number of essential events.
From the run $\rho$ in $\textsf{PS 2.0}\xspace$, we construct a $K$ bounded
run $\rho'$ in $\textsf{PS 2.0}\xspace$ where the processes run in the order of generation
of essential events. So, the process which generates the first essential event is run first, till that event happens,
then the second process which generates the second essential event
is run, and so on. This continues till $K+n$ contexts : the $K$ bounds the number of essential events, and the $n$ is to ensure all processes are run to completion.
The bound on the number of essential events gives a bound on the number of
timestamps that need to be maintained.
As observed in \cite{pldi2019}, one view altering read requires two timestamps; additionally, each promise/reservation requires one timestamp. Since we have $K$ such essential events, $2K$ time stamps suffice. We choose $\mathsf{Time}=\{0,1,2, \dots, 2K\}$ as the set of timestamps.
\noindent {\bf{Data Structures}}.
We mention the significant ones.
The \goldf{\texttt{message}}~ data structure represents a message generated as a write or a promise and has 4 fields (i) $\mathit{var}$, the address of the memory location written to;
(ii) the timestamp $t$ in the view associated with the message;
(iii) $v$, the value written; and
(iv) $\mathit{flag}$, that keeps track of whether it is a message or a promise; and, in case of a promise, which process it belongs to.
The \goldf{\textsf{View}} data structure stores, for each memory location $x$, (i) a timestamp $t \in \mathsf{Time}$,
(ii) a value $\mathit{v}$ written to $x$,
(iii) a Boolean $l \in \{\tt{true}, \tt{false}\}$ representing whether $t$ is an exact timestamp (which can be used for essential events) or an abstract timestamp (which corresponds to non-essential events).
\noindent{\bf {Global Variables}}.
\label{para:globvars}
The \goldf{\texttt{Memory}}~ is an array of size $K$ holding elements
of type \goldf{\texttt{message}}~. This array
is populated with the view switching messages, promises and reservations generated by the program. We maintain counters for
(1)
the number of elements in \goldf{\texttt{Memory}}~;
(2)
the number of context switches that have occurred; and
(3)
the number of essential events that have occurred.
\noindent{\bf{Local Variables}}.
In addition to its local registers, each process has local variables including
\begin{itemize}
\item a local variable $\goldf{\texttt{view}}~$, which stores a local instance of the view function (this is of type
\goldf{\textsf{View}}),
\item
$\mathit{active}$: a boolean variable which is set when the
process is running in the current context, and
\item $\mathit{checkMode}$: a boolean denoting whether the process is in the certification phase. We implement the certification phase as a function call, and hence store the process state and return address, while entering it.
\end{itemize}
\noindent {\bf{Subroutines}}. We use certain helper subroutines as follows:
\begin{itemize}
\item genMessage is a subroutine which generates an instance of the \goldf{\texttt{message}}~ data structure;
\item saveState($p$) is a subroutine which saves the values of the global variables and the local states (instruction labels and local variables) of process $p$. This is used when switching into check mode.
\item loadState($p$) is a subroutine which loads the
the values of global variables and local states of $p$ which was saved using saveState($p$). This is use when switching out of check mode.
\end{itemize}
\subsection{Translation Maps}
\begin{wrapfigure}{r}{0.45\textwidth}
\footnotesize
\begin{algorithm}[H]
\DontPrintSemicolon
\SetCustomAlgoRuledWidth{0.45\textwidth}
\caption{$\mathsf{CSO}$}
\tcc{nondeterministically enter check mode and exit context}
\If{nondet()}{
\uIf{$\neg$checkMode}{
\tcc{enter consistency check}
\If{not in context}{
enter context \;
}
checkMode $\leftarrow \texttt{true}$ \;
save localstate \;
returnAddr $\leftarrow \lambda$ \;
}
\Else{
\tcc{consistency check successful!}
ensure all Promises for process are certified \;
\tcc{for next context}
mark all Promises as uncertified \;
$\mathit{checkMode} \leftarrow \texttt{false}$ \;
load localstate \;
goto $\mathit{returnAddr}$ \;
exit context \;
}
}
\label{alg:cso}
\end{algorithm}
\end{wrapfigure}
In what follows we illustrate how the translation simulates a run under $\textsf{PS 2.0}\xspace$. At the outset, recall that each process alternates, in its execution, between two modes: a \emph{normal} mode (\texttt{n} in Figure \ref{fig:scrun}) at the beginning of each context and
the \emph{check} mode
at the end of the current context (\texttt{cc} in Figure \ref{fig:scrun}), where it may make new promises and certify them before switching out of context.
\noindent \textbf{Context Switch Out ($CSO^{p, \lambda}$).}
We describe the \textsc{CSO} module (Algorithm \ref{alg:cso} provides its pseudocode). \textsc{CSO}$^{p, \lambda}$ is placed after each instruction $\lambda$ in the original program and serves as an entry and exit point for the consistency check phase of the process. When in normal mode (\texttt{n}) after some instruction $\lambda$, \textsc{CSO} non-deterministically guesses whether the process should exit the context at this point, and sets the \textit{checkMode} flag to true and subsequently, saves its local state and the return address (to mark where to resume execution from, in the next context).
The process then continues its execution in the consistency check mode
(\texttt{cc}) from the current instruction label ($\lambda$) itself. Now the process may generate new promises (see Algorithm \ref{alg:write}) and certify these as well as earlier made promises. In order to conclude the check mode phase, the process will enter the \textsc{CSO} block at some different instruction label $\lambda'$. Now since the \textit{checkMode} flag is true, the process enters the else branch, verifies that there are no outstanding promises of $p$ to be certified. Since the promises are not yet fulfilled, when $p$ switches out of context, it has to mark all its promises uncertified.
When the context is back to $p$ again, this will be used
to fulfil the promises or to certify them again before the context switches out of $p$ again.
Then it exits the check mode phase, setting \textit{checkMode} to false. Finally it loads the saved state, and returns to the instruction label $\lambda$ (where it entered check mode) and exits the context.
\noindent \textbf{Write Statements}.
We now discuss the translation of a write instruction $\llbracket x\coloneqq\reg\rrbracket_o$, where $o \in\{{\tt{rlx}},{\tt{ra}}\}$ of a process $\proc$, the intuitive pseudocode for which is given in Algorithm \ref{alg:write}.
This is the general psuedo code for both kinds of memory accesses, with specific details
pertaining to the particular access mode omitted.
Let us first consider execution in the normal mode (i.e., $\mathit{checkMode}$ is false).
First, the process updates its local state with the value that it will write.
Then, the process non-deterministically chooses one of three possibilities for the write, it either
(i) does not assign a fresh timestamp (non-essential event),
(ii) assigns a fresh timestamp and adds it to memory, or
(iii) fulfils some outstanding promise.
\begin{wrapfigure}{r}{0.45\textwidth} \footnotesize
\begin{algorithm}[H]
\DontPrintSemicolon
\SetCustomAlgoRuledWidth{0.45\textwidth}
\caption{$\texttt{Write}$}
update localstate with write \;
\uIf(\tcc*[f]{(i) no fresh timestamp}){nondet()}{
\uIf{checkMode}{
\tcc{since write is not a promise}
certify message with reservation or splitting
}
}
\uElseIf(\tcc*[f]{(ii) fresh timestamp}){nondet()}{
generate a view; generate a message \;
\eIf{checkMode}{
insert message into Memory as Promise and certify \;
}{
insert message into Memory as concrete message \;
}
}
\Else(\tcc*[f]{(iii) fulfill old promise}){
get Promise from Memory \;
check variable, value and view match \;
\eIf{checkMode}{
mark message as certified \;
}{
mark message as fulfilled \;
}
replace message into Memory \;
}
\label{alg:write}
\end{algorithm}
\end{wrapfigure}
Let us now consider a write executing when $\mathit{checkMode}$ is true, and highlight differences with the normal mode.
In case (i), non essential events exclude promises and reservations.
Then, while in certification phase, since we use a capped memory,
the process can make a write if either (1) the write interval can be generated through splitting insertion or (2) the write can be certified with the help of a reservation.
Basically the writes we make either split an existing interval (and add this to the left of a promise), or forms a part of a reservation.
\setlength\intextsep{0pt}
\begin{figure}{r}
\footnotesize
\begin{algorithm}[H]
\DontPrintSemicolon
\SetCustomAlgoRuledWidth{0.45\textwidth}
\caption{$\texttt{Read}$}
\uIf(\tcc*[f]{local read}){nondet()}{
check local state is valid \;
update local state with read \;
}
\Else(\tcc*[f]{nonlocal (view-switching) read}){
check that local state allows read \;
get message from Memory \;
check variable, value, view are allowed \;
update local state with message view \;
}
\label{alg:read}
\end{algorithm}
\end{figure}
Thus,
the time stamp of a neighbour is used.
In case (ii) when a fresh time stamp is used, the write is made as a promise, and then certified before switching out of context.
The analogue of case (iii) is the certification of promises for the current context; promise fulfilment happens only in the normal mode.
To help a process decide the value of a promise,
we use the fact that CBMC allows us to assign a non-deterministic value of a variable. On top of that, we have implemented an optimization that checks the set of possible values to be written in the future.
\noindent \textbf{Read Statements.}
The translation of a read instruction $\llbracket\reg\coloneqq x\rrbracket_o$, $o \in \{{\tt{rlx}},{\tt{ra}}\}$ of process $\proc$ is given in Algorithm \ref{alg:read}.
The process first guesses, whether it will read from a view altering message in the memory of from its local view. If it is the latter, the process must first verify whether it can read from the local view ;
for instance, reading from the local view may not be possible after execution of a \texttt{fence} instruction when the timestamp of a variable $x$ gets incremented from the local view $t$ to $t' > t$. In the case of a view altering read, we first check that we have not reached the context switching/essential event bound. Then the new \goldf{\texttt{message}}~ is fetched from \goldf{\texttt{Memory}}~ and we check the view (timestamps) in the acquired \goldf{\texttt{message}}~ satisfy the conditions imposed by the access type $\in \{{\tt{ra}}, {\tt{rlx}}\}$. Finally, the process updates its view with that of the new message and increments the counters for the context switches and the essential events. Theorem \ref{thm:s2s} proves the correctness
of our translation.
\begin{theorem}
Given a program $\prog$ under $\textsf{PS 2.0}\xspace$, and $K \in \mathbb{N}$, the source to source translation
constructs a program $\sem{prog}$ whose size is polynomial in $\prog$ and $K$ such that, for every $K$-bounded run of $\prog$ under $\textsf{PS 2.0}\xspace$ reaching a set of instruction labels, there is a $K+n$-bounded context run of $\sem{prog}$
under SC that reaches the same set of instruction labels.
\label{thm:s2s}
\end{theorem}
\section{Implementation and Experimental Results}
\label{sec:eval}
In order to check the efficiency of the source-to-source translation, we implement a prototype tool, $\mathsf{PS2SC}${} which is the first tool to handle \textsf{PS 2.0}\xspace. $\mathsf{PS2SC}$~ takes as input a C program and a bound $K$ and translates it to a program $\mathit{Prog}'$ to be run under SC. We use CBMC version 5.10 as backend to verify $\mathit{Prog}'$. CBMC takes as input $L$, the loop unrolling parameter for bounded model checking of $\mathit{Prog}'$.
We supply the bound on \textit{Essential Events}, $K$ as a parameter to $\mathsf{PS2SC}$. $\mathsf{PS2SC}$~ then considers the subset of executions respecting the bounds $K$ and $L$ provided as input. If it returns \textit{unsafe}, then the program has an unsafe execution. Conversely, if it returns \textit{safe} then none of the executions within the subset violate any assertion. $K$ may be iteratively incremented to increase the number of executions explored. We provide a functionality with which the user optionally selects a subset of processes for which promises and reservations will be enabled. While in the extreme cases we can run $\mathsf{PS2SC}$~ in the promise-full (all processes can promise) and promise-free modes, \textit{partial promises} (allowing subsets of processes to promise) turns out to be an effective technique.
We now report the results of experiments we have performed with $\mathsf{PS2SC}$. We have two objectives: (1) studying the performance of $\mathsf{PS2SC}$~ on benchmarks which are unsafe only if promises are enabled and (2) comparing $\mathsf{PS2SC}$~ with other model checkers when operating in the promise-free mode (since they can not handle promises).
In the first case, we show that $\mathsf{PS2SC}$~ is able to uncover bugs in examples with low interaction (reads and writes) with the shared memory. When this interaction increases, however, $\mathsf{PS2SC}$~ does not scale, owing to the huge non-determinism in \textsf{PS 2.0}\xspace. However, with partial promises, $\mathsf{PS2SC}$~ is once again able to uncover bugs in reasonable amounts of time.
In the second case, our observations highlight the ability to detect hard to find bugs with small $K$ for unsafe benchmarks, and scalability by altering $K$ as discussed earlier in case of safe benchmarks. We compare $\mathsf{PS2SC}$ with three state-of-the-art stateless model checking tools, $\cdsc$ \cite{cdsc}, $\genmc$ \cite{genmc} and $\rcmc$ \cite{rcmc} that support the promise-free subset of the \textsf{PS 2.0}\xspace semantics.
In the tables that follow we
provide the value of $K$ (for $\mathsf{PS2SC}${} only) and the value of $L$ (for all tools).
We do not consider compilation time for any tool while reporting the results. For $\mathsf{PS2SC}$, the time reported is the time taken by the CBMC backend for analysis. The timeout used is 1 hour for all benchmarks. All experiments are conducted on a machine with a 3.00 GHz Intel Core i5-3330 CPU and 8GB RAM running a Ubuntu 16 64-bit operating system. We denote timeout by `TO', and memory limit exceeded by `MLE'.
\subsection{Experimenting with Promises}
In this section we check the efficiency of the source-to-source translation in handling promises for \textsf{PS 2.0}\xspace (which is the most difficult part due to the non-determinism).
We first test $\mathsf{PS2SC}${} on litmus-tests adapted from \cite{promising,promising2,thinair,jmm}. These examples are small programs that serve as barebones thin-air tests for the C11 memory model. Consistency tests based on the Java Memory Model are proposed in \cite{jmm}. These were also experimented on in \cite{mrder} with the MRDer tool. Like MRDer, $\mathsf{PS2SC}${} is able to verify most of these tests within 1 minute which shows its ability to handle typical programming idioms of \textsf{PS 2.0}\xspace.
\setlength\intextsep{-6pt}
\captionsetup[table]{font=scriptsize}
\begin{table}[t]
\vspace{-0.3cm}
\scriptsize
\centering
\resizebox{0.25\columnwidth}{!}{
\begin{tabular}{cccc}
\hline
\textbf{testcase} & $K$ & \textbf{$\mathsf{PS2SC}$}
\\
\hline\hline
ARM\_weak & 4 & 0.765s \\
Upd-Stuck & 4 & 1.252s \\
split & 4 & 25.737s \\ \hline
LBd & 3 & 1.481s \\
LBfd & 3 & 1.512s \\ \hline
CYC & 5 & 1.967s \\
Coh-CYC & 5 & 42.67s \\ \hline\hline
Pugh2 & 3 & 13.725s \\
Pugh3 & 3 & 12.920s \\
Pugh8 & 3 & 1.67s \\ \hline
Pugh5 & 5 & 4.811s \\
Pugh10 & 5 & 3.868s \\
Pugh13 & 5 & 3.345s \\ \hline
\end{tabular}}
\caption{Litmus Tests}
\label{tab:litmus}
\end{table}
\begin{table}[t]
\small
\centering
\resizebox{0.27\columnwidth}{!}{
\begin{tabular}{cccc}
\hline
\textbf{testcase} & $K$ & \textbf{$\mathsf{PS2SC}$}
\\
\hline\hline
fib\_local\_3 & 4 & 0.742s \\
fib\_local\_4 & 4 & 0.761s \\ \hline
fib\_local\_cas\_3 & 4 & 1.132s \\
fib\_local\_cas\_4 & 4 & 1.147s \\ \hline
\end{tabular}}
\caption{Performance of $\mathsf{PS2SC}$~ on cases with local computation}
\label{tab:prom1}
\end{table}
\begin{table}[t]
\resizebox{0.3\columnwidth}{!}{
\begin{tabular}{cccc}
\hline
\textbf{testcase} & $K$ & \textbf{$\mathsf{PS2SC}$}[1p]
\\
\hline\hline
fib\_global\_2 & 4 & 55.972s \\
fib\_global\_3 & 4 & 2m4s \\
fib\_global\_4 & 4 & 4m20s \\ \hline
exp\_global\_1 & 4 & 19m37s \\
exp\_global\_2 & 4 & 41m12s \\ \hline
\end{tabular}}
\caption{Performance of $\mathsf{PS2SC}$~ on cases with global computation}
\label{tab:prom2}
\end{table}
In Table \ref{tab:prom1} we consider unsafe examples in which a process is required to generate a promise (speculative write) with value as the $i^{\mathit{th}}$ fibonacci number (\texttt{Fibonacci}-based benchmarks for SV-COMP 2019 \cite{beyer2019automatic}). This promise is certified using computations local to the process. Thus though the parameter $i$ increases the interaction of the promising process with the memory remains constant. The ${\mathbf{CAS}}$ variant requires the process to make use of reservations. We note that $\mathsf{PS2SC}$~ uncovers the bugs effectively in all these cases.
Now we consider the case where promises require some interaction between processes. We consider an example adapted from the \texttt{Fibonacci}-based benchmarks for SV-COMP 2019 \cite{beyer2019automatic}, where two processes compute the $i^{\text{th}}$ fibonacci number in a distributed fashion. Unlike the previous case, here, the amount of interaction increases with $i$. Here however, our tool times out.
\textit{How do we recover tractable analysis in this case?} We tackle this problem by a modular approach of allowing partial-promises, i.e. subsets of processes are allowed to generate promises/reservations. In the experiments, we allowed only a single process to do so. The results obtained are in Table \ref{tab:prom2}, where $\mathsf{PS2SC}$[1p] denotes that only one process is permitted to perform promises. We then repeat our experiments on two other unsafe benchmarks - \texttt{ExponentialBug} from Fig. 2 of \cite{huang} and have similar observations. With this modular approach $\mathsf{PS2SC}$~ uncovers the bug. To summarize, we note that the source to source approach performs well on programs requiring limited global memory interaction. When this interaction increases, $\mathsf{PS2SC}$~ times out, owing to the huge non-determinism of \textsf{PS 2.0}\xspace. However, the modular approach of partial-promises enables us to recover effectiveness.
\subsection{Comparing Performance with Other Tools}
In this section we compare performance of $\mathsf{PS2SC}$~ in promise-free mode with $\cdsc$ (\cite{cdsc}), $\genmc$ (\cite{genmc}) and $\rcmc$ (\cite{rcmc}) on safe and unsafe benchmarks. We provide a subset of the experimental results, the remaining can be found in the full version. The results of this section indicate that the source-to-source translation with essential event bounding is effective at uncovering hard to find bugs in non-trivial programs. We will observe that in most examples discussed below, we had $K \leq 10$. Additionally, the bound $K$ allows incremental verification of safe programs in cases where the other tools timeout.
\paragraph{Parameterized Benchmarks}
\label{para:exp2}
In Table~\ref{tab:parab} we compare the performance of these tools on two parametrized benchmarks: $\texttt{ExponentialBug}$ (from Fig. 2 of \cite{huang}) and $\texttt{Fibonacci}$
(from SV-COMP 2019). In $\texttt{ExponentialBug}(N)$
$N$ represents the number of times a process writes to a variable. We note that in $\texttt{ExponentialBug}(N)$ the number of executions grows as $N!$, while the processes have to follow a specific interleaving to uncover the hard to find bug.
In $\texttt{Fibonacci}(N)$, two processes compute the value of the $n^{th}$ fibonacci number in a distributed fashion.
Our tool performs better than the other tools on the $\texttt{ExponentialBug}$ and competes well on $\texttt{Fibonacci}$ for larger values of the parameter. These results show the ability of our tool to uncover bugs with a small value of $K$.
\begin{table}[t]
\resizebox{0.7\columnwidth}{!}{
\begin{tabular}{ccccccc}
\hline
\textbf{benchmark} & $L$ & $K$ & \textbf{$\mathsf{PS2SC}$} & \textbf{CDSChecker} & \textbf{GenMC} & \textbf{RCMC} \\ \hline\hline
exponential\_10\_unsafe & 10 & 10 & 1.854s & 1.921s & 0.367s & 3m41s \\
exponential\_25\_unsafe & 25 & 10 & 3.532s & 7.239s & 3.736s & TO \\
exponential\_50\_unsafe & 50 & 10 & 6.128s & 36.361s & 39.920s & TO \\
\hline
fibonacci\_2\_unsafe & 2 & 20 & 2.746s & 2.332s & 0.084s & 0.086s \\
fibonacci\_3\_unsafe & 3 & 20 & 9.392s & 46m8s & 0.462s & 0.544s \\
fibonacci\_4\_unsafe & 4 & 20 & 34.019s & TO & 12.437s & 18.953s \\ \hline
\end{tabular}}
\caption{Comparison on a set of parameterized benchmarks}
\label{tab:parab}
\end{table}
\paragraph{Concurrent data structures based benchmarks}
\label{para:exp3}
We compare the tools in Table \ref{tab:ds} on benchmarks based on concurrent data structures. The first of these is a concurrent locking algorithm originating from \citet{hehner}. The second, $\texttt{LinuxLocks(N)}$ is adapted from evaluations of $\cdsc$ \cite{cdsc}. We note that if not completely fenced, it is unsafe. We fence all but one lock access. \texttt{Queue} is a \textit{safe} benchmark adapted from SV-COMP 2018, parameterized by the number of processes. We note the ability of the tool to uncover bugs with a small value of $K$.
\begin{table}[t]
\small
\resizebox{0.7\columnwidth}{!}{
\begin{tabular}{ccccccc}
\hline
\textbf{benchmark} & $L$ & $K$ & \textbf{$\mathsf{PS2SC}$} & \textbf{CDSChecker} & \textbf{GenMC} & \textbf{RCMC} \\ \hline\hline
hehner2\_unsafe & 4 & 5 & 7.207s & 0.033s & 0.094s & 0.087s \\
hehner3\_unsafe & 4 & 5 & 28.345s & 0.036s & 2m53s & 1m13s \\ \hline
linuxlocks2\_unsafe & 2 & 4 & 0.547s & 0.032s & 0.073s & 0.078s \\
linuxlocks3\_unsafe & 2 & 4 & 1.031s & 0.031s & 0.083s & 0.081s \\ \hline
queue\_2\_safe & 4 & 4 & 0.180s & 0.031s & 0.082s & 0.085s \\
queue\_3\_safe & 4 & 4 & 0.347s & 0.037s & 0.090s & 0.092s \\ \hline
\end{tabular}}
\caption{Comparison on concurrent data structures}
\label{tab:ds}
\end{table}
\paragraph{Variations of mutual exclusion protocols}
We now consider safe and unsafe variants of mutual exclusion protocols from SV-COMP 2018. The fully fenced versions of the protocols are \textit{safe}. We modify these protocols by introducing bugs and comparing the performance of $\mathsf{PS2SC}$~ for bug detection with the other tools. These benchmarks are parameterized by the number of processes.
\begin{table}[t]
\small
\resizebox{0.7\columnwidth}{!}{
\begin{tabular}{ccccccc}
\hline
\textbf{benchmark} & $L$ & $K$ & \textbf{$\mathsf{PS2SC}$} & \textbf{CDSChecker} & \textbf{GenMC} & \textbf{RCMC} \\ \hline\hline
peterson1U(4) & 1 & 6 & 1.408s & 0.039s & TO & 9.129s \\
peterson1U(8) & 1 & 6 & 47.786s & TO & TO & TO \\
\hline
szymanski1U(4) & 1 & 2 & 1.015s & 0.043s & MLE & TO \\
szymanski1U(8) & 1 & 2 & 6.176s & TO & TO & TO \\
\hline
\end{tabular}}
\caption{Comparison of performance on mutual exclusion benchmarks with a single unfenced process}
\label{tab:mutex1}
\end{table}
In Table \ref{tab:mutex1}, we unfence a single process of the \texttt{Peterson} and \texttt{Szymanski} protocols making them \textit{unsafe}. For $\mathsf{PS2SC}$, the value of $K$ taken is 6 and 2 respectively, asserting that bugs can be found (even for non-trivial examples) with small $K$. We note that the other tools eventually timeout for larger values of $n$.
\setlength\intextsep{10pt}
In Table~\ref{tab:mutex2} we keep all processes fenced but introduce a bug into the critical section of a process (write a value to a shared variable and read a different value from it). We note that all other tools timeout, while $\mathsf{PS2SC}$~ is able to detect the bug within one minute, showing that essential event-bounding is an effective technique for bug-finding. Additionally in \texttt{Peterson2C}, we vary the example by changing the process in which we add the bug. We note that $\cdsc$, can uncover the bug in \texttt{Peterson2C(5)} in around two minutes, while for \texttt{Peterson1C(5)} it timed out. Thus, $\cdsc$ algorithm is sensitive to changes in the position of the bug due to its DPOR exploration strategy.
\begin{table}[h]
\small
\resizebox{0.7\columnwidth}{!}{
\begin{tabular}{ccccccc}
\hline
\textbf{benchmark} & $L$ & $K$ & \textbf{$\mathsf{PS2SC}$} & \textbf{CDSChecker} & \textbf{GenMC} & \textbf{RCMC} \\ \hline\hline
peterson1C(3) & 1 & 2 & 0.487s & 0.053s & 0.083s & 0.087s \\
peterson1C(5) & 1 & 2 & 2.713s & TO & TO & TO \\
peterson1C(7) & 1 & 2 & 11.008s & TO & TO & TO \\ \hline
peterson2C(3) & 1 & 2 & 0.481s & 0.032s & 0.099s & 0.091s \\
peterson2C(5) & 1 & 2 & 2.801s & 1m47s & TO & TO \\
peterson2C(7) & 1 & 2 & 11.030s & TO & TO & TO \\ \hline
\end{tabular}}
\caption{Comparison of performance on completely fenced peterson mutual exclusion benchmarks with a bug introduced in the critical section of a single process}
\label{tab:mutex2}
\vspace{-0.6cm}
\end{table}
We consider in Table~\ref{app-tab:mutex4} completely fenced versions of the mutual exclusion protocols.
In this experiment, we increase the loop unwinding bound and with it, the value of $K$. These examples exhibit the practicality of iterative increments in $K$. The other tools eventually timeout, while $\mathsf{PS2SC}$~ is able to provide atleast partial guarantees.
\begin{table}[h]
\small
\vspace{0.1cm}
\resizebox{0.7\columnwidth}{!}{
\begin{tabular}{ccccccc}
\hline
\textbf{benchmark} & $L$ & $K$ & $\mathsf{PS2SC}$ & \textbf{CDSChecker} & \textbf{GenMC} & \textbf{RCMC} \\ \hline\hline
peterson(3) & 1 & 2 & 0.878s & TO & 9.665s & 26.208s \\
peterson(2) & 1 & 2 & 0.321s & 0.325s & 0.087s & 0.068s \\ \hline
peterson(3) & 2 & 4 & 1.695s & TO & MLE & TO \\
peterson(2) & 2 & 4 & 0.539s & 15m22s & 0.039s & 0.428s \\ \hline
peterson(3) & 4 & 4 & 15.900s & TO & MLE & TO \\
peterson(2) & 4 & 4 & 3.412s & TO & TO & TO \\ \hline
\end{tabular}}
\caption{Evaluation using safe mutual exclusion protocols}
\label{app-tab:mutex4}
\vspace{-0.6cm}
\end{table}
\section{Conclusion}
In this paper, we investigate decidability
of the promising semantics, $\textsf{PS 2.0}\xspace$ from \citet{promising2}. The release-acquire (${\tt{ra}}$) fragment of $\textsf{PS 2.0}\xspace$ with RMW operations is known to be undecidable \cite{pldi2019}. However, the decidability of the fragment of $\textsf{PS 2.0}\xspace$ with only relaxed (${\tt{rlx}}$) accesses (denoted $\textsf{PS 2.0-rlx}$)
was open. We started with this fragment, and obtained undecidability of the reachability
problem, when there is no bound on the number of promises. In the quest for decidability, we
considered an underapproximation of $\textsf{PS 2.0-rlx}$
where we bound the number of promises in any execution. The fragment of $\textsf{PS 2.0-rlx}$ with bounded promises is denoted as $\textsf{bdPS 2.0-rlx}$.
We showed that reachability is decidable for $\textsf{bdPS 2.0-rlx}$.
Our decidability proof includes the introduction of a new memory model ${\mathsf{LoHoW}}$, and
proving the equivalence of $\textsf{PS 2.0-rlx}$ and ${\mathsf{LoHoW}}$.
The decidability of $\textsf{bdPS 2.0-rlx}$ is shown using the theory of well structured transition systems. This also gives non-primitive recursive complexity of $\textsf{bdPS 2.0-rlx}$, with a proof similar to RMW-free fragment of release-acquire \cite{pldi2019}.
Having explored the decidability landscape of $\textsf{PS 2.0}\xspace$ thoroughly, we moved towards practical verification techniques for $\textsf{PS 2.0}\xspace$. Motivated
by the success of context bounded reachability in SC \cite{DBLP:conf/tacas/QadeerR05}, and subsequent notions in weak memory models, we
introduced a notion of essential events bounded reachability for $\textsf{PS 2.0}\xspace$, which bounds the number
of promises and view altering messages in any execution. We provide a source to source translation from a concurrent program
under $\textsf{PS 2.0}\xspace$ with this bounded notion to
a bounded context SC program, and implemented this in a tool $\mathsf{PS2SC}${}. $\mathsf{PS2SC}${} is the first tool capable of handling the promising framework, $\textsf{PS 2.0}\xspace$ from \citet{promising2} and the $\mathsf{PS}$ model from \citet{promising}. $\mathsf{PS2SC}${} allows modularity with respect
to allowing/disallowing promises on a thread-by-thread basis. We exhibit the efficacy of this modular technique in the face of non-determinism induced by $\textsf{PS 2.0}\xspace$. We also compare the performance of $\mathsf{PS2SC}${} with existing tools which do not support promises by operating it in the \textit{promise-free} mode (in which no threads are allowed to promise). In this case, we exhibit the effectiveness of the bounding technique in uncovering hard-to find bugs.
\subsection{Correctness of Translation (Proof of Theorem \ref{thm:s2s})}
\label{app:bv}
The proof is in two parts. In the first part, we show that that every $K+n$ context bounded run of $\prog'$ in SC corresponds to a $K$-bounded run of $\prog$ under $\ps$, and in the second part, we show that for every $K$-bounded run in $\ps$, there is a $K+n$ context bounded run in SC.
At the outset we review a high level description of the translation. We denote by \emph{normal}
and $\mathit{checkMode}$, the two phases
respectively where
$\mathit{checkMode}$ is false and $\mathit{checkMode}$ is true. These are the two phases in which a process functions. Each process executes instructions in the \emph{normal} phase by skipping over the $CSO$ blocks of code. When a process needs to switch out, it enters the $CSO$ block following the most recent instruction executed and sets $\mathit{checkMode}$ to true. Now, it makes a ``ghost'' run in $\mathit{checkMode}$, a terminology to indicate that this phase of the run does not change the the global state and local state of the process permanently (this is facilitated by the saveState and loadState functions). One exception to this is the writes that the process makes as reservations, and published promises which are maintained permanently. Hence, this part of the run is equivalent to the process making fresh promises after a \emph{normal} execution; providing a witness for consistency and then switching out of context. The run then is a sequence of interleaved \emph{normal} and $\mathit{checkMode}$ phases. Moreover, the local states of the process is identical at the start and end of any given $\mathit{checkMode}$ phase.
We request the reader to refer to the glossary [\ref{app:ctc}] of the variables used which will aid in better understanding of the translation.
We give the proof of correctness of the translation through two
sections.
\textbf{Intuition} The translation relies on the fact that in a run of the $K$-bounded $\textsf{PS 2.0}\xspace$ program, it suffices to store the relative order only between $K$ totally ordered timestamps for each variable. Additionally, these $K$-timestamps are precisely those corresponding to the $K$ essential events - promises, reservations, view-altering reads. While we maintain an exact ordering between essential events, those of non-essential events (which are none of view-altering reads, reservations or promises) are abstracted in the SC run. Thus in the original run under $\ps$, all timestamps are exact, while in the run under SC, the non-essential timestamps are abstracted away.
The correctness of the translation then relies on being able to faithfully concretize the abstract timestamps from the SC run. We account for these concretizations by separating the essential timestamps by sufficiently large intervals, so that, the non-essential timestamps can be inserted in between, respecting their order.
\subsection*{SC to $\ps$}
\textbf{Details} We start from SC to $\ps$. We show that every $K+n$ context bounded run of $\prog'$ under SC corresponds to a $K$-bounded run of $\prog$ under $\textsf{PS 2.0-rlx}$. Keeping in mind the description above, we split this proof into two parts.
\begin{enumerate}
\item First, we consider only runs in \emph{normal} mode and prove that they have an analog in $\ps$.
\item Second, we prove that any run in $\mathit{checkMode}$ is indeed an analog of a process making fresh promises and reservations and certifying them along with previous unfulfilled promises, before switching out of context.
\end{enumerate}
Combining these two, indeed, we will have a run under $\ps$.
We begin by defining some terminology. Consider a run $\tau$ of program $\prog'$. Each event of the run $\tau$ is an execution of either a read, write, ${\mathbf{CAS}}$ or $\mathsf{SC\text{-}fence}$.
A read in this run is called \emph{global} (and otherwise \emph{local}) if the process decides to read from the global array $\mathit{messageStore}$. Only global reads can be view-altering in the corresponding run under $\ps$. A write can be of three types - publishedS, publishedF and local. These represent, `simple published', `fulfilling published', and `timestamp not assigned writes' respectively. Note that each of these types can be performed in \emph{normal} as well $\mathit{checkMode}$. A ${\mathbf{CAS}}$ can therefore be of 6 types since it involves a read and write. At a high level this translation is facilitated by the following two \textbf{key observations}:
\begin{itemize}
\item The number of publishedS, publishedF writes are bounded due to the bound $K$, and hence the requisite data-structure for these can be maintained using bounded space.
\item Local writes are unbounded, however, these writes are only used (read-from) locally by the writing process and need not be stored permanently by the algorithm.
\end{itemize}
Let $w_1$ be the number of $write$ events in the \emph{normal} mode of run $\tau$, $w_2$ be the maximum number of $write$ events, maximum being taken over all $\mathit{checkMode}$ phases of the run, $u-1$ be the number of ${\mathbf{CAS}}$ events in the run, and let $\ell = w_1 + w_2 + u$. Let $\mathsf{M_x}$, for each shared variable $x$, be an increasing function from $[2K]$ to $\mathsf{N}$ representing a mapping from the notion of time-stamps in SC to time-stamps in $\ps$. For each variable $x$, and each process $p$, let $\mathsf{View_{SC}(x)} = \mathit{view}[x].t$ (defined above) and $\mathsf{View_{\ps}(x)}$ be the time stamp of $x$ in the view of $p$ in $\rho$. Given a run $\tau$, we will construct a $K$ bounded run $\rho$ of $\prog$ which reaches the same set of labels after $i$ events, for any $i$.
We will first treat the \emph{normal} (non-$\mathit{checkMode}$) part of the run. While going through the steps, we will also construct the increasing functions $\mathsf{M_x}$. In addition to the invariants in $\ref{app:ctc}$, we maintain the following timestamp-based invariants for all processes $p$ and variables $x$.
\begin{enumerate}
\item If $\mathit{view}[x].l$ is true for a process in $\tau$, then $\mathsf{M_x}(\mathsf{View_{SC}(x))} = \mathsf{View_{\ps}(x)}$.
\item If $\mathit{view}[x].l$ is true and the time-stamp $\mathit{view}[x].t$ corresponds to a write message instead of a message added due to a ${\mathbf{CAS}}$, then $\mathsf{M_x}$($\mathit{view}[x].t$) = $\mathit{view}[x].t \cdot \ell \cdot u$
\item If $\mathit{view}[x].l$ is false, then $\mathsf{M_x}(\mathit{view}[x].t) < \mathsf{View_{\ps}(x)} < (\mathit{view}[x].t+1) \cdot \ell \cdot u$. Moreover, if the last event to assign false to $\mathit{view}[x].l$ was a write, then $\mathsf{View_{\ps}(x)}$ is a multiple of $u$.
\item If a message is of type ${\mathbf{CAS}}$, then its time-stamp $t$ in $\rho$ satisfies $t \not \equiv 0 \mod u$
\item The sum of view-switch points and promises is $\leq K$ in $\rho$.
\item The time-stamps of essential messages in $\tau$ and the corresponding message in $\rho$ are related by $\mathsf{M_x}$. That is, $\mathsf{M_x}(\mathsf{View_{SC}(x)}) = \mathsf{View_{\ps}(x)}$.
\end{enumerate}
The base case, that is, after 0 events ($i=0$) is trivial since the configurations are semantically equivalent and we define $\mathsf{M_x}(0) = 0$ for all variables, which satisfies the invariants. We make the following three cases depending on the $i^{th}$ event of $\tau$. \\
\begin{itemize}
\item Case 1. $e_i$ is an execution of a write for process $p$, variable $x$ and value $v$.
\begin{itemize}
\item
If the write is of publishedS or publishedF type, then $\mathit{view}[x].t$ is updated from $t$ to a new time-stamp $t'$ (which in the case of publishedF is the timestamp of the retrieved message) and $\mathit{view}[x].l$ is assigned true. In $\rho$, if we can make $\mathsf{View_{\ps}(x)}$ = $t'' = t' \cdot \ell \cdot u$ then the invariants are satisfied. It is not possible for $t''$ to have been assigned already to some write message in $\rho$ since $t'$ was not assigned to some message in $\tau$ (checked using $\mathit{avail}[x][t']$). A ${\mathbf{CAS}}$ message could not have been assigned $t''$ either, by the fourth invariant. Since $t<t'$, $\mathsf{View_{\ps}(x)} < t''$ (by invariants 2 and 3). Hence, $\mathsf{View_{\ps}(x)}$ can be updated to $t''$ since it is available and is greater than the current view. If the write is published, then the message is added to $\mathit{messageStore}$. This is done to maintain invariant (6). Note how, if the write is of publishedF type, the message flag is set to 0, effectively removing it from the promise bag and maintaining the $\mathit{flag}$ invariant (refer to [\ref{app:ctc}]).
\item If the write is local, then we pick the smallest available multiple of $u$ between $\mathsf{M_x}(\mathit{view}[x].t)$ and $(\mathit{view}[x].t+1) \cdot \ell \cdot u$. This can always be done since there are $\ell-1$ multiples of $u$ between $\mathit{view}[x].t \cdot \ell \cdot u$ and $(\mathit{view}[x].t+1) \cdot \ell \cdot u$ and there are $\leq (\ell-1)$ messages (even considering those produced in $\mathit{checkMode}$) in total. Notice that multiples of $u$ have been reserved for writes by invariant 4.
\end{itemize}
\item Case 2. $e_i$ is an execution of a read for process $p$, variable $x$.
\begin{itemize}
\item If the read is local in $\tau$, then the process is either reading a local message written by itself or a useful message (a useful message is one which is read by a process, but does not create a change of view). In either case, this read can be performed in $\rho$ without any change in time-stamps. Note that this cannot be a view-switching event. Moreover note that the local value in $\mathit{view}[x].v$ has been ascertained to be usable.
\item If the read is global, then $numEE < K$ before the read and therefore $numEE \leq K$ afterwards. In this case, a message is fetched from $\mathit{messageStore}$ and the process view is updated according to this message. Since $\mathsf{M_x}$ is an increasing function, the results of comparisons in SC will be the same as in $\ps$ and the read operation has the same effect on values and time-stamps of the variables. Moreover $\mathit{view}[x].f$ is set to true maintaining the $\mathit{view}[x].f$ invariant [\ref{app:ctc}].
\end{itemize}
\item Case 3. $e_i$ is an execution of a ${\mathbf{CAS}}$ for process $p$, variable $x$ and values $v$, $v'$.
\begin{itemize}
\item If the read here is local, and $\mathit{view}[x].u$ is true then we need to ensure that the timestamp chosen for the write immediately follows $\mathsf{M_x}(\mathit{view}[x].t)$. It is first checked if $\mathit{view}[x].t$ has been used for an update earlier or not. If it has not been, then the time-stamp $\mathsf{M_x}(\mathit{view}[x].t) + 1$ is available in $\ps$ since all messages that come from writes have time-stamps in multiples of $u$ and $\mathsf{M_x}(\mathit{view}[x].t)$ is a multiple of $u$. Note, that we also ensure that $\mathit{view}[x].f$ is true in this case, which implies that the local value is usable.
\item If the read here is local and $\mathit{view}[x].u$ is false (and hence so is $\mathit{view}[x].l$), then it definitely has not been used for an update (${\mathbf{CAS}}$) in $\tau$ since the process reading the message is the only one that knows of its existence. Now, if this message was a result of a local write, then its time-stamp $t$ in $\ps$ is a multiple of $u$ and $t+1$ is available for the update message. Otherwise, this message was a result of a ${\mathbf{CAS}}$ whose write was local and has a time-stamp of the form $a \cdot u + b$ where $b<u$. Note that this implies $b-1$ consecutive ${\mathbf{CAS}}$s were made to get here since all the messages that are a result of (non-${\mathbf{CAS}}$) write operations get time-stamps that are multiples of $u$. Since $u-1$ is the total number of ${\mathbf{CAS}}$s in $\tau$, $b < u-1$ (at most $u-2$ ${\mathbf{CAS}}$s have taken place before this one). This implies $a \cdot u+b+1$ is available and can be used for the write.
\item If the read is global, then it is done correctly as explained in Case 2. The write part of the ${\mathbf{CAS}}$ goes through as explained above.
\end{itemize}
\item Case 4: $e_i$ is an $\mathsf{SC\text{-}fence}$
\begin{itemize}
\item We iterate over the variables, updating $\mathit{globalTimeMap}[x]$ and $\mathit{view}[x].t$ to the maximum of the two.
\item In case the former was greater, we set $\mathit{view}[x].l$ to true, signifying that $\mathit{view}[x].t$ is valid and maintaining invariant (1) above. Moreover we set $\mathit{view}[x].f$ to false. This is necessary since, the timestamp of the message corresponding to $\mathit{view}[x].v$ is now less than $\mathit{view}[x].t$ and hence the locally stored value is unusable.
\item If the latter is greater, we check whether $\mathit{view}[x].l$ is true (which signifies that $\mathit{view}[x].t$ is valid). If it is we can set $\mathit{globalTimeMap}[x]$ to it. If not, then the $\mathsf{M_x}(\mathit{view}[x].t) < \mathsf{View_{\ps}(x)}$ (by invariant (6)), and hence we set it to $\mathit{view}[x].t + 1$. Finally we note that $\mathsf{View_{\ps}(x)} < (\mathit{view}[x].t + 1)\cdot \ell \cdot u$ and hence $\mathsf{M_x}(\mathit{globalTimeMap}[x])$ now matches the essential event immediately following the event with timestamp $\mathit{view}[x].t$.
\end{itemize}
\end{itemize}
We now briefly justify the $\mathit{checkMode}$ phase of the run. For any such phase, we need to ascertain that the run has analogous run in $\ps$ which respects the notion of consistency. The management of timestamps is identical to the \emph{normal} phase explained above so we only highlight the special aspects. First we recall some invariants:
\begin{enumerate}
\item $\mathit{liveChain}[x]$ is true only when the most recent write made in the \textit{current} $\mathit{checkMode}$ phase was unpublished (was not a promise) and neither was it certified using a reservation.
\item $\mathit{extView}[x]$ is true if $\mathit{view}[x].v$ corresponds to a message from outside $\mathit{checkMode}$.
\item For the process $p$ currently in $\mathit{checkMode}$, $message\_flag$ is -1 for temporarily (only within current $\mathit{checkMode}$ phase) certified promises and is $p$ for as yet uncertified promises. If it is $p' \neq p$, then the message is in the promise bag of some other process. Additionally if it is 0, it is not in the promise bag of any process. Note how this is maintained in the write, ${\mathbf{CAS}}$ sections above.
\end{enumerate}
We review how these invariants are maintained and used throughout the code. When entering $\mathit{checkMode}$, $\mathit{liveChain}[x]$ is false. For any write happening in \emph{normal} phase we set $\mathit{extView}[x]$ to true. Otherwise we set it to false. Once again we consider cases for a particular event $e_i$:
\begin{itemize}
\item Case 1. $e_i$ is a write event.
\begin{itemize}
\item In the case, the process performs a local write, the process can either set $\mathit{liveChain}[x]$ is set to true, maintaining the invariant or it can generate a reservation which will be used to certify the write. In this case the reservation is marked as used.
\item In the case the process decides to publish a write it must publish it as a promise, incrementing $numEE$ (after checking that the bound of $K$ has not been crossed), setting the promise flag to -1, maintaining invariant (3) above (leading to a publishedS write). Also, if it decides to certify a previous promise , it does so, similar to the \emph{normal} phase, though it now sets the timestamp to -1, indicating that the certification is local to the current phase and must be reset when normal phase resumes. Moreover (publishedF write) note that $\mathit{liveChain}[x]$ is set to false maintaining invariant (1).
\item Also, note that $\mathit{extView}[x]$ is set to true maintaining invariant (2).
\end{itemize}
\item Case 2. $e_i$ is a read event.
\begin{itemize}
\item The main highlight of read events in $\mathit{checkMode}$, is that we ascertain that $\mathit{liveChain}[x]$ is false while making a global read. This is to ensure that we forbid additive insertion. Indeed, following invariant (1) above, if $\mathit{liveChain}[x]$ were true during a global read, it would mean that the interval corresponding to the previous message (which caused $\mathit{liveChain}[x]$ to be true) is additively.
\end{itemize}
\item Case 3. $e_i$ is a ${\mathbf{CAS}}$ event.
\begin{itemize}
\item Once again similar to \emph{normal} phase we guess whether we make a local or a global read. Crucially however, we note that we forbid making a local write for a ${\mathbf{CAS}}$ when $\mathit{extView}[x]$ is true. Considering the invariant (2) above, this is done precisely to forbid ${\mathbf{CAS}}$ where, the promised interval containing the write is non-adjacent to the message being read from. The remainder book keeping is identical to previous cases.
\end{itemize}
\item Case 4. $e_i$ is a $\mathsf{SC\text{-}fence}$ event. This case does not arise since a process in $\mathit{checkMode}$ may not execute a $\mathsf{SC\text{-}fence}$ instruction, as otherwise the run will not be consistent \cite{promising,promising2}.
\end{itemize}
To conclude, note due to loadState and saveState functions, only used reservations and promises are retained after the $\mathit{checkMode}$ phase. Moreover due to the check of message flags after termination of a $\mathit{checkMode}$ phase, it is ensured that the process is in a consistent state while switching contexts. Noting that we keep track of promises as well as view-switches using $numEE$ we may only generate a run in which the sum of the two is bounded by $K$.
Next, we consider the converse direction from $\textsf{PS 2.0-rlx}$ to SC.
\subsection*{$\ps$ to SC}
We now prove the second part, from $\ps$ to SC. We prove that for every $K$-bounded run $\rho$ in $\ps$, there is a $K+n$ context bounded run $\tau$ in SC. We will show this in two steps.
\begin{itemize}
\item Given the $K$-bounded $\rho$, first we will construct a run $\rho''$ which is $K$-bounded and $K+n$ context bounded that reaches the same configuration as $\rho$.
\item We will then construct a run $\tau$ of SC using $\rho''$.
\end{itemize}
\textbf{Intuition} While we concretized the abstract (non-essential) timestamps when going from SC to $\ps$ earlier now we do the opposite. However, we will additionally show that $K+n$ SC contexts suffice for the translation. The way we account for the $K+n$ contexts is as follows - $n$ contexts for the process initializations and (atmost) one context for each essential event.
Hence, we ensure that atleast one essential event occurs in each context. This is possible for the following reason. Consider a run with $K$ essential events occuring in some order executed by processes $p_1$ to $p_K$. If we schedule the processes $p_i$ in the run under SC in the same order, then we will get a valid run under SC. Since view-switches account for all the external reads-from dependencies, the runw which we obtain is also valid.
More concretely, we ensure that each process only switches out of context only when it is awaiting a message for an external read from another process or when it has made atleast one promise or reservation. Since the total number of such essential events along a \emph{normal} phase + additional messages in all $\mathit{checkMode}$ phases is bounded above by $K$, we need at most $K+n$ context switches. We add $n$ for the concluding contexts required to reach the $\terminated$ configurations.
\newline\textbf{Details}
Let $\mathit{rf}$ (called $reads$-$from$) be a binary relation on events such that $(e_a, e_b) \in \mathit{rf}$ iff $e_b$ reads from a message $\textit{published}$ by $e_a$. Note that every run under $\ps$ semantics defines a $\mathit{rf}$ relation as the reads are executed. For construction of $\rho''$, the intuition is that a context switch is required only when the current process has reached $\terminated$ or it needs a message that is yet to be published by some other process. At a configuration $\mathsf{conf}_i$ of $\rho$, we say that an event of $\rho$ is a \emph{requesting} event if it is a view-altering event in $\rho$ and it reads a message that is not in the message pool at $\mathsf{conf}_i$. Also, we call the events that publish messages for these events as \emph{servicing} events ($\mathfrak{write}$ or ${\mathbf{CAS}}$, either simple or promises). Note that the set of servicing and requesting events is dependent on the configuration $\mathsf{conf}_i$. The two sets change along the run $\rho$. Specifically, an event is removed from the requesting event set as soon as the servicing event corresponding to it is executed. Let the size of the set of requesting events be $r$. At $\initconf$, $r = K$. We will prove by induction that given a set of processes ($n$), the $\mathit{rf}$ relation, and a run $\rho$ in $\ps$ that maintains the $\mathit{rf}$ relation, there is a run which uses at most $r+n$ context switches and defines the same $\mathit{rf}$ relation.
\noindent{\bf{The Base Case}}.
For $r+n=1$, there is only one process so the number of context switches is $0$ and the run $\rho$ itself uses 0 context switches.
\noindent{\bf{The Inductive Step}}.
Assume the hypothesis for $r+n=\ell$ and we prove the claim for $r+n=\ell+1$. Clearly at $\initconf$, there is at least one process which either has no requesting events, or has a servicing event before any requesting events in its instruction sequence. Otherwise, the run $\rho$ will not be able to execute all the events since no process will be able to move past its requesting event.
If we have a process that can reach termination directly, then in $\rho''$, we run that process and reduce $r+n$. Otherwise, consider the instructions of the process ($p_j$) that has a servicing event before any of its requesting events.
The instructions of $p_j$, till the first requesting event, can be executed since all the messages they need are already in the pool and hence we can create a new run $\rho_t$ in which these instructions are executed first and the remaining ones follow the same order as $\rho$. Note that $\rho_t$ reduces $r$ by at least $1$ while executing the instructions of $p_j$. By applying the hypothesis on the remaining sequence of instructions, we have a run that uses $r-1+n$ context switches and that maintains $\mathit{rf}$ of the remaining instructions. This can now be combined by the instructions of $p_j$ that have already been executed to give $\rho''$. \\
We now construct the run $\tau$ from $\rho''$. As explained in the text above, at most $2K$ time-stamps are needed to simulate the $\rho''$. Let the set of such time-stamps be $U\_x$ for each variable $x$. Let $\mathsf{M_x}$ be an increasing (mapping) function for each variable from $U\_x \cup \{0\}$ to $\{0, \dots 2K\}$ such that $\mathsf{M_x}(0)=0$.
We will construct the run $\tau$ in SC from $\rho''$, event by event, while maintaining the following invariants
\begin{enumerate}
\item All the time-stamps, in a particular message in $\mathit{messageStore}$, are related to the time-stamps in the corresponding essential messages in $\ps$ by $M_x$.
\item For a process $p$, $\mathsf{View_{\ps}(x)} \in U\_x$ iff $\mathit{view}[x].l$ is true at that point in SC and $\mathit{view}[x].t$ = $\mathsf{M_x}(\mathsf{View_{\ps}(x)}))$
\end{enumerate}
The $i^{th}$ event of $\rho''$ can be one of the following:
\begin{itemize}
\item Case 1. $e_i$ is a write to variable $x$ with value $v$.
\begin{itemize}
\item If the time-stamp $t$ of this write belongs to $U\_x$, then we first allocate $M_x(t)$ in SC to this write and make $\mathit{view}[x].l$ true. This maintains invariant (2).
\item If the event is a servicing event, then the time-stamp of this message satisfies the requirements of invariant (1) and hence it can be added to $\mathit{messageStore}$.
Otherwise, we do not update the $\mathsf{View_{SC}(x)}$ of the process and make $\mathit{view}[x].l$ false.
\end{itemize}
\item Case 2. $e_i$ is a read of variable $x$. \\
If this event is a view-altering event, then the current timestamp in the
$\mathsf{View_{\ps}}$ will be used for comparison. The effect of the read in SC will be same as in $\ps$ since $V\_x$ is an increasing function. All the invariants will still hold after this, since all the messages in $\mathit{messageStore}$ satisfy the invariants.
\item Case 3. $e_i$ is a ${\mathbf{CAS}}$ to variable $x$ with values $v, v'$.
If this event is not view-altering, then the process either reads some other process's message again or reads its own. If it reads its own message, then no change to the $\mathsf{View_{SC}(x)}$ has to be done for the read part and the new message is added to $\mathit{messageStore}$ if $e_i's$ message is essential. If it reads some other processes' message again, then $\mathit{view}[x].l$ is true, and since this message has not been used for a ${\mathbf{CAS}}$ yet, the check of $upd\_x[\mathit{view}[x].t]$ will go through in $Prog'$. Now, it needs to be decided if the new message is essential. If the read is view-altering, then it is similar to Case 2 followed by the decision of adding the new message to $\mathit{messageStore}$.
\item Case 4. $e_i$ is an $\mathsf{SC\text{-}fence}$
If $\mathit{globalTimeMap}[x]$ is greater than $\mathit{view}[x].t$, we maintain invariants (2) by setting $\_\mathit{view}[x].l$ to true and the $\mathit{view}[x].f$ invariant [\ref{app:ctc}] by setting it to $\mathit{view}[x].f$. On the other hand, if $\mathit{view}[x].t$ is greater, we set $\mathit{globalTimeMap}[x]$ to the smallest member $t \in \mathsf{Time}$, which satisfies $t \geq \mathsf{M_x}(\mathsf{View_{\ps}}(x))$. In case $\mathit{view}[x].l$ is true, $t$ is $\mathit{view}[x].t$ itself by invariant (2). If not, then we set it to $\mathit{view}[x].t + 1$, since we note that $\mathit{view}[x].t$ is the largest member of $\mathsf{Time}$, that $p$ has had as $\mathsf{View_{\ps}}(x)$, and currently the former is lower than $\mathsf{M_x}(\mathsf{View_{\ps}}(x))$.
\end{itemize}
|
1,116,691,497,543 | arxiv | \section{#1}}
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{\Large \bf Holographic Confinement/Deconfinement Phase Transitions
of AdS/QCD in Curved Spaces}
\vspace{10mm}
{\large Rong-Gen Cai\footnote{Email address: [email protected]} and
Jonathan P. Shock\footnote{Email address:
[email protected]}}\\
\vspace{5mm}
{ \em Institute of Theoretical Physics, Chinese Academy of Sciences, \\
P.O. Box 2735, Beijing 100080, China}\\
\end{center}
\vspace{5mm} \centerline{{\bf{Abstract}}}
\vspace{5mm}
Recently Herzog has shown that deconfinement of AdS/QCD can
be realized, in the hard-wall model where the small radius region
is removed in the asymptotically AdS space, via a first order
Hawking-Page phase transition between a low temperature phase given
by a pure AdS geometry and a high temperature phase given by the AdS
black hole in Poincare coordinates. In this paper we first extend
Herzog's work to the hard wall AdS/QCD model in curved spaces by
studying the thermodynamics of AdS black holes with spherical or
negative constant curvature horizon, dual to a non-supersymmetric
Yang-Mills theory on a sphere or hyperboloid respectively. For the
spherical horizon case, we find that the temperature of the phase
transition increases by introducing an infrared cutoff, compared to
the case without the cutoff; For the hyperbolic horizon case, there
is a gap for the infrared cutoff, below which the Hawking-Page phase
transition does not occur. We also discuss charged AdS black holes
in the grand canonical ensemble, corresponding to a Yang-Mills
theory at finite chemical potential, and find that there is always a
gap for the infrared cutoff due to the existence of a minimal
horizon for the charged AdS black holes with any horizon topology.
\end{titlepage}
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\sect{Introduction}
The remarkable AdS/CFT correspondence~\cite{AdS} conjectures that
string/M theory in an anti-de Sitter space (AdS) times a compact
manifold is dual to a large $N$ strongly coupling conformal field
theory (CFT) residing on the boundary of the AdS space. A special
example of the AdS/CFT correspondence is that type IIB string theory
in $AdS_5\times S^5$ is dual to a four dimensional ${\cal N}=4$
supersymmetric Yang-Mills theory on the boundary of $AdS_5$. At
finite temperature, in the spirit of the AdS/CFT correspondence,
Witten~\cite{Witten} has argued that the thermodynamics of black
holes in AdS space can be identified with that of the dual strongly
coupling field theory in the high temperature limit. Therefore one
can discuss the thermodynamics and phase structure of strongly
coupling field theories by studying the thermodynamics of various
kinds of black holes in AdS space. Indeed, it is well-known that
there exists a phase transition between the Schwarzschild-AdS black
hole and thermal AdS space, the so-called Hawking-Page phase
transition~\cite{HP}: the black hole phase dominates the partition
function in the high temperature limit, while the thermal AdS space
dominates in the low temperature limit. This phase transition is of
first order, and is interpreted as the confinement/deconfinement
phase transition in the dual field theory~\cite{Witten}. Note that
the dual conformal field theory to the Schwarzschild-AdS black hole
configuration resides on a sphere. For example, for the five
dimensional Schwarzschild-AdS black holes, the dual field theory is
${\cal N}=4$ supersymmetric Yang-Mills theory at finite
temperature~\cite{Witten}. That is, by discussing the thermodynamics
of AdS black holes with a spherical horizon, one can study
thermodynamics of the dual non-supersymmetric Yang-Mills theory on a
sphere at finite temperature.
It is interesting to note that in AdS space the black hole horizon
is not necessarily a sphere~\cite{Topo}, the black hole horizon can
also be a Ricci flat surface \cite{Flat} or a negative constant
curvature surface~\cite{Negative}. Thus using those AdS black hole
solutions, one can study dual strongly coupled field theories
residing on Ricci flat or hyperbolic spaces. These so-called
topological black holes have been investigated in higher dimensions
\cite{Birm,high,GB,Love} and in dilaton gravity~\cite{CJS,CZ}. It
was found that the Hawking-Page phase transition, which happens for
spherical AdS black holes, does not occur for Ricci flat and
negative curvature AdS black holes, the latter two being not only
locally stable (heat capacity is always positive), but also globally
stable (see for example, \cite{Birm}). Here it is worth mentioning
that if some directions of the horizon surface are compact for Ricci
flat AdS black holes, similar to the case of spherical AdS black
holes, the Hawking-Page phase transition can
occur~\cite{SSW,Page,Myers,CKW,BD} due to the existence of so-called
AdS soliton~\cite{HM}. The absence of a Hawking-Page phase
transition for Ricci flat AdS black holes without compact directions
is consistent with the result that the AdS space and the AdS black
holes in Poincare coordinates without compact directions are both in
the deconfinement phases~\cite{Witten}. This can be confirmed by
calculating quark-anti-quark potential through
Wilson-loop~\cite{Wilson}. In addition, the black hole entropy is
proportional to $N^2$, where $N$ is the rank of gauge group $SU(N)$
for the dual gauge field.
Using the AdS/CFT correspondence, it is expected that we can obtain
some more qualitative understanding of QCD and the nature of
confinement. The authors of papers~\cite{KS,MN,PS} are able to
realize confinement of related supersymmetric field theories by
finding dual gravitational configurations where the geometry in the
infrared at small radius is capped off in a smooth way. The author
of \cite{PS2} proposed a simpler model, $AdS_5$ in Poincare
coordinates without compact directions where the small radius region
is removed, to realize confinement. The field theory dual to this is
a non-supersymmetric Yang-Mills theory in four dimensions. Although
such a model is somewhat rough, the results obtained in
\cite{EKSS,DP} show that one can get some realistic,
semiquantitative descriptions of low energy QCD, by using this hard
wall model.
Recently, Herzog~\cite{Herz} (see also \cite{BBBZ}) has shown that
in this simple hard wall model of AdS/QCD, the
confinement/deconfinement phase transition occurs via the first
order Hawking-Page phase transition between the low temperature
thermal AdS space and high temperature AdS black hole in Poincare
coordinates. Note the facts that removing the small radius region of
$AdS_5$ is dual to introducing an infrared (IR) cutoff in the dual
field theory and that the Hawking-Page phase transition does not
occur for the negative constant curvature AdS black holes without a
cutoff. It is quite interesting to revisit the thermodynamics and
the Hawking-Page phase transitions for spherical AdS black holes and
negative constant curvature AdS black holes by removing the small
radius regions. That is, we are interested in the
confinement/deconfinement phase transition for the hard wall QCD
model in curved spaces. Recall that charged AdS black hole
configurations are dual to supersymmetric field theories with
so-called R-charges~\cite{CEJM,CG,Cai}. Introducing R-charges to the
AdS black holes is equivalent to introducing a chemical potential in
the dual field theory. In this paper, therefore, we will also
discuss the case of charged AdS black holes.
The paper is organized as follows. In the next section we will
revisit the Hawking-Page phase transition for spherical AdS black
holes with an IR cutoff. The case for the negative constant
curvature black holes will be discussed in Sec.~3. In. Sec.~4 we
will investigate the phase transition for charged AdS black holes
with an IR cutoff. Sec.~5 will be devoted to conclusions and
discussions.
\sect{Hawking-Page Phase Transition for Spherical Black Holes with
an IR Cutoff}
Let us start with the action of five-dimensional general relativity
with a negative cosmological constant
\begin{equation}
\label{2eq1}
{\cal S}= \frac{1}{2\kappa^2} \int d^5x \sqrt{-g} \left(
R+\frac{12}{l^2}\right),
\end{equation}
where $k^2= 8\pi G$ and $l$ is the radius of the five-dimensional
AdS space.
The action (\ref{2eq1}) admits the AdS spherical black hole
solution with the metric
\begin{equation}
\label{2eq2}
ds^2= -V(r) dt^2 + V(r)^{-1} dr^2 + r^2 d\Omega_3^2,
\end{equation}
where
\begin{equation}
\label{2eq3}
V(r) = 1+ \frac{r^2}{l^2} -\frac{r^2_s}{r^2},
\end{equation}
$d\Omega_3^2$ is the line element of a three-dimensional sphere
with unit radius and $r_s^2$ is an integration constant,
the Schwarzschild mass parameter. The black hole horizon is determined
by $V(r_+)=0$, which gives $r_+^2 = l^2
(-1+\sqrt{1+4r_s^2/l^2})/2$. The black hole has an inverse temperature $1/T$
\begin{equation}
\beta = 1/T=\frac{2\pi r_+}{1+ 2r_+^2/l^2}.
\end{equation}
In the global coordinate
(\ref{2eq2}), the AdS boundary is located at $r\rightarrow
\infty$. In Poincare coordinates the radial direction is dual to the energy scale of
the dual field theory. If one introduces a regulating UV cutoff at $r=R \gg r_+$, then
the range from $r=R$ to $ \infty$ should be removed from the bulk
geometry. On the other hand, one further introduces an IR
cutoff at $r=r_0$ in the dual field theory (The IR cutoff is
equivalent to a mass gap in the dual field theory), the range from $r=0$ to $r_0$
also should be removed from the bulk geometry~\cite{PS2}. This corresponds to a hard wall
truncation of the space, independent of the radius of the horizon of the black
hole. Namely, we are
considering the bulk geometry from $r=r_0$ to $r=R$. The UV cutoff
is necessary in order to regularize the action. At the end of
calculation, we will remove the UV cutoff by letting
$R\rightarrow \infty$.
To see the phase structure of QCD in the hard wall AdS/QCD
model, let us calculate the Euclidean action of the AdS
black hole, by choosing the AdS vacuum solution (\ref{2eq2}) with
\begin{equation}
\label{2eq5}
V_b=1+\frac{r^2}{l^2}
\end{equation}
as the reference background. As a result, the Euclidean action of
the reference background is
\begin{equation}
\label{2eq6}
{\cal S}_b= \frac{4\Omega_3}{\kappa^2 l^2} \beta_b \int^R _{r_0}
r^3 dr,
\end{equation}
while the Euclidean action of the black hole is
\begin{equation}
\label{2eq7}
{\cal S}_{bh} = \frac{4\Omega_3}{\kappa^2l^2}\beta
\int^R_{r_{\rm max}}r^3dr,
\end{equation}
where $\Omega_3$ is the volume of the unit three-sphere and
$r_{\rm max}= {\rm max}(r_0,r_+)$ is the IR cutoff in the high temperature phase.
If $r_0<r_+$, one has $r_{\rm max}=r_+$ while $r_{\rm max}=r_0$ as
$r_0>r_+$. In addition, let us mention here that usually the
Hilbert-Einstein action of general relativity should be supplemented
by the Gibbons-Hawking surface term in order to have a well-defined
variational principle. However, for the case of the asymptotically AdS spaces, such
terms have no contribution to the difference of two Euclidean
actions between the configuration under consideration and the
reference background~\cite{Witten}.
In order that the black hole solution (\ref{2eq2}) can be embedded
into the background consistently, at the boundary $r=R$ the period
$\beta_b$ of the Euclidean time for the reference background has to
obey the relation
\begin{equation}
\label{2eq8}
\beta_b \sqrt{V_b(R)}=\beta \sqrt{V(R)}.
\end{equation}
Using (\ref{2eq8}), calculating the difference between (\ref{2eq7})
and (\ref{2eq6}), and taking the limit $ R\rightarrow \infty $, we
obtain
\begin{equation}
\label{2eq9}
{\cal I} = -\frac{\Omega_3 \beta }{2\kappa^2 l^2} \left ( 2r^4_{\rm
max }-r_+^4 -2r_0^4 -r_+^2l^2 \right ).
\end{equation}
Without the IR cutoff, namely $r_0=0$, the action (\ref{2eq9})
reduces to
\begin{equation}
\label{2eq10}
{\cal I} =-\frac{\Omega_3 \beta }{2\kappa^2 l^2}r_+^2 ( r_+^2 -l^2
).
\end{equation}
Clearly when $r_+=l$, the Euclidean action alters its sign, a
first-order phase transition happens. This is just the well-known
Hawking-Page phase transition. The phase transition temperature is
\begin{equation}
\label{2eq11}
T_{\rm HP}=\frac{3}{2\pi l}.
\end{equation}
When $r_+>l$, or $T>T_{\rm HP}$, the black hole phase is dominant
in the partition function, while the thermal AdS space is dominant
as $T<T_c$ in the low temperature phase. With the action
(\ref{2eq10}), we obtain the mass of the AdS black hole, via
$E=\partial {\cal I}/\partial \beta$,
\begin{equation}
\label{2eq12}
E= \frac{3\Omega_3r_+^2}{2\kappa^2}\left(
1+\frac{r_+^2}{l^2}\right)=M,
\end{equation}
and the entropy of the black hole, via $S=\beta E-{\cal I}$,
\begin{equation}
\label{2eq13}
S = \frac{\Omega_3r_+^3}{4G},
\end{equation}
satisfying the area formula of black hole entropy.
Now we consider the case with an IR cutoff, $r_0$. There are two
quite different cases.
(i) $r_0 \ge r_+$. In this case, one has $r_{\rm max}=r_0$. And the
action (\ref{2eq9}) turns out to be
\begin{equation}
{\cal I}= \frac{\Omega_3 \beta }{2\kappa^2 l^2} \left (
r_+^4 +r_+^2l^2 \right ).
\end{equation}
Clearly in this case, the action is always positive and therefore
no phase transition will occur. The thermal AdS space is globally stable and
is dominant in the partition function of the dual field theory.
(ii) $r_0<r_+$. In this case one has $r_{\rm max}=r_+$. And the
action (\ref{2eq9}) becomes
\begin{equation}
\label{2eq15}
{\cal I} = -\frac{\Omega_3 \beta }{2\kappa^2 l^2} \left (
r_+^4 -2r_0^4 -r_+^2l^2 \right ).
\end{equation}
Obviously, the action changes its sign at
\begin{equation}
r_+^2 = \frac{l^2}{2}\left( 1+\sqrt{1+8r_0^4/l^4} \right).
\end{equation}
Therefore in this case, the phase transition can occur and the
critical temperature is
\begin{equation}
T_c = \frac{2+\sqrt{1+8r_0^4/l^4}}{\sqrt{2}\pi l
\sqrt{1+\sqrt{1+8r_0^4/l^4}}} >T_{\rm HP}.
\end{equation}
This critical temperature is higher than the one for the case
without the IR cutoff. The thermal energy of dual QCD is
\begin{equation}
\label{2eq18}
E= M+\frac{\Omega_3r_0^4}{\kappa^2l^2},
\end{equation}
where $M$ is given in (\ref{2eq12}), and the entropy is still
given by (\ref{2eq13}). We can clearly see from (\ref{2eq18}) the
relation between the IR cutoff $r_0$ and the mass gap in the
holographic QCD..
As a result, we have seen that by introducing an IR cutoff, the
critical temperature for the corresponding Hawking-Page phase
transition increases, compared to the case without the cutoff. The
critical temperature is determined by the ratio $r_0/l$. In
addition, we stress here that the condition $r_0 <r_+$ in the
gravity side corresponds to the one that the mass gap is less than
temperature in the field theory side. Similar statement holds in
the following discussions.
\sect{Hawking-Page Phase Transition for Hyperbolic Black Holes with
an IR Cutoff}
In this section we will consider the so-called negative curvature
black holes whose horizon is a negative constant curvature surface.
In this case, the AdS black hole has the form
\begin{equation}
\label{3eq1}
ds^2 =-V(r) dt^2 +V(r)^{-1} dr^2 +r^2 d\Sigma_3^2,
\end{equation}
where
\begin{equation}
\label{3eq2}
V(r)= -1+\frac{r^2}{l^2}-\frac{r^2_s}{r^2},
\end{equation}
and $d\Sigma^2_3$ is the line element for the three-dimensional
surface with curvature $-6$. With suitable identification, one can
construct closed horizon surfaces with high genus. Such black holes
are called topological black holes in some of the literature. In this case,
the dual field theory resides on a hyperboloid described by $d\Sigma_3^2$.
In this
negative curvature case, there exist two remarkable features of
the solution. One is that even when the mass parameter $r_s^2$
vanishes, the metric (\ref{3eq1}) still has a black hole causal
structure. The metric function $V(r)$ is replaced
by
\begin{equation}
\label{3eq3}
V_b= -1 +\frac{r^2}{l^2}.
\end{equation}
The black hole is called a ``massless'' black hole with a
horizon at $r_+=l$. The other is that when $r_s^2$ is negative, there
exists a ``negative mass" black hole with
\begin{equation}
\label{3eq4}
V_b(r) =-1 +\frac{r^2}{l^2} +\frac{r^2_b}{r^2},
\end{equation}
provided $0< r_b^2 \le l^2/4$. When $r_b=l/2$, the black hole solution
(\ref{3eq4}) is an extremal one with vanishing temperature. For
the black hole (\ref{3eq2}), the inverse Hawking temperature is
\begin{equation}
\label{3eq5}
\beta = \frac{2\pi r_+}{2r_+^2/l^2-1},
\end{equation}
where $r_+$ is the black hole horizon satisfying $V(r_+)=0$. Once
again, in order to regularize the Euclidean action, one has to
choose a suitable reference background. One may consider the AdS space
(\ref{3eq3}) as the reference background. But its shortcoming is that the period of the
Euclidean time of the solution is fixed as $2\pi l$ and cannot be chosen as an arbitrary
value,
otherwise the reference background has a conical singularity.
Another choice is the solution (\ref{3eq4}) with $r_b=l/2$. In
this case, the Euclidean time can be arbitrary since it describes
an extremal black hole. Of course, when $r_b=0$, the solution
(\ref{3eq4}) reduces to (\ref{3eq3}). Therefore in what follows,
we will consider the solution (\ref{3eq4}) with $r_b=l/2$ as the
reference background. When $r_b=0$, it becomes the case
(\ref{3eq3}). That is, in the following discussions,
either $r_b=l/2$ or $r_b=0$.
Considering the period $\beta_b$ of the Euclidean time for the reference background
obeying
\begin{equation}
\beta_b\sqrt{V_b(R)}=\beta \sqrt{V(R)},
\end{equation}
at the boundary, and introducing an IR cutoff $r_0$ (here the meaning of $r_0$ is
the same as the one in the previous section, the relation of $r_0$ to the mass
gap in the dual field theory can be obtained by calculating the mass from the
Euclidean action below), we obtain
the Euclidean action difference between the black hole and the
background
\begin{equation}
\label{3eq7}
{\cal I} = -\frac{\Sigma_3 \beta}{2\kappa^2 l^2} \left( 2r^4_{\rm
max}-2r^4_0 -r_+^4+ r_+^2l^2 -r_b^2l^2\right),
\end{equation}
where $\Sigma_3$ is the volume of the closed horizon surface with unit
radius. First let us consider the case without the cutoff. In
that case, $r_{\rm max}=r_+$, and the action reduces to
\begin{equation}
\label{3eq8}
{\cal I} =-\frac{\Sigma_3 \beta}{2\kappa^2 l^2} \left(r_+^4+ r_+^2l^2
-r_b^2l^2\right).
\end{equation}
Note that the minimal horizon radius for the black hole
(\ref{3eq2})
is $r_{\rm min}=l/\sqrt{2}$. If one chooses $r_b=l/2$ and
(\ref{3eq4}) as the reference background, or $r_b=0$ and
(\ref{3eq3}) as the reference background, then the action (\ref{3eq8})
is always negative and the dual field theory is in the deconfinement phase.
Therefore the usual Hawking-Page phase
transition will not occur in this case.
Next we consider the case with an IR cutoff $r_0$. Here there are
also two different cases.
(i) $r_0>r_+$. In this case, one has $r_{\rm max}=r_0$, and
the action becomes
\begin{equation}
\label{3eq9}
{\cal I} = \frac{\Sigma_3 \beta}{2\kappa^2 l^2} \left(r_+^4-
r_+^2l^2 + r_b^2l^2\right).
\end{equation}
If $r_b=l/2$, we find that the action is always positive. Thus
no phase transition occurs. If $r_b=0$, however, an interesting
phenomenon appears. For those ``negative mass" black holes
with horizon radius $l/\sqrt{2} \le r_+ <l$, the action is
negative, while it is positive for black holes with horizon
$r_+>l$. The action changes its sign at $r_+=l$. This implies
that in the low temperature phase $0<T<1/2\pi l$, the system is globally stable,
while it becomes unstable as $T>1/2\pi l$; a Hawking-Page phase
transition happens at $r_+=l$. This is an anti-intuitive result.
This seemingly indicates that choosing the AdS black hole
solution (\ref{3eq3}) is not suitable. Indeed, the result from
the surface counterterm method indicates that one should choose
$r_b=l/2$ as the reference background~\cite{EMP}.
(ii) $r_0<r_+$. In this case, we have $r_{\rm max}=r_+$, and the
action is given by
\begin{equation}
\label{3eq10} {\cal I} = -\frac{\Sigma_3 \beta}{2\kappa^2 l^2}
\left( r^4_+-2r^4_0 + r_+^2l^2 -r_b^2l^2\right).
\end{equation}
When $r^2_+ > r_c^2 $ with
\begin{equation}
\label{3eq11}
r_c^2=
\frac{l^2}{2}\left(-1+\sqrt{1+\frac{8r_0^4}{l^4}+\frac{4r_b^2}{l^2}}
\right),
\end{equation}
the action is negative, while it is positive for $r_{\rm min}^2
\le r_+^2 < r_c^2$. The Hawking-Page phase transition happens at $r_+=r_c$.
The critical temperature is
\begin{equation}
\label{3eq12}
T_c= \frac{2r_c^2/l^2-1}{2\pi r_c}.
\end{equation}
Note that when $r_0=0$, one has $r_c<r_{\rm
min}$, the action is always negative. In order to have $r_c
>r_{\rm min}$, there exists therefore a gap for the IR cutoff
$r_0$:
\begin{equation}
\label{3eq13}
\frac{r_0^4}{l^4} > \frac{1}{8} \left( 3
-\frac{4r_b^2}{l^2}\right)=\frac{1}{4}.
\end{equation}
In the above calculation we have taken $r_b=l/2$. Therefore the IR
cutoff must be larger than the minimal horizon radius $r_{\rm
min}=l/\sqrt{2}$.
On the other hand, if we take $r_b=0$, the critical radius is
\begin{equation}
\label{3eq14} r_c^2=
\frac{l^2}{2}\left(-1+\sqrt{1+\frac{8r_0^4}{l^4}}
\right).
\end{equation}
The Hawking-Page phase transition happens when the temperature crosses the critical
value (\ref{3eq12}). In this case, there is no gap for the IR cutoff
$r_0$.
\sect{Hawking-Page Phase Transition for Charged AdS Black Holes with
an IR Cutoff}
In this section we consider the cases where the dual gravity
configurations are charged AdS black holes with different topology
horizons. In this case, the dual field theory is a Yang-Mills
theory with chemical potential at finite temperature. We start
with the Einstein-Maxwell action with a negative cosmological
constant in five dimensions
\begin{equation}
\label{4eq1}
{\cal S}= \frac{1}{16\pi G}\int d^5x\sqrt{-g} \left( R
+\frac{12}{l^2}- F^2\right),
\end{equation}
where $F_{\mu\nu}$ is the Maxwell field strength. This action can come
from the spherical ($S^5$) reduction of type IIB supergravity~\cite{CEJM,CG,Cai}.
The charged AdS black holes have the
metric form
\begin{equation}
\label{4eq2}
ds^2 =- V(r) dt^2 +V(r)^{-1}dr^2 + r^2 d\Omega_3^2,
\end{equation}
where
\begin{equation}
\label{4eq3}
V(r) = k -\frac{m}{r^2} +\frac{q^2}{r^4}+\frac{r^2}{l^2},
\end{equation}
where $d\Omega_3^2$ is the line element for a three-dimensional
surface with constant curvature $6k$. Without loss of generality,
one may take $k=1$, $0$ or $-1$. In addition, $m$ and $q$ are two
integration constants, which are related to the mass and charge
of the solution, respectively.
The mass parameter $m$ can be expressed in terms of the
horizon, $r_+$, as $m = r_+^2 (k +q^2/r_+^4+r_+^2/l^2)$.
In order that the metric (\ref{4eq3}) describes a black hole with
horizon $r_+$, the potential must have a positive radial derivative leading
to the constraint on the horizon radius
\begin{equation}
\label{4eq4}
2r_+^6+ kr_+^4l^2 -q^2l^2 \ge 0,
\end{equation}
otherwise, the solution describes a naked singularity. This
condition can also be obtained from following Hawking
temperature of the black hole (to keep the positiveness of
the Hawking temperature).
The Hawking temperature of the black hole is
\begin{equation}
\label{4eq5}
T =\frac{1}{\beta} =\frac{1}{2\pi r_+}\left( k
+\frac{2r_+^2}{l^2} - \frac{q^2}{r_+^4}\right).
\end{equation}
For the charged solution, the associated electric potential is
\begin{equation}
\label{4eq6}
{\cal A} = \left( - \frac{q}{cr^2} +\Phi\right) dt,
\end{equation}
where $c=2/\sqrt{3}$, and $\Phi$ is a constant. We choose a gauge
with $\Phi= q/cr_+^2$ so that the potential vanishes at the black
hole horizon. Note that this provides us with a gauge invariant
quantity because $\Phi$ describes a potential
difference between at the horizon and at the infinity. Namely, if
one chooses another gauge, $\Phi$ appearing in the following
equations represents the potential difference between at the
horizon and at the infinity. As a result, $\Phi$ is gauge
invariant.
To discuss the thermodynamics and phase structure of dual QCD,
one has to choose a suitable reference background. For the
charged solution (\ref{4eq3}), one suitable background is the
solution (\ref{4eq2}) with
\begin{equation}
\label{4eq7}
V_b(r) = k +\frac{r^2}{l^2} +\frac{r_b^2}{r^2}\delta_{k,-1},
\end{equation}
where $r_b=l/2$ or $0$, based on the discussion in the previous
section. The choice of the background implies that we are going
to discuss the thermodynamics of the charged black holes in a
grand canonical ensemble, where the temperature and electric
(chemical) potential are fixed at the boundary. Note that although the
solution (\ref{4eq7}) is the one without charge, we are
discussing the thermodynamics in grand canonical ensemble, thus,
the solution (\ref{4eq7}) with a constant potential $\Phi$ is
still a solution of the action (\ref{4eq1}). Therefore the
solution (\ref{4eq7}) with a constant potential is a good
reference background, in order to analyze the thermodynamics of
charged black holes in grand canonical ensemble~\cite{CEJM}.
Considering the solution (\ref{4eq7}) as the reference background,
we get the Euclidean action difference between the charged black
hole and the background
\begin{equation}
\label{4eq8}
{\cal I}= \frac{\Omega_3 \beta}{16\pi Gl^2} \left( 2r_0^4
-2r^4_{\rm max} +kr_+^2l^2 +\frac{q^2l^2}{r_+^2} +r_+^4
-\frac{2q^2l^2}{r^2_{\rm max}}+ r^2_bl^2\delta_{k,-1}\right),
\end{equation}
where $\Omega_3$ is the volume of the three-dimensional closed
surface $d\Omega_3^2$.
We first discuss the case without the IR cutoff. In this case,
$r_0=0$, and then $r_{\rm max}=r_+$. The action becomes
\begin{eqnarray}
\label{4eq9}
{\cal I} &=& -\frac{\Omega_3 \beta}{16\pi Gl^2} \left( r_+^4
-kr_+^2l^2 +\frac{q^2l^2}{r_+^2} - r^2_bl^2\delta_{k,-1}\right),
\nonumber \\
&=& -\frac{\Omega_3 \beta}{16\pi Gl^2} \left( r_+^4
-r_+^2l^2(k-c^2\Phi^2) - r^2_bl^2\delta_{k,-1}\right).
\end{eqnarray}
Obviously, when $k=0$ or $-1$, the action is always negative.
Therefore the dual field theory is in the deconfinement phase and
the Hawking-Page phase transition does not occur here. On the other
hand, when $k=1$, the action changes its sign at
\begin{equation}
\label{4eq10}
r_+^2 =r_c^2= l^2(1-c^2\Phi^2),
\end{equation}
provided $\Phi^2 <1/c^2$, which implies that the Hawking-Page
phase transition happens at the critical temperature
\begin{equation}
\label{4eq11}
T_{\rm HP}=\frac{3}{2\pi l}\sqrt{1 - c^2\Phi^2}.
\end{equation}
When $\Phi^2 >1/c^2$, the action is also always negative even if
$k=1$. In that case, the dual field theory is in the deconfinement phase
and the confinement/deconfinement phase transition will not
occur.
Next we consider the case with an IR cutoff $r_0$. And we will
discuss separately the cases with $k=1$, $0$ and $-1$ below.
\subsection{ Case $k=0$: Ricci flat black holes}
The dual field theory to this gravitational background is a
Yang-Mills theory at finite chemical potential and at finite
temperature living on a flat four dimensional spacetime. For the
gravitational computation in this case one has two subcases:
$r_0>r_+$ or
$r _0<r_+$.
(i) When $r_0>r_+$, one has $r_{\rm max}=r_0$. In this case, the
action reduces to
\begin{equation}
\label{4eq12}
{\cal I}= \frac{\Omega_3 \beta}{16\pi Gl^2} \left (r_+^4 +c^2\Phi^2 l^2r_+^2(1-2r_+^2/r_0^2)
\right).
\end{equation}
The condition for the existence (\ref{4eq4}) of the horizon requires
$r_+ >c\Phi l/\sqrt{2}$. We find that when $c\Phi l/\sqrt{2} <r_+
<r_0$, the action (\ref{4eq12}) is always positive. As a result,
no phase transition happens in this case.
(ii) When $r_0<r_+$, we have $r_{\rm max}=r_+$, and the action
is
\begin{equation}
\label{4eq13}
{\cal I}= \frac{\Omega_3 \beta}{16\pi Gl^2} \left( 2r_0^4
-r^4_+ -c^2\Phi^2 l^2 r_+^2 \right).
\end{equation}
The action changes its sign from positive to negative when the horizon radius
crosses
\begin{equation}
\label{4eq14}
r_c^2 = \frac{-c^2\Phi^2 l^2+\sqrt{c^4\Phi^4l^4 +8r_0^4}}{2}.
\end{equation}
Note from (\ref{4eq4}) that there exists a minimal horizon radius
$r_{\rm min}= c\Phi l/\sqrt{2}$. we see from (\ref{4eq14}) that there is
a gap for the IR cutoff. Namely the cutoff $r_0$ must satisfy
\begin{equation}
\label{in1}
\frac{r_0^4}{l^4} > \frac{3}{8}c^4\Phi^4.
\end{equation}
Clearly, this gap disappears when the charge is
absent~\cite{Herz}. The Hawking-Page phase transition
temperature is
\begin{equation}
\label{4eq15}
T_c= \frac{1}{2\pi r_c l^2}\left( -2 c^2\Phi^2 l^2
+\sqrt{c^4\Phi^4 l^4 +8r_0^4}\right).
\end{equation}
Clearly, the phase transition
occurs due to the introduction of the IR cutoff.
\subsection{Case $k=1$: spherical black hole}
(i) When $r_0>r_+$, one has $r_{\rm max}=r_0$ and the reduced
action is
\begin{equation}
\label{4eq16} {\cal I}=\frac{\Omega_3 \beta}{16\pi Gl^2} \left(
r_+^4 +r_+^2l^2 +c^2\Phi^2 l^2 r_+^2 (1-2r_+^2/r_0^2)\right).
\end{equation}
Considering the condition (\ref{4eq4}) for the existence of the black hole
horizon, once again, we find that the action is always positive.
Therefore, no phase transition will occur in this case.
(ii) When $r_0<r_+$, we have $r_{\rm max}=r_+$, and the Euclidean
action is
\begin{equation}
\label{4eq17}
{\cal I}= \frac{\Omega_3 \beta}{16\pi Gl^2} \left( 2r_0^4
-r^4_+ + r_+^2l^2(1-c^2\Phi^2)\right).
\end{equation}
From the action, we can see that the Hawking-Page phase transition
happens when the
horizon radius crosses
\begin{equation}
\label{4eq18}
r_c^2 =\frac{l^2(1-c^2\Phi^2) +\sqrt{(1-c^2\Phi^2)^2l^4
+8r_0^4}}{2}.
\end{equation}
And the corresponding temperature at which the phase transition occurs is
\begin{equation}
\label{4eq19}
T_c= \frac{1}{2\pi r_c}\left(
2(1-c^2\Phi^2)+\sqrt{(1-c^2\Phi^2)l^4 +8r_0^4}\right).
\end{equation}
This temperature is higher than the one (\ref{4eq11}) for the
case without the IR cutoff. Again, there is a gap for the IR
cutoff because of the existence of the minimal horizon radius:
the IR cutoff must be
\begin{equation}
\label{in2}
\frac{r_0^4}{l^4} > \frac{3}{8} (1-c^2\Phi^2)^2.
\end{equation}
\subsection{Case $k=-1$: hyperbolic black hole}
(i) When $r_0>r_+$, we have $r_{\rm max}=r_0$, and the action
reduces to
\begin{equation}
\label{4eq20} {\cal I}= \frac{\Omega_3 \beta}{16\pi Gl^2} \left(
r_+^4 -r_+^2l^2 +c^2\Phi^2 r_+^2 l^2 (1-2r_+^2/r_0^2)+
r^2_bl^2\right).
\end{equation}
According to the analysis in the previous section, we see that
choosing the solution (\ref{3eq4}) with $r_b=0$ as the reference
background is not reasonable. Therefore in this subsection we
focus on the case with $r_b=l/2$. Once again, within the range
$r_{\rm min}= l\sqrt{(1+c^2\Phi^2)/2} <r_+ <r_0$, the action is found to be always positive and
no phase transition will occur, dual QCD is in the confinement
phase.
(ii) When $r_0<r_+$, one has $r_{\rm max}=r_+$. The action
reduces to
\begin{equation}
\label{4eq21}
{\cal I}= \frac{\Omega_3 \beta}{16\pi Gl^2} \left( 2r_0^4
-r^4_+-r_+^2l^2 -c^2\Phi^2 l^2 r_+^2+ r^2_bl^2\right).
\end{equation}
Clearly when $r_0=0$, no phase transition will occur. However,
when $r_0 \ne 0$, the Hawking-Page phase transition happens when
the horizon crosses $r_c$, satisfying
\begin{equation}
\label{4eq22}
r_c^2=\frac{(1+c^2\Phi^2)l^2}{2} \left( -1 +\sqrt{1+\frac{8r_0^4
+4r_b^2l^2}{(1+c^2\Phi^2)l^4}}\right).
\end{equation}
Since the critical radius must be larger than the minimal horizon
radius $r_{\rm min}$, this leaves us with a gap for the IR cutoff
\begin{equation}
\label{4eq23}
\frac{r_0^4}{l^4} > \frac{2+3c^2\Phi^2}{8}.
\end{equation}
Compared to the case (\ref{3eq13}) without charge, the IR cutoff
gap increases. Furthermore, the critical temperature of the
phase transition is
\begin{equation}
\label{4eq24}
T_c=\frac{1}{2\pi r_c}\left(-2 (1+c^2\Phi^2)l^2 +\sqrt{1+\frac{8r_0^4
+4r_b^2l^2}{(1+c^2\Phi^2)^2l^4}} \right ).
\end{equation}
\sect{Conclusions}
In this work we have studied the thermodynamics and Hawking-Page
phase transition of a hard wall AdS/QCD model in curved spaces by
introducing an IR cutoff, generalizing the work in \cite{Herz,BBBZ},
where the authors have discussed the phase transition between the
low temperature AdS space and the high temperature AdS black holes
in Poincare coordinates, which implies that the dual field theory
resides on a Ricci flat space. In our case, the dual field theory
lives in a curved space with positive or negative constant
curvature, dual to black hole configurations having spherical or
hyperbolic horizons.
In the case of the spherical AdS black holes, introducing the IR
cutoff leads the Hawking-Page phase transition temperature to
increase, compared to the case without the IR cutoff. For the case
of the hyperbolic black hole, the Hawking-Page phase transition will
not occur when one does not introduce an IR cutoff, while the
transition happens once the IR cutoff is introduced. However, there
is a gap for the IR cutoff in this case. Below that gap, the
Hawking-Page phase transition still does not occur.
For the charged AdS black holes with any horizon topology, like the
case without charge, the Hawking-Page phase transition becomes
possible due to the introduction of an IR cutoff. A remarkable
feature in this case is that for any horizon topology, a gap for the
IR cutoff always exists due to the existence of a minimal black hole
horizon.
\section*{Acknowledgments}
RGC thanks C. Herzog for helpful email correspondence and Y.Q. Chen,
C. Liu, J.P. Ma and F. Wu for useful discussions. The work of
R.G.C. was supported in part by a grant from Chinese Academy of
Sciences, and by the NSFC under grants No. 10325525 and No.
90403029. The work of J.S was funded by the NSFC under grants No.
10475105, 10491306 and PHY99-07949.
|
1,116,691,497,544 | arxiv | \section{Introduction}
Elastic strips and rods are fundamental physical structures that can be found in many contexts on many different length-scales. Often, they appear in the theoretical description of physical phenomena where the interplay between physics and geometry plays a key role. The collapse of an elastic sheet adhered to a curved substrate is certainly one of them. This discontinuous buckling instability appears in various problems in biology (shapes and morphogenesis of the mitochondrion \citep{Sakashita:2014}), physics (packing of thin films \citep{Boue:2006}, delamination of thin elastic films \citep{Vella:2009, Wagner:2013}, design of stretchable electronic devices \citep{Sun:2006}, elastocapillary snapping \citep{Fargette:2014}) and engineering (collapse of buried steel pipelines \citep{Omara:1997,Vasilikis:2009}).
A simple example illustrates the physical phenomenon we are concerned with. Let us imagine to insert a rectangular piece of paper inside a rigid cylindrical substrate. We hold the two edges with the hands so that the sheet of paper is kept in contact with the cylinder. Subsequently, we apply an increasing tangential compressive force. Due to presence of the surrounding container, the sheet of paper cannot freely deform to the outside. Initially it appears to be undeformed or slightly compressed. However, when the applied force crosses a suitable threshold the strip abruptly buckles and forms an inward hump. What is the critical force at which the collapse occurs? How are the geometric features of the hump at the transition related to the material constants and to the geometry of the container? Although the buckling of slender elastic structures is one of the oldest fundamental problems, nobody seems to have addressed these points. Furthermore, it is worth noticing that classical bifurcation analysis
fails in this problem due to the presence of the confinement.
A strictly related problem deals with the packing of a flexible cylinder inside a rigid circular tube of smaller radius. This is widely explored in the literature, both theoretically \citep{Cerda:2005} and experimentally \citep{Boue:2006}. Recent papers have also included the possibility of adherence by capillarity \citep{DePascalis:2014, Rim:2014} or the extension to the spherical geometry \citep{Kahraman:2012}. In the usual theoretical treatment, the strip is modelled as an inextensible Euler beam, whose energy is only associated to the bending mode. The inextensibility assumption, first suggested by \cite{Rayleigh}, is based on energetic considerations. The bending energy and the stretching energy scale differently with the beam thickness. When the structure thickness is diminished without limit, stretching is energetically prohibitively expensive and the distortion is pure bending. However, the validity of this assumption is challenged when the thickness becomes comparable to some other length-
scales in the problem. {Moreover, axial compressibility is known to change the nature of the phase transition. For example, the usual supercritical bifurcation point in the classical inextensible Euler beam becomes a subcritical bifurcation \citep{01magn,Audoly:2010}.
However in our confined problem, the assumption of an axially inextensible beam shows its inadequacy in a more fundamental way. For instance, in the limit of arbitrarily small hump, the inextensible Euler's Elastica model predicts the unphysical result of an infinite pressure exerted by the beam on the external container \citep{Cerda:2005, DePascalis:2014}. This prevents the transition to a buckled state as it would require an infinite force to induce the buckling.}
{To avoid this unacceptable behaviour, in this paper we revise this effect by allowing the sheet to both stretch and bend. Within this framework, the energy landscape exhibits a first order phase transition between a state where compression energy dominates, to another where bending energy prevails. From the mechanical point of view, a {\it snap through} buckling occurs: as the sheet grows, it suddenly switches from a completely adhered solution to a buckled solution showing a symmetric inward hump. The transition is studied numerically and, in the limit of {\it weak stretchability}, we provide the analytic expressions for the main quantities at the critical point. Interestingly, we find that for homogeneous beams the threshold does not depend on the material parameters but only on the geometric features of the problem.}
\section{The model}
We assume a planar configuration so that the longitudinal profile of the strip can be modelled as a stretchable and flexible rod. This is represented by a parametric curve ${\bf r}(S)$, with $S\in [-L/2,L/2]$, where $L$ denotes the length of the strip in the stress-free configuration and $S$ is the referential arclength. The governing nonlinear Kirchhoff equations express the balance of linear and angular momentum. When an external distributed force ${\bf f}$ is applied and in absence of external distributed torques, they read
\begin{eqnarray}
{\bf T}' + {\bf f} ={\bf 0},
\label{b1}
\end{eqnarray}
\begin{eqnarray}
{\bf M}' + {\bf r}' \times {\bf T} = {\bf 0},
\label{b2}
\end{eqnarray}
where ${\bf T}(S)$ and ${\bf M}(S)$ represent the internal forces and the internal torques, respectively. The prime denotes differentiation with respect to $S$; hence, ${\bf r}' = \lambda {\bf t}$, where $\lambda$ is the local stretch and ${\bf t}$ is the unit tangent. The inextensibility condition corresponds to $ \lambda = 1$.
Balance equations are complemented by the linear constitutive laws \citep{Dill:1992}:
\begin{eqnarray}
{\bf T} = b (\lambda -1) {\bf t} + T_n {\bf n},
\end{eqnarray}
\begin{eqnarray}
{\bf M} = k {\bf t} \times {\bf t}' ,
\label{momento}
\end{eqnarray}
where the positive constants $b$ and $k$ represent the stretching and the bending rigidity, respectively. The quantity $\ell = \sqrt{k/b}$ defines an intrinsic characteristic length. Since $b= E A$ and $k = E I$, where $E$ is the elastic modulus of the material, $A$ is the area of the film cross-section and $I$ is the second moment of area of the strip cross-section, the intrinsic length $\ell$ turns out to be of the order of the strip thickness. Only the tension is given constitutively; the shear internal force $T_n$ is related to the derivative of the internal torque by Eq. \eqref{b2}. We parametrize the tangent and the normal unit vectors by
\[
{\bf t} = \cos \theta(S) {\bf e}_x + \sin \theta(S) {\bf e}_y, \qquad {\bf n} = -\sin \theta(S) {\bf e}_x + \cos \theta(S) {\bf e}_y
\]
and, hence,
$
{\bf e}_z = {\bf t} \times {\bf n}.
$
Consequently, Eqn. \eqref{momento} reduces to
$
{\bf M} = k \theta' {\bf e}_z.
\label{c2}
$
Furthermore, we posit a symmetric equilibrium configuration of the rod and assume that the \emph{displacement} of its end-points can be controlled: the point with referential coordinate $S=L/2$ undergoes a tangential compressive displacement $r \Delta \alpha$, $\Delta \alpha \geq 0$. The strip can both stretch and bend, therefore there are at least two equilibrium configurations with the same imposed $\Delta \alpha$: (i) the {\it adhered solution} where the strip, uniformly compressed, totally adheres to the wall, and (ii) the {\it collapsed solution} with an inward symmetric hump. The adhered solution is a uniformly compressed arc of circumference with
\[
\lambda_{adh}= 1 - 2 r \Delta \alpha/L, \qquad \theta'_{adh} = - \lambda_{adh}/r.
\]
On the other hand, the collapsed solution comprises two parts: a free (non-adhered) curve for $S \in (-{\bar S}, {\bar S})$ and two adhered pieces for $S \in[-L/2,-{\bar S}] \cup [{\bar S}, L/2]$ (see Figure \ref{blister}). The symmetry of the strip implies that the function $\theta(S)$ is odd and allows us to restrict the study of the solution in the range $S \in [0,L/2]$.
\begin{figure}[ht]
\centerline{\includegraphics[scale=1.2]{fig_Geometry.pdf} }
\caption{\label{blister} Schematic representation of the strip deformation. The gray curve represents the referential configuration, while the blue curve shows the buckled configuration with a hump due to the displacement $r\Delta \alpha$ of the end points.}
\end{figure}
Since the adhered part, $S \in [{\bar S},L/2]$, is an arc of circumference with $\theta' (S)=-\lambda/r$, the internal torque is ${\bf M} = -k (\lambda/r) {\bf e}_z$. Therefore, the equilibrium equations for this imposed geometry necessarily give $T_n=0$ and $\lambda' = 0$, whence $\lambda(S) = \bar{\lambda}= {\rm const}$. However, the compressive tangential force ${\bf F}= -F {\bf t}$ for a given $\Delta \alpha$ is not known \emph{a-priori} but it is related to the stretch by $F = b(1-\bar \lambda)$.
In the non-adhered region, $S \in [0,{\bar S}]$, there is no applied distributed load (${\bf f}= {\bf 0}$). Furthermore, since $\theta(S)$ is odd, the vertical component of the tension vanishes, while the horizontal component, denoted by $h$, is constant and satisfies the identity
$
b(\lambda-1) = h \cos \theta.
$
The balance of the torque yields the second order differential equation for the free part
\begin{eqnarray}
\theta'' + \lambda \tau \sin \theta =0.
\label{eul}
\end{eqnarray}
where $\tau = - h/k$.
The deflection angle $\theta(S)$ should satisfy the Dirichlet boundary conditions $\theta(0)=0, \; \theta({\bar S}) = \bar{\theta}$. Further boundary conditions are provided by requiring the continuity of the tangential component of the internal force and the continuity of the torque at the detachment point $S={\bar S}$. These yield
\[
h \cos \bar{\theta} =-b(1-\bar \lambda), \qquad \theta'({\bar S}) = - \bar \lambda r^{-1},
\] respectively. One more constraint translates the geometrical condition that the detachment point must lie on the circumference of radius $r$
\begin{eqnarray}
\int_0^ {{\bar S}} \lambda \cos \theta {\rm d} S = - r \sin \bar{\theta}.
\label{vinco}
\end{eqnarray}
This establishes a relationships between ${\bar S}$ and $\bar{\theta}$. The displacement of the end-point, $r \Delta \alpha$, is related to $\bar\lambda$, $\bar{S}$ and $\bar{\theta}$ by (Figure \ref{blister})
\begin{eqnarray}
r\Delta \alpha = r \bar{\theta} + (1-\bar \lambda) \frac{L}{2} + \bar \lambda \bar{S}.
\label{geo1}
\end{eqnarray}
{This geometrical identity is simply obtained by requiring that the total length is the sum of the hump length and the adhered length.}
\section{Results}
{We start by observing that} for a perfectly unstretchable strip the compression modulus diverges and the intrinsic length vanishes, $\ell=0$. The tangential component of the internal force, $T_t$, then becomes a Lagrange multiplier associated to the inextensibility constraint $\lambda=1$ and the adhered solution is admissible only for $\Delta \alpha=0$.
By contrast, when both compression and bending are allowed, the energy must attain its absolute minimum in the observed solution, either adhered or collapsed, which is then expected to be stable. The other possible solution is either non-existing, unstable or metastable, depending on the energy landscape. The total elastic energy consists of two terms, related respectively to the bending mode and to the stretching (or compression) mode:
\begin{eqnarray}
W = \frac{k}{2}\int_{-L/2}^{L/2} (\theta')^2 {\rm d} S + \frac{b}{2}\int_{-L/2}^{L/2} (1-\lambda)^2 {\rm d} S.
\label{energia}
\end{eqnarray}
Equation \eqref{energia} assumes a particularly simple analytic form when evaluated in the adhered solution
\begin{eqnarray}
W_{adh} = \frac{1}{2} \frac{k L}{r^2} + 2 \frac{k}{r} \Delta \alpha \left[- 1 + \frac{r}{L} (1+\xi^{-2}) \Delta \alpha \right],
\label{en_adh}
\end{eqnarray}
where $\xi = \ell/r>0$ measures the compressibility strength at fixed radius.
\begin{figure}[ht]
\centerline{\includegraphics[scale=1]{fig_energy.pdf}}
\caption{\label{figura_energie} {Plot of the equilibrium free energy $W$ for the adhered solution (red) and the collapsed solution (blue) with $L=\tfrac{3}{2}\pi r$ and $\xi=0.01$. Solid lines represent solutions with least energy; dashed lines represent metastable states or maxima. For small $\Delta \alpha$ only the adhered solution is admissible. The dotted line represents the energy of a perfectly inextensible strip as given in Eq.\eqref{eq:W_approx}.}}
\end{figure}
On the other hand, the energy associated with the collapsed solution, $W_{col}$, is not amenable for a simple analytical approximation and is best calculated with a numerical simulation. Figure \ref{figura_energie} shows the comparison between the energies associated with each branch as a function of $\Delta \alpha$, for fixed $\xi=0.01$. The adhered branch depends quadratically on $\Delta \alpha$ (see equation \eqref{en_adh}). The collapsed solution comprises two branches. The upper branch, with higher energy, corresponds to small humps, while the lower branch corresponds to equilibrium solutions with larger humps. Finally, it is worth noticing that for sufficiently small values of $\Delta \alpha$ only the adhered solution is admissible.
Figure \ref{figura_energie} clearly shows the first-order transition that occurs between the two solutions. For small displacements of the end-points, the adhered solution is energetically favoured. However, when the displacement crosses a critical value $r(\Delta \alpha)_{cr}$, the energy attains its absolute minimum at the collapsed solution \emph{with larger hump}. Therefore, as $\Delta \alpha$ is increased, the strip undergoes a discontinuous transition, abruptly passing from an adhered configuration to a buckled solution with an inward hump. The presence of compressibility implies a lower bound for the hump size: humps whose dimensions are small compared to the characteristic length $\ell$ cannot be observed and the strip is simply compressed. The critical threshold increases as the strip becomes softer (see Figure \ref{figura_dacritico}). In particular, in the inextensible limit the critical threshold vanishes, as expected.
\begin{figure}[ht]
\centerline{\includegraphics[scale=1]{fig_Da_critico}}
\caption{\label{figura_dacritico} {Critical threshold $(\Delta \alpha)_{cr}$ as a function of the reduced compressibility. The solid line represents the numerical results, while the dotted line represents the analytical approximate result as given in Eq.\eqref{soglia_critica}. Typical shapes are shown in the inset.}}
\end{figure}
In the {\it nearly unstretchable} regime $(\xi \ll 1)$ one can estimate the critical threshold analytically. Indeed, at low $\xi$ and close to the critical point, it has been observed numerically that the collapsed solution is nearly equivalent to the buckled profile of a perfectly inextensible strip. Figure \ref{figura_energie} clearly shows this agreement in terms of elastic energies. From the analytic approximation we also learn that the relevant variable for scaling the critical threshold is the {\it reduced compressibility}, defined as
\[
\xi_0 := \xi \sqrt{L/2r}.
\]
This indicates an unexpected non-trivial dependence on the geometrical and material parameters. {For homogeneous materials $\xi_0$ is just a geometrical dimensionless parameter, since both the compression and the bending rigidity are linear functions of the Young modulus. By contrast, the material moduli could play a role in the case of composite materials where the dependence of $b$ and $k$ on the Young modulus is more complex.}
When the compression modulus is large ($\xi \ll 1$), compression becomes energetically prohibitively expensive for small $\Delta\alpha$. The strip then prefers to buckle and pay some bending energy provided it can relax its compression energy. Therefore, it is natural to approximate the collapsed strip as inextensible. The advantage of this approximation is that it is now possible to perform an analytic treatment. In particular, Eq.\eqref{eul} simply reduces to the nonlinear pendulum equation with first integral
\begin{equation}
(\theta')^2 = 2\tau(\cos \theta - \cos \theta_0) ,
\label{eq:first_integral}
\end{equation}
where $\theta_0 = \theta({S_0}) \in [0,\pi]$ is the maximum value of $\theta(S)$ in $(0,\bar{S})$.
The boundary condition for $\theta'$ at $\bar{S}$ becomes $\theta'(\bar{S})=-1/r$. This is used together with the first integral to eliminate $\tau$ in favour of $\theta_0$. Thus, we arrive at the first order differential equation
\begin{equation}
\theta' = \pm \frac{1}{r} \sqrt{\frac{\cos \theta - \cos \theta_0}{\cos \bar{\theta} - \cos \theta_0}},
\label{tetas}
\end{equation}
where the sign $+$ (respectively, $-$) is to be used in the interval $S\in(0,{S_0})$ (respectively, $S\in({S_0},\bar{S})$). By symmetry $\theta(0)=0$ and Eq.\eqref{tetas} evaluated at $S=0$ shows that $\cos\bar{\theta} -\cos{\theta_0} >0$. This gives a restriction on the possible values of $\bar{\theta}$: $|\bar{\theta}|<{\theta_0}$. Furthermore, Eq.\eqref{tetas} is an ordinary differential equation which can be solved by separation of variables in $(0,\bar{S})$ (see \cite[Sect. 3]{DePascalis:2014})
\begin{equation}
2{\rm F}(q_0)-{\rm F}(\bar q) = \frac{\bar{S}}{2r} {\sqrt \frac{1-\cos\theta_0}{\cos \bar{\theta} - \cos \theta_0}},
\label{prima}
\end{equation}
where ${\rm F}$ denotes the incomplete elliptic integral of first kind \citep{Abramowitz:1970} and, for ease of notation, we set
\[
q_0 := \{{{\theta_0}}/{2}, \csc^2 ({{\theta_0}}/{2}) \}, \qquad \bar q := \{{\bar{\theta}}/{2}, \csc^2 ({{\theta_0}}/{2}) \}.
\]
Similarly, we simplify Eq.\eqref{vinco} (with $\lambda=1$) as follows
\begin{eqnarray}
2 {\rm E} (q_0)- {\rm E} (\bar q) = - \frac{\bar{S} \cos {\theta_0} + r\sin \bar \theta}{2 r \left(1-\cos\theta_0\right)} \sqrt\frac{{1-\cos\theta_0}}{\cos \bar{\theta} - \cos \theta_0},
\label{seconda}
\end{eqnarray}
where ${\rm E}$ represents the incomplete elliptic integral of second kind. Equations \eqref{prima}, \eqref{seconda}, and the inextensible version of Eq.\eqref{geo1}, $r \Delta \alpha = r \bar{\theta} + \bar{S} $, provide $\bar{S}$, ${\theta_0}$ and $\bar{\theta}$ as functions of $\Delta \alpha$. In the limit $\Delta \alpha \ll 1$, after some tedious but straightforward calculations which we omit for brevity, it is possible to find the asymptotic expansions of ${\theta_0}$ and $\bar{\theta}$ {\citep{DePascalis:2014}
\begin{align}
{\theta_0} & \approx 2.3454 (\Delta\alpha)^{1/3} - 0.78762 \Delta\alpha + 0.68361 (\Delta\alpha)^{5/3} , \\
\bar{\theta} & \approx -2.2894 (\Delta\alpha)^{1/3} + 0.83072 \Delta\alpha - 0.62489 (\Delta\alpha)^{5/3}.
\end{align}
The energy of an inextensible buckled solution has a particularly simple expression when it is written in terms of ${\theta_0}$, $\bar{\theta}$ ans $\bar{S}$. To this end, we insert Eq.\eqref{tetas}, Eq.\eqref{vinco} and $\lambda=1$ into Eq.\eqref{energia} to get
\begin{align}
W_{col} & = k \int_{0}^{\bar{S} } (\theta')^2 {\rm d} S + k \int_{\bar{S}}^{L/2} (\theta')^2 {\rm d} S
= \frac{k}{r^2} \int_{0}^{\bar{S} } \frac{\cos\theta(S) - \cos{\theta_0}}{\cos\bar{\theta} - \cos{\theta_0}} {\rm d} S + \frac{k}{r^2} \Big(\frac{L}{2}-\bar{S}\Big) \notag \\
& = \frac{1}{2}\frac{k L}{r^2} - \frac{k}{r^2}\frac{\bar{S} \cos\bar{\theta} + r \sin \bar{\theta}}{\cos\bar{\theta}-\cos{\theta_0}}.
\label{eq:IncBuckled}
\end{align}
The asymptotic expansions of ${\theta_0}$ and $\bar{\theta}$ are then inserted into Eq.\eqref{eq:IncBuckled} to yield the approximate expression for the energy of the collapsed solution
\begin{eqnarray}
W_{col}\! \approx \!\frac{1}{2}\frac{k L}{r^2} \!+ \!\frac{k}{r}\big(w_0 (\Delta \alpha)^{1/3}
+ w_1 \Delta \alpha + w_2 (\Delta \alpha)^{5/3} \big),
\label{eq:W_approx}
\end{eqnarray}
where $w_0 \approx 23.113, \; w_1 \approx -18.532, \; w_2 \approx 1.6131$. }
Thus, we are able to give an approximate analytic solution of the transition equation $W_{col} =W_{adh}$, which to leading order reads
\begin{equation}
(\Delta \alpha)_{cr} \approx 6.5813 \; \xi_0^{6/5}.
\label{soglia_critica}
\end{equation}
As shown in Figure \ref{figura_dacritico}, this expression provides a very good account of the numerical results, in the range of compressibility considered.
Figure \ref{fig_tb_xi} sketches the critical behaviour of the detachment angle. This angle is directly related to the hump dimension and is easily accessible experimentally. {For a given $\xi$, $|\bar{\theta}_{cr}|$ represents the minimum stable detachment angle. Lower values of $\bar{\theta}$ corresponds to metastable or unstable solution. Thus, we can conclude that there is a limiting size below which the hump cannot develop. The dotted line in Figure \ref{fig_tb_xi} represents the asymptotic approximation}
$ |\bar{\theta}_{cr}| \approx 4.29 \; \xi_0^{2/5}. $
The lower bound for the blister size corresponds to a higher bound for the applied tangential force and thus for the pressure exerted on the delimiting container. Figure \ref{force} reports the behaviour at the transition of the tangential force for the adhered (red line) and collapsed (blue line) solutions, as functions of the reduced compressibility. Since at the transition the solution passes from one to the other, the tangential force undergoes a discontinuous jump. This is consistent with the idea of buckling as the mechanism through which the system relaxes its internal stress. {For fixed $\xi_0$, the correspondent value of the red line shows the maximum internal force acceptable by the system under pure compression. The critical load at which the buckling occurs is given by \mbox{$F_{cr} = 2 b r (\Delta \alpha)_{cr}/L$}. Furthermore, when $\Delta \alpha$ is below its critical threshold the internal force is simply proportional to the strain and hence it vanishes as $\Delta \alpha$ tends to zero}.
\begin{figure}[t]
\centerline{\includegraphics[scale=1]{fig_Theta_critico}}
\caption{\label{fig_tb_xi} Detachment angle at the transition as a function of $\xi$. The dotted line represents the asymptotic approximation.}
\end{figure}
\begin{figure}[t]
\centerline{\includegraphics[scale=1]{fig_F_critico}}
\caption{\label{force} Tangential compressive force at the transition in the adhered solution (red line) and collapsed solution (blue line) as a function of $\xi$. The tangential force has a finite jump at the transition. The dotted lines show the asymptotic approximations.}
\end{figure}
\section{Concluding remarks}
In many applied contexts it is useful to study the slightly different problem where the sheet is closed, {\it i.e.} the lateral ends of the strip are glued together. For instance, the elastic sheet can be made of a growing biological membrane confined by a rigid cylinder of radius $r$. In this case, the circumferential growth, suitably measured by $\epsilon:= (L - 2 \pi r)/(2 \pi r)$, may trigger the instability. For moderate growth, the excess material can simply result in a uniform compression of the adhered strip and in an increase of the hoop stress. By contrast, a further increase of the growth (when the critical threshold has been crossed) leads to the buckling of the membrane and the sheet is then only in partial contact with the container. Our results can be simply adapted to this closed problem. As expected we again find that there is a critical growth $\epsilon_{cr}$ above which the delamination is preferred with respect to the adhered solution. Indeed, the equations governing the post-buckling behaviour
are
the same except for Eq.\eqref{geo1} that is to be replaced with
\[
\pi r \epsilon = r \bar{\theta} + (1-\bar \lambda) {L}/{2} + \bar \lambda \bar{S}.
\]
It is worth noticing explicitly that even the energy expression \eqref{energia} remains unchanged. By contrast, the adhered solution is characterized by a uniform stretch $\lambda_{adh} = 1/(1 + \epsilon) $ and its elastic energy is
\[
W_{adh}= \pi k r^{-1} (1 + \epsilon)^{-1}(1+ \epsilon^2 \xi^{-2}).
\]
However, it turns out that the two problems are mathematically equivalent provided we substitute $\Delta \alpha$ with $\pi \epsilon$. We finally find the following approximation for the critical growth: $\epsilon_{cr} \approx 41.0912\; \xi^{6/5}$.
As an application of our results, we look at the experiment on the packing of flexible films reported in \cite{Boue:2006}. {In the first part of their paper, the Authors studied the pressure exerted by the sheet on the external container as a function of $\epsilon$, here named {\it the confinement parameter}. Their experimental data exhibit an increasing pressure as the confinement parameter $\epsilon$ is decreased. Numerical studies of the elastica model indicate indeed a divergent pressure in the limit of $\epsilon \rightarrow 0$. This is also consistent with the theoretical results \citep{Cerda:2005,DePascalis:2014}, which in addition show an asymptotic dependence of the internal forces, and hence the pressure, on $\epsilon^{-2/3}$}. However, the thickness of the sheet used in the experiment is $h = 0.1\:$mm, while the container radius is $r=26\;$mm. Since for a rectangular section $\ell = h/\sqrt{12}$, we obtain that $\xi = 1.11 \times 10^{-3}$ and $\epsilon_{cr}=1.17 \times 10^{-2}$. Unfortunately,
the smallest experimental value of $\epsilon$ reported in Figure 3 of \cite{Boue:2006} is just above this critical value and hence they are not able to see the transition experimentally. {On the one hand, this justify the validity of the Elastica model above the critical threshold}. On the other hand, our results predict that, by further decreasing $\epsilon$, the hump should suddenly disappear. At the transition, the pressure (per unit length) on the container undergoes a jump to $P_{cr} \approx b r^{-1} \epsilon_{cr} $, while below $\epsilon_{cr}$ the pressure is proportional to $\epsilon$, hence it goes to zero as $\epsilon \to 0$. This solve the apparent paradox, described in \cite{Cerda:2005} , \cite{DePascalis:2014} and \cite{Boue:2006}, of a diverging pressure in the low confinement regime.
In conclusion, the present analysis recognizes the key role played by stretchability for a correct description of the mechanical collapse of a loaded or growing elastic thin sheet adhered to a curved substrate. Even when the sheet is nearly unstretchable, the energetic contribution of compression cannot be neglected when the length scales in the problem are comparable with the sheet thickness (i.e., small humps). Our model yields analytic approximations for the critical threshold and the minimum size for the humps. {This threshold depends on the reduced compressibility that, for homogeneous materials, is a purely geometrical dimensionless parameter}. Furthermore, it sets an upper bound to the exerted pressure on the container.
Future works can consider the effects of other physical quantities (i.e. capillary adhesion, intrinsic curvature and gravity) or the coupling of growth with internal stress \citep{Tiero:2014} on the morphology and the critical parameters.
\section*{Acknowledgements}
This research has been carried out within the Young Researchers Project ``Collasso Meccanico di Membrane Biologiche Confinate'', supported by the Italian `Gruppo Nazionale per la Fisica Matematica' (GNFM).
|
1,116,691,497,545 | arxiv | \section{Introduction}
Let $f: \{0,1\}^n \to \{0,1\}$ be a total Boolean function. The \emph{conflict complexity} of $f$, denoted by $\chi(f)$, is a complexity measure of $f$ that is recently defined in \cite{random2} and appears implicitly in \cite{random1}. Using this notion, \cite{random2} and \cite{random1} independently show an improved composition theorem for randomized decision tree complexity, a topic that is intensively studied in recent years, see \cite{anshu2017composition,ben2016randomized,random2, random1, ben2020tight}. Let $R(f)$ denote the randomized decision tree complexity of $f$ with a bounded error. Let $g: \{0,1\}^m \to \{0,1\}$ be another Boolean function. The composed function $f\circ g: \{0,1\}^{nm} \to \{0,1\}$ is defined by $(f\circ g)(x_1, \ldots, x_{nm}) = f(g(x_1,\ldots, x_m), g(x_{m+1}, \ldots, x_{2m}), \ldots, g(x_{(n-1)m+1}, \ldots, x_{nm}))$. Priori to \cite{random2, random1}, it is shown in \cite{ben2016randomized} that $R(f\circ g) \ge \Omega\Big(R(f) \sqrt{R(g)/\log R(g)} \Big)$. Introducing the conflict complexity, it is proven in \cite{random2} that $R(f\circ g) \ge \Omega(R(f) \chi(g))$ and $\chi(g) \ge \Omega(\sqrt{R(g)})$. Hence, $R(f\circ g) \ge \Omega(R(f) \sqrt{R(g)})$, removing the logarithmic factor in the previous lower bound.
Another important complexity measure of a Boolean function $f$ is its \emph{block sensitivity}, denoted by $\bs(f)$. Block sensitivity is firstly defined in \cite{bs}. Block sensitivity, its variants, and the famous sensitivity conjecture (now a theorem by \cite{huang2019induced}) are widely studied and applied in complexity theory and combinatorics, see e.g. \cite{rubinstein1995sensitivity, survey, hatami2010variations, goos2014communication,ambainis2014tighter,kulkarni2016fractional} etc.
The relation of block sensitivity to other frequently used complexity measures of a Boolean function such as sensitivity, degree, decision tree complexity and certificate complexity etc, is relatively well understood. In particular, they are polynomially related, see the survey \cite{survey} and \cite{huang2019induced}. Explicit polynomial relations among different complexity measures are very useful. In particular, it allows one to transfer lower or upper bounds based on one complexity measure (e.g. degree) to another (e.g., block sensitivity). For example, \cite{huang2019induced} solves the sensitivity conjecture, which states that block sensitivity is upper bounded by a polynomial of sensitivity, by showing that degree is upper bounded by a polynomial of sensitivity.
Let $D(f)$ denote the deterministic decision tree complexity of $f$, see definition in \cite{survey}. Later after we formally define $\chi(f)$, it will be clear that $\chi(f) \le D(f)$. Hence, by \cite{random2}, $\Omega(\sqrt{R(f)}) \le \chi(f) \le D(f)$. By \cite{survey}, $D(f)^{1/3} \le \bs(f) \le R(f)$. Hence, $\Omega(\sqrt{\bs(f)}) \le \chi(f) \le \bs(f)^3$. We improve this to a linear lower bound \footnote{Shortly after this work, \cite{gavinsky2018composition} shows the \emph{max conflict complexity}, which is an upper bound of conflict complexity, is larger than the block sensitivity.} with an explicit coefficient.
\begin{theorem} \label{thm:my}
For every non-constant total Boolean function $f: \{0,1\}^n \to \{0,1\}$, $\chi(f) \ge (\bs(f)+1)/2$.
\end{theorem}
The conflict complexity and block sensitivity are formally defined in Section~\ref{sec:def}. Theorem~\ref{thm:my} is proved in Section~\ref{sec:proof}. A question on comparing conflict complexity with certificate complexity and its possible consequences are discussed in Section \ref{sec:discussion}.
\section{Conflict Complexity and Block Sensitivity} \label{sec:def}
Let $f: \{0,1\}^n \to \{0,1\}$ be a Boolean function. Let $\mu_0, \mu_1$ be two probabilisitic distributions on $f^{-1}(0)$ and $f^{-1}(1)$, respectively. Given $\mathcal T$ as a deterministic decision tree\footnote{For more background on the definition of a decision tree and related complexity measures, see the survey \cite{survey}.}, which is a binary tree, that computes $f$ correctly. Let $v$ be a node of $\mathcal T$, by an abuse of notation we also use $v$ to denote the Boolean variable that is queried at node $v$. Every node $v \in \mathcal T$ corresponds to a subcube of $\{0,1\}^n $ that is uniquely determined by the path leading from the root of $\mathcal T$ to $v$. Denote
\begin{equation} \label{eq:mu-v}
\mu^v_{0} = \mu_0|_v, \quad \mu^v_{1} = \mu_1|_v.
\end{equation}
That is, they are the distributions of $\mu_0$ and $\mu_1$ conditioned on the subcube corresponding to $v$. At node $v$, the decision tree $\mathcal T$ branches to left or right according to $v = 0$ or $v = 1$, respectively. Denote
\begin{equation} \label{eq:ab}
\alpha_v = \Pr_{x \sim \mu^v_0} [v = 0], \quad
\beta_v = \Pr_{x \sim \mu^v_1} [v = 0],
\end{equation}
where $x \sim \mu^v_0$ (resp. $x \sim \mu^v_1$) means to sample $x \in \{0,1\}^n$ according to the distribution $\mu^v_0$ (resp. $\mu^v_1$).
Consider the following random walk on the tree $\mathcal T$ as follows: at node $v$, it goes to its left child (where $v=0$) with probability $\min\{\alpha_v, \beta_v\}$; it goes to its right child (where $v=1$) with probability $1 - \max\{\alpha_v, \beta_v\}$; it stops at $v$ with probability $|\alpha_v - \beta_v|$. We call a node $v \in \mathcal T$ a \emph{last query node} if after the value of $v$ is queried, the tree outputs accordingly. Alternatively, the two nodes after a last query node are two leaves of the tree $\mathcal T$. It is easy to see that $|\alpha_v - \beta_v| = 1$ if $v$ is a last query node. In particular, the random walk always stops once it reaches a last query node. Intuitively, conflict complexity denotes the expected number of nodes the random walk has visited when it stops. Note that we count the node at which the random walk stops.
\begin{definition}[\cite{random2}] \label{def:conflictcomplexity}
Let $\mu_0, \mu_1, \mathcal T$ be as defined before. Let $X = X(\mu_0, \mu_1, \mathcal T)$ be the random variable taking values in $\mathbb{N} = \{1,2, \ldots, n\}$ that represents the number of nodes the random walk has visited when it stops. The \emph{conflict complexity} of $f$, denoted by $\chi(f)$, is defined as
\begin{equation} \label{eq:def-chi}
\chi(f) = \max_{\mu_0, \mu_1} \min_{\mathcal T} \Ex X.
\end{equation}
\end{definition}
Since the random walk always stops if it reaches a last query node, $\chi(f) \le D(f)$.
We proceed to define the block sensitivity. Let $f: \{0,1\}^n \to \{0,1\}$, $x \in \{0,1\}^n$, and $B \subseteq \{1,2,\ldots,n\}$ be a subset. Denote $x^B \in \{0,1\}^n$ as the $n$-bit string obtained from $x$ by flipping all bits whose indices are in the subset $B$. A subset $B$ is said to be a \emph{sensitive block} for $x$ if $f(x) \neq f(x^B)$. Let $\bs(f,x)$ denote the maximal number of disjoint sensitive blocks of $x$. The \emph{block sensitivity} of $f$, denoted by $\bs(f)$, is defined as $\bs(f) = \max_{x\in \{0,1\}^n} \bs(f,x)$.
\section{Proof of Theorem~\ref{thm:my}} \label{sec:proof}
\begin{proof}
We exhibit distributions $\mu_0$ and $\mu_1$, such that for every deterministic decision tree $\mathcal T$ that computes $f$ correctly, $\Ex X \ge (\bs(f)+1)/2$. Let $k = \bs(f)$. One has $k \ge 1$ since $f$ is not a constant function. Let $z \in \{0,1\}^n$ be an input string that achieves the block sensitivity of $f$, and $B_1, \ldots, B_k \subseteq \{1,2,\ldots,n\}$ are the disjoint sensitive blocks of $z$. Denote $y_i = z^{B_i} \in \{0,1\}^n$ for $i=1, \ldots, k$. Without loss of generality assume $f(z) = 0$, then $f(y_1) = \cdots = f(y_k) = 1$. Let $\mu_0$ be the distribution that is supported on the single point $z$, and $\mu_1$ be the uniform distribution over $Y = \{y_1, \ldots, y_k\}$. That is,
\begin{equation} \label{eq:distributions}
\mu_0(z) = 1, \quad
\mu_1(y_i) = 1/k, \ i=1, \ldots, k.
\end{equation}
Let $X$ be the random variable defined as in Definition \ref{def:conflictcomplexity}. Let $P$ denote the unique path that is travelled by the input $z$ in the decision tree $\mathcal T$. Let $\ell$ denote the length (i.e., number of nodes) of $P$, obviously $\ell \ge k$ since at least one bit from each sensitive block of $z$ must be queried. Renaming variables if necessary, we may assume $x_1, \ldots, x_\ell$ are the successive nodes in the path $P$ where $x_1$ is the root and $x_\ell$ is the last query node. Note that $\mu_0^{x_1} = \mu_0$ and $\mu_1^{x_1} = \mu_1$.
We will analyze $\alpha_v$ and $\beta_v$, for $v=x_1, \ldots, x_\ell$. Consider $\alpha_v$ first. We claim that $\alpha_v = 1$ (resp. $\alpha_v = 0$) if $v= 0$ (resp. $v= 1$) in the path $P$. Indeed, since $P$ is the path travelled by $z$ and $\mu_0(z) = 1$, for every node $v$ in the path $P$ it holds that $\mu^v_0(z) = 1$. Hence, the claim follows by definition \eqref{eq:ab}.
We proceed to analyze $\beta_v$. It is illuminating to firstly analyze $\beta_v$ at the root $v = x_1$. There are two cases according to whether $1 \in \cup_{j=1}^k B_j$. Note that here the number $1$ is the index of $x_1$.
\begin{itemize}
\item[(i)] $1 \not\in \cup_{j=1}^k B_j$. This implies $y_{i,1}= z_1$ for all $y_i \in Y$. Hence, $\beta_{x_1} = \alpha_{x_1}$, showing that
$\Pr[X=1] = | \alpha_{x_1} - \beta_{x_1}| = 0$, $\Pr[\text{the random walk reaches}\ x_2] = 1$, and $\mu^{x_2}_1 = \mu_1$.
\item[(ii)] $1 \in B_j$ for some $j \in \{1, \ldots, k\}$. Since $B_j \cap B_i =\emptyset$ for every $i\neq j$, there is exactly one such $B_j$. Without loss of generality, we assume $z_1 = 0$. Hence $\alpha_{x_1} = 1$. Since $z_1 = 0$, one has $y_{j,1} = 1$ and $y_{i,1} = 0$ for all other $i \neq j$. Hence,
\[
\beta_{x_1} = \Pr_{x \sim \mu_1} [x_1 = 0] = \Pr_{y_i \sim Y}[y_{i,1} = 0] = (k-1)/k.
\]
Therefore,
\[
\Pr[X=1] = |\alpha_{x_1} - \beta_{x_1}| = 1/k.
\]
Besides, $\Pr[\text{the random walk reaches}\ x_2] = (k-1)/k$, and $\mu^{x_2}_1$ is a uniform distribution over $Y - y_j$, a set of $k-1$ elements.
\end{itemize}
We claim that the phenomenon discussed in (i) and (ii) is true in general: for every $r = 1, \ldots, \ell$, either $\Pr[X=r] = 0$ or $\Pr[X=r] = 1/k$. Since $\ell \ge k$, this immediately implies the desired bound
\[
\Ex X \ge \sum_{i=1}^k i \cdot (1/k) = (k+1)/2.
\]
We proceed to show the claim. Consider an arbitrary node $v = x_r$ in the path $P$. Let
\[
\mathcal A^v = \{j: B_j \cap \{1, \ldots, r-1\} = \emptyset\}, \quad Y^v = \cup_{j \in \mathcal A^v} \{y_j\}.
\]
Intuitively, $Y^v$ is the set of $y_j$ that are still ``active'' at node $v$ (i.e., not deviated from the path $P$ before $v$). By a similar analysis as we did for the root, we know that $\mu^v_1$ is a uniform distribution on $Y^v$. Hence, at node $v=x_r$, the random walk stops with probability,
\begin{equation} \label{eq:prob-stop}
|\alpha_v - \beta_v| =
\begin{cases}
0, &\quad r \not\in \cup_{j \in \mathcal A^v} B_j; \\
1/ |\mathcal A^v|, &\quad r \in \cup_{j \in \mathcal A^v} B_j.
\end{cases}
\end{equation}
On the other hand, for any two successive nodes $x_i, x_{i+1}$ in $P$,
\[
\Pr[\text{the random walk branches from\ } x_i \text{ to } x_{i+1}]
=
\begin{cases}
1, &\quad i \not\in \cup_{j \in \mathcal A^{x_i}} B_j; \\
( |\mathcal A^{x_i}| - 1)/ |\mathcal A^{x_i}|, &\quad i \in \cup_{j \in \mathcal A^{x_i}} B_j.
\end{cases}
\]
This implies
\begin{equation} \label{eq:prob-reach}
\Pr[\text{the random walk reaches $v$}] = |\mathcal A^v| / k.
\end{equation}
Apply \eqref{eq:prob-stop} and \eqref{eq:prob-reach}, we get
$\Pr[X=r] = \Pr[\text{the random walk reaches $v$}] \cdot |\alpha_v - \beta_v|$,
is either $0$ or $1/k$ as claimed.
\end{proof}
\section{Discussion} \label{sec:discussion}
It seems an interesting problem to compare conflict complexity with certificate complexity, as explained below. Given $B\subseteq \{1,2,\ldots,n\}$ and $x \in \{0,1\}^n$. Let $x|_B$ denote the $|B|$-bit substring obtained by restricting $x$ to the indices in $B$. For every Boolean function $f: \{0,1\}^n \to \{0,1\}$, let $C_x(f)$ denote the minimal integer $0\le k \le n$ such that there exists a subset $B\subseteq \{1,2,\ldots,n\}$ with $|B|=k$ satisfying $f(y)= f(x)$ whenever $y|_B = x|_B$. The \emph{certificate complexity} of $f$, denoted by $C(f)$, is defined as $C(f) = \max_{x\in \{0,1\}^n} C_x(f)$. The following holds,
\[
(\sqrt{C(f)} +1)/2 \le \chi(f) \le C(f)^2.
\]
Indeed, $\sqrt{C(f)} \le \bs(f) \le D(f) \le C(f)^2$ where $D(f)$ is the deterministic decision tree complexity of $f$, see \cite{survey}. Hence, the inequality follows from Theorem \ref{thm:my} and the fact that $\chi(f) \le D(f)$.
\noindent {\bf Question}: $\chi(f) \ge \Omega(C(f))$ or $\chi(f) \le O(C(f))$\ ?
The answer could be neither. However, a positive answer in either case has interesting consequences.
\begin{itemize}
\item The case $\chi(f) \ge \Omega(C(f))$. Let $R(f)$ denote the randomized decision tree complexity of $f$ with a bounded error (see definition in \cite{survey}). Via $C(f) \ge \sqrt{D(f)}$ (see \cite{survey}), by \cite{random2} one has $R(f \circ g) \ge \Omega\Big(R(f) \chi(g) \Big) \ge \Omega\Big(R(f) \sqrt{D(g)}\Big)$, improving the lower bound $R(f\circ g) \ge \Omega\Big(R(f) \sqrt{R(g)}\Big)$ in \cite{random2, random1, gavinsky2018composition}.
\item The case $\chi(f) \le O(C(f))$. This has two consequences.
\begin{itemize}
\item[(1)] Consider the $\AND$ of $\OR$ tree $\AND_n \circ \OR_n: \{0,1\}^{n^2} \to \{0,1\}$, which is the composition of $n$-bit $\AND$ function with $n$-bit $\OR$ function. It is easy to see that $C(\AND_n \circ \OR_n) = n$ and $\bs(\AND_n) = \bs(\OR_n) = n$, see \cite{survey}. Hence, Theorem \ref{thm:my} implies $\chi(\AND_n) \chi(\OR_n) \ge \Omega(n^2)$, but $\chi(\AND_n \circ \OR_n) \le O\Big( C(\AND_n \circ \OR_n) \Big) = O(n)$. This implies $\chi(\AND_n \circ \OR_n) \le O\Big(\sqrt{\chi(\AND_n) \chi(\OR_n)}\Big)$. In particular, it shows that $\chi(f\circ g) = \Theta\Big(\chi(f) \chi(g)\Big)$ can not hold.
\item[(2)] In \cite{jain2010partition} it is shown $R(\AND_n \circ \OR_n) = \Theta(n^2)$. Hence, $\chi(\AND_n \circ \OR_n) \le O\Big(\sqrt{R(\AND_n \circ \OR_n)}\Big)$. This implies the lower bound $\chi(f) \ge \Omega(\sqrt{R(f)})$ in \cite{random2,gavinsky2018composition} is tight.
\end{itemize}
\end{itemize}
The same question can be asked for the \emph{max conflict complexity} as introduced in \cite{gavinsky2018composition}.
\section*{Acknowledgement}
The author thanks the host of Simons Institute for the Theory of Computing where this work was done.
|
1,116,691,497,546 | arxiv | \section{Introduction}\label{sec_intro}
Let $N$ be a positive integer, $\mathbb{N}^+:=\{1,2,\ldots\}$ and $\mathbb{R}^+:=(0,+\infty)$. Consider the linear diffusion equation over $\mathbb{R}^N$:
\begin{equation}\label{yu-6-24-1}
\left\{
\begin{array}{lll}
\partial_t u(x,t)-\sum\limits_{j,k=1}^N\partial_j\left(a_{jk}(x)\partial_ku(x,t)\right)=0&\mbox{in}\;\;\mathbb R^N\times\mathbb R^+,\\
u(x,0)=u_0(x) &\mbox{in}\;\;\mathbb R^N,
\end{array}\right.
\end{equation}
where $u_0\in L^2(\mathbb R^N)$, $a_{jk}(x)=a_{kj}(x)$ for all $j,k=1,\dots,N$, and all $x\in\mathbb R^N$. We assume the matrix-valued function $A(\cdot)=(a_{jk}(\cdot))_{j,k=1}^N$ is Lipschitz continuous and satisfies the uniform ellipticity condition, i.e.,
there is a constant $\lambda\geq1$ such that
\begin{equation}\label{yu-11-28-2}
|a_{jk}(x)-a_{jk}(y)|\leq \lambda|x-y|\quad\text{and}\quad
\lambda^{-1}|\xi|^2\leq \sum_{j,k=1}^N a_{jk}(x)\xi_j\xi_k \leq \lambda|\xi|^2
\end{equation}
for all $j,k=1,\dots,N$, $x, y\in\mathbb{R}^N$, and all $\xi=(\xi_1,\xi_2\cdots,\xi_N)\in\mathbb{R}^N$.
According to \cite[Theorem 10.9]{Brezis}, \eqref{yu-6-24-1} has a unique solution
$u\in L^2(\mathbb{R}^+; H^1(\mathbb{R}^N))\cap C([0,+\infty); L^2(\mathbb{R}^N))$.
Furthermore, if $u_0\in H^1(\mathbb R^N),$ then $u\in C([0,+\infty); H^1(\mathbb{R}^N))$
and $\partial_t u\in L^2(\mathbb{R}^+; L^2(\mathbb{R}^N))$.
Throughout the paper, $B_R(x_0)$ stands for an open ball in $\mathbb R^N$ with the center $x_0$ and of radius $R>0$, while
$B_R(x_0,0)$ stands for an open ball in $\mathbb R^{N+1}$ with the center $(x_0,0)\in\mathbb R^{N}\times\mathbb R$ and of radius $R>0$.
We denote the usual inner product and norm in $L^2(B_R(x_0))$ by $\langle\cdot,\cdot\rangle_{L^2(B_R(x_0))}$ and $\|\cdot\|_{L^2(B_R(x_0))}$, respectively. {Define the cube with center $x_0 =(x_{0,1},x_{0,2},\cdots,x_{0,N})$ and side length $R>0$ as $Q_R(x_0):=\{v=(v_1,v_2,\cdots,v_N):|v_i-x_{0,i}|\leq R,\;\forall i=1,2,\ldots,N\}$. Denote by $\mathrm{int}(Q_{R}(x))$ the interior of $Q_{R}(x)$, and by $\partial B_R(x_0)$ the boundary of $B_R(x_0)$.
Write $\bar z$ for the conjugate of a complex number $z\in\mathbb C$, and $|\omega|$
for the Lebesgue measure for a measurable set $\omega\subset\mathbb R^N$.
\subsection{Main results}
The main results of this paper concerning the quantitative estimate of unique continuation and the observability inequality for solutions of \eqref{yu-6-24-1} are stated as follows.
\begin{theorem}\label{yu-theorem-7-10-6}
Let $T>0$, $\rho>0$ and $0<r<+\infty$. Let $\{x_i\}_{i\in\mathbb{N}^+}\subset\mathbb R^N$ be arbitrarily fixed so that
\begin{equation*}
\mathbb{R}^{N}=\bigcup_{i\in\mathbb{N}^+}Q_{r}(x_{i})
\quad \text{with}\quad \mathrm{int}(Q_{r}(x_{i}))\bigcap \mathrm{int}(Q_{r}(x_{j}))=\emptyset\quad \text{for each}\quad i\neq j\in\mathbb N^+.
\end{equation*}
Let $\{\omega_i\}_{i\in\mathbb{N}^+}$ be some N-dimensional Lebesgue measurable sets in $\mathbb{R}^N$ so that
$$\omega_{i} \subset B_{\frac r2}(x_{i})\quad\text{and}\quad \frac{|\omega_{i}|}{|B_{r}(x_i)|}\geq\rho \quad\text{for each}\quad i\in\mathbb N^+. $$
Then there exist constants $C=C(r,N,\lambda,\rho)>0$ and $\sigma=\sigma(r,N,\lambda,\rho)\in(0,1)$ such that for any $u_{0}\in L^{2}(\mathbb R^N)$, the corresponding solution $u$ of \eqref{yu-6-24-1} satisfies
\begin{equation*}\label{yu-7-10-2}
\int_{\mathbb R^N}|u(x,T)|^2dx\leq Ce^{C(T+\frac{1}{T})}\left(\int_{\omega}|u(x,T)|^2dx\right)^{\sigma}\left(\int_{\mathbb R^N}|u_{0}(x)|^2dx\right)^{1-\sigma},
\end{equation*}
where $\omega:=\cup_{i\in\mathbb{N}^+} \omega_i$.
\end{theorem}
\medskip
\begin{theorem}\label{jiudu4}
Let $E$ be a subset of positive Lebesgue measure in $(0,T)$. Under the same assumptions in Theorem \ref{yu-theorem-7-10-6}, then there exist constants $C=C(r,N,\lambda,\rho)>0$ and $\tilde{C}=\tilde{C}(E, r,N,\lambda,\rho)>0$ such that for any $u_{0}\in L^{2}(\mathbb R^N)$, the corresponding solution $u$ of \eqref{yu-6-24-1} satisfies
\begin{equation*
\int_{\mathbb R^N}|u(x,T)|^2dx\leq e^{\tilde{C}+CT}\int_{\omega\times E}|u(x,t)|^2dxdt.
\end{equation*}
\end{theorem}
\medskip
Several remarks are given below.
\begin{remark}
When $E=(0,T)$, the constant $\widetilde{C}(E, r, N,\lambda,\rho)$ in
Theorem \ref{jiudu4} can take the form $\widetilde{C}( r,N,\lambda,\rho)/T$.
\end{remark}
\begin{remark}
When $A(\cdot)\equiv I$ (the $N\times N$ order identity matrix) in the equation \eqref{yu-6-24-1}, the above two theorems have been already established in \cite{WangZhangZhang} for solutions of the heat equation in $\mathbb R^N$, when the observation set is thick. The proofs there are based on quantitative estimates from measurable sets for real analytic functions, and uncertainty principle. To the best of our knowledge, however, this approach cannot be extended to the case of variable coefficients.
In the present paper, we built a new approach to obtain the
observability inequality for the diffusion equation \eqref{yu-6-24-1}.
\end{remark}
\begin{remark}\label{remark409}
Recall that a subset $\mathcal{O}\subset\mathbb R^N$ is called thick with parameters $L>0$ and $\gamma>0$ means that in each cube $Q_L\subset\mathbb R^N$ with the length $L$, the $N$-dimensional Lebesgue measure of $\mathcal{O}\cap Q_L$ is bigger than or equals to $\gamma L^N$.
As the thick set $\mathcal{O}$ is almost dense in the whole space, one may imagine that such kind of thick set has a close
relationship with the equidistributed set $\omega$ as defined in
Theorem \ref{yu-theorem-7-10-6}.
Indeed, an equidistributed set $\omega$ is clearly a thick set.
On the other hand, if $\mathcal{O}$ is a
thick set, then one could construct an equidistributed
set $\omega$ to be such that $\omega \subset \mathcal{O}$.
\end{remark}
\begin{remark}\label{re522}
By \cite[Lemma 2.5]{WangZhangZhang}, one can obtain that the equidistributed observation set is also a necessary condition such that the observability inequality holds for the diffusion equation \eqref{yu-6-24-1}.
\end{remark}
\begin{remark}
It is also worth mentioning that the authors in \cite{WangZhangZhang, EV17} have proved that the observability inequality for the heat equation holds if and only if the observation set is thick. Combined with Remarks \ref{remark409} and \ref{re522} above, one can finally conclude that the observability inequality for the diffusion equation \eqref{yu-6-24-1} holds if and only if the observation set is thick or equidistributed in $\mathbb R^N$.
\end{remark}
\subsection{Motivation and novelty}
The observability inequality for linear parabolic equations on bounded domains are previously studied in a large number of publications. When $E$ is the whole time interval and the observation region $\omega$ is a nonempty open subset, we refer the readers to \cite{FI,fz} and a vast number of references therein for the observability inequality for parabolic equations. Their approach is mainly based on the Carleman inequality method.
When $E$ is only a subset of positive Lebesgue measure in the time interval
and the observation region $\omega$ is a non-empty open subset, we refer the readers to \cite{Apraiz-Escauriaza-Wang-Zhang,Phung-Wang-2013, Phung-Wang-Zhang,wang-zhang1} for the observability inequality for parabolic equations. More generally,
when the observation subdomain is a measurable subset of positive measure in the space and time variables, we refer the readers to \cite{Escauriaza-Montaner-Zhang,Escauriaza-Montaner-Zhang2,Apraiz-Escauriaza22} for the observability inequality for analytic parabolic equations. The proofs of those inequalities are mainly based on the parabolic frequency function method, and the propagation of smallness estimate for real analytic functions.
Note that the solution to \eqref{yu-6-24-1} is obviously not spatial real-analyticity any more, and thus we cannot derive the observability inequality from measurable sets by the real analytic technique.
The studies on the observability inequality for parabolic equations on unbounded domains are rather few in last decades. The observability inequality may not be true when the heat equation is evolving in the whole space and the observation subdomain is only a bounded and open subset (see, e.g., \cite{MZ,MZb,M05a}). We would like to mention the work \cite{CMZ} for sufficient conditions so that the observability inequalities hold true for heat equations in unbounded domains. It showed that, for some parabolic equations in an unbounded domain
$\Omega \subset \mathbb{R}^N$, the observability inequality holds when observations are made over an open subset $\omega \subset\Omega$, with $\Omega \backslash \omega$ bounded. The proofs are based on Carleman estimates. For other similar results, we refer the reader to \cite{B,CMV,Gde,RM,Z16}.
Recently, \cite{WangZhangZhang} and \cite{EV17} independently obtained the observability inequality for the pure heat equation on the whole space, where the observation is a thick set (see Remark \ref{remark409}). This could be extended to
the time-independent parabolic equation associated to the Schr\"odinger operator with analytic coefficients (see \cite{EV19,leb,nttv}).
The methods utilized in these papers are all based on the spectral inequality (or uncertainty principle). Unfortunately, they are not valid any more for the case that the coefficients
in parabolic equations are non-analytic.
A H$\ddot{\mathrm{o}}$lder type propagation of smallness for solutions, as well as
for their gradients, on a possibly lower dimensional Hausdorff subset to general second order elliptic equations has been established in \cite{Logunov-Malinnikova}. The approach in
\cite{Logunov-Malinnikova} does not base on the analyticity of solutions.
This result stimulates our motivation and opens
a possible way to establish the observability inequality from measurable sets for solutions of general parabolic equations, getting rid of the above-mentioned technique of analyticity.
Recently, quantitative estimates of unique continuation for solutions of second order parabolic equations, such as the doubling property, as well as the two-ball and one-cylinder inequality, have been well understood (see, e.g., \cite{Apraiz-Escauriaza-Wang-Zhang}, \cite{Canuto-Rosset-Vessella}, \cite{Duan-Wang-Zhang}, \cite{Escauriaza-Fernandez-Vessella-2006}, \cite{LO}, \cite{Lin90}, \cite{Phung-Wang-2010}, \cite{Phung-Wang-2013}).
The main effort of this paper is to prove a locally quantitative estimate of strong unique continuation for solutions to the diffusion equation, by utilizing the propagation of smallness estimate for gradients of solutions to elliptic equations.
Combing with the above local result and the geometry of the equidistributed observation, we obtain a globally quantitative estimate at one time point for solutions of the diffusion equation. We finally apply the telescoping series method to prove the desired observability inequality.
\subsection{Plan of this paper}
The rest of this paper is organized as follows. In Section \ref{sc2}, we give several auxiliary lemmas. Sections \ref{kaodu3} and \ref{finalproof} prove Theorem \ref{yu-theorem-7-10-6} and Theorem \ref{jiudu4}, respectively.
\section{Some auxiliary lemmas}\label{sc2}
In this section, we first show two exponential estimates (see Lemmas \ref{yu-lemma-6-10-1} and \ref{yu-lemma-6-18-1} below) in Section \ref{kaodu1}, and then give a quantitative estimate of Cauchy uniqueness (see Lemma \ref{yu-proposition-7-1-1} below) in Section \ref{yu-section-7-26-3}.
\subsection{Local energy estimates}\label{kaodu1}
Let $T>0$ and $\tau\in(0,T)$ be arbitrarily fixed.
Let $\eta\in C^\infty(\mathbb{R}^+;[0,1])$ be a cutoff function satisfying
\begin{equation}\label{yu-6-6-6}
\begin{cases}
\eta= 1 &\mbox{in}\;\;[0,\tau],\\
\eta=0 &\mbox{in}\;\; [T,+\infty),\\
|\eta_t|\leq \frac{C_{1}}{T-\tau}&\mbox{in}\;\;(\tau,T),
\end{cases}
\end{equation}
where the positive constant $C_{1}$ is independent of $\tau$ and $T$.
Let $R>0$ and $x_0\in\mathbb R^N$. Let $v$ be the solution to
\begin{equation}\label{yu-11-29-4}
\begin{cases}
v_t-\mbox{div}(A(x)\nabla v)=0&\mbox{in}\;\;B_{R}(x_0)\times\mathbb{R}^+,\\
v=\eta u&\mbox{on}\;\;\partial B_R(x_0)\times\mathbb{R}^+,\\
v(\cdot,0)=0&\mbox{in}\;\; B_R(x_0),
\end{cases}
\end{equation}
where $u$ satisfies \eqref{yu-6-24-1} with the initial value $u_0\in H^1(\mathbb{R}^N)$ and $\eta$ verifies \eqref{yu-6-6-6}.
Then, we have the following exponential decay result.
\begin{lemma}\label{yu-lemma-6-10-1}
There exist positive constants $\hat{C}_{1}=\hat{C}_{1}(\lambda)$ and $\hat{C}_{2}=\hat{C}_{2}(\lambda,N)$ such that
\begin{equation}\label{yu-6-18-1}
\|v(\cdot,t)\|_{H^1(B_R(x_0))}\leq \hat{C}_{1}\left(1+\frac{1}{T}\right)^{\frac{1}{2}}
e^{\frac{\hat{C}_{1}T}{T-\tau}-\hat{C}_{2}R^{-2}(t-T)^+}F(R)\quad\text{for all}\;\; t\in\mathbb{R}^+,
\end{equation}
where $(t-T)^+:=\max\{0,t-T\}$ and
$F(R):=\sup\limits_{s\in[0,T]}\|u(\cdot,s)\|_{H^1(B_R(x_0))}$.
\end{lemma}
\begin{proof}
We proceed the proof into two steps as follows.
\par
\vskip 5pt
\textit{Step 1. To prove \eqref{yu-6-18-1} when $t\in[0,T]$.}
Setting $w=v-\eta u$ in $B_R(x_0)\times\mathbb R^+$, we see that $w$ verifies
\begin{equation}\label{yu-6-6-7}
\begin{cases}
w_s-\mbox{div}(A(x)\nabla w)=-\eta_su
&\mbox{in}\;\;B_R(x_0)\times\mathbb R^+,\\
w=0&\mbox{on}\;\;\partial B_R(x_0)\times\mathbb R^+,\\
w(\cdot,0)=-u_{0}&\mbox{in}\;\;B_R(x_0).
\end{cases}
\end{equation}
We next prove that for each $t\in[0,T]$,
\begin{equation}\label{yu-6-7-6}
\|w(\cdot,t)\|_{L^2(B_R(x_0))}\leq e^{\frac{C_{1}T}{T-\tau}}
\sup_{s\in[0,T]}\|u(\cdot,s)\|_{L^2(B_R(x_0))}.
\end{equation}
Indeed, multiplying first (\ref{yu-6-6-7}) by $w$ and then integrating by parts over $B_R(x_0)\times(0,t)$ lead to
\begin{eqnarray*}\label{yu-6-7-1}
&\;&\| w(\cdot,t)\|_{L^2(B_R(x_0))}^2+2\int_0^t\int_{B_R(x_0)}A\nabla
w\cdot\nabla w\,dxds\nonumber\\
&\leq&\| u_{0}\|_{L^2(B_R(x_0))}^2+\frac{C_{1}t}{T-\tau}\sup_{s\in[0,T]}\|u(\cdot,s)\|^2_{L^2(B_R(x_0))}+\frac{C_{1}}{T-\tau}\int_0^t\int_{B_R(x_0)}|w|^2dxds\nonumber\\
&\leq&\left(1+\frac{C_{1}T}{T-\tau}\right)\sup_{s\in[0,T]}\|u(\cdot,s)\|^2_{L^2(B_R(x_0))}+\frac{C_{1}}{T-\tau}\int_0^t\int_{B_R(x_0)}|w|^2dxds.
\end{eqnarray*}
Here, we used (\ref{yu-6-6-6}). This, together with the uniform ellipticity condition (see (\ref{yu-11-28-2})), implies that
$$\| w(\cdot,t)\|_{L^2(B_R(x_0))}^2\leq \left(1+\frac{C_{1}T}{T-\tau}\right)\sup_{s\in[0,T]}\|u(\cdot,s)\|^2_{L^2(B_R(x_0))}+\frac{C_{1}}{T-\tau}\int_0^t\int_{B_R(x_0)}|w|^2dxds.$$
By the Gronwall inequality, we get (\ref{yu-6-7-6}) immediately.
From (\ref{yu-6-7-6}) and the definition of $w$, we know that for any $t\in[0,T]$,
\begin{equation}\label{yu-6-7-11-1}
\|v(\cdot,t)\|_{L^2(B_R(x_0))}\leq 2e^{\frac{C_{1}T}{T-\tau}}
\sup_{s\in[0,T]}\|u(\cdot,s)\|_{L^2(B_R(x_0))}.
\end{equation}
\par
We next claim that
\begin{eqnarray}\label{yu-6-8-3}
\|\nabla w(\cdot,t)\|_{L^2(B_R(x_0))}^2
\leq \frac{4C_{1}^{2}\lambda T}{(T-\tau)^{2}}
\sup_{s\in[0,T]}\|u(\cdot,s)\|^2_{L^2(B_R(x_0))}+\lambda^{2}\|\nabla u_{0}\|_{L^2(B_R(x_0))}^2.
\end{eqnarray}
If it holds true, along with the definition of $w$, we have that, for each $t\in[0,T]$,
\begin{eqnarray}\label{yu-6-8-4}
\|\nabla v(\cdot,t)\|_{L^2(B_R(x_0))}^2&\leq& 2\|\nabla w(\cdot,t)\|^2_{L^2(B_R(x_0))}+2\|\nabla u(\cdot,t)\|_{L^2(B_R(x_0))}^2\nonumber\\
&\leq&2\lambda^{2}\|\nabla u_{0}\|_{L^2(B_R(x_0))}^2+2\|\nabla u(\cdot,t)\|_{L^2(B_R(x_0))}^2\nonumber\\
&&+ \frac{8C_{1}^{2}\lambda T}{(T-\tau)^{2}}
\sup_{s\in[0,T]}\|u(\cdot,s)\|^2_{L^2(B_R(x_0))}\nonumber\\
&\leq& \left[2+2\lambda^{2}+\frac{8C_{1}^{2}\lambda T}{(T-\tau)^{2}}\right]\sup_{s\in[0,T]}\|u(\cdot,s)\|^2_{H^1(B_R(x_0))}.
\end{eqnarray}
Hence, the desired estimate \eqref{yu-6-18-1} follows from $\lambda\geq1$, (\ref{yu-6-7-11-1}) and (\ref{yu-6-8-4})
when $t\in[0,T]$ with $\hat{C}_1=2(3+\lambda^2)(1+2C_1)$.
The rest is devoted to prove the claim (\ref{yu-6-8-3}). In fact,
multiplying first (\ref{yu-6-6-7}) by $w_s$ and then integrating by parts over $B_R(x_0)\times(0,t)$,
we find
\begin{eqnarray*}\label{yu-6-7-12}
&\;&\int_0^t\int_{B_R(x_0)}|w_s|^2dxds
+\frac{1}{2}\int_0^t\int_{B_R(x_0)}[\nabla w\cdot(A\nabla w)]_sdxds\nonumber\\
&=&\int_0^t\int_{B_R(x_0)}|w_s|^2dxds+\frac{1}{2}\int_0^t\int_{B_R(x_0)}\left[(\nabla w)_{s}\cdot (A\nabla w)+\nabla w\cdot(A\nabla w)_{s}\right]dxds\nonumber\\
&=&\int_0^t\int_{B_R(x_0)}|w_s|^2dxds+\int_0^t\int_{B_R(x_0)}\nabla w_{s}\cdot(A\nabla w) dxds\nonumber\\
&=&\int_0^t\int_{B_R(x_0)}|w_s|^2dxds-\int_0^t\int_{B_R(x_0)} w_{s}\mathrm{div}(A\nabla w) dxds\nonumber\\
&=&-\int_0^t\int_{B_R(x_0)}w_s\eta_sudxds\nonumber\\
&\leq&\frac{C_{1}}{\epsilon_{1}(T-\tau)}\int_0^t\int_{B_R(x_0)}
|w_{s}|^2dxds+\frac{C_{1}\epsilon_{1}}{T-\tau}\int_0^t\int_{B_R(x_0)}|u|^2dxds\qquad \text{for any}\;\;\epsilon_{1}>0.
\end{eqnarray*}
Here, we used (\ref{yu-6-6-6}). Letting
$\epsilon_{1}=2C_{1}/(T-\tau)$
in the inequality above, combined with the uniform ellipticity condition \eqref{yu-11-28-2}, we get that
\begin{eqnarray*}\label{yu-6-8-1}
&\;&\int_{B_R(x_0)}\nabla w(x,t)\cdot(A(x)\nabla w(x,t))dx\nonumber\\
&\leq&\int_{B_R(x_0)}\nabla u_{0}\cdot(A(x)\nabla u_{0})dx+\frac{4C_{1}^{2}}{(T-\tau)^{2}}\int_0^t\int_{B_R(x_0)}|u|^2dxds\nonumber\\
&\leq& \lambda\int_{B_R(x_0)}|\nabla u_{0}|^{2}dx+\frac{4C_{1}^{2}}{(T-\tau)^{2}}\int_0^t\int_{B_R(x_0)}|u|^2dxds.
\end{eqnarray*}
By the uniform ellipticity condition \eqref{yu-11-28-2} again, this means that
\begin{eqnarray*}\label{yu-6-8-2}
\|\nabla w(\cdot,t)\|_{L^2(B_R(x_0))}^2&\leq&
\lambda^{2}\int_{B_R(x_0)}|\nabla u_{0}|^{2}dx+\frac{4\lambda C_{1}^{2}}{(T-\tau)^{2}}\int_0^t\int_{B_R(x_0)}|u|^2dxds\\
&\leq& \lambda^{2}\int_{B_R(x_0)}|\nabla u_{0}|^{2}dx+\frac{4C_{1}^{2}\lambda T}{(T-\tau)^{2}}
\sup_{s\in[0,T]}\|u(\cdot,s)\|^2_{L^2(B_R(x_0))}.
\end{eqnarray*}
Thus, (\ref{yu-6-8-3}) is true.
\medskip
\vskip 5pt
\textit{Step 2. To prove \eqref{yu-6-18-1} when $t\geq T$.}
\par
Denoting $\mathcal{A}(\cdot):=-\mbox{div}(A(\cdot)\nabla)$ with domain $D(\mathcal{A}):=H_{0}^1(B_R(x_0))\cap H^2(B_R(x_0))$,
we claim that
there is a generic constant $C_{2}=C_{2}(\lambda,N)>0$ such that
\begin{equation}\label{yu-6-7-9}
\langle\mathcal{A}f,f\rangle_{{L}^2(B_R(x_0))}\geq C_{2}R^{-2}\|f\|^2_{L^2(B_R(x_0))}
\;\;\mbox{for each}\;\;f\in D(\mathcal{A}).
\end{equation}
In fact, by the Poincar\'e inequality
\begin{equation*}\label{yu-11-30-b-1}
\int_{B_R(x_0)}|f|^2dx\leq \left(\frac{2R}{N}\right)^2\int_{B_R(x_0)}|\nabla f|^2dx\;\;\mbox{for each}\;\;f\in H_0^1(B_R(x_0)),
\end{equation*}
we derive
\begin{equation*}\label{yu-10-12-1}
\langle\mathcal{A}f,f\rangle_{{L}^2(B_R(x_0))}\geq\lambda^{-1}\int_{B_R(x_0)}
|\nabla f|^2dx\geq \lambda^{-1}\left(\frac{N}{2R}\right)^{2}
\int_{B_R(x_0)}
|f|^2dx.
\end{equation*}
We can conclude the claim (\ref{yu-6-7-9}).
\par
As a consequence of \eqref{yu-6-7-9}, we see that the inverse of $(\mathcal A, D(\mathcal{A}))$ is positive, self-adjoint and compact in $L^2(B_R(x_0))$.
By the spectral theorem for compact self-adjoint operators, there are eigenvalues
$\{\mu_i\}_{i\in\mathbb{N}^+}\subset \mathbb{R}^+$ and eigenfunctions
$\{f_i\}_{i\in\mathbb{N}^+}\subset H_0^1(B_R(x_0))$, which make up a complete orthogonal basis of $L^2(B_R(x_0))$,
such that
\begin{equation}\label{yu-6-7-10}
\begin{cases}
-\mathcal{A}f_i=\mu_if_i\;\;\mbox{and}\;\;\|f_i\|_{{L}^2(B_R(x_0))}=1&\mbox{for each}\;\;i\in\mathbb{N}^+,\\
C_{2}R^{-2}\leq \mu_1\leq \mu_2\leq \cdots
\leq \mu_i\to+\infty&\mbox{as}\;\;i\to+\infty.
\end{cases}
\end{equation}
Then, by the formula of Fourier decomposition,
the solution $w$ of (\ref{yu-6-6-7}) in $[T,+\infty)$ is given by
\begin{equation*}\label{yu-6-12-3}
w(\cdot,t)=\sum_{i=1}^\infty\langle w(\cdot,T),f_i\rangle_{{L}^2(B_R(x_0))}e^{-\mu_i(t-T)}f_i \quad\text{in}\;\;B_R(x_0)\;\;\mbox{for each}\;\;t\in[T,+\infty).
\end{equation*}
Hence, we deduce that for each $t\in[T,+\infty)$,
\begin{eqnarray}\label{yu-6-12-4}
\|w(\cdot,t)\|^2_{{L}^2(B_R(x_0))}
\leq e^{-C_{2}R^{-2}(t-T)}\|w(\cdot,T)\|_{{L}^2(B_R(x_0))}^2
\end{eqnarray}
and
\begin{eqnarray*
w_t(\cdot,t)=-\sum_{i=1}^\infty\mu_i\langle w(\cdot,T),f_i\rangle_{{L}^2(B_R(x_0))}e^{-\mu_i(t-T)}f_i.
\end{eqnarray*}
It follows that
\begin{eqnarray}\label{yu-6-13-1}
-\langle w(\cdot,t),w_t(\cdot,t)\rangle_{{L}^2(B_R(x_0))}
=\sum_{i=1}^\infty\mu_i|\langle w(\cdot,T),f_i\rangle_{{L}^2(B_R(x_0))}|^2
e^{-2\mu_i(t-T)},
\end{eqnarray}
for each $t\in[T,+\infty)$. In particular, by taking $t=T$ in the above identify, we have
\begin{equation}\label{yu-6-13-2}
-\langle w(\cdot,T),w_t(\cdot,T)\rangle_{{L}^2(B_R(x_0))}
=\sum_{i=1}^\infty\mu_i|\langle w(\cdot,T),f_i\rangle_{{L}^2(B_R(x_0))}|^2.
\end{equation}
Meanwhile, it follows from (\ref{yu-6-6-7}), (\ref{yu-6-6-6}) and \eqref{yu-11-28-2} that
\begin{eqnarray}\label{yu-6-13-3}
-\langle w(\cdot,T),w_t(\cdot,T)\rangle_{{L}^2(B_R(x_0))}\leq \lambda\|\nabla w(\cdot,T)\|^2_{L^2(B_R(x_0))}.
\end{eqnarray}
From (\ref{yu-6-13-2}) and (\ref{yu-6-13-3}), we have
\begin{equation*}\label{yu-6-14-1}
\sum_{i=1}^\infty\mu_i|\langle w(\cdot,T),f_i\rangle_{{L}^2(B_R(x_0))}|^2
\leq \lambda\|\nabla w(\cdot,T)\|^2_{L^2(B_R(x_0))}.
\end{equation*}
This, together with (\ref{yu-6-13-1}) and (\ref{yu-6-7-10}), gives
\begin{equation}\label{yu-6-14-2}
-\langle w(\cdot,t),w_t(\cdot,t)\rangle_{{L}^2(B_R(x_0))}
\leq \lambda e^{-2C_{2}R^{-2}(t-T)}\|\nabla w(\cdot,T)\|^2_{L^2(B_R(x_0))},
\end{equation}
for each $t\in[T,+\infty)$. On the other hand, by (\ref{yu-6-6-6}), (\ref{yu-6-6-7}) and \eqref{yu-11-28-2}, we see that for each $t\in[T,+\infty)$,
\begin{equation}\label{yu-6-14-3}
-\langle w(\cdot,t),w_t(\cdot,t)\rangle_{{L}^2(B_R(x_0))}=\langle w(\cdot,t),\mathcal{A}w(\cdot,t)\rangle_{{L}^2(B_R(x_0))}
\geq \lambda^{-1}\|\nabla w(\cdot,t)\|^2_{L^2(B_R(x_0))}.
\end{equation}
By (\ref{yu-6-14-2}) and (\ref{yu-6-14-3}), we find that for each $t\in[T,\infty)$,
\begin{equation*}\label{yu-6-14-4}
\|\nabla w(\cdot,t)\|^2_{L^2(B_R(x_0))}
\leq \lambda^{2}
e^{-2C_{2}R^{-2}(t-T)}\|\nabla w(\cdot,T)\|^2_{L^2(B_R(x_0))}.
\end{equation*}
This, together with (\ref{yu-6-12-4}) and $\lambda\geq1$, means that
\begin{equation*}\label{yu-6-18-2}
\|w(\cdot,t)\|_{H^1(B_R(x_0))}^2\leq \lambda^{2}e^{-C_{2}R^{-2}(t-T)}\|w(\cdot,T)\|^2_{H^1(B_R(x_0))}.
\end{equation*}
By (\ref{yu-6-6-6}), we have that $w(\cdot,t)=v(\cdot,t)$ for each $t\geq T$. Thus, by (\ref{yu-6-18-1}) for the case $t\in[0,T]$, we conclude the desired result with
$\hat{C}_1=2\lambda^2(3+\lambda^2)(1+2C_1)$ and $\hat{C}_2=C_{2}/2$.
\end{proof}
Define
\begin{equation*}\label{yu-6-18-5}
\tilde{v}(\cdot,t):=
\begin{cases}
v(\cdot,t)&\mbox{if}\;\;t\geq 0,\\
0&\mbox{if}\;\;t<0,
\end{cases}
\end{equation*}
where $v$ is the solution of \eqref{yu-11-29-4}.
By Lemma \ref{yu-lemma-6-10-1}, we see the Fourier transform of $\tilde{v}$ with respect to the time variable $t\in\mathbb R$ is meaningful
\begin{equation*}\label{yu-6-18-6}
\hat{v}(x,\mu)=\int_{\mathbb{R}}e^{-i\mu t}\tilde{v}(x,t)dt\quad\text{for}\;\;(x,\mu)\in B_R(x_0)\times\mathbb R.
\end{equation*}
We further have
\begin{lemma}\label{yu-lemma-6-18-1}
There exist positive constants $\hat{C}_{3}=\hat{C}_{3}(\lambda,N)$ and $\hat{C}_{4}=\hat{C}_{4}(\lambda,N)$ such that for each $\mu\in\mathbb{R}$,
the following two estimates hold:
\begin{equation}\label{yu-6-23-5}
\|\nabla \hat{v}(\cdot,\mu)\|^{2}_{L^2(B_r(x_0))}\leq \frac{\hat{C}_{3}}{(R-2r)^{2}}\|\hat{v}(\cdot,\mu)\|^{2}_{L^2(B_{\frac{R}{2}}(x_0))}\quad \text{for all} \;\;0<r <\frac{R}{2}
\end{equation}
and
\begin{equation}\label{yu-6-22-16}
\|\hat{v}(\cdot,\mu)\|_{L^2(B_{\frac{R}{2}}(x_0))}\leq \hat{C}_{4}
\left(T+R^{2}\right)e^{\frac{\hat{C}_{1}T}{T-\tau}-\frac{\sqrt{|\mu|}R}{4e\hat{C}_{5}}}
F(R),
\end{equation}
where $\hat{C}_{5}=\sqrt{8\pi^2\sqrt{2+\lambda^{6}}}$, $\hat{C}_1>0$ and $F(R)$ are given by Lemma \ref{yu-lemma-6-10-1}.
\end{lemma}
\begin{proof}
By \eqref{yu-11-29-4}, we have that for each $\mu\in\mathbb R$,
\begin{equation}\label{yu-6-18-7}
i\mu\hat{v}(x,\mu)-\mbox{div}(A(x)\nabla \hat{v}(x,\mu))=0
\;\;\mbox{in}\;\;B_{R}(x_0).
\end{equation}
Take arbitrarily $r\in(0,R/2)$ and define a cutoff function $\psi \in C^\infty(\mathbb{R}^N;[0,1])$ verifying
\begin{equation}\label{yu-6-23-1}
\begin{cases}
\psi=1&\mbox{in}\;\;\overline{B_r(x_0)},\\
\psi=0&\mbox{in}\;\;\mathbb{R}^N\backslash B_{\frac{R}{2}}(x_0),\\
|\nabla\psi|\leq \frac{C_{3}}{R-2r}&\mbox{in}\;\;\mathbb{R}^N,
\end{cases}
\end{equation}
where $C_{3}=C_{3}(N)$ is a positive constant. The rest proof is divided into two steps.
\vskip 5pt
\textit{Step 1. The proof of (\ref{yu-6-23-5}).}
\par
Multiplying first (\ref{yu-6-18-7}) by $\bar{\hat{v}}\psi^2$ and then integrating by parts over
$B_{\frac{R}{2}}(x_0)$, we have
\begin{eqnarray*}\label{yu-6-23-2}
\int_{B_{\frac{R}{2}}(x_0)}\nabla\bar{\hat{v}}\cdot (A\nabla\hat{v})\psi^2dx
+2\int_{B_{\frac{R}{2}}(x_0)}\nabla\psi\cdot(A\nabla\hat{v})\bar{\hat{v}}\psi dx
=-i\int_{B_{\frac{R}{2}}(x_0)}\mu |\hat{v}|^2\psi^2dx.
\end{eqnarray*}
By (\ref{yu-11-28-2}) and the Young inequality, we derive that for each $\epsilon_2>0$,
\begin{eqnarray}\label{yu-6-23-3}
&\;&\lambda^{-1}\int_{B_{\frac{R}{2}}(x_0)}|\nabla\hat{v}|^2\psi^2dx
=\lambda^{-1}\mbox{Re}\int_{B_{\frac{R}{2}}(x_0)}|\nabla\hat{v}|^2\psi^2dx\nonumber\\
&\leq&-2\mbox{Re}
\int_{B_{\frac{R}{2}}(x_0)}\nabla\psi\cdot(A\nabla\hat{v})\bar{\hat{v}}\psi dx
\leq 2\lambda
\int_{B_{\frac{R}{2}}(x_0)}|\nabla\psi||\nabla\hat{v}||\bar{\hat{v}}||\psi| dx\nonumber\\
&\leq&\epsilon_2\int_{B_{\frac{R}{2}}(x_0)}|\nabla \hat{v}|^2\psi^2dx
+\frac{\lambda^2}{\epsilon_2}\int_{B_{\frac{R}{2}}(x_0)}|\hat{v}|^2|\nabla\psi|^2dx.
\end{eqnarray}
Taking $\epsilon_2=1/(2\lambda)$ in the above inequality, by (\ref{yu-6-23-1}) and (\ref{yu-6-23-3}), we derive
(\ref{yu-6-23-5}).
\vskip 5pt
\textit{Step 2. The proof of (\ref{yu-6-22-16}).}
\par
Note that when $\mu=0$, by (\ref{yu-6-7-11-1}), (\ref{yu-6-12-4}) and the definitions of $\hat{v}, \tilde{v}, w$, we have
\begin{eqnarray}\label{yu-6-22-15}
&&\|\hat{v}(\cdot,0)\|_{L^2(B_{R}(x_0))}\nonumber\\
&=&
\left\|\int_{\mathbb{R}}\tilde{v}(x,t)dt\right\|_{L^2(B_{R}(x_0))}
=\left\|\int_{0}^{+\infty}v(x,t)dt\right\|_{L^2(B_{R}(x_0))}\nonumber\\
&\leq&\int_{0}^{T}\|v(x,t)\|_{L^2(B_{R}(x_0))}dt+\int_{T}^{+\infty}\|v(x,t)\|_{L^2(B_{R}(x_0))}dt\nonumber\\
&\leq&2Te^{\frac{C_{1}T}{T-\tau}}
\sup_{s\in[0,T]}\|u(\cdot,s)\|_{L^2(B_R(x_0))}+\int_{T}^{+\infty}e^{-\frac{C_{2}(t-T)}{2R^2}}\|v(T)\|_{L^2(B_{R}(x_0))}dt\nonumber\\
&\leq&2e^{\frac{C_{1}T}{T-\tau}}
\sup_{s\in[0,T]}\|u(\cdot,s)\|_{H^1(B_R(x_0))}\left[T+\int_{T}^{+\infty}e^{-\frac{C_{2}(t-T)}{2R^2}}dt\right]\nonumber\\
&\leq&2e^{\frac{C_{1}T}{T-\tau}}
\left[T+\frac{2R^{2}}{C_{2}}\right]\sup_{s\in[0,T]}\|u(\cdot,s)\|_{H^1(B_R(x_0))}.
\end{eqnarray}
This yields that \eqref{yu-6-22-16} is true when $\mu=0$. Thus it suffices to prove \eqref{yu-6-22-16} in the case that $\mu\neq0$.
To this end, define for each $\mu\in\mathbb{R}\setminus\{0\}$,
\begin{equation*}\label{yu-6-19-1}
p(x,\xi,\mu)=e^{i\sqrt{|\mu|}\xi}\hat{v}(x,\mu) \;\; \mbox{for a.e.}\;\; (x,\xi)\in B_R(x_0)\times\mathbb R.
\end{equation*}
Then, by (\ref{yu-6-18-7}), one can check easily that, for each fixed $\mu\in\mathbb{R}\setminus\{0\}$, $p(\cdot,\cdot,\mu)$ verifies
\begin{equation*}\label{yu-6-19-2}
\mbox{div}(A(x)\nabla p(x,\xi,\mu))+i\mbox{sign}(\mu)\partial_{\xi\xi}p(x,\xi,\mu)
=0\;\;\mbox{in}\;\;B_R(x_0)\times\mathbb{R}.
\end{equation*}
Here
\begin{equation*}\label{yu-6-19-3}
\mbox{sign}(\mu)=
\begin{cases}
1&\mbox{if}\;\;\mu>0,\\
-1&\mbox{if}\;\;\mu<0.
\end{cases}
\end{equation*}
As for the equation verified by $p(\cdot,\cdot,\mu)$, it is elliptic with complex coefficients and its coefficients are independent of the $\xi$-variable. These facts imply by standard energy methods ($m$ times localized Cacciopoli’s inequalities, see also \cite{Canuto-Rosset-Vessella}) that
\begin{equation}\label{yu-6-22-4}
\int_{B_{\frac{R}{2}}(x_0)\times(-\frac{R}{2},\frac{R}{2})}|p^{(m)} |^2dxd\xi
\leq2R\left[\frac{C_{4}m^2}{R^2}\right]^m\int_{B_R(x_0)}|\hat{v}(x,\mu)|^2dx
\end{equation}
with $p^{(m)}=\partial_\xi^m p$ and $C_{4}=8\pi^2\sqrt{2+\lambda^{6}}$ (For the sake of completion, a detail proof for \eqref{yu-6-22-4} is presented in Appendix).
By the similar arguments in (\ref{yu-6-22-15}) and the definition of $\hat{v}$, we get that for each $\mu\in\mathbb{R}$,
\begin{eqnarray}\label{yu-6-22-5}
\|\hat{v}(\cdot,\mu)\|_{L^2(B_{R}(x_0))}
&\leq&\int_{0}^{+\infty}\|v(x,t)\|_{L^2(B_{R}(x_0))}dt\nonumber\\
&\leq&2e^{\frac{C_{1}T}{T-\tau}}
\left[T+\frac{2R^{2}}{C_{2}}\right]\sup_{s\in[0,T]}\|u(\cdot,s)\|_{H^1(B_R(x_0))}.
\end{eqnarray}
Therefore, by (\ref{yu-6-22-4}), (\ref{yu-6-22-5}) and the definition of $F(R)$, we get that for each $m\in\mathbb{N}^+$,
\begin{eqnarray}\label{yu-6-22-6}
\int_{B_{\frac{R}{2}}(x_0)\times(-\frac{R}{2},\frac{R}{2})}|p^{(m)}|^2dxd\xi
\leq8R \left[\frac{C_{4}m^2}{R^2}\right]^m e^{\frac{2C_{1}T}{T-\tau}}
\left[T+\frac{2R^{2}}{C_{2}}\right]^{2}
F^2(R).
\end{eqnarray}
\par
For any $\varphi\in L^2(B_{R/2}(x_0);\mathbb{C})$, we define
\begin{equation*}\label{yu-6-22-7}
P_{\mu}(\xi):=\int_{B_{\frac{R}{2}}(x_0)}p(x,\xi,\mu)\bar{\varphi}(x)dx, \;\;\;\xi\in\left(-\frac{R}{2},\frac{R}{2}\right).
\end{equation*}
It is well known that the following interpolation inequality holds
\begin{equation}\label{yu-6-22-8}
\|f\|_{L^\infty(I)}\leq C_{5}\left(|I|\|f'\|_{L^2(I)}^2+\frac{1}{|I|}\|f\|_{L^2(I)}^2\right)^{\frac{1}{2}}
\;\;\mbox{for each}\;\;f\in H^1(I),
\end{equation}
where $C_{5}>0$, $I$ is an bounded nonempty interval of $\mathbb{R}$ and $|I|$ is the length. Therefore, by
(\ref{yu-6-22-6}) and (\ref{yu-6-22-8}), we have that for any $\xi\in(-R/2, R/2)$ and $m\in\mathbb{N}^+$,
\begin{eqnarray}\label{yu-6-22-9}
&&|P_{\mu}^{(m)}(\xi)|\nonumber\\
&\leq& C_{5}\left(R\int_{-\frac{R}{2}}^{\frac{R}{2}}|P_{\mu}^{(m+1)}(\xi)|^2d\xi
+\frac{1}{R}\int_{-\frac{R}{2}}^{\frac{R}{2}}|P_{\mu}^{(m)}(\xi)|^2d\xi\right)^{\frac{1}{2}}\nonumber\\
&=&C_{5}\left(R\int_{-\frac{R}{2}}^{\frac{R}{2}}\left|\int_{B_{\frac{R}{2}}(x_0)}p^{(m+1)}(x,\xi,\mu)\bar{\varphi}(x)dx\right|^2d\xi
+\frac{1}{R}\int_{-\frac{R}{2}}^{\frac{R}{2}}\left|\int_{B_{\frac{R}{2}}(x_0)}p^{(m)}(x,\xi,\mu)\bar{\varphi}(x)dx\right|^2d\xi\right)^{\frac{1}{2}}\nonumber\\
&\leq&C_{5}\left(R\int_{B_{\frac{R}{2}}(x_0)\times(-\frac{R}{2},\frac{R}{2})}
|p^{(m+1)}|^2dxd\xi+\frac{1}{R}\int_{B_{\frac{R}{2}}(x_0)\times(-\frac{R}{2},\frac{R}{2})}|p^{(m)}|^2dxd\xi\right)
^{\frac{1}{2}}\|\varphi\|_{L^2(B_{\frac{R}{2}}(x_0))}\nonumber\\
&\leq&4C_{5}e^{\frac{C_{1}T}{T-\tau}}
\left[T+\frac{2R^{2}}{C_{2}}\right] F(R)\frac{C_{4}^{\frac{m+1}{2}}(m+1)^{m+1}}{R^{m}}\|\varphi\|_{L^2(B_{\frac{R}{2}}(x_0))}.
\end{eqnarray}
This implies that $P_{\mu}(\cdot)$ can be
analytically extended to the complex ball (still denoted by the same notation)
\begin{equation*}\label{yu-6-22-10}
E_0:=\left\{\xi\in\mathbb{C}:|\xi|<\frac{R}{2\sqrt{C_{4}}e} \right\}.
\end{equation*}
Then,
\begin{equation*}\label{yu-6-22-11}
|P_{\mu}(\xi)|\leq \sum_{m=0}^\infty\frac{|P_{\mu}^{(m)}(0)|}{m!}|\xi|^m,
\end{equation*}
when $\xi\in i\mathbb{R}\cap E_0$.
Taking $\xi_0=-iR/(4\sqrt{C_{4}}e)$, by (\ref{yu-6-22-9}), we get that
\begin{equation}\label{yu-6-22-12}
|P_{\mu}(\xi_0)|\leq 4C_{5}\sqrt{C_{4}}e^{\frac{C_{1}T}{T-\tau}}
\left[T+\frac{2R^{2}}{C_{2}}\right]
F(R)\sum_{m=0}^\infty\frac{(m+1)^{m+1}}{m!(4e)^m}
\|\varphi\|_{L^2(B_{\frac{R}{2}}(x_0))}.
\end{equation}
While, by the definitions of $P_{\mu}(\cdot)$ and $\xi_0$,
\begin{eqnarray*}\label{yu-6-22-13}
P_{\mu}(\xi_0)=e^{\frac{\sqrt{|\mu|}R}{4\sqrt{C_{4}}e}}\int_{B_{\frac{R}{2}}(x_0)}\hat{v}(x,\mu)\bar{\varphi}(x)dx.
\end{eqnarray*}
This, together with (\ref{yu-6-22-12}) and the arbitrariness of $\varphi$, means that,
\begin{equation}\label{yu-6-22-14}
\|\hat{v}(\cdot,\mu)\|_{L^2(B_{\frac{R}{2}}(x_0))}\leq 4C_{5}\sqrt{C_{4}}
\left[T+\frac{2R^{2}}{C_{2}}\right]e^{\frac{C_{1}T}{T-\tau}-\frac{\sqrt{|\mu|}R}{4\sqrt{C_{4}}e}}
\sum_{m=0}^\infty\frac{(m+1)^{m+1}}{m!(4e)^m}
F(R).
\end{equation}
By (\ref{yu-6-22-14}) and (\ref{yu-6-22-15}), we derive (\ref{yu-6-22-16}) and complete the proof.
\end{proof}
\medskip
\subsection{Stability estimate for elliptic equations}\label{yu-section-7-26-3}
For our purpose and the convenience of the reader, we first quote from \cite[Theorem 5.1]{Logunov-Malinnikova} the following propagation estimate of smallness (for sets on hyperplanes) for gradients of solutions to elliptic equations in the way we understood.
\begin{lemma}\label{auxiliary-lemma}
Let $0<r<r'<R<+\infty$ and $\rho>0.$ Suppose a positive measurable set $\omega\subset B_{r/2}(x_0)$ satisfy $|\omega|/|B_{r}(x_0)|>\rho$. If $g$ is a solution to $\mathrm{div}(A(x)\nabla g)+g_{x_{N+1}x_{N+1}}=0$ in $B_R(x_0,0),$ with $A(\cdot)$ satisfying the condition (\ref{yu-11-28-2}), then there exists $\hat{C}_{6}=\hat{C}_{6}(r,r'\rho,\lambda)>0$ and $\alpha_{1}=\alpha_{1}(r,r'\rho,\lambda)\in(0,1)$ such that
{\begin{equation}\label{auxiliary-lemma11}
\sup_{B_{r}(x_0,0)}|(\nabla g,\partial_{N+1}g)|\leq \hat{C}_{6}\left(\sup_{\omega}| (\nabla g(\cdot,0),\partial_{N+1}g(\cdot,0))|\right)^{\alpha_{1}}\left(\sup_{B_{r'}(x_0,0)}|(\nabla g,\partial_{N+1}g)|\right)^{1-\alpha_{1}}.
\end{equation}}
\end{lemma}
\begin{remark}
We refer the interesting reader to \cite{Logunov-Malinnikova} for more general statements on propagation of
smallness on possibly lower dimensional subsets for solutions to elliptic equations.
\end{remark}
\begin{remark}
Regarding the dependence of $\omega$, we here emphasize that the constants $\hat{C}_{6}$ and $\alpha_{1}$ in (\ref{auxiliary-lemma11})
depend only on the $r, r'$ and $N$-dimensional Lebesgue measure $|\omega|$, but not on the shape or position of $\omega$. This
uniform dependence is very important in our approach to prove the main result of this paper (see
also \cite{Apraiz-Escauriaza-Wang-Zhang,Escauriaza-Montaner-Zhang,Escauriaza-Montaner-Zhang2}).
\end{remark}
Let $0<2r<R<+\infty$ and $\rho>0.$ Suppose a positive measurable set $\omega\subset B_{r/2}(x_0)$ satisfy $|\omega|/|B_{r}(x_0)|>\rho$.
Let $g\in H^1(B_R(x_0,0))$ be a solution of
\begin{equation}\label{yu-6-23-9}
\begin{cases}
\mbox{div}(A(x)\nabla g)+g_{x_{N+1}x_{N+1}}=0&\;\;\;\text{in}\;\; B_R(x_0,0),\\
g(x,0)=0&\;\;\;\text{in}\;\; B_R(x_0),\\
g_{x_{N+1}}(x,0)=z(x)&\;\;\;\text{in}\;\; B_R(x_0),
\end{cases}
\end{equation}
where $z\in L^2(B_R(x_0))$.
\begin{lemma}\label{yu-proposition-7-1-1}
There exist constants $\hat{C}_{7}=\hat{C}_{7}(N,\rho,r,\lambda)>0$ and $\alpha_{2}=\alpha_{2}(N,\rho,r,\lambda)\in(0,1)$ such that
{\begin{equation}\label{yu-7-3-b-2}
\|g\|_{H^1(B_r(x_0,0))}\leq \hat{C}_{7}\|g\|_{H^1(B_{\frac{3r}{2}}(x_0,0))}^{\alpha_{2}}\|z\|^{1-\alpha_{2}}_{L^2(\omega)}.
\end{equation}}
\end{lemma}
\begin{proof}
We quote from \cite[Lemma 4.3]{Lin-1991}, \cite[Lemma A]{Lebeau-Zuazua} and \cite[Theorem 5.1]{LU-2013} that the following stability estimate holds: there exist constants $C_{6}=C_{6}(\lambda,N)>0$ and $\gamma=\gamma(r,\lambda,N)\in(0,1)$ such that
{\begin{equation}\label{yu-7-3-b-2000}
\|g\|_{H^1(B_r(x_0,0))}\leq C_{6}r^{-2}\| g\|_{H^1(B_{\frac{7r}{6}}(x_0,0))}^{\gamma}\|z\|^{1-\gamma}_{L^2(B_{\frac{7r}{6}}(x_0))}.
\end{equation}}
We first assume $z\in C(\overline{B_R(x_0)})$.
Applying Lemma \ref{auxiliary-lemma} to gradients of solutions for (\ref{yu-6-23-9}), we have
{\begin{equation}\label{yu-7-3-b-2000111}
\|(\nabla g,\partial_{N+1}g)\|_{L^\infty(B_{\frac{7r}{6}}(x_0,0))}\leq C_{7}\| (\nabla g(\cdot,0),\partial_{N+1}g(\cdot,0))\|_{L^\infty(\omega)}^{\theta}\|(\nabla g,\partial_{N+1}g)\|^{1-\theta}_{L^{\infty}(B_{\frac{4r}{3}}(x_0,0))}
\end{equation}}
with $C_{7}=C_{7}(\rho,r,\lambda)>0$ and $\theta=\theta(\rho,r,\lambda)\in(0,1).$
Next, we observe from the second line in (\ref{yu-6-23-9}) that $\nabla g(\cdot,0)=0$ in $ B_{R}(x_0)$. By (\ref{yu-7-3-b-2000111}) and the standard elliptic regularity (see, e.g., \cite[Theorem 8.32]{Gilbarg-Trudinger})
$$\|(\nabla g,\partial_{N+1}g)\|_{L^\infty(B_{\frac{4r}{3}}(x_0,0))}\leq C_{8}\| g\|_{H^1(B_{\frac{3r}{2}}(x_0,0))}$$
with $C_{8}=C_{8}(r,\lambda,N)>0$,
we have
$$\|z\|_{L^\infty(B_{\frac{7r}{6}}(x_0))}\leq \|(\nabla g,\partial_{N+1}g)\|_{L^\infty(B_{\frac{7r}{6}}(x_0,0))}\leq C_{7}C^{1-\theta}_{8}\|z\|_{L^\infty(\omega)}^{\theta}\| g\|^{1-\theta}_{H^1(B_{\frac{3r}{2}}(x_0,0))}.$$
This, combined with (\ref{yu-7-3-b-2000}), leads to
\begin{eqnarray}\label{yu-7-3-b-2000222}
\|g\|_{H^1(B_r(x_0,0))}&\leq& C_{6}r^{-2}\| g\|_{H^1(B_{\frac{7r}{6}}(x_0,0))}^{\gamma}\|z\|^{1-\gamma}_{L^2(B_{\frac{7r}{6}}(x_0))}\nonumber\\
&\leq& C_{6}r^{-2}|B_{\frac{7r}{6}}|^{1-\gamma}\|g\|_{H^1(B_{\frac{7r}{6}}(x_0,0))}^{\gamma}\|z\|^{1-\gamma}_{L^\infty(B_{\frac{7r}{6}}(x_0))}\nonumber\\
&\leq& C_{6}r^{-2}|B_{\frac{7r}{6}}|^{1-\gamma}\|g\|_{H^1(B_{\frac{4}{3}r}(x_0,0))}^{\gamma}\left[C_{7}C^{1-\theta}_{8}\|z\|_{L^\infty(\omega)}^{\theta}\| g\|^{1-\theta}_{H^1(B_{\frac{3r}{2}}(x_0,0))}\right]^{1-\gamma}\nonumber\\
&\leq& C_{9}\|z\|_{L^\infty(\omega)}^{\beta}\| g\|^{1-\beta}_{H^1(B_{\frac{3r}{2}}(x_0,0))}
\end{eqnarray}
with $C_{9}=C_{9}(r,\rho,N,\lambda)>0$ and $\beta=\beta(r,\rho,N,\lambda)\in(0,1).$
Finally, we complete the proof by replacing the $L^\infty$-norm in (\ref{yu-7-3-b-2000222}) with $L^2$-norm. To this end, we define a new Lebesgue measurable set
{\begin{equation}\label{yu-7-3-b-2000333}
\tilde{\omega}:=\left\{x\in \omega: \left|z(x)\right|\leq\left(\frac{2}{|\omega|}\int_{\omega}|z|^{2}dx\right)^{\frac{1}{2}}\right\}.
\end{equation}}
It is clear that $$\int_{\omega}|z|^{2}dx\geq\int_{\omega\backslash \tilde{\omega}}|z|^{2}dx\geq\frac{2|\omega\backslash \tilde{\omega}|}{|\omega|}\int_{\omega}|z|^{2}dx,$$
where $\omega\backslash \tilde{\omega}=\omega\cap \tilde{\omega}^{c}$.
This implies that $|\omega\backslash \tilde{\omega}|\leq |\omega|/2$, and hence $\tilde{\omega}\geq |\omega|/ 2$ . Applying (\ref{yu-7-3-b-2000222}) with $\omega$ replaced by $\tilde{\omega}$
leads to
$$\|g\|_{H^1(B_r(x_0,0))}\leq C_{10}\| g\|_{H^1(B_{\frac{3r}{2}}(x_0,0))}^{\beta_{1}}\|z\|^{1-\beta_{1}}_{L^\infty(\tilde{\omega})}$$
for some new constant $C_{10}=C_{10}(r,\rho,N,\lambda)>0$ and $\beta_{1}=\beta_{1}(r,\rho,N,\lambda)\in(0,1).$
This, together with (\ref{yu-7-3-b-2000333}), indicates the desired estimate
(\ref{yu-7-3-b-2}) for the case $z\in C(\overline{B_R(x_0)})$. This, together with (\ref{yu-7-3-b-2000}) and the fact
$C(\overline{B_R(x_0)})$ is dense in $L^2(B_R(x_0))$, yields that
(\ref{yu-7-3-b-2}) holds for any $z\in L^2(B_R(x_0))$. This ends the proof.
\end{proof}
\section{Proof of Theorem \ref{yu-theorem-7-10-6}}\label{kaodu3}
In order to give the proof of Theorem \ref{yu-theorem-7-10-6}, we need the following the interpolation inequality (i.e., Lemma \ref{lemma-2A2}), whose proof below is based on a reduction method \cite{LO} (see also \cite{Lin90}). It is worth mentioning that the interpolation inequality established in Lemma \ref{lemma-2A2}
is related to the so-called two spheres and one cylinder inequality obtained in \cite[Theorem 3.1.1$'$]{Canuto-Rosset-Vessella} (see also \cite[Theorem 2]{Escauriaza-Fernandez-Vessella-2006}) which has some applications in the study of inverse problem
(see, for instance, \cite{Canuto-Rosset-Vessella}). Comparing with them, however, there are two main differences: $(i)$ the radius $r$ of the observation sphere is here allowed to be independent of the observation time $\tau$; $(ii)$ we specify the dependence of the observability constant on the observation time $\tau$.
\begin{lemma}\label{lemma-2A2}
Let $T>0$, $0<r<+\infty$ and $\rho>0$. Suppose a positive measurable set $\omega\subset B_{r/2}(x_0)$ satisfy $|\omega|/|B_{r}(x_0)|>\rho$. There exist constants $\hat{C}_{8}=\hat{C}_{8}(r,N,\lambda,\rho)>0$ and $\sigma=\sigma(r,N,\lambda,\rho)\in(0,1)$ such that for each
$\tau\in(0,T/2)$, the corresponding solution $u$ of \eqref{yu-6-24-1} with the initial value $u_{0}\in H^{1}(\mathbb R^N)$ satisfies
\begin{equation*}\label{yu-7-10-2}
\|u(\cdot,\tau)\|_{L^2(B_r(x_0))}\leq \hat{C}_{8}
\left[T^{2}e^{\frac{\hat{C}_{1}T}{T-\tau}}+e^{\frac{\hat{C}_{8}}{\tau}}\right]\|u(\cdot,\tau)\|_{L^2(\omega)}^\sigma
\left(\sup_{s\in[0,T]}\|u(\cdot,s)\|^{2}_{H^1(B_{\varrho r}(x_0))}\right)^{\frac{1-\sigma}{2}},
\end{equation*}
where $\varrho:=16\hat{C}_{5}e$, the constant $\hat{C}_1>0$ is given by Lemma \ref{yu-lemma-6-10-1} and the constant $\hat{C}_5=\sqrt{8\pi^2\sqrt{2+\lambda^6}}$ is given by Lemma \ref{yu-lemma-6-18-1}.
\end{lemma}
\begin{proof}
Taking $u_0\in H^1(\mathbb{R}^N)$ arbitrarily. Let $u$ be the solution to the equation (\ref{yu-6-24-1}) with the initial value $u_{0}\in H^{1}(\mathbb R^N)$, and $R:=\varrho r$.
Let $u_1$ and $u_2$ be accordingly the solutions to
\begin{equation*}\label{yu-7-4-4}
\begin{cases}
\partial_tu_{1}-\mbox{div}(A(x)\nabla u_1)=0&\mbox{in} \;B_R(x_0)\times(0,2T),\\
u_1=u&\mbox{on}\;\;\partial B_R(x_0)\times(0,2T),\\
u_1(\cdot,0)=0 &\mbox{in}\;\;B_{R}(x_0)
\end{cases}
\end{equation*}
and
\begin{equation*}\label{yu-7-4-5}
\begin{cases}
\partial_tu_{2}-\mbox{div}(A(x)\nabla u_2)=0&\mbox{in}\;\; B_R(x_0)\times(0,2T),\\
u_2=0&\mbox{on}\;\;\partial B_R(x_0)\times(0,2T),\\
u_2(\cdot,0)=u_0&\mbox{in}\;\; B_{R}(x_0).
\end{cases}
\end{equation*}
It is clear that $u=u_1+u_2$ in $B_R(x_0)\times[0,2T]$.
By the standard energy estimate for solutions of parabolic equations, we have
\begin{equation}\label{yu-7-4-7}
\sup_{t\in[0,T]}\|u_2(\cdot,t)\|_{H^1(B_R(x_0))}\leq C_{11} \|u_0\|_{H^1(B_R(x_0))}
\end{equation}
with $C_{11}=C_{11}(N,\lambda)>0.$
Arbitrarily fix $\tau\in(0,T/2)$. Let $v_1$ be the solution to
\begin{equation*}\label{yu-11-29-4-jia}
\begin{cases}
\partial_tv_1-\mbox{div}(A(x)\nabla v_1)=0&\mbox{in}\;\;B_{R}(x_0)\times\mathbb{R}^+,\\
v_1=\eta u&\mbox{on}\;\;\partial B_R(x_0)\times\mathbb{R}^+,\\
v_1(\cdot,0)=0&\mbox{in}\;\; B_R(x_0),
\end{cases}
\end{equation*}
where $\eta$ is given by \eqref{yu-6-6-6}.
It is clear that
$u=v_1+u_2$ in $B_R(x_0)\times[0,\tau]$.
We extend $v_1$ to $\mathbb R^-\times B_R(x_0)$ by zero, and still denote it by the same way.
Define
\begin{equation*}\label{yu-6-18-6jia}
\hat{v}_1(x,\mu)=\int_{\mathbb{R}}e^{-i\mu t}v_1(x,t)dt
\quad\text{for}\;\;(x,\mu)\in B_R(x_0)\times\mathbb R.
\end{equation*}
Note from Lemma \ref{yu-lemma-6-10-1} that $\hat{v}_1$ is well defined.
\par
Let
\begin{equation*}\label{yu-7-5-bb-1}
\kappa:=\frac{\sqrt{2}}{4e\hat{C}_{5}} \quad\text{with}\;\;\hat{C}_{5}=\sqrt{8\pi^2\sqrt{2+\lambda^{6}}}\;\;\text{given in Lemma
\ref{yu-lemma-6-18-1}}.
\end{equation*}
We define
$$V=V_1+V_2\quad\mbox{in}\;\; B_R(x_0)\times(-\kappa R,\kappa R)
$$
with
\begin{equation}\label{yu-6-23-6jia}
V_1(x,y)=\frac{1}{2\pi}\int_{\mathbb{R}}e^{i\tau\mu}\hat{v}_1(x,\mu)
\frac{\sinh(\sqrt{-i\mu}y)}{\sqrt{-i\mu}}d\mu\quad\mbox{in}\;B_R(x_0)\times(-\kappa R,\kappa R),
\end{equation}
\begin{equation}\label{yu-7-5-7}
V_2(x,y)=\sum_{k=1}^\infty\alpha_ke^{-\mu_k\tau}f_k(x)\frac{\sinh(\sqrt{\mu_k}y)}
{\sqrt{\mu_k}}\;\;\mbox{in}\;\;B_R(x_0)\times(-\kappa R,\kappa R)
\end{equation}
where $\{\mu_k\}_{k=1}^\infty$, $\{f_k\}_{k=1}^{\infty}$ are given by $(\ref{yu-6-7-10})$, and
$\alpha_k=\langle u_2(\cdot,0),f_k\rangle_{L^2(B_R(x_0))}$.
Note from
Lemma \ref{yu-lemma-6-18-1} that $V_1$ is also well defined.
One can readily check that
\begin{equation}\label{yu-7-5-9}
\begin{cases}
\mbox{div}(A(x)\nabla V(x,y))+V_{yy}(x,y)=0&\mbox{in}\;\;
B_{\frac{R}{2}}(x_0)\times(-\kappa R,\kappa R),\\
V(x,0)=0&\mbox{in}\;\;B_{\frac{R}{2}}(x_0),\\
V_y(x,0)=u(x,\tau)&\mbox{in}\;\;B_{\frac{R}{2}}(x_0).
\end{cases}
\end{equation}
Let $r\in (0, \kappa R/2)$. By the three-ball inequality for elliptic operators (cf., e.g., \cite[Theorem 3.1]{MV}), there exist constants
$C_{13}=C_{13}(\lambda,N,r)>0$ and $\beta_{2}=\beta_{2}(\lambda,N,r)\in(0,1)$ such that
\begin{equation}\label{yu-7-4-10}
\|V\|_{L^2(B_{\frac{7r}{4}}(x_0,0))}\leq C_{13}\|V\|^{\beta_{2}}_{L^2(B_{r}(x_0,0))}\|V\|_{L^2(B_{2r}(x_0,0))}^{1-\beta_{2}}.
\end{equation}
Since $V_y$ satisfies the first equation of (\ref{yu-7-5-9}), by the interior estimate of elliptic equations
we find
\begin{eqnarray}\label{yu-7-4-11}
\int_{B_{\frac{7r}{4}}(x_0,0)}|V|^2dxdy&\geq &C_{14}r^2\int_{B_{\frac{3r}{2}}(x_0,0)}(|\nabla V|^2+|V_y|^2)dxdy\nonumber
\\&\geq&\frac{C_{14}r^2}{2}
\left(\int_{B_{\frac{3r}{2}}(x_0,0)}|V_y|^2dxdy+\int_{B_{\frac{3r}{2}}(x_0,0)}|V_y|^2dxdy\right)\nonumber\\
&\geq&C_{14}r^2\left(\int_{B_{\frac{3r}{2}}(x_0,0)}|V_y|^2dxdy+r^2\int_{B_{\frac{5r}{4}}(x_0,0)}(|\nabla V_y|^2+|V_{yy}|^2)dxdy\right)\nonumber\\
&\geq& C_{14}r^3\left(\frac{1}{r}\int_{B_{\frac{5r}{4}}(x_0,0)}|V_y|^2dxdy+r\int_{B_{\frac{5r}{4}}(x_0,0)}|V_{yy}|^2dxdy\right)
\end{eqnarray}
with $C_{14}=C_{14}(\lambda,N)>0$.
As a simple corollary of \cite[Lemma 9.9, Page 315]{Brezis}, we have the following trace theorem
\begin{equation*}\label{yu-7-4-12}
\int_{B_{r}(x_0)}|f(x,0)|^2dx\leq C_{15}(N)\left(\frac{1}{r}\int_{B_{\frac{5r}{4}}(x_0,0)}|f|^2dxdy+r\int_{B_{\frac{5r}{4}}
(x_0,0)}|f_y|^2dxdy\right)
\end{equation*}
for any $f\in H^1(B_{5r/4}(x_0,0))$. Hence, by (\ref{yu-7-5-9}) and (\ref{yu-7-4-11}) we have
\begin{eqnarray}\label{yu-7-4-13}
C_{16}r^3\int_{B_{r}(x_0)}|u(x,\tau)|^2dx\leq \int_{B_{\frac{7r}{4}}(x_0,0)}|V|^2dxdy
\end{eqnarray}
with $C_{16}=C_{16}(\lambda,N)>0$.
By Lemma \ref{yu-proposition-7-1-1}, we obtain that there is $C_{17}=C_{17}(\rho,N,r,\lambda)>0$ and $\beta_{3}=\beta_{3}(\rho,N,r,\lambda)\in(0,1)$ such that
\begin{equation}\label{yu-7-4-2}
\|V\|_{L^2(B_r(x_0,0))}\leq C_{17}\|V\|_{H^1(B_{\frac{3r}{2}}(x_0,0))}^{\beta_{3}}\|u(\cdot,\tau)\|_{L^2(\omega)}^{1-\beta_{3}}.
\end{equation}
Again, by the interior estimate, there is a constant $C_{18}=C_{18}(N)>0$ such that
\begin{equation}\label{yu-7-4-3}
\|V\|_{H^1(B_{\frac{3r}{2}}(x_0,0))}\leq C_{18}r^{-1} \|V\|_{L^2(B_{2r}(x_0,0))}.
\end{equation}
Hence, it follows from (\ref{yu-7-4-2}) and (\ref{yu-7-4-3}) that
\begin{equation}\label{yu-7-4-1}
\|V\|_{L^2(B_r(x_0,0))}\leq C_{19}\|V\|_{L^2(B_{2r}(x_0,0))}^{\beta_{3}}\|u(\cdot,\tau)\|_{L^2(\omega)}^{1-\beta_{3}}
\end{equation}
with $C_{19}=C_{19}(\rho,N,r,\lambda)>0$.
It follows from (\ref{yu-7-4-13}), (\ref{yu-7-4-10}) and \eqref{yu-7-4-1} that
\begin{equation}\label{yu-7-5-1}
\|u(\cdot,\tau)\|_{L^2(B_{r}(x_0))}\leq C_{20}\|u(\cdot,\tau)\|_{L^2(\omega)}^{(1-\beta_{3})\beta_{2}}
\|V\|^{1-(1-\beta_{3})\beta_{2}}_{L^2(B_{2r}(x_0,0))}
\end{equation}
with $C_{20}=C_{20}(\rho,N,r,\lambda)>0$.
\par
\medskip
To finish the proof, it suffices to bound the term $\|V\|_{L^2(B_{2r}(x_0,0))}$. Recall that $V=V_1+V_2$, we will treat
$V_1$ and $V_2$ separately.
In fact, we derive from (\ref{yu-6-23-6jia}) that
for each $x\in B_{2r}(x_0)\subset B_{R}(x_0)$ and $|y|<\kappa R/(4\sqrt{2})$,
\begin{eqnarray*}\label{yu-7-5-2}
|V_1(x,y)|
&=&\left|\frac{1}{2\pi}\int_{\mathbb{R}}e^{i\tau\mu}\hat{v}_1(x,\mu)\int_{-y}^ye^{\sqrt{-i\mu}s}dsd\mu\right|
\leq \frac{1}{2\pi}\int_{\mathbb{R}}|\hat{v}_1(x,\mu)|\int_{-y}^y|e^{\sqrt{-i\mu}s}|dsd\mu
\nonumber\\
&\leq&\frac{\kappa R}{4\sqrt{2}\pi}\int_{\mathbb{R}}|\hat{v}_1(x,\mu)|e^{\frac{1}{4\sqrt{2}}\kappa\sqrt{|\mu|}R}d\mu\nonumber\\
&\leq &\frac{\kappa R}{4\sqrt{2}\pi}\left(\int_{\mathbb{R}}|\hat{v}_1(x,\mu)|^2e^{\frac{1}{\sqrt{2}}\kappa\sqrt{|\mu|}R}d\mu\right)^{\frac{1}{2}}
\left(\int_{\mathbb{R}}e^{-\frac{1}{2\sqrt{2}}\kappa \sqrt{|\mu|}R}d\mu\right)^{\frac{1}{2}}\nonumber\\
&=&\frac{1}{2\pi}\left(\int_{\mathbb{R}}|\hat{v}_1(x,\mu)|^2e^{\frac{1}{\sqrt{2}}\kappa\sqrt{|\mu|}R}d\mu\right)^{\frac{1}{2}}.
\end{eqnarray*}
Hence, by Lemma \ref{yu-lemma-6-18-1}, we have for each $r<\kappa R/(4\sqrt{2})$,
\begin{eqnarray}\label{yu-7-5-3}
\int_{B_{2r}(x_0,0)}|V_1|^2dxdy
&\leq& \frac{1}{2\pi^2}\hat{C}_{4}^2Re^{\frac{2\hat{C}_{1}T}{T-\tau}}
\left[T+R^{2}\right]^{2}F^2(R)
\int_{\mathbb{R}}e^{-\frac{1}{\sqrt{2}}\kappa \sqrt{|\mu|}R}d\mu\nonumber\\
&\leq&\frac{32e}{\pi^2}\hat{C}_{5}^2\hat{C}_{4}^2R^{-1}e^{\frac{2\hat{C}_{1}T}{T-\tau}}
\left[T+R^{2}\right]^{2}F^2(R).
\end{eqnarray}
While, by (\ref{yu-7-5-7}) and (\ref{yu-7-4-7}) we obtain
\begin{eqnarray}\label{yu-7-5-10}
\int_{B_{2r}(x_0,0)}|V_2|^2dxdy&\leq&\int_{-2r}^{2r}\int_{B_{R}(x_0)}|V_2|^2dxdy
\leq \int_{-2r}^{2r}\sum_{k=1}^\infty\alpha_k^2e^{-2\mu_k\tau}\left|\frac{\sinh(\sqrt{\mu_k}y)}{\sqrt{\mu_k}}\right|^2dy\nonumber\\
&\leq&\frac{e^{\frac{2r^{2}}{\tau}}}{4\hat{C}_{2}^{3/2}}R^{3}\sum_{k=1}^\infty\alpha_k^2=
\frac{e^{\frac{2r^{2}}{\tau}}}{4\hat{C}_{2}^{3/2}}R^{3}\int_{B_R(x_0)}|u(x,0)|^2dx\nonumber\\
&\leq& C_{21}R^{3}e^{\frac{C_{21}}{\tau}}F^2(R)
\end{eqnarray}
with $C_{21}=C_{21}(N,r,\lambda)>0$.
Therefore, by (\ref{yu-7-5-3}) and (\ref{yu-7-5-10}), we conclude that
\begin{equation*}\label{yu-7-5-11}
\|V\|_{L^2(B_{2r}(x_0,0))}\leq C_{22}\left[e^{\frac{\hat{C}_{1}T}{T-\tau}}R^{-1}
\left(T+R^{2}\right)^{2}+e^{\frac{C_{21}}{\tau}}R^{3}\right]^{\frac{1}{2}}F(R)
\end{equation*}
with $C_{22}=C_{22}(N,r,\lambda)>0$.
This, together with (\ref{yu-7-5-1}), means that
\begin{eqnarray*}\label{yu-7-5-12}
\|u(\cdot,\tau)\|_{L^2(B_{r}(x_0))}
&\leq& C_{20}\|u(\cdot,\tau)\|_{L^2(\omega)}^{(1-\beta_{3})\beta_{2}}
\|V\|^{1-(1-\beta_{3})\beta_{2}}_{L^2(B_{2r}(x_0,0))}\nonumber\\
&\leq& C_{20}C_{22}^{1-(1-\beta_{3})\beta_{2}}\left[e^{\frac{\hat{C}_{1}T}{T-\tau}}R^{-1}
\left(T+R^{2}\right)^{2}+e^{\frac{C_{21}}{\tau}}R^{3}\right]^{\frac{1-(1-\beta_{3})\beta_{2}}{2}}\nonumber\\
&&\times \|u(\cdot,\tau)\|_{L^2(\omega)}^{(1-\beta_{3})\beta_{2}}F(R)^{1-(1-\beta_{3})\beta_{2}}.
\end{eqnarray*}
Taking $\sigma=(1-\beta_{3})\beta_{2}$, the proof is immediately achieved by the arbitrariness of $u_0$.
\end{proof}
\begin{lemma}\label{yu-LEMMA-7-10-6}
Let $T>0$, $0<r<+\infty$ and $\rho>0$. Assume that there is a sequence $\{x_i\}_{i\in\mathbb{N}^+}\subset\mathbb R^N$ so that
\begin{equation*}\mathbb{R}^{N}=\bigcup_{i\in\mathbb{N}^+}Q_{r}(x_{i})
\quad \text{with}\quad \mathrm{int}(Q_{r}(x_{i}))\bigcap \mathrm{int}(Q_{r}(x_{j}))=\emptyset\quad \text{for each}\quad i\neq j\in\mathbb N^+.
\end{equation*}
Let $$\widetilde{\omega}\triangleq\bigcup_{i\in\mathbb{N}^+}\widetilde{\omega}_{i} \quad\text{with} \quad \widetilde{\omega}_{i} \subset B_{r/2}(x_{i})\quad\text{satisfy}\quad \frac{|\widetilde{\omega}_{i}|}{|B_{r}(x_i)|}\geq\rho \quad\text{for each}\quad i\in\mathbb N^+ $$
where, for each $i\in\mathbb{N}^+$, $\widetilde{\omega}_{i}$ is a $N$-dimensional Lebesgue measurable set of positive measure.
There exist constants $C=C(r,N,\lambda,\rho)>0$ and $\sigma=\sigma(r,N,\lambda,\rho)\in(0,1)$ such that for any $u_{0}\in H^{1}(\mathbb R^N)$, the corresponding solution $u$ of \eqref{yu-6-24-1} satisfies
\begin{equation*}\label{yu-7-10-2}
\|u(\cdot,T)\|_{L^2(\mathbb R^N)}\leq C
\left(T^{3}+e^{\frac{C}{T}}\right)\|u(\cdot,T)\|_{L^2(\tilde{\omega})}^\sigma
\left(\sup_{s\in[0,2T]}\|u(\cdot,s)\|_{H^1(\mathbb R^N)}\right)^{1-\sigma}.
\end{equation*}
\end{lemma}
\begin{proof}
By Lemma~\ref{lemma-2A2} (where $r$ and $\omega$ are replaced by $ \sqrt{N}r$ and $\widetilde{\omega}_{i}$ ($i\in\mathbb{N}^+$), respectively), we get that
\begin{eqnarray*}
\int_{Q_{r}(x_{i})}|u(x,\tau)|^{2}\mathrm{d}x
&\leq& \int_{B_{\sqrt{N}r}(x_{i})}|u(x,\tau)|^{2}\mathrm{d}x\\
&\leq&\hat{C}_{9}\left(T^{2}e^{\frac{\hat{C}_{1}T}{T-\tau}}
+e^{\frac{\hat{C}_{9}}{\tau}}\right)^{2}\left(\int_{\widetilde{\omega}_{i}}
|u(x,\tau)|^{2}\mathrm{d}x\right)^\theta\\
&&\times\left(\sup_{s\in[0,T]}\|u(\cdot,s)\|^{2}_{H^1(B_{\sqrt{N}\varrho r}(x_i))}\right)^{1-\theta},
\end{eqnarray*}
where $\hat{C}_{9}=\hat{C}_{9}(r,N,\lambda,\rho)>0$ and $\theta=\theta(r,N,\lambda,\rho)\in(0,1)$. This, along with Young's inequality,
implies that for each $\varepsilon>0,$
\begin{eqnarray*}
\int_{Q_{r}(x_{i})}|u(x,\tau)|^{2}\mathrm{d}x
&\leq&\hat{C}^{2}_{9}
\biggl(T^{2}e^{\frac{\hat{C}_{1}T}{T-\tau}}+e^{\frac{\hat{C}_{9}}{\tau}}\biggl)^{2}
\biggl(\varepsilon\theta\sup_{s\in[0,T]}\|u(\cdot,s)\|^{2}_{H^1(B_{\sqrt{N}\varrho r}(x_i))}\\
&&\;\;\;\;\;\;\;\;+\varepsilon^{-\frac{\theta}{1-\theta}}(1-\theta)
\int_{\widetilde{\omega}_{i}}|u(x,\tau)|^{2}\mathrm{d}x\biggl).
\end{eqnarray*}
Then
\begin{equation}\label{3.44444}
\begin{array}{lll}
&&\displaystyle{}\int_{\mathbb{R}^{N}}|u(x,\tau)|^{2}\mathrm{d}x
=\sum_{i\in\mathbb{N}^+}\int_{Q_{r}(x_{i})}|u(x,\tau)|^{2}\mathrm{d}x\\
&\leq&\hat{C}^{2}_{9}\left(T^{2}e^{\frac{\hat{C}_{1}T}{T-\tau}}+e^{\frac{\hat{C}_{9}}{\tau}}\right)^{2}
\left(\varepsilon\theta \displaystyle\sum_{i\in\mathbb{N}^+}\sup_{s\in[0,T]}\|u(\cdot,s)\|^{2}_{H^1(B_{\sqrt{N}\varrho r}(x_i))}+\varepsilon^{-\frac{\theta}{1-\theta}}(1-\theta)
\int_{\widetilde{\omega}}|u(x,\tau)|^{2}\mathrm{d}x\right).
\end{array}
\end{equation}
\par
Next, let $R:=\varrho r$, we show that
\begin{equation}\label{yu-4-9-4}
\sum_{i\in\mathbb{N}^+}\sup_{s\in[0,T]}\|u(\cdot,s)\|^{2}_{H^1(B_{\sqrt{N}R}(x_i))}\leq \hat{C}_{10}(1+T)\sup_{s\in[0,T]}\|u(\cdot,s)\|^2_{H^1(\mathbb{R}^{N})},
\end{equation}
where $\hat{C}_{10}(\lambda,R,N)>0$. Indeed, we take a cut-off function $\psi\in C_0^\infty(\mathbb{R}^N;[0,1])$ so that $\mbox{supp}\psi\Subset Q_{2\sqrt{N}R}(0)$ and $\psi\equiv 1$ in
$B_{\sqrt{N}R}(0)$. For each $i\in\mathbb{N}^+$, we let $\psi_i(x):=\psi(x-x_i)$ for any
$x\in\mathbb{R}^+$. It is clear that $\mbox{supp}\psi_i\Subset Q_{2\sqrt{N}R}(x_i)$ and
$\psi_i\equiv 1$ in $B_{\sqrt{N}R}(x_i)$. We suppose that
$$
\begin{cases}
\mbox{Card}\{j\in\mathbb{N}^+: Q_{r}(x_1)\cap Q_{2\sqrt{N}R}(x_j)\neq \emptyset\}=m^*,\\
\mbox{Card}\{j\in\mathbb{N}^+: Q_{2\sqrt{N}R}(x_1)\cap Q_{r}(x_j)\neq \emptyset\}=n^*.
\end{cases}
$$
This means that, for each $k\in\mathbb{N}^+$,
$$
\begin{cases}
\mbox{Card}\{j\in\mathbb{N}^+: Q_{r}(x_k)\cap Q_{2\sqrt{N}R}(x_j)\neq \emptyset\}=m^*,\\
\mbox{Card}\{j\in\mathbb{N}^+: Q_{2\sqrt{N}R}(x_k)\cap Q_{r}(x_j)\neq \emptyset\}=n^*.
\end{cases}
$$
For each $k$, we suppose $\{y_{k,i}\}_{i=1}^{n^*}
=\{x_i|Q_{2\sqrt{N}R}(x_k)\cap Q_r(x_i)\neq \emptyset\}$.
Thus, for any $f\in H^1(\mathbb{R}^N)$,
\begin{eqnarray}\label{yu-4-9-1}
\sum_{k\in\mathbb{N}^+}\|f\|^{2}_{H^1(Q_{2\sqrt{N}R}(x_k))}
&\leq&\sum_{k\in\mathbb{N}^+}\sum_{i=1}^{n^*}\|f\|^{2}_{H^1(Q_r(y_{k,i})}
\leq \sum_{i=1}^{n^*}\sum_{k\in\mathbb{N}^+}\|f\|^2_{H^1(Q_r(y_{k,i}))}\nonumber\\
&\leq&m^*n^*\sum_{k\in\mathbb{N}^+}\|f\|^2_{H^1(Q_r(x_k))}=m^*n^*\|f\|^2_{H^1(\mathbb{R}^N)}.
\end{eqnarray}
For each $i\in\mathbb{N}^+$, we let $v_i=\psi_iu$. It is obvious that $v_i$ verifies
\begin{equation}\label{yu-4-9-2}
\begin{cases}
(v_i)_t-\mbox{div}(A(x)\nabla v_i)=-2\nabla\psi_i\cdot (A(x)\nabla u)-u\mbox{div}(A(x)\nabla \psi_i)
&\mbox{in}\;\;\; Q_{2\sqrt{N}R}(x_i)\times [0,T],\\
v_i=0&\mbox{on}\;\; \partial Q_{2\sqrt{N}R}(x_i)\times [0,T],\\
v_i(0)=\psi_iu_0 &\mbox{in}\;\;\;Q_{2\sqrt{N}R}(x_i).
\end{cases}
\end{equation}
By the standard energy estimate for (\ref{yu-4-9-2}), one can easily check that
\begin{eqnarray*}
&\;&\sup_{s\in[0,T]}\|u(\cdot,s)\|_{H^1(B_{\sqrt{N}R}(x_i))}^2
\leq \sup_{s\in[0,T]}\|v_i(\cdot,s)\|_{H^1(Q_{2\sqrt{N}R}(x_i))}^2\nonumber\\
&\leq& C(\lambda,N)(1+R^{-4})\left(\|u_0\|_{H^1(Q_{2\sqrt{N}R}(x_i))}^2+\int_0^T
\|u(\cdot, s)\|_{H^1(Q_{2\sqrt{N}R}(x_i))}^2ds\right),
\end{eqnarray*}
where $C(\lambda,N)>0$. This, along with (\ref{yu-4-9-1}), yields that
\begin{eqnarray*}
&&\sum_{i\in\mathbb{N}^+}\sup_{s\in[0,T]}\|u(\cdot,s)\|_{H^1(B_{\sqrt{N}R}(x_i))}^2\\
&\leq& C(\lambda,N)(1+R^{-4})\sum_{i\in\mathbb{N}^+}\left(\|u_0\|_{H^1(Q_{2\sqrt{N}R}(x_i))}^2+\int_0^T
\|u(\cdot, s)\|_{H^1(Q_{2\sqrt{N}R}(x_i))}^2ds\right)\nonumber\\
&\leq&m^*n^*C(\lambda,N)(1+R^{-4})(1+T)\sup_{s\in[0,T]}\|u(\cdot,s)\|^2_{H^1(\mathbb{R}^N)}.
\end{eqnarray*}
Thus, (\ref{yu-4-9-4}) is true.
By (\ref{yu-4-9-4}) and (\ref{3.44444}), we obtain that
\begin{eqnarray*}
\int_{\mathbb{R}^{N}}|u(x,\tau)|^{2}\mathrm{d}x
&\leq&\displaystyle{}\hat{C}^{2}_{9}\displaystyle{}\left(T^{2}
e^{\frac{\hat{C}_{1}T}{T-\tau}}+e^{\frac{\hat{C}_{9}}{\tau}}\right)^{2}
\biggl(\varepsilon\theta \hat{C}_{10}(1+T)\sup_{s\in[0,T]}\|u(\cdot,s)\|^2_{H^1(\mathbb{R}^{N})}
\\
&&\;\;\;\;\;\;\;\;\;+\varepsilon^{-\frac{\theta}{1-\theta}}(1-\theta)
\int_{\widetilde{\omega}}|u(x,\tau)|^{2}\mathrm{d}x\biggl)\\
\end{eqnarray*}
for each $ \varepsilon>0.$
This implies that
\begin{equation*}\label{3.444440}
\begin{array}{lll}
\displaystyle{}\int_{\mathbb{R}^{N}}|u(x,\tau)|^{2}\mathrm{d}x
&\leq&\displaystyle{}\hat{C}^{2}_{9}\displaystyle{} \left(T^{2}e^{\frac{\hat{C}_{1}T}{T-\tau}}+e^{\frac{\hat{C}_{9}}{\tau}}\right)^{2}
\left(\hat{C}_{10}(1+T)\sup_{s\in[0,T]}\|u(\cdot,s)\|^2_{H^1(\mathbb{R}^{N})}\right)^\theta\\
&&\times\left(\int_{\widetilde{\omega}}
|u(x,\tau)|^{2}\mathrm{d}x\right)^{1-\theta}.
\end{array}
\end{equation*}
By Lemma~\ref{lemma-2A2} (where $\tau$ and $T$ are replaced by $T$ and $2T$, respectively), we finish the proof of Lemma \ref{yu-LEMMA-7-10-6}.
\end{proof}
\vskip 5pt
\noindent\textbf{Proof of Theorem \ref{yu-theorem-7-10-6}.}
First, by standard energy estimates of solutions to \eqref{yu-6-24-1} we have
\begin{equation}\label{yu-7-12-2}
\|u(\cdot,t)\|_{H^1(\mathbb R^N)}\leq \frac{C_{23}e^{C_{23}t}}{\sqrt{t}}\|u_0\|_{L^2(\mathbb R^N)}
\end{equation}
with $C_{23}=C_{23}(N,\lambda)>0$, for each $t\in(0,6T]$. Moreover, if $u_0\in H^1(\mathbb R^N)$, then
\begin{equation}\label{yu-7-12-3}
\|u(\cdot,t)\|_{H^1(\mathbb R^N)}\leq C_{24}e^{C_{24}t}\|u_0\|_{H^1(\mathbb R^N)}\;\;\mbox{for each}\;\;
t\in[0,6T],
\end{equation}
where $C_{24}=C_{24}(N,\lambda)>0$.
Second, we consider the following equation
\begin{equation*}\label{yu-7-12-12}
\begin{cases}
v_t-\mbox{div}(A(x)\nabla v)=0&\mbox{in}\;\;\mathbb R^N\times(0,4T),\\
v(\cdot,0)=u(\cdot,\frac{T}{2})&\mbox{in}\;\;\mathbb R^N.
\end{cases}
\end{equation*}
It is obvious that $v(\cdot,t)=u(\cdot,t+T/2)$ when $t\in[0,4T]$. Moreover, by (\ref{yu-7-12-2}) we have
$u(\cdot,T/2)\in H^1(\mathbb R^N)$,
which means that $v\in C([0,4T];H^1(\mathbb R^N))$.
From Lemma \ref{yu-LEMMA-7-10-6} (where $T$ and $\tilde{\omega}$ are replaced by $T/2$ and $\omega$, respectively), it follows that there are $C>0$ and $\sigma\in(0,1)$ such that
\begin{equation*}\label{yu-7-13-1}
\left\|v\left(\cdot,\frac{T}{2}\right)\right\|_{L^2(\mathbb R^N)}\leq C\left(T^{3}+e^{\frac{C}{T}}\right) \left\|v\left(\cdot,\frac{T}{2}\right)\right\|^\sigma_{L^2(\omega)}\left(\sup_{s\in[0,T]}\|v(\cdot,s)\|_{H^1(\mathbb R^N)}\right)^{1-\sigma}.
\end{equation*}
This, along with (\ref{yu-7-12-3}), gives that
\begin{equation*}\label{yu-7-13-2}
\left\|v\left(\cdot,\frac{T}{2}\right)\right\|_{L^2(\mathbb R^N)}\leq C C_{24}e^{C_{24}T}\left(T^{3}+e^{\frac{C}{T}}\right) \left\|v\left(\cdot,\frac{T}{2}\right)\right\|^\sigma_{L^2(\omega)}\|v(\cdot,0)\|^{1-\sigma}_{H^1(\mathbb R^N)}. \end{equation*}
Which is
\begin{equation*}\label{yu-7-13-3}
\left\|u\left(\cdot,T\right)\right\|_{L^2(\mathbb R^N)}\leq C C_{24}e^{C_{24}T}\left(T^{3}+e^{\frac{C}{T}}\right) \left\|u\left(\cdot,T\right)\right\|^\sigma_{L^2(\omega)}\left\|u\left(\cdot,\frac{T}{2}\right)\right\|^{1-\sigma}_{H^1(\mathbb R^N)}.
\end{equation*}
This, together with (\ref{yu-7-12-2}), completes the proof.\qed
\section{Proof of Theorem \ref{jiudu4}}\label{finalproof}
Now, we are able to present the proof of Theorem \ref{jiudu4}.\\
\noindent\textbf{Proof of Theorem \ref{jiudu4}.}
By Theorem \ref{yu-theorem-7-10-6} (where $r, x_{i}$ and $w_{i}$ are replaced by
$r, x_{i}$ and $w_{i}$, respectively) and Young's inequality,
for any $0\leq t_{1}<t_{2}\leq T$, we see that
\begin{equation}\label{2019-7-9}
\|u(t_{2})\|^{2}_{L^{2}(\mathbb{R}^{N})}\leq\varepsilon
\|u(t_{1})\|^{2}_{L^{2}(\mathbb{R}^{N})}+
\frac{C_{25}}{\varepsilon^{\alpha}}e^{\frac{C_{26}}{t_{2}-t_{1}}}
\|u(t_{2})\|^{2}_{L^{2}(\omega)} \ \ \ \mathrm{for\ each}\ \varepsilon>0,
\end{equation}
where $C_{25}:=\left(Ce^{CT}\right)^{2/(1-\sigma)}$,
$C_{26}:=2C/(1-\sigma)$
and $\alpha:= \sigma/(1-\sigma)$.
Let $l$ be a density point of $E$. According to Proposition 2.1 in \cite{Phung-Wang-2013},
for each $\kappa>1$, there exists $l_{1}\in (l,T)$, depending on $\kappa$ and $E$,
so that the sequence $\{l_{m}\}_{m\in\mathbb{N}^+}$, given by
$$
l_{m+1}=l+\frac{1}{\kappa^{m}}(l_{1}-l)\;\;\mbox{for each}\;\;m\in\mathbb{N}^+,
$$
satisfies that
\begin{equation}\label{3.2525251}
l_{m}-l_{m+1}\leq 3|E\cap(l_{m+1},l_{m})|.
\end{equation}
\par
Next, let $0<l_{m+2}<l_{m+1}\leq t<l_{m}<l_{1}<T$. It follows from (\ref{2019-7-9}) that
\begin{equation}\label{3.2525252}
\|u(t)\|^{2}_{L^{2}(\mathbb{R}^{N})}\leq \varepsilon\|u(l_{m+2})\|^{2}_{L^{2}(\mathbb{R}^{N})}
+\frac{C_{25}}{\varepsilon^{\alpha}}
e^{\frac{C_{26}}{t-l_{m+2}}}\|u(t)\|^{2}_{L^{2}(\omega)} \ \mathrm{for\ each}\ \varepsilon>0.
\end{equation}
By a standard energy estimate, we have that
$$
\|u(l_{m})\|_{L^{2}(\mathbb{R}^{N})}\leq C_{27}\|u(t)\|_{L^{2}(\mathbb{R}^{N})},
$$
where $C_{27}=C_{27}(\lambda,N)\geq1.$
This, along with (\ref{3.2525252}), implies that
$$
\|u(l_{m})\|^{2}_{L^{2}(\mathbb{R}^{N})}\leq C^{2}_{27}\left(\varepsilon\|u(l_{m+2})
\|^{2}_{L^{2}(\mathbb{R}^{N})}+
\frac{C_{25}}{\varepsilon^{\alpha}}
e^{\frac{C_{26}}{t-l_{m+2}}}\|u(t)\|^{2}_{L^{2}(\omega)}\right)
\ \mathrm{for\ each}\ \varepsilon>0,
$$
which indicates that
$$\|u(l_{m})\|^{2}_{L^{2}(\mathbb{R}^{N})}
\leq \varepsilon\|u(l_{m+2})\|^{2}_{L^{2}(\mathbb{R}^{N})}
+\frac{C_{28}}{\varepsilon^{\alpha}}
e^{\frac{C_{26}}{t-l_{m+2}}}\|u(t)\|^{2}_{L^{2}(\omega)}
\ \mathrm{for\ each}\ \varepsilon>0,
$$
where $C_{28}=C_{27}^{2(1+\alpha)}C_{25}$.
Integrating the latter inequality over $ E\cap(l_{m+1},l_{m})$, we get that
\begin{equation}\label{3.2525253}
\begin{array}{lll}
\displaystyle{}|E\cap(l_{m+1},l_{m})|\|u(l_{m})\|^{2}_{L^{2}(\mathbb{R}^{N})}
&\leq&\displaystyle{}\varepsilon |E\cap(l_{m+1},l_{m})|\|u(l_{m+2})\|^{2}_{L^{2}(\mathbb{R}^{N})}\\
&&\displaystyle{}+\frac{C_{28}}
{\varepsilon^{\alpha}}e^{\frac{C_{26}}{l_{m+1}-l_{m+2}}}
\int_{l_{m+1}}^{l_{m}}\chi_{E}\|u(t)\|^{2}_{L^{2}(\omega)}\mathrm{d}t
\ \mathrm{for\ each}\ \varepsilon>0.
\end{array}
\end{equation}
Here and throughout the proof of Theorem~\ref{jiudu4}, $\chi_{E}$ denotes the characteristic function of $E$. Since $l_{m}-l_{m+1}=(\kappa-1)(l_{1}-l)/\kappa^{m},$ by (\ref{3.2525253}) and (\ref{3.2525251}), we obtain that
\begin{eqnarray*}
\|u(l_{m})\|^{2}_{L^{2}(\mathbb{R}^{N})}&\leq& \varepsilon \|u(l_{m+2})\|^{2}_{L^{2}(\mathbb{R}^{N})}+\frac{1}{|E\cap(l_{m+1},l_{m})|}
\frac{C_{28}}{\varepsilon^{\alpha}}
e^{\frac{C_{26}}{l_{m+1}-l_{m+2}}}
\int_{l_{m+1}}^{l_{m}}\chi_{E}\|u(t)\|^{2}_{L^{2}(\omega)}\mathrm{d}t\\
&\leq&\frac{3\kappa^{m}}{(l_{1}-l)(\kappa-1)}
\frac{C_{28}}{\varepsilon^{\alpha}}
e^{C_{26}\left(\frac{1}{l_{1}-l}\frac{\kappa^{m+1}}{\kappa-1}\right)}
\int_{l_{m+1}}^{l_{m}}\chi_{E}\|u(t)\|^{2}_{L^{2}(\omega)}\mathrm{d}t+
\varepsilon \|u(l_{m+2})\|^{2}_{L^{2}(\mathbb{R}^{N})}
\end{eqnarray*}
for each $\varepsilon>0$.
This yields that
\begin{equation}\label{3.2525254}
\begin{array}{lll}
\displaystyle{}\|u(l_{m})\|^{2}_{L^{2}(\mathbb{R}^{N})}&\leq& \displaystyle{}
\frac{1}{\varepsilon^{\alpha}}\frac{3}{\kappa}
\frac{C_{28}}{C_{26}}
e^{2C_{26}\left(\frac{1}{l_{1}-l}
\frac{\kappa^{m+1}}{\kappa-1}\right)}\int_{l_{m+1}}^{l_{m}}\chi_{E}
\|u(t)\|^{2}_{L^{2}(\omega)}\mathrm{d}t\displaystyle{}+
\varepsilon \|u(l_{m+2})\|^{2}_{L^{2}(\mathbb{R}^{N})}
\end{array}
\end{equation}
for each $\varepsilon>0$.
Denote $d:= 2C_{26}/[\kappa(l_{1}-l)(\kappa-1)]$.
It follows from (\ref{3.2525254}) that
\begin{eqnarray*}
\varepsilon^{\alpha}e^{-d\kappa^{m+2}}\|u(l_{m})\|^{2}_{L^{2}(\mathbb{R}^{N})}
-\varepsilon^{1+\alpha}e^{-d\kappa^{m+2}}\|u(l_{m+2})\|^{2}_{L^{2}(\mathbb{R}^{N})}
\leq\frac{3}{\kappa}\frac{C_{28}}
{C_{26}}\int_{l_{m+1}}^{l_{m}}\chi_{E}\|u(t)\|^{2}_{L^{2}(\omega)}\mathrm{d}t
\end{eqnarray*}
for each $\varepsilon>0$.
Choosing $\varepsilon=e^{-d\kappa^{m+2}}$ in the latter inequality, we observe that
\begin{equation}\label{3.25252555}
\begin{array}{lll}
\displaystyle{}e^{-(1+\alpha)d\kappa^{m+2}}\|u(l_{m})\|^{2}_{L^{2}(\mathbb{R}^{N})}
-e^{-(2+\alpha)d\kappa^{m+2}}\|u(l_{m+2})\|^{2}_{L^{2}(\mathbb{R}^{N})}
\leq\displaystyle{}\frac{3}{\kappa}\frac{C_{28}}
{C_{26}}\int_{l_{m+1}}^{l_{m}}\chi_{E}\|u(t)\|^{2}_{L^{2}(\omega)}\mathrm{d}t.
\end{array}
\end{equation}
Take $\kappa=\sqrt{(\alpha+2)/(\alpha+1)}$ in (\ref{3.25252555}). Then we have that
\begin{eqnarray*}
e^{-(2+\alpha)d\kappa^{m}}\|u(l_{m})\|^{2}_{L^{2}(\mathbb{R}^{N})}
-e^{-(2+\alpha)d\kappa^{m+2}}\|u(l_{m+2})\|^{2}_{L^{2}(\mathbb{R}^{N})}
\leq \frac{3}{\kappa}\frac{C_{28}}{C_{26}}
\int_{l_{m+1}}^{l_{m}}\chi_{E}\|u(t)\|^{2}_{L^{2}(\omega)}\mathrm{d}t.
\end{eqnarray*}
Changing $m$ to $2m'$ and summing the above inequality from $m'=1$ to infinity give the desired result. Indeed,
\begin{eqnarray*}
&&C_{27}^{-2}e^{-(2+\alpha)d\kappa^{2}}\|u(T)\|^{2}_{L^{2}(\mathbb{R}^{N})}
\leq e^{-(2+\alpha)d\kappa^{2}}\|u(l_{2})\|^{2}_{L^{2}(\mathbb{R}^{N})}\\
&\leq&\sum_{m'=1}^{+\infty}\left(e^{-(2+\alpha)d\kappa^{2m'}}\|u(l_{2m'})\|_{L^{2}(\mathbb{R}^{N})}
-e^{-(2+\alpha)d\kappa^{2m'+2}}\|u(l_{2m'+2})\|^{2}_{L^{2}(\mathbb{R}^{N})}\right)\\
&\leq& \frac{3}{\kappa}\frac{C_{28}}{C_{26}}\sum_{m'=1}^{+\infty}
\int_{l_{2m'+1}}^{l_{2m'}}\chi_{E}\|u(t)\|^{2}_{L^{2}(\omega)}\mathrm{d}t
\leq \frac{3}{\kappa}\frac{C_{28}}{C_{26}}\int_{0}^{T}\chi_{E}\|u(t)\|^{2}_{L^{2}(\omega)}\mathrm{d}t.
\end{eqnarray*}
In summary, we finish the proof of Theorem~\ref{jiudu4}.\qed
\section{Appendix: Proof of (\ref{yu-6-22-4})}
Let $m\in\mathbb{N}^+$ and $a_j=1-j/(2m)$ for $j=0,1,\dots,m+1$. For each $j\in\{0,1,\cdots, m\}$, we define a cutoff function
\begin{equation*}\label{yu-6-19-4}
h_j(s):=
\begin{cases}
0&\mbox{if}\;\;|s|>a_j,\\
\frac{1}{2}\left[1+\cos\left(\frac{\pi(a_{j+1}-|s|)}{a_{j+1}-a_j}\right)\right]
&\mbox{if}\;\;a_{j+1}\leq |s|\leq a_j,\\
1&\mbox{if}\;\;|s|<a_{j+1}.
\end{cases}
\end{equation*}
Clearly,
\begin{equation*}\label{yu-6-20-2}
|h_j'(s)|\leq m\pi \;\;\mbox{for any}\;\;s\in\mathbb{R}.
\end{equation*}
Denote
$p_j=\partial^jp/\partial\xi^j$, $j=0,1,\dots,m$. Then $p_j$ verifies
\begin{equation}\label{yu-6-19-6}
\mbox{div}(A\nabla p_j(\cdot,\cdot,\mu))+i\mbox{sign}(\mu)p_{j+2}(\cdot,\cdot,\mu)
=0\;\;\mbox{in}\;\;B_R(x_0)\times\mathbb{R}.
\end{equation}
Let
\begin{equation*}\label{yu-6-19-7}
\eta_j(x,\xi)=h_j\left(\frac{|x-x_0|}{R}\right)h_j\left(\frac{\xi}{R}\right)\quad\text{for}\;\;(x,\xi)\in B_R(x_0)\times\mathbb{R}.
\end{equation*}
Multiplying first (\ref{yu-6-19-6}) by
$\bar{p}_j\eta_j^2$ and then integrating by parts over $D_j=B_{a_jR}(x_0)\times(-a_jR,a_jR)$, we obtain
\begin{eqnarray}\label{yu-6-19-8}
&\;&-\int_{D_j}\nabla\bar{p}_j\cdot(A\nabla p_j)\eta_j^2dxd\xi
-i\mbox{sign}(\mu)\int_{D_j}|p_{j+1}|^2\eta_j^2dxd\xi\nonumber\\
&=&\int_{D_j}\nabla\eta_j^2\cdot(A\nabla p_i)\bar{p}_jdxd\xi+i\mbox{sign}(\mu)\int_{D_j}p_{i+1}\partial_\xi\eta^2_j\bar{p}_jdxd\xi.
\end{eqnarray}
Since $\nabla\bar{p}_j\cdot(A\nabla p_j)$ and $|p_{j+1}|^2$ are real-valued,
by (\ref{yu-6-19-8}) we get
\begin{eqnarray}\label{yu-6-19-9}
&\;&\left(\int_{D_j}\nabla\bar{p}_j\cdot(A\nabla p_j)\eta_j^2dxd\xi\right)^2
+|\mbox{sign}(\mu)|^2\left(\int_{D_j}|p_{j+1}|^2\eta_j^2dxd\xi\right)^2\nonumber\\
&\leq&2\left(\int_{D_j}|\nabla \eta_j^2\cdot(A\nabla p_j)||p_j|dxd\xi\right)^2
+2\left(\int_{D_j}|p_{j+1}||p_j||\partial_{\xi}\eta^2_j|dxd\xi\right)^2
:=2\sum_{i=1}^2I_i.
\end{eqnarray}
\par
Next, we will estimate $I_i$ $(i=1,2)$ one by one. By Young's inequality, we have
\begin{eqnarray*}\label{yu-6-21-1}
&\;&\int_{D_j}|\nabla \eta_j^2\cdot(A\nabla p_j)||p_j|dxd\xi
\leq 2\lambda\int_{D_j}|\nabla \eta_j||\eta_j||\nabla p_j||p_j|dxd\xi\nonumber\\
&\leq&\epsilon_1\lambda\int_{D_j}|\nabla p_j|^2\eta_j^2dxd\xi
+\frac{\lambda}{\epsilon_1}\int_{D_j}|p_j|^2|\nabla\eta_j|^2dxd\xi\nonumber\\
&\leq&\epsilon_1\lambda\int_{D_j}|\nabla p_j|^2\eta_j^2dxd\xi
+\frac{\lambda\pi^2m^2}{R^2\epsilon_1}\int_{D_j}|p_j|^2dxd\xi.
\end{eqnarray*}
Thus, we get that
\begin{equation}\label{yu-6-21-2}
I_1\leq 2\lambda^2\epsilon_1^2\left(\int_{D_j}|\nabla p_j|^2\eta_j^2dxd\xi\right)^2
+\frac{2\lambda^2\pi^4m^4}{R^4\epsilon^2_1}\left(\int_{D_j}|p_j|^2dxd\xi\right)^2, \;\;\forall\ \epsilon_1>0.
\end{equation}
Furthermore,
\begin{eqnarray*}\label{yu-6-21-3}
\int_{D_j}|p_{j+1}|p_j||\partial_{\xi}\eta^2_j|dxd\xi
&\leq&\epsilon_2\int_{D_j}|p_{j+1}|^2\eta_j^2dxd\xi
+\frac{1}{\epsilon_2}\int_{D_j}|p_j|^2|\partial_\xi\eta_j|^2dxd\xi\nonumber\\
&\leq&\epsilon_2\int_{D_j}|p_{j+1}|^2\eta_j^2dxd\xi
+\frac{m^2\pi^2}{R^2\epsilon_2}\int_{D_j}|p_j|^2dxd\xi.
\end{eqnarray*}
Thus, we have that
\begin{equation}\label{yu-6-21-4}
I_2\leq 2\epsilon_2^2\left(\int_{D_j}|p_{j+1}|^2\eta_j^2dxd\xi\right)^2
+\frac{2\pi^4m^4}{R^4\epsilon_2^2}\left(\int_{D_j}|p_j|^2dxd\xi\right)^2, \;\;\forall \ \epsilon_2>0.
\end{equation}
Taking $\epsilon_1=\sqrt{2}/(4\lambda^2)$, $\epsilon_2=1/4$
in (\ref{yu-6-21-2})
and (\ref{yu-6-21-4}), respectively,
we derive that
\begin{eqnarray}\label{yu-6-21-8}
\sum_{i=1}^2I_i&\leq&\frac{\lambda^{-2}}{4}\left(\int_{D_j}|\nabla p_j|^2\eta_j^2dxd\xi
\right)^2+\frac{1}{8}\left(\int_{D_j}|p_{j+1}|^2\eta_j^2dxd\xi\right)^2\nonumber\\
&\;&+\frac{(2+\lambda^{6})4^{2}\pi^4m^{4}}{R^{4}}
\left(\int_{D_j}|p_j|^2dxd\xi\right)^2.
\end{eqnarray}
On the other hand, by the uniform ellipticity condition (\ref{yu-11-28-2}), we find that
\begin{equation*}\label{yu-6-22-2}
\left(\int_{D_j}\nabla \bar{p}_j\cdot(A\nabla p_j)\eta_j^2dxd\xi\right)^2
\geq \lambda^{-2}\left(\int_{D_j}|\nabla p_j|^2\eta_j^2dxd\xi\right)^2.
\end{equation*}
This, together with (\ref{yu-6-19-9}) and (\ref{yu-6-21-8}), gives that for each $j\in\{0,1,\ldots,m-1\}$,
\begin{eqnarray}\label{yu-6-22-3}
\int_{D_{j+1}}|p_{j+1}|^2dxd\xi&=&\int_{D_{j+1}}|p_{j+1}|^2\eta^2_jdxd\xi
\leq \sqrt{\frac{4^{3}(2+\lambda^{6})\pi^4m^{4}}{3R^{4}}}
\int_{D_j}|p_j|^2dxd\xi\nonumber\\
&\leq&\frac{C_{4}m^2}{R^2}\int_{D_j}|p_j|^2dxd\xi,
\end{eqnarray}
where $C_{4}=8\pi^2\sqrt{2+\lambda^{6}}$.
Here, we used the definition of $D_j$. Iterating (\ref{yu-6-22-3}) for each
$j\in\{0,1,\ldots,m-1\}$, by the fact that $p_0=p=\hat{v}$ we obtain \eqref{yu-6-22-4}
and complete the proof.
\qed
\bigskip
\noindent \textbf{Acknowledgement}. The authors thank the financial support by
the National Natural Science Foundation of China under grants 11971363.
|
1,116,691,497,547 | arxiv |
\section{Introduction}
Speaking rate control is a speech processing task of changing the speaking rate of a given speech~\cite{src_4,src_0,src_1,src_2,mpm}.
This technology has been widely applied in many real-world applications.
Since speaking rate changes speech intelligibility and accessibility, users may need different speaking rates depending on the situation.
For instance, a high speaking rate may be needed by visually-impaired people~\cite{blind,bragg2018large} and students who take online courses~\cite{dodson2021will} since they are used to listening to speech with a higher speed.
Instead, people who are in an environment with high reverberation~\cite{nabvelek1989reverberant} or people who have aphasia~\cite{hux2020effect} and elderly adults~\cite{sr,rudner2011working} may need a low speaking rate to understand the content of speech.
The most prevailing method for speaking rate control is time-scale modification (TSM) algorithms \cite{driedger2016review, muller2015fundamentals} based on waveform warping.\footnote{Although speaking rate control can also be realized by source filter vocoders~\cite{straight,world} and time-domain pitch synchronous
overlap-add (TDPSOLA)~\cite{tdpsola,sinu,ahm}, the
synthesis quality of these methods is highly dependent on the
accuracy of fundamental frequency analysis~\cite{ndf,harvest} and pitch marking~\cite{pitch_marking}.}
Basically, such an algorithm first decomposes a waveform into several frames, then warps each frame to the target rate and finally reconstructs the waveform.
However, although TSM algorithms have acceptable quality and efficiency, their performance is still not satisfactory.
Moreover, recent advances in applying deep neural networks (DNNs) for speech synthesis like neural vocoder~\cite{wn_vocoder,wavernn,waveglow,valin2019lpcnet,kumar2019melgan,pwg,kong2020hifi} demonstrate superior performance in synthesizing speech with high fidelity and efficiency, but since TSM algorithms modify time-domain waveforms directly, it is difficult to combine a DNN-based speech synthesis model with a TSM algorithm.
Therefore, a powerful and efficient speaking rate control method that can be seamlessly implemented in DNN-based speech synthesis models becomes necessary.
A DNN-based speaking rate control method with multi-speaker WaveNet vocoder~\cite{si_wn_vocoder} had been initially provided and it outperformed the conventional TSM-based method and source-filter vocoder~\cite{okamoto2022spcom}.
However, the inference speed of the method was quite slow due to the auto-regressive structure and the huge size of the WaveNet model~\cite{wn_vocoder}.
Most recently, Morrison {\it et al.} \cite{morrison2021neural} proposed a speaking rate control method using LPCNet for real-time inference~\cite{valin2019lpcnet}.
However, their method was mainly designed for pitch shifting, and the neural vocoder used in their work is different from the one used in this work.
Another issue in the evaluations of previous work is a lack of ground-truth speaking-rate-controlled speech.
The goal of the control is to synthesize speech as if the speaker has uttered the speech with the specified speaking rate.
However, since past studies using existing corpora~\cite{okamoto2022spcom, morrison2021neural} always compared speaking-rate-controlled speech with original speech, we cannot state how far those control methods are from the goal.
\begin{figure}[t]
\centering
\includegraphics[width=0.98\linewidth]{figures_takamichi/diagram.pdf}
\caption{Diagram of our idea. An interpolation layer is inserted into HiFi-GAN to warp the length of mel-spectrograms or hidden features and control the speaking rate. In this work, we use a TSM algorithm WSOLA as the baseline method.}
\label{fig:proposed}
\vspace{-4mm}
\end{figure}
In this paper, we investigate the possibility of using a neural vocoder to control speaking rate.
We use a recent neural vocoder HiFi-GAN \cite{kong2020hifi} based on generative-adversarial-network (GAN) as vocoder, which can synthesize high-fidelity speech with $0.01$ of the real-time factor.
Our idea is illustrated in Figure~\ref{fig:proposed}.
The proposed method interpolates mel-spectrograms or hidden features in the possible inner layers of HiFi-GAN to control the speaking rate without fundamental frequency analysis as~\cite{okamoto2022spcom}.
When the interpolation layer is inserted before the generated waveform, the proposed method degrades to a TSM algorithm, which is used as the baseline method of this work.
We consider both a bandlimited signal resampling and an image scaling interpolation methods in the experiments.
To evaluate the proposed method, we built a corpus with three kinds of speaking rates of unique speakers and conducted experiments on it.
Results of comprehensive objective and subjective evaluations demonstrated that the proposed method can synthesize speech with higher fidelity and efficiency than the baseline TSM algorithm.
The key contributions of this work are as follows:
\begin{enumerate} \leftskip -5mm \itemsep -0mm
\item We propose a speaking rate control method by a neural vocoder with high fidelity and efficiency. The implementation is open-sourced in our website\footnote{\url{https://github.com/Aria-K-Alethia/speaking-rate-controllable-hifi-gan}}.
\item We design a corpus for speaking rate control and give insights through experiments upon this corpus. The corpus is also open-sourced in our website\footnote{\url{https://ast-astrec.nict.go.jp/en/release/speedspeech_ja_2022/download.html}}.
\end{enumerate}
\section{Proposed Method and Corpus}
\subsection{Method}
\subsubsection{Overview}
We use HiFi-GAN \cite{kong2020hifi} as the vocoder. Compared with the existing models~\cite{wn_vocoder,wavernn,waveglow,valin2019lpcnet,kumar2019melgan,pwg},
HiFi-GAN utilizes adversarial training to discriminate synthetic and genuine speech in parallel, thus can synthesize speech with high fidelity and efficiency.
As Figure~\ref{fig:proposed} illustrates, the basic idea of the proposed method is to use an interpolation layer to change data length so that the speaking rate of the synthesized speech will be different from the original mel-spectrogram\footnote{
Yet another approach is to change upsampling intervals~\cite{morrison2021neural} and can be implemented by changing the stride width of the transposed convolution layers of HiFi-GAN. A preliminary experiment confirmed that the proposed method was better than this method.
}.
Since the generator of HiFi-GAN has several similar blocks to upsample the data, it is natural and intuitive to insert an interpolation layer between these blocks.
We use the waveform similarity overlapping-add (WSOLA) algorithm \cite{verhelst1993overlap} as the baseline method.
WSOLA can change the speaking rate of the speech while maintaining the periodic patterns of the signal by finding a frame that has the maximal similarity to the current frame.
Benefiting from this property, WSOLA is suitable for modifying the time-scale of human speech.
\begin{figure}[t]
\centering
\includegraphics[width=0.65\linewidth]{figures_takamichi/interpolation.pdf}
\caption{Examples of bandlimited resampling and linear interpolation. Bandlimited resampling is usually used to interpolate one-dimensional waveforms, while linear interpolation is designed to interpolate two-dimensional data like mel-spectrograms.}
\label{fig:interpolation}
\vspace{-5mm}
\end{figure}
\subsubsection{Interpolation Methods}
The interpolation layer should change the feature length while maintaining the semantic information within it.
In this work we consider two interpolation methods: bandlimited resampling based on the kaiser window~\cite{eldar2015sampling} and linear interpolation for image scaling.
Examples of these two methods are illustrated in Figure~\ref{fig:interpolation}.
Bandlimited interpolation is a commonly used resampling method in digital signal processing.
Since the outputs of the hidden layers of HiFi-GAN can be regarded as hidden representations of waveforms, we consider that it is reasonable to use this method to interpolate them.
Besides, it is also intuitive to interpret hidden features including mel-spectrograms as images that contain two axes time and frequency.
Therefore, we also use linear interpolation, a geometric interpolation method.
In the experiments we compare these two methods.
\subsection{Corpus}
To enhance the evaluation approaches of this task, we further design a speech corpus for speaking rate control.
The proposed corpus includes parallel texts spoken by a male and a female speaker with three kinds of speaking rates (slow, normal, fast).
The text is balanced in phonemes.
The speakers are professional so that all utterances of each speaking rate are intelligible.
With this corpus, we can compare speaking-rate-controlled speech with ground-truth, e.g., comparing speech changed from ``slow'' to ``normal'' with actual ``normal'' speech.
\section{Experiments}
\subsection{Corpus Specification}
We hired a male and a female Japanese professional speaker to construct our corpus.
We used texts of the ITA corpus\footnote{\url{https://github.com/mmorise/ita-corpus}}, which is an open-sourced phoneme-balanced Japanese corpus.
Each speaker was required to speak $325$ utterances at three different rates: slow, normal, and fast, so each speaker has $975$ utterances.
We name this dataset \textit{SpeedSpeech-JA-2022}. Table~\ref{tab:speaking_rate} lists the exact speaking rate measured by mora per second.
This was computed by dividing the mora number by the voiced length of each utterance~\cite{mpm_2,mpm}.
The voiced frames were detected by using voice activity detection\footnote{\url{https://github.com/wiseman/py-webrtcvad}}.
It can be seen that the speaking rate of the fast speech is almost 2-time faster than the one of the slow speech.
Also, the male speaker speaks faster than the female speaker.
\input{tables/speaking_rate}
\subsection{Experimental Setup}
We randomly picked up $45$ utterances for each speaker in which each rate has $15$ utterances.
The texts of the $15$ utterances of each speaker between different rates were kept to be the same so that we could measure the mel-cepstral distortion in the later experiments.
For all speech of each speaking rate, we converted them to the rest two rates and regarded the speech of the target rate as the ground-truth data.
Since every utterance has a different speaking rate, we computed the conversion factor for each utterance separately.
Denoting the durations of the source and the target utterances with the same text as $t_{\mathrm{src}}$ and $t_{\mathrm{tgt}}$, the conversion factor $f$ is defined as ${t_{\mathrm{src}}}/{t_{\mathrm{tgt}}}$.
In the subjective evaluations, we also converted the speech with standard factors $\{0.25, 0.50, 0.75, 1.25, 1.50, 1.75, 2.00\}$ that are widely used in real-world applications.
We used a pretrained universal HiFi-GAN model\footnote{\url{https://github.com/jik876/hifi-gan}} to convert mel-spectrograms into time-domain waveforms.
This model was trained on multilingual datasets so it can synthesize Japanese without further training.
Since the generator of the pretrained HiFi-GAN has four upsampling layers, it is possible to insert the interpolation layer into $5$ places (mel-spectrogram and layer $1, 2, 3, 4$).
All combinations of interpolation methods and places were investigated.
As a result, there were $11$ methods including $10$ proposed methods ($5$ places and $2$ interpolation methods) and $1$ baseline method in the experiments.
We used the bandlimited resampling class in torchaudio\footnote{\url{https://pytorch.org/audio/stable/transforms.html#torchaudio.transforms.Resample}} and the linear interpolation function of pytorch\footnote{\url{https://pytorch.org/docs/stable/generated/torch.nn.functional.interpolate.html}} as the implementations.
For the WSOLA algorithm we used audiotsm\footnote{\url{https://github.com/Muges/audiotsm}} package as the implementation.
In addition, we also evaluated the proposed method with a speech synthesis model.
We trained text-to-speech (TTS) models FastSpeech2~\cite{ren2020fastspeech} followed by the aforementioned universal HiFi-GAN model.
We followed the model architecture and hyperparameters of the open-sourced implementation\footnote{\url{https://github.com/ming024/FastSpeech2}}.
We trained a FastSpeech2 model using the JSUT corpus~\cite{jsut-jvs} and fine-tuned it using the SpeedSpeech-JA-2022 corpus.
The fine-tuning was performed with $10$k steps. The pretrained model was fine-tuned on the utterances of different speakers and speaking rates separately, i.e., there were six FastSpeech2 models in total.
The test data size was the same as above, and the remaining data were used for fine-tuning.
\subsection{Objective Evaluations}
\subsubsection{Mel-cepstral Distortions}
We first measured mel-cepstral distortions (MCDs) between the converted and ground truth speech to evaluate each method.
Utterance pairs were aligned by dynamic time warping.
The result is shown in Table~\ref{tab:mcd}.
The mel-spectrogram linear interpolation method (``Linear" and ``mel") obtained the best performance.
Besides, it can be observed that interpolating mel-spectrograms is better than interpolating hidden features of inner layers.
The baseline method (WSOLA) also obtained relatively good performance but is still worse than the two mel-spectrogram interpolation methods.
Surprisingly, all the methods of interpolating hidden features of inner layers have poor performance.
Here we cannot list all results due to the page limit, but the MCDs degraded as the interpolation layer was placed close to the output layer.
After a preliminary analysis, we found that the fundamental frequency (F0) of the utterances converted by these methods were destructed, though the semantic information was well preserved.
We assume this is because the convolutional neural network architecture of HiFi-GAN makes it depend on the hidden feature length to work, thus changing the feature size will influence the behavior without further training.
\input{tables/mcd}
\subsubsection{Real-Time Factors}
To evaluate the efficiency of each method, we then computed real-time factors (RTFs) for each method using an NVIDIA GeForce RTX 2080 Ti GPU card.\footnote{HiFi-GAN can also realize real-time inference on CPUs~\cite{kong2020hifi,asru2021okamoto}.}
The computation time is defined as the summation of the generation time of HiFi-GAN and the speaking rate conversion time of each method.
For simplicity in this evaluation we only used standard factors to convert all slow, normal, and fast utterances but did not convert between them.
The result is shown in Figure~\ref{fig:rtf}.
It can be seen that all the proposed methods have better efficiency than the baseline WSOLA method.
The mel-spectrogram interpolation methods which obtained the best performance in the MCDs evaluation have the best performance when the conversion factor is greater than $1.0$ but obtained the worst performance among the proposed methods when the conversion factor is less than $1.0$.
This is because the four upsampling blocks in HiFi-GAN will magnify the data length changing effect, therefore the shortened or elongated data will become shorter or longer after being processed by the blocks, and further increase or reduce the computation time.
\begin{figure}[t]
\centering
\includegraphics[width=0.98\linewidth]{figures_takamichi/rtf.pdf}
\caption{Results of efficiency evaluation. ``HiFi-GAN'' represents the efficiency of the model itself without changing the speaking rate.}
\label{fig:rtf}
\vspace{-3mm}
\end{figure}
\subsection{Subjective Evaluations}
\subsubsection{Speech naturalness: fixed source rates}
In the subjective evaluations we evaluated the naturalness of the converted speech by mean opinion score (MOS) tests~\cite{p830}.
On the basis of the results of the objective evaluations, we only selected three methods: WSOLA, mel-spec interpolation using linear interpolation and bandlimited resampling with kaiser window in this test.
In the first MOS test we intended to clarify the influences of the source speaking rates on the MOS.
Totally $1,500$ listeners joined in the test.
Each utterance was evaluated by $30$ listeners.
The result is illustrated in Figure~\ref{fig:mos}.
It can be observed that if the speaking rate is in a common range $[4, 10]$, the two mel-spectrogram interpolation methods are better than WSOLA and the linear interpolation method is slightly better than the kaiser window method, which is consistent with the conclusions we got from the objective evaluations.
This implies that it is easy to slow down the utterances with high speaking rates or speed up the utterances with low speaking rates.
Second, when the speaking rates become extremely slow or fast, the MOS scores become very low and it is hard to distinguish which method is better for the listeners.
We assume that there are two possible reasons for this problem.
One possible reason is that the pretrained HiFi-GAN model was trained on a corpus with normal speaking rates, so it cannot synthesize speech well with slow or fast speaking rates.
Another possible reason is that the listeners are not used to listening to speech with uncommon speaking rates.
But since in the later experiments it can be observed that the listeners rated low scores even for natural speech with uncommon speaking rates, (Section~\ref{subsubsection:comp_with_gt}), we conclude that the latter reason is more appropriate to explain the result.
\begin{figure}[t]
\centering
\includegraphics[width=0.98\linewidth]{figures_takamichi/mos.pdf}
\caption{Results of naturalness MOS evaluation.}
\label{fig:mos}
\vspace{-5mm}
\end{figure}
\subsubsection{Speech naturalness: fixed target rates}
\label{subsubsection:comp_with_gt}
In the second test we instead aimed to know the performance of the proposed method compared to the ground truth under specified speaking rates.
Based on the first evaluation, we only used the proposed method using mel-spectrogram linear interpolation.
We specified three speaking rates (slow, normal, fast) for each speaker.
For each specification, we prepared four types of speech: ground-truth raw speech (i.e., the speaker's natural speech with the specified speaking rate); the analysis-synthesized speech using HiFi-GAN without speaking rate control; and two kinds of rate-controlled speech converted from the rest two speaking rates, e.g. ``From normal" and ``From fast" for slow speech.
We conducted 6 MOS tests on naturalness for all combinations of speakers and speaking rates.
Each test had $35$ listeners; each listener evaluated $20$ samples.
Figure ~\ref{fig:mos_per_speed} shows the result.
It can be seen that when the source and target speaking rates are quite different, e.g. slow to fast, the performance becomes bad.
Also, the listeners tend to rate high scores for utterances with normal speaking rates, which is consistent with the conclusion of the previous evaluation.
In the task of normal-to-slow and slow-to-normal conversion of the female speaker's speech, the performance of the proposed method even surpasses HiFi-GAN, which demonstrates the effectiveness of the proposed method.
However, when the target speaking rate is slow or fast, the performance of the proposed method is not as good as HiFi-GAN.
Finally, we conducted a test using TTS-synthesized speech.
The speaking rate control function was added to the speech synthesis system.
Specifically, mel-spectrograms output from FastSpeech2 were converted into waveforms by HiFi-GAN after being interpolated.
The experimental setup is almost the same as above except that (1) we use TTS-synthesized speech without speaking rate control to replace ``HiFi-GAN" and (2) add WSOLA for comparison.
Each test had $35$ listeners; each listener evaluated $25$ samples.
Figure~\ref{fig:mos_per_speed_tts} shows the result.
First, the proposed method always outperforms the baseline.
The overall tendencies are similar to the result of the previous test.
The normal-to-slow and slow-to-normal conversion of the female speaker's speech are comparable to slow TTS and normal TTS, respectively.
In other cases, the rate-controlled speech is still worse than the speech without rate control.
\begin{figure}[t]
\centering
\includegraphics[width=0.98\linewidth]{figures_takamichi/mos_per_speed.pdf}
\caption{Results of naturalness MOS evaluation using analysis-synthesized speech with fixed target speaking rate. ``HiFi-GAN'' represents speech without rate changing. Scores with ``*'' are not significantly different, i.e., $p > 0.05$.}
\label{fig:mos_per_speed}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.98\linewidth]{figures_takamichi/mos_per_speed_tts.pdf}
\caption{Results of naturalness MOS evaluation using TTS speech with fixed target speaking rate. ``TTS'' represents speech without rate changing. Scores with ``*'' are not significantly different, i.e., $p > 0.05$.}
\label{fig:mos_per_speed_tts}
\vspace{-3mm}
\end{figure}
\section{Conclusions}
This paper described a method for speaking rate control by HiFi-GAN using feature interpolation.
The idea of the proposed method is to insert a feature interpolation layer into the model to change the data length and control the speaking rate.
A Japanese corpus with three speaking rates was designed to evaluate the proposed method.
Results of subjective and objective evaluations demonstrated that the proposed method of mel-spectrogram interpolation using linear interpolation had better efficiency and quality performance than the baseline TSM algorithm WSOLA.
The future work will be testing the generalization ability of the proposed method on other neural vocoders.
|
1,116,691,497,548 | arxiv | \section{Introduction}
In recent years significant progress has been made towards understanding the excitation spectrum of strings moving in five-dimensional anti-de Sitter space-time and, accordingly,
the spectrum of scaling dimensions of composite operators in planar ${\cal N} = 4$ supersymmetric gauge theory. This progress became possible due to the fundamental insight that strings propagating in
AdS space can be described by an integrable model.
In certain aspects, however, the deep origin of this exact solvability has not yet been unraveled, mainly because of
tremendous complexity of the corresponding model. A related question concerns robustness of integrability in the context of the gauge-string correspondence \cite{M},
as well as the relationship between integrability and the amount of global (super)symmetries preserved by the target space-time in which strings propagate.
To shed further light on these important issues,
one may attempt to search for new examples of integrable string backgrounds that can be solved by similar techniques. One such instance, where this program is largely promising to succeed, is to study various deformations of the string target space that preserve the integrability of the two-dimensional quantum field theory on the world sheet. Simultaneously, this should provide interesting new information about integrable string models and their dual gauge theories.
\smallskip
There are two known classes of integrable deformations of the ${\rm AdS}_5\times {\rm S}^5\ $ superstring. The first of these is a class of backgrounds obtained either by orbifolding ${\rm AdS}_5\times {\rm S}^5\ $ by a discrete subgroup of the corresponding isometry group
\cite{Kachru:1998ys,Lawrence:1998ja} or
by applying a sequence of T-duality -- shift -- T-duality transformations (also known as $\gamma$-deformations) to this space, giving a string theory on a TsT-transformed background \cite{LM,F05}.
Eventually all deformations of this class can be conveniently described in terms of the original string theory, where the deformations result into quasi-periodic but still integrable
boundary conditions for the world-sheet fields.
\smallskip
The second class of deformations affects the ${\rm AdS}_5\times {\rm S}^5\ $ model on a much more fundamental level and is related to deformations of the underlying symmetry algebra.
In the light-cone gauge this symmetry algebra constitutes two copies of the centrally extended Lie superalgebra $\alg{psu}(2|2)$ with the same central extension for each copy.
It appears that this centrally extended $\alg{psu}(2|2)$, or more precisely its universal enveloping algebra,
admits a natural deformation $\alg{psu}_q(2|2)$ in the sense of quantum groups \cite{Beisert:2008tw,Beisert:2011wq}. This
algebraic structure is the starting point for the construction of a $\alg{psu}_q(2|2)\oplus \alg{psu}_q(2|2)$-invariant S-matrix, giving
a quantum deformation of the ${\rm AdS}_5\times {\rm S}^5\ $ world-sheet S-matrix \cite{Beisert:2008tw, Ben,deLeeuw:2011jr}. The deformation parameter
$q$ can be an arbitrary complex number, but in
physical applications is typically taken to be either real or a root of unity.
\smallskip
Since these quantum group deformations modify the dispersion relation and the scattering matrix, to solve the corresponding model by means of the mirror Thermodynamic Bethe Ansatz (TBA), for a recent review see \cite{Stijn}, one has to go through the entire procedure of first deriving the TBA equations for the ground state and then extending them to include excited states. While this program has been successfully carried out for deformations with $q$ being a root of unity \cite{Arutyunov:2012zt,Arutyunov:2012ai}, the corresponding string background remains unknown. There is a conjecture that in the limit of infinite 't Hooft coupling the q-deformed S-matrix tends to that of the Pohlmeyer-reduced version of the ${\rm AdS}_5\times {\rm S}^5\ $ superstring \cite{Beisert:2010kk,Hoare:2013ysa}. It is not straightforward, however, to identify the S-matrix of the latter theory as one has to understand whether the elementary excitations that scatter in that model are solitons or kinks.
\smallskip
The case of real deformation parameter considered in this paper is not less compelling. Recently there was an interesting proposal on how to deform the sigma-model for strings on ${\rm AdS}_5\times {\rm S}^5\ $ with a real deformation parameter
$\eta$ \cite{Delduc:2013qra}.
Deformations of this type constitute a general class of deformations
governed by solutions of the classical Yang-Baxter equation \cite{Cherednik:1981df,Klimcik:2008eq}. This class is not solely restricted to the string model in question but can be applied to a
large variety of two-dimensional integrable models based on (super)groups or their cosets \cite{Klimcik:2002zj}-\cite{Kawaguchi:2012gp}.
\smallskip
The aim of the present work is to compute the $2\to 2$ scattering matrix for the $\eta$-deformed model in the limit of large string tension $g$ and to compare the corresponding result with the known q-deformed S-matrix found from quantum group symmetries, unitarity and crossing \cite{Beisert:2008tw, Ben}. In the context of the undeformed model a computation of this type has been carried out in \cite{Klose:2006zd}.
\smallskip
The $\eta$-deformed model appears to be rather involved, primarily because of fermionic degrees of freedom.
Our strategy is therefore to switch off fermions and proceed by studying the corresponding bosonic action.
Of course, the perturbative S-matrix computed from this action will not coincide with the full world-sheet S-matrix but nevertheless
will give a sufficient part of the scattering data to provide a non-trivial test for both integrability (the Yang-Baxter equation) and a comparison
with the q-deformed S-matrix. The contribution of fermions is of secondary concern and will be discussed elsewhere \cite{ABF}.
\smallskip
Let us summarize the results of this paper. Coming back to the bosonic action, we find that it corresponds to a string background which in addition to the metric also supports a non-vanishing $B$-field.
The deformation breaks ${\rm AdS}_5\times {\rm S}^5\ $ isometries down to ${\rm U}(1)^3\times {\rm U}(1)^3$, where the first and second factors refer to the deformed AdS and five-sphere, respectively.
Thus, only isometries corresponding to the Cartan elements of the isometry algebra of the ${\rm AdS}_5\times {\rm S}^5\ $ survive, very similar to the case of generic $\gamma$-deformations.
As for the metric, its AdS part exhibits a singularity whose nature is currently unclear. Computed in a string frame the metric includes the contribution of a dilaton, and
to extract the latter one needs to know the RR-fields which requires considering fermions.
\smallskip
With the bosonic action at hand it is straightforward to compute the corresponding tree-level S-matrix. We then show that it matches
perfectly with the q-deformed S-matrix taken in the large tension limit and restricted to the scattering of bosons, provided we identify the deformation parameters as
$$
q=e^{-\nu/g}\, ,\quad \nu = {2\eta\over 1+\eta^2}\,.
$$
This is the main result of our work which makes it quite credible that the $\eta$-deformed model indeed may enjoy hidden $\alg{psu}_q(2|2)\oplus \alg{psu}_q(2|2)$ symmetry for finite
values of the coupling constant $g$. If true, this implies that despite the singular behaviour of the metric the quantum string sigma model would be well defined. In particular it would be possible to compute its exact spectrum by means of the mirror TBA.
\smallskip
The paper is organized as follows. In the next section we recall the general form of the action for the $\eta$-deformed model and use it to derive an explicit form of the Lagrangian for bosonic
degrees of freedom. In section 3, upon fixing the uniform light-cone gauge, the corresponding Hamiltonian is derived up to quartic order in fields and further used to compute
the tree-level S-matrix. This result is subsequently compared to the one arising from the q-deformed S-matrix (which includes the dressing phase) in the large $g$ limit. We conclude by outlining
interesting open problems. Finally some technical details on the derivation of the bosonic Lagrangian, the perturbative expansion of the q-deformed dressing phase and the form of the q-deformed
S-matrix are collected in three appendices.
\section{Superstrings on $\eta$-deformed ${\rm AdS}_5\times {\rm S}^5\ $}
According to \cite{Delduc:2013qra}, the action for superstrings on the deformed ${\rm AdS}_5\times {\rm S}^5\ $ is
\begin{eqnarray}\nonumber
S=\int {\rm d\sigma}{\rm d\tau} \mathscr{L}\, ,
\end{eqnarray}
where the Lagrangian density depending on a real deformation parameter $\eta$ is given by\footnote{Note that our $\eta$-dependent prefactor differs from the one in \cite{Delduc:2013qra}. Our choice is necessary to match the perturbative world-sheet scattering matrix with the q-deformed one.}
\begin{eqnarray}
\label{defLag}
{\mathscr L}=-\frac{g}{4}(1+\eta^2)\big(\gamma^{\alpha\beta}-\epsilon^{\alpha\beta}\big)\, {\rm str}\Big[\tilde{d}(A_{\alpha})\frac{1}{1-\eta R_\alg{g} \circ d}(A_{\beta}) \Big]\, .
\end{eqnarray}
Here and in what follows we use the notations and conventions from \cite{Arutyunov:2009ga}, in particular $\epsilon^{\tau\sigma}=1$ and $\gamma^{\alpha\beta}=h^{\alpha\beta}\sqrt{-h}$ that is the Weyl invariant combination of the world-sheet metric $h_{\alpha\beta}$; the component $\gamma^{\tau\tau}<0$.
The coupling constant $g$ is the effective string tension.
Further, $A_{\alpha}=-\alg{g}^{-1}\partial_{\alpha}\alg{g}$, where $\alg{g}\equiv \alg{g}(\tau,\sigma)$ is a coset representative from ${\rm PSU}(2,2|4)/{\rm SO}(4,1)\times {\rm SO}(5)$.
To define the operators
$d$ and $\tilde{d}$ acting on the currents $A_{\alpha}$, we need to recall that the Lie superalgebra $\mathscr{G}=\alg{psu}(2,2|4)$ admits a ${\mathbb Z}_4$-graded decomposition
$$
\mathscr{G}=\mathscr{G}^{(0)}\oplus \mathscr{G}^{(1)}\oplus \mathscr{G}^{(2)}\oplus \mathscr{G}^{(3)}\, .$$
Here $\mathscr{G}^{(0)}$ coincides with $\alg{so}(4,1)\times \alg{so}(5)$. Denoting by $P_i$, $i=0,1,2,3$, projections on the corresponding components of the graded decomposition above,
operators $d$ and $\tilde{d}$ are defined as
\begin{eqnarray}
\nonumber
d&=&P_1+\frac{2}{1-\eta^2}P_2-P_3,\\
\nonumber
\tilde{d}&=&-P_1+\frac{2}{1-\eta^2}P_2+P_3\, .
\end{eqnarray}
Finally, the action of the operator $R_\alg{g}$ on $M\in {\mathscr G}$ is given by
\begin{equation}\label{Rgop}
R_\alg{g}(M) = \alg{g}^{-1}R(\alg{g} M\alg{g}^{-1})\alg{g}\, ,
\end{equation}
where $R$ is a linear operator on $\mathscr{G}$ satisfying the modified classical Yang-Baxter equation. In the following we define the action of $R$ on an arbitrary $8\times 8$ matrix $M$
as
\begin{equation}\label{Rop}
R(M)_{ij} = -i\, \epsilon_{ij} M_{ij}\,,\quad \epsilon_{ij} = \left\{\begin{array}{ccc} 1& \rm if & i<j \\
0&\rm if& i=j \\
-1 &\rm if& i>j \end{array} \right.\,,
\end{equation}
In the limit $\eta\to 0$ one recovers from (\ref{defLag}) the Lagrangian density of the ${\rm AdS}_5\times {\rm S}^5\ $ superstring.
\smallskip
Our goal now is to obtain an explicit form for the corresponding bosonic action.
With fermionic degrees of freedom switched off, formula (\ref{defLag}) simplifies to
\begin{eqnarray}
\label{defLagbos}
{\mathscr L}=- \frac{g}{2}(1+\varkappa^2)^{1\ov2}\, \big(\gamma^{\alpha\beta}-\epsilon^{\alpha\beta}\big){\rm str}\, \Big[A_{\alpha}^{(2)}\frac{1}{1-\varkappa R_\alg{g} \circ P_2}(A_{\beta}) \Big]\, ,
\end{eqnarray}
where we have introduced
$$
\varkappa=\frac{2\eta}{1-\eta^2}\,,
$$
which as we see in a moment is a convenient deformation parameter.
To proceed, we need to choose a representative $\alg{g}$ of a bosonic coset ${\rm SU}(2,2|4)\times {\rm SU}(4)/{\rm SO}(4,1)\times {\rm SO}(5)$ and invert the operator
$1-\eta R_\alg{g} \circ d$.
A convenient choice of a coset representative and the inverse of $1-\eta R_\alg{g} \circ d$ are discussed in appendix \ref{IRgd}. Making use of the inverse operator, one can easily compute the corresponding bosonic Lagrangian.
It is given by the sum of the AdS and sphere parts
\begin{equation}\label{Lfull}
\mathscr L = \mathscr L_{\alg{a}} +\mathscr L_{\alg{s}}= \mathscr L_{\alg{a}}^{G} +\mathscr L_{\alg{a}}^{WZ} +\mathscr L_{\alg{s}}^{G} +\mathscr L_{\alg{s}}^{WZ}\, ,
\end{equation}
where we further split each part into the contribution of the metric and Wess-Zumino pieces. Accordingly, for the metric pieces we obtain
\begin{eqnarray}\nonumber
\mathscr L_{\alg{a}}^{G} &=&-{g\ov2}(1+\varkappa^2)^{1\ov2}\, \gamma^{\alpha\beta}\Big(
-\frac{\partial_\alpha t\partial_\beta t\left(1+\rho ^2\right)}{ 1-\varkappa ^2 \rho ^2}
+\frac{\partial_\alpha \rho\partial_\beta \rho}{ \left(1+\rho ^2\right) \left(1-\varkappa ^2 \rho ^2\right)}
+\frac{\partial_\alpha \zeta\partial_\beta\zeta \rho ^2}{1+ \varkappa ^2 \rho ^4 \sin ^2\zeta }
\\\label{LaG}
&&\qquad\qquad\qquad+\frac{\partial_\alpha \psi_1\partial_\beta\psi_1\rho ^2 \cos
^2\zeta}{ 1+\varkappa ^2 \rho ^4 \sin ^2\zeta}+\partial_\alpha \psi_2\partial_\beta\psi_2
\rho ^2 \sin ^2\zeta \Big)\,,
\end{eqnarray}
\begin{eqnarray}\nonumber
\mathscr L_{\alg{s}}^{G} &=&-{g\ov2}(1+\varkappa^2)^{1\ov2}\, \gamma^{\alpha\beta}\Big(\frac{\partial_\alpha \phi\partial_\beta \phi
\left(1-r^2\right)}{1+\varkappa ^2 r^2}+\frac{\partial_\alpha r\partial_\beta r
}{ \left(1-r^2\right) \left(1+\varkappa ^2 r^2\right)}
+\frac{\partial_\alpha \xi\partial_\beta \xi r^2}{1+ \varkappa ^2 r^4 \sin ^2\xi}
\\\label{LsG}
&&\qquad\qquad\qquad+\frac{\partial_\alpha \phi_1\partial_\beta \phi_1 r^2 \cos ^2\xi }{1+ \varkappa ^2
r^4 \sin ^2\xi } +\partial_\alpha \phi_2\partial_\beta \phi_2 r^2 \sin^2\xi \Big)\,,
\end{eqnarray}
while the Wess-Zumino parts $\mathscr L_{\alg{a}}^{WZ}$ and $\mathscr L_{\alg{s}}^{WZ}$ read
\begin{eqnarray}\label{LaWZ}
\mathscr L_{\alg{a}}^{WZ} &=&{g\ov2} \varkappa (1+\varkappa^2)^{1\ov2}\, \epsilon^{\alpha\beta}\frac{ \rho ^4 \sin 2 \zeta}{1+ \varkappa ^2 \rho ^4 \sin ^2\zeta}\partial_\alpha\psi_1\partial_\beta\zeta\,,
\end{eqnarray}
\begin{eqnarray}\label{LsWZ}
\mathscr L_{\alg{s}}^{WZ} &=&-{g\ov2} \varkappa (1+\varkappa^2)^{1\ov2}\, \epsilon^{\alpha\beta}\frac{ r^4 \sin 2 \xi }{1+ \varkappa ^2 r^4 \sin^2\xi}\partial_\alpha\phi_1\partial_\beta\xi\, .
\end{eqnarray}
Here the coordinates $t\,,\,\psi_1\,,\,\psi_2\,,\, \zeta\,,\, \rho$ parametrize the deformed AdS space, while the coordinates $\phi\,,\,\phi_1\,,\,\phi_2\,,\, \xi\,,\, r$ parametrize the deformed five-sphere. Switching off the deformation, one finds that
the AdS$_5$ coordinates are related to the embedding
coordinates $Z^A,$ $A = 0,1,\ldots,5$ obeying the constraint $\eta^{AB}Z_AZ_B=-1$ where $\eta^{AB}=(-1,1,1,1,1,-1)$ as
\begin{eqnarray}
Z_1+iZ_2 = \rho\cos\zeta\,e^{i\psi_1}\,,\quad Z_3+iZ_4 = \rho\sin\zeta\,e^{i\psi_2}\,,\quad Z_0+iZ_5 = \sqrt{1+\rho^2}\,e^{it}\,,
\end{eqnarray}
while the $S_5$ coordinates are related to the embedding
coordinates $Y^A,$ $A = 1,\ldots,6$ obeying $Y_A^2=1$ as
\begin{eqnarray}
Y_1+iY_2 = r\cos\xi\,e^{i\phi_1}\,,\quad Y_3+iY_4 = r\sin\xi\,e^{i\phi_2}\,,\quad Y_5+iY_6 = \sqrt{1-r^2}\,e^{i\phi}\,.
\end{eqnarray}
It is obvious that the deformed action is invariant under ${\rm U}(1)^3\times {\rm U}(1)^{3}$ corresponding to the shifts of $t$, $\psi_k$, $\phi$, $\phi_k$. One also finds the ranges of $\rho$ and $r$: $0\le \rho\le{1\over\varkappa}$ and $0\le r\le1$.
The (string frame) metric of the deformed AdS is singular at $\rho=1/\varkappa$. Since we do not know the dilaton it is unclear if the Einstein frame metric exhibits the same singularity. The bosonic Wess-Zumino terms signify the presence of a non-trivial background $B$-field which is absent in the undeformed case.
In the next section we are going to impose the light-cone gauge, take the decompactification limit and compute the bosonic part of the four-particle world-sheet scattering matrix. To this end, we first expand the Lagrangian \eqref{Lfull} up to quartic order in $\rho$, $r$ and their derivatives, and then make the shifts of $\rho$ and $r$ as described in appendix \ref{IRgd}, {\it c.f.} (\ref{shift}).
Since we are interested in the perturbative expansion in powers of fields around $\rho=0$,
the final step consists in changing the spherical coordinates to $(z_i,y_i)_{i=1,\ldots,4}$ as
\begin{eqnarray}
\begin{aligned}
\label{flatcoord}
\frac{z_1+iz_2}{1-\tfrac{1}{4}z^2}=\rho\cos\zeta e^{i\psi_1}\, , ~~~~~~~\frac{z_3+iz_4}{1-\tfrac{1}{4}z^2}=\rho\sin\zeta e^{i\psi_2} \, , ~~~~~z^2\equiv z_i^2\, ,\\
\frac{y_1+iy_2}{1+\tfrac{1}{4}y^2}=r\cos\xi e^{i\phi_1}\, , ~~~~~~~\frac{y_3+iy_4}{1+\tfrac{1}{4}y^2}=r\sin\xi e^{i\phi_2} \, ,~~~~~~y^2\equiv y_i^2 \, ,
\end{aligned}
\end{eqnarray}
with further expanding the resulting action up to the quartic order in $z$ and $y$ fields. In this way we find the following quartic Lagrangian
\begin{equation}\label{Lquart}
\begin{aligned}
\mathscr L_{\alg{a}} &= -\frac{g}{2} (1+\varkappa^2)^{1\ov2} \, \gamma^{\alpha \beta} \Bigg[ -\left( 1 + (1+\varkappa^2) z^2 +\frac{1}{2}(1+\varkappa^2)^2(z^2)^2\right) \partial_{\alpha}t \partial_{\beta}t \\
& + \left(1+(1-\varkappa^2)\frac{z^2}{2}\right) \partial_{\alpha}z_i\partial_{\beta}z_i \Bigg]+2g \varkappa(1+\varkappa^2)^{1\ov2} (z_3^2+z_4^2) \epsilon^{\alpha\beta} \partial_{\alpha}z_1 \partial_{\beta}z_2 \, , \\
\mathscr L_{\alg{s}}&= -\frac{g}{2} (1+\varkappa^2)^{1\ov2}\, \gamma^{\alpha \beta} \Bigg[ \left( 1 - (1+\varkappa^2) y^2 +\frac{1}{2}(1+\varkappa^2)^2(y^2)^2\right) \partial_{\alpha}\phi \partial_{\beta}\phi \\
& + \left(1-\frac{1}{2}(1-\varkappa^2)y^2 \right) \partial_{\alpha}y_i\partial_{\beta}y_i \Bigg] - 2 g\varkappa(1+\varkappa^2)^{1\ov2} (y_3^2+y_4^2) \epsilon^{\alpha\beta} \partial_{\alpha}y_1 \partial_{\beta}y_2\, .
\end{aligned}
\end{equation}
We point out that the metric part of this Lagrangian has a manifest ${\rm SO}(4)\times {\rm SO(4)}$ symmetry which is however broken by the Wess-Zumino terms.
\section{Perturbative bosonic world-sheet S-matrix}
\subsection{Light-cone gauge and quartic Hamiltonian}
To fix the light-cone gauge and compute the scattering matrix, it is advantageous to use the Hamiltonian formalism. For the reader's convenience we start with a general discussion on how to construct the Hamiltonian for the world-sheet action
of the form
\begin{equation}
S=-\frac{g}{2} \int_{-r}^r \, {\rm d}\sigma {\rm d} \tau \left( \, \gamma^{\alpha\beta} \partial_\alpha X^M \partial_\beta X^N G_{MN} -\epsilon^{\alpha\beta} \partial_\alpha X^M \partial_\beta X^N B_{MN} \right),
\end{equation}
where $G_{MN}$ and $B_{MN}$ are the background metric and $B$-field respectively.
In the first order formalism we introduce conjugate momenta
\begin{equation}
p_M = \frac{\delta S}{\delta \dot{X}^M} = - g \gamma^{0\beta} \partial_\beta X^N G_{MN} + g X^{'N} B_{MN}.
\end{equation}
The action can be rewritten as
\begin{equation}
S= \int_{-r}^r \, {\rm d}\sigma {\rm d} \tau \left( p_M \dot{X}^M + \frac{\gamma^{01}}{\gamma^{00}} C_1 + \frac{1}{2g \gamma^{00}} C_2 \right),
\end{equation}
where $C_1, C_2$ are the Virasoro constraints.
They are given by
\begin{eqnarray}
C_1 &=& p_M X'^{M}, \\
C_2 &=& G^{MN} p_M p_N - 2 g p_M X'^{Q} G^{MN} B_{NQ} + g^2 X'^{P} X'^{Q} B_{MP} B_{NQ} G^{MN} + g^2 X'^{M} X'^{N} G_{MN}. \nonumber
\end{eqnarray}
The first Virasoro constraint has the same form as in the undeformed case. In particular, the solution for $x_-'$ in terms of $p_\mu,x_\mu$ will still be the same.
When expressed in terms of the conjugate momenta, the second constraint gets an explicit dependence on the B-field.
To impose light-cone gauge, one first introduces light-cone coordinates
\begin{equation}\label{lcg}
x_-=\phi-t, \qquad x_+= (1-a)t +a \phi.
\end{equation}
The second Virasoro constraint can be written as
\begin{equation}
\begin{aligned}
C_2 &= G^{--} p_-^2 +2 G^{+-} p_+ p_- + G^{++} p_+^2 \\
& + g^2 G_{++} x_-'^2 + 2 g^2 G_{+-} x_+' x_-' + g^2 G_{--} x_+'^2 + \mathcal{H}_x\, ,
\end{aligned}
\end{equation}
where
\begin{equation}
\begin{aligned}
\nonumber
G^{--} &= a^2 G_{\phi\phi}^{-1} - (a-1)^2 G_{tt}^{-1}, \qquad &G^{+-}& = a G_{\phi\phi}^{-1} - (a-1) G_{tt}^{-1}, \qquad G^{++} = G_{\phi\phi}^{-1} - G_{tt}^{-1}, \\
G_{++} &= (a-1)^2 G_{\phi\phi} - a^2 G_{tt}, \qquad &G_{+-} &= -(a-1) G_{\phi\phi} + a G_{tt}, \qquad G_{--} = G_{\phi\phi} - G_{tt},
\end{aligned}
\end{equation}
and $\mathcal{H}_x$ is the part that depends on the transverse fields only
\begin{equation}
\mathcal{H}_x = G^{\mu\nu} p_\mu p_\nu + g^2 X'^\mu X'^\nu G_{\mu\nu} -2 g p_\mu X'^{\rho} G^{\mu\nu} B_{\nu\rho} + g^2 X'^\lambda X'^\rho B_{\mu\lambda} B_{\nu\rho} G^{\mu\nu}.
\end{equation}
Notice that the $B$-field is contained only in $\mathcal{H}_x$, since in the action it does not couple to the derivatives of $x_\pm$.
We impose the uniform light-cone gauge
\begin{equation}
x_+= \tau, \qquad p_+=1.
\end{equation}
Solving $C_2=0$ for $p_-$ gives the Hamiltonian
\begin{equation}
\mathcal{H} = -p_-(p_\mu,x^\mu,x'^\mu).
\end{equation}
Formally the solution for the Hamiltonian is still given by eq. (2.16) of the review \cite{Arutyunov:2009ga}, with the only difference that now the components of the metric are deformed and that $\mathcal{H}_x$ has also the $B$-field contribution.
Rescaling the fields with powers of $g$ and expanding in $g$ one can find $\mathcal{H}_n$, namely the part of the Hamiltonian that is of order $n$ in the fields.
Then the action acquires the form
\begin{equation}
S= \int {\rm d}\tau {\rm d} \sigma \, \left( p_\mu \dot{x}^\mu - \mathcal{H}_2 - \frac{1}{g} \mathcal{H}_4 - \cdots \right),
\end{equation}
where the quadratic Hamiltonian is given by
\begin{equation}
\mathcal{H}_2 = \frac{1}{2} p_\mu^2 + \frac{1}{2} (1+\varkappa^2) x_\mu^2 + \frac{1}{2} (1+\varkappa^2) x'^2_\mu.
\end{equation}
The quartic Hamiltonian in a general $a$-gauge is
\begin{equation}
\begin{aligned}
\mathcal{H}_4 &= \frac{1}{4} \Bigg( (2 \varkappa^2 z^2 -(1+\varkappa^2) y^2 ) p_z^2 - (2 \varkappa^2 y^2 -(1+\varkappa^2) z^2 ) p_y^2 \\
&+\left(1+\varkappa ^2\right) \left(\left(2 z^2-\left(1+\varkappa ^2\right) y^2\right)z'^2 + \left(\left(1+\varkappa ^2\right) z^2-2
y^2\right)y'^2\right)\Bigg) \\
&- 2 \varkappa \left(1+\varkappa ^2\right)^{1\ov2} \left(\left(z_3^2+z_4^2\right) \left(p_{z_1} z_2'-p_{z_2} z_1'\right) - \left(y_3^2+y_4^2\right) \left(p_{y_1} y_2'-p_{y_2} y_1'\right) \right) \\
&+\frac{(2a-1)}{8} \Bigg( (p_y^2+p_z^2)^2 -(1+\varkappa^2)^2 (y^2+z^2)^2 \\
&+2 (1+\varkappa^2)(p_y^2+p_z^2)(y'^2+z'^2)+(1+\varkappa^2)^2 (y'^2+z'^2)^2 -4 (1+\varkappa^2) (x_-')^2\Bigg).
\end{aligned}
\end{equation}
Here we use the notation $p_z^2\equiv p_{z_i}^2, \ p_y^2\equiv p_{y_i}^2$, where sum over $i$ is assumed.
To simplify the quartic piece, we
can remove the terms of the form $p_z^2y^2$ and $p_y^2z^2$ by performing a canonical transformation generated by
\begin{equation}
V= \frac{(1+\varkappa^2)}{4} \int {\rm d}\sigma \Big( p_y y z^2 -p_z z y^2 \Big) ,
\end{equation}
where the shorthand notation $p_y y \equiv p_{y_i} {y_i}, \ p_z z \equiv p_{z_i} {z_i}$ was used. After this is done the quartic Hamiltonian is
\begin{eqnarray}
\mathcal{H}_4 &=& \frac{(1+\varkappa^2)}{2} ( z^2 z'^2- y^2 y'^2 ) + \frac{(1+\varkappa^2)^{2}}{2} (z^2 y'^2-y^2 z'^2) +\frac{\varkappa^2}{2} ( z^2 p_z^2 - y^2 p_y^2 ) \nonumber \\
&-& 2\varkappa(1+\varkappa^2)^{1\ov2} \left[\left(z_3^2+z_4^2\right) \left(p_{z_1} z_2'-p_{z_2} z_1'\right) - \left(y_3^2+y_4^2\right) \left(p_{y_1} y_2'-p_{y_2} y_1'\right) \right] \nonumber \\
&+&\frac{(2a-1)}{8}\Bigg( (p_y^2+p_z^2)^2 - (1+\varkappa^2)^2 (y^2+z^2)^2 \\
&+&2 (1+\varkappa^2) (p_y^2+p_z^2)(y'^2+z'^2)+ (1+\varkappa^2)^2 (y'^2+z'^2)^2 -4 (1+\varkappa^2) (x_-')^2\Bigg). \nonumber
\end{eqnarray}
We recall that in the undeformed case the corresponding theory is invariant with respect to the two copies of the centrally extended superalgebra $\alg{psu}(2|2)$, each containing two
$\alg{su}(2)$ subalgebras. To render invariance under $\alg{su}(2)$ subalgebras manifest, one can introduce two-index notation for the world-sheet fields. It is also convenient to adopt the same
notation for the deformed case\footnote{This parameterisation is different from the one used in \cite{Arutyunov:2009ga} and the difference is the exchange of the definitions for $Y^{1\dot{1}}$ and $Y^{2\dot{2}}$. This does not matter in the undeformed case but is needed here in order to correctly match the perturbative S-matrix with the q-deformed one computed from symmetries.}
\begin{equation}
\begin{aligned}
&Z^{3\dot{4}} =\tfrac{1}{2} (z_3-i z_4), \qquad &Z^{3\dot{3}} =\tfrac{1}{2} (z_1-i z_2), \\
& Z^{4\dot{3}}=-\tfrac{1}{2} (z_3+i z_4), \qquad &Z^{4\dot{4}}=\tfrac{1}{2} (z_1+i z_2),
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
&Y^{1\dot{2}}=\tfrac{1}{2} (y_3-i y_4), \qquad &Y^{1\dot{1}}=\tfrac{1}{2} (y_1+i y_2), \\
&Y^{2\dot{1}}=-\tfrac{1}{2} (y_3+i y_4), \qquad &Y^{2\dot{2}}=\tfrac{1}{2} (y_1-i y_2)\, .
\end{aligned}
\end{equation}
In terms of two-index fields the quartic Hamiltonian becomes $\mathcal{H}_4 = \mathcal{H}^G_4 + \mathcal{H}^{WZ}_4$,
where $\mathcal{H}^G_4$ is the contribution coming from the spacetime metric and $\mathcal{H}^{WZ}_4 $ from the $B$-field
{\small
\begin{eqnarray}
\mathcal{H}^G_4 &=&2(1+\varkappa^2) \left( Z_{\alpha\dot{\alpha}} Z^{\alpha\dot{\alpha}} Z'_{\beta\dot{\beta}} Z'^{\beta\dot{\beta}} -Y_{a\dot{a}}Y^{a\dot{a}} Y'_{b\dot{b}}Y'^{b\dot{b}} \right) \nonumber \\
&+& 2(1+\varkappa^2)^{2} \left( Z_{\alpha\dot{\alpha}} Z^{\alpha\dot{\alpha}} Y'_{b\dot{b}}Y'^{b\dot{b}} - Y_{a\dot{a}}Y^{a\dot{a}} Z'_{\beta\dot{\beta}} Z'^{\beta\dot{\beta}} \right) \nonumber \\
& +&\frac{\varkappa^2}{2} \left( Z_{\alpha\dot{\alpha}} Z^{\alpha\dot{\alpha}} P_{\beta\dot{\beta}} P^{\beta\dot{\beta}} - Y_{a\dot{a}}Y^{a\dot{a}} P_{b\dot{b}}P^{b\dot{b}} \right) \nonumber \\
&+&\frac{(2a-1)}{8} \Bigg( \frac{1}{4}(P_{a\dot{a}}P^{a\dot{a}}+P_{\alpha\dot{\alpha}}P^{\alpha\dot{\alpha}})^2 -4 (1+\varkappa^2)^2 (Y_{a\dot{a}}Y^{a\dot{a}}+Z_{\alpha\dot{\alpha}}Z^{\alpha\dot{\alpha}})^2 \\
&+&2 (1+\varkappa^2) (P_{a\dot{a}}P^{a\dot{a}}+P_{\alpha\dot{\alpha}}P^{\alpha\dot{\alpha}})(Y'_{a\dot{a}}Y'^{a\dot{a}}+Z'_{\alpha\dot{\alpha}}Z'^{\alpha\dot{\alpha}})+4 (1+\varkappa^2)^2 (Y'_{a\dot{a}}Y'^{a\dot{a}}+Z'_{\alpha\dot{\alpha}}Z'^{\alpha\dot{\alpha}})^2 \nonumber \\
&-&4 (1+\varkappa^2) (P_{a\dot{a}}Y'^{a\dot{a}} +P_{\alpha\dot{\alpha}}Z'^{\alpha\dot{\alpha}})^2\Bigg), \nonumber \\
\mathcal{H}^{WZ}_4 &=& 8 i\varkappa(1+\varkappa^2)^{1\ov2} \left( Z^{3\dot{4}} Z^{4\dot{3}} ( P_{3\dot{3}} Z'^{3\dot{3}} -P_{4\dot{4}} Z'^{4\dot{4}} ) + Y^{1\dot{2}} Y^{2\dot{1}} ( P_{1\dot{1}} Y'^{1\dot{1}} -P_{2\dot{2}} Y'^{2\dot{2}} ) \right)\, . \nonumber
\end{eqnarray}
}
\noindent
Note that we have used the Virasoro constraint $C_1$ in order to express $x'_-$ in terms of the two index fields.
The gauge dependent terms multiplying $(2a-1)$ are invariant under SO(8) as in the underformed case.
\subsection{Tree level bosonic S-matrix}
The computation of the tree level bosonic S-matrix follows the route reviewed in \cite{Arutyunov:2009ga}, and we also use the same notations.
It is convenient to rewrite the tree-level S-matrix as a sum of two terms $\mathbb{T}=\mathbb{T}^G + \mathbb{T}^{WZ}$, coming from $\mathcal{H}^G_4$ and $\mathcal{H}^{WZ}_4$ respectevely. The reason is that $\mathbb{T}^G$ preserves the $\alg{so}(4)\oplus \alg{so}(4)$ symmetry, while $\mathbb{T}^{WZ}$ breaks it.
To write the results we always assume that $p>p'$.
Then, one finds that the action of $\mathbb{T}^G$ on the two-particle states is given by
\begin{equation}
\label{Tmatrix}
\begin{aligned}
\mathbb{T}^G \, \ket{Y_{a\dot{c}} Y_{b\dot{d}}'} &= \left[ \frac{1-2a}{2}(p \omega' - p' \omega) +\frac{1}{2} \frac{ (p-p')^2 +\nu^2 (\omega-\omega')^2}{p \omega' - p' \omega} \right] \, \ket{Y_{a\dot{c}} Y_{b\dot{d}}'} \\
& + \frac{p p' + \nu^2 \omega \omega' }{p \omega' - p' \omega} \left( \ket{Y_{a\dot{d}} Y_{b\dot{c}}'} + \ket{Y_{b\dot{c}} Y_{a\dot{d}}'} \right), \\ \\
\mathbb{T}^G \, \ket{Z_{\alpha\dot{\gamma}} Z_{\beta\dot{\delta}}'} &= \left[ \frac{1-2a}{2}(p \omega' - p' \omega) -\frac{1}{2} \frac{ (p-p')^2 +\nu^2 (\omega-\omega')^2 }{p \omega' - p' \omega} \right] \, \ket{Z_{\alpha\dot{\gamma}} Z_{\beta\dot{\delta}}'} \\
& - \frac{ p p' + \nu^2 \omega \omega' }{p \omega' - p' \omega} \left( \ket{Z_{\alpha\dot{\delta}} Z_{\beta\dot{\gamma}}'} + \ket{Z_{\beta\dot{\gamma}} Z_{\alpha\dot{\delta}}'} \right), \\ \\
\mathbb{T}^G \, \ket{Y_{a\dot{b}} Z_{\alpha\dot{\beta}}'} &= \left[ \frac{1-2a}{2}(p \omega' - p' \omega) -\frac{1}{2}\frac{\omega^2-\omega'^2}{p \omega' - p' \omega} \right] \, \ket{Y_{a\dot{b}} Z_{\alpha\dot{\beta}}'}, \\ \\
\mathbb{T}^G \, \ket{Z_{\alpha\dot{\beta}} Y_{a\dot{b}}'} &= \left[ \frac{1-2a}{2}(p \omega' - p' \omega) +\frac{1}{2}\frac{\omega^2-\omega'^2}{p \omega' - p' \omega} \right] \, \ket{Z_{\alpha\dot{\beta}} Y_{a\dot{b}}'},
\end{aligned}
\end{equation}
and the action of $\mathbb{T}^{WZ}$ on the two-particle states is
\begin{equation}
\begin{aligned}
\mathbb{T}^{WZ} \, \ket{Y_{a\dot{c}} Y_{b\dot{d}}'} &
= i \nu\left(\epsilon_{ab} \ket{Y_{b\dot{c}} Y_{a\dot{d}}'}
+\epsilon_{\dot{c}\dot{d}} \ket{Y_{a\dot{d}} Y_{b\dot{c}}'}\right)
, \\
\mathbb{T}^{WZ} \, \ket{Z_{\alpha\dot{\gamma}} Z_{\beta\dot{\delta}}'} &
= i \nu \left( \epsilon_{\alpha\beta} \ket{Z_{\beta\dot{\gamma}} Z_{\alpha\dot{\delta}}'}
+ \epsilon_{\dot{\gamma}\dot{\delta}} \ket{Z_{\alpha\dot{\delta}} Z_{\beta\dot{\gamma}}'}\right)\,,
\end{aligned}
\end{equation}
where on the r.h.s. we obviously do not sum over the repeated indices. In the formulae the frequency $\omega$ is related to the momentum $p$ as
\begin{equation}\label{omega}
\omega=(1+\varkappa^2)^{1\ov2} \sqrt{1+p^2} = \sqrt{1+p^2\over 1-\nu^2}\,,
\end{equation}
and we have introduced the parameter
\begin{equation}
\nu = {\varkappa\over (1+\varkappa^2)^{1\ov2}}={2\eta\over 1+\eta^2}\,,
\end{equation}
which, as one can see from the expressions above, is the natural deformation parameter. In fact, as we discuss in the next subsection, it is related in a very simple way to the parameter $q$ of the q-deformed S-matrix: $q=e^{-\nu/g}$.
The S-matrix ${\mathbb S}$ computed in perturbation theory is related to the ${\mathbb T}$-matrix as
\begin{equation}\label{Tpert}
{\mathbb S}=\mathbbm{1} +\frac{i}{g}{\mathbb T}\, .
\end{equation}
In the undeformed case, as a consequence of invariance of ${\mathbb S}$ with respect to two copies of the centrally extended superalgebra $\alg{psu}(2|2)$, the corresponding ${\mathbb T}$-matrix
admits a factorization
\begin{eqnarray}
{\mathbb T}^{P\dot{P},Q\dot{Q}}_{M\dot{M},N\dot{N}}=(-1)^{\epsilon_{\dot M}(\epsilon_{N}+\epsilon_{Q})}{\cal T}_{MN}^{PQ}\delta_{\dot{M}}^{\dot{P}}\delta_{\dot{N}}^{\dot{Q}}
+(-1)^{\epsilon_Q(\epsilon_{\dot{M}}+\epsilon_{\dot{P}})}\delta_{M}^{P}\delta_{N}^{Q} {\cal T}_{\dot{M}\dot{N}}^{\dot{P}\dot{Q}}\, .
\end{eqnarray}
Here $M=(a, \alpha)$ and $\dot{M}=(\dot{a},\dot{\alpha})$, and dotted and undotted indices are referred to two copies of $\alg{psu}(2|2)$, respectively, while
$\epsilon_{M}$ and $\epsilon_{\dot{M}}$ describe statistics of the corresponding indices, {\it i.e.} they are zero for bosonic (Latin) indices and equal to one for fermionic (Greek) ones.
The factor ${\mathcal T}$ can be regarded as $16\times 16$ matrix.
\smallskip
It is not difficult to see that the same type of factorization persists in the deformed case as well. Indeed, from the formulae (\ref{Tmatrix}) we extract the following elements for the ${\mathcal T}$-matrix
\begin{eqnarray}\label{cTmatr}
\begin{aligned}
&{\mathcal T}_{ab}^{cd}=
A\,\delta _a^c\delta _b^d+B\,\delta _a^d\delta _b^c+W\, \epsilon_{ab}\delta _a^d\delta _b^c\, , \\
&{\mathcal T}_{\alpha\beta}^{\gamma\delta }=
D\,\delta _\alpha^\gamma\delta _\beta^\delta +E\,\delta _\alpha^\delta \de_\beta^\gamma+W\, \epsilon_{\alpha\beta}\, \delta _{\alpha}^{\delta}\delta _{\beta}^{\gamma}\,, \\
&{\mathcal T}_{a\beta}^{c\delta }= G\,\delta _a^c\delta _\beta^\delta \,,\qquad
~{\mathcal T}_{\alpha b}^{\gamma d}= L\,\delta _\alpha^\gamma\delta _b^d\,,
\end{aligned}
\end{eqnarray} where the
coefficients are given by
\begin{eqnarray}
\begin{aligned}
\label{Tmatrcoef}
&A(p,p')= \frac{1-2a}{4}(p \omega' - p' \omega) +\frac{1}{4} \frac{ (p-p')^2 +\nu^2 (\omega-\omega')^2}{p \omega' - p' \omega} \,,\\
&B(p,p')=-E(p,p')= \frac{p p' + \nu^2 \omega \omega' }{p \omega' - p' \omega} \,, \\
&D(p,p')=\frac{1-2a}{4} (p \omega' - p' \omega) -\frac{1}{4} \frac{ (p-p')^2 +\nu^2 (\omega-\omega')^2 }{p \omega' - p' \omega} \,,\\
&G(p,p')=-L(p',p)=\frac{1-2a}{4}(p \omega' - p' \omega) -\frac{1}{4} \frac{\omega^2-\omega'^2}{p \omega' - p' \omega} \,,\\
&W(p,p')= i\nu \, .
\end{aligned}
\end{eqnarray}
Here $W$ corresponds to the contribution of the Wess-Zumino term and it does not actually depend on the particle momenta.
All the four remaining coefficients ${\mathcal T}_{ab}^{\gamma\delta },{\mathcal T}_{\alpha\beta}^{cd},{\mathcal T}_{a\beta}^{\gamma d},{\mathcal T}_{\alpha b}^{\gamma d}$ vanish in the bosonic case but will be switched on once fermions are
taken into account. The matrix ${\mathcal T}$ is recovered from its matrix elements as follows
\begin{eqnarray}
\nonumber
{\mathcal T}={\cal T}_{MN}^{PQ}\, E_P^M\otimes E_Q^N={\mathcal T}_{ab}^{cd}\, E_c^a\otimes E_d^b+{\mathcal T}_{\alpha\beta}^{\gamma\delta }\, E_\gamma^\alpha\otimes E_\delta^\beta+
{\mathcal T}_{a\beta}^{c\delta }\, E_c^a\otimes E_\delta ^\beta+{\mathcal T}_{\alpha b}^{\gamma d}\, E_\gamma^\alpha\otimes E_d^b\, ,
\end{eqnarray}
where $E_M^N$ are the standard matrix unities. For the reader convenience we present ${\mathcal T}$ as an explicit $16\times 16$ matrix\footnote{See appendix 8.5 of
\cite{Arutyunov:2006yd}
for the corresponding matrix in the undeformed case.}
{\scriptsize
\begin{eqnarray}
{\mathcal T}\equiv \left( \begin{array}{ccccccccccccccccccc}
{\cal A}_1&0&0&0&|&0&0&0&0&|&0&0&0&0&|&0&0&0&0\\
0&{\cal A}_2&0&0&|&{\cal A}_4&0&0&0&|&0&0&0&0&|&0&0&0 &0\\
0&0&{\cal A}_3&0&|&0&0&0&0&|&0&0&0&0&|&0&0&0&0\\
0&0&0&{\cal A}_3&|&0&0&0&0&|&0&0&0&0&|&0&0&0&0\\
-&-&-&-&-&-&-&-&-&-&-&-&-&-&-&-&-&-&-\\
0&{\cal A}_5&0&0&|& {\cal A}_2&0&0&0&|&0&0&0&0&|&0&0& 0 &0\\
0&0&0&0&|&0&{\cal A}_1&0&0&|&0&0&0&0&|&0&0&0&0\\
0&0&0&0&|&0&0&{\cal A}_3&0&|&0&0&0&0&|&0&0&0&0\\
0&0&0&0&|&0&0&0&{\cal A}_3&|&0&0&0&0&|&0&0&0&0\\
-&-&-&-&-&-&-&-&-&-&-&-&-&-&-&-&-&-&-\\
0&0&0&0&|&0&0&0&0&|&{\cal A}_8&0&0&0&|&0&0&0&0\\
0&0&0&0&|&0&0&0&0&|&0&{\cal A}_8&0&0&|&0&0&0&0\\
0&0&0&0&|&0&0&0&0&|&0&0& {\cal A}_6 &0&|&0&0&0&0\\
0&0&0&0&|& 0&0&0&0&|&0&0&0&{\cal A}_7&|&0&0&{\cal A}_9&0\\
-&-&-&-&-&-&-&-&-&-&-&-&-&-&-&-&-&-&-\\
0&0&0&0&|&0&0&0&0&|&0&0&0&0&|&{\cal A}_8&0&0&0\\
0&0&0&0&|&0&0&0&0&|&0&0&0&0&|&0&{\cal A}_8&0&0\\
0&0&0&0&|&0&0&0&0&|&0&0&0&{\cal A}_{10}&|&0&0&{\cal A}_7&0\\
0&0&0&0&|&0&0&0&0&|&0&0&0&0&|&0&0&0&{\cal A}_6\\
\end{array} \right) \, .\nonumber
\end{eqnarray}
}
Here the non-trivial matrix elements of ${\mathcal T}$ are given by
\begin{eqnarray}
&&{\cal A}_1=A+B\, ,\quad
{\cal A}_2=A\, ,\quad
{\cal A}_4=B-W\, ,\quad
{\cal A}_5=B+W\,, \quad{\cal A}_6=D+E\, ,
\\\nonumber
&&{\cal A}_6=D+E\, ,\quad
{\cal A}_7=D\, , \quad
{\cal A}_8=L\, ,\quad
{\cal A}_9=E-W=-{\cal A}_5 \, ,\quad
{\cal A}_{10}=E+W=-{\cal A}_4\, .
\end{eqnarray}
We conclude this section by pointing out that the found matrix ${\mathcal T}$ satisfies the classical Yang-Baxter equation
\begin{eqnarray}
[{\mathcal T}_{12}(p_1,p_2),{\mathcal T}_{13}(p_1,p_3)+{\mathcal T}_{23}(p_2,p_3)]+[{\mathcal T}_{13}(p_1,p_3),{\mathcal T}_{23}(p_2,p_3)]=0\,
\end{eqnarray}
for any value of the deformation parameter $\nu$.
\subsection{Comparison with the q-deformed S-matrix}
In this subsection we show that the perturbative bosonic world-sheet S-matrix coincides with the first nontrivial term in the large $g$ expansion of the q-deformed ${\rm AdS}_5\times {\rm S}^5\ $ S-matrix, in other words with the corresponding classical $r$-matrix\footnote{The difference with the expansion performed in \cite{Beisert:2010kk} is that we include the dressing factor in the definition of the S-matrix.}.
Let us recall that up to an overall factor the q-deformed ${\rm AdS}_5\times {\rm S}^5\ $ S-matrix is given by a tensor product of two copies of the $\alg{psu}(2|2)_q$-invariant S-matrix \cite{Beisert:2008tw} which is reviewed in appendix \ref{app:matrixSmatrix}.
Including the overall factor $S_{\alg{su}(2)}$ which is the scattering matrix in the $\alg{su}(2)$ sector, the complete S-matrix can be written in the form \cite{Ben}
\begin{eqnarray}
&&\hspace{-1.5cm}
{\mathbf S}=S_{\alg{su}(2)} S\, \hat{\otimes} \, S\, , ~~~S_{\alg{su}(2)}=\frac{e^{i a(p_2{\cal E}_1-p_1{\cal E}_2)}}{\sigma_{12}^2}\frac{x_1^++\xi}{x_1^-+\xi}\frac{x_2^-+\xi}{x_2^++\xi}\cdot
\frac{x_1^--x_2^+}{x_1^+-x_2^-}\frac{1-\frac{1}{x_1^-x_2^+}}{1-\frac{1}{x_1^+x_2^-}}\, ,
\end{eqnarray}
where $S$ is the $\alg{psu}(2|2)_q$-invariant S-matrix \eqref{Sqmat}, $\hat{\otimes} $ stands for the graded tensor product, $a$ is the parameter of the light-cone gauge \eqref{lcg}, $\sigma$ is the dressing factor, and ${\cal E}$ is the q-deformed dispersion relation \eqref{qdisp} whose large $g$ expansion starts with $\omega$. The dressing factor can be found by solving the corresponding crossing equation, and it is given by \cite{Ben}
\begin{equation}\label{eq:def-theta}
\sigma_{12}=e^{i\theta_{12}}\,,\quad \theta_{12} = \chi(x^+_1,x^+_2) + \chi(x^-_1,x^-_2)- \chi(x^+_1,x^-_2) - \chi(x^-_1,x^+_2),
\end{equation}
where
\begin{equation}\label{chi12}
\chi(x_1,x_2) = i \oint \frac{dz}{2 \pi i} \frac{1}{z-x_1} \oint \frac{dz'}{2 \pi i} \frac{1}{z'-x_2} \log\frac{\Gamma_{q^2}(1+\frac{i g}{2} (u(z)-u(z')))}{\Gamma_{q^2}(1-\frac{i g}{2} (u(z)-u(z')))}.
\end{equation}
Here $\Gamma_{q}(x)$ is the q-deformed Gamma function which for complex $q$
admits an integral representation \eqref{lnGovG} \cite{Ben}.
To develop the large $g$ expansion of the q-deformed ${\rm AdS}_5\times {\rm S}^5\ $ S-matrix, one has to assume that $q=e^{-\upsilon/g}$ where $\upsilon$ is a deformation parameter which is kept fixed in the limit $g\to\infty$, and should be related to $\nu$. Then, due to the factorisation of the perturbative bosonic world-sheet S-matrix and the q-deformed ${\rm AdS}_5\times {\rm S}^5\ $ S-matrix, it is sufficient to compare the ${\cal T}$-matrix \eqref{cTmatr} with the ${\mathbf T}$-matrix appearing in the expansion of the ``square root'' of $\mathbf S$
\begin{equation}\label{qTpert}
S_{\alg{su}(2)}^{1/2}\, \mathbbm{1}_g\, S=\mathbbm{1} +\frac{i}{g}{\mathbf T}\, ,
\end{equation}
where $\mathbbm{1}_g$ is the graded identity which is introduced so that the expansion starts with $\mathbbm{1}$.
The only term which is not straightforward to expand is the $S_{\alg{su}(2)}$ scalar factor because it contains the dressing phase $\theta_{12}$.
The scalar factor obviously can contribute only to the part of the ${\mathbf T}$-matrix proportional to the identity matrix. Since in the expansion of the $\alg{psu}(2|2)_q$-invariant S-matrix \eqref{Sqmat}, $\mathbbm{1}_gS=\mathbbm{1}+{i\over g}r$, the element $r_{11}^{11}$ is equal to 0 (because $a_1=1$) it is convenient to subtract ${\cal T}_{11}^{11}\mathbbm{1} = {\cal A}_1\mathbbm{1}$ from the ${\cal T}$-matrix and compare the resulting matrix with the classical $r$-matrix. One should obviously remove the off-diagonal terms from the classical $r$-matrix which appear due to the presence of fermions in the full superstring action \eqref{defLag}.
With this done, one finds that they are equal to each other provided $\upsilon = \nu$, and therefore $q=e^{-\nu/g}$ is real. Thus, to show that ${\mathbf T}={\cal T}$ one should demonstrate that
$ {\cal A}_1$ is equal to the $1/g$ term in the expansion of $S_{\alg{su}(2)}^{1/2}$.
To this end one should find the large $g$ expansion of the dressing phase $\theta_{12}$ which is done by first expanding the ratio of $\Gamma_{q^2}$-functions in \eqref{chi12} with $u(z)$ and $u(z')$ being kept fixed. This is done in appendix \ref{app:qGamma}, see \eqref{qGamma1}.
Next, one combines it with the expansion of the $\frac{1}{z-x_1^\pm} \frac{1}{z'-x_2^\pm} $ terms which appear in the integrand of \eqref{eq:def-theta}. As a result one finds that the dressing phase is of order $1/g$ just as it was in the undeformed case \cite{AFS}. We have not tried to compute the resulting double integrals analytically but we have checked numerically that the element
${\cal A}_1$ is indeed equal to the $1/g$ term in the expansion of $S_{\alg{su}(2)}^{1/2}$ if the deformation parameter $\nu$ satisfies $\nu<1/\sqrt 2$. At $\nu=1/\sqrt 2$ the integral representation for the dressing factor breaks down but it is unclear to us if it is a signal of a genuine problem with the q-deformed S-matrix. In fact it is not difficult to extract from ${\cal A}_1$ the leading term in the large $g$ expansion of the dressing phase which appears to be very simple
\begin{equation}
\theta_{12}= \frac{\nu^2 \left(\omega _1-\omega _2\right)+p_2^2 \left(\omega _1-1\right)-p_1^2 \left(\omega _2-1\right)}{2g
\left(p_1+p_2\right)} +\cdots\,.
\end{equation}
It would be curious to derive this expression from the double integral representation. Note that doing this double integral one also could get the full AFS order of the phase.
\section{Conclusions}
In this work we successfully matched in the large tension limit the tree-level bosonic S-matrix arising from the sigma-model on the deformed ${\rm AdS}_5\times {\rm S}^5\ $ space with the
q-deformed S-matrix obtained from symmetries. There are many other important issues to be addressed.
\smallskip
We identified NSNS background fields in the string frame. More studies are needed however to extract RR fields since the latter couple directly to fermionic degrees of freedom.
Rather intricate field redefinitions should be performed to bring the deformed action to the standard one for Type IIB superstring in an arbitrary supergravity background,
thus allowing the identification of the full bosonic background. It might be easier in fact just to use the NSNS background fields and the type IIB supergravity equations of motion to find the full supergravity background \cite{ABF}.
\smallskip
Next, the matching of S-matrices, successful at tree level, can be further extended by computing admittedly more complicated loop corrections to the tree-level scattering matrix
of the light-cone sigma-model; this also requires taking fermions into account. It is natural to expect that the deformation parameter $\nu$ undergoes a non-trivial renormalization to fit the parameter $q$ entering the exact, {\it i.e.}~all-loop,
q-deformed S-matrix.
\smallskip
We also showed that in the large tension limit the conjectured dispersion relation \eqref{qdisp} turns into the perturbative one \eqref{omega}. It would be interesting to find an $\eta$-deformed giant magnon solution \cite{HM} which would provide additional evidence in favour of \eqref{qdisp}. In the case of the finite angular momentum the corresponding solution would also provide important information about the structure of the finite size corrections \cite{AFZ} in the $\eta$-deformed theory.
\smallskip
It is also interesting to find explicit spinning string solutions of the $\eta$-deformed bosonic sigma-model. Due to the singularity of the $\eta$-deformed AdS a particularly interesting solution to analyse would be the GKP string and its generalisation \cite{GKP,FT02}. Then, in the case of $\text{AdS}$, substituting the spinning Ansatz in the sigma-model
equations of motion leads to the emergence \cite{Arutyunov:2003uj} of the Neumann model, a famous finite-dimensional integrable system. One may hope that studies of the $\eta$-deformed sigma-model in this context may reveal new integrable finite-dimensional systems which can be described as deformations of the Neumann model.
Furthermore, known finite-gap integration techniques can be applied to obtain a wider class of solutions that generalize the solutions of the Neumann system. Normally they are
described by a certain algebraic curve which is supposed to emerge from the Bethe Ansatz based on the exact q-deformed S-matrix in the semi-classical limit. This would serve as another
non-trivial check that the two models, one based on the explicitly known deformed action and the other based on the exact quantum group symmetry, have a good chance to describe the same physics.
\smallskip
One can also adopt the logic of the undeformed case construction and use the exact q-deformed S-matrix to engineer the mirror TBA equations for real $q$; a solution of this problem is under way
\cite{Arutyunov:2014ota}.
\smallskip
With the knowledge of a complete supergravity background and its symmetries for the deformed case,
one can approach perhaps the most interesting question about the dual gauge theory.
Since the deformation affects the isometries of the AdS space, the theory will be neither conformal nor Lorentz invariant. Since there is a $B$-field on the string theory side,
one may expect that this theory is a non-commutative deformation of ${\cal N}=4$ super Yang-Mills in the sense of the Moyal star product
with a hidden quantum group symmetry which would include the two copies of the
$\alg{psu}_q(2|2)$ algebra.
It would be fascinating to construct such a theory explicitly.
\bigskip
\section*{Acknowledgments}
\noindent
We thank Marius de Leeuw, Stijn van Tongeren and Benoit Vicedo for useful discussions.
G.A. and R.B. acknowledge support by the Netherlands Organization for Scientific Research (NWO) under the VICI grant 680-47-602.
The work by G.A. and R.B. is also a part of the ERC Advanced grant research programme No. 246974, {\it ``Supersymmetry: a
window to non-perturbative physics"} and of the D-ITP consortium, a program of the NWO that is funded by the Dutch Ministry of Education, Culture and Science (OCW).
S.F. is supported by a DFG grant in the framework of the SFB 647 ``Raum - Zeit - Materie. Analytische und Geometrische Strukturen''
and by the Science Foundation Ireland under Grant 09/RFP/PHY2142.
|
1,116,691,497,549 | arxiv | \section{Cross-metric Correlation Analysis}\label{sec:cross_metrics_corelation}
\paragraph{Correlations across meaning preservation metrics} Figure~\ref{fig:cross_metric_meaning} presents a cross-metric correlation-based analysis of the different approaches for measuring meaning preservation. We observe consistent trends across languages: methods that are similar in nature
correlate well with each other. Concretely, across settings, $n$-gram based methods (i.e., \textsc{bleu}\xspace, \textsc{meteor}, and chr\textsc{f}) yield $0.8-0.9$ correlation scores. The latter also holds when looking at correlations within the group of embedding-based methods (cosine and \textsc{wmd}) and and group of \textsc{sts}\xspace approaches for \textsc{en}\xspace, \textsc{fr}\xspace, and \textsc{it}\xspace, while for \textsc{br-pt}\xspace we observe that the correlation between \textsc{xlm-r} and m\textsc{bert} based approaches is smaller ($0.7$ vs. $0.8$ for other languages).
Finally, $n$-gram approaches correlate better with \textsc{sts}\xspace methods (with correlations in the range of $0.7-0.8$) across languages, while the lowest correlations ($0.5-0.6$) are observed between embedding-based methods (i.e., cosine, \textsc{wmd}) and each of the rest metrics.
\paragraph{Correlations within and across formality-fluency metrics} Figure~\ref{fig:cross_metric_fluency_formality} presents results of cross-metric correlations for the studied approaches that capture formality transfer and fluency. For formality, each of the translate-based approaches (i.e., \textsc{translate-train} and \textsc{translate-test}) yields high correlations ($0.8-0.9$) between models that fine-tune \textsc{xlm-r} vs. m\textsc{bert}, while their correlations decrease ($0.7$) for \textsc{it}\xspace and \textsc{br-pt}\xspace in the zero-shot setting. Finally, pseudo-perplexity metrics extracted from \textsc{xlm-r}---that consists
the best correlated metric with human judgments for fluency---yield positive correlations with all formality metrics.
\fi
\section{System-level Analysis}\label{sec:pairwise_results}
Table~\ref{tab:system_pairise} presents the number of correct system-level pair-wise comparisons of automatic metrics based on human judgments. For \textsc{sts}, chr\textsc{f}, \textsc{f.reg*}, \textsc{f.class*}, and \textsc{pseudo-lkl*}, system-level scores are extracted via averaging sentence-level scores. For s-\textsc{bleu}\xspace and r-\textsc{bleu}\xspace the system scores are extracted at the corpus-level. The total number of pairwise comparisons for each language is $10$ (given access to $5$ systems). Among the meaning preservation metrics (i.e., \textsc{sts}, s-\textsc{bleu}\xspace, and chr\textsc{f}), chr\textsc{f} yields the highest number of correct comparisons (i.e., $37$ out of $40$ for all languages). The formality regression models (i.e., \textsc{f.reg*}) result in correct rankings more frequently than the formality classifiers (i.e., \textsc{f.class*}) yielding $35$ out of $40$ correct comparisons. Reference-\textsc{bleu}\xspace (i.e., r-\textsc{bleu}\xspace) is compared with overall ranking judemnts. It ranks $8$ out of $10$ systems correctly for \textsc{en}\xspace, \textsc{fr}\xspace, and \textsc{br-pt}\xspace and only $6$ for \textsc{it}\xspace. Finally, perplexity (i.e., \textsc{ppl}) results in the fewest correct rankings at system-level (i.e., $22$ out of $40$), despite correlating well with human judgments at the segment-level.
Additionally, in Figure~\ref{fig:system_ranking_per_dimenstion} we visualize the differences between relative rankings induced by human judgments and the best segment-level correlated metrics for each dimension, averaged per system.
\begin{figure}[!t]
\centering
\includegraphics[width=0.4\textwidth]{figs/overall_xformal.pdf}
\caption{Difference in relative ranking between human judgments and automatic metrics across systems (i.e, represented by different markers) for different evaluation dimensions.
\textsc{sts}\xspace, s-\textsc{bleu}\xspace and ch\textsc{rf} are compared with meaning rankings, r-\textsc{bleu}\xspace (reference-\textsc{bleu}\xspace) with overall, \textsc{xlm-r} classifiers (*\textsc{f.class}) and regression (*\textsc{f.ref}) models with formality, and \textsc{xlm-r} pseudo-perplexity (*\textsc{ppl}) with fluency.
}
\label{fig:system_ranking_per_dimenstion}
\end{figure}
\begin{table*}[!t]
\centering
\scalebox{0.8}{
\begin{tabular}{lrrrrrrrrrrr}
\rowcolor{gray!10}
& \multicolumn{4}{c}{\textsc{meaning}} & \multicolumn{2}{c}{\textsc{formality}} & \textsc{fluency} & \multicolumn{4}{c}{\textsc{overall}}\\
Language & \textsc{sts} & s-\textsc{bleu} & r-\textsc{bleu} & chr\textsc{f} & \textsc{f.reg*} & \textsc{f.class*} & \textsc{ppl} & r-\textsc{bleu} & \textsc{f.reg*} & chr\textsc{f} & pseudo-\textsc{ll} \\
\textsc{en} & 6 & 7 & 8 & 7 & 9 & 9 & 7 & 8 & 6 & 6 & 3\\
\textsc{br-pt}\xspace & 10 & 9 & 7 & 9 & 7 & 7 & 5 & 8 & 8 & 4 & 5\\
\textsc{fr}\xspace & 9 & 7 & 10 & 10 & 10 & 9 & 5 & 8 & 6 & 3 & 7\\
\textsc{it}\xspace & 10 & 10 & 8 &10 & 9 & 8 & 5 & 6 & 5 & 9 & 4\\
\textsc{all} & 35 & 33 & 35 & 36 & 35 & 33 & 22 & 30 & 25 & 22 & 19\\
\end{tabular}}
\caption{Number of correct system level pair-wise comparisons between $5$ systems for each language.}
\label{tab:system_pairise}
\end{table*}
\section{Evaluated Systems Details}\label{sec:evaluated_systems}
For each of \textsc{br-pt}\xspace, \textsc{it}\xspace, and \textsc{fr}\xspace,
outputs are sampled from:
\begin{enumerate}
\item Rule-based systems consisting of hand-crafted transformations (e.g., fixing casing, normalizing punctuation, expanding contractions, etc.);
\item Round-trip translation models that pivot to \textsc{en}\xspace and backtranslate to the original language;
\item Bi-directional neural machine translation~(\textsc{mt}) models that employ side constraints to perform style transfer for both directions of formality (i.e., informal$\leftrightarrow$formal)---trained on (machine) translated informal-formal pairs of an English parallel corpus (i.e., \textsc{gyafc});
\item Bi-directional \textsc{nmt} models that augment the training data of 3. via backtranslation of informal sentences;
\item A multi-task variant of 3. that augments the training data with parallel-sentences from bilingual resources (i.e., OpenSubtitles) and learns to translate jointly between and across languages.
\end{enumerate}
For \textsc{en}\xspace, the outputs were sampled from:
\begin{enumerate}
\item A rule-based system of similar transformations to ones for \textsc{br-pt}\xspace, \textsc{fr}\xspace, and \textsc{it}\xspace;
\item A phrase-based machine translation model trained on informal-formal pairs of \textsc{gyafc};
\item An \textsc{nmt} model trained on \textsc{gyafc} to perform style transfer uni-directionaly;
\item A variant of 3. that incorporates a copy-enriched mechanism that enables direct copying of words from input;
\item A variant of 4. trained on additional back-translated data of target style sentences using 2.
\end{enumerate}
In general, neural models performed best for all languages according to overall human judgments, while
the simpler baselines perform closer to the more advanced neural models for \textsc{br-pt}\xspace, \textsc{fr}\xspace, and \textsc{it}\xspace.
For each evaluation dimension $500$ outputs are evaluated for \textsc{en}\xspace and $100$ outputs per system for \textsc{br-pt}\xspace, \textsc{fr}\xspace, and \textsc{it}\xspace.
\section{Meaning Preservation Metrics (reference-based)}
Table~\ref{tab:meaning_preservation_metrics_ref_based_correlations_with_human} presents supplemental results on meaning preservation metrics for reference-based settings.
\begin{table}[!ht]
\centering
\scalebox{0.75}{
\begin{tabular}{lrrrr}
\rowcolor{gray!10}
\textbf{\textsc{metric}} & \textbf{\textsc{en}} & \textbf{\textsc{it}} & \textbf{\textsc{pt}} & \textbf{\textsc{fr}} \\
r-\textsc{bleu} & 0.306 &0.004 &0.047 &0.122 \\
\textsc{meteor} & 0.279 &-0.005 &0.061 &0.124 \\
chr\textsc{f} & 0.319 &0.065 &0.044 &0.174 \\
\textsc{wmd} & 0.316 &0.039 &0.098 &0.198 \\
Cosine & 0.218 &0.027 &0.048 &0.161 \\
\textsc{bert}-score & 0.359 &-0.023 &0.054 &0.112 \\
m\textsc{bert} (\textsc{translate-train}) & 0.369 &0.077 &0.167 &0.165 \\
m\textsc{bert} (\textsc{zero-shot}) & - &0.124 &\textbf{0.197} &0.179 \\
\textsc{xlm-r} (\textsc{translate-train}) & \textbf{0.385} &\textbf{0.183} &0.136 &\textbf{0.259} \\
\textsc{xlm-r} (\textsc{zero-shot}) & - &0.179 &0.153 &0.258 \\
\end{tabular}}
\caption{Spearman's $\rho$ correlation of meaning preservation metrics for reference-based meaning.}
\label{tab:meaning_preservation_metrics_ref_based_correlations_with_human}
\end{table}
\section{Conclusions}
Automatic (and human) evaluation processes are well-known problems for the field of Natural Language Generation \cite{howcroft-etal-2020-twenty, clinciu-etal-2021-study} and the burgeoning subfield of \textsc{st}\xspace is not immune. \textsc{st}\xspace, in particular, has suffered from a lack of standardization of automatic metrics, a lack of agreement between human judgments and automatics metrics, as well as a blindspot to developing metrics for languages other than English.
We address these issues by conducting the first controlled multilingual evaluation for leading \textsc{st}\xspace metrics with a focus on formality, covering metrics for $3$ evaluation dimensions and overall ranking for $4$ languages. Given our findings, we recommend the formality style transfer community adopt the following best practices:
\paragraph{1. Formality} \textsc{xlm-r} formality regression models in the \textsc{zero-shot} cross-lingual transfer setting yields the clear best metrics across all four languages as it correlates very well with human judgments.
However, the commonly used binary classifiers do not generalize across languages (due to misleadingly over-predicting formal labels). We propose that the field use regression models instead since they are designed to capture a wide spectrum of formality rates.
\paragraph{2. Meaning Preservation} We recommend using chr\textsc{f} as it exhibits strong correlations with human judgments for all four languages.
We caution against using \textsc{bleu}\xspace for this dimension, despite its overwhelming use in prior work as both its reference and self variants do not correlate as strongly as other more recent metrics.
\paragraph{3. Fluency} \textsc{xlm-r} is again the best metric (in particular for French). However, it does not correlate well with human judgments as compared to the other two dimensions.
\paragraph{4. System-level Ranking} chr\textsc{f} and \textsc{xlm-r} are the best metrics using a pairwise comparison evaluation. However, an ideal evaluation would be to have a large number of systems with which to draw reliable correlations.
\paragraph{5. Cross-lingual Transfer} Our results support using
\textsc{zero-shot} transfer
instead of machine translation to port metrics from English to other languages for formality transfer tasks.\\
We view this work as a strong point of departure for future investigations of \textsc{st}\xspace evaluation. Our work first calls for
exploring how these evaluation metrics generalize to other \textit{styles} and \textit{languages}. Across the different ways of defining style evaluation (either automatic or human), prior work has mostly focused on the three main dimensions covered in our study. As a result, although our meta-evaluation on \textsc{st} metrics focuses on formality as a case study, it can inform the evaluation of other style definitions (e.g., politeness, sentiment, gender, etc.). However, more empirical evidence is needed to test the applicability of the best performing metrics for evaluating style transfer beyond formality.
Our work suggests that the top metrics based on \textsc{xlm-r} and chr\textsc{f} are robust across $4$ Romance languages; yet, our conclusions and recommendations are currently limited to \textit{this} set of languages. We hope that future work in multilingual style transfer will allow for testing their generalization to a broader spectrum of languages and style definitions.
Furthermore, our study highlights that more research is needed on automatically ranking systems. For example, one could build a metric that combines metrics' outputs for the three dimensions, or one could develop a singular metric.
In line with \citet{briakou-etal-2021-review}, our study also calls for releasing more human evaluations and more system outputs to enable robust evaluation.
Finally, there is still room for improvement in assessing how fluent a rewrite is.
Our study provides a framework to address these questions systematically and calls for \textsc{st}\xspace papers to standardize and release data to support larger-scale evaluations.
\section*{Acknowledgements}
We thank
Sudha Rao for providing references and materials of the \textsc{gyafc} dataset,
Jordan Boyd-Graber, Pedro Rodriguez, the \textsc{clip} lab at \textsc{umd}, and the
\textsc{emnlp} reviewers for their helpful and constructive comments.
\section{Experiment Settings}\label{sec:conditions}
\begin{table*}[!t]
\centering
\scalebox{0.83}{
\begin{tabular}{lllllr}
\rowcolor{gray!10}
\textbf{\textsc{dimension}}& \textbf{\textsc{language code}} & \textbf{\textsc{dataset}} & \textbf{\textsc{lineage}} & \textbf{\textsc{labels}} & \textbf{\textsc{size}} \\
\addlinespace[0.2cm]
%
\multirow{2}{*}{Formality} & \multirow{2}{*}{\textsc{en}\xspace} & \textsc{gyafc} & \citet{rao-tetreault-2018-dear} & informal vs. formal & $105$K\\
& & Formality ratings & \citet{pavlick-tetreault-2016-empirical} & $[$ $-3$, $-2$, $-1$, $0$, $1$, $2$, $3$ $]$ & $5$K\\
%
\addlinespace[0.2cm]
%
\addlinespace[0.2cm]
\multirow{1}{*}{Meaning} & \textsc{en}\xspace & \textsc{sts} & \citet{cer-etal-2017-semeval} & $[$ $1$, $2$, $3$, $4$, $5$ $]$ & $5$K \\
\addlinespace[0.2cm]
\multirow{1}{*}{Fluency} & \textsc{en}\xspace, \textsc{br-pt}, \textsc{it}, \textsc{fr} & OpenSubtitles & \citet{lison-tiedemann-2016-opensubtitles2016} & \textit{none} & $1$M \\
\end{tabular}}
\caption{Details on training data used for model-based metrics across the three \textsc{st} evaluation aspects.}
\label{tab:training_data}
\end{table*}
\paragraph{Supervised Metrics} For all supervised model-based approaches,
we experiment with fine-tuning two multilingual pre-trained language models:
\begin{inparaenum}
%
\item multilingual \textsc{bert}, dubbed \textbf{m\textsc{bert}}~\cite{devlin-etal-2019-bert}---a transformer-based model pre-trained
with a masked language model objective on the concatenation of monolingual Wikipedia corpora from the $104$ languages with the largest Wikipedias.
%
\item \textbf{\textsc{xlm-r}}~\cite{conneau-etal-2020-unsupervised}---a transformer-based masked language model trained on $100$ languages using monolingual CommonCrawl data.
%
\end{inparaenum}
All models are based on the HuggingFace Transformers~\cite{wolf-etal-2020-transformers}\footnote{\url{https://github.com/huggingface/transformers}} library. We fine-tune with the Adam optimizer~\cite{KingmaB14}, a batch size of $32$, and a learning rate of $5\mathrm{e}{-5}$ for $3$ and $5$ epochs for classification and regression tasks, respectively. We perform a grid search on held-out validation sets over learning rate with values: $2\mathrm{e}{-3}$, $2\mathrm{e}{-4}$, $2\mathrm{e}{-5}$, and $5\mathrm{e}{-5}$ and over number of epochs with values: $3$, $5$, and $8$.
\paragraph{Cross-lingual Transfer} For supervised model-based methods that rely on the availability of human-annotated instances to train dedicated models for specific tasks, we experiment with three standard cross-lingual transfer approaches~(e.g., \citet{pmlr-v119-hu20b}):
\begin{inparaenum}
\item \textsc{zero-shot} trains a single model on the \textsc{en}\xspace training data and evaluates it on the original test data for each language;
\item \textsc{translate-train} uses machine translation~(\textsc{mt}) to obtain training data in each language through translating the \textsc{en}\xspace training set---and trains independent systems for each language;
\item \textsc{translate-test} trains a single model on the \textsc{en}\xspace training data and evaluates it on the test data that are translated into \textsc{en}\xspace using \textsc{mt}.
\end{inparaenum}
\paragraph{Unsupervised Metrics} For meaning preservation metrics, we use the open-sourced implementations of:
\citet{post-2018-call} for \textsc{bleu}\xspace~\cite{papineni-etal-2002-bleu};
\citet{banerjee-lavie-2005-meteor} for \textsc{meteor};
\citet{popovic-2015-chrf} for chr\textsc{f}.\footnote{\url{https://github.com/mjpost/sacrebleu}}\textsuperscript{,}\footnote{\url{https://www.cs.cmu.edu/~alavie/METEOR/}}\textsuperscript{,}\footnote{\url{https://github.com/m-popovic/chrF}}
For \textsc{bert}-score we use the implementation of~\citet{bertscore};\footnote{\url{https://github.com/Tiiiger/bert_score}}
non-contextualized embeddings-based approaches are based on \texttt{fastText} pre-trained embeddings.\footnote{\url{https://fasttext.cc}}
For fluency metrics, we use the implementation
of~\citet{salazar-etal-2020-masked} for computing pseudo-likelihood.\footnote{\url{https://github.com/awslabs/mlm-scoring}}
\textsc{ppl} and \textsc{ll} scores are extracted from a
$5$-gram Ken\textsc{lm}\xspace model \citep{heafield-2011-kenlm}.\footnote{\url{https://github.com/kpu/kenlm}}
\paragraph{Training Data} Table~\ref{tab:training_data} presents
statistics on the training data used for supervised and unsupervised models across the $3$ \textsc{st} evaluation aspects.
For datasets that are only available for \textsc{en}\xspace, we use
the already available \textit{machine} translated resources for \textsc{sts} \footnote{\url{https://github.com/PhilipMay/stsb-multi-mt}} and formality datasets \cite{xformal}. The former employs the \texttt{DeepL} service (no information of translation quality is available) while the latter uses the \textsc{aws} translation service\footnote{\url{https://aws.amazon.com/translate}} (with reported \textsc{bleu} scores of $37.16$ (\textsc{br-pt}\xspace), $33.79$ (\textsc{fr}\xspace), and $32.67$ (\textsc{it}\xspace)).\footnote{\textsc{bleu}\xspace scores were computed on $5{,}000$ randomly sampled data from OpenSubtitles.} The Ken\textsc{lm}\xspace models for all the languages are trained on $1$M randomly sampled sentences from the OpenSubtitles dataset \cite{lison-tiedemann-2016-opensubtitles2016}.
\section{Introduction}
Textual style transfer (\textsc{st}\xspace) is defined as a generation task where a text sequence is paraphrased while controlling one aspect of its style~\cite{jin2021deep}.
For instance, the informal sentence in Italian \textit{``in bocca al lupo!''} (i.e., ``good luck'') is rewritten to the formal version \textit{``Ti rivolgo un sincero augurio!''} (i.e., ``I send you a sincere wish!'').
Despite the growing attention on \textsc{st}\xspace in the \textsc{nlp}\xspace literature \citep{jin2021deep}, progress is hampered by a lack of standardized and reliable automatic evaluation metrics.
Standardizing the latter would allow for quicker development of new methods and comparison to prior art without relying on time and cost-intensive human evaluation that is currently employed by more than $70\%$ of \textsc{st}\xspace papers~\cite{briakou-etal-2021-review}.
\textsc{st}\xspace is usually evaluated across three dimensions: style transfer (i.e., has the style of the generated output changed as intended?), meaning preservation (i.e., are the semantics of the input preserved?), and fluency (i.e., is the output well-formed?). As we will see, a wide range of automatic evaluation metrics and models has been used to quantify each of these dimensions.
For example, prior work has employed as many as nine different automatic systems to rate formality alone (see Table~\ref{tab:status}).
However, it is not clear how different automatic metrics compare to each other and how well they agree with human judgments.
Furthermore, previous studies of automatic evaluation have exclusively focused on the English language ~\cite{yamshchikov2020styletransfer,pang-2019-towards, Pang2019UnsupervisedEM, tikhonov-etal-2019-style, mir-etal-2019-evaluating}; yet, \textsc{st}\xspace requires evaluation methods that generalize reliably beyond English.
We address these limitations by conducting a controlled empirical comparison of commonly used automatic evaluation metrics. Concretely, for all three evaluation dimensions, we compile a list of different automatic evaluation approaches used in prior \textsc{st}\xspace work and study
how well they correlate with human judgments. We choose to build on available resources as collecting human judgments across the evaluation dimensions is a costly process that requires recruiting fluent speakers in each language addressed in evaluation. While there are many stylistic transformations in \textsc{st}\xspace, we conduct our study through the lens of formality style transfer (\textsc{f}o\textsc{st}\xspace), which is one of the most popular style dimensions considered by past \textsc{st}\xspace work \citep{jin2021deep,briakou-etal-2021-review} and for which reference outputs and human judgments are available for four languages: English, Brazilian-Portuguese, French, and Italian.
\begin{itemize}
\item We contribute a meta-evaluation study that is not only the first \textit{large-scale} comparison of automatic metrics for \textsc{st}\xspace but is also the first work to investigate the robustness of these metrics in \textit{multilingual} settings.
%
\item We show that automatic evaluation approaches based on a formality regression model fine-tuned on \textsc{xlm-r} and the chr\textsc{f} metric correlate well with human judgments for style transfer and meaning preservation, respectively, and propose that the field adopts their usage. These metrics are shown to work well \textit{across} languages, and not just in English.
%
\item We show that framing style transfer evaluation as a binary classification task is problematic and propose that the field treats it as a regression task to better mirror human evaluation.
%
\item Our analysis code and meta-evaluation files with system outputs are made public to facilitate further work in developing better automatic metrics for \textsc{st}\xspace: \url{https://github.com/Elbria/xformal-FoST-meta}.
\end{itemize}
\section{Evaluating Evaluation Metrics}
We evaluate evaluation metrics (described in~\S\ref{sec:automatic_details}) for multilingual \textsc{f}o\textsc{st}\xspace, in four languages
for which human evaluation judgments (described in~\S\ref{sec:human_details}) on \textsc{f}o\textsc{st}\xspace system outputs are available.
\subsection{Human Judgments}\label{sec:human_details}
We use human judgments collected by prior work of ~\citet{rao-tetreault-2018-dear} for \textsc{en}\xspace and \citet{xformal} for \textsc{br-pt}\xspace, \textsc{fr}\xspace, and \textsc{it}\xspace. We include details on their annotation frameworks, the quality of human judges, and the evaluated systems below.
\paragraph{Human Annotations} We briefly describe the annotation frameworks employed by~\citet{rao-tetreault-2018-dear} and \citet{xformal} to collect human judgments for each evaluation aspect:
\begin{inparaenum}
\item \textbf{formality} ratings are collected---for each system output---on a $7$-point discrete scale, ranging from $-3$ to $+3$, as per~\citet{DBLP:journals/corr/Lahiri15}
\textit{
(Very informal, Informal, Somewhat Informal, Neutral, Somewhat Formal, Formal. Very Formal)};
\item \textbf{meaning preservation} judgments adopt the Semantic Textual Similarity annotation scheme of~\citet{agirre-etal-2016-semeval}, where an
informal input and its corresponding formal system output are rated on a scale from $1$ to $6$ based on their similarity
\textit{
(Completely dissimilar, Not equivalent but on same topic, Not equivalent but share some details, Roughly equivalent, Mostly equivalent, Completely equivalent)};
\item \textbf{fluency} judgments are collected for each system output on a discrete scale of $1$ to $5$, as per~\citet{heilman-etal-2014-predicting}
\textit{
(Other, Incomprehensible, Somewhat Comprehensible, Comprehensible, Perfect)};
\item \textbf{overall} judgments are collected following a \textit{relative ranking} approach: all system outputs are ranked in order of their formality, taking into account both meaning preservation and fluency.
\end{inparaenum}
\paragraph{Human Annotators} Both studies recruited workers
from the Amazon Mechanical Turk platform after employing quality control methods to exclude poor quality workers (i.e., manual checks for \textsc{en}\xspace, and qualification tests for \textsc{br-pt}\xspace, \textsc{fr}\xspace, and \textsc{it}\xspace). For all human evaluations and languages \citet{xformal} report at least moderate inter-annotator agreement.
\paragraph{Evaluated Systems} The evaluated system outputs were sampled from $5$ \textsc{f}o\textsc{st}\xspace models for each language, spanning a range of simple baselines to neural architectures \cite{rao-tetreault-2018-dear, xformal}. We also include detailed descriptions of them in Appendix~\ref{sec:evaluated_systems}. For each evaluation dimension $500$ outputs are evaluated for \textsc{en}\xspace and $100$ outputs per system for \textsc{br-pt}\xspace, \textsc{fr}\xspace, and \textsc{it}\xspace.
\subsection{Evaluation Metrics}\label{sec:automatic_details}
For the \textsc{f}o\textsc{st}\xspace evaluation aspects described below, we cover a broad spectrum of approaches that range from dedicated models for the tasks at hand to more lightweight methods relying on unsupervised approaches and automated metrics.
\paragraph{Formality} We benchmark model-based approaches that fine-tune multilingual pre-trained language models (i.e., \textsc{xlm-r}, m\textsc{bert}), where the task of formality detection is modeled either as a \textbf{binary classification} task (i.e., formal vs. informal), or as a \textbf{regression} task that predicts different formality levels on an ordinal scale.
\paragraph{Meaning Preservation} We evaluate the \textsc{bleu}\xspace score ~\cite{papineni-etal-2002-bleu} of the system output compared to the reference rewrite (r-\textsc{\textsc{bleu}\xspace}) since it is the dominant metric in prior work.
Prior reviews of meaning preservation metrics for paraphrase and sentiment \textsc{st}\xspace tasks in \textsc{en}\xspace \citep{yamshchikov2020styletransfer} cover $n$-gram metrics and embedding-based approaches.
We consider three additional metric classes to compare system outputs with inputs, as human annotators do:
\begin{enumerate}
%
\item \textbf{n-gram based metrics} include: s-\textsc{bleu} (self-\textsc{bleu}\xspace that compares system outputs with their inputs as opposed to references, i.e., r-\textsc{bleu}\xspace), \textsc{meteor}~\cite{banerjee-lavie-2005-meteor} based on the harmonic mean of unigram precision and recall while accounting for synonym matches, and
chr\textsc{F}~\cite{popovic-2015-chrf} based on the character $n$-gram F-score;
%
\item \textbf{embedding-based methods} fall under the category of unsupervised evaluation approaches that rely on either
\textit{contextual} word representations extracted from pre-trained language models or
\textit{non-contextual pre-trained word embeddings}
(e.g., word2vec~\cite{w2v}; Glove~\cite{pennington-etal-2014-glove}).
For the former, we use \textsc{bert}-score~\cite{bertscore}
which computes the similarity between each output token and each reference token based on \textsc{bert} contextual embeddings. For the latter, we experiment with two similarity metrics: the first is the cosine distance between the
sentence-level feature representations of the compared texts extracted via averaging their word embeddings; the second is the
\textit{Word Mover's Distance} (\textsc{wmd}) metric of~\citet{pmlr-v37-kusnerb15} that measures the dissimilarity between two texts as the minimum amount of distance that the embedded words of one text need to ``travel" to reach the word embeddings of the other;
%
\item \textbf{semantic textual similarity (\textsc{sts}) models}
constitute supervised methods that we model via fine-tuning multilingual pre-trained language models (i.e., \textsc{xlm-r}, m\textsc{bert}) to predict a semantic similarity score for a pair of texts on an ordinal scale.
\end{enumerate}
\paragraph{Fluency} We experiment with \textbf{perplexity}~(\textsc{ppl}) and \textbf{likelihood}~(\textsc{ll}) scores based on probability scores of language models trained from scratch (e.g., Ken\textsc{lm}~\cite{heafield-2011-kenlm}), as well as \textbf{pseudo-likelihood scores}~(\textsc{pseudo-ll}) extracted from pre-trained masked language models similarly to \citet{salazar-etal-2020-masked}, by masking sentence tokens one by one.
\section{Experimental Results}
We analyze the results of comparing the outputs from the several automatic metrics to their human-generated counterparts for formality style transfer (\S\ref{sec:formality_results}), meaning preservation (\S\ref{sec:meaning_results}), fluency (\S\ref{sec:fluency_results}) via conducting segment-level analysis---and then, turn into analyzing system-level rankings to evaluation overall task success (\S\ref{sec:overall_results}).
\input{tables/formality_classifier}
\input{tables/formality_regression}
\subsection{Formality Transfer Metrics}
\label{sec:formality_results}
The field is divided on the best way to evaluate the style dimension \---\ formality in our case.
Practitioners use either a binary approach (is the new sentence formal or informal?) or a regression approach (how formal is the new sentence?). We discuss the first approach and its limitations in \S~\ref{sec:binary_classifiers}, before moving to regression in \S~\ref{sec:regression_models}.
\subsubsection{Evaluating Binary Classifiers}\label{sec:binary_classifiers}
As discussed in~\S\ref{sec:status}, the vast majority of \textsc{f}o\textsc{st}\xspace works evaluate style transfer based on the accuracy of a binary classifier trained to predict whether human-written segments are formal or informal.
Yet, as Table~\ref{tab:status} indicates, this approach fails to identify the best system in this dimension 59\% of the time.
To better understand this issue, we evaluate these classifiers on human-written texts versus \textsc{st} system outputs.
\paragraph{Human Written Texts} Table~\ref{tab:binary_classifiers_results} presents F$1$ scores when testing the binary formality classifiers on the task they are trained on: predicting whether human-written sentences from \textsc{gyafc}\xspace and \textsc{xformal}\xspace are formal or informal. First, the last column (i.e., $\delta$(\textsc{xlm-r}, m\textsc{bert})) shows that \textsc{xlm-r} is a better model than m\textsc{bert} for this task, across languages, with the largest improvements in the \textsc{zero-shot} setting where \textsc{xlm-r} beats m\textsc{bert} by $+3$, $+2$, $+1$ for \textsc{br-pt}\xspace, \textsc{fr}\xspace, and \textsc{it}\xspace respectively.
Second, \textsc{zero-shot} is surprisingly the best strategy to port \textsc{en}\xspace models to other languages. \textsc{translate-train} and \textsc{translate-test} hurt F1 by $3$ and $9$ points on average compared to \textsc{zero-shot}, despite exploiting more resources in the form of machine translation systems and their training data. However, transfer accuracy is likely affected by regular translation errors (as suggested by larger F1 drops for languages with lower \textsc{mt} \textsc{bleu}\xspace scores) and by formality-specific errors. Machine translation has been found to produce outputs that are more formal than its inputs \citep{xformal}, which yields noisy training signals for \textsc{translate-train} and alters the formality of test samples for \textsc{translate-test}.
\paragraph{System Outputs} We now evaluate the best performing binary classifier (i.e., \textsc{xlm-r} in \textsc{zero-shot} setting) on real system outputs---a setup in line with automatic evaluation frameworks.
Figure~\ref{fig:counts_regression} presents a breakdown of the number of formal vs. informal predictions of the classifiers binned by human-rated formality levels. Across languages, the performance of the classifier deteriorates as we move away from extreme formality ratings (i.e., very informal ($-3$) and very formal ($+3$)). This lack of sensitivity to different formality levels is problematic since system outputs across languages are concentrated around neutral formality values. In addition, when testing on \textsc{br-pt}\xspace, \textsc{fr}\xspace, and \textsc{it}\xspace (\textsc{zero-shot} settings), the classifier is more biased towards the formal class, which leads one to question its ability to correctly evaluate more formal outputs in multilingual settings.
Taken together, these results suggest that validating the classifiers against human rewrites rather than system outputs is unrealistic and potentially misleading.
\subsubsection{Regression Models}\label{sec:regression_models}
Table~\ref{tab:regression_results} presents Spearman’s $\rho$ correlation of
regression models' predictions with human judgments. Again, \textsc{xlm-r} with \textsc{zero-shot} transfer yields the highest correlation across languages. More specifically, the trends across different transfer approaches and different pre-trained language models are similar to the ones observed on evaluation of binary classifiers:
\textsc{xlm-r} outperforms m\textsc{bert} for almost all settings, while \textsc{zero-shot} is the most successful
transfer approach, followed by \textsc{translate-train}, with \textsc{translate-test} yielding the lowest correlations across languages. Interestingly, regression models highlight the differences between the generalization abilities of \textsc{xlm-r} and m\textsc{bert} more clearly than the previous analysis on binary predictions: \textsc{zero-shot} transfer on \textsc{xlm-r} yields $8\%$, $8\%$, and $10\%$ higher correlations than m\textsc{bert} for \textsc{br-pt}\xspace, \textsc{fr}\xspace, and \textsc{it}\xspace---while both models yield similar correlations for \textsc{en}\xspace.
\subsection{Meaning Preservation Metrics}
\label{sec:meaning_results}
\input{tables/meaning}
Table~\ref{tab:meaning_free} presents Spearman's $\rho$ correlation of meaning preservation metrics with human judgments. chr\textsc{F}
consistently yields the highest correlations across languages---this result is in line with prior observations on evaluating meaning preservation metrics for \textsc{en}\xspace \textsc{st}\xspace tasks~\cite{yamshchikov2020styletransfer} and is now confirmed in a multilingual setting.
This trend might be explained by chr\textsc{f}'s ability to match spelling errors within words via character $n$-grams.
\textsc{xlm-r} trained on \textsc{sts}\xspace with zero-shot transfer is a close second to chr\textsc{f}, consistent with this model's top-ranking behavior as a formality transfer metric.
However, chr\textsc{f} outperforms the remaining more complex and expensive metrics, including \textsc{bert}-score and m\textsc{bert} models.
In contrast to \citet{yamshchikov2020styletransfer},
embedding-based methods (i.e., cosine, \textsc{wmd}) show no advantage over $n$-gram metrics, perhaps due to differences in word embedding quality across languages. Finally, it should be noted that r\textsc{-bleu} is the worst performing metric across languages, and its correlation with human scores is particularly poor for languages other than English. This is remarkable because it has been used in $75\%$ of automatic evaluations for \textsc{f}o\textsc{st}\xspace meaning preservation evaluation (as seen in Table~\ref{tab:status}). We, therefore, recommend discontinuing its use.
\subsection{Fluency Metrics}
\label{sec:fluency_results}
Table~\ref{tab:fluency_correlations_with_human} presents Spearman's $\rho$ correlation of various fluency metrics with human judgments.
Pseudo-likelihood (\textsc{pseudo-ll}) scores obtained from \textsc{xlm-r} correlate with human fluency ratings best across languages. Their correlations are strong across languages, while other methods only yield weak (i.e., Ken\textsc{lm}, m\textsc{bert}) to moderate correlations (i.e, Ken\textsc{lm-ppl}) for \textsc{it}\xspace. We, therefore, recommend evaluating fluency using Pseudo-likelihood scores derived from \textsc{xlm-r} to help standardize fluency evaluation across languages.
\begin{table}[!t]
\centering
\scalebox{0.91}{
\begin{tabular}{lcccc}
\rowcolor{gray!10}
\multicolumn{1}{l}{\textbf{\textsc{method}}} & \textbf{\textsc{en}} & \textbf{\textsc{it}} &
\textbf{\textsc{pt}} & \textbf{\textsc{fr}} \\
\addlinespace[0.5em]
Ken\textsc{lm}\xspace (\textsc{ll}) & $0.33$ & $0.27$ & $0.43$ & $0.39$\\
Ken\textsc{lm}\xspace (\textsc{ppl}) & $0.40$ & $0.35$ & $0.45$ & $0.41$\\
\addlinespace[0.1cm]
m\textsc{bert} (\textsc{pseudo-ll}) & $0.42$ & $0.28$ & $0.43$ & $0.41$ \\
\textsc{xlm-r} (\textsc{pseudo-ll}) & $\mathbf{0.50}$ & $\mathbf{0.46}$& $\mathbf{0.55}$ & $\mathbf{0.61}$\\
\end{tabular}}
\caption{Spearman's $\rho$ correlation of fluency metrics with human judgments.}
\label{tab:fluency_correlations_with_human}\vspace{-0.3cm}
\end{table}
\subsection{System-level Rankings}
\label{sec:overall_results}
Finally, we turn to predict the overall ranking of systems by focusing on how many correct pairwise system comparisons each metric gets correct. For each language, there are $5$ systems, which means there are $10$ pairwise comparisons, for a total of $40$ given the $4$ languages. We analyze corpus-level r\textsc{-bleu}, commonly used for this dimension, along with leading metrics from the other dimensions: \textsc{xlm-r} formality regression models, chr\textsc{f} and \textsc{xlm-r} pseudo-likelihood. r\textsc{-bleu} gets $30$ out of $40$ comparisons correct while the other metrics get $25$, $22$, and $19$ respectively. This indicates that r\textsc{-bleu} correlates with human judgments better at the corpus-level than at the sentence-level, as in machine translation evaluation \citep{mathur-etal-2020-results}. We caution that these results are not definitive but rather suggestive of the best performing metric, given the
ideal evaluation would be a larger number of systems with which to perform a rank correlation. The complete analysis for each language is in Appendix~\ref{sec:pairwise_results}.
\section{Background}
\label{sec:status}
\input{tables/status_updated}
\subsection{Limitations of Automatic Evaluation}
Recent work highlights the need for research to improve evaluation practices for \textsc{st}\xspace along multiple directions. Not only does \textsc{st}\xspace lack standardized evaluation practices \cite{yamshchikov2020styletransfer}, but commonly used methods have major drawbacks which hamper progress in this field.
\citet{pang-2019-towards} and \citet{Pang2019UnsupervisedEM} show that the most widely adopted automatic metric, \textsc{bleu}\xspace, can be gamed. They observe that untransferred text achieves the highest \textsc{bleu}\xspace score for the task of sentiment transfer, questioning complex models' ability to surpass this trivial baseline. \citet{mir-etal-2019-evaluating} discuss the inherent trade-off between \textsc{st}\xspace evaluation aspects and propose that models are evaluated at specific points of their trade-off plots.
\citet{tikhonov-etal-2019-style} argue that, despite their cost, human-written references
are needed for future experiments with style transfer. They also show
that comparing models without reporting error margins can lead to incorrect conclusions as state-of-the-art models sometimes end up within error margins from one another.
\subsection{Structured Review of \textsc{st}\xspace Evaluation}\label{sec:lit_review}
We systematically review automatic evaluation practices in \textsc{st}\xspace with formality as a case study. We select \textsc{f}o\textsc{st}\xspace for this work since it is one of the most frequently studied styles~\cite{jin2021deep} and there is human annotated data including human references available for these evaluations \cite{rao-tetreault-2018-dear,xformal}.
Tables \ref{tab:status} and \ref{tab:status_mapping} summarize evaluation details for all \textsc{f}o\textsc{st}\xspace methods in papers from the \textsc{st}\xspace survey by \citet{jin2021deep}.\footnote{The complete list is hosted at: \url{https://github.com/fuzhenxin/Style-Transfer-in-Text}}
Most works employ automatic evaluation for \textit{style}~($87\%$) and \textit{meaning} preservation ($83\%$). \textit{Fluency} is the least frequently evaluated dimension ($43\%$), while $74\%$
of papers employ automatic metrics to assess the \textit{overall} quality of system outputs that captures all desirable aspects.
Across dimensions, papers also frequently rely on human evaluation ($55\%$, $58\%$, $60\%$, and $40\%$ for style, meaning, fluency, and overall). However, human judgments and automatic metrics do not always agree on the best-performing system. In $60\%$ of evaluations, the top-ranked system is the same according to human and automatic evaluation (marked as \cmark \ in Table~\ref{tab:status}), and their ranking disagrees in $40\%$ of evaluations (marked as \xmark \ in Table~\ref{tab:status}). When there is a disagreement, human evaluation is trusted more and viewed as the standard. This highlights the need for a systematic evaluation of automatic evaluation metrics.
Finally, almost all papers~($91\%$) consider \textsc{f}o\textsc{st}\xspace for English (\textsc{en}\xspace), as summarized in Table~\ref{tab:status_mapping}. There are only two exceptions:
\citet{korotkova} study \textsc{f}o\textsc{st}\xspace for Latvian (\textsc{lv}) and Estonian (\textsc{et}) in addition to \textsc{en}\xspace,
while \citet{xformal}
study \textsc{f}o\textsc{st}\xspace for $3$ Romance languages: Brazilian Portuguese (\textsc{br-pt}\xspace), French (\textsc{fr}\xspace
), and Italian (\textsc{it}\xspace). The former provides system output samples as a means of evaluation, and the latter employs human evaluations, highlighting the challenges of automatic evaluation in multilingual settings.
Next, we review the automatic metrics used for each dimension of evaluation in \textsc{f}o\textsc{st}\xspace papers. As we will see, a wide range of approaches is used. Yet, it remains unclear how they compare to each other, what their respective strengths and weaknesses are, and how they might generalize to languages other than English.
\subsection{Automatic Metrics for \textsc{f}o\textsc{st}\xspace}
\paragraph{Formality} Style transfer is often evaluated using model-based approaches. The most frequent method consists of training a binary classifier on human written formal vs. informal pairs. The classifier is later used to predict the percentage of generated outputs that match the
desired attribute per evaluated system---the system with the highest percentage is considered the best
performing with respect to style.
Across methods, the corpus used to train the classifier is the \textsc{gyafc} parallel-corpus~\cite{rao-tetreault-2018-dear} consisting of $105$K parallel informal-formal human-generated excerpts. This corpus is curated for \textsc{f}o\textsc{st}\xspace in \textsc{en}\xspace, while similar resources
are not available for other languages.
Different model architectures have been used by prior work (e.g., \textsc{cnn}\xspace, \textsc{lstm}\xspace, \textsc{gru}\xspace, fine-tuning on pre-trained language models such as Ro\textsc{bert}a\xspace and \textsc{bert}\xspace; Table~\ref{tab:status}).
In most papers, the resulting classifier is evaluated on the test side of the \textsc{gyafc} corpus, reporting accuracy scores in the range of $80-90$\%. Despite the high accuracy scores, the best ranking system under the classifier is very often in disagreement with human evaluations~(marked as \xmark \ \ under the third subcolumn of style of Table~\ref{tab:status}). A few works train regression-based models instead, using the training data of \citet{pavlick-tetreault-2016-empirical} that are human-annotated for formality on a $7$-point scale---while, again, this resource is only available for \textsc{en}\xspace.
\paragraph{Meaning Preservation} Evaluation of this dimension is performed using a wider spectrum of approaches, as presented in the third column of Table~\ref{tab:status}. The most frequently used metric is reference-\textsc{bleu}\xspace (r-\textsc{bleu}\xspace), which is based on the $n$-gram precision of the system output compared to human rewrites of the desired formality. Other approaches include self-\textsc{bleu}\xspace (s-\textsc{bleu}\xspace), where the system output is compared to its input, measuring the semantic similarity between the system input and its output, or regression models (e.g., \textsc{cnn}\xspace, \textsc{bert}\xspace) trained on data annotated for similarity-based tasks, such as the Semantic Textual Similarity task (\textsc{sts}\xspace)~\cite{agirre-etal-2016-semeval}.
\paragraph{Fluency} Fluency is typically evaluated with model-based approaches (see fourth column of Table~\ref{tab:status}).
Among those, the most frequent method is that of computing perplexity (\textsc{ppl}\xspace) under a language model.
The latter is either trained from scratch on the same corpus used to train the \textsc{f}o\textsc{st}\xspace models (i.e., \textsc{gyafc})
using different underlying architectures
(e.g., Ken\textsc{lm}\xspace, \textsc{lstm}\xspace), or employ large pre-trained language models (e.g., \textsc{gpt}\xspace). A few other works train models on \textsc{en}\xspace data annotated for grammaticality \citep{heilman-etal-2014-predicting} or linguistic acceptability \citep{warstadt-etal-2019-neural} instead.
\paragraph{Overall} Systems' overall quality (see fifth column of Table~\ref{sec:status}) is mostly evaluated using r-\textsc{bleu}\xspace or by combining independently computed metrics into a single score (e.g., geometric mean - \textsc{gm}(.), harmonic mean - \textsc{hm}(.), \textsc{f}$1$(.)). Moreover, $6$ out of $8$ approaches that rely on combined scores do not include fluency scores in their overall evaluation.
\paragraph{English Focus} Since most of the current work on \textsc{f}o\textsc{st}\xspace and \textsc{st}\xspace is in \textsc{en}\xspace, prior work relies heavily on \textsc{en}\xspace resources for designing automatic evaluation methods. For instance, resources for training stylistic classifiers or regression models are not available for other languages.
For the same reason, it is unclear whether model-based approaches for measuring meaning preservation and fluency can be ported to multilingual settings. Furthermore, reference-based evaluations (e.g., r-\textsc{bleu}\xspace) require human rewrites that are only available for \textsc{en}\xspace, \textsc{br-pt}\xspace, \textsc{it}\xspace, and \textsc{fr}\xspace. Finally, even though perplexity does not rely on annotated data, without standardizing the data language models are trained on, we cannot make meaningful cross-system comparisons.
\subsection{Summary}
Reviewing the literature shows the lack of standardized metrics
for \textsc{st}\xspace evaluation, which hampers comparisons across papers, the lack of agreement between human judgments and automatic metrics, which hampers system development, and the lack of portability to languages other than English which severely limits the impact of the work. These issues motivate the controlled multilingual evaluation of evaluation metrics in our paper.
|
1,116,691,497,550 | arxiv | \section{Introduction}
Let $W$ be a finite subgroup of $ O(n)$ generated by reflections.
The algebra of $W$-invariant polynomials is generated by $n$
algebraically independent $W$-invariant homogeneous polynomials and
the degrees of these basic invariants are uniquely determined. This
theorem and its converse were first stated by Shephard and Todd in
\cite{20} where the direct statement was proved case by case. Soon
after, in \cite{6} Chevalley gave a beautiful unique proof of this
direct statement, which is often called `Chevalley's theorem'.
A $W$-invariant complex analytic function may be written as a
complex analytic function of the basic invariants (\cite{21}).
Glaeser's theorem (\cite{11}) shows that real $W$-invariant
functions of class ${\mathcal C}^{\infty}$, may be expressed as
${\mathcal C}^{\infty}$ functions of the basic invariants.
In finite class of differentiability, Newton's theorem in class
${\mathcal C}^r$ (\cite{2}) dealt with symmetric functions and as a
consequence with the Weyl group of $A_n$. This particular case shows
a loss of differentiability as already did Whitney's even function
theorem (\cite{22}) which established the result for the Weyl group
of $A_1$. A first attempt to study the general case may be found in
the first part of \cite{4} where the best result was obtained for
the Weyl groups of $A_n, B_n$ and the dihedral groups $I_2(k)$ by a
method which was on the right track but needed an additional
ingredient to deal with the general case. The behavior of the
partial derivatives of the functions of the basic invariants on the
critical image was studied in \cite{12} for $A_n,B_n, D_n$ and
$I_2(k)$.
Here we give for any reflection group a result which is the best
possible as shown by a general counter example. Let $p_1, \ldots ,
p_n$ be an integrity basis, we define the `Chevalley' mapping
$P:{\mathbf R}^n \ni x\mapsto P(x)=(p_1(x), \ldots , p_n(x))\in
{\mathbf R}^n$. The loss of differentiability is governed by the
highest degree of the basic invariant polynomials. More precisely we
have:
\begin{thm}\label{thm1}
Let $W$ be a finite group generated by reflections acting
orthogonally on ${\mathbf R}^n$ and let $f$ be a $W$-invariant
function of class ${\mathcal C}^r$ on ${\mathbf R}^n$. There exists
a function $F$ of class ${\mathcal C}^{[r/h]}$ on ${\mathbf R}^n$
such that $f=F\circ P$, where $P$ is a Chevalley polynomial mapping
associated with $W$ and $h$ is the highest degree of the coordinate
polynomials in $P$, equal to the greatest Coxeter number of the
irreducible components of $W$.
\end{thm}
Since a change of basic invariants is an invertible polynomial map
on the target, the statement does not depend on the choice of the
set of basic invariants.
\section{ The Chevalley mapping}
A detailed study may be found in \cite{5} or \cite{8}.
When $W$ is reducible, it is a direct product of its irreducible
components, say $W= W^0\times W^1 \times \ldots \times W^s $ and we may
write $\mathbf{R}^n$ as an orthogonal direct sum $\mathbf{R}^{n_0}\oplus
\mathbf{R}^{n_1}\oplus \ldots \oplus \mathbf{R}^{n_s}$ where $W^0$ is
the identity on $\mathbf{R}^{n_0}$,
subspace of $W$-invariant vectors, and for $i=1,\ldots, s$, $W^i$
is an irreducible finite Coxeter group acting on ${\bf R}^{n_i}$.
We may choose coordinates that fit with this orthogonal direct sum.
If $w=w_1\ldots w_s\in W$ with $w_i\in W^i, \; 1\le i\le s$
we have $w(x)= w(x^0, x^1,\ldots , x^s)= (x^0, w_1(x^1),\ldots,
w_s(x^s))$ for all $x\in \mathbf{R}^{n}$. The
direct product of the identity on ${\bf R}^{n_0}$ and Chevalley
mappings $P^i$ associated with $W^i$ acting on $\mathbf {R}^{n_i},\; 1
\le i\le s$, is a Chevalley map $P= Id_0\times P^1\times \ldots
\times P^s$ associated with the action of $W$ on $\mathbf{R}^{n}$.
For an irreducible $W$ (or for an irreducible component)
we will assume as we may that the degrees of the coordinate
polynomials $p_1,\ldots ,p_n$ are in increasing order:
$2=k_1\le \ldots \le k_n=h$, Coxeter number of $W$.
In the reducible case we will denote by $h$ the highest degree of
the coordinate polynomials which is the maximal Coxeter number of
the irreducible components.
Let $\mathcal{R}$ be the set of reflections different from identity
in $W$. The number of these reflections is
$\mathcal{R}^{\#}=d=\sum_{i=1}^n(k_i-1)$. For each $\tau \in
\mathcal{R}$, let $ \lambda_{\tau}$ be a linear form the kernel of
which is the hyperplane $ H_{\tau}=\{ x\in {\bf R}^n |\tau (x) = x
\} $. The jacobian of $P$ is $J_P=c \prod_{\tau \in \mathcal{R}}
\lambda_{\tau}$ for some constant $c\neq 0$. The critical set is the
union of the $ H_{\tau}$ when $\tau$ runs through $ \mathcal{R}$.
A Weyl Chamber $C$ is a connected component of the regular
set. The other connected components are obtained by the
action of $W$ and the regular set is $\bigcup_{w\in W} w(C)$.
There is a stratification of ${\mathbf R}^n$ by the regular set, the
reflecting hyperplanes $ H_{\tau}$ and their intersections.
The mapping $P$ induces an analytic diffeomorphism of $C$ onto
the interior of $P({\mathbf R}^n)$ and an homeomorphism
that carries the stratification from the
fundamental domain $\overline{C}$ onto $P(\mathbf{R}^n)$.
The Chevalley mapping is neither injective nor surjective. Actually
the fiber over each point of the image is a $W$-orbit. The mapping $P$
is proper and separates the orbits (\cite{19}). It is the restriction
to $\mathbf{R}^n$ of a complex $W$-invariant mapping from $\mathbf{C}^n$
onto (\cite{13}) $\mathbf{C}^n$, still denoted by $P$.
\noindent On its regular set, the complex $P$ is a local analytic isomorphism.
Its critical set, where the jacobian vanishes, is the union of the
complex hyperplanes $H_{\tau} =\{ z\in {\mathbf C}^n | \tau(z)=z \}$,
kernels of the complex forms $\lambda_{\tau}$.
The critical image is the algebraic set $\{ u\in {\mathbf C}^n |
\Delta(u)= J^2_P(z)=0 \}$, on which $P$ carries the stratification.
The complex $P$ is proper, it is a $W^{\#}$-fold covering of ${\bf C}^n$
ramified over the critical image.
Finally, there are only finitely many types of irreducible finite Coxeter groups
defined by their connected graph types. Even when these groups are Weyl
groups of root systems or Lie algebras, we will follow the general usage
and denote them with upper case letters.
\section{ Whitney Functions and $r$-regular, $m$-continuous jets.}
A complete study of Whitney functions may be found in
\cite{21}.\vskip 5pt
A jet of order $m\in {\mathbf N}$, on a locally closed set $E\subset
{\mathbf R}^n$ is a collection $A=(a_k)_{k\in {\bf N^n}\atop \mid
k\mid \le m}$ of real valued functions $a_k$ continuous on $E$. At
each point $x\in E$ the jet $A$ determines a polynomial $A_x(X)$,
and we sometimes speak of continuous polynomial fields instead of
jets (\cite{15}). As a function, $A_x$ acts upon vectors $x'-x$ tangent
to ${\bf R}^n$ at $x$. To avoid introducing the notation $T_x^r A$,
we write somewhat inconsistently:
\[ A_x: x' \mapsto A_x(x')=\sum_k {1\over
k!} a_k(x)\; (x'-x)^k.\]
\noindent By formal derivation of $A$ of order $q\in {\mathbf
N}^n,\; \mid q\mid \le m$, we get jets of the form $(a_{q+k})_{\mid
k\mid \le m-\mid q\mid}$ inducing polynomials \[(D^qA)_x(x')= \left(
{\partial^{\mid q\mid} A\over
\partial x^q}\right)_x (x')=a_q(x)+\sum_{k>q\atop |k|\le m}{1\over (k-q)!}
a_k(x)\; (x'-x)^{k-q}.\]
\noindent For $0\le \mid q\mid \leq r\le m$, we put:
$$(R_xA)^q(x')=(D^q A)_{x'}(x')-(D^q A)_x(x').$$
\begin{defn} Let $A$ be an $m$-jet on $E$. For $r\le m , A$ is
$r$-regular on $E$, if and only if for all compact set $K$ in $E$,
for $(x, x')\in K^2$, and for all $q\in {\bf N}^n$ with $\mid q\mid
\leq r$, it satisfies the Whitney conditions.
\centerline {$({\mathcal W}_q^r) \qquad (R_xA)^q(x') = o (\mid
x'-x\mid^{r-\mid q\mid})$, when $\mid x-x'\mid \to 0$.}
\end{defn}
\begin{rem} Even if $m> r$ there is no need to
consider the truncated field $A^r$ in stead of $A$ in the conditions
$({\mathcal W}_q^r)$. Actually $(R_xA^r)^q(x')$ and $(R_xA)^q(x')$
differ by a sum of terms $[a_k(x)/(k-q)!] \, (x'-x)^{k-q}$, with
$a_k$ uniformly continuous on $K$ and $ \vert k\vert -\vert q\vert
> r-\vert q\vert $.\end{rem}
The space of $r$-regular jets of order $m$ on $E$, is naturally
provided with the Fr\'echet topology defined by the family of
semi-norms: \[\Vert A\Vert_{K_n}^{r,m}=\sup_{x\in K_n \atop \mid
k\mid \leq m} \mid {1\over k!}a_k(x)\mid +\sup_{(x,x')\in K_n^2
\atop x\neq x', \mid k\mid \leq r} \left( {\mid (R_xA)^k(x')\mid
\over \mid x-x'\mid^{r-\mid k\mid}}\right)\] where $K_n$ runs
through a countable exhaustive collection of compact sets of $E$.
Provided with this topology the space of $r$-regular, $m$-continuous
polynomial fields on $E$ is a Fr\'echet space denoted by ${\mathcal
E}^{r,m}(E)$. If $r=m,\; {\mathcal E}^r(E)$ is the space of Whitney
fields of order $r$ or Whitney functions of class ${\mathcal C}^r$
on $E$.
\begin{thm} \label{Wextsion} {\rm Whitney extension theorem (\cite{23}}).
The restriction mapping of the space ${\mathcal E}^r({\mathbf R}^n)$
of functions of class ${\mathcal C}^r$ on ${\mathbf R}^n $ to the
space ${\mathcal E}^r(E)$ of Whitney fields of order $r$ on $E$, is
surjective. There is a linear section, continuous when the spaces
are provided with their natural Fr\'echet topologies. \end{thm}
Let $E$ be a closed subset of ${\mathbf C}^n\simeq {\mathbf
R}^{2n}$, we consider jets $A$ on $E$ with complex valued
coefficients $a_k$. They induce in $z\in E$ the polynomials:
\[A_z(X,Y)= \sum_{\mid k \mid +\mid l\mid \leq m} {1\over {k!l!}}
a_{k,l}(z)\; X^kY^l \in {\bf C}[X,Y],\] and we may define the
Fr\'echet space ${\mathcal E}^{r}(E;{\mathbf C})$ of complex valued
Whitney functions of class ${\mathcal C}^r$.
\begin{defn}(\cite{15}) A Whitney function $A\in {\mathcal E}^{r}(E;{\bf
C})$ is formally holomorphic if it satisfies the Cauchy-Riemann
equalities: \[i{\partial A\over\partial {X_j}}= {\partial A\over
\partial {Y_j}}, \; j=1, ... ,n.\] \end{defn}
Let $Z=(Z_1,\ldots ,Z_n), \; Z_j=X_j+iY_j,\; j=1, .\ldots ,n$.
The field $A$ is formally holomorphic if and only if
$ \partial A/\partial \overline{Z_j} =0, \; j=1, ... ,n $.
Thus for all $z\in E$ the polynomial $A_z$ belongs to ${\bf C}[Z]$
and is of the form $\displaystyle{A_z(Z)=\sum_k {1\over k!} a_k(z)
Z^k} $.\vskip 5pt
The algebra of formally holomorphic Whitney functions of class
${\mathcal C}^r$ on the (locally) closed set $E$ of ${\mathbf C}^n$
will be denoted by ${\mathcal H}^{r}(E)$. It is a closed sub-algebra
of ${\mathcal E}^{r}(E;{\mathbf C})$ and therefore a Fr\'echet space
when provided with the induced topology. In practice we define the
semi-norms $\Vert A\Vert^{K_n}_{r}$ on ${\mathcal H}^{r}(E)$ by the
same formulas as in ${\mathcal E}^{r}(E;{\mathbf R})$, only using
modulus instead of absolute value.
To take advantage of compensation phenomenons, it may be convenient
to consider Fr\'echet spaces ${\mathcal H}^{r,m}(E)$ of formally
holomorphic $r$-regular jets of order $m \ge r$ on $E$.
\begin{defn} A real form (\cite{17}) or real
situated subspace (\cite{15}) of ${\bf C}^n$ is a real
vector subspace $E$ of real dimension $n$ such that $E \oplus i E
={\bf C}^n$.\end{defn}
\noindent A real form is a real subspace
$E_S= \{z\in {\mathbf C}^n| Sz=z \}$, where $S$ is an anti-involution.
Example. Let $\alpha$ be an involution of $\{1,\ldots,n\}$,
$\Gamma_{\alpha}=\{ z\in {\mathbf C}^n | z_{\alpha (j)}=
\overline{z_j},\; j=1,\ldots,n\}$ is a real form of ${\mathbf C}^n$ defined by
the anti-involution $z\mapsto \overline{\alpha( z)}$.
Let $W$ be a finite reflection group acting orthogonally on ${\bf
R}^n$ and $P$ be its Chevalley polynomial mapping as above.
Since $P$ is defined over ${\mathbf R}$ (its coefficients are real),
$P^{-1}({\mathbf R}^n)$ is the union of real forms $\Gamma_{S_w}
\subset{\mathbf C}^n$, where $w$ runs through the involutions of $W$
and $S_w$ is the anti-involution defined by $S_w(u+iv)= wu-iwv$.
\vskip 5pt
Let $f\in {\mathcal C}^r({\mathbf R}^n)^W$ be a $W$-invariant function of class
${\mathcal C}^r$. It induces on ${\bf R}^n$ a $W$-invariant Whitney
field of order $r$ and
a formally holomorphic field in ${\mathcal H}^{r}({\mathbf R}^n)^W$ which
will still be denoted by $f$. By using Whitney's extension theorem,
one may show (\cite{2}) that there is a linear and continuous extension
\footnote{This extension will allow us to get a field $\tilde F$ on ${\bf R}^n$
with $\tilde f=\tilde F\circ P$ and derive its regularity from its continuity
by using the below lemma \ref{lem3}. This process might be avoided if there was an
available proof of the Whitney regularity property (\cite {24}) of $P({\bf R}^n)$,
a most likely conjecture proved for $A_n$ in \cite {14}, easy to show for some
lower dimensional reflection groups (\emph{e.g.} $I_2(k), H_3$). Unfortunately there is none, although in \cite{9} a key point for the proof of this conjecture is studied and a sketchy proof of it is given for several Coxeter groups.}:
\[{\mathcal H}^r({\mathbf R}^n)^W \ni f \mapsto \tilde f \in {\mathcal H}^r
(P^{-1}({\mathbf R}^n))^W .\]
\section{Some multiplication and division properties.}
\begin{lem}\label{lem1} Let $\Gamma$ be a finite union of real forms of ${\bf
C}^n$, $A$ be in $\mathcal{H}^{r}(\Gamma)$, and $Q$ be a polynomial
$(s-1)$-flat on $S$. Let $z\in \Gamma$ and $z_0\in S\cap \Gamma$,
then for all $q\in {\bf N}^n, \mid q\mid \le r$:
\[(R_{z_0}QA)^q(z)=(D^q QA)_{z}(z)-(D^q QA)_{z_0}(z)\in o(\mid
z-z_0\mid^{r-\mid q\mid +s}).\] Moreover $QA\in \mathcal
{H}^{r+s}(S\cap \Gamma)$ and is $(s-1)$-flat on $S\cap \Gamma$. For
all compact $K\subset S\cap\Gamma$,
there exists a numerical constant $c$ such that $\|QA\|^{r+s}_K \le
c\|Q\|^{r+s}_K\|A\|^r_K$. \end{lem}
{\it Proof.} Let $z_0\in S\cap \Gamma$. For all $z\in \Gamma$, all
$q\in {\bf N}^n, \mid q\mid \le r$, and $p\le q$, we consider:
\[(D^p Q)_{z}(z)(D^{q-p} A)_{z}(z)-(D^{ p} Q)_{z_0}(z)(D^{q-p} A
)_{z_0}(z).\] By Taylor's formula for polynomials $( D^{p}
Q)_{z}(z)=(D^{p} Q)_{z_0}(z)$, and this difference is: \[(D^{p}
Q)_z(z)\left[(D^{q-p} A)_{z}(z)-(D^{ q-p} A)_{z_0}(z) \right].\] By
assumption $(D^{q-p} A)_{z}(z)-( D^{q-p} A )_{z_0}(z)\in o( \mid
z-z_0 \mid^{r-\mid q\mid +\mid p \mid } )$, and for $\mid p\mid <
s$ $(D^{p} Q)_z(z)\in O(\mid z-z_0 \mid^{s- \mid p\mid })$. The
product is in $o(\mid z-z_0 \mid^{ r-\mid q\mid +s})$ either because
$|p|<s$ and $r-|q|+|p|+s-|p|=r-|q|+s$ or because $|p|\geq s$ and
$r-|q|+|p|\geq r-|q|+s$.\vskip 5pt
\noindent The behavior of $(R_{z_0}QA)^q(z)$ is now a consequence of
Leibniz' derivation formula.\vskip 5pt
Actually $QA\in \mathcal{H}^{r,r+s}$. On $S\cap \Gamma$, $\mid p\mid
< s \Rightarrow (D^{p} Q)_{z_0}(z_0)=0$, therefore in the
derivatives of $QA$ of order $\le r+s$ the only derivatives of $A$
that are not multiplied by a derivative of $Q$ that vanishes, are of
order $\le r$. Then the above estimates show that when $|q|\le r+s$,
the field $QA$ satisfies Whitney conditions $\mathcal{W}_q^{r+s}$ on
$S\cap \Gamma$. Thus $QA\in \mathcal {H}^{r+s}(S\cap \Gamma)$, and
clearly it is $(s-1)$-flat on $S\cap \Gamma$.
This situation was already noticed in \cite{10}: when multiplying a field
$r_1$-regular, $(s_1-1)$-flat by a field $r_2$-regular,
$(s_2-1)$-flat, the product is $ \min (r_1+s_2, r_2+s_1)$-regular
and $(s_1+s_2-1)$ flat. Here, on $S\cap \Gamma$, we have $r_1=r,\;
s_1=0$ for $A$ and $\;r_2= +\infty,\; s_2=s$ for $Q$. $\;\Box$
\vskip 10pt
Example. Let $Q$ be an homogeneous polynomial of degree $s$. It
vanishes at the origin with all its derivatives of order $\le s-1$.
If $A\in \mathcal{H}^{r}(\Gamma)$, for all $z\in \Gamma$ and all
$q\in {\bf N}^n, \mid q\mid \le r$: \[(R_{0}QA)^q(z)=(D^{ q}
QA)_{z}(z)-(D^{q} QA)_{0}(z)\in o(\mid z\mid^{r+s-\mid q\mid}).\]
\noindent The same result holds if instead of a product $QA$ we have
a sum $\sum_{i=1}^{n}Q_iA_i$, with homogeneous polynomials $Q_i$ of
degree $s_i\geq s$ and $A_i\in \mathcal{ H}^{r}(\Gamma)$.
Let us recall the following division lemma:
\begin{lem} [\cite{2}] \label{lem2} Let $\Gamma$ be a finite union of real forms of
${\bf C}^n$, and $\lambda\neq 0$ be a complex linear form with
kernel $H$. If $A\in \mathcal{H}^{r}(\Gamma)$ is such that $A_z(Z)$
is divisible by $\lambda_z(Z)$ whenever $z\in \Gamma \cap H$ then
there exists a field $B\in \mathcal{H}^{r-1}(\Gamma)$ such that $A^r
= (\lambda B )^r$. For all compact $K\subset \Gamma$, there exists a
constant $c$ such that $\|B\|^{r-1}_K \le c\|A\|^r_K$.\end{lem}
\noindent Actually $B\in \mathcal{H}^{r}(\Gamma\setminus H )$ and if
$\mid s\mid = r$, then $\displaystyle{\lambda (z)(D^{s} B)_{z}(z)}$
tends to zero with $\lambda (z)$.\vskip 5pt
\begin{rem} The lemma still holds if we replace $\Gamma$ by its
intersection with one or several hyperplanes distinct from $H$.
\end{rem}
The proof of lemma \ref{lem2} relies upon a consequence of the mean
value theorem that will be instrumental in what follows: \vskip 5pt
\begin{lem}[\cite{1},\cite{15}]\label{lem3} Let $\Gamma$ be a finite union of
real forms of ${\bf C}^n$, $\Delta\ne 0$ be a polynomial,
and $X= \{x\in {\bf C}^n \mid \Delta(x)=0 \}$.
If $f\in \mathcal{H}^{r}(\Gamma \setminus X)$ is $r$-continuous on
$\Gamma$, then $f\in \mathcal{H}^{r}(\Gamma)$. \end{lem}
Let $(\lambda_{\tau})_{\tau\in \mathcal{D}}$ be
$\mathcal{D}^{\#}=d$ non zero complex linear forms with kernels
$(H_{\tau})_{\tau\in \mathcal{D}}$ all distinct. The hyperplanes
$(H_{\tau})_{\tau\in \mathcal{D}}$ and their intersections induce a
stratification on $\Gamma$. Let $S_p$ be a stratum, connected component
of the intersection of $\Gamma$ and exactly $p$ of these hyperplanes, say
$ (H_{\tau})_{\tau\in \mathcal{B}_p}$, $\mathcal{B}_p^{\#}=p$.
The border $\overline{S_p}\setminus S_p$ is a union
$\bigcup S_{p+l}$ of strata of lower
dimensions, containing $S_d=\Gamma\cap (\bigcap_{\tau\in \mathcal{
D}}H_{\tau})$. Using these notations we have:\vskip 5pt
\begin{lem}\label{lem4} For $i=1,\ldots,n$, let $A_i$ be in $\mathcal{H}^r(\Gamma)$ and $Q_i$
be an homogeneous polynomial $(s_p-1)$-flat on
$S_p$ and more generally $(s_{p+l}-1)$-flat on each of the
$S_{p+l}$. Assume $p+l-s_{p+l}$ is an increasing function of $l$
and that $A=\sum_{i=1}^n Q_iA_i=(\prod_{\tau \in \mathcal{D}}
\lambda_{\tau}) C$, meaning that:
\[\forall \mathcal{U}\subseteq \mathcal{D}, \; \; A_z(Z) {\it is\; divisible\; by}
\prod_{\tau \in \mathcal{U} }\lambda_{\tau}(Z)\;{\it when}\; z\in \Gamma \cap
(\bigcap_{\tau \in \mathcal{U}} H_{\tau}).\] The field $C$
is in $\mathcal{H}^{r+s_p-p} (S_p)$ and its coefficients of order $\le r+s_d-d$
are continuous on $ \overline{S_p}$. \footnote{Actually since $\overline{S_p}$ is
convex and thus Whitney 1-regular (\cite{24}), lemma \ref{lem4} yields that $C$
is $(r+s_d-d)$-regular on $\overline{S_p}$.}
\end{lem}
{\it Proof.} By lemma \ref{lem1}, $\sum_{i=1}^n Q_iA_i$ is in ${\mathcal H}^{r+s_p}(S_p)$
and in ${\mathcal H}^{r+s_{p+l}}(S_{p+l})$.
By lemma \ref{lem2}, the field $C$ is in
${\mathcal H}^{r+s_p-p}(S_p)$ and in ${\mathcal H}^{r+s_{p+l}-(p+l)}(S_{p+l})$.
We are just to show the continuity on $\overline{S_p}$ of the coefficients
of order $\le r+s_d-d$ in $C$.
Let $S_{p+q}$ be one of the strata of largest dimension in $\overline{S_p}
\setminus S_p$, and let ${\mathcal B}_q$ with ${\mathcal B}_q^{\#}=q$,
be the subset of ${\mathcal D}$
such that $S_{p+q}$ is a connected component of the intersection of
$\Gamma$ and the hyperplanes $(H_{\tau})_{\tau\in {\mathcal B}_p\cup{\mathcal B}_q}$,
but no other. We may have $q=1$ but not necessarily since the addition of one hyperplane
may automatically entail the addition of some more. \vskip 5pt
We put:
$A= (\prod_{\tau\in {\mathcal B}_p} \lambda_\tau)( \prod_{\tau\in {\mathcal B}_q}
\lambda_\tau)( \prod_{\tau\in {\mathcal D}\setminus({\mathcal B}_p\cup{\mathcal B}_q)}
\lambda_\tau) \; C$
and define:
\centerline{$C^1=(\prod_{\tau\in {\mathcal D}\setminus({\mathcal B}_p\cup{\mathcal B}_q)}
\lambda_\tau) \; C ,\quad {\rm and}
\quad B=(\prod_{\tau\in {\mathcal B}_q} \lambda_\tau) \; C^1.$}
\noindent On $S_p$, $B$ is in ${\mathcal H}^{r+s_p-p}$ and so are $C$ and
$C^1$. On $S_{p+q}$, $C$ and $C^1$ are in ${\mathcal H}^{r+s_{p+q}-(p+q)}$.
\noindent Let $z_0$ be the orthogonal projection on $S_{p+q}$ of some $z\in
\Gamma$. By lemma \ref{lem1}, we have:
\[A_z(z)-A_{z_0}(z)=[\prod_{\tau \in {\mathcal B}_p}\lambda_{\tau}(z)]
[B_z(z)-B_{z_0}(z)] \in o(\mid z-z_0\mid^{r+s_{p+q}}).\]
Let $\pi$ be a derivation of order $p$, by Leibniz derivation formula:
\[D^{\pi}A_z(z)-D^{\pi}A_{z_0}(z)=[\prod_{\tau \in {\mathcal
B}_p}\lambda_{\tau}(z)]
[D^{\pi}B_z(z)-D^{\pi}B_{z_0}(z)]+\ldots \] \[\ldots+
c \;[B_z(z)-B_{z_0}(z)]\in o(\mid z-z_0\mid^{r+s_{p+q}-p})\] for
some constant $c\neq 0$.
For the remaining part of the proof, we assume that $z$ is in $S_p$.
Then for all $\tau\in {\mathcal B}_p,\; \lambda_\tau (z)=0$ and we get:
\[B_z(z)-B_{z_0}(z) \in o(\mid z-z_0\mid^{r+s_{p+q}-p}).\]
More generally, by considering derivations of order $\pi +\kappa$,
with $\mid \kappa \mid =k \le r+s_p-p$ \;we would get in the same way:
\[D^{\kappa}B_z(z)-D^{\kappa}B_{z_0}(z)\in o(\mid
z-z_0)\mid^{r+s_{p+q}-p-k}).\]
Let us put $E=\Gamma\cap (\bigcap_{\tau\in {\mathcal B}_p}
H_{\tau})$ and $F = H_{\tau}$ for some $\tau \in {\mathcal B}_q$.
Since $E$ and $F$ are ${\bf R}$-linear subspaces, there exists a
constant $a> 0$ such that for all $ x ,\; d(x,E)+d(x,F) \ge a\; d(x,
E\cap F)$ or equivalently, a constant $b> 0$ such that for all $
x\in E$, $d(x,F) \ge b\; d(x, E \cap F)$. We say that $E$ and $F$
are regularly separated. \footnote{This is a trivial case of regular
separation and does not need a general study. The interested reader
may take a look at \cite{16} or \cite{21}. The regular separation of
real forms was implicitly used in the above extension of $f$ to
$P^{-1}(\mathbf{R}^n)$}
\noindent The regular separation brings the existence of constants
$c_{\tau}$ such that: \[\forall\tau\in {\mathcal B}_q,\quad
|z-z_0|\le c_{\tau}\; d(z,H_{\tau})=
c_{\tau}^1|\lambda_{\tau}(z)|.\] Therefore, since
\[B_z(z)-B_{z_0}(z)= [\prod_{\tau \in {\mathcal
B}_q}\lambda_{\tau}(z)] [C^1_z(z)-C^1_{z_0}(z)]\in o(|
z-z_0|^{r+s_{p+q}-p}),\] we get that $C^1_z(z)-C^1_{z_0}(z) \in
o(\mid z-z_0\mid^{r+s_{p+q}-(p+q)})$.\vskip 5pt
\noindent Let us assume by induction that for $\mid l\mid \;\le k
-1\; <r+s_{p+q}-(p+q)$ : \[D^l C^1_z(z)-D^lC^1_{z_0}(z) \in o(\mid
z-z_0\mid^{r+s_{p+q}-(p+q)-\mid l\mid}).\] By Leibniz derivation
formula, for $j,\;|j|=k$: \[D^jB_z(z)-D^jB_{z_0}(z) =[\prod_{\tau
\in {\mathcal B}_q}\lambda_{\tau}(z)](D^j C^1_z(z)-D^j
C^1_{z_0}(z))+\]
\[+ \sum_{k-q\le |j_i|= k-l \le k-1}
a_{q-l}[\prod_{q-l}\lambda_{\tau}(z)]
(D^{j_i}C^1_z(z)-D^{j_i}C^1_{z_0}(z))\in o(\mid
z-z_0\mid^{r+s_{p+q}-p- k}).\]
\noindent The $[\prod_{q-l}\lambda_{\tau}(z)]$ stand for $
D^{j-j_i}(\prod_{\tau \in {\mathcal B}_q}\lambda_{\tau})(z)$, up to
a constant factor included in $a_{q-l}$. Applying the induction
assumption to the derivations $D^{j_i}$ of order $\le k -1$, we see
that each term of the sum is in \[o(\mid
z-z_0\mid^{r+s_{p+q}-(p+q)-( k-l)+q-l}) = o(\mid
z-z_0\mid^{r+s_{p+q}-p-k})\] and thus, that the first term also is.
Then, using the regular separation as above, we obtain: \[D^j
C^1_z(z)-D^j C^1_{z_0}(z)\in o(\mid z-z_0\mid^{r+s_{p+q}-(p+q)-
k}).\] This completes the induction, and shows that the coefficients
of $C^1$ of order $\le r+s_{p+q}-(p+q)$ are continuous in $z_0$.
Since they are continuous on $S_{p+q}$, by using the triangular
inequality, we get their continuity on $S_p \cup S_{p+q}$.
\noindent Any $z_1\in S_{p+q}$ has a neighborhood which does not
meet any of the $H_{\tau}$ but those containing $z_1$ and
in this neighborhood, the continuity of the
coefficients of $C$ and $C^1$ are the same. Hence the continuity on
$S_p\cup S_{p+q}$ of the coefficients of $C$ of order $\le r+s_{p+q}-(p+q)$.
\noindent We can get an analogous result for each
stratum $S_{p+q'}$ of maximal dimension in
$\overline{S_p}\setminus S_p$ ($q $ and $q'$ may be equal but ${\mathcal B}_q $ and
${\mathcal B}_{q'}$ are distinct).
\noindent We then proceed with the strata of highest dimension in
$\overline{S_{p+q}}-S_{p+q}$ to get the continuity on $S_{p+q}\cup
S_{(p+q)+l}$ of the coefficients of order at most $r+s_{p+q+l}-(p+q+l)$.
The continuity on $S_p\cup S_{p+q}\cup
S_{(p+q)+l}$ when $z\in S_p$ tends to $z_1\in S_{(p+q)+l}$ is obtained by the
triangular inequality, considering the orthogonal projection $z_0$
of $z$ on $S_{p+q}$. We can do the same for each of the
$\overline{S_{p+q'}}-S_{p+q'}$, and continue
the processes until we reach $S_d$ through all possible paths, getting the
continuity of coefficients of order at most $r+s_d-d$. Thus we
have the continuity of these coefficients between any two points
in $S_{p+l}$ and $S_{(p+l)+m}$.
\noindent About the continuity between points in $\overline{S_{p+q}}$ and
$\overline{S_{p+q'}}$, if they are in the intersection we already have
the continuity, but if they are not, we consider their orthogonal
projections on the intersection and thanks to the regular separation of
the strata, we can get the continuity by the triangular inequality.
\noindent Then the global $(r+s_d-d)$-continuity on $\overline{S_p}$ is
clear. $\quad \Box$
\begin{rem} When $p=0$, the strata of type $S_0$ are open
and $s_0=0$. For $q=1$, the first step is given by
lemma \ref{lem2}.\end{rem}
\section{ Proof of Theorem 1.1.}
We consider an invariant function
$ f\in \mathcal{C}^r({\bf R}^n)^W$. This function or rather the formally
holomorphic field it induces on ${\bf R}^n$ has a linear and
continuous extension $\tilde f \in \mathcal{H}^r(P^{-1}({\bf R}^n))^W$.
\subsection*{Pointwise solution.}
\begin{lem} [\cite{4}]\label{lem5} For all $W$-invariant, formally
holomorphic polynomial field $\tilde f$ of degree $r$ on $P^{-1}({\bf
R}^n)$, there exists a formally holomorphic polynomial field
$\widetilde F$ of degree at most $r$ such that for all $z\in P^{-1}
({\bf R}^n),\; \tilde f_z= (\widetilde F_{P(z)}\circ P)^r_z $.
\end{lem}
{\it Proof.} On the complement of
$\Gamma \cap \bigcup_{\tau\in \mathcal{R}} H_{\tau}$ in $\Gamma $,
the mapping $P$ is a local analytic isomorphism and this yields the
construction of $\widetilde F= (\tilde f\circ P^{-1})^r$,
unambiguously since both $ \tilde f$ and $ P$ are $ W$-invariant. On
the regular image of $P$, $\widetilde F$ verifies $\tilde
f^r=(\widetilde F\circ P)^r$.
Let $x\in \Gamma \cap (\bigcup_{\tau\in \mathcal{R}}H_{\tau})$ and
let $W_x$ be the isotropy subgroup of $W$ at $x$. The polynomial
$\tilde f_x$ is $W_x$-invariant since for all $w_0\in W_x\subset W$:
$ \tilde f_x(X)=\tilde f_{w_0x}(w_0X)=\tilde f_x(w_0X)$ where the
first equality results from the $W$-invariance of the field $\tilde
f$ and the second from $w_0x=x$.
As a consequence, $\tilde f_x$ is a polynomial in the
$W_x$-invariant generators $v=(v_1, \ldots , v_n)$ of the subalgebra
of $W_x$-invariant polynomials, and we have $\tilde f_x=Q\circ v$.
There exists a neighborhood of $x$ in ${\bf C}^n$ which does not
meet any of the hyperplanes $H_{\tau}$ but those containing $x$. In
this neighborhood we may write $P= q\circ v$ for some polynomial
$q$, since $P$ is $W_x$-invariant. Up to a multiplicative constant
the jacobian of $q$ at $ v(x)$ is the product
$\prod_{\lambda_s(x)\neq 0} \lambda_s$ and $q$ is an analytic
isomorphism in a neighborhood of $v(x)$.
We define the jet at $P(x)$ by $\widetilde F_{P(x)}= [Q\circ
q^{-1}]^r$ and get:
$[\widetilde F\circ P]^r_x= [(Q\circ q^{-1})^r\circ (q\circ v)]^r_x=
[(Q\circ q^{-1})\circ (q\circ v)]^r_x=(Q\circ v)_x= \tilde f_x .
\quad \Box$
\begin{rem} When the isotropy subgroup of $x_0$ is $W$ itself,
$\; \forall w\in W, \; \tilde f_{x_0}(X)=\tilde f_{wx_0}(wX)=\tilde
f_{x_0}(wX)$. This means that $\tilde f_{x_0}(X)$ is $W$-invariant
and by the polynomial Chevalley's theorem, that $\tilde
f_{x_0}(X)=Q_0(P(X))$. The polynomial $Q_0=\widetilde F_{P(x_0)}$ of
weight $r$ is of degree $[r/h]$ in the invariant polynomial $p$ of
highest degree $h$. The result announced in theorem \ref{thm1} fits
with the formal computation.\end{rem}
\subsection*{A criterion of regularity for $\tilde{F}$}
When $\tilde f\in \mathcal{H}^r(P^{-1}(\mathbf{R}^n))$, the above
proof of lemma \ref{lem5} shows that $\widetilde F= (\tilde f\circ P^{-1})^r$ is
r-regular on the complement in ${\bf R}^n$ of
the critical image $\{u\in {\bf C}^n\mid \Delta (u)=0\}$. The
discriminant $\Delta $ is a polynomial, therefore by lemma \ref{lem4}
it will be sufficient to prove that $\widetilde F$ is $[r/h]$-continuous
on ${\bf R}^n$ to get its $[r/h]$-regularity.
Since $P$ is proper the continuity of $\widetilde F_{\alpha}\circ P$,
entails the continuity of the coefficient $\widetilde F_{\alpha}$.
So let us check the continuity of the $\widetilde F_{\alpha}\circ P$
when $\mid \alpha \mid \le [r/h]$.
Clearly $\widetilde F_0\circ P=\tilde f_0$ is continuous.
For the first derivatives, it is natural to consider the partial
derivatives of $\tilde f$, and get the system:
\begin{equation}
\left( {\partial \tilde f\over \partial z}\right) = \left(\left(
{\partial p_i\over \partial z_j}\right)_{1\le i\le n\atop 1\le j\le n}\right)
\left({\partial \widetilde F\over \partial p} \circ P\right). \label{sys1}
\tag{I}
\end{equation}
\noindent If we show that the loss of differentiability from
$\tilde f=\widetilde F\circ P$ to $(\displaystyle{{\partial \widetilde
F/ \partial p}) \circ P}$ when solving ~\eqref{sys1} is of $h$ units, applying
the same process to $\tilde g_j=({\displaystyle{\partial \widetilde
F}/ \displaystyle{\partial p_j}})\circ P$ instead of $\tilde f=\widetilde F \circ
P$, at the next step again there will be a loss of differentiability
of $h$ units. An induction would show that for
$|\alpha|\le \displaystyle{[{r\over h}]}$, the mappings
$({\displaystyle{\partial^{|\alpha|} \widetilde F}/
\displaystyle{\partial p^{\alpha}}})\circ P$ are continuous on
$P^{-1}({\bf R}^n)$ and since $P$ is proper, that the derivatives
${\displaystyle{\partial^{|\alpha|} \widetilde F}/
\displaystyle{\partial p^{\alpha}}}$ of $\widetilde F$
are continuous on ${\bf R}^n$.\vskip 5pt
\noindent \emph{Conclusion:} To complete the proof, we just have to show that
when solving ~\eqref{sys1}, $\forall j=1, \ldots,n$ we get:
$({\displaystyle{\partial \widetilde F}/ \displaystyle{\partial
p_j}})\circ P \in \mathcal{H}^{r-h}(P^{-1}({\bf R}^n))$.
\subsection*{Solving ~\eqref{sys1}, Reduction to the irreducible
case}
\noindent Using Cramer's method, we multiply both sides by the adjoint matrix of the system.
Since the jacobian determinant is
$ c \;\displaystyle{(\prod_{\tau\in \mathcal{R}}\lambda_{\tau})}$, we get:
\begin{equation}
\left \{ c \;\displaystyle{(\prod_{\tau\in \mathcal{R}}
\lambda_{\tau})\; \;{\partial \widetilde F\over
\partial p_j}\circ P =\sum_{i=1}^n
(-1)^{i+j}M_{i,j}{\partial \tilde f \over \partial
z_i.}},\; j=1\ldots ,n \right. \label{soleq2}\tag{II}
\end{equation}
\noindent From ~\eqref{soleq2}, $\forall \tau \in \mathcal{R}$, if
$\lambda_{\tau}(z)=0$ the polynomial
$\left(\sum_{i=1}^n(-1)^{i+j}M_{i,j}(\partial \tilde f/
\partial z_j)\right)_z(Z)$ is divisible by
$\displaystyle{ \lambda_{\tau}(Z)} $, and since the $\lambda_{\tau}$
are pairwise relatively prime, $\forall \mathcal{U}\subseteq
\mathcal{R}$ if $z\in \bigcap_{\tau\in \mathcal{U}} H_{\tau}$, then
$\left(\sum_{i=1}^n(-1)^{i+j}M_{i,j}(\partial \tilde f/
\partial z_j)\right)_z(Z)$ is divisible by
$ \prod_{\tau \in \mathcal{U}}\lambda_{\tau}(Z)$.\vskip 5pt
In the reducible case, in convenient bases, the jacobian matrix of $P$ is block
diagonal. We have $J_P=J_{P^1}\ldots J_{P^s}$ and
$M_{i,j}$is of the form $J_{P^1}\ldots J_{P^{l-1}}M_{i,j}^lJ_{P^{l+1}}\ldots J_{P^s}$
where $M_{i,j}^l$
is a minor of the $n_l\times n_l$ block associated with $P^l$, Chevalley map of $W^l$.
After simplification we see that it is sufficient to study each block.
For any finite reflection group: \[h=k_n=\sum_{1\le j\le
n}(k_j-1)-\sum_{1\le j\le n-1}(k_j-1)+1,\] where the first sum
$\sum_{1\le j\le n}(k_j-1)$ is the degree of the jacobian
determinant, equal to the number of linear forms $\lambda_{\tau}$
which is the number $d={\mathcal R}^{\#}$ of reflections in $W$. The
second sum $\sum_{1\le j\le n-1}(k_j-1)$ is the least degree $s$ of
the minors $M_{i,j}$ of the jacobian determinant of system
~\eqref{sys1}.
If $W$ is reducible, the formula also holds for each irreducible
component. In particular for any component with the greatest Coxeter
number $h$, we have $h=1+d'-s'$ where $d'$ and $s'$ are those
associated with this irreducible component. Accordingly, we may and
will assume from now on, without loss of generality, that $W$ is an
irreducible Coxeter group.
\subsection*{Stratification and Compensation by the $M_{i,j}$.}
We denote by $S_p$ a stratum in $P^{-1}({\bf R}^n)$, which is a connected component
of the intersection of $\Gamma$ and exactly $p$ of the reflecting hyperplanes.
The points of each stratum are stabilized by the same isotropy group,
subgroup of $W$ generated by reflections about the hyperplanes containing the
stratum.
The different possible isotropy subgroups and strata
types may be determined from the Dynkin diagram. The stratum of dimension
$0$ is the origin. The strata of dimension 1 are those determined by
removing only one point in the Dynkin diagram, they are strata $S_q$ such
that their closure is $\overline{S_q}=S_q\cup\{0\}$. At the other end the
strata of dimension $n$ are the connected components of the regular set
in $\Gamma$. \medskip
On any stratum $S_p$, by lemma \ref{lem1}
and lemma \ref{lem2}, $({\displaystyle{\partial \widetilde
F}/ \displaystyle{\partial p_j}})\circ P \in \mathcal{H}^{r-1+s_p-p}(S_p)$,
if $M_{i,j}$ is at least $(s_p-1)$-flat on $S_p$.
In a neighborhood of $z\in S_p$ we have $P=q\circ v$
with $q$ invertible and $v$ the Chevalley mapping of $W_z$,
isotropy subgroup of $z$ (and of any point in $S_p$). Observe that $W_z$ is
not irreducible: it is not essential since it stabilizes $S_p$, and
it may also have several irreducible components.
The adjoint of the jacobian matrix of $P$ which is the transpose of its
comatrix is the product in this order of the adjoint of the jacobian
matrices $V$ and $Q$ of $v$ and $q$ respectively.
We have $M_{i,j}=\sum_{k=1}^n V_{k,j}Q_{i,k}$. The $V_{k,j}$ and accordingly
the $M_{i,j}$ are $(s_z-1)$-flat on $S_p$. So $s_p=s_z$ and,
since $p$ is the number $d_z$ of reflections in
$W_z,\; 1-s_p+p$ is the Coxeter number $h_z=1-s_z+d_z$ of $W_z$.
The isotropy group of the points $z\in S_p$ is a subgroup of
the isotropy group of the points $z'\in S_{p+q}$. Therefore $h_z\le
h_{z'}$, or $1+p-s_p\le 1+p+q-s_{p+q}$.
Also $1+d-s_d=h$ is larger than
$h_z=1+d_z-s_z$ for any $z\neq 0$.
Finally, the minors $M_{i,j}$ are homogeneous polynomials of
degree: \[s_j=\displaystyle{\sum_{1\le u\le n, u\neq j} (k_u-1)} \ge
s=\displaystyle{ \sum_{1\le u\le n-1} (k_u-1)}.\] They are at least
$(s-1)$-flat on the intersection of the $\mathcal{R}^{\#}=d$
reflecting hyperplanes. (Observe that this might be used to get
$s_p=s_z$ by induction).
In ~\eqref{soleq2}, lemma \ref{lem4} applies to the closure
of each connected component of the regular set, with $\displaystyle{A_i=
{\partial \tilde f / \partial z_i} \in \mathcal{H}^{r-1}(P^{-1}({\bf
R}^n))}$, $Q_i= (-1)^{i+j}M_{i,j}$,
and gives the $[(r-1)+s-d]$-continuity of the ${\displaystyle{(\partial
\widetilde F}/ \displaystyle{\partial p_j})}\circ P$ on their union
$P^{-1}({\bf R}^n)$.
The result we needed to complete the proof:
\[{\displaystyle{\partial \widetilde F}\over \displaystyle{\partial
p_j}}\circ P \in \mathcal{H}^{r-1-d+s}(P^{-1}({\bf R}^n)),\qquad 1+d-s=h\]
is now a consequence of lemma \ref{lem3}.$ \footnote{The closure of each connected
component of the regular set being convex and thus Whitney
1-regular (\cite{24}), lemma \ref{lem4} could directly give
the result.}.\qquad \square$
\begin{rem} The above result gives a loss
of differentiability of $1+d-s$, where $s$ is the least degree
of the $M_{i,j}$. Actually $M_{i,j}$ is the jacobian of the polynomial
mapping:
\[(z_1, \ldots , z_{i-1}, z_{i+1},\ldots , z_n;\; z_i)\mapsto
(p_1(z),\ldots, p_{j-1}(z), p_{j+1}(z),\ldots,p_n(z);\; z_i).\]
\noindent This mapping is invariant by the subgroup $W_i$ of $W$
that leaves invariant the $i^{th}$ coordinate axis in ${\bf R}^n$,
say ${\bf R\; e}_i$ (\cite{3}). This subgroup $W_i$ is generated by
the subset $\mathcal{R}_i\subset \mathcal{R} $ of the reflections it
contains. These are the reflections $\alpha$ in $W$ such that
$\alpha({\bf e}_i)={\bf e}_i$, about the hyperplanes $H_{\alpha}$
containing ${\bf e}_i$ \footnote{The description of $W_i$ given in
\cite{3} was correct. Unfortunately in \cite{4} it was not. Although
not essential in the reasoning it was misleading. The explicit
computations were correct however and gave the best result in the
cases of $A_n$, $B_n$ and $I_2(k)$.}. The $M_{i,j},\; j=1,\ldots,
l,$ as jacobians of $W_i$-invariant polynomial mappings are
polynomial multiples of $(\prod_{\tau\in \mathcal{R}_i}
\lambda_{\tau})$. In \cite{4} the formula for the loss of
differentiability at each step was also of the form $1+d-s$, but
$s$ was $\min_{1\le i\le n}{\mathcal R}_i^{\#}$. Clearly
$\min_{1\le i\le n}{\mathcal R}_i^{\#}\le \min_{1\le i,j\le n}
{{\rm degree}M_{i,j}}$. In some cases ($A_n, B_n, I_2(k)$)
the equality holds but in general the loss of differentiability
given by \cite{4} was not the best possible. Considering $H_3$ for an example,
we now have $s= 6$ instead of $2$, and
the class of differentiability of $F$ is $[r/10]$ instead of $[r/14]$.
\end{rem}
All the operations from $f\in \mathcal{C}^r({\bf
R}^n)^W$ up to $F\in \mathcal{C}^{[r/h]}({\bf R}^n)$ are linear and
continuous when using the natural Fr\'echet topologies. A modulus of
continuity for the Whitney conditions could be followed from
$\|f\|^r$ to $\|F\|^{[r/h]}$. So Chevalley's theorem in class
$\mathcal{C}^r $ may be restated as:
{\bf Theorem 1.1.} {\it Let $W$ be a finite group generated by
reflections acting orthogonally on ${\bf R}^n$, $P$ the Chevalley
polynomial mapping associated with $W$, and $h=k_n$ the highest
degree of the coordinate polynomials in $P$ (equal to the greatest
Coxeter number of the irreducible components of $W$). There exists a
linear and continuous mapping:
$$\mathcal{C}^r({\bf R}^n)^W \ni f \to F\in \mathcal{C}^{[r/h]}
({\bf R}^n)$$ such that $f=F\circ P$.}
\begin{rem} Theorem \ref{thm1} gives global results. As it is clear
from the proof, in the neighborhood of each point $x$ the loss of
differentiability is governed by the isotropy group $W_x$ and its
Coxeter number $h_x$.
About the partial derivatives at the origin (or a point $x$ where
$W_x=W$), since the $M_{i,j}$ are homogeneous of degree $s_j=
\sum_{1\le u\le n, u\neq j}(k_u-1)$, we see that $({\partial
\widetilde F/\partial p_j})\circ P$ is of class ${\mathcal
H}^{r-k_j}$. Reasoning as above, we could show that the partial
derivatives $\displaystyle{{\partial^{|m|} \widetilde F/
\partial P^m}}$ of order $m=(m_1,\ldots,m_n)$ are continuous if
$m_1k_1+\ldots +m_nk_n\le r$. For instance the partial derivatives
in $W$-invariant directions are continuous up to the order $r$.
\end{rem}
\section{Counter Example.}
Let us give a counter example which applies to almost every finite
reflection group. It is sufficient to consider essential
irreducible groups.
We consider $F:{\bf R}^n \to {\bf R}$ defined by $
F(y)=y_n^{s+\alpha}$ for some integer $s$ and an $\alpha\in ]0,1[$.
$F$ is of class $\mathcal{C}^s$ but not of class $\mathcal{C}^{s+1}$
in any neighborhood of $\{y | y_n = 0\}$. Let $P$ be the Chevalley
mapping associated with some finite irreducible Coxeter group $W$
acting on ${\bf R}^n$ and consider the composite mapping $F\circ
P(x)= p_n^{s+\alpha}(x)$. We study the differentiability of this
mapping when $p_n(x)=0$.
A set of basic invariants is available in \cite{18} for any finite
Coxeter group. Disregarding $D_n$, for any other group there exists
an invariant set of linear forms $\{L_1,\ldots ,L_v\}$ the kernels
of which intersect only at the origin, and such that for
$i=1,\ldots,n,\;p_i(X)=\sum_{j=1}^v [L_j(X)]^{k_i}$ with $k_i$s as
determined in [7]. With the two exceptions of $A_{2n}$ and
$I_2(2p+1)$, $k_n$ is even and therefore $p_n(x)$ vanishes only at
the origin. We will not study the two exceptional cases, but a
fairly general counter example is given in [1] for symmetric
functions and thus for $A_n$ (including $A_2=I_2(3)$). As usual,
$D_n$ does not follow the general line but we may choose $p_n(x)
=\sum_1^n x_i^{2(n-1)}$ and the results of the general case apply.
We have $p_n(x)=\sum_1^v[L_i(x)]^{k_n}$, and since $|L_i(x)|\le
a_i|x|,\; i=1,\ldots , v$ for some numerical constants $a_i$, we
have the estimate $|p_n(x)|\le (\sum_1^v
a_i^{k_n})|x|^{k_n}=A|x|^{k_n}$.
Analogously, since $|D^1L_i(x)|\le b_i$ for some numerical constants
$b_i$, we get: $$|D^jp_n(x)|\le \sum_1^v b_i^j{k_n\choose
j}|L_i(x)|^{k_n-j}= B_j|x|^{k_n-j}.$$
The derivatives of the composite mapping $p_n^{s+\alpha}(x)$ are
given by the Faa di Bruno formula: \[D^k p_n^{s+\alpha}(x)= \sum
{k!\over \mu_1!\ldots \mu_q!}
D^py_n^{s+\alpha}(p_n(x))\big({D^1p_n(x)\over 1!}\big)^{\mu_1}
\ldots \big({D^qp_n(x)\over q!}\big)^{\mu_q},\] where the sum is
over all the $q$-tuples $(\mu_1,\ldots\mu_q)\in {\bf N}^q$ such that
$1\mu_1+\ldots +q\mu_q=k$, with $p=\mu_1+\ldots +\mu_q$. There are
constants $C_{(\mu_1,\ldots,\mu_q)}$ such that:
$$ |\big({D^1p_n(x)\over 1!}\big)^{\mu_1} \ldots
\big({D^qp_n(x)\over q!}\big)^{\mu_q}|\le
C_{(\mu_1,\ldots,\mu_q)}|x|^{(k_n-1)\mu_1+\ldots
+(k_n-q)\mu_q}=C_{(\mu_1,\ldots,\mu_q)}|x|^{k_np-k},$$ and therefore
constants $A_{(\mu_1,\ldots\mu_q)}$ and $A$ such that: \[|D^k
p_n^{s+\alpha}(x)|\le \sum
A_{(\mu_1,\ldots\mu_q)}|x|^{k_n(s+\alpha-p)}|x|^{k_np-k} \le A
|x|^{k_ns+k_n\alpha-k}.\] This shows that the derivatives of order
$k\le k_ns$ tend to $0$ at the origin while the derivatives of order
$k_ns+1$ will not if $\alpha<1/k_n$. This means that the composite
mapping $f=F\circ P$ is of class $\mathcal{C}^{k_n s}$ but not of
class $\mathcal{C}^{k_n s+1}$ at $x=0$ and it factors through $F$
which is of class $\mathcal{C}^s$ and not of class
$\mathcal{C}^{s+1}$. The loss of differentiability is as given in
theorem \ref{thm1} and cannot be reduced.
\bibliographystyle{amsplain}
|
1,116,691,497,551 | arxiv | \section*{Abstract}
Understanding tumor invasion and metastasis is of crucial importance
for both fundamental cancer research and clinical practice.
\textit{In vitro} experiments have established that the invasive growth
of malignant tumors is characterized by the dendritic invasive branches composed of chains
of tumor cells emanating from the primary tumor mass. The preponderance
of previous tumor simulations focused on non-invasive (or proliferative) growth.
The formation of the invasive cell chains and their interactions
with the primary tumor mass and host microenvironment are not well understood.
Here, we present a novel cellular automaton (CA) model that enables one to efficiently
simulate invasive tumor growth in a heterogeneous host microenvironment.
By taking into account a variety of microscopic-scale tumor-host interactions,
including the short-range mechanical interactions between
tumor cells and tumor stroma, degradation of extracellular matrix by the
invasive cells and oxygen/nutrient gradient driven cell motions,
our CA model predicts a rich spectrum of growth dynamics and emergent behaviors of invasive tumors.
Besides robustly reproducing the salient features
of dendritic invasive growth, such as least resistance and intrabranch homotype attraction,
we also predict nontrivial coupling of the growth dynamics of
the primary tumor mass and the invasive cells. In addition, we show that
the properties of the host microenvironment can significantly
affect tumor morphology and growth dynamics,
emphasizing the importance of understanding the tumor-host interaction.
The capability of our CA model suggests that well-developed \textit{in silico} tools
could eventually be utilized in clinical situations to predict neoplastic progression
and propose individualized optimal treatment strategies.
\section*{Author Summary}
The goal of the present work is to develop an efficient single-cell based
cellular automaton (CA) model that enables one to investigate the growth dynamics
and morphology of invasive solid tumors. Recent experiments have shown that highly malignant
tumors develop dendritic branches composed of tumor cells that follow each other,
which massively invade into the host microenvironment and ultimately lead
to cancer metastasis. Previous theoretical/computational
cancer modeling neither addressed the question of how such chain-like
invasive branches form nor how they interact with the host microenvironment
and the primary tumor. Our CA model, which incorporates a variety of
microscopic-scale tumor-host interactions (e.g., the mechanical interactions
between tumor cells and tumor stroma, degradation of extracellular matrix by
the tumor cells and oxygen/nutrient gradient driven cell motions), can robustly reproduce
experimentally observed invasive tumor evolution and predict a
wide spectrum of invasive tumor growth dynamics and emergent behaviors in various different heterogeneous
environments. Further refinement of our CA model could eventually lead to the
development of a powerful simulation tool that can be utilized in the clinic
to predict neoplastic progression
and propose individualized optimal treatment strategies.
\section*{Introduction}
Cancer is not a single disease,
but rather a highly complex and heterogeneous set of diseases that can
adapt in an opportunistic manner, even under a variety of stresses.
It is now well accepted that genome level changes in cells, resulting in
the gain of function of oncoproteins or the loss of function of tumor suppressor
proteins, initiate the transformation of normal cells into malignant ones
and neoplastic progression \cite{Co98, hanahan00}.
In the most aggressive form, malignant cells can leave the primary tumor,
invade into surrounding tissues, find their way into the
circulatory system (through vascular network)
and be deposited at certain organs in the body, leading to the
development of secondary tumors (i.e., metastases) \cite{fearon90}.
The emergence of invasive behavior in cancer is fatal. For example,
the malignant cells that invade into the surrounding host tissues can quickly
adapt to various environmental stresses and develop resistance to therapies.
The invasive cells that are left behind after resection are responsible for
tumor recurrence and thus an ultimately fatal outcome. Therefore,
significant effort has been expended to understand the mechanisms evolved in the invasive
growth of malignant tumors \cite{hanahan00, deisboeck01, fidler03, kerbel90, liotta03}
and their treatment \cite{gillies04, gatenby09}.
It is generally accepted that the invasive behavior of cancer
is the outcome of many complex interactions occurring between
the tumor cells, and between a tumor and the host
microenvironment \cite{fearon90}. Tumor invasion itself
is a complex multistep process involving homotype detachment,
enzymatic matrix degradation, integrin-mediated heterotype adhesion,
as well as active, directed and random motility \cite{deisboeck01}.
In recent \textit{in vitro} experiments involving glioblastoma multiforme (GBM),
the most malignant brain cancer, it has been observed that
dendritic invading branches composed of chains
of tumor cells are emanating from the primary tumor mass; see Fig.~\ref{fig_MTS}.
Such invasive behaviors are characterized by intrabranch homotype
attraction and least resistance \cite{deisboeck01}.
Although recently progress has been made in understanding certain
aspects of the complex tumor-host interactions that may be responsible
for invasive cancer behaviors \cite{deisboeck01, fred91, brand00, kitano04},
many mechanisms are either not fully understood or are unknown at the moment.
Even if all of the mechanisms for cancer invasion could be identified,
it is still not clear that progress in understanding neoplastic progression
and proposing individualized optimal treatment strategies could be made
without the knowledge of how these different mechanisms couple
to one another and to the heterogeneous host microenvironment in which tumor grows \cite{To11}.
Theoretical/computational cancer modeling that integrates apart
mechanisms, when appropriately linked with experimental
and clinical data, offers a promising avenue for a better understanding
of tumor growth, invasion and metastasis.
A successful model would enable one to broaden the conclusions drawn
from existing medical data, suggest new experiments, test hypotheses,
predict behavior in experimentally unobservable situations, and be
employed for early detection and prognosis.
Indeed, cancer modeling has been a very active area of research for
the last two decades (see Refs.~\cite{Byrne10} and \cite{To11} for recent reviews).
A variety of interactions between tumor cells and between tumor and its
host microenvironment have been investigated \cite{chaplain98, kansal00a, kansal00b, kansal02,
gevertz06, jana08, jana09, anderson05, anderson06,
bankhead07, gatenby96, gatenby96b, gatenby06, bellomo00, scalerandi01, scalerandi02, kim10},
via continuum \cite{gatenby96, gatenby96b, gatenby06, gatenby06b, kim10}, discrete \cite{kansal00a, jana08, stein07}
or hybrid \cite{gevertz06, jana09, anderson05, anderson06} mathematical models.
Very recently, multiscale mathematical models \cite{anderson05, anderson06, gatenby06b}
have been employed to study the effects of
the host microenvironment on the morphology and phenotypic evolution of invasive tumors
and it has been shown that microenvironmental heterogeneity can dramatically
affect the growth dynamics of invasive tumors. Although the simulated
tumors showed certain invasive characteristics (e.g., developing protruding
surfaces), no dendritic invasive branches emerged.
In response to the challenge to develop an ``Ising'' model for
cancer growth \cite{To11}, we generalize a cell-based discrete
cellular automaton (CA) model that we have developed
\cite{kansal00a, kansal00b, kansal02, jana08, jana09} to
investigate the invasive growth of malignant tumors in
heterogeneous host microenvironments. To the best of our knowledge,
this generalized CA model is the first to investigate the
formation of invasive cell chains and their interactions with the
primary tumor mass and the host microenvironment. Our cellular
automaton model takes into account a variety of microscopic-scale
tumor-host interactions, including the short-range mechanical
interactions between tumor cells and tumor stroma, the degradation
of extracellular matrix by the invasive cells and oxygen/nutrient
gradient driven cell motions and thus, it can predict a wide range
of growth dynamics and emergent behaviors of invasive tumors. In
particular, our CA model robustly reproduces the salient features
of dendritic invasive growth observed in experiments, which is
characterized by least resistance and intrabranch homotype
attraction. The model also predicts nontrivial coupling of the
growth dynamics of the primary tumor mass and the invasive cells,
e.g., the invasive cells can facilitate the growth of primary
tumor in harsh microenvironment. Moreover, we show that the
properties of the host microenvironment can significantly affect
tumor growth dynamics and lead to a variety of tumor morphology.
These emergent behaviors naturally arise due to various
microscopic-scale tumor-host interactions, which emphasizes the
importance of taking into account microenvironmental heterogeneity
in understanding cancer. Further refinement of our model could
eventually lead to the development of a powerful \textit{in
silico} tool that can be utilized in the clinic. As a
demonstration of the capability and versatility of our CA model,
we mainly consider invasive tumor growth in two dimensions,
although the model is easily extended to three dimensions. Indeed,
the algorithmic details of the model are given for any spatial
dimension.
\section*{Materials and Methods}
\subsection*{Biophysical Background of the CA Model}
\subsubsection*{Voronoi Tessellation: The Underlying Cellular Structure}
The underlying cellular structure is modeled using a
Voronoi tessellation of the space into polyhedra \cite{torquato},
based on centers of spheres in a packing generated by a random sequential
addition (RSA) process \cite{kansal00a, jana09} (see Fig.~\ref{fig_Vor}).
In particular, nonoverlapping spheres are randomly and sequentially
placed in a prescribed region until there is no void space for additional
spheres, i.e., saturation is achieved. Such a saturated RSA packing possesses
relative small variations in its Voronoi polyhedra and thus, has
served as models for many biological systems \cite{To_RMP, Patel09}.
We refer to the polyhedra associated with the Voronoi
tessellation as automaton cells. These automaton cells may
correspond to real cells or tumor stroma (e.g., clusters of the ECM macromolecules). In previous studies, such
automaton cells have represented clusters of real cells of
various sizes or have implicitly represented healthy tissues \cite{kansal00a}.
Thus, the Voronoi tessellation associated with RSA sphere centers provides a
highly flexible model for real-cell aggregates with a high degree of shape isotropy.
For example, one can use a variable automaton cell size to
simulate avascular tumor growth from a few malignant cells
to its macroscopic size \cite{kansal00a}. In addition, such a Voronoi tessellation
can reduce the undesired growth bias due to the anisotropy
of ordered tessellations based on square and simple cubic lattices.
Since our new CA model explicitly takes into account the
interactions between a single cell and its neighbors and
microenvironment, each automaton cell here represents either a
single tumor cell or a region of tumor stroma. Thus, the linear
size of a single automaton cell is approximately $15 - 20~\mu$m
and the linear size of the 2D simulation domain is approximately 5
mm, which contains $\sim 100~000$ automaton cells. In the current
model, we mainly focus on the effects of ECM
macromolecule density and ECM degradation by malignant cells on tumor growth.
Henceforth, we will refer to the host microenvironment (or tumor
stroma) as ``ECM'' for simplicity. Each ECM associated
automaton cell is assigned a particular density
$\rho_{\mbox{\tiny{ECM}}}$, representing the density of the ECM
molecules within the automaton cell. A tumor cell can occupy an
ECM associated automaton cell only if the density of this
automaton cell $\rho_{\mbox{\tiny{ECM}}} = 0$, which means that
either the ECM is degraded or it is deformed (pushed away) by the
proliferating tumor cells.
\subsubsection*{Microenvironment Heterogeneity}
The microenvironment in which tumor grows is usually highly heterogeneous, composed of various types of
stromal cells and ECM structures. The ECM is a complex mixture of
macromolecules that provide mechanical supports for the tissue (such as collagens) and
those that play an important role for cell adhesion and motility (such as laminin and fibronectin) \cite{anderson05, jana_brm, ECM}.
For different individuals with tumors, the ECM in
the host microenvironments generally possess distinct mechanical and transport properties.
By explicitly representing the ECM using automaton cells with different macromolecule densities,
the effects of microenvironment heterogeneity
on tumor growth can be very well explored. For example,
various distributions of the ECM densities (i.e., the densities of the
ECM macromolecules) can be employed to mimic
the actual heterogeneous host microenvironment of the tumor. Certain
tumor stroma contains fibroblasts, which actively produce ECM macromolecules
leading to a higher ECM density around these cells.
The automaton cells representing the ECM
with larger densities are considered to be more rigid and more difficult
to degrade. Since each automaton cell associated with the ECM has its own density, this
allows a variation of ECM characteristics on the length scale comparable
to that of a single tumor cell.
In addition, the tumor in our model is only allowed to grow in a compact
growth-permitting region. This is to mimic the physical confinement of the host microenvironment,
such as the boundary of an organ. In other words, only automaton
cells within this region can be occupied by the cells of the tumor as it grows.
In general, the growth-permitting
region can be of any shape that best models the organ shape. Here
we simply choose a spherical region to study the effects of the heterogeneous
ECM on tumor growth. More sophisticated growth-region shapes have
been employed to investigate the effects of physical confinement on tumor growth \cite{jana08, jana09}.
Furthermore, we assume a constant radially symmetric nutrient/oxygen
gradient in the growth-permitting region with the highest nutrition
concentration at the boundary of this region, i.e., it is a vascular boundary.
However, this assumption can also be relaxed.
\subsubsection*{Tumor Cell Phenotypes and Interactions with the Host Microenvironment}
For highly malignant tumors, we consider the cells to be of one of
the two classes of phenotypes: either invasive or non-invasive.
Following Ref.\cite{kansal00a}, the non-invasive
cells remain in the primary tumor and can be proliferative, quiescent
or necrotic, depending on the nutrition supply they get. For avascular
tumor growth, as we focus here, the nutrition the tumor cells can get
are essentially those diffuse into the tumor through its surface. As
the tumor grows, the amount of nutrition supply,
which is proportional to the surface area of the tumor, cannot
meet the needs of all cells whose number increases with the tumor volume,
leading to the development of necrotic and quiescent regions.
Following Ref.\cite{kansal00a}, characteristic diffusion lengths are employed to
determine the states of a non-invasive cell. For example,
quiescent cells more than $\delta_n$ away from
the tumor surface become necrotic (see details in the next section).
The diffusion length $\delta_n$ (also the characteristic thickness of
the rim of living tumor cells) depends on
the size of the primary tumor.
As a proliferative cell divides, its daughter cell effectively
pushes away/degrades the surrounding ECM and occupies the
automaton cell originally associated with the ECM \cite{mech1,
mech2, degrad1}. It is easier for a tumor cell to take up an ECM
associated automaton cell with lower density (i.e., less rigid ECM
regions) than that with higher density (i.e., more rigid ECM
regions) and thus, the tumor growth is affected by the ECM
heterogeneity through the local mechanical interaction between
tumor cells and the ECM. If there is no space available for the
placement of a daughter cell within a distance $\delta_p$ from the
proliferative cell, the proliferative cell turns quiescent.
The invasive cells are considered to be mutant daughters of the
proliferative cells \cite{mutant1}, which can gain a variety of degrees of ECM
degradation ability $\chi$ (i.e., the matrix-degradative enzymes)
and motility $\mu$ that allow them to leave the
primary tumor and invade into surrounding microenvironment \cite{degrad2}.
We consider the invasive cells can move from one automaton cell
to another only if the ECM in the target automaton cell
is completely degraded (i.e., with $\rho_{\mbox{\tiny{ECM}}} = 0$). Each trial
move of an invasive cell involves the degradation of the ECM in its neighbor
automaton cells, followed by a possible move to one of the automaton cells whose ECM
is completely degraded; otherwise the invasive cell does not move.
The number of trial moves of an invasive cell and to what extent it
degrades the ECM are respectively determined by $\mu$ and $\chi$
(see the following section for details).
The oxygen/nutrient gradient also drives the invasive cells to move as far as
possible from the primary tumor \cite{mot}, which takes up the majority of oxygen/nutrients.
The motility $\mu$ is the maximum possible number of such trial moves.
In addition, we assume that the invasive cells do not divide as they migrate.
\subsection*{Algorithmic Details}
We now provide specific details for the CA model to study invasive tumor growth in
confined heterogeneous microenvironment. In what follows, we will simply refer to the
primary tumor as ``the tumor'' and explicitly use ``invasive'' when considering invasive cells.
After generating the automaton cells by Voronoi tessellation of RSA sphere centers,
an ECM macromolecule density $\rho_{\mbox{\tiny{ECM}}} \in (0, ~1)$ is assigned to
each automaton cell within the growth-permitting region, which represents
the heterogeneous host microenvironment. Then a tumor is introduced by designating any
one or more of the automaton cells as proliferative
cancer cells. Time is then discretized into units that represent one real
day. At each time step:
\begin{itemize}
\item Each automaton cell is checked for type: invasive, proliferative, quiescent,
necrotic or ECM associated. Invasive cells degrade and migrate into the ECM surrounding the
tumor. Proliferative cells are actively dividing cancer cells, quiescent
cancer cells are those that are alive, but do not have enough oxygen and
nutrients to support cellular division and necrotic cells are dead cancer cells.
\item All ECM associated automaton cells and tumorous necrotic cells are inert
(i.e., they do not change type).
\item Quiescent cells more than a certain distance $\delta_n$ from the tumor's
edge are turned necrotic. The tumor's edge, which is assumed to be the
source of oxygen and nutrients, consists of all ECM associated automaton cells that border
the neoplasm. The critical distance $\delta_n$ for quiescent cells to turn
necrotic is computed as follows:
\begin{equation*}
\label{eq_delta}
\delta_n = aL_t^{(d-1)/d},
\end{equation*}
where $a$ is prescribed parameter (see Table \ref{tab_Param}), $d$ is the spatial dimension and
$L_t$ is the distance between the
geometric centroid ${\bf x}_c$ of the tumor and the tumor edge cell that is closest to the
quiescent cell under consideration. The position of the tumor centroid ${\bf x}_c$ is given by
\begin{equation*}
\label{eq_xc}
{\bf x}_c = \frac{{\bf x}_1 + {\bf x}_2 + \cdots + {\bf x}_N}{N},
\end{equation*}
where $N$ is the total number of noninvasive cells contained in the tumor, which
is updated when a new noninvasive daughter cell is added to the tumor.
\item Each proliferative cell will attempt to divide with probability $p_{div}$
into the surrounding ECM (i.e., the automaton cells associated with the ECM) by
degrading and pushing away the ECM in that automaton cell.
We consider that $p_{div}$ depends on both the physical confinement imposed by the
boundary of the growth-permitting region and the local mechanical interaction
between the tumor cells and the ECM, i.e.,
\begin{equation*}
\label{eq_pdiv}
p_{div} = \left\{
\begin{array}{ll}
& \mbox{if any ECM associated automaton cell within} \\
\frac{p_0}{2}[(1-\frac{r}{L_{max}})+(1-\rho_{\mbox{\tiny{ECM}}})] & \mbox{the predefined growth distance is in the growth-}\\
& \mbox{permitting microenvironment} \\ \\
& \mbox{if no ECM associated automaton cell within} \\
0 & \mbox{the predefined growth distance is in the growth-}\\
& \mbox{permitting microenvironment}
\end{array}
\right.
\end{equation*}
where $p_0$ is the base probability of division (see Table \ref{tab_Param}),
$r$ is the distance of the dividing cell from the tumor centroid,
$L_{max}$ is the distance between the closest growth-permitting
boundary cell in the direction of tumor growth and the tumor's
geometric centroid ${\bf x}_c$ and $\rho_{\mbox{\tiny{ECM}}}$ is the ECM density of the
automaton cell to be taken by the new tumor cell. When a
ECM associated automaton cell is taken by a tumor cell, it density is set to be zero.
The predefined growth distance ($\delta_p$) is described in a following bullet point.
\item If a proliferative cell divides, it can produce a mutant daughter cell
possessing an invasive phenotype with a prescribed probability $\gamma$ (i.e., the
mutation rate). The invasive daughter cell gains ECM degradation ability
$\chi$ and motility $\mu$, which enables it to leave the
tumor and invade into surrounding ECM. The rules for updating invasive
cells are given in a following bullet point. If the daughter cell is noninvasive,
it is designated as a new proliferative cell.
\item A proliferative cell turns quiescent if there is no space available for
the placement of a daughter cell within a distance $\delta_p$ from the
proliferative cell, which is given by
\begin{equation*}
\label{eq_deltap}
\delta_p = bL_t^{(d-1)/d},
\end{equation*}
where $b$ a nutritional parameter (see Table \ref{tab_Param}), $d$ is the
spatial dimension and $L_t$ is the
distance between the geometric tumor centroid ${\bf x}_c$ and the tumor edge cell that is
closest to the proliferative cell under consideration.
\item An invasive cell degrades the surrounding ECM (i.e., those in the neighboring
automaton cells of the invasive cell) and can move from
one automaton cell to another if the associated ECM is completely degraded.
For an invasive cell with motility $\mu$ and ECM degradation ability $\chi$,
it will make $m$ attempts to degrade the ECM in the neighboring automaton cells
and jump to these automaton cells, where $m$ is an arbitrary integer in $[0,~\mu]$.
For each attempt, the surrounding ECM density $\rho_{\mbox{\tiny{ECM}}}$ is
decreased by $\delta\rho$, where $\delta\rho$ is an arbitrary number in $[0,~\chi]$.
Using random numbers for ECM degradation ability and cellular motility
is to take into account tumor genome heterogeneity, which is manifested as
heterogeneous phenotypes (such as different $m$ and $\delta\rho$).
When the ECM in multiple neighboring automaton cells of
the invasive cell are completely degraded (i.e., $\rho_{\mbox{\tiny{ECM}}} = 0$), the invasive cell moves
in a direction that maximizes the nutrients
and oxygen supply. Here we assume that the migrating invasive cells do not
divide. The degraded ECM shows the invasive path of the tumor.
\end{itemize}
\noindent The aforementioned automaton rules are briefly illustrated in
Fig.~\ref{fig_CA}. We note that non-invasive tumor growth can be studied by
imposing a mutation rate $\gamma = 0$. This enables us to compare the growth dynamics
of invasive and non-invasive tumors and in turn to investigate
the effects of coupling growth of the primary tumor mass and the invasive cells.
Although we only consider spherical-growth-permitting regions here,
the CA rules given above allow growth-permitting regions with arbitrary shapes.
The important parameters mentioned in the bullet points above are summarized in Table \ref{tab_Param}.
In the following, we will employ our CA model to investigate the growth
dynamics of malignant tumors with different degrees of invasiveness
in variety of different heterogeneous microenvironments.
\subsection*{Quantitative Metrics for Tumor Morphology}
To characterize quantitatively the morphology of simulated tumors,
we present several scalar metrics that capture the salient
geometric features of the primary tumor, dendritic invasive
branches or the entire invasive pattern. These metrics include the
ratio $\beta$ of invasive area over tumor area (defined below),
the specific surface $s$ of the invasive pattern, the asphericity
$\alpha$ of the primary tumor and the angular anisotropy metric
$\psi$ for the invasive branches. The metrics are computed for all
simulated tumors and compared to available experimental data. We
note that the invasive pattern associated with a neoplasm includes
both the primary tumor and the invasive branches.
Following Ref.~\cite{deisboeck01}, the tumor area $A_T$ is defined as
the area of the circumcircle of the primary tumor (see Fig.~\ref{Metric}(a))
and the invasive area $A_I$ is the area of the region between the effective circumcircle
of the invasive pattern and the circumcircle of the primary tumor (see Fig.~\ref{Metric}(a)).
The radius of the effective circumcircle of the invasive pattern is
defined to be the average distance from the invasive branch tip to the tumor center.
The ratio $\beta = A_I/A_T$ as a function of time $t$
reflects the degree of coupling between the primary tumor and the invasive cells.
If $\beta(t)$ is linear in $t$, there is no coupling; otherwise the two are coupled.
The specific surface $s$ \cite{torquato} for the invasive pattern is defined as the ratio of
the total length of the perimeter of the invasive pattern over its total area.
In general, $s$ is inversely proportional to the size of the tumor and thus, large
tumors have small $s$ values. Moreover, given the tumor size, tumors with a large number of
long dendritic invasive branches possess a large value of $s$. And $s$ is minimized
for perfectly circular tumors with $s=2/R_T$, where $R_T$ is the radius.
Since $s$ depends on the size of the tumor, which makes it difficult to
compare tumors with different sizes, in the calculations that follow we employ a normalized $s$ with respect to $2/R_T$
for an arbitrary-shaped tumor with effective radius $R_T$ (i.e., the average distance from tumor edge to tumor center).
For simplicity, we will still refer to the normalized specific surface as
``specific surface'' and designate it with symbol $s$.
The asphericity $\alpha$ of the primary tumor is defined as the ratio
of the radius of circumcircle $R_c$ of the primary tumor over its incircle
radius $R_{in}$ \cite{ToJi09}, i.e., $\alpha = R_c/R_{in}$ (see Fig.~\ref{Metric}(b)).
A large $\alpha$ value indicates a large deviation of the shape of
primary tumor from that of a perfect circle, i.e., the tumor is more anisotropic.
To quantify the degree of anisotropy of the invasive branches, we
introduce the angular anisotropy metric $\psi$. In particular, the
entire invasive pattern is evenly divided into $n_a$ sectors with lines
emanating from the tumor center (see Fig.~\ref{Metric}(c)).
The angular anisotropy metric $\psi$ is defined as
\begin{equation}
\label{eq_psi}
\psi = \frac{\sum_{i=1}^{n_a}|\overline{\ell}(i)-\ell _{ave}|}{n_a \cdot \ell_{ave}},
\end{equation}
where $\overline{\ell}(i)$ is the average length of the invasive branches
within the $i$th sector and
\begin{equation}
\ell_{ave} = \frac{\sum_{i=1}^{n_a}\overline{\ell}(i)}{n_a},
\end{equation}
is the average length of all invasive branches. For tumors with
invasive branches of similar lengths that are uniformly angularly
distributed, the metric $\psi$ is small. Large fluctuations of
both invasive branch length and angular distribution can lead to
large $\psi$ values. In the following, we use $n_a = 16$ to
compute $\psi$ for the simulated invasive tumors.
\section*{Results}
\subsection*{Model Validation}
To verify the robustness and predictive capacity of our CA model,
we first employ it to quantitatively reproduce the observed invasive
growth of a GBM multicellular tumor spheroid (MTS) in vitro \cite{deisboeck01}.
In particular, the boundary of the growth-permitting region is considered to be
vascularized, i.e., a growing tumor can receive oxygen and nutrients
from the growth-permitting region. A constant radially symmetric nutrient/oxygen
gradient in the growth-permitting region with the highest nutrient/oxygen
concentration at the vascular boundary is used. Initially, approximately 250 proliferative
tumor cells are introduced at the center of the growth-permitting
region with homogeneous ECM and tumor growth is started.
This corresponds to an initial MTS with diameter $D_{\mbox{\tiny MTS}} \approx 310~ \mu$m
which is consistent with the in vitro experiment set-up \cite{deisboeck01}.
The following values of the growth and invasiveness parameters are used: $p_0 = 0.384$,
$a = 0.58$ mm$^{1/2}$, $b = 0.30$ mm$^{1/2}$, $\gamma = 0.05$, $\chi = 0.55$, $\mu = 3$.
Note that the value of $p_0$ corresponds to a cell doubling time of 40 hours, which
is consistent with the reported experimental data \cite{deisboeck01}.
A small value of ECM density $\rho_{\mbox{\tiny ECM}} = 0.15$
is used, which corresponds to the soft DMEM medium used in the experiment \cite{deisboeck01}.
In the visualizations of the tumor that follow, we use the following convention:
the ECM in the growth-permitting region is white, and gray outside this region.
The ECM degraded by the tumor cells is blue.
In the primary tumor, necrotic cells are black, quiescent cells are yellow
and proliferative cells are red. The invasive tumor cells are green.
Figure \ref{fig_SimuMTS}(a) and (b) respectively show the morphology
of simulated MTS and a magnification of its invasive branches with
increasing branch width towards the proliferative core. Specifically,
one can clearly see that within the branches, chains of cells are formed
as observed in experiments \cite{deisboeck01} (see Fig.~1). The invasive
cells tend to follow one another (which is termed ``homotype attraction'') since paths of degraded ECM are
formed by pioneering invasive cells and it is easier for other cells
to follow and enhance such paths than degrading ECM to create new paths by themselves.
In other words, invasive cells tend to take paths with ``least resistance''.
We note that no CA rules are imposed to force such
cellular behaviors. Instead, they are emergent properties that arise
from our simulations.
The ratio of invasion area over primary tumor area $\beta = A_{I}/A_{T}$
as a function of time for the simulated tumor is computed and compared to reported experimental
data \cite{deisboeck01} (see Fig.~\ref{fig_SimuMTS}(c)). One can clearly see that our simulation
results agree with experimental data very well. Moreover, the
deviation of $\beta(t)$ from a linear function of $t$ indicates
that the growth of primary tumor and the invasive branches are strongly coupled \cite{deisboeck01}
Other metrics for tumor morphology such as the
specific surface $s$ of the invasive pattern, the sphericity $\alpha$
of the primary tumor and the angular anisotropy metric $\psi$
for the invasive branches are computed from our simulation results
and from the image of invasive MTS in Fig.~1(a) at 24 hours after initialization.
The values are given in Table \ref{tab_MTS}, from which one can see again a good agreement.
Thus, we have shown that our CA model is both robust and quantitatively accurate with properly
selected parameters.
\subsection*{Simulated Invasive Growth in Heterogeneous Miroenvironments}
Having verified the robustness and predictive capacity of our CA
model, we now consider three types of ECM density distributions,
i.e., homogeneous, random and sine-like, to systematically study
the effects of microenvironment heterogeneity on invasive tumor
growth (see Fig.~\ref{fig_ECM}). These ECM density distributions
represent real host microenvironments in which a tumor grows.
(Details about these ECM distributions are given in the following
sections.) Again, the boundary of the growth-permitting region is
considered to be vascularized with a constant radially symmetric
nutrient/oxygen gradient in the growth-permitting region pointing
to the tumor center. We note that although generally the
nutrient/oxygen concentration field in vivo is more complicated,
previously numerical studies that considered the exact evolution
of nutrient/oxygen concentrations have shown a decay of the
concentrations toward the tumor center \cite{anderson05, anderson06}. Since the directions of
cell motions are determined only by the nutrient/oxygen gradient, our
constant-gradient approximation is a very reasonable one.
In the beginning, a proliferative tumor cell is introduced at the
center of the growth-permitting region and tumor growth is
initiated. The growth parameters for the primary tumors in all
cases studied here are the same and are given in Table
\ref{tab_Param}. The invasiveness parameters and ECM densities are
variable and specified in each case separately. The values of the
growth parameters for the CA model were chosen to be consistent
with GBM data from the medical literature \cite{kansal00a,
jana08}. Specifically, the value of the base probability of
division is $p_0 = 0.192$, which corresponds to a cell doubling
time of 4 days \cite{hoshino79, pertuiset85}. This value is used
for all of the cases of invasive growth that follow. Since our CA
model takes into account general microscopic tumor-host
interactions, we expect that the general growth dynamics and
emergent behaviors predicted by the model will qualitatively apply
to other solid tumors. We note that all of the reported growth dynamics and emergent
properties of the simulated tumors for any specific set of growth
and invasiveness parameters are repeatedly observed in 25
independent simulations.
\subsubsection*{Effects of Cellular Motility}
We first simulate the growth of malignant tumors with different degrees of
invasiveness in a homogeneous ECM with $\rho _{\mbox{\tiny{ECM}}} = 0.45$. In particular,
we consider three invasive cases with the same mutation rate $\gamma = 0.05$
and ECM degradation ability $\chi = 0.9$, but different cell motility
$\mu = 1, 2, 3$. A non-invasive growth case (i.e., $\gamma = 0$)
in the same microenvironment ($\rho_{\mbox{\tiny{ECM}}} = 0.45$) is also studied for comparison purposes.
Figure~\ref{fig_homo} shows the simulated growing tumors 100 days
after initiation (plots showing the full growth history of the
tumors are given in the Supplementary Information). The computed
metrics for tumor morphology are given in Table \ref{tab_homo}.
The primary tumors for both invasive [Figs.~\ref{fig_homo}(b),(c)
and (d)] and non-invasive [Fig.~\ref{fig_homo}(a)] cases develop
necrotic and quiescent regions. For invasive tumors, when the cell
motility is small (i.e., $\mu=1$), the invasive cells do not form
dendritic invasive branches but rather clump near the outer border
of the proliferative rim [see Fig.~\ref{fig_homo}(b)], forming
bumpy invasive concentric-like shells with relatively small
specific surface (e.g., $s = 1.09$ on day 100). Such invasive
shells significantly enhance the growth the primary tumor, i.e.,
the size of the primary tumor in Fig.~\ref{fig_homo}(b) is much
larger than in Figs.~\ref{fig_homo}(a), (c) and (d). A
quantitative comparison of the tumor sizes is given in the
Supplementary Information.
By contrast, for larger cell motility, long dendritic invasive
branches are developed as manifested by the large specific surface
(e.g., $s = 7.89$ on day 100 and $s = 9.73$ on day 120 for
$\mu=3$). In particular, one can clearly see that within the
branches the cells tend to follow one another to form chains, as
observed in experiments \cite{deisboeck01}. We emphasize that no
rules are imposed to force the cells to follow on another in our
CA model. This homotype attraction is purely due to the mechanical
interaction between the invasive cells and the ECM, i.e., once a
path of invasion is established by a leading invasive cell (by
degrading the ECM), other invasive cells nearby turn to follow and
enhance this path since the resistance for migration is minimized
on a existing path. Furthermore, we can see that larger cell
motility (i.e., high malignancy) leads to more invasive branches
[see Figs.~\ref{fig_homo}(c) and (d)] and thus, a larger specific
area of the invasive pattern.
\subsubsection*{Effects of the ECM Rigidity}
It is not very surprising that isotropic tumor shapes and
and invasive patterns are developed in a homogeneous ECM with relative low density
(i.e.,the ECM is soft) compared
to the ECM degradation ability of the invasive tumor cells. However,
real tumors are rarely isotropic, primarily due to the host
microenvironment in which they grow, which we now explore.
Consider the invasive growth of a tumor in a much more rigid ECM than
that in the previous section, i.e., $\rho_{\mbox{\tiny{ECM}}} =
0.85$. The invasiveness parameters used are $\gamma = 0.05$,
$\mu=3$ and $\chi = 0.9$. The snap shots of the growing tumor are
shown in Fig.~\ref{fig_rho} and the tumor morphology metrics are
given in Table \ref{tab_homo}. It can be clearly seen that both
the size of the primary tumor and the extent of its invasive
branches are much smaller than those of the tumors growing in a
softer ECM (see Fig.~\ref{fig_homo}). Importantly, although the
ECM is still homogeneous, due to its high rigidity, the primary
tumor develops an anisotropic shape with protrusions in the
proliferation rim caused by the invasive branches (e.g., $\alpha =
1.40$ and $\psi = 1.02$ on day 100). Since the invasive cells have
degraded the ECM either completely or partially along the invasive
branches, it is easier for the proliferative cells in the primary
tumor to take these ``weak spots'' than to push against the rigid
ECM themselves. Again, we emphasize that we do not force the cells
to behave this way by imposing special CA rules; this behavior
results purely from the mechanical interaction between the tumor
and its host and the coupling dynamics of invasive and
non-invasive tumor cells.
\subsubsection*{Effects of the ECM Heterogeneity: Random Distribution of ECM Density}
The real host microenvironment for tumors are far from homogeneous
in general. To investigate how heterogeneity of ECM affects the
tumor growth dynamics, we use a random distribution of ECM
density, i.e., for each ECM associated automaton cell, it density
$\rho_{\mbox{\tiny{ECM}}}$ is a random number uniformly chosen
from the interval $[0, ~1]$ (see Fig.~\ref{fig_ECM}(b)). The
invasiveness parameters used are $\gamma = 0.05$, $\mu=3$ and
$\chi = 0.9$ and snap shots of the growing tumor are shown in
Fig.~\ref{fig_rand}. The tumor morphology metrics are given in
Table \ref{tab_hetero}. Note that the primary tumor develops a
rough surface and slightly anisotropic shape in the early growth
stages (e.g., $\alpha = 1.32$ on day 50 and $\alpha = 1.34$ on
day 80), which reflects the ECM heterogeneity
[Figs.~\ref{fig_rand}(a) and (b)]. Since the characteristic
heterogeneity length scale is comparable to a single cell, its
effects are diminished (e.g., $\alpha = 1.18$ on day 100 and
$\alpha = 1.15$ on day 120) as the tumor grows larger and larger
[Fig.~\ref{fig_rand}(c) and (d)]. (In other words, on large length
scales, the ECM is still effectively homogeneous.) However, the
anisotropy in the invasive pattern (i.e., the extents of invasive
branches in different directions) still persists (e.g., $\psi =
0.64$ on day 100) even though the primary tumors almost resumes an
isotropic shape.
\subsubsection*{Effects of the ECM Heterogeneity: Sine-like Distribution of ECM Density}
To represent large-scale heterogeneities in the ECM, we use a sine-like
distribution of ECM density, i.e., for an automaton cell with centroid
$(x_1, x_2, \ldots, x_d)$, the associated ECM density is given by
\begin{equation}
\label{eq_sine}
\rho_{\mbox{\tiny{ECM}}}(x_1, \ldots, x_d) = \frac{1}{2^d}[\sin(\frac{2\pi x_1}{L})+1][\sin(\frac{2\pi x_2}{L})+1]\cdots[\sin(\frac{2\pi x_d}{L})+1],
\end{equation}
where $d$ is the spatial dimension and $L$ is the edge length of the
$d$-dimensional cubic simulation box. A two-dimensional sine-like
ECM distribution is shown in Fig.~\ref{fig_ECM}(c). The red spots correspond
to large $\rho_{\mbox{\tiny{ECM}}}$ and high ECM rigidity; they can be considered
as effective obstacles (e.g., brain ventricles) that hinder tumor growth.
Figures~\ref{fig_sine}(a),(b) and (c),(d) show the snap shots of
invasive tumors growing in the aforementioned ECM on day 80 and
day 120, with invasiveness parameters $\gamma = 0.05$, $\mu=1$ and
$\chi = 0.9$ and $\gamma = 0.05$, $\mu=3$ and $\chi = 0.9$,
respectively. The plots showing the full growth history is given
in Supplementary Information and the tumor morphology metrics are
given in Table \ref{tab_hetero}. We can see that in the early
growth stage, both the primary tumor and invasive pattern in the
two cases are significantly affected by the ECM heterogeneity. In
particular, the tumors are highly anisotropic in shape and the
invasive branches clearly favor two orthogonal directions
associated with low ECM densities (e.g., $\alpha = 1.61$, $\psi =
1.18$ for $\mu =1$ on day 80; and $\alpha = 1.67$, $\psi = 1.33$
for $\mu =3$ on day 80). For the case with large cellular
motility, anisotropy effects are diminished in later growth stages
($\alpha = 1.21$, $\psi = 0.23$ for $\mu =3$ on day 120). For
small cellular motility, anisotropy in both primary tumor shape
and the invasive pattern persists ($\alpha = 1.26$, $\psi = 0.98$
for $\mu =1$ on day 120). Furthermore, one can see that again
invasive cells with low motility significantly facilitate the
growth of the primary tumor. However, instead of forming ``bumpy''
concentric-like shells as in homogeneous ECM, the invasive cells
form large invasive cones, protruding into the ECM. These invasive
cones are followed by weak protrusion of the proliferative rim,
leading to bumpy surface of the primary tumor. The fact that such
complex growth dynamics are only observed for tumors growing
heterogeneous ECM emphasizes the crucial importance of
understanding the effects of physical heterogeneity in cancer
research.
\section*{Discussion}
We have developed a novel cellular automaton (CA) model which, with just a few
parameters, can produce a rich spectrum of growth dynamics for invasive
tumors in heterogeneous host microenvironment. Besides robustly reproducing the salient features
of branched invasive growth, such as least resistance and intrabranch homotype attraction
observed in \textit{in vitro} experiments, our model also enables us
to systematically investigate the effects of microenvironment heterogeneity on
tumor growth as well as the coupling growth of the primary tumor and the invasive cells.
In particular, we have shown that in homogeneous ECM with low densities (i.e., soft microenvironment),
both the shape of the primary tumor and invasive pattern are isotropic. For high cellular
motility cases, the invasive cells form extended dendritic invasive branches; while for low
cellular motility cases, the invasive cells clump near the primary tumor surface
and form a bumpy concentric-like shell that facilitates the growth of the primary tumor.
Tumors growing in a highly rigid homogeneous ECM can develop anisotropic shapes,
facilitated by the invasive cells that degrade the ECM; both the tumor size and
the extent of invasive branches are much smaller.
In heterogeneous ECM, both the primary tumor and invasive pattern are
significantly affected during the early growth stages, i.e., anisotropic shapes
and patterns are developed to avoid high density/rigid regions of the ECM.
If the characteristic length scale of the heterogeneities is comparable to the macroscopic tumor size,
such effects can persist in later growth stages. In addition, invasive cells
with large motility can significant diminish the anisotropy effects by their
ECM degradation activities. We emphasize that we did not manipulate the behavior of cells
by imposing artificial CA rules to give rise to these complex and rich growth dynamics.
Instead, these are emergent behaviors that naturally arise due to
various microscopic-scale tumor-host interactions that are incorporated into our CA model,
including the short-range mechanical interaction between
tumor cells and tumor stroma, and the degradation of extracellular matrix by the
invasive cells.
Note that the growth dynamics of tumors growing in a heterogeneous
microenvironment is distinctly different than those in a
homogeneous microenvironment. This emphasizes the importance of
understanding the effects of physical heterogeneity of the host
microenvironment in modeling tumor growth. Here we just make a
first attempt to take into account a simple level of host
heterogeneity, i.e., by considering the ECM with variable
density/rigidity. Currently, the invasion of the malignant cells
into the host microenivronment is considered to be a consequence
of invasive cell phenotype gained by mutation, and is not
triggered by environmental stresses. However, the effects of
environmental stresses can be easily taken into account. For
example, a CA rule can be imposed that if the division probability
of a malignant cell is significantly reduced by ECM rigidity,
i.e., it is extremely difficult to push away/degrade ECM to make
room for daughter cells, the malignant cell leaves the primary
tumor and invades into soft regions of surrounding ECM. This would
lead to reduced tumor invasion (i.e., development of the dendritic
invasive branches) in soft microenvironments but enhanced invasion
in rigid microenvironments \cite{guiot07}.
Moreover, the spatial-temporal evolution of more complicated and
realistic nutrient/ oxygen fields can be incorporated into our CA
model. This can be achieved by solving the coupled nonlinear
partial differential equations governing the evolution of the
nutrient/oxygen concentrations as was done in
Refs.~\cite{gevertz06} and \cite{jana09}. Since the CA rules are
given for any spatial dimension, our model is readily generalized
to three dimensions. In addition, the model can be easily modified
to incorporate other host heterogeneities, such as stromal cells,
blood vessels and the shape anisotropy of the host organ
\cite{jana08, jana09}. As currently implemented, a single 2D
simulation takes less than 0.5 hours on a 32-bit 1.56Gb Memory
1.44GHz dual core Dell Workstation. We expect that a 3D simulation
will take no longer than 24 hours on a supercomputer when a proper
parallel implementation is used.
Such an \textit{in silica} tool not only enables one to investigate
tumor growth in complex heterogeneous microenvironment that closely represents
the real host microenvironments but also allows one to infer and even
reconstruct individual host microenvironment given limited growth data
of tumors (such as shape and size at various times). Such
microstructural information of the individual host would be extremely
valuable for developing individualized treatment strategies.
For example, based on the host microstructure one can design
special encapsulation and transport agents that maximize drug delivery efficiency \cite{To11}.
In our current CA model, the microscopic parameters governing
tumor invasion are variable and can be arbitrarily chosen within
a feasible range as given in Table \ref{tab_Param}. Given sufficient and reliable experimental data
of invasive tumor growth, the parameters in our CA model
can be uniquely determined and thus, the model can produce robust
predictions on the neoplastic progression.
Although the current CA model is specifically implemented to reproduce and
predict the growth dynamics of invasive solid tumors \textit{in vitro},
further refinement of the model could eventually lead to the development of a powerful
simulation tool that can be utilized clinically. For example, more complicated and
realistic host heterogeneities such as the vascular structure, various
stromal cells, the corresponding spatial-temporal evolution of the
nutrient/oxygen concentrations as well as environmental stress-induced
mutations should be incorporated as we described earlier.
If the robustness of the refined model could be validated clinically, we would
expect it to produce quantitative predictions for {\it in vivo} tumor growth, which are
valuable for tumor prognosis and proposing individualized treatment strategies.
\section*{Acknowledgments}
The authors are grateful to Robert Gatenby and Bob Austin for
valuable comments on our manuscript. The authors are also
grateful to the anonymous reviewers for their valuable comments.
|
1,116,691,497,552 | arxiv | \section{Introduction}
\vspace{0in}The representation-valued equivariant index of a transversally
elliptic operator is an important invariant in $K$-theory (see \cite{A}).
There are few known nontrivial examples in the literature where this
invariant is explicitly computed. Part of the motivation of this paper is to
provide an interesting and sufficiently general class of examples of
transversally elliptic differential operators for which such computations
are possible.
It is well-known that each compactly supported $K$-theory class of the
cotangent bundle over an even-dimensional spin$^{c}$ manifold is represented
by the symbol of a Dirac-type operator. This implies that Dirac-type symbols
map onto the image of the $K$-theory index homomorphism. In this paper, we
will generalize the second fact to the case of transversally elliptic
operators over a compact manifold endowed with a compact Lie group action.
The role of the Dirac-type operators will be played by a new class of
transversally elliptic differential operators introduced in this paper. To
construct these operators, we lift the group action to a principal bundle so
that all orbits in the principal bundle have the same dimension. There is a
natural transversal Dirac operator associated to this action. This operator
induces a transversally elliptic differential operator on the base manifold
with the desired properties. In the case when all orbits have the same
dimension, the orbits on the base manifold form a Riemannian foliation, and
our construction produces a transversal Dirac operator as studied by \cite%
{GlK}, \cite{Hab}, \cite{BrKRi}, \cite{DouglasGlK}, \cite{Jung1}, \cite%
{Jung2}, and others. This new operator will generate all possible values of
the representation-valued equivariant index. Further, we show that the
decomposition of this equivariant index representation into irreducible
components may be computed by means of equivariant indices of elliptic
operators. Thus, the techniques of Atiyah and Segal \cite{ASe} for elliptic
operators become applicable to transversally elliptic operators as well.
Now we describe the content of the paper. Let $M$ be a closed Riemannian
manifold. Let $Q\subset TM$ be a smooth distribution over $M$; we do not
assume that $Q$ or its normal bundle are involutive. Section \ref%
{RestrCliffBundlesSection} contains preliminary results about connections
associated to restrictions of Clifford structures. In Section \ref%
{transverseDiracDistributions}, we assume only that $E\rightarrow M$ is a $%
\mathbb{C}\mathrm{l}\left( Q\right) $-bundle with corresponding compatible
Clifford connection. Such a connection always exists if $E$ is in addition a
$\mathbb{C}\mathrm{l}\left( TM\right) $-bundle; see Section \ref%
{RestrCliffBundlesSection}. Using this $\mathbb{C}\mathrm{l}\left( Q\right) $
connection, we construct an operator $D_{Q}$ whose principal symbol $\sigma
\left( D_{Q}\right) \left( \xi _{x}\right) $ is invertible for all $\xi
_{x}\in Q_{x}\setminus \left\{ 0\right\} $ and prove that it is essentially
self-adjoint. In the case of a Riemannian foliation with normal bundle $Q$,
this construction produces the well-known self-adjoint version of the
transversal Dirac operator (see \cite{GlK}, \cite{Jung1}, etc.).
In Section \ref{EquivariantOpsFrameBndleSection}, we assume that there is an
isometric action of a compact Lie group $G$ on $M$. The action of $G$ lifts
to the orthonormal frame bundle $F_{O}$ of $M$. Given an equivariant
transversal Dirac operator on $F_{O}$ and irreducible representation of the
orthogonal group, we show how to construct a transversally elliptic operator
on $M$. Similarly, using the transversal Dirac operator and an irreducible
representation of $G$, we construct an elliptic operator on $F_{O}\diagup G$%
. The precise relationship between the eigenspaces of these two operators is
stated in Proposition \ref{SpectraSame}.
In Section \ref{topPropertiesSection}, we study the equivariant index of the
$K$-theory class of operators constructed in Section \ref%
{EquivariantOpsFrameBndleSection}. To do this, we first derive a
multiplicative property of the equivariant index on associated fiber bundles
with compact fibers. This property, stated in Theorem \ref%
{EquivariantMultiplicativeTheorem}, is a generalization of the
multiplicative property of the index for sphere bundles shown by Atiyah and
Singer in \cite{ASi1}. The main result of Section \ref{topPropertiesSection}
is Theorem \ref{indexClassGivenByTransvDiracThm}, in which we show that the
symbols of the lifted transversal Dirac operators generate all the possible
equivariant indices, if the $F_{O}$ is $G$-transversally spin$^{c}$.
In Section \ref{ExampleSection}, we demonstrate our constructions of the
lifted transversal Dirac operator and verify our results by explicit
calculations on the two-sphere.
The reader may consult \cite{A} for the basic properties of transversally
elliptic equivariant operators and their equivariant indices. Interesting
and relevant results also appear in \cite{ASe}, \cite{B-G-V}, \cite{Be-V1},
\cite{Be-V2}, \cite{Bra3}, \cite{BrKRi}, and \cite{Paradan}.
\section{Restrictions of Clifford structures\label{RestrCliffBundlesSection}}
Let $M$ be a closed Riemannian manifold with metric $\left\langle ~\cdot
~,~\cdot ~\right\rangle $, and let $E$ be a Clifford bundle over $M$. Recall
that a Clifford bundle $E$ is a complex Hermitian vector bundle endowed with
a Clifford action $c:TM\otimes \mathbb{C}\rightarrow \mathrm{End}\left(
E\right) $ and a connection $\nabla ^{E}$ compatible with this action and
the metric. Let $Q$ be a subbundle of $TM$, and let $L$ be the orthogonal
complement of $Q$ in $TM$. The Levi-Civita connection $\nabla ^{M}$ induces
a connection $\nabla ^{Q}$ on $Q$ by the following formula. Given any
section $Y\in \Gamma Q$ and any vector field $X\in \Gamma \left( TM\right) $%
, define%
\begin{equation}
\nabla _{X}^{Q}Y=\pi \nabla _{X}^{M}Y, \label{Qconnection}
\end{equation}%
where $\pi :TM\rightarrow Q$ is the orthogonal bundle projection. It is
elementary to check that Formula (\ref{Qconnection}) yields a metric
connection on $Q$ (with the restricted metric).
\begin{remark}
Note that this connection is not generally $Q$-torsion-free, because the
torsion-free property is equivalent to the integrability of $Q$ (i.e. $\left[
\Gamma Q,\Gamma Q\right] \subset \Gamma Q$).
\end{remark}
We now modify the connection $\nabla ^{E}$ so that it has the desired
compatibility with $\nabla ^{Q}$. Every metric connection $\widetilde{\nabla
^{E}}$ on $E$ satisfies%
\begin{equation*}
\widetilde{\nabla _{X}^{E}}=\nabla _{X}^{E}+B_{X},
\end{equation*}%
where $B_{X}$ is a skew-Hermitian endomorphism of $E$ that is $C^{\infty
}\left( M\right) $-linear in $X$. In order that $\widetilde{\nabla ^{E}}$ is
a $\mathbb{C}\mathrm{l}\left( Q\right) $-connection compatible with $\nabla
^{Q}$, we must have that if $X\in \Gamma \left( TM\right) $, $Y\in \Gamma Q$%
, $s\in \Gamma E$,%
\begin{equation*}
\widetilde{\nabla _{X}^{E}}c\left( Y\right) s=c\left( \nabla
_{X}^{Q}Y\right) s+c\left( Y\right) \widetilde{\nabla _{X}^{E}}s\text{.}
\end{equation*}%
By (\ref{Qconnection}) we see that%
\begin{equation*}
B_{X}c\left( Y\right) s=-c\left( \left( 1-\pi \right) \nabla
_{X}^{M}Y\right) s+c\left( Y\right) B_{X}s,
\end{equation*}%
or%
\begin{equation}
c\left( \left( 1-\pi \right) \nabla _{X}^{M}Y\right) s=\left[ c\left(
Y\right) ,B_{X}\right] s. \label{cliffordRequirement}
\end{equation}%
Computing with a local orthonormal frame $\left\{ e_{1},...,e_{p}\right\} $
for $L$, we have
\begin{eqnarray*}
\left( 1-\pi \right) \nabla _{X}^{M}Y &=&\sum_{m=1}^{p}\left\langle \nabla
_{X}^{M}Y,e_{m}\right\rangle e_{m}=-\sum_{m=1}^{p}\left\langle Y,\nabla
_{X}^{M}e_{m}\right\rangle e_{m} \\
&=&-\sum_{m=1}^{p}\left\langle Y,\pi \nabla _{X}^{M}e_{m}\right\rangle e_{m}
\\
&=&\frac{1}{2}\sum_{m=1}^{p}\left( c\left( Y\right) c\left( \pi \nabla
_{X}^{M}e_{m}\right) +c\left( \pi \nabla _{X}^{M}e_{m}\right) c\left(
Y\right) \right) e_{m}~.
\end{eqnarray*}%
Then (\ref{cliffordRequirement}) implies that%
\begin{equation*}
\frac{1}{2}\sum_{m=1}^{p}\left( c\left( Y\right) c\left( \pi \nabla
_{X}^{M}e_{m}\right) c\left( e_{m}\right) -c\left( \pi \nabla
_{X}^{M}e_{m}\right) c\left( e_{m}\right) c\left( Y\right) \right) =\left(
c\left( Y\right) B_{X}-B_{X}c\left( Y\right) \right) ,
\end{equation*}%
or%
\begin{equation*}
\left( B_{X}-\frac{1}{2}\sum_{m=1}^{p}c\left( \pi \nabla
_{X}^{M}e_{m}\right) c\left( e_{m}\right) \right) c\left( Y\right) =c\left(
Y\right) \left( B_{X}-\frac{1}{2}\sum_{m=1}^{p}c\left( \pi \nabla
_{X}^{M}e_{m}\right) c\left( e_{m}\right) \right) .
\end{equation*}%
We conclude that the requirement that $\widetilde{\nabla ^{E}}$ is a $%
\mathbb{C}\mathrm{l}\left( Q\right) $-connection defines $B_{X}$ (and thus $%
\widetilde{\nabla _{X}^{E}}$ ) up to a skew-adjoint endomorphism of $E$ that
commutes with Clifford multiplication by vectors in $Q$. We may always take%
\begin{equation*}
B_{X}=\frac{1}{2}\sum_{m=1}^{p}c\left( \pi \nabla _{X}^{M}e_{m}\right)
c\left( e_{m}\right) .
\end{equation*}%
This choice of $B_{X}$ (and thus $\widetilde{\nabla _{X}^{E}}$ ) is
well-defined and canonical, since the formula is independent of the local
orthonormal frame for $L$.
We now show how to express $B_{X}$ in terms of any local orthonormal frame $%
f_{1},...,f_{q}$ for $Q$.
\begin{eqnarray*}
B_{X} &=&\frac{1}{2}\sum_{m=1}^{p}c\left( \pi \nabla _{X}^{M}e_{m}\right)
c\left( e_{m}\right) =\frac{1}{2}\sum_{m=1}^{p}\sum_{j=1}^{q}\left\langle
\nabla _{X}^{M}e_{m},f_{j}\right\rangle c\left( f_{j}\right) c\left(
e_{m}\right) \\
&=&-\frac{1}{2}\sum_{m=1}^{p}\sum_{j=1}^{q}\left\langle e_{m},\nabla
_{X}^{M}f_{j}\right\rangle c\left( f_{j}\right) c\left( e_{m}\right) \\
&=&\frac{1}{4}\sum_{m=1}^{p}\sum_{j=1}^{q}\left\{ c\left( e_{m}\right)
c\left( \nabla _{X}^{M}f_{j}\right) +c\left( \nabla _{X}^{M}f_{j}\right)
c\left( e_{m}\right) \right\} c\left( f_{j}\right) c\left( e_{m}\right) \\
&=&\frac{1}{4}\sum_{j=1}^{q}\sum_{m=1}^{p}\left\{ -c\left( e_{m}\right)
c\left( \nabla _{X}^{M}f_{j}\right) c\left( e_{m}\right) c\left(
f_{j}\right) +c\left( \nabla _{X}^{M}f_{j}\right) c\left( f_{j}\right)
\right\} \\
&=&\frac{1}{4}\sum_{j=1}^{q}\left\{ -pc\left( \pi \nabla
_{X}^{M}f_{j}\right) c\left( f_{j}\right) -\left( p-2\right) c\left( \left(
1-\pi \right) \nabla _{X}^{M}f_{j}\right) c\left( f_{j}\right) +pc\left(
\nabla _{X}^{M}f_{j}\right) c\left( f_{j}\right) \right\} \\
&=&\frac{1}{2}\sum_{j=1}^{q}c\left( \left( 1-\pi \right) \nabla
_{X}^{M}f_{j}\right) c\left( f_{j}\right) .
\end{eqnarray*}%
Observe that this expression for $B_{X}$ is the same as the original
expression for $B_{X}$ with $Q$ replaced by $L$. We have shown the following.
\begin{proposition}
Let $M$ be a closed Riemannian manifold, and let $\left( E,\nabla
^{E},c\right) $ be a Hermitian Clifford bundle over $M$. Let $Q$ be a
subbundle of $TM$, and let $\nabla ^{Q}$ denote the metric connection on $Q$
defined by $\nabla _{X}^{Q}Y=\pi \nabla _{X}^{M}Y$, where $\pi
:TM\rightarrow Q$ is the orthogonal bundle projection. Then the connection $%
\widetilde{\nabla ^{E}}$ defined by%
\begin{equation*}
\widetilde{\nabla _{X}^{E}}=\nabla _{X}^{E}+\frac{1}{2}\sum_{m=1}^{p}c\left(
\pi \nabla _{X}^{M}e_{m}\right) c\left( e_{m}\right)
\end{equation*}%
for all $X\in \Gamma \left( TM\right) $ is a well-defined $\mathbb{C}\mathrm{%
l}\left( Q\right) $-connection and a metric connection on $E$ with respect
to the connection $\nabla ^{Q}$. Furthermore, $\widetilde{\nabla ^{E}}$ is a
$\mathbb{C}\mathrm{l}\left( L\right) $-connection and a metric connection on
$E$ with respect to the connection $\nabla ^{L}=\left( 1-\pi \right) \nabla
^{M}$.
\end{proposition}
\section{Transverse Dirac Operators for Distributions\label%
{transverseDiracDistributions}}
We showed in Section \ref{RestrCliffBundlesSection} that, for a given
distribution $Q\subset TM$, it is always possible to obtain a bundle of $%
\mathbb{C}\mathrm{l}\left( Q\right) $-modules with Clifford connection from
a bundle of $\mathbb{C}\mathrm{l}\left( TM\right) $-Clifford modules. In
this section, we will assume more generally that a $\mathbb{C}\mathrm{l}%
\left( Q\right) $-module structure on a complex Hermitian vector bundle $E$
is given and will define transverse Dirac operators on sections of $E$. As
in Section \ref{RestrCliffBundlesSection}, $M$ is a closed Riemannian
manifold with metric $\left\langle ~\cdot ~,~\cdot ~\right\rangle $, $%
c:Q\rightarrow \mathrm{End}\left( E\right) $ is the Clifford multiplication
on $E$, and $\nabla ^{E}$ is a $\mathbb{C}\mathrm{l}\left( Q\right) $
connection that is compatible with the metric on $M$; that is, Clifford
multiplication by each vector is skew-Hermitian, and we require%
\begin{equation*}
\nabla _{X}^{E}\left( c\left( V\right) s\right) =c\left( \nabla
_{X}^{Q}V\right) s+c\left( V\right) \nabla _{X}^{E}s
\end{equation*}%
for all $X\in \Gamma \left( TM\right) $, $V\in \Gamma Q$, and $s\in \Gamma E$%
. Note that the connection $\widetilde{\nabla ^{E}}$ from Section \ref%
{RestrCliffBundlesSection} is an example of such a connection, but not all
such $\mathbb{C}\mathrm{l}\left( Q\right) $ connections are of that type.
Let $L=Q^{\bot }$, let $\left( f_{1},...,f_{q}\right) $ be a local
orthonormal frame for $Q$, and let $\pi :TM\rightarrow Q$ be the orthogonal
projection. We define the Dirac operator $A_{Q}$ corresponding to the
distribution $Q$ as
\begin{equation}
A_{Q}=\sum_{j=1}^{q}c\left( f_{j}\right) \nabla _{f_{j}}^{E}. \label{AQdef}
\end{equation}%
This definition is independent of the choices made; in fact it is the
composition of the maps
\begin{equation*}
\Gamma \left( E\right) \overset{\nabla ^{E}}{\rightarrow }\Gamma \left(
T^{\ast }M\otimes E\right) \overset{\cong }{\rightarrow }\Gamma \left(
TM\otimes E\right) \overset{\pi }{\rightarrow }\Gamma \left( Q\otimes
E\right) \overset{c}{\rightarrow }\Gamma \left( E\right) .
\end{equation*}%
We calculate the formal adjoint. Letting $\left( s_{1},s_{2}\right) $ denote
the pointwise inner product of sections of $E$, we have that
\begin{eqnarray*}
\left( A_{Q}s_{1},s_{2}\right) -\left( s_{1},A_{Q}s_{2}\right)
&=&\sum_{j=1}^{q}\left( c\left( f_{j}\right) \nabla
_{f_{j}}^{E}s_{1},s_{2}\right) -\left( s_{1},c\left( f_{j}\right) \nabla
_{f_{j}}^{E}s_{2}\right) \\
&=&\sum_{j=1}^{q}\left( \nabla _{f_{j}}^{E}\left( c\left( f_{j}\right)
s_{1}\right) ,s_{2}\right) -\left( c\left( \pi \nabla
_{f_{j}}^{M}f_{j}\right) s_{1},s_{2}\right) +\left( c\left( f_{j}\right)
s_{1},\nabla _{f_{j}}^{E}s_{2}\right) \\
&=&\sum_{j=1}^{q}\nabla _{f_{j}}^{M}\left( c\left( f_{j}\right)
s_{1},s_{2}\right) -\left( c\left( \sum_{j=1}^{q}\pi \nabla
_{f_{j}}^{M}f_{j}\right) s_{1},s_{2}\right) \\
&=&-\sum_{j=1}^{q}\nabla _{f_{j}}^{M}i_{f_{j}}\omega +\omega \left(
\sum_{j=1}^{q}\pi \nabla _{f_{j}}^{M}f_{j}\right) ,
\end{eqnarray*}%
where $\omega $ is the one-form defined by $\omega \left( X\right) =-\left(
c\left( X\right) s_{1},s_{2}\right) $ for $X\in \Gamma Q$ and is zero for $%
X\in \Gamma L$. Continuing,
\begin{eqnarray*}
\left( A_{Q}s_{1},s_{2}\right) -\left( s_{1},A_{Q}s_{2}\right)
&=&-\sum_{j=1}^{q}\left( i_{f_{j}}\nabla _{f_{j}}^{M}+i_{\nabla
_{f_{j}}^{M}f_{j}}\right) \omega +\omega \left( \sum_{j=1}^{q}\pi \nabla
_{f_{j}}^{M}f_{j}\right) \\
&=&-\sum_{j=1}^{q}\left( i_{f_{j}}\nabla _{f_{j}}^{M}+i_{\left( \pi \nabla
_{f_{j}}^{M}f_{j}\right) }\right) \omega +\omega \left( \sum_{j=1}^{q}\pi
\nabla _{f_{j}}^{M}f_{j}\right) \\
&=&-\sum_{j=1}^{q}i_{f_{j}}\nabla _{f_{j}}^{M}\omega
=-\sum_{j=1}^{q}i_{f_{j}}\pi \nabla _{f_{j}}^{M}\omega ,
\end{eqnarray*}%
where the orthogonal projection $T^{\ast }M\rightarrow Q^{\ast }$ is denoted
by $\pi $ as well. In what follows, let $\left( e_{1},...,e_{p}\right) $ be
an orthonormal frame of $L$, and let $\nabla ^{M}=$ $\nabla ^{Q}+\nabla
^{L}=\pi \nabla ^{M}+\left( 1-\pi \right) \nabla ^{M}$ on forms. The
divergence of a general one-form $\beta $ that is zero on $L$ is
\begin{eqnarray*}
\delta \beta &=&-\sum_{j=1}^{q}i_{f_{j}}\nabla _{f_{j}}^{M}\beta
-\sum_{m=1}^{p}i_{e_{m}}\nabla _{e_{m}}^{M}\beta \\
&=&-\sum_{j=1}^{q}i_{f_{j}}\pi \nabla _{f_{j}}^{M}\beta
-\sum_{m=1}^{p}i_{e_{m}}\nabla _{e_{m}}^{L}\beta ,
\end{eqnarray*}%
Letting $\beta =\sum_{k=1}^{q}\beta _{k}f_{k}^{\ast },$ then
\begin{eqnarray*}
\delta \beta +\sum_{j=1}^{q}i_{f_{j}}\pi \nabla _{f_{j}}^{M}\beta
&=&-\sum_{m=1}^{p}i_{e_{m}}\nabla _{e_{m}}^{L}\left( \sum_{k=1}^{q}\beta
_{k}f_{k}^{\ast }\right) \\
&=&-\sum_{k=1}^{q}\sum_{m=1}^{p}\beta _{k}i_{e_{m}}\nabla _{e_{m}}^{L}\left(
f_{k}^{\ast }\right) \\
&=&-\sum_{k=1}^{q}\sum_{m=1}^{p}\beta _{k}i_{e_{m}}\left(
\sum_{j=1}^{p}\left( \nabla _{e_{m}}^{M}\left( f_{k}^{\ast }\right)
,e_{j}^{\ast }\right) e_{j}^{\ast }\right) \\
&=&\sum_{k=1}^{q}\sum_{m=1}^{p}\beta _{k}i_{e_{m}}\left(
\sum_{j=1}^{p}\left( \nabla _{e_{m}}^{M}\left( e_{j}^{\ast }\right)
,f_{k}^{\ast }\right) e_{j}^{\ast }\right) \\
&=&\sum_{k=1}^{q}\beta _{k}\left( \sum_{m=1}^{p}\left( \nabla
_{e_{m}}^{M}\left( e_{m}^{\ast }\right) ,f_{k}^{\ast }\right) \right) \\
&=&i_{H^{L}}\beta ,
\end{eqnarray*}%
where $H^{L}$ is the mean curvature vector field of $L$. Thus, for every
one-form $\beta $ that is zero on $L$ ,
\begin{equation*}
-\sum_{j=1}^{q}i_{f_{j}}\pi \nabla _{f_{j}}^{M}\beta =\delta \beta
-i_{H^{L}}\beta .
\end{equation*}%
Applying this result to the form $\omega $ defined above, we have
\begin{eqnarray*}
\left( A_{Q}s_{1},s_{2}\right) -\left( s_{1},A_{Q}s_{2}\right) &=&\delta
\omega -i_{H^{L}}\omega \\
&=&\delta \omega -\left( s_{1},c\left( H^{L}\right) s_{2}\right) .
\end{eqnarray*}%
Thus, the formal adjoint $A_{Q}^{\ast }$ of $A_{Q}$ is%
\begin{equation*}
A_{Q}^{\ast }=A_{Q}-c\left( H^{L}\right) ,
\end{equation*}%
and the operator%
\begin{equation}
D_{Q}=A_{Q}-\frac{1}{2}c\left( H^{L}\right) \label{DQdef}
\end{equation}%
is formally self-adjoint.
A quick look at \cite{C} yields the following.
\begin{theorem}
For each distribution $Q\subset TM$ and every bundle $E$ of $\mathbb{C}%
\mathrm{l}\left( Q\right) $-modules, the transversally elliptic operator $%
D_{Q}$ defined by (\ref{AQdef}) and (\ref{DQdef}) is essentially
self-adjoint.
\end{theorem}
It is not necessarily the case that the spectrum of $D_{Q}$ is discrete, as
the following example shows.
\begin{example}
We consider the torus $M=\left( \mathbb{R}\diagup 2\pi \mathbb{Z}\right)
^{2} $ with the metric $e^{2g\left( y\right) }dx^{2}+dy^{2}$ for some $2\pi $%
-periodic smooth function $g$. Consider the orthogonal distributions $L=%
\mathrm{span}\left\{ \partial _{y}\right\} $ and $Q=\mathrm{span}\left\{
\partial _{x}\right\} $. Let $E$ be the trivial complex line bundle over $M$%
, and let $\mathbb{C}\mathrm{l}\left( Q\right) $ and $\mathbb{C}\mathrm{l}%
\left( L\right) $ both act on $E$ via $c\left( \partial _{y}\right)
=i=c\left( e^{-g\left( y\right) }\partial _{x}\right) $. The connections $%
\nabla ^{L}$ and $\nabla ^{Q}$ satisfy
\begin{eqnarray*}
\nabla _{\partial _{y}}^{L}\partial _{y} &=&\nabla _{\partial
_{x}}^{L}\partial _{y}=0; \\
\nabla _{\partial _{x}}^{Q}\partial _{x} &=&0;~\nabla _{\partial
_{y}}^{Q}\partial _{x}=g^{\prime }\left( y\right) \partial _{x}.
\end{eqnarray*}
The trivial connection $\nabla ^{E}$ is a $\mathbb{C}\mathrm{l}\left(
L\right) $ connection with respect to $\nabla ^{L}$ and is also a $\mathbb{C}%
\mathrm{l}\left( Q\right) $ connection with respect to $\nabla ^{Q}$.
Observe that the mean curvatures of these distributions are%
\begin{eqnarray*}
H^{Q} &=&\left( 1-\pi \right) \nabla _{e^{-g\left( y\right) }\partial
_{x}}^{M}e^{-g\left( y\right) }\partial _{x}=-g^{\prime }\left( y\right)
\partial _{y}\text{ and} \\
H^{L} &=&\pi \nabla _{\partial _{y}}^{M}\partial _{y}=\nabla _{\partial
_{y}}^{M}\partial _{y}=0
\end{eqnarray*}%
From formulas (\ref{AQdef}) and (\ref{DQdef}),%
\begin{eqnarray*}
A_{L} &=&i\partial _{y},\text{ and} \\
D_{L} &=&i\left( \partial _{y}+\frac{1}{2}g^{\prime }\left( y\right) \right)
.
\end{eqnarray*}%
The spectrum $\sigma \left( D_{L}\right) =\mathbb{Z}$ is a set consisting of
eigenvalues of infinite multiplicity, and thus $\sigma \left( D_{L}\right) $
consists entirely of pure point spectrum. The eigenspace $E_{n}$
corresponding to the eigenvalue $n$ is
\begin{equation*}
E_{n}=\left\{ e^{-iny-\frac{g\left( y\right) }{2}}f\left( x\right) :f\in
L^{2}\left( S^{1}\right) \right\} ,
\end{equation*}%
and $\bigcup\limits_{n\in \mathbb{Z}}E_{n}$ is dense in $L^{2}\left(
M\right) $.
On the other hand, the operator%
\begin{equation*}
D_{Q}=A_{Q}-\frac{1}{2}c\left( H^{L}\right) =A_{Q}=ie^{-g\left( y\right)
}\partial _{x}
\end{equation*}%
has only one eigenvalue, $0$, corresponding to the eigenspace $\left\{
h\left( y\right) :h\in L^{2}\left( S^{1}\right) \right\} $. Next, note that $%
F_{n}=\left\{ e^{-inx}\psi \left( y\right) :\psi \in L^{2}\left(
S^{1}\right) \right\} $ is an invariant subspace for $D_{Q}$ . The spectrum
of the restriction of $D_{Q}$ to $F_{n}$ is $n\left[ a,b\right] $, where $%
\left[ a,b\right] \subset \left( 0,\infty \right) $ is the range of $%
e^{-g\left( y\right) }$. Thus, the spectrum $\sigma \left( D_{Q}\right) $ is%
\begin{equation*}
\sigma \left( D_{Q}\right) =\bigcup\limits_{n\in \mathbb{Z}}n\left[ a,b%
\right] ,
\end{equation*}%
and the pure point spectrum of $D_{Q}$ is $\left\{ 0\right\} $.
\end{example}
\begin{example}
\label{RiemFoliationTrDiracExample}Suppose a closed manifold $M$ is endowed
with a Riemannian foliation $\mathcal{F}$ such that the metric is
bundlelike, meaning that the leaves are locally equidistant. If the orbits
of a $G$-manifold have the same dimension, then they form a Riemannian
foliation. In such foliations, there is a natural construction of
transversal Dirac operators (see \cite{BrKRi} , \cite{GlK} , \cite{La}),
which is a special case of the construction in this section. Choose a local
adapted frame field $\left\{ e_{1},...,e_{n}\right\} $ for the tangent
bundle of $M$ , such that $\left\{ e_{1},...,e_{q}\right\} $ is a local
basis of the normal bundle $N\mathcal{F}$ for the foliation and such that
each $e_{j}$ is a basic vector field for $1\leq j\leq q$. The word \emph{%
basic} means that the flows of those vector fields map leaves to leaves, and
such a basis can be chosen near every point if and only if the foliation is
Riemannian. Next, assume that we have a complex Hermitian vector bundle $%
E\rightarrow M$ that is a bundle of $\mathbb{C}\mathrm{l}\left( N\mathcal{F}%
\right) $ modules that is equivariant with respect to the $G$ action, and
let $\nabla $ be the corresponding equivariant, metric, Clifford connection.
We define the transversal Dirac operator by
\begin{equation*}
A_{N\mathcal{F}}=\sum_{j=1}^{q}c\left( e_{j}\right) \nabla _{e_{j}},
\end{equation*}%
as in the notation of this section. As before, the operator
\begin{equation*}
D_{N\mathcal{F}}=A_{N\mathcal{F}}-\frac{1}{2}c\left( H\right)
\end{equation*}%
is an essentially self-adjoint operator, where $H$ is the mean curvature
vector field of the orbits.
\end{example}
\section{Equivariant operators on the frame bundle\label%
{EquivariantOpsFrameBndleSection}}
\subsection{Equivariant structure of the orthonormal frame bundle\label%
{equivariantStructureSection}}
\vspace{0in}Given a complete, connected $G$-manifold, the action of $g\in G$
on $M$ induces an action of $dg$ on $TM$, which in turn induces an action of
$G$ on the principal $O\left( n\right) $-bundle $F_{O}\overset{p}{%
\rightarrow }M$ of orthonormal frames over $M$.
\begin{lemma}
\vspace{0in}The action of $G$ on $F_{O}$ is regular, i.e. the isotropy
subgroups corresponding to any two points of $M$ are conjugate.
\end{lemma}
\begin{proof}
Let $H$ be the isotropy subgroup of a frame $f\in F_{O}$. Then $H$ also
fixes $p\left( f\right) \in M$, and since $H$ fixes the frame, its
differentials fix the entire tangent space at $p\left( f\right) $. Since it
fixes the tangent space, every element of $H$ also fixes every frame in $%
p^{-1}\left( p\left( f\right) \right) $; thus every frame in a given fiber
must have the same isotropy subgroup. Since the elements of $H$ map
geodesics to geodesics and preserve distance, a neighborhood of $p\left(
f\right) $ is fixed by $H$. Thus, $H$ is a subgroup of the isotropy subgroup
at each point of that neighborhood. Conversely, if an element of $G$ fixes a
neighborhood of a point $x$ in $M$, then it fixes all frames in $%
p^{-1}\left( x\right) $, and thus all frames in the fibers above that
neighborhood. Since $M$ is connected, we may conclude that every point of $%
F_{O}$ has the same isotropy subgroup $H$, and $H$ is the subgroup of $G$
that fixes every point of $M$.
\end{proof}
\begin{remark}
Since this subgroup $H$ is normal, we often reduce the group $G$ to the
group $G/H$ so that our action is effective, in which case the isotropy
subgroups on $F_{O}$ are all trivial.
\end{remark}
In any case, the $G$ orbits on $F_{O}$ are diffeomorphic and form a
Riemannian fiber bundle, in the natural metric on $F_{O}$ defined as
follows. The Levi-Civita connection on $M$ determines the horizontal
subbundle $\mathcal{H}$ of $TF_{O}$. We construct the local product metric
on $F_{O}$ using a biinvariant fiber metric and the pullback of the metric
on $M$ to $\mathcal{H}$; with this metric, $F_{O}$ is a compact Riemannian $%
G\times O\left( n\right) $-manifold. The lifted $G$-action commutes with the
$O\left( n\right) $-action. Let $\mathcal{F}$ denote the foliation of $G$%
-orbits on $F_{O}$, and observe that $F_{O}\overset{\pi }{\rightarrow }%
F_{O}\diagup G=F_{O}\diagup \mathcal{F}$ is a Riemannian submersion of
compact $O\left( n\right) $-manifolds.
Let $E\rightarrow F_{O}$ be a Hermitian vector bundle that is equivariant
with respect to the $G\times O\left( n\right) $ action. Let $\rho
:G\rightarrow U\left( V_{\rho }\right) $ and $\sigma :O\left( n\right)
\rightarrow U\left( W_{\sigma }\right) $ be irreducible unitary
representations. We define the bundle $\mathcal{E}^{\sigma }\rightarrow M$
by
\begin{equation*}
\mathcal{E}_{x}^{\sigma }=\Gamma \left( p^{-1}\left( x\right) ,E\right)
^{\sigma },
\end{equation*}%
where the superscript $\sigma $ is defined for a $O\left( n\right) $-module $%
Z$ by%
\begin{equation*}
Z^{\sigma }=\mathrm{eval}\left( \mathrm{Hom}_{O\left( n\right) }\left(
W_{\sigma },Z\right) \otimes W_{\sigma }\right) ,
\end{equation*}%
where $\mathbb{\mathrm{eval}}:\mathrm{Hom}_{O\left( n\right) }\left(
W_{\sigma },Z\right) \otimes W_{\sigma }\rightarrow Z$ is the evaluation map
$\phi \otimes w\mapsto \phi \left( w\right) $. The space $Z^{\sigma }$ is
the vector subspace of $Z$ on which $O\left( n\right) $ acts as a direct sum
of representations of type $\sigma $. The bundle $\mathcal{E}^{\sigma }$ is
a Hermitian $G$-vector bundle of finite rank over $M$. The metric on $%
\mathcal{E}^{\sigma }$ is chosen as follows. For any $v_{x}$,$w_{x}\in
\mathcal{E}_{x}^{\sigma }$, we define%
\begin{equation*}
\,\left\langle v_{x},w_{x}\right\rangle :=\int_{p^{-1}\left( x\right)
}\left\langle v_{x}\left( y\right) ,w_{x}\left( y\right) \right\rangle
_{y,E}~d\mu _{x}\left( y\right) ,
\end{equation*}%
where $d\mu _{x}$ is the measure on $p^{-1}\left( x\right) $ induced from
the metric on $F_{O}$. See \cite{BrKRi} for a similar construction.
Similarly, we define the bundle $\mathcal{T}^{\rho }\rightarrow F_{O}\diagup
G$ by%
\begin{equation*}
\mathcal{T}_{y}^{\rho }=\Gamma \left( \pi ^{-1}\left( y\right) ,E\right)
^{\rho },
\end{equation*}%
and $\mathcal{T}^{\rho }\rightarrow F_{O}\diagup G$ is a Hermitian $O\left(
n\right) $-equivariant bundle of finite rank. The metric on $\mathcal{T}%
^{\rho }$ is%
\begin{equation*}
\left\langle v_{z},w_{z}\right\rangle :=\int_{\pi ^{-1}\left( y\right)
}\left\langle v_{z}\left( y\right) ,w_{z}\left( y\right) \right\rangle
_{z,E}~dm_{z}\left( y\right) ,
\end{equation*}%
where $dm_{z}$ is the measure on $\pi ^{-1}\left( z\right) $ induced from
the metric on $F_{O}$.
The vector spaces of sections $\Gamma \left( M,\mathcal{E}^{\sigma }\right) $
and $\Gamma \left( F_{O},E\right) ^{\sigma }$ can be identified via the
isomorphism%
\begin{equation*}
i_{\sigma }:\Gamma \left( M,\mathcal{E}^{\sigma }\right) \rightarrow \Gamma
\left( F_{O},E\right) ^{\sigma },
\end{equation*}%
where for any section $s\in \Gamma \left( M,\mathcal{E}^{\sigma }\right) $, $%
s\left( x\right) \in \Gamma \left( p^{-1}\left( x\right) ,E\right) ^{\sigma
} $ for each $x\in M$, and we let%
\begin{equation*}
i_{\sigma }\left( s\right) \left( f_{x}\right) :=\left. s\left( x\right)
\right\vert _{f_{x}}
\end{equation*}%
for every $f_{x}\in p^{-1}\left( x\right) \subset F_{O}$. Then $i_{\sigma
}^{-1}:\Gamma \left( F_{O},E\right) ^{\sigma }\rightarrow \Gamma \left( M,%
\mathcal{E}^{\sigma }\right) $ is given by%
\begin{equation*}
i_{\sigma }^{-1}\left( u\right) \left( x\right) =\left. u\right\vert
_{p^{-1}\left( x\right) }.
\end{equation*}%
Observe that $i_{\sigma }:\Gamma \left( M,\mathcal{E}^{\sigma }\right)
\rightarrow \Gamma \left( F_{O},E\right) ^{\sigma }$ extends to an $L^{2}$
isometry. Given $u,v\in $ $\Gamma \left( M,\mathcal{E}^{\sigma }\right) $,%
\begin{eqnarray*}
\left\langle u,v\right\rangle _{M} &=&\int_{M}\left\langle
u_{x},v_{x}\right\rangle ~dx=\int_{M}\int_{p^{-1}\left( x\right)
}\left\langle u_{x}\left( y\right) ,v_{x}\left( y\right) \right\rangle
_{y,E}~d\mu _{x}\left( y\right) ~dx \\
&=&\int_{M}\left( \int_{p^{-1}\left( x\right) }\left\langle i_{\sigma
}\left( u\right) ,i_{\sigma }\left( v\right) \right\rangle _{E}~d\mu
_{x}\left( y\right) \right) ~dx \\
&=&\int_{F_{O}}\left\langle i_{\sigma }\left( u\right) ,i_{\sigma }\left(
v\right) \right\rangle _{E}~=\left\langle i_{\sigma }\left( u\right)
,i_{\sigma }\left( v\right) \right\rangle _{F_{O}},
\end{eqnarray*}%
where $dx$ is the Riemannian measure on $M$; we have used the fact that $p$
is a Riemannian submersion. Similarly, we let%
\begin{equation*}
j_{\rho }:\Gamma \left( F_{O}\diagup G,\mathcal{T}^{\rho }\right)
\rightarrow \Gamma \left( F_{O},E\right) ^{\rho }
\end{equation*}%
be the natural identification, which extends to an $L^{2}$ isometry.
Let%
\begin{equation*}
\Gamma \left( M,\mathcal{E}^{\sigma }\right) ^{\alpha }=\mathrm{eval}\left(
\mathrm{Hom}_{G}\left( V_{\alpha },\Gamma \left( M,\mathcal{E}^{\sigma
}\right) \right) \otimes V_{\alpha }\right) .
\end{equation*}%
Similarly, let%
\begin{equation*}
\Gamma \left( F_{O}\diagup G,\mathcal{T}^{\rho }\right) ^{\beta }=\mathrm{%
eval}\left( \mathrm{Hom}_{G}\left( W_{\beta },\Gamma \left( F_{O}\diagup G,%
\mathcal{T}^{\rho }\right) \right) \otimes W_{\beta }\right) .
\end{equation*}
\begin{theorem}
\label{IsomorphismsOfSectionsTheorem}For any irreducible representations $%
\rho :G\rightarrow U\left( V_{\rho }\right) $ and $\sigma :O\left( n\right)
\rightarrow U\left( W_{\sigma }\right) $, the map $j_{\rho }^{-1}\circ
i_{\sigma }:\Gamma \left( M,\mathcal{E}^{\sigma }\right) ^{\rho }\rightarrow
\Gamma \left( F_{O}\diagup G,\mathcal{T}^{\rho }\right) ^{\sigma }$ is an
isomorphism (with inverse $i_{\sigma }^{-1}\circ j_{\rho }$) that extends to
an $L^{2}$-isometry.
\end{theorem}
\begin{proof}
Observe that $i_{\sigma }$ implements the isomorphism
\begin{eqnarray*}
\Gamma \left( M,\mathcal{E}^{\sigma }\right) &=&\Gamma \left( M,\Gamma
\left( p^{-1}\left( \cdot \right) ,\left. E\right\vert _{p^{-1}\left( \cdot
\right) }\right) ^{\sigma }\right) \\
&\cong &\Gamma \left( M,\Gamma \left( p^{-1}\left( \cdot \right) ,\left.
E\right\vert _{p^{-1}\left( \cdot \right) }\right) \right) ^{\sigma }=\Gamma
\left( F_{O},E\right) ^{\sigma }
\end{eqnarray*}%
to the space of sections of $E$ of $O\left( n\right) $ representation type $%
\sigma $. Its restriction to $\Gamma \left( M,\mathcal{E}^{\sigma }\right)
^{\rho }$ is%
\begin{eqnarray*}
\Gamma \left( M,\mathcal{E}^{\sigma }\right) ^{\rho } &=&\Gamma \left(
M,\Gamma \left( p^{-1}\left( \cdot \right) ,\left. E\right\vert
_{p^{-1}\left( \cdot \right) }\right) ^{\sigma }\right) ^{\rho } \\
&\cong &\left( \Gamma \left( M,\Gamma \left( p^{-1}\left( \cdot \right)
,\left. E\right\vert _{p^{-1}\left( \cdot \right) }\right) \right) ^{\sigma
}\right) ^{\rho } \\
&=&\left( \Gamma \left( F_{O},E\right) ^{\sigma }\right) ^{\rho }=\Gamma
\left( F_{O},E\right) ^{\sigma ,\rho },
\end{eqnarray*}%
where the superscript $\sigma ,\rho $ denotes restriction first to sections
of $O\left( n\right) $-representation type $\left[ \sigma \right] $ and then
to the subspace of sections of $G$-representation type $\left[ \rho \right] $%
. Since the $O\left( n\right) $ and $G$ actions commute, we may do this in
the other order, so that%
\begin{eqnarray*}
\Gamma \left( F_{O},E\right) ^{\sigma ,\rho } &=&\left( \Gamma \left(
F_{O},E\right) ^{\rho }\right) ^{\sigma } \\
&\cong &\Gamma \left( F_{O}\diagup G,\Gamma \left( \pi ^{-1}\left( y\right)
,\left. E\right\vert _{\pi ^{-1}\left( \cdot \right) }\right) ^{\rho
}\right) ^{\sigma } \\
&=&\Gamma \left( F_{O}\diagup G,\mathcal{T}^{\rho }\right) ^{\sigma },
\end{eqnarray*}%
where the isomorphism is the inverse of the restriction of $j_{\rho }$ to $%
\Gamma \left( F_{O}\diagup G,\mathcal{T}^{\rho }\right) ^{\sigma }$. Since $%
i_{\sigma }$ and $j_{\rho }$ are $L^{2}$ isometries, the result follows.
\end{proof}
\subsection{Dirac-type operators on the frame bundle\label{DiracFrameBundle}}
Let $E\rightarrow F_{O}$ be a Hermitian vector bundle of $\mathbb{C}\mathrm{l%
}\left( N\mathcal{F}\right) $ modules that is equivariant with respect to
the $G\times O\left( n\right) $ action. With notation as in Example \ref%
{RiemFoliationTrDiracExample}, we have the transversal Dirac operator $A_{N%
\mathcal{F}}$ defined by the composition
\begin{equation*}
\Gamma \left( F_{O},E\right) \overset{\nabla }{\rightarrow }\Gamma \left(
F_{O},T^{\ast }F_{O}\otimes E\right) \overset{\mathrm{proj}}{\rightarrow }%
\Gamma \left( F_{O},N^{\ast }\mathcal{F}\otimes E\right) \overset{c}{%
\rightarrow }\Gamma \left( F_{O},E\right) .
\end{equation*}%
As explained previously, the operator
\begin{equation*}
D_{N\mathcal{F}}=A_{N\mathcal{F}}-\frac{1}{2}c\left( H\right)
\end{equation*}%
is a essentially self-adjoint $G\times O\left( n\right) $-equivariant
operator, where $H$ is the mean curvature vector field of the $G$-orbits in $%
F_{O}$.
From $D_{N\mathcal{F}}$ we now construct equivariant differential operators
on $M$ and $F_{O}\diagup G$, as follows. We define the operators%
\begin{equation*}
D_{M}^{\sigma }:=i_{\sigma }^{-1}\circ D_{N\mathcal{F}}\circ i_{\sigma
}:\Gamma \left( M,\mathcal{E}^{\sigma }\right) \rightarrow \Gamma \left( M,%
\mathcal{E}^{\sigma }\right) ,
\end{equation*}%
and%
\begin{equation*}
D_{F_{O}\diagup G}^{\rho }:=j_{\rho }^{-1}\circ D_{N\mathcal{F}}\circ
j_{\rho }:\Gamma \left( F_{O}\diagup G,\mathcal{T}^{\rho }\right)
\rightarrow \Gamma \left( F_{O}\diagup G,\mathcal{T}^{\rho }\right) .
\end{equation*}%
For an irreducible representation $\alpha :G\rightarrow U\left( V_{\alpha
}\right) $, let
\begin{equation*}
\left( D_{M}^{\sigma }\right) ^{\alpha }:\Gamma \left( M,\mathcal{E}^{\sigma
}\right) ^{\alpha }\rightarrow \Gamma \left( M,\mathcal{E}^{\sigma }\right)
^{\alpha }
\end{equation*}%
be the restriction of $D_{M}^{\sigma }$ to sections of $G$-representation
type $\left[ \alpha \right] $. Similarly, for an irreducible representation $%
\beta :G\rightarrow U\left( W_{\beta }\right) $, let
\begin{equation*}
\left( D_{F_{O}\diagup G}^{\rho }\right) ^{\beta }:\Gamma \left(
F_{O}\diagup G,\mathcal{T}^{\rho }\right) ^{\beta }\rightarrow \Gamma \left(
F_{O}\diagup G,\mathcal{T}^{\rho }\right) ^{\beta }
\end{equation*}%
be the restriction of $D_{F_{O}\diagup G}^{\rho }$ to sections of $O\left(
n\right) $-representation type $\left[ \beta \right] $. The proposition
below follows from Theorem \ref{IsomorphismsOfSectionsTheorem}.
\begin{proposition}
\label{SpectraSame}The operator $D_{M}^{\sigma }$ is transversally elliptic
and $G$-equivariant, and $D_{F_{O}\diagup G}^{\rho }$ is elliptic and $%
O\left( n\right) $-equivariant, and the closures of these operators are
self-adjoint. The operators $\left( D_{M}^{\sigma }\right) ^{\rho }$ and $%
\left( D_{F_{O}\diagup G}^{\rho }\right) ^{\sigma }$ have identical discrete
spectrum, and the corresponding eigenspaces are conjugate via Hilbert space
isomorphisms.
\end{proposition}
Thus, questions about the transversally elliptic operator $D_{M}^{\sigma }$
can be reduced to questions about the elliptic operators $D_{F_{O}\diagup
G}^{\rho }$ for each irreducible $\rho :G\rightarrow U\left( V_{\rho
}\right) $.
\section{Topological properties of the lifted Dirac operators\label%
{topPropertiesSection}}
In this section, we will prove that if $F_{O}$ is $G$-transversally spin$^{c}
$, then the symbols of the lifted transversal Dirac operators generate all
the possible equivariant indices. To show this, we generalize the standard
multiplicative property of $K$-theory to the equivariant setting of our
paper.
\subsection{Equivariant Multiplicative Properties of $K$-theory\label%
{Multiplicative PropertySection}}
Let $H$ be a compact Lie group. Suppose that $P$ is a principal $H$-bundle
over a compact manifold $M$. Suppose that the compact Lie group $G$ acts on $%
M$ and lifts to $P$, such that the $G$-action on $P$ commutes with the $H$%
-action. Let $Z\overset{\pi }{\rightarrow }M$ be a fiber bundle associated
to $P$ with $H$-fiber $Y$; that is,%
\begin{equation*}
Z=P\times _{H}Y=P\times Y\diagup \left( p,y\right) \sim \left(
ph,h^{-1}y\right) .
\end{equation*}%
Then $G$ acts on $Z$ via $g\left[ \left( p,y\right) \right] =\left[ \left(
gp,y\right) \right] $.
For any $v\ $in the Lie algebra $\mathfrak{g}$ of $G$, let $\overline{v}$
denote the fundamental vector field on $M$ associated to $v$. As in \cite{A}%
, let
\begin{equation*}
T_{G}^{\ast }M=\left\{ \xi \in T^{\ast }M:\xi \left( \overline{v}\right) =0%
\text{ for all }v\in \mathfrak{g}\right\} ,
\end{equation*}%
and let $T_{G}^{\ast }Z$ be defined similarly. Let $K_{cpt,G}\left(
T_{G}^{\ast }M\right) $ denote the $G$-equivariant, compactly supported
K-group of $T_{G}^{\ast }M$, which is isomorphic to the group of stable $G$%
-equivariant homotopy classes of transversally elliptic first-order symbols
under direct sum. Likewise, $K_{cpt,H}\left( T^{\ast }Y\right) $ is
isomorphic to the group of the stable $H$-equivariant homotopy classes of
first order elliptic symbols over $Y$.
We define a multiplication%
\begin{equation*}
K_{cpt,G}\left( T_{G}^{\ast }M\right) \otimes K_{cpt,H}\left( T^{\ast
}Y\right) \rightarrow K_{cpt,G}\left( T_{G}^{\ast }Z\right)
\end{equation*}%
as follows. Let $u$ be a transversally elliptic, $G$-equivariant symbol over
$M$ taking values in $\mathrm{\mathrm{Hom}}\left( E^{+},E^{-}\right) $, and
let $v$ be a $H$-equivariant elliptic symbol over $Y$ taking values in $%
\mathrm{\mathrm{Hom}}\left( F^{+},F^{-}\right) $. First, we lift the symbol $%
u$ to the $H\times G$-equivariant symbol $\widehat{u}$ on $P$. Let $\widehat{%
u}\ast v$ be the standard K-theory multiplication (similar to \cite[Lemma 3.4%
]{A})
\begin{equation*}
K_{cpt,H\times G}\left( T_{G}^{\ast }P\right) \otimes K_{cpt,H}\left(
T^{\ast }Y\right) \rightarrow K_{cpt,H\times G}\left( T_{H\times G}^{\ast
}\left( P\times Y\right) \right) .
\end{equation*}%
An element $\left( h,g\right) \in H\times G$ acts on $\left( p,y\right) \in
P\times Y$ by%
\begin{equation}
\left( h,g\right) \left( p,y\right) =\left( phg,h^{-1}y\right) =\left(
pgh,h^{-1}y\right) . \label{HxGaction}
\end{equation}%
Since the action of $H\times \left\{ e\right\} $ is free, we have%
\begin{equation*}
K_{cpt,H\times G}\left( T_{H\times G}^{\ast }\left( P\times Y\right) \right)
\cong K_{cpt,G}\left( T_{G}^{\ast }\left( P\times _{H}Y\right) \right)
=K_{cpt,G}\left( T_{G}^{\ast }Z\right) .
\end{equation*}%
Finally we define
\begin{equation*}
u\cdot v=\widetilde{\widehat{u}\ast v}
\end{equation*}%
to be the image of $\widehat{u}\ast v$ in $K_{cpt,G}\left( T_{G}^{\ast
}Z\right) $ under the isomorphism above.
Given any finite-dimensional unitary virtual $H$-representation $\tau $ on $%
V $, we may form the associated $G$-virtual bundle $\widetilde{V_{\tau }}%
=P\times _{\tau }V$ over $M$, defining a class in $K_{G}\left( M\right) $.
The tensor product makes $K_{cpt,G}\left( T_{G}^{\ast }M\right) $ naturally
into a $K_{G}\left( M\right) $-module; for each $\left[ u\right] \in
K_{cpt,G}\left( T_{G}^{\ast }M\right) $, the symbol $u\otimes \tau
:=u\otimes \mathbf{1}_{\widetilde{V_{\tau }}}$ defines an element of $%
K_{cpt,G}\left( T_{G}^{\ast }M\right) $.
We let $\mathrm{ind}^{H}\left( \cdot \right) $ denote the virtual
representation-valued index as explained in \cite{A}; note that the result
is a finite-dimensional virtual representation if the input is a symbol of
an elliptic operator.
\begin{theorem}
\label{EquivariantMultiplicativeTheorem}Let $Z=P\times _{H}Y$ as above, with
$P$ a $H$-bundle over $M$. Let $u$ be a transversally elliptic, $G$%
-equivariant symbol over $M$ taking values in $\mathrm{\mathrm{Hom}}\left(
E^{+},E^{-}\right) $, and let $v$ be a $H$-equivariant elliptic symbol over $%
Y$ taking values in $\mathrm{\mathrm{Hom}}\left( F^{+},F^{-}\right) $, so
that $u$ and $v$ define classes $\left[ u\right] $ and $\left[ v\right] $ in
$K_{cpt,G}\left( T_{G}^{\ast }M\right) $ and $K_{cpt,H}\left( T^{\ast
}Y\right) $, respectively. Then $u\cdot v$ defines an element of $%
K_{cpt,G}\left( T_{G}^{\ast }Z\right) $, and
\begin{equation*}
\mathrm{ind}^{G}\left( u\cdot v\right) =\mathrm{ind}^{G}\left( u\otimes
\mathrm{ind}^{H}\left( v\right) \right) .
\end{equation*}
\begin{proof}
We adopt the argument in \cite[13.6]{L-M} to our situation. Let $L$ be a
transversally elliptic, $G$-equivariant first order operator representing $u$%
, and let $Q$ be an elliptic, $H$-equivariant first-order operator
representing $v$. Let $\widehat{u}$ be the lift of $u$ to a $H\times G$%
-transversely elliptic symbol over $P$, and let $\widehat{L}$ be a
transversally elliptic, $H\times G$-equivariant first order operator
representing $\widehat{u}$. Next, consider operator product $D=\widehat{L}%
\ast Q$ over $P\times Y$, which represents $\widehat{u}\ast v$. This
operator is $H\times G$ equivariant with respect to the action (\ref%
{HxGaction}). Then%
\begin{equation*}
\ker \left( D^{\ast }D\right) =\left[ \ker \left( \widehat{L}\otimes \mathbf{%
1}\right) \cap \ker \left( \mathbf{1}\otimes Q\right) \right] \oplus \left[
\ker \left( \widehat{L}^{\ast }\otimes \mathbf{1}\right) \cap \ker \left(
\mathbf{1}\otimes Q^{\ast }\right) \right]
\end{equation*}%
and%
\begin{equation*}
\ker \left( DD^{\ast }\right) =\left[ \ker \left( \widehat{L}^{\ast }\otimes
\mathbf{1}\right) \cap \ker \left( \mathbf{1}\otimes Q\right) \right] \oplus %
\left[ \ker \left( \widehat{L}\otimes \mathbf{1}\right) \cap \ker \left(
\mathbf{1}\otimes Q^{\ast }\right) \right]
\end{equation*}%
Let $\widetilde{D}$ and $\widetilde{D}^{\ast }$ be the restrictions of the
operators $D$ and $D^{\ast }$ to sections that are pullbacks of sections
over the base $Z=P\times _{H}Y$, i.e. those that are $H$-invariant. Let $%
\tau ^{+}$ denote the $H$-representation $\ker $ $Q$, and let $\tau ^{-}$ be
the representation $\ker $ $Q^{\ast }$. By the definition of the $H$-action
in (\ref{HxGaction}), the decomposition yields the associated kernels%
\begin{eqnarray*}
\ker \left( \widetilde{D}^{\ast }\widetilde{D}\right) &=&\ker \left(
\widehat{L}\otimes \tau ^{+}\right) \oplus \ker \left( \widehat{L}^{\ast
}\otimes \tau ^{-}\right) , \\
\ker \left( \widetilde{D}\widetilde{D}^{\ast }\right) &=&\ker \left(
\widehat{L}^{\ast }\otimes \tau ^{+}\right) \oplus \ker \left( \widehat{L}%
\otimes \tau ^{-}\right) .
\end{eqnarray*}%
We next decompose the above as $G$-representations, and we obtain%
\begin{equation*}
\mathrm{ind}^{G}\left( \widetilde{D}\right) =\mathrm{ind}^{G}\left( \widehat{%
L}\otimes \mathrm{ind}^{H}\left( Q\right) \right) .
\end{equation*}%
The result follows, since $u\cdot v=\widetilde{\widehat{u}\ast v}$ is stably
homotopic to the principal symbol of $\widetilde{D}$.
\end{proof}
\end{theorem}
\subsection{Index of Lifted Dirac operators}
Suppose that $D:\Gamma \left( M,E^{+}\right) \rightarrow \Gamma \left(
M,E^{-}\right) $ is any transversally elliptic, $G$-equivariant operator
with transversally elliptic symbol $u\in \Gamma \left( M,\mathrm{\mathrm{Hom}%
}\left( E^{+},E^{-}\right) \right) $, so that $\left[ u\right] \in
K_{cpt,G}\left( T_{G}^{\ast }M\right) $. If $\mathbf{1}$ denotes the trivial
$O\left( n\right) $ representation over the identity, then let $\ v$ be any
element of the class $i!\left( \mathbf{1}\right) \in K_{cpt,O\left( n\right)
}\left( T^{\ast }O\left( n\right) \right) $ induced from the inclusion of
the identity in $O\left( n\right) $ via an extension of the Thom isomorphism
(see \cite{ASi1}). Observe that the equivariant index $\mathrm{ind}^{O\left(
n\right) }\left( v\right) $ of the elliptic symbol $v$ is equal to one copy
of the trivial representation (see axioms of the equivariant index in \cite[%
13.6]{ASi1}). By Theorem \ref{EquivariantMultiplicativeTheorem}, $G$%
-equivariant transversally elliptic symbol $u\cdot v$ defines an element of $%
K_{cpt,G}\left( T_{G}^{\ast }F_{O}\right) $ such that%
\begin{eqnarray*}
\mathrm{ind}^{G}\left( u\cdot v\right) &=&\mathrm{ind}^{G}\left( u\otimes
\mathrm{ind}^{O\left( n\right) }\left( v\right) \right) \\
&=&\mathrm{ind}^{G}\left( u\otimes \mathbf{1}\right) \\
&=&\mathrm{ind}^{G}\left( u\right) .
\end{eqnarray*}
Suppose further that $F_{O}$ is $G$-transversally spin$^{c}$. Then the class
$\left[ u\cdot v\right] \in K_{cpt,G}\left( T_{G}^{\ast }F_{O}\right) $ may
be represented by the symbol of a transversally-elliptic, $G$-equivariant
operator $D_{N\mathcal{F}}$ of Dirac type.
Thus, the operator $D_{M}^{\mathbf{1}}=i_{\mathbf{1}}^{-1}\circ D_{N\mathcal{%
F}}\circ i_{\mathbf{1}}$ satisfies%
\begin{eqnarray*}
\mathrm{ind}^{G}\left( D_{M}^{\mathbf{1}}\right) &=&\mathrm{ind}^{G}\left(
D_{N\mathcal{F}}\right) =\mathrm{ind}^{G}\left( u\cdot v\right) \\
&=&\mathrm{ind}^{G}\left( u\right) =\mathrm{ind}^{G}\left( D\right) .
\end{eqnarray*}
The result below follows.
\begin{theorem}
Suppose that $F_{O}$ is $G$-transversally spin$^{c}$. Then for every
transversally elliptic symbol class $\left[ u\right] \in K_{cpt,G}\left(
T_{G}^{\ast }M\right) $, there exists an operator of type $D_{M}^{\mathbf{1}%
} $ such that $\mathrm{ind}^{G}\left( u\right) =\mathrm{ind}^{G}\left(
D_{M}^{\mathbf{1}}\right) $.\label{indexClassGivenByTransvDiracThm}
\end{theorem}
\section{Example\label{ExampleSection}}
\subsection{A transversal Dirac operator on the sphere}
Let $G=S^{1}$ act on $S^{2}\subset \mathbb{R}^{3}$ by rotations about the $z$%
-axis. Let $p:F_{O}\rightarrow S^{2}$ be the oriented orthonormal frame
bundle. We will identify $F_{O}$ with $SO\left( 3\right) $ by letting the
first row denote the point on $S^{2}$ and the last two rows denote the
framing of the tangent space. We choose the metric on $F_{O}$ to be
\begin{equation*}
\left\langle A,B\right\rangle =\mathrm{tr}\left( A^{t}B\right) .
\end{equation*}%
The action of $S^{1}$ lifted to $F_{O}$ is given by multiplication on the
right:
\begin{equation*}
R_{t}\left( A\right) =A\left(
\begin{array}{lll}
\cos t & -\sin t & 0 \\
\sin t & \cos t & 0 \\
0 & 0 & 1%
\end{array}%
\right) .
\end{equation*}%
Tangent vectors to $F_{O}$ are elements of the Lie algebra
\begin{equation*}
\mathfrak{o}\left( 3\right) =\left\{ \left. \left(
\begin{array}{lll}
0 & a & b \\
-a & 0 & c \\
-b & -c & 0%
\end{array}%
\right) \,\right\vert \,a,b,c\in \mathbb{R}\right\} ,
\end{equation*}%
and the tangent space to the $S^{1}$ action is the span of the
left-invariant vector field $T$ induced by $\left(
\begin{array}{lll}
0 & 1 & 0 \\
-1 & 0 & 0 \\
0 & 0 & 0%
\end{array}%
\right) $ at the identity. Thus, the normal bundle of the corresponding
foliation on $F_{O}$ is trivial. It is the subbundle $N_{S^{1}}$ of $TF_{O}$
that is given at $A\in F_{O}$ by
\begin{equation*}
\left. N_{S^{1}}\right\vert _{A}=\left\{ \left. A\left(
\begin{array}{lll}
0 & 0 & b \\
0 & 0 & c \\
-b & -c & 0%
\end{array}%
\right) \,\right\vert \,b,c\in \mathbb{R}\right\} .
\end{equation*}%
The vectors $V_{1}=A\left(
\begin{array}{lll}
0 & 0 & 1 \\
0 & 0 & 0 \\
-1 & 0 & 0%
\end{array}%
\right) $, $V_{2}=A\left(
\begin{array}{lll}
0 & 0 & 0 \\
0 & 0 & 1 \\
0 & -1 & 0%
\end{array}%
\right) $ and the orbit direction $T=A\left(
\begin{array}{lll}
0 & 1 & 0 \\
-1 & 0 & 0 \\
0 & 0 & 0%
\end{array}%
\right) $ are mutually orthogonal. Let $E=F_{O}\times \mathbb{C}%
^{2}\rightarrow F_{O}$ be the trivial bundle. The action of $\mathbb{C}%
\mathrm{l}\left( N_{S^{1}}\right) \cong \mathbb{C}\mathrm{l}\left( \mathbb{R}%
^{2}\right) $ on fibers of $E$ is defined by $c\left( V_{1}\right) =\left(
\begin{array}{ll}
0 & -1 \\
1 & 0%
\end{array}%
\right) ,$ $c\left( V_{2}\right) =\left(
\begin{array}{ll}
0 & i \\
i & 0%
\end{array}%
\right) $. We identify the vectors $\left(
\begin{array}{c}
1 \\
0%
\end{array}%
\right) $ and $\left(
\begin{array}{c}
0 \\
1%
\end{array}%
\right) \in \mathbb{C}^{2}$ with the left-invariant fields $V_{1}$ and $%
V_{2} $ in $\Gamma \left( N_{S^{1}}\right) $. We assume that the $S^{1}$%
-action on $E$ is trivial. As in Example \ref{RiemFoliationTrDiracExample},
the transversal Dirac operator is
\begin{equation*}
D_{N\mathcal{F}}=A_{N\mathcal{F}}=\sum_{j=1}^{2}c\left( V_{j}\right) \nabla
_{V_{j}},
\end{equation*}%
where $\nabla _{V_{j}}$ is the directional derivative in the direction $%
V_{j} $. Since the length of each orbit of the $S^{1}$ action is constant,
the mean curvature vector is zero.
The bundle $F_{O}\rightarrow S^{2}$ is an $SO\left( 2\right) $ principal
bundle and comes equipped with an action of $SO\left( 2\right) $ on the
frames over a point. The left action of%
\begin{equation*}
\left(
\begin{array}{cc}
\cos \alpha & -\sin \alpha \\
\sin \alpha & \cos \alpha%
\end{array}%
\right) \in SO\left( 2\right)
\end{equation*}%
on a frame $A\in SO\left( 3\right) $ is given by%
\begin{equation*}
L_{\alpha }\left( A\right) =\left(
\begin{array}{ccc}
1 & 0 & 0 \\
0 & \cos \alpha & \sin \alpha \\
0 & -\sin \alpha & \cos \alpha%
\end{array}%
\right) A.
\end{equation*}%
Again, we extend this action trivially to the $\mathbb{C}^{2}$ bundle. Note
that $D_{N\mathcal{F}}$ is equivariant with respect to both the above left $%
SO\left( 2\right) $ action and the right $S^{1}$ action.
We choose the standard spherical coordinates $x\left( \theta ,\phi \right)
\in S^{2}$. Let $P_{\theta ,\phi }$ denote parallel transport in the tangent
bundle from the north pole along the minimal geodesic connected to $x\left(
\theta ,\phi \right) $. Then%
\begin{equation*}
P_{\theta ,\phi }v=\left(
\begin{array}{ccc}
\cos \theta & -\sin \theta & 0 \\
\sin \theta & \cos \theta & 0 \\
0 & 0 & 1%
\end{array}%
\right) \left(
\begin{array}{ccc}
\cos \phi & 0 & \sin \phi \\
0 & 1 & 0 \\
-\sin \phi & 0 & \cos \phi%
\end{array}%
\right) \left(
\begin{array}{ccc}
\cos \theta & \sin \theta & 0 \\
-\sin \theta & \cos \theta & 0 \\
0 & 0 & 1%
\end{array}%
\right) v.
\end{equation*}%
We parallel transport the standard frame $\left( e_{1},e_{2}\right) $ at the
north pole to get $X_{\theta ,\phi }=P_{\theta ,\phi }e_{1}$, $Y_{\theta
,\phi }=P_{\theta ,\phi }e_{2}$, and the we rotate by $\alpha $ to get all
possible frames. The result is a coordinate chart $U^{1}:\left[ 0,2\pi %
\right] \times \left[ 0,\frac{\pi }{2}\right] \times \left[ 0,2\pi \right]
\rightarrow SO\left( 3\right) $ defined by
\begin{equation*}
U^{1}\left( \theta ,\phi ,\alpha \right) =L_{\alpha }\left(
\begin{array}{c}
x\left( \theta ,\phi \right) \\
X_{\theta ,\phi } \\
Y_{\theta ,\phi }%
\end{array}%
\right) .
\end{equation*}
A section $u$ is defined to be of irreducible representation type $\sigma
_{n}:SO\left( 2\right) \rightarrow \mathbb{C}$ if it satisfies $\left(
\left( L_{\beta }\right) _{\ast }u\right) =e^{in\beta }u$. Since the action
of $SO\left( 2\right) $ on the fibers of $E$ is trivial, we have
\begin{eqnarray}
\left( \left( L_{\beta }\right) _{\ast }u\right) \left( \theta ,\phi ,\alpha
\right) &=&\left( u\circ \left( L_{\beta }\right) ^{-1}\right) \left( \theta
,\phi ,\alpha \right) \notag \\
&=&u\left( \theta ,\phi ,\alpha -\beta \right) =e^{in\beta }u\left( \theta
,\phi ,\alpha \right) . \label{leftMultiplicationFormula}
\end{eqnarray}%
Thus, it suffices to calculate $u\left( \theta ,\phi ,0\right) =u$ at $%
U^{1}\left( \theta ,\phi ,0\right) $.
The lower hemisphere coordinates of the point and vectors would have the
opposite third coordinate, and the sign of $X_{\theta ,\phi }$ is reversed
in addition to ensure that the frame is oriented. Note that the $\phi $ in
the lower hemisphere is $\left( \pi -\phi \right) $ in the upper hemisphere.
Thus the second chart is $U^{2}:\left[ 0,2\pi \right] \times \left[ 0,\frac{%
\pi }{2}\right] \times \left[ 0,2\pi \right] $%
\begin{equation*}
U^{2}\left( \theta ,\phi ,\alpha \right) =U_{\alpha }^{2}\left(
\begin{array}{c}
x\left( \theta ,\phi \right) \\
-X_{\theta ,\phi } \\
Y_{\theta ,\phi }%
\end{array}%
\right) \mathrm{diag}\left( 1,1,-1\right)
\end{equation*}%
One can check that
\begin{equation*}
U^{1}\left( \theta ,\frac{\pi }{2},\alpha \right) =U^{2}\left( \theta ,\frac{%
\pi }{2},\alpha -2\theta \right) =L_{-2\theta }U^{2}\left( \theta ,\frac{\pi
}{2},\alpha \right) .
\end{equation*}%
Thus, the clutching function for the frame bundle is multiplication on the
left by $e^{2\theta i}$.
Next, suppose that $u$ is a section such that $\left( \left( L_{\beta
}\right) _{\ast }u\right) \left( M\right) =u\left( L_{-\beta }M\right)
=e^{in\beta }u\left( M\right) $. This means in fact that $u\left( \theta
,\phi ,\alpha -\beta \right) =e^{in\beta }u\left( \theta ,\phi ,\alpha
\right) $ in both charts. Thus we may trivialize the bundle by restricting
to $\left( \theta ,\phi ,0\right) $ in each chart. We observe
\begin{equation*}
u^{1}\left( \theta ,\frac{\pi }{2},0\right) =u^{2}\left( L_{-2\theta }\left(
\theta ,\frac{\pi }{2},0\right) \right) =e^{i2n\theta }u^{2}\left( \theta ,%
\frac{\pi }{2},0\right) ,
\end{equation*}%
and thus the clutching function for $\mathbb{E}^{\sigma _{n}}$ is $%
e^{2n\theta i}$.
One may express the vector fields $V_{1}$, $V_{2}$ in terms of the
coordinate vector fields $\partial _{\alpha }$,$\partial _{\theta }$, $%
\partial _{\phi }$. In the upper hemisphere,%
\begin{eqnarray*}
V_{1}^{1} &=&\frac{\sin \theta \left( \cos \phi -1\right) }{\sin \phi }%
\partial _{\alpha }+\frac{\sin \theta \cos \phi }{\sin \phi }\partial
_{\theta }-\cos \theta ~\partial _{\phi } \\
V_{2}^{1} &=&\frac{\cos \theta \left( 1-\cos \phi \right) }{\sin \phi }%
\partial _{\alpha }-\frac{\cos \theta \cos \phi }{\sin \phi }\partial
_{\theta }-\sin \theta ~\partial _{\phi }
\end{eqnarray*}
Now we wish to consider the operator%
\begin{equation*}
D_{S^{2}}^{\sigma _{n}}=i_{\sigma _{n}}^{-1}\circ D_{N\mathcal{F}}\circ
i_{\sigma _{n}}:\Gamma \left( S^{2},\mathcal{E}^{\sigma _{n}}\right)
\rightarrow \Gamma \left( S^{2},\mathcal{E}^{\sigma _{n}}\right) ,
\end{equation*}%
where $i_{\sigma _{n}}:\Gamma \left( S^{2},\mathcal{E}^{\sigma _{n}}\right)
\rightarrow \Gamma \left( F_{O},E\right) ^{\sigma _{n}}$. We have%
\begin{eqnarray*}
\sqrt{2}D_{N\mathcal{F}}^{1}\left(
\begin{array}{c}
u_{1} \\
u_{2}%
\end{array}%
\right) &=&\left(
\begin{array}{ll}
0 & -1 \\
1 & 0%
\end{array}%
\right) \nabla _{V_{1}^{1}}\left(
\begin{array}{c}
u_{1} \\
u_{2}%
\end{array}%
\right) +\left(
\begin{array}{ll}
0 & i \\
i & 0%
\end{array}%
\right) \nabla _{V_{2}^{1}}\left(
\begin{array}{c}
u_{1} \\
u_{2}%
\end{array}%
\right) \\
&=&\left(
\begin{array}{c}
-\overline{\left( V_{1}^{1}+iV_{2}^{1}\right) }u_{2} \\
\left( V_{1}^{1}+iV_{2}^{1}\right) u_{1}%
\end{array}%
\right) .
\end{eqnarray*}%
There is a similar formula for $\sqrt{2}D_{N\mathcal{F}}^{2}$ in the lower
hemisphere chart. Observe that%
\begin{equation*}
\left( V_{1}^{1}+iV_{2}^{1}\right) =-ie^{i\theta }\left( \cot \phi -\csc
\phi \right) \partial _{\alpha }+-ie^{i\theta }\cot \phi \partial _{\theta
}+-e^{i\theta }\partial _{\phi }
\end{equation*}
We easily check that right multiplication by $\beta \in S^{1}=\mathbb{R~}%
\mathrm{mod}\,2\pi $ on $U^{1}\left( \theta ,\phi ,\alpha \right) $ satisfies
\begin{equation*}
R_{\beta }U^{1}\left( \theta ,\phi ,\alpha \right) =U^{1}\left( \theta
+\beta ,\phi ,\alpha +\beta \right) .
\end{equation*}
If $\psi ^{1}\left( \theta ,\phi ,0\right) $ is a section of $F_{O}\times
\mathbb{C}^{2}\rightarrow F_{O}$ of type $\sigma _{n}$ (with respect to the
fiberwise action of $SO\left( 2\right) $) over the upper hemisphere, then%
\begin{eqnarray*}
\left( \left( R_{\beta }\right) _{\ast }\psi ^{1}\right) \left( \theta ,\phi
,0\right) &=&\psi ^{1}\circ R_{-\beta }\left( \theta ,\phi ,0\right) \\
&=&\psi ^{1}\left( \theta -\beta ,\phi ,-\beta \right) =e^{in\beta }\psi
^{1}\left( \theta -\beta ,\phi ,0\right) ,
\end{eqnarray*}%
using the upper hemisphere trivialization $U^{1}\left( \theta ,\phi ,\alpha
\right) $ and equation (\ref{leftMultiplicationFormula}).
\vspace{0in}If we assume that $\psi :F_{O}\rightarrow \mathcal{E}^{\sigma
_{n}}$ is a section of type $\rho _{m}$ with respect to the lifted $S^{1}$
action, then $\left( \left( R_{\beta }\right) _{\ast }\psi ^{1}\right)
=e^{im\beta }\psi ^{1}$. Thus,
\begin{equation*}
\left( \left( R_{\beta }\right) _{\ast }\psi ^{1}\right) \left( \theta ,\phi
,0\right) =e^{im\beta }\psi ^{1}\left( \theta ,\phi ,0\right) =e^{in\beta
}\psi ^{1}\left( \theta -\beta ,\phi ,0\right) ,
\end{equation*}%
which implies%
\begin{equation*}
\psi ^{1}\left( \theta ,\phi ,0\right) =e^{i\left( n-m\right) \theta }\psi
^{1}\left( 0,\phi ,0\right) .
\end{equation*}%
The analogous calculation in the lower hemisphere chart yields%
\begin{equation*}
\psi ^{2}\left( \theta ,\phi ,0\right) =e^{i\left( -n-m\right) \theta }\psi
^{2}\left( 0,\phi ,0\right) .
\end{equation*}%
\vspace{0in}
\subsection{Calculation of $\ker $ $D_{S^{2}}^{\protect\sigma _{n}}$}
Since $D_{S^{2}}^{\sigma _{n}}=i_{\sigma _{n}}^{-1}\circ D_{N\mathcal{F}%
}\circ i_{\sigma _{n}}$, we seek solutions to the equation%
\begin{equation*}
\sqrt{2}D_{N\mathcal{F}}^{1}\left(
\begin{array}{c}
\psi _{1} \\
\psi _{2}%
\end{array}%
\right) =\left(
\begin{array}{c}
-\overline{\left( V_{1}^{1}+iV_{2}^{1}\right) }\psi _{2} \\
\left( V_{1}^{1}+iV_{2}^{1}\right) \psi _{1}%
\end{array}%
\right) =\left(
\begin{array}{c}
0 \\
0%
\end{array}%
\right)
\end{equation*}%
in the upper hemisphere chart. From the equations $\left(
V_{1}^{1}+iV_{2}^{1}\right) \psi _{1}=0$, $\partial _{\alpha }\psi
_{1}=-ni\psi _{1}$ (since $\Psi \in \Gamma \left( S^{2},\mathcal{E}^{\sigma
_{n}}\right) $ ), and $\partial _{\theta }\psi _{1}=i\left( n-m\right) \psi
_{1}$ (since $\Psi \in \Gamma \left( S^{2},\mathcal{E}^{\sigma _{n}}\right)
^{\rho _{m}}$), we have%
\begin{eqnarray*}
0 &=&\left( V_{1}^{1}+iV_{2}^{1}\right) \psi _{1}=\left( -ie^{i\theta
}\left( \cot \phi -\csc \phi \right) \partial _{\alpha }+-ie^{i\theta }\cot
\phi \partial _{\theta }+-e^{i\theta }\partial _{\phi }\right) \psi _{1} \\
&=&\left( -me^{i\theta }\left( \cot \phi \right) +ne^{i\theta }\left( \csc
\phi \right) -e^{i\theta }\partial _{\phi }\right) \psi _{1}.
\end{eqnarray*}%
Solving this equation, we obtain%
\begin{equation*}
\psi _{1}\left( 0,\phi \right) =C_{2}\frac{\left( \sin \phi \right) ^{n-m}}{%
\left( \cos \phi +1\right) ^{n}}.
\end{equation*}%
This implies that $\psi _{1}$ is%
\begin{eqnarray*}
\psi _{1}\left( \theta ,\phi \right) &=&C_{2}\frac{\left( \sin \phi \right)
^{n-m}}{\left( \cos \phi +1\right) ^{n}}e^{i\left( n-m\right) \theta },\text{
or} \\
\psi _{1}\left( z\right) &=&C_{2}\frac{z^{n-m}}{\left( \sqrt{1-\left\vert
z\right\vert ^{2}}+1\right) ^{n}}\text{ }
\end{eqnarray*}%
in the complex coordinates of the projection of the upper hemisphere to the $%
xy$ plane. Thus, $\psi _{1}$ is smooth in the upper hemisphere only if $%
n\geq m$. Similarly, $-\overline{\left( V_{1}^{1}+iV_{2}^{1}\right) }\psi
_{2}=0$, $\partial _{\alpha }\psi _{2}=-ni\psi _{2}$, and $\partial _{\theta
}\psi _{2}=i\left( n-m\right) \psi _{2}$ implies%
\begin{eqnarray*}
\psi _{2}\left( \theta ,\phi \right) &=&C\left( \cos \phi +1\right)
^{n}\left( \sin \phi \right) ^{m-n}e^{i\left( n-m\right) \theta },\text{ or}
\\
\psi _{2}\left( z\right) &=&C_{2}\left( \sqrt{1-\left\vert z\right\vert ^{2}}%
+1\right) ^{n}\overline{z}^{m-n}.
\end{eqnarray*}%
Hence, $\psi _{2}$ is smooth in the upper hemisphere only if $m\geq n$.
\vspace{0in}We need to see if the solutions $\psi _{1}$ and $\psi _{2}$
extend to solutions over the entire sphere. In the lower hemisphere, we have
the equation%
\begin{equation*}
\sqrt{2}D_{N\mathcal{F}}^{2}\Psi =\sqrt{2}D_{N\mathcal{F}}^{2}\left(
\begin{array}{c}
\psi _{1} \\
\psi _{2}%
\end{array}%
\right) =\left(
\begin{array}{c}
\left( -V_{1}^{2}+iV_{2}^{2}\right) \psi _{2} \\
\left( V_{1}^{2}+iV_{2}^{2}\right) \psi _{1}%
\end{array}%
\right) =\left(
\begin{array}{c}
0 \\
0%
\end{array}%
\right) .
\end{equation*}%
Similar computations show that%
\begin{eqnarray*}
\psi _{1}\left( z\right) &=&C_{2}\left( \sqrt{1-\left\vert z\right\vert ^{2}}%
+1\right) ^{n}z^{-n-m}, \\
\psi _{2}\left( z\right) &=&C_{2}\left( \sqrt{1-\left\vert z\right\vert ^{2}}%
+1\right) ^{-n}\overline{z}^{m+n}
\end{eqnarray*}%
in the complex coordinates of the projection of the lower hemisphere to the $%
xy$ plane. Thus, $\psi _{1}$ is smooth in the lower hemisphere only if $%
n+m\leq 0$, and $\psi _{2}$ is smooth in the lower hemisphere only if $%
m+n\geq 0$.
In summary, we seek solutions to $\sqrt{2}D_{N\mathcal{F}}^{2}\Psi =\sqrt{2}%
D_{N\mathcal{F}}^{1}\Psi =0$ restricted to sections of $\mathcal{E}^{\sigma
_{n}}$ of type $\rho _{m}$. The clutching function of $\mathcal{E}^{\sigma
_{n}}$ is multiplication by $e^{2n\theta i}$ (i.e. $z^{2n}$ or $\overline{z}%
^{-2n}$), so that $\Psi ^{1}\left( \theta ,\frac{\pi }{2},0\right) =\Psi
^{2}\left( A_{2\theta }\left( \theta ,\frac{\pi }{2},0\right) \right)
=e^{i2n\theta }\Psi ^{2}\left( \theta ,\frac{\pi }{2},0\right) $.
The function $\psi _{1}$ is continuous if and only if $\psi _{1}^{1}\left(
\theta ,\frac{\pi }{2},0\right) =e^{i2n\theta }\psi _{1}^{2}\left( \theta ,%
\frac{\pi }{2},0\right) $ and $m\leq -\left\vert n\right\vert $. Similarly, $%
\psi _{2}$ is continuous if and only if $\psi _{2}^{1}\left( \theta ,\frac{%
\pi }{2},0\right) =e^{i2n\theta }\psi _{2}^{2}\left( \theta ,\frac{\pi }{2}%
,0\right) $ and $m\geq \left\vert n\right\vert $.
\vspace{0in}From the equations in the previous section, the index of $%
D_{S^{2}}^{\sigma _{n}}$ restricted to sections of type $\rho _{m}$ is
\begin{equation}
\mathrm{ind}^{\rho _{m}}\left( D_{S^{2}}^{\sigma _{n}}\right) =\left\{
\begin{array}{cc}
-1 & \text{if }m>\left\vert n\right\vert \text{ or }m=\left\vert
n\right\vert ~\text{and }n\neq 0 \\
0 & \text{if }-\left\vert n\right\vert <m<\left\vert n\right\vert \text{ or }%
m=n=0 \\
1 & \text{if }m<-\left\vert n\right\vert \text{ or }m=-\left\vert
n\right\vert ~\text{and }n\neq 0%
\end{array}%
\right. . \label{indexFormula}
\end{equation}%
Note that the kernel of $D_{S^{2}}^{\sigma _{n}}$ is infinite-dimensional.
The operator $D_{S^{2}}^{\sigma _{n}}$ fails to be elliptic precisely at the
points where $\cot \phi =0$; that is, at the equator.
\subsection{The operator on $F_{O}\diagup G$}
We now construct the operator $D_{F_{O}\diagup G}^{\rho _{m}}:\Gamma \left(
F_{O}\diagup G,\mathcal{T}^{\rho _{m}}\right) \rightarrow \Gamma \left(
F_{O}\diagup G,\mathcal{T}^{\rho _{m}}\right) $. First, observe that $%
F_{O}\diagup G=SO\left( 3\right) \diagup S^{1}$ is again the sphere $S^{2}$.
The orbits of the action of $G$ on $F_{O}$ are of the form $\left\{
R_{t}M:t\in \left[ 0,2\pi \right] \right\} $, with $M\in F_{O}$. The map $%
R_{t}$ rotates the first and second columns of the matrix, so that the map $%
F_{O}\rightarrow S^{2}$ is the map to the third column. Thus, the projection
of the vector $V_{1}$ to $TS^{2}$ is
\begin{equation*}
V_{1}=M\left(
\begin{array}{lll}
0 & 0 & 1 \\
0 & 0 & 0 \\
-1 & 0 & 0%
\end{array}%
\right) \mapsto \text{first column of }M
\end{equation*}
Similarly,%
\begin{equation*}
V_{2}=M\left(
\begin{array}{lll}
0 & 0 & 0 \\
0 & 0 & 1 \\
0 & -1 & 0%
\end{array}%
\right) \mapsto \text{second column of }M
\end{equation*}
A section of type $\rho _{m}$ of $F_{O}\times \mathbb{C}^{2}\rightarrow
F_{O} $ is one for which the partial derivative in direction
\begin{equation*}
T=M\left(
\begin{array}{lll}
0 & 1 & 0 \\
-1 & 0 & 0 \\
0 & 0 & 0%
\end{array}%
\right)
\end{equation*}%
is multiplication by $im$. In the $U^{1}$ coordinate chart (corresponding to
$0\leq \phi \leq \frac{\pi }{2}$, $0\leq \theta \leq 2\pi $, $0\leq \alpha
\leq 2\pi $), the quotient to $F_{O}\diagup G$ goes to $\left(
\begin{array}{c}
\cos \phi \\
-\cos \left( \theta -\alpha \right) \sin \phi \\
-\sin \left( \theta -\alpha \right) \sin \phi%
\end{array}%
\right) $, mapping to the entire upper hemisphere $x\geq 0$, with fibers of
the form $\left( \phi ,\theta ,\alpha \right) =\left( \phi _{0},\theta
_{0}+t,\alpha _{0}+t\right) $. Thus, we may fix $\alpha =0$ for the sake of
argument and allow $\theta =\theta -\alpha $ and $\phi $ to vary. The group $%
S^{1}$ acts as before by $U^{1}\left( \theta ,\phi ,\alpha \right) \overset{%
R_{\beta }}{\mapsto }U^{1}\left( \theta +\beta ,\phi ,\alpha +\beta \right) $%
. In particular, if $\psi $ is a section of type $\rho _{m}$, $\left(
R_{\beta }\psi ^{1}\right) \left( \theta ,\phi ,\alpha \right) =\psi
^{1}\circ m_{-\beta }\left( \theta ,\phi ,\alpha \right) =\psi ^{1}\left(
\theta -\beta ,\phi ,\alpha -\beta \right) =e^{im\beta }\psi ^{1}\left(
\theta ,\phi ,\alpha \right) $. Thus, setting $\alpha =0$, we have%
\begin{equation*}
\psi ^{1}\left( \theta -\beta ,\phi ,-\beta \right) =e^{im\beta }\psi \left(
\theta ,\phi ,0\right) .
\end{equation*}%
Thus,%
\begin{eqnarray*}
\frac{\partial }{\partial \beta }\left[ \psi ^{1}\left( \theta -\beta ,\phi
,-\beta \right) \right] &=&im\psi ^{1}\left( \theta -\beta ,\phi ,-\beta
\right) \\
&=&\left( \left( -\frac{\partial }{\partial \theta }-\frac{\partial }{%
\partial \alpha }\right) \psi ^{1}\right) \left( \theta -\beta ,\phi ,-\beta
\right)
\end{eqnarray*}%
So all sections of type $\rho _{m}$ satisfy $\left( -\frac{\partial }{%
\partial \theta }-\frac{\partial }{\partial \alpha }\right) \psi ^{1}=im\psi
^{1},$ or $\partial _{\alpha }=-\partial _{\theta }-im$.
Restricted to this space of sections, we have%
\begin{equation*}
\left( V_{1}^{1}+iV_{2}^{1}\right) =-ie^{i\theta }\csc \phi \partial
_{\theta }-e^{i\theta }\partial _{\phi }-me^{i\theta }\left( \cot \phi -\csc
\phi \right)
\end{equation*}%
The interested reader may check that this operator is elliptic at all points
of the hemisphere $x\geq 0$. A similar statement is true in the other
hemisphere. Thus, $D_{F_{O}\diagup G}^{\rho _{m}}:\Gamma \left( F_{O}\diagup
G,\mathcal{T}^{\rho _{m}}\right) \rightarrow \Gamma \left( F_{O}\diagup G,%
\mathcal{T}^{\rho _{m}}\right) $ is elliptic, as expected.
The kernel of $D_{F_{O}\diagup G}^{\rho _{m}}$ restricted to $\Gamma \left(
F_{O}\diagup G,\mathcal{T}^{\rho _{m}}\right) ^{\sigma _{n}}$ has dimension
\begin{eqnarray*}
\dim \left( \ker ^{\sigma _{n}}\left( D_{F_{O}\diagup G}^{\rho _{m}}\right)
\right) &=&\dim \left( \ker ^{\rho _{m}}\left( D_{S^{2}}^{\sigma
_{n}}\right) \right) \\
&=&\left\{
\begin{array}{ll}
2~ & \text{if }m=n=0 \\
1 & \text{if }\left\vert n\right\vert \leq \left\vert m\right\vert ,~m\neq 0
\\
0 & \text{if }\left\vert n\right\vert >\left\vert m\right\vert%
\end{array}%
\right. ~,
\end{eqnarray*}%
using the results preceding formula (\ref{indexFormula}). As expected, for a
given bundle $\mathcal{T}^{\rho _{m}}$, only a finite number of the
representation types $\sigma _{n}$ occur, and the kernel of $D_{F_{O}\diagup
G}^{\rho _{m}}$ on the space of all sections is finite-dimensional.
|
1,116,691,497,553 | arxiv | \section{Introduction}
\begin{figure*}[t!]
\centering
\setlength{\tabcolsep}{1pt}
\resizebox{\linewidth}{!}
{
\begin{tabular}{cccccc}
Bicubic Upscaling & Team ZTE VIP & Team Noah\_TerminalVision & Team Rainbow & Team Diggers & High-Resolution Target \\
\\
\includegraphics[width=0.3\linewidth]{Figures/scene_3/5.png}&
\includegraphics[width=0.3\linewidth]{Figures/scene_3/4.png}&
\includegraphics[width=0.3\linewidth]{Figures/scene_3/3.png}&
\includegraphics[width=0.3\linewidth]{Figures/scene_3/2.png}&
\includegraphics[width=0.3\linewidth]{Figures/scene_3/1.png}&
\includegraphics[width=0.3\linewidth]{Figures/scene_3/0.png}\\
\includegraphics[width=0.3\linewidth]{Figures/scene_2/5.png}&
\includegraphics[width=0.3\linewidth]{Figures/scene_2/4.png}&
\includegraphics[width=0.3\linewidth]{Figures/scene_2/3.png}&
\includegraphics[width=0.3\linewidth]{Figures/scene_2/2.png}&
\includegraphics[width=0.3\linewidth]{Figures/scene_2/1.png}&
\includegraphics[width=0.3\linewidth]{Figures/scene_2/0.png}\\
\includegraphics[width=0.3\linewidth]{Figures/scene_4/5.png}&
\includegraphics[width=0.3\linewidth]{Figures/scene_4/4.png}&
\includegraphics[width=0.3\linewidth]{Figures/scene_4/3.png}&
\includegraphics[width=0.3\linewidth]{Figures/scene_4/2.png}&
\includegraphics[width=0.3\linewidth]{Figures/scene_4/1.png}&
\includegraphics[width=0.3\linewidth]{Figures/scene_4/0.png}\\
\includegraphics[width=0.3\linewidth]{Figures/scene_5/5.png}&
\includegraphics[width=0.3\linewidth]{Figures/scene_5/4.png}&
\includegraphics[width=0.3\linewidth]{Figures/scene_5/3.png}&
\includegraphics[width=0.3\linewidth]{Figures/scene_5/2.png}&
\includegraphics[width=0.3\linewidth]{Figures/scene_5/1.png}&
\includegraphics[width=0.3\linewidth]{Figures/scene_5/0.png}\\
\end{tabular}
}
\vspace{0cm}
\caption{Sample crops from the original video frames, results obtained by challenge participants and the target high-resolution frames.}
\label{fig:qualitative}
\vspace{-0.2cm}
\end{figure*}
An increased popularity of various video streaming services and a widespread of mobile devices have created a strong need for efficient and mobile-friendly video super-resolution approaches. Over the past years, many accurate deep learning-based solutions have been proposed for this problem~\cite{nah2019ntire,nah2019ntireChallenge,wang2019edvr,kappeler2016video,shi2016real,sajjadi2018frame,fuoli2019efficient}. The biggest limitation of these methods is that they were primarily targeted at achieving high fidelity scores while not optimized for computational efficiency and mobile-related constraints, which is essential for tasks related to image~\cite{ignatov2017dslr,ignatov2018wespe,ignatov2020replacing} and video~\cite{nah2020ntire} enhancement on mobile devices. In this challenge, we take one step further in solving this problem by using a popular REDS~\cite{nah2019ntire} video super-resolution dataset and by imposing additional efficiency-related constraints on the developed solutions.
When it comes to the deployment of AI-based solutions on mobile devices, one needs to take care of the particularities of mobile NPUs and DSPs to design an efficient model. An extensive overview of smartphone AI acceleration hardware and its performance is provided in~\cite{ignatov2019ai,ignatov2018ai}. According to the results reported in these papers, the latest mobile NPUs are already approaching the results of mid-range desktop GPUs released not long ago. However, there are still two major issues that prevent a straightforward deployment of neural networks on mobile devices: a restricted amount of RAM, and a limited and not always efficient support for many common deep learning layers and operators. These two problems make it impossible to process high resolution data with standard NN models, thus requiring a careful adaptation of each architecture to the restrictions of mobile AI hardware. Such optimizations can include network pruning and compression~\cite{chiang2020deploying,ignatov2020rendering,li2019learning,liu2019metapruning,obukhov2020t}, 16-bit / 8-bit~\cite{chiang2020deploying,jain2019trained,jacob2018quantization,yang2019quantization} and low-bit~\cite{cai2020zeroq,uhlich2019mixed,ignatov2020controlling,liu2018bi} quantization, device- or NPU-specific adaptations, platform-aware neural architecture search~\cite{howard2019searching,tan2019mnasnet,wu2019fbnet,wan2020fbnetv2}, \etc.
While many challenges and works targeted at efficient deep learning models have been proposed recently, the evaluation of the obtained solutions is generally performed on desktop CPUs and GPUs, making the developed solutions not practical due to the above mentioned issues. To address this problem, we introduce the first \textit{Mobile AI Workshop and Challenges}, where all deep learning solutions are developed for and evaluated on real mobile devices.
In this competition, the participating teams were provided with the original high-quality and downscaled by a factor of 4 videos from the REDS~\cite{nah2019ntire} dataset to train their networks. Within the challenge, the participants were evaluating the runtime and tuning their models on the OPPO Find X2 smartphone featuring the Qualcomm Adreno 650 GPU that can efficiently accelerate floating-point neural networks.
The final score of each submitted solution was based on the runtime and fidelity results, thus balancing between the image reconstruction quality and efficiency of the proposed model. Finally, all developed solutions are fully compatible with the TensorFlow Lite framework~\cite{TensorFlowLite2021}, thus can be deployed and accelerated on any mobile platform providing AI acceleration through the Android Neural Networks API (NNAPI)~\cite{NNAPI2021} or custom TFLite delegates~\cite{TFLiteDelegates2021}.
\smallskip
This challenge is a part of the \textit{MAI 2021 Workshop and Challenges} consisting of the following competitions:
\small
\begin{itemize}
\item Learned Smartphone ISP on Mobile NPUs~\cite{ignatov2021learned}
\item Real Image Denoising on Mobile GPUs~\cite{ignatov2021fastDenoising}
\item Quantized Image Super-Resolution on Mobile NPUs~\cite{ignatov2021real}
\item Real-Time Video Super-Resolution on Mobile GPU
\item Single-Image Depth Estimation on Mobile Devices~\cite{ignatov2021fastDepth}
\item Quantized Camera Scene Detection on Smartphones~\cite{ignatov2021fastSceneDetection}
\item High Dynamic Range Image Processing on Mobile NPUs
\end{itemize}
\normalsize
\noindent The results obtained in the other competitions and the description of the proposed solutions can be found in the corresponding challenge papers.
\begin{figure*}[t!]
\centering
\setlength{\tabcolsep}{1pt}
\resizebox{0.96\linewidth}{!}
{
\includegraphics[width=1.0\linewidth]{Figures/ai_benchmark_custom.png}
}
\vspace{0.2cm}
\caption{Loading and running custom TensorFlow Lite models with AI Benchmark application. The currently supported acceleration options include Android NNAPI, TFLite GPU, Hexagon NN, Samsung Eden and MediaTek Neuron delegates as well as CPU inference through TFLite or XNNPACK backends. The latest app version can be downloaded at \url{https://ai-benchmark.com/download}}
\vspace{-0.2cm}
\label{fig:ai_benchmark_custom}
\end{figure*}
\section{Challenge}
To develop an efficient and practical solution for mobile-related tasks, one needs the following major components:
\begin{enumerate}
\item A high-quality and large-scale dataset that can be used to train and evaluate the solution;
\item An efficient way to check the runtime and debug the model locally without any constraints;
\item An ability to regularly test the runtime of the designed neural network on the target mobile platform or device.
\end{enumerate}
This challenge addresses all the above issues. Real training data, tools, and runtime evaluation options provided to the challenge participants are described in the next sections.
\subsection{Dataset}
In this challenge, we use the REDS~\cite{nah2019ntire} dataset that serves as a benchmark for traditional video super-resolution task as it contains a large diversity of content and dynamic scenes. Following the standard procedure, we use 240 videos for training, 30 videos for validation, and 30 videos for testing. Each video has sequences of length 100, where every sequence contains video frames of 1280$\times$720 resolution at 24 fps. To generate low-resolution data, the videos were bicubically downsampled with a factor of 4. The low-resolution video data is then considered as input, and the high-resolution~--- are the target.
\begin{table*}[t!]
\centering
\resizebox{\linewidth}{!}
{
\begin{tabular}{l|c|cc|cc|ccc|c}
\hline
Team \, & \, Author \, & \, Framework \, & Model Size, & \, PSNR$\uparrow$ \, & \, SSIM$\uparrow$ \, & \multicolumn{2}{c}{\, Runtime per 10 frames $\downarrow$ \,} & Speed-Up & \, Final Score \\
& & & KB & & & \, CPU, ms \, & \, GPU, ms \, & \\
\hline
\hline
Diggers & \, chenyuxiang \, & Keras / TensorFlow & 230 & \textBF{28.33} & \textBF{0.8112} & 916 & 199 & 4.6 & \textBF{8.13} \\
ZTE VIP & jieson\_zheng & \, PyTorch / TensorFlow \, & 50 & 27.85 & 0.7983 & 163 & \textBF{113} & 1.4 & 7.36 \\
Rainbow & Zheng222 & TensorFlow & 204 & 27.99 & 0.8021 & 429 & 180 & 2.4 & 5.61 \\
Noah\_TerminalVision \, & JeremieG & TensorFlow & 30 & 27.97 & 0.8017 & 448 & \footnotesize{\textcolor{red}{Error $^*$}} & - & 2.19 \\
\rowcolor{grayhighlight} Bicubic Upscaling & Baseline & & & 26.50 & 0.7508 & & & & - \\
\end{tabular}
}
\vspace{2.6mm}
\caption{\small{Mobile AI 2021 Real-Time Video Super-Resolution challenge results and final rankings. During the runtime measurements, the models were upscaling 10 subsequent video frames from 180$\times$320 to 1280$\times$720 pixels on the OPPO Find X2 smartphone. Team \textit{Diggers} is the challenge winner. $^*$~The solution from \textit{Noah\_TerminalVision} was not parsed correctly by the TFLite GPU delegate.}}
\label{tab:results}
\end{table*}
\subsection{Local Runtime Evaluation}
When developing AI solutions for mobile devices, it is vital to be able to test the designed models and debug all emerging issues locally on available devices. For this, the participants were provided with the \textit{AI Benchmark} application~\cite{ignatov2018ai,ignatov2019ai} that allows to load any custom TensorFlow Lite model and run it on any Android device with all supported acceleration options. This tool contains the latest versions of \textit{Android NNAPI, TFLite GPU, Hexagon NN, Samsung Eden} and \textit{MediaTek Neuron} delegates, therefore supporting all current mobile platforms and providing the users with the ability to execute neural networks on smartphone NPUs, APUs, DSPs, GPUs and CPUs.
\smallskip
To load and run a custom TensorFlow Lite model, one needs to follow the next steps:
\begin{enumerate}
\setlength\itemsep{0mm}
\item Download AI Benchmark from the official website\footnote{\url{https://ai-benchmark.com/download}} or from the Google Play\footnote{\url{https://play.google.com/store/apps/details?id=org.benchmark.demo}} and run its standard tests.
\item After the end of the tests, enter the \textit{PRO Mode} and select the \textit{Custom Model} tab there.
\item Rename the exported TFLite model to \textit{model.tflite} and put it into the \textit{Download} folder of the device.
\item Select mode type \textit{(INT8, FP16, or FP32)}, the desired acceleration/inference options and run the model.
\end{enumerate}
\noindent These steps are also illustrated in Fig.~\ref{fig:ai_benchmark_custom}.
\subsection{Runtime Evaluation on the Target Platform}
In this challenge, we use the \textit{OPPO Find X2} smartphone with the \textit{Qualcomm Snapdragon 865} mobile SoC as our target runtime evaluation platform. The considered chipset demonstrates very decent AI Benchmark scores~\cite{AIBenchmark202104} and can be found in the majority of flagship Android smartphones released in 2020. It can efficiently accelerate floating-point networks on its Adreno 650 GPU with a theoretical FP16 performance of 2.4 TFLOPS. Within the challenge, the participants were able to upload their TFLite models to an external server and get a feedback regarding the speed of their model: the runtime of their solution on the above mentioned OPPO device or an error log if the network contains some incompatible operations. The models were parsed and accelerated using the TensorFlow Lite GPU delegate~\cite{lee2019device} demonstrating the best performance on this platform according to AI Benchmark results. The same setup was also used for the final runtime evaluation.
\subsection{Challenge Phases}
The challenge consisted of the following phases:
\vspace{-0.8mm}
\begin{enumerate}
\item[I.] \textit{Development:} the participants get access to the data and AI Benchmark app, and are able to train the models and evaluate their runtime locally;
\item[II.] \textit{Validation:} the participants can upload their models to the remote server to check the fidelity scores on the validation dataset, to get the runtime on the target platform, and to compare their results on the validation leaderboard;
\item[III.] \textit{Testing:} the participants submit their final results, codes, TensorFlow Lite models, and factsheets.
\end{enumerate}
\vspace{-0.8mm}
\subsection{Scoring System}
All solutions were evaluated using the following metrics:
\vspace{-0.8mm}
\begin{itemize}
\setlength\itemsep{-0.2mm}
\item Peak Signal-to-Noise Ratio (PSNR) measuring fidelity score,
\item Structural Similarity Index Measure (SSIM), a proxy for perceptual score,
\item The runtime on the target OPPO Find X2 smartphone.
\end{itemize}
\vspace{-0.8mm}
The goal of this challenge was to produce an efficient solution balancing between the fidelity scores and latency. For the fidelity evaluation, we compute the PSNR and SSIM measures between the target sharp high-resolution and the produced super-resolved videos, both scores were averaged over the entire sequence of frames. Different to common VSR methods~\cite{wang2019edvr,chan2020basicvsr} where the input of the model is a 5-dimensional tensor including the video sequence information, this challenge encouraged the participants to build models that receive mobile-friendly 4-dimensional tensors.
The input model tensor should accept 10 subsequent video frames and have a size of [$1 \times 180 \times 320 \times 30$], where the first dimension is the batch size, the second and third dimensions are the height and width of the input frames, and the last dimension is the number of channels (3 color channels $\times$ 10 frames). The size of the output tensor of the model should be [$1 \times 720 \times 1280 \times 30$].
The score of each final submission was evaluated based on the next formula ($C$ is a constant normalization factor):
\smallskip
\begin{equation*}
\text{Final Score} \,=\, \frac{2^{2 \cdot \text{PSNR}}}{C \cdot \text{runtime}},
\end{equation*}
\smallskip
During the final challenge phase, the participants did not have access to the test dataset. Instead, they had to submit their final TensorFlow Lite models that were subsequently used by the challenge organizers to check both the runtime and the fidelity results of each submission under identical conditions. This approach solved all the issues related to model overfitting, reproducibility of the results, and consistency of the obtained runtime/accuracy values.
\iffalse
\begin{table*}[t!]
\centering
\resizebox{\linewidth}{!}
{
\begin{tabular}{l|cc|cc|cc|c}
\hline
Team \, & \, Snapdragon 888 \, & \, Snapdragon 855 \, & \, Dimensity 1000 \, & \, Dimensity 820 \, & \, Exynos 2100 \, & \, Exynos 990 \, & \, Kirin 990 5G \, \\
& \,Hexagon 780 DSP, ms \, & \, Hexagon 690 DSP, ms \, & \, APU 3.0, ms \, & \, APU 3.0, ms \, & \, NPU, ms \, & \, Mali-G77 GPU, ms \, & \, Da Vinci NPU, ms \, \\
\hline
MCG & 4471 & 47 & 75.1 & 61.4 & 192 & 220 & 297 \\
Aselsan Research & \footnotesize{\textcolor{red}{Failed}} & 407 & \footnotesize{\textcolor{red}{Failed}} & \footnotesize{\textcolor{red}{Failed}} & 711 & 1065 & 770 \\
Noah\_TerminalVision \, & 4316 & 38 & 147 & 42.1 & 122 & 136 & Failed \\
ALONG & \footnotesize{\textcolor{red}{Failed}} & 64 & 129 & 157 & 210 & 208 & 243 \\
EmbededAI & \footnotesize{\textcolor{red}{Failed}} & 69 & \footnotesize{\textcolor{red}{Failed}} & \footnotesize{\textcolor{red}{Failed}} & \footnotesize{\textcolor{red}{Failed}} & \footnotesize{\textcolor{red}{Failed}} & \footnotesize{\textcolor{red}{Failed}} \\
mju\_gogogo & \footnotesize{\textcolor{red}{Failed}} & 676 & 298 & 419 & 1716 & 1906 & \footnotesize{\textcolor{red}{Failed}} \\
\end{tabular}
}
\vspace{2.6mm}
\caption{\small{Text}}
\label{tab:runtime_results}
\end{table*}
\fi
\begin{figure*}[t!]
\centering
\resizebox{1.0\linewidth}{!}
{
\includegraphics[width=1.0\linewidth]{Figures/diggers.png}
}
\caption{\small{Team Diggers proposes a bidirectional RNN with efficient feature extractors (FEB) to exploit the temporal dependencies.}}
\label{fig:Diggers}
\end{figure*}
\section{Challenge Results}
From above 125 registered participants, 4 teams entered the final phase and submitted valid results, TFLite models, codes, executables and factsheets. Table~\ref{tab:results} summarizes the final challenge results and reports PSNR, SSIM and runtime numbers for each submitted solution on the final test dataset and on the target evaluation platform, while Fig.~\ref{fig:qualitative} shows the obtained qualitative results. The proposed methods are described in section~\ref{sec:solutions}, and the team members and affiliations are listed in Appendix~\ref{sec:apd:team}.
\subsection{Results and Discussion}
All submitted solutions demonstrated a very high efficiency: the first three models can upscale video frames from 180$\times$320 to 1280$\times$720 resolution at more than 50 FPS on the target Snapdragon 865 chipset. The solution proposed by team \textit{Noah\_TerminalVision} can potentially achieve even higher frame rates, though it is currently not compatible with the TFLite GPU delegate due to \textit{split} operations and thus was tested on Snapdragon's CPU only. Team \textit{Diggers} is the challenge winner~--- the model proposed by the authors achieves the best fidelity results while demonstrating good runtime values. This is the only solution in this challenge that applied recurrent connections to make use of inter-frame dependencies for getting better reconstruction results. While the other methods were performing only a standard per-frame upscaling, the image crops shown in Fig.~\ref{fig:qualitative} demonstrate that the visual quality of the reconstructed video frames obtained in their cases is just slightly behind the one of the winning solution, while all results are significantly better compared to the simple bicubic video interpolation. Therefore, we can conclude that, from the practical aspect, all proposed solutions can be applied for video super-resolution task on real mobile devices~--- the final choice will depend on the set of supported ops (\eg, the first model might not be compatible with some mobile NPUs~\cite{ignatov2021real}) and the target FPS and runtime values.
\section{Challenge Methods}
\label{sec:solutions}
\noindent This section describes solutions submitted by all teams participating in the final stage of the MAI 2021 Real-Time Video Super-Resolution challenge.
\subsection{Diggers}
\label{sec:diggers}
Team Diggers proposed a bi-directional recurrent model for the considered video super-resolution task that uses feature maps computed for the previous and future video frames as an additional information while super-resolving the current frame (Fig.~\ref{fig:Diggers}). The model architecture is generally based on the ideas proposed in~\cite{isobe2020revisiting} and~\cite{hui2019lightweight}: for each input frame, two feature extraction blocks (FEBs) are applied to generate the corresponding feature maps: forward [blue] and backward [orange]. The forward feature maps of the current and previous frames are then combined and passed to another feature extraction block to generate the final forward feature map for the current frame. As for the backward frames, the procedure is exactly the same, though the sequence is reversed. The obtained final forward and backward features are fed to the selection units layer (SEL) module~\cite{choi2017deep}, one IMDB~\cite{hui2019lightweight} module and two convolutional and image resizing layers performing final frame upscaling.
During the training process, sequences of 21 subsequent low- and high-resolution video frames were used as model's inputs and targets. First, the model was trained for 31 epochs with a batch size of 16 and an initial learning rate of $4e-3$ multiplied by $0.7$ each 2 epochs starting from the 7th one. Next, it was trained for another 31 epochs with a batch size of 32 and the same learning rate policy. $L_2$ loss was used as a target loss function, model parameters were optimized with Adam~\cite{kingma2014adam}. The images were additionally flipped randomly during the training for data augmentation.
\subsection{ZTE VIP}
\label{sec:zte}
\begin{figure}[h!]
\centering
\resizebox{1.0\linewidth}{!}
{
\includegraphics[width=1.0\linewidth]{Figures/ZTE.png}
}
\caption{\small{Feed-forward CNN proposed by ZTE VIP team. Each basic module (BM) consists of a residual block with two convolutional layers.}}
\label{fig:ZTE}
\end{figure}
Team ZTE VIP proposed a model performing per-frame upscaling without taking into account any inter-frame dependencies (Fig.~\ref{fig:ZTE}) which can significantly speed-up the inference. In its first layer, the input is resized so that the batch size is equal to the number of video frames, and then they are processed separately by several residual blocks~\cite{he2016deep} and a \textit{depth-to-space} layer performing final frame upsampling. The number of residual blocks and their size was found using the Neural Architecture Search (NAS)~\cite{kim2019fine}, where the target metric was composed of the fidelity loss and the number of model FLOPS. The final model contains five residual blocks, each one consisting of two 3$\times$3 convolutions with eight feature maps. The network was trained to minimize $L_1$ loss with a batch size of 4 for 1000 epochs using Adam optimizer with a learning rate of $2e-4$ down-scaled by a factor of 0.5 till 400 epochs. A more detailed description of the model, design choices and training procedure is provided in~\cite{liu2021EVSRNet}.
\subsection{Rainbow}
\label{sec:rainbow}
\begin{figure}[h!]
\centering
\resizebox{1.0\linewidth}{!}
{
\includegraphics[width=0.22\linewidth]{Figures/Rainbow_IMDN.pdf}
\includegraphics[width=0.5\linewidth]{Figures/Rainbow_IMDB_s.pdf}
}
\caption{\small{The overall model architecture proposed by Rainbow team (left). The structure of the information multi-distillation blocks (IMDB\_s) is presented on the right side; 12, 9 and 3 denote the size of the output channels of each convolutional layer.}}
\label{fig:Rainbow}
\end{figure}
Similarly to the previous solution, Rainbow team has developed a pure CNN model performing per-frame video upscaling (Fig.~\ref{fig:Rainbow}). The authors presented a network consisting of three information multi-distillation blocks (IMDB\_s)~\cite{hui2019lightweight} followed by a \textit{depth-to-space} upsampling layer processing each video frame separately. A global skip connection is used to improve the fidelity results of the model. $L_1$ loss was used as a target fidelity measure, network parameters were optimized using Adam with an initial learning rate of $2e-4$ halved every 50K iterations. A batch size of 8 was used during the training, horizontal and vertical flipping was used to augment the training data.
\subsection{Noah\_TerminalVision}
\label{sec:noah}
\begin{figure}[h!]
\centering
\resizebox{1.0\linewidth}{!}
{
\includegraphics[width=\linewidth]{Figures/Noah.PNG}
}
\caption{\small{The solution proposed by Noah\_TerminalVision consists of a light-weight architecture with three residual blocks and asymmetric convolutions.}}
\label{fig:Noah}
\end{figure}
Team Noah\_TerminalVision presented a TinyVSRNet architecture that contains three residual blocks (each consisting of 2 convolutions with 16 channels) followed by \textit{depth-to-space} upsampling layer and one global skip connection performing bilinear image upscaling (Fig.~\ref{fig:Noah}). The authors have also proposed a ``single-frame'' solution by converting 10 video frames from the channel dimension to the batch dimension with \textit{split} and \textit{concat} layers. To boost the model performance, they adopted the approach developed in~\cite{ding2019acnet}, where three asymmetric convolution kernels (of size 3$\times$3, 1$\times$3, and 3$\times$1) are used during the training and then fused into one single convolution op during the inference. With this modification, the results of the TinyVSRNet were improved by around 0.05 dB. The network was trained to minimize $L_1$ loss, its parameters were optimized using Adam for one million iterations with a cyclic learning rate starting from $5e-4$ and decreased to $1e-6$ each 200K iterations.
\section{Additional Literature}
An overview of the past challenges on mobile-related tasks together with the proposed solutions can be found in the following papers:
\begin{itemize}
\item Video Super-Resolution:\, \cite{nah2019ntireChallenge,nah2020ntire}
\item Image Super-Resolution:\, \cite{ignatov2018pirm,lugmayr2020ntire,cai2019ntire,timofte2018ntire}
\item Learned End-to-End ISP:\, \cite{ignatov2019aim,ignatov2020aim}
\item Perceptual Image Enhancement:\, \cite{ignatov2018pirm,ignatov2019ntire}
\item Bokeh Effect Rendering:\, \cite{ignatov2019aimBokeh,ignatov2020aimBokeh}
\item Image Denoising:\, \cite{abdelhamed2020ntire,abdelhamed2019ntire}
\end{itemize}
\section*{Acknowledgements}
We thank OPPO Mobile Co., Ltd., AI Witchlabs, ETH Zurich (Computer Vision Lab) and Seoul National University (SNU), the organizers and sponsors of this Mobile AI 2021 challenge.
|
1,116,691,497,554 | arxiv | \section{Introduction}
The mass loss mechanism of AGB stars and red supergiants is a
long-standing astrophysical problem. In the carbon-rich case, an
extraordinary condensate exists (amorphous carbon) which is very
stable, i.\,e.\ it can exist already close the star, and is very opaque in
the optical and near IR spectral region. Detailed dynamical models
with time-dependent dust formation theory (Winters${\rm \hspace*{0.8ex}et\hspace*{0.7ex}al.\hspace*{0.65ex}}$
2000)\nocite{wlj2000} show that the formation of amorphous carbon can
provide sufficient radiation pressure to drive massive outflows,
consistent with the basic characteristics of C-star winds. This result has
been confirmed by dynamical models with frequency-dependent radiative
transfer by H{\"o}fner${\rm \hspace*{0.8ex}et\hspace*{0.7ex}al.\hspace*{0.65ex}}$(2003)\nocite{hga2003}.
However, in the oxygen-rich case, no such condensate exists. The most
stable metal-oxides like Al$_2$O$_3$ are too rare. The abundant pure
silicates like Mg$_2$SiO$_4$ are already less stable and almost
completely transparent around $1\,\mu$m where most of the stellar flux
escapes. Solid Fe and Mg-Fe-silicates are opaque but even less
stable. Stationary models of dust-driven O-rich AGB star winds with
grey radiative transfer (Ferrarotti${\rm \hspace*{0.7ex}\&\hspace*{0.7ex}}$ Gail 2006) and dynamical
models with pulsation and grey radiative transfer (Jeong${\rm \hspace*{0.8ex}et\hspace*{0.7ex}al.\hspace*{0.65ex}}$ 2003)
nevertheless came to the conclusion that the winds of O-rich AGB stars
are dust-driven, where the stellar pulsation helps to provide the
necessary density conditions to form the dust close to the star
(``pulsation-enhanced'').
In contrast, the a posteori frequency-dependent radiative transfer
analysis of non-linear pulsation models with simplified dust
formation theory by Ireland${\rm \hspace*{0.7ex}\&\hspace*{0.7ex}}$ Scholz (2006) did not find much
radiation pressure on dust (Al$_2$O$_3$ and
Mg$_{2x}$Fe$_{2-2x}$SiO$_4$) in O-rich Mira variables, with radiative
accelerations as small as 0.08 to 0.29 times the gravitational
deceleration.
Recent mid-IR observations of O-rich AGB stars in globular
clusters with {\sc Spitzer} (Lebzelter${\rm \hspace*{0.8ex}et\hspace*{0.7ex}al.\hspace*{0.65ex}}$ 2006) and of galactic
bulge AGB stars with ISO (Blommaert et al.\ 2006) show a clear
correlation between the kind of condensate and the mass loss rate
$\dot{M}$, called the ``observational dust condensation sequence'': stars
with low $\dot{M}$ show mainly Al$_2$O$_3$, whereas stars with higher
$\dot{M}$ show increasing amounts of Mg-Fe-silicates. From {\sc Midi}
interferometry of the red supergiant $\alpha$\,Ori,
Verhoelst et al.\ (2006)\nocite{vdmhc2006} concluded that Al$_2$O$_3$ grains
are already present at radial distances as small as
$r\!=\!1.5\,R_\star$.
\section{The model}
\noindent{\bf Hydrodynamics} is solved by using the {\sc
Flash}-solver (Fryxell et al.\ 2000)\nocite{for2000} in spherical
symmetry, including gravity and self-developed modules for radiation
pressure on dust${\rm \hspace*{0.7ex}\&\hspace*{0.7ex}}$ molecules and radiative heating\,/\,cooling
(see Woitke 2006a for details)\nocite{woi2006a}. We use
here, however, a piston approximation as inner boundary condition to
simulate the pulsation of the star, and a new equation of state for a
mixture of H$^+$, e$^-$, H, H$_2$, He and other atomic metals in LTE,
including ionisation and dissociation potentials, and vibrational
and rotational excitation energies of H$_2$.
\smallskip
\noindent{\bf Radiative transfer:} For the models presented in this
letter, we have developed a new frequency-dependent Monte Carlo
radiative transfer technique which allows for arbitrarily high optical
depths (Woitke 2006b)\nocite{woi2006b}. In frequency-dependent
stellar atmospheres, the gas is always optically thick at least in some
wavelengths ($\sim\!10^{\,5}$), which is a big problem for standard MC
techniques. The method is coupled to the hydrodynamics and can be used
also for 2D models.
\smallskip
\noindent{\bf Opacities:} The basis for our radiative transfer
treatment are monochromatic molecular gas opacities from the {\sc
Marcs} stellar atmosphere code (J{\o}rgensen${\rm \hspace*{0.8ex}et\hspace*{0.7ex}al.\hspace*{0.65ex}}$ 1992)\nocite{j92},
extracted by Helling${\rm \hspace*{0.8ex}et\hspace*{0.7ex}al.\hspace*{0.65ex}}$ (2000)\nocite{hws2000}. The frequency space
is subdivided into five spectral bands with two opacity distribution
points in each band, resulting in altogether $5 \times 2$ effective
wavelengths sampling points (for the traditional ODF
approximation see e.\,g.\ Carbon 1979)\nocite{car79}. High and low mean opacity
values are tabulated for each spectral band during the initialisation
phase of the program in such a way that they simultaneously result in
the correct Planck {\it and} Rosseland band-mean gas opacities. The
details will be explained in another paper (Woitke 2006b).
\begin{figure}
\epsfig{file=kappa_dust.eps,width=8.8cm,height=8cm}
\caption{Extinction efficiencies over particle radius $Q_{\rm ext}/a$
in the Rayleigh-limit of Mie theory according to the Jena optical
data base. The data is partly log-log extrapolated.}
\vspace*{-1mm}
\label{fig:kappa_dust}
\end{figure}
Dust opacities are calculated in the Rayleigh limit of Mie theory
according to the Jena optical data base, kindly provided by Th.~Posch
(see Fig.~\ref{fig:kappa_dust}). The total dust extinction coefficient
$\rm[cm^2/g]$ is assumed to be given by\footnote{According to our
assumption of dirty grains, an application of effective medium theory
would be more appropriate, which will be examined in a future
paper. First results show that the effective extinction is stronger
than the simple volume-means used in this paper.}
\begin{equation}
\hat{\kappa}_{\lambda,\rm\,ext}^{\rm\;dust}
= \frac{3}{4}\,L_3 \sum_s \frac{V_s}{V_{\rm tot}} Q^s_{\rm ext}(a,\lambda)/a
\end{equation}
where $L_3$ is the third moment of the dust size distribution function
and $V_s/V_{\rm tot}$ is the volume fraction of solid material $s$ in the
dust component. The extinction efficiencies over particle radius of the pure
solids $s$ are shown in Fig.~\ref{fig:kappa_dust}. Note that most
oxygen-rich condensates have a ``glassy'' character. They are almost
transparent in the optical and near IR but opaque in the mid IR where
the strong vibrational resonances are situated. In contrast, solid Fe
and Fe-rich silicates are opaque even in the optical. The
monochromatic dust opacities are subject to the same averaging
procedure to result in high/low band-mean dust opacities as described
above for the gas opacities.
\smallskip
\noindent{\bf Dust formation} is described by a system of differential
moment equation explained in (Helling${\rm \hspace*{0.7ex}\&\hspace*{0.7ex}}$ Woitke 2006)\nocite{hw2006}
considering the growth and evaporation of inhomogeneous dust grains
composed of a mixture of Mg$_2$SiO$_4$, SiO$_2$, Al$_2$O$_3$, TiO$_2$,
and solid Fe (13 growth/evaporation reactions). The nucleation rate of
$\rm(TiO_2)_N$ clusters is adopted from Jeong
(2000)\nocite{jeo2000}. The molecular concentrations entering into the
calculation of the nucleation and growth rates are calculated by a
small neutral equilibrium chemistry for 11 atoms (H, He, C, O, N, Mg,
Al, Si, S, Ti, Fe) and 33 molecules (H2, CO, CO$_2$, OH, H$_2$O,
CH$_4$, N$_2$, CN, HCN, NH$_3$, H$_2$S, SiS, SO, HS, SiO, SiH,
SiH$_4$, SiO$_2$, SiN, SO$_2$, MgH, MgS, MgO, MgOH, Mg(OH)$_2$, FeO,
Fe(OH)$_2$, AlOH, AlO$_2$H, Al$_2$O, AlH, TiO, TiO$_2$).
\section{The static solution}
\label{sec:static}
\begin{figure}
\hspace*{-1mm}\epsfig{file=T2800_rstruc.eps,width=9.2cm}
\caption{Hydrostatic initial model (full) in
comparison to a spherical {\sc Marcs} model
(dashed) and a grey model (grey $T_{\rm g}$-line) for
$M_{\star}\!=\!1\,M_{\odot}$,
$T_{\star}\!=\!2800\,$K, $\log g\!=\!-0.6$
($L_{\star}\!=\!6048\,L_{\odot}$), $Z\!=\!1$. $S_{\rm{\!TiO2}}$
shows the supersaturation ratio of $\rm TiO_2$, indicating that
nucleation is already possible very close to the star.
Long-dashed graphs belong to the r.h.s.\ axis.}
\vspace*{-1mm}
\label{fig:Tstruc}
\end{figure}
The hydrostatic, dust-free solution (see Woitke 2006a for equations
and further details) used as initial model for the dynamical models is
shown in Fig.~\ref{fig:Tstruc}. The obtained degree of agreement with
the {\sc Marcs} model is quite remarkable for such a rough
10-wavelengths-point treatment of radiative transfer. Similar results
have been obtained by H{\"o}fner${\rm \hspace*{0.8ex}et\hspace*{0.7ex}al.\hspace*{0.65ex}}$(2003) and Helling${\rm \hspace*{0.7ex}\&\hspace*{0.7ex}}$
J{\o}rgensen (1998)\nocite{hj98}, using a sparse opacity sampling
technique with about 100 sampling points. The gas temperature
structure $T_{\rm g}(r)$ shows a pronounced ``step'' of almost 1000\,K
around $1.25\,R_\star$ (the {\sc Marcs} model is actually not extended
enough to reveal this step completely). It is clear, however, that
this is not just a ``surface effect''. Outside of this $T_{\rm g}$-step even
the strongest molecular lines become optically thin (in the
hydrostatic case) and the line blanketing effect works at full
strength. The grey model fails completely in predicting this step
which is very meaningful for the dust formation. The agreement with
the {\sc Marcs} results concerning the mass density $\rho(r)$, the gas
pressure $p(r)$, the mean molecular weight $\mu(r)$, and the
acceleration by radiation pressure on molecules divided by gravity
$\Gamma_{\rm gas}=a_{\rm rad}/g(r)$ is also satisfying.
\section{Rough estimates of the dust acceleration}
The calculation of the spectral mean intensities $J_{\lambda\,}(r)$ and
spectral fluxes $F_{\!\lambda\,}(r)$ in the static model (see
Fig.~\ref{fig:Tstruc}) allows for a quick estimate of the maximum
possible radiative acceleration by dust if the dust is still optically
thin. Table~\ref{tab:dust_estimates} shows the resulting dust
temperatures $T_{\rm d}$ of several pure condensates
\begin{equation}
\int\!\hat{\kappa}^{\rm\;dust}_{\lambda,\rm\,abs}
\,J_{\lambda\,}(r) \,d\lambda
\;= \int\!\hat{\kappa}^{\rm\;dust}_{\lambda,\rm\,abs}
\,B_\lambda(T_{\rm d}) \,d\lambda
\end{equation}
and the radiative acceleration by dust divided by gravity
\begin{equation}
\Gamma_{\rm dust}(r) \,=\,
\frac{\frac{1}{c} \int \hat{\kappa}^{\rm\;dust}_{\lambda,\rm\,ext}
\,F_{\!\lambda\,}(r) \,d\lambda}
{\frac{G M(r)}{r^2}}
\end{equation}
at three selected distances from the star. For each condensate
we take the maximum possible dust volume per mass
$L_{3,\rm\,max}$\linebreak\\*[-2.3ex]
given by element conservation constraints, e.\,g.\
$L_{3,\rm\,max}^{\rm Mg_2SiO_4}=$\linebreak\\*[-2.3ex]
${\rm Min}\big\{\frac{1}{2}\epsilon_{\rm Mg},\epsilon_{\rm Si},
\frac{1}{4}\epsilon_{\rm O}\big\}\,V_{\rm Mg_2SiO_4}\,n_{\langle{\rm H}\rangle}/\rho$
where $n_{\langle{\rm H}\rangle}$ is the hydrogen nuclei density, $\epsilon_k$ the abundance
of element $k$ and $V_{\rm Mg_2SiO_4}$ the monomer volume of Mg$_2$SiO$_4$
(see Helling${\rm \hspace*{0.7ex}\&\hspace*{0.7ex}}$ Woitke 2006)\nocite{hw2006}.
The results shown in Table~\ref{tab:dust_estimates} demonstrate that
the dust temperatures $T_{\rm d}$ are strongly material-dependent, with
differences as large as 1000\,K at the same distance from the star,
which is a remarkable result. All condensates (except solid Fe) have
strongly peaked mid-IR resonances which are situated just around the
maximum of the local Planck function -- they work perfect for
radiative cooling. In contrast, the glassy character of the oxides and
pure silicates like Al$_2$O$_3$, SiO$_2$, Mg$_2$SiO$_4$ and MgSiO$_3$
(the low absorption efficiencies at optical and near-IR wavelengths,
see Fig.~\ref{fig:kappa_dust}) prevent efficient heating by the
star. Consequently, the pure glassy condensates can exist
astonishingly close to the star (see also Woitke 1999).\nocite{woi99}
\begin{table}
\caption{Calculated dust temperatures $T_{\rm d}$ (first row) and dust radiative
accelerations $\Gamma_{\rm dust}=a^{\rm dust}_{\rm rad}/g$
(second row) in case of full condensation of several
condensates into small particles
in the static model (see Fig.~\ref{fig:Tstruc}). The resulting
dust-to-gas ratio $\rho_{\rm dust}/\rho_{\rm gas}$ is shown
in the middle column. Temperatures values with $^\star$ mark
thermally unstable condensates.}
\label{tab:dust_estimates}
\begin{tabular}{c|c|ccc}
solid material
& $\frac{\rho_{\rm dust}}{\rho_{\rm gas}}\,[10^{-3}]$
& $r=1.5\,R_\star$
& $r=2\,R_\star$
& $r=5\,R_\star$ \\
\hline
& & & & \\*[-1.7ex]
{\sf TiO$_2$} & 0.0061
& 1030\,K
& 750\,K
& 380\,K \\
& & 0.00004
& 0.00004
& 0.00005 \\
{\sf Al$_2$O$_3$} & 0.11
& 1090\,K
& 810\,K
& 420\,K \\
& & 0.0013
& 0.0014
& 0.0015 \\
{\sf SiO$_2$} & 1.6
& 1000\,K
& 740\,K
& 380\,K \\
& & 0.032
& 0.034
& 0.036 \\
{\sf Mg$_2$SiO$_4$} & 1.9
& 1150\,K
& 850\,K
& 430\,K \\
& & 0.022
& 0.024
& 0.025 \\
{\sf MgFeSiO$_4$} & 4.0
& 1930\,K$^\star$
& 1710\,K$^\star$
& {\bf 1170\,K} \\
& & {1.3}
& {1.4}
& {\bf 1.4} \\
{\sf MgSiO$_3$} & {2.3}
& {1010\,K}
& { 740\,K}
& { 380\,K} \\
& & {0.025}
& {0.027}
& {0.029} \\
{\sf Mg$_{\,0.5}$Fe$_{\,0.5}$SiO$_3$} & {3.0}
& {1880\,K}$^\star$
& {1580\,K}$^\star$
& {690\,K} \\
& & {0.21}
& {0.21}
& {0.18} \\
{\sf Fe} & {1.3}
& {1980\,K}$^\star$
& {1770\,K}$^\star$
& {\bf 1280\,K} \\
& & {0.85}
& {0.89}
& {\bf 0.88} \\
{\sf am.\,carbon} & {3.0}
& {1870\,K}$^\star$
& {\bf 1640\,K}
& {\bf 1130\,K} \\
(C/O\,=\,1.5) & & {20}
& {\bf 21}
& {\bf 21}
\end{tabular}
\end{table}
For the same reasons, radiative pressure on all glassy condensates is
negligible! It is without effect for the wind acceleration
mechanism whether for example Mg$_2$SiO$_4$ condenses out or not. The
only dust species that can potentially drive a stellar outflow are
solid Fe and Fe-rich silicates like MgFeSiO$_4$.
The unavoidable consequence of the spectral characteristics of oxygen-rich
dust is that radiative acceleration must be paid for by radiative
heating, i.\,e.\ dust species capable of driving a stellar wind
($\Gamma\!\ga\!1$) are hot and can only exist at a relatively large
distance from the star (e.\,g.\ $r\!\ga\!5\,R_\star$, marked in bold in
Table~\ref{tab:dust_estimates}). For comparison, amorphous carbon is so
opaque and stable (in a C-rich gas) that it could accelerate the gas
already from $2\,R_\star$ onwards, with $20\,\times$ the local gravity
in this model.
\section{Results of the dynamical models}
The first results of the dynamical models showed almost no mass loss
($\dot{M}\!\la\!10^{-10}\rm\,M_\odot/yr$), just some erratic
large-scale and long-term excursions for which the mass loss rate is
actually difficult to measure. We then approached rather extreme stellar
parameters ($M_{\star}\!=\!1\,M_{\odot}$, $T_{\star}\!=\!2500\,$K,
$L_{\star}\!=\!10000\,L_{\odot}$), still without success. Finally, in
order to see how a dust-driven wind could look like, we arbitrarily
enhanced the radiative acceleration by
\begin{equation}
\Gamma = \Gamma_{\rm gas} + 5\,\Gamma_{\rm dust} \ .
\end{equation}
The results of this simulation is shown in Fig.~\ref{fig:dynamical}.
\begin{figure}
\epsfig{file=dynamical.eps,width=9.1cm}
\caption{Dynamical wind model after 100 years of simulation time.
Parameter: $M_{\star}\!=\!1\,M_{\odot}$,
$T_{\star}\!=\!2500\,$K, $L_{\star}\!=\!10000\,L_{\odot}$
($\log g\!=\!-1.015$), $Z\!=\!1$, piston period $P\!=\!600$
days and velocity amplitude $\Delta{\rm
v}\!=\!2\,$km/s. The dotted curves show the hydrostatic
solution, and the $\Gamma=1$ level. $\Gamma$ is arbitrarily
enhanced (see text).}
\vspace*{-1mm}
\label{fig:dynamical}
\end{figure}
The models (also those without enhanced $\Gamma$) show extended warm
molecular layers, truncated by the $T_{\rm g}$-step described in
Sect.~\ref{sec:static} due to the line blanketing effect. The
pulsation of the star leads to a time-dependent extension of these
layers to roughly $1.5-2\,R_\star$. If dust forms, it fills in the gas
opacity gaps in frequency space which reduces the line blanketing
effect.
The formation of seed particles (see the nucleation rate $J_\star/n_{\langle{\rm H}\rangle}$ in
Fig.~\ref{fig:dynamical}) happens right above these molecular layers,
i.\,e.\ very close to the star. The gas is cold here
($T_{\rm g}\!\approx\!700-900\,{\rm K}$) which is not revealed by models
using grey radiative transfer (Jeong${\rm \hspace*{0.8ex}et\hspace*{0.7ex}al.\hspace*{0.65ex}}$ 2003, Ferrarotti${\rm \hspace*{0.7ex}\&\hspace*{0.7ex}}$ Gail 2006).
According to the model, two dust layers develop: almost pure glassy
{Al$_2$O$_3$} grains close to the star ($r\!\ga\!1.5\,R_\star$, even
partly inside the warm molecular layers up to
$T_{\rm g}\!\approx\!T_{\rm d}\!\approx\!1500\,K$), and further out the more opaque
{Mg-Fe-silicates} grains which grow on top of the Al$_2$O$_3$ particles
with a very small, steadily increasing Fe content.
The temperature of the dirty dust grains is controlled by the
iron content, which has already been noted by Tielens et al.\
(1998)\nocite{twmj98}. The volume fraction of solid iron inclusions relaxes
quickly to a level where a further increase would cause too much
radiative heating and thus thermal re-evaporation of the solid iron
inclusions
\begin{equation}
T_{\rm d} \approx T_S^{\rm Fe}(\rho) \ ,
\end{equation}
where $T_S^{\rm Fe}$ is the sublimation temperature of solid
iron. This is a robust and recurrent result concerning many
simulations, because there is a stable self-regulation mechanism: The
iron content adapts quickly to any changes in the ambient medium
(density, radiation field) unless the density gets too small out in
the wind (here at $r\!\ga\!5\,R_\star$) where the degree of
condensation of Fe freezes in on a relatively low final level of
$\sim\!17\%$. It is this factor that kills the mass loss (in
comparison Mg: $\sim\!65\%$, Al: $\sim\!100\%$). Noteworthy,
silicates in AGB star winds are in fact observed to be Fe-poor
(Bowey${\rm \hspace*{0.8ex}et\hspace*{0.7ex}al.\hspace*{0.65ex}}$ 2002)\nocite{bow2002}.
The radiative acceleration just even approaches the gravitational
deceleration around
$r\!\approx\!3.5-4\,R_\star$, which can be considered as an analog of
the sonic point in stationary winds. Determined by long-term averages
at the outer boundary, the mean mass loss rate, the mean outflow
velocity and the mean dust-to-gas ratio are
$\langle\dot{M}\rangle\!\approx\!2.3\times 10^{-9}\rm\,M_\odot/yr$,
$\langle v_\infty\rangle\!\approx\!2.6\,$km/s and $\langle \rho_{\rm
d}/\rho_{\rm g}\rangle\!\approx\!1.6\times 10^{-3}$, respectively (for
$\Gamma\!=\!\Gamma_{\rm gas}\!+\!5\,\Gamma_{\rm dust}$).
\section{Conclusions}
\begin{enumerate}
\item This letter reports on a {\it negative result}. Detailed
dynamical models with frequency-dependent Monte Carlo radiative
transfer and time-dependent formation of dirty dust grains
cannot explain the observed magnitude of mass loss rates from
oxygen-rich AGB stars, even in case of extreme stellar
parameters (i.\,e.\ high $L_\star/M_\star$ ratios).
\item The {\it role of solid iron and Fe-rich silicates} is crucial
for the wind driving mechanism. These condensates are the only
ones that are opaque around $1\,\mu$m and, thus, only these
condensates can efficiently absorb the stellar light. Since the
Fe containing condensates are not particularly stable, they form
at too large distances from the star in order to provide an efficient
mass loss mechanism.
\item Previous {\it grey models} (Jeong${\rm \hspace*{0.8ex}et\hspace*{0.7ex}al.\hspace*{0.65ex}}$ 2003, Ferrarotti${\rm \hspace*{0.7ex}\&\hspace*{0.7ex}}$
Gail 2006) have calculated the radiation pressure on dust with
Rosseland mean opacities which leads to a severe overestimation
in the O-rich case, because the mid-IR dust absorption
resonances are strongly peaked in a spectral region where the
stellar flux is low. The local flux $F_{\!\lambda}$ is not
$B_\lambda(T_{\rm d})$-like, because the dust features are optically thick
which makes the radiation field more isotrop and reduce
$F_\lambda$ just where $\kappa_{\lambda,\rm\,ext}^{\rm\;dust}$
is large.
\item The {\it dust condensation sequence} is strongly affected by
radiative transfer effects. Pure, glassy condensates like
Al$_2$O$_3$ have lower dust temperatures than solid Fe or
Fe-rich silicates. The differences are as large as
1000\,K, which favours the formation of the glassy condensates
and prevents the formation of Fe-inclusions close to the
star. The results in this paper are consistent with the
observational finding of Al$_2$O$_3$ at radial distances as
small as $1.5\,R_\star$ around $\alpha$\,Ori (Verhoelst${\rm \hspace*{0.8ex}et\hspace*{0.7ex}al.\hspace*{0.65ex}}$
2006) as well as with the observed dust condensation sequence in
O-rich AGB stars (Blommaert${\rm \hspace*{0.8ex}et\hspace*{0.7ex}al.\hspace*{0.65ex}}$ 2006, Lebzelter${\rm \hspace*{0.8ex}et\hspace*{0.7ex}al.\hspace*{0.65ex}}$ 2006),
because Al$_2$O$_3$ can already exist in an extended atmosphere
without mass loss, whereas the Mg-Fe-silicates form in the more
distant wind regions which require mass loss.
\item The {\it mass loss mechanism} of oxygen-rich AGB stars and red
supergiants is still a puzzle. Pulsations alone
cannot drive an outflow because the radiative cooling of the gas
is too efficient, even in non-LTE (Woitke 2003, Schirrmacher et al.\
2003)\nocite{woi2003,sws2003}. According to the results of this
letter, even a combination of stellar pulsation and radiation pressure
on dust is insufficient to drive the mass loss. Do we have to
re-visit Alfv{\'e}n-waves (e.\,g.\ Vidotto${\rm \hspace*{0.7ex}\&\hspace*{0.7ex}}$ Jatenco-Pereira 2006)?
\nocite{vp2006}
\end{enumerate}
\begin{acknowledgements}
This work is part of the {\sc AstroHydro3D} initiative supported by
the {\sc NWO Computational Physics programme}, grant
614.031.017. The computations have been done on the parallel
Xeon cluster {\sc Lisa} in Almere, the Netherlands, {\sc Sara} grant
MP-103. The software used in this work was in part developed by the
DOE-supported ASCI/Alliance Center for Astrophysical Thermonuclear
Flashes at the University of Chicago.
\end{acknowledgements}
\vspace*{-5mm}
|
1,116,691,497,555 | arxiv | \section{Introduction}
Tidal disruption events (TDEs) occur when stars in the center of a galaxy that orbit close to the supermassive black hole (SMBH) get close enough that the tidal forces acting on them exceed their own self gravity, causing the star to be disrupted. In this case a large fraction of the star's mass can be accreted onto the black hole producing a flare of electromagnetic radiation \citep[e.g.][]{rees88}.
TDEs provide uniquely powerful tools for determining black hole demographics and investigating super-Eddington accretion. TDE rates are generally skewed to lower mass black holes, since the tidal disruption radius is interior of the Schwarzschild radius for \mbh$>10^{8}$ \msol, and therefore TDEs provide a useful signpost of lower mass SMBHs. Furthermore, for \mbh$<10^{7}$ \msol, TDEs can emit above the Eddington luminosity \citep{strubbe09}, making them laboratories for extreme accretion.
Distinguishing TDEs from flares of more common accretion onto an SMBH can be challenging \citep{auchettl18}. One defining feature of TDEs is that their luminosities decline monotonically, often with a power-law profile approximately following $t^{-5/3}$, determined by the time in which the stellar debris gets accreted \citep{evans89}.
While TDEs are regularly being discovered by wide-field optical surveys such as the Zwicky Transient Facility \citep[ZTF, e.g.][]{vanvelzen19} and the All-Sky Automated Survey for Supernovae \citep[ASAS-SN, e.g.][]{holoien19}, TDEs discovered in the X-rays are currently comparatively rare, although \erosita\ is set to change this, and has already identified a handful of candidate events \citep[e.g.][]{khabibullin20}. In general, optical/UV events have cooler spectra ($10^4$ K) and X-ray events have hotter ones ($10^5$ K) \citep{komossa15}.
We have recently begun a program to search through public \swiftxrt\ observations for transient sources. The {\it Neil Gehrels Swift Observatory} \citep[hereafter \swift,][]{burrows05} observes several tens of targets every day, many of which are monitoring observations with cadences of a few days, well suited to finding transient sources. With a field of view of 560 arcmin$^2$, \swiftxrt\ provides a great potential for serendipitously discovering X-ray transients in the fields of view of other targets \citep[e.g.][]{soderberg09}. Furthermore, since most \swift\ data are downloaded from the satellite and made public within hours of the observation, this allows the opportunity to follow up promptly in real time with other observatories.
On 2020 February 5, we serendipitously detected an X-ray source in the field of view (FoV) of a \swiftxrt\ observation of SN~2020bvc, a broad-lined Type Ic supernova in the galaxy UGC09379 \citep{ho20}, where no previous X-ray source had been detected. The position of the X-ray source was RA=14h 33m 58.96s, Decl.=+40{$^\circ$}\ 06\arcsec\ 33.5\arcmin, with a positional uncertainty of 3.5\arcsec\ (90\% confidence). This is $\sim8$\arcmin\ from the supernova. The position of the X-ray source placed it in or near the galaxy \galaxy, different from SN~2020bvc. \galaxy\ has a spectroscopic redshift of $z=0.099$ (Section \ref{sec_keck}). Here we report on follow up and subsequent observations of the source which lead us to conclude that it was likely a X-ray TDE.
Throughout this paper we assume the cosmological parameters $H_{0}=70$~km\,s$^{-1}$\,Mpc$^{-1}$, $\Omega_{\rm m}=0.27$, and $\Omega_{\rm \Lambda}=0.73$. Under this assumed cosmology, the luminosity distance to \galaxy\ at $z=0.099$ is 456~Mpc. All uncertainties are quoted at the 90\% level unless otherwise stated.
\section{X-ray data analysis}
\subsection{Swift}
\label{sec_swift}
After the initial detection of the X-ray source, we requested follow up observations with \swift\ with both the XRT and Ultraviolet/Optical Telescope (UVOT) instruments, initially with a cadence of a few days, then a few times a month. In addition to the initial detection in the first XRT observation (obsID 00032818012), \swift\ has observed and detected the transient in the X-rays 27\ times, all in photon counting mode. Previous to this, \swift\ observed the position of the source 17 times, 12 times in 2013 and 5 times in 2016 where the source was not detected in X-rays. We analyze all \swift\ observations here.
In order to obtain an X-ray lightcurve of the source, we used the online tool provided by the University of Leicester\footnote{https://www.swift.ac.uk/user\_objects/} \citep{evans07,evans09}. All products from this tool are fully calibrated and corrected for effects such as pile-up and the bad columns on the CCD. The XRT lightcurve is shown in Figure \ref{fig_ltcrv}. For observations where a source has zero total counts, we estimate the 90\% upper limit on the count rate using a typical background count rate of 7$\times10^{-5}$ counts\,s$^{-1}$\ and Poisson statistics. At peak, the transient event was detected at a brightness two orders of magnitude greater than these upper limits.
We also used the online tool as described above to build a stacked spectrum of the source. Furthermore, for each individual observation, we extracted events of the source using the {\sc heasoft} v6.25 tool {\sc xselect} \citep{arnaud96}. Source events were selected from a circular region with a 25\arcsec\ radius centered on the above coordinates, and a background region consisting of a larger circle external to the source region was used to extract background events. For each source spectrum, we constructed the auxiliary response file (ARF) using {\tt xrtmkarf}. The relevant response matrix file (RMF) from the CALDB was used. All spectra were grouped with a minimum of 1 count per bin.
The stacked spectrum has a total exposure time of 55 ks, and the average count rate of the source is (3.04$\pm0.08)\times10^{-2}$ counts\,s$^{-1}$. We initially fitted the spectrum with an absorbed power-law model, {\tt tbabs*ztbabs*powerlaw} in {\sc xspec}, where the {\tt tbabs} model accounts for absorption in our Galaxy, fixed at 9.8$\times10^{19}$ \cmsq\ \citep{HI4PI16}, and {\tt ztbabs} accounts for absorption at the redshift of the source and is left as a free parameter. We find that \nh$=(7\pm3)\times10^{20}$ \cmsq, and the photon index $\Gamma=3.0\pm$0.2, where $C=252.93$ with 283 DoFs. We also tested a {\tt diskbb} model in place of the {\tt powerlaw} model, but it does not provide a good fit, where $C=489.66$ with 283 DoFs. However, the {\it addition} of a {diskbb} component to the {\tt tbabs*ztbabs*powerlaw} model does present a small improvement to the spectral fit, yielding \nh$=(1.1\pm0.1)\times10^{21}$ \cmsq, the temperature of the inner disk $kT=0.13^{+0.09}_{-0.03}$ keV, and photon index $\Gamma=2.8\pm$0.3, where $C=245.85$ with 283 DoFs.
Subsequently, we fitted the spectra from the individual obsIDs with the absorbed power-law model. We do not fit the more complicated model due to the low count nature of the individual spectra. Figure \ref{fig_gamma} shows the variation in $\Gamma$ over time, overplotted with binned averages (bins contain 5 observations each). There is no evidence of X-ray spectral evolution from the observations reported here.
The observed (absorbed) 0.3--10 keV flux as measured by XRT is 5$\times10^{-12}$ \mbox{\thinspace erg\thinspace cm$^{-2}$\thinspace s$^{-1}$}, which corresponds to a luminosity of 1$\times10^{44}$ \mbox{\thinspace erg\thinspace s$^{-1}$} at a distance of 456~Mpc. Assuming this model, the upper limit on the X-ray luminosity prior to the transient was $\sim10^{42}$ \mbox{\thinspace erg\thinspace s$^{-1}$}, corresponding to a $>2$ order-of-magnitude increase in the X-ray luminosity.
In addition to the XRT data, \swift\ also observed the source with its UVOT instrument, which has six filters, {\it UVW2} (central wavelength $\lambda=1928$ \AA), {\it UVM2} ($\lambda=2246$ \AA), {\it UVW1} ($\lambda=2600$ \AA), {\it U} ($\lambda=3465$ \AA), {\it B} ($\lambda=4392$ \AA), and {\it V} ($\lambda=5468$ \AA). In order to extract the photometry from the UVOT data, we used the tool {\tt uvotsource}, using circular regions with a 5\arcsec\ radius. Not every observation is taken with all six filters, however. We show the XRT and UVOT lightcurves in Figure \ref{fig_ltcrv}. While a UVOT source was clearly seen prior to the 2020 observations, likely emission from the host galaxy, a small increase in brightness measured by UVOT can be seen in the 2020 observations, though it is much weaker than seen in the X-rays. Also shown in Figure \ref{fig_ltcrv} are data from ZTF, which are described in Section \ref{sec_ztf}.
\begin{figure}
\begin{center}
\includegraphics[width=90mm]{all_ltcrv.pdf}
\caption{Long-term lightcurve of \galaxy, from all \swiftxrt, \swiftuvot, and ZTF observations. On 2020 February 5 a bright X-ray source was detected with a count rate 2 orders of magnitude greater than previous upper limits (shown by downward pointing arrows). The host galaxy was seen in the UVOT data prior to the transient, so only a small increase in brightness was measured, and it was much less than seen in the X-rays. The ZTF data are from difference imaging, hence the host galaxy has been subtracted, and show the transient was detected in the optical $\sim60$ days before \swift\ detected it in the X-rays.}
\label{fig_ltcrv}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=90mm]{gamma.pdf}
\caption{The power-law index, $\Gamma$, of the fit to the \swiftxrt\ data as a function of time (black data points). The solid red line shows the value $\Gamma$ from the integrated spectrum. Also shown are binned averages where bins contain 5 observations each (red data points).}
\label{fig_gamma}
\end{center}
\end{figure}
\subsection{NuSTAR}
\label{sec_nustar}
In order to study the hard X-ray emission from the transient, we obtained Director's Discretionary Time observation on the {\it Nuclear Spectroscopic Telescope Array} \citep[\nustar, obsID 90601606002,][]{harrison13}, which took place on 2020 February 13, 8 days after the X-ray transient was first detected by \swift. We used the {\sc heasoft} (v6.27) tool {\tt nuproducts} with default parameters to extract the \nustar\ spectrum. We used a circular region with a radius of 50\arcsec, centered on the peak of the emission to extract the source and a region with 100\arcsec\ radius to extract the background. The exposure time after filtering was 51.9~ks, from which the source was detected above background in each detector up to $\sim$15 keV, with a count rate of 0.01 counts\,s$^{-1}$\ in the 3--15 keV band.
\subsection{Chandra}
\label{sec_chandra}
On 2020 February 16 and 29, 11 and 24 days after the initial \swift\ detection respectively, \galaxy\ was also serendipitously observed by \chandra\ \citep[obsIDs 23171 and 23172,][]{weisskopf99} for 10\,ks each exposure with ACIS-S at the aimpoint. These observations also targeted SN~2020bvc \citep{ho20}. This allowed us to obtain a better position of the source than \swiftxrt\ provided, and a higher signal-to-noise spectrum.
In order to determine the position of the transient, we first ran the {\sc ciao} tool {\tt wavdetect} on the observations to obtain lists of positions for all sources in the \chandra\ FoV. Wavelet scales of 1, 2, 4, 8, and 16 pixels and a significance threshold of $10^{-5}$ were used. A total of 41 and 40 X-ray sources were detected in each observation, respectively.
We then cross-correlated the \chandra\ source lists with the {\it Gaia} DR2 catalog \citep{gaia18} to obtain the astrometric shifts. First we filtered to \gaia\ sources within 1\arcsec\ of the X-ray sources, excluding the transient itself, which left five \chandra/\gaia\ sources from both obsIDs. We define the astrometric shifts as the mean difference in RA and Dec between these matched sources. For obsID 23171, $\delta$RA$=-0.10\pm0.33$\arcsec\ and $\delta$Dec$=+0.55\pm0.28$\arcsec, and for obsID 23172, $\delta$RA$=+0.28\pm0.40$\arcsec\ and $\delta$Dec$=+0.01\pm0.38$\arcsec.
Having applied the astrometric shifts to the \chandra\ source catalog, the position of the X-ray source from obsID 23171 is R.A. = 14h 33m 59.170s, Decl.=+40{$^\circ$}\ 06\arcmin\ 36.18\arcsec\ (J2000), with an astrometric uncertainty of 0$\farcs$41 from the residual offsets with the {\it Gaia} catalog. From obsID 23172 the position is R.A. = 14h 33m 59.170s, Decl.=+40{$^\circ$}\ 06\arcmin\ 36.10\arcsec\ (J2000), with an astrometric uncertainty of 0$\farcs$37 from the residual offsets with the {\it Gaia} catalog. The \gaia\ position of the nucleus is R.A. = 14h 33m 59.170s, Decl.=+40{$^\circ$}\ 06\arcmin\ 36.05\arcsec\ (J2000). Figure \ref{fig_img} shows the PanSTARRS image of \galaxy, with the \gaia\ position of the nucleus shown with respect to the \chandra\ position of the X-ray source, which is coincident.
Also shown in Figure \ref{fig_img} is the position of ZTF19acymzwg, a candidate optical transient source detected in the $g$, $r$, and $i$ bands by the ZTF on 2019 December 14, 53 days prior to the detection of the X-ray transient by \swiftxrt. We describe the analysis of the ZTF data fully in Section \ref{sec_ztf}, including an updated position for the transient of RA=14h 33m 59.17s and Dec=+40{$^\circ$}\ 06\arcmin\ 36.1\arcsec\ with a 1$\sigma$ positional uncertainty of 0.29\arcsec. ZTF19acymzwg is likely related to the X-ray transient one since their positions consistent with each other within the uncertainties.
\begin{figure}
\begin{center}
\includegraphics[width=90mm]{panstarrs_image.pdf}
\caption{PanSTARRS $i$-band image of the galaxy SDSS J143359.16+400636.0, where the green circle shows the \gaia\ position of the nucleus. The position of the X-ray transient detected by \swiftxrt\ is shown with a blue circle where the radius represents the 3.5\arcsec\ uncertainty (90\% confidence), which does not clearly place the source in the galaxy. The more accurate position provided by \chandra\ obsID 23172 is shown with a magenta circle (1$\sigma$ confidence), and identifies the transient with the nucleus of the galaxy. The orange circle shows the position of the related ZTF transient (1$\sigma$ confidence).}
\label{fig_img}
\end{center}
\end{figure}
Due to the relatively high count rate and readout time of the ACIS detectors, we check for pileup of the source using the {\sc ciao} v4.11 tool {\sc pileup\_map}. We find that the pileup fraction is only $\sim2$ \% and therefore negligible. We then proceed to extract the spectrum of the source from both obsIDs, using the {\sc ciao} tool {\sc specextract}, and an elliptical region with a semi-major axis of 7.7\arcsec\ and a semi-minor axis of 4.4\arcsec. We used this shape and size due to the source being off axis where the PSF is larger and elongated. Background events were extracted from a nearby region. The source was detected in the $\sim$10 ks observations with a count rate of $1.45\pm0.03\times10^{-1}$ counts\,s$^{-1}$\ and $7.9\pm0.3\times10^{-2}$ counts\,s$^{-1}$\ respectively in the 0.5--8 keV band in the ACIS-S detector. There is clear evidence for a drop in flux over the 13-day period between \chandra\ observations. Intra-observational lightcurves of the \chandra\ observations were also extracted, binning on various time scales, though none of these showed significant count rate variability during the observations.
We jointly fitted the \nustar\ spectra with both \chandra\ spectra in {\sc xspec} using the C-statistic and a cross-calibration constant included to account for cross-calibration uncertainties and flux variability. The spectra are plotted in Figure \ref{fig_spec}, which shows that they are well described by a simple absorbed power-law ({\tt constant*zTbabs*powerlaw}) over the 0.5--15 keV range, with \nh$=(9\pm5)\times10^{20}$ \cmsq\ and $\Gamma=2.9\pm0.1$, where $C=847.66$ with 883 DoFs, consistent with the integrated \swift\ spectrum. The absorption measured is in excess of the Galactic value 9.8$\times10^{19}$ \cmsq\ and is therefore attributable to the host.
As with the integrated \swift\ spectrum in Section \ref{sec_swift}, we fitted other spectral models to the joint \chandra\ and \nustar\ spectra. Again a fit with a {\tt diskbb} model instead of a {\tt powerlaw} model does not fit the spectrum well, with $C=1371.92$ with 883 DoFs. The addition of a {\tt diskbb} model to the {\tt powerlaw} model does not produce an improvement to the fit, where $C=847.12$ with 881 DoFs, at odds with the \swift\ data. Since the \swift\ data were integrated over all exposures, which covered a larger and later time span than the Chandra data, it's possible that this {\tt diskbb} component emerged at later times. We checked this by creating a \swift\ spectrum which covered the same time period as the \chandra\ observations. As with the full \swift\ dataset, the addition of the {\tt diskbb} model to the {\tt powerlaw} model produces an improvement to the fit, arguing against the above hypothesis and leaving the \chandra\ and \swift\ data at odds with each other. Since the \chandra\ data have a higher number of counts with higher signal-to-noise than the \swift\ data, we defer to the \chandra\ results, concluding that there is no evidence for a {\tt diskbb} component in addition to the powerlaw one.
The cross-calibration constant for \nustar\ FPMA, $C_{\rm FPMA}$, is fixed to unity, while $C_{\rm FPMB}$ is fixed to 1.04 \citep{madsen15}. The constants for \chandra\ are $C_{\rm 23171}=1.36^{+0.18}_{-0.16}$ and $C_{\rm 23172}=0.91\pm0.05$. The 0.5--15 keV flux, as measured 2020 February 13, 8 days after the X-ray transient was first detected by \swift, is 4.0$\times10^{-12}$ \mbox{\thinspace erg\thinspace cm$^{-2}$\thinspace s$^{-1}$}, which corresponds to a luminosity of 9.8$\times10^{43}$ \mbox{\thinspace erg\thinspace s$^{-1}$}\ at a distance of 456~Mpc. These X-ray spectral fitting results are summarized in Table \ref{table_fitxray}.
\begin{figure}
\begin{center}
\includegraphics[width=90mm]{spectral_fig.pdf}
\caption{\chandra\ obsID 23171 (brown), obsID 23172 (magenta), \nustar\ FPMA (blue) and FPMB (cyan) spectra of the X-ray transient in \galaxy, taken 8--24 days after the \swiftxrt\ detection. The data are consistent with an absorbed power-law, with a constant to account for flux variability between data sets, plotted here as solid lines. The data have been binned for plotting clarity, where each bin has a minimum $5\sigma$ detection.}
\label{fig_spec}
\end{center}
\end{figure}
\begin{table}
\centering
\caption{\nustar\ and \chandra\ X-ray spectral fitting results.}
\label{table_fitxray}
\begin{center}
\begin{tabular}{l l l l l}
\hline
Parameter & Result \\
\hline
\nh & $(9\pm5)\times10^{20}$ \cmsq\ \\
$\Gamma$ & $2.9\pm0.1$ \\
Normalization & $(1.2\pm0.2)\times10^{-3}$ \\
\fx\ (0.5--15 keV) & $4.0^{+0.2}_{-0.4}\times10^{-12}$ \mbox{\thinspace erg\thinspace cm$^{-2}$\thinspace s$^{-1}$}\ \\
\lx\ (0.5--15 keV) & $9.8^{+0.2}_{-0.4}\times10^{43}$ \mbox{\thinspace erg\thinspace s$^{-1}$}\ \\
$C_{\rm FPMA}$ & 1.0 (fixed) \\
$C_{\rm FPMB}$ & 1.04 (fixed) \\
$C_{\rm 23171}$ & $1.36^{+0.18}_{-0.16}$ \\
$C_{\rm 23172}$ & $0.91\pm0.05$ \\
C-statistic & 847.66 \\
DoFs & 883 \\
\hline
\end{tabular}
\tablecomments{Results from the fit of an absorbed powerlaw to the \nustar\ and \chandra\ spectra of the X-ray transient in \galaxy\ as measured 2020 February 13, 8 days after the X-ray transient was first detected by \swift.}
\end{center}
\end{table}
\subsection{eROSITA}
\cite{khabibullin20} reported via The Astronomer's Telegram (\#13494) the detection by {\it Spectrum-Roentgen-Gamma} ({\it SRG})/eROSITA of a very bright X-ray source, SRGet J143359.25+400638.5, centered on \galaxy\ on 2019 December 27, 40 days prior to the detection of the transient with \swiftxrt. The reported 0.3--8 keV flux was $6.5\times10^{-12}$ \mbox{\thinspace erg\thinspace cm$^{-2}$\thinspace s$^{-1}$}, with no reported variability over the 11 individual scans with an interval of 4 hours. This reported flux is almost the same flux that \swiftxrt\ measured, suggesting that the X-ray flux of the source remained approximately constant for at least 40 days prior to the detection by \swiftxrt, before declining, or rose and fell, or vice versa. The X-ray spectrum was reported to be soft and described by a disk black body spectrum with a temperature of 0.29 keV. We simulate a spectrum with these model parameters and fit with a power-law model, which yields $\Gamma=2.9$, which is the same as measured by \swiftxrt, indicating that no spectral evolution took place between the \erosita\ detection and the \swiftxrt\ one. The authors suggested an association with ZTF19acymzwg which we confirm here.
\subsection{ROSAT}
{\galaxy\ was not detected by the {\it ROentgen SATellite} ({\it ROSAT}) in its all-sky survey performed in 1990 \citep[RASS,][]{voges99,boller16}. We calculate an X-ray flux upper limit for \galaxy\ from the {\it ROSAT} data using the {\tt SOSTA} (source statistics) tool available in the {\sc heasoft} {\tt XIMAGE} image processing package. The 3-$\sigma$ upper limit on the 0.1--2.4 keV count rate calculated using this method is 0.07 counts\,s$^{-1}$. Assuming the spectral shape measured by \nustar\ and \chandra\ above, this corresponds to a 0.3--10 keV flux of 1.1$\times10^{-12}$ \mbox{\thinspace erg\thinspace cm$^{-2}$\thinspace s$^{-1}$}, which is slightly less than the peak flux of the event. This limit is also similar to that of the XMM Slew Survey \citep{saxton12a}.}
\section{Zwicky Transient Facility}
\label{sec_ztf}
ZTF is an optical time-domain survey that uses the Palomar 48-inch Schmidt telescope with a 48 deg$^2$ field of view and scans more than 3750 deg$^2$ an hour to a depth of 20.5 mag \citep{bellm19,graham19,masci19}. As described in Section \ref{sec_chandra}, the candidate optical transient ZTF19acymzwg was detected in the $g$, $r$, and $i$ bands by ZTF on 2019 December 14, 53 days before the detection with \swiftxrt. Previous to this date, the field was observed on 2019 October 5 and the transient was not detected in any filter.
First, in order to determine the position of the transient we use \texttt{The Tractor} \citep{lang16} to forward model the host galaxy profile and the transient point source position. \texttt{The Tractor} forward models in pixel space by parametrizing a sky noise and point spread function model for each image and modeling this simultaneously with each source's shape, flux and position. We apply the modeling to 49 $g$, $r$ and $i$-band ZTF images with limiting magnitude $> 21.5$ taken from 2019 December 29 to 2020 March 28 when the transient is bright in these bands. We find that the galaxy is better modeled by a de Vaucoleur profile than an exponential profile and that the transient point source position is given by RA=14h 33m 59.17s and Dec=+40{$^\circ$}\ 06\arcmin\ 36.1\arcsec\ with a 3$\sigma$ positional uncertainty of 0.61\arcsec.
Once we obtained the position of the transient, we produced ZTF lightcurves using the ZTF forced-photometry service \citep{masci19} to produce difference-imaging photometry at the best-fit transient position across all ZTF images of the field taken between 2018 March 21 and 2020 May 11. We found no evidence for nuclear activity before the flare. The ZTF difference magnitudes are plotted in Figure \ref{fig_ltcrv}, along with the \swift\ lightcurve.
\section{Keck/LRIS optical spectroscopy}
\label{sec_keck}
We obtained an optical spectrum of the host galaxy nucleus with Keck/LRIS \citep{oke95} on 2020 February 18, 13 days after the initial \swift\ detection. The data were acquired using a standard long slit mode using a 1\arcsec\ slit on both the red and blue sides when the seeing was 1.01\arcsec\ in $i$ band. The spectra were reduced using standard long slit reduction procedures, including flat-fielding, wavelength calibration using arcs and flux calibration using a standard star as implemented in the \texttt{lpipe} package \citep{perley19}. The spectrum in shown in Figure \ref{fig_keck_spec}.
We proceeded to fit the Keck/LRIS spectrum in order to determine the velocity dispersion from the stellar absorption lines and the fluxes of the emission lines. We applied Penalized Pixel-Fitting \citep{cappellari04,cappellari17} to the spectrum which finds the velocity dispersion of stellar absorption lines using a large sample of high spectral resolution templates of single stellar populations adjusted to match the resolution of the input spectrum. We simultaneously fitted the narrow H$\alpha$, H$\beta$, H$\gamma$, H$\delta$, [S\,{\sc ii}] 6717, 6731, [N\,{\sc ii}] 6550, 6585, [O\,{\sc i}] 6302, 6366 and [O\,{\sc iii}] 5007, 4959 emission lines during template fitting. The emission line fluxes were each fit as free parameters but the line widths of the Balmer series were tied to each other, as were the line widths of the forbidden lines. We show the best fit model to the Keck/LRIS spectrum, including both the emission line component and the stellar continuum component, in Figure \ref{fig_keck_spec}. equivalent widths of the lines are presented in Table \ref{table_ews}. The The redshift of the galaxy was also determined to be 0.099.
\begin{figure}
\begin{center}
\includegraphics[width=90mm]{keck_spec_dan2.pdf}
\caption{Keck/LRIS spectrum of the nucleus of \galaxy\ (top) taken on 2020 February 18 (black), 13 days after the X-ray transient was detected by \swift. Key emission lines are labelled. The model fit to the spectrum is underplotted (red), consisting of a stellar continuum component (middle) and an emission line component (bottom).}
\label{fig_keck_spec}
\end{center}
\end{figure}
\begin{table}
\centering
\caption{Equivalent widths of the narrow lines observed in the Keck/LRIS spectrum.}
\label{table_ews}
\begin{center}
\begin{tabular}{l l l}
\hline
Line & EW (\AA) & EW error (\AA) \\
\hline
{\rm{H$\delta$}} & 0.832 & 0.257 \\
{\rm{H$\gamma$}} & 0.793 & 0.209 \\
{\rm{H$\beta$}} & 2.575 & 0.429 \\
{\rm{H$\alpha$}} & 8.627 & 0.365 \\
{\rm{[S\,\sc{ii}]}}6716 & 1.523 & 0.208 \\
{\rm{[S\,\sc{ii}]}}6731 & 1.125 & 0.213 \\
\oiii5007 & 6.009 & 0.408 \\
{\rm{[O\,\sc{i}]}}6300 & 0.363 & 0.230 \\
\nii6583 & 10.286 & 0.409 \\
\hline
\end{tabular}
\end{center}
\end{table}
The velocity dispersion of the stellar absorption lines was determined to be $213\pm12$ km\,s$^{-1}$. We used this to calculate the black hole mass from the \mbh-$\sigma_{*}$ relation, using the fit to the reverberation-mapped AGN sample from \cite{woo13}, and the following formula, log(\mbh/\msol) = $\alpha + \beta $log$(\sigma_{*}/200$ km\,s$^{-1}$), where $\alpha =7.31\pm0.15$ and $\beta = 3.46\pm0.61$. The intrinsic scatter of this relation is $\epsilon = 0.41\pm0.05$. This yielded log(\mbh/\msol)$=7.41\pm0.41$. Alternatively, if we use the quiescent galaxy + AGN sample from \cite{woo13}, where $\alpha =8.36\pm0.05$ and $\beta = 4.93\pm0.28$ with intrinsic scatter $\epsilon = 0.43\pm0.04$, the black hole mass estimate is log(\mbh/\msol)$=8.49\pm0.43$, an order of magnitude more massive.
We then plotted the emission line ratios {\rm{[O\,\sc{iii}]}}/{\rm{H$\beta$}}\ and {\rm{[N\,\sc{ii}]}}/{\rm{H$\alpha$}}\ in Figure \ref{fig_bpt}, along with the diagnostic lines from \cite{kewley01} and {\cite{kauffmann03} to determine the excitation mechanism of the narrow lines. The line ratios place the nucleus of \galaxy\ in the LINER region of this diagnostic diagram, almost at the border of the Seyfert region. The stellar absorption template fitting predicted strong {\rm{H$\beta$}}\ absorption, which is why we see a high {\rm{[O\,\sc{iii}]}}/{\rm{H$\beta$}}\ ratio in the initial spectrum. The lack of broad lines classifies the nucleus as a type 2 LINER. Also plotted on Figure \ref{fig_bpt} are the line ratios of nine optically- and radio-selected TDE hosts \citep{lawsmith17,french16,french17,mattila18,anderson19} along with SDSS galaxies for comparison. The TDE-host and galaxy emission line flux data have been taken from the SDSS DR7 MPI-JHU catalog\footnote{https://wwwmpa.mpa-garching.mpg.de/SDSS/DR7/}, where the stellar absorption-line spectra have also been subtracted before measurement \citep{kauffmann03,tremonti04}.
\begin{figure}
\begin{center}
\includegraphics[width=90mm]{BPT_Kunal_labels.pdf}
\caption{Emission line ratio diagnostic diagram showing where \galaxy\ (red star) lies with respect to the Seyfert, LINER and star-forming (HII) regions. The nucleus lies in the LINER region, indicating that AGN activity was present, at least at a low level, before the onset of the transient. For comparison, data from SDSS on optically- and radio-selected TDE hosts are shown as blue circles and labelled, and other galaxies are shown in gray, where the stellar absorption-line spectrum has been subtracted.}
\label{fig_bpt}
\end{center}
\end{figure}
Since the narrow lines are produced in the narrow line region, which can be kiloparsecs from the SMBH \citep[e.g.][]{chen19}, this tells us that \galaxy\ had low-level AGN activity some time before the onset of the X-ray transient. To determine the spatial extent of the narrow line region in \galaxy, we analyzed the 2-dimensional Keck/LRIS spectrum. This shows the galaxy emission had a spatial extent of $\sim$5.2\arcsec. We extracted a spectrum from each edge of the galaxy which were separated by a 2.4\arcsec\ gap and each extraction region had a width of 1.4\arcsec. In both edge spectra, we located narrow line emission from the {\rm{[O\,\sc{iii}]}}\ 5007\AA, 4959\AA\ doublet. This suggests that the narrow line emission has a spatial extent of $\sim2.4$\arcsec. Given the scale of 1.831 kpc/\arcsec\ at this redshift under our assumed cosmology, this implies the narrow lines were produced at a projected distance of 4.4~kpc, and that they were illuminated at least 10,000 years prior to the transient.
The flux of the {\rm{[O\,\sc{iii}]}}\ line is $3.78\pm0.15\times10^{-16}$ \mbox{\thinspace erg\thinspace cm$^{-2}$\thinspace s$^{-1}$}. From an investigation of the relationship between X-ray and optical line emission in 340 \swift/BAT-selected AGN \citep{berney15}, the {\rm{[O\,\sc{iii}]}}\ flux expected from the 2--10 keV flux of $10^{-12}$ \mbox{\thinspace erg\thinspace cm$^{-2}$\thinspace s$^{-1}$}, the peak X-ray flux measured by \swiftxrt, is in the range of $10^{-15}$--$10^{-13}$ \mbox{\thinspace erg\thinspace cm$^{-2}$\thinspace s$^{-1}$}, higher than what we measure. The lower than expected {\rm{[O\,\sc{iii}]}}\ flux we measure indicates that the AGN was at a low luminosity prior to the transient. This is also consistent with the upper limits on the X-ray luminosity of the nucleus prior to the transient, which at $\sim10^{42}$ \mbox{\thinspace erg\thinspace s$^{-1}$}, is relatively low for an AGN.
\section{Karl G. Jansky Very Large Array}
\label{sec_vla}
We carried out radio observations with the Karl G. Jansky Very Large Array (VLA) through Director's Discretionary Time (project code VLA/20A-579, PI: Mooley) on 2020 August 2, 180 days after the detection by \swift. Data were obtained at C band in the 3-bit mode of the WIDAR correlator to get a contiguous frequency coverage between 4--8 GHz. Standard VLA calibrator sources 3C286 and J1416+3444 were used to calibrate the flux/bandpass and phases respectively. The data were processed using the NRAO CASA pipeline and imaged using the {\tt clean} task in CASA.
We did not detect any radio source at the location of the transient, and place a 3$\sigma$ upper limit of 28~$\mu$Jy on the 6 GHz flux density. We can therefore place an upper limit of $4 \times 10^{37}$\,erg\,s$^{-1}$ on the radio luminosity at a distance of 456~Mpc. The closest X-ray observation in time to the VLA one was by \swiftxrt\ on 2020 July 27 (obsID 00013265017), where we measured a 0.3--10 keV luminosity of $3.8\times10^{42}$ \mbox{\thinspace erg\thinspace s$^{-1}$}. The X-ray-to-radio luminosity ratio is therefore $>10^5$. Comparing our radio upper limit with the radio emission seen in jetted TDEs \citep[e.g.][]{alexander2020}, we can rule out the presence of a relativistic jet.
\section{Lightcurve fitting}
\label{sec_ltcrv}
After the initial detection by \swift, the lightcurve of the transient appeared to monotonically decline in flux, shown by \swift, \nustar, \chandra, and ZTF. In order to infer more details regarding the nature of the source, we fitted the lightcurve of the source in each band with a power-law model, $F=A(t-t_{\rm 0})^n+C$, where $F$ is the observed flux density of the source, $A$ is a normalization constant, $t$ is the time in days since the transient was first detected by \swift\ (2020 February 5), $t_{\rm 0}$ is the inferred start time of the event in days, and $n$ is the power-law index. $C$ is a constant which represents the underlying emission from the galaxy in UVOT data only, and set to zero for the XRT data since no X-ray emission is seen from the galaxy, and set to zero for the ZTF data since the galaxy has already been subtracted in these data. We determine the underlying emission from the galaxy in the UVOT data by averaging over the photometry measured previous to the detection of the transient.
We calculate the 2 keV monochromatic flux density as measured from the power-law model fit to the \nustar\ and \chandra\ data with the \nh\ and $\Gamma$ parameters fixed to their best-fit values. We use the UVOT flux densities as produced by {\tt uvotsource}, and the ZTF difference imaging fluxes. We present these measurements in Tables \ref{table_fluxes} and \ref{table_ztf}.
\begin{table*}
\centering
\caption{\swift/XRT and UVOT fluxes.}
\label{table_fluxes}
\begin{center}
\begin{tabular}{c c c c c c c c c}
\hline
Time & obsID & 2 keV & UVW2 & UVM2 & UVW1 & U & B & V\\
(Days) & & ($\mu Jy$) & ($mJy$) & ($mJy$) & ($mJy$) & ($mJy$) & ($mJy$) & ($mJy$)\\
\hline
0&00032818012 & 0.260$\pm$ 0.084& 0.023$\pm$ 0.004& 0.024$\pm$ 0.004& 0.023$\pm$ 0.006& 0.040$\pm$ 0.016& 0.065$\pm$ 0.031& 0.004$\pm$ 0.063\\
1&00032818014 & 0.261$\pm$ 0.098& 0.021$\pm$ 0.004& 0.017$\pm$ 0.004& 0.026$\pm$ 0.008& 0.012$\pm$ 0.014& 0.082$\pm$ 0.033& -\\
2&00032818015 & 0.340$\pm$ 0.098& 0.022$\pm$ 0.004& 0.026$\pm$ 0.005& 0.016$\pm$ 0.006& 0.031$\pm$ 0.016& 0.015$\pm$ 0.030& 0.029$\pm$ 0.068\\
3&00032818016 & 0.208$\pm$ 0.084& 0.020$\pm$ 0.005& 0.020$\pm$ 0.005& 0.022$\pm$ 0.007& 0.034$\pm$ 0.017& -& -\\
4&00032818017 & 0.145$\pm$ 0.064& 0.021$\pm$ 0.004& 0.019$\pm$ 0.004& 0.030$\pm$ 0.007& 0.045$\pm$ 0.017& 0.045$\pm$ 0.031& 0.070$\pm$ 0.071\\
7&00032818018 & 0.333$\pm$ 0.068& 0.017$\pm$ 0.003& 0.024$\pm$ 0.004& 0.014$\pm$ 0.004& 0.026$\pm$ 0.011& 0.030$\pm$ 0.022& 0.034$\pm$ 0.060\\
9&00089025001 & 0.182$\pm$ 0.042& 0.017$\pm$ 0.001& -& -& -& -& -\\
9&00032818019 & 0.233$\pm$ 0.066& -& -& -& -& -& -\\
11&00032818020 & 0.187$\pm$ 0.055& 0.015$\pm$ 0.004& 0.017$\pm$ 0.005& 0.027$\pm$ 0.008& 0.052$\pm$ 0.019& 0.027$\pm$ 0.032& 0.087$\pm$ 0.080\\
18&00013265001 & 0.208$\pm$ 0.057& 0.019$\pm$ 0.003& 0.025$\pm$ 0.005& 0.025$\pm$ 0.005& 0.011$\pm$ 0.008& 0.029$\pm$ 0.017& 0.058$\pm$ 0.042\\
24&00013265002 & 0.215$\pm$ 0.072& 0.022$\pm$ 0.004& 0.004$\pm$ 0.003& 0.015$\pm$ 0.005& 0.027$\pm$ 0.010& 0.011$\pm$ 0.021& -\\
31&00013265003 & 0.159$\pm$ 0.056& 0.009$\pm$ 0.002& 0.002$\pm$ 0.003& 0.009$\pm$ 0.005& 0.020$\pm$ 0.009& 0.022$\pm$ 0.020& 0.084$\pm$ 0.049\\
36&00013265004 & 0.163$\pm$ 0.077& 0.015$\pm$ 0.004& -& 0.017$\pm$ 0.005& 0.014$\pm$ 0.009& 0.034$\pm$ 0.020& -\\
42&00013265005 & 0.055$\pm$ 0.028& 0.014$\pm$ 0.002& 0.010$\pm$ 0.003& 0.019$\pm$ 0.005& 0.030$\pm$ 0.009& 0.009$\pm$ 0.016& -\\
48&00013265006 & 0.049$\pm$ 0.037& 0.009$\pm$ 0.003& 0.008$\pm$ 0.005& 0.006$\pm$ 0.005& 0.003$\pm$ 0.010& -& 0.022$\pm$ 0.053\\
55&00013265007 & 0.067$\pm$ 0.027& 0.012$\pm$ 0.003& 0.006$\pm$ 0.003& 0.010$\pm$ 0.004& 0.020$\pm$ 0.009& 0.025$\pm$ 0.019& -\\
62&00013265008 & 0.061$\pm$ 0.048& 0.010$\pm$ 0.003& 0.008$\pm$ 0.005& 0.008$\pm$ 0.005& 0.016$\pm$ 0.011& 0.009$\pm$ 0.022& 0.026$\pm$ 0.058\\
66&00013265009 & 0.038$\pm$ 0.021& 0.014$\pm$ 0.004& 0.009$\pm$ 0.004& 0.010$\pm$ 0.006& 0.015$\pm$ 0.012& -& -\\
72&00013265010 & 0.045$\pm$ 0.028& 0.006$\pm$ 0.002& 0.004$\pm$ 0.003& 0.013$\pm$ 0.004& 0.017$\pm$ 0.008& 0.010$\pm$ 0.021& 0.002$\pm$ 0.039\\
101&00013265011 & 0.011$\pm$ 0.026& 0.007$\pm$ 0.002& -& 0.009$\pm$ 0.005& 0.032$\pm$ 0.011& -& 0.009$\pm$ 0.045\\
114&00013265012 & 0.024$\pm$ 0.038& 0.012$\pm$ 0.003& 0.010$\pm$ 0.004& 0.008$\pm$ 0.005& 0.017$\pm$ 0.010& 0.002$\pm$ 0.018& 0.001$\pm$ 0.044\\
119&00013265013 & 0.018$\pm$ 0.018& 0.007$\pm$ 0.002& 0.016$\pm$ 0.004& 0.009$\pm$ 0.005& 0.007$\pm$ 0.009& -& -\\
128&00013265014 & 0.029$\pm$ 0.020& 0.008$\pm$ 0.002& 0.004$\pm$ 0.003& 0.005$\pm$ 0.007& -& 0.023$\pm$ 0.023& -\\
142&00013265015 & 0.001$\pm$ 0.000& 0.011$\pm$ 0.002& -& 0.010$\pm$ 0.006& 0.014$\pm$ 0.007& -& 0.025$\pm$ 0.033\\
156&00013265016 & 0.016$\pm$ 0.013& -& 0.003$\pm$ 0.003& 0.012$\pm$ 0.005& 0.028$\pm$ 0.010& 0.019$\pm$ 0.019& -\\
172&00013265017 & 0.007$\pm$ 0.012& 0.008$\pm$ 0.001& -& -& -& -& -\\
200&00013265019 & 0.002$\pm$ 0.009& 0.005$\pm$ 0.001& -& -& -& -& -\\
214&00013265020 & 0.022$\pm$ 0.022& 0.004$\pm$ 0.003& -& -& -& -& -\\
228&00013265021 & 0.001$\pm$ 0.004& 0.005$\pm$ 0.001& -& -& -& -& -\\
\hline
\end{tabular}
\tablecomments{Time is in days since 2020 February 5, the date on which the transient was first detected by \swiftxrt.}
\end{center}
\end{table*}
\begin{table}
\centering
\caption{ZTF fluxes.}
\label{table_ztf}
\begin{center}
\begin{tabular}{c c c c }
\hline
Time & g & r & i\\
(Days) & ($mJy$) & ($mJy$) & ($mJy$) \\
\hline
-53& 0.060$\pm$ 0.012& -& 0.046$\pm$ 0.006\\
-50& -& 0.053$\pm$ 0.010& -\\
-48& -& -& 0.050$\pm$ 0.004\\
-38& 0.033$\pm$ 0.002& 0.035$\pm$ 0.004& -\\
-35& -& -& 0.039$\pm$ 0.006\\
-30& 0.036$\pm$ 0.007& -& 0.045$\pm$ 0.008\\
-29& 0.031$\pm$ 0.007& 0.023$\pm$ 0.004& -\\
-28& -& 0.022$\pm$ 0.003& -\\
-24& -& -& 0.024$\pm$ 0.008\\
-23& 0.024$\pm$ 0.007& -& -\\
-22& 0.023$\pm$ 0.007& -& 0.027$\pm$ 0.004\\
-21& 0.023$\pm$ 0.005& 0.018$\pm$ 0.004& 0.025$\pm$ 0.003\\
-20& -& -& 0.026$\pm$ 0.003\\
-18& 0.019$\pm$ 0.003& 0.014$\pm$ 0.004& -\\
-14& 0.013$\pm$ 0.003& -& -\\
-13& 0.014$\pm$ 0.003& 0.014$\pm$ 0.004& 0.017$\pm$ 0.005\\
-12& -& 0.018$\pm$ 0.003& 0.028$\pm$ 0.004\\
-9& -& -& 0.020$\pm$ 0.003\\
-8& 0.012$\pm$ 0.002& 0.009$\pm$ 0.002& 0.018$\pm$ 0.003\\
-7& 0.010$\pm$ 0.003& 0.009$\pm$ 0.003& 0.017$\pm$ 0.004\\
-6& 0.015$\pm$ 0.003& 0.015$\pm$ 0.004& 0.017$\pm$ 0.005\\
-5& 0.010$\pm$ 0.002& -& -\\
-4& -& 0.008$\pm$ 0.003& 0.016$\pm$ 0.003\\
-2& -& -& 0.023$\pm$ 0.005\\
-1& 0.021$\pm$ 0.005& 0.022$\pm$ 0.004& 0.016$\pm$ 0.005\\
0& 0.016$\pm$ 0.005& -& -\\
1& 0.016$\pm$ 0.005& 0.010$\pm$ 0.003& 0.014$\pm$ 0.004\\
2& 0.026$\pm$ 0.006& 0.012$\pm$ 0.004& -\\
3& -& 0.015$\pm$ 0.005& 0.022$\pm$ 0.006\\
7& -& -& 0.014$\pm$ 0.003\\
8& -& 0.012$\pm$ 0.004& -\\
9& -& -& 0.012$\pm$ 0.003\\
10& -& 0.012$\pm$ 0.004& -\\
11& -& -& 0.013$\pm$ 0.003\\
19& -& -& 0.011$\pm$ 0.003\\
23& -& -& 0.013$\pm$ 0.003\\
27& -& -& 0.011$\pm$ 0.003\\
78& 0.006$\pm$ 0.002& -& -\\
79& -& 0.010$\pm$ 0.003& -\\
80& 0.008$\pm$ 0.002& -& -\\
\hline
\end{tabular}
\tablecomments{Time is in days since 2020 February 5, the date on which the transient was first detected by \swiftxrt.}
\end{center}
\end{table}
We show a fit to the \swiftxrt, \swiftuvot, and ZTF lightcurves in Figure \ref{fig_fit_swift_ltcrv}, and Figure \ref{con_fit_swift_ltcrv} shows the \chisq\ contours of $t_{\rm0}$ vs. $n$. We find that in X-rays, to 1$\sigma$, the power-law index is consistent with $-1.1>n>-1.9$, with a best fit of $n=-1.7$. In the UV and optical bands, the data are not as constraining and are consistent with the X-ray with e.g. $-1.1>n>-2.2$ for {\it UVW2}. There are indications that the transient in the {\it UVW2} band started prior to the X-rays, where $-45<t_{\rm0}<-5$ for X-rays and $-100<t_{\rm0}<-20$ for {\it UVW2}, although their 1$\sigma$ confidence intervals are overlapping. While the $t_{\rm0}$ constraints from the X-rays are consistent with the \erosita\ detection at $t=-40$ days, the \erosita\ flux measurement is clearly not consistent with the fit to the \swift\ lightcurve, as seen in Figure \ref{fig_fit_swift_ltcrv}.
For the ZTF data, we find that the power-law index is consistent with $-2.0<n<-1.2$, and therefore with the X-ray and UV constraints, but $-100<t_{\rm0}<-70$, which is consistent with the UV constraints, but not the X-ray ones. The transient was first detected by ZTF at $t=-53$, but could have started as early as $t=-123$ due to an observing gap. The average best fit of $t_{\rm0}$ in the $g$ and $r$ bands is $-70$ days. If the start time of the optical transient was $t=-53$, then it would be marginally consistent with the X-ray constraints for $t_{\rm0}$, but in conclusion, we do not have good constraints on when the transient started, neither in X-ray nor in the optical/UV. We summarize the lightcurve fitting results in Table \ref{tab_fitltcrv}.
\begin{table}
\centering
\caption{Lightcurve fitting results}
\label{tab_fitltcrv}
\begin{center}
\begin{tabular}{l l l l l}
\hline
Band & $n$ & $t_{\rm 0}$ (days) \\
\hline
X-ray (2 keV) & $-1.7_{-0.1}^{+0.2}$ & $-30^{+10}_{-5}$ \\
UV (UVW2) & $-1.9_{-0.1}^{+0.5}$ & $-80^{+40}_{-20}$ \\
Optical (i) & $-1.6_{-0.1}^{+0.2}$ & $<-$90 \\
\hline
\end{tabular}
\tablecomments{Results from the fit of a powerlaw decline model to the X-ray, UV, and optical lightcurves of the transient in \galaxy.}
\end{center}
\end{table}
We then assume that the optical, UV and X-ray transients had the same start time. We do this by fixing $t_{\rm0}$ to $-70$ days in all our lightcurve fits which is the best constraint from ZTF. This best-fit is shown as a dashed line in Figure \ref{fig_fit_swift_ltcrv} which shows it as fitting the UVOT data well. In the X-rays, it under-predicts the XRT data between 0--50 days, with a flatter power-law index, $n=-1.5$. Interestingly, this model matches the eROSITA flux better.
\begin{figure}
\begin{center}
\includegraphics[width=80mm]{fit_all_ltcrv.pdf}
\caption{\swiftxrt\ (2 keV), \swiftuvot\ ({\it UVW2}, {\it UVM2}, {\it UVW1}, {\it U}, {\it B}, and {\it V}), and ZTF ($g$, $r$, and $i$) lightcurves of the X-ray transient in \galaxy. Solid black lines represent fits to the data with a power-law model where all fit parameters are free to vary. Dashed black lines represent fits where the start time of the transient has been fixed to $-70$ days. Open squares in the X-ray lightcurves are the data points from \erosita, \nustar, and \chandra\ which were not used to fit the lightcurve. Black dotted lines show the quiescent flux from the galaxy in the \swiftuvot\ filters before detection of the X-ray transient.}
\label{fig_fit_swift_ltcrv}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=90mm]{con_fit_all_ltcrv.pdf}
\caption{1, 2, and 3$\sigma$ \chisq\ contours of the fits to the \swiftxrt, \swiftuvot\ and ZTF lightcurves. Crosses mark the \chisq\ minimum. The X-ray contours, plotted with black lines, are over-plotted on the optical/UV ones for comparison.}
\label{con_fit_swift_ltcrv}
\end{center}
\end{figure}
\section{SED fitting}
\label{sec_sed}
The \swiftuvot\ and ZTF data in combination with the \chandra\ and \nustar\ spectra allow us to construct a broadband SED of the source. Since the \swiftuvot\ data include emission from the host galaxy, we used the photometry inferred by the model fitting described above in Section \ref{sec_ltcrv}. This naturally accounts for the host galaxy emission underlying the source which is assumed to be constant. The photometric errors were calculated by fixing all model parameters with the exception of the normalization. The SED is shown in Figure \ref{fig_fit_sed}.
\begin{figure*}
\begin{center}
\includegraphics[width=180mm]{fit_sed_nufnu.pdf}
\caption{The SED of the transient in \galaxy\ (blue data points), 8 days after it was detected by \swift, showing data from VLA, ZTF, \swiftuvot, \chandra, and \nustar. The VLA data are from 172 days after the optical--X-ray data. Upper limits are shown with downward pointing arrows. The best-fit deredened and unabsorbed disk blackbody ({\tt diskbb}, shown with a dashed line) plus powerlaw model is shown as a solid black line. }
\label{fig_fit_sed}
\end{center}
\end{figure*}
In order to fit the broadband SED, we converted the UVOT and ZTF fluxes into a PHA (pulse height amplitude) file using the tool {\sc ftflx2xsp} so that it can be loaded into {\sc xspec}. We used the time of the \nustar\ observation to calculate the UVOT photometry and take the closest ZTF data. Using {\sc xspec} and the \chisq\ statistic for spectral fitting, we find that the ZTF and \swiftuvot\ data alone can be well described by a powerlaw model, $F_{\gamma}=AE^{-\Gamma}$ where $F_{\gamma}$ is the photon flux in units of photons\,cm$^{-2}$\,s$^{-1}$\,keV$^{-1}$, $A$ is a normalization constant, $E$ is the photon energy in keV, and $\Gamma$ is the powerlaw index. For this model, we find $\Gamma=1.01^{+0.41}_{-0.56}$, where \chisq=2.42 with 6 degrees of freedom. The 0.002--0.01 keV flux is $2.2\times10^{-13}$ \mbox{\thinspace erg\thinspace cm$^{-2}$\thinspace s$^{-1}$}, which corresponds to a luminosity of $5.1\times10^{42}$ \mbox{\thinspace erg\thinspace s$^{-1}$}\ at a distance of 456~Mpc, and is a factor of $\sim20$ lower than the 0.5--15 keV luminosity measured at the same time (Section \ref{sec_nustar}).
The $\Gamma=1.0$ observed in the UVOT data is much flatter than the $\Gamma=3.0$ observed in the X-ray band. For fitting the full SED, we use the \chisq\ statistic and therefore grouped the X-ray data with a minimum of 20 counts per bin. Furthermore, as with the X-ray only data, we include a Galactic absorption component, fixed at 9.8$\times10^{19}$ \cmsq\ \citep{HI4PI16}, and an intrinsic absorption component, {\tt ztbabs} accounts for absorption at the redshift of the source and is left as a free parameter. The simplest model to fit the full SED is a broken powerlaw model where the break occurs at 1 keV, which yields a good fit where \chisq=114.62 with 113 DoFs.
We then tried fitting a more physically motivated models, specifically a standard accretion disk model, {\tt diskbb} in {\sc xspec} \citep[e.g.][]{mitsuda84,makishima86}. However this model does not produce a good fit, where \chisq/DoF=2093.68/120, fitting the ZTF and \swiftuvot\ data well, but severely under-predicting the X-ray data.
We then introduced a scattered powerlaw in addition to the {\tt diskbb} model, using the {\tt simpl} model \citep{steiner09}. The {\tt simpl} model is an empirical convolution model of Comptonization in which a fraction of the photons in an input seed spectrum, in this case the disk black body model, is up-scattered into a power-law component. In {\sc xspec} this is written as {\tt simpl*diskbb}. This model accounts for the excess X-rays well, which significantly improves the model fit to \chisq/DoF=129.40/118. The best fit parameters of the disk model are an inner disk temperature of $T_{\rm in}=0.063^{+0.003}_{-0.007}$ keV ($7.3^{+0.3}_{-0.8}\times10^{5}$ K) and a normalization of $N=3.7^{+1.4}_{-1.0}\times10^4$. The parameters of the scattered powerlaw are $\Gamma=3.2\pm0.1$ with a scattered fraction, $f_{\rm scatt}>0.35$ (unconstrained at the upper end). The absorption intrinsic to the source is measured to be \nh$=(1.6\pm0.7)\times10^{21}$ \cmsq}. We summarize the SED fitting results in Table \ref{tab_fitsed}.
\begin{table}
\centering
\caption{SED fitting results}
\label{tab_fitsed}
\begin{center}
\begin{tabular}{l l l l l}
\hline
Parameter & Result \\
\hline
\nh & $(1.6\pm0.7)\times10^{21}$ \cmsq\ \\
$T_{\rm in}$ & $0.063^{+0.003}_{-0.007}$ keV \\
Normalization & $3.7^{+1.4}_{-1.0}\times10^4$ \\
$\Gamma$ & $3.2\pm0.1$ \\
$f_{\rm scatt}$ & $>0.35$ \\
Flux (bolometric) & ($2.3\pm0.3)\times10^{-11}$ \mbox{\thinspace erg\thinspace cm$^{-2}$\thinspace s$^{-1}$}\ \\
Luminosity (bolometric) & $(5.7\pm0.1)\times10^{44}$ \mbox{\thinspace erg\thinspace s$^{-1}$}\ \\
\chisq\ & 129.40 \\
DoFs & 118 \\
\hline
\end{tabular}
\tablecomments{Results from the fit of a disk black body plus scattered powerlaw model to the ZTF, \swiftuvot, \nustar\ and \chandra\ data on the transient in \galaxy\ as measured 2020 February 13, 8 days after the X-ray transient was first detected by \swift.}
\end{center}
\end{table}
The normalization of the {\tt diskbb} model described above is related to the apparent inner disk radius, where $N=(R_{\rm in}/D_{\rm 10})^2 cos\theta$. $N$ is the normalization measured, $D_{\rm 10}$ is the distance to the source in units of 10 kpc, and $\theta$ the angle of the disk ($\theta= 0$ is face-on). Assuming a face-on disk and a distance of 456 Mpc to the source, the normalization of $3.7^{+1.4}_{-1.0}\times10^4$ measured corresponds to an inner disk radius of $8.3\times10^{9}$ km, or $3.8\times10^{10}$ km for an edge-on disk ($\theta=$ 87{$^\circ$}). The gravitational radius of a black hole with a mass of log(\mbh/\msol)$=7.41\pm0.41$ is $3.8\times10^{10}$ km. Therefore, the implied inner disk radius is consistent with the gravitational radius of the SMBH, given the uncertainty in the mass and the unknown disk inclination.
In addition to ruling out a relativistic jet from this source from the non-detection of radio emission, models of synchrotron emission, such as {\tt srcut} and {\tt sresc} in {\sc xspec} can reproduce the ZTF and \swiftuvot\ data, but have too much curvature in the X-ray band to fit the overall SED well. A Bremsstrahlung model, such as {\tt bremss}, also does not fit the spectrum well, being too steep for the ZTF and \swiftuvot\ data and with too much curvature in the X-ray band. We therefore adopt the {\tt simpl*diskbb} as our best-fit model.
In order to calculate the bolometric luminosity of the event, we integrated the flux of the unabsorbed/deredened disk blackbody plus scattered power-law model over the 0.001--10 keV range. For the data taken at 8 days after the X-ray transient was detected by \swift\ described above, this yields $2.3\pm0.3\times10^{-11}$ \mbox{\thinspace erg\thinspace cm$^{-2}$\thinspace s$^{-1}$}, which corresponds to a luminosity of $5.7\pm0.1\times10^{44}$ \mbox{\thinspace erg\thinspace s$^{-1}$}\ at a distance of 456~Mpc. Given the black hole mass of log(\mbh/\msol)$=7.41\pm0.41$ as measured from the stellar velocity dispersion, the Eddington luminosity of the SMBH is $3.1\times10^{45}$ \mbox{\thinspace erg\thinspace s$^{-1}$}, therefore the Eddington fraction at this time was $\sim$10\%. However, if we extrapolate the data back to when ZTF first detected the transient, when it was approximately five times more luminous in the optical bands, this implies that the Eddington fraction could have reached as high as 50\%, if not greater.
\section{The host galaxy \galaxy}
\galaxy\ is listed in SDSS with magnitudes $u=20.76$, $g=19.23$, $r=18.56$, $i=18.21$, and $z=17.98$ \citep{alam15}, and in PanSTARRS with magnitudes $g=18.72$, $r=19.36$, $i=18.97$, $z=18.87$, and $y=18.49$ \citep{chambers16}. In the infrared, {\it WISE} measured $W1=15.67$, $W2=15.43$, $W3=12.45$, and $W4<8.86$, and in the UV {\it GALEX} measured NUV$=22.31$ \citep{bianchi11}.
As described in Section \ref{sec_vla}, neither the transient nor the galaxy were detected in the radio, with a 3$\sigma$ upper limit of 28~$\mu$Jy on the 6 GHz flux density. The VLA Faint Images of the Radio Sky at Twenty-cm \citep[FIRST][]{becker95} survey, which covered the region with a sensitivity of 1~mJy at 1.4 GHz, also did not detect the galaxy.
No morphological type for the galaxy is reported in the literature, however, our Tractor fitting of the ZTF images finds that its brightness profile was better fit by a deVaucoleur profile than an exponential profile, implying that it is an elliptical galaxy.
The {\it WISE} colors of W1$-$W2=0.24 are less than the W1$-$W2$\geq0.8$ selection criterion of \cite{stern12} for AGN, meaning there was no evidence for the presence of a powerful AGN from the infrared in the galaxy prior to the X-ray transient. However, as described in Section \ref{sec_keck}, the optical line ratios revealed LINER activity in the nucleus.
\galaxy also has a companion galaxy, SDSS J143357.57+400647.3, which has an angular separation of 21\arcsec\ and has spectroscopic redshift of 0.0990 from SDSS. This angular distance corresponds to a projected separation of 38 kpc at this redshift meaning that the two galaxies are likely interacting. The companion is brighter and visually larger on the sky, implying it is the more massive of the two.
\section{The nature of the X-ray transient in \galaxy}
The X-ray transient in \galaxy, with a peak luminosity of $\sim10^{44}$ \mbox{\thinspace erg\thinspace s$^{-1}$}\ and spatially coincident with the nucleus of the galaxy, is likely caused by an AGN flare or a TDE. Such events can be challenging to distinguish from each other \citep{auchettl18}. One specific example is IC 3599 which exhibited multiple X-ray flares and were interpreted both as AGN flares \citep{gruppe15} and multiple tidal disruption flares \citep{campana15}. We explore the likelihood of an AGN flare or a TDE in the following sections.
\subsection{An AGN flare in \galaxy?}
One of the distinguishing features of the X-ray transient in \galaxy\ is that the X-ray spectrum is soft, with $\Gamma\sim3$, and the spectral shape does not appear to vary with time, even as the source luminosity dropped by an order of magnitude (Figure \ref{fig_gamma}). These properties are in contrast to typical AGN properties, where the mean spectral index is $\Gamma=1.8$ \cite[e.g.][]{ricci17b}, i.e. harder than observed for this transient. Furthermore, luminous AGN usually show spectral evolution with a softer when brighter behaviour \citep[e.g.][]{sobolewska09,auchettl18}, not seen for this source.
This softer when brighter behaviour for AGN also reveals itself in studies of the correlation between the X-ray power-law index, $\Gamma$, and the Eddington ratio, \lamedd, \citep[e.g.][]{shemmer06,shemmer08,risaliti09}, but see \cite{trakhtenbrot17}. For example, from a sample of 69 X-ray bright sources in the \chandra\ Deep Field South and COSMOS surveys, \cite{brightman13} found that $\Gamma=(0.32\pm0.05)$log$_{10}$\lamedd$+(2.27\pm0.06)$. Given the observed peak Eddington ratio of 10\% that we have calculated, $\Gamma$ is expected to be $\sim1.8$, much lower than the value of 3 observed. We illustrate this in Figure \ref{fig_lam_edd} which shows the variation of $\Gamma$ with \lamedd\ for the X-ray transient in \galaxy\ along with the AGN data from \cite{brightman13}. While narrow-line Seyfert 1 galaxies show steeper spectral slopes than other broad-lined AGN \citep{brandt97}, similar to our source, they also show high Eddington ratios \citep[e.g.][]{pounds95} unlike our source.
\begin{figure}
\begin{center}
\includegraphics[width=90mm]{lambda_gamma.pdf}
\caption{The X-ray power-law index, $\Gamma$, of the X-ray transient in \galaxy\ plotted against its Eddington ratio, \lamedd, and how it has varied over time (red data points). Data from a sample of AGN presented in \cite{brightman13} are plotted for comparison (black data points), along with the statistically significant correlation found between these quantities (black line). This shows that $\Gamma$ is not consistent with this property of AGN, being too large for its \lamedd.}
\label{fig_lam_edd}
\end{center}
\end{figure}
Furthermore, for AGN the bright quasar-like X-ray emission should be accompanied by bright UV emission, as predicted by the tight relationship between the X-ray and UV luminosities of quasars \cite[e.g.][]{steffen06,lusso10,lusso16}. Studies of this relationship usually parameterize these quantities by the monochromatic flux densities at 2 keV and 2500 \AA. We use our fits to the lightcurve in Section \ref{sec_ltcrv} to calculate these quantities as a function of time and plot them on Figure \ref{fig_luv_lx}. Also plotted are data from 743 quasars selected from SDSS and 3XMM \citep{lusso16}, along with the relation log$L_{\rm 2 keV}=0.642L_{\rm 2500}+6.965$ derived from them.
At the observed peak of the transient, the rest frame monochromatic flux at 2 keV was 3.4$\times10^{-30}$ \mbox{\thinspace erg\thinspace cm$^{-2}$\thinspace s$^{-1}$ \thinspace Hz$^{-1}$}, corresponding to a luminosity of 8.4$\times10^{25}$ \mbox{\thinspace erg\thinspace s$^{-1}$ \thinspace Hz$^{-1}$}\ at $z=0.099$, whereas the flux density at 2500\AA\ as determined from our SED fit is 1.7$\times10^{-28}$ \mbox{\thinspace erg\thinspace cm$^{-2}$\thinspace s$^{-1}$ \thinspace Hz$^{-1}$}, corresponding to a luminosity of 4.3$\times10^{27}$ \mbox{\thinspace erg\thinspace s$^{-1}$ \thinspace Hz$^{-1}$}\ at this redshift. Given the relation log$L_{\rm 2 keV}=0.642L_{\rm 2500}+6.965$ from 743 quasars selected from SDSS and 3XMM \citep{lusso16}, the expected 2 keV luminosity of the X-ray transient in \galaxy\ given the measured 2500\AA\ one is 5.1$\times10^{24}$ \mbox{\thinspace erg\thinspace s$^{-1}$ \thinspace Hz$^{-1}$}\ which an order or magnitude less luminous than measured, indicating that the X-ray transient in \galaxy\ does not exhibit the UV--X-ray properties of AGN.
However, the data from \cite{lusso16} are from single epochs observations of mostly steady-state AGN which may not capture the properties of a flaring AGN which may be more appropriate. \cite{auchettl18} conducted a comparison between a sample of X-ray TDEs and a sample of flaring AGN. The flaring AGN with most in common to \galaxy\ is Mrk 335, a narrow-line Seyfert galaxy at $z=0.025$, whose flaring activity was revealed through long-term \swift\ observations \citep[e.g.][]{gallo18}. In order to compare the X-ray to UV properties of the transient in \galaxy\ to an AGN flare, we take the \swift\ data presented in \cite{gallo18}, and plot them on Figure \ref{fig_luv_lx}. Here we have converted the XRT count rates to the 2 keV monochromatic flux density by assuming a power-law spectrum with $\Gamma=2$, and we have used the UVW1 photometry to calculate the 2500\AA\ monochromatic fluxes. The range in X-ray luminosity of the flare from Mrk 335 is comparable to that observed from \galaxy, however the UV luminosity of the flare from Mrk 335 is $\sim2$ orders of magnitude higher. This indicates that the X-ray transient in \galaxy\ does not exhibit the UV--X-ray properties of this flaring AGN.
\begin{figure}
\begin{center}
\includegraphics[width=90mm]{luv_lx.pdf}
\caption{The 2 keV luminosity of the X-ray transient in \galaxy, plotted against its luminosity at 2500\AA\ as a function of time (red data points representing data taken on 2020 Feb 5 and every 30 days after that). Data from 743 quasars selected from SDSS and 3XMM presented in \cite{lusso16} are plotted for comparison (black data points), along with the statistically significant correlation they found between these quantities (black line). Also shown are data from a flare from the AGN Mrk 335 (blue points). These show that the X-ray luminosity of the transient is not consistent with the X-ray--UV properties of quasars, being too large for its UV luminosity.}
\label{fig_luv_lx}
\end{center}
\end{figure}
The presence of narrow emission lines in the optical spectrum with flux ratios common to LINER galaxies suggests that a low accretion rate AGN was present in this galaxy, at least $10^{4}$ years prior to the transient this is how long it would have taken to illuminate the narrow line region located on kpc-scales from the SMBH. The galaxy would also not be selected as an AGN with its {\it WISE} colors of W1$-$W2=0.24, which is less than the W1$-$W2$\geq0.8$ criterion of \cite{assef13}. We also checked for historical AGN variability in the W1 and W2 bands by building a neoWISE \citep{mainzer11} light curve between 2014 January 8 and 2019 June18 and found no evidence of prior variability. Furthermore, the AGN luminosity inferred from the {\rm{[O\,\sc{iii}]}}\ flux is lower than expected from the current X-ray luminosity. Therefore while a low-luminosity AGN may have existed before the onset of this new activity, it is difficult to reconcile the X-ray and UV properties of this transient with the properties of the general AGN population, or indeed an AGN flare.
\subsection{A TDE in \galaxy?}
The alternative solution is that this transient was a TDE. \cite{auchettl17} presented a comprehensive analysis of the X-ray emission from TDEs, and in \cite{auchettl18} they conducted a comparison between the X-ray properties of X-ray TDEs to flaring AGN. \cite{auchettl17} stipulated several criteria for identifying an X-ray transient as a TDE. The ones which \galaxy\ satisfies are that the X-ray light curve has a well defined shape and observable trend with several observations prior to the flare; the general shape of the X-ray light curve decay is monotonically declining; the maximum luminosity detected from the event is at least two orders of magnitude larger than the X-ray upper limit immediately preceding the discovery of the flare; over the full time range of X-ray data available for the source of interest, the candidate TDE shows evidence of X-ray emission from only the flare, while no other recurrent X-ray activity is detected; the X-ray flare is coincident with the nucleus of the host galaxy.
One further criterion states that the X-ray light curve shows a rapid increase in X-ray luminosity, which then declines on time-scales of months to years. While the decline on time-scales of months was observed, the rise of the X-ray transient in \galaxy\ was not. \erosita\ detected the transient 40 days prior to \swift, but prior to that, the nearest X-ray observation to that was 4 years earlier, also by \swift. The \erosita\ measurement is also not consistent with the $t^{-5/3}$ as measured by \swift, but it is possible that this was part of the rise, and that the source peaked and declined before \swift\ detected it, or the lightcurve initially exhibited a plateau. This was seen in ASASSN-14li, where the X-ray lightcurve was constant for the first $\sim$100 days, after which is followed the $t^{-5/3}$ decline \citep{vanvelzen16,holoien16,brown17}.
Furthermore, \cite{auchettl17} stipulate that based on its optical spectrum or other means, one finds no evidence of AGN activity arising from its host galaxy. We find LINER-like line ratios in the optical spectrum of \galaxy, indicating low-level AGN activity prior to the event, so \galaxy\ does not strictly satisfy this criterion. However, we note that several other TDEs have shown indications of prior AGN activity, including ASASSN-14li, as determined from a radio detection and a narrow {\rm{[O\,\sc{iii}]}}\ line \citep{vanvelzen16}, and those shown in Figure \ref{fig_bpt}.
Finally, we compare the optical/UV and X-ray luminosities of the X-ray transient in \galaxy\ to those presented for the X-ray TDEs in \cite{auchettl17} in Figure \ref{fig_auchettl17}. This shows that the X-ray luminosity with respect to the optical/UV luminosity for \galaxy\ is consistent with other X-ray TDEs, albeit that these events present more diverse properties than AGN.
In their comparison between the X-ray properties of X-ray TDEs to flaring AGN, \cite{auchettl18} noted the lack of X-ray spectral evolution in TDEs, whereas AGN often show significant spectral evolution, as we showed in the previous section. We therefore find that since the source satisfies most of the criteria for classifying X-ray TDEs set out by \cite{auchettl17}, and that the X-ray and UV properties of the X-ray transient in \galaxy\ are more comparable to known TDEs than AGN, we conclude that the transient likely is powered by a TDE.
\begin{figure}
\begin{center}
\includegraphics[width=90mm]{lx_luv_auchettl17.pdf}
\caption{The X-ray luminosity of the X-ray transient in \galaxy, plotted against its optical/UV luminosity (red data point). Data from a sample of X-ray TDEs presented in \cite{auchettl17} is plotted for comparison (black data points). The dashed black line marks where the two quantities are equal. The X-ray luminosity with respect to the optical/UV luminosity for \galaxy\ is consistent with other X-ray TDEs.}
\label{fig_auchettl17}
\end{center}
\end{figure}
\section{The X-ray transient in \galaxy\ in the context of TDEs}
Of the 13 transients classified as X-ray TDEs or likely X-ray TDEs from the sample of \cite{auchettl17}, most (10) were first detected in the X-ray band, either from \xmm\ slews, serendipitously in \chandra\ or \xmm\ pointed observations, or from hard X-ray monitors such as \swift/BAT. The other three were detected in optical surveys. Therefore \galaxy\ adds to the number of TDEs first detected in the X-rays.
In comparison to these other TDEs, we find that SDSS J1201+30 is the event which shows most similarity to \galaxy\ in terms of its X-ray and optical/UV luminosities. It was also powered by a black hole of similar mass, \citep[$10^{7.2}$ \msol,][]{wevers19}. SDSS J1201+30 was first detected by \xmm\ during a slew with \lx$\sim3\times10^{44}$ \mbox{\thinspace erg\thinspace s$^{-1}$}, which was 56 times brighter than a previous {\it ROSAT} upper limit and decayed with a with $t^{-5/3}$ profile \citep{saxton12}. A power-law fit to the X-ray spectrum of the source yielded $\Gamma=3.38\pm0.04$. The optical/UV emission from this source was also weak, with 0.002--0.1 keV luminosity of $2.64\pm0.31\times10^{42}$ \mbox{\thinspace erg\thinspace s$^{-1}$}\ \citep{auchettl17}. The source also did not present broad or coronal optical lines. The X-ray spectrum could be reproduced with a Bremsstrahlung or double-power-law model. These characteristics are similar to \galaxy.
One property of SDSS J1201+30 that we do not see in \galaxy\ is variability on timescales of days in addition to the monotonic flux decline. SDSS J1201+30 became invisible to \swift\ between 27 and 48 days after discovery, which \cite{saxton12} suggested could be due to self-absorption by material driven from the system by radiation pressure during an early super-Eddington accretion phase. Alternatively, \cite{liu14_tde} suggested that a supermassive black hole binary lies at the heart of SDSS J1201+30, and that the dips in the lightcurve were due to disruption of the accretion flow by the secondary SMBH. \galaxy, however, does not show evidence for excess variability from the powerlaw decline.
In terms of how the X-ray lightcurve of \galaxy\ compares with the well sampled X-ray light curves of other X-ray TDEs, \galaxy\ appears to have shown a plateau of emission before declining, similar to ASSASN-14li \citep{vanvelzen16}, while XMMSL1 J0740-85 declined monotonically without evidence for a plateau \citep{saxton17}.
Having compared the properties of the TDE in \galaxy\ to other X-ray TDEs, it is useful to compare the optical emission from the TDE in \galaxy\ to that of optically selected TDEs. For this we use the recent sample of 17 ZTF-discovered TDEs presented in \cite{vanvelzen20}. Here the authors use a simple blackbody model to fit the optical/UV data of their sample. We proceed to fit the optical/UV data described in Section \ref{sec_sed}, finding that these can be described by a blackbody with log($T$/K)$=4.3^{+0.2}_{-0.1}$, where the $g$-band luminosity is log($L_{g}$/\mbox{\thinspace erg\thinspace s$^{-1}$})=41.0$\pm$0.1, and the total blackbody luminosity is log($L_{\rm bb}$/\mbox{\thinspace erg\thinspace s$^{-1}$})=42.8$\pm$0.1. While the temperature is comparable to the sample of \cite{vanvelzen20}, which has the range log($T$/K)=4.1--4.6, the luminosities are much lower, where the ZTF TDEs have log($L_{g}$/\mbox{\thinspace erg\thinspace s$^{-1}$})=42.8--43.6 and log($L_{\rm bb}$/\mbox{\thinspace erg\thinspace s$^{-1}$})=43.2--44.7.
The black hole mass inferred from the stellar velocity dispersion of \galaxy\ is $\sim10^{7.4}$ \msol, which is around the peak of the observed distribution of black hole masses of TDEs \citep{stone16}, although for optical events this was found to be lower, $\sim10^{6}$ \msol\ \citep{wevers17}. We calculated that the Eddington fraction of the event near peak was only $\sim10$\%. This is naturally explained since the SMBH has a mass of $\sim10^{7.4}$ \msol, meaning that a very massive star would have been needed to reach Eddington luminosities. \cite{strubbe09} stated that TDEs can emit above the Eddington luminosity for a BH with \mbh$<10^{7}$ \mbh. Indeed \cite{stone16} concluded that Eddington-limited emission channels of TDEs dominate the rates.
Finally, we noted that \galaxy\ has a companion galaxy, SDSS J143357.57+400647.3, which has a projected separation of 38 kpc. This may be important since a companion galaxy that may be undergoing an interaction with the host could be relevant to the fueling of TDEs \citep{french20}.
\section{Implications and Conclusions}
Only 13 transients were classified as X-ray TDEs or likely X-ray TDEs from the sample of \cite{auchettl17}, so the number of known X-ray TDEs is still small. Therefore finding more events of this nature are important for understanding this population, even just one event as we have reported here.
This TDE was one of a few identified where previous AGN activity in the galaxy was known, albeit at a low-level. Other TDEs with known AGN activity prior to the flare include ASSASN-14li \citep{vanvelzen16}, where archival radio data and narrow {\rm{[O\,\sc{iii}]}}\ emission showed a low-luminosity AGN existed prior to the event. As can be seen in Fig \ref{fig_bpt}, several other TDE hosts showed similar evidence for prior AGN activity from their narrow line ratios, including CNSS J0019+00 \citep[Sy2,][]{anderson19}. Furthermore, \cite{ricci20} postulated that a TDE caused the changing-look behaviour of the AGN 1ES 1927+654, and \cite{merloni15} suggested that TDEs may be drivers of these changing-look events.
While we used ZTF data to determine the optical evolution of this TDE, this event was not identified as a TDE by wide field optical surveys such as ZTF or ASAS-SN, possibly due to its low optical luminosity. We note, however, that ZTF was not observing the field of \galaxy\ when the optical luminosity was at its peak, which may be the reason it was missed. This TDE was also not classified as a TDE from its optical spectrum. Taken together, this suggests many more events like it are being missed, and ultimately only wide field UV or X-ray surveys will catch events like these. \erosita\ is currently conducting an all-sky survey in the 0.2--10 keV band and will likely identify a large number of them \citep{merloni12}.
In conclusion, we have reported on an X-ray transient, observed to peak at a 0.3--10 keV luminosity of $10^{44}$ \mbox{\thinspace erg\thinspace s$^{-1}$}, originating in the nucleus of the galaxy \galaxy\ at $z=0.099$. The X-ray transient was also accompanied by a less powerful optical/UV transient. A soft X-ray spectrum with $\Gamma=3$ and the low UV/X-ray ratio disfavor an AGN flare scenario. The source was observed to decline monotonically in all bands, consistent with a $t^{-5/3}$ profile favoring a TDE scenario. Since this event was not identified as a TDE by wide-field optical surveys, or by optical spectroscopy, we are lead to the conclusion that a significant fraction of X-ray TDEs may be going unnoticed.
\facilities{Swift (XRT, UVOT), NuSTAR, CXO, Keck:I (LRIS), PO:1.2m PO:1.5m, VLA}
\software{{\tt CASA} \citep{mcmullin07}, {\tt CIAO} \citep{fruscione06}, {\tt lpipe} \citep{perley19}, {\tt The Tractor} \citep{lang16}, {\tt XSPEC} \citep{arnaud96}}
\acknowledgements{
The majority of this research and manuscript preparation took place during the COVID-19 global pandemic. The authors would like to thank all those who risked their lives as essential workers in order for us to safely continue our work from home.
We wish to thank the \swift\ PI, Brad Cenko, for approving the target of opportunity requests we made to observe \galaxy, as well as the rest of the \swift\ team for carrying the observations out. We also acknowledge the use of public data from the \swift\ data archive.
We also wish to thank the \nustar\ PI, Fiona Harrison, for approving the DDT request we made to observe \galaxy, as well as the \nustar\ SOC for carrying out the observation. This work was also supported under NASA Contract No. NNG08FD60C. \nustar\ is a project led by the California Institute of Technology, managed by the Jet Propulsion Laboratory, and funded by the National Aeronautics and Space Administration. This research has made use of the NuSTAR Data Analysis Software (NuSTARDAS) jointly developed by the ASI Science Data Center (ASDC, Italy) and the California Institute of Technology (USA).
ZTF is supported by the National Science Foundation under Grant No. AST-1440341 and a collaboration including Caltech, IPAC, the Weizmann Institute for Science, the Oskar Klein Center at Stockholm University, the University of Maryland, the University of Washington, Deutsches Elektronen-Synchrotron and Humboldt University, Los Alamos National Laboratories, the TANGO Consortium of Taiwan, the University of Wisconsin at Milwaukee, and Lawrence Berkeley National Laboratories. Operations are conducted by COO, IPAC, and UW.
This paper is based on observations obtained with the Samuel Oschin Telescope 48-inch and the 60-inch Telescope at the Palomar Observatory as part of the Zwicky Transient Facility project. ZTF is supported by the National Science Foundation under Grant No. AST-1440341 and a collaboration including Caltech, IPAC, the Weizmann Institute for Science, the Oskar Klein Center at Stockholm University, the University of Maryland, the University of Washington, Deutsches Elektronen-Synchrotron and Humboldt University, Los Alamos National Laboratories, the TANGO Consortium of Taiwan, the University of Wisconsin at Milwaukee, and Lawrence Berkeley National Laboratories. Operations are conducted by COO, IPAC, and UW. SED Machine is based upon work supported by the National Science Foundation under Grant No. 1106171 .
The ZTF forced-photometry service was funded under the Heising-Simons Foundation grant \#12540303 (PI: Graham).}
|
1,116,691,497,556 | arxiv | \section{Introduction}
The nuclear Equation of State (EoS) around the saturation density of symmetric matter ($n_{sat}$) can be accurately characterized by the so-called empirical parameters, defined as the set of successive derivatives of the energy functional~\cite{Myers1969}. Since the density of atomic nuclei is not far from $n_{sat}$, nuclear experiments essentially probe the low order empirical parameters, e.g. up to order two or so. The better known the empirical parameters, the more precise the EoS around $n_{sat}$. With the continuous improvement of theoretical modeling and of the nuclear data, the knowledge of these parameters as well as of the correlations among them have considerably progressed in the recent years, inducing tighter constraints on the nuclear EoS. Typical examples are given by the determination of the saturation energy from mass measurements~\cite{Goriely2015,Steiner2015,Bertsch2017,McDonnel2015}, the nuclear incompressibility from the Giant Monopole Resonance (GMR)~\cite{Blaizot1980,Colo2004,Khan2012,Khan2013,Colo2014}, and the symmetry energy properties ($E_{sym}$ and $L_{sym}$) from various experiments such as mass measurements~\cite{Kortelainen2012,Danielewicz2014}, isovector giant resonances~\cite{Trippa2008,Colo2014}, the correlation between $L_{sym}$ and the surface stiffness parameter for the determination of the neutron skin~\cite{Warda2009,Centelles2010,Mondal2016}, or more recently the constraints on $K_{sym}$ induced by the unitary limit in neutron matter~\cite{Tews2017}.
These constraints are summarized in various reviews, such as Refs.~\cite{Lattimer2013,Newton2014,BALi2014} for instance.
Most of the experimental determination of the empirical parameters so far relies on the assumed linear correlation between an experimental observable and a single empirical parameter. For such analyses, the final uncertainty on each empirical parameter -- and therefore on the global EoS -- crucially depends on the quality of the correlation. In addition, the quality of the correlation depends on the variation range of the other parameters of the set, which can be model dependent. A way to settle the model dependence is to look for specific correlations between empirical parameters which would be induced by the specific functional form of the chosen model. This was for instance shown to be the case for the experimental determination of $K_{sat}$. The correlation between $K_{sat}$ and $Q_{sat}$ typically found for Skyrme and Gogny interactions, is related to the presence of a single density dependent term in the nuclear force~\cite{Khan2012,Khan2013}. The density dependence of the energy per particle in symmetric matter being controlled by $K_{sat}$ and $Q_{sat}$ at first orders, it is important to understand such model dependence, as discussed in Ref.~\cite{Margueron2018a,Margueron2018b}.
Since experimental measurements are often sensitive to both the bulk and surface properties of finite nuclei, they probe a continuous range in density, implying that there is no one-to-one correspondence between observables and EoS parameters. Each experimental probe is sensitive to a set of parameters, possibly leading to some model dependence in the determination of single parameters. The determination of the EoS thus comes through the intersection between the different correlation plots among empirical parameters, as established through the comparison of density functional calculations to the different observables~\cite{Tsang2012,Lattimer2013,Lattimer2014,Dutra2014,Fortin2016}.
In the absence of a global analysis involving most of the possible model dependence, the question of the EoS uncertainties remains however unsolved.
To progress on the question of the model dependence, a meta-modeling was proposed~\cite{Margueron2018a}, where the variation of the empirical parameters is set to be free and only constrained by the physical requirements imposed to the meta-modeling, e.g., existence of the saturation point, stability of the EoS, positiveness of the symmetry energy, causality, constraints given by ab-initio calculations at low density, etc.... Based on this global analysis satisfying a set of physical requirements, generic correlations among low-order empirical parameters have been analyzed, see for instance Refs.~\cite{Margueron2018b}. These correlations are usually found to be weaker than the ones deduced using both a specific model and a direct fit to nuclear properties, e.g. Skyrme or relativistic mean field, see Refs.~\cite{Kortelainen2012,Danielewicz2014}.
In this paper, we focus on simple cases where the correlations among empirical parameters can be analyzed from general nuclear matter properties. We try to estimate how much the correlations between low order empirical parameters are blurred by the uncertainties on higher-order ones. To this aim, we estimate the propagation of the high-order parameter uncertainties down to the lower order ones, based on a simple Taylor expansion of the EoS around the saturation density $n_{sat}$, presented in Sec.~\ref{sec:def}. In Sec.~\ref{sec:results} the quality of these error estimations is then checked against the prediction of a set of $\sim$50 different realistic nuclear functionals. Finally, in Sec.~\ref{sec:MM} a more complete analysis of the correlations is performed within a meta-model of the equation of state~\cite{Margueron2018a}, in which several hundreds of thousand different functionals are generated assuming full independence among the empirical parameters, and subsequently filtered through many-body perturbation theory (MBPT) predictions based on chiral effective field theory ($\chi$EFT) interactions, stability of the EoS and causality conditions.
\section{Definitions and strategy}
\label{sec:def}
Given a generic functional for the energy per particle of homogeneous nuclear matter $e(n_n,n_p)$, simply expressed as the sum of an isoscalar $e_{sat}(n)$ and isovector $e_{sym}(n)$ terms ($n=n_n+n_p$, $\delta=(n_n-n_p)/n$),
\begin{equation}
e(n_n,n_p) = e_{sat}(n) + e_{sym}(n) \delta^2 + ...,
\label{eq:energy}
\end{equation}
where the small contribution from non-quadratic terms are neglected here, the isoscalar empirical parameters are defined as
the successive density derivatives of $e_{sat}(n)$,
\begin{equation}
P_{IS}^{(k)}= (3n_{sat})^k \frac{\partial^k e_{sat}}{\partial n^k}|_{\delta=0,n=n_{sat}},
\label{eq:empIS}
\end{equation}
We will note $P_{IS}^{(0)}=E_{sat}$ the saturation energy,
$P_{IS}^{(2)}=K_{sat}$ the incompressibility, $P_{IS}^{(3)}=Q_{sat}$ the skewness, and $P_{IS}^{(4)}=Z_{sat}$ the kurtosis.
In Eq.~(\ref{eq:energy}), $e_{sym}(n)$ is the symmetry energy function of the density $n$ and defined as $e_{sym}=\nicefrac{1}{2}\partial^2e/\partial\delta^2|_{\delta=0}$ in symmetric matter.
The isovector parameters measure the density derivatives of the symmetry energy as,
\begin{equation}
P_{IV}^{(k)}= (3n_{sat})^k \frac{\partial^k e_{sym}}{\partial n^k}|_{\delta=0,n=n_{sat}} .
\label{eq:empIV}
\end{equation}
We will note $P_{IV}^{(0)}=E_{sym}$ the symmetry energy at saturation, $P_{IV}^{(1)}=L_{sym}$ the symmetry energy slope,
$P_{IV}^{(2)}=K_{sym}$ the isovector incompressibility, $P_{IV}^{(3)}=Q_{sym}$ the isovector skewness, and $P_{IV}^{(4)}=Z_{sym}$ the isovector kurtosis.
A Taylor expansion around the saturation density $n_{sat}$ is naturally suggested by the definition of the empirical parameters (\ref{eq:empIS})-(\ref{eq:empIV}), and depending on the truncation of the Taylor series we will have different approximations for the functional $e(n_n,n_p)$ as:
\begin{eqnarray}
e_{sat,2}(x)&=&E_{sat}+\frac 1 2 K_{sat} x^2 \, , \label{eq:esat2}\\
e_{sat,3}(x)&=&E_{sat}+\frac 1 2 K_{sat} x^2 + \frac 1 6 Q_{sat} x^3 \, , \\
e_{sat,4}(x)&=&E_{sat}+\frac 1 2 K_{sat} x^2 + \frac 1 6 Q_{sat} x^3 + \frac 1 {24} Z_{sat} x^4 \, , \label{eq:esat4}
\end{eqnarray}
and
\begin{eqnarray}
e_{sym,2}(x)&=&E_{sym}+L_{sym} x + \frac 1 2 K_{sym} x^2 \, , \label{eq:esym2}\\
e_{sym,3}(x)&=&E_{sym}+L_{sym} x + \frac 1 2 K_{sym} x^2 + \frac 1 6 Q_{sym} x^3 \, , \\
e_{sym,4}(x)&=&E_{sym}+L_{sym} x + \frac 1 2 K_{sym} x^2 + \frac 1 6 Q_{sym} x^3
+ \frac 1 {24} Z_{sym} x^4 \, . \nonumber \\ \label{eq:esym4}
\end{eqnarray}
where the parameter $x$ is introduced for convenience and is defined as $x=(n-n_{sat})/(3n_{sat})$. The empirical parameters (\ref{eq:empIS})-(\ref{eq:empIV})
can be identified as the coefficients of the expansion in Eqs.~(\ref{eq:esat2})-(\ref{eq:esym4}), where we adopt the naming usage for the empirical parameters.
Note however that the convention may depend on the authors, see the appendix of Ref.~\cite{Piekarewicz2009} for a detailed discussion.
In principle, both the isospin expansion (\ref{eq:energy}) and the density expansions (\ref{eq:empIS})-(\ref{eq:empIV}) could be performed beyond the orders we considered here. For the characterization of the nuclear EoS between 0 and $n_{sat}$ that we analyze here, the proposed expansions (\ref{eq:esat2})-(\ref{eq:esym4}) are found to be sufficient.
From the series expansion of the functional $e(n_n,n_p)$, it is clear that any direct measurement or physical constraint on the functional will naturally produce some correlations among the empirical parameters $P^{(k)}$. Let us consider, for instance, an observable $\langle O(x)\rangle=f(e_{sym})$ that we suppose to be both sensitive to the isovector part of the functional and independent of the terms $\propto x^k$, $k\geq 2$, where this last condition will be met if, e.g. the observable is defined sufficiently close to saturation density. The constraint of reproducing the observable $\langle O\rangle$ would naturally produce an exact linear correlation between the parameters $E_{sym}$ and $L_{sym}$. However, in a realistic application, the higher order terms $k\geq 2$ are never fully negligible, and might blur such correlation.
In the following sections, we will work out the different correlations among empirical parameters implied in Eqs.~(\ref{eq:esat2})-(\ref{eq:esym4}),
when the value of the energy functional $e(n_n,n_p)$ is imposed by some experimental measurements or some physical constraints at some densities.
The uncertainties of the correlations will be extracted from the impact of the higher order parameters not included in the correlation itself. The effect of the latter terms will be estimated from a set of $\sim$50 realistic EoS models, that have been successfully compared to a large set of observables in the literature~\cite{Margueron2018a}.
This chosen set comprises Skyrme, Relativistic Mean Field (RMF), Relativistic Hartree-Fock (RHF), as well as many-body perturbation theory (MBPT) based on 7 chiral N3LO EFT interactions~\cite{Drischler2016} ($\chi$EFT 2016), see Ref.~\cite{Margueron2018a} for the complete list and references.
The correlations can be extracted from the different truncation orders defined in Eqs.~(\ref{eq:esat2})-(\ref{eq:esym4}), as well as from the set of realistic functionals.
Comparing the results of these different correlations will show the relative importance of the high order parameters in the blurring of the expected correlations.
\section{Results}
\label{sec:results}
In this section, we employ the simple functional~(\ref{eq:energy}) to estimate the strength of various correlations between empirical parameters, such as the well-known correlation between $E_{sym}$ and $L_{sym}$, as well as some other correlations such as the one between $K_{sym}$ and $3E_{sym}-L_{sym}$ recently proposed in Ref.~\cite{Mondal2017}, and we discuss the one between $K_{sat}$ and $Q_{sat}$.
\subsection{Correlation between $E_{sym}$ and $L_{sym}$}
\label{sec:elsym}
\begin{figure}[t]
\begin{center}
\includegraphics[angle=0,width=1.0\linewidth]{plot_ELsym-1.pdf}
\end{center}
\caption{(Color online) Expectation values for the parameter $\alpha_{EL,i}$ as function of $E_{sym}$ for a set of different type of nuclear interactions considered in Ref.~\cite{Margueron2018a}: Skyrme, RMF, and RHF. The points labelled $\chi$EFT 2016 stand for the MBPT based on N3LO EFT interaction~\cite{Drischler2016}. See text for more details.
}
\label{fig:EL1}
\end{figure}
From the analysis of the giant dipole resonance (GDR) of $^{208}$Pb, a well-constrained estimate of $e_{sym}$ at $n_a=0.1\approx\nicefrac{2}{3}n_{sat}$~fm was proposed~\cite{Trippa2008}.
Original ideas suggesting that finite nuclei data could reveal nuclear properties at the nuclear average density 0.10-0.12~fm$^{-3}$ can also be found in Refs.~\cite{Furnstahl2002,Niksic2008}.
Considering the following condition, $e_{sym}(x=x_a)\equiv E_{sym}^{a}=24.1\pm0.8$~MeV~\cite{Trippa2008,Colo2014}, where $x_a=x(n_a)=-\nicefrac{1}{9}$,
and using Eqs.~(\ref{eq:esym2})-(\ref{eq:esym4}), one can obtain the following correlation between $E_{sym}$ and $L_{sym}$,
\begin{equation}
L_{sym,i} = \beta_{EL} E_{sym} + \alpha_{EL,i}
\label{eq:corrEL}
\end{equation}
with $\beta_{EL}=-x_a^{-1}$ and the value of the $\alpha_{EL,i}$ parameter depends on the truncation order ($i=2$-4) of the Taylor expansion as:
\begin{eqnarray}
\alpha_{EL,2} &=& -x_a^{-1} E_{sym}^{a} - \frac {x_a} {2}K_{sym} \, , \\
\alpha_{EL,3} &=& -x_a^{-1} E_{sym}^{a} - \frac {x_a} {2}K_{sym}-\frac {x_a^2} {6} Q_{sym} \, , \\
\alpha_{EL,4} &=& -x_a^{-1} E_{sym}^{a} - \frac {x_a} {2}K_{sym}-\frac {x_a^2} {6} Q_{sym}-\frac {x_a^3} {24} Z_{sym} \, .
\end{eqnarray}
\begin{figure}[t]
\begin{center}
\includegraphics[angle=0,width=1.0\linewidth]{plot_ELsym-2.pdf}
\end{center}
\caption{(Color online) Correlation between the empirical parameters $L_{sym}$ and $E_{sym}$
for different kind of nuclear interactions described in the text (Skyrme, RMF, RHF, and $\chi$EFT 2016~\cite{Drischler2016}) are also plotted. The bands stands for the uncertainty in the correlation estimated from $\alpha_{EL,i}$ at orders $i=2$-4. The solid line is the correlation obtained by Farine et al. 1978~\cite{Farine1978}, Dashed line by Oyamatsu \& Iida 2003~\cite{Oyamatsu2003}, and the ellipses the 68\% and 95\% confidence intervals of Kortelainen et al. 2010~\cite{Kortelainen2010}.}
\label{fig:EL2}
\end{figure}
Both $\beta_{EL}$ and $\alpha_{EL,i}$ are affected by some uncertainties. The uncertainty on $\beta_{EL}$ depends on the width of the density domain
which is effectively explored in the experiment. This information is difficult to evaluate and is not provided from the analysis of the GDR~\cite{Trippa2008}.
In the present case, we therefore fix $\beta_{EL}$ to be $\beta_{EL}=9$ without uncertainty. The uncertainty on the $\alpha_{EL,i}$ parameter explicitly depends on the uncertainty on the high order empirical parameters, which are not strongly constrained by empirical observations. We can estimate this uncertainties by considering a set of $\sim$50 chosen realistic phenomenological functional (Skyrme, RMF and RHF) and many-body perturbation theory (MBPT) based on 7 chiral N3LO EFT interactions~\cite{Drischler2016} ($\chi$EFT 2016), for which the empirical parameters are all given in Ref.~\cite{Margueron2018a}.
Using the predicted empirical parameters for these $\sim$50 functionals, the coefficients $\alpha_{EL,i}$ for the different orders $i=2$-4 are represented in Fig.~\ref{fig:EL1}, as a function of $E_{sym}$.
If we now select the models for which $28<E_{sym}<36$~MeV~\cite{BALi2013} (rectangles in Fig.~\ref{fig:EL1}), we obtain the following estimation for the coefficients $\alpha_{EL,i}$:
$\alpha_{EL,2}=-221.5\pm 17.5$~MeV, $\alpha_{EL,3}=-222\pm 17$~MeV, and $\alpha_{EL,4}=-222\pm 17$~MeV.
The contribution of the uncertainty in the estimated value of $E_{sym}^{a}$~\cite{Trippa2008} accounts for $\sim$7~MeV of the total uncertainty in $\alpha_{EL,i}$, while the rest of the uncertainty accounts for the contribution of the high order parameters estimated from the selected models. The estimation of $\alpha_{EL,i}$ at different orders $i$ closely agree, indicating that the value of $\alpha_{EL,i}$ is essentially determined by $E_{sym}^{a}$, $n_c$ and the isovector incompressibility $K_{sym}$, while the higher order parameters play a negligible role.
A better knowledge of the empirical parameter $K_{sym}$ will therefore lead to an improvement of the $E_{sym}$-$L_{sym}$ correlation.
The correlation~(\ref{eq:corrEL}) between $E_{sym}$ and $L_{sym}$ is shown in Fig.~\ref{fig:EL2} varying the coefficient $\alpha_{EL,i}$ within the boundaries obtained from the analysis of Fig.~\ref{fig:EL1}.
The gray band corresponds to $\alpha_{EL,2}$ and the pink one to $\alpha_{EL,4}$.
For comparison, the $E_{sym}-L_{sym}$ correlation for the $\sim$50 considered models is also plotted in Fig.~\ref{fig:EL2}.
There is a good overlap between our predicted correlation band and the values $E_{sym}$-$L_{sym}$ predicted by the $\sim$50 considered models, indicating that i) our simple analytical model for the symmetry energy~(\ref{eq:esym2})-(\ref{eq:esym4}) can efficiently map the $E_{sym}$-$L_{sym}$ correlation and ii) our error estimate for the $E_{sym}$-$L_{sym}$ correlation is satisfactory.
Considering only the $\sim$50 models sampling, the correlation coefficient between $L_{sym}$ and $E_{sym}$ is found to be $\sim$0.80 and $\sim$0.55 if we consider the reduced sample of models for which $28<E_{sym}<36$~MeV.
In summary, the dispersion of the $E_{sym}$-$L_{sym}$ correlation can be understood as partially coming from the experimental uncertainty in $E_{sym}^{a}$ and partially due to the uncertainty on the poorly known empirical parameter $K_{sym}$.
The $E_{sym}$-$L_{sym}$ correlation was discussed in earlier works, so we report in Fig.~\ref{fig:EL2} on other $E_{sym}$-$L_{sym}$ correlations, as in Ref.~\cite{Lattimer2013}. One of the first studies of the correlation between $E_{sym}$ and $L_{sym}$ from a Skyrme mass formula was presented in Ref.~\cite{Farine1978}, called 'Farine et al. 1978 in Fig.~\ref{fig:EL2}. Based on a macroscopic nuclear model, the $E_{sym}$-$L_{sym}$ correlation was later re-examined~\cite{Oyamatsu2003}, 'Oyamastu \& Iida 2003'.
The 68\% and 95\% confidence intervals of 'Kortelainen et al. 2010'~\cite{Kortelainen2010} are also plotted in Fig.~\ref{fig:EL2}.
Our analysis agrees well with more recent investigations: the correlation coefficient was found to be $\sim$0.71 in Ref.~\cite{Kortelainen2012} and 0.9-0.95 in Ref.~\cite{Nazarewicz2014} (the variation in the correlation coefficient reflects the dispersion of the models). Based on a different sampling of models a sizeable $E_{sym}$-$L_{sym}$ correlation was also found in Ref.~\cite{Ducoin2011}. Concerning the experimental probe to be chosen to determine $E_{sym}^{a}$, an alternative choice was proposed from an analysis of the isobaric analog state (IAS) and neutron skin radius~\cite{Danielewicz2014}.
In summary, we found a satisfactory agreement between the $E_{sym}$-$L_{sym}$ correlation suggested from our analysis and the dispersion of the $\sim$50 models considered here, as well as with previous investigations. In addition, our analysis suggests that the better knowledge of the empirical parameter $K_{sym}$ will reduce the blurring of the correlation.
\subsection{Correlation between $K_{sym}$ and $3E_{sym}-L_{sym}$}
A recent analysis of 500 different density functional models has revealed a general correlation between the empirical parameter $K_{sym}$ and the linear combinaison $3E_{sym}-L_{sym}$ as,
\begin{equation}
K_{sym} = \beta (3E_{sym}-L_{sym}) + \alpha \, ,
\label{eq:corrK}
\end{equation}
where the fit gives $\alpha=66.80\pm 2.14$~MeV and $\beta=-4.97\pm0.07$~MeV~\cite{Mondal2017}.
The origin of such a correlation was however not explained in Ref.~\cite{Mondal2017}.
In this section, we propose a simple explanation for the correlation~(\ref{eq:corrK}).
Defining the energy of neutron matter (NM) from Eq.~(\ref{eq:energy}) as $e_{NM}(n)=e_{sat}(n)+e_{sym}(n)$, we impose the very general constraint that the neutron energy per particle should be zero at zero density,
\begin{eqnarray}
e_{NM}(x=-\nicefrac{1}{3})&=&0 \hbox{ MeV} \, ,
\end{eqnarray}
which gives the following linear combination among the empirical parameters,
\begin{eqnarray}
e_{NM,4}(x=-\nicefrac{1}{3})&=&0=E_{sat}+E_{sym}-\frac 1 3 L_{sym} + \nonumber \\
&& \hspace{-3cm}\frac 1 {18}(K_{sat}+K_{sym} ) - \frac 1 {162} Q_{NM} + \frac 1 {1944} Z_{NM} + ... \, ,
\end{eqnarray}
where $Q_{NM}=Q_{sat}+Q_{sym}$ and $Z_{NM}=Z_{sat}+Z_{sym}$.
This condition can be expressed as a correlation between $K_{sym}$ and the linear combinaison $3E_{sym}-L_{sym}$ -- which naturally appears here -- as,
\begin{equation}
K_{sym,i} = \beta_{K_{sym}} (3E_{sym}-L_{sym}) + \alpha_{K_{sym},i} \, .
\label{eq:ksymi}
\end{equation}
with $\beta_{K_{sym}}=-6$ and $\alpha_{K_{sym},i}$ at different orders,
\begin{eqnarray}
\alpha_{K_{sym},2} &=& -18E_{sat}-K_{sat} \, , \\
\alpha_{K_{sym},3} &=& -18E_{sat}-K_{sat}+Q_{NM}/9 \, , \\
\alpha_{K_{sym},4} &=& -18E_{sat}-K_{sat}+Q_{NM}/9-Z_{NM}/108 \, .
\end{eqnarray}
$\alpha_{K_{sym},2}$ is the expectation for the constant $\alpha_{K_{sym},i}$ assuming a Taylor expansion up to second order only, while $\alpha_{K_{sym},3}$ and $\alpha_{K_{sym},4}$
take into account the uncertainties induced by the unknown higher orders terms.
Assuming $E_{sat}=-16\pm0.5$~MeV and $K_{sat}=230\pm20$~MeV~\cite{Khan2012,Bertsch2017,Margueron2018a}, one can get a rough estimation for $\alpha_{K_{sym},2} \approx 58\pm30$~MeV, which is compatible with the coefficient $\alpha$ fitted in Ref.~\cite{Mondal2017}.
As in the previous section, the average values of $\alpha_{K_{sym},i}$ ($i=2$-4) and their uncertainties can be estimated from a set of the same $\sim$50 functionals. The result is given in Fig.~\ref{fig:alpha} as function of the linear combinaison $3E_{sym}-L_{sym}$.
\begin{figure}[t]
\begin{center}
\includegraphics[angle=0,width=1.0\linewidth]{plot_Ksym-1.pdf}
\end{center}
\caption{(Color online) Expectations values for $\alpha_{K_{sym},i}$ as function of the linear combinaison $3E_{sym}-L_{sym}$ for a set of different types of energy functionals, as in Fig.\ref{fig:EL1}.
The value for $\alpha$ given in Ref.~\cite{Mondal2017} is also shown (purple band).}
\label{fig:alpha}
\end{figure}
Note that, at variance with the $E_{sym}$-$L_{sym}$ correlation, there is no "experimental" uncertainty on the energy or the density here. The blurring of the correlation can only come from the role of the high order empirical parameters. Comparing the results for different truncation orders in Fig.~\ref{fig:alpha} we can see that there is a noticeable correction in $\alpha_{K_{sym},i}$ induced by the third order parameter $Q_{NM}$, while the correction induced by the forth order parameter is less important.
Rectangular boxes are also drawn into Fig.~\ref{fig:alpha}.
They comprise the results obtained from models which correspond to values for the linear combinaison $-6\leq 3E_{sym}-L_{sym}\leq 80$.
This interval is obtained considering the conservative estimation for $E_{sym}$ and $L_{sym}$: $28\leq E_{sym}\leq 36$ and $30\leq L_{sym} \leq 90$~\cite{BALi2013,Margueron2018a}.
Note that this box contains all our $\sim$50 models, but it is possible to find other models out of this box, see for instance Ref.~\cite{Mondal2017}.
We deduce the following uncertainties for $\alpha_{K_{sym,}i}$ at different orders:
$\alpha_{K_{sym},2}=8\pm83$~MeV, $\alpha_{K_{sym},3}=50\pm95$~MeV, and $\alpha_{K_{sym},4}=58\pm93$~MeV.
We now represent in Fig.~\ref{fig:Ksym} the correlation between $K_{sym}$ and $3E_{sym}-L_{sym}$ within different cases: the different bands correspond to the correlation~(\ref{eq:ksymi}) at different orders $i=2$, 4 while the thinner band labelled 'Mondal2017' shows the result of the fit from Ref.~\cite{Mondal2017}. The points show the position of the $\sim$50 models as in previous figures.
We remark from Fig.~\ref{fig:Ksym} that the correlation~(\ref{eq:ksymi}) deduced from the condition $e_{NM}(x=-\nicefrac{1}{3})=0$~MeV is very consistent with the behavior of the $\sim$50 models as well as with the fit from Ref.~\cite{Mondal2017} (Mondal2017) where a larger number of models has been considered.
The dispersion in the fit Mondal2017 is however smaller than in our case, and Fig.~\ref{fig:Ksym} shows that many models are indeed out of the fit Mondal2017.
It was already clear from the results presented in Ref.~\cite{Mondal2017} that the dispersion of the fit was underestimating the one of the model sample.
The estimation of the dispersion of the correlation~(\ref{eq:ksymi}) obtained in our case is closer to the one of our models, as shown in Fig.~\ref{fig:Ksym}.
It seems to reproduce also very well the larger sample of model shown in Ref.~\cite{Mondal2017}.
The correlation~(\ref{eq:ksymi}) and its dispersion at orders $i=3$-4 are very close -- we have therefore represented only $i=4$ -- but they are slightly different from the correlation at order $i=2$, where most of the dispersion is generated by the uncertainty in $K_{sat}$.
The impact of adding the skewness parameter $Q_{NM}=Q_{sat}+Q_{sym}$ (at order $i=3$) is to shift up the correlation, improving the overlap with the $\sim$50 models.
We can therefore conclude that while most of the correlation~(\ref{eq:ksymi}) relies on the knowledge of $E_{sat}$ and $K_{sat}$ (isoscalar parameters), the role of the skewness parameter $Q_{NM}$ is also important to better reproduce the datum while the higher order parameter $Z_{NM}$ can here be neglected.
\begin{figure}[t]
\begin{center}
\includegraphics[angle=0,width=1.0\linewidth]{plot_Ksym-2.pdf}
\end{center}
\caption{(Color online) Correlation between the empirical parameters $K_{sym}$ and the variable $3E_{sym}-L_{sym}$
for different kind of nuclear interactions, as in Fig.~\ref{fig:EL2}.
The fit from Ref.~\cite{Mondal2017} is shown as well as our analytical expression taken with different order corrections.}
\label{fig:Ksym}
\end{figure}
Given the rather large dispersion in the correlation~(\ref{eq:ksymi}), the correlation coefficient extracted directly from the points largely depends on the model sampling.
While it is found to be $-0.91$ for the $\sim$50 models considered here, it decreases to $-0.48$ if we reduce the range of the x-axis to $30$-$60$~MeV as suggested by $\chi$EFT analyses~\cite{Tews2017,Drischler2016}: $28\leq E_{sym}\leq 36$~MeV and $40\leq L_{sym} \leq 60$~MeV.
To summarize, the correlation proposed in Ref.~\cite{Mondal2017} can be related to the very general condition $e_{NM}(x=-\nicefrac{1}{3})=0$~MeV and the dispersion is related to our uncertainty in the empirical parameter $K_{sat}$ in symmetric matter as well the skewness parameter $Q_{NM}$ in neutron matter, which is almost unknown. In the present case, we guesstimated its value and dispersion from a set of "realistic" models. The correlation coefficient depends largely on the dispersion of the model prediction for the variable $3E_{sym}-L_{sym}$.
The correlation proposed in Ref.~\cite{Mondal2017} is therefore very interesting but not very constraining within the present knowledge of nuclear physics.
\subsection{Correlation between $K_{sat}$ and $Q_{sat}$}
We have discussed in the previous sections that the physical correlations between empirical EoS parameters, such as the well-known correlation between
$E_{sym}$ and $L_{sym}$, or the more recently observed \cite{Mondal2017} correlation between $K_{sym}$ and $3E_{sym}-L_{sym}$, are largely blurred by our present poor knowledge on the high order parameters, both in the isoscalar and in the isovector sector.
A way to reduce this uncertainty could be to pin down these high order parameters from some existing correlation with the low order ones, which are more effectively constrained by experimental data.
For this reason we examine in the present section the correlation between $K_{sat}$ and $Q_{sat}$.
From the observation of our representative set of $\sim$50 EoS models, it comes out that this correlation is weak.
We found for the considered models a coefficient of 0.52, and if we reduce the sampling to the more realistic models for which $210<K_{sat}<250$~MeV, then the correlation coefficient drops down to 0.23.
We want here to understand what are the physical reasons of such an absence of correlation.
The density dependent incompressibility in symmetric matter (SM) is defined as $K_v(n)=9\,n\,\partial^2\epsilon/\partial n^2\,(\delta=0)$, with $\epsilon(n)=n \,e_{sat}(n)$.
Using the Taylor expansion Eqs.(\ref{eq:esat2})-(\ref{eq:esat4}), it reads,
\begin{eqnarray}
\frac{K_{v}(x)}{1+3x} &=& \left ( {1+9x}\right ) K_{sat} +x \left ({1+6x}\right ) Q_{sat}
+\frac{x^2}{2} \left ( {1+5x}\right ) Z_{sat} + ... \, \nonumber \\
\label{eq:Kv}
\end{eqnarray}
It was recently observed that the incompressibility $K_v$ calculated for different models crosses at a density of about $n_c = (0.71\pm0.01) n_{sat} = 0.114\pm0.002$~fm$^{-3}$, for a value which is
$K_{v,c}=37\pm 8$~MeV~\cite{Khan2012}, where the systematic dispersion between Skyrme and Gogny type models is included in the error-bars.
The reason of this behavior was understood from the fact that these different models have been calibrated to reproduce the experimental value of the GMR, which provides a constraint at the average density of finite nuclei. It is therefore not surprising that the value of the crossing density $n_c$ is close to the average density in nuclei $n_a$ previously introduced in Sec.~\ref{sec:elsym}.
Indeed the experimental value of the GMR turns out to be well correlated with the parameter $M_c$ defined as
$M_c=3 n_c \, \partial K_{v}/ \partial n\,(n=n_c)$, and an experimental value for $M_c=1050\pm100$~MeV was deduced from the correlation of this parameter with the ISGMR energy of Sn and Pb~\cite{Khan2012}.
The parameter $M_c$ can be deduced from Eq.~(\ref{eq:Kv}) as,
\begin{eqnarray}
\frac{M_{c}}{1+3x_c} &=& 6\left( 2 + 9x_c \right ) K_{sat} + \left ({1+18x_c+54x_c^2}\right ) Q_{sat}\nonumber \\
&&\hspace{1cm}+ x_c \left ({1+12x_c+30x_c^2}\right ) Z_{sat} + ... \, ,
\label{eq:Mc}
\end{eqnarray}
which, for a typical value $x_c\sim-0.1$, gives
\begin{eqnarray}
M_{c} \approx 4.6 \,K_{sat} - 0.18 \, Q_{sat} - 0.007 \, Z_{sat} + ... \, .
\label{eq:Mcapprox}
\end{eqnarray}
There is therefore still a very strong correlation between $M_c$ and $K_{sat}$, and the influence of $Q_{sat}$ is non-negligible but it remains small ($Q_{sat}$ is not well known, but typical values of nuclear models are of the order of a few $\pm K_{sat}$~\cite{Margueron2018a}).
For densities below $n_{sat}$, the coefficient in front of $Q_{sat}$ is bounded between $-0.5$ and $1$ and is even passing by zero at two densities: $n\approx0.12$-$0.13$~fm$^{-3}$ and $n\approx0.03$-$0.04$~fm$^{-3}$.
The impact of $Q_{sat}$ on the parameter $M_c$ is therefore quenched around $n_c$, explaining why the coefficient in front of $Q_{sat}$ in Eq.~(\ref{eq:Mcapprox}) is so small.
Since the energy of the ISGMR is very well correlated with $M_c$~\cite{Khan2012,Khan2013}, we can understand \textsl{a posteriori} why there is still a good correlation between the energy of the ISGMR and the empirical parameter $K_{sat}$. Such a correlation has been widely used to estimate the value of $K_{sat}$ from experimental measurement of the ISGMR since the seminal work of Blaizot~\cite{Blaizot1980}, see for instance Refs.~\cite{Colo2004,Colo2014,Khan2012}.
The impact of the other empirical parameters $Q_{sat}$ and $Z_{sat}$ remains small, but they are important for accurate and model independent determination of empirical parameters~\cite{Khan2012,Khan2013}.
As a consequence, a better knowledge of $Q_{sat}$ is necessary to reduce the uncertainty in the determination of $K_{sat}$.
Even if the impact of $Q_{sat}$ is small in Eq.~(\ref{eq:Mc}), it is still possible to use Eq.~(\ref{eq:Mc}) to express the following correlation between $K_{sat}$ and $Q_{sat}$,
\begin{equation}
Q_{sat,i} = \beta_{KQ} K_{sat} + \alpha_{KQ,i} \,
\label{eq:corrKQ2}
\end{equation}
where
\begin{equation}
\beta_{KQ} = -\frac{6(2 + 9x_c) }{ 1 + 18 x_c+54x_c^2} \, ,
\end{equation}
and
\begin{eqnarray}
\alpha_{KQ,3} = \alpha_1 M_c \, , \hspace{0.5 cm} \alpha_{KQ,4} &=& \alpha_1 M_c+\alpha_2 Z_{sat} \, ,
\end{eqnarray}
with
\begin{eqnarray}
\alpha_1 &=& \frac{1}{\left (1+3x_c\right )\left ( 1 + 18 x_c+54x_c^2\right ) } \, , \\
\alpha_2 &=& - x_c \frac{1+12x_c+30x_c^2}{1+18x_c+54x_c^2} \, .
\end{eqnarray}
Considering the uncertainty in $n_c$, we obtain $\beta_{KQ}=29\pm4$, $\alpha_1=-6.06\pm0.57$ and $\alpha_2=-0.0505\pm0.012$.
Considering in addition the uncertainty in $M_c$, we find $\alpha_{KQ,3}=-6300\pm1200$~MeV.
The value of $\alpha_{KQ,4}$ is a more difficult to calculate since it implies the parameter $Z_{sat}$ which is unknown.
Similar to the strategy of the previous sections, we evaluate $\alpha_{KQ,4}$ from our set of $\sim$50 nuclear models.
The result is shown in Fig.~\ref{fig:KQ3}, and a rectangle sets the most realistic boundaries under the assumption that $210<K_{sat}<250$~MeV,
giving $\alpha_{KQ,4}=-6650\pm1450$~MeV. The values allowed for $\alpha_{KQ,3}$ are also shown in Fig.~\ref{fig:KQ3}.
\begin{figure}[t]
\begin{center}
\includegraphics[angle=0,width=1.0\linewidth]{plot_KQsat-3.pdf}
\end{center}
\caption{(Color online) Expectations values for $\alpha_{KQ,i}$ as function of $K_{sat}$ for a set of different type of nuclear interactions, as in Fig.\ref{fig:EL1}. }
\label{fig:KQ3}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[angle=0,width=1.0\linewidth]{plot_KQsat-4.pdf}
\end{center}
\caption{(Color online) Correlation between the empirical parameters $Q_{sat}$ and $K_{sat}$
for different kind of nuclear interactions, as in Fig.\ref{fig:EL2}.}
\label{fig:KQ4}
\end{figure}
Combining the uncertainties in $\beta_{KQ}$, $\alpha_{KQ,3}$ and $\alpha_{KQ,4}$, we compare the correlation (\ref{eq:corrKQ2}) to the values of the $\sim$50 nuclear models in Fig.~\ref{fig:KQ4}.
We can see that the estimated band is wide enough to contain the predictions of all $\sim$50 nuclear models.
At variance with the previous correlations, the band of the $Q_{sat}$-$K_{sat}$ correlation is even wider than the actual spreading among the models.
From our analysis, this wide width comes from both the uncertainty in the crossing density $n_c$ and in the parameter $M_c$.
Since $\alpha_1\approx 6$, the uncertainty in $M_c$ is largely amplified for $Q_{sat}$.
There could be at least two reasons why the $\sim$50 models seem to have a smaller width than our prediction based on Eq.~(\ref{eq:Mc}).
The first reason is that there may be another constraint satisfied by the $\sim$50 models which tights the band's width, and which is not included in our analysis.
One may think for instance of the surface energy which provides constraint in the density dependence of the energy per particle and which is not included in our analysis.
It is however also known that phenomenological models exhibit spurious correlations in the $Q_{sat}$-$K_{sat}$ diagram, see for instance Refs.~\cite{Khan2012,Margueron2018a}.
So the other reason could be that the dispersion among the $\sim$50 models is artificially smaller than it should be in reality.
\begin{figure*}[tb]
\begin{center}
\includegraphics[angle=0,width=0.8\linewidth]{run8-p-plot-correlation-EFT2-eos-paper.pdf}
\end{center}
\vspace{-1.5cm}
\caption{(Color online) Correlation among the empirical parameters, completed with the effective mass $m_{sat}^*$, the effective mass splitting $\Delta m_{sat}^*$, the parameter $b$ and the variable $3E_{sym}-L_{sym}$. The correlation above the diagonal corresponds to the fit of the $\chi$EFT~\cite{Drischler2016} predictions only, while below the diagonal, the stability and causality conditions are added up to $0.4$~fm$^{-3}$. See text for discussion.}
\label{fig:MM}
\end{figure*}
Let us mention that we have also explored the $Q_{sat}$-$K_{sat}$ correlation generated by the crossing value $K_{v,c}$, as well as the one emerging from the spinodal condition -- though there is no experimental measurement of it. Our conclusion on the band width in these two cases are the same as the one found here based on the experimental measurement of $M_c$.
In summary, while the width of the $Q_{sat}$-$K_{sat}$ correlation is quite large from our analysis, we are able to determine an upper and lower bound in this correlation simply related to the experimental determination of the parameter $M_c$. The reason for the $Q_{sat}$-$K_{sat}$ correlation to be very weak is the small contribution of the parameter $Q_{sat}$ to the incompressibility in the region of densities around $n_c$. In the future, it would be interesting to estimate the reduction of the width of the $Q_{sat}$-$K_{sat}$ correlation induced by additional constraints, such as for instance the one provided by the surface energy.
\section{Meta-modelling analysis of the correlations}
\label{sec:MM}
In Sec.~\ref{sec:results}, we have employed the simple and analytical model presented in Sec.~\ref{sec:def} to perform various correlation analyses among empirical parameters, considering a given experimental constraints: $e_{sym}(n=\nicefrac{2}{3}n_{sat})$ for the $E_{sym}$-$L_{sym}$ correlation, $e_{NM}(n=0)$ for the $K_{sym}$-$(3E_{sym}-L_{sym})$ correlation and $M_c$ for the $K_{sat}$-$Q_{sat}$ correlation. These correlations are blurred by the presence of other empirical parameters, which aren't known for most of them. The estimation of the importance of the blurring is therefore not an easy task. In order to circumvent this issue, we used a set of $\sim$50 nuclear functionals to estimate the dispersion among the unknown empirical parameters. The parameters of these $\sim$50 functionals have been optimized on different nuclear structure data, meaning that we can consider that experimental constraints from low energy nuclear experiments are somehow implicitly accounted in the choice of parameters. These low density constraints are however not always sufficient to pin down the behavior of high order parameters, both via direct measurement or via the exploitation of correlations with low order parameters. Indeed, the correlations involving high order parameters observed in existing phenomenological nuclear models are essentially induced by the assumed functional form and do not reflect physical constraints. For this reason, the correlations analyzed in Sec.~\ref{sec:results} may still potentially contain some model dependence.
To overcome this problem, we have recently proposed~\cite{Margueron2018a,Margueron2018b} a meta-modelling formulation of the EoS employing i) a functional form flexible enough to be able to reproduce within its parameter space, most relativistic and non-relativistic functionals, including ab-initio ones; ii) no \textsl{a priori} correlation among the empirical parameters -- such that we can consider a portion of the parameter space which is not explored by existing models; iii) an \textsl{a posteriori} filtering of the huge parameter space with basic physical requirements (stability and causality) and the existing constraints from \textsl{ab initio} approaches, such as the MBPT based on $\chi$EFT interactions~\cite{Drischler2016}.
We consider in this section the metamodel -- version ELF-c -- of Ref.~\cite{Margueron2018a} which is determined from a given set of empirical parameters, see Eqs.(\ref{eq:empIS})-(\ref{eq:empIV}), from the effective mass $m_{sat}^*$ defined at $n_{sat}$ in symmetric matter, from the effective mass splitting $\Delta m_{sat}^*/m = m_{n}^*/m-m_{p}^*/m$ defined at $n_{sat}$ in neutron matter, and from the parameter $b$ which incorporates, at low density, the effects of the neglected high order terms in the series expansion, see Ref.~\cite{Margueron2018a} for more details. These parameters are sampled as in Ref.~\cite{Margueron2018b} and they are first filtered against the MBPT predictions based on $\chi$EFT interactions~\cite{Drischler2016} in symmetric and neutron matter, in a similar way as it has been done in Ref.~\cite{Tews2018}.
Since we may want to control the behaviour of the selected models above saturation density -- within a reasonable range -- we have additionally imposed the stability and causality condition up to 0.4~fm$^{-3}$.
We have calculated the correlation coefficient among the 13 parameters of the model, plus the combinaison $3E_{sym}-L_{sym}$, for the following two selection conditions: i) the models selected only from the MBPT predictions in symmetric and neutron matter based on six chiral EFT interactions~\cite{Drischler2016}, and ii) the models additionally filtered against stability and causality. The bayesian selection mentioned in i) assumes that the theoretical MBPT predictions could be used in the definition of a likelihood probability where the theoretical centroid and uncertainty for the binding energy and the baryon pressure define a $\chi^2$. Each model set is weighted with the likelihood probability $p=\exp[-\chi^2 / (2 N_{dof})]$ where $N_{dof}=N_{tot}-13$, $N_{tot}=32$ for 8 density points from 0.04 to 0.20~fm$^{-3}$. Note that more evolved bayesian analyses could be perform, see for instance Ref.~\cite{Melendez:2017}. While the details of the marginalized posterior probabilities certainly depend on the bayesian prescription, the gross correlation properties shown in this study are much less impacted.
The results are shown in Fig.~\ref{fig:MM}, where the correlation coefficients above the diagonal are obtained from the selection condition i), and the ones below the diagonal from the selection condition ii).
There is a general agreement for the correlation coefficients obtained from conditions i) and ii), with some exceptions. For instance, the $Q_{sat}$-$Z_{sat}$ correlation is very weak in the case i) while it is very large in the case ii). It is simply due to the stability and causality conditions which bring strong constraints above saturation density, as expected. $K_{sym}$ is more correlated with $Q_{sat}$ and $Z_{sat}$ in the case ii) than in the case i). The $K_{sym}$-$Q_{sym}$ correlation is weaker in case ii) compared to case i). Despite these few exceptions, the correlation coefficients are rather stable and independent of the additional filtering against stability and causality.
The $E_{sym}$-$L_{sym}$, the $K_{sym}$-$(3E_{sym}-L_{sym})$, and the $K_{sat}$-$Q_{sat}$ correlation coefficients shown in Fig.~\ref{fig:MM} essentially confirm our previous analysis in Sec.~\ref{sec:results}. The correlation coefficient for the $E_{sym}$-$L_{sym}$ correlation is estimated to be 0.52-0.53, which is not so different from the one deduced from the $\sim$50 functionals and imposing
$28<E_{sym}<36$~MeV. The blurring of the $E_{sym}$-$L_{sym}$ correlation shown in Fig.~\ref{fig:EL2} can therefore be considered as realistic of the model dependence of this correlation.
The anticorrelation coefficient for the $K_{sym}$-$(3E_{sym}-L_{sym})$ correlation is estimated to be -0.47-(-0.61), which is also similar to the one deduced from the $\sim$50 functionals and imposing $28\leq E_{sym}\leq 36$~MeV and $40\leq L_{sym} \leq 60$~MeV.
The extremely weak correlation coefficient for the $Q_{sat}$-$K_{sat}$ correlation -- 0.17-(-0.06) -- shown in Fig.~\ref{fig:MM} reflects our conclusions from our previous analysis as well: the $Q_{sat}$ and $K_{sat}$ empirical parameters are very weakly correlated by either the experimental parameter $M_c$ or by the MBPT predictions in symmetric matter below saturation density.
\section{Conclusions and outlooks}
In this paper, we have examined the quality of the correlations among the EoS empirical parameters, coming from the existence of general physics constraints on the EoS, as well as from empirical measurements.
Specifically, we have analyzed the origin and the model dependence of the correlation between $E_{sym}$ and $L_{sym}$, largely observed in the literature, as well as the correlation between $K_{sym}$ and $3E_{sym}-L_{sym}$, recently proposed~\cite{Mondal2017}, and we have further analyzed the reason of the very weak $Q_{sat}$-$K_{sat}$ correlation.
Within a simple analytical Taylor expansion of the EoS around saturation, we have confirmed that the $E_{sym}$-$L_{sym}$ correlation arises from the empirical knowledge of the symmetry energy at density slightly below saturation, as obtained for example from the IVGDR measurement, and we have estimated its width coming from model dependence. We have found that the main source of uncertainties in this correlation is coming from $K_{sym}$ while the contribution of the higher order empirical parameters ($Q_{sym}$ and $Z_{sym}$) are negligible.
Concerning the correlation $K_{sym}$-$(3E_{sym}-L_{sym})$, we have shown that it trivially emerges from the boundary condition on the neutron matter energy density, which explains why it is universally respected. However, when only the functionals corresponding to realistic values of $E_{sym}$ and $L_{sym}$ are retained, the quality of the correlation considerably worsens.
Finally, we found that $Q_{sat}$ and $K_{sat}$ are weakly correlated, as expected from previous studies~\cite{Khan2012,Margueron2018a}.
These results have been confirmed within a more evolved meta-modeling of the EoS.
We have explained the origin of the dispersion among these correlations from the effect of the high order EoS parameters.
Indeed, while the values of $E_{sym}$ and $L_{sym}$ are relatively close among the different functionals and functional families, high order parameters
such as the isovector incompressibility $K_{sym}$ and the skewness and kurtosis $Q_{sym},Q_{sat},Z_{sym},Z_{sat}$ are largely model dependent.
For phenomenological approaches, this model dependence is mainly due to the small number of free parameters and to the absence of experimental constraints.
The high order empirical parameters are functions of the same model-coefficient as the low order ones, inducing such kind of spurious correlations.
We have shown that the dispersion of the $E_{sym}$-$L_{sym}$ correlation can be nicely understood from the propagation of the uncertainties of $K_{sym}$,
while the $K_{sym}$-$(3E_{sym}-L_{sym})$ correlation is mostly affected by the uncertainty on $Q_{sat}$ and $Q_{sym}$.
The weak $Q_{sat}$-$K_{sat}$ correlation induced by either the experimental parameter $M_c$ or the MBPT predictions can be explained from the small contribution of $Q_{sat}$ to the incompressibility below saturation density. The determination of $Q_{sat}$ shall therefore be better constrained by experiments probing matter properties above saturation density, such as for instance heavy-ion collisions.
In conclusion, we have illustrated the complexity in determining the empirical parameters of nuclear matter from correlations between an observable and a single empirical parameter. In the present cases, we have shown the important contribution of the unknown empirical parameter $K_{sym}$ (resp. $Q_{NM}$) on the blurring of the $E_{sym}$-$L_{sym}$ (resp. $K_{sym}$-$(3E_{sym}-L_{sym})$) correlation.
This complexity suggests that in the future, multi-parameter correlation analyses -- satisfying a set of experimental contraints -- shall better be performed to provide better posterior probabilities for the empirical parameters. A meta-modeling, such as the one employed here, is a well adapted tool to perform such statistical analyses.
\begin{acknowledgments}
This work was partially supported by the IN2P3 Master Project MAC, "NewCompStar" COST Action MP1304, PHAROS COST Action MP16214.
\end{acknowledgments}
|
1,116,691,497,557 | arxiv | \section{Introduction}
\label{Sect:Intro}
The hadronic contribution to the muon anomalous magnetic moment
$a_{\mu} = (g_{\mu}-2)/2$ represents one of the long--standing
challenging issues of elementary particle physics, which engages
the entire pattern of interactions within the Standard Model.
The~experimental measurements~\cite{BNL06, FNAL21} and theoretical
evaluations (see a recent comprehensive review~\cite{WP20}, which
is mainly based
on Refs.~\cite{Davier:2010nc, Davier:2017zfy, Keshavarzi:2018mgv,
Colangelo:2018mtw, Hoferichter:2019mqg, Davier:2019can, Keshavarzi:2019abf,
Kurz:2014wya, FermilabLattice:2017wgj, Budapest-Marseille-Wuppertal:2017okr,
RBC:2018dos, Giusti:2019xct, Shintani:2019wai, FermilabLattice:2019ugu,
Gerardin:2019rua, Aubin:2019usy, Giusti:2019hkz, Melnikov:2003xd,
Masjuan:2017tvw, Colangelo:2017fiz, Hoferichter:2018kwz, Gerardin:2019vio,
Bijnens:2019ghy, Colangelo:2019uex, Pauk:2014rta, Danilkin:2016hnh,
Jegerlehner:2017gek, Knecht:2018sci, Eichmann:2019bqf, Roig:2019reh,
Colangelo:2014qya, Blum:2019ugy, Aoyama:2012wk, Aoyama:2019ryr,
Czarnecki:2002nt, Gnendiger:2013pva}) of this quantity have achieved
an impressive
accuracy, and the remaining discrepancy of the order of a few standard
deviations between them may be an evidence for the existence of a new
fundamental physics beyond the Standard Model. The~uncertainty of
theoretical estimation of~$a_{\mu}$ is largely dominated by the
hadronic contribution, which involves the tangled dynamics of colored
fields in the infrared domain inaccessible within perturbation theory.
There are basically two approaches to the theoretical assessment
of the hadronic vacuum polarization contributions to the muon
anomalous magnetic moment~$a^{\text{HVP}}_{\mu}$.
Specifically, in the framework
of the first (``spacelike'') approach~$a^{\text{HVP}}_{\mu}$ is commonly
represented as the integral of the hadronic vacuum polarization
function~$\bar\Pi(Q^2)$ [or~the related Adler function~$D(Q^2)$]
with corresponding kernel functions~$K_{\Pi}(Q^2)$ [or~$K_{D}(Q^2)$]
over the entire kinematic interval. Here the perturbative results
for the involved functions~$\bar\Pi(Q^2)$ and~$D(Q^2)$ have to be
supplemented with the relevant nonperturbative inputs. The latter can
be provided~by,~e.g., lattice simulations~\cite{Lattice1, Lattice2, BMW21}
(which, being capable of delivering valuable insights into the
underlying mechanisms, have a large scientific potential), highly
anticipated MUonE measurements~\cite{MUonE1, MUonE2, MUonE3},
and other methods.
Alternatively, in the framework of the second (``timelike'')
approach~$a^{\text{HVP}}_{\mu}$ can also be represented as
the integral of the
function~$R(s)$ with respective kernel functions~$K_{R}(s)$.
Here the perturbative results for the function~$R(s)$ are usually
complemented~by the low--energy experimental data on
the~\mbox{$R$--ratio} of electron--positron annihilation into
hadrons, that constitutes the data--driven method of evaluation
of~$a^{\text{HVP}}_{\mu}$. In~turn, the ``spacelike'' and
``timelike'' kernel
functions can be calculated within various techniques, such~as
the mass operator approach~\cite{Schwinger, Milton74a, Milton74b},
the hyperspherical approach~\cite{HS1, HS2, HS3, HS4},
the dispersive method~\cite{DispMeth1, DispMeth2, DispMeth3, DispMeth4,
DispMeth5}, and the asymptotic expansion method~\cite{Smirnov12,
Krause96, Kurz:2014wya}. The~``timelike'' kernel functions have been
extensively studied over the past decades, whereas
the corresponding ``spacelike'' kernel functions remain largely
unavailable.
The primary objective of this paper is to derive the complete set of
relations, which mutually express the~``spacelike'' and~``timelike''
kernel functions~$K_{\Pi}(Q^2)$, $K_{D}(Q^2)$, and~$K_{R}(s)$ in terms
of each other, and to calculate the explicit expression for the
next--to--leading order ``spacelike'' kernel function~$\KG{\Pi}{3a}(Q^2)$
by making use of the obtained relations.
The layout of the paper is as follows. Section~\ref{Sect:Methods}
recaps the essentials of the dispersion relations for the hadronic
vacuum polarization function~$\bar\Pi(Q^2)$, the Adler function~$D(Q^2)$,
and~the function~$R(s)$, and expounds the basics of the hadronic
vacuum polarization contributions to the muon anomalous magnetic
moment. In~Sect.~\ref{Sect:Results} the complete set of relations,
which mutually express the kernel functions~$K_{\Pi}(Q^2)$, $K_{D}(Q^2)$,
and~$K_{R}(s)$ in terms of each other, is~obtained, and the explicit
expression for the ``spacelike'' kernel function~$\KG{\Pi}{3a}(Q^2)$
is~calculated. Section~\ref{Sect:Concl} summarizes the basic results.
The ``timelike'' kernel function~$\KG{R}{3a}(s)$ is given in
the App.~\ref{Sect:KR3aExpl}.
\section{Methods}
\label{Sect:Methods}
\subsection{General dispersion relations}
\label{Sect:GDR}
Let us begin by briefly elucidating the essentials of dispersion
relations for the hadronic vacuum polarization function~$\Pi(q^2)$,
the Adler function~$D(Q^2)$, and the function~$R(s)$ (the~detailed
description of this issue can be found in,~e.g., Chap.~1 of
Ref.~\cite{Book} and references therein). The theoretical
exploration of a certain class of the strong interaction
processes is primarily based on the hadronic vacuum polarization
function~$\Pi(q^2)$, which is defined as the scalar part of the
hadronic vacuum polarization tensor
\begin{equation}
\label{P_Def}
\Pi_{\mu\nu}(q^2) = i\!\int\!d^4x\,e^{i q x} \bigl\langle 0 \bigl|\,
T\!\left\{J_{\mu}(x)\, J_{\nu}(0)\right\} \bigr| 0 \bigr\rangle =
\frac{i}{12\pi^2} (q_{\mu}q_{\nu} - g_{\mu\nu}q^2) \Pi(q^2).
\end{equation}
As discussed in, e.g., Ref.~\cite{Feynman}, the
function~$\Pi(q^2)$~(\ref{P_Def}) has the
only cut along the positive semiaxis of real~$q^2$ starting at the
hadronic production threshold~$q^2 \ge s_{0}$, that leads~to
\begin{equation}
\label{PDisp}
\Delta\Pi(q^{2},q_{0}^{2}) =
(q^2-q_{0}^{2})\int\limits_{s_{0}}^{\infty}
\frac{R(\sigma)}{(\sigma-q^2)(\sigma-q_0^2)}\, d\sigma,
\end{equation}
where
\begin{equation}
\label{PSubDef}
\Delta\Pi(q^{2},q_{0}^{2}) = \Pi(q^2) - \Pi(q_0^2),
\qquad
\Pi(0)=0,
\qquad
\Delta\Pi(0,-p^2)=-\Pi(-p^2)=\bar\Pi(p^2),
\end{equation}
and
\begin{equation}
\label{RDefP}
R(s) = \frac{1}{2 \pi i} \lim_{\varepsilon \to 0_{+}}
\Bigl[\Pi(s + i \varepsilon) - \Pi(s - i \varepsilon)\Bigr].
\end{equation}
The function~$R(s)$~(\ref{RDefP}) is commonly identified with the
so--called $R$--ratio of electron--positron annihilation into hadrons
\begin{equation}
\label{RDefExp}
R(s) =
\frac{\sigma(e^{+}e^{-} \to \text{hadrons}; s)}{\sigma(e^{+}e^{-} \to
\mu^{+}\mu^{-}; s)}\,,
\end{equation}
where \mbox{$s=q^2>0$} is the timelike
kinematic variable, namely, the center--of--mass energy squared.
At~the same time, in~practical applications it proves to be
convenient to deal with the Adler function~\cite{Adler}
\begin{equation}
\label{GDR_DP}
D(Q^2) = -\,\frac{d\, \Pi(-Q^2)}{d \ln Q^2},
\end{equation}
where~$Q^2=-q^2>0$ stands for the spacelike kinematic variable. The
corresponding dispersion relation~\cite{Adler}
\begin{equation}
\label{GDR_DR}
D(Q^2) =
Q^2 \int\limits_{s_{0}}^{\infty}
\frac{R(\sigma)}{(\sigma+Q^2)^2}\, d\sigma
\end{equation}
immediately follows from Eqs.~(\ref{PDisp}) and~(\ref{GDR_DP}).
\subsection{Hadronic vacuum polarization contributions to~$a_{\mu}$}
\label{AmuHVP}
In the framework of the ``timelike'' (or~data--driven) method of
assessment of the hadronic vacuum polarization contributions to
the muon anomalous magnetic moment the latter is commonly represented in
terms of the $R$--ratio of electron--positron annihilation into
hadrons~(\ref{RDefP}). In~the leading order of perturbation theory
(namely, in the second order in the electromagnetic coupling)
the corresponding contribution is given by the diagram displayed
in Fig.~\ref{Plot:Amu2}, that yields~\cite{BKK56, BM61, KO67}
\begin{equation}
\label{Amu2Def}
\amu{2} = \frac{1}{3} \Bigl(\frac{\alpha}{\pi}\Bigr)^{\!2}
\!\int\limits_{s_{0}}^{\infty}
\frac{G_{2}(s)}{s} R(s) ds,
\end{equation}
where
\begin{equation}
\label{K2RInt}
G_{2}(s) = \int\limits_{0}^{1}\!
\frac{x^2 (1-x)}{x^2 + (1-x) s/m_{\mu}^2}\,dx
\end{equation}
and~$s=q^2 \ge 0$ denotes the timelike kinematic variable.
The kernel function~$G_{2}(s)$~(\ref{K2RInt}) can also be
represented in explicit form~\cite{BKK56, D6263, BdR67, LdR68} and
the expression appropriate for the practical
applications reads
\begin{equation}
\label{K2RExpl}
G_{2}(s) = \frac{1}{2} + 4\eta\Bigl[(2\eta-1)\ln(4\eta)-1\Bigr]
-2\Bigl[2(2\eta-1)^2-1\Bigr]
\frac{A(\eta)}{\psi(\eta)},
\end{equation}
where
\begin{equation}
\label{DefAux1}
\psi(\eta) = \frac{\sqrt{\eta-1}}{\sqrt{\eta}},
\qquad
A(\eta) = {\rm arctanh}\Bigl[\psi(\eta)\Bigr],
\qquad
\eta=\frac{s}{4m_{\mu}^2}.
\end{equation}
\begin{figure}[t]
\centerline{\includegraphics[height=50mm,clip]{AmuHVP1.pdf}}
\caption{The leading--order hadronic vacuum polarization
contribution to the muon anomalous magnetic moment~(\ref{Amu2Def}).}
\label{Plot:Amu2}
\end{figure}
Factually, the specific form of the kernel
function~$G_{2}(s)$~(\ref{K2RInt})
makes it possible to express the leading--order
contribution~(\ref{Amu2Def})
in terms of the ``spacelike'' hadronic vacuum polarization
function~$\bar\Pi(Q^2)$ [Eqs.~(\ref{PDisp}),~(\ref{PSubDef})]
and the Adler function~$D(Q^2)$~(\ref{GDR_DP}),
but only in this particular case
(a~discussion of this issue can
be found in, e.g.,~Ref.~\cite{EdR17}). Namely, Eqs.~(\ref{Amu2Def}),
(\ref{K2RInt}),~(\ref{PDisp}), and~(\ref{PSubDef}) can be
reduced~to~\cite{LPdR71}
\begin{align}
\label{Amu2Px}
\amu{2} & = \frac{1}{3} \Bigl(\frac{\alpha}{\pi}\Bigr)^{\!2}
\!\int\limits_{0}^{1}\! d x (1-x)
\!\!\int\limits_{s_{0}}^{\infty}\!
\frac{d s}{s}
\frac{m_{\mu}^2 x^2 (1-x)^{-1}}{s + m_{\mu}^2 x^2 (1-x)^{-1}} R(s)
= \nonumber \\ &
= \frac{1}{3} \Bigl(\frac{\alpha}{\pi}\Bigr)^{\!2}
\!\int\limits_{0}^{1}\! (1-x)
\bar\Pi\biggl(\!m_{\mu}^2\frac{x^2}{1-x}\biggr) d x,
\end{align}
where~$\bar\Pi(Q^2)=\Delta\Pi(0,-Q^2)$ and~$Q^2 = -q^2 \ge 0$
stands for the spacelike kinematic variable.
In~turn, Eq.~(\ref{Amu2Px}) can also be represented in terms
of the Adler function~(\ref{GDR_DP}) by making use of the
integration by parts, that eventually yields~\cite{Knecht2004, EdR17}
\begin{equation}
\label{Amu2Dx}
\amu{2} = \frac{1}{3} \Bigl(\frac{\alpha}{\pi}\Bigr)^{\!2}
\!\int\limits_{0}^{1}\! (1-x) \biggl(\!1-\frac{x}{2}\biggr)
D\biggl(\!m_{\mu}^2\frac{x^2}{1-x}\biggr) \frac{d x}{x}.
\end{equation}
It~is necessary to emphasize here that this way of the
derivation of the ``spacelike'' expressions~(\ref{Amu2Px})
and~(\ref{Amu2Dx})
from the ``timelike'' one~(\ref{Amu2Def}) entirely relies on the
particular form of the leading--order kernel
function~$G_{2}(s)$~(\ref{K2RInt})
and cannot be performed in any other case.
\bigskip
\begin{figure}[t]
\centerline{%
\begin{tabular}{lcr}
\includegraphics[height=50mm,clip]{AmuHVP2a1.pdf} & &
\includegraphics[height=50mm,clip]{AmuHVP2a2.pdf} \\
\end{tabular}}
\caption{Two of the diagrams contributing to~$\amu{3a}$~(\ref{Amu3aDef}).}
\label{Plot:Amu3a}
\end{figure}
In~the next--to--leading order of perturbation theory
(namely, in the third order in the electromagnetic coupling)
the hadronic vacuum polarization contribution to
the muon anomalous magnetic moment~$\amu{3}$ is composed
of three parts. Specifically, the first part~$\amu{3a}$ corresponds
to the diagrams, which include (in~addition to the hadronic insertion
shown in~Fig.~\ref{Plot:Amu2}) one photon line or closed muon
loop, see~Fig.~\ref{Plot:Amu3a}. In~turn, the second part~$\amu{3b}$
corresponds to the diagrams, which additionally include one closed
electron (or~\mbox{$\tau$--lepton}) loop, whereas the
third part~$\amu{3c}$ corresponds to the diagram with double
hadronic insertion. In what follows we shall primarily focus on the
first part of~$\amu{3}$, which can be represented~as
\begin{equation}
\label{Amu3aDef}
\amu{3a} = \frac{2}{3} \Bigl(\frac{\alpha}{\pi}\Bigr)^{\!3}
\!\int\limits_{s_{0}}^{\infty}
\frac{G_{3a}(s)}{s} R(s) ds.
\end{equation}
The ``timelike'' kernel function~$G_{3a}(s)$ entering this equation
has been calculated explicitly in Ref.~\cite{DispMeth5},
see~App.~\ref{Sect:KR3aExpl}. However, the explicit form of the
corresponding kernel functions required for the assessment
of~$\amu{3a}$ within ``spacelike'' methods is still unavailable.
\section{Results and discussion}
\label{Sect:Results}
\subsection{Relations between the kernel functions}
\label{Sect:Rels}
First of all, for practical purposes it is convenient to represent the
hadronic vacuum polarization contribution to the muon
anomalous magnetic moment, which corresponds to the
\mbox{$\ell$--th}~order in the electromagnetic coupling,
in the following form
\begin{subequations}
\label{Amu}
\begin{align}
\label{AmuP}
\amu{\ell} & = \APF{\ell}\!\!\int\limits_{0}^{\infty}\!
\KG{\Pi}{\ell}(Q^2) \bar\Pi(Q^2) \frac{d Q^2}{4m_{\mu}^2} =
\APF{\ell}\!\!\int\limits_{0}^{\infty}\!
\KGt{\Pi}{\ell}(\zeta) \bar\Pi(4\zetam_{\mu}^2) d \zeta =
\\[1.25mm]
\label{AmuD}
& = \APF{\ell}\!\!\int\limits_{0}^{\infty} \!
\KG{D}{\ell}(Q^2) D(Q^2) \frac{d Q^2}{4m_{\mu}^2} =
\APF{\ell}\!\!\int\limits_{0}^{\infty} \!
\KGt{D}{\ell}(\zeta) D(4\zetam_{\mu}^2) d \zeta = \\[1.25mm]
\label{AmuR}
& = \APF{\ell}\!\!\int\limits_{s_{0}}^{\infty} \!
\KG{R}{\ell}(s) R(s) \frac{d s}{4m_{\mu}^2} =
\APF{\ell}\!\!\int\limits_{\chi}^{\infty} \!
\KGt{R}{\ell}(\eta) R(4\etam_{\mu}^2) d \eta.
\end{align}
\end{subequations}
In~this equation~$\APF{\ell}$ is a constant prefactor,
$\bar\Pi(Q^2)=\Delta\Pi(0,-Q^2)$ is~defined in~Eq.~(\ref{PSubDef}),
$Q^2 = -q^2 \ge 0$ and~$s = q^2 \ge 0$ stand, respectively,
for the spacelike and timelike kinematic variables,
$\zeta = Q^2/(4m_{\mu}^2)$ and~$\eta = s/(4m_{\mu}^2)$ denote
the dimensionless kinematic variables, and~\mbox{$\chi=s_{0}/(4m_{\mu}^2)$}.
For~example, for the leading--order hadronic vacuum polarization
contribution~(\ref{Amu2Def})
\begin{equation}
\label{KR2eta}
\APF{2} = \frac{1}{3} \Bigl(\frac{\alpha}{\pi}\Bigr)^{\!2},
\qquad
\KGt{R}{2}(\eta) = \KG{R}{2}(4\etam_{\mu}^2) =
G_{2}(4\etam_{\mu}^2) \frac{1}{\eta}.
\end{equation}
\begin{figure}[t]
\centerline{\includegraphics[width=75mm,clip]{contour1.pdf}}
\caption{The closed integration contour~$C$ in the complex~$q^2$--plane
in Eq.~(\ref{IntC1}). The physical cut~$q^2 \ge s_{0}$ of
the hadronic vacuum polarization
function~$\Pi(q^2)=-\bar\Pi(-q^2)$~(\ref{PSubDef})
is shown along the positive semiaxis of real~$q^2$,
whereas the physical cut~\mbox{$q^2 \le 0$} of
the ``timelike'' kernel function~$K_{R}(q^2)$~(\ref{AmuR})
is shown along the negative
semiaxis of real~$q^2$.}
\label{Plot:Contour1}
\end{figure}
In~fact, the kernel functions~$K_{\Pi}(Q^2)$, $K_{D}(Q^2)$,
and~$K_{R}(s)$ appearing in Eq.~(\ref{Amu}) can all be
expressed in terms of each other. Let us begin by
expressing the ``spacelike'' kernel
function~$K_{\Pi}(Q^2)$~(\ref{AmuP})
in terms of the ``timelike'' one~$K_{R}(s)$~(\ref{AmuR}).
As mentioned earlier, the hadronic vacuum polarization
function~$\Pi(q^2)=-\bar\Pi(-q^2)$~(\ref{PSubDef})
possesses the only cut along the positive semiaxis
of real~$q^2$ starting at the hadronic production
threshold~$q^2 \ge s_{0}$, whereas the kernel function
$K_{R}(q^2)$~(\ref{AmuR}) has the only cut along the negative
semiaxis of real~$q^2$ starting at the origin~$q^2 \le 0$,
see, e.g., Ref.~\cite{DispMeth5}. Therefore, the integral of
the product of the functions~$K_{R}(q^2)$~(\ref{AmuR})
and~$\bar\Pi(-q^2)$~(\ref{PSubDef}) along
the contour displayed in Fig.~\ref{Plot:Contour1} vanishes,
namely
\begin{equation}
\label{IntC1}
\oint_{C} K_{R}(q^2) \bar\Pi(-q^2) d q^2 = 0,
\end{equation}
that can also be represented~as
\begin{align}
\label{IntC2}
& \int\limits_{\infty-i\varepsilon}^{s_{0}-i\varepsilon}
K_{R}(q^2) \bar\Pi(-q^2) d q^2
+ \int\limits_{s_{0}+i\varepsilon}^{\infty+i\varepsilon}
K_{R}(q^2) \bar\Pi(-q^2) d q^2
+ \nonumber \\ &
+ \int\limits_{-\infty+i\varepsilon}^{i\varepsilon}
K_{R}(q^2) \bar\Pi(-q^2) d q^2
+ \int\limits_{-i\varepsilon}^{-\infty-i\varepsilon}
K_{R}(q^2) \bar\Pi(-q^2) d q^2 = 0.
\end{align}
The change of the integration variables~$q^2 = p^2 - i\varepsilon$
in the first and fourth terms of Eq.~(\ref{IntC2})
and~$q^2 = p^2 + i\varepsilon$ in its second and third terms
casts Eq.~(\ref{IntC2})~to
\begin{equation}
\label{IntC3}
- \frac{1}{2 \pi i} \lim_{\varepsilon \to 0_{+}}
\int\limits_{0}^{-\infty} \bar\Pi(-p^2)
\Bigl[ K_{R}(p^2+i\varepsilon) - K_{R}(p^2-i\varepsilon)\Bigr]
d p^2 =
\int\limits_{s_{0}}^{\infty} K_{R}(p^2) R(p^2) d p^2,
\end{equation}
where the limit~$\varepsilon \to 0_{+}$
is assumed and Eq.~(\ref{RDefP}) is employed.
Then the change of the integration variables~$p^2=-Q^2$
on the left--hand side of Eq.~(\ref{IntC3}) and~$p^2=s$
on its right--hand side leads~to
\begin{equation}
\label{IntC4}
\int\limits_{0}^{\infty} \bar\Pi(Q^2) K_{\Pi}(Q^2) d Q^2 =
\int\limits_{s_{0}}^{\infty} K_{R}(s) R(s) d s,
\end{equation}
where
\begin{equation}
\label{KRelPR}
K_{\Pi}(Q^2) = \frac{1}{2 \pi i} \lim_{\varepsilon \to 0_{+}}
\Bigl[ K_{R}(-Q^2+i\varepsilon) - K_{R}(-Q^2-i\varepsilon)\Bigr],
\qquad
Q^2 \ge 0.
\end{equation}
This relation has also been independently derived in a different way
in~Ref.~\cite{BLP}.
In turn, the relation inverse to Eq.~(\ref{KRelPR}) directly follows
from Eqs.~(\ref{AmuP}) and~(\ref{PDisp}), specifically
\begin{equation}
\int\limits_{0}^{\infty}
K_{\Pi}(Q^2) \bar\Pi(Q^2) \frac{d Q^2}{4m_{\mu}^2} =
\int\limits_{0}^{\infty} \frac{d Q^2}{4m_{\mu}^2}
K_{\Pi}(Q^2)\, Q^2\!\! \int\limits_{s_{0}}^{\infty}
\frac{d s}{s}\, \frac{R(s)}{s+Q^2} =
\int\limits_{s_{0}}^{\infty}
K_{R}(s) R(s) \frac{d s}{4m_{\mu}^2},
\end{equation}
where
\begin{equation}
\label{KRelRP}
K_{R}(s) = \frac{1}{s} \int\limits_{0}^{\infty}
K_{\Pi}(Q^2) \frac{Q^2}{s + Q^2}\, d Q^2,
\qquad
s \ge 0.
\end{equation}
The ``timelike'' kernel function~$K_{R}(s)$~(\ref{AmuR}) can be
expressed in terms of the ``spacelike'' one~$K_{D}(Q^2)$~(\ref{AmuD})
in a similar way. In~particular, Eqs.~(\ref{AmuD}) and~(\ref{GDR_DR})
imply
\begin{equation}
\int\limits_{0}^{\infty}
K_{D}(Q^2) D(Q^2) \frac{d Q^2}{4m_{\mu}^2} =
\int\limits_{0}^{\infty} \frac{d Q^2}{4m_{\mu}^2}
K_{D}(Q^2)\, Q^2\!\! \int\limits_{s_{0}}^{\infty}
\frac{R(s)}{(s+Q^2)^2}\, d s =
\int\limits_{s_{0}}^{\infty}
K_{R}(s) R(s) \frac{d s}{4m_{\mu}^2},
\end{equation}
where
\begin{equation}
\label{KRelRD}
K_{R}(s) = \int\limits_{0}^{\infty}
K_{D}(Q^2) \frac{Q^2}{(s + Q^2)^2}\, d Q^2,
\qquad
s \ge 0.
\end{equation}
In turn, the corresponding relation between the kernel
functions~$K_{\Pi}(Q^2)$~(\ref{AmuP})
and $K_{D}(Q^2)$~(\ref{AmuD}) can be derived from Eqs.~(\ref{GDR_DP})
and~(\ref{AmuD}), namely
\begin{align}
\label{KRelPDaux1}
& \int\limits_{0}^{\infty}
K_{D}(Q^2) D(Q^2) d Q^2 =
-\int\limits_{0}^{\infty} d Q^2
K_{D}(Q^2)\, Q^2 \,\frac{d\,\Pi(-Q^2)}{d\, Q^2} =
\nonumber \\[1.25mm] &
= K_{D}(Q^2)\, Q^2 \,\bar\Pi(Q^2) \Biggr|_{0}^{\infty}
- \int\limits_{0}^{\infty} \bar\Pi(Q^2)
\biggl[K_{D}(Q^2) + \frac{d\,K_{D}(Q^2)}{d\,\ln Q^2}\biggr] d Q^2,
\end{align}
with the integration by parts being used. Since the first term of
this equation vanishes (see also remarks given below),
Eqs.~(\ref{AmuP}) and~(\ref{KRelPDaux1}) yield
\begin{equation}
\label{KRelPD}
K_{\Pi}(Q^2) = - \biggl[K_{D}(Q^2)
+ \frac{d\,K_{D}(Q^2)}{d\,\ln Q^2}\biggr],
\qquad
Q^2 \ge 0.
\end{equation}
\begin{figure}[t]
\centerline{\includegraphics[width=75mm,clip]{contour2.pdf}}
\caption{The integration contour in the complex~$p^2$--plane
in Eq.~(\ref{KRelDR}). The physical cut $p^2 \ge 0$ of the
``timelike'' kernel function~$K_{R}(-p^2)$~(\ref{AmuR}) is shown
along the positive semiaxis of real~$p^2$.}
\label{Plot:Contour2}
\end{figure}
The kernel function~$K_{D}(Q^2)$~(\ref{AmuD}) can be expressed in terms
of~$K_{\Pi}(Q^2)$~(\ref{AmuP}) in the following way. The solution to the
differential equation~(\ref{KRelPD})
\begin{equation}
\label{KRelDPaux1}
K_{D}(Q^2) + \frac{d\,K_{D}(Q^2)}{d\,\ln Q^2} = - K_{\Pi}(Q^2)
\end{equation}
reads
\begin{equation}
\label{KRelDPaux2}
K_{D}(Q^2) = \frac{1}{Q^2}
\biggl[-\!\int\!\! K_{\Pi}(Q^2)\, d Q^2 + c_{0} \biggr]\!,
\end{equation}
where~$c_{0}$ denotes an arbitrary integration constant.
The latter has to be chosen in the way that prevents the
appearance of the mutually canceling divergences at the
upper limit of both terms in the second line of
Eq.~(\ref{KRelPDaux1}). Specifically, the integration
constant~$c_{0}$ has to subtract the value of the
antiderivative of the function~$K_{\Pi}(Q^2)$
at~$Q^2 \to \infty$, that eventually results~in
\begin{equation}
\label{KRelDP}
K_{D}(Q^2) = \frac{1}{Q^2}
\int\limits_{Q^2}^{\infty}\!\! K_{\Pi}(\xi)\, d \xi,
\qquad
\xi = -p^2 \ge 0,
\end{equation}
where~$\xi$ stands for a spacelike kinematic variable.
It~is worthwhile to note also that Eqs.~(\ref{KRelDP})
and~(\ref{KRelRP}) imply that the functions~$Q^2 K_{D}(Q^2)$
and~$sK_{R}(s)$ acquire the same value in the infrared limit,
namely
\begin{equation}
\label{KRDlim}
\lim_{Q^2 \to 0_{+}} Q^2 K_{D}(Q^2) =
\lim_{s \to 0_{+}} sK_{R}(s) =
\int\limits_{0}^{\infty}\!\! K_{\Pi}(\xi)\, d \xi.
\end{equation}
Finally, the relation inverse to Eq.~(\ref{KRelRD}) can be derived
by making use of Eqs.~(\ref{KRelDP}) and~(\ref{KRelPR}), namely
\begin{equation}
\label{KRelDRaux1}
K_{D}(Q^2) = - \frac{1}{2 \pi i} \lim_{\varepsilon \to 0_{+}}
\frac{1}{Q^2}
\int\limits_{Q^2}^{\infty}
\Bigl[ K_{R}(-\xi-i\varepsilon) - K_{R}(-\xi+i\varepsilon)\Bigr] d \xi.
\end{equation}
The change of the integration variables~$\xi = p^2 - i\varepsilon$
and~$\xi = p^2 + i\varepsilon$ in, respectively, the first and the
second terms in the square brackets in Eq.~(\ref{KRelDRaux1})
eventually leads~to
\begin{equation}
\label{KRelDR}
K_{D}(Q^2) = - \frac{1}{2 \pi i} \lim_{\varepsilon \to 0_{+}}
\frac{1}{Q^2}
\int\limits_{Q^2 + i\varepsilon}^{Q^2 - i\varepsilon}
K_{R}(-p^2) d p^2,
\end{equation}
where the integration contour on the right--hand side of this equation
lies in the region of analyticity of the function~$K_{R}(-p^2)$,
see Fig.~\ref{Plot:Contour2}.
Thus, the derived equations~(\ref{KRelPR}), (\ref{KRelRP}),
(\ref{KRelRD}), (\ref{KRelPD}), (\ref{KRelDP}), and~(\ref{KRelDR})
constitute the complete set of relations, which mutually express
the ``spacelike'' and ``timelike'' kernel
functions~$K_{\Pi}(Q^2)$, $K_{D}(Q^2)$, and~$K_{R}(s)$
entering Eq.~(\ref{Amu}) in terms of each other.
\subsection{The ``spacelike'' kernel functions}
\label{Sect:KP3aExpl}
The relations obtained in Sect.~\ref{Sect:Rels} enable one to
calculate the unknown kernel functions~(\ref{Amu})
by making use of the known ones. To~exemplify this method,
let us first address the hadronic vacuum polarization contribution
to the muon anomalous magnetic moment in the leading~order.
Specifically, the explicit form of the ``spacelike'' kernel
function~$\KG{\Pi}{2}(Q^2)$~(\ref{AmuP}) can be obtained directly
from the corresponding ``timelike''
one~$\KG{R}{2}(s)$ [Eqs.~(\ref{K2RExpl}),~(\ref{KR2eta})]
by making use of the derived relation~(\ref{KRelPR}),
that eventually results~in
\begin{equation}
\label{KP2expl}
\zeta\KGt{\Pi}{2}(\zeta) = \frac{1}{\zeta^2} \,
\frac{y^{5}(\zeta)}{1-y(\zeta)},
\qquad
y(\zeta) = \zeta \Bigl(\sqrt{1+\zeta^{-1}}-1\Bigr),
\qquad
\zeta = \frac{Q^2}{4m_{\mu}^2}.
\end{equation}
It is worth noting that Eq.~(\ref{KP2expl}) is identical
to the result of the mapping the integration range~$0 \le x < 1$
in~Eq.~(\ref{Amu2Px}) onto the kinematic
interval~\mbox{$0 \le Q^2 < \infty$}
reported~in Refs.~\cite{Pivovarov2002, JPG42, EdR17}
and to the result of straightforward calculation~\cite{Blum2003}
performed within the technique~\cite{HS1, HS2, HS3, HS4}.
In turn, the explicit form of the kernel
function~$\KG{D}{2}(Q^2)$~(\ref{AmuD}) can be derived
from Eq.~(\ref{KP2expl}) by making use of the obtained
relation~(\ref{KRelDP}), that leads~to
\begin{equation}
\label{KD2expl}
\zeta\KGt{D}{2}(\zeta) = (2\zeta+1)^{2}
- 2(2\zeta+1)\sqrt{\zeta(\zeta+1)} - \frac{1}{2}.
\end{equation}
This equation coincides with the result of the mapping the
integration range~$0 \le x < 1$ in~Eq.~(\ref{Amu2Dx}) onto
the kinematic interval~\mbox{$0 \le Q^2 < \infty$},
see~Refs.~\cite{Pivovarov2002, EdR17}.
\begin{figure}[t]
\centerline{\includegraphics[width=120mm,clip]{Kernels2.pdf}}
\caption{The kernel functions
$\zeta\KGt{\Pi}{2}(\zeta)$ [Eq.~(\ref{KP2expl}), solid curve],
$\zeta\KGt{D}{2}(\zeta)$ [Eq.~(\ref{KD2expl}), dashed curve],
and
$\eta\KGt{R}{2}(\eta)$ [Eqs.~(\ref{K2RExpl}),~(\ref{KR2eta}),
dot--dashed curve]
in the
spacelike [$Q^2 = -q^2 \ge 0$, \mbox{$\zeta = Q^2/(4m_{\mu}^2)$}]
and timelike [$s = q^2 \ge 0$, $\eta = s/(4m_{\mu}^2)$]
domains.}
\label{Plot:K2PRD}
\end{figure}
The ``spacelike'' [$\KG{\Pi}{2}(Q^2)$, Eq.~(\ref{KP2expl})
and $\KG{D}{2}(Q^2)$, Eq.~(\ref{KD2expl})]
and ``timelike'' [$\KG{R}{2}(s)$, Eqs.~(\ref{K2RExpl}),~(\ref{KR2eta})]
kernel functions
satisfy all six relations~(\ref{KRelPR}), (\ref{KRelRP}),
(\ref{KRelRD}), (\ref{KRelPD}), (\ref{KRelDP}),
and~(\ref{KRelDR}) derived in Sect.~\ref{Sect:Rels}.
The plots of the kernel functions (\ref{KP2expl}),
(\ref{KD2expl}), and~(\ref{K2RExpl}) are displayed
in~Fig.~\ref{Plot:K2PRD}. In particular, as~one can
infer from this figure, in~the infrared limit
the kernel functions
$\zeta\KGt{D}{2}(\zeta)$~(\ref{KD2expl})
and
$\eta\KGt{R}{2}(\eta)$~(\ref{K2RExpl})
assume the same value~$1/2$, as
determined by the relation~(\ref{KRDlim}).
\bigskip
In the next--to--leading~order the explicit form of the uncalculated
yet ``spacelike'' kernel function $\KG{\Pi}{3a}(Q^2)$ appearing in
Eq.~(\ref{AmuP}) can be obtained in a similar way. Specifically,
the derived relation~(\ref{KRelPR}) and
Eq.~(\ref{KR3aExpl}) eventually result~in
\begin{align}
\label{KP3aExpl}
\zeta \tilde{K}\ind{(3a)}{$\Pi$}(\zeta) & =
- \biggl[
\frac{19}{12} + \frac{7}{9}\zeta
+ \frac{23}{9}\zeta^2 - \frac{1}{4(\zeta+1)}
\biggr]
+ \nonumber \\[1mm] &
+ \biggl(
\frac{1}{3\zeta} + \frac{127}{36}
+ \frac{115}{18}\zeta + \frac{23}{9}\zeta^2
\biggr) \psi(\zeta+1)
- \nonumber \\[1mm] &
- \frac{5}{3} \zeta^2 \ln(4\zeta)
- \biggl( \frac{14}{3}\zeta + 1 \biggr)
(\zeta+1)
\psi(\zeta+1)
\times \nonumber \\[1mm] &
\times
\biggl\{
\frac{1}{2}\ln(4\zeta) + 3A(\zeta+1) + 2\ln\Bigl[1+B(\zeta+1)\Bigr]\!
\biggr\}
+ \nonumber \\[1mm] &
+ \biggl(
-\frac{19}{6} + \frac{53}{3}\zeta + \frac{58}{3}\zeta^2
-\frac{1}{3\zeta} + \frac{2}{\zeta+1}
\biggr)
A(\zeta+1)
+ \nonumber \\[1mm] &
+ \biggl[
\frac{13}{12\,\zeta} + \frac{7}{6} + \zeta
+\frac{8}{3}\zeta^2 + \frac{1}{4\zeta(\zeta+1)}
\biggr]
\psi(\zeta+1)
A(\zeta+1)
- \nonumber \\[1mm] &
- \biggl(
\frac{1}{2} + \frac{14}{3}\zeta + 8\zeta^2
\biggr)
\times \nonumber \\[1mm] &
\times
\biggl\{
2A(\zeta+1)
\Bigl\{
2\ln\bigl[1+B(\zeta+1)\bigr] + \ln\bigl[1-B(\zeta+1)\bigr]\!
\Bigr\}
-
\nonumber \\[1mm] &
-2\Bigl\{
\Li{2}\bigl[B(\zeta+1)\bigr]
+2\Li{2}\bigl[-B(\zeta+1)\bigr]\!
\Bigr\}\!
\biggr\}.
\end{align}
An~equivalent form of this equation has been independently
derived in~Ref.~\cite{BLP}. In~Eq.~(\ref{KP3aExpl})
$\zeta=Q^2/(4m_{\mu}^2)$,
$Q^2 = -q^2 \ge 0$ stands for the spacelike kinematic
variable, the functions~$\psi(\zeta)$ and~$A(\zeta)$
are defined in~Eq.~(\ref{DefAux1}),
\begin{equation}
\label{DefAux2}
B(\zeta) = \frac{1-\psi(\zeta)}{1+\psi(\zeta)},
\end{equation}
and
\begin{equation}
\label{Li2Def}
\Li{2}(y) = -\int\limits_{0}^{y}\ln(1-t)\frac{d\,t}{t}
\end{equation}
denotes the dilogarithm function.
\begin{figure}[t]
\centerline{\includegraphics[width=120mm,clip]{Kernels3a.pdf}}
\caption{The kernel functions
$\zeta\KGt{\Pi}{3a}(\zeta)$ [Eq.~(\ref{KP3aExpl}), solid curve],
$\zeta\KGt{D}{3a}(\zeta)$ [Eqs.~(\ref{KRelDP}),~(\ref{KP3aExpl}),
dashed curve],
and
$\eta\KGt{R}{3a}(\eta)$ [Eq.~(\ref{KR3aExpl}), dot--dashed curve]
in the
spacelike [$Q^2 = -q^2 \ge 0$, \mbox{$\zeta = Q^2/(4m_{\mu}^2)$}]
and timelike [$s = q^2 \ge 0$, $\eta = s/(4m_{\mu}^2)$]
domains.}
\label{Plot:K3aPR}
\end{figure}
It is straightforward to verify that the
``spacelike'' [$\KG{\Pi}{3a}(Q^2)$, Eq.~(\ref{KP3aExpl})]
and ``timelike'' [$\KG{R}{3a}(s)$, Eq.~(\ref{KR3aExpl})]
kernel functions satisfy the corresponding
relations~(\ref{KRelPR})
and~(\ref{KRelRP}) obtained in Sect.~\ref{Sect:Rels}.
The~plots of the kernel
functions~[$\KG{\Pi}{3a}(Q^2)$, Eq.~(\ref{KP3aExpl})],
[$\KG{D}{3a}(Q^2)$, computed numerically by making use
of~Eqs.~(\ref{KRelDP}) and~(\ref{KP3aExpl})],
and~[$\KG{R}{3a}(s)$, Eq.~(\ref{KR3aExpl})]
are displayed in~Fig.~\ref{Plot:K3aPR}.
As~one can infer from this figure, in~the infrared limit
the kernel functions
$\zeta\KGt{D}{3a}(\zeta)$~[Eqs.~(\ref{KRelDP}),~(\ref{KP3aExpl})]
and
$\eta\KGt{R}{3a}(\eta)$~[Eq.~(\ref{KR3aExpl})] acquire the same
value determined by the relation~(\ref{KRDlim}), specifically
\begin{equation}
\lim_{\zeta \to 0_{+}} \zeta\KGt{D}{3a}(\zeta) =
\lim_{\eta \to 0_{+}} \eta\KGt{R}{3a}(\eta) =
\frac{197}{144} +\frac{1}{2}\zeta_{2}
-3\zeta_{2}\ln(2) +\frac{3}{4}\zeta_{3}
\simeq -0.328479,
\end{equation}
where
\begin{equation}
\label{RiemannZeta}
\zeta_{t} = \sum_{n=1}^{\infty} \frac{1}{n^{t}}
\end{equation}
stands for the Riemann $\zeta$~function.
\section{Conclusions}
\label{Sect:Concl}
The complete set of relations [Eqs.~(\ref{KRelPR}), (\ref{KRelRP}),
(\ref{KRelRD}), (\ref{KRelPD}), (\ref{KRelDP}),~(\ref{KRelDR})],
which mutually express the~``spacelike'' [$K_{\Pi}(Q^2)$, Eq.~(\ref{AmuP})
and $K_{D}(Q^2)$, Eq.~(\ref{AmuD})] and~``timelike'' [$K_{R}(s)$,
Eq.~(\ref{AmuR})] kernel functions in terms of each other, is obtained.
By~making use of the derived relations the explicit expression for the
next--to--leading order ``spacelike'' kernel function~$\KG{\Pi}{3a}(Q^2)$
is calculated [Eq.~(\ref{KP3aExpl})] and the kernel
function~$\KG{D}{3a}(Q^2)$ is computed numerically
[Eqs.~(\ref{KRelDP}),~(\ref{KP3aExpl}) and~Fig.~\ref{Plot:K3aPR}].
The~obtained results can be employed in the assessments
of the hadronic vacuum polarization contributions to the muon
anomalous magnetic moment in the framework of the spacelike methods,
such as lattice studies~\cite{Lattice1, Lattice2}, MUonE
project~\cite{MUonE1, MUonE2, MUonE3}, and others.
|
1,116,691,497,558 | arxiv | \section{Introduction}
By using a large-scale antenna array at the base station (BS), massive multiple-input multiple-output (MIMO) can serve multiple users at the same frequency band simultaneously and thus boost the system spectral efficiency \cite{Marzetta10noncooperative,Larsson14massive,Sun15Beam,Wang18spatial}. Therefore, massive MIMO has become one of the key technologies in the fifth-generation (5G) wireless communication systems and is also anticipated to play an important role in the sixth-generation (6G) wireless communication systems \cite{Zhang196G}.
However, the large-scale antenna array in massive MIMO systems also brings tremendous hardware complexity and power consumption if each radio frequency (RF) chain is connected to one antenna \cite{Shepard13Argos,Yang18Digital}\footnote{Note that in time division duplex-based full digital systems, an independent RF chain pair, which consists of a transmit RF chain and a receive RF chain, is connected to each antenna via a switch.}.
Huge baseband signal processing pressure has also been caused by such a large-scale antenna array and numerous RF chains. As a result, these hardware constraints severely impact the wide deployment of massive MIMO systems \cite{Yang18Digital}.
Recently, several methods have been proposed to overcome these constraints. For example, hybrid (i.e., the analog and digital combined) beamforming \cite{Ayach14Spatially,Liang14Low,Payami16Hybrid,Payami19Phase,Sungwoo17Dynamic} is leveraged in massive MIMO systems to reduce the number of RF chains. Nevertheless, hybrid beamforming usually incurs additional restrictions on the relevant signal processing, such as constant modulus and a heavy beam training overhead for channel estimation.
Low-resolution analog-to-digital converters (ADCs) and digital-to-analog converters (DACs) have also been introduced in \cite{Wang18Finite,Kong17Full,Zhang16ON,Dong20spatially} to reduce the hardware cost and power consumption. {However, in low-resolution massive MIMO systems, part of the signal information is inevitably lost. Therefore, complex iterative algorithms must be used to recover data at the receiver.}
Apart from that, antenna selection \cite{Li14Energy,Rodriguez17Reduced,Gao18MassiveSwitch,Asaad18antennaselection,Chen19Intell} has also been extended to massive MIMO systems to alleviate the requirement on the number of RF chains. Whereas, the RF switching network in such systems becomes less power-efficient due to the increasing MIMO dimension. More importantly, similar to the hybrid beamforming systems, the numbers of transmit and receive RF chains in massive MIMO systems with antenna selection are also reduced, {thereby degrading the data transmission capability at both uplink and downlink.}
In general, the demand on uplink data transmission is different from that of the downlink in practical scenarios. For example, many popular applications receive more data in downlink than in uplink \cite{Ericsson16}. On the other hand, the data transmission rate is highly related to the number of RF chains \cite{Molisch05capacity}. The more RF chains, the higher the transmission rate we can achieve; however, the higher the circuit complexity and cost we need to address. With massive MIMO, the hardware complexity of RF chains has been greatly increased in 5G while the budgets for size, power, and cost for the system nearly remain the same \cite{Shepard20what}.
In addition, from \cite{Yang18Digital}, the cost and power consumption of the massive MIMO transceiver circuit for the fully digital beamforming architecture and the hybrid beamforming architecture are almost the same. The major gap on cost and power consumption is contributed by the ADCs, DACs, and baseband processing field-programmable gate arrays (FPGAs).
{Therefore, an alternatively feasible way to alleviate the cost and hardware challenges is by decoupling the transmit and the receive RF chains so that the numbers of the receive and transmit RF chains can be set different.}
Based on this idea, a full digital system architecture with nonreciprocal antenna arrays has been proposed in \cite{Guo20Design}, {where only a few antennas in the center part of the antenna array at the base station} are connected with both transmit and receive RF chains while the others only connect with transmit RF chains. Owing to this asymmetry in the numbers of transmit and receive antennas, we call such an architecture as the asymmetrical transceiver-based architecture in this paper. Benefited from this flexibly unequal numbers of transmit and receive RF chains, the system cost and hardware complexity of the asymmetrical transceiver-based massive MIMO systems can be substantially decreased and the baseband data processing pressure can be alleviated. Consequently, we can achieve a better energy and cost efficiencies, which are the key performance indicators of 6G \cite{You20Towards}.
Inspired by the nonreciprocal array design proposed in \cite{Guo20Design}, we investigate the detailed uplink and downlink transmission procedures and the system performance of the asymmetrical transceiver-based massive MIMO system in this paper.
In particular, we first provide a generic system architecture for the asymmetrical transceiver, where the number of receive RF chains is fewer than that of transmit RF chains. Then, the overall transmission procedure of the asymmetrical transceiver-based massive MIMO system is presented. Afterwards, we design two uplink-to-downlink channel transfer algorithms. Finally, we develop cost and power consumption models for the asymmetrical transceiver-based massive MIMO system and investigate the spectral efficiency (SE), cost, and energy efficiency (EE) of the asymmetrical transceiver-based system.
The main contributions can be summarized as follows:
\begin{itemize}
\item We provide a general architecture and an overall transmission procedure for the asymmetrical transceiver-based massive MIMO system. The impact of the channel inconsistency due to the unequal number of transmit and receive RF chains is also investigated. Owing to the asymmetrical architecture, the asymmetrical transceiver can significantly reduce the overall system hardware complexity while the signal processing flexibility and superior downlink performance can still be maintained.
\item We propose two uplink-to-downlink channel transfer algorithms, i.e., the discrete Fourier transform (DFT)-based algorithm and the modified Newtonized orthogonal matching pursuit (mNOMP)-based algorithm, to deal with the unequal number of transmit and receive RF chains. {The computational complexity and convergence of the two uplink-to-downlink channel transfer algorithms are also discussed.} Collaborated with the random receive antenna array, these two algorithms can well recover the downlink channel information, which makes the asymmetrical transceiver demonstrate excellent downlink transmission capability.
\item We develop {cost and power consumption models} for the asymmetrical transceiver-based massive MIMO system. The SE and EE comparisons with other conventional symmetrical transceivers are performed as well. Numerical results demonstrate that the proposed asymmetrical transceiver can achieve a superior downlink SE while maintaining a well EE simultaneously. The asymmetrical transceiver architecture is a promising alternative solution for future massive MIMO systems.
\end{itemize}
The rest of this paper is organized as follows. In Section II, we present the system architecture and the channel model for the asymmetrical transceiver-based massive MIMO system. Section III investigates the overall transmission procedure. Two uplink-to-downlink channel transfer algorithms are proposed in Section IV and the cost and EE are analyzed in Section V. Section VI provides the numerical results. We conclude the paper in Section VIII.
\emph{Notation:} Bold lowercase $\mathbf{x}$ and bold uppercase $\mathbf{X}$ are used to denote vectors and matrices, respectively. The inverse, transpose, conjugate and conjugate-transpose operations of matrix are denoted by $(\cdot)^{-1}$, $(\cdot)^{T}$, $(\cdot)^{*}$ and $(\cdot)^{H}$, respectively. $\mathbf{I}_M$ is an identity matrix with dimension $M\times M$. $\left\| \mathbf{x} \right\|$ represents the Euclidean norm of the vector $\mathbf{x}$ and ${\left| \mathcal{A} \right|}$ stands for the ensemble of the set $\mathcal{A}$.
$[\mathbf{x}]_{m}$ indicates the $m$th element of the vector $\mathbf{x}$ while $[\mathbf{x}]_{\mathcal{A}}$ represents a new vector with $[\mathbf{x}]_{\mathcal{A}}$'s elements coming from $\mathbf{x}$ whose selected elements are specified by the set ${\mathcal{A}}$.
\section{System Model}
In this section, we will first introduce the asymmetrical transceiver-based massive MIMO system and then describe the uplink and downlink channel models for the asymmetrical transceiver.
\subsection{System Architecture}
The architecture of an asymmetrical transceiver-based multi-user massive MIMO system is shown in Fig.\,\ref{Fig:system_architecture}. Different from the conventional symmetrical transceivers where each transmit RF chain is coupled with a receive RF chain, the BS in Fig.\,\ref{Fig:system_architecture} adopts an asymmetrical transceiver and serves $K$ single-antenna users simultaneously.
For simplicity, we assume that an $M$-element uniform linear array (ULA) is equipped at the BS and the adjacent antenna element spacing is half carrier wavelength.\footnote{The developed hardware architecture can also be easily extended to other array topologies, such as uniform planar antenna array (UPA) and uniform cylinder antenna array (UCA). Note that the subsequent analyses including the system model and the uplink-to-downlink channel transfer algorithms are primarily designed for ULA. When UPA or UCA is employed, the following system model and channel transfer algorithms may necessitate appropriate modifications or reconsideration.} We also assume that this system operates in time-division duplex (TDD) mode and the uplink and downlink transmission can be performed in a coherent interval such that the channel reciprocity between uplink and downlink can be leveraged.
\begin{figure}[!t]
\centering
\includegraphics[scale= 0.15]{Fig1.eps}
\caption{The system architecture of an asymmetrical transceiver-based massive MIMO system, where a BS is equipped with an $M$-element ULA and serves $K$ single-antenna users simultaneously. An asymmetrical transceiver is adopted at the BS with $N$ out of $M$ antennas connecting with both transmit and receive RF chains, while the remaining $M-N$ antennas connect with only transmit RF chains.}\label{Fig:system_architecture}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[scale= 0.17]{Fig2.eps}
\caption{Receive antenna arrays constructed by three receive antenna selection methods: (a) random antenna selection, (b) successive antenna selection, and (c) comb antenna selection. The blue filled one is selected for both transmitting and receiving while the red one is only used for transmitting. The total number of the BS antenna array elements is $M$, and the number of antenna elements utilized for both transmitting and receiving is $N$, which is consistent with Fig.\,\ref{Fig:system_architecture}.}\label{Fig:antenna_selection}
\end{figure}
Note that in an asymmetrical transceiver-based system, the number of receive RF chains is different from that of transmit RF chains, depending on whether we want to boost the system uplink transmission rate or reduce the uplink baseband processing pressure. By considering that the amount of required data is relatively smaller than that of the downlink \cite{ITU17Minimum,3GPP19overall}, we assume the number of the receive RF chains is fewer than that of the transmit RF chains in this paper. As shown in Fig.\,\ref{Fig:system_architecture}, each of the $M$ antennas at the BS connects with a transmit RF chain, but only $N$ ($N \ge K$) out of $M$ antennas are connected with the receive RF chains. In other words, only $N$ antennas connect with both transmit and receive RF chains. Besides, it is worth mentioning that the proposed asymmetrical transceiver-based BS utilizes a full digital beamforming architecture. Therefore, the BS can always employ full digital beamforming/combining at downlink/uplink transmission and thus achieves a higher signal processing flexibility than the hybrid beamforming architecture. The hybrid beamforming architecture usually leverages phase shifters and thereby suffers from unitary modulus constraints. In addition, because of the reduction of both the expensive RF components and the high-rate ADCs, the system cost and hardware complexity of the asymmetrical transceiver-based massive MIMO systems can be greatly reduced and the high uplink baseband data processing pressure can also be alleviated at the cost of moderate uplink transmission capacity reduction.
As is known that the topology of receive antenna array has a great influence on the system performance, thus selecting appropriate antenna elements to connect with those limited number of receive RF chains is important.
Although antenna selection is not a new topic and there are various antenna selection algorithms for antenna selection systems \cite{Asaad18antennaselection,Gao13Antenna,Dua06Receive}, those exiting antenna selection algorithms proposed for symmetrical transceivers cannot be directly applied to asymmetrical transceivers owing to the different system architecture.
Therefore, to uncover the potential performance of asymmetrical transceivers, we propose three intuitive antenna selection methods in this paper\footnote{The sophisticated antenna selection criteria, such as maximizing the angular resolution of the receive antenna array, minimizing the interference signal level from other directional users, and so on, are left for our future research direction.} as shown in Fig.\,\ref{Fig:antenna_selection}, i.e., (a) random antenna selection method \cite{Lo64Mathematical}, (b) successive antenna selection method, and (c) comb antenna selection method.
For the random antenna selection method, the antenna elements used for signal reception are randomly selected; while for the successive selection method, $N$ consecutive antenna elements are selected for signal receiving. In contrast, the receive antenna elements selected by the comb selection method are uniformly distributed across the whole BS antenna array.
\subsection{Channel Model}
We adopt the block-fading parametric channel model\footnote{Although the statistical channel model \cite{Ngo13energy} is widely utilized in wireless communication systems, especially in sub-6G systems, we mainly focus on the parametric channel modeling in this paper. The research considering the statistical channel model is left in our future work and thus is beyond the scope of this paper.} in this paper. Due to the channel reciprocity\footnote{In this paper, we assume that the hardware components are ideally calibrated such that we can leverage the channel reciprocity and ignore the hardware mismatch between the transmit RF chains and the receive RF chains.}, the uplink and downlink channel of user $k$ in the asymmetrical transceiver-based multi-user massive MIMO system can be expressed as \cite{Yin13acoordinated,Wang19anoverview}
\begin{equation}\label{eq:hku}
{\bf{h}}_{k,{\rm{A}}}^{\rm{U}} = \sqrt {\frac{N}{P_k}} \sum\limits_{i = 1}^{P_k} {{g_{k,i}}{{\bf{a}}_{\rm{U}}}({\theta _{k,i}})},
\end{equation}
and
\begin{equation}\label{eq:hkd}
{\mathbf{h}}_{k}^{\rm{D}} = \sqrt {\frac{M}{P_k}} \sum\limits_{i = 1}^{P_k} {{{g}_{k,i}}{{\mathbf{a}}_{\rm{D}}}({{\theta }_{k,i}})},
\end{equation}
respectively, where $P_k$ denotes the number of paths; ${g_{k,i}}$ is the complex gain of the $i$th path for the $k$th user; $\theta_{k,i}$ represents the angle of arrival/departure (AoA/AoD) of the $i$th path at uplink/downlink. ${{\bf{a}}_{\rm{U}}}(\cdot)\in\mathbb{C}^{N\times 1}$ and ${{\mathbf{a}}_{\rm{D}}}(\cdot)\in\mathbb{C}^{1\times M}$ are the uplink and downlink steering vectors, respectively, and can be expressed as
\begin{equation}\label{eq:steeringVectorau}
{{\bf{a}}_{\rm{U}}}({\theta _{k,i}}) = \frac{1}{{\sqrt N }}{\left[ {{e^{ - \frac{{j2\pi {a_1}d\sin \theta_{k,i} }}{\lambda }}}, \ldots ,{e^{ - \frac{{j2\pi {a_N}d\sin \theta_{k,i} }}{\lambda }}}} \right]^T},
\end{equation}
and
\begin{equation}\label{eq:steeringVectorad}
{{\bf{a}}_{\rm{D}}}({\theta _{k,i}}) = \frac{1}{{\sqrt M }}{\left[1,{{e^{ - \frac{{j2\pi d\sin \theta_{k,i} }}{\lambda }}}, \ldots ,{e^{ - \frac{{j2\pi (M-1)d\sin \theta_{k,i} }}{\lambda }}}} \right]},
\end{equation}
where $\lambda$ is the carrier wavelength and $d$ is the adjacent antenna element spacing of the original BS antenna array (i.e., the $M$-antenna ULA). Denote ${a_n}\in\{1,2,\ldots,M\},\forall n=1,\ldots,N,$ as the index of the BS's antennas selected for uplink receiving and ${\cal A} \buildrel \Delta \over = \left\{ {{a_1},{a_2}, \ldots {a_N}} \right\}$ as the receive antenna index set.
Due to the unequal numbers of transmit and receive antennas in the asymmetrical transceiver, the expression form of the steering vector in the uplink i.e., (\ref{eq:steeringVectorau}), is different from that in the downlink i.e., (\ref{eq:steeringVectorad}), even for the same user. Note that the steering vectors at uplink and downlink remain the same form for the same user at conventional symmetrical transceiver-based massive MIMO systems.
The following remark is provided to highlight this different feature.
\begin{Remark}\label{Remark_1}
Unlike the existing massive MIMO systems where symmetrical transceivers are generally employed, the channels at the uplink and the downlink in the asymmetrical transceiver-based massive MIMO system are not consistent.
\end{Remark}
In particular, the channel inconsistency between the uplink and the downlink mainly exhibits in two respects. First, the dimensions of the uplink and downlink channel vectors are unequal.
For example, in Fig.\,\ref{Fig:system_architecture}, the dimension of the uplink channel vectors is $N$ while the dimension of the downlink channel vector is $M$ ($M>N$). Owing to this unequal channel vector dimension, the uplink channel estimate cannot be directly used for the downlink precoding. Additional signal processing, called uplink-to-downlink channel transfer, is needed before performing the downlink precoding.
Second, although there are totally $P_k$ paths for user $k$ in the propagation environment, the number of paths the BS can distinguish for user $k$ at uplink and downlink may be different.
This is primarily incurred by the different antenna array apertures the BS used for uplink and downlink.
Thanks to the asymmetrical transceiver architecture, the transmit antenna array (i.e., the $M$-antenna ULA) of the BS always has a larger aperture and thus acquires a better angular resolution than the receive antenna array (i.e., the selected $N$-antenna array).
For example, for the successive antenna selection under our system configuration, we have the angular resolution of the receive antenna array $\Delta^s_r=2/N$, while the angular resolution of the transmit antenna array is $\Delta_t=2/M$ and $\Delta^s_r>\Delta_t$.
Hence, if the AoA difference of the $i$th path and the $j$th path $\Delta_{\theta_{ij}}$ is smaller than $\Delta^s_r$ but larger than $\Delta_t$, i.e., $\Delta_t<\Delta_{\theta_{ij}}<\Delta^s_r$, the $i$th path and the $j$th path can be distinguished at the BS in downlink but may not be distinguished in uplink.
In this situation, the uplink channel of user $k$ from the view of BS becomes
\begin{equation}\label{eq:hkus}
{\bf{h}}_{k,{\rm{A}}}^{\rm{U,s}} = \sqrt {\frac{N}{P_k}} \sum\limits_{i = 1}^{P^s_k} {{g^{{s}}_{k,i}}{{\bf{a}}^s_{\rm{U}}}({\theta^s_{k,i}})},
\end{equation}
where ${g^{{s}}_{k,i}}$ and ${\theta^s_{k,i}}$ denote the complex gain and the AoA of the $i$th identified path, respectively, from the composition of the unresolvable paths, and the steering vector of the selected antenna array can be written as
\begin{equation}\label{eq:steeringVectoraus}
{{\bf{a}}^s_{\rm{U}}}({\theta^s_{k,i}}) = \frac{1}{{\sqrt N }}{\left[1,{{e^{ - \frac{{j2\pi d\sin {\theta^s_{k,i}} }}{\lambda }}}, \ldots ,{e^{ - \frac{{j2\pi (N-1)d\sin {\theta^s_{k,i}} }}{\lambda }}}} \right]},
\end{equation}
$P^s_k(\leq P_k)$ represents the actual number of paths that the BS can resolve.
To better illustrate this phenomenon, Fig.\,\ref{Fig:unresolved_paths} provides a detailed example. We assume that there are two dominant paths existing in the wireless propagation environment, whose angles are $51.3^o$ and $54.3^o$, respectively. At the BS, a ULA is used and the number of transmit antennas is $M=256$ (i.e., $\Delta_t=0.45^o$). For the receive antenna array at the BS, we employ the successive antenna selection and assume $N=32$ (i.e., $\Delta^s_r=3.58^o$). As presented in Fig.\,\ref{Fig:unresolved_paths}, since $0.45^o<3^o<3.58^o$, those two paths can be well resolved at downlink while only one dominant path whose angle is $52.8^o$ can be resolved at uplink.
\begin{figure}[!t]
\centering
\includegraphics[scale= 0.4]{Fig3.eps}
\caption{An example for the channel inconsistency at the path number/angle in asymmetrical transceiver-based massive MIMO systems. Because of $M=256$ and $N=32$, we have $\Delta_t=0.45^o$ and $\Delta^s_r=3.58^o$. Therefore, those two paths are well resolved at downlink but only one dominant path whose compositive angle is $52.8^o$ can be resolved at uplink.}\label{Fig:unresolved_paths}
\end{figure}
Note that the best angular resolution of the receive antenna array in (\ref{eq:hku}) can be obtained as $\Delta_r=\Delta_t$ via the appropriate antenna selection.
Therefore, despite the unequal channel vector dimension, the unequal angular resolutions of the receive and the transmit antenna arrays should also be considered. The receive antenna array topology design at the BS in terms of the effective array aperture is nontrivial for the asymmetrical transceiver-based massive MIMO systems.
\section{Transmission Procedure}
In this section, we illustrate the overall transmission procedure of the asymmetrical transceiver-based massive MIMO system, including the uplink transmission, the uplink-to-downlink channel transfer, and the downlink transmission.
\subsection{Overall Transmission Procedure}
Fig.\,\ref{Fig:frame}(a) presents the frame structure of the asymmetrical transceiver-based massive MIMO system. Since the system operates in TDD mode, there is a guard symbol when switching from the downlink to the uplink and the transmission procedure is demonstrated in Fig.\,\ref{Fig:frame}(b).
\begin{figure}[!t]
\centering
\includegraphics[scale= 0.17]{Fig4.eps}
\caption{The wireless frame structure and the transmission procedure of the asymmetrical transceiver-based massive MIMO system.}\label{Fig:frame}
\end{figure}
During the synchronization and initial access symbols, the system synchronization and the user initial access are performed. First, the BS broadcasts the synchronization sequences to the users so that they can accomplish their frequency and time synchronization upon the received sequences.
After that, users start the initial access and then move into the uplink transmission when they acquire the acknowledgement from the BS. During the uplink transmission period, uplink pilots are first sent by the $K$ users to perform uplink channel estimation.
Since the full digital architecture is employed for data reception, the BS can theoretically estimate the uplink channels for $K$ users after $K$ measurements. This significantly reduces the training overhead when compared with the hybrid architecture. For the conventional hybrid transceiver, the number of channel estimation measurements is usually proportional to the product of the number of BS's receive antennas and the number of users \cite{heath2016overview}. Moreover, due to the employed much fewer receive RF chains in the asymmetrical transceiver, the uplink baseband data throughput and processing pressure are also considerably reduced at the same time.
After the uplink training, uplink data transmission begins and the acquired uplink channel estimates are used to recover the uplink data. In the conventional symmetrical transceivers, the uplink channel estimates are simultaneously used for the downlink precoding matrix calculation. However, in the asymmetrical transceiver, this cannot be directly performed due to the aforementioned channel inconsistency. To overcome this channel inconsistency, we introduce an additional step, i.e., the uplink-to-downlink channel transfer, before the downlink transmission. In the uplink-to-downlink channel transfer, the uplink channel estimates are extended and refined to obtain the downlink channel estimates. Then, the downlink precoding is accomplished based on these recovered downlink channel estimates and the downlink pilot/data transmission is subsequently performed.
In the following subsections, we present the details of the transmission procedure.
\subsection{Uplink Transmission}
The uplink transmission includes the uplink pilot training and the uplink data transmission.
During the uplink pilot training, we assume that users transmit orthogonal pilot sequences. Denote the length of the pilots as $\tau(\tau\geq K)$, we have
\begin{equation}\label{eq:Y}
{{\bf{Y}} } = \sqrt {{\rho_\tau }} {\bf{H}}_{\rm{A}}^{\rm{U}}{{\bf{P}} } + {\bf{N}},
\end{equation}
where ${{\bf{Y}} } \in {\mathbb{C}^{N \times \tau }}$ is the received signal at the BS, $\rho_\tau$ denotes the pilot transmit power by each user, ${{\bf{P}} } \in {\mathbb{C}^{K \times \tau }}$ is the normalized pilot matrix, and ${\bf{N}} \in {\mathbb{C}^{N \times \tau }}$ represents an additive white, zero-mean complex Gaussian noise matrix with element's variance being $\sigma_n^2=1$ without loss of generality. ${\bf{H}}_{\rm{A}}^{\rm{U}}\in {\mathbb{C}^{N \times K}}$ denotes the composite uplink channel matrix and can be expressed as ${\bf{H}}_{\rm{A}}^{\rm{U}} = \left[ {{\bf{h}}_{1{\rm{,A}}}^{\rm{U}}, \cdots ,{\bf{h}}_{K{\rm{,A}}}^{\rm{U}}} \right]$.
When the least-square (LS) channel estimation is utilized, we obtain
\begin{equation}\label{eq:HAuestimatedLS}
{\bf{\hat H}}_{\rm{A}}^{\rm{U}} = \frac{1}{{\sqrt {{\rho_\tau }} }}{\bf{Y}}{\bf{P}}^H,
\end{equation}
where the columns of ${\bf{\hat H}}_{\rm{A}}^{\rm{U}}$ denote the uplink channel estimates for different users, i.e., ${\bf{\hat H}}_{\rm{A}}^{\rm{U}} = \left[ {{\bf{\hat h}}_{1{\rm{,A}}}^{\rm{U}}, \cdots ,{\bf{\hat h}}_{K{\rm{,A}}}^{\rm{U}}} \right]$.
When the linear minimum mean-squared error (LMMSE) channel estimation is employed, the uplink channel can be estimated as
\begin{equation}\label{eq:HAuestimatedLMMSE2}
{\bf{\bar H}}_{\rm{A}}^{\rm{U}} = \frac{1}{{\sqrt {{\rho _\tau }} }}{\bf{Y}}{{\bf{P}}^H}{\left( {\frac{1}{{{\rho _\tau }}}{\bf{R}}_{{\bf{H}}_{\rm{A}}^{\rm{U}}}^{ - 1} + {{\bf{I}}_K}} \right)^{ - 1}},
\end{equation}
where ${{\bf{R}}_{{{\bf{H}}^{\rm{U}}_{\rm{A}}}}} \buildrel \Delta \over = \mathbb{E}\left\{ {{{\left( {{\bf{H}}_{\rm{A}}^{\rm{U}}} \right)}^H}{\bf{H}}_{\rm{A}}^{\rm{U}}} \right\}$ with ${\bf{\bar H}}_{\rm{A}}^{\rm{U}} = \left[ {{\bf{\bar h}}_{1{\rm{,A}}}^{\rm{U}}, \cdots ,{\bf{\bar h}}_{K{\rm{,A}}}^{\rm{U}}} \right]$ is the correlation matrix of the uplink channel.
After the uplink pilot training, uplink data is transmitted. Denote ${\bf{x}} = {\left[ {{x_1}, \ldots ,{x_K}} \right]^T}$ as the transmitted uplink data, where $x_k$ represents the symbol transmitted by the $k$th user and $\mathbb{E}\left\{ {{{\left| {{x_k}} \right|}^{{2}}}} \right\} = 1,\forall k = 1, \ldots ,K$, then, we have
\begin{equation}\label{eq:rMRC}
{{\bf{r}}_{{\rm{MRC}}}} = \sqrt {{\rho_u}} {\left( {{\bf{\tilde H}}_{\rm{A}}^{\rm{U}}} \right)^H}{\bf{H}}_{\rm{A}}^{\rm{U}}{\bf{x}} + {\left( {{\bf{\tilde H}}_{\rm{A}}^{\rm{U}}} \right)^H}{\bf{n}}_u,
\end{equation}
for the MRC receiver\footnote{To simplify the analysis and better highlight the insights on how the system parameters impact the uplink ergodic achievable SE, we have adopted the MRC detector here. Notice that other detectors, such as zero-forcing and LMMSE detectors, can also be utilized for uplink data transmission.}, where ${{\bf{r}}_{{\rm{MRC}}}} \in {\mathbb{C}^{K \times {{1}}}}$ is the recovered signal vector, $\rho_u$ represents the transmit data power by each user, and ${\bf{n}}_u\sim\mathcal{CN}({\bf{0}},{{\bf{I}}_N})$ denotes the additive white Gaussian noise. ${\bf{\tilde H}}_{\rm{A}}^{\rm{U}} = \left[ {{\bf{\tilde h}}_{1{\rm{,A}}}^{\rm{U}}, \cdots ,{\bf{\tilde h}}_{K{\rm{,A}}}^{\rm{U}}} \right]$ is the channel estimate from the uplink pilot training. When the LS channel estimation is adopted, ${\bf{\tilde H}}_{\rm{A}}^{\rm{U}} = {\bf{\hat H}}_{\rm{A}}^{\rm{U}}$, and when the LMMSE channel estimation is adopted, we have ${\bf{\tilde H}}_{\rm{A}}^{\rm{U}} = {\bf{\bar H}}_{\rm{A}}^{\rm{U}}$.
Therefore, the ergodic uplink achievable SE of the $k$th user under maximum ratio combining (MRC) can be expressed as
\begin{align}\label{eq:rMRCkthSE}
&R_{{\rm{MRC}},k}^{\rm{U}}= \nonumber \\
&\mathbb{E} \left\{ {{{\log }_2}\left( {1 + \frac{{{\rho_u}{{\left| {{{\left( {{\bf{\tilde h}}_{k,{\rm{A}}}^{\rm{U}}} \right)}^H}{\bf{h}}_{k,{\rm{A}}}^{\rm{U}}} \right|}^2}}}{{{\rho_u}\sum\nolimits_{i = 1,i \ne k}^K {{{\left| {{{\left( {{\bf{\tilde h}}_{k,{\rm{A}}}^{\rm{U}}} \right)}^H}{\bf{h}}_{i,{\rm{A}}}^{\rm{U}}} \right|}^2} + {{\left\| {{\bf{\tilde h}}_{k,{\rm{A}}}^{\rm{U}}} \right\|}^2}} }}} \right)} \right\},
\end{align}
and the ergodic uplink system achievable SE can be given by
\begin{equation}\label{eq:rMRCsysSE}
\eta _{{\rm{SE}}}^{\rm{U}} = \sum\limits_{i = 1}^K {R_{{\rm{MRC}},i}^{\rm{U}}}\quad{\rm{ (bit/s/Hz)}}.
\end{equation}
From (\ref{eq:hku}), (\ref{eq:steeringVectorau}), and (\ref{eq:hkus}), the ergodic uplink SE, i.e., $R_{{\rm{MRC}},k}^{\rm{U}}$ in (\ref{eq:rMRCkthSE}), is affected by both the antenna number and the topology of the receive antenna array.
Since the angular resolution of the receive antenna array from the comb antenna selection is $\Delta^c_r=2/M$, which is equal to $\Delta_t=2/M$, the channel model used in the analysis of $R_{{\rm{MRC}},k}^{\rm{U}}$ is ${\bf{h}}_{k,{\rm{A}}}^{\rm{U}}$ in (\ref{eq:hku}). While for the successive and the random antenna selections, the angular resolution of the realized receive antenna array is either proportional to the number of antenna elements or to the aperture dimension the selected elements spread. Therefore, the impact of the angular mismatch due to the degraded angular resolution should be considered in analyzing $R_{{\rm{MRC}},k}^{\rm{U}}$.
Define the normalized uplink SNR loss as
\begin{equation}
{\rm{SNR}_{{\rm{loss}}}}\buildrel \Delta \over =1-\frac {{{{\left| {{{\left( {{\bf{\tilde h}}_{k,{\rm{A}}}^{\rm{U}}} \right)}^H}{\bf{h}}_{k,{\rm{A}}}^{\rm{U}}} \right|}^2}}}{{ {{\left\| {{\bf{\tilde h}}_{k,{\rm{A}}}^{\rm{U}}} \right\|}^2}}{ {{\left\| {\bf{h}}_{k,{\rm{A}}}^{\rm{U}} \right\|}^2}} }.
\end{equation}
We can find the uplink SNR loss due to the degraded angular resolution under single-user scenarios in the following.
\begin{Proposition}\label{Pro1}
Considering the degraded angular resolution from successive antenna selection, when the two resolvable equal-power paths\footnote{Since these two paths come from adjacent angles, we assume the power of these two paths are equal for simplicity. However, the following analysis can be easily extended to unequal-power paths.} (i.e., $P_k=2$, ${\Delta_r=\Delta_t<|\theta_{k,1}-\theta_{k,2}| \leq \Delta^s_r}$) can only be resolved as one dominant path at BS (i.e., ${P_k^s=1}$ with the compositive angle being $\theta^s_{k,1}$), the normalized uplink SNR loss of user $k$ with no channel estimation error from white noise can be calculated as
\begin{align}
{\rm{SNR}_{{\rm{loss}}}} & = 1 - \frac{{\Lambda _1^2 + \Lambda _2^2 + 2{\Lambda _1}{\Lambda _2}\cos \Gamma}}{{2{N^2} + 2N\Lambda \cos \Gamma}}, \label{eq:SNRloss}
\end{align}
where ${\Lambda _1} \buildrel \Delta \over = \frac{{\sin \left( {\pi dN{\Theta _1}/\lambda } \right)}}{{\sin \left( {\pi d{\Theta _1}/\lambda } \right)}}$, ${\Lambda _2} \buildrel \Delta \over = \frac{{\sin \left( {\pi dN{\Theta _2}/\lambda } \right)}}{{\sin \left( {\pi d{\Theta _2}/\lambda } \right)}}$, $\Lambda \buildrel \Delta \over = \frac{{\sin \left( {\pi dN\Theta /\lambda } \right)}}{{\sin \left( {\pi d\Theta /\lambda } \right)}}$, $\Theta \buildrel \Delta \over = \sin {\theta _{k,1}} - \sin {\theta _{k,2}}$, ${\Theta _1} \buildrel \Delta \over = \sin \theta _{k,1}^s - \sin {\theta _{k,1}}$, ${\Theta _2} \buildrel \Delta \over = \sin \theta _{k,1}^s - \sin {\theta _{k,2}}$, $\Gamma \buildrel \Delta \over = {\phi _1} - {\phi _2} - \pi d(N - 1)\Theta /\lambda$, $\phi_1$ and $\phi_2$ are the complex gains' phases of the original two paths, and we have
\begin{align}
0< {\rm{SNR}_{{\rm{loss}}}} < 1. \label{eq:SNRlossrange}
\end{align}
\end{Proposition}
\begin{IEEEproof}
See Appendix A.
\end{IEEEproof}
From (\ref{eq:SNRloss}), the normalized uplink SNR loss is not only related to the number of receive antennas and the angle difference between the two paths, but also related to the phase difference between the two paths' complex gains. With the fixed number of receive antennas and angle difference and $\Theta_1=\Theta_2$, the normalized uplink SNR loss decreases with the increase of the cosine function of the phase difference, i.e., $\cos \Gamma$. It should also be mentioned that when $\cos\Gamma<0$, there is a high probability that two paths can be still resolved even if $|\theta_{k,1}-\theta_{k,2}| < \Delta^s_r$. Interestingly, due to such destructive superposition, the angle difference of these resolved two effective paths will be larger than the angular resolution of the receive antenna array, i.e., $|\theta_{k,1}'-\theta_{k,2}'| >\Delta^s_r$. The numerical results in Section VI will demonstrate this phenomenon.
Additionally, thanks to the accomplished different angular resolutions by different antenna selection methods, the numerical results presented later also indicate that the random antenna selection outperforms the successive antenna selection in terms of the ergodic uplink system SE.
This means that uplink transmission can benefit from the flexible receive antenna topology design in the asymmetrical transceiver-based massive MIMO systems. Furthermore, although the comb antenna selection also achieves a superior angular resolution as the random antenna selection, it suffers from angle ambiguity incurred by the grating lobe, which consequently degrades the channel estimation performance and the achievable SE.
\subsection{Uplink-to-Downlink Channel Transfer}
Due to the difference between the numbers of receive and transmit antennas at the BS, the uplink channel information cannot be directly utilized for downlink data precoding even in the TDD mode and with physical channel reciprocity. Therefore, it is necessary to figure out how to deduce the downlink channel information from the uplink channel estimates in asymmetrical transceivers.
According to the reciprocity between the uplink and downlink channels in TDD systems, we have
\begin{equation}\label{eq:hdl}
{\left[ {{\mathbf{h}}_k^{\rm D}} \right]_{\mathcal{A}}} = {\left( {{\mathbf{h}}_{k,{\rm A}}^{\rm U}} \right)^T},
\end{equation}
where ${\mathbf{h}}_k^{\rm D} \in {\mathbb{C}^{1 \times M}}$ is the downlink channel of user $k$.
Since $|\mathcal{A}|=N$ and ${\mathbf{h}}_{k,{\rm A}}^{\rm U} \in {\mathbb{C}^{N \times 1}}$, ${\mathbf{h}}_{k,{\rm A}}^{\rm U}$ only contains $N$ elements of ${\mathbf{h}}_k^{\rm D}$. Hence, $M-N$ elements in ${\mathbf{h}}_k^{\rm D}$ need to be recovered.
Unfortunately, in the asymmetrical system, the effective aperture of the uplink receive antenna array may be smaller than that of the downlink transmit antenna array. Thus, the angular resolution of the transceiver at the uplink is worse than that of the downlink. Hence, to effectively acquire the large transmit antenna array gain at downlink, it is essential for the uplink-to-downlink channel transfer algorithms to not only fully exploit the channel information, e.g., AoA, based on the received small-dimension signals at the uplink, but also use this uplink channel information to recover and refine the channel estimates for the downlink.\footnote{ Apart from the uplink-to-downlink channel transfer algorithms, a subarray-based switch network collaborated with a round-robin scheme can be also introduced in the asymmetrical transceiver-based BS to acquire the whole channel information. However, undesired insertion loss will also be brought simultaneously, and the round-robin scheme will result in additional pilot training overhead, which makes the system inefficient and may become unacceptable when the number of receive RF chains is relatively small.}
Substituting (\ref{eq:hku}) into (\ref{eq:hdl}), we obtain
\begin{equation}\label{eq:hdl2}
{\left[ {{\mathbf{h}}_k^{\rm D}} \right]_{\mathcal{A}}} = \sqrt {\frac{N}{P_k}} \sum\limits_{i = 1}^{P_k} {{g_{k,i}}{{\bf{a}}^T_{\rm{U}}}({\theta _{k,i}})}.
\end{equation}
Since there is a known relative topology shared by the transmit and receive antenna array, i.e., the relationship between ${{\bf{a}}_{\rm{U}}}({\theta _{k,i}})\in\mathbb{C}^{N\times 1}$ and ${{\bf{a}}_{\rm{D}}}({\theta _{k,i}})\in\mathbb{C}^{1\times M}$, we only need to determine $P_k$, $g_{k,i}$, and $\theta_{k,i}$ to recover user $k$'s downlink channel. Note that we can merely obtain the uplink channel estimate ${\mathbf{\tilde h}}_{k,{\rm A}}^{\rm U}$ instead of the perfect uplink channel ${{\mathbf{h}}_{k,{\rm A}}^{\rm U}}$. Then, the objective of the uplink-to-downlink channel transfer algorithms has been transformed as recovering ${\mathbf{h}}_k^{\rm D}$ from ${\mathbf{\tilde h}}_{k,{\rm A}}^{\rm U}$.
In order to acquire the potential downlink performance of the asymmetrical transceivers, we exploit uplink-to-downlink channel transfer algorithms in Section IV.
\subsection{Downlink Transmission}
When the downlink channel information is obtained via the uplink-to-downlink channel transfer algorithms, downlink pilot/data transmission begins.
Assume the obtained downlink channel estimate of the $k$th user from the uplink-to-downlink channel transfer algorithms is ${\mathbf{\tilde h}}_k^{\rm D} \in {\mathbb{C}^{1 \times M}}$, then, the downlink channel estimation matrix for these $K$ users can be written as
\begin{equation}\label{eq:HdownlinkK}
{\mathbf{\tilde H}}_{\text{A}}^{\text{D}} = {\left[ {{{\left( {{\mathbf{\tilde h}}_1^{\text{D}}} \right)}^T}, \cdots ,{{\left( {{\mathbf{\tilde h}}_K^{\text{D}}} \right)}^T}} \right]^T} \in {\mathbb{C}^{K \times M}}.
\end{equation}
Let ${\mathbf{s}} = {\left[ {{s_1}, \ldots ,{s_K}} \right]^T}$ be the signal vector transmitted by the BS to these $K$ single-antenna users and $\mathbb{E}\{ {{{\left| {{s_k}} \right|}^{{2}}}} \}{{ = 1,}}\forall k = 1, \ldots ,K$, then the received signal vector by users can be expressed as
\begin{equation}
{\mathbf{v}} = \sqrt {{\rho_d}} {\mathbf{H}}_{\text{A}}^{\text{D}}{\mathbf{Ws}} + {\mathbf{n}}_d\in {\mathbb{C}^{K \times 1}},
\end{equation}
where $\rho_d$ is the downlink transmit power for each user; ${\mathbf{H}}_{\text{A}}^{\text{D}} = {\left[ {{{\left( {{\mathbf{h}}_1^{\text{D}}} \right)}^T}, \cdots ,{{\left( {{\mathbf{h}}_K^{\text{D}}} \right)}^T}} \right]^T} \in {\mathbb{C}^{K \times M}}$ denotes the composite downlink channel; ${\mathbf{W}} = \left[ {{{\mathbf{w}}_1}, \ldots ,{{\mathbf{w}}_K}} \right] \in {\mathbb{C}^{M \times K}}$ is the column-normalized downlink precoding matrix, whose detailed expression is based on ${\mathbf{\tilde H}}_{\text{A}}^{\text{D}}$ and the precoding scheme we selected; ${\mathbf{n}}_d \in {\mathbb{C}^{K \times 1}}$ denotes the additive Gaussian white noise with each of its element satisfying $n_{d,k}\sim\mathcal{CN}(0,1)$. Hence, the received signal at the $k$th user can be expressed as
\begin{equation}
{v_k} = \sqrt {{\rho_d}} {\mathbf{h}}_k^{\text{D}}{{\mathbf{w}}_k}{s_k} + \sum\limits_{i = 1,i \ne k}^K {\sqrt {{\rho_d}} {\mathbf{h}}_k^{\text{D}}{{\mathbf{w}}_i}{s_i}} + {n_{d,k}},
\end{equation}
and the ergodic downlink SE of the $k$th user can be given by
\begin{equation}\label{eq:HdownlinkRk}
R_k^{\rm D} = \mathbb{E}\left\{ {{{\log }_2}\left( {1 + \frac{{{{\left| {{\mathbf{h}}_k^{\text{D}}{{\mathbf{w}}_k}} \right|}^2}}}{{\sum\nolimits_{i = 1,i \ne k}^K {{{\left| {{\mathbf{h}}_k^{\text{D}}{{\mathbf{w}}_i}} \right|}^2} + {1 \mathord{\left/
{\vphantom {1 {{\rho_d}}}} \right.
\kern-\nulldelimiterspace} {{\rho_d}}}} }}} \right)} \right\}.
\end{equation}
When the maximal-ratio transmitting (MRT) precoding is employed, we have
\begin{equation}\label{eq:Hdownlinkwk}
{{\mathbf{w}}_k} = {{\left( {{\mathbf{\tilde h}}_k^{\rm D}} \right)}^H}/{\xi _k},
\end{equation}
where ${\xi _k} = \| {{\mathbf{\tilde h}}_k^{\rm D}} \|$ is utilized for normalization.
Thus, the downlink system SE is
\begin{equation}\label{eq:HdownlinksysSE}
\eta _{{\text{SE}}}^{\text{D}} = \sum\limits_{k = 1}^K {R_{k}^{\text{D}}} {\text{ (bit/s/Hz)}}.
\end{equation}
From (\ref{eq:HdownlinkRk}) and (\ref{eq:Hdownlinkwk}), the downlink SE of the asymmetrical transceiver-based system is similar to that of the conventional symmetrical transceiver-based full digital massive MIMO system, except for the downlink channel estimate ${{\mathbf{\tilde h}}_k^D}$, where the former endures a signal dimension extension (i.e., the uplink-to-downlink channel transfer) while the latter directly comes from the uplink channel estimates.\footnote{We assume the conventional symmetrical transceiver-based full digital massive MIMO system also operates in TDD mode, and the uplink-to-downlink channel reciprocity maintain perfectly.} Therefore, the uplink-to-downlink channel transfer algorithms, as well as the receive antenna array topology, have a fundamental impact on the downlink performance of the asymmetrical transceiver-based system.
\section{Uplink-to-Downlink Channel Transfer}
In this section, we propose two algorithms for the uplink-to-downlink channel transfer, i.e., the DFT-based channel transfer algorithm and the mNOMP-based channel transfer algorithm. The DFT-based algorithm is mainly designed for the line-of-sight (LOS) path-dominant scenarios for fast channel transfer while the mNOMP-based algorithm is more suitable for multi-path scenarios.
\subsection{DFT-based Channel Transfer}
Based on (\ref{eq:hkd}) and (\ref{eq:steeringVectorad}), the steering vector of paths in ${\left( {{\mathbf{h}}_k^{\rm D}} \right)^T} \in {\mathbb{C}^{M \times 1}}$ asymptotically becomes orthogonal with the columns of the $M$-dimensional DFT matrix when the number of BS antennas continuously increases due to the employment of ULA. Hence, DFT matrix can be used as a spatial matched filter to perform the fast path detection. The underlying principle of the DFT-based channel transfer algorithm is that we first extends the uplink channel estimate ${\mathbf{\tilde h}}_{k,{\rm A}}^{\rm U}$ to an $M$-dimension vector according to the array topology and then executes the spatial matched filtering to recover the downlink channel information. As shown in {\bf Algorithm\,\ref{Alogrithm1}}, the proposed DFT-based channel transfer algorithm has the following steps.
\begin{enumerate}[(i)]
\item {\em Dimension extension}: Pad ${\mathbf{\tilde h}}_{k,{\rm A}}^{\rm U} \in {\mathbb{C}^{N \times 1}}$ with $M-N$ zeros to obtain an $M-$dimensional vector ${\mathbf{\tilde h}}_{k,{\rm S}}^{\rm U}$, i.e.,
\begin{equation}\label{eq:husmissingelement}
{\left[ {{\mathbf{\tilde h}}_{k,{\rm S}}^{\rm U}} \right]_{\mathcal{A}}} = {\mathbf{\tilde h}}_{k,{\rm A}}^{\rm U};
\end{equation}
\item {\em DFT matrix construction}: Set the oversampling factor as $\zeta$ and construct an $M\zeta-$dimensional DFT matrix ${{\mathbf{F}}_{M\zeta }}$ with its $m$th row and $n$th column element being
\begin{equation}
{\left[ {{{\mathbf{F}}_{M\zeta }}} \right]_{mn}} = {e^{ - j2\pi (m - 1)(n - 1)/M\zeta }};
\end{equation}
\item {\em Spatial matched filtering}: Utilize ${{\mathbf{F}}_{M\zeta}}$ to perform the spatial matched filtering and obtain the complex path gains, i.e.,
\begin{equation}\label{eq:dftcalculate}
{{\bf{g}}_{{\rm{DFT}}}} = \frac{{1 }}{ N}{{\bf{F}}_{M\zeta }}{\left[ {{{\left( {{\bf{\tilde h}}_{k,{\rm{S}}}^{\rm{U}}} \right)}^H},{\bf{0}}_{M(\zeta - 1)}^T} \right]^T}.
\end{equation}
\item {\em Path detection}: Denote ${\tilde g_{[1]}}, \cdots ,{\tilde g_{[N_{\rm peak}]}}$ as the peak elements\footnote{A peak element is a element belonging to the element sequence whose absolute value is larger than its two neighboring elements, i.e., for ${\tilde g_{[i]}},i=1,\ldots,N_{\rm peak}$, we have $|{\tilde g_{[i]}}|\ge |{\tilde g_{[i]-1}}|$ and $|{\tilde g_{[i]}}|\ge |{\tilde g_{[i]+1}}|$.} of the elements of ${{\mathbf{g}}_{\rm{DFT}}}$ and $|{\tilde g_{[1]}}|\ge \cdots \ge|{\tilde g_{[N_{\rm peak}]}}|$.
Then, the number of paths $P_k$, the complex path gains $g_{k,i}$, and the AoAs $\theta_{k,i}$ for user $k$ can be estimated as
\begin{equation}\label{eq:pDft}
\hat P_k=\mathop {\min }\limits_{\hat P_k}\left\{{\left\| {{\bf{\tilde h}}_{k,{\rm{A}}}^{\rm{U}}} \right\|^2} - \sum\limits_{i = 1}^{{{\hat P}_k}} {{{N\left| {{{\tilde g}_{[i]}}} \right|}^2}} \le {{\cal T}_{\rm DFT}}\right\},
\end{equation}
\begin{equation}\label{eq:gDft}
{\hat g_{k,i}}={\sqrt{\hat{P_k}}}{\tilde g_{[i]}^*}, i=1,\ldots,\hat P_k
\end{equation}
and
\begin{equation}\label{eq:thetaDft}
{\hat \theta _{k,i}} = \arcsin\{{{2 {I_i}} \mathord{\left/ {\vphantom {{2\pi {I_i}} {M\zeta }}} \right. \kern-\nulldelimiterspace} {M\zeta }}\}, i=1,\ldots,\hat P_k
\end{equation}
respectively, where $\mathcal{T}_{\rm DFT}$ is the energy threshold and ${I_1}, \cdots ,{I_{N_{\rm peak}}}$ are the original indexes of ${\tilde g_{[1]}}, \cdots ,{\tilde g_{[N_{\rm peak}]}}$ in ${{\mathbf{g}}_{\rm{DFT}}}$. Generally, the smaller $\mathcal{T}_{\rm DFT}$, the more paths we will obtain.
\item {\em Downlink channel recovery}: Based on (\ref{eq:pDft}), (\ref{eq:gDft}), (\ref{eq:thetaDft}) and (\ref{eq:hkd}), we acquire the downlink channel estimate at the BS for user $k$ as
\begin{equation}\label{eq:hdDft}
{\mathbf{\tilde h}}_k^{\rm{D}} = \sqrt {\frac{M}{\hat P_k}} \sum\limits_{i = 1}^{\hat P_k} {{{\hat g}_{k,i}}{{\mathbf{a}}_{\rm{D}}}({{\hat \theta }_{k,i}})}.
\end{equation}
\end{enumerate}
In the \emph{spatial matched filtering} step, we scan the space with an oversampling factor $\zeta$ to obtain a better path estimation accuracy. However, this may result in many fake large path gains in the subsequent detection step due to the limited array aperture. To solve this problem, we find the peak elements of ${{\bf{g}}_{{\rm{DFT}}}}$ first and then sort them to detect the true paths in the {\emph{path detection}} step. The energy threshold in the {\emph{path detection}} step is set as
\begin{equation}\label{eq:thresholdDFT}
{{\cal T}_{\rm DFT}}=N/\rho_{\tau},
\end{equation}
whose detailed derivation has been demonstrated in Appendix B. ${{\cal T}_{\rm DFT}}$ in \eqref{eq:thresholdDFT} indicates that the energy threshold for path detection increases with the number of receive antennas but decreases with the pilot transmit SNR.
Note that the proposed DFT-based channel transfer algorithm can be used for $K$ users in parallel to obtain their downlink channel estimation. Regardless of the specific antenna selection methods, this algorithm can also be easily extended to UPA by constructing an appropriate ${{{\mathbf{F}}_{M\zeta }}}$ for UPA.
\begin{algorithm}[t]
\small
\caption{DFT-based Channel Transfer Algorithm}\label{Alogrithm1}
\begin{algorithmic}[1]
\REQUIRE ${\mathbf{\tilde h}}_{k,{\rm A}}^{\rm{U}}$, ${{\mathbf{\tilde h}}_{k,{\rm S}}^{\rm{U}}}={\mathbf{0}}_{M \times 1}$, ${{{\mathbf{F}}_{M\zeta }}}$, ${{\bf{a}}_{\rm{D}}}(\cdot)$, $\hat P_k=1$, $\mathcal{T}_{\rm DFT}$, $M$, $N$, $\mathcal{A}$. \\
\STATE set ${\left[{{\mathbf{\tilde h}}_{k,{\rm S}}^{\rm U}} \right]_{\mathcal{A}}} = {\mathbf{\tilde h}}_{k,{\rm A}}^{\rm U}$;
\STATE ${{\bf{g}}_{{\rm{DFT}}}} = \frac{{1 }}{ N}{{\bf{F}}_{M\zeta }}{\left[ {{{\left( {{\bf{\tilde h}}_{k,{\rm{S}}}^{\rm{U}}} \right)}^H},{\bf{0}}_{M(\zeta - 1)}^T} \right]^T}$;
\STATE $\left([{\tilde g_{1}}, \cdots ,{\tilde g_{N_{\rm peak}}}],[{I_1}, \cdots ,{I_{N_{\rm peak}}}]\right)=\text{findpeaks}({\mathbf{g}}_{\rm{DFT}})$;
\STATE $\left([{\tilde g_{[1]}}, \cdots ,{\tilde g_{[N_{\rm peak}]}}]\right)=\text{sort}([{\tilde g_{1}}, \cdots ,{\tilde g_{N_{\rm peak}}}],\text{`descend'})$;
\WHILE {${\left\| {{\bf{\tilde h}}_{k,{\rm{A}}}^{\rm{U}}} \right\|^2} - \sum\limits_{i = 1}^{{{\hat P}_k}} {{{N\left| {{{\tilde g}_{[i]}}} \right|}^2}} \le {\cal T}_{\rm DFT}$}
\STATE ${\hat g_{k,{\hat P_k}}}={\sqrt{\hat{P_k}}}{\tilde g_{[\hat P_k]}^*}$;
\STATE ${\hat \theta _{k,{\hat P_k}}} = \arcsin\{{2 {I_{\hat P_k}}} \mathord{\left/ {\vphantom {{2\pi {I_{\hat P_k}}} {M\zeta }}} \right. \kern-\nulldelimiterspace} {M\zeta }\}$;
\STATE $\hat P_k = \hat P_k +1$;
\ENDWHILE
\STATE ${\mathbf{\tilde h}}_k^{\rm{D}} = \sqrt {\frac{M}{\hat P_k}} \sum\limits_{i = 1}^{\hat P_k} {{{\hat g}_{k,i}}{{\mathbf{a}}_{\rm{D}}}({{\hat \theta }_{k,i}})}$;
\ENSURE ${\mathbf{\tilde h}}_k^{\rm{D}}$.
\end{algorithmic}
\end{algorithm}
Additionally, since there is no iteration or matrix inversion calculation, the DFT-based channel transfer algorithm has low computational complexity. The dominant computational complexity of the DFT-based algorithm is resulted by the spatial matched filtering step, which, however, can also be efficiently executed through a fast Fourier transform (FFT). Nevertheless, the performance of the DFT-based algorithm is constrained by the angle mismatch between ${\mathbf{\tilde h}}_{k,{\rm S}}^{\rm U}$ and the DFT columns and the element absence of ${\mathbf{\tilde h}}_{k,{\rm S}}^{\rm U}$, especially when the number of receive antennas is small or there are multiple paths in the propagation channel. Path detection error may occur when these paths are close to each other.
\subsection{mNOMP-based Channel Transfer}
To overcome the path detection error problem and improve the estimation accuracy, we develop another channel transfer algorithm based on Newtonized orthogonal matching pursuit (NOMP).
From (\ref{eq:hdl2}), the uplink-to-downlink channel transfer problem is similar to that of the line spectrum estimation problem \cite{Mamandipoor16Newtonized} but with the absence of elements. Therefore, we propose to modify the NOMP algorithm to restore the downlink channel information. By enhancing the NOMP algorithm with the capability of dealing with the absence of majority elements, the obtained mNOMP algorithm can well solve the uplink-to-downlink channel transfer problem. Furthermore, we also derive an energy threshold for the termination condition step of the mNOMP-based algorithm. By considering the residual noise power into the channel estimates, the proposed mNOMP algorithm can refine the recovered downlink channel estimates as a noise filter. The algorithm is summarized in {\bf Algorithm\,\ref{Alogrithm2}} and the detailed procedure is listed as follows.
\noindent (i) {\em Element completion and initialization}: Pad ${\mathbf{\tilde h}}_{k,{\rm A}}^{\rm U} \in {\mathbb{C}^{N \times 1}}$ with $M-N$ zeros to obtain ${\mathbf{\tilde h}}_{k,{\rm S}}^{\rm U}$ and ${\left[ {{\mathbf{\tilde h}}_{k,{\rm S}}^{\rm U}} \right]_{\mathcal{A}}} = {\mathbf{\tilde h}}_{k,{\rm A}}^{\rm U}$. Initialize the residual vector ${{\mathbf{y}}_r} = {\mathbf{\tilde h}}_{k,{\rm S}}^{\rm U}$, the complex path gain sets $\mathcal{G}=\mathcal{G}_k = \emptyset $, the AoA angle sets $\Omega=\Omega_k = \emptyset $, and the threshold $\mathcal{T}_{\rm m}$;
\noindent (ii) {\em New path detection}: Detect the path with the maximum gain by performing FFT on the residual vector ${{\mathbf{y}}_r}$ and obtain the complex gain $g_{max}$ and the spatial angle $w_{max}$ \footnote{We define $w_{max} \buildrel \Delta \over = \sin\theta_{max}$ and in this section we use ${\bf{a}}(w)$ to denote the steering vector ${\bf{a}}(\arcsin w)$ for notation simplicity.}, i.e.,
\begin{align}
({g_{\max}},{w_{\max}}) &= \mathop {\max }\limits_{|{g_i}|} \left\{ g_i|{{\bf{g}}_{{\rm{m}}}}={\frac{1}{{ N }}{{\bf{F}}_{M\zeta }}{{\left[ {{\bf{y}}_r^H,{\bf{0}}_{M(\zeta - 1)}^T} \right]}^T}} \right\},
\end{align}
where $g_i$ denotes the elements of ${{\bf{g}}_{{\rm{m}}}}$.
Then, we update the residual vector as
\begin{align}
{{{\bf{\dot y}}}_r} &= {{\mathbf{y}}_r} - {\sqrt N}{g_{\max }}{{\mathbf{a}}_{\rm S}}({w_{max}}),
\end{align}
where ${\left[ {{{\mathbf{a}}_{\rm S}}({w_{max}})} \right]_{\mathcal{A},:}} = {{\mathbf{a}}_{\rm U}}({w_{max}})$, ${\left[ {{{\mathbf{a}}_{\rm S}}({ w_{max}})} \right]_{\mathcal{M}\backslash \mathcal{A},:}} = {\mathbf{0}}$ and $\mathcal{M} = \{ 1,2, \cdots ,M\} $;
\noindent (iii) {\em Termination condition}: ${\left\| {{{\mathbf{\dot y}}_r}} \right\|^2}$ is compared with the threshold $\mathcal{T}_{\rm m}$. When ${\left\| {{{\mathbf{\dot y}}_r}} \right\|^2}<\mathcal{T}_{\rm m}$, $\mathcal{G}_k =\mathcal{G}$, $\Omega_k = \Omega $. The algorithm terminates and outputs the downlink channel estimate for user $k$ as
\begin{equation}
{\mathbf{\hat h}}_k^{\rm D} = \sqrt{M} \sum\limits_{i = 1}^{\tilde P_k} {{{\tilde g}_i}{{\mathbf{a}}_{\rm D}}({{\tilde w }_i})},
\end{equation}
where ${{\tilde g}_i} \in \mathcal{G}_k$, ${{\tilde w }_i}\in \Omega_k$. The estimated number of paths for user $k$ is $\tilde P_k={\left| \mathcal{G}_k \right|}$;
When ${\left\| {{{\mathbf{\dot y}}_r}} \right\|^2}\geq\mathcal{T}_{\rm m}$, proceed to the next step;
\noindent (iv) {\em Local gain and angle refinement}: To alleviate the estimation error due to the on-grid angular constraint in the new path detection step, we perform the Newton updates for $ g_{max}$ and $ w_{max}$ with ${{\mathbf{\dot y}}_r}$ by $R_s$ times to obtain optimized values, i.e., for each update we first have
\begin{equation}\label{eq:w_update}
{\tilde w_{\max }} = {w_{\max }} - {{J'_w(w_{max})} \mathord{\left/ {\vphantom {{J'_w(w_{max})} {J^{''}_w(w_{max})}}} \right. \kern-\nulldelimiterspace} {J^{''}_w(w_{max})}},
\end{equation}
\begin{equation}\label{eq:g_update}
{\tilde g_{\max }} = \frac{{{{\left( {{{\bf{a}}_{\rm{S}}}({{\tilde w}_{\max }})} \right)}^H}}}{{{{\left\| {{{\bf{a}}_{\rm{S}}}({{\tilde w}_{\max }})} \right\|}^2}}}\left( {{{{\bf{\dot y}}}_r} + \sqrt N {g_{\max }}{{\bf{a}}_{\rm{S}}}({w_{\max }})} \right),
\end{equation}
where
\begin{equation}
J'_w(w) = - 2\operatorname{Re} \left\{ {\sqrt N}{{g_{\max }}{\mathbf{\dot y}}_r^H\frac{{\partial {{\mathbf{a}}_{\rm S}}(w )}}{{\partial w }}} \right\},
\end{equation}
and
\begin{align}\label{eq:Jdouble_update}
J^{''}_w(w)=& - 2\operatorname{Re} \left\{ {\sqrt N}{{g_{\max }}{\mathbf{\dot y}}_r^H\frac{{{\partial ^2}{{\mathbf{a}}_{\rm S}}(w)}}{{\partial {w ^2}}}} \right\} \nonumber \\
&+ 2N{\left| {{g_{\max }}} \right|^2}{\left( {\frac{{\partial {{\mathbf{a}}_{\rm S}}(w )}}{{\partial w }}} \right)^H}\frac{{\partial {{\mathbf{a}}_{\rm S}}(w )}}{{\partial w }},
\end{align}
and then update
\begin{equation}\label{eq:ydot_update1}
{{{\bf{\dot y}}}_r^{\text{temp}}}={{\mathbf{\dot y}}_r} + {\sqrt N}{g_{\max }}{{\mathbf{a}}_{\rm S}}({w _{\max }}) - {\sqrt N}{\tilde g_{\max }}{{\mathbf{a}}_{\rm S}}({\tilde w _{\max }}),
\end{equation}
and
\begin{equation}\label{eq:ydot_update2}
{{{\bf{\dot y}}}_r}={{\mathbf{\dot y}}_r^{\text{temp}}}.
\end{equation}
Note that we update $ w_{max}$ in \eqref{eq:w_update}, $ g_{max}$ in \eqref{eq:g_update} and ${{{\bf{\dot y}}}_r}$ in \eqref{eq:ydot_update2} only when ${J^{''}_w(w_{max})}>0$ and ${\left\| {{{\bf{\dot y}}}_r^{\text{temp}}} \right\|^2} \le {\left\| {{{\bf{\dot y}}}_r} \right\|^2}$.
\noindent (v) {\em Global refinement}: Let ${{{\bf{\ddot y}}}_r} = {{\mathbf{\dot y}}_r}$ and update $\mathcal{G} = \{ \mathcal{G},{\tilde g_{\max }}\} $ and $\Omega = \{ \Omega ,{\tilde w _{\max }}\} $.
Then, for each $\tilde w \in \Omega$, cyclically perform the local gain and angle refinement step by $R_c$ times to update the angle estimates in $\Omega$;
\noindent (vi) {\em Global gain corrections}: Based on the $\Omega$ from the global refinement, update all the estimated complex gains in $\mathcal{G}$ by means of the LS method:\\
Let ${\mathbf{A}} = \left[ {{{\mathbf{a}}_{\rm S}}({{\tilde \theta }_1}), \ldots ,{{\mathbf{a}}_{\rm S}}({{\tilde \theta }_{\left| \Omega \right|}})} \right]$, then
\begin{equation}
{\mathbf{\tilde g}} = {\left( {{{\mathbf{A}}^H}{\mathbf{A}} + {\sigma ^2}{\mathbf{I}_{\left| \Omega \right|}}} \right)^{ - 1}}{{\mathbf{A}}^H}{\mathbf{\tilde h}}_{k,{\rm S}}^{\rm U}/{\sqrt N},
\end{equation}
where ${\mathbf{\tilde g}} \triangleq {\left[ {{{\tilde g}_1}, \ldots ,{{\tilde g}_{\left| \mathcal{G} \right|}}} \right]^T}$, ${\left| \Omega \right|}={\left| \mathcal{G} \right|}$, and $\sigma^2\mathbf{I}_{\left| \Omega \right|}$ is introduced for retaining full rank of ${\left( {{{\mathbf{A}}^H}{\mathbf{A}} + {\sigma ^2}{\mathbf{I}_{\left| \Omega \right|}}} \right)^{ - 1}}$. We set ${\sigma ^2} = {10^{ - 4}}$ in the simulation. Then, after updating
\begin{equation}
{{\mathbf{y}}_r} = {\mathbf{\tilde h}}_{k,{\rm S}}^{\rm U} - {\sqrt N}{\mathbf{A\tilde g}},
\end{equation}
we proceed to the new path detection step for the next round of processing until the termination condition is satisfied.
Similar to the DFT-based algorithm, in the proposed mNOMP-based algorithm, we first introduce ${\mathbf{\tilde h}}_{k,{\rm S}}^{\rm U}$ into the element completion step to deal with the element absence. The power variation owing to the absence of the $M-N$ elements is also considered in the algorithm.
In addition, the objective function of the Newton update in the local gain and angle refinement step is
\begin{equation}
\left( {{{\tilde w}_{\max }},{{\tilde g}_{\max }}} \right) = \mathop {\arg \min }\limits_{{w},{g}} {{{\left\| {{{{\bf{\dot y}}}_r} - \sqrt N {g}{{\bf{a}}_{\rm{S}}}({w})} \right\|}^2}},
\end{equation}
and
\begin{equation}
{J_w(w)}={{{\left\| {{{{\bf{\dot y}}}_r} - \sqrt N {g_{\max}}{{\bf{a}}_{\rm{S}}}({w})} \right\|}^2}}.
\end{equation}
$J'_w(w)=\partial {J_w(w)}/\partial w$ and $J^{''}_w(w)=\partial^2 {J_w(w)}/\partial w^2$ are the first and the second-order partial derivatives of ${J_w(w)}$ with respect to $w$, respectively.
The threshold, $\mathcal{T}_{\rm m}$, in the termination condition step is obtained according to the residual noise power. From Appendix B, we have
\begin{equation}
\mathcal{T_{\rm m}}=N/\rho_\tau.
\end{equation}
Note that $\mathcal{T_{\rm m}}$ is different from those in \cite{Mamandipoor16Newtonized,Han19Efficient} where the thresholds are based on the false alarm rate.
Numerical results in Section VI reveal that the proposed mNOMP-based algorithm achieves great performance under both LOS-dominant and multi-path scenarios. Moreover, the antenna array's topology significantly impacts the performance of the proposed mNOMP-based algorithm.
\subsection{Discussion}
As mentioned before, the dominant computational complexity of the DFT-based algorithm comes from \eqref{eq:dftcalculate} in the spatial matched filtering step. Since \eqref{eq:dftcalculate} can be calculated via FFT, the computational complexity of the DFT-based algorithm is $\mathcal{O}({M\zeta }\log{M\zeta })$. As for the mNOMP-based algorithm, the computational complexity mainly comes from the new path detection step, the global refinement step, and the gain correction step, whose computational complexity are $\mathcal{O}(\tilde P_k{M\zeta }\log{M\zeta })$, $\mathcal{O}(M{\tilde P_k^2}R_c R_s)$, and $\mathcal{O}({\tilde P_k^4}+M{\tilde P_k^3})$, respectively. Hence, the computational complexity of the mNOMP-based algorithm is $\mathcal{O}(\tilde P_k{M\zeta }\log{M\zeta }+M{\tilde P_k^2}R_c R_s+M{\tilde P_k^3}+{\tilde P_k^4})$.
Although the mNOMP-based algorithm has much higher computational complexity than the DFT-based algorithm, we always have $\tilde P_k <<M$ and $R_c, R_s <<M$.
In addition, as we emphasized in step (iv) of the mNOMP-based algorithm, the updates of $ w$, $ g$ and ${{{\bf{\dot y}}}_r}$ are only performed when ${J^{''}_w(w)}>0$ and ${\left\| {{{\bf{\dot y}}}_r^{\text{temp}}} \right\|^2} \le {\left\| {{{\bf{\dot y}}}_r} \right\|^2}$. Therefore, at each iteration we only retain the detected new path (i.e., $w_{\max }$ and $g_{\max }$) that decreases the residual signal energy. Furthermore, since the LS update in step (vi) of the mNOMP-based algorithm can also only lead to a residual signal energy reduction \cite{Mamandipoor16Newtonized}, the convergence of the mNOMP-based algorithm is guaranteed.
\begin{algorithm}[h]
\small
\caption{mNOMP-based Channel Transfer Algorithm}\label{Alogrithm2}
\begin{algorithmic}[1]
\REQUIRE ${{\mathbf{y}}_r} = {\mathbf{\tilde h}}_{k,{\rm S}}^{\rm U}$, ${{\mathbf{a}}_{\rm S}}(\cdot)$, ${{{\mathbf{F}}_{M\zeta }}}$, $\mathcal{G},\mathcal{G}_k = \emptyset $, $\Omega, \Omega_k = \emptyset $, $R_c$, $R_s$, $\mathcal{T}_{\rm m}$, $M$, $N$, $\mathcal{A}$. \\
\WHILE {${\left\| {{{\mathbf{y}}_r}} \right\|^2}\geq\mathcal{T}_{\rm m}$}
\STATE $({g_{\max}},{w_{\max}}) = \mathop {\max }\limits_{|{g_i}|} \left\{ g_i|{{\bf{g}}_{{\rm{m}}}}={\frac{1}{{ N }}{{\bf{F}}_{M\zeta }}{{\left[ {{\bf{y}}_r^H,{\bf{0}}_{M(\zeta - 1)}^T} \right]}^T}} \right\}$;
\STATE ${{\mathbf{\dot y}}_r} = {{\mathbf{y}}_r} - {\sqrt N}{g_{\max }}{{\mathbf{a}}_{\rm S}}({w _{\max }})$;
\STATE Refine $g_{\max}$ and $w_{\max}$ with ${{\mathbf{\dot y}}_r}$ by $R_s$ times via \eqref{eq:w_update}-\eqref{eq:ydot_update2}.
\STATE Let ${{\mathbf{\ddot y}}_r} = {{\mathbf{\dot y}}_r}$, $\mathcal{G} = \{ \mathcal{G},{\tilde g_{\max }}\} $, $\Omega = \{ \Omega ,{\tilde w _{\max }}\} $;
\FOR{$i=1:R_c$}
\FOR{each $\tilde w \in \Omega$ and $\tilde g \in \mathcal{G}$}
\STATE Refine $\tilde w$ and $\tilde g$ with ${{\mathbf{\ddot y}}_r}$ by $R_s$ times via \eqref{eq:w_update}-\eqref{eq:ydot_update2}.
\ENDFOR
\ENDFOR
\STATE ${\mathbf{A}} = [ {{{\mathbf{a}}_{\rm S}}({{\tilde w }_1}), \ldots ,{{\mathbf{a}}_{\rm S}}({{\tilde w }_{\left| \Omega \right|}})} ]$;\\
\STATE ${\mathbf{\tilde g}} = {\left( {{{\mathbf{A}}^H}{\mathbf{A}} + {\sigma ^2}{\mathbf{I}}} \right)^{ - 1}}{{\mathbf{A}}^H}{\mathbf{\tilde h}}_{k,{\rm S}}^{\rm U}/{\sqrt N}$;\\
\STATE ${{\mathbf{y}}_r} = {\mathbf{\tilde h}}_{k,{\rm S}}^{\rm U} - {\sqrt N}{\mathbf{A\tilde g}}$;\\
\ENDWHILE
\STATE $\mathcal{G}_k =\mathcal{G}$, $\Omega_k = \Omega $, $\tilde P_k={\left| \mathcal{G}_k \right|}$;
\STATE ${\mathbf{\hat h}}_k^{\rm D} = \sqrt {{M}} \sum\limits_{i = 1}^{\tilde P_k} {{{\tilde g}_i}{{\mathbf{a}}_{\rm D}}({{\tilde w}_i})}$;
\ENSURE $\mathcal{G}_k, \Omega_k, {\mathbf{\hat h}}_k^{\rm D}$.
\end{algorithmic}
\end{algorithm}
\section{Cost and Energy Comparison}
In this section, we first provide the cost and the energy consumption models for the asymmetrical transceiver-based massive MIMO systems and then the cost and energy consumption comparisons with other conventional symmetrical transceiver-based massive MIMO systems, e.g., the full digital massive MIMO systems and the hybrid massive MIMO systems, are performed.
\begin{table*}[th]
\caption{The cost and power consumption of hardware components.}\label{tab:parameters}
\scriptsize
\centering
\begin{tabular}{m{1.4cm}m{0.55cm}m{0.85cm}<{\centering}m{0.55cm}<{\centering}m{0.65cm}<{\centering}m{0.65cm}<{\centering}m{0.65cm}m{0.65cm}<{\centering}m{0.75cm}m{0.75cm}m{0.5cm}b{0.5cm}<{\centering}}
\toprule[1.2pt]
{Component} & {PA}& {PA driver}& {LNA}& {Switch} & {Mixer} & {LO amp.} & {Phase shifter}& {IF Tx chain}& {IF Rx chain} & {DAC} & {ADC}\\[3pt]
\hline
{Reference Cost (USD)} & $50$& $30$& $27$& $27$& $24$ & $30$& {170}& {140}& {140} & {55} & {451}\\[3pt]
\rowcolor{mygray}
{Power Consump. (W)} & $3.68$ & $0.85$ & $0.33$ & $0.1$ &$0$ & $0.6$&{0}& {1.75}& {1.25} & {2.07} & {2.82}\\[3pt]
\bottomrule[1.0pt]
\hline
\end{tabular}
\end{table*}
Table\,\ref{tab:parameters} presents a reference cost and power consumption table of the hardware components that are used to implement a massive MIMO testbed operating at $28$\,GHz with $500$\,MHz bandwidth \cite{Yang18Digital}.
Then, based on the block diagram of the asymmetrical transceiver presented in Fig.\,\ref{Fig:system_architecture} and the architectures of the hybrid transceivers provided in \cite{heath2016overview}, we have
\begin{align}
{C_{{\text{ADBN}}}} &= M\big( {{C_{{\text{PA}}}} + {C_{{\text{PA driver}}}} + {C_{{\text{Mixer}}}} + {C_{{\text{LO amp}}}} + {C_{{\text{IF Tx chain}}}} } \nonumber \\
&\quad + {C_{{\text{DAC}}}}\big)+ N\left( {{C_{{\text{LNA}}}} + {C_{{\text{Switch}}}} + {C_{{\text{IF Rx chain}}}} + {C_{{\text{ADC}}}}} \right),
\end{align}
\begin{align}
{C_{{\text{DBM}}}}& = M( {C_{{\text{PA}}}} + {C_{{\text{PA driver}}}} + {C_{{\text{LNA}}}} + {C_{{\text{Switch}}}} + {C_{{\text{Mixer}}}} \nonumber \\
&\quad + {C_{{\text{LO amp}}}} + {C_{{\text{IF Tx chain}}}}+ {C_{{\text{IF Rx chain}}}} + {C_{{\text{ADC}}}} + {C_{{\text{DAC}}}} ),
\end{align}
\begin{align}
{C_{{\text{HBFN}}}} &= M\left( {{C_{{\text{PA}}}} + {C_{{\text{PA driver}}}} + {C_{{\text{LNA}}}} + 2{C_{{\text{Switch}}}}} \right) \nonumber \\
&\quad + MN{C_{{\text{Phase shifter}}}}+ N\big( {C_{{\text{Mixer}}}} + {C_{{\text{LO amp}}}} \nonumber \\
&\quad + {C_{{\text{IF Tx chain}}}}+ {C_{{\text{IF Rx chain}}}} + {C_{{\text{ADC}}}} + {C_{{\text{DAC}}}} \big),
\end{align}
\begin{align}
{C_{{\text{HBSN}}}} &= M\left( {{C_{{\text{PA}}}} + {C_{{\text{PA driver}}}} + {C_{{\text{LNA}}}} + 2{C_{{\text{Switch}}}} + {C_{{\text{Phase shifter}}}}} \right) \nonumber \\
&\quad + N\big( {C_{{\text{Mixer}}}} + {C_{{\text{LO amp}}}} + {C_{{\text{IF Tx chain}}}} + {C_{{\text{IF Rx chain}}}} \nonumber \\
&\quad + {C_{{\text{ADC}}}} + {C_{{\text{DAC}}}}\big),
\end{align}
where ${C_{{\text{ADBN}}}}$, ${C_{{\text{DBM}}}}$, ${C_{{\text{HBFN}}}}$, and ${C_{{\text{HBSN}}}}$ represent the cost of the asymmetrical transceiver-based, the conventional full digital symmetrical transceiver-based, the full-connected hybrid transceiver-based, and the subarray hybrid transceiver-based BSs, respectively.\footnote{For the two hybrid transceivers, conventional symmetrical architectures are employed and $N$ denotes the number of RF chains.} The $C_{\{\cdot\}}$ in the right-hand side denotes the cost of the corresponding hardware components.
Consequently, combined with Table\,\ref{tab:parameters}, we obtain
\begin{equation}\label{eq:costComparison}
{C_{{\text{HBFN}}}} > {C_{{\text{DBM}}}} > {C_{{\text{HBSN}}}} > {C_{{\text{ADBN}}}},
\end{equation}
when $M=128$ and $N=16$.
The above inequality indicates that the proposed asymmetrical transceiver achieves the lowest cost while the full-connected hybrid transceiver achieves the highest. This is because, on one hand, fewer ADCs are needed for the asymmetrical transceivers when compared with the full digital symmetrical transceivers; On the other hand, the phase shifters, which are generally utilized in hybrid transceivers (e.g. $MN$ phase shifters are needed for the full-connected hybrid transceiver), are in fact expensive.
Similar to the cost model, we also provide the energy consumption model of the asymmetrical transceiver-based BS as
\begin{align}
{P_{{\text{ADBN}}}} &= (1 - \varepsilon )M\big( {P_{{\text{PA}}}} + {P_{{\text{PA driver}}}} + {P_{{\text{Mixer}}}} + {P_{{\text{LO amp}}}} \nonumber \\
& \quad + {P_{{\text{IF Tx chain}}}} + {P_{{\text{DAC}}}} \big)+ \varepsilon N\big( {P_{{\text{LNA}}}} + {P_{{\text{Switch}}}} + {P_{{\text{Mixer}}}}\nonumber \\
&\quad+ {P_{{\text{LO amp}}}} + {P_{{\text{IF Rx chain}}}} + {P_{{\text{ADC}}}} \big),
\end{align}
where $P_{\{\cdot\}}$ in the right hand side denotes the power of the corresponding hardware components; $\varepsilon$ is the ratio of the slots assigned for the uplink to that of the downlink and we set $\varepsilon =1/3$.
Thus, we can also obtain ${P_{{\text{DBM}}}}$, ${P_{{\text{HBFN}}}}$ and ${P_{{\text{HBSN}}}}$, and
\begin{equation}\label{eq:powerComparison}
{P_{{\text{DBM}}}} > {P_{{\text{ADBN}}}}>{P_{{\text{HBSN}}}}= {P_{{\text{HBFN}}}}.
\end{equation}
This result comes from the fact that the power consumption of phase shifters in the hybrid transceivers are nearly zero and there are less power-hungry ADCs and IF Rx chains in the asymmetrical transceivers.
Moreover, we define the energy efficiency as
\begin{equation}\label{eq:EE}
{\text{EE}} = \frac{{\varepsilon {{\eta _{{\text{SE}}}^{\text{U}}}} + (1 - \varepsilon ){{\eta _{{\text{SE}}}^{\text{D}}}}}}{{{P_{{\text{BS}}}}}} \times {\text{Bandwidth}},
\end{equation}
where ${P_{{\text{BS}}}}$ is one of ${P_{{\text{ADBN}}}}$, ${P_{{\text{DBM}}}}$, ${P_{{\text{HBFN}}}}$ and ${P_{{\text{HBSN}}}}$.
Due to the channel model adopted in (\ref{eq:hku}) and (\ref{eq:hkd}), it is difficult to obtain closed-form expressions for the EE above. Hence, we evaluate the EEs of different transceivers via the Monte-Carlo simulation.
The simulation results in Section VI indicate that, when the numbers of users served at downlink and uplink are the same, the hybrid transceivers achieve a better EE than the asymmetrical transceiver. Nevertheless, thanks to the employed large number of transmit RF chains, the asymmetrical transceivers can simultaneously support more data streams at downlink than that of the uplink. Hence, the downlink EE of the asymmetrical transceivers can be further enhanced when unequal number of users are served at uplink and downlink.
\section{Numerical Results}
In this section, we first evaluate the beam pattern of different receive antenna arrays and uplink SNR loss. Then, the performance of the proposed uplink-to-downlink channel transfer algorithms is investigated.
After that, the uplink and downlink SEs and the corresponding EEs are presented to demonstrate the system performance and the superiority of the proposed asymmetrical transceiver-based systems in contrast to conventional symmetrical massive MIMO systems.
\subsection{Beam Pattern and Uplink SNR Loss}
We evaluate the angular resolution of different receive antenna arrays via their beam patterns.
Fig.\,\ref{fig:staticDirectional} presents the static directional diagram of the receive antenna array from different antenna selections (i.e., the random antenna selection, the successive antenna selection, and the comb antenna selection).
From Fig.\,\ref{fig:staticDirectional}, a broader beam as well as a lower angular resolution are resulted in the successive antenna array due to the successive antenna element arrangement and consequently a limited effective aperture. In contrast, owing to the distributed antenna element arrangement which in fact spans across the whole antenna array ($M=128$), the receive antenna array with a random-element pattern attains a smaller beamwidth and thus a higher angular resolution \cite{Lo64Mathematical}. Therefore, the random antenna array ($N=32$) acquires the same main lobe width as that of $N=M=128$ and is capable of distinguishing more users or more paths. Despite that, thanks to the large effective array aperture, the array with a comb-element pattern achieves a sharp beam as well. However, because of the larger adjacent element spacing, e.g., $2\lambda$ when $N=32$, grating lobes occur in the comb antenna array and there will be angular ambiguity when interfering paths come from the angles of the grating lobes.
\begin{figure}[h]
\centering
\includegraphics[scale= 0.35]{Fig5.eps}
\caption{ The static directional diagram of different receive antenna arrays when $M=128$ and $N=32$.}\label{fig:staticDirectional}
\end{figure}
\begin{figure}[htbp]
\centering
\subfigure[The normalized uplink SNR loss.]{
\begin{minipage}[t]{1\linewidth}
\centering
\includegraphics[scale= 0.45]{Fig6a.eps}
\end{minipage
\vspace{0.1pt}
\subfigure[The resolved paths.]{
\begin{minipage}[t]{1\linewidth}
\centering
\includegraphics[scale= 0.45]{Fig6b.eps}
\end{minipage
\centering
\caption{The uplink SNR loss incurred by the degraded angular resolution from successive antenna selection. Two equal-power paths are assumed in the environment. $\Delta_t=\Delta_r=0.45^o$, $\Delta^s_r=3.58^o$, $|\theta_{1}-\theta_{2}|=2.97^o$, and ${\Delta_r=\Delta_t<|\theta_{1}-\theta_{2}| \leq \Delta^s_r}$.}\label{fig:SNRloss}
\end{figure}
Fig.\,\ref{fig:SNRloss} investigates the normalized uplink SNR loss provided in Section III.B. Note that the phase difference is defined as $\phi_2-\phi_1$. From Fig.\,\ref{fig:SNRloss}(a) and (b), it can be observed that, with the increasing phase difference, the $\rm SNR_{loss}$ first decreases and then increases and there is only one dominant path that can be resolved when the phase difference is between $0$ and $\pi$. Nearly $20\%$ SNR loss can be achieved when $\phi_2-\phi_1=\pi$. Besides, Fig.\,\ref{fig:SNRloss}(b) also indicates that two weak paths instead of one dominant path will be resolved even if $|\theta_{1}-\theta_{2}| < \Delta^s_r$ when the phase difference continuously increases (i.e., larger than $\pi$ but less than $2\pi$). This is because $\cos\Gamma<0$ and thus these two paths are in destructive superposition in the anticipated compositive angle, which finally results in two effective paths with a larger angular separation, i.e., $|\theta_{1}'-\theta_{2}'| >\Delta^s_r$.
\subsection{Uplink-to-Downlink Channel Transfer}
To examine the performance of the uplink-to-downlink channel transfer algorithms, the normalized mean-squared error (NMSE) is utilized as the metric and is defined as
\begin{equation}\label{eq:NMSE}
{\rm NMSE} = \mathbb{E}\left\{ {{{{{\left\| {{\mathbf{\tilde h}}_k^{\rm D} - {\mathbf{h}}_k^{\rm D}} \right\|}^2}}}/{{{{\left\| {{\mathbf{h}}_k^{\rm D}} \right\|}^2}}}} \right\}.
\end{equation}
Note that ${\mathbf{\bar h}}_{k,{\rm A}}^{\rm U}$, i.e., the uplink LMMSE channel estimate, is used as the input of the channel transfer algorithms.
Figs.\,\ref{fig:NMSELOS} and \ref{fig:NMSENLOS} present the NMSE for user 1 under different antenna array topologies for LOS and multi-path scenarios, respectively. The angles of paths are uniformly randomly generated over $[-60^o,60^o]$.
As can be observed from Figs.\,\ref{fig:NMSELOS} and \ref{fig:NMSENLOS}, both the two algorithms benefit from the increasing number of receive antennas and the increasing aperture of array topologies.
With the receive antenna array from the random antenna selection, the mNOMP-based algorithm can acquire nearly $4$\,dB performance improvement when the number of receive antennas $N$ increases from $16$ to $32$ at high SNR regime.
When $N=16$, the random-element pattern achieves the best performance due to its larger array aperture than the successive-element pattern and its better grating lobe suppression than the comb-element pattern. The better grating lobe suppression of the random-element pattern comes from the unequal sized element spacing, which makes the grating lobes replaced by unequal amplitude side lobes all less than the main lobe \cite{King60unequally,Mailloux84array}. In contrast, the comb-element pattern achieves the worst performance due to the emergence of grating lobes in the visible region, i.e., $[-60^o,60^o]$.
The comb-element pattern can only perform well under extremely narrow angular spread scenarios owing to these grating lobes.
In addition, at the cost of the additional computational complexity, the mNOMP-based algorithm always outperforms the DFT-based algorithm no matter at the LOS scenarios or at the multi-path scenarios.
The DFT-base algorithm is more suitable for fast channel transfer or limited available computational resource scenarios.
\begin{figure}[h]
\centering
\includegraphics[scale= 0.5]{Fig7.eps}
\caption{ The NMSE of the estimated downlink channel information for user $1$ under LOS path-dominant scenarios. $M=128$ and $R_s=R_c=2$. $P=2$ and the power allocation for paths is $[0.9\quad 0.1]$. $\zeta$ for the DFT- and mNOMP-based algorithms are $8$ and $4$, respectively. }\label{fig:NMSELOS}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[scale= 0.5]{Fig8.eps}
\caption{ The NMSE of the estimated downlink channel information for user $1$ under multi-path scenarios. $M=128$, $P=3$ and $R_s=R_c=2$. $\zeta$ for the DFT- and mNOMP-based algorithms are $8$ and $4$, respectively.}\label{fig:NMSENLOS}
\end{figure}
\subsection{SE and EE}
Fig.\,\ref{fig:SEUL} shows the uplink and downlink SEs. The receive antennas are selected based on random antenna selection and the mNOMP-based uplink-to-downlink channel transfer algorithm is employed for the asymmetrical transceiver.
For the purpose of comparison, we also provide the SEs of the conventional full digital symmetrical transceiver-based and the hybrid transceiver-based massive MIMO systems. Note that $M=128$, $N=32$, $K=10$, and $P_1=\ldots=P_K=3$ in Fig.\,\ref{fig:SEUL}. LMMSE channel estimation and zero-forcing (ZF) detection are employed for all the systems at the uplink, and ZF precoding is utilized for asymmetrical and conventional full digital systems at the downlink.
As for the hybrid transceiver-based systems, eigenvectors of the uplink channel correlation matrices are used for analog combining at the uplink. While for the downlink precoding, the phased-ZF method proposed by \cite{Liang14Low} is employed for the full-connected hybrid transceiver-based systems and the SIC-based hybrid precoding method proposed by \cite{Gao16Energy} is utilized for the subarray-based hybrid transceiver-based systems, respectively.
Both of these two hybrid precoding methods use the full-dimensional (i.e., $M \times K$) instantaneous channel state information.
\begin{figure}[htbp]
\centering
\subfigure[Uplink.]{
\begin{minipage}[t]{1\linewidth}
\centering
\includegraphics[scale= 0.45]{Fig9a.eps}
\end{minipage
\vspace{0.1pt}
\subfigure[Downlink.]{
\begin{minipage}[t]{1\linewidth}
\centering
\includegraphics[scale= 0.45]{Fig9b.eps}
\end{minipage
\centering
\caption{The spectral efficiency comparison.}\label{fig:SEUL}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[scale= 0.45]{Fig10.eps}
\caption{ The energy efficiency comparison.}\label{fig:EE}
\end{figure}
Fig.\,\ref{fig:SEUL}(a) shows that the proposed asymmetrical transceiver achieves a similar uplink SE to the subarray-based $M$-antenna hybrid transceiver. Additionally, as we expected, the proposed asymmetrical transceiver achieves a larger uplink SE than the conventional $N$-antenna full digital symmetrical transceiver owing to its flexible receive antenna topology design.
Furthermore, Fig.\,\ref{fig:SEUL}(b) indicates that the proposed asymmetrical transceiver can even outperform the conventional $M$-antenna full digital symmetrical transceiver in terms of the downlink spectral efficiency, especially at the low SNR regime. This is because, by fully exploiting the channel architecture, the proposed mNOMP-based channel transfer algorithm can well recover the key parameters of the downlink channel (e.g., AoA and complex path gains) from the $N$-dimensional uplink LMMSE channel estimates with $N$ receive RF chains.
As a consequence, a higher downlink spectral efficiency can be achieved via the estimates from the proposed mNOMP-based algorithm than that from the pure uplink LMMSE channel estimation with $M(>>N)$ receive RF chains, and thus the downlink transmission capability of the asymmetrical transceiver-based massive MIMO systems can be well maintained.
The energies efficiency of different systems are demonstrated in Fig.\,\ref{fig:EE}. From the figure, the full-connected hybrid transceiver achieves the best energy efficiency. However, due to the employment of a large and complicated phase shifter network, the full-connected hybrid transceiver severely suffers from the tremendous hardware cost and complexity. Additionally, as anticipated, the proposed asymmetrical transceiver achieves a better EE than the conventional $M$-antenna full digital symmetrical transceiver. Moreover, the proposed asymmetrical transceiver also outperforms the subarray-based hybrid transceiver at the low SNR regime. This result mainly comes from the benefits of the full digital architecture of the asymmetrical transceiver and the proposed channel transfer algorithm. Hence, the proposed asymmetrical transceiver-based architecture can be a promising alternative solution for the next generation massive MIMO systems.
\section{Conclusion}
This paper proposed an asymmetrical transceiver-based massive MIMO system by releasing the RF chain pairs and allowing asymmetric transmitter and receiver architecture at the BS. Based on this architecture, the detailed uplink and downlink transmission procedures were investigated and system spectral efficiency and energy efficiency analyses were performed. Besides, to adapt to characteristics of the asymmetrical transceiver-based systems, the uplink-to-downlink channel transfer step was introduced with two channel transfer algorithms, i.e., the DFT-based and the mNOMP-based channel transfer algorithms. The proposed asymmetrical transceiver-based massive MIMO systems could achieve an excellent downlink spectral efficiency and a good system energy efficiency simultaneously. The asymmetrical transceiver-based system architecture provides a promising alternative solution, especially for extra-large scale massive MIMO systems in the future.
\appendices
\section{Proof of Proposition 1}
Since the successive antenna selection is employed, the receive antenna index is set as ${a_n}=n,\forall n=1,\ldots,N$ in (\ref{eq:hku}) for simplicity. In addition, because the single-user scenario and the perfect channel estimation are considered, we have the uplink SNR for user $k$ when these two paths are perfectly resolved as
\begin{align}\label{eq:uplinkSNRUU}
{\rm{SNR}_{\Delta_r}} &= {\rho_u}{{\left\| {{\bf{h}}_{k,{\rm{A}}}^{\rm{U}}} \right\|}^2} ,\nonumber\\
& = {\frac{{\rho_u}N}{2}}{\left\| \sum\limits_{i = 1}^{2} {{g_{k,i}}{{\bf{a}}_{\rm{U}}}({\theta _{k,i}})}\right\|}^2.
\end{align}
When there is only one dominant path that has been resolved due to the degraded angular resolution, the uplink SNR becomes
\begin{align}\label{eq:uplinkSNRSS}
&{\rm{SNR}_{\Delta_r^s}} \nonumber \\
&= {\rho_u} {{{{{\left| {{{\left( {\bf{h}}_{k,{\rm{A}}}^{\rm{U,s}} \right)}^H}{\bf{h}}_{k,{\rm{A}}}^{\rm{U}}} \right|}^2}}}/{{ {{\left\| {\bf{h}}_{k,{\rm{A}}}^{\rm{U,s}} \right\|}^2}} }} ,\nonumber\\
& = {{{\rho_u}}}{\left| {{{\left( {g_{k,1}^s{\bf{a}}_{\rm{U}}^s(\theta _{k,1}^s)} \right)}^H}\left( {\sum\limits_{i = 1}^2 {{g_{k,i}}{{\bf{a}}_{\rm{U}}}({\theta _{k,i}})} } \right)} \right|^2}/{\left\| {{g^{{s}}_{k,1}}{{\bf{a}}^s_{\rm{U}}}({\theta^s_{k,1}})}\right\|}^2.
\end{align}
Considering that those two paths are equal power, we assume $g_{k,1}=e^{j\phi_1}$ and $g_{k,2}=e^{j\phi_2}$ in (\ref{eq:hku}) without loss of generality and $|g_{k}^s|=\sqrt 2$ according to the energy conservation law. Then, the uplink SNRs in (\ref{eq:uplinkSNRUU}) and (\ref{eq:uplinkSNRSS}) can be calculated as
\begin{align}\label{eq:uplinkSNRUU2}
{\rm{SNR}_{\Delta_r}} &={\rho_u}N + {\rho_u}\Lambda \cos \left[ {{\phi _1} - {\phi _2} - \frac{{\pi d(N - 1)\Theta }}{\lambda}} \right],
\end{align}
and
\begin{align}\label{eq:uplinkSNRSS2}
&{\rm{SNR}_{\Delta_r^s}} \nonumber \\
&=\frac{{{\rho_u}(\Lambda _1^2 + \Lambda _2^2)}}{{2N}} + \frac{{\rho_u}{{\Lambda _1}{\Lambda _2}}}{N}\cos \left[ {{\phi _1} - {\phi _2} - \frac{{\pi d(N - 1)\Theta }}{\lambda}} \right],
\end{align}
respectively. Therefore, the normalized SNR loss can be written as
\begin{align}\label{eq:SNRlossDerivation}
&{\rm{SN}}{{\rm{R}}_{{\rm{loss}}}} \nonumber \\
&= \frac{{{\rm{SN{R}}_{{\Delta _{\rm{r}}}}} - {\rm{SN{R}}_{\Delta _{\rm{r}}^{\rm{s}}}}}}{{{\rm{SN{R}}_{{\Delta _{\rm{r}}}}}}},\nonumber\\
& = 1 - \frac{{\Lambda _1^2 + \Lambda _2^2 + 2{\Lambda _1}{\Lambda _2}\cos \left[ {{\phi _1} - {\phi _2} - \pi d(N - 1)\Theta /\lambda} \right]}}{{2{N^2} + 2N\Lambda \cos \left[ {{\phi _1} - {\phi _2} - \pi d(N - 1)\Theta /\lambda} \right]}}.
\end{align}
Substituting $\Gamma \buildrel \Delta \over = {\phi _1} - {\phi _2} - \pi d(N - 1)\Theta /\lambda$ into (\ref{eq:SNRlossDerivation}), we arrive at (\ref{eq:SNRloss}).
Furthermore, according to (\ref{eq:uplinkSNRUU}), (\ref{eq:uplinkSNRSS}) and the Cauchy-Schwarz inequality, we have ${\rm{SNR}_{\Delta_r^s}}\le{\rm{SNR}_{\Delta_r}}$ and ${\rm{SN{R}}_{{\rm{loss}}}}\ge 0$.
Additionally, since $|\sin (Nx)/\sin (x)| \le N$ and the equality holds when $x=k\pi,k = 0, \pm 1, \pm 2,\ldots$, we have ${\rm{SN{R}}_{{\rm{loss}}}}=0$ when $\Lambda_1=\Lambda_2=\Lambda=N$. This means that when the minimum of ${\rm{SN{R}}_{{\rm{loss}}}}$ is achieved, we have $\pi d\Theta /\lambda = k\pi$ and $\pi d\Theta_i /\lambda = k\pi$ for $i=1,2$. Considering that $d=\lambda/2$, $|\Theta|<2$ and $|\Theta_i|<2$ for $i=1,2$, ${\rm{SN{R}}_{{\rm{loss}}}}=0$ indicates $k=0$ and $\theta_{k,1}=\theta_{k,2}$. This result actually contradicts the premise that there are two paths from different angles. Therefore, we have ${\rm{SN{R}}_{{\rm{loss}}}}> 0$.
On the other hand, since $|\cos\Gamma|<1$ and $\Lambda< N$, we have $2N^2+2N\Lambda\cos \Gamma>0$ and ${\Lambda _1^2 + \Lambda _2^2 + 2{\Lambda _1}{\Lambda _2}\cos \Gamma}>(\Lambda_1-\Lambda_2)^2>0$. Thus, the second term of (\ref{eq:SNRlossDerivation}) is larger than zero. Finally, we obtain $0<{\rm{SN{R}}_{{\rm{loss}}}}<1$ and the proof is concluded.
\section{The derivation of ${{\cal T}_{\rm DFT}}$ and $\mathcal{T}_{\rm m}$}
From (\ref{eq:HAuestimatedLS}), we have
\begin{equation}\label{eq:hestimatewithnoise}
{\mathbf{\tilde h}}_{k,{\rm A}}^{\rm U} = {\mathbf{h}}_{k,{\rm A}}^{\rm U}+\frac{1}{\sqrt{\rho_{\tau}}}{\mathbf{\tilde n}}_k,
\end{equation}
and ${\mathbf{\tilde n}}_k\sim\mathcal{CN}({\bf{0}},{{\bf{I}}_N})$ for the LS channel estimation. Assume the number of paths, the complex path gains and the AoAs in the channel transfer algorithms are perfectly recovered. Then for the path detection step in the DFT-based algorithm and the termination condition in the mNOMP-based algorithm, we have
\begin{align}\label{eq:thresholderivation}
\mathbb{E}\left\{{\left\| {{\bf{\tilde h}}_{k,{\rm{A}}}^{\rm{U}}} \right\|^2} - \sum\limits_{i = 1}^{{{\hat P}_k}} {{{N\left| {{{\tilde g}_{[i]}}} \right|}^2}} \right\}
&=\mathbb{E}\left\{{\left\| {{\bf{\tilde h}}_{k,{\rm{A}}}^{\rm{U}}} \right\|^2} - {\left\| {{\bf{ h}}_{k,{\rm{A}}}^{\rm{U}}} \right\|^2} \right\},\nonumber\\
&\mathop = \limits^{(a)} \mathbb{E}\left\{{\left\| {{{{\bf{\tilde n}}}_k}} \right\|^2}/{\rho _\tau }\right\},\nonumber\\
&=N/{\rho _\tau },
\end{align}
where (a) utilizes the independence between ${\mathbf{h}}_{k,{\rm A}}^{\rm U}$ and ${\mathbf{\tilde n}}_k$.
(\ref{eq:thresholderivation}) indicates that the channel estimate error due to the channel noise is $N/{\rho _\tau }$. Hence, to detect all the paths, we set ${{\cal T}_{\rm DFT}}=\mathcal{T}_{\rm m}=N/{\rho _\tau }$.
Note that from (\ref{eq:HAuestimatedLMMSE2}), it can be observed that (\ref{eq:hestimatewithnoise}) also holds for the LMMSE channel esimation when SNR is high. Therefore, the derived ${{\cal T}_{\rm DFT}}$ and $\mathcal{T}_{\rm m}$ can also be applied for LMMSE channel estimation when $\rho_{\tau}$ is large.
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
\footnotesize
|
1,116,691,497,559 | arxiv | \section{Basic definitions}
A {\it tournament} $T=(V(T),A(T))$ or $(V,A)$ consists of a finite
{\it vertex} set $V$ with an {\it arc} set $A$ of ordered pairs of
distinct vertices satisfying: for $x, y \in V$, with $x \neq y$,
$(x, y) \in A$ if and only if $(y, x) \notin A$. The {\it
cardinality} of T is that of $V(T)$ denoted by $\mid\!V(T)\!\mid$.
For two distinct vertices $x$ and $y$ of a tournament $T$, $x
\longrightarrow y$ means that $(x,y) \in A(T)$. For $x \in V(T)$ and
$Y \subset V(T)$, $x \longrightarrow Y$ (rep. $Y \longrightarrow x$)
signifies that for every $y \in Y$, $x \longrightarrow y$ (resp. $y
\longrightarrow x$). Given a vertex $x$ of a tournament $T = (V,
A)$, $N_{T}^{+}(x)$ denotes the set $\{y \in V: x \longrightarrow
y \}$. The {\it score} of $x$ (in $T$), denoted by $s_{T}(x)$, is
the cardinality of $N_{T}^{+}(x)$. A tournament is {\it regular} if
all its vertices share the same score. A {\it transitive} tournament
or {\it total order} is a tournament $T$ such that for $x$, $y$,
$z\in V(T)$, if $x \longrightarrow y$ and $y \longrightarrow z$,
then $x \longrightarrow z$. For two distinct vertices $x$ and $y$ of
a total order $T$, $x < y$ means that $x \longrightarrow y$. We
write $T = a_{0} < \cdots < a_{n}$ to mean that $T$ is the total
order defined on $V(T) = \{a_{0}, \ldots, a_{n}\}$ by $A(T) =
\{(a_{i}, a_{j}) : i < j\}$.
The notions of isomorphism, of subtournament and of embedding are
defined in the following manner. First, let $T = (V, A)$ and $T' =
(V', A')$ be two tournaments. A one-to-one correspondence $f$ from
$V$ onto $V'$ is an {\it isomorphism} from $T$ onto $T'$ provided
that for $x, y \in V$, $(x, y) \in A$ if and only if $(f(x), f(y))
\in A'$. The tournaments $T$ and $T'$ are then said to be {\it
isomorphic}, which is denoted by $T \simeq T'$. Moreover, an
isomorphism from a tournament $T$ onto itself is called an {\it
automorphism} of $T$. The automorphisms of $T$ form a subgroup of
the permutation group of $V(T)$, called the {\it automorphism group}
of $T$. Second, given a tournament $T = (V, A)$, with each subset
$X$ of $V$ is associated the {\it subtournament} $T(X) = (X, A \cap
(X \times X))$ of $T$ {\it induced} by $X$. For $x \in V $, the
subtournament $T(V - \{x\})$ is denoted by $T-x$. For tournaments
$T$ and $T'$, if $T'$ is isomorphic to a subtournament of $T$, then
we say that $T'$ {\it embeds} into $T$. Otherwise, we say that $T$
{\it omits} $T'$. The {\it dual} of a tournament $T=(V, A)$ is the
tournament obtained from $T$ by reversing all its arcs. This
tournament is denoted by $T^{\star}=(V, A^{\star})$, where
$A^{\star} = \{(x, y): \ (y,x) \in A\}$. A tournament $T$ is then
said to be {\it self-dual} if $T$ and $T^{\star}$ are isomorphic.
The indecomposability plays an important role in this paper. Given a
tournament $T = (V, A)$, a subset $I$ of $V$ is an \emph{interval}
(\cite{F1}, \cite{I}, \cite{ST}) (or a {\it clan} \cite{E} or an
{\it homogeneous subset} \cite{G}) of $T$ provided that for every $x
\in V-I$, $x \rightarrow I$ or $I \rightarrow x$. This definition
generalizes the notion of interval of a total order. Given a
tournament $T = (V, A)$, $\emptyset $, $V$ and $\{x\}$, where $x \in
V$, are clearly intervals of $T$, called {\it trivial} intervals. A
tournament is then said to be {\it indecomposable} (\cite{I},
\cite{ST}) (or {\it primitive} \cite{E}) if all of its intervals are
trivial, and is said to be {\it decomposable} otherwise. For
instance, the 3-cycle $C_{3} = (\{0, 1, 2\}, \{(0, 1), (1, 2), (2,
0)\})$ is indecomposable whereas a total order of cardinality $\geq
3$ is decomposable. Let us mention the following relationship
between indecomposability and duality. The tournaments $T$ and
$T^{\star}$ have the same intervals and, thus, $T$ is indecomposable
if and only if $T^{\star}$ is indecomposable.
\section{The critical tournaments}
An indecomposable tournament $T = (V, A)$ is said to be {\it
critical} if $\mid\!V\!\mid > 1$ and for all $x \in V$, $T-x$ is
decomposable. In order to present our main results and to present
the characterization of the critical tournaments due to J.H.~
Schmerl and W.T. Trotter \cite{ST}, we introduce the tournaments
$T_{2n + 1}$, $U_{2n + 1}$ and $W_{2n + 1}$ defined on $2n + 1$
vertices, where $n\geq2$, as follows:
\begin{itemize}
\item The tournament $T_{2n+1}$ is the tournament defined on
$\mathbb{Z}/(2n+1)\mathbb{Z}$ by $A(T_{2n+1}) = \{(i,j): j-i \in
\{1, \ldots, n\}\}$, so that, $T_{2n+1}(\{0,\ldots, n\}) = 0< \cdots
<n$, $T_{2n+1}(\{n+1, \ldots, 2n\}) = n+1< \cdots < 2n$ and for $i
\in \{0, \ldots, n-1\},\ \{i+1, \ldots, n\} \longrightarrow i+n+1
\longrightarrow\{0, \ldots, i\}$ (see Figure 1).
\item The tournament $U_{2n+1}$ is obtained from $T_{2n+1}$ by reversing the arcs of $T_{2n+1}(\{n+1,
\ldots, 2n\})$. Therefore, $U_{2n+1}$ is defined on $\{0,\ldots,
2n\}$ as follows: $U_{2n+1}(\{0,\ldots, n\}) = 0< \cdots <n$,
$U_{2n+1}^{\star}(\{n+1, \ldots, 2n\}) = n+1< \cdots < 2n$ and for
$i \in \{0, \ldots, n-1\},\ \{i+1, \ldots, n\} \longrightarrow i+n+1
\longrightarrow\{0, \ldots, i\}$ (see Figure 2).
\item The tournament $W_{2n+1}$ is defined on $\{0,\ldots, 2n\}$ in the following manner: $W_{2n+1}-2n = 0< \cdots <2n-1$
and $\{1, 3, \ldots, 2n-1\} \longrightarrow 2n \longrightarrow \{0,
2, \ldots, 2n-2 \}$ (see Figure 3).
\end{itemize}
\begin{figure}[ht]
\begin{center}
\setlength{\unitlength}{1cm}
\begin{picture}(11,5)
\put(0,1){0} \put(1.5,1){1} \put(3.5,1){i} \put(2,1.1){.}
\put(2.5,1.1){.} \put(3,1.1){.} \put(5,1){i+1} \put(5.9,1.1){.}
\put(6.2,1.1){.} \put(6.5,1.1){.} \put(7,1){$n-1$}
\put(9.2,1){$n$} \put(1.6,0.8){\line(2,-1){0.5}}
\put(2.1,0.55){\vector(1,0){3.2}} \put(0.1,0.8){\line(3,-2){1.14}}
\put(1.24,0.04){\vector(1,0){6.23}} \put(0.3,1.1){\vector(1,0){1}}
\put(3.8,1.1){\vector(1,0){1}} \put(8,1.1){\vector(1,0){1}}
\put(0.3,2.5){n+1} \put(3.8,2.5){i+n+1} \put(8.3,2.5){2n}
\put(1.9,2.6){\vector(1,0){1}} \put(1.15,2.6){.} \put(1.4,2.6){.}
\put(1.65,2.6){.} \put(3.1,2.6){.} \put(3.32,2.6){.}
\put(3.54,2.6){.}
\put(5.8,2.6){\vector(1,0){1.5}} \put(5,2.6){.} \put(5.25,2.6){.}
\put(5.5,2.6){.} \put(7.55,2.6){.} \put(7.8,2.6){.}
\put(8.05,2.6){.}
\put(4.2,2.9){\line(2,1){0.52}} \put(4.7,3.15){\vector(1,0){1.77}}
\put(0.65,2.9){\line(2,1){1.32}}
\put(1.95,3.55){\vector(1,0){1.5}}
\put(0.3,2.3){\vector(-1,-4){0.23}}
\put(1.55,1.4){\vector(-1,1){0.9}}
\put(2.3,1.3){\vector(-3,2){1.4}}
\put(3.8,2.3){\vector(-1,-1){1}} \put(4,2.3){\vector(-1,-3){0.3}}
\put(5.25,1.4){\vector(-1,1){0.9}}
\put(6.1,1.3){\vector(-3,2){1.5}}
\put(8.45,2.3){\vector(-1,-1){1}}
\put(8.2,2.3){\vector(-3,-2){1.5}}
\put(9.57,1.4){\vector(-1,1){0.9}}
\end{picture}
\end{center}
\caption{$T_{2n+1}$.}
\end{figure}
\begin{figure}[ht]
\begin{center}
\setlength{\unitlength}{1cm}
\begin{picture}(11,5)
\put(0,1){0} \put(1.5,1){1} \put(3.5,1){i} \put(2,1.1){.}
\put(2.5,1.1){.} \put(3,1.1){.} \put(5,1){i+1} \put(5.9,1.1){.}
\put(6.2,1.1){.} \put(6.5,1.1){.} \put(7,1){$n-1$}
\put(9.2,1){$n$} \put(1.6,0.8){\line(2,-1){0.5}}
\put(2.1,0.55){\vector(1,0){3.2}} \put(0.1,0.8){\line(3,-2){1.14}}
\put(1.24,0.04){\vector(1,0){6.23}} \put(0.3,1.1){\vector(1,0){1}}
\put(3.8,1.1){\vector(1,0){1}} \put(8,1.1){\vector(1,0){1}}
\put(0.3,2.5){n+1} \put(3.8,2.5){i+n+1} \put(8.3,2.5){2n}
\put(2.9,2.6){\vector(-1,0){1}} \put(1.15,2.6){.} \put(1.4,2.6){.}
\put(1.65,2.6){.} \put(3.1,2.6){.} \put(3.32,2.6){.}
\put(3.54,2.6){.}
\put(7.3,2.6){\vector(-1,0){1.5}} \put(5,2.6){.} \put(5.25,2.6){.}
\put(5.5,2.6){.} \put(7.55,2.6){.} \put(7.8,2.6){.}
\put(8.05,2.6){.}
\put(4.2,2.9){\line(-2,1){0.5}} \put(3.7,3.15){\vector(-1,0){1.3}}
\put(8.5,2.9){\line(-2,1){1.32}}
\put(7.2,3.55){\vector(-1,0){1.9}}
\put(0.3,2.3){\vector(-1,-4){0.23}}
\put(1.55,1.4){\vector(-1,1){0.9}}
\put(2.3,1.3){\vector(-3,2){1.4}}
\put(3.8,2.3){\vector(-1,-1){1}} \put(4,2.3){\vector(-1,-3){0.3}}
\put(5.25,1.4){\vector(-1,1){0.9}}
\put(6.1,1.3){\vector(-3,2){1.5}}
\put(8.45,2.3){\vector(-1,-1){1}}
\put(8.2,2.3){\vector(-3,-2){1.5}}
\put(9.57,1.4){\vector(-1,1){0.9}}
\end{picture}
\end{center}
\caption{$U_{2n+1}$.}
\end{figure}
\begin{figure}[ht]
\begin{center}
\setlength{\unitlength}{1cm}
\begin{picture}(11,5)
\put(0,1){0} \put(1.5,1){1} \put(3.5,1){2i} \put(2,1.1){.}
\put(2.5,1.1){.} \put(3,1.1){.} \put(5,1){2i+1} \put(6,1.1){.}
\put(6.3,1.1){.} \put(6.6,1.1){.} \put(7,1){$2n-2$}
\put(9.2,1){$2n-1$} \put(1.6,0.8){\line(2,-1){0.5}}
\put(2.1,0.55){\vector(1,0){3.2}} \put(0.1,0.8){\line(3,-2){1.14}}
\put(1.24,0.04){\vector(1,0){6.23}} \put(0.3,1.1){\vector(1,0){1}}
\put(3.9,1.1){\vector(1,0){1}} \put(8.1,1.1){\vector(1,0){1}}
\put(4.2,3){2n}
\put(3.8,3){\vector(-2,-1){3.4}}
\put(1.9,1.4){\vector(3,2){2.1}}
\put(4.25,2.8){\vector(-1,-3){0.475}}
\put(5.35,1.4){\vector(-2,3){0.95}}
\put(4.7,2.8){\vector(2,-1){2.9}}
\put(9.7,1.4){\vector(-3,1){4.8}}
\end{picture}
\end{center}
\caption{$W_{2n+1}$.}
\end{figure}
\begin{thm}[\cite{ST}]\label{t1} Up to isomorphism, the critical tournaments of cardinality $\geq 5$ are
the tournaments $T_{2n+1}$, $U_{2n+1}$ and $W_{2n+1}$, where $n
\geq 2$.
\end{thm}
Notice that the critical tournaments are self-dual.
\section{The tournaments $T_{5}$, $U_{5}$ and $W_{5}$ in an indecomposable tournament}
We study the indecomposable tournaments according to their
indecomposable subtournaments on 5 vertices. A recent result on our
topic is a characterization of the indecomposable tournaments
omitting $W_{5}$ obtained by B.J. Latka \cite{BJL}. In order to
recall this characterization, we introduce the {\it Paley}
tournament $P_{7}$ defined on $\mathbb{Z}/7\mathbb{Z}$ by $A(P_{7})
=\{(i, j): j-i \in \{1, 2, 4\}\}$. Notice that the tournaments
obtained from $P_{7}$ by deleting one vertex are isomorphic and
denote $P_{7} - 6$ by $B_{6}$.
\begin{thm}[\cite{BJL}]\label{t2} Given a tournament $T$
of cardinality $\geq 5$, $T$ is indecomposable and omits $W_{5}$ if
and only if $ T $ is isomorphic to an element of $\{ B_{6},
P_{7}\}\cup \{T_{2n+1}: n \geq 2\}\cup \{U_{2n+1}: n \geq 2\}$.
\end{thm}
A {\it diamond} is a tournament on 4 vertices admitting only one
interval of cardinality 3. Up to isomorphism, there are exactly two
diamonds $D_{4}$ and $D_{4}^{\star}$, where $D_{4}$ is the
tournament defined on $\{0, 1, 2, 3\}$ by $D_{4}(\{0, 1,2\}) =
C_{3}$ and $3\longrightarrow \{0, 1,2\}$.
The following theorem is the main result. This theorem is presented
in \cite{HI} without a detailed proof.
\begin{thm}\label{t3} Given an indecomposable tournament $T$, if a diamond and $T_{5}$ embed into $T$, then $U_{5}$ and $W_{5}$ embed into $T$.
\end{thm}
C. Gnanvo and P. Ille
\cite{GI} and G. Lopez and C. Rauzy \cite{LR} characterized the
tournaments omitting diamonds. In the indecomposable case they
obtained the following characterization.
\begin{prop}[\cite{GI,LR}]\label{p4}
Given an indecomposable tournament $T$ of cardinality $\geq 5$,
$T$ omits the diamonds $D_{4}$ and $D_{4}^{\star}$ if and only if
$T$ is isomorphic to $T_{2n+1}$ for some $n \geq 2$.
\end{prop}
\section{Proof of Theorem~\ref{t3}}
Before proving Theorem~\ref{t3}, we introduce some notations and
definitions.
\begin{defn}\label{d5} Given a tournament $T=(V,A)$, with each subset $X$ of $V$,
such that $\mid\!X\!\mid \geq 3$ and $T(X)$ is indecomposable, are
associated the following subsets of $V-X$.
\begin{itemize}
\item $Ext(X) = \{x \in V-X: \ T(X \cup \{x\})$ is indecomposable$\}$.
\item $[X] = \{x \in V-X: \ x \rightarrow X$ or $X \rightarrow x \}$.
\item For every $u \in X$, $X(u) = \{x \in V-X: \ \{u, x\}$ is an interval
of $T(X \cup \{x\})\}$.
\end{itemize} \end{defn}
\begin{lem}[\cite{E}]\label{l6} Let $T=(V,A)$ be a tournament and let $X$ be a subset
of $V$ such that $\mid\!X\!\mid \geq 3$ and $T(X)$ is
indecomposable.
\begin{enumerate}
\item The family $\{X(u): u \in X\} \cup \{Ext(X), [X]\}$
constitutes a partition of $V-X$.
\item Given $u\in X$, for all $x\in X(u)$ and for all $y\in V-(X\cup
X(u))$, if $T(X\cup \{x, y\})$ is decomposable, then $\{u, x\}$ is
an interval of $T(X\cup\{x, y\})$.
\item For every $x\in [X]$ and for every $y\in V-(X\cup [X])$, if $T(X\cup\{x,
y\})$ is decomposable, then $X\cup \{y\}$ is an interval of
$T(X\cup\{x, y\})$.
\item Given $x, y \in Ext(X)$, with $x \neq y$, if $T(X\cup\{x,
y\})$ is decomposable, then $\{x, y\}$ is an interval of
$T(X\cup\{x, y\})$. \end{enumerate}
\end{lem}
The below result follows from Lemma~\ref{l6}.
\begin{prop}[\cite{E}]\label{p7} Let $T=(V,A)$ be an indecomposable tournament. If
$X$ is a subset of $V$, such that $\mid\!X\!\mid \geq 3$,
$\mid\!V-X\!\mid \geq 2$ and $T(X)$ is indecomposable, then there
are distinct elements $x$ and $y$ of $V-X$ such that $T(X\cup\{x,
y\})$ is indecomposable.
\end{prop}
\begin{cor}\label{c8}
Let $T = (V, A)$ be an indecomposable tournament such that $\mid V
\mid$ is even and $\mid V \mid \geq 6$. For each $x \in V$, there
is $y \in V - \{x\}$ such that $T - y$ is indecomposable.
\end{cor}
\bpr
As $T$ is indecomposable, there is $X \subset V$ such that $x \in X$
and $T(X) \simeq C_{3}$. Otherwise, $N_{T}^{+}(x)$ or $V- (\{x\}\cup
N_{T}^{+}(x))$ would be non trivial intervals of $T$. Since $\mid V
\mid$ is even, by applying several times Proposition~\ref{p7} from the
indecomposable subtournament $T(X)$, we get a vertex $y \in V - X$
such that $T - y$ is indecomposable.
\epr
The 3-cycle $C_{3}$ is indecomposable and embeds into any
indecomposable tournament of cardinality $\geq3$ as observed in the
preceding proof. It follows, by Proposition~\ref{p7}, that any
indecomposable tournament $T$ of cardinality $\geq5$, admits an
indecomposable subtournament on 5 vertices. The indecomposable
tournaments on 5 vertices are critical because the four tournaments
on 4 vertices are decomposable. So let us mention the following
facts.
\begin{rem}\label{r9}\
\begin{itemize}
\item The indecomposable tournaments on 5 vertices are, up to
isomorphism, the three critical tournaments $T_{5}$, $U_{5}$ and
$W_{5} $.
\item There is no indecomposable tournament of cardinality $\geq5$
omitting each of the tournaments $T_{5}$, $U_{5}$ and $W_{5} $.
\end{itemize}
\end{rem}
The tournaments $T_{2n+1}$ play an important role in the proof of
Theorem~\ref{t3}. We recall some of their properties.
\begin{rem}\label{r10}\
\begin{itemize}
\item The tournaments $T_{2n+1}$ are regular: for all $i \in \{0, \ldots,
2n\}$, $s_{T_{2n+1}}(i) = n$;
\item For $0 \leq i \leq 2n$, the unique non trivial interval of
$T_{2n+1} -i$ is \{i+n, i+n+1\};
\item The automorphism group of $T_{2n+1}$ is generated by the permutation $\sigma: i \mapsto i+1$;
\item The permutation $\pi: i \mapsto -i$, is an isomorphism from $T_{2n+1}$ onto
its dual.
\end{itemize}
\end{rem}
Now we are ready to prove Theorem~\ref{t3}.
~
\bprA
Let $T = (V,A)$ be an indecomposable
tournament into which
a diamond and $T_{5}$ embed. Consider a
minimal subset $X$ of $V$ such that $T(X)$ is indecomposable and a
diamond and $T_{5}$ embed into $T(X)$. Now, let $Y$ be a maximal
subset of $X$ such that $T(Y) \simeq T_{2n+1}$ for some $n \geq 2$.
We establish that $\mid X \mid = 6$ by using the following
observation. Consider a subset $Z$ of $X$ such that $T(Z) \simeq
T_{2n+1}$ and assume that $Ext(Z) \cap X \neq \emptyset$. Let $x\in Ext(Z) \cap X
$. We have $T(Z\cup\{x\})$ is indecomposable. Furthermore,
as $\mid Z\cup\{x\} \mid$ is even, a diamond embeds into
$T(Z\cup\{x\})$ by Proposition~\ref{p4}. Since $T_{5}$ embeds into
$T(Z\cup\{x\})$ as well, it follows from the minimality of $X$ that
$X= Z\cup\{x\}$. As an immediate consequence, we have: if $Z$ is a
subset of $X$ such that $T(Z) \simeq T_{2n+1}$ and $\mid X- Z \mid \geq 2$,
then $Ext(Z) \cap X = \emptyset$. By Lemma~\ref{l6}, for every $x\in X-Z$,
either $x\in [Z]$ or there is $u\in Z$ such that $x\in Z(u)$.
For a contradiction, suppose that $Ext(Y) \cap X = \emptyset$. By
Proposition~\ref{p7}, there are $x \neq y \in X-Y$ such that $T(Y \cup \{x,
y \})$ is indecomposable. Clearly, if $\{x, y\}\subseteq [Y]$, then
$Y$ would be a non trivial interval of $T(Y \cup \{x, y \})$. For
instance, assume that there is $v\in Y$ such that $y\in Y(v)$. By
Lemma~\ref{l6}, either there is $u\in Y$ such that $x\in Y(u)$ or $x\in
[Y]$. In each of both instances, we obtain a contradiction.
First, suppose that there is $u \in Y$ such that $x \in Y(u)$. We
have $u\neq v$, otherwise $\{u, x, y\}$ would be a non trivial
interval of $T(Y \cup \{x, y \})$. By Remark~\ref{r10}, the automorphism
group of $T_{2n+1}$ is generated by $\sigma : i\mapsto i+1$.
Therefore, by interchanging $x$ and $y$, we can denote the element
of $Y$ by $0,\ldots, 2n$ in such a way that $T(Y)= T_{2n+1}$, $u =
0$ and $1\leq v\leq n$. Since $T(Y \cup \{x, y \})$ is
indecomposable and $0\longrightarrow v$, we get $y\longrightarrow x$
by Lemma~\ref{l6}. Consider $Z = (Y-\{0\}) \cup \{x\}$. We have $T(Z)
\simeq T_{2n+1}$ and, by the preceding observation, either $y\in
[Z]$ or there is $w \in Z$ such that $y \in Z(w)$. The first
instance is not possible because $\{v-2, v-1\} \cap Z \neq
\emptyset$ and $\{v-2, v-1\} \longrightarrow y \longrightarrow x$.
So assume that there is $w \in Z$ such that $y \in Z(w)$. As $y
\longrightarrow x \longrightarrow v$, $w \neq v$. Moreover, if $w =
x$, then $\{x, y\}$ is an interval of $T(Z \cup \{y\})$. Since $\{v,
y\}$ is an interval of $T((Z \cup \{y\})-\{x\})$, we would obtain
that $\{x, y, v\}$ is an interval of $T(Z \cup \{y\})$ so that $\{x,
v\}$ would be an interval of $T(Z)$. Therefore, $w \notin \{v, x\}$
and hence $\{v, w\}$ is an interval of $T(Z) - x$. As $x \in Y(0)$,
it follows from Remark~\ref{r10} that $\{v, w\} = \{n, n+1\}$ so that $v=
n$ and $n \longrightarrow y$. By considering the automorphism
$\sigma^{n+1}$ of $T(Y)$ defined by $ \sigma^{n+1}(i)= i+n+1$, we
obtain that $y\in Y(0)$ and $x\in Y(n+1)$. By considering
$T^{\star}$ instead of $T$, we get $y \in Y(0)$ and $x\in Y(n)$
because the permutation $\pi: i \mapsto -i$ is an isomorphism from
$T(Y)$ onto $T(Y)^{\star}$ by Remark~\ref{r10}. Lastly, by interchanging
$x$ and $y$ in the foregoing, we obtain $n \longrightarrow x$ in
$T^{\star}$ which means that initially $x\longrightarrow 0$ in $T$.
It follows that the function $Y\cup\{x,
y\}\longrightarrow\{0,\ldots, 2n+2\}$, defined by $ x \mapsto 2n+2$,
$ y \mapsto n+1$, $ i \mapsto i$ for $0\leq i\leq n$ and $ i \mapsto
i+1$ for $n+1\leq i\leq 2n$, realizes an isomorphism from $T(Y \cup
\{x, y\})$ onto $T_{2n+3}$. Consequently, $T(Y \cup \{x, y\}) \simeq
T_{2n+3}$, with $Y \cup \{x, y\} \subseteq X$, which contradicts the
maximality of $Y$.
Second, suppose that $x\in [Y]$. By interchanging $T$ and
$T^{\star}$, assume that $y \longrightarrow x \longrightarrow Y$.
Consider $Z = (Y - \{v\}) \cup \{y\}$. We have $T(Z) \simeq
T_{2n+1}$ and, by the previous observation, either $x\in [Z]$ or
there is $w\in Z$ such that $x\in Z(w)$. The first instance is not
possible because $y \longrightarrow x \longrightarrow Z-\{y\}$.
Since $y \longrightarrow x \longrightarrow Z-\{y\}$ and hence
$s_{T(Z \cup \{x\})}(x) = 2n$, the second is not possible either.
Indeed, given $w\in Z$, if $x\in Z(w)$, then $s_{T(Z \cup \{x\})}(x)
\in \{n, n+1\}$ because $s_{T(Z)}(w) = n$.
It follows that $Ext(Y) \cap X \neq
\emptyset$. Set $T(Y)= T_{2n +1}$. By the preceding observation,
$X = Y \cup \{x\}$, where $x\in Ext(Y) \cap X$. As $\mid X\mid$ is
even, it follows from Corollary~\ref{c8} that there is $j\in X-\{x\}$ such
that $T(X)-j$ is indecomposable. By considering the automorphism $\sigma^{2n
+1-j}$ of $T(Y)$, we can assume that $j=0$. For a contradiction,
suppose that $T(X) - ~ 0 ~ \simeq ~ T_{2n+1}$. We would have $s_{(T(X) - 0)}(x) =
n$. Since $s_{(T(Y) - 0)}(i) = n$ for $ 1\leq i \leq n $ and $s_{(T(Y) -
0)}(i) = n-1$ for $n+1\leq i \leq 2n$, we would obtain that
$N^{+}_{(T(X)-0)}(x) = \{1, \ldots, n\} $ so that $\{0, x\}$ would
be a non trivial interval of $T(X)$. Consequently, $T(X)-0$ is not
isomorphic to $T_{2n +1}$. By Proposition~\ref{p4}, a diamond embeds into
$T(X)-0$. It follows from the minimality of $T(X)$ that $T(X)-0$ and
hence $T(Y)-0$ omit $T_{5}$. As $T_{5}$ embeds into $T_{2m +1}-0$
for $m\geq 3$, we get $n=2$.
It remains to verify that $U_{5}$ and $W_{5}$ embed into $T(X)$.
Since $x \notin [Y]$, $s_{T(X)}(x) \in \{1,2,3,4\}$. By
interchanging $T$ and $T^{\star}$, assume that $s_{T(X)}(x) =1$ or
2. First, assume that there is $i\in \mathbb{Z}/5\mathbb{Z}$ such
that $N^{+}_{T(X)}(x) = \{i\}$. By considering the automorphism
$j\mapsto j-i$ of $T_{5}$, assume that $i=0$. The function
$\mathbb{Z}/5\mathbb{Z}\longrightarrow X-\{3\}$, which fixes $0$,
$1$, $2$, $4$ and which maps $3$ to $x$, is an isomorphism from
$U_{5}$ onto $T(X)-3$. Furthermore, the function
$\mathbb{Z}/5\mathbb{Z}\longrightarrow X-\{2\}$, defined by
$0\mapsto 3$, $1\mapsto 4$, $2\mapsto x$, $3\mapsto 0$ and $4\mapsto
1$, is an isomorphism from $W_{5}$ onto $T(X)-2$. Finally, assume
that there is $i\in \mathbb{Z}/5\mathbb{Z}$ such that
$N^{+}_{T(X)}(x) = \{i, i+1\}$ or $\{i, i+2\}$. If $N^{+}_{T(X)}(x)
= \{i, i+1\}$, then $\{i-1, x\}$ would be an interval of $T(X)$. So,
by considering the automorphism $k\mapsto k-i$ of $T_{5}$, assume
that $N^{+}_{T(X)}(x) = \{0, 2\}$. The function
$\mathbb{Z}/5\mathbb{Z}\longrightarrow X-\{0\}$, defined by
$0\mapsto 2$, $1\mapsto 3$, $2\mapsto 4$, $3\mapsto x$ and
$4\mapsto 1$, is an isomorphism from $U_{5}$ onto $T(X)-0$.
Furthermore, the function $\mathbb{Z}/5\mathbb{Z}\longrightarrow
X-\{2\}$, defined by $0\mapsto 3$, $1\mapsto 4$, $2\mapsto x$,
$3\mapsto 0$ and $4\mapsto 1$, is an isomorphism from $W_{5}$ onto
$T(X)-2$.
\epr
\section{A new characterization of the critical tournaments}
In this section we discuss some other questions concerning the
indecomposable subtournaments on 5 and 7 vertices of an
indecomposable tournament. In particular, we obtain a new
characterization of the critical tournaments. In that order, we
recall the following two results concerning the critical
tournaments.
\begin{lem}[\cite{ST}]\label{l11} The indecomposable subtournaments of $T_{2n+1}$ on at least $5$ vertices, where $n \geq 2$,
are isomorphic to $T_{2m+1}$, where $2 \leq m\leq n$. The same holds
for the indecomposable subtournamants of $U_{2n+1}$ and of
$W_{2n+1}$.
\end{lem}
\begin{lem}[\cite{Y}]\label{l12} Given an indecomposable tournament $T$
of cardinality $\geq5$, $T$ is critical if and only if $T$ omits
any indecomposable tournament on six vertices.\end{lem}
Let $T$ be an indecomposable tournament of cardinality $\geq
5$. We denote by $I_{5}(T)$ the set of the elements of $\{T_{5},
U_{5}, W_{5}\}$ embedding in $T$. By Remark~\ref{r9}, $I_{5}(T) \neq
\emptyset$. By Theorem~\ref{t3}, $I_{5}(T) \neq \{T_{5}, U_{5}\}$ and
$I_{5}(T) \neq \{T_{5}, W_{5}\}$. We characterize the
indecomposable tournaments $T$ such that $I_{5}(T) = \{T_{5}\}$
(resp. $I_{5}(T) = \{U_{5}\}$). The following remark completes
this discussion.
\begin{rem}\label{r13} For $J = \{W_{5}\}$, $\{U_{5}, W_{5}\}$ or $\{T_{5}, U_{5},
W_{5}\}$ and for $n \geq 6$, there exists an indecomposable
tournament $T$ of cardinality $n$ such that $I_{5}(T) = J$.
\\
For $n \geq 5$, the tournaments $E_{n+1}$, $F_{n+1}$ and $G_{n+1}$
defined below on $\{0,\ldots,$ $n\}$ are indecomposable and satisfy
$I_{5}(E_{n+1}) = \{T_{5}, U_{5}, W_{5}\}$, $I_{5}(F_{n+1})
=\{W_{5}\}$ and $I_{5}(G_{n+1}) =\{U_{5}, W_{5}\}$.
\begin{itemize}
\item $E_{n+1}(\{0, \ldots, 4\}) = T_{5}$ and, for all $5 \leq k \leq n$,
$N_{E_{n+1}(\{0, \ldots, k\})}^{+}(k) = \{k-1\}$;
\item $A(F_{n+1}) = \{(i,j):i+1 < j\ or\ i = j+1\}$;
\item $G_{n}(\{0, \ldots, n-1 \}) = F_{n}$ and $N^{+}_{G_{n+1}}(n) = \{0\}$.
\end{itemize}
\end{rem}
The following is an easy consequence of Theorem~\ref{t2} and of Lemma~\ref{l11}.
\begin{cor}\label{c14} The next two assertions are satisfied by any
indecomposable tournament $T$ of cardinality $\geq 5$.
\begin{enumerate}
\item $T$ is isomorphic to $T_{2n+1}$ for some $n\geq 2$ if and only
if the indecomposable subtournaments of $T$ on 5 vertices are
isomorphic to $T_{5}$.
\item $T$ is isomorphic to $B_{6}$, $P_{7}$ or to $U_{2n+1}$ for
some $n\geq 2$ if and only if the indecomposable subtournaments of
$T$ on 5 vertices are isomorphic to $U_{5}$.
\end{enumerate}
\end{cor}
For all $n \geq 6$, the tournament $F_{n}$ defined in Remark~\ref{r13} is
an indecomposable non critical tournament all the indecomposable
subtournaments of which are isomorphic to $W_{5}$. This leads us to
the following characterization of the tournaments $W_{2n+1}$ and to
the problem below.
\begin{prop}\label{p15}
Given an indecomposable tournament $T$ of cardinality $\geq 7$, $T$
is isomorphic to $W_{2n+1}$ for some $n \geq 3$ if and only if the
indecomposable subtournaments on 7 vertices of $T$ are isomorphic to
$W_{7}$.
\end{prop}
\bpr
By Lemma~\ref{l11}, if $T \simeq W_{2n+1}$, where $n \geq 3$, then the
indecomposable subtournaments of $T$ on 7 vertices are isomorphic
to $W_{7}$. Conversely, assume that the indecomposable
subtournaments of $T$ on 7 vertices are isomorphic to $W_{7}$. By
Lemma~\ref{l11}, it suffices to show that $T$ is critical. Clearly, if
$\mid \!V(T)\! \mid = 7$, then $T \simeq W_{7}$. So assume that
$\mid \!V(T) \!\mid \geq 8$. For a contradiction, suppose that $T$
is not critical. It follows from Lemma~\ref{l12} that there exists
$X\subset V(T)$ such that $\mid \!X\! \mid = 6$ and $T(X)$ is
indecomposable. By Proposition~\ref{p7}, there is $Y\subseteq V(T)$ such
that $X\subset Y$, $\mid \!Y\! \mid = 8$ and $T(Y)$ is
indecomposable. As $\mid \!Y\! \mid$ is even, $T(Y)$ is not
critical. Consider $x \in Y$ such that $T(Y) - x$ is indecomposable.
We have $ T(Y) - x \simeq W_{7}$ and hence we can denote the
elements of $Y$ by $0,\ldots, 7$ in such a way that $x = 7$ and
$T(Y)-7 = W_{7}$. By Corollary~\ref{c8}, there is $y \in \{0,\ldots, 6\}$
such that $T(Y)- y$ is indecomposable and thus $T(Y)- y\simeq
W_{7}$. To obtain a contradiction, we verify that $\{y, 7\}$ would
be a non trivial interval of $T(Y)$. By interchanging $T$ and
$T^{\star}$, we can assume that $y\in \{0, 1, 2\}\cup \{6\}$ because
the permutation of $\mathbb{Z}/7\mathbb{Z}$, which fixes $6$ and
which exchanges $i$ and $5- i$ for $0\leq i\leq 5$, is an
isomorphism from $W_{7}$ onto its dual. First, assume that $y= 6$.
We have $ T(Y)-\{6, 7\}= 0<\cdots< 5$. Since $\{1,\ldots, 5\}\cup
\{7\}$ is not an interval of $T(Y)-6$, $7\longrightarrow 0$. As
$\{i, i+1\}$ is not an interval of $T(Y)-6$ for $0\leq i\leq 4$, we
obtain successively that $1\longrightarrow 7$, $7\longrightarrow 2$,
$3\longrightarrow 7$, $7\longrightarrow 4$ and $5\longrightarrow 7$.
Second, assume that $y\in \{0, 1, 2\}$. For $z\in \{0,\ldots,
7\}-\{y, 6\}$, $C_{3}$ embeds into $T(Y)-\{y, z\}$ because $T(\{2i,
2i+1, 6\})\simeq C_{3}$ for $i\in \{0, 1, 2\}$. It follows that the
isomorphism from $ W_{7}$ onto $T(Y)-y$ fixes $6$. Consequently,
$T(Y)-\{y, 6\}$ is transitive. We have only to check that $T(Y)-\{y,
6\}$ is obtained from the usual total order on $\{0, \ldots, 5\}$ by
replacing $y$ by $7$. If $y =0$, then $7\longrightarrow 1$ because
$1\longrightarrow \{2,\ldots, 6\}$. Thus $T(Y)-\{y, 6\}= 7< 1<\cdots
< 5$. If $y =1$ or $2$, then $\{y-1, y+1\}$ is an interval of
$T(Y)-\{y, 7\}$. Therefore, $\{y-1, y+1\}$ is not an interval of
$T(\{y-1, y+1, 7\})$ and hence $T(Y)-\{y, 6\}= \cdots < y-1 < 7 <
y+1<\cdots< 5 $.
\epr
From Corollary~\ref{c14} and Proposition~\ref{p15}, we obtain the following
recognition of the critical tournaments from their indecomposable
subtournaments on 7 vertices.
\begin{cor}\label{c16}
Given an indecomposable tournament $T$, with $\mid\!V(T)\!\mid \geq
7$, $T$ is critical if and only if the indecomposable subtournaments on 7 vertices of $T$ are isomorphic to one and only one of the tournaments $T_{7}$, $U_{7}$ and $W_{7}$.
\end{cor}
\begin{prob}\label{p17}
Characterize the indecomposable tournaments all of whose
indecomposable subtournaments on 5 vertices are isomorphic to
$W_{5}$.
\end{prob}
|
1,116,691,497,560 | arxiv | \section{Introduction}
There has been a growing interest in green wireless communications to reduce the power consumption of wireless networks over the past decade \cite{8014295}. Various technologies for green communications have been proposed including cloud radio access networks (C-RANs), energy harvesting, and cognitive radio (CR) networks \cite{7264986}. However, these existing approaches share two common disadvantages. First, the deployment of centralized baseband unit (BBU) pools in C-RANs, equipping energy harvesting transceivers, and the signaling overhead for spectrum sensing in CR inevitably cause additional power consumption. Second, while signal processing and energy resources can be improved with these approaches, the wireless channels are treated as a ``black box" which cannot be controlled as would be desirable for green communications.
Thanks to the rapid evolution of radio frequency (RF) micro-electro-mechanical systems (MEMS), the integration of intelligent reflecting surfaces (IRSs) into wireless communication systems has been recently proposed \cite{di2019smart,xu2020resource}. In particular, with programmable reflecting elements, IRSs are able to provide reconfigurable reflections of the impinging wireless signals \cite{8466374}. This unique property creates the possibility of customizing favorable wireless propagation environments, which can be exploited to further reduce the power consumption of wireless systems. More importantly, typical IRSs consume no power for operation as the reflecting elements are implemented by \emph{passive} devices, e.g., dipoles and phase shifters \cite{8990007}.
Furthermore, IRSs can be fabricated as artificial thin films attached to existing infrastructures, such as the facades of buildings, which greatly reduces the implementation cost. To sum up, IRSs are promising candidates for power-efficient green wireless communications, and, more remarkably, are cost-effective devices with the ability to manipulate the radio propagation environment \cite{di2019smart}.
Nevertheless, to further reduce the power consumption of IRS-assisted wireless systems, the IRSs have to be delicately designed and integrated with conventional communication techniques, such as the transmit beamforming at access points (APs).
There are several works on the design of green IRS-assisted communication systems.
For instance, the energy efficiency was maximized in \cite{8741198}, where suboptimal zero-forcing beamforming was assumed at the AP. Hence, a significant performance loss
is expected as the joint design of the beamformers and reflecting elements was not considered. Besides, the transmit power minimization problem was investigated for multiuser multiple-input single-output (MISO) systems \cite{8811733}, Internet-of-Things (IoT) applications \cite{9013643}, and simultaneous wireless information and power transfer (SWIPT) systems \cite{xujie}. In \cite{8811733,9013643,xujie}, based on alternating minimization (AltMin) and semidefinite relaxation (SDR) methods, the total transmit power was minimized while taking into account the minimum required quality-of-service (QoS) of the users.
However, the combination of AltMin and SDR techniques does not guarantee the local optimality of the corresponding algorithms. In particular, the solutions generated by the Gaussian randomization process needed when applying SDR are not guaranteed to satisfy the QoS constraints and to monotonically decrease the transmit power during AltMin.
In this paper, we study the power-efficient resource allocation design for IRS-assisted multiuser MISO systems. The IRS is assumed to be implemented by programmable phase shifters. We investigate the joint design of the beamforming vectors at the AP and the phase shifts at the IRS for minimization of the total transmit power while guaranteeing a minimum required signal-to-interference-plus-noise ratio (SINR) at each user. Instead of employing the AltMin and SDR approaches, the inner approximation (IA) method is proposed for tackling the non-convexity of the formulated optimization problem.
By convexifying the non-convex constraints, a sequence of approximating convex programs are solved in the IA algorithm.
The proposed IA algorithm is guaranteed to converge to a Karush-Kuhn-Tucker (KKT) solution of the original optimization problem, which is the main difference compared to existing algorithms that cannot guarantee local optimality \cite{8811733,9013643,xujie}. Our simulation results reveal that the proposed IA algorithm outperforms the state-of-the-art SDR-based AltMin algorithms in terms of transmit power consumption. In addition, the deployment of IRSs is shown to be more energy-efficient than equipping multiple antennas at the AP.
\emph{Notations:} In this paper, $\jmath=\sqrt{-1}$ denotes the imaginary unit of a complex number.
Vectors and matrices are denoted by boldface lower-case and capital letters, respectively.
$\mathbb{C}^{m\times n}$ stands for the set of all $m\times n$ complex-valued matrices;
$\mathbb{H}^{m}$ represents the set of all $m\times m$ Hermitian matrices;
$\mathbf{1}_m$ denotes the $m\times1$ all-ones vector; $\mathbf{I}_m$ is the $m$-dimensional identity matrix.
$\mathbf{A}^H$ stands for the conjugate transpose of matrix $\mathbf{A}$.
The $\ell_2$-norm of vector $\mathbf{a}$ is denoted as $||\mathbf{a}||_2$. The spectral norm, nuclear norm, and Frobenius norm of matrix $\mathbf{A}$ are represented as $\left\Vert\mathbf{A}\right\Vert_2$, $\left\Vert\mathbf{A}\right\Vert_*$, and $\left\Vert\mathbf{A}\right\Vert_F$, respectively.
$\mathrm{diag}(\mathbf{a})$ represents a diagonal matrix whose main diagonal elements are extracted from vector $\mathbf{a}$;
$\mathrm{Diag}(\mathbf{A})$ denotes a vector whose elements are extracted from the main diagonal elements of matrix $\mathbf{A}$.
The eigenvector associated with the maximum eigenvalue of matrix $\mathbf{A}$ is denoted by $\boldsymbol{\lambda}_{\max}(\mathbf{A})$.
$\Rank(\mathbf{A})$ and $\Tr(\mathbf{A})$ denote the rank and trace of matrix $\mathbf{A}$;
$\mathbf{A}\succeq\mathbf{0}$ indicates that $\mathbf{A}$ is a positive semidefinite (PSD) matrix.
For a real-valued continuous function $f(\mathbf{A})$, $\nabla_\mathbf{A}f$ denotes the gradient of $f$ with respect to matrix $\mathbf{A}$.
$\mathbb{E}[\cdot]$ and $\Re(\cdot)$ stand for statistical expectation and the real part of a complex number, respectively.
$\mathbf{A}^\star$ denotes the optimal value of an optimization variable $\mathbf{A}$.
\section{System Model}
In this section, we first present the considered IRS-assisted multiuser MISO system and formulate the optimization problem. Then, we discuss the existing approach for solving the problem and its main limitations.
\subsection{IRS-Assisted System Model}
We consider downlink transmission in an IRS-assisted multiuser MISO wireless communication system, which consists of an ${N_\mathrm{t}}$-antenna AP, $K$ single-antenna users, and an IRS, as shown in Fig. \ref{model}.
The IRS is implemented by $M$ programmable phase shifters.
The baseband signal received at user $k$ is given by
\begin{equation}
y_k=\left(\mathbf{h}_k^H\mathbf{\Phi}\mathbf{F}+\mathbf{g}_k^H\right)\sum_{j\in\mathcal{K}}\mathbf{w}_js_j+n_k,\quad\forall k\in\mathcal{K},
\end{equation}
where $\mathcal{K}=\{1,\cdots,K\}$. The IRS-user $k$ channel, AP-IRS channel, and AP-user $k$ channel are represented by $\mathbf{h}_k^H\in\mathbb{C}^{1\times M}$, $\mathbf{F}\in\mathbb{C}^{M\times{N_\mathrm{t}}}$, and $\mathbf{g}_k^H\in\mathbb{C}^{1\times{N_\mathrm{t}}}$, respectively. Since the IRS employs $M$ phase shifters, the phase shift matrix at the IRS is given by $\mathbf{\Phi}=\mathrm{diag}\left(e^{\jmath\theta_1},\cdots,e^{\jmath\theta_M}\right)$, where $\theta_m\in[0,2\pi]$, $\forall m\in\{1,\cdots,M\}$, represents the phase shift of the $m$-th reflecting element. The information-carrying signal transmitted to user $j$ is denoted by $s_j$, where $\mathbb{E}\left[|s_j|^2\right]=1$, $\forall j\in\mathcal{K}$, without loss of generality. The beamforming vector for user $j$ is denoted by $\mathbf{w}_j$. Variable $n_k$ represents the additive white Gaussian noise at user $k$ with zero mean and variance $\sigma_k^2$.
Therefore, the received SINR at user $k$ is given by
\begin{equation}\label{sinr}
\mathrm{SINR}_k=\frac{\left|\left(\mathbf{h}_k^H\mathbf{\Phi F}+\mathbf{g}_k^H\right)\mathbf{w}_k\right|^2}
{\sum_{j\in\mathcal{K}\backslash\{k\}}\left|\left(\mathbf{h}_k^H\mathbf{\Phi F}+\mathbf{g}_k^H\right)\mathbf{w}_j\right|^2+\sigma_k^2}.
\end{equation}
Our goal in this paper is to minimize the transmit power while ensuring a minimum required QoS of the users. The proposed power-efficient design of the beamformers and reflecting elements is obtained by solving the following optimization problem:
\begin{equation}
\begin{aligned}
&\underset{\mathbf{w}_k,\mathbf{\Phi}}{\mathrm{minimize}} && f\left(\mathbf{w}_k;\mathbf{\Phi}\right)=\sum_{k\in\mathcal{K}}\left\Vert\mathbf{w}_k\right\Vert_2^2\\
&\mathrm{subject\thinspace to}&&\mathrm{SINR}_k\ge\gamma_k,\quad\forall k,
\\
&&&\mathbf{\Phi}=\mathrm{diag}\left(
e^{\jmath\theta_1},e^{\jmath\theta_2},\cdots,e^{\jmath\theta_M}\right),
\end{aligned}\label{problem}
\end{equation}
where $\gamma_k$ is the predefined minimum required SINR of user $k$.
\begin{figure}[t]
\centering\includegraphics[height=4.5cm]{./figure/model}
\caption{IRS-assisted multiuser MISO system consisting of $K=3$ users.}
\label{model}
\end{figure}
\emph{Remark 1:} There are two main challenges in solving problem \eqref{problem}. First, each IRS reflecting element in $\mathbf{\Phi}$ has unit modulus, i.e., $\left|e^{\jmath\theta_m}\right|=1$, $\forall m\in\{1,\cdots,M\}$, which intrinsically is a non-convex constraint. Second, the optimization variables $\{\mathbf{w}_k\}_{k\in\mathcal{K}}$ and $\mathbf{\Phi}$ are coupled in the QoS constraint. These two facts make problem \eqref{problem} not jointly convex with respect to the optimization variables, and hence, in general difficult to solve optimally.
\subsection{Existing Approach}
To tackle the difficulties in solving problem \eqref{problem}, SDR-based AltMin algorithms have been widely adopted in the literature \cite{8811733,9013643,xujie}. In particular, the optimization of $\{\mathbf{w}_k\}_{k\in\mathcal{K}}$ and $\mathbf{\Phi}$ is decoupled and performed alternately by capitalizing on AltMin. For a given $\mathbf{\Phi}$, the optimization of the beamformers $\{\mathbf{w}_k\}_{k\in\mathcal{K}}$ is formulated as
\begin{equation}\label{p1}
\begin{aligned}
&\underset{\mathbf{w}_k}{\mathrm{minimize}} && f\left(\mathbf{w}_k\right)=\sum_{k\in\mathcal{K}}\left\Vert\mathbf{w}_k\right\Vert_2^2\\
&\mathrm{subject\thinspace to}&&\mathrm{SINR}_k\ge\gamma_k,\quad\forall k,
\end{aligned}
\end{equation}
which is identical to the corresponding problem in conventional wireless systems without IRSs, and therefore can be optimally solved via second-order cone programming (SOCP) \cite{8811733}.
On the other hand, the phase shift matrix $\mathbf{\Phi}$ can be optimized by solving the following feasibility check problem:
\begin{equation}\label{eq4}
\begin{aligned}
&\underset{\mathbf{\Phi}}{\mathrm{minimize}} && 1\\
&\mathrm{subject\thinspace to}&&\mathrm{SINR}_k\ge\gamma_k,\quad\forall k,
\\
&&&\mathbf{\Phi}=\mathrm{diag}\left(
e^{\jmath\theta_1},e^{\jmath\theta_2},\cdots,e^{\jmath\theta_M}\right).
\end{aligned}
\end{equation}
According to \cite[eq. (44)]{8811733}, problem \eqref{eq4} can be reformulated as
\begin{equation}\label{eq5}
\begin{aligned}
&\underset{\mathbf{V}\in\mathbb{H}^{M+1}}{\mathrm{minimize}} && 1\\
&\mathrm{subject\thinspace to}&&\Tr\left(\mathbf{R}_k\mathbf{V}\right)\le\hat{\gamma}_k,\quad\forall k,
\\
&&&\Diag\left(\mathbf{V}\right)=\mathbf{1}_{M+1},\\
&&&\Rank\left(\mathbf{V}\right)=1,\\
&&&\mathbf{V}\succeq\mathbf{0},
\end{aligned}
\end{equation}
where $\hat{\gamma}_k=\left|\mathbf{g}_k^H\mathbf{w}_k\right|^2-\gamma_k\left(\sigma_k^2+\sum_{j\in\mathcal{K}\backslash\{k\}}\left|\mathbf{g}_k^H\mathbf{w}_j\right|^2\right)$, $|x|^2=1$, $\mathbf{v}=\left[e^{\jmath\theta_1},\cdots,e^{\jmath\theta_M},x\right]^H$, and $\mathbf{V}=\mathbf{vv}^H$. In addition, $\mathbf{R}_k=-\mathbf{T}_{k,k}+\gamma_k\sum_{j\in\mathcal{K}\backslash\{k\}}\mathbf{T}_{k,j}$, where $\mathbf{T}_{k,j}$ is given by $\mathbf{T}_{k,j}=$
\begin{equation}
\begin{bmatrix}
\mathrm{diag}\left(\mathbf{h}_k^H\right)\mathbf{F}\mathbf{w}_j\mathbf{w}_j^H\mathbf{F}^H\mathrm{diag}\left(\mathbf{h}_k\right)&\mathrm{diag}\left(\mathbf{h}_k^H\right)\mathbf{F}\mathbf{w}_j\mathbf{w}_j^H\mathbf{g}_k\\
\mathbf{g}_k^H\mathbf{w}_j\mathbf{w}_j^H\mathbf{F}^H\mathrm{diag}\left(\mathbf{h}_k\right)&0
\end{bmatrix}.
\end{equation}
One common approach to handle problem \eqref{eq5} is to first drop the non-convex rank-one constraint. The relaxed problem is then convex with respect to $\mathbf{V}$ and can be solved by standard convex program solvers such as CVX \cite{grant2008cvx}.
Unfortunately, there is no guarantee that the obtained optimal solution $\mathbf{V}^\star$ is a rank-one matrix. A Gaussian randomization approach is therefore adopted to generate a unit modulus solution\footnote{Note that optimization variable $\mathbf{\Phi}$ in problems \eqref{problem} and \eqref{eq4} can be determined once $\mathbf{v}$ has been obtained.} $\mathbf{v}$, i.e., $|v_i|=1$, $\forall i\in\{1,\cdots,M+1\}$, \cite{8811733,9013643,xujie}.
Although the optimal solution $\mathbf{V}^\star$ of the relaxed problem satisfies the QoS constraint, there is no guarantee that the randomized solution $\mathbf{v}$ also fulfills the constraint. In other words, the randomized solution $\mathbf{v}$ is not necessarily feasible for problem \eqref{eq4}.
More importantly, in the AltMin procedure, we have
\begin{equation}
f\left(\mathbf{w}_k^{(t)};\mathbf{v}^{(t)}\right)\overset{(a)}{=}f\left(\mathbf{w}_k^{(t)};\mathbf{v}^{(t+1)}\right)\overset{(b)}{\not\geq} f\left(\mathbf{w}_k^{(t+1)};\mathbf{v}^{(t+1)}\right),
\end{equation}
where $t$ is the iteration index. The equality in (a) is because problem \eqref{eq4} only finds a $\mathbf{v}$ that does not affect the objective value. On the other hand, the uncertainty in (b) means that the objective function does not necessarily decrease after solving problem $\eqref{p1}$ based on the $\mathbf{v}^{(t+1)}$ that has been obtained form problem \eqref{eq4}.
In particular, for problem $\eqref{p1}$, $f\left(\mathbf{w}_k^{(t)};\mathbf{v}^{(t)}\right)$ and $f\left(\mathbf{w}_k^{(t+1)};\mathbf{v}^{(t+1)}\right)$ are optimal objective values based on the two different sets of parameters $\mathbf{v}^{(t)}$ and $\mathbf{v}^{(t+1)}$, respectively.
When the parameter is updated from $\mathbf{v}^{(t)}$ to $\mathbf{v}^{(t+1)}$, the feasible set of problem \eqref{p1} changes. However, the relation between the two feasible sets cannot be quantified as both $\mathbf{v}^{(t)}$ and $\mathbf{v}^{(t+1)}$ may be infeasible for problem \eqref{eq4}.
Hence, there is no guarantee how the optimal objective value $f$ of problem \eqref{p1} improves when the parameter is updated from $\mathbf{v}^{(t)}$ to $\mathbf{v}^{(t+1)}$.
For the above mentioned two reasons, the state-of-the-art SDR-based AltMin algorithm is not guaranteed to converge to a locally optimal solution, as will be also verified in Section IV-A. This motives us to develop a novel algorithm design for solving problem \eqref{problem} in the next section.
\section{Design of IRS-Assisted Multiuser MISO Wireless Systems}
In the SDR-based AltMin algorithm, only one set of the optimization variables is updated in each iteration, such that local optimality cannot be guaranteed. Therefore, we develop an algorithm that optimizes all optimization variables in each iteration. To this end, we first reformulate problem \eqref{problem} as follows.
The numerator of the SINR in \eqref{sinr} is rewritten as
\begin{equation}
\begin{split}
&\left|\left(\mathbf{h}_k^H\mathbf{\Phi F}+\mathbf{g}_k^H\right)\mathbf{w}_k\right|^2=
2\Re\left[\tilde{\mathbf{v}}^H\mathrm{diag}\left(\mathbf{h}^H_k\right)\mathbf{FW}_k\mathbf{g}_k\right]\\
&+\tilde{\mathbf{v}}^H\mathrm{diag}\left(\mathbf{h}^H_k\right)\mathbf{FW}_k\mathbf{F}^H\mathrm{diag}\left(\mathbf{h}_k\right)\tilde{\mathbf{v}}+\mathbf{g}^H_k\mathbf{W}_k\mathbf{g}_k\\
&=\mathbf{v}^H\mathbf{G}_k^H\mathbf{W}_k\mathbf{G}_k\mathbf{v},
\end{split}
\end{equation}
where $\mathbf{G}_k=\begin{bmatrix}
\mathbf{F}^H\mathrm{diag}\left(\mathbf{h}_k\right)&\mathbf{g}_k
\end{bmatrix}$, $\mathbf{W}_k=\mathbf{w}_k\mathbf{w}_k^H$, and $\tilde{\mathbf{v}}=\left[
e^{\jmath\theta_1},e^{\jmath\theta_2},\cdots,e^{\jmath\theta_M}\right]^H$. Hence, the denominator can be rewritten in a similar manner and problem \eqref{problem} is reformulated as
\begin{align}
&\underset{\mathbf{W}_k\in\mathbb{H}^{{N_\mathrm{t}}},\mathbf{V}\in\mathbb{H}^{M+1}}{\mathrm{minimize}} && f(\mathbf{W}_k)=\sum_{k\in\mathcal{K}}\Tr\left(\mathbf{W}_k\right)\notag\\
&\mathrm{\quad\,\,\, subject\thinspace to}&&\mbox{C1:}\,\gamma_k\sigma_k^2+\gamma_k\sum_{k\in\mathcal{K}}\Tr\left(\mathbf{W}_j\mathbf{G}_k\mathbf{V}\mathbf{G}_k^H\right)\notag\\
&&&\relphantom{\mbox{C1}}-\Tr\left(\mathbf{W}_k\mathbf{G}_k\mathbf{V}\mathbf{G}_k^H\right)\le0,\quad\forall k,\notag\\
&&&\mbox{C2:}\,\mathrm{Diag}\left(\mathbf{V}\right)=\mathbf{1}_{M+1},\label{refor}\\
&&&\mbox{C3:}\,\mathrm{Rank}\left(\mathbf{V}\right)=1,\notag\\
&&&\mbox{C4:}\,\mathrm{Rank}\left(\mathbf{W}_k\right)=1,\quad\forall k,\notag\\
&&&\mbox{C5:}\,\mathbf{V}\succeq\mathbf{0},\quad\mbox{C6:}\,\mathbf{W}_k\succeq\mathbf{0},\quad\forall k.\notag
\end{align}
Next, we leverage the IA method to tackle the non-convex constraints $\mbox{C1}$ and $\mbox{C3}$ in problem \eqref{refor}. In particular, the general IA algorithm optimizes a sequence of approximating convex programs. In each iteration of the algorithm, the non-convex constraints are approximated by their convex counterparts.
\subsection{IA Method for QoS Constraint $\mbox{C1}$}
We take the term $\Tr\left(\mathbf{W}_j\mathbf{G}_k\mathbf{V}\mathbf{G}_k^H\right)$ as an example to explain how we construct a convex constraint approximating the non-convex QoS constraint $\mbox{C1}$. The term is rewritten as
\begin{equation}\label{eq10}
\begin{split}
&\Tr\left(\mathbf{W}_j\mathbf{G}_k\mathbf{V}\mathbf{G}_k^H\right)=\frac{1}{2}\left\Vert\mathbf{W}_j+\mathbf{G}_k\mathbf{V}\mathbf{G}_k^H\right\Vert_F^2\\
&-\frac{1}{2}\Tr\left(\mathbf{W}_j^H\mathbf{W}_j\right)-\frac{1}{2}\Tr\left(\mathbf{G}_k\mathbf{V}^H\mathbf{G}_k^H\mathbf{G}_k\mathbf{V}\mathbf{G}_k^H\right).
\end{split}
\end{equation}
Now, constraint ${\mbox{C1}}$ can be rewritten in form of a difference of convex (d.c.) functions, where the last two terms in \eqref{eq10} are non-convex with respect to $\mathbf{W}_k$ and $\mathbf{V}$, respectively.
To facilitate IA, we construct a global underestimator for the non-convex terms by first-order Taylor approximation. Specifically, we have
\begin{align}
&\Tr\left(\mathbf{W}_j^H\mathbf{W}_j\right)\ge-\left\Vert\mathbf{W}_j^{(t)}\right\Vert_F^2+2\Tr\left(\left(\mathbf{W}_j^{(t)}\right)^H\mathbf{W}_j\right)\text{and}\notag\\
&\Tr\left(\mathbf{G}_k\mathbf{V}^H\mathbf{G}_k^H\mathbf{G}_k\mathbf{V}\mathbf{G}_k^H\right)\ge-\left\Vert\mathbf{G}_k\mathbf{V}^{(t)}\mathbf{G}_k^H\right\Vert_F^2\label{eq11}\\
&+2\Tr\left(\left(\mathbf{G}_k^H\mathbf{G}_k\mathbf{V}^{(t)}\mathbf{G}_k^H\mathbf{G}_k\right)^H\mathbf{V}\right),\notag
\end{align}
where
\begin{figure*}
\begin{equation}\label{long}
\begin{split}
\overline{\mbox{C1}}\mbox{:}\,&\frac{1}{2}\left\Vert\mathbf{W}_k-\mathbf{G}_k\mathbf{V}\mathbf{G}_k^H\right\Vert_F^2+\frac{\gamma_k}{2}\sum_{j\in\mathcal{K}\backslash\{k\}}\left\Vert\mathbf{W}_j+\mathbf{G}_k\mathbf{V}\mathbf{G}_k^H\right\Vert_F^2-\gamma_k\sum_{j\in\mathcal{K}\backslash\{k\}}\Tr\left(\left(\mathbf{W}_j^{(t)}\right)^H\mathbf{W}_j\right)\\
&-\Tr\left(\left(\mathbf{W}_k^{(t)}\right)^H\mathbf{W}_k\right)-\left[1+\gamma_k(K-1)\right]\Tr\left(\left(\mathbf{G}_k^H\mathbf{G}_k\mathbf{V}^{(t)}\mathbf{G}_k^H\mathbf{G}_k\right)^H\mathbf{V}\right)\\
&+\gamma_k\sigma_k^2+\frac{1}{2}\left\Vert\mathbf{W}_k^{(t)}\right\Vert_F^2
+\left[\frac{1}{2}+\frac{\gamma_k}{2}(K-1)\right]\left\Vert\mathbf{G}_k\mathbf{V}^{(t)}\mathbf{G}_k^H\right\Vert_F^2
+\frac{\gamma_k}{2}\sum_{j\in\mathcal{K}\backslash\{k\}}\left\Vert\mathbf{W}_j^{(t)}\right\Vert_F^2\le0,\quad\forall k.
\end{split}
\end{equation}
\hrule
\end{figure*}
$\mathbf{W}_j^{(t)}$ and $\mathbf{V}^{(t)}$ are the solutions obtained in the $t$-th iteration, at which the Taylor expansions are performed.
In addition, the term $-\Tr\left(\mathbf{W}_k\mathbf{G}_k\mathbf{V}\mathbf{G}_k^H\right)$ in constraint $\mbox{C1}$ is upper bounded in a similar manner as $\eqref{eq10}$ and \eqref{eq11}, and therefore the non-convex constraint $\mbox{C1}$ is approximated by constraint $\overline{\mbox{C1}}$ in \eqref{long}, shown at the top of this page.
Note that compared to constraint $\mbox{C1}$, where the optimization variables are coupled, the optimization variables are decoupled in constraint $\overline{\mbox{C1}}$, which is also jointly convex with respect to $\{\mathbf{W}_k\}_{k\in\mathcal{K}}$ and $\mathbf{V}$.
\subsection{IA Method for Rank-One Constraint $\mbox{C3}$}
Since it is difficult to directly derive an upper bound for the rank-one constraint $\mbox{C3}$, we first rewrite the rank-one constraint in equivalent form via the following lemma.
\begin{lem}
The rank-one constraint $\mbox{C3}$ is equivalent to constraint $\widetilde{\mbox{C3}}$, given by
\end{lem}
\begin{equation}\label{force}
\widetilde{\mbox{C3}}\mbox{:}\,\left\Vert\mathbf{V}\right\Vert_*-\left\Vert\mathbf{V}\right\Vert_2\le0.
\end{equation}
\begin{IEEEproof}
For any $\mathbf{X}\in\mathbb{H}^{m}$, the inequality $\left\Vert\mathbf{X}\right\Vert_*=\sum_i{\sigma_i}\ge\left\Vert\mathbf{X}\right\Vert_2=\underset{i}{\max}\{\sigma_i\}$ holds, where $\sigma_i$ is the $i$-th singular value of $\mathbf{X}$. Equality holds if and only if $\mathbf{X}$ has unit rank.
\end{IEEEproof}
Now, constraint $\widetilde{\mbox{C3}}$ is written in form of d.c. functions. Therefore, by deriving the first-order Taylor approximation of $\left\Vert\mathbf{V}\right\Vert_2$ as
\begin{equation}
\begin{split}
\left\Vert\mathbf{V}\right\Vert_2&\ge
\left\Vert\mathbf{V}^{(t)}\right\Vert_2+\Tr\Big[\boldsymbol{\lambda}_{\max}\left(\mathbf{V}^{(t)}\right)\\
&\relphantom{\ge}\times\boldsymbol{\lambda}_{\max}^H\left(\mathbf{V}^{(t)}\right)\left(\mathbf{V}-\mathbf{V}^{(t)}\right)\Big],
\end{split}
\end{equation}
we obtain a convex approximation of constraint $\widetilde{\mbox{C3}}$, which is given by constraint $\overline{\mbox{C3}}$ as follows:
\begin{equation}
\begin{split}
\overline{\mbox{C3}}\mbox{:}\,&\left\Vert\mathbf{V}\right\Vert_*-\Tr\Big[\boldsymbol{\lambda}_{\max}\left(\mathbf{V}^{(t)}\right)\boldsymbol{\lambda}_{\max}^H\left(\mathbf{V}^{(t)}\right)\\
&\times\left(\mathbf{V}-\mathbf{V}^{(t)}\right)\Big]-\left\Vert\mathbf{V}^{(t)}\right\Vert_2\le0.
\end{split}
\end{equation}
Therefore, a convex approximation $\overline{\mbox{C3}}$ of the non-convex rank-one constraint $\mbox{C3}$ is constructed and this constraint ensures that $\mbox{C3}$ is satisfied when the IA algorithm converges.
\subsection{Overall IA Algorithm}
With the approximated convex constraints $\overline{\mbox{C1}}$ and $\overline{\mbox{C3}}$ at hand, the optimization problem that has to be solved in the $(t+1)$-th iteration of the overall IA algorithm is given by
\begin{equation}\label{overall}
\begin{aligned}
&\underset{\mathbf{W}_k\in\mathbb{H}^{{N_\mathrm{t}}},\mathbf{V}\in\mathbb{H}^{M+1}}{\mathrm{minimize}} &&f(\mathbf{W}_k)= \sum_{k\in\mathcal{K}}\Tr\left(\mathbf{W}_k\right)\\
&\mathrm{\quad\,\,\, subject\thinspace to}&&\overline{\mbox{C1}},\mbox{C2},\overline{\mbox{C3}},\mbox{C4},\mbox{C5},\mbox{C6}.
\end{aligned}
\end{equation}
We note that the remaining non-convexity of problem \eqref{overall} stems from the $K$ rank-one constraints in \mbox{C4}. To tackle this issue, we remove constraint \mbox{C4} by applying SDR where the relaxed version of \eqref{overall} can be efficiently solved via standard convex program solvers such as CVX \cite{grant2008cvx}. The tightness of this SDR is revealed in the following theorem.
\begin{thm}
An optimal beamforming matrix $\mathbf{W}_k$ satisfying $\Rank\left(\mathbf{W}_k\right)=1$ can always be obtained for problem \eqref{overall}.
\end{thm}
\begin{IEEEproof}
Please refer to the Appendix.
\end{IEEEproof}
\begin{algorithm}[t]
\caption{Inner Approximation (IA) Algorithm}
\begin{algorithmic}[1]
\STATE Initialize $\mathbf{V}^{(0)}$ with random phases and obtain $\mathbf{W}_k^{(0)}$ by solving problem $\eqref{p1}$. Set the convergence tolerance $\varepsilon$ and iteration index $t=0$;
\REPEAT
\STATE For given $\mathbf{W}_k^{(t)}$ and $\mathbf{V}^{(t)}$, update $\mathbf{W}_k^{(t+1)}$ and $\mathbf{V}^{(t+1)}$ as the optimal solution of problem \eqref{overall} without $\mbox{C4}$;
\STATE $t\leftarrow t+1$;
\UNTIL $\frac{f\left(\mathbf{W}_k^{(t)}\right)-f\left(\mathbf{W}_k^{(t+1)}\right)}{f\left(\mathbf{W}_k^{(t+1)}\right)}\le\varepsilon$
\end{algorithmic}
\end{algorithm}
The overall IA algorithm is summarized in \textbf{Algorithm 1}. According to \cite[Th. 1]{marks1978general}, the objective function $f$ in \eqref{refor} is non-increasing in each iteration and the proposed algorithm is guaranteed to converge to a KKT solution of problem \eqref{problem}.
The computational complexity of each iteration of the proposed IA algorithm is given by $\mathcal{O}\left(\log\frac{1}{\varepsilon}\left(K{N_\mathrm{t}}^{\frac{7}{2}}+M^{\frac{7}{2}}\right)\right)$, where $\mathcal{O}\left(\cdot\right)$ is the big-O notation \cite[Th. 3.12]{polik2010interior}.
\section{Simulation Results}
In this section, we evaluate the performance of the proposed IA algorithm. The system carrier center frequency is $2.4$ GHz while the noise power at each user is set to $\sigma_k^2=-90$ dBm, $\forall k$. The AP serves one sector of a cell with radius $R$, where $K$ users are randomly and uniformly distributed in this sector and the IRS is deployed at the edge of the cell.
The channel matrix $\mathbf{F}$ between AP and IRS is modeled as
\begin{equation}
\mathbf{F}=\sqrt{L_0d^{-\alpha}}\left(\sqrt{\frac{\beta}{1+\beta}}\mathbf{F}^\mathrm{L}+\sqrt{\frac{1}{1+\beta}}\mathbf{F}^\mathrm{N}\right),
\end{equation}
where $L_0=\left(\frac{\lambda_{c}}{4\pi}\right)^2$ is a constant with $\lambda_{c}$ being the wavelength of the carrier frequency. The distance between AP and IRS is denoted by $d$ and $\alpha=2$ is the path loss exponent. The small-scale fading is assumed to be Ricean fading with Ricean factor $\beta=1$. $\mathbf{F}^\mathrm{L}$ and $\mathbf{F}^\mathrm{N}$ are the line-of-sight (LoS) and non-LoS components, respectively. The LoS component is the product of the receive and transmit array response vectors while the non-LoS component is modeled by Rayleigh fading. The channel vectors $\left\{\mathbf{h}_k\right\}_{k\in\mathcal{K}}$ are generated in a similar way as $\mathbf{F}$. In addition, the direct links $\left\{\mathbf{g}_k\right\}_{k\in\mathcal{K}}$ between AP and users are modeled as pure non-LoS channels, i.e., $\alpha=4$ and $\beta=0$, since one of the motivations for deploying IRSs is that the direct links are shadowed by obstacles. For the ease of presentation, we assume that the SINR thresholds for all users are identical $\gamma_k=\gamma$, $\forall k$. The number of random vectors generated by the Gaussian randomization in the existing SDR-based AltMin approach is $50$ and the convergence tolerance in the proposed IA algorithm is set to $\varepsilon=10^{-5}$.
\subsection{Convergence Performance}
\begin{figure}[t]
\centering\includegraphics[height=5.6cm]{figure/fig0}
\caption{Convergence of different algorithms for $R=200$ m, ${N_\mathrm{t}}=M=10$, $K=3$, and $\gamma=2$ dB.} \label{fig0}
\end{figure}
The convergence of the SDR-based AltMin algorithm and the proposed IA algorithm is investigated for a typical snapshot and averaged over $500$ realizations in Fig. \ref{fig0}, respectively. As can be observed for the snapshot (upper half of Fig. \ref{fig0}), the objective function fluctuates significantly during AltMin, which confirms the analysis in Section II-B. In contrast, the proposed IA algorithm guarantees a monotonic convergence, which shows its superiority compared to the existing approach. While the oscillation is smoothed over a large number of realizations, the SDR-based AltMin algorithm still cannot guarantee the convergence of the average objective value, as shown in the lower half of Fig. \ref{fig0}. On the contrary, the proposed IA algorithm converges within 100 iterations on average. These results clearly show the motivation and importance of the proposed IA algorithm.
\subsection{Transmit Power Minimization}
In Fig. \ref{fig1}, the average transmit power at the AP is plotted for different algorithms. To show the effectiveness of the approach proposed in this paper, besides the SDR-based AltMin algorithm, two additional baseline schemes are considered.
For baseline scheme 1, we evaluate the transmit power when an IRS is not deployed.
For baseline scheme 2, we adopt an IRS implemented with random phases and optimize the beamformers by solving problem \eqref{p1}.
Since the SDR-based AltMin algorithm cannot guarantee convergence, for a fair comparison, we set the maximum iteration number equal to the number of iterations required by the IA algorithm to converge.
First, we observe that the required average transmit power is significantly reduced by deploying an IRS in the considered multiuser MISO system.
This shows the ability of IRSs to establish favorable channel conditions, which facilitates achieving the QoS of the users at lower transmit powers. Hence, deploying IRSs is a promising approach for power-efficient wireless systems.
In addition, we note that the proposed IA algorithm outperforms both the SDR-based AltMin algorithm and the baseline scheme with random phases. This reveals the effectiveness of the proposed optimization methodology for jointly optimizing the beamformers and reflecting elements in IRS-assisted systems.
\begin{figure}[t]
\centering\includegraphics[height=5.6cm]{figure/fig1}
\caption{Average transmit power achieved by different algorithms when $R=100$ m, ${N_\mathrm{t}}=4$, $M=20$, and $K=4$.} \label{fig1}
\end{figure}
\subsection{Energy Efficiency Evaluation}
IRSs are recognized as energy-efficient devices for improving communication performance.
In Fig. \ref{fig2}, we show the energy efficiency versus the number of antenna elements at the AP and the number of reflecting elements at the IRS. The energy efficiency is defined as \cite[eq. (32)]{my}
\begin{equation}
\eta=\frac{\sum_{k\in\mathcal{K}}\log_2\left(1+\mathrm{SINR}_k\right)}{\frac{1}{\mu}\sum_{k\in\mathcal{K}}\left\Vert\mathbf{w}_k\right\Vert_2^2+P_\mathrm{s}+{N_\mathrm{t}} P_\mathrm{t}},
\end{equation}
where $0<\mu\le1$ is the power amplifier efficiency, $P_\mathrm{s}$ is the static power consumed by the AP and IRS controller, and $P_\mathrm{t}$ accounts for the circuit power consumption introduced by deploying one antenna element.
We evaluate the average energy efficiency versus the number of reflecting elements for ${N_\mathrm{t}}=4$ transmit antennas (blue curves) and versus the number of transmit antennas for $M=4$ reflecting elements (red curves). As can be observed, the energy efficiency monotonically increases with the number of reflecting elements. In particular, additional reflecting elements at the IRS provide more degrees of freedom for creating a more favorable propagation environment which allows a further reduction of the transmit power. Moreover, deploying more reflecting elements does not consume additional power as they are passive devices. On the other hand, the energy efficiency of the system decreases as the number of transmit antennas equipped at the AP becomes large.
This is because more circuit power is consumed if additional RF chains are deployed for driving the additional transmit antennas, which outweighs the transmit power reduction facilitated by employing more antennas.
This observation strongly encourages the application of IRSs as power-efficient communication devices in next-generation green wireless communication systems.
\section{Conclusions}
This paper studied the joint design of the beamforming vectors at the AP and the phase shifts at the IRS in an IRS-assisted multiuser MISO communication system. It was shown that the proposed IA algorithm is an effective design approach that effectively tackles the non-convexity of the formulated power minimization problem. Different from existing algorithms that do not guarantee local optimality, one particular contribution of this paper is that the proposed IA algorithm is guaranteed to converge to a KKT solution.
Our simulation results revealed that IRSs have significant potential for the establishment of power-efficient green wireless communication systems.
\begin{figure}[t]
\centering\includegraphics[height=5.6cm]{figure/fig2}
\caption{Average energy efficiency versus the number of transmit antennas, ${N_\mathrm{t}}$, or reflecting elements, $M$, when $R=100$ m, $K=4$, $\mu=0.32$, $P_\mathrm{s}=54$ mW, and $P_\mathrm{t}=100$ mW.}\label{fig2}
\end{figure}
|
1,116,691,497,561 | arxiv |
\chapter{A Novel Construction Algorithm\label{ch:algo}}
In this chapter, we propose a novel method for the construction of orthogonal arrays using existing orthogonal arrays and unit column vectors with the help of the Kronecker Product operator. The method proposed herein serves towards forming orthogonal arrays of larger dimensions from orthogonal arrays of smaller dimensions. We primarily deal with linear orthogonal arrays, however as we will show later, non-linear orthogonal arrays can also be constructed using this approach.
We then investigate the correctness of the construction and other possible extensions to the algorithm.
\section{The Construction}
Let $A$ denote a linear seed $OA(N,k,s,t)$ with levels from $GF(s)$. The construction provides an approach to generate another $OA(N^{2}, k^{2}+2k, s, t)$.
Let $C = \{ c_{1}, c_{2}, \hdots, c_{k} \}$ denote the set of all the factors of $A$. Now let us define another zero column vector $c_{k+1}$. Let $U$ denote a unit column vector of dimensions $N \times 1$. See $Algorithm \ 1$ for the detailed construction.
\begin{algorithm}
\DontPrintSemicolon
\SetKwFunction{Construct}{Construct}
\SetKwFunction{return}{return}
\Begin{
$ i \leftarrow 1$ \\
$ j \leftarrow 1$ \\
$C = C \cup \{ c_{k+1} \}$\\
$ C^{\prime} = \phi $ \\
\While{$i \leq k+1 \ $}
{\While{$j \leq k+1 \ $}{
\If{$i \neq k+1$ or $j \neq k+1$}{
$c^{\prime} = c_{i} \otimes U + U \otimes c_{j}$ \\
$c^{\prime} = c^{\prime} ( $mod$ \ s ) $ \\
$C^{\prime} = C^{\prime} \cup \{ c^{\prime} \} $\\
$ j \leftarrow j+1 $ \\
}
}
$ i \leftarrow i+1 $
}
return $C^{\prime}$
}
\caption{The Construction Procedure on (A,C,U,s)}
\end{algorithm}
$Algorithm \ 1$ returns a set of column vectors $C^{\prime}$. Then, $C^{\prime}$ forms the set of all the factors of an $OA(N^{2}, k^{2}+2k, s, t)$. Let us denote this $OA(N^{2}, k^{2}+2k, s, t)$ as $B$.
\section{Correctness of the Algorithm}
Now let us discuss the proof of correctness of the construction proposed above.
We shall state the result formally in the form of the following theorem.\\
\textbf{Theorem 3.1:} The existence of a linear orthogonal array $OA(N,k,s,t)$ with levels from $GF(s)$ implies the existence of an orthogonal array $OA(N^2, k^2 + 2k, s, t).$ \\
\textbf{Proof.} \ We proceed with the proof by first showing that the $OA(N^2, k^2 + 2k, s, t)$ generated from the seed linear orthogonal array $OA(N,k,s,t)$ is itself linear. Recall the denotations used in the previous section since they will be used again in this proof. \\
We can write $B=[B_1 | B_2 | \hdots | B_k | B_{k+1} | B_{k+2} ]$, where $B_i$ is defined as follows:
\[
B_i = \left\{
\begin{array}{l l}
\{ c^{\prime}_{i} | c^{\prime}_{i} = ( c_{i} \otimes U + U \otimes c_{j} ) mod \ s, \ \forall j = 1 \ to \ k \} & \quad \text{for $i=1$ \ to \ $k$}\\
\{ c^{\prime}_{k+1} | c^{\prime}_{k+1} = ( U \otimes c_{j} ) mod \ s , \ \forall j = 1 \ to \ k \} & \quad \text{if $i= k + 1$}\\
\{ c^{\prime}_{k+2} | c^{\prime}_{k+2} = ( c_{j} \otimes U ) mod \ s , \ \forall j = 1 \ to \ k \} & \quad \text{if $i= k + 2$}\\
\end{array} \right.
\]
Let $R_j=[b_{1}^{j}|b_{2}^{j}| \hdots |b_{k+2}^{j}]$ be the $j^{th}$ row of $B$, such that $b_{1}^{j},b_{2}^{j},\hdots,b_{k+2}^{j}$ are the $j^{th}$ rows of $B_1, B_2 , \hdots , B_{k+2}$ respectively.
Then due to the definition of the construction, it can be seen that the following results hold:
\begin{itemize}
\item for $i=1 \ to \ k$, $b_{i}=r+\beta_{i} U_{1 \times k}$
\item $b_{k+1} = r$
\item $b_{k+2}=[\beta_{1}|\beta_{2}| \hdots |\beta_{k}]$
\end{itemize}
where $r$ and $[\beta_{1}|\beta_{2}| \hdots |\beta_{k}]$ are rows of $A$ and $U_{1 \times k}$ is a unit row vector.
Thus we can find rows $r_1$, $r_2$, $[\beta_{1}|\beta_{2}| \hdots |\beta_{k}]$ and $[\beta_{1}^{\prime}|\beta_{2}^{\prime}| \hdots |\beta_{k}^{\prime}]$ of $A$ which give the the $i^{th}$ and $j^{th}$ rows of $B$ as follows:
\[R_i=[b_{1}^{i}|b_{2}^{i}| \hdots |b_{k+2}^{i}]= [ r_1+\beta_{1}U_{1 \times k}|r_1+\beta_{2}U_{1 \times k}| \hdots |r_1+\beta_{k}U_{1 \times k}|r_1|\beta_{1}|\beta_{2}| \hdots |\beta_{k}]\]
\[R_j=[b_{1}^{j}|b_{2}^{j}| \hdots |b_{k+2}^{j}]= [ r_2+\beta_{1}^{\prime}U_{1 \times k}|r_2+\beta_{2}^{\prime}U_{1 \times k}| \hdots |r_2+\beta_{k}^{\prime}U_{1 \times k}|r_2|\beta_{1}^{\prime}|\beta_{2}^{\prime}| \hdots |\beta_{k}^{\prime}]\]
A linear combination of the above two rows using scalars $h_1$ and $h_2$ from the $GF(s)$ will result in the following: \\
$h_1 R_i + h_2 R_j = [ h_1 r_1+ h_2 r_2+(h_1 \beta_1+h_2 \beta_{1}^{\prime})U_{1 \times k}|h_1 r_1+ h_2 r_2+(h_1 \beta_2+h_2 \beta_{2}^{\prime})U_{1 \times k}| \hdots |h_1 r_1+ h_2 r_2+(h_1 \beta_k+ h_2\beta_{k}^{\prime})U_{1 \times k}|h_1 r_1+ h_2 r_2|h_1 \beta_1+h_2 \beta_{1}^{\prime}|h_1 \beta_2+h_2 \beta_{2}^{\prime}| \hdots |h_1 \beta_k+h_2 \beta_{k}^{\prime}]$\\
Now, since A is known to be linear, and $r_1$ and $r_2$ are rows of $A$, $h_1 r_1 + h_2 r_2$ is also a row of A. Similarly, $[h_1 \beta_1+h_2 \beta_{1}^{\prime} | h_1 \beta_2+h_2 \beta_{2}^{\prime} | \hdots | h_1 \beta_k+h_2 \beta_{k}^{\prime}]$ is a row of A. Hence, by definition of the construction and its properties discussed earlier in this proof, $h_1 R_i + h_2 R_j$ is a row of $B$. It can also be easily noted that one row of $B$ is a zero vector of $1 \times (k^2+2k)$ size. Hence, it can be concluded that the rows of $B$ form a linear subspace of $GF(s)^{k}$.
So, let $N^2=s^n$ and let $G$ be the generator matrix for $B$ of dimensions $n \times (k^2+2k)$ such that the rows of $B$ consist of all $k$-tuples $\eta G$, where $\eta \in GF(s)^n$. Now, suppose there exist $t$ columns of $B$ that are linearly dependent over $GF(s)$. Then, due to the nature of the construction, there exist $x$ $(1 \leq x \leq t)$ columns in $A$ that are linearly dependent, which is a contradiction. Hence, every selection of $t$ columns of $B$ are linearly independent over $GF(s)$. So, let us choose $t$ columns of $B$ and let $G_1$ be the corresponding submatrix of generator matrix $G$. Then, the columns of $G_1$ will be linearly independent. Also, the number of times a $t$-tuple $\tau$ is present as a row in these $t$ columns of $B$ is determined by and is equal to the number of $\eta$ such that $\eta G_1 = \tau$.
Now, since $G_1$ has rank $t$, the number of such $\eta$ is $s^{(n-t)}$, for all $\tau$. Hence, $B$ is an orthogonal array of strength $t$. This concludes the proof. \hfill
\section{Binary Orthogonal Arrays}
In this section we propose the following lemma, wherein we show that the linearity condition in Theorem 1 can be dropped for binary seed orthogonal arrays (i.e. OAs of strength 2).
\textbf{Lemma 3.1:} The existence of an $OA(N,k,s,2)$ implies the existence of an $OA(N^{2}, k^{2}+2k, s, 2)$.
\textbf{Proof.} Let $A$ be $OA(N,k,s,2)$ as described before and let $c^{\prime}_{1}$ and $c^{\prime}_{2}$ be any two distinct factors of $B$. Also, let us assume $c^{\prime}_{1}$ and $c^{\prime}_{2}$ were generated from the factors $c_{i_{1}}$, $c_{j_{1}}$, $c_{i_{2}}$, $c_{j_{2}}$ of $A$ using the following equations:
\[ c^{\prime}_{1} = ( c_{i_{1}} \otimes U + U \otimes c_{j_{1}} ) mod \ s \]
\[ c^{\prime}_{2} = ( c_{i_{2}} \otimes U + U \otimes c_{j_{2}} ) mod \ s \]
Let $c^{\prime}_{1}$ and $c^{\prime}_{2}$ be denoted by $[X_{1} : X_{2} : \hdots : X_{N}]^{\prime}$ and $[Y_{1} : Y_{2} : \hdots : Y_{N}]^{\prime}$, where the dimensions of $X_{i}$ and $Y_{i}, \ \forall i = 1 \ to \ n$, is $N \times 1$.
In order to show the validity of the algorithm we split the analysis into four cases as follows.\\
$Case \ 1: \ $ $c_{j_{1}} \neq c_{j_{2}}$ such that $c_{j_{1}} , c_{j_{2}} \neq c_{k+1}$ \\
We know that $X_{i} = (c_{j_{1}} + p_{i} U)mod \ s$ and $Y_{i} = (c_{j_{2}} + q_{i} U)mod \ s$, where $p_{i}, q_{i} \in \{ 0, 1, 2, \hdots , s-1 \}, \ $ for all $i = 1 \ to \ N$. Also $c_{j_{1}} \neq c_{j_{2}}$ and $c_{j_{1}}, c_{j_{2}} \in C$ implies that $c_{j_{1}}$ and $c_{j_{2}}$ form the factors of an $OA(N,2,s,2)$. Since, for all $i = 1 \ to \ N, \ X_{i}$ and $Y_{i}$ are just cyclic permutations of symbols of $c_{j_{1}}$ and $c_{j_{2}}$, the sets $\{ X_{i}, Y_{i} \}$ form the factors of Orthogonal Arrays with parameters $(N,2,s,2)$. Therefore, $Z = [c^{\prime}_{1} : c^{\prime}_{2}]$ being a vertical juxtaposition of the matrices $[X_{i} : Y_{i}]$ for $i = 1 \ to \ N$, is an $OA(N^2,2,s,2)$. Hence, $c^{\prime}_{1}$ and $c^{\prime}_{2}$ form the factors of an $OA(N^2,2,s,2)$. \\
$Case \ 2: \ $ $c_{j_{1}} = c_{j_{2}}$ such that $ c_{j_{1}} , c_{j_{2}} \neq c_{k+1}$ \\
$X_{i} = (c_{j} + p_{i} U)mod \ s$ and $Y_{i} = (c_{j} + q_{i} U)mod \ s$, where $c_{j_{1}} = c_{j_{2}} = c_{j}$ (say) and $p_{i}, q_{i} \in \{ 0, 1, 2, \hdots , s-1 \}, \ $ for all $i = 1 \ to \ N$. Since we know that $c_{i_{1}} \neq c_{i_{2}}$, and also the fact that $c_{i_{1}}$ and $c_{i_{2}}$ together form the factors of an $OA(N,2,s,2)$, we therefore can conclude that all possible 2-tuples on $ \{0,1,2, \hdots ,s-1 \} $ belong to the set $ \{ (p_i,q_i) : i = 1 \ to \ N \} $ and also that each tuple occurs exactly $N/s$ times. Hence, $Z = [c^{\prime}_{1} : c^{\prime}_{2}]$ which is a vertical juxtaposition of the matrices $[X_{i} : Y_{i}]$ for $i = 1 \ to \ N$, is an $OA(N^2,2,s,2)$and thus $c^{\prime}_{1}$ and $c^{\prime}_{2}$ form the factors of an $OA(N^2,2,s,2)$.\\
$Case \ 3: \ $ $c_{j_{1}} = c_{k+1}$ and $c_{j_{2}} \neq c_{k+1}$ \\
$c^{\prime}_{1}$ is a juxtaposition of $\alpha U, \alpha \in \{0,1,2, \hdots ,s-1\}$ with each $\alpha U$ appearing $N/s$ times. $c^{\prime}_{2}$ is a juxtaposition of $c_{j_{2}} + \beta U, \beta \in \{0,1,2, \hdots ,s-1\}$ where each $c_{j_{2}} + \beta U$ has symbols from $\{0,1,2, \hdots ,s-1\}$ an equal number of times (since $A$ is orthogonal of order $1$). Hence $c^{\prime}_{1}$ and $c^{\prime}_{2}$ contain all possible 2-tuples over $ \{0,1,2, \hdots ,s-1 \} $ an equal number of times.\\
$Case \ 4: \ $ $c_{j_{1}} = c_{j_{2}} = c_{k+1}$ \\
$c_{j_{1}} = c_{j_{2}}$ implies $c_{i_{1}} \neq c_{i_{2}} $. Also since $c_{i_{1}} $ and $ c_{i_{2}} $ contain all possible 2-tuples over $\{0,1,2,\hdots,s-1\}$ (since $A$ is orthogonal of order $2$), $c^{\prime}_{1}$ and $c^{\prime}_{2}$ contain all possible 2-tuples over $ \{0,1,2,\hdots,s-1 \} $ equal number of times.\\
Since $c^{\prime}_{1}$ and $c^{\prime}_{2}$ were chosen arbitrarily, $C^{\prime}$ forms the set of all the factors of an $OA(N^{2}, k^{2}+2k, s, 2)$. Hence, the construction given in $Algorithm \ 1$ gives an $OA(N^2,k^2+2k,s,2)$ for every existing $OA(N,k,s,2)$.\\
It can be noticed that the linearity constraint is not necessary for proving that the method gives an $OA(N^{2}, k^{2}+2k, s, 2)$. Hence the above lemma is proved.\\
\section{Generating Non-Linear Orthogonal Arrays}
Although we originally proposed the method for generating linear orthogonal arrays, it is possible to generate non-linear orthogonal arrays using the same construction. For accomplishing this we must take a linear orthogonal array to begin with, choose any one of its columns and cyclically permute its symbols. Then clearly the resulting orthogonal array will loose its linearity property. However, even then the construction will work well for this kind of seed orthogonal array and generate a non-linear orthogonal array. We discuss further about this approach and its correctness.
Given a linear orthogonal array $A(N,k,s,t)$, we randomly choose a column $c_i$ of which we shall cyclically permute the symbols. After undergoing a cyclic permutation, the new column $c_i^{\prime}$ can be represented as follows:
\[ c_i^{\prime} = c_i + \alpha U\]
where $\alpha$ is a non-zero element of $GF(s)$ and $U$ is a unit column vector of dimensions $N \times 1$.
Then the columns of the orthogonal array $B$ generated using $c_i^{\prime}$ from the seed array $A$ may be of the form:
\[ c^{\prime} = ((c_i + \alpha U) \otimes U + U \otimes c_j ) mod s = (c_i \otimes U + U \otimes c_j + \alpha U \otimes U) mod s\]
Irrespective of whether $c_j$ is a zero column vector or is equal or not equal to $c_i^{\prime}$, we always get a symbol-wise cyclic permutation of the column of $B$ we would have generated if $A$ were linear. Similarly, the result holds for
\[ c^{\prime} = ((c_i + \alpha U) \otimes U + U \otimes c_j ) mod s \]
Since the the columns of the array we have obtained is proved to be equal to or a cyclic permutation of symbols of the columns of the linear orthogonal array which we would otherwise have generated, the array thus produced is orthogonal. Further, since it does not contain a zero vector for a row, it is not linear. Hence, we can successfully generate non-linear orthogonal arrays using the aforementioned construction.
\chapter{Conclusion and Future Work\label{ch:conclusion}}
In this work, we presented a novel construction algorithm for generating orthogonal arrays from seed linear orthogonal arrays. The proposed method works well for seed linear orthogonal arrays of all strengths and levels. We also discussed the proof of correctness of the construction and also the possibility of generating non-linear orthogonal arrays using the same construction. The results show that the proposed construction is indeed capable of contributing to the existing libraries of orthogonal arrays. Lists of new orthogonal arrays generated using this method have been provided in the Results section.\\
Extensive experimental observations suggest that the proposed construction works well even if the required linearity condition is dropped. This gives us reason to conjecture that the construction would work well for any seed orthogonal array. We have already proved this result for seed orthogonal arrays of strength 2. Our future work will be primarily directed towards proving the correctness of the construction for non-linear seed orthogonal arrays of any strength.
\chapter{Future Work\label{ch:futurework}}
Future work should include options in the template for a masters thesis or an undergraduate senior thesis. It should also support running headings in the headers using the `headings' pagestyle. The print mode and proquest mode included in the template might also be candidates to include in the class itself.
\chapter{Introduction\label{ch:intro}}
Construction of orthogonal arrays (OAs) is an important problem in combinatorial design which holds great significance for design of experiments in statistical analysis. In the past, several construction methods for generating orthogonal arrays have been proposed and analyzed. Hedayat et al. (1999) provide a comprehensive study of orthogonal arrays. The significance of the Kronecker Product and the Kronecker Sum operations in the context of generating orthogonal arrays is well established, as seen in the works of Shrikhande (1964), Wang and Wu (1991), Zhang, Weiguo, Mao and Zheng (2006) and Sinha, Vellaisamy and Sinha (2008). Sinha et al. (2009) used the Kronecker Sum operation on ternary orthogonal arrays and Balanced Incomplete Block Designs to construct new symmetrical ternary orthogonal arrays. In the current work we propose a novel construction approach for orthogonal arrays using unit column vectors and the Kronecker Product operations on existing orthogonal arrays.
\section{Preliminaries and Definitions}
\subsection{Balanced Arrays}
\par A balanced array, denoted by $BA(N,m,s,t)$ $ \{ \mu_{x_1,....,x_t} \} $, is defined as an $N \times m$ matrix $B$ with elements belonging to the set $S = \{0,1,....,s-1\}$ of $s$ symbols, $m$ factors, $N$ runs and strength $t$ such that every $N \times t$ sub-matrix of $B$ contains the ordered row vector $(x_1,....,x_t)$, $ \mu_{x_1,....,x_t}$ times, where $\mu_{x_1,....,x_t}$ is invariant under any permutation of $x_1,....,x_t$.
\subsection{Orthogonal Arrays}
\par An $N \times k$ array $A$ with entries from $S$ is said to be an orthogonal array with $s$ levels, strength $t \ ( 0 \leq t \leq k )$ and index $\lambda$ if every $N \times t$ sub-array of $A$ contains each $t$-tuple based on $S$ exactly $\lambda$ times as a row.
\par If $ \mu_{x_1,....,x_t} =\mu$(constant) $ \forall \ t-tuples \ (x_1,\hdots,x_t) \in S$ in a balanced array, then the balanced array becomes an orthogonal array with index $\mu$.
\subsection{Kronecker Sum}
\par Sinha et al. (1979) defined the Kronecker sum of matrices $A$ (order $m \times n$) and $B$ (order $p \times q$) as $A\otimes B = A \otimes J + J \otimes B$ , where $\otimes$ denotes the usual Kronecker product and $J$ is a matrix with all its elements unity but is of dimension $p \times q$ in the first term and $m \times n$ in the second term. The Kronecker sum of binary orthogonal arrays was defined by Sinha et al. (2008).
\subsection{Galois Field}
\par A Galois field is a field that contains finite number of elements and is denoted by $GF(s)$, where $s$ is its order. The order of a Galois field is the number of elements in the field and is of the form $p^n$, where $p$ is a prime number known as characteristic of the field and n is a positive integer. We shall denote the elements of the $GF(s)$ by $\{0, 1, 2,...., s-1 \}$ and the set of all $n$-tuples with entries from $GF(s)$ by $GF(s)^n$.
\subsection{Simple Orthogonal Arrays}
\par An orthogonal array is said to be simple if all its runs are distinct.
\subsection{Linear Orthogonal Arrays}
\par Let $s=p^n$, where $p$ is a prime and $n$ is a positive integer. Then, the orthogonal array $OA(N,k,s,t)$ with levels from $GF(s)$ is linear if it satisfies the following two conditions:
\begin{itemize}
\item it is simple
\item when its rows are considered as $k$-tuples from the $GF(s)$, its $N$ runs form a vector space over $GF(s)$.
\end{itemize}
\chapter{Related Work\label{ch:pastwork}}
The problem of generating orthogonal arrays has been of interest to researchers since more than half a century. A number of algorithms have been proposed and constructions given. The orthogonal arrays that have been successfully constructed using such constructions are stored in libraries. In this chapter we shall discuss some important results in this area of research so as to build a basic framework before discussing the work presented in this thesis.
We shall discuss the theorems and constructions briefly without digressing by going into the details of the proofs. For detailed proofs the reader is directed to the references.
\section{The Rao-Hamming Construction}
As the name indicates, this construction was given by Rao (1947, 1949) and Hamming (1950), both of whom had found this algorithm independently. Rao had originally introduced a special case of the idea of orthogonal arrays in a rather implicit sense in his concepts of hypercube of strength $t$. His construction of hypercubes of strength $2$ find relevance and correspondence with orthogonal arrays $OA(s^n,(s^n-1)/(s-1),s,2)$, $s$ being a prime power. This brings us to the following theorem.
\textbf{Theorem 2.1:} If $s$ is a prime power then an $OA(s^n,(s^n-1)/(s-1),s,2)$ exists whenever $n \geq 2$.
\textbf{Construction:} Let us consider an $s^n \times n$ array whose rows are all possible $n$-tuples over $GF(s)$. Let, $C_1, C_2,.....,C_n$ be the columns of this array. The columns of the orthogonal array then consist of all the columns of the form
\[ z_1 C_1 + z_2 C_2 + ..... + z_n C_n = [C_1, C_2,.....,C_n] z , \]
where $z=(z_1, z_2, ......, z_n)^T$ is an $n$-tuple from $GF(s)$, not all the $z_i$ are $0$, and the first $z_i$ is 1. Then, there are $(s^n-1)/(s-1)$ such columns.
\section{Bush's Construction}
Bush's (1952) research on orthogonal arrays of index 1 is a well known and important result. The theorem proposed by Bush goes as follows.
\textbf{Theorem 2.2:} If $s \geq 2$ is a prime power then an $OA(s^t,s+1,s,t)$ of index unity exists whenever $s \geq t-1 \geq 0$.
The result stated in the above theorem can be improved for some values of $s$ and $t$ as can be seen below.
\textbf{Theorem 2.3:} If $s=2^m, m \geq 1,$ and $t=3$ then there exists an $OA(s^3,s+2,s,t).$
The orthogonal arrays constructed in Theorems 2.2 and 2.3 are simple and linear.
\textbf{Theorem 2.4:} If $s$ is a prime power and a linear array $OA(s^t,k,s,t)$ exists, then there also exists a linear array $OA(s^{k-t},k,s,k-t).$
\section{Bose and Bush's Recursive Construction}
This construction was proposed by Bose and Bush (1952). It allows for the construction of orthogonal arrays of strength two with a large number of factors and possibly the the maximal number, provided that the number of symbols $s$ and the index $\lambda$ are powers of the same prime.
The theorem is stated as follows:
\textbf{Theorem 2.5:} Let $s = p^v$ and $\lambda = p^u$, where $p$ is a prime and $u$ and $v$ are integers with $u \geq 0, v \geq 1$. Let $d = \lfloor u/v \rfloor$. Then there exists an
\[ OA(\lambda s^2, \lambda (s^{d+1}-1)/(s^d - s^{d-1}) + 1, s, 2)\]
\section{Hadamard Matrices and Orthogonal Arrays}
A Hadamard matrix is a square matrix which takes only two symbols +1 and -1 as its entries such that for every two different rows there are matching entries in exactly half of the cases and non-matching entries in the remaining half. It can be easily noticed that Hadamard matrices are difference schemes with two symbols. Hadamard matrices and orthogonal arrays have close resemblances in their combinatorial properties. Hadamard matrices can be generated using recurrence relations. One such method is called Sylvester's method which goes as follows:
Let $H_i$ denote a Hadamard matrix of order $i$. Then,
\[ H_{1} = \begin{bmatrix} 1 \end{bmatrix}\]
\[ H_2 = \begin{bmatrix} 1 & 1 \\ 1 & -1 \\ \end{bmatrix} \]
\[ \vdots \]
\[ H_{2^k} = \begin{bmatrix} H_{2^{k-1}} & H_{2^{k-1}} \\ H_{2^{k-1}} & -H_{2^{k-1}} \\ \end{bmatrix} \]
An important result that illustrates the close connection between Hadamard matrices and orthogonal arrays is mentioned below.
\textbf{Theorem 2.6:} Orthogonal arrays $OA(4 \lambda, 4 \lambda -1, 2, 2)$ and $OA(8 \lambda, 4 \lambda, 2, 3)$ exist if and only if there exists a Hadamard matrix of order $4 \lambda$.
\chapter{Results\label{ch:results}}
The significance of a new construction lies in its ability to generate new orthogonal arrays which can be contributed to the existing libraries. Hence, it is important to show that the construction proposed in this work is actually capable of generating new orthogonal arrays. Thereby, in this chapter we shall discuss about the new orthogonal arrays that can be constructed using the proposed method. Comprehensive libraries of orthogonal arrays can be found in Hedayat $et \ al.$ (1999) or on the following websites:
\begin{center}
\textit{http://www2.research.att.com/~njas/oadir/} \\
\textit{http://support.sas.com/techsup/technote/ts723.html} \\
\end{center}
\section{List of New Contributions}
In this section we shall list out tables of orthogonal arrays which are generated using the proposed construction but are not found in any of the above mentioned libraries.
\begin{table}[ht]
\caption{Orthogonal Arrays with Two Levels and Strength 2}
\centering
\begin{tabular}{c c c}
\hline\hline
Sl. No. & Seed OA & Generated OA \\ [0.5ex]
\hline
! & OA(8,5,2,2) & OA(64,35,2,2) \\
2 & OA(8,7,2,2) & OA(64,63,2,2) \\
3 & OA(12,11,2,2) & OA(144,143,2,2) \\
4 & OA(16,15,2,2) & OA(256,255,2,2) \\
5 & OA(20,19,2,2) & OA(400,399,2,2) \\ [1ex]
\hline
\end{tabular}
\label{table:nonlin}
\end{table}
\begin{table}[ht]
\caption{Orthogonal Arrays with Two Levels and Strength 3}
\centering
\begin{tabular}{c c c}
\hline\hline
Sl. No. & Seed OA & Generated OA \\ [0.5ex]
\hline
1 & OA(24,12,2,3) & OA(576,168,2,3) \\
2 & OA(32,16,2,3) & OA(1024,288,2,3) \\
3 & OA(40,20,2,3) & OA(1600,440,2,3) \\
4 & OA(48,24,2,3) & OA(2304,624,2,3) \\
5 & OA(56,28,2,3) & OA(3136,840,2,3) \\
6 & OA(64,32,2,3) & OA(4096,1088,2,3) \\
7 & OA(72,36,2,3) & OA(5184,1368,2,3) \\ [1ex]
\hline
\end{tabular}
\label{table:nonlin}
\end{table}
\begin{table}[ht]
\caption{Orthogonal Arrays with Two Levels and Strength $>$ 3}
\centering
\begin{tabular}{c c c}
\hline\hline
Sl. No. & Seed OA & Generated OA \\ [0.5ex]
\hline
1 & OA(80,6,2,4) & OA(6400,48,2,4) \\
2 & OA(128,9,2,5) & OA(16384,99,2,5) \\
3 & OA(64,7,2,6) & OA(4096,63,2,6) \\ [1ex]
\hline
\end{tabular}
\label{table:nonlin}
\end{table}
\begin{table}[ht]
\caption{Orthogonal Arrays with Three Levels and Strength $\geq$ 2}
\centering
\begin{tabular}{c c c}
\hline\hline
Sl. No. & Seed OA & Generated OA \\ [0.5ex]
\hline
1 & OA(27,13,3,2) & OA(729,195,3,2) \\
2 & OA(81,40,3,2) & OA(6561,1680,3,2) \\
3 & OA(54,5,3,3) & OA(2196,35,3,3) \\ [1ex]
\hline
\end{tabular}
\label{table:nonlin}
\end{table}
\begin{table}[ht]
\caption{Orthogonal Arrays with More Than Three Levels}
\centering
\begin{tabular}{c c c}
\hline\hline
Sl. No. & Seed OA & Generated OA \\ [0.5ex]
\hline
1 & OA(16,5,4,2) & OA(256,35,4,2) \\
2 & OA(64,21,4,2) & OA(4096,483,4,2) \\
3 & OA(64,6,4,3) & OA(4096,48,4,3) \\
4 & OA(25,6,5,2) & OA(625,48,5,2) \\
5 & OA(49,8,7,2) & OA(2401,80,7,2) \\
6 & OA(64,9,8,2) & OA(4096,99,8,2) \\
7 & OA(81,10,9,2) & OA(6561,120,9,2) \\
8 & OA(121,12,11,2) & OA($11^4$,168,11,2) \\
9 & OA(169,14,13,2) & OA($13^4$,224,13,2) \\
10 & OA(256,17,16,2) & OA($2^{16}$,288,16,2) \\
11 & OA(289,18,17,2) & OA($17^4$,360,17,2) \\ [1ex]
\hline
\end{tabular}
\label{table:nonlin}
\end{table}
|
1,116,691,497,562 | arxiv | \section{Introduction}
A fundamental question in discrete tomography is whether a binary
image can be reconstructed from a small number of projections. As a
special case, one might restrict attention to permutation matrices,
and try to determine which vectors of antidiagonal sums appear only
once. This problem, considered by Bebeacua, Mansour, Postnikov and
Severini~\cite{MR2301096}, is apparently still open.
In this note, we consider the analogous problem for alternating sign
matrices. An alternating sign matrix is a square matrix of $0$s,
$1$s and $-1$s such that the sum of each row and each column is $1$,
and the nonzero entries in each row and in each column alternate in
sign. For an $n\times n$-alternating sign matrix $A$, the $k$-th
(antidiagonal) sum is $x_k = \sum_{i+j=k+1} A_{i,j}$ and the
(antidiagonal) \Dfn{X-ray} is the vector $x_1,\dots,x_{2n-1}$. For
example, the alternating sign matrices of size three together with
their X-rays are as follows:
\begin{center}
\scalebox{0.8}{\mbox{%
$\begin{array}{ccccccc}
\begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{pmatrix}& \begin{pmatrix}
0 & 1 & 0 \\
1 & 0 & 0 \\
0 & 0 & 1
\end{pmatrix}& \begin{pmatrix}
1 & 0 & 0 \\
0 & 0 & 1 \\
0 & 1 & 0
\end{pmatrix}& \begin{pmatrix}
0 & 1 & 0 \\
1 & -1 & 1 \\
0 & 1 & 0
\end{pmatrix}& \begin{pmatrix}
0 & 0 & 1 \\
0 & 1 & 0 \\
1 & 0 & 0
\end{pmatrix}& \begin{pmatrix}
0 & 0 & 1 \\
1 & 0 & 0 \\
0 & 1 & 0
\end{pmatrix}& \begin{pmatrix}
0 & 1 & 0 \\
0 & 0 & 1 \\
1 & 0 & 0
\end{pmatrix}\\[16pt]
1/ 0/ 1/ 0/ 1&
0/ 2/ 0/ 0/ 1&
1/ 0/ 0/ 2/ 0&
0/ 2/-1/ 2/ 0&
0/ 0/ 3/ 0/ 0&
0/ 1/ 1/ 1/ 0&
0/ 1/ 1/ 1/ 0
\end{array}$
}}
\end{center}
Note that all X-rays except $0/ 1/ 1/ 1/ 0$ occur precisely once.
Thus, there are five alternating sign matrices determined by there
X-rays. We can now state our main result:
\begin{thm}
There is an explicit bijection between Dyck paths of semilength $n$
and $n\times n$-alternating sign matrices which are determined by
their X-rays.
\end{thm}
The coincidence described by the theorem was observed when submitting
the statistic\footnote{\url{http://www.findstat.org/St000889}}
counting the number of alternating sign matrices with the same X-rays
to the online database of combinatorial statistics
FindStat~\cite{FindStat2017} and looking at the first few generating
functions automatically produced there. We currently have no
explanation for any of the other terms in the distribution.
\section{The bijection}
The map $\mathcal A$ from Dyck paths to alternating sign matrices is
defined as follows, see Figure~\ref{fig:exampleA} for an example.
For more visual clarity in the pictures, we use ($+$)s and ($-$)s
instead of $1$s and $-1$s and omit $0$s.
\begin{itemize}
\item Draw the Dyck path in an $n\times n$ square, beginning in the
top left corner, taking east and south steps and terminating in the
bottom right corner, never going below the main diagonal of the
matrix.
\item Add the reflection through the main diagonal of the Dyck path
to the picture.
\item For each peak of the Dyck path, fill the cells lying between
the peak and its mirror image on the antidiagonal with $1$s.
\item For each valley of the Dyck path, fill the cells lying between
the valley and its mirror image on the antidiagonal with $-1$s.
\item Fill the remaining cells with $0$s.
\end{itemize}
\begin{figure}[h]
\centering
\begin{tikzpicture}[scale=0.5]
\draw [line width=0.5] (0,0) -- (0,8) -- (8,8) -- (8,0) -- cycle;
\draw[dotted] (0,8)--(8,0);
\draw[rounded corners=1, line width=1]%
(0,8)--(4,8)--%
(4,7)--(6,7)--%
(6,5)--(7,5)--%
(7,4)--(8,4)--%
(8,0);%
\draw[rounded corners=1, line width=1, dotted]%
(0,8)--(0,4)--%
(1,4)--(1,2)--%
(3,2)--(3,1)--%
(4,1)--(4,0)--%
(8,0);%
\node[below left] at (4,8) {$1$};
\node[below left] at (3,7) {$1$};
\node[below left] at (2,6) {$1$};
\node[below left] at (1,5) {$1$};
%
\node[below left] at (6,7) {$1$};
\node[below left] at (5,6) {$1$};
\node[below left] at (4,5) {$1$};
\node[below left] at (3,4) {$1$};
\node[below left] at (2,3) {$1$};
%
\node[below left] at (7,5) {$1$};
\node[below left] at (6,4) {$1$};
\node[below left] at (5,3) {$1$};
\node[below left] at (4,2) {$1$};
%
\node[below left] at (8,4) {$1$};
\node[below left] at (7,3) {$1$};
\node[below left] at (6,2) {$1$};
\node[below left] at (5,1) {$1$};
\node[below left] at (4,7) {$-1$};
\node[below left] at (3,6) {$-1$};
\node[below left] at (2,5) {$-1$};
%
\node[below left] at (6,5) {$-1$};
\node[below left] at (5,4) {$-1$};
\node[below left] at (4,3) {$-1$};
%
\node[below left] at (7,4) {$-1$};
\node[below left] at (6,3) {$-1$};
\node[below left] at (5,2) {$-1$};
%
\foreach\x in {1,2,3,5,6,7,8}
\node[below left] at (\x,8) {$0$};
\foreach\x in {1,2,5,7,8}
\node[below left] at (\x,7) {$0$};
\foreach\x in {1,4,6,7,8}
\node[below left] at (\x,6) {$0$};
\foreach\x in {3,5,8}
\node[below left] at (\x,5) {$0$};
\foreach\x in {1,2,4}
\node[below left] at (\x,4) {$0$};
\foreach\x in {1,3,8}
\node[below left] at (\x,3) {$0$};
\foreach\x in {1,2,3,7,8}
\node[below left] at (\x,2) {$0$};
\foreach\x in {1,2,3,4,6,7,8}
\node[below left] at (\x,1) {$0$};
\end{tikzpicture}
\quad
\begin{tikzpicture}[scale=0.5]
\draw [line width=0.5] (0,0) -- (0,8) -- (8,8) -- (8,0) -- cycle;
\draw[dotted] (0,8)--(8,0);
\draw[rounded corners=1, line width=1]%
(0,8)--(4,8)--%
(4,7)--(6,7)--%
(6,5)--(7,5)--%
(7,4)--(8,4)--%
(8,0);%
\draw[rounded corners=1, line width=1, dotted]%
(0,8)--(0,4)--%
(1,4)--(1,2)--%
(3,2)--(3,1)--%
(4,1)--(4,0)--%
(8,0);%
\node[below left] at (4,8) {$+$};
\node[below left] at (3,7) {$+$};
\node[below left] at (2,6) {$+$};
\node[below left] at (1,5) {$+$};
%
\node[below left] at (6,7) {$+$};
\node[below left] at (5,6) {$+$};
\node[below left] at (4,5) {$+$};
\node[below left] at (3,4) {$+$};
\node[below left] at (2,3) {$+$};
%
\node[below left] at (7,5) {$+$};
\node[below left] at (6,4) {$+$};
\node[below left] at (5,3) {$+$};
\node[below left] at (4,2) {$+$};
%
\node[below left] at (8,4) {$+$};
\node[below left] at (7,3) {$+$};
\node[below left] at (6,2) {$+$};
\node[below left] at (5,1) {$+$};
\node[below left] at (4,7) {$-$};
\node[below left] at (3,6) {$-$};
\node[below left] at (2,5) {$-$};
%
\node[below left] at (6,5) {$-$};
\node[below left] at (5,4) {$-$};
\node[below left] at (4,3) {$-$};
%
\node[below left] at (7,4) {$-$};
\node[below left] at (6,3) {$-$};
\node[below left] at (5,2) {$-$};
\end{tikzpicture}
\caption{The image of A Dyck path}
\label{fig:exampleA}
\end{figure}
\section{A map on diagonally symmetric alternating sign matrices}
Because transposing a matrix preserves the X-ray, only diagonally
symmetric alternating sign matrices may be reconstructible from their
X-ray. We now present a map $\mathcal M$ on diagonally symmetric
alternating sign matrices that preserves the X-ray and is the
identity precisely on the matrices in the image of the map
$\mathcal A$ from the previous section. Let $A$ be a diagonally
symmetric alternating sign matrix, then $\mathcal M(A)$ is obtained
as follows, see Figure~\ref{fig:exampleM} for an example.
\begin{itemize}
\item Imagine a sun in the north-east, such that the $1$s in $A$ cast
shadows, and trace out a Dyck path by following the shadow line.
\item Reflect the entries of $A$ which are strictly south-west of the
entries just below the Dyck path through the subdiagonal.
\item Into each cell just south-west of a valley of the Dyck path
which is not on the subdiagonal and which contains a $0$, place a
$-1$, and place a $1$ in the cell reflected through the
subdiagonal.
\end{itemize}
Note that the Dyck path constructed in the first step returns to the
main diagonal exactly once for each direct summand of $A$, regarding
$A$ as a block diagonal matrix. Thus, the map $\mathcal M$ is such
that it can be applied to each direct summand of $A$ individually.
\begin{figure}[h]
\centering
\begin{tikzpicture}[scale=0.40]
\draw [line width=0.5] (0,0) -- (0,8) -- (8,8) -- (8,0) -- cycle;
\draw [fill=gray, opacity=0.3]
(0,7) -- (3,7) -- (3,6) -- (5,6) -- (5,4) -- (6,4) -- (6,3) --
(7,3) -- (7,0) -- (4,0) -- (4,1) -- (3,1) -- (3,2) -- (1,2) --
(1,4) -- (0,4) -- cycle;
\draw[dotted] (0,7)--(7,0);
\draw[rounded corners=1, line width=1]%
(0,8)--(4,8)--%
(4,7)--(6,7)--%
(6,5)--(7,5)--%
(7,4)--(8,4)--%
(8,0);%
\draw[rounded corners=1, line width=1, dotted]%
(0,8)--(0,4)--%
(1,4)--(1,2)--%
(3,2)--(3,1)--%
(4,1)--(4,0)--%
(8,0);%
%
\node[below left] at (2,7) {$+$};
%
\node[below left] at (4,8) {$+$};
\node[below left] at (1,5) {$+$};
%
\node[below left] at (6,7) {$+$};
\node[below left] at (5,6) {$+$};
\node[below left] at (3,4) {$+$};
\node[below left] at (2,3) {$+$};
%
\node[below left] at (7,5) {$+$};
\node[below left] at (4,2) {$+$};
%
\node[below left] at (8,4) {$+$};
\node[below left] at (7,3) {$+$};
\node[below left] at (6,2) {$+$};
\node[below left] at (5,1) {$+$};
\node[below left] at (4,7) {$-$};
\node[below left] at (2,5) {$-$};
%
\node[below left] at (7,4) {$-$};
\node[below left] at (6,3) {$-$};
\node[below left] at (5,2) {$-$};
%
\end{tikzpicture}
\quad\raisebox{50pt}{$\mapsto$}\quad
\begin{tikzpicture}[scale=0.40]
\draw [line width=0.5] (0,0) -- (0,8) -- (8,8) -- (8,0) -- cycle;
\draw [fill=gray, opacity=0.3]
(0,7) -- (3,7) -- (3,6) -- (5,6) -- (5,4) -- (6,4) -- (6,3) --
(7,3) -- (7,0) -- (4,0) -- (4,1) -- (3,1) -- (3,2) -- (1,2) --
(1,4) -- (0,4) -- (0,7);
\draw[dotted] (0,7)--(7,0);
\draw[rounded corners=1, line width=1]%
(0,8)--(4,8)--%
(4,7)--(6,7)--%
(6,5)--(7,5)--%
(7,4)--(8,4)--%
(8,0);%
\draw[rounded corners=1, line width=1, dotted]%
(0,8)--(0,4)--%
(1,4)--(1,2)--%
(3,2)--(3,1)--%
(4,1)--(4,0)--%
(8,0);%
%
\node[below left] at (1,6) {$+$};
%
\node[below left] at (4,8) {$+$};
\node[below left] at (3,7) {$+$};
%
\node[below left] at (6,7) {$+$};
\node[below left] at (5,6) {$+$};
\node[below left] at (4,5) {$+$};
\node[below left] at (2,3) {$+$};
%
\node[below left] at (7,5) {$+$};
\node[below left] at (6,4) {$+$};
%
\node[below left] at (8,4) {$+$};
\node[below left] at (7,3) {$+$};
\node[below left] at (6,2) {$+$};
\node[below left] at (5,1) {$+$};
\node[below left] at (4,7) {$-$};
\node[below left] at (3,6) {$-$};
%
\node[below left] at (7,4) {$-$};
\node[below left] at (6,3) {$-$};
\node[below left] at (5,2) {$-$};
\end{tikzpicture}
\quad\raisebox{50pt}{$\mapsto$}\quad
\begin{tikzpicture}[scale=0.40]
\draw [line width=0.5] (0,0) -- (0,8) -- (8,8) -- (8,0) -- cycle;
\draw[dotted] (0,7)--(7,0);
\draw[rounded corners=1, line width=1]%
(0,8)--(4,8)--%
(4,7)--(6,7)--%
(6,5)--(7,5)--%
(7,4)--(8,4)--%
(8,0);%
\draw[rounded corners=1, line width=1, dotted]%
(0,8)--(0,4)--%
(1,4)--(1,2)--%
(3,2)--(3,1)--%
(4,1)--(4,0)--%
(8,0);%
%
\node[below left] at (1,6) {$+$};
\node[below left, red] (A) at (6,5) {$-$};
\node[draw=black, circle, minimum size=12pt] at (A) {};
\node[below left, red] (B) at (3,2) {$+$};
\node[draw=black, circle, minimum size=12pt] at (B) {};
%
\node[below left] at (4,8) {$+$};
\node[below left] at (3,7) {$+$};
%
\node[below left] at (6,7) {$+$};
\node[below left] at (5,6) {$+$};
\node[below left] at (4,5) {$+$};
\node[below left] at (2,3) {$+$};
%
\node[below left] at (7,5) {$+$};
\node[below left] at (6,4) {$+$};
%
\node[below left] at (8,4) {$+$};
\node[below left] at (7,3) {$+$};
\node[below left] at (6,2) {$+$};
\node[below left] at (5,1) {$+$};
\node[below left] at (4,7) {$-$};
\node[below left] at (3,6) {$-$};
%
\node[below left] at (7,4) {$-$};
\node[below left] at (6,3) {$-$};
\node[below left] at (5,2) {$-$};
\end{tikzpicture}
\caption{A diagonally symmetric alternating sign matrix and its
image}
\label{fig:exampleM}
\end{figure}
\begin{lem}
The map $\mathcal M$, applied to a diagonally symmetric alternating
sign matrix, produces an alternating sign matrix.
\end{lem}
\begin{proof}
Let $A$ be a diagonally symmetric alternating sign matrix. Let us
call the region in $A$ symmetric with respect to the subdiagonal,
whose south-west border is the reflected Dyck path, the \Dfn{shade}
of $A$. This is the shaded region in Figure~\ref{fig:exampleM}.
Consider a column $c$ of $A$, and its reflection $r$ through the
subdiagonal. Thus, when $c$ is the first column, $r$ is the second
row of $A$.
Suppose first that the top most nonzero entry of $c$ and the right
most nonzero entry of $r$ within the shade of $A$ are both $1$.
This is the case when $c$ (or $r$) does not contain a peak of the Dyck
path, or the valley below (or to the left of) the peak contains a $-1$.
In this case, column $c$ and row $r$ of $\mathcal M(A)$ satisfy the
alternating sign matrix conditions, because after reflecting
through the subdiagonal the top most nonzero entry of all columns,
and the right most nonzero entry of all rows within the shade is
$1$.
Let us now consider the second scenario, where the top most nonzero
entry of $c$ within the shade of the original matrix $A$ is $-1$.
In the example of Figure~\ref{fig:exampleM}, this happens in the
sixth column.
In this case, the Dyck path must have a peak in this column. Let
$v$ be the cell just south-west of the valley below the peak. Note
that $v$ must contain a $0$.
The cell $v$ must be strictly above the diagonal, because otherwise
the reflection of the $-1$ below it through the main diagonal would
lie on or above the Dyck path. Thus, by definition of
$\mathcal M$, we place a $-1$ into the cell $v$. The effect of
this is that column $c$ of $\mathcal M(A)$ is alternating.
Furthermore, we place a $1$ in the cell $v'$ corresponding to $v$
reflected through the subdiagonal. This satisfies the alternating
sign matrix conditions, because after reflecting through the
subdiagonal, the row containing $v'$ begins with a $-1$.
\end{proof}
\begin{lem}
The map $\mathcal M$, applied to a diagonally symmetric alternating
sign matrix $A$, is the identity if and only if $A$ is in the image
of $\mathcal A$.
\end{lem}
\begin{proof}
If $A$ is in the image of $\mathcal A$, the shade of $A$ is
symmetric. Moreover, the cells just south-west of the valleys
which are above the subdiagonal all contain $-1$s. Thus,
$\mathcal M(A) = A$.
Otherwise, since $A$ is symmetric, and the shade of $A$ is
reflected through the subdiagonal, $\mathcal M(A)$ cannot be
symmetric.
\end{proof}
\section{Reconstructing the alternating sign matrix}
To complete the proof of the main theorem, we have to show the following:
\begin{lem}
The X-ray corresponding to an alternating sign matrix in the image
of $\mathcal A$ determines the matrix unambiguously.
\end{lem}
\begin{proof}
Consider the antidiagonal sums beginning at the north-west corner.
Suppose that the entries of the first $k$ antidiagonals are
uniquely determined by their X-rays $x_1/x_2/\dots/x_k$, and
suppose that $x_k\neq 0$, $x_{k+1}=\dots=x_{\ell-1}=0$ and
$x_{\ell}\neq 0$.
For simplicity, assume that $x_k > 0$. By hypothesis, the
alternating sign matrix then has the following form:
\begin{center}
\begin{tikzpicture}[scale=0.4]
\draw[dotted] (0,8)--(8,0);
\draw[rounded corners=1, line width=1]%
(0,8)--(4,8)--%
(4,7)--(6,7);%
\draw[rounded corners=1, line width=1, dotted]%
(0,8)--(0,4)--%
(1,4)--(1,2);%
\draw [fill=gray, opacity=0.3] (6,6)--(6,2)--(2,2)--cycle;
\draw [fill=red, opacity=0.3] (6,2)--(9,2)--(6,-1)--cycle;
\node[below left] at (4,8) {$1$};
\node[below left] at (3,7) {$1$};
\node[below left] at (2,6) {$1$};
\node[below left] at (1,5) {$1$};
\node[below left] at (4,7) {$-1$};
\node[below left] at (3,6) {$-1$};
\node[below left] at (2,5) {$-1$};
\node[below left] at (6,7) {$1$};
\node[below left] at (5,6) {$1$};
\node[below left] at (4,5) {$1$};
\node[below left] at (3,4) {$1$};
\node[below left] at (2,3) {$1$};
%
\foreach\x in {1,2,3,5,6,7}
\node[below left] at (\x,8) {$0$};
\foreach\x in {1,2,5}
\node[below left] at (\x,7) {$0$};
\foreach\x in {1,4}
\node[below left] at (\x,6) {$0$};
\foreach\x in {3}
\node[below left] at (\x,5) {$0$};
\foreach\x in {1,2}
\node[below left] at (\x,4) {$0$};
\foreach\x in {1}
\node[below left] at (\x,3) {$0$};
\foreach\x in {1}
\node[below left] at (\x,2) {$0$};
\end{tikzpicture}
\end{center}
Let us first note that there cannot be any nonzero entries on the
antidiagonals $k+1,\dots,\ell-1$, since all these have sum zero.
More precisely, suppose for the sake of contradiction that there is
such an antidiagonal and consider the first of these. Because
every row and every column of an alternating sign matrix must begin
with a $1$, this antidiagonal can have $-1$s only in the triangular
region south-east of the sequence of $1$s in the $k$-th
antidiagonal - shaded grey in the example above. However, there
cannot be any $1$s on the same antidiagonal, necessarily outside of
this triangular region: any such $1$ below the main diagonal would
be followed by another $1$ in the same column above it.
We now distinguish two cases: if $x_\ell < 0$, by hypothesis
$x_\ell$ is so large that all cells of the antidiagonal within the
triangular region defined above are filled with $-1$s. Thus, in
this case the entries on the $\ell$-th antidiagonal are also
uniquely determined by the antidiagonal sum.
On the other hand, if $x_\ell > 0$, by hypothesis $x_\ell$ is so
large that all cells of the antidiagonal that lie within the
triangular region shaded red in the example above are filled with
$1$s. Since there cannot be any $1$s on the same antidiagonal
outside of the red triangular region, also in this case the entries
of the $\ell$-th antidiagonal are uniquely determined by their sum.
\end{proof}
\printbibliography
\end{document}
|
1,116,691,497,563 | arxiv | \section{Introduction}
In this paper we deal with the problem of comparing the lifetimes of $k$-out-of-$n$ systems with respect to ageing properties, by relying on the theory of \textit{stochastic orders}, Shaked and Shanthikumar~\cite{shaked2007}. We recall that the lifetime of a $k$-out-of-$n$ system is represented by the waiting time until fewer than $k$ components remain functioning in a system of $n$ components. Within a probabilistic framework, if we assume that the lifetime of each component is distributed according to a common, or \textit{parent}, cumulative distribution function (CDF), say $F$, then the lifetime of the system is represented by the \textit{order statistic} $X_{k:n}$, corresponding to a random sample of size $n$ from $F$.
For this reason, stochastic comparisons of order statistics represent a major issue in reliability theory. Engineering is typically concerned with choosing the system which may provide the best performance, according to some characteristics. Similarly to most decision problems in other research fields (e.g., economics, finance, etc.), the ``best'' performance of a $k$-out-of-$n$ system is generally understood as i) larger \textit{magnitude}, to be understood as the tendency of one random variable (RV) to take larger values, and ii) smaller risk or dispersion, since lifetime predictability is always preferable in such a context.
In reliability, the main stochastic orders used for comparisons of order statistics are \textit{likelihood ratio order, hazard rate order, first-order stochastic dominance} (FSD), with regard to \textit{magnitude} problems, Lillo~\cite{lillo2001}, Shaked and Shanthikumar~\cite{shaked2007} or Kochar~\cite{kochar2012}, and \textit{convex transform order, star order, Lorenz order, dispersive order}, Arnold and Villase\~{n}or~\cite{arnold1991}, Arnold and Nagaraja~\cite{arnold1991exp}, Wilfling~\cite{wilfling1996c}, Kochar~\cite{kochar2006}, Kochar and Xu~\cite{kochar2014} or Wu etal.~\cite{wu2020}, to deal with dispersion characterizations.
Recently, Lando and Bertoli-Barsotti~\cite{lando2019} considered the problem of ranking order statistics via \textit{second-order stochastic dominance} (SSD), that is the most widely used stochastic order in areas such as economics, finance, decision science and management. As well known, SSD, also referred to as generalized Lorenz dominance, is a scale-dependent version of the Lorenz order, which enables comparisons of RVs in terms of both magnitude and dispersion, therefore combining aspects i) and ii) into a single preorder. In this paper, we focus on the derivation of SSD for order statistics, dealing with both the one-sample (same parent distribution) and the two-sample problems (different parent distributions). For technical reasons, most results in the aforementioned literature about stochastic comparisons of order statistics are obtained by imposing restrictive constraints on the parent's distribution shape, or by focusing on particular parametric families of parent distributions.
However, restrictive shape assumptions are rather inconsistent with modern nonparametric statistical approach, in which the parent distribution is supposed to be unknown and has to be estimated from the data, with no prior constraint on its mathematical form. To address this issue, we propose a general method to derive SSD conditions for order statistics, according to different assumptions on the parent distribution.
Let $F$ and $H$ be continuous CDFs.
Renaming the definition of the convex transform order of van Zwet~\cite{zwet1964}, we shall say that $F$ is $H^{-1}$--\textit{convex} iff $H^{-1} \circ F$ is convex.
For the one-sample problem, we propose a method to compare, with respect to SSD, the order statistics $X_{i:n}$ and $X_{j:m}$ with common parent distribution $F$, by assuming that $F$ is $H^{-1}$--convex w.r.t. some suitable function $H^{-1}$. By focusing on four convenient choices of $H^{-1}$, we determine four partially nested classes of parent distributions that are relevant in terms of reliability properties, among which we can mention the well known \textit{increasing failure rate} (IFR) class Barlow et al.~\cite{barlow1963}. Correspondingly, we obtain four different sets of conditions implying the SSD between the order statistics, that are expressed in terms of the ranks $i$, $j$ and the sample sizes $m$, $n$. Moreover, this approach to finding conditions implying the SSD order may be extended to the two-sample problem by assuming that the two parent distributions are ordered w.r.t. a fractional-degree stochastic dominance relation recently introduced by Lando and Bertoli-Barsotti~\cite{landodist}. This method provides a flexible framework for SSD comparisons according to the information available on the parent distribution's shape. Finally, to draw such information from data, we propose statistical tests to evaluate whether the parent distribution is $H^{-1}$--convex.
\section{Preliminaries}
\noindent We consider absolutely continuous RVs with finite means. Let us begin with some notations. Let $X$ be an RV with CDF $F_X$ and probability density function (PDF) $f_X$. For any ordering relation $\succ$ we shall write $X\succ Y$ or $F_X \succ F_Y$ interchangeably. Let $X_{k:n}$ the $k$-th order statistic corresponding to an i.i.d. random sample of size $n$ from $X\sim F_X$. It is well known that the CDF of $X_{k:n}$ is given by $F_{B} \circ F_X$, where $B\sim beta(k,n-k+1)$, Jones~\cite{jones2004}. Expressions are strongly simplified if we consider sample minima and maxima, whose CDFs reduce to $1-(1-F_X)^k$ and $F_X^k$, respectively. But, in general, investigating stochastic orders between order statistics is quite complicated, owing to the number of parameters and non-closed functional forms. We recall the basic definitions of FSD and SSD.
\begin{definition}\label{FSD}
We say that $X$ dominates $Y$ w.r.t. FSD and we write $X \ge_1 Y$ iff $F_{X} (x)\le F_{Y} (x)$, for every $x \in \mathbb{R}$. Equivalently, $X \ge_1 Y$ iff $E(g(X))\geq E(g(Y))$ for every increasing function $g$.
\end{definition}
\begin{definition}
We say that $X$ dominates $Y$ w.r.t. SSD and we write $X \ge_2 Y$ iff $\int_{-\infty}^{x}{F_{X}(t)dt}\leq \int_{-\infty}^{x}{F_{Y}(t)dt}$, for every $x\in \mathbb{R}$. Equivalently, $X \ge_2 Y$ iff $E(g(X))\geq E(g(Y))$ for every increasing concave function $g$.
\end{definition}
Intuitively, FSD represents preference for the RV with larger magnitude, as Definition~\ref{FSD} says that $X$ is less likely than $Y$ to take values in any left tail. On the other hand, SSD represents preference for the RV with larger magnitude or smaller dispersion: in particular, if $X$ dominates $Y$ w.r.t. SSD then $E(X)\geq E(Y)$, and, in case of equality, $\mathrm{Var}(X)\leq \mathrm{Var}(Y)$ and $\mathrm{\Gamma}(X)\leq \mathrm{\Gamma}(Y)$, where $\mathrm{\Gamma}$ is the Gini coefficient, Fishburn~\cite{fishburn1980} or Muliere and Scarsini~\cite{muliere1989}.
As discussed in the literature, in many practical situations it is convenient to use orders that interpolate FSD and SSD, defining a family of fractional-degree dominance relations between these two. We refer to Fishburn~\cite{fishburn1976}, M\"uller adn Scarsini~\cite{muller2017} or Huang et al.~\cite{huang2020} and, in particular, to Lando and Bertoli-Barsotti~\cite{landodist}, who achieved this objective by comparing sample maxima through SSD for a fixed sample size, that is, $X_{k:k}\geq_2 Y_{k:k}$, for some positive integer $k$. For technical reasons, such an order proves useful in deriving SSD conditions for order statistics in the two-sample problem, as we show in Section 3. Moreover, \cite{landodist} proved that $X \ge_{1} Y$ iff $X_{k:k}\geq_2 Y_{k:k}$, for every positive integer $k$, (whereas, clearly, $X \ge_{2} Y$ iff $X_{1:1}\geq_2 Y_{1:1}$). In general, assuming $k\geq h\geq 1$, the following relations hold
\begin{equation}
X \geq_{1} Y\quad\Rightarrow\quad X_{k:k}\geq_2 Y_{k:k} \quad\Rightarrow\quad X_{h:h}\geq_2 Y_{h:h}\quad\Rightarrow\quad X \geq_{2} Y.
\end{equation}
The implications above mean that, if $X\geq_2 Y$ is satisfied, the relation $X_{h:h}\geq_2 Y_{h:h}$ may hold for every $h$ up to some value $k$. Moreover, as larger values of $h$ correspond to stronger orders, this justifies the definition of a fractional-degree stochastic dominance relation, introduced in \cite{landodist},
\begin{equation}
X\geq_{1+\frac{1}{k}} Y \quad\Leftrightarrow\quad
k=\sup\{h\geq 1: X_{h:h}\geq_2 Y_{h:h}\}.
\end{equation}
For instance, $k=10$ gives $X\geq_{1.1} Y$, meaning that
$X_{h:h}\geq_2 Y_{h:h}$ for $h\leq10$, and $X_{h:h}\not\geq_2 Y_{h:h}$ for $h\geq 11$.
The conditions for SSD are particularly simple if CDFs are single-crossing or, alternatively, if PDFs are double-crossing.
Let us denote the number of sign changes of a function, $u$, defined on an interval, $I$, with
\begin{center}$S^{-}(u)=\sup{\{S^{-}[u(x_{1}),\ldots
,u(x_{\ell})], x_{1}<x_{2}<\ldots<x_{\ell}\}}$,
\end{center}
where $\ell<\infty$, $x_{i}\in I$, $i=1,\ldots,\ell$, and $S^{-}[y_{1},\ldots ,y_{\ell}]$ is the number of sign changes of
the sequence, $y_{1},\ldots ,y_{\ell}$, where the zero terms are
omitted \cite{shaked1982}.
Sufficient conditions for SSD can be derived as follows.
\begin{lemma}\label{LEM1}
If $S^{-}(F_X-F_Y)\le 1$ and the sign sequence
starts with $-$, then $X\ge _{2} Y$ iff $E(X)\ge
E(Y)$ (Hanoch and Levy~\cite{hanoch1969}).
If $S^{-}(f_X-f_Y)\le 2$ and the sign sequence
starts with $-$, then $X\ge _{2} Y$ iff $E(X)\ge
E(Y)$ (Ramos et al.~\cite{ramos2000}).
\end{lemma}
Throughout the paper we shall frequently use the following preservation result.
\begin{lemma}\label{LEM2}
Let $\phi$ be an increasing and concave function. If $X \geq_2 Y$ then $\phi(X) \geq_2 \phi(Y)$.
\end{lemma}
\begin{proof}
For every increasing concave function $g$, the functional composition $g \!\circ\! \phi$ is increasing concave as well. Then, $X \geq_2 Y$ implies $E(g \!\circ\! \phi(X))\geq E(g \!\circ\! \phi (Y))$, for every increasing concave $g$, that is, $\phi(X) \geq_2 \phi(Y)$, by characterization of SSD.
\end{proof}
We shall also need the following orders. The first one is due to Chan et al.~\cite{chan1990}, whereas the latter is ascribable to van Zwet~\cite{zwet1964}.
\begin{definition}
\label{def:twoorders}
Let $H,G$ be a pair of CDFs.
\begin{enumerate}
\item
Let $H$ be absolutely continuous w.r.t. $G$. We say that $H$ is \textit{more convex}
than $G$ and write $H \geq^*_c G$ iff $H \!\circ\! G^{-1}$ is convex.
\item
We say that $H$ dominates $G$ w.r.t. the \textit{convex transform order} and write $H \geq_c G$ iff $H^{-1} \!\circ\! G$ is convex on the support of $G$ or, equivalently, $G$ is $H^{-1}$--convex.
\end{enumerate}
\end{definition}
In fact, the notion of being more convex mentioned above is a translation into a geometrical interpretation of the relative convexity introduced by Hardy et al.~\cite{HLP52}.
The two orders in Definition~\ref{def:twoorders} are different but closely related: if $H$ and $G$ are defined on the unit interval, $H \geq_c F$ iff $H^{-1} \geq^*_c G^{-1}$. In turn, it can be easily seen that $>^*_c$ is equivalent to the likelihood ratio order, Chan et al.~\cite{chan1990}. Moreover, if $F_X\geq_c F_Y$, then Y is said to be a \textit{convex transform} of $X$, since $F_Y^{-1}\!\circ\! F_X(X)$ has the same distribution as $Y$.
Finally, we present two technical lemmas which will be useful in the next section.
\begin{lemma}\label{LEM3}
Let $B_1\sim beta(a_1,b_1)$ and $B_2\sim beta(a_2,b_2)$.
\begin{enumerate}
\item If $a_1\geq a_2$ and $b_1\leq b_2$ then $\frac{f_{B_1}(x)}{f_{B_2}(x)}$ is increasing.
\item If $a_1\geq a_2$ then $S^{-}(f_{B_1}-f_{B_2})\leq 2$, where the sign sequence starts with $-$.
\end{enumerate}
\end{lemma}
\begin{proof}
Let $\ell(x)=\frac{f_{B_1}(x)}{f_{B_2}(x)}=\beta x^{a_1-a_2}(1-x)^{b_1-b_2}$, where $\beta=\frac{B(a_2,b_2)}{B(a_1,b_1)}$. Differentiating we get $\ell'(x)= \beta x^{a_1-a_2-1}(1-x)^{b_1-b_2-1}((a_2-a_1+b_2-b_1)x+a_1-a_2)$. Thus, $S^{-}(\ell'(x))=S^{-}((a_2-a_1+b_2-b_1)x+a_1-a_2).$
\begin{enumerate}
\item
If $a_1\geq a_2$ and $b_1\leq b_2$, then, $\ell'(x)\geq 0$, meaning that $\ell$ is increasing.
\item
If $a_1\geq a_2$, then $S^{-}(\ell'(x))\leq 1$ and the sign sequence starts with +, meaning that $\ell$ is either increasing or increasing and then decreasing. Consequently, the conclusion follows from $S^{-}(f_{B_1}-f_{B_2})=S^{-}(\ell-1)$.
\end{enumerate}
\end{proof}
\begin{lemma}\label{P2}
Let $H^{-1}$ be a quantile function and $B_{i,n}\sim beta(i,n-i+1)$, where
$1\leq i \leq n$.
\begin{enumerate}
\item If $H^{-1}(p)=p$ then $E(H^{-1} \!\circ\! B_{i,n})=\frac{i}{n+1}$.
\item If $H^{-1}(p)=\log{\frac{p}{1-p}}$ then $E(H^{-1} \!\circ\! B_{i,n})=\psi(i) - \psi(n-i + 1)$, where $\psi$ is the digamma function.
\item If $H^{-1}(p)=-\log{(1-p)}$ then $E(H^{-1} \!\circ\! B_{i,n})=\sum_{k=n-i+1}^{n}{\frac{1}{k}}$.
\item If $H^{-1}(p)=\frac{p}{1-p}$ then $E(H^{-1} \!\circ\! B_{i,n})=\frac{i}{n-i}$.
\end{enumerate}
\end{lemma}
\begin{proof}
Expression (1) is trivial. Expression (2) can be found in Birnbaum and Dudman~\cite{birnbaum}. Expression (3) is given in Arnold and Nagaraja~\cite{arnold1991exp}. As for (4), $H^{-1} \!\circ\! B_{i,n}$ is a beta distribution of the second type and the expression of the mean is straightforward.
\end{proof}
\section{Main results}
In this section, we enable SSD comparisons of order statistics based on different decompositions of the CDF of $X_{k:n}$, where the decomposition chosen determines the range of application of the corresponding SSD conditions, as we establish in the following theorems.
\begin{theorem}\label{THM1}
\noindent Let $X \sim F$, $B_1\sim beta(i,n-i+1)$ and $B_2\sim beta(j,m-j+1)$. Let $H$ be a CDF. If $F$ is $H^{-1}$--convex, $i\geq j$ and $E(H^{-1} \!\circ\! B_1)\geq E(H^{-1} \!\circ\! B_2)$, then $X_{i:n}{\ge }_2 X_{j:m}$.
\end{theorem}
\begin{proof}
From Lemma~\ref{LEM3}, it follows that $i\geq j$ implies $S^{-}(f_{B_1}-f_{B_2})\le 2$, where the sign sequence starts with $-$. Consequently $S^{-}(F_{B_1}-F_{B_2})\leq 1$, where the sign sequence starts also with $-$ \cite{ramos2000}. Similarly, since $H$ is increasing we have $S^{-}(F_{B_1}\!\circ\! H-F_{B_2}\!\circ\! H)\le 1$, where $F_{B_i}\!\circ\! H$ is the CDF of $H^{-1} \!\circ\! B_i$, for $i=1,2$. Taking into account that $i\geq j$ and $E(H^{-1} \!\circ\! B_1)\geq E(H^{-1} \!\circ\! B_2)$, it follows from Lemma~\ref{LEM1} that $H^{-1} \!\circ\! B_1\geq_2 H^{-1} \!\circ\! B_2$. Since $H^{-1} \!\circ\! F$ is convex, $F^{-1} \!\circ\! H$ is concave, Lemma~\ref{LEM2} yields
\begin{equation}\label{decomposition}
F^{-1} \!\circ\! H \!\circ\! H^{-1} \!\circ\! B_1 \geq_2 F^{-1} \!\circ\! H \!\circ\! H^{-1} \!\circ\! B_2,
\end{equation}
where the RVs $F^{-1} \!\circ\! B_i$, for $i=1,2$, have CDFs $F_{B_i} \!\circ\! F$, meaning that $X_{i:n}{\ge }_2 X_{j:m}$.
\end{proof}
As different choices of $H$ may lead to different conditions for SSD, it is natural to wonder whether ordered choices of $H$ (in the sense of $\geq_c$) yield ordered (i.e., stronger/weaker) SSD conditions. The next theorem answers to this question.
\begin{theorem}\label{THM2}
Let $X \sim F$, $B_1\sim beta(i,n-i+1)$ and $B_2\sim beta(j,m-j+1)$, where $i\geq j$. Let $H,G$ be a pair of CDFs such that $H \geq_c G$.
\begin{enumerate}
\item $F$ is $G^{-1}$--convex $\quad\Rightarrow\quad$ $F$ is $H^{-1}$--convex.
\item $E(H^{-1} \!\circ\! B_1)\geq E(H^{-1} \!\circ\! B_2)$ $\quad\Rightarrow\quad$ $E(G^{-1} \!\circ\! B_1)\geq E(G^{-1} \!\circ\! B_2)$.
\end{enumerate}
\end{theorem}
\begin{proof}
\begin{enumerate}
\item
The implication is a straightforward consequence of the transitivity of the convex transform order $\geq_c$. In fact, since $H^{-1} \!\circ\! G$ and $G^{-1} \!\circ\! F$ are convex, also the composition $H^{-1} \!\circ\! G\!\circ\! G^{-1} \!\circ\! F=H^{-1}\!\circ\! F$ is convex.
\item
The implication follows from the single-crossing argument of Lemma~\ref{LEM1}, that is, $i \geq j$ and $E(H^{-1} \!\circ\! B_1)\geq E(H^{-1} \!\circ\! B_2)$ imply $H^{-1} \!\circ\! B_1\geq_2 H^{-1} \!\circ\! B_2$. Taking into account that $G^{-1} \!\circ\! H$ is concave, Lemma~\ref{LEM2} yields $G^{-1} \!\circ\! B_1\geq_2 G^{-1} \!\circ\! B_2$ and, in particular, $E(G^{-1} \!\circ\! B_1)\geq E(G^{-1} \!\circ\! B_2)$.
\end{enumerate}
\end{proof}
In spite of its simplicity, Theorem~\ref{THM2} provides many useful ways to apply Theorem~\ref{THM1}. The general concept can be summarized as follows. Ideally, we wish to be able to compare as many pairs $(X_{i:n}$,$X_{j:m})$ as possible, according to our knowledge of the parent distribution. If only partial information about the $F$ is available, we can check its $H^{-1}$--convexity w.r.t. some suitable $H^{-1}$ and then apply Theorem~\ref{THM2}. Clearly, the more convex $H^{-1}$ is (in the sense of Definition~\ref{def:twoorders}), the more likely we can apply the method, in that we might choose $H$ so that basically every distribution satisfies $H \geq_c F$. On the other hand, choosing an $H^{-1}$ with a higher degree of convexity yields weaker conditions on the parent distribution but, at the same time, stronger conditions on $i,j,n,m$. In other words, the larger the set of comparable families, the smaller the set of comparable order statistics, and vice versa. As a limiting case, we may consider the situation in which $F$ is assumed to be known, which is clearly the most restrictive condition. However, in this case we can directly compute the expectations of the order statistics (at least numerically) and check whether $i \geq j$ and $E(X_{i:n}){\geq} E(X_{j:m})$: this enables us to rank the largest possible set of pairs $(X_{i:n}$,$X_{j:m})$.
We focus on four partially ordered choices of $H$.
\begin{enumerate}
\item Uniform (trivial case): $H(x)=x$, for $x\in[0,1]$, and $H^{-1}(p)=p$, for $p\in[0,1]$.
\item Logistic: $H(x)=\frac{1}{1 + \mathrm{e}^{-x}}$, for $x\in(-\infty,\infty)$, and $H^{-1}(p)=\log{\frac{p}{1-p}}$, for $p\in(0,1)$ (log-odds, or logit function).
\item Exponential: $H(x)=1 - \mathrm{e}^{-x}$, for $x\in[0,\infty)$, and $H^{-1}(p)=-\log{(1-p)}$, for $p\in[0,1)$.
\item Log-logistic: $H(x)=\frac{x}{1+x}$, for $x\in[0,\infty)$, and $H^{-1}(p)={\frac{p}{1-p}}$, for $p\in[0,1)$ (odds function).
\end{enumerate}
Correspondingly, the sets of $H^{-1}$--convex distributions, that is, those satisfying $H\geq_c F$, determine four classes, defined as follows.
\begin{definition}\label{classes} Let $F$ be a CDF. We say that:
\begin{enumerate}
\item $F\in \mathcal{F}_C$ iff $F$ is convex.
\item $F\in \mathcal{F}_{CL}$ iff $F$ is \textit{logit--convex}, i.e., $\log\frac{F}{1-F}$ is convex.
\item $F\in \mathcal{F}_{IFR}$ iff $-\log(1-F)$ is convex.
\item $F\in \mathcal{F}_{CO}$ iff $F$ is \textit{odds--convex}, i.e., $\frac{F}{1-F}$ is convex.
\end{enumerate}
\end{definition}
Let $X\sim F$ represent the failure time. All the four classes defined above are of interest in terms of reliability properties:
\begin{description}
\item[$\mathcal{F}_{IFR}$] is referred to as the IFR class as $-\log(1-F)$, namely, the \textit{hazard function} of $F$, is convex iff $r(x)=\frac{f(x)}{1-F(x)}=\lim_{\Delta x \rightarrow 0} {\frac{P(X\in(x,x+\Delta x]|X>x)}{\Delta x}} $, namely, the \textit{failure rate} of $F$, is increasing. $\mathcal{F}_{IFR}$ is an important class in reliability theory and contains many relevant models (see Shaked and Shanthikumar~\cite{shaked2007} and references therein).
\item[$\mathcal{F}_C$] contains just distributions with bounded support, as $F\in \mathcal{F}_C$ iff the corresponding PDF, $f$, is increasing. Moreover, the probability of failure within a fixed-width interval increases with time, i.e., the function $P(X\in(x,x+\Delta])$ is increasing in $x$ for every positive $\Delta$.
\item[$\mathcal{F}_{CL}$] contains distributions with unbounded support of the form $(-\infty, c)$, where $c\leq +\infty$ (as the limit of logit function at $0$ is $-\infty$). $F\in \mathcal{F}_{CL}$ iff the \textit{log-odds rate}, that is, the ratio between the failure rate and the CDF, $\frac{r}{F}$, is increasing. Equivalently, note that $\frac{r}{F}=r+r^*$, where $r^*=\frac{f}{F}$ is the \textit{reversed failure rate} of $F$. This class has been studied by Zimmer et al.~\cite{zimmer1998}, Wang et al.~\cite{wang2003}, Sankaran and Jayakumar~\cite{sankaran2008} or Navarro et al.~\cite{navarro2008}.
\item[$\mathcal{F}_{CO}$] is characterized by the convexity of the odds for failure, that is, the ratio between failure and survival probability, where it can be seen that
\begin{center}
$\frac{F(x)}{1-F(x)}=\frac{P(X\in(x,x+\Delta x]|X>x)}{P(X\in(x-\Delta x,x]|X\leq x)},$
\end{center}
for arbitrarily small $\Delta x>0$, Kirmani and Gupta~\cite{kirmani2001}. Equivalently, $F\in \mathcal{F}_{CO}$ iff the ratio between the failure rate and the survival function, $\frac{r}{1-F}$, is increasing. As shown in the sequel, $\mathcal{F}_{CO}$ is the widest class considered, in that it contains all the classes above and also some heavy tailed families.
\end{description}
The relations between these four classes can be derived straightforwardly. First, it can be readily seen that $F$ convex implies $-\log{(1-F)}$ convex. In turn, $\frac{r}{F}$ increasing implies $r$ increasing. Finally, $r$ increasing implies $\frac{r}{1-F}$ increasing. This can be summarized as follows
\begin{equation}\label{class}
\mathcal{F}_C\subset\mathcal{F}_{IFR}\subset\mathcal{F}_{CO}\qquad\text{and}\qquad \mathcal{F}_{CL}\subset\mathcal{F}_{IFR}\subset\mathcal{F}_{CO},
\end{equation}
whereas clearly $\mathcal{F}_C\cap \mathcal{F}_{CL}=\emptyset$, because of the different support assumptions.
The classifications of some basic models are given it Table 1.
\begin{table}[h]
\caption{Different popular distributions and corresponding convexity conditions.}
{\scriptsize\begin{tabular}{|l|l|c|c|c|c|c|c|} \hline
\textbf{Distribution} &\multicolumn{1}{c|}{\textbf{CDF}}&\textbf{Parameters}&\textbf{Support}& $\boldsymbol{\mathcal{F}_C}$ & $\boldsymbol{\mathcal{F}_{CL}}$ & $\boldsymbol{\mathcal{F}_{IFR}}$&$\boldsymbol{\mathcal{F}_{CO}}$ \\ \hline
Uniform & $\displaystyle\frac{x-a}{b-a}$&$b>a$ &$[a,b]$& yes&no&yes&yes \\ \hline
Power function &$\displaystyle{\left(\frac{x}{b}\right)}^a$&$a,b>0$&$[0,b]$& yes &no&yes &yes \\
& & & & ($a\geq 1$)& & ($a\geq 1$)&($a\geq 1$) \\ \hline
Logistic & $\displaystyle\frac{\mathrm{1}}{\mathrm{1+}{\mathrm{exp} \left(\frac{\mu -x}{\sigma }\right)\ }}$&$\sigma >0$&$\mathbb{R}$ & no&yes&yes&yes \\ \hline
Gumbel& $\displaystyle1-\exp{(-\mathrm{e}^{\frac{x-\mu}{\sigma }} )}$&$\sigma >0$&$\mathbb{R}$ & no&yes&yes&yes \\ \hline
Exponential & $\displaystyle1-\mathrm{e}^{-ax}$&$a>0$&$[0,\infty)$ & no&no&yes&yes \\ \hline
Normal & $\displaystyle\frac{1}{2}\mathrm{erf}\left(\frac{x-\mu}{\sqrt{2}\sigma} \right)$ & $\sigma >0$&$\mathbb{R}$ & no&no&yes&yes \\ \hline
Beta& $\displaystyle\int_0^x{\frac{(1 - t)^{ b-1} t^{a-1}}{B(a,b)}}dt$&$a,b>0$&$[0,1]$ & yes &no&yes &yes \\
& & & & ($a,b\leq 1$) & & ($a\geq1$) & ($a\geq 1$)\\
\hline
Gamma& $\displaystyle\int_0^x{\frac{\mathrm{e}^{-x/b} x^{a-1}}{b^{a} \Gamma(a)}}dt$&$a,b>0$&$[0,\infty)$ & no&no&yes &yes \\
& & & & & & ($a\geq1$)& ($a\geq1$)\\ \hline
Weibull & $\displaystyle1-\exp{-(\frac{x}{b})^{a}}$&$a,b>0$&$[0,\infty)$& no&no&yes &yes \\
& & & & & & ($a\geq1$)& ($a\geq1$)\\\hline
Cauchy & $\displaystyle\frac{1}{2}+\frac{1}{\pi}\arctan{\frac{x-\mu}{\sigma }}$ & $\sigma>0$&$\mathbb{R}$ & no&no&no&yes \\ \hline
Lognormal & $\displaystyle\frac{1}{2}\mathrm{erf}\left(\frac{\log{x}-\mu}{\sqrt{2}\sigma} \right)$ & $\sigma>0$&$[0,\infty) $& no&no&no&yes \\ \hline
Log-logistic & $\displaystyle\frac{1}{1+{(x/b)}^{-a}}$&$a,b>0$&$(0,\infty)$ & no&no&no&yes \\
& & & & & & & ($a\geq 1$) \\ \hline
Pareto & $\displaystyle1-\left(\frac{b}{x}\right)^{a}$&$a,b>0$&$(b,\infty)$ & no&no&no&yes \\
& & & & & & & ($a\geq 1$)\\ \hline
\end{tabular}}
\end{table}
According to the classification of Definition~\ref{classes}, we determine four methods to derive SSD, in the one-sample and in the two-sample problems, as stated in the following corollaries. Note that part (4) of Corollary~\ref{COR1} is already proved in Lando and Bertoli-Barsotti~\cite{lando2019}.
\begin{corollary}\label{COR1}
Let $X \sim F$. If $i \geq j$, the following conditions imply $X_{i:n}{\ge }_2 X_{j:m}$
\begin{enumerate}
\item
$F\in\mathcal{F}_C$ and $\frac{i}{n+1} \geq \frac{j}{m+1}$.
\item
$F\in\mathcal{F}_{CL}$ and $\psi(i) - \psi(n - i + 1)\geq \psi(j) - \psi(m - j+ 1)$, where $\psi$ is the digamma function.
\item
$F\in\mathcal{F}_{IFR}$ and $\sum_{k=n-i+1}^{n}{\frac{1}{k}}\geq \sum_{k=m-j+1}^{m}{\frac{1}{k}}$.
\item
$F\in\mathcal{F}_{CO}$ and $i\geq j$ and $\frac{i}{n} \geq \frac{j}{m}$.\\
\end{enumerate}
\end{corollary}
\begin{proof}
As usual, let $B_1\sim beta(i,n-i+1)$ and $B_2\sim beta(j,m-j+1)$. The results can be proved by repeated application of Theorem~\ref{THM1}, each one based on a different choice for the function $H$ in
(\ref{decomposition}), and, for each case, deriving the expressions of $E(H^{-1} \!\circ\! B_1)$ and $E(H^{-1} \!\circ\! B_2)$ through Lemma~\ref{P2}. According to Definition~\ref{classes}, each of the stated four cases correspond to choosing $H$ as the CDF of the uniform, logistic, exponential and log-logistic, respectively.
\end{proof}
Corollary~\ref{COR1} yields SSD between order statistics by imposing conditions both on i) the shape of the parent distribution (different classes) and ii) the ranks and the sample sizes, $i,j,n,m$. As for i), the relations between different classes are depicted in \eqref{class}. On what regards ii), if $i\geq j$ the following implications follow straightforwardly from Theorem~\ref{THM2}:
\begin{eqnarray*}
\frac{i}{n}\geq\frac{j}{m} & \quad\Rightarrow\quad & \sum_{k=n-i+1}^{n}{\frac{1}{k}}\geq \sum_{k=m-j+1}^{m}{\frac{1}{k}};\\
\sum_{k=n-i+1}^{n}{\frac{1}{k}}\geq \sum_{k=m-j+1}^{m}{\frac{1}{k}} & \quad\Rightarrow\quad & \frac{i}{n+1} \geq \frac{j}{m+1};\\
\sum_{k=n-i+1}^{n}{\frac{1}{k}}\geq \sum_{k=m-j+1}^{m}{\frac{1}{k}} & \quad\Rightarrow\quad & \psi(i) - \psi(n - i + 1)\geq \psi(j) - \psi(m - j+ 1).
\end{eqnarray*}
Corollary~\ref{COR1} is useful in a nonparametric context, in which the functional form of $F$ is supposed to be unknown, but some general assumptions on its shape can be made, which can be verified by means of statistical testing, as we discuss in Section~4. Corollary \ref{COR1} can be used \textit{a fortiori} in a parametric context, if the parameters are supposed to be unknown. Finally, if the exact form of $F$ is known (that is a quite unrealistic assumption in statistics), we can directly compare $E(X_{i:n})$ and $E(X_{j:m})$. The next example illustrates some possible applications of Corollary \ref{COR1}.
\begin{example}
Let $F\in \mathcal{F}_{CO}$. Take for instance $n = 200, j = 43, m = 44$. Suppose we need to determine the smallest value of $i$ such that $X_{i:n}{\ge }_2X_{j:m}$. Corollary \ref{COR1} yields $X_{i:200}{\ge }_2X_{43:44}$ for $i\geq\left\lceil {\mathrm{max} \{j,\frac{nj}{m}\}\ }\right\rceil =196$, where $\left\lceil \bullet \right\rceil $ denotes the ceiling function, whereas the values $i=194,i=195$ are not sufficient to guarantee SSD. Now, assume we have additional information about the parent distribution; say $F\in \mathcal{F}_{IFR}$. Since $\sum_{k=l}^{200}{\frac{1}{k}}$ is decreasing in $l$, it is easy to check that $ \sum_{k=7}^{200}{\frac{1}{k}}\geq \sum_{k=2}^{44}{\frac{1}{k}}> \sum_{k=8}^{200}{\frac{1}{k}}$, meaning that $X_{i:200}{\ge }_2X_{43:44}$, for every $i\geq 194$ while we cannot ensure $X_{i:200}{\ge }_2X_{43:44}$ for $i\leq 193$. In a parametric context, if we assume that $F$ is a Gamma distribution with unknown shape parameter $a\geq 1$ and unknown scale parameter $b$, we know that $X_{i:200}{\ge }_2X_{43:44}$ for $i\geq 194$, as $F\in \mathcal{F}_{IFR}$.
Moreover, if we assume that the parameters are known, e.g. $a=b=2$, we can compute the expectations of the order statistics and surprisingly, we obtain again that SSD holds just for $i\geq 194$, meaning that strong additional assumptions on the parent distribution do not necessarily weaken the conditions on $i$, obtained through Corollary \ref{COR1}.
\end{example}
Let $X_1,\ldots ,X_n$ denote a sample of i.i.d. RVs from an RV $X$ and $Y_1,\ldots ,Y_m$ denote a sample of i.i.d. RVs from another RV $Y$. The following corollary enables the determination of the sample sizes $n\ \mathrm{and}\ m$ and the ranks $i$ and $j$ such that $X_{i:n}{\geq}_2Y_{j:m}$ by introducing an extra condition on $X$ and $Y$. In particular, we need that $X$ dominates $Y$ w.r.t. an order which is easy to verify and is stronger than SSD, namely $X\geq_{1+\frac{1}{k}} Y$, where $k\geq i$ (\textit{a fortiori}, FSD is clearly sufficient).
\begin{corollary}\label{COR2}
Assume that $i\geq j$ and $X\geq_{1+\frac{1}{k}} Y$, where $k\geq i$. Each one of the following conditions imply $X_{i:n}{\geq}_2Y_{j:m}$.
\begin{enumerate}
\item
$F_Y\in\mathcal{F}_C$ and $\frac{i}{n+1} \geq \frac{j}{m+1}$.
\item
$F_Y\in\mathcal{F}_{CL}$ and $\psi(i) - \psi(n - i + 1)\geq \psi(j) - \psi(m - j+ 1)$.
\item
$F_Y\in\mathcal{F}_{IFR}$ and $\sum_{k=n-i+1}^{n}{\frac{1}{k}}\geq \sum_{k=m-j+1}^{m}{\frac{1}{k}}$.
\item
$F_Y\in\mathcal{F}_{CO}$ and $i\geq j$ and $\frac{i}{n} \geq \frac{j}{m}$.\\
\end{enumerate}
\end{corollary}
\begin{proof}
Let $B_1\sim beta(i,n-i+1)$ and $B_2\sim beta(j,m-j+1)$. First, we prove that $i\geq j$ and $n-i\leq m-j$ implies $F_{B_1}\geq^*_c F_{B_2}$. Indeed, $F_{B_1}\geq^*_c F_{B_2}$ iff the likelihood ratio $\ell(x)=\frac{f_{B_1}(x)}{f_{B_2}(x)}$ is increasing, Chan et al.~\cite{chan1990}, which is implied by $i\geq j$ and $n-i\leq m-j$ (see Lemma~\ref{LEM3}).
Condition $X\geq_{1+\frac{1}{k}} Y$ is equivalent to $\int_0^x{(F_X (t))^k}\leq \int_0^x{(F_Y (t))^k}$, for every $x\geq 0$. Therefore, if $i\leq k$, then $P_k>_c^* F_{B_1}$, where $P_k$ is a CDF such that $P_k(t)=t^k$ on the support $[0,1]$. Hence, we apply Theorem 1 of Lando and Bertoli-Barsotti~\cite{landodist}, which establishes that $X_{k:k}\geq_2 Y_{k:k}\Rightarrow X_{i:n}\geq_2 Y_{i:n}$, for every $i\leq k$. Now, if $ i\geq j$ and if any of the conditions 1., 2., 3. or 4. hold, we can apply Corollary~\ref{COR1}, which yields $Y_{i:n}\geq_2 Y_{j:m}$, and the conclusion follows by transitivity.
\end{proof}
Basically, we can compare order statistics with different parent distributions according to the strength of the dominance relation between them. Such strength, determined by $k$, imposes constraints on $i$ ($i \leq k$). FSD enables the comparisons for every value of $i \geq j$ but, on the other hand, it is the strongest order. Put otherwise, all sets of conditions are well balanced: if we relax some constraints, we need to compensate by strengthening some of the others.
Similarly to Corollary~\ref{COR1}, Corollary~\ref{COR2} is suitable for those situations in which the parent distributions are supposed to be unknown: in such cases, the dominance relation between them can be tested nonparametrically. Tests for stochastic dominance of degree $1+1/k$ may be obtained by readapting SSD tests, using the relation $F_X\geq_{1+\frac{1}{k}}F_Y\Leftrightarrow F^k_X\geq_{2}F^k_Y$, however, this is beyond the scope of our paper. In any case, we can always rely on existing tests for FSD.
\begin{example}
Let $X$ be a Dagum RV with CDF $F_X(x)=(1 + \frac{27}{x^3})^{-2}$, for $x>0$ and let $Y$ be a log-logistic RV with parameters $a=b=2$. First we determine the degree of the dominance relation between $X$ and $Y$. By studying the function $F_X-F_Y$, we find that $S^{-}(F_X-F_Y)\le 1$ and the sign sequence starts with $-$ (the two CDFs cross at $x\approx 13.57$). Since the power function is increasing, we have also $S^{-}(F_X^k-F_Y^k)\le 1$ and the sign sequence starts with $-$, so that $X\geq_{1+\frac{1}{k}} Y$ iff $E(X_{k:k})\geq E(Y_{k:k})$, where $E(X_{k:k})=\frac{- \Gamma(-\frac13)\Gamma(\frac13 + 2 k)}{\Gamma(2 k)}$ and $E(Y_{k:k})=\frac{2 \sqrt{\pi} \Gamma(\frac12 + k)}{\Gamma(k)}$, Lin~\cite{lin2000}. Therefore, we obtain $E(X_{k:k})\geq E(Y_{k:k})$ iff $k\leq9$, that is, $X\geq_{1+\frac{1}{9}} Y$. Using the information about $F_Y$, that is $F_Y\in\mathcal{F}_{CO}$ (note that $F_Y$ belongs only to this class), it follows from Corollary~\ref{COR2} that $X_{i:n}{\ge }_2Y_{j:m}$ for $9\geq i\geq j$ and $\frac{i}{n}\geq\frac{j}{m}$. For instance, letting $n=30$, $j=4$, $m=25$, we get $X_{i:30}{\geq}_2Y_{4:25}$ for $5\leq i\leq 9$, although we cannot guarantee SSD for $i>9$. The constraint $i\leq 9$ can be removed only by strengthening the dominance relation between the parent distributions. For instance, if we take $Z$ to be a log-logistic with parameters $a=2$, $b=3$ then it is easy to see that $Z\geq_1 Y$, therefore $X_{i:30}{\geq}_2Y_{5:20}$ holds for every $i\geq 8$.
\end{example}
The following example illustrates that despite having some distribution with unknown parameters, one can derive SSD between their corresponding order statistics. Moreover, we show that additional information about the shape of the parent distribution has an effective impact in determining SSD.
\begin{example}
Let $X\geq_1 Y$, where $Y$ is a logistic RVs with unknown parameters. Let $i = 18$, $n = 200$, $j = 4$, $m = 44$. We can derive SSD from conditions 2., 3. or 4. of Corollary~\ref{COR2} (the support of the logistic is unbounded, so definitely it does not belong to $\mathcal{F}_C$). Nevertheless, $F_Y\in\mathcal{F}_{CO}$ but $\frac{18}{200}<\frac{4}{44}$ (condition 4. does not hold); $F_Y\in\mathcal{F}_{IFR}$ but again $\sum_{k=183}^{200}{\frac{1}{k}} < \sum_{k=41}^{44}{\frac{1}{k}}$ (condition 3. does not hold). Finally, since $F_Y\in\mathcal{F}_{CL}$ and $\psi(18) - \psi(183) >\psi(4) - \psi(41)$, we can ensure that $X_{18:200}{\geq}_2Y_{4:44}$ only through condition 2..
\end{example}
Sufficient SSD conditions described in Corollary~\ref{COR2} can be of multiple use, for instance they can be applied to determine the range of an unknown parameter of a parent distribution, based on dominance constraints, as shown in the example below.
\begin{example}
Suppose we have a $j$-out-of-$m$ system with log-logistic parent distribution $F_Y$, with parameters $a_Y=3$, $b_Y=1$, and we are looking for an $i$-out-of-$n$ system with a log-logistic parent distribution $F_X$ that dominates $F_Y$. Assume that only the scale parameter $b_X$ is known, say $b_X=2$. We need to determine the values of the shape parameter $a_X$ such that $X_{i:n} \geq_2 Y_{j:m}$ for some given instances $i,j,n,m$. Simple algebra shows that $S^- (F_X-F_Y )=1$ with sign sequence $-,+$, for $ a_X\geq a_Y$, hence $X\geq_{1+\frac{1}{i}} Y$ iff $E(X_{i:i})\geq E(Y_{i:i})$, where $E(X_{i:i})=b_X\frac{i\Gamma(\frac{a_X-1}{a_X})\Gamma(\frac1{a_X}+i)}{\Gamma(1+i)}$ (and similarly for $Y$), Lin~\cite{lin2000}. Let, for instance, $i=30$, $j=10$, $n=110$ and $m=100$. Since $F_X,F_Y\in F_{CO}$ and $\frac{i}{n} \geq \frac{j}{m}$. Numerical computation gives $3=a_Y\leq a_X\leq 5.58 \Rightarrow X\geq_{1+\frac{1}{30}} Y$, which, in turn, implies $X_{30:110}\geq_2 Y_{20:100}$.
\end{example}
\section{Testing $H^{-1}$--convexity}
In the literature, various methods have been proposed to test failure rate properties of distributions, with particular reference to the IFR property, Barlow and Proschan~\cite{barlow1969}, Tenga and Santner~\cite{tenga1984}, Bickel~\cite{bickel1969}, Bickel and Doksum~\cite{bickeldoksum}, Proscha nd Pyke~\cite{proschan1967}, or Sahoo and Sengupta~\cite{sahoo2017}. In this section we study a rather general method to test $H^{-1}$--convexity, where, for technical reasons, $H^{-1}$ is a quantile function such that $H^{-1}(0)=H(0)=0$. Denote with $\mathcal{F}_H$ the family of those CDFs that are $H^{-1}$--convex, that is, such that $H^{-1}\!\circ\! F$ is convex. We aim at testing the null hypothesis $\mathcal{H}_0:F \in \mathcal{F}_H$ against the alternative $\mathcal{H}_1:F \notin \mathcal{F}_H$. In particular, as the logit function does not fit the assumption $H^{-1}(0)=0$, we may be interested in checking whether $F\in \mathcal{F}_C$, $F\in \mathcal{F}_{IFR}$ or $F\in \mathcal{F}_{CO}$, in order to derive SSD between order statistics through a nonparameteric approach.
Denote by $F_n$ the empirical CDF of a random sample $\mathbf{X}=(X_1,\ldots,X_n)$ from $F$, that is, $F_n(t)=\frac1n\sum_{i=1}^{n}{\mathbf{1}_{X_i \leq t}}$. Then $H^{-1} \!\circ\! F_n$ converges almost surely to $H^{-1}\!\circ\! F$ on $[0,F^{-1}(1))$. Denote by $\mathbf{x}=(x_1,\ldots,x_n)$ an ordered realization of the random sample $\mathbf{X}$. Following Tenga and Santner~\cite{tenga1984}, our test is based on the distance between $H^{-1} \!\circ\! F_n$ and its greatest convex minorant (GCM) $g$, that is, the largest convex function that does not exceed $H^{-1} \!\circ\! F_n$, or, formally,
\begin{equation*}
g(x)=\sup{\{\phi(x): \phi \mbox{ is convex and }\phi(y)\leq H^{-1} \!\circ\! F_n(y),\forall y\in [x_1,x_n]\}},
\end{equation*}
To get a more concrete description of the GCM, the sample $\mathbf{x}$ determines a step function $H^{-1} \!\circ\! F_n$. For notational purposes, henceforth we set $h_k=H^{-1} \!\circ\! F_n(X_{k:n})=H^{-1}(\frac{k}{n})$. Let $i<j<n$ be positive integers, and $L_n^{i,j}$ the straight line connecting $(x_{i},h_{i-1})$ and $(x_{j},h_{j-1})$, that is,
\begin{equation*}
L_n^{i,j}(t)=h_{i-1}+\frac{t-x_{i}}{x_{j-1}-x_{i-1}} \left(h_{j-1}-h_{i-1}\right).
\end{equation*}
The GCM $g$ of the step function $H^{-1} \!\circ\! F_n$, corresponding to $\mathbf{x}$, is a piecewise linear function on $[x_1,x_n]$ defined by
\begin{equation*}
g(t)=
\begin{cases} h_1& t=x_1 \\
\min\left\{h_{j-1},\min \{L_n^{i,k}(x_j):1\leq i<j<k\leq n\}\right\} &t=x_j,\,2\leq j<n \\
h_{n-1} &t=x_n
\end{cases}
\end{equation*}
and by linear interpolation for $t\in(x_{j-1},x_j)$.
Intuitively, the value of $g$ at $x_j$ is the minimum of the heights of all segments connecting the nodes $(x_i, h_{i-1})$ and $(x_k, h_{k_1})$, where $i<j\leq k$ (see for instance Figure~\ref{f1}).
In a probabilistic setting, the GCM associated to the random sample $\mathbf{X}$ is an estimator of $H^{-1} \!\circ\! F$ under the assumption of convexity.
The test statistic is based on a distance between $H^{-1} \!\circ\! F_n$ and its GCM, $g$. In particular, we consider a weighted Kolmogorov-Smirnov test statistic, that is,
\begin{equation*}
{\rm KS}_n(X_1,...,X_n)={\rm KS}_n(\mathbf{X})=\max_{j\in (1,n)}\{{w_j(h_{j-1}-g(X_{j:n}))}\},
\end{equation*}
where the weights $w_j$ are suitably chosen according to $H^{-1}.$
If $H^{-1}$ is the identity we set $w_j=1$. If $H^{-1}$ is convex (that is, if we test odds-convexity or the IFR property), $h_{j-1}\geq g(X_{j:n})$ and the distance $h_j-h_{j-1}$ is increasing, for $1\leq j\leq n$. Therefore, the weights are tailored to downsize the effect of larger differences due to larger $j$'s. In particular, we set $w_j=\frac{1}{h_{j-1}}$, which provided the best performance in our analysis. Note also that ${\rm KS}_n$ is scale invariant.
We reject the null hypothesis for large values of ${\rm KS}_n$. The critical values or the $p$-values of the test may be obtained via the least favorable distribution of the test statistic under $\mathcal{H}_0$, which can be determined following the same approach of Tenga and Santner~\cite{tenga1984}.
\begin{proposition}\label{P1}
Let $\mathbf{Y}=(Y_{1},\ldots,Y_{n})$ be a random sample from $Y\sim H$, where $H(0)=0$. Under $\mathcal{H}_0:F \in \mathcal{F}_H$, ${\rm KS}_n(\mathbf{Y})\geq_1 {\rm KS}_n(\mathbf{X})$.
\end{proposition}
\begin{proof}
Let $\mathbf{y}=(y_1,\ldots,y_n)$ be an ordered random sample from $H$ and denote the corresponding empirical CDF by $H_n$. Subsequently, the GCM of $H^{-1} \!\circ\! H_n$ is denoted by $g^*$.
Let $x_i=v(y_i)=F^{-1}\!\circ\! H(y_i)$, for $i=1,\ldots,n$. Then $\mathbf{x}=(x_1,\ldots,x_n)$ is an ordered random sample from $F$. It is sufficient to show ${\rm KS}_n(\mathbf{y})\geq{\rm KS}_n(\mathbf{x})$ for any ordered vector $\mathbf{y}$. For both vectors, we have $H^{-1}\!\circ\! H_n(y_i)=H^{-1}\!\circ\! F_n(x_i)=H^{-1}(\frac{i}{n})=h_i$, for $i=1,\ldots,n$.
Under $\mathcal{H}_0$, the function $v=F^{-1}\!\circ\! H$ is increasing and concave with $v(0)=0$. Hence, Theorem~2.2 of Tenga and Santner~\cite{tenga1984} yields $g(y_{i}) \geq g^*(x_{i})$, for $i=1,\ldots,n$, which implies the conclusion of the statement.
\end{proof}
The least favorable distribution of ${\rm KS}_n$ under $\mathcal{H}_0$ can be computed through simulation. Let $\mathbf{x}$ be a realization of $\mathbf{X}$. We reject $\mathcal{H}_0$ when ${\rm KS}_n(\mathbf{x}) \geq c_{\alpha,n}$, where $c_{\alpha,n}$ is the solution of $P({\rm KS}_n(\mathbf{Y})\geq c_{\alpha,n}) \geq \alpha$ and $\alpha$ is the size of the test. Alternatively, we can compute the $p$-value of the test, that is, $p=P({\rm KS}_n(\mathbf{Y})\geq {\rm KS}_n(\mathbf{x}))$.
\subsection{Simulations}
In this subsection we focus on the null hypothesis $\mathcal{H}_0:F\in \mathcal{F}_{CO}$. Likewise, we may obtain tests for $\mathcal{H}_0:F\in \mathcal{F}_{C}$ or $\mathcal{H}_0:F\in \mathcal{F}_{IFR}$ (as for the IFR class, the reader is referred to Tenga and Santner~\cite{tenga1984} and to the aforementioned literature).
The computational work has been performed in Mathematica~\cite{mathematica}.
Compatibly with our computing capacities, we generated 3000 random samples of sample sizes $n=10,15,20,25,30$ and 1000 random samples of size $n=40,50,75,100$. Correspondingly, we obtained quantiles of the test statistic ${\rm KS}_n$, reported in Table~2, which may be used to determine (approximately) critical values and $p$-values. These tests are conservative, thus, as rule of thumb, values of the test statistic above 0.9 may provide evidence against $\mathcal{H}_0$ (for all the sample sizes considered). Simulation studies confirm the validity of the test.
\begin{enumerate}
\item
We simulated 100 random samples from a gamma distribution with different shape parameters $a=2,1,0.5$ and $b=1$ (the test is scale invariant). The test yields large $p$-values and 100\% acceptance rate for $a=2$ and in the limit case $a=1$. Nevertheless, for $a=0.5$, although $p$-values decrease, the rejection rate is quite low (always less than 50\% for $\alpha=0.1$): this is due to the shape of the function $\frac{F}{1-F}$, which is actually convex for $x\gtrapprox 0.16$ (as it can be seen from the second derivative). Therefore, the performance of the test is satisfactory, even in the latter case, as the median of a gamma(0.5,1) is approximately $0.22$, which implies that more than half of the observations are expected to exceed 0.16 (so that it is likely that most of the nodes of $\frac{F_n}{1-F_n}$ lay on a convex curve).
\item
We simulated 100 random samples from a Pareto distribution and set $a=2,1,0.5$. The test yields large $p$-values and high acceptance rates for $a=2$. For $a=0.5$, $p$-values are low and the rejection rate is high (always more than 65\% for $\alpha=0.1$) and increases with sample size. Therefore, the simulated power of the test has an increasing trend, as $n$ increases. Moreover, in the limiting case $a=1$, the $p$-values average around 0.5 and exhibit largest standard deviation (around 0.3). This is quite logical: in fact, the CDF of the Pareto(1,1), $F$, is just a shifted version of $H(x)=\frac{x}{1+x}$, that is, $F(x)=H(x+1)$, thus $F$ and $H$ are equivalent w.r.t. $\geq_c$.
\end{enumerate}
The results of the simulations are reported in Table~3. Some examples of the construction of GCM are depicted in Figures~1~and~2.
\begin{table}[h]
\caption{Simulated quantiles of ${\rm KS}_n$ for the test $\mathcal{H}_0:F\in \mathcal{F}_{CO}$. For each value of $n$, the number of simulation runs is given at the top.}
{\scriptsize
\begin{tabular}{|l|r|r|r|r|r|r|r|r|r|} \hline
$Runs$ & \multicolumn{1}{c|}{$3000$} & \multicolumn{1}{c|}{$3000$} & \multicolumn{1}{c|}{$3000$} & \multicolumn{1}{c|}{$3000$} & \multicolumn{1}{c|}{$3000$} & \multicolumn{1}{c|}{$1000$} & \multicolumn{1}{c|}{$1000$} & \multicolumn{1}{c|}{$1000$} & \multicolumn{1}{c|}{$1000$} \\ \hline
\backslashbox{$p$}{$n$}& \multicolumn{1}{c|}{$n=10$} & \multicolumn{1}{c|}{$n=15$} & \multicolumn{1}{c|}{$n=20$} & \multicolumn{1}{c|}{$n=25$} & \multicolumn{1}{c|}{$n=30$} & \multicolumn{1}{c|}{$n=40$} & \multicolumn{1}{c|}{$n=50$} & \multicolumn{1}{c|}{$n=75$} & \multicolumn{1}{c|}{$n=100$} \\ \hline
$p=0.1$&$0.510$&$0.561$&$0.597$&$0.599$&$0.605$&$ 0.585 $&$0.582$&$0.603$&$0.600$ \\ \hline
$p=0.2$&$0.639$&$0.677$&$0.698$&$0.700$&$0.704$&$ 0.694 $&$0.694$&$ 0.715 $&$0.713$ \\ \hline
$p=0.3$&$0.734$&$0.752$&$0.777$&$0.777$&$0.768$&$ 0.776 $&$0.711$&$ 0.776 $&$0.775$ \\ \hline
$p=0.4$&$0.805$&$0.819$&$0.834$&$0.836$&$0.828$&$ 0.833$&$0.838$&$0.840 $&$0.838$ \\ \hline
$p=0.5$&$0.860$&$0.870$&$0.876$&$0.882$&$0.874$&$0.877 $&$0.885$&$ 0.881 $&$0.882$ \\ \hline
$p=0.6$&$0.901$&$0.910$&$0.914$&$0.919$&$0.912$&$ 0.916 $&$0.921$&$ 0.926 $&$0.921$ \\ \hline
$p=0.7$&$0.938$&$0.942$&$0.944$&$0.949$&$0.943$&$ 0.946 $&$0.948$&$0.953 $&$0.949$ \\ \hline
$p=0.8$&$0.967$&$0.966$&$0.969$&$0.971$&$0.969$&$ 0.969 $&$0.971$&$0.973 $&$0.972$ \\ \hline
$p=0.9$&$0.987$&$0.987$&$0.988$&$0.988$&$0.987$&$0.987 $&$0.989$&$0.989 $&$0.989$ \\ \hline
$p=0.95$&$0.994$&$0.994$&$0.995$&$0.995$&$0.995$&$0.995$&$0.995$&$0.995 $&$0.995$ \\ \hline
\end{tabular}}
\end{table}
\begin{table}
\caption{Simulation results ($\mathcal{H}_0:F\in \mathcal{F}_{CO}$). The data refers to 100 simulated samples from 6 different distributions. The cells contain: average $p$-values; standard deviations and acceptance rates ($\alpha=0.1$), separated by semicolons.}
{\footnotesize\begin{tabular}{|l|r|r|r|r|} \hline
\backslashbox{$F$}{$n$}& \multicolumn{1}{c|}{$n=25$} & \multicolumn{1}{c|}{$n=50$} & \multicolumn{1}{c|}{$n=75$} & \multicolumn{1}{c|}{$n=100$} \\ \hline
$gamma(2,1)$&$0.89 ;0.15; 100\%$&$0.94 ;0.12; 100\%$&$0.89 ;0.18; 99\%$&$0.91 ;0.15; 100\%$\\ \hline
$gamma(1,1)$&$0.81 ;0.2; 100\%$&$0.82 ;0.23; 100\%$&$0.82 ;0.21; 99\%$&$0.82 ;0.2; 100\%$\\ \hline
$gamma(0.5,1)$&$0.48 ;0.3; 87\%$&$0.37 ;0.25; 87\%$&$0.27 ;0.25; 72\%$&$0.2 ;0.2; 59\%$\\ \hline
$Pareto(2,1)$&$0.73 ;0.22; 95\%$&$0.81 ;0.23; 99\%$&$0.8 ;0.23; 99\%$&$0.71 ;0.26; 98\%$\\ \hline
$Pareto(1,1)$&$0.5 ;0.3; 89\%$&$0.49 ;0.28; 91\%$&$0.48 ;0.31; 85\%$&$0.53 ;0.32; 89\%$\\ \hline
$Pareto(0.5,1)$&$0.13 ;0.2; 35\%$&$0.09 ;0.12; 25\%$&$0.04 ;0.07; 14\%$&$0.04 ;0.1; 11\%$\\ \hline
\end{tabular}}
\end{table}
\begin{figure}\label{f1}
\begin{center}
\includegraphics{order}
\caption{$F\sim gamma(2,1)$ and $n=20$. The figure shows $ \frac{ F_n}{1-F_n}$ and its GCM. ${\rm KS}_n=0.36$, $p$-value=0.996.}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics{order2}
\caption{$F\sim Pareto(0.5,1)$ and $n=20$. The figure shows $\frac{ F_n}{1-F_n}$ and its GCM. ${\rm KS}_n=0.996$, $p$-value=0.06.}
\end{center}
\end{figure}
\section{Discussion}
The method proposed in this paper enables derivation of SSD according to different assumptions on the parent's distribution shape and different conditions to be verified on ranks and sample sizes. We find that stronger (weaker) conditions on $F$ yield weaker (stronger) conditions on $(i,j,n,m)$.
Whereas most of the results in the literature regarding stochastic comparisons of order statistics rely on parametric assumptions, SSD conditions can be easily obtained by testing the hypothesis $F\in \mathcal{F}_{H}$. In the two-sample problem, we can determine whether the order statistics preserve an existing dominance relation between the parent distributions by using the same approach. The method has a wide range of application, as the four classes analyzed make it possible to compare most basic models. As shown in some examples, there are different ways to apply our results.
Finally, it should be stressed that the present study can be easily extended, in order to derive the \textit{increasing and convex order} Shaked and Santhikumar~\cite{shaked2007}, an order that is somewhat complementary to SSD. In doing so, we would deal with different classes of parent distributions, such as the \textit{decreasing failure rate} class.
\bibliographystyle{plain}
|
1,116,691,497,564 | arxiv | \section{Introduction}
\label{sec.Introduction}
The Peccei-Quinn (PQ) mechanism~\cite{Peccei:1977hh,Peccei:1977ur} represents the most elegant solution to the strong CP problem in Quantum Chromodynamics~\cite{Baker:2006ts,Pendlebury:2015lrz,Abel:2020pzs}. The key ingredient consists of a new dynamical pseudo-scalar field -- the \textit{axion}~\cite{Wilczek:1977pj,Weinberg:1977ma} -- which is driven towards its minimal energy configuration by the QCD dynamics, restoring the CP-invariance of strong interactions~\cite{Vafa:1984xg}.
The implications of a cosmic axion background crucially depend on the underlying production mechanism~\cite{DiLuzio:2020wdo}. If axions are produced via non-thermal channels, (\emph{e.g.} by the vacuum realignment mechanism~\cite{Abbott:1982af,Dine:1982ah,Preskill:1982cy,Linde:1985yf,Seckel:1985tj,Lyth:1989pb,Linde:1990yj} and/or by topological defects decay~\cite{Vilenkin:2000jqa,Kibble:1976sj,Kibble:1982dd,Vilenkin:1981kz,Davis:1986xc,Vilenkin:1982ks, Sikivie:1982qv,Vilenkin:1982ks,Huang:1985tt}) they should be considered natural cold dark matter candidates~\cite{Preskill:1982cy,Abbott:1982af,Dine:1982ah}.
Conversely, a thermal population of axions produced by scattering and decays of particles can provide additional radiation energy-density contributing to the hot dark matter component of the Universe, similarly to massive neutrinos. Notice also that, while the axion cold dark matter density is a decreasing function of the mass, the axion couplings are proportional to the mass itself. In order to have a significant thermal population, axions must represent a sub-dominant component of the total cold dark matter abundance and these two scenarios can be analyzed separately.
In this work we shall focus on the thermal axion mass limits from cosmology. A mandatory first step is the calculation of the axion relic abundance.
While most of the cosmological analyses carried out in the literature~\cite{Hannestad:2005df,Melchiorri:2007cd,Hannestad:2007dd,Hannestad:2008js,Hannestad:2010yi,Archidiacono:2013cha,Giusarma:2014zza,DiValentino:2015zta,DiValentino:2015wba,Archidiacono:2015mda,Ferreira:2020bpb} have been based on chiral perturbation theory, in Ref.~\cite{DiLuzio:2021vjd} it was pointed out that this approach can be safely extended only up to a temperature $T\lesssim 60$ MeV (see also Ref.~\cite{Giare:2020vzo,DiLuzio:2022tbb}), since the perturbative scheme breaks down. A practical solution to settle this issue employs an interpolation of the thermalization rate to cover the gap between the highest safe temperature reachable by chiral perturbation theory and the regime above the confinement scale, where the axion production rate is instead dominated by the axion-gluon scattering~\cite{DEramo:2021psx,DEramo:2021lgb}. Improved cosmological bounds~\cite{DEramo:2022nvb} have been derived for the two most accredited models of QCD axion interactions namely, the KSVZ~\cite{Kim:1979if,Shifman:1979if} and the DFSZ one~\cite{Dine:1981rt,Zhitnitsky:1980tq} (see also Ref.~\cite{Caloni:2022uya}). The current $95\%$~CL upper limits obtained in mixed hot dark matter scenarios in which massive neutrino species are also considered are $m_{\rm a}\lesssim 0.2$ eV and $\sum m_\nu\lesssim 0.15$~eV~\cite{DEramo:2022nvb}, using the most recent Cosmic Microwave Background (CMB) data released by the \textit{Planck} satellite~\cite{Aghanim:2019ame,Aghanim:2018eyx,Akrami:2018vks,Aghanim:2018oex}, the astrophysical observations of primordial light elements forged during the Big Bang Nucleosynthesis (BBN) epoch~\cite{Aver:2015iza,Peimbert:2016bdg,Cooke:2017cwo,Aghanim:2018eyx}, and the large scale structure information of the Universe in the form of Baryon Acoustic Oscillation (BAO) measurements~\cite{Beutler:2011hx,Ross:2014qpa,Alam:2016hwk}, see also the recent~\cite{Notari:2022zxo,DiLuzio:2022gsc}.
However, these limits have been obtained under two \emph{standardized} assumptions in cosmological parameter analyses. Namely, \textit{(1)} that the underlying model describing our universe is the minimal flat $\Lambda$CDM, and, \textit{(2)} restricting CMB measurements to those from the \textit{Planck} satellite observations. Concerning the first assumption, one should realize that a number of several intriguing tensions and anomalies have emerged at different statistical levels~\cite{Abdalla:2022yfr,Perivolaropoulos:2021jda,DiValentino:2022fjm,DiValentino:2021izs}, questioning the validity of the canonical flat $\Lambda$CDM picture. A small curvature component, or a more general dark energy fluid, are some examples of very promising scenarios that should be carefully explored. From what regards the second assumption, analyses should also include the recent small-scale measurements of the CMB angular power spectra released by the Atacama Cosmology Telescope~\cite{ACT:2020frw,ACT:2020gnv} (ACT) and the South Pole Telescope~\cite{SPT-3G:2014dbx,SPT-3G:2021eoc} (SPT) Collaborations.
It is therefore clear that the role of parameterizations, priors and models may lead to different constraints on the cosmological neutrino and axion masses.
Quantifying the impact resulting from the parameterization adopted for the cosmological model~\cite{diValentino:2022njd,Gariazzo:2018meg} and also from including independent CMB observations~\cite{DiValentino:2022oon,DiValentino:2022rdg} is the main goal of the present study, where we derive new, and robust, model-marginalized limits on hot dark matter scenarios involving axions $\&$ massive neutrinos.
The paper is organized as follows: in \autoref{sec.Methods} we explain our statistical, computational and data analysis methods, in \autoref{sec.Results} we present the bounds on the hot dark matter masses in the different cosmological scenarios, together with the model-marginalized limits. We conclude in \autoref{sec.Conclusions}.
\section{Methodology}
\label{sec.Methods}
\subsection{Bayesian statistics}
The first aim of this study is to test how the results change when
an extended cosmological model is considered as the underlying theory,
instead of the simple $\Lambda$CDM scenario.
In order to do that, we proceed by performing a marginalization over a number of different models.
Given a set of models ($\mathcal{M}_i$), in order to compute the model-marginalized posterior, one starts defining the posterior probabilities, $p_i$, of the model $\mathcal{M}_i$ over all the possible models:
\begin{equation}
\label{eq:modelposterior}
p_i = \frac{\pi_i Z_i}{\sum _j\pi_j Z_j}\,,
\end{equation}
where $\pi_i$ and $Z_i$ indicates the prior probability and the Bayesian evidence of model $\mathcal{M}_i$.
The model-marginalized posterior $p(\theta|d)$
for a set of parameters $\theta$, given some data $d$, reads as
\begin{equation}
\label{eq:mmposterior_def}
p(\theta|d)
\equiv
\sum_i
p(\theta|d,\mathcal{M}_i)
p_i
\,,
\end{equation}
where $p(\theta|d,\mathcal{M}_i)$ is the parameter posterior within the model $\mathcal{M}_i$. If all models have the same prior and using the Bayes factors $B_{i0}=Z_i/Z_0$ with respect to the favored model $\mathcal{M}_0$, the model-marginalized posterior is:
\begin{equation}
\label{eq:mmposterior}
p(\theta|d)
=
\frac{\sum_i
p(\theta|d,\mathcal{M}_i)
B_{i0}
}{
\sum_j B_{j0}
}
\,.
\end{equation}
Notice that if the Bayes factors are large in favor of the simplest and usually preferred model, extensions of the minimal picture will not contribute significantly to the model-marginalized posterior.
In order to perform Bayesian model comparison using the Bayes factors,
we follow the Jeffreys' scale to evaluate the strength of preference in favor of the best model, see~\autoref{tab:bayes}, extracted from Ref.~\cite{Trotta:2008qt}.
\begin{table*}[htbp!]
\begin{center}
\renewcommand{\arraystretch}{2}
\begin{tabular}{c | c | c | c }
\hline
\textbf{$|\ln B_0|$}
& \textbf{Odds}
& \textbf{Probability}
& \textbf{Strength of evidence}
\\
\hline
$<0.1$& $\lesssim 3:1$ & $<0.750$ & Inconclusive\\
$1$& $\sim 3:1$ & $0.750$ & Weak\\
$2.5$& $\sim 12:1$ & $0.923$ & Moderate\\
$5$& $\sim 150:1$ & $0.993$ & Strong\\
\hline\hline
\end{tabular}
\end{center}
\caption{Jeffreys’ empirical scale to establish the strength of evidence when comparing two competing models.}
\label{tab:bayes}
\end{table*}
In order to avoid the dependency on prior in determining credible intervals, one can make use of the method of Ref.~\cite{Gariazzo:2019xhx}. Given some model $\mathcal{M}$ which contains a parameter $x$ (for instance, the axion or the neutrino mass),
the \emph{relative belief updating ratio} $\mathcal{R}(x_1, x_2|d,\mathcal{M})$ is defined as:
\begin{equation}
\label{eq:R_def}
\mathcal{R}(x_1, x_2|d,\mathcal{M})
\equiv
\frac{Z^{x_1}_\mathcal{M}}{Z^{x_2}_\mathcal{M}}
\,,
\end{equation}
where $Z^{x}_\mathcal{M}$
is defined as the Bayesian evidence of model $\mathcal{M}$ but fixing $x$ to a specific value%
\footnote{We also assume that the prior on $x$ is independent on the other parameters and viceversa.}:
\begin{equation}
\label{eq:Zx_def}
Z^{x}_\mathcal{M}
=
\int_{\Omega_\psi}
d\psi
\pi(\psi|\mathcal{M})
\mathcal{L}_\mathcal{M}(x,\psi)
\,,
\end{equation}
where $\psi$ represents all the parameters in model $\mathcal{M}$ except $x$,
which can vary in a parameter space $\Omega_\psi$,
$\pi(\psi|\mathcal{M})$ is their prior (notice that the $x$ prior is not included here)
and
$\mathcal{L}$ is the likelihood.
From Eq.~\eqref{eq:R_def}, we easily understand that the relative belief updating ratio
does not represent a probability,
as it is the ratio of two evidences. Importantly, the function $\mathcal{R}(x_1, x_2|d,\mathcal{M})$ is completely prior-independent.
Using the Bayes theorem,
it is possible to obtain a different expression for the former function:
\begin{equation}
\label{eq:R}
\mathcal{R}(x_1, x_2|d,\mathcal{M})
=
\frac{
p(x_1|d,\mathcal{M})/\pi(x_1|\mathcal{M})
}{
p(x_2|d,\mathcal{M})/\pi(x_2|\mathcal{M})
}\,,
\end{equation}
where $\pi(x|\mathcal{M})$ is the unidimensional prior on $x$.
This formulation is extremely useful in Monte Carlo Markov Chain (MCMC) runs where one can calculate these functions directly from the run chains.
The definition of $\mathcal{R}(x_1,x_2|d)$
can be easily extended to perform a model marginalization:
\begin{equation}
\label{eq:mmR_def}
\mathcal{R}(x_1, x_2|d)
\equiv
\frac{
\sum_i Z^{x_1}_{\mathcal{M}_i}\pi(\mathcal{M}_i)
}{
\sum_j Z^{x_2}_{\mathcal{M}_j}\pi(\mathcal{M}_j)
}
\,,
\end{equation}
where now the evidences $Z^{x}_{\mathcal{M}_j}$ are computed within a specific model and
$\pi(\mathcal{M}_j)$ is the model prior.
In order to write $\mathcal{R}(x_1, x_2|d)$ using the parameter prior and posterior,
the simplest assumption is to consider the same prior $\pi(x)$ within all the models.
In such case, Eq.~\eqref{eq:mmR_def} becomes:
\begin{equation}
\label{eq:mmR}
\mathcal{R}(x_1, x_2|d)
=
\frac{
p(x_1|d)/\pi(x_1)
}{
p(x_2|d)/\pi(x_2)
}\,,
\end{equation}
where $p(x|d)$ is the model-marginalized posterior in Eq.~\eqref{eq:mmposterior_def}.
\subsection{Axion Modeling}
We address the effects induced by a relic population of thermal axions by employing a modified version of the Boltzmann integrator code \textsc{CAMB} \citep{Lewis:1999bs,Howlett:2012mh}. The code has been modified to accommodate the effects of QCD axions on cosmological scales only in terms of the axion mass which we employ as an additional cosmological parameters in our analysis, see Ref.~\cite{DEramo:2022nvb} for a detailed calculation. We vary the axion mass in the range between $0.01$ eV and $10$ eV and focus exclusively on the KSVZ model of axion-hadron interactions.
As long as the axion remains relativistic ($T_a\gg m_{\rm a}$), it behaves as radiation in the early Universe and its cosmological effects are those produced via their contribution to the effective number of neutrino species $N_{\rm eff}$. As detailed in Ref.~\cite{DEramo:2022nvb}, such a contribution is precisely evaluated by solving the Boltzmann equation for the axion number density in the early Universe. Indeed, very light axions ($m_{\rm a} \lesssim 0.1$ eV) are still relativistic at recombination and thus modify the CMB angular power spectrum via $N_{\rm eff}$. While such corrections are typically very small ($\Delta N_{\rm eff}\sim 0.03$), they are relevant for the next generation of CMB experiments~\cite{Abazajian:2016yjj}. On the other hand, heavier axions with masses larger than $0.1$ eV are highly non-relativistic at the recombination epoch. In this case, their impact on the CMB angular power spectra is both direct (through their impact on the early integrated Sachs-Wolfe effect, similarly to massive neutrinos) and indirect (by modifying the primordial helium abundance during the BBN). In this regard, it is worth stressing that the axion starts behaving as cold dark matter much earlier than massive neutrinos, leading to a significant impact on structure formation. This feature, not only allows to distinguish massive neutrinos from massive axions through their effect on structure formation but it also allows to set stringent constraints on the axion mass exploiting large scale structure data, as well. Nonetheless, when the two species have similar masses, the evolution of their energy densities prevent to reach constraints on their masses lower than $ \sim 0.1$ eV, see also Refs.~\cite{Giare:2021cqr,DEramo:2022nvb}. Allowing for a prior on the axion mass spanning three orders of magnitude we can properly take into account all these effect of light and heavy axions on the CMB anisotropies, see ~\autoref{tab.Priors}.
\subsection{Cosmological Model Parameterization}
\begin{table}[htbp]
\begin{center}
\renewcommand{\arraystretch}{1.5}
\begin{tabular}{l@{\hspace{0. cm}}@{\hspace{1.5 cm}} c}
\hline
\textbf{Parameter} & \textbf{Prior} \\
\hline\hline
$\Omega_{\rm b} h^2$ & $[0.005\,,\,0.1]$ \\
$\Omega_{\rm b} h^2$ & $[0.005\,,\,0.1]$ \\
$\Omega_{\rm k} $ & $[-0.3\,,\,0.3]$\\
$w_0$ & $[-3\,,\,1]$ \\
$w_a$ & $[-3\,,\,2]$ \\
$100\,\theta_{\rm {MC}}$ & $[0.5\,,\,10]$ \\
$\log(10^{10}A_{\rm S})$ & $[2.91\,,\,3.91]$ \\
$n_{\rm s}$ & $[0.8\,,\, 1.2]$ \\
$\alpha_{\rm s}$ & $[-1\,,\, 1]$ \\
$\sum m_{\nu}$ [eV] & $[0.06\,,\,5]$\\
$m_{\rm a}$ [eV] & $[0.01\,,\,10]$\\
\hline\hline
\end{tabular}
\caption{List of the parameter priors.}
\label{tab.Priors}
\end{center}
\end{table}
As pointed out in the \hyperref[sec.Introduction]{Introduction}, a key point in our analysis is to derive robust bounds on the hot dark matter sector marginalizing over a plethora of possible background cosmologies. Therefore, along with the six $\Lambda$CDM parameters (the amplitude $A_s$ and the spectral index $n_s$ of scalar perturbations, the baryon $\Omega_b h^2$ and the cold dark matter $\Omega_c h^2$ energy densities, the angular size of the sound horizon at recombination $\theta_{\rm MC}$ and the reionization optical depth, $\tau$), we also include the sum of neutrino masses $\sum m_\nu$ and the axion mass $m_{\rm a}$. We then explore several extensions of this minimal model, enlarging the parameter space including one or more parameters, such as
a running of the scalar index ($\alpha_s$),
a curvature component ($\Omega_k$),
and
the dark energy equation of state parameters ($w_0$ and $w_a$)
(see \autoref{tab.Priors} for the priors adopted in the cosmological parameters).
\begin{itemize}
\item The running of scalar spectral index, $\alpha_s$. In simple inflationary models, the running of the spectral index is a second order perturbation and it is typically very small. However, specific models can produce a large running over a range of scale accessible to CMB experiments. Indeed, a non-zero value of $\alpha_s$ alleviates the $\sim 2.7\sigma$ discrepancy in the value of the scalar spectral index $n_{\rm s}$ measured by \emph{Planck} ($n_{\rm s}=0.9649\pm 0.0044$)~\cite{Planck:2018vyg,Forconi:2021que} and by the \emph{Atacama Cosmology Telescope} (ACT) ($n_{\rm s}=1.008\pm 0.015$)~\cite{ACT:2020gnv}, see Refs.~\cite{DiValentino:2022rdg, DiValentino:2022oon,Giare:2022rvg}.
\item Curvature density, $\Omega_k$. Recent data analyses of the CMB temperature and polarization spectra from Planck 2018 team exploiting the baseline \emph{Plik} likelihood suggest that our Universe could have a closed geometry at more than three standard deviations~\cite{Planck:2018vyg,Handley:2019tkm,DiValentino:2019qzk,Semenaite:2022unt}. These hints mostly arise from TT observations, that would otherwise show a lensing excess~\cite{DiValentino:2020hov,Calabrese:2008rt,DiValentino:2019dzu}.
In addition, analyses exploiting the \emph{CamSpec} TT likelihood~\cite{Efstathiou:2019mdh,Rosenberg:2022sdy} point to a closed geometry of the Universe with a significance above 99\% CL. Furthermore, an indication for a closed universe is also present in the BAO data, using Effective Field Theories of Large Scale Structure~\cite{Glanville:2022xes}. These recent findings strongly motivate to leave the curvature of the Universe as a free parameter~\cite{Anselmi:2022uvj} and obtain limits on the neutrino and axion masses in this context.
\item Dynamical Dark Energy equation of state.
Cosmological neutrino and axion mass bounds become weaker if the dark energy equation of state is taken as a free parameter. Even if current data fits well with the assumption of a cosmological constant within the minimal $\Lambda$CDM scenario, the question of having an equation of state parameter different from $ -1 $ remains certainly open. Along with constant dark energy equation of state models, in this paper we also consider the possibility of having a time-varying $ w(a) $ described by the Chevalier-Polarski-Linder parametrizazion (CPL)~\cite{Chevallier:2000qy,Linder:2002et}:
\begin{equation}
\label{eq:cpl}
w(a) = w_0 + (1-a)w_a
\end{equation}
where $ a $ is the scale factor and is $ a_0 = 1 $ at the present time, $ w(a_0)=w_0 $ is the value of the equation of state parameter today. Dark energy changes the distance to the CMB consequently pushing it further (closer) if $w < -1$ ($w > -1$) from us. This effect can be balanced by having a larger matter density or, equivalently, by having more massive hot relics, leading to less stringent bounds on both the neutrino and axion masses. Accordingly, the mass bounds of cosmological neutrinos and axions become weaker if the dark energy equation of state is taken as a free parameter.
\end{itemize}
\subsection{Statistical Analyses and Likelihoods}
In order to study the constraints achievable by current CMB and large scale structure probes, we make use of the publicly available code \textsc{COBAYA}~\citep{Torrado:2020xyz}. The code explores the posterior distributions of a given parameter space using the Monte Carlo Markov Chain (MCMC) sampler developed for \textsc{CosmoMC}~\cite{Lewis:2002ah} and tailored for parameter spaces with speed hierarchy implementing the ''fast dragging'' procedure developed in~\cite{Neal:2005}. The prior distributions for the parameters involved in our analysis are chosen to be uniform along the range of variation (see \autoref{tab.Priors}) with the exception of the optical depth for which the prior distribution is chosen accordingly to the CMB datasets as discussed below.
To perform model comparison, we compute the Bayesian Evidence of the different models and estimate the Bayes factors through the publicly available package \texttt{MCEvidence},\footnote{\href{https://github.com/yabebalFantaye/MCEvidence}{github.com/yabebalFantaye/MCEvidence}~\cite{Heavens:2017hkr,Heavens:2017afc}.} properly modified to be compatible with \textsc{COBAYA}.
Concerning the cosmological and astrophysical observations, our baseline data-sets and likelihoods include:
\begin{itemize}[leftmargin=*]
\item Planck 2018 temperature and polarization (TT TE EE) likelihood, which also includes low multipole data ($\ell < 30$)~\citep{Aghanim:2019ame,Aghanim:2018eyx,Akrami:2018vks}. We refer to this combination as ``Planck 2018''.
\item Planck 2018 temperature and polarization (TT TE EE) likelihood up to multipole $\ell=650$, to use in combination with the alternative ground-based small-scales CMB experiments. We refer to this combination as "Planck650".
\item Planck 2018 lensing likelihood~\citep{Aghanim:2018oex}, reconstructed from measurements of the power spectrum of the lensing potential. We refer to this dataset as ``lensing''.
\item Atacama Cosmology Telescope DR4 temperature and polarization (TT TE EE) likelihood, with a Gaussian prior on the optical depth at reionization $\tau = 0.065 \pm 0.015$, as done in~\cite{Aiola:2020azj}. We refer to this dataset combination as "ACT."
\item South Pole Telescope polarization (TE EE) measurements SPT-3G~\cite{SPT-3G:2021eoc} combined with a Gaussian prior on the optical depth at reionization $\tau = 0.065 \pm 0.015$. We refer to this dataset combination as "SPT-3G."
\item Baryon Acoustic Oscillations (BAO) measurements extracted from data from the 6dFGS~\cite{Beutler:2011hx}, SDSS MGS~\cite{Ross:2014qpa} and BOSS DR12~\cite{Alam:2016hwk} surveys. We refer to this dataset combination as ``BAO''.
\end{itemize}
\section{Results}
\label{sec.Results}
We start by discussing the limits in the mixed hot dark matter scenario assuming the standard $\Lambda$CDM cosmology, see \autoref{tab.lCDM+mnu+ma}.
The tightest constraints are obtained when combining Planck650 temperature, polarization and lensing measurements with ACT-CMB and BAO data: the limits we get on the hot dark matter relic masses are $m_{\rm a}<0.18$~eV and $m_\nu<0.16$~eV, both at $95\%$~CL. Adding ACT CMB observations therefore considerably improves the limit on hot relics, as with Planck plus BAO data alone the $95\%$~CL bounds are $m_{\rm a}<0.28$~eV and $m_\nu<0.16$~eV, in perfect agreement with the results for the KSVZ model of Ref.~\cite{DEramo:2022nvb} (see also the recent~\cite{Notari:2022zxo}). Concerning the remaining cosmological parameters, notice that both ACT and SPT observations (either alone or combined with BAO) prefer $n_s\simeq 1$, pointing to a Harrison-Zel'dovich primordial power spectrum, as can also be noticed from the left panel of ~\autoref{fig:whisker_all} (see also Ref.~\cite{Giare:2022rvg}).
Enlarging the minimal $\Lambda$CDM picture with a curvature component $\Omega_k$ only degrades mildly the limits on $\sum m_\nu$, while the limit on $m_{\rm a}$ remains unchanged.
From the results in \autoref{tab.omkCDM+mnu+ma} one can notice that the most constraining bounds are $m_{\rm a}<0.18$~eV and $m_\nu<0.20$~eV, both at $95\%$~CL for ACT plus Planck650 plus BAO observations. The preference for $n_s \simeq 1$ from SPT and ACT still persists, see the left panel of \autoref{fig:whisker_all}. Notice that \emph{all} CMB data prefer a value of $\Omega_k<0$ with a significance above the $\sim 2\sigma$ level for most of the cases. When CMB observations are combined with BAO measurements such a preference is however diluted. This behavior is shown in the bottom right panel of ~\autoref{fig:whisker_all}.
\autoref{tab.mnu+ma+nrun} instead depicts the constraints on the mixed hot dark matter scenario when including a running ($\alpha_s$) of the scalar spectral index $n_s$. In this case, the $95\%$~CL bounds for the most powerful data set combination (i.e. Planck650 plus ACT and BAO) are $m_{\rm a}<0.25$~eV and $m_\nu<0.17$~eV, limiting the constraining power of these observations within the minimal $\Lambda$CDM scenario. Interestingly, the preference for $n_s\simeq 1$ from either SPT and/or ACT is not as strong as in the previous two background cosmologies (see the left panel of ~\autoref{fig:whisker_all}) and it is instead translated into a mild preference for a non-zero value of $\alpha_s$ in the case of SPT. However, ACT observations shows a $\sim 5 \sigma $ preference for a positive value of $\alpha_s$, see the whisker plot in the right panel of ~\autoref{fig:whisker_all}, that corresponds to a preference for a positive neutrino mass.
Leaving freedom in the dark sector of the background cosmology leads to the results shown in \autoref{tab.w0} and \autoref{tab.w0wa}. We start by discussing the simplest dark energy model with a constant dark energy equation of state $w_0$. First of all, notice from the results depicted in \autoref{tab.w0} that \emph{all} CMB measurements prefer a phantom dark energy universe, that is, a universe in which $w_0<-1$. The significance is larger than $2\sigma$ when considering Planck measurements, either alone or in combination with other CMB data sets. The larger negative value of $w_0$ is associated to a very large value of $H_0$, due to their strong degeneracy. Indeed, it has been shown that a phantom-like dark energy component can solve the current tension between high-redshift estimates and local universe measurements of the Hubble constant~\cite{DiValentino:2016hlg}. The addition of BAO observations leads however the value of $w_0$ very close to its cosmological constant expectation of $w_0=-1$ and the mean value of the Hubble constant is notably reduced, $H_0\sim 69$~km/s/Mpc. The results for $w_0$ are illustrated in the top right panel of ~\autoref{fig:whisker_all}. Concerning the limits on the hot relic masses, we obtain $m_{\rm a}<0.18$~eV for the axion mass and $\sum m_\nu <0.23$~eV for the neutrino masses, both at $95\%$~CL for the most powerful data set combination, which is, as in the previous background cosmologies, the one exploiting Planck650 plus ACT plus BAO observations. While the axion mass bound barely changes from the standard $\Lambda$CDM case, the neutrino mass limit is degraded to $\sum m_\nu <0.23$~eV, due to the strong degeneracy between the neutrino mass and the dark energy equation of state: if $w_0$ is allowed to freely change including also the phantom region, $\Omega_m$ can take very high values and also the neutrino mass can be much higher than in standard cosmological backgrounds. \autoref{tab.w0wa} shows the constraints in the widely exploited, two-parameter CPL parameterization for the dark energy component, see Eq.~(\ref{eq:cpl}). The results for $w_0$ are very similar to those previously described, preferring all CMB observations values of $w_0<-1$ albeit with a mild significance. The corresponding $H_0$ value is also considerably larger than within the $\Lambda$CDM scenario (with hot relics). However, in this case, the addition of BAO data shifts the mean value of $w_0$ to the non-phantom region, with a very mild preference ($\sim 1.5\sigma$) for $w_0>-1$. Notice that CMB data alone is unable to measure the time derivative of the dark energy equation of state $w_a$, providing only an upper bound on this parameter. When BAO information is also considered in the analyses, a mean value of $w_a\sim -1$ is preferred. The mean value of the Hubble constant after the inclusion of BAO observations is much closer to the value measured by the Planck collaboration in a standard cosmology. The results above for the dark energy parameters are illustrated by means of the whisker plots for the $w_0$ and $w_a$ parameters depicted in ~\autoref{fig:whisker_all}.
Concerning the hot relics, notice that this background cosmology, having two extra parameters largely degenerated with the neutrino masses, leads to the least constraining hot relic mass bounds: the most powerful combination sets $95\%$~CL limits of $m_{\rm a}<0.20$~eV for the axion mass and $\sum m_\nu <0.33$~eV for the total neutrino mass.
\autoref{tab:BFs} presents the Bayes factors with respect to the best model for each of the five possible background cosmologies considered here and for the different data combinations. Interestingly, the best background cosmology is never found within the minimal $\Lambda$CDM plus two hot dark matter relics, regardless of the data set combinations. The combination of Planck or Planck650 with either BAO, SPT or ACT prefers a universe with a non-zero value of the running in the primordial power spectrum with strong evidence. Ground-based small-scale CMB probes, both alone and combined with BAO, prefer either non-flat universes, as in the case of SPT, or a model with a time varying dark energy component, as in the case of ACT. Such evidences are substantial when including BAO measurements.
\autoref{fig:Rratio} shows the model-marginalized relative belief updating ratio $\mathcal{R}$, Eq.~(\ref{eq:mmR}), for both the axion mass $m_{\rm a}$ (left) and for the sum of the neutrino masses $\sum m_\nu$ (right), considering the extensions of the $\Lambda$CDM model considered and using ACT + Planck650 + BAO (SPT + Planck650 + BAO) data. The horizontal lines show the significance levels $\exp(-1)$ and $\exp(-3)$.
The vertical lines
indicate the value $0.1$~eV, corresponding to the approximate lower limits for $\sum m_\nu$ in the inverted ordering case.
The quantity $\mathcal{R}$ is independent of the shape and normalization of the prior and it is statistically equivalent to a Bayes factor between a model where $m_{\rm a}$ ($m_\nu$) has been fixed to some value and another model where $m_{\rm a}=0$ ($m_\nu=0$). The red curve shows the model-marginalized function $\mathcal{R}$, from which we derive the limits in~\autoref{tab:marglims}. The black and gray lines show the $\mathcal{R}$ function within each model, where the darker lines are those that contribute most to the model marginalization, that is, they have the best Bayesian evidences. For instance, for the case of ACT + Planck650+ BAO, the $95\%$~CL marginalized limit is $0.21$~eV, for both $m_{\rm a}$ and $\sum m_\nu$. Those bounds are led by the models which have the best Bayesian evidences, which, for this particular data combination, are the
$\Lambda$CDM + $m_{\rm a}$ + $\sum m_\nu$ + $\Omega_k$ and the
$\Lambda$CDM + $m_{\rm a}$ + $\sum m_\nu$ + $\alpha_s$ ones, see ~\autoref{tab:BFs}, corresponding to the $95\%$~CL upper bounds of $m_{\rm a}<0.176$~eV, $\sum m_\nu <0.205$~eV and $m_{\rm a}<0.248$~eV, $\sum m_\nu <0.172$~eV, respectively.
Instead, for the other data combination illustrated in ~\autoref{fig:Rratio}, that is, SPT + Planck650 + BAO,
the $95\%$~CL marginalized limits are $0.35$~eV and $0.23$, for $m_{\rm a}$ and $\sum m_\nu$ respectively. Those bounds are led by the models which have the best Bayesian evidences, which, for this particular data combination, are the
$\Lambda$CDM + $m_{\rm a}$ + $\sum m_\nu$ + $\alpha_s$ and the
$\Lambda$CDM + $m_{\rm a}$ + $\sum m_\nu$ + $\Omega_k$ ones, see ~\autoref{tab:BFs}, corresponding to the $95\%$~CL upper bounds of $m_{\rm a}<0.308$~eV, $\sum m_\nu <0.168$~eV and $m_{\rm a}<0.356$~eV, $\sum m_\nu <0.224$~eV, respectively.
Interestingly, the minimal $\Lambda$CDM cosmology never provides the best Bayesian evidence, for any of these two data combinations.
Notice also that, while the ACT + Planck650+ BAO data combination provides more powerful limits on $m_{\rm a}$ than the SPT + Planck650+ BAO one, these two data sets are equally powerful when constraining the neutrino mass, as can be noticed from the results shown in ~\autoref{tab:marglims}.
We conclude this section by summarizing our results in the whisker plots shown in ~\autoref{fig:whisker_ma_mnu}, illustrating the $95\%$~CL upper bounds on the axion mass $m_{\rm a}$ and on the total neutrino mass $\sum m_\nu$ arising for different data combinations in each of the five background cosmologies here. We also depict the model-marginalized limits on these two quantities. For the data combination Planck650 + ACT+ BAO, the most constraining bound for $m_{\rm a}$ is obtained within the $\Lambda$CDM + $m_{\rm a}$ + $\sum m_\nu$ + $\Omega_k$ scenario ($m_{\rm a} < 0.176$~eV at $95\%$~CL). For the total neutrino mass, the tightest $95\%$~CL upper bound ($m_{\rm a} < 0.163$~eV at $95\%$~CL) is found in the $\Lambda$CDM + $m_{\rm a}$ + $\sum m_\nu$ canonical scheme. For the data set SPT + Planck650+ BAO, the tightest limits on the hot thermal relic masses are those derived in the $\Lambda$CDM + $m_{\rm a}$ + $\sum m_\nu$ +
$\alpha_s$ cosmological background, and correspond to $m_{\rm a} < 0.301$~eV and $\sum m_\nu < 0.168$~eV (both at $95\%$~CL).
\begin{table*}[htbp!]
\begin{center}
\renewcommand{\arraystretch}{2}
\resizebox{1 \textwidth}{!}{\begin{tabular}{c | c c| c c| c c| c c| c c }
\hline
\textbf{Parameter} & \textbf{\nq{Planck}} & \textbf{\nq{\\ +BAO}} & \textbf{\nq{ACT}} & \textbf{\nq{\\ +BAO}} & \textbf{\nq{SPT}} & \textbf{\nq{\\ +BAO}} & \textbf{ACT+Planck650} & \nq{\\ \textbf{+BAO}} & \textbf{SPT+Planck650} & \nq{\\ \textbf{+ BAO}} \\
\hline\hline
$ \Omega_\mathrm{b} h^2 $ & $ 0.02243\pm 0.00015 $ & $ 0.02250\pm 0.00015 $ & $ 0.02165\pm 0.00033 $ & $ 0.02170\pm 0.00032 $ & $ 0.02251\pm 0.00033 $ & $ 0.02252\pm 0.00032 $ & $ 0.02238\pm 0.00014 $ & $ 0.02245\pm 0.00013 $ & $ 0.02246\pm 0.00014 $ & $ 0.02252\pm 0.00013 $ \\
$ \Omega_\mathrm{c} h^2 $ & $ 0.1225^{+0.0016}_{-0.0022} $ & $ 0.1208^{+0.0012}_{-0.0014} $ & $ 0.1242\pm 0.0046 $ & $ 0.1190^{+0.0024}_{-0.0020} $ & $ 0.1195^{+0.0060}_{-0.0046} $ & $ 0.1175^{+0.0022}_{-0.0020} $ & $ 0.1214\pm 0.0015 $ & $ 0.1204^{+0.0011}_{-0.0012} $ & $ 0.1216^{+0.0014}_{-0.0018} $ & $ 0.1204^{+0.0012}_{-0.0014} $ \\
$ \tau $ & $ 0.0564\pm 0.0076 $ & $ 0.0581^{+0.0070}_{-0.0080} $ & $ 0.072\pm 0.015 $ & $ 0.070\pm 0.015 $ & $ 0.067\pm 0.015 $ & $ 0.069\pm 0.014 $ & $ 0.0562^{+0.0072}_{-0.0082} $ & $ 0.0567^{+0.0070}_{-0.0081} $ & $ 0.0548\pm 0.0081 $ & $ 0.0556\pm 0.0078 $ \\
$ 100\theta_\mathrm{MC} $ & $ 1.04058\pm 0.00036 $ & $ 1.04081\pm 0.00032 $ & $ 1.04120\pm 0.00081 $ & $ 1.04208\pm 0.00064 $ & $ 1.03854\pm 0.00082 $ & $ 1.03939\pm 0.00066 $ & $ 1.04087\pm 0.00031 $ & $ 1.04101\pm 0.00027 $ & $ 1.04044\pm 0.00032 $ & $ 1.04060\pm 0.00029 $ \\
$ n_\mathrm{s} $ & $ 0.9681\pm 0.0049 $ & $ 0.9703\pm 0.0043 $ & $ 1.005^{+0.022}_{-0.018} $ & $ 1.022\pm 0.015 $ & $ 0.987^{+0.031}_{-0.024} $ & $ 1.013\pm 0.018 $ & $ 0.9705\pm 0.0043 $ & $ 0.9727^{+0.0037}_{-0.0041} $ & $ 0.9700\pm 0.0048 $ & $ 0.9721\pm 0.0043 $ \\
$ \log(10^{10} A_\mathrm{s}) $ & $ 3.055^{+0.015}_{-0.016} $ & $ 3.055^{+0.014}_{-0.016} $ & $ 3.077\pm 0.032 $ & $ 3.063\pm 0.031 $ & $ 3.047\pm 0.035 $ & $ 3.035\pm 0.033 $ & $ 3.059\pm 0.016 $ & $ 3.057^{+0.014}_{-0.016} $ & $ 3.047\pm 0.017 $ & $ 3.047\pm 0.016 $ \\
$ m_\mathrm{a}$ [eV]& $ < 0.888 $ & $ < 0.282 $ & $ < 2.27 $ & $ < 1.12 $ & $ < 4.93 $ & $ < 0.987 $ & $ < 0.190 $ & $ < 0.180 $ & $ < 0.388 $ & $ < 0.310 $ \\
$ \sum m_\nu$ [eV] & $ < 0.278 $ & $ < 0.156 $ & $ < 2.63 $ & $ < 0.351 $ & $ < 2.18 $ & $ < 0.339 $ & $ < 0.305 $ & $ < 0.163 $ & $ < 0.300 $ & $ < 0.169 $ \\
$ H_0 $ & $ 66.9^{+1.2}_{-0.73} $ & $ 67.90\pm 0.53 $ & $ 60^{+7}_{-4} $ & $ 67.97\pm 0.77 $ & $ 61.1^{+6.3}_{-3.9} $ & $ 68.50\pm 0.74 $ & $ 67.0^{+1.1}_{-0.73} $ & $ 67.81\pm 0.52 $ & $ 67.1^{+1.1}_{-0.72} $ & $ 67.95\pm 0.54 $ \\
$ \sigma_8 $ & $ 0.793^{+0.023}_{-0.011} $ & $ 0.8052^{+0.0099}_{-0.0075} $ & $ 0.656^{+0.11}_{-0.076} $ & $ 0.779^{+0.032}_{-0.026} $ & $ 0.611^{+0.10}_{-0.081} $ & $ 0.755^{+0.031}_{-0.024} $ & $ 0.801^{+0.020}_{-0.011} $ & $ 0.809^{+0.011}_{-0.0084} $ & $ 0.791^{+0.020}_{-0.011} $ & $ 0.799^{+0.012}_{-0.0087} $ \\
\hline \hline
\end{tabular}}
\end{center}
\caption{95\%~CL upper bounds on the QCD axion mass and on the sum of neutrino masses and 68\% CL cosmological parameter errors in the minimal $\Lambda$CDM picture.}
\label{tab.lCDM+mnu+ma}
\end{table*}
\begin{table*}[htbp!]
\begin{center}
\renewcommand{\arraystretch}{2}
\resizebox{1 \textwidth}{!}{\begin{tabular}{c | c c| c c| c c| c c| c c }
\hline
\textbf{Parameter} & \textbf{\nq{Planck}} & \textbf{\nq{\\ +BAO}} & \textbf{\nq{ACT}} & \textbf{\nq{\\ +BAO}} & \textbf{\nq{SPT}} & \textbf{\nq{\\ +BAO}} & \textbf{ACT+Planck650} & \nq{\\ \textbf{+BAO}} & \textbf{SPT+Planck650} & \nq{\\ \textbf{+ BAO}} \\
\hline\hline
$ \Omega_\mathrm{b} h^2 $ & $ 0.02256\pm 0.00020 $ & $ 0.02248\pm 0.00016 $ & $ 0.02178\pm 0.00033 $ & $ 0.02166\pm 0.00032 $ & $ 0.02253\pm 0.00034 $ & $ 0.02244\pm 0.00033 $ & $ 0.02249^{+0.00017}_{-0.00021} $ & $ 0.02242\pm 0.00013 $ & $ 0.02261\pm 0.00018 $ & $ 0.02248\pm 0.00015 $ \\
$ \Omega_\mathrm{c} h^2 $ & $ 0.1214\pm 0.0020 $ & $ 0.1214^{+0.0016}_{-0.0019} $ & $ 0.1182\pm 0.0047 $ & $ 0.1237\pm 0.0048 $ & $ 0.1163\pm 0.0063 $ & $ 0.1232\pm 0.0052 $ & $ 0.1203\pm 0.0016 $ & $ 0.1209\pm 0.0014 $ & $ 0.1210\pm 0.0019 $ & $ 0.1211^{+0.0015}_{-0.0018} $ \\
$ \tau $ & $ 0.0497\pm 0.0077 $ & $ 0.0589\pm 0.0070 $ & $ 0.065\pm 0.015 $ & $ 0.071\pm 0.014 $ & $ 0.065\pm 0.015 $ & $ 0.068\pm 0.015 $ & $ 0.0475\pm 0.0079 $ & $ 0.0566\pm 0.0077 $ & $ 0.0488\pm 0.0083 $ & $ 0.0552\pm 0.0079 $ \\
$ 100\theta_\mathrm{MC} $ & $ 1.04058\pm 0.00037 $ & $ 1.04072\pm 0.00034 $ & $ 1.04131\pm 0.00077 $ & $ 1.04157\pm 0.00075 $ & $ 1.03868\pm 0.00085 $ & $ 1.03886\pm 0.00081 $ & $ 1.04075\pm 0.00033 $ & $ 1.04094\pm 0.00029 $ & $ 1.04037\pm 0.00035 $ & $ 1.04052\pm 0.00032 $ \\
$ n_\mathrm{s} $ & $ 0.9727\pm 0.0055 $ & $ 0.9697\pm 0.0049 $ & $ 1.004\pm 0.020 $ & $ 1.014\pm 0.017 $ & $ 0.994\pm 0.030 $ & $ 0.999\pm 0.022 $ & $ 0.9727\pm 0.0058 $ & $ 0.9716\pm 0.0043 $ & $ 0.9738\pm 0.0056 $ & $ 0.9709\pm 0.0049 $ \\
$ \log(10^{10} A_\mathrm{s}) $ & $ 3.037\pm 0.017 $ & $ 3.058\pm 0.014 $ & $ 3.046\pm 0.033 $ & $ 3.075\pm 0.032 $ & $ 3.032\pm 0.039 $ & $ 3.049\pm 0.036 $ & $ 3.036\pm 0.017 $ & $ 3.058\pm 0.016 $ & $ 3.034\pm 0.018 $ & $ 3.047\pm 0.016 $ \\
$ \Omega_k $ & $ -0.029^{+0.016}_{-0.011} $ & $ 0.0010\pm 0.0021 $ & $ -0.169^{+0.070}_{-0.091} $ & $ 0.0069^{+0.0057}_{-0.0065} $ & $ -0.057^{+0.075}_{-0.035} $ & $ 0.0079^{+0.0060}_{-0.0068} $ & $ -0.091^{+0.047}_{-0.032} $ & $ 0.0011\pm 0.0020 $ & $ -0.047^{+0.029}_{-0.017} $ & $ 0.0014^{+0.0019}_{-0.0022} $ \\
$ m_\mathrm{a}$ [eV]& $ < 1.62 $ & $ < 0.359 $ & $ < 2.18 $ & $ < 1.71 $ & $ < 4.34 $ & $ < 1.71 $ & $ < 1.37 $ & $ < 0.176 $ & $ < 1.96 $ & $ < 0.356 $ \\
$ \sum m_\nu$ [eV] & $ < 0.736 $ & $ < 0.183 $ & $ 2.0^{+1.4}_{-1.4} $ & $ < 0.604 $ & $ < 2.73 $ & $ < 0.567 $ & $ < 1.23 $ & $ < 0.205 $ & $ < 0.808 $ & $ < 0.224 $ \\
$ H_0 $ & $ 56.3\pm 4.3 $ & $ 68.14\pm 0.73 $ & $ 35.8^{+3.3}_{-6.6} $ & $ 68.45\pm 0.94 $ & $ 50^{+9}_{-10} $ & $ 69.07\pm 0.91 $ & $ 45.0^{+4.6}_{-6.6} $ & $ 68.02\pm 0.69 $ & $ 52.7\pm 5.4 $ & $ 68.25\pm 0.73 $ \\
$ \sigma_8 $ & $ 0.701\pm 0.047 $ & $ 0.805^{+0.011}_{-0.0085} $ & $ 0.459^{+0.037}_{-0.069} $ & $ 0.768^{+0.035}_{-0.030} $ & $ 0.535^{+0.084}_{-0.12} $ & $ 0.745^{+0.034}_{-0.030} $ & $ 0.627^{+0.051}_{-0.077} $ & $ 0.808^{+0.013}_{-0.0094} $ & $ 0.688^{+0.067}_{-0.053} $ & $ 0.797^{+0.015}_{-0.0098} $ \\
\hline \hline
\end{tabular}}
\end{center}
\caption{95\%~CL upper bounds on the QCD axion mass and on the sum of neutrino masses and 68\% CL cosmological parameter errors in the presence of a non-zero curvature component.}
\label{tab.omkCDM+mnu+ma}
\end{table*}
\newpage
\begin{table*}[htbp!]
\begin{center}
\renewcommand{\arraystretch}{2}
\resizebox{1 \textwidth}{!}{\begin{tabular}{c | c c| c c| c c| c c| c c }
\hline
\textbf{Parameter} & \textbf{\nq{Planck}} & \textbf{\nq{\\ +BAO}} & \textbf{\nq{ACT}} & \textbf{\nq{\\ +BAO}} & \textbf{\nq{SPT}} & \textbf{\nq{\\ +BAO}} & \textbf{ACT+Planck650} & \nq{\\ \textbf{+BAO}} & \textbf{SPT+Planck650} & \nq{\\ \textbf{+ BAO}} \\
\hline\hline
$ \Omega_\mathrm{b} h^2 $ & $ 0.02245\pm 0.00016 $ & $ 0.02252\pm 0.00015 $ & $ 0.02137\pm 0.00033 $ & $ 0.02152\pm 0.00032 $ & $ 0.02249\pm 0.00034 $ & $ 0.02251\pm 0.00032 $ & $ 0.02234\pm 0.00014 $ & $ 0.02239\pm 0.00014 $ & $ 0.02247\pm 0.00015 $ & $ 0.02252\pm 0.00014 $ \\
$ \Omega_\mathrm{c} h^2 $ & $ 0.1223^{+0.0016}_{-0.0021} $ & $ 0.1208^{+0.0011}_{-0.0014} $ & $ 0.1240^{+0.0051}_{-0.0045} $ & $ 0.1185^{+0.0024}_{-0.0021} $ & $ 0.1186^{+0.0060}_{-0.0047} $ & $ 0.1174^{+0.0023}_{-0.0019} $ & $ 0.1217^{+0.0014}_{-0.0018} $ & $ 0.1205^{+0.0011}_{-0.0014} $ & $ 0.1218^{+0.0017}_{-0.0020} $ & $ 0.1204^{+0.0012}_{-0.0014} $ \\
$ \tau $ & $ 0.0572^{+0.0073}_{-0.0082} $ & $ 0.0598\pm 0.0070 $ & $ 0.064\pm 0.015 $ & $ 0.066\pm 0.015 $ & $ 0.066\pm 0.015 $ & $ 0.066\pm 0.014 $ & $ 0.0543\pm 0.0078 $ & $ 0.0551\pm 0.0077 $ & $ 0.0549\pm 0.0079 $ & $ 0.0556\pm 0.0080 $ \\
$ 100\theta_\mathrm{MC} $ & $ 1.04061\pm 0.00035 $ & $ 1.04082\pm 0.00031 $ & $ 1.04079\pm 0.00075 $ & $ 1.04222\pm 0.00064 $ & $ 1.03864\pm 0.00084 $ & $ 1.03943\pm 0.00065 $ & $ 1.04083\pm 0.00032 $ & $ 1.04098\pm 0.00028 $ & $ 1.04041\pm 0.00033 $ & $ 1.04060\pm 0.00028 $ \\
$ n_\mathrm{s} $ & $ 0.9677\pm 0.0052 $ & $ 0.9700\pm 0.0045 $ & $ 0.916^{+0.024}_{-0.027} $ & $ 0.986\pm 0.019 $ & $ 0.969\pm 0.034 $ & $ 1.001^{+0.025}_{-0.029} $ & $ 0.9724^{+0.0044}_{-0.0050} $ & $ 0.9740^{+0.0040}_{-0.0046} $ & $ 0.9703\pm 0.0052 $ & $ 0.9722\pm 0.0046 $ \\
$ \log(10^{10} A_\mathrm{s}) $ & $ 3.057\pm 0.016 $ & $ 3.058\pm 0.015 $ & $ 3.068\pm 0.031 $ & $ 3.055\pm 0.030 $ & $ 3.041\pm 0.035 $ & $ 3.029\pm 0.034 $ & $ 3.053\pm 0.017 $ & $ 3.051\pm 0.016 $ & $ 3.048\pm 0.017 $ & $ 3.046\pm 0.017 $ \\
$ \alpha_s $ & $ -0.0027\pm 0.0071 $ & $ -0.0029\pm 0.0067 $ & $ 0.133\pm 0.028 $ & $ 0.086\pm 0.029 $ & $ 0.048\pm 0.054 $ & $ 0.031\pm 0.054 $ & $ 0.0083\pm 0.0064 $ & $ 0.0080\pm 0.0063 $ & $ 0.0004\pm 0.0068 $ & $ 0.0003\pm 0.0067 $ \\
$ m_\mathrm{a}$ [eV] & $ < 0.661 $ & $ < 0.294 $ & $ < 3.92 $ & $ < 1.53 $ & $ < 4.86 $ & $ < 1.04 $ & $ < 0.580 $ & $ < 0.248 $ & $ < 0.753 $ & $ < 0.308 $ \\
$ \sum m_\nu$ [eV] & $ < 0.279 $ & $ < 0.155 $ & $ 2.3^{+1.9}_{-1.8} $ & $ < 0.375 $ & $ < 2.06 $ & $ < 0.329 $ & $ < 0.325 $ & $ < 0.172 $ & $ < 0.318 $ & $ < 0.168 $ \\
$ H_0 $ & $ 67.0^{+1.1}_{-0.76} $ & $ 67.93\pm 0.52 $ & $ 51.0^{+3.5}_{-5.2} $ & $ 68.00^{+0.74}_{-0.66} $ & $ 61.0^{+5.8}_{-4.1} $ & $ 68.54\pm 0.72 $ & $ 66.9^{+1.2}_{-0.72} $ & $ 67.79\pm 0.53 $ & $ 67.0^{+1.2}_{-0.77} $ & $ 67.96\pm 0.54 $ \\
$ \sigma_8 $ & $ 0.794^{+0.021}_{-0.010} $ & $ 0.8057^{+0.0098}_{-0.0072} $ & $ 0.489^{+0.045}_{-0.073} $ & $ 0.756\pm 0.031 $ & $ 0.600^{+0.097}_{-0.083} $ & $ 0.753^{+0.031}_{-0.026} $ & $ 0.795^{+0.026}_{-0.012} $ & $ 0.807^{+0.012}_{-0.0086} $ & $ 0.786^{+0.027}_{-0.013} $ & $ 0.799^{+0.012}_{-0.0088} $ \\
\hline \hline
\end{tabular}}
\end{center}
\caption{95\%~CL upper bounds on the QCD axion mass and on the sum of neutrino masses and 68\% CL cosmological parameter errors in the presence of a running of the scalar tilt in the primordial power spectrum.}
\label{tab.mnu+ma+nrun}
\end{table*}
\begin{table*}[htbp!]
\begin{center}
\renewcommand{\arraystretch}{2}
\resizebox{1 \textwidth}{!}{\begin{tabular}{c | c c| c c| c c| c c| c c }
\hline
\textbf{Parameter} & \textbf{\nq{Planck}} & \textbf{\nq{\\ +BAO}} & \textbf{\nq{ACT}} & \textbf{\nq{\\ +BAO}} & \textbf{\nq{SPT}} & \textbf{\nq{\\ +BAO}} & \textbf{ACT+Planck650} & \nq{\\ \textbf{+BAO}} & \textbf{SPT+Planck650} & \nq{\\ \textbf{+ BAO}} \\
\hline\hline
$ \Omega_\mathrm{b} h^2 $ & $ 0.02247\pm 0.00017 $ & $ 0.02248\pm 0.00015 $ & $ 0.02166\pm 0.00032 $ & $ 0.02169\pm 0.00032 $ & $ 0.02249\pm 0.00033 $ & $ 0.02252\pm 0.00033 $ & $ 0.02241\pm 0.00014 $ & $ 0.02242\pm 0.00013 $ & $ 0.02249\pm 0.00015 $ & $ 0.02251\pm 0.00014 $ \\
$ \Omega_\mathrm{c} h^2 $ & $ 0.1221^{+0.0018}_{-0.0023} $ & $ 0.1215^{+0.0013}_{-0.0018} $ & $ 0.1243\pm 0.0045 $ & $ 0.1191\pm 0.0026 $ & $ 0.1198^{+0.0062}_{-0.0048} $ & $ 0.1179\pm 0.0025 $ & $ 0.1212\pm 0.0015 $ & $ 0.1208\pm 0.0013 $ & $ 0.1216^{+0.0016}_{-0.0020} $ & $ 0.1210^{+0.0014}_{-0.0017} $ \\
$ \tau $ & $ 0.0547\pm 0.0077 $ & $ 0.0566\pm 0.0077 $ & $ 0.072\pm 0.015 $ & $ 0.070\pm 0.015 $ & $ 0.067\pm 0.015 $ & $ 0.068\pm 0.014 $ & $ 0.0556\pm 0.0079 $ & $ 0.0559^{+0.0073}_{-0.0081} $ & $ 0.0542\pm 0.0075 $ & $ 0.0548\pm 0.0078 $ \\
$ 100\theta_\mathrm{MC} $ & $ 1.04060\pm 0.00037 $ & $ 1.04071\pm 0.00032 $ & $ 1.04114\pm 0.00080 $ & $ 1.04201\pm 0.00068 $ & $ 1.03846\pm 0.00083 $ & $ 1.03932\pm 0.00068 $ & $ 1.04088\pm 0.00030 $ & $ 1.04096\pm 0.00029 $ & $ 1.04043\pm 0.00034 $ & $ 1.04052\pm 0.00031 $ \\
$ n_\mathrm{s} $ & $ 0.9695\pm 0.0050 $ & $ 0.9697\pm 0.0046 $ & $ 1.003^{+0.022}_{-0.019} $ & $ 1.023\pm 0.015 $ & $ 0.983^{+0.029}_{-0.026} $ & $ 1.013\pm 0.018 $ & $ 0.9708\pm 0.0044 $ & $ 0.9718\pm 0.0041 $ & $ 0.9705\pm 0.0049 $ & $ 0.9716\pm 0.0046 $ \\
$ \log(10^{10} A_\mathrm{s}) $ & $ 3.050\pm 0.016 $ & $ 3.053\pm 0.015 $ & $ 3.078\pm 0.033 $ & $ 3.062\pm 0.031 $ & $ 3.049\pm 0.036 $ & $ 3.034\pm 0.034 $ & $ 3.056\pm 0.016 $ & $ 3.056\pm 0.016 $ & $ 3.046\pm 0.016 $ & $ 3.046\pm 0.016 $ \\
$ w_{0} $ & $ -1.64^{+0.22}_{-0.33} $ & $ -1.052^{+0.061}_{-0.047} $ & $ -1.70\pm 0.63 $ & $ -1.035^{+0.098}_{-0.074} $ & $ -1.55^{+0.80}_{-0.67} $ & $ -1.051^{+0.098}_{-0.075} $ & $ -1.62^{+0.24}_{-0.34} $ & $ -1.039^{+0.057}_{-0.046} $ & $ -1.58^{+0.25}_{-0.36} $ & $ -1.044^{+0.062}_{-0.047} $ \\
$ m_\mathrm{a}$ [eV] & $ < 0.858 $ & $ < 0.466 $ & $ < 2.31 $ & $ < 1.41 $ & $ < 4.96 $ & $ < 1.44 $ & $ < 0.192 $ & $ < 0.181 $ & $ < 0.755 $ & $ < 0.442 $ \\
$ \sum m_\nu $ [eV] & $ < 0.343 $ & $ < 0.221 $ & $ < 2.67 $ & $ < 0.455 $ & $ < 2.30 $ & $ < 0.438 $ & $ < 0.400 $ & $ < 0.232 $ & $ < 0.376 $ & $ < 0.242 $ \\
$ H_0 $ & $ > 83.1 $ & $ 69.0^{+1.2}_{-1.4} $ & $ 72^{+10}_{-20} $ & $ 68.5^{+1.6}_{-1.8} $ & $ 71^{+10}_{-20} $ & $ 69.4^{+1.6}_{-1.9} $ & $ > 82.1 $ & $ 68.6^{+1.1}_{-1.3} $ & $ > 80.5 $ & $ 68.9^{+1.2}_{-1.4} $ \\
$ \sigma_8 $ & $ 0.945^{+0.083}_{-0.053} $ & $ 0.812\pm 0.014 $ & $ 0.73\pm 0.14 $ & $ 0.775^{+0.031}_{-0.028} $ & $ 0.66^{+0.12}_{-0.14} $ & $ 0.751^{+0.031}_{-0.028} $ & $ 0.955^{+0.090}_{-0.057} $ & $ 0.815\pm 0.015 $ & $ 0.926^{+0.092}_{-0.062} $ & $ 0.804\pm 0.016 $ \\
\hline \hline
\end{tabular}}
\end{center}
\caption{95\%~CL upper bounds on the QCD axion mass and on the sum of neutrino masses and 68\% CL cosmological parameter errors varying the dark energy equation of state.}
\label{tab.w0}
\end{table*}
\begin{table*}[htbp!]
\begin{center}
\renewcommand{\arraystretch}{2}
\resizebox{1 \textwidth}{!}{\begin{tabular}{c | c c| c c| c c| c c| c c }
\hline
\textbf{Parameter} & \textbf{\nq{Planck}} & \textbf{\nq{\\ +BAO}} & \textbf{\nq{ACT}} & \textbf{\nq{\\ +BAO}} & \textbf{\nq{SPT}} & \textbf{\nq{\\ +BAO}} & \textbf{ACT+Planck650} & \nq{\\ \textbf{+BAO}} & \textbf{SPT+Planck650} & \nq{\\ \textbf{+ BAO}} \\
\hline\hline
$ \Omega_\mathrm{b} h^2 $ & $ 0.02248\pm 0.00017 $ & $ 0.02245\pm 0.00015 $ & $ 0.02166\pm 0.00033 $ & $ 0.02168\pm 0.00032 $ & $ 0.02249\pm 0.00033 $ & $ 0.02249\pm 0.00033 $ & $ 0.02241\pm 0.00014 $ & $ 0.02239\pm 0.00013 $ & $ 0.02252\pm 0.00016 $ & $ 0.02247\pm 0.00014 $ \\
$ \Omega_\mathrm{c} h^2 $ & $ 0.1220^{+0.0018}_{-0.0022} $ & $ 0.1223^{+0.0015}_{-0.0020} $ & $ 0.1241\pm 0.0045 $ & $ 0.1207\pm 0.0029 $ & $ 0.1198^{+0.0059}_{-0.0046} $ & $ 0.1191\pm 0.0028 $ & $ 0.1212^{+0.0014}_{-0.0016} $ & $ 0.1214^{+0.0012}_{-0.0014} $ & $ 0.1217^{+0.0018}_{-0.0022} $ & $ 0.1216^{+0.0015}_{-0.0018} $ \\
$ \tau $ & $ 0.0547\pm 0.0076 $ & $ 0.0557^{+0.0071}_{-0.0082} $ & $ 0.072\pm 0.015 $ & $ 0.070\pm 0.015 $ & $ 0.067\pm 0.015 $ & $ 0.067\pm 0.015 $ & $ 0.0554\pm 0.0077 $ & $ 0.0556\pm 0.0079 $ & $ 0.0542\pm 0.0078 $ & $ 0.0545^{+0.0070}_{-0.0078} $ \\
$ 100\theta_\mathrm{MC} $ & $ 1.04062\pm 0.00037 $ & $ 1.04060\pm 0.00034 $ & $ 1.04117\pm 0.00081 $ & $ 1.04185\pm 0.00068 $ & $ 1.03846\pm 0.00080 $ & $ 1.03921\pm 0.00070 $ & $ 1.04088\pm 0.00031 $ & $ 1.04087\pm 0.00029 $ & $ 1.04035\pm 0.00035 $ & $ 1.04044\pm 0.00031 $ \\
$ n_\mathrm{s} $ & $ 0.9697\pm 0.0051 $ & $ 0.9691\pm 0.0047 $ & $ 1.004^{+0.022}_{-0.019} $ & $ 1.021\pm 0.015 $ & $ 0.984^{+0.029}_{-0.025} $ & $ 1.010\pm 0.019 $ & $ 0.9711\pm 0.0045 $ & $ 0.9707\pm 0.0042 $ & $ 0.9702^{+0.0058}_{-0.0052} $ & $ 0.9705\pm 0.0047 $ \\
$ \log(10^{10} A_\mathrm{s}) $ & $ 3.050\pm 0.016 $ & $ 3.053^{+0.015}_{-0.017} $ & $ 3.076\pm 0.032 $ & $ 3.066\pm 0.031 $ & $ 3.048\pm 0.036 $ & $ 3.035\pm 0.034 $ & $ 3.056\pm 0.016 $ & $ 3.057\pm 0.016 $ & $ 3.048\pm 0.017 $ & $ 3.047^{+0.015}_{-0.016} $ \\
$ w_{0} $ & $ -1.32^{+0.43}_{-0.55} $ & $ -0.70\pm 0.21 $ & $ -1.46^{+0.72}_{-1.1} $ & $ -0.80\pm 0.25 $ & $ -1.30^{+0.83}_{-1.1} $ & $ -0.88\pm 0.24 $ & $ -1.30^{+0.44}_{-0.58} $ & $ -0.70\pm 0.22 $ & $ -1.10^{+0.47}_{-0.70} $ & $ -0.74\pm 0.22 $ \\
$ w_{a} $ & $ < -0.693 $ & $ -1.14^{+0.79}_{-0.65} $ & $ < -0.325 $ & $ -0.96^{+1.0}_{-0.85} $ & $ < -0.284 $ & $ -0.70^{+1.0}_{-0.74} $ & $ < -0.671 $ & $ -1.10^{+0.80}_{-0.64} $ & $ < -1.19 $ & $ -0.99^{+0.80}_{-0.63} $ \\
$ m_{\rm a}$ [eV] & $ < 0.972 $ & $ < 0.716 $ & $ < 2.36 $ & $ < 1.66 $ & $ < 4.85 $ & $ < 1.59 $ & $ < 0.222 $ & $ < 0.204 $ & $ < 2.60 $ & $ < 0.544 $ \\
$ \sum m_\nu$ [eV]& $ < 0.337 $ & $ < 0.291 $ & $ < 2.66 $ & $ < 0.549 $ & $ < 2.26 $ & $ < 0.505 $ & $ < 0.378 $ & $ < 0.326 $ & $ < 0.420 $ & $ < 0.305 $ \\
$ H_0 $ & $ > 80.4 $ & $ 66.7^{+1.7}_{-2.0} $ & $ 71^{+10}_{-20} $ & $ 67.3^{+1.9}_{-2.3} $ & $ 70^{+10}_{-20} $ & $ 68.4^{+2.1}_{-2.3} $ & $ > 79.6 $ & $ 66.2\pm 1.9 $ & $ > 75.3 $ & $ 66.8\pm 1.9 $ \\
$ \sigma_8 $ & $ 0.929^{+0.10}_{-0.060} $ & $ 0.791\pm 0.019 $ & $ 0.72\pm 0.14 $ & $ 0.766^{+0.034}_{-0.030} $ & $ 0.65^{+0.12}_{-0.14} $ & $ 0.745\pm 0.031 $ & $ 0.941^{+0.11}_{-0.063} $ & $ 0.796\pm 0.020 $ & $ 0.880^{+0.14}_{-0.070} $ & $ 0.789\pm 0.020 $ \\
\hline \hline
\end{tabular}}
\end{center}
\caption{95\%~CL upper bounds on the QCD axion mass and on the sum of neutrino masses and 68\% CL cosmological parameter errors varying the dark energy equation of state using the CPL two-parameter parameterization, see Eq.~(\ref{eq:cpl}).}
\label{tab.w0wa}
\end{table*}
\begin{figure*}
\includegraphics[width =\textwidth]{whisker_all.pdf}
\caption{Whisker plot with the mean values and their $68\%$~CL associated errors on $n_s$, $w_0$, $w_a$, $\alpha_s$ and $\Omega_k$ for different data combinations. The darker (lighter) circles depict the CMB limits with (without) the addition of BAO measurements. In the case of $n_s$ (left panel), we show the results for different background cosmologies, and the blue (red) vertical region refers to the value of $n_s$ as measured by Planck (ACT) within the baseline $\Lambda$CDM model.}
\label{fig:whisker_all}
\end{figure*}
\begin{figure*}
\begin{tabular}{cc}
\includegraphics[width = 0.5\textwidth]{Rmarg_ACT_Planck_BAO_ma.pdf} &
\includegraphics[width = 0.5\textwidth]{Rmarg_ACT_Planck_BAO_mnu.pdf}\\
\includegraphics[width = 0.5\textwidth]{Rmarg_SPT_Planck_BAO_ma.pdf} &
\includegraphics[width = 0.5\textwidth]{Rmarg_SPT_Planck_BAO_mnu.pdf}\\
\end{tabular}
\caption{Model-marginalized relative belief updating ratio $\mathcal{R}$ for $m_{\rm a}$ (left) and $\sum m_\nu$ (right), considering the extensions of the $\Lambda$CDM model considered here. Black and gray lines show the $\mathcal{R}$ function within each model, where the darker lines are those that contribute most to the model marginalization, that is, they have the best Bayesian evidences. Horizontal lines show the significance levels $\exp(-1)$ and $\exp(-3)$. The upper (lower) panel refers to the ACT + Planck650+BAO (Planck + SPT+ BAO) data analyses. Vertical lines
indicate the value 0.1 eV, corresponding to the approximate lower limits for $\sum m_\nu$ in the inverted ordering case.}
\label{fig:Rratio}
\end{figure*}
\begin{table*}[htbp!]
\begin{center}
\renewcommand{\arraystretch}{2}
\resizebox{1 \textwidth}{!}{
\begin{tabular}{c | c c | c c | c c | c c | c c }
\hline
\textbf{Model}
& \textbf{\nq{Planck}} & \textbf{\nq{\\ +BAO}}
& \textbf{\nq{ACT}} & \textbf{\nq{\\ +BAO}}
& \textbf{\nq{SPT}} & \textbf{\nq{\\ +BAO}}
& \textbf{\nq{ACT+Planck650}} & \textbf{\nq{\\ +BAO}}
& \textbf{\nq{SPT+Planck650}} & \textbf{\nq{\\ +BAO}}
\\
\hline\hline
$\Lambda$CDM + $m_{\rm a}$ + $\sum m_\nu$& 6.73 & 6.46 & 0.25 & 3.06 & 1.43 & 3.38 & 4.71 & 5.06 & 4.41 & 5.29 \\
$\Lambda$CDM + $m_{\rm a}$ + $\sum m_\nu$ + $\alpha_s$& 0.00 & 0.00 & 6.35 & 4.39 &
0.22 & 0.79 & 0.00 & 0.45 & 0.00 & 0.00 \\
$\Lambda$CDM + $m_{\rm a}$ + $\sum m_\nu$ +$\Omega_k$ & 5.15 & 0.13 & 1.84 & 0.80 & 0.00 & 0.00 & 7.38 & 0.00 & 5.51 & 0.82 \\
$w\Lambda$CDM + $m_{\rm a}$ + $\sum m_\nu$ & 5.50 & 1.64 & 0.26 & 0.78 & 2.75 & 1.66 & 5.37 & 2.37 & 5.55 & 1.82 \\
$w_0 w_a\Lambda$CDM + $m_{\rm a}$ + $\sum m_\nu$ & 5.35 & 1.62 & 0.00 & 0.00 & 2.36 & 0.70 & 5.68 & 2.37 & 7.29 & 1.51 \\
\hline \hline
\end{tabular}
}
\end{center}
\caption{Logarithms of the Bayes factors with respect to the best model for different data combinations.}
\label{tab:BFs}
\end{table*}
\begin{table*}[htbp!]
\begin{center}
\renewcommand{\arraystretch}{2}
\begin{tabular}{c | c c |c c | c c| c c| c c}
\hline
\textbf{Parameter}
& \textbf{\nq{Planck}} & \textbf{\nq{\\ +BAO}}
& \textbf{\nq{ACT}} & \textbf{\nq{\\ +BAO}}
& \textbf{\nq{SPT}} & \textbf{\nq{\\ +BAO}}
& \textbf{\nq{ACT+Planck650}} & \textbf{\nq{\\ +BAO}}
& \textbf{\nq{SPT+Planck650}} & \textbf{\nq{\\ +BAO}}
\\
\hline\hline
$m_a$ (68 \%) & 0.18 & 0.14 & 1.01 & 0.71 & 2.15 & 0.69 & 0.13 & 0.09 & 0.20 & 0.14 \\
$m_a$ (95 \%) & 0.70 & 0.38 & 2.33 & 1.62 & 4.06 & 1.55 & 0.64 & 0.21 & 0.79 & 0.35 \\
$\sum m_\nu$ (68 \%) & 0.16 & 0.12 & 1.42 & 0.29 & 1.29 & 0.29 & 0.16 & 0.12 & 0.17 & 0.13 \\
$\sum m_\nu$ (95 \%) & 0.31 & 0.21 & 2.79 & 0.55 & 2.59 & 0.53 & 0.33 & 0.21 & 0.34 & 0.23 \\
\hline \hline
\end{tabular}
\end{center}
\caption{Marginalized upper bounds on $m_{\rm a}$ and $\sum m_{\nu}$ in eV for different data combinations.}
\label{tab:marglims}
\end{table*}
\begin{figure*}
\includegraphics[width =\textwidth]{whisker_ma_mnu.pdf}
\caption{Whisker plot with the $95\%$~CL upper bounds on the axion mass $m_{\rm a}$ (left) and on the total neutrino mass $\sum m_\nu$ (right) for different data combinations. The darker (lighter) lines depict the CMB limits with (without) the addition of BAO measurements. The top panels refer to constraints in each of the five possible background cosmologies explored here, while the lower panels show the model-marginalized ones derived here, see the main text of the manuscript for details.}
\label{fig:whisker_ma_mnu}
\end{figure*}
\section{Conclusions}
\label{sec.Conclusions}
Axions provide the most elegant solution to the strong CP problem in Quantum Chromodynamics. In the early universe, axions can be produced via thermal or non thermal processes. Indeed, an axion population produced by scattering and decays of particles can provide additional radiation energy-density contributing to the hot dark matter component of the Universe, similarly to massive neutrinos. Therefore, it is certainly possible to set thermal axion mass limits from cosmology.
Previous works in the literature have computed the current thermal axion population based on chiral perturbation theory. However, these limits can not be extended to high temperatures in the early universe because the underlying perturbation theory would not longer be valid. A possible method to overcome this problem makes use of an interpolation of the thermalization rate in order to cover the gap between the highest safe temperature reachable by chiral perturbation theory and the regime above the confinement scale, where the axion production rate is instead dominated by the axion-gluon scattering~\cite{DEramo:2021psx,DEramo:2021lgb}.
Nevertheless, all previous axion mass bounds in the literature assume the minimal flat $\Lambda$CDM and neglect the other ground-based small-scale CMB measurements than those of \textit{Planck} satellite observations.
Here we relax the two above assumptions and present strong, model-marginalized limits on mixed hot dark matter scenarios, which consider both thermal neutrinos and thermal QCD axions. A novel aspect of our analyses is the inclusion of small-scale Cosmic Microwave Background (CMB) observations from the Atacama Cosmology Telescope (ACT) and the South Pole Telescope (SPT), together with those from the Planck satellite and Baryon Acoustic Oscillation (BAO) data.
The tightest $95\%$~CL marginalised limits are $0.21$~eV, for both $\sum m_\nu$ and $m_{\rm a}$, from the combination of ACT, Planck650 and BAO measurements. Restricting the analyses to the standard $\Lambda$CDM picture extended with free neutrino and axion masses, we find $\sum m_\nu<0.16$~eV and $m_{\rm a}<0.18$~eV, both at $95\%$~CL. Interestingly, the best background cosmology is never found within the minimal $\Lambda$CDM plus hot relics, regardless of the data sets exploited in the analyses. The combination of Planck or Planck 650 with either BAO, SPT or ACT prefers a universe with a non-zero value of the running in the primordial power spectrum with strong evidence. Ground-based small-scale CMB probes, both alone and combined with BAO, prefer either with substantial evidence for non-flat universes, as in the case of SPT, or a model with a time varying dark energy component, as in the case of ACT.
If the existence of an axion which may be thermally produced in the early universe and neutrino masses will be independently confirmed by other probes, upcoming cosmological observations may strengthen the evidence against the minimal cosmological framework, pointing to possible exciting new ingredients in the theory.
\begin{acknowledgments}
\noindent
EDV is supported by a Royal Society Dorothy Hodgkin Research Fellowship.
This article is based upon work from COST Action CA21136 Addressing observational tensions in cosmology with systematics and fundamental physics (CosmoVerse) supported by COST (European Cooperation in Science and Technology). AM and WG are supported by the TASP INFN initiative.
We acknowledge IT Services at The University of Sheffield for the provision of services for High Performance Computing.
This work has been partially supported by the MCIN/AEI/10.13039/501100011033 of Spain under grant PID2020-113644GB-I00, by the Generalitat Valenciana of Spain under grant PROMETEO/2019/083 and by the European Union’s Framework Programme for Research and Innovation Horizon 2020 (2014–2020) under grant agreement 754496 (FELLINI) and 860881 (HIDDeN).
\end{acknowledgments}
\clearpage
\section{Introduction}
\label{sec.Introduction}
The Peccei-Quinn (PQ) mechanism~\cite{Peccei:1977hh,Peccei:1977ur} represents the most elegant solution to the strong CP problem in Quantum Chromodynamics~\cite{Baker:2006ts,Pendlebury:2015lrz,Abel:2020pzs}. The key ingredient consists of a new dynamical pseudo-scalar field -- the \textit{axion}~\cite{Wilczek:1977pj,Weinberg:1977ma} -- which is driven towards its minimal energy configuration by the QCD dynamics, restoring the CP-invariance of strong interactions~\cite{Vafa:1984xg}.
The implications of a cosmic axion background crucially depend on the underlying production mechanism~\cite{DiLuzio:2020wdo}. If axions are produced via non-thermal channels, (\emph{e.g.} by the vacuum realignment mechanism~\cite{Abbott:1982af,Dine:1982ah,Preskill:1982cy,Linde:1985yf,Seckel:1985tj,Lyth:1989pb,Linde:1990yj} and/or by topological defects decay~\cite{Vilenkin:2000jqa,Kibble:1976sj,Kibble:1982dd,Vilenkin:1981kz,Davis:1986xc,Vilenkin:1982ks, Sikivie:1982qv,Vilenkin:1982ks,Huang:1985tt}) they should be considered natural cold dark matter candidates~\cite{Preskill:1982cy,Abbott:1982af,Dine:1982ah}.
Conversely, a thermal population of axions produced by scattering and decays of particles can provide additional radiation energy-density contributing to the hot dark matter component of the Universe, similarly to massive neutrinos. Notice also that, while the axion cold dark matter density is a decreasing function of the mass, the axion couplings are proportional to the mass itself. In order to have a significant thermal population, axions must represent a sub-dominant component of the total cold dark matter abundance and these two scenarios can be analyzed separately.
In this work we shall focus on the thermal axion mass limits from cosmology. A mandatory first step is the calculation of the axion relic abundance.
While most of the cosmological analyses carried out in the literature~\cite{Hannestad:2005df,Melchiorri:2007cd,Hannestad:2007dd,Hannestad:2008js,Hannestad:2010yi,Archidiacono:2013cha,Giusarma:2014zza,DiValentino:2015zta,DiValentino:2015wba,Archidiacono:2015mda,Ferreira:2020bpb} have been based on chiral perturbation theory, in Ref.~\cite{DiLuzio:2021vjd} it was pointed out that this approach can be safely extended only up to a temperature $T\lesssim 60$ MeV (see also Ref.~\cite{Giare:2020vzo,DiLuzio:2022tbb}), since the perturbative scheme breaks down. A practical solution to settle this issue employs an interpolation of the thermalization rate to cover the gap between the highest safe temperature reachable by chiral perturbation theory and the regime above the confinement scale, where the axion production rate is instead dominated by the axion-gluon scattering~\cite{DEramo:2021psx,DEramo:2021lgb}. Improved cosmological bounds~\cite{DEramo:2022nvb} have been derived for the two most accredited models of QCD axion interactions namely, the KSVZ~\cite{Kim:1979if,Shifman:1979if} and the DFSZ one~\cite{Dine:1981rt,Zhitnitsky:1980tq} (see also Ref.~\cite{Caloni:2022uya}). The current $95\%$~CL upper limits obtained in mixed hot dark matter scenarios in which massive neutrino species are also considered are $m_{\rm a}\lesssim 0.2$ eV and $\sum m_\nu\lesssim 0.15$~eV~\cite{DEramo:2022nvb}, using the most recent Cosmic Microwave Background (CMB) data released by the \textit{Planck} satellite~\cite{Aghanim:2019ame,Aghanim:2018eyx,Akrami:2018vks,Aghanim:2018oex}, the astrophysical observations of primordial light elements forged during the Big Bang Nucleosynthesis (BBN) epoch~\cite{Aver:2015iza,Peimbert:2016bdg,Cooke:2017cwo,Aghanim:2018eyx}, and the large scale structure information of the Universe in the form of Baryon Acoustic Oscillation (BAO) measurements~\cite{Beutler:2011hx,Ross:2014qpa,Alam:2016hwk}, see also the recent~\cite{Notari:2022zxo,DiLuzio:2022gsc}.
However, these limits have been obtained under two \emph{standardized} assumptions in cosmological parameter analyses. Namely, \textit{(1)} that the underlying model describing our universe is the minimal flat $\Lambda$CDM, and, \textit{(2)} restricting CMB measurements to those from the \textit{Planck} satellite observations. Concerning the first assumption, one should realize that a number of several intriguing tensions and anomalies have emerged at different statistical levels~\cite{Abdalla:2022yfr,Perivolaropoulos:2021jda,DiValentino:2022fjm,DiValentino:2021izs}, questioning the validity of the canonical flat $\Lambda$CDM picture. A small curvature component, or a more general dark energy fluid, are some examples of very promising scenarios that should be carefully explored. From what regards the second assumption, analyses should also include the recent small-scale measurements of the CMB angular power spectra released by the Atacama Cosmology Telescope~\cite{ACT:2020frw,ACT:2020gnv} (ACT) and the South Pole Telescope~\cite{SPT-3G:2014dbx,SPT-3G:2021eoc} (SPT) Collaborations.
It is therefore clear that the role of parameterizations, priors and models may lead to different constraints on the cosmological neutrino and axion masses.
Quantifying the impact resulting from the parameterization adopted for the cosmological model~\cite{diValentino:2022njd,Gariazzo:2018meg} and also from including independent CMB observations~\cite{DiValentino:2022oon,DiValentino:2022rdg} is the main goal of the present study, where we derive new, and robust, model-marginalized limits on hot dark matter scenarios involving axions $\&$ massive neutrinos.
The paper is organized as follows: in \autoref{sec.Methods} we explain our statistical, computational and data analysis methods, in \autoref{sec.Results} we present the bounds on the hot dark matter masses in the different cosmological scenarios, together with the model-marginalized limits. We conclude in \autoref{sec.Conclusions}.
\section{Methodology}
\label{sec.Methods}
\subsection{Bayesian statistics}
The first aim of this study is to test how the results change when
an extended cosmological model is considered as the underlying theory,
instead of the simple $\Lambda$CDM scenario.
In order to do that, we proceed by performing a marginalization over a number of different models.
Given a set of models ($\mathcal{M}_i$), in order to compute the model-marginalized posterior, one starts defining the posterior probabilities, $p_i$, of the model $\mathcal{M}_i$ over all the possible models:
\begin{equation}
\label{eq:modelposterior}
p_i = \frac{\pi_i Z_i}{\sum _j\pi_j Z_j}\,,
\end{equation}
where $\pi_i$ and $Z_i$ indicates the prior probability and the Bayesian evidence of model $\mathcal{M}_i$.
The model-marginalized posterior $p(\theta|d)$
for a set of parameters $\theta$, given some data $d$, reads as
\begin{equation}
\label{eq:mmposterior_def}
p(\theta|d)
\equiv
\sum_i
p(\theta|d,\mathcal{M}_i)
p_i
\,,
\end{equation}
where $p(\theta|d,\mathcal{M}_i)$ is the parameter posterior within the model $\mathcal{M}_i$. If all models have the same prior and using the Bayes factors $B_{i0}=Z_i/Z_0$ with respect to the favored model $\mathcal{M}_0$, the model-marginalized posterior is:
\begin{equation}
\label{eq:mmposterior}
p(\theta|d)
=
\frac{\sum_i
p(\theta|d,\mathcal{M}_i)
B_{i0}
}{
\sum_j B_{j0}
}
\,.
\end{equation}
Notice that if the Bayes factors are large in favor of the simplest and usually preferred model, extensions of the minimal picture will not contribute significantly to the model-marginalized posterior.
In order to perform Bayesian model comparison using the Bayes factors,
we follow the Jeffreys' scale to evaluate the strength of preference in favor of the best model, see~\autoref{tab:bayes}, extracted from Ref.~\cite{Trotta:2008qt}.
\begin{table*}[htbp!]
\begin{center}
\renewcommand{\arraystretch}{2}
\begin{tabular}{c | c | c | c }
\hline
\textbf{$|\ln B_0|$}
& \textbf{Odds}
& \textbf{Probability}
& \textbf{Strength of evidence}
\\
\hline
$<0.1$& $\lesssim 3:1$ & $<0.750$ & Inconclusive\\
$1$& $\sim 3:1$ & $0.750$ & Weak\\
$2.5$& $\sim 12:1$ & $0.923$ & Moderate\\
$5$& $\sim 150:1$ & $0.993$ & Strong\\
\hline\hline
\end{tabular}
\end{center}
\caption{Jeffreys’ empirical scale to establish the strength of evidence when comparing two competing models.}
\label{tab:bayes}
\end{table*}
In order to avoid the dependency on prior in determining credible intervals, one can make use of the method of Ref.~\cite{Gariazzo:2019xhx}. Given some model $\mathcal{M}$ which contains a parameter $x$ (for instance, the axion or the neutrino mass),
the \emph{relative belief updating ratio} $\mathcal{R}(x_1, x_2|d,\mathcal{M})$ is defined as:
\begin{equation}
\label{eq:R_def}
\mathcal{R}(x_1, x_2|d,\mathcal{M})
\equiv
\frac{Z^{x_1}_\mathcal{M}}{Z^{x_2}_\mathcal{M}}
\,,
\end{equation}
where $Z^{x}_\mathcal{M}$
is defined as the Bayesian evidence of model $\mathcal{M}$ but fixing $x$ to a specific value%
\footnote{We also assume that the prior on $x$ is independent on the other parameters and viceversa.}:
\begin{equation}
\label{eq:Zx_def}
Z^{x}_\mathcal{M}
=
\int_{\Omega_\psi}
d\psi
\pi(\psi|\mathcal{M})
\mathcal{L}_\mathcal{M}(x,\psi)
\,,
\end{equation}
where $\psi$ represents all the parameters in model $\mathcal{M}$ except $x$,
which can vary in a parameter space $\Omega_\psi$,
$\pi(\psi|\mathcal{M})$ is their prior (notice that the $x$ prior is not included here)
and
$\mathcal{L}$ is the likelihood.
From Eq.~\eqref{eq:R_def}, we easily understand that the relative belief updating ratio
does not represent a probability,
as it is the ratio of two evidences. Importantly, the function $\mathcal{R}(x_1, x_2|d,\mathcal{M})$ is completely prior-independent.
Using the Bayes theorem,
it is possible to obtain a different expression for the former function:
\begin{equation}
\label{eq:R}
\mathcal{R}(x_1, x_2|d,\mathcal{M})
=
\frac{
p(x_1|d,\mathcal{M})/\pi(x_1|\mathcal{M})
}{
p(x_2|d,\mathcal{M})/\pi(x_2|\mathcal{M})
}\,,
\end{equation}
where $\pi(x|\mathcal{M})$ is the unidimensional prior on $x$.
This formulation is extremely useful in Monte Carlo Markov Chain (MCMC) runs where one can calculate these functions directly from the run chains.
The definition of $\mathcal{R}(x_1,x_2|d)$
can be easily extended to perform a model marginalization:
\begin{equation}
\label{eq:mmR_def}
\mathcal{R}(x_1, x_2|d)
\equiv
\frac{
\sum_i Z^{x_1}_{\mathcal{M}_i}\pi(\mathcal{M}_i)
}{
\sum_j Z^{x_2}_{\mathcal{M}_j}\pi(\mathcal{M}_j)
}
\,,
\end{equation}
where now the evidences $Z^{x}_{\mathcal{M}_j}$ are computed within a specific model and
$\pi(\mathcal{M}_j)$ is the model prior.
In order to write $\mathcal{R}(x_1, x_2|d)$ using the parameter prior and posterior,
the simplest assumption is to consider the same prior $\pi(x)$ within all the models.
In such case, Eq.~\eqref{eq:mmR_def} becomes:
\begin{equation}
\label{eq:mmR}
\mathcal{R}(x_1, x_2|d)
=
\frac{
p(x_1|d)/\pi(x_1)
}{
p(x_2|d)/\pi(x_2)
}\,,
\end{equation}
where $p(x|d)$ is the model-marginalized posterior in Eq.~\eqref{eq:mmposterior_def}.
\subsection{Axion Modeling}
We address the effects induced by a relic population of thermal axions by employing a modified version of the Boltzmann integrator code \textsc{CAMB} \citep{Lewis:1999bs,Howlett:2012mh}. The code has been modified to accommodate the effects of QCD axions on cosmological scales only in terms of the axion mass which we employ as an additional cosmological parameters in our analysis, see Ref.~\cite{DEramo:2022nvb} for a detailed calculation. We vary the axion mass in the range between $0.01$ eV and $10$ eV and focus exclusively on the KSVZ model of axion-hadron interactions.
As long as the axion remains relativistic ($T_a\gg m_{\rm a}$), it behaves as radiation in the early Universe and its cosmological effects are those produced via their contribution to the effective number of neutrino species $N_{\rm eff}$. As detailed in Ref.~\cite{DEramo:2022nvb}, such a contribution is precisely evaluated by solving the Boltzmann equation for the axion number density in the early Universe. Indeed, very light axions ($m_{\rm a} \lesssim 0.1$ eV) are still relativistic at recombination and thus modify the CMB angular power spectrum via $N_{\rm eff}$. While such corrections are typically very small ($\Delta N_{\rm eff}\sim 0.03$), they are relevant for the next generation of CMB experiments~\cite{Abazajian:2016yjj}. On the other hand, heavier axions with masses larger than $0.1$ eV are highly non-relativistic at the recombination epoch. In this case, their impact on the CMB angular power spectra is both direct (through their impact on the early integrated Sachs-Wolfe effect, similarly to massive neutrinos) and indirect (by modifying the primordial helium abundance during the BBN). In this regard, it is worth stressing that the axion starts behaving as cold dark matter much earlier than massive neutrinos, leading to a significant impact on structure formation. This feature, not only allows to distinguish massive neutrinos from massive axions through their effect on structure formation but it also allows to set stringent constraints on the axion mass exploiting large scale structure data, as well. Nonetheless, when the two species have similar masses, the evolution of their energy densities prevent to reach constraints on their masses lower than $ \sim 0.1$ eV, see also Refs.~\cite{Giare:2021cqr,DEramo:2022nvb}. Allowing for a prior on the axion mass spanning three orders of magnitude we can properly take into account all these effect of light and heavy axions on the CMB anisotropies, see ~\autoref{tab.Priors}.
\subsection{Cosmological Model Parameterization}
\begin{table}[htbp]
\begin{center}
\renewcommand{\arraystretch}{1.5}
\begin{tabular}{l@{\hspace{0. cm}}@{\hspace{1.5 cm}} c}
\hline
\textbf{Parameter} & \textbf{Prior} \\
\hline\hline
$\Omega_{\rm b} h^2$ & $[0.005\,,\,0.1]$ \\
$\Omega_{\rm b} h^2$ & $[0.005\,,\,0.1]$ \\
$\Omega_{\rm k} $ & $[-0.3\,,\,0.3]$\\
$w_0$ & $[-3\,,\,1]$ \\
$w_a$ & $[-3\,,\,2]$ \\
$100\,\theta_{\rm {MC}}$ & $[0.5\,,\,10]$ \\
$\log(10^{10}A_{\rm S})$ & $[2.91\,,\,3.91]$ \\
$n_{\rm s}$ & $[0.8\,,\, 1.2]$ \\
$\alpha_{\rm s}$ & $[-1\,,\, 1]$ \\
$\sum m_{\nu}$ [eV] & $[0.06\,,\,5]$\\
$m_{\rm a}$ [eV] & $[0.01\,,\,10]$\\
\hline\hline
\end{tabular}
\caption{List of the parameter priors.}
\label{tab.Priors}
\end{center}
\end{table}
As pointed out in the \hyperref[sec.Introduction]{Introduction}, a key point in our analysis is to derive robust bounds on the hot dark matter sector marginalizing over a plethora of possible background cosmologies. Therefore, along with the six $\Lambda$CDM parameters (the amplitude $A_s$ and the spectral index $n_s$ of scalar perturbations, the baryon $\Omega_b h^2$ and the cold dark matter $\Omega_c h^2$ energy densities, the angular size of the sound horizon at recombination $\theta_{\rm MC}$ and the reionization optical depth, $\tau$), we also include the sum of neutrino masses $\sum m_\nu$ and the axion mass $m_{\rm a}$. We then explore several extensions of this minimal model, enlarging the parameter space including one or more parameters, such as
a running of the scalar index ($\alpha_s$),
a curvature component ($\Omega_k$),
and
the dark energy equation of state parameters ($w_0$ and $w_a$)
(see \autoref{tab.Priors} for the priors adopted in the cosmological parameters).
\begin{itemize}
\item The running of scalar spectral index, $\alpha_s$. In simple inflationary models, the running of the spectral index is a second order perturbation and it is typically very small. However, specific models can produce a large running over a range of scale accessible to CMB experiments. Indeed, a non-zero value of $\alpha_s$ alleviates the $\sim 2.7\sigma$ discrepancy in the value of the scalar spectral index $n_{\rm s}$ measured by \emph{Planck} ($n_{\rm s}=0.9649\pm 0.0044$)~\cite{Planck:2018vyg,Forconi:2021que} and by the \emph{Atacama Cosmology Telescope} (ACT) ($n_{\rm s}=1.008\pm 0.015$)~\cite{ACT:2020gnv}, see Refs.~\cite{DiValentino:2022rdg, DiValentino:2022oon,Giare:2022rvg}.
\item Curvature density, $\Omega_k$. Recent data analyses of the CMB temperature and polarization spectra from Planck 2018 team exploiting the baseline \emph{Plik} likelihood suggest that our Universe could have a closed geometry at more than three standard deviations~\cite{Planck:2018vyg,Handley:2019tkm,DiValentino:2019qzk,Semenaite:2022unt}. These hints mostly arise from TT observations, that would otherwise show a lensing excess~\cite{DiValentino:2020hov,Calabrese:2008rt,DiValentino:2019dzu}.
In addition, analyses exploiting the \emph{CamSpec} TT likelihood~\cite{Efstathiou:2019mdh,Rosenberg:2022sdy} point to a closed geometry of the Universe with a significance above 99\% CL. Furthermore, an indication for a closed universe is also present in the BAO data, using Effective Field Theories of Large Scale Structure~\cite{Glanville:2022xes}. These recent findings strongly motivate to leave the curvature of the Universe as a free parameter~\cite{Anselmi:2022uvj} and obtain limits on the neutrino and axion masses in this context.
\item Dynamical Dark Energy equation of state.
Cosmological neutrino and axion mass bounds become weaker if the dark energy equation of state is taken as a free parameter. Even if current data fits well with the assumption of a cosmological constant within the minimal $\Lambda$CDM scenario, the question of having an equation of state parameter different from $ -1 $ remains certainly open. Along with constant dark energy equation of state models, in this paper we also consider the possibility of having a time-varying $ w(a) $ described by the Chevalier-Polarski-Linder parametrizazion (CPL)~\cite{Chevallier:2000qy,Linder:2002et}:
\begin{equation}
\label{eq:cpl}
w(a) = w_0 + (1-a)w_a
\end{equation}
where $ a $ is the scale factor and is $ a_0 = 1 $ at the present time, $ w(a_0)=w_0 $ is the value of the equation of state parameter today. Dark energy changes the distance to the CMB consequently pushing it further (closer) if $w < -1$ ($w > -1$) from us. This effect can be balanced by having a larger matter density or, equivalently, by having more massive hot relics, leading to less stringent bounds on both the neutrino and axion masses. Accordingly, the mass bounds of cosmological neutrinos and axions become weaker if the dark energy equation of state is taken as a free parameter.
\end{itemize}
\subsection{Statistical Analyses and Likelihoods}
In order to study the constraints achievable by current CMB and large scale structure probes, we make use of the publicly available code \textsc{COBAYA}~\citep{Torrado:2020xyz}. The code explores the posterior distributions of a given parameter space using the Monte Carlo Markov Chain (MCMC) sampler developed for \textsc{CosmoMC}~\cite{Lewis:2002ah} and tailored for parameter spaces with speed hierarchy implementing the ''fast dragging'' procedure developed in~\cite{Neal:2005}. The prior distributions for the parameters involved in our analysis are chosen to be uniform along the range of variation (see \autoref{tab.Priors}) with the exception of the optical depth for which the prior distribution is chosen accordingly to the CMB datasets as discussed below.
To perform model comparison, we compute the Bayesian Evidence of the different models and estimate the Bayes factors through the publicly available package \texttt{MCEvidence},\footnote{\href{https://github.com/yabebalFantaye/MCEvidence}{github.com/yabebalFantaye/MCEvidence}~\cite{Heavens:2017hkr,Heavens:2017afc}.} properly modified to be compatible with \textsc{COBAYA}.
Concerning the cosmological and astrophysical observations, our baseline data-sets and likelihoods include:
\begin{itemize}[leftmargin=*]
\item Planck 2018 temperature and polarization (TT TE EE) likelihood, which also includes low multipole data ($\ell < 30$)~\citep{Aghanim:2019ame,Aghanim:2018eyx,Akrami:2018vks}. We refer to this combination as ``Planck 2018''.
\item Planck 2018 temperature and polarization (TT TE EE) likelihood up to multipole $\ell=650$, to use in combination with the alternative ground-based small-scales CMB experiments. We refer to this combination as "Planck650".
\item Planck 2018 lensing likelihood~\citep{Aghanim:2018oex}, reconstructed from measurements of the power spectrum of the lensing potential. We refer to this dataset as ``lensing''.
\item Atacama Cosmology Telescope DR4 temperature and polarization (TT TE EE) likelihood, with a Gaussian prior on the optical depth at reionization $\tau = 0.065 \pm 0.015$, as done in~\cite{Aiola:2020azj}. We refer to this dataset combination as "ACT."
\item South Pole Telescope polarization (TE EE) measurements SPT-3G~\cite{SPT-3G:2021eoc} combined with a Gaussian prior on the optical depth at reionization $\tau = 0.065 \pm 0.015$. We refer to this dataset combination as "SPT-3G."
\item Baryon Acoustic Oscillations (BAO) measurements extracted from data from the 6dFGS~\cite{Beutler:2011hx}, SDSS MGS~\cite{Ross:2014qpa} and BOSS DR12~\cite{Alam:2016hwk} surveys. We refer to this dataset combination as ``BAO''.
\end{itemize}
\section{Results}
\label{sec.Results}
We start by discussing the limits in the mixed hot dark matter scenario assuming the standard $\Lambda$CDM cosmology, see \autoref{tab.lCDM+mnu+ma}.
The tightest constraints are obtained when combining Planck650 temperature, polarization and lensing measurements with ACT-CMB and BAO data: the limits we get on the hot dark matter relic masses are $m_{\rm a}<0.18$~eV and $m_\nu<0.16$~eV, both at $95\%$~CL. Adding ACT CMB observations therefore considerably improves the limit on hot relics, as with Planck plus BAO data alone the $95\%$~CL bounds are $m_{\rm a}<0.28$~eV and $m_\nu<0.16$~eV, in perfect agreement with the results for the KSVZ model of Ref.~\cite{DEramo:2022nvb} (see also the recent~\cite{Notari:2022zxo}). Concerning the remaining cosmological parameters, notice that both ACT and SPT observations (either alone or combined with BAO) prefer $n_s\simeq 1$, pointing to a Harrison-Zel'dovich primordial power spectrum, as can also be noticed from the left panel of ~\autoref{fig:whisker_all} (see also Ref.~\cite{Giare:2022rvg}).
Enlarging the minimal $\Lambda$CDM picture with a curvature component $\Omega_k$ only degrades mildly the limits on $\sum m_\nu$, while the limit on $m_{\rm a}$ remains unchanged.
From the results in \autoref{tab.omkCDM+mnu+ma} one can notice that the most constraining bounds are $m_{\rm a}<0.18$~eV and $m_\nu<0.20$~eV, both at $95\%$~CL for ACT plus Planck650 plus BAO observations. The preference for $n_s \simeq 1$ from SPT and ACT still persists, see the left panel of \autoref{fig:whisker_all}. Notice that \emph{all} CMB data prefer a value of $\Omega_k<0$ with a significance above the $\sim 2\sigma$ level for most of the cases. When CMB observations are combined with BAO measurements such a preference is however diluted. This behavior is shown in the bottom right panel of ~\autoref{fig:whisker_all}.
\autoref{tab.mnu+ma+nrun} instead depicts the constraints on the mixed hot dark matter scenario when including a running ($\alpha_s$) of the scalar spectral index $n_s$. In this case, the $95\%$~CL bounds for the most powerful data set combination (i.e. Planck650 plus ACT and BAO) are $m_{\rm a}<0.25$~eV and $m_\nu<0.17$~eV, limiting the constraining power of these observations within the minimal $\Lambda$CDM scenario. Interestingly, the preference for $n_s\simeq 1$ from either SPT and/or ACT is not as strong as in the previous two background cosmologies (see the left panel of ~\autoref{fig:whisker_all}) and it is instead translated into a mild preference for a non-zero value of $\alpha_s$ in the case of SPT. However, ACT observations shows a $\sim 5 \sigma $ preference for a positive value of $\alpha_s$, see the whisker plot in the right panel of ~\autoref{fig:whisker_all}, that corresponds to a preference for a positive neutrino mass.
Leaving freedom in the dark sector of the background cosmology leads to the results shown in \autoref{tab.w0} and \autoref{tab.w0wa}. We start by discussing the simplest dark energy model with a constant dark energy equation of state $w_0$. First of all, notice from the results depicted in \autoref{tab.w0} that \emph{all} CMB measurements prefer a phantom dark energy universe, that is, a universe in which $w_0<-1$. The significance is larger than $2\sigma$ when considering Planck measurements, either alone or in combination with other CMB data sets. The larger negative value of $w_0$ is associated to a very large value of $H_0$, due to their strong degeneracy. Indeed, it has been shown that a phantom-like dark energy component can solve the current tension between high-redshift estimates and local universe measurements of the Hubble constant~\cite{DiValentino:2016hlg}. The addition of BAO observations leads however the value of $w_0$ very close to its cosmological constant expectation of $w_0=-1$ and the mean value of the Hubble constant is notably reduced, $H_0\sim 69$~km/s/Mpc. The results for $w_0$ are illustrated in the top right panel of ~\autoref{fig:whisker_all}. Concerning the limits on the hot relic masses, we obtain $m_{\rm a}<0.18$~eV for the axion mass and $\sum m_\nu <0.23$~eV for the neutrino masses, both at $95\%$~CL for the most powerful data set combination, which is, as in the previous background cosmologies, the one exploiting Planck650 plus ACT plus BAO observations. While the axion mass bound barely changes from the standard $\Lambda$CDM case, the neutrino mass limit is degraded to $\sum m_\nu <0.23$~eV, due to the strong degeneracy between the neutrino mass and the dark energy equation of state: if $w_0$ is allowed to freely change including also the phantom region, $\Omega_m$ can take very high values and also the neutrino mass can be much higher than in standard cosmological backgrounds. \autoref{tab.w0wa} shows the constraints in the widely exploited, two-parameter CPL parameterization for the dark energy component, see Eq.~(\ref{eq:cpl}). The results for $w_0$ are very similar to those previously described, preferring all CMB observations values of $w_0<-1$ albeit with a mild significance. The corresponding $H_0$ value is also considerably larger than within the $\Lambda$CDM scenario (with hot relics). However, in this case, the addition of BAO data shifts the mean value of $w_0$ to the non-phantom region, with a very mild preference ($\sim 1.5\sigma$) for $w_0>-1$. Notice that CMB data alone is unable to measure the time derivative of the dark energy equation of state $w_a$, providing only an upper bound on this parameter. When BAO information is also considered in the analyses, a mean value of $w_a\sim -1$ is preferred. The mean value of the Hubble constant after the inclusion of BAO observations is much closer to the value measured by the Planck collaboration in a standard cosmology. The results above for the dark energy parameters are illustrated by means of the whisker plots for the $w_0$ and $w_a$ parameters depicted in ~\autoref{fig:whisker_all}.
Concerning the hot relics, notice that this background cosmology, having two extra parameters largely degenerated with the neutrino masses, leads to the least constraining hot relic mass bounds: the most powerful combination sets $95\%$~CL limits of $m_{\rm a}<0.20$~eV for the axion mass and $\sum m_\nu <0.33$~eV for the total neutrino mass.
\autoref{tab:BFs} presents the Bayes factors with respect to the best model for each of the five possible background cosmologies considered here and for the different data combinations. Interestingly, the best background cosmology is never found within the minimal $\Lambda$CDM plus two hot dark matter relics, regardless of the data set combinations. The combination of Planck or Planck650 with either BAO, SPT or ACT prefers a universe with a non-zero value of the running in the primordial power spectrum with strong evidence. Ground-based small-scale CMB probes, both alone and combined with BAO, prefer either non-flat universes, as in the case of SPT, or a model with a time varying dark energy component, as in the case of ACT. Such evidences are substantial when including BAO measurements.
\autoref{fig:Rratio} shows the model-marginalized relative belief updating ratio $\mathcal{R}$, Eq.~(\ref{eq:mmR}), for both the axion mass $m_{\rm a}$ (left) and for the sum of the neutrino masses $\sum m_\nu$ (right), considering the extensions of the $\Lambda$CDM model considered and using ACT + Planck650 + BAO (SPT + Planck650 + BAO) data. The horizontal lines show the significance levels $\exp(-1)$ and $\exp(-3)$.
The vertical lines
indicate the value $0.1$~eV, corresponding to the approximate lower limits for $\sum m_\nu$ in the inverted ordering case.
The quantity $\mathcal{R}$ is independent of the shape and normalization of the prior and it is statistically equivalent to a Bayes factor between a model where $m_{\rm a}$ ($m_\nu$) has been fixed to some value and another model where $m_{\rm a}=0$ ($m_\nu=0$). The red curve shows the model-marginalized function $\mathcal{R}$, from which we derive the limits in~\autoref{tab:marglims}. The black and gray lines show the $\mathcal{R}$ function within each model, where the darker lines are those that contribute most to the model marginalization, that is, they have the best Bayesian evidences. For instance, for the case of ACT + Planck650+ BAO, the $95\%$~CL marginalized limit is $0.21$~eV, for both $m_{\rm a}$ and $\sum m_\nu$. Those bounds are led by the models which have the best Bayesian evidences, which, for this particular data combination, are the
$\Lambda$CDM + $m_{\rm a}$ + $\sum m_\nu$ + $\Omega_k$ and the
$\Lambda$CDM + $m_{\rm a}$ + $\sum m_\nu$ + $\alpha_s$ ones, see ~\autoref{tab:BFs}, corresponding to the $95\%$~CL upper bounds of $m_{\rm a}<0.176$~eV, $\sum m_\nu <0.205$~eV and $m_{\rm a}<0.248$~eV, $\sum m_\nu <0.172$~eV, respectively.
Instead, for the other data combination illustrated in ~\autoref{fig:Rratio}, that is, SPT + Planck650 + BAO,
the $95\%$~CL marginalized limits are $0.35$~eV and $0.23$, for $m_{\rm a}$ and $\sum m_\nu$ respectively. Those bounds are led by the models which have the best Bayesian evidences, which, for this particular data combination, are the
$\Lambda$CDM + $m_{\rm a}$ + $\sum m_\nu$ + $\alpha_s$ and the
$\Lambda$CDM + $m_{\rm a}$ + $\sum m_\nu$ + $\Omega_k$ ones, see ~\autoref{tab:BFs}, corresponding to the $95\%$~CL upper bounds of $m_{\rm a}<0.308$~eV, $\sum m_\nu <0.168$~eV and $m_{\rm a}<0.356$~eV, $\sum m_\nu <0.224$~eV, respectively.
Interestingly, the minimal $\Lambda$CDM cosmology never provides the best Bayesian evidence, for any of these two data combinations.
Notice also that, while the ACT + Planck650+ BAO data combination provides more powerful limits on $m_{\rm a}$ than the SPT + Planck650+ BAO one, these two data sets are equally powerful when constraining the neutrino mass, as can be noticed from the results shown in ~\autoref{tab:marglims}.
We conclude this section by summarizing our results in the whisker plots shown in ~\autoref{fig:whisker_ma_mnu}, illustrating the $95\%$~CL upper bounds on the axion mass $m_{\rm a}$ and on the total neutrino mass $\sum m_\nu$ arising for different data combinations in each of the five background cosmologies here. We also depict the model-marginalized limits on these two quantities. For the data combination Planck650 + ACT+ BAO, the most constraining bound for $m_{\rm a}$ is obtained within the $\Lambda$CDM + $m_{\rm a}$ + $\sum m_\nu$ + $\Omega_k$ scenario ($m_{\rm a} < 0.176$~eV at $95\%$~CL). For the total neutrino mass, the tightest $95\%$~CL upper bound ($m_{\rm a} < 0.163$~eV at $95\%$~CL) is found in the $\Lambda$CDM + $m_{\rm a}$ + $\sum m_\nu$ canonical scheme. For the data set SPT + Planck650+ BAO, the tightest limits on the hot thermal relic masses are those derived in the $\Lambda$CDM + $m_{\rm a}$ + $\sum m_\nu$ +
$\alpha_s$ cosmological background, and correspond to $m_{\rm a} < 0.301$~eV and $\sum m_\nu < 0.168$~eV (both at $95\%$~CL).
\begin{table*}[htbp!]
\begin{center}
\renewcommand{\arraystretch}{2}
\resizebox{1 \textwidth}{!}{\begin{tabular}{c | c c| c c| c c| c c| c c }
\hline
\textbf{Parameter} & \textbf{\nq{Planck}} & \textbf{\nq{\\ +BAO}} & \textbf{\nq{ACT}} & \textbf{\nq{\\ +BAO}} & \textbf{\nq{SPT}} & \textbf{\nq{\\ +BAO}} & \textbf{ACT+Planck650} & \nq{\\ \textbf{+BAO}} & \textbf{SPT+Planck650} & \nq{\\ \textbf{+ BAO}} \\
\hline\hline
$ \Omega_\mathrm{b} h^2 $ & $ 0.02243\pm 0.00015 $ & $ 0.02250\pm 0.00015 $ & $ 0.02165\pm 0.00033 $ & $ 0.02170\pm 0.00032 $ & $ 0.02251\pm 0.00033 $ & $ 0.02252\pm 0.00032 $ & $ 0.02238\pm 0.00014 $ & $ 0.02245\pm 0.00013 $ & $ 0.02246\pm 0.00014 $ & $ 0.02252\pm 0.00013 $ \\
$ \Omega_\mathrm{c} h^2 $ & $ 0.1225^{+0.0016}_{-0.0022} $ & $ 0.1208^{+0.0012}_{-0.0014} $ & $ 0.1242\pm 0.0046 $ & $ 0.1190^{+0.0024}_{-0.0020} $ & $ 0.1195^{+0.0060}_{-0.0046} $ & $ 0.1175^{+0.0022}_{-0.0020} $ & $ 0.1214\pm 0.0015 $ & $ 0.1204^{+0.0011}_{-0.0012} $ & $ 0.1216^{+0.0014}_{-0.0018} $ & $ 0.1204^{+0.0012}_{-0.0014} $ \\
$ \tau $ & $ 0.0564\pm 0.0076 $ & $ 0.0581^{+0.0070}_{-0.0080} $ & $ 0.072\pm 0.015 $ & $ 0.070\pm 0.015 $ & $ 0.067\pm 0.015 $ & $ 0.069\pm 0.014 $ & $ 0.0562^{+0.0072}_{-0.0082} $ & $ 0.0567^{+0.0070}_{-0.0081} $ & $ 0.0548\pm 0.0081 $ & $ 0.0556\pm 0.0078 $ \\
$ 100\theta_\mathrm{MC} $ & $ 1.04058\pm 0.00036 $ & $ 1.04081\pm 0.00032 $ & $ 1.04120\pm 0.00081 $ & $ 1.04208\pm 0.00064 $ & $ 1.03854\pm 0.00082 $ & $ 1.03939\pm 0.00066 $ & $ 1.04087\pm 0.00031 $ & $ 1.04101\pm 0.00027 $ & $ 1.04044\pm 0.00032 $ & $ 1.04060\pm 0.00029 $ \\
$ n_\mathrm{s} $ & $ 0.9681\pm 0.0049 $ & $ 0.9703\pm 0.0043 $ & $ 1.005^{+0.022}_{-0.018} $ & $ 1.022\pm 0.015 $ & $ 0.987^{+0.031}_{-0.024} $ & $ 1.013\pm 0.018 $ & $ 0.9705\pm 0.0043 $ & $ 0.9727^{+0.0037}_{-0.0041} $ & $ 0.9700\pm 0.0048 $ & $ 0.9721\pm 0.0043 $ \\
$ \log(10^{10} A_\mathrm{s}) $ & $ 3.055^{+0.015}_{-0.016} $ & $ 3.055^{+0.014}_{-0.016} $ & $ 3.077\pm 0.032 $ & $ 3.063\pm 0.031 $ & $ 3.047\pm 0.035 $ & $ 3.035\pm 0.033 $ & $ 3.059\pm 0.016 $ & $ 3.057^{+0.014}_{-0.016} $ & $ 3.047\pm 0.017 $ & $ 3.047\pm 0.016 $ \\
$ m_\mathrm{a}$ [eV]& $ < 0.888 $ & $ < 0.282 $ & $ < 2.27 $ & $ < 1.12 $ & $ < 4.93 $ & $ < 0.987 $ & $ < 0.190 $ & $ < 0.180 $ & $ < 0.388 $ & $ < 0.310 $ \\
$ \sum m_\nu$ [eV] & $ < 0.278 $ & $ < 0.156 $ & $ < 2.63 $ & $ < 0.351 $ & $ < 2.18 $ & $ < 0.339 $ & $ < 0.305 $ & $ < 0.163 $ & $ < 0.300 $ & $ < 0.169 $ \\
$ H_0 $ & $ 66.9^{+1.2}_{-0.73} $ & $ 67.90\pm 0.53 $ & $ 60^{+7}_{-4} $ & $ 67.97\pm 0.77 $ & $ 61.1^{+6.3}_{-3.9} $ & $ 68.50\pm 0.74 $ & $ 67.0^{+1.1}_{-0.73} $ & $ 67.81\pm 0.52 $ & $ 67.1^{+1.1}_{-0.72} $ & $ 67.95\pm 0.54 $ \\
$ \sigma_8 $ & $ 0.793^{+0.023}_{-0.011} $ & $ 0.8052^{+0.0099}_{-0.0075} $ & $ 0.656^{+0.11}_{-0.076} $ & $ 0.779^{+0.032}_{-0.026} $ & $ 0.611^{+0.10}_{-0.081} $ & $ 0.755^{+0.031}_{-0.024} $ & $ 0.801^{+0.020}_{-0.011} $ & $ 0.809^{+0.011}_{-0.0084} $ & $ 0.791^{+0.020}_{-0.011} $ & $ 0.799^{+0.012}_{-0.0087} $ \\
\hline \hline
\end{tabular}}
\end{center}
\caption{95\%~CL upper bounds on the QCD axion mass and on the sum of neutrino masses and 68\% CL cosmological parameter errors in the minimal $\Lambda$CDM picture.}
\label{tab.lCDM+mnu+ma}
\end{table*}
\begin{table*}[htbp!]
\begin{center}
\renewcommand{\arraystretch}{2}
\resizebox{1 \textwidth}{!}{\begin{tabular}{c | c c| c c| c c| c c| c c }
\hline
\textbf{Parameter} & \textbf{\nq{Planck}} & \textbf{\nq{\\ +BAO}} & \textbf{\nq{ACT}} & \textbf{\nq{\\ +BAO}} & \textbf{\nq{SPT}} & \textbf{\nq{\\ +BAO}} & \textbf{ACT+Planck650} & \nq{\\ \textbf{+BAO}} & \textbf{SPT+Planck650} & \nq{\\ \textbf{+ BAO}} \\
\hline\hline
$ \Omega_\mathrm{b} h^2 $ & $ 0.02256\pm 0.00020 $ & $ 0.02248\pm 0.00016 $ & $ 0.02178\pm 0.00033 $ & $ 0.02166\pm 0.00032 $ & $ 0.02253\pm 0.00034 $ & $ 0.02244\pm 0.00033 $ & $ 0.02249^{+0.00017}_{-0.00021} $ & $ 0.02242\pm 0.00013 $ & $ 0.02261\pm 0.00018 $ & $ 0.02248\pm 0.00015 $ \\
$ \Omega_\mathrm{c} h^2 $ & $ 0.1214\pm 0.0020 $ & $ 0.1214^{+0.0016}_{-0.0019} $ & $ 0.1182\pm 0.0047 $ & $ 0.1237\pm 0.0048 $ & $ 0.1163\pm 0.0063 $ & $ 0.1232\pm 0.0052 $ & $ 0.1203\pm 0.0016 $ & $ 0.1209\pm 0.0014 $ & $ 0.1210\pm 0.0019 $ & $ 0.1211^{+0.0015}_{-0.0018} $ \\
$ \tau $ & $ 0.0497\pm 0.0077 $ & $ 0.0589\pm 0.0070 $ & $ 0.065\pm 0.015 $ & $ 0.071\pm 0.014 $ & $ 0.065\pm 0.015 $ & $ 0.068\pm 0.015 $ & $ 0.0475\pm 0.0079 $ & $ 0.0566\pm 0.0077 $ & $ 0.0488\pm 0.0083 $ & $ 0.0552\pm 0.0079 $ \\
$ 100\theta_\mathrm{MC} $ & $ 1.04058\pm 0.00037 $ & $ 1.04072\pm 0.00034 $ & $ 1.04131\pm 0.00077 $ & $ 1.04157\pm 0.00075 $ & $ 1.03868\pm 0.00085 $ & $ 1.03886\pm 0.00081 $ & $ 1.04075\pm 0.00033 $ & $ 1.04094\pm 0.00029 $ & $ 1.04037\pm 0.00035 $ & $ 1.04052\pm 0.00032 $ \\
$ n_\mathrm{s} $ & $ 0.9727\pm 0.0055 $ & $ 0.9697\pm 0.0049 $ & $ 1.004\pm 0.020 $ & $ 1.014\pm 0.017 $ & $ 0.994\pm 0.030 $ & $ 0.999\pm 0.022 $ & $ 0.9727\pm 0.0058 $ & $ 0.9716\pm 0.0043 $ & $ 0.9738\pm 0.0056 $ & $ 0.9709\pm 0.0049 $ \\
$ \log(10^{10} A_\mathrm{s}) $ & $ 3.037\pm 0.017 $ & $ 3.058\pm 0.014 $ & $ 3.046\pm 0.033 $ & $ 3.075\pm 0.032 $ & $ 3.032\pm 0.039 $ & $ 3.049\pm 0.036 $ & $ 3.036\pm 0.017 $ & $ 3.058\pm 0.016 $ & $ 3.034\pm 0.018 $ & $ 3.047\pm 0.016 $ \\
$ \Omega_k $ & $ -0.029^{+0.016}_{-0.011} $ & $ 0.0010\pm 0.0021 $ & $ -0.169^{+0.070}_{-0.091} $ & $ 0.0069^{+0.0057}_{-0.0065} $ & $ -0.057^{+0.075}_{-0.035} $ & $ 0.0079^{+0.0060}_{-0.0068} $ & $ -0.091^{+0.047}_{-0.032} $ & $ 0.0011\pm 0.0020 $ & $ -0.047^{+0.029}_{-0.017} $ & $ 0.0014^{+0.0019}_{-0.0022} $ \\
$ m_\mathrm{a}$ [eV]& $ < 1.62 $ & $ < 0.359 $ & $ < 2.18 $ & $ < 1.71 $ & $ < 4.34 $ & $ < 1.71 $ & $ < 1.37 $ & $ < 0.176 $ & $ < 1.96 $ & $ < 0.356 $ \\
$ \sum m_\nu$ [eV] & $ < 0.736 $ & $ < 0.183 $ & $ 2.0^{+1.4}_{-1.4} $ & $ < 0.604 $ & $ < 2.73 $ & $ < 0.567 $ & $ < 1.23 $ & $ < 0.205 $ & $ < 0.808 $ & $ < 0.224 $ \\
$ H_0 $ & $ 56.3\pm 4.3 $ & $ 68.14\pm 0.73 $ & $ 35.8^{+3.3}_{-6.6} $ & $ 68.45\pm 0.94 $ & $ 50^{+9}_{-10} $ & $ 69.07\pm 0.91 $ & $ 45.0^{+4.6}_{-6.6} $ & $ 68.02\pm 0.69 $ & $ 52.7\pm 5.4 $ & $ 68.25\pm 0.73 $ \\
$ \sigma_8 $ & $ 0.701\pm 0.047 $ & $ 0.805^{+0.011}_{-0.0085} $ & $ 0.459^{+0.037}_{-0.069} $ & $ 0.768^{+0.035}_{-0.030} $ & $ 0.535^{+0.084}_{-0.12} $ & $ 0.745^{+0.034}_{-0.030} $ & $ 0.627^{+0.051}_{-0.077} $ & $ 0.808^{+0.013}_{-0.0094} $ & $ 0.688^{+0.067}_{-0.053} $ & $ 0.797^{+0.015}_{-0.0098} $ \\
\hline \hline
\end{tabular}}
\end{center}
\caption{95\%~CL upper bounds on the QCD axion mass and on the sum of neutrino masses and 68\% CL cosmological parameter errors in the presence of a non-zero curvature component.}
\label{tab.omkCDM+mnu+ma}
\end{table*}
\newpage
\begin{table*}[htbp!]
\begin{center}
\renewcommand{\arraystretch}{2}
\resizebox{1 \textwidth}{!}{\begin{tabular}{c | c c| c c| c c| c c| c c }
\hline
\textbf{Parameter} & \textbf{\nq{Planck}} & \textbf{\nq{\\ +BAO}} & \textbf{\nq{ACT}} & \textbf{\nq{\\ +BAO}} & \textbf{\nq{SPT}} & \textbf{\nq{\\ +BAO}} & \textbf{ACT+Planck650} & \nq{\\ \textbf{+BAO}} & \textbf{SPT+Planck650} & \nq{\\ \textbf{+ BAO}} \\
\hline\hline
$ \Omega_\mathrm{b} h^2 $ & $ 0.02245\pm 0.00016 $ & $ 0.02252\pm 0.00015 $ & $ 0.02137\pm 0.00033 $ & $ 0.02152\pm 0.00032 $ & $ 0.02249\pm 0.00034 $ & $ 0.02251\pm 0.00032 $ & $ 0.02234\pm 0.00014 $ & $ 0.02239\pm 0.00014 $ & $ 0.02247\pm 0.00015 $ & $ 0.02252\pm 0.00014 $ \\
$ \Omega_\mathrm{c} h^2 $ & $ 0.1223^{+0.0016}_{-0.0021} $ & $ 0.1208^{+0.0011}_{-0.0014} $ & $ 0.1240^{+0.0051}_{-0.0045} $ & $ 0.1185^{+0.0024}_{-0.0021} $ & $ 0.1186^{+0.0060}_{-0.0047} $ & $ 0.1174^{+0.0023}_{-0.0019} $ & $ 0.1217^{+0.0014}_{-0.0018} $ & $ 0.1205^{+0.0011}_{-0.0014} $ & $ 0.1218^{+0.0017}_{-0.0020} $ & $ 0.1204^{+0.0012}_{-0.0014} $ \\
$ \tau $ & $ 0.0572^{+0.0073}_{-0.0082} $ & $ 0.0598\pm 0.0070 $ & $ 0.064\pm 0.015 $ & $ 0.066\pm 0.015 $ & $ 0.066\pm 0.015 $ & $ 0.066\pm 0.014 $ & $ 0.0543\pm 0.0078 $ & $ 0.0551\pm 0.0077 $ & $ 0.0549\pm 0.0079 $ & $ 0.0556\pm 0.0080 $ \\
$ 100\theta_\mathrm{MC} $ & $ 1.04061\pm 0.00035 $ & $ 1.04082\pm 0.00031 $ & $ 1.04079\pm 0.00075 $ & $ 1.04222\pm 0.00064 $ & $ 1.03864\pm 0.00084 $ & $ 1.03943\pm 0.00065 $ & $ 1.04083\pm 0.00032 $ & $ 1.04098\pm 0.00028 $ & $ 1.04041\pm 0.00033 $ & $ 1.04060\pm 0.00028 $ \\
$ n_\mathrm{s} $ & $ 0.9677\pm 0.0052 $ & $ 0.9700\pm 0.0045 $ & $ 0.916^{+0.024}_{-0.027} $ & $ 0.986\pm 0.019 $ & $ 0.969\pm 0.034 $ & $ 1.001^{+0.025}_{-0.029} $ & $ 0.9724^{+0.0044}_{-0.0050} $ & $ 0.9740^{+0.0040}_{-0.0046} $ & $ 0.9703\pm 0.0052 $ & $ 0.9722\pm 0.0046 $ \\
$ \log(10^{10} A_\mathrm{s}) $ & $ 3.057\pm 0.016 $ & $ 3.058\pm 0.015 $ & $ 3.068\pm 0.031 $ & $ 3.055\pm 0.030 $ & $ 3.041\pm 0.035 $ & $ 3.029\pm 0.034 $ & $ 3.053\pm 0.017 $ & $ 3.051\pm 0.016 $ & $ 3.048\pm 0.017 $ & $ 3.046\pm 0.017 $ \\
$ \alpha_s $ & $ -0.0027\pm 0.0071 $ & $ -0.0029\pm 0.0067 $ & $ 0.133\pm 0.028 $ & $ 0.086\pm 0.029 $ & $ 0.048\pm 0.054 $ & $ 0.031\pm 0.054 $ & $ 0.0083\pm 0.0064 $ & $ 0.0080\pm 0.0063 $ & $ 0.0004\pm 0.0068 $ & $ 0.0003\pm 0.0067 $ \\
$ m_\mathrm{a}$ [eV] & $ < 0.661 $ & $ < 0.294 $ & $ < 3.92 $ & $ < 1.53 $ & $ < 4.86 $ & $ < 1.04 $ & $ < 0.580 $ & $ < 0.248 $ & $ < 0.753 $ & $ < 0.308 $ \\
$ \sum m_\nu$ [eV] & $ < 0.279 $ & $ < 0.155 $ & $ 2.3^{+1.9}_{-1.8} $ & $ < 0.375 $ & $ < 2.06 $ & $ < 0.329 $ & $ < 0.325 $ & $ < 0.172 $ & $ < 0.318 $ & $ < 0.168 $ \\
$ H_0 $ & $ 67.0^{+1.1}_{-0.76} $ & $ 67.93\pm 0.52 $ & $ 51.0^{+3.5}_{-5.2} $ & $ 68.00^{+0.74}_{-0.66} $ & $ 61.0^{+5.8}_{-4.1} $ & $ 68.54\pm 0.72 $ & $ 66.9^{+1.2}_{-0.72} $ & $ 67.79\pm 0.53 $ & $ 67.0^{+1.2}_{-0.77} $ & $ 67.96\pm 0.54 $ \\
$ \sigma_8 $ & $ 0.794^{+0.021}_{-0.010} $ & $ 0.8057^{+0.0098}_{-0.0072} $ & $ 0.489^{+0.045}_{-0.073} $ & $ 0.756\pm 0.031 $ & $ 0.600^{+0.097}_{-0.083} $ & $ 0.753^{+0.031}_{-0.026} $ & $ 0.795^{+0.026}_{-0.012} $ & $ 0.807^{+0.012}_{-0.0086} $ & $ 0.786^{+0.027}_{-0.013} $ & $ 0.799^{+0.012}_{-0.0088} $ \\
\hline \hline
\end{tabular}}
\end{center}
\caption{95\%~CL upper bounds on the QCD axion mass and on the sum of neutrino masses and 68\% CL cosmological parameter errors in the presence of a running of the scalar tilt in the primordial power spectrum.}
\label{tab.mnu+ma+nrun}
\end{table*}
\begin{table*}[htbp!]
\begin{center}
\renewcommand{\arraystretch}{2}
\resizebox{1 \textwidth}{!}{\begin{tabular}{c | c c| c c| c c| c c| c c }
\hline
\textbf{Parameter} & \textbf{\nq{Planck}} & \textbf{\nq{\\ +BAO}} & \textbf{\nq{ACT}} & \textbf{\nq{\\ +BAO}} & \textbf{\nq{SPT}} & \textbf{\nq{\\ +BAO}} & \textbf{ACT+Planck650} & \nq{\\ \textbf{+BAO}} & \textbf{SPT+Planck650} & \nq{\\ \textbf{+ BAO}} \\
\hline\hline
$ \Omega_\mathrm{b} h^2 $ & $ 0.02247\pm 0.00017 $ & $ 0.02248\pm 0.00015 $ & $ 0.02166\pm 0.00032 $ & $ 0.02169\pm 0.00032 $ & $ 0.02249\pm 0.00033 $ & $ 0.02252\pm 0.00033 $ & $ 0.02241\pm 0.00014 $ & $ 0.02242\pm 0.00013 $ & $ 0.02249\pm 0.00015 $ & $ 0.02251\pm 0.00014 $ \\
$ \Omega_\mathrm{c} h^2 $ & $ 0.1221^{+0.0018}_{-0.0023} $ & $ 0.1215^{+0.0013}_{-0.0018} $ & $ 0.1243\pm 0.0045 $ & $ 0.1191\pm 0.0026 $ & $ 0.1198^{+0.0062}_{-0.0048} $ & $ 0.1179\pm 0.0025 $ & $ 0.1212\pm 0.0015 $ & $ 0.1208\pm 0.0013 $ & $ 0.1216^{+0.0016}_{-0.0020} $ & $ 0.1210^{+0.0014}_{-0.0017} $ \\
$ \tau $ & $ 0.0547\pm 0.0077 $ & $ 0.0566\pm 0.0077 $ & $ 0.072\pm 0.015 $ & $ 0.070\pm 0.015 $ & $ 0.067\pm 0.015 $ & $ 0.068\pm 0.014 $ & $ 0.0556\pm 0.0079 $ & $ 0.0559^{+0.0073}_{-0.0081} $ & $ 0.0542\pm 0.0075 $ & $ 0.0548\pm 0.0078 $ \\
$ 100\theta_\mathrm{MC} $ & $ 1.04060\pm 0.00037 $ & $ 1.04071\pm 0.00032 $ & $ 1.04114\pm 0.00080 $ & $ 1.04201\pm 0.00068 $ & $ 1.03846\pm 0.00083 $ & $ 1.03932\pm 0.00068 $ & $ 1.04088\pm 0.00030 $ & $ 1.04096\pm 0.00029 $ & $ 1.04043\pm 0.00034 $ & $ 1.04052\pm 0.00031 $ \\
$ n_\mathrm{s} $ & $ 0.9695\pm 0.0050 $ & $ 0.9697\pm 0.0046 $ & $ 1.003^{+0.022}_{-0.019} $ & $ 1.023\pm 0.015 $ & $ 0.983^{+0.029}_{-0.026} $ & $ 1.013\pm 0.018 $ & $ 0.9708\pm 0.0044 $ & $ 0.9718\pm 0.0041 $ & $ 0.9705\pm 0.0049 $ & $ 0.9716\pm 0.0046 $ \\
$ \log(10^{10} A_\mathrm{s}) $ & $ 3.050\pm 0.016 $ & $ 3.053\pm 0.015 $ & $ 3.078\pm 0.033 $ & $ 3.062\pm 0.031 $ & $ 3.049\pm 0.036 $ & $ 3.034\pm 0.034 $ & $ 3.056\pm 0.016 $ & $ 3.056\pm 0.016 $ & $ 3.046\pm 0.016 $ & $ 3.046\pm 0.016 $ \\
$ w_{0} $ & $ -1.64^{+0.22}_{-0.33} $ & $ -1.052^{+0.061}_{-0.047} $ & $ -1.70\pm 0.63 $ & $ -1.035^{+0.098}_{-0.074} $ & $ -1.55^{+0.80}_{-0.67} $ & $ -1.051^{+0.098}_{-0.075} $ & $ -1.62^{+0.24}_{-0.34} $ & $ -1.039^{+0.057}_{-0.046} $ & $ -1.58^{+0.25}_{-0.36} $ & $ -1.044^{+0.062}_{-0.047} $ \\
$ m_\mathrm{a}$ [eV] & $ < 0.858 $ & $ < 0.466 $ & $ < 2.31 $ & $ < 1.41 $ & $ < 4.96 $ & $ < 1.44 $ & $ < 0.192 $ & $ < 0.181 $ & $ < 0.755 $ & $ < 0.442 $ \\
$ \sum m_\nu $ [eV] & $ < 0.343 $ & $ < 0.221 $ & $ < 2.67 $ & $ < 0.455 $ & $ < 2.30 $ & $ < 0.438 $ & $ < 0.400 $ & $ < 0.232 $ & $ < 0.376 $ & $ < 0.242 $ \\
$ H_0 $ & $ > 83.1 $ & $ 69.0^{+1.2}_{-1.4} $ & $ 72^{+10}_{-20} $ & $ 68.5^{+1.6}_{-1.8} $ & $ 71^{+10}_{-20} $ & $ 69.4^{+1.6}_{-1.9} $ & $ > 82.1 $ & $ 68.6^{+1.1}_{-1.3} $ & $ > 80.5 $ & $ 68.9^{+1.2}_{-1.4} $ \\
$ \sigma_8 $ & $ 0.945^{+0.083}_{-0.053} $ & $ 0.812\pm 0.014 $ & $ 0.73\pm 0.14 $ & $ 0.775^{+0.031}_{-0.028} $ & $ 0.66^{+0.12}_{-0.14} $ & $ 0.751^{+0.031}_{-0.028} $ & $ 0.955^{+0.090}_{-0.057} $ & $ 0.815\pm 0.015 $ & $ 0.926^{+0.092}_{-0.062} $ & $ 0.804\pm 0.016 $ \\
\hline \hline
\end{tabular}}
\end{center}
\caption{95\%~CL upper bounds on the QCD axion mass and on the sum of neutrino masses and 68\% CL cosmological parameter errors varying the dark energy equation of state.}
\label{tab.w0}
\end{table*}
\begin{table*}[htbp!]
\begin{center}
\renewcommand{\arraystretch}{2}
\resizebox{1 \textwidth}{!}{\begin{tabular}{c | c c| c c| c c| c c| c c }
\hline
\textbf{Parameter} & \textbf{\nq{Planck}} & \textbf{\nq{\\ +BAO}} & \textbf{\nq{ACT}} & \textbf{\nq{\\ +BAO}} & \textbf{\nq{SPT}} & \textbf{\nq{\\ +BAO}} & \textbf{ACT+Planck650} & \nq{\\ \textbf{+BAO}} & \textbf{SPT+Planck650} & \nq{\\ \textbf{+ BAO}} \\
\hline\hline
$ \Omega_\mathrm{b} h^2 $ & $ 0.02248\pm 0.00017 $ & $ 0.02245\pm 0.00015 $ & $ 0.02166\pm 0.00033 $ & $ 0.02168\pm 0.00032 $ & $ 0.02249\pm 0.00033 $ & $ 0.02249\pm 0.00033 $ & $ 0.02241\pm 0.00014 $ & $ 0.02239\pm 0.00013 $ & $ 0.02252\pm 0.00016 $ & $ 0.02247\pm 0.00014 $ \\
$ \Omega_\mathrm{c} h^2 $ & $ 0.1220^{+0.0018}_{-0.0022} $ & $ 0.1223^{+0.0015}_{-0.0020} $ & $ 0.1241\pm 0.0045 $ & $ 0.1207\pm 0.0029 $ & $ 0.1198^{+0.0059}_{-0.0046} $ & $ 0.1191\pm 0.0028 $ & $ 0.1212^{+0.0014}_{-0.0016} $ & $ 0.1214^{+0.0012}_{-0.0014} $ & $ 0.1217^{+0.0018}_{-0.0022} $ & $ 0.1216^{+0.0015}_{-0.0018} $ \\
$ \tau $ & $ 0.0547\pm 0.0076 $ & $ 0.0557^{+0.0071}_{-0.0082} $ & $ 0.072\pm 0.015 $ & $ 0.070\pm 0.015 $ & $ 0.067\pm 0.015 $ & $ 0.067\pm 0.015 $ & $ 0.0554\pm 0.0077 $ & $ 0.0556\pm 0.0079 $ & $ 0.0542\pm 0.0078 $ & $ 0.0545^{+0.0070}_{-0.0078} $ \\
$ 100\theta_\mathrm{MC} $ & $ 1.04062\pm 0.00037 $ & $ 1.04060\pm 0.00034 $ & $ 1.04117\pm 0.00081 $ & $ 1.04185\pm 0.00068 $ & $ 1.03846\pm 0.00080 $ & $ 1.03921\pm 0.00070 $ & $ 1.04088\pm 0.00031 $ & $ 1.04087\pm 0.00029 $ & $ 1.04035\pm 0.00035 $ & $ 1.04044\pm 0.00031 $ \\
$ n_\mathrm{s} $ & $ 0.9697\pm 0.0051 $ & $ 0.9691\pm 0.0047 $ & $ 1.004^{+0.022}_{-0.019} $ & $ 1.021\pm 0.015 $ & $ 0.984^{+0.029}_{-0.025} $ & $ 1.010\pm 0.019 $ & $ 0.9711\pm 0.0045 $ & $ 0.9707\pm 0.0042 $ & $ 0.9702^{+0.0058}_{-0.0052} $ & $ 0.9705\pm 0.0047 $ \\
$ \log(10^{10} A_\mathrm{s}) $ & $ 3.050\pm 0.016 $ & $ 3.053^{+0.015}_{-0.017} $ & $ 3.076\pm 0.032 $ & $ 3.066\pm 0.031 $ & $ 3.048\pm 0.036 $ & $ 3.035\pm 0.034 $ & $ 3.056\pm 0.016 $ & $ 3.057\pm 0.016 $ & $ 3.048\pm 0.017 $ & $ 3.047^{+0.015}_{-0.016} $ \\
$ w_{0} $ & $ -1.32^{+0.43}_{-0.55} $ & $ -0.70\pm 0.21 $ & $ -1.46^{+0.72}_{-1.1} $ & $ -0.80\pm 0.25 $ & $ -1.30^{+0.83}_{-1.1} $ & $ -0.88\pm 0.24 $ & $ -1.30^{+0.44}_{-0.58} $ & $ -0.70\pm 0.22 $ & $ -1.10^{+0.47}_{-0.70} $ & $ -0.74\pm 0.22 $ \\
$ w_{a} $ & $ < -0.693 $ & $ -1.14^{+0.79}_{-0.65} $ & $ < -0.325 $ & $ -0.96^{+1.0}_{-0.85} $ & $ < -0.284 $ & $ -0.70^{+1.0}_{-0.74} $ & $ < -0.671 $ & $ -1.10^{+0.80}_{-0.64} $ & $ < -1.19 $ & $ -0.99^{+0.80}_{-0.63} $ \\
$ m_{\rm a}$ [eV] & $ < 0.972 $ & $ < 0.716 $ & $ < 2.36 $ & $ < 1.66 $ & $ < 4.85 $ & $ < 1.59 $ & $ < 0.222 $ & $ < 0.204 $ & $ < 2.60 $ & $ < 0.544 $ \\
$ \sum m_\nu$ [eV]& $ < 0.337 $ & $ < 0.291 $ & $ < 2.66 $ & $ < 0.549 $ & $ < 2.26 $ & $ < 0.505 $ & $ < 0.378 $ & $ < 0.326 $ & $ < 0.420 $ & $ < 0.305 $ \\
$ H_0 $ & $ > 80.4 $ & $ 66.7^{+1.7}_{-2.0} $ & $ 71^{+10}_{-20} $ & $ 67.3^{+1.9}_{-2.3} $ & $ 70^{+10}_{-20} $ & $ 68.4^{+2.1}_{-2.3} $ & $ > 79.6 $ & $ 66.2\pm 1.9 $ & $ > 75.3 $ & $ 66.8\pm 1.9 $ \\
$ \sigma_8 $ & $ 0.929^{+0.10}_{-0.060} $ & $ 0.791\pm 0.019 $ & $ 0.72\pm 0.14 $ & $ 0.766^{+0.034}_{-0.030} $ & $ 0.65^{+0.12}_{-0.14} $ & $ 0.745\pm 0.031 $ & $ 0.941^{+0.11}_{-0.063} $ & $ 0.796\pm 0.020 $ & $ 0.880^{+0.14}_{-0.070} $ & $ 0.789\pm 0.020 $ \\
\hline \hline
\end{tabular}}
\end{center}
\caption{95\%~CL upper bounds on the QCD axion mass and on the sum of neutrino masses and 68\% CL cosmological parameter errors varying the dark energy equation of state using the CPL two-parameter parameterization, see Eq.~(\ref{eq:cpl}).}
\label{tab.w0wa}
\end{table*}
\begin{figure*}
\includegraphics[width =\textwidth]{whisker_all.pdf}
\caption{Whisker plot with the mean values and their $68\%$~CL associated errors on $n_s$, $w_0$, $w_a$, $\alpha_s$ and $\Omega_k$ for different data combinations. The darker (lighter) circles depict the CMB limits with (without) the addition of BAO measurements. In the case of $n_s$ (left panel), we show the results for different background cosmologies, and the blue (red) vertical region refers to the value of $n_s$ as measured by Planck (ACT) within the baseline $\Lambda$CDM model.}
\label{fig:whisker_all}
\end{figure*}
\begin{figure*}
\begin{tabular}{cc}
\includegraphics[width = 0.5\textwidth]{Rmarg_ACT_Planck_BAO_ma.pdf} &
\includegraphics[width = 0.5\textwidth]{Rmarg_ACT_Planck_BAO_mnu.pdf}\\
\includegraphics[width = 0.5\textwidth]{Rmarg_SPT_Planck_BAO_ma.pdf} &
\includegraphics[width = 0.5\textwidth]{Rmarg_SPT_Planck_BAO_mnu.pdf}\\
\end{tabular}
\caption{Model-marginalized relative belief updating ratio $\mathcal{R}$ for $m_{\rm a}$ (left) and $\sum m_\nu$ (right), considering the extensions of the $\Lambda$CDM model considered here. Black and gray lines show the $\mathcal{R}$ function within each model, where the darker lines are those that contribute most to the model marginalization, that is, they have the best Bayesian evidences. Horizontal lines show the significance levels $\exp(-1)$ and $\exp(-3)$. The upper (lower) panel refers to the ACT + Planck650+BAO (Planck + SPT+ BAO) data analyses. Vertical lines
indicate the value 0.1 eV, corresponding to the approximate lower limits for $\sum m_\nu$ in the inverted ordering case.}
\label{fig:Rratio}
\end{figure*}
\begin{table*}[htbp!]
\begin{center}
\renewcommand{\arraystretch}{2}
\resizebox{1 \textwidth}{!}{
\begin{tabular}{c | c c | c c | c c | c c | c c }
\hline
\textbf{Model}
& \textbf{\nq{Planck}} & \textbf{\nq{\\ +BAO}}
& \textbf{\nq{ACT}} & \textbf{\nq{\\ +BAO}}
& \textbf{\nq{SPT}} & \textbf{\nq{\\ +BAO}}
& \textbf{\nq{ACT+Planck650}} & \textbf{\nq{\\ +BAO}}
& \textbf{\nq{SPT+Planck650}} & \textbf{\nq{\\ +BAO}}
\\
\hline\hline
$\Lambda$CDM + $m_{\rm a}$ + $\sum m_\nu$& 6.73 & 6.46 & 0.25 & 3.06 & 1.43 & 3.38 & 4.71 & 5.06 & 4.41 & 5.29 \\
$\Lambda$CDM + $m_{\rm a}$ + $\sum m_\nu$ + $\alpha_s$& 0.00 & 0.00 & 6.35 & 4.39 &
0.22 & 0.79 & 0.00 & 0.45 & 0.00 & 0.00 \\
$\Lambda$CDM + $m_{\rm a}$ + $\sum m_\nu$ +$\Omega_k$ & 5.15 & 0.13 & 1.84 & 0.80 & 0.00 & 0.00 & 7.38 & 0.00 & 5.51 & 0.82 \\
$w\Lambda$CDM + $m_{\rm a}$ + $\sum m_\nu$ & 5.50 & 1.64 & 0.26 & 0.78 & 2.75 & 1.66 & 5.37 & 2.37 & 5.55 & 1.82 \\
$w_0 w_a\Lambda$CDM + $m_{\rm a}$ + $\sum m_\nu$ & 5.35 & 1.62 & 0.00 & 0.00 & 2.36 & 0.70 & 5.68 & 2.37 & 7.29 & 1.51 \\
\hline \hline
\end{tabular}
}
\end{center}
\caption{Logarithms of the Bayes factors with respect to the best model for different data combinations.}
\label{tab:BFs}
\end{table*}
\begin{table*}[htbp!]
\begin{center}
\renewcommand{\arraystretch}{2}
\begin{tabular}{c | c c |c c | c c| c c| c c}
\hline
\textbf{Parameter}
& \textbf{\nq{Planck}} & \textbf{\nq{\\ +BAO}}
& \textbf{\nq{ACT}} & \textbf{\nq{\\ +BAO}}
& \textbf{\nq{SPT}} & \textbf{\nq{\\ +BAO}}
& \textbf{\nq{ACT+Planck650}} & \textbf{\nq{\\ +BAO}}
& \textbf{\nq{SPT+Planck650}} & \textbf{\nq{\\ +BAO}}
\\
\hline\hline
$m_a$ (68 \%) & 0.18 & 0.14 & 1.01 & 0.71 & 2.15 & 0.69 & 0.13 & 0.09 & 0.20 & 0.14 \\
$m_a$ (95 \%) & 0.70 & 0.38 & 2.33 & 1.62 & 4.06 & 1.55 & 0.64 & 0.21 & 0.79 & 0.35 \\
$\sum m_\nu$ (68 \%) & 0.16 & 0.12 & 1.42 & 0.29 & 1.29 & 0.29 & 0.16 & 0.12 & 0.17 & 0.13 \\
$\sum m_\nu$ (95 \%) & 0.31 & 0.21 & 2.79 & 0.55 & 2.59 & 0.53 & 0.33 & 0.21 & 0.34 & 0.23 \\
\hline \hline
\end{tabular}
\end{center}
\caption{Marginalized upper bounds on $m_{\rm a}$ and $\sum m_{\nu}$ in eV for different data combinations.}
\label{tab:marglims}
\end{table*}
\begin{figure*}
\includegraphics[width =\textwidth]{whisker_ma_mnu.pdf}
\caption{Whisker plot with the $95\%$~CL upper bounds on the axion mass $m_{\rm a}$ (left) and on the total neutrino mass $\sum m_\nu$ (right) for different data combinations. The darker (lighter) lines depict the CMB limits with (without) the addition of BAO measurements. The top panels refer to constraints in each of the five possible background cosmologies explored here, while the lower panels show the model-marginalized ones derived here, see the main text of the manuscript for details.}
\label{fig:whisker_ma_mnu}
\end{figure*}
\section{Conclusions}
\label{sec.Conclusions}
Axions provide the most elegant solution to the strong CP problem in Quantum Chromodynamics. In the early universe, axions can be produced via thermal or non thermal processes. Indeed, an axion population produced by scattering and decays of particles can provide additional radiation energy-density contributing to the hot dark matter component of the Universe, similarly to massive neutrinos. Therefore, it is certainly possible to set thermal axion mass limits from cosmology.
Previous works in the literature have computed the current thermal axion population based on chiral perturbation theory. However, these limits can not be extended to high temperatures in the early universe because the underlying perturbation theory would not longer be valid. A possible method to overcome this problem makes use of an interpolation of the thermalization rate in order to cover the gap between the highest safe temperature reachable by chiral perturbation theory and the regime above the confinement scale, where the axion production rate is instead dominated by the axion-gluon scattering~\cite{DEramo:2021psx,DEramo:2021lgb}.
Nevertheless, all previous axion mass bounds in the literature assume the minimal flat $\Lambda$CDM and neglect the other ground-based small-scale CMB measurements than those of \textit{Planck} satellite observations.
Here we relax the two above assumptions and present strong, model-marginalized limits on mixed hot dark matter scenarios, which consider both thermal neutrinos and thermal QCD axions. A novel aspect of our analyses is the inclusion of small-scale Cosmic Microwave Background (CMB) observations from the Atacama Cosmology Telescope (ACT) and the South Pole Telescope (SPT), together with those from the Planck satellite and Baryon Acoustic Oscillation (BAO) data.
The tightest $95\%$~CL marginalised limits are $0.21$~eV, for both $\sum m_\nu$ and $m_{\rm a}$, from the combination of ACT, Planck650 and BAO measurements. Restricting the analyses to the standard $\Lambda$CDM picture extended with free neutrino and axion masses, we find $\sum m_\nu<0.16$~eV and $m_{\rm a}<0.18$~eV, both at $95\%$~CL. Interestingly, the best background cosmology is never found within the minimal $\Lambda$CDM plus hot relics, regardless of the data sets exploited in the analyses. The combination of Planck or Planck 650 with either BAO, SPT or ACT prefers a universe with a non-zero value of the running in the primordial power spectrum with strong evidence. Ground-based small-scale CMB probes, both alone and combined with BAO, prefer either with substantial evidence for non-flat universes, as in the case of SPT, or a model with a time varying dark energy component, as in the case of ACT.
If the existence of an axion which may be thermally produced in the early universe and neutrino masses will be independently confirmed by other probes, upcoming cosmological observations may strengthen the evidence against the minimal cosmological framework, pointing to possible exciting new ingredients in the theory.
\begin{acknowledgments}
\noindent
EDV is supported by a Royal Society Dorothy Hodgkin Research Fellowship.
This article is based upon work from COST Action CA21136 Addressing observational tensions in cosmology with systematics and fundamental physics (CosmoVerse) supported by COST (European Cooperation in Science and Technology). AM and WG are supported by the TASP INFN initiative.
We acknowledge IT Services at The University of Sheffield for the provision of services for High Performance Computing.
This work has been partially supported by the MCIN/AEI/10.13039/501100011033 of Spain under grant PID2020-113644GB-I00, by the Generalitat Valenciana of Spain under grant PROMETEO/2019/083 and by the European Union’s Framework Programme for Research and Innovation Horizon 2020 (2014–2020) under grant agreement 754496 (FELLINI) and 860881 (HIDDeN).
\end{acknowledgments}
\clearpage
|
1,116,691,497,565 | arxiv | \section{Introduction}
The following parabolic initial-boundary value problem is considered
for a singularly perturbed linear system of second order
differential equations
\begin{equation}\label{BVP}
\frac{\partial\vec{u}}{\partial t}
-E\frac{\partial^2\vec{u}}{\partial
x^2}+A\vec{u}=\vec{f},\;\;\mathrm{on}\;\;
\Omega,\;\;\vec{u}\;\;\mathrm{given\;\; on}\;\;\Gamma,
\end{equation}
where $\Omega=\{(x,t): 0 < x < 1, 0 < t \leq T\}$,\;\;
$\overline{\Omega}=\Omega \cup \Gamma,\;\; \Gamma = \Gamma_L \cup
\Gamma_B \cup \Gamma_R$ with $ \vec u(0,t)=\vec \phi_L(t)\;
\mathrm{on}\;\Gamma_L=\{(0,t): 0 \leq t \leq T\},\;\; \vec
u(x,0)=\vec \phi_B(x)\;\mathrm{on}\;\Gamma_B=\{(x,0): 0 \leq x \leq
1\},\;\; \vec u(1,t)=\vec \phi_R(t)\; \mathrm{on}\;\Gamma_R=\{(1,t):
0 \leq t \leq T\}.$ Here, for all $(x,t) \in \overline{\Omega},$
$\;\vec{u}(x,t)$ and $\vec{f}(x,t)$ are column
$n-\mathrm{vectors},\;E\;$ and $\;A(x,t)\;$ are $\;n\times n\;$
matrices, $\;E = \mathrm{diag}(\vec{\eps}),\;\vec\eps =
(\eps_1,\;\cdots,\;\eps_n)\;$ with $\;0\;<\;\eps_i\;\le\;1\;$ for
all $\;i=1,\ldots,n$. The $\eps_i$ are assumed to be distinct and,
for convenience, to have the ordering\[\eps_1\;<\;\cdots\;<\;\eps_n.\] Cases with some of the parameters coincident are not considered here.\\
\noindent The problem can also be written in the operator form
\[\vec L\vec u\; = \;\vec{f}\;\;\mathrm{on}\;\;
\Omega,\;\;\vec{u}\;\;\mathrm{given\;\; on}\;\;\Gamma,\] where the
operator $\;\vec{L}\;$ is defined by
\[\vec{L}\;=\;\frac{\partial}{\partial t}-E\frac{\partial^2}{\partial x^2} + A.\]
For all $(x,t) \in \overline{\Omega}$ it is assumed
that the components $a_{ij}(x,t)$ of $A(x,t)$
satisfy the inequalities \\
\begin{eqnarray}\label{a1} a_{ii}(x,t) > \displaystyle{\sum_{^{j\neq
i}_{j=1}}^{n}}|a_{ij}(x,t)| \; \; \rm{for}\;\; 1 \le i \le n, \;\; \rm{and} \;\;a_{ij}(x,t) \le 0 \;\; \rm{for} \; \; i \neq j\end{eqnarray} and, for some $\alpha$,
\begin{eqnarray}\label{a2} 0 <\alpha <
\displaystyle{\min_{^{(x,t)\in \overline\Omega}_{1 \leq i \leq
n}}}(\sum_{j=1}^n a_{ij}(x,t)).
\end{eqnarray}
It is also assumed, without loss of generality, that
\begin{eqnarray}\label{a3}
\max_{1 \leq i \leq n} \sqrt{\eps_i} \leq \frac{\sqrt{\alpha}}{6}.
\end{eqnarray}
The reduced problem corresponding to \eqref{BVP} is defined by
\begin{equation}
\frac{\partial\vec{u}_0}{\partial t}
+A\vec{u}_0=\vec{f},\;\;\mathrm{on}\;\;
\Omega,\;\;\vec{u}_0=\vec{u}\;\;\mathrm{on}\;\;\Gamma_B.
\end{equation}
The norms $\parallel \vec{V} \parallel =\max_{1 \leq k \leq n}|V_k|$
for any n-vector $\vec{V}$, $\parallel y
\parallel_D =\sup\{|y(x,t)|: (x,t)\in D\}$ for any
scalar-valued function $y$ and domain $D$, and $\parallel \vec{y}
\parallel=\max_{1 \leq k \leq n}\parallel y_{k}
\parallel$ for any vector-valued function $\vec{y}$ are introduced.
When $D=\overline{\Omega}$ or $\Omega$ the subscript $D$ is usually
dropped. Throughout the paper $C$ denotes a generic positive
constant, which is independent of $x, t$ and of all singular
perturbation and discretization parameters. Furthermore,
inequalities between vectors
are understood in the componentwise sense. Whenever necessary the required smoothness of the problem data is assumed.\\
\noindent For a general introduction to parameter-uniform numerical methods for singular perturbation problems, see \cite{MORS}, \cite{RST} and \cite{FHMORS}. The
piecewise-uniform Shishkin meshes $\Omega^{M,N}$ in the present paper have the
elegant property that they reduce to uniform meshes when
the parameters are not small. The problem posed in the present paper
is also considered in \cite{GLOR}, where parameter uniform
convergence is proved, which is first order in time and essentially
first order in space. The meshes used there do not have the above
typical property of Shishkin meshes. The main result of the present
paper is well known in the scalar case, when $n=1$. It is
established in \cite{L} for the case $n=2$. The proof below of first
order convergence in the time variable and essentially second order
convergence in the space variable, for general $n$, draws heavily
on the analogous result in \cite{PVM} for a reaction-diffusion
system.
The plan of the paper is as follows. In the next two sections both standard and novel bounds on the smooth and singular components of the exact solution are
obtained. The sharp estimates for the singular component in Lemma \ref{lsingular}
are proved by mathematical induction, while interesting orderings of the points $x_{i,j}$ are
established in Lemma \ref{layers}. In Section 4
piecewise-uniform Shishkin meshes are introduced, in Section 5 the
discrete problem is defined and the discrete maximum principle, the discrete stability properties and a comparison result are established. In Section 6 an
expression for the local truncation error is derived and standard estimates are stated. In Section 7 parameter-uniform estimates for the local truncation error of
the smooth and singular components are obtained in a sequence of theorems. The section culminates with the statement and proof of the essentially second order
parameter-uniform error
estimate.\\
\section{Standard analytical results}
The operator $\vec L$ satisfies the following maximum principle
\begin{lemma}\label{max} Let $A(x,t)$ satisfy (\ref{a1}) and (\ref{a2}). Let
$\;\vec{\psi}\;$ be any function in the domain of $\;\vec L\;$ such
that $\vec{\psi}\ge \vec{0}$ on $\Gamma$. Then $\;\vec
L\vec{\psi}(x)\ge\vec{0}\;$ on $\Omega$ implies that
$\;\vec{\psi}(x)\ge\vec{0}\;$ on $\overline{\Omega}$.
\end{lemma}
\begin{proof}Let $i^*, x^*, t^*$ be such that $\psi_{i^*}(x^{*}, t^*)=\min_{i}\min_{\overline{\Omega}}\psi_i(x,t)$
and assume that the lemma is false. Then $\psi_{i^*}(x^{*},t^*)<0$ .
From the hypotheses we have $(x^*,t^*) \not\in\;\Gamma$ and
$\frac{\partial^2 \psi_{i^*}}{\partial x^2}(x^*,t^*)\geq 0$. Thus
\begin{equation*}(\vec{L}\vec \psi(x^*,t^*))_{i^*}= \frac{\partial \psi_{i^*}}{\partial t}(x^*,t^*)
-\eps_{i^*}\frac{\partial^2\psi_{i^*}}{\partial
x^2}(x^*,t^*)+\sum_{j=1}^n a_{i^*,j}(x^{*},t^*)\psi_j(x^*,t^*)<0,
\end{equation*} which contradicts the assumption and proves the
result for $\vec{L}$.\eop \end{proof}
Let $\tilde{A}(x,t)$ be any principal sub-matrix of $A(x,t)$ and $\vec{\tilde{L}}$ the corresponding operator. To see that any $\vec{\tilde{L}}$ satisfies the same
maximum principle as $\vec{L}$, it suffices to observe that the elements of $\tilde{A}(x,t)$ satisfy \emph{a fortiori} the same inequalities as those of $A(x,t)$.
\begin{lemma}\label{stab} Let $A(x,t)$ satisfy (\ref{a1}) and (\ref{a2}). If $\vec{\psi}$ is any function in the domain of $\;\vec L,\;$
then, for each $i, \; 1 \leq i \leq n $ and $(x,t)\in \overline{\Omega}$, \[|\psi_i(x,t)| \le\;
\max\ds\left\{\parallel\vec{\psi}\parallel_{\Gamma}, \dfrac{1}{\alpha}\parallel \vec L\vec \psi\parallel\right\}.\]
\end{lemma}
\begin{proof}Define the two functions \[\vec \theta^\pm(x,t)\;=\;
\max\ds\left\{\parallel\vec{\psi}\parallel_{\Gamma},\;\dfrac{1}{\alpha}\parallel\vec{L}\vec \psi\parallel\right\}\vec e\;\pm\;\vec \psi(x,t)\] where $\;\vec
e\;=\;(1,\;\ldots,\;1)^T\;$ is the unit column vector. Using the properties of $\;A\;$ it is not hard to verify that $\vec \theta^\pm\;\ge\;\vec 0$ on $\Gamma$ and
$\;\vec L\vec \theta^\pm\;\ge\;\vec 0$ on $\Omega.$ It follows from Lemma \ref{max} that $\;\vec \theta^\pm\;\ge\;\vec 0\;$ on $\overline{\Omega}$ as required.\eop
\end{proof}
A standard estimate of the exact solution and its derivatives is
contained in the following lemma.
\begin{lemma}\label{lexact} Let $A(x,t)$ satisfy (\ref{a1}) and (\ref{a2})
and let $\vec u$ be the exact solution of (\ref{BVP}). Then, for all $(x,t)\in \oln \Omega$ and each $i=1,\dots,n$,
\begin{equation*}\label{e1}
\begin{array}{lcl}
|\frac{\partial^l u_i}{\partial t^l}(x,t)| &\leq& C(||\vec{u}||_\Gamma+\sum_{q=0}^{l}||\frac{\partial^q\vec{f}}{\partial t^q}||),\;\; l= 0,1,2\\\\
|\frac{\partial^l u_i}{\partial x^l}(x,t)| &\leq& C\eps_i^{\frac{-l}{2}}(||\vec{u}||_\Gamma+||\vec{f}||+||\frac{\partial\vec{f}}{\partial t}||), \;\; l=1,2\\\\
|\frac{\partial^l u_i}{\partial x^l}(x,t)| &\leq&
C\eps^{-1}_i \eps^{\frac{-(l-2)}{2}}_1 (||\vec{u}||_\Gamma+||\vec{f}||+
||\frac{\partial\vec{f}}{\partial t}||+||\frac{\partial^2 \vec f}{\partial t^2}||+
\eps^{\frac{l-2}{2}}_1||\frac{\partial^{l-2} \vec f}{\partial x^{l-2}}||), \;\; l=3,4\\\\
|\frac{\partial^l u_i}{\partial^{l-1}x \partial t}(x,t)| &\leq& C\eps_i^{\frac{1-l}{2}}(||\vec{u}||_\Gamma+||\vec{f}||+||\frac{\partial\vec{f}}{\partial
t}||+||\frac{\partial^2 \vec f}{\partial t^2}||), \;\; l=2,3.
\end{array}
\end{equation*}
\end{lemma}
\begin{proof}
The bound on $\vec{u}$ is an immediate consequence of Lemma \ref{stab}.
Differentiating (\ref{BVP}) partially with respect to '$t$' once and twice, and applying Lemma \ref{stab} the bounds $ \frac{\partial \vec u}{\partial t}$, and $
\frac{\partial^2 \vec u}{\partial t^2} $ are obtained.
To bound $\frac{\partial u_i}{\partial x}$, for all $i$ and any $(x,t)$, consider an interval $I=(a, a+\sqrt \eps_i)$ such that $x \in I$.\\
Then for some $y\in I$ and $t \in (0,T]$
\[\s\frac{\partial u_i}{\partial x}(y,t)=\frac{u_i(a+\sqrt{\eps_i},t)-u_i(a,t)}{\sqrt{\eps_i}}\]
\begin{equation} \label{e2}
|\frac{\partial u_i}{\partial x}(y,t)|\leq C {\eps_i}^{\frac{-1}{2}}||\vec u||.\s\s\;\;\;
\end{equation}
Then for any $ x \in I $
\[\frac{\partial u_i}{\partial x}(x,t) =\frac{\partial u_i}{\partial x}(y,t)+\int_y^x\frac{\partial ^2 u_i(s,t)}{\partial x^2}ds \]
\[\frac{\partial u_i}{\partial x}(x,t) =\frac{\partial u_i}{\partial x}(y,t)+\eps_i^{-1}\int_y^x\left(\frac{\partial u_i(s,t)}{\partial t} - f_i(s,t)+\sum_{j=1}^{n} a_{ij}(s,t)u_j(s,t)\right)ds \]
\[|\frac{\partial u_i}{\partial x}(x,t)| \leq |\frac{\partial u_i}{\partial x}(y,t)|+C\eps_i^{-1}\int_y^x(||\vec u||_\Gamma + ||\vec f||+||\frac{\partial \vec f}{\partial t}||)ds.\]
Using (\ref{e2}) in the above equation
\[|\frac{\partial u_i}{\partial x}(x,t)|\leq C {\eps_i}^{\frac{-1}{2}}(||\vec u||_\Gamma+||\vec f||+||\frac{\partial \vec f}{\partial t}||).\]
Rearranging the terms in (\ref{BVP}), it is easy to get
\[|\frac{\partial^2 u_i}{\partial x^2}|\leq C \eps_i^{-1}(||\vec u||_{\Gamma}+||\vec f||+||\frac{\partial \vec f}{\partial t}||).\]
Analogous steps are used to get the rest of the estimates. \s\s\s\eop \end{proof}
The Shishkin decomposition of the exact solution $\;\vec{u}\;$ of
(\ref{BVP}) is $\;\vec{u}=\vec{v}+\vec{w}\;$ where the smooth
component $\;\vec v\;$ is the solution of
\begin{equation}\label{smoothcomp}\;\vec L\vec v = \vec f\; \; \mathrm{in} \; \Omega,
\;\vec
v = \vec u_0\; \; \mathrm{on} \; \Gamma
\end{equation} and the singular component $\;\vec w\;$
is the solution of \begin{equation}\label{singularcomp} \vec L\vec
w\;=\;\vec 0 \; \mathrm{in} \; \Omega, \;\vec w = \vec u-\vec v\; \;
\mathrm{on} \; \Gamma.
\end{equation} For convenience the left and right boundary layers of
$\vec w$ are separated using the further decomposition $\vec w
=\vec{w}^L+\vec{w}^R$ where $\vec{L}\vec{w}^L=\vec{0}$\;on
$\Omega,\; \vec{w}^L=\vec w$ \; on $\Gamma_L,\; \vec{w}^L=\vec{0}$
on $\Gamma_B \cup \Gamma_R$ and $\vec{L}\vec{w}^R= \vec{0}$ \; on
$\Omega,\; \vec{w}^R=\vec{w}$\;on $\Gamma_R$, $\vec{w}^R=\vec{0}$
on $\Gamma_L \cup \Gamma_B$.\\
Bounds on the smooth component and its derivatives are contained in
\begin{lemma}\label{lsmooth1} Let $A(x,t)$ satisfy (\ref{a1}) and (\ref{a2}).
Then the smooth component $\vec v$ and its derivatives satisfy, for all $(x,t) \in \overline \Omega$ and each $i=1,\dots,n$,
\begin{equation*} \label{e5}
\begin{array}{lcl}
|\frac{\partial^l v_i}{\partial t^l}(x,t)| &\leq& C \; \mathrm{for}\; l=0,1,2\\\\
|\frac{\partial^l v_i}{\partial x^l}(x,t)| &\leq& C(1+\eps_i^{1-\frac{l}{2}})\; \mathrm{for}\; l=0,1,2,3,4\\\\
|\frac{\partial^l v_i}{\partial x^{l-1} \partial t}(x,t)| &\leq& C\;\; \text{for}\;\; l=2,3.
\end{array}
\end{equation*}
\end{lemma}
\begin{proof}
The bound on $\vec{v}$ is an immediate consequence of the defining
equations for $\vec{v}$ and Lemma (\ref{stab}). Differentiating the
equation \eqref{smoothcomp} twice partially with respect to $x$ and
applying Lemma \ref{stab} for $\frac{\partial^2 v_i}{\partial x^2}$,
we get
\begin{equation}\label{e5a}
|\frac{\partial^2v_i}{\partial x^2}|\leq C(1+||\frac{\partial \vec v}{\partial x}||).
\end{equation}
Let
\begin{equation}\label{e6}
\frac{\partial v_{i^*}}{\partial x}(x^*,t*) = ||\frac{\partial
\vec v}{\partial x}|| \s \text{for some}\; i=i^* , \; x=x^*,\;
t=t^*.
\end{equation}
Using Taylor expansion, it follows that, for some $ y \in [0,1-x^*]$ and some $\eta \in (x^*, x^*+y)$
\begin{equation}\label{e7} v_{i^*}(x^*+y,t^*)=v_{i^*}(x^*,t^*)+y\frac{\partial v_{i^*}}{\partial x}(x^*,t^*)+\frac{y^2}{2} \frac{\partial^2 v_{i^*}}{\partial x^2}(\eta,t^*).
\end{equation}
Rearranging (\ref{e7}) yields
\[ \frac{\partial v_{i^*}}{\partial x}(x^*,t^*)= \frac{v_{i^*}(x^*+y,t^*)-v_{i^*}(x^*,t^*)}{y}-\frac{y}{2} \frac{\partial^2v_{i^*}}{\partial x^2}(\eta,t^*) \]
\begin{equation} \label{e8}
|\frac{\partial v_{i^*}}{\partial x}(x^*,t^*)| \leq
\frac{2}{y}||\vec v||+\frac{y}{2}||\frac{\partial^2 \vec
v}{\partial x^2}||.
\end{equation}
Using (\ref{e6}) and (\ref{e8}) in (\ref{e5a}),
\[|\frac{\partial^2v_i}{\partial x^2}|\leq C(1+\frac{2}{y}||\vec v||+\frac{y}{2}||\frac{\partial ^2 \vec v}{\partial x^2}||).\]
This leads to
\[(1-\frac{Cy}{2}) ||\frac{\partial^2 \vec v}{\partial x^2}||\leq C(1+\frac{2}{y}||\vec v||)\]
or\\
\begin{equation}\label{e9}
||\frac{\partial^2 \vec v}{ \partial x^2}|| \leq C.
\end{equation}
Using (\ref{e9}) in (\ref{e8}) yields
\[||\frac{\partial \vec v}{\partial x}|| \leq C. \]
Repeating the above steps with $\frac{\partial v_i}{\partial t}$,
it is easy to get the required bounds on the mixed derivatives.
The bounds on $\dfrac{\partial^3 \vec v}{\partial
x^3},\;\;\dfrac{\partial^4 \vec v}{\partial x^4} $ are derived by
a similar argument.\s\s\s\s\s\s\s\s\s\s\s\s\s\s\eop
\end{proof}
\section{Improved estimates}
The layer functions $B^{L}_{i}, \; B^{R}_{i}, \; B_{i}, \; i=1,\;
\dots , \; n,\;$, associated with the solution $\;\vec u$, are
defined on $[0,1]$ by
\[B^{L}_{i}(x) = e^{-x\sqrt{\alpha/\eps_i}},\;B^{R}_{i}(x) =
B^{L}_{i}(1-x),\;B_{i}(x) = B^{L}_{i}(x)+B^{R}_{i}(x).\] The
following elementary properties of these layer functions, for all $1
\leq i < j \leq n$ and $0 \leq x < y \leq
1$, should be noted:\\
(a)\;$B^{L}_i(x)\; <\; B^{L}_j(x),\;\;B^{L}_i(x)\;
>\; B^{L}_i(y), \;\;0\;<\;B^{L}_i(x)\;\leq\;1$.\\
(b)\;$B^{R}_i(x)\; <\; B^{R}_j(x),\;\;B^{R}_i(x)\; <\;
B^{R}_i(y), \;\;0\;<\;B^{R}_i(x)\;\leq\;1$.\\
(c)\;$B_{i}(x)$ is monotone decreasing (increasing) for
increasing $x \in [0,\frac{1}{2}] ([\frac{1}{2},1])$.\\
(d)\;$B_{i}(x) \leq 2B_{i}^L(x)$ for $x \in [0,\frac{1}{2}]$,
\;$B_{i}(x) \leq 2B_{i}^R(x)$ for $x \in [\frac{1}{2},1]$.
\begin{definition}
For $B_i^L$, $B_j^L$, each $i,j, \;\;1 \leq i \neq j \leq n$ and
each $s, s>0$, the point $x^{(s)}_{i,j}$ is defined by
\begin{equation}\label{x1}\frac{B^L_i(x^{(s)}_{i,j})}{\varepsilon^s _i}=
\frac{B^L_j(x^{(s)}_{i,j})}{\varepsilon^s _j}. \end{equation}
\end{definition}
It is remarked that
\begin{equation}\label{x2}\frac{B^R_i(1-x^{(s)}_{i,j})}{\varepsilon^s _i}=
\frac{B^R_j(1-x^{(s)}_{i,j})}{\varepsilon^s _j}. \end{equation} In the next lemma the existence and uniqueness of the points $x^{(s)}_{i,j}$ are shown. Various
properties are also established.
\begin{lemma}\label{layers} For all $i,j$, such that $1 \leq i < j \leq
n$ and $0<s \leq 3/2$, the points $x_{i,j}$ exist, are uniquely
defined and satisfy the following inequalities
\begin{equation}\label{x3}
\frac{B^L_{i}(x)}{\eps^s _i} > \frac{B^L_{j}(x)}{\eps^s _j},\;\; x \in [0,x^{(s)}_{i,j}),\;\; \frac{B^L_{i}(x)}{\eps^s _i} <
\frac{B^L_{j}(x)}{\eps^s _j}, \; x \in (x^{(s)}_{i,j}, 1].\end{equation}\\
Moreover
\begin{equation}\label{x4}x^{(s)}_{i,j}< x^{(s)}_{i+1,j}, \; \mathrm{if} \;\; i+1<j \;\;
\mathrm{and} \;\; x^{(s)}_{i,j}<
x^{(s)}_{i,j+1}, \;\; \mathrm{if} \;\; i<j. \end{equation} \\
Also
\begin{equation}\label{x5}
x^{(s)}_{i,j}< 2s\sqrt{\frac{\eps_j}{\alpha}}\;\; and \;\;
x^{(s)}_{i,j} \in
(0,\frac{1}{2})\;\; \mathrm{if} \;\; i<j. \end{equation}\\
Analogous results hold for the $B^R_i$, $B^R_j$ and the points $1-x^{(s)}_{i,j}.$\\
\end{lemma}
\begin{proof} Existence, uniqueness and (\ref{x3}) follow
from the observation that the ratio of the two sides of (\ref{x1}),
namely
\[\frac{B^L_{i}(x)}{\eps^s _i}\frac{\eps^s _j}{B^L_{j}(x)}=
\frac{\eps^s _j}{\eps^s _i} \exp{(-\sqrt{\alpha}
x(\frac{1}{\sqrt{\eps_i}}-\frac{1}{\sqrt{\eps_j}}))},\] is
monotonically decreasing from the value $\frac{\eps^s_j}{\eps^s_i}
>1$ as $x$ increases
from $0$.\\
The point $x^{(s)}_{i,j}$ is the unique point $x$ at which this
ratio has the value $1.$ Rearranging (\ref{x1}), and using the
inequality $\ln x <x-1$ for all $x>1$, gives
\begin{equation}\label{x5}x^{(s)}_{i,j} = 2s\ds\left[
\frac{\ln(\frac{1}{\sqrt{\eps_i}})-
\ln(\frac{1}{\sqrt{\eps_j}})}{\sqrt{\alpha}(\frac{1}{\sqrt{\eps_i}}-
\frac{1}{\sqrt{\eps_j}})}\right]=
\frac{2s\;\ln(\frac{\sqrt{\eps_j}}{\sqrt{\eps_i}})}{\sqrt{\alpha}(\frac{1}{\sqrt{\eps_i}}-
\frac{1}{\sqrt{\eps_j}})}<
2s\sqrt{\frac{\eps_j}{\alpha}},\end{equation} which
is the first part of (\ref{x5}). The second part follows immediately from this and (\ref{a3}).\\
To prove (\ref{x4}), writing $\sqrt{\eps_k} = \exp(-p_k)$, for some
$p_k
> 0$ and all $k$, it follows that
\[x^{(s)}_{i,j}=\frac{2s(p_i -p_j)}{\sqrt{\alpha}(\exp{p_i} -\exp{p_j})}.\] The
inequality $x^{(s)}_{i,j}< x^{(s)}_{i+1,j}$ is equivalent to
\[\frac{p_i -p_j}{\exp{p_i} -\exp{p_j}}<\frac{p_{i+1} -p_j}{\exp{p_{i+1}} -\exp{p_j}}, \]
which can be written in the form
\[(p_{i+1}-p_j)\exp(p_i-p_j)+(p_{i}-p_{i+1})-(p_{i}-p_j)\exp(p_{i+1}-p_j)>0. \]
With $a=p_i-p_j$ and $b=p_{i+1}-p_j$ it is not hard to see that
$a>b>0$ and $a-b=p_i-p_{i+1}$. Moreover, the previous inequality is
then equivalent to
\[\frac{\exp{a}-1}{a}>\frac{\exp{b}-1}{b}, \] which is true because $a>b$ and proves
the first part of (\ref{x4}). The second part is proved by a similar argument.\\ The analogous results for the $B^R_i$, $B^R_j$ and the points $1-x^{(s)}_{i,j}$ are
proved by a similar argument.\eop
\end{proof}
In the following lemma sharper estimates of the smooth component are
presented.
\begin{lemma}\label{lsmooth2}
Let $\;A(x,t)\;$ satisfy (\ref{a1}) and (\ref{a2}). Then the smooth component $\;\vec v\;$ of the solution $\;\vec u\;$ of \eqref{BVP} satisfies for all
$\;i=1,\cdots,n$ and all $\;(x,t)\;\in\oln\Omega$
\[ |\frac{\partial^l v_i}{\partial x^l}(x,t)| \leq C\left (1+\sum_{q=i}^{n}\frac{B_q(x)}{\eps_q ^{\frac{l}{2}-1}}\right) \; \mathrm{for}\; l=0,1,2,3.\]
\end{lemma}
\begin{proof}Define two barrier functions
\[\vec\psi^\pm(x,t)\;=\;C[1+B_n(x)]\vec e\;\pm\;\frac{\partial^l\vec v}{\partial x^l}(x,t),\;\;l=0,1,2\;\;\;\text{and}\;\;\;(x,t)\in\overline\Omega.\]
We find that, for a proper choice of C,
\[\psi_i^\pm(0,t)\;=\; C\;\pm\;\frac{\partial^l u_{0,i}}{\partial x^l}(0,t)=C \geq 0\]
\[\psi_i^\pm(1,t)\;=\; C\;\pm\;\frac{\partial^l u_{0,i}}{\partial x^l}(1,t)=C \geq 0\]
\[\psi_i^\pm(x,0)\;=\; C[1+B_n(x)]\;\pm\;\frac{\partial^l \phi_{B,i}(x)}{\partial x^l}= C[1+B_n(x)]\;\pm\; C \geq\ 0\]
as $\phi_b(x) \in C^{(2)}(\Gamma_b)$
and $(\vec L \vec\psi^{\pm})_i(x,t)\geq 0$.\\
Using Lemma \ref{max}, we conclude that
\begin{equation}\label{e15}
|\frac{\partial^l v_i}{\partial x^l}(x,t)| \leq C[1+B_n(x)]\;\; \text{for}\;\;\; l=0,1,2.
\end{equation}
Consider the equation
\begin{equation}\label{e16}(\vec L(\frac{\partial^2 \vec v}{\partial x^2}))_i\;=\; \frac{\partial^2 f_i}{\partial x^2}-2 \frac{\partial \sum_{j=1}^{n} a_{ij}}{\partial x} \frac{\partial v_j}{\partial x}-\frac{\partial^2 \sum_{j=1}^{n}a_{ij}}{\partial x^2}v_j
\end{equation}
with\\
\begin{equation}\label{e17}
\frac{\partial^2 v_i}{\partial x^2}(0,t)\;=\;0,\frac{\partial^2 v_i}{\partial x^2}(1,t)\;=\;0, \frac{\partial^2 v_i}{\partial x^2}(x,0)\;=\;\frac{\partial^2
\phi_{b,i}(x)}{\partial x^2}.
\end{equation}
For convenience, let $\vec p$ denote $\frac{\partial^2\vec v}{\partial x^2}$. Then
\begin{equation}\label{e18}
\vec \L\vec p =\vec g \;\;\text{with}\;\; \vec p(0,t) =\vec 0,\;\; \vec p(1,t) =\vec 0,\;\; \vec p(x,0) =\vec s
\end{equation}
where \[g_i = \frac{\partial^2 f_i}{\partial x^2} -2\frac{\partial \sum_{j=1}^{n} a_{ij}}{\partial x}\frac{\partial v_j}{\partial x}-\sum_{j=1}^{n}\frac{\partial^2
a_{ij}}{\partial x^2}v_j \;\;\text{and}\;\; s_i = \frac{\partial^2 \phi_{b,i}}{\partial x^2}(x).\] Let $\vec q$ and $\vec r$ be the smooth and singular components
of $\vec p$ given by
\begin{equation}\label{e19}
\vec L \vec q = \vec g \;\;\text{with}\;\; \vec q(0,t) =\vec p_0(0,t),\;\; \vec q(1,t) =\vec p_0(1,t),\;\; \vec q(x,0) =\vec p(x,0)
\end{equation}
where $\vec p_0$ is the solution of the reduced problem
\[\frac{\partial \vec p_0}{\partial t} + A \vec p_0=\vec g \;\;\text{with}\;\; \vec p_0(x,0) = \vec p(x,0)= \vec s. \]\\
Now,
\begin{equation}\label{e20}
\vec L \vec r = \vec 0,\;\;\text{with}\;\; \vec r(0,t) = - \vec q (0,t),\; \vec r(1,t) = -\vec q(1,t),\; \vec r(x,0) = \vec 0.
\end{equation}
Using Lemma \ref{lsmooth1} and Lemma \ref{lsingular}, we have for $i=1,\dots,n$ and $(x,t) \in \overline \Omega$
\[ |\frac{\partial q_i }{\partial x}(x,t)| \leq C\]
and \[|\frac{\partial r_i }{\partial x}(x,t)| \leq C[\frac{B_i(x)}{\sqrt {\eps_i}}+\dots+\frac{B_n(x)}{\sqrt {\eps_n}}].\]\\
Hence, for $(x,t) \in \overline \Omega\;$and $i=1,\dots,n$,
\begin{equation}\label{e21}
|\frac{\partial^3 v_i }{\partial x^3}| =|\frac{\partial p_i }{\partial x}| \leq C[1+\frac{B_i(x)}{\sqrt {\eps_i}}+\dots+\frac{B_n(x)}{\sqrt {\eps_n}}].
\end{equation}
From (\ref{e15}) and (\ref{e21}), we find that for $l=0,1,2,3$ and $(x,t)\in \overline \Omega$
\[|\frac{\partial^l v_i }{\partial x^l}| \leq C[1+\eps_i^{1-\frac{l}{2}}B_i(x)+\dots+
\eps_i^{1-\frac{l}{2}}B_n(x)].\s\s\s\eop\] \end{proof}
{\bf Remark :} It is interesting to note that the above estimate
reduces to the estimate of the smooth component of the solution of
the scalar problem given in \cite{MORS} when $\;n=1.\;$\\
Bounds on the singular components $\vec{w}^L,\; \vec{w}^R$ of $\vec{u}$ and their derivatives are contained in\\
\begin{lemma}\label{lsingular} Let $A(x,t)$ satisfy (\ref{a1}) and
(\ref{a2}). Then there exists a constant $C,$ such that, for each $(x,t) \in \oln \Omega$ and $i=1,\; \dots , \; n$,
\begin{equation*} \begin{array} {l}
|\frac{\partial^l w^L_i}{\partial t^l}(x,t)| \le C B^L_{n}(x),\;\mathrm{for} \;\; l=0,1,2. \\\\
|\frac{\partial^l w^L_i}{\partial x^l}(x,t)| \le C\sum_{q=i}^n \frac{B^L_{q}(x)}{\eps_q^{\frac{l}{2}}},\;\mathrm{for}\;\; l=1,2.\\\\
|\frac{\partial^3 w^L_i}{\partial x^3}(x,t)| \le C\sum_{q=1}^n \frac{B^L_{q}(x)}{\eps_q^{\frac{3}{2}}}.\\\\
|\frac{\partial^4 w^L_i}{\partial x^4}(x,t)| \le C\frac{1}{\eps_i} \sum_{q=1}^n \frac{B^L_{q}(x)}{\eps_q}.
\end{array} \end{equation*}
Analogous results hold for $w^R_i$ and its derivatives.
\end{lemma}
\begin{proof}To obtain the bound of $\vec{w}^L$, define the functions
${\psi_i}^\pm(x,t)\;=\;Ce^{\alpha t}B_n^L(x)\;\pm\;w_i^L(x,t)$, for each $i=1,\dots,n$. It is clear that
${\psi_i}^\pm(0,t)$, ${\psi_i}^\pm(x,0)$, ${\psi_i}^\pm(1,t)$ and $(\vec L\vec\psi^\pm)_i(x,t)$ are non-negative. By Lemma 1, ${\psi_i}^\pm(x,t)\geq0$. It follows that $|w_i^L|\leq Ce^{\alpha t}B_n^L(x)$ \\
or
\begin{equation}\label{e22}
|w_i^L|\leq CB_n^L(x).
\end{equation}
To obtain the bound for $\frac{\partial w_i^L}{\partial t}$, define
the two functions
${\theta_i}^\pm(x,t)\;=\;CB_n^L(x)\;\pm\;\frac{\partial
w_i^L}{\partial t}(x,t)$ for each $i=1,\dots,n$. Differentiating the
homogeneous equation satisfied by $w_i^L$, partially with respect to
$t$, and rearranging yields
\[\frac{\partial^2 w_i^L}{\partial t^2} - \eps_i \frac{\partial^3 w^L_i}{\partial x^2 \partial t} + \sum_{j=1}^{n} a_{ij}\frac{\partial w_j^L}{\partial t} = \frac{-\partial\sum_{j=1}^{n}a_{ij}}{\partial t}w_j^L,\]
and we get
\[|L\frac{\partial w_i^L}{\partial t}|\leq C{B_n}^L(x)\]
\[|\frac{\partial w_i^L}{\partial t}(0,t)|\leq |\frac{\partial {u_i}}{\partial t}(0,t)|+|\frac{\partial {v_i}}{\partial t}(0,t)|=|\frac{\partial {\phi}_{L,i}(t)}{\partial t}|\leq C\]
\[|\frac{\partial w_i^L}{\partial t}(1,t)|\leq |\frac{\partial {u_i}}{\partial t}(1,t)|+|\frac{\partial {v_i}}{\partial t}(1,t)|=|\frac{\partial {\phi}_{R,i}(t)}{\partial t}|\leq C\]
\[|\frac{\partial w_i^L}{\partial t}(x,0)|\leq |\frac{\partial {\phi}_{B,i}(x)}{\partial t}|=0.\]
By Lemma \ref{stab}, it follows that
\begin{equation}\label{e23}
|\frac{\partial w_i^L}{\partial t}|\leq C{B_n}^L(x).
\end{equation}
Now the bound for $\frac{\partial^2 w^L_i}{\partial x \partial t}$ is obtained by using Lemma (\ref{lexact}) and Lemma (\ref{lsmooth1})
\[ |\frac{\partial^2 w^L_i}{\partial x \partial t}|\leq |\frac{\partial^2 u_i}{\partial x \partial t}|+|\frac{\partial^2 v_i}{\partial x \partial t}|\]
\[ |\frac{\partial^2 w^L_i}{\partial x \partial t}| \leq C {\eps_i}^{\frac{-1}{2}}({||\vec u||_\Gamma}+||\vec f||+|| \frac{\partial \vec f}{\partial t}||+||\frac{\partial^2 \vec f}{\partial t^2}||).\]
Similarly,
\[ |\frac{\partial^3 w^L_i}{\partial x^2 \partial t}| \leq C \eps_i^{-1}({||\vec u||_\Gamma}+||\vec f||+|| \frac{\partial \vec f}{\partial t}||+||\frac{\partial^2 \vec f}{\partial t^2}||). \]
The bounds on $\frac{\partial^l w_i^L}{\partial x^l}, l=1,2,3,4$ and
$i=1,\dots,n$ are derived by the method of induction on $n$. It is
assumed that the bounds $\frac{\partial w_i^L}{\partial x} ,
\frac{\partial^2 w_i^L}{\partial x^2},\frac{\partial^3
w_i^L}{\partial x^3}$ and $\frac{\partial^4 w_i^L}{\partial x^4}$
hold for all systems up to $n-1$. Define
$\vec{\tilde{w}}^L=(w_1^L,\dots,w_{n-1}^L)$, then
$\vec{\tilde{w}}^L$
satisfies the system \[\frac{\partial \vec{\tilde{w}}^L}{\partial t}-\tilde{E}\frac{\partial^2 \vec{\tilde{w}}^L}{\partial x^2}+\tilde A \vec{\tilde{w}}^L = \vec g, \]
with
\[\vec{\tilde{w}}^L(0,t) = \vec{\tilde{u}}(0,t)-\vec{\tilde{u}}_0(0,t), \vec{\tilde{w}}^L(1,t) =\vec{\tilde{0}},\]
\[\vec{\tilde{w}}^L(x,0)=\vec{\tilde{u}}(x,0)-\vec{\tilde{u}}_0(x,0) = \vec{\tilde{\phi}}_B(x)-\vec{\tilde{\phi}}_B(x)=\vec{\tilde{0}}.\]
Here, $\tilde{E}$ and $\tilde{A}$ are the matrices obtained by
deleting the last row and column from $E,A$ respectively, the
components of $\vec g$ are $g_i = -a_{i,n}w_n^L$ for $1\leq i \leq
n-1$ and $\vec{\tilde{u}}_0$ is the solution of the reduced problem.
Now decompose $\vec{\tilde{w}}^L$ into smooth and singular
components to get $\vec{\tilde{w}}^L = \vec{q} +
\vec{r},\;\frac{\partial \vec{\tilde{w}}^L}{\partial x} =
\frac{\partial \vec q}{\partial x} + \frac{\partial \vec r}{\partial
x} $. By induction, the bounds on the derivatives of ${ \vec{\tilde
w}}^L$ hold. That is for $i=1,\dots,n-1$
\begin{equation} \label{e24}
\ds \left.
\begin{array}{lcl}
|\dfrac{\partial w_i^L}{\partial x}| &\leq& C \sum_{q=i}^{n-1} \eps_q^{\frac{-1}{2}}B_q^L(x) \\\\
|\dfrac{\partial^2 w_i^L}{\partial x^2}| &\leq& C \sum_{q=i}^{n-1} \eps_q^{-1}{B_q}^L(x) \\\\
|\dfrac{\partial^3 w_i^L}{\partial x^3}| &\leq& C \sum_{q=1}^{n-1} \eps_q^{\frac{-3}{2}}B_q^L(x) \\\\
|\eps_i \dfrac{\partial^4 w_i^L}{\partial x^4}| &\leq& C \sum_{q=1}^{n-1} \eps_q^{-1}B_q^L(x)
\end{array}
\right \}
\end {equation}
Rearranging the $n^{th}$ equation of the system satisfied by $w_n^L$
yields
\[\eps_n \frac{\partial^2 w_n^L}{\partial x^2 }=\frac{\partial w_n^L}{\partial t} + \sum_{j=1}^{n} a_{nj}{w_j^L}.\]
Using (\ref{e22}) and (\ref{e23}) gives
\begin{equation} \label{e25}
|\frac{\partial^2 w_n^L}{\partial x^2 }| \leq C \eps_n^{-1} B_n^L(x).
\end{equation}
Applying the mean value theorem to $w_n^L$ at some $y$, $a < y< a+\sqrt \eps_n$
\[ \frac{\partial w_n^L}{\partial x}(y,t)= \frac{w_n^L(a+\sqrt \eps_n,t)-w_n^L(a,t)}{\sqrt \eps_n} \]
Using (\ref{e22}) gives
\[|\frac{\partial w_n^L}{\partial x}(y,t)|\leq \frac{C}{\sqrt \eps_n}(B_n^L(a+\sqrt \eps_n)+B_n^L(a)). \]
So
\begin{equation} \label{e26}
|\frac{\partial w_n^L}{\partial x}(y,t)|\leq \frac{C}{\sqrt \eps_n}B_n^L(a).
\end{equation}
Again
\begin{equation} \label{e27}
\frac{\partial w_n^L}{\partial x}(x,t) = \frac{\partial w_n^L}{\partial x}(y,t) + (y-x) \frac{\partial^2 w_n^L}{\partial x^2}(\eta,t),\;\;\; y<\eta<x.
\end{equation}
Using (\ref{e25}) and (\ref{e26}) in (\ref{e27}) yields
\[\begin{array}{lcl}
|\frac{\partial w_n^L}{\partial x}(x,t)| &\leq& C[\eps_n^{\frac{-1}{2}}B_n^L(a)+ \eps_n^{\frac{-1}{2}}B_n^L(\eta)] \\
&\leq& C \eps_n^{\frac{-1}{2}}B_n^L(a)\\
&=& C \eps_n^{\frac{-1}{2}}B_n^L(x)\frac{B_n^L(a)}{B_n^L(x)}\\
&=& C \eps_n^{\frac{-1}{2}}B_n^L(x) e^{(x-a)\sqrt \alpha / \sqrt \eps_n}\\
&=& C \eps_n^{\frac{-1}{2}}B_n^L(x) e^{\sqrt \eps_n \sqrt \alpha / \sqrt \eps_n}.\\
\end{array}\]
Therefore
\begin{equation} \label{e28}
|\frac{\partial w_n^L}{\partial x}(x,t)|\leq C \eps_n^{\frac{-1}{2}}B_n^L(x).\;\;\s\s\s
\end{equation}
Now, differentiating the equation satisfied by $w_n^L$ partially with respect to $x$, and rearranging, gives
\[\eps_n \frac{\partial^3 w_n^L}{\partial x^3} = \frac{\partial^2 w_n^L}{\partial x \partial t} + \sum_{q=1}^{n-1}a_{nq}\frac{\partial w_q^L}{\partial x } + a_{nn}\frac{\partial w_n^L}{\partial x} + \sum_{q=1}^{n} \frac{\partial a_{nq}}{\partial x} w_q^L. \]
The bounds on $w_n^L$ and (\ref{e24}) then give
\[|\frac{\partial^3 w_n^L}{\partial x^3}| \leq C \sum_{q=1}^{n} {\eps_q}^{\frac{-3}{2}}B_q^L(x). \]
Similarly
\[|\eps_n \frac{\partial^4 w_n^L}{\partial x^4}| \leq C \sum_{q=1}^{n} \eps_q^{-1}B_q^L(x).\]
Using the bounds on $w_n^L, \frac{\partial w_n^L}{\partial x},\frac{\partial^2 w_n^L}{\partial x^2},\frac{\partial^3 w_n^L}{\partial x^3}$ and $\frac{\partial^4
w_n^L}{\partial x^4}$, it is seen that $\vec g,\;\frac{\partial \vec g}{\partial x},$ $\frac{\partial^2 \vec g}{\partial x^2},$ $\frac{\partial^3 \vec g}{\partial
x^3},$ $\frac{\partial^4 \vec g}{\partial x^4}$ are bounded by $ CB_n^L(x),$ $C\frac{B_n^L(x)}{\sqrt \eps_n},$ $C\frac{B_n^L(x)}{\eps_n},$ $\sum_{q=1}^{n}
\frac{B_q^L(x)}{\eps_q^{\frac{3}{2}}},$ $C \eps_n^{-1} \sum_{q=1}^{n} \frac{B_q^L(x)}{\eps_q} $ respectively.
Introducing the functions ${\vec \psi}^\pm(x,t)\;=\;CB_n^L(x)\vec e\;\pm\;\vec q(x,t)$, it is easy to see that ${\vec \psi}^\pm(0,t)=C\vec e\;\pm\;\vec q(0,t) \geq
\vec0$, ${\vec \psi}^\pm(1,t)=CB_n^L(1)\vec e\;\pm\;\vec 0 \geq \vec 0$, ${\vec \psi}^\pm(x,0)=CB_n^L(x)\vec e\;\pm\;\vec 0 \geq \vec 0$ and
\[(\vec L {\vec \psi}^\pm)_i(x,t)=C(-\eps_i \frac{\alpha}{\eps_n}+\sum_{j=1}^{n}a_{ij})B_n^L(x)\;\pm\;CB_n^L(x) \geq 0.\]
Applying Lemma 1, it follows that $||\vec q(x,t)|| \leq C B_n^L(x)$. Defining barrier functions ${\vec \theta}^\pm (x,t) = C \eps_n^{\frac{-1}{2}}B_n^L(x) \vec
e\;\pm\; \frac{\partial \vec q}{\partial x} $ and using Lemma 3 for the problem satisfied by $\vec q$, the bound required for $\frac{\partial \vec q}{\partial x} $
and $\frac{\partial^2 \vec q}{\partial x^2} $ is obtained. By induction, the following bounds for \vec r are obtained for $i=1,\dots,n-1,$
\[ |\frac{\partial r_i}{\partial x} |\leq [\frac{B_i^L(x)}{{\sqrt \eps_i}}+\dots+\frac{B_{n-1}^L(x)}{{\sqrt \eps_{n-1}}} ],\]
\[ |\frac{\partial^2 r_i}{\partial x^2} |\leq C[\frac{B_i^L(x)}{{ \eps_i}}+\dots+\frac{B_{n-1}^L(x)}{{ \eps_{n-1}}}],\]
\[ |\frac{\partial^3 r_i}{\partial x^3} |\leq C[\frac{B_1^L(x)}{{ \eps_1^{\frac{3}{2}}}}+\dots+\frac{B_{n-1}^L(x)}{{ \eps_{n-1}^\frac{3}{2}}}],\]
\[ |\eps_i\frac{\partial^4 r_i}{\partial x^4} |\leq C[\frac{B_1^L(x)}{{ \eps_1}}+\dots+\frac{B_{n-1}^L(x)}{{ \eps_{n-1}}}].\]
Combining the bounds for the derivatives of $q_i$ and $r_i$ it follows that, for $i= 1,2,\dots,n$
\[ \begin{array}{rcl}
|\frac{\partial^l w^L_i}{\partial x^l}|&\leq& |\frac{\partial^l q_i}{\partial x^l}|+|\frac{\partial^l r_i}{\partial x^l }|\\\\
|\frac{\partial^l w^L_i}{\partial x^l}|&\leq& C \sum_{q=i}^{n} \frac{B_q^L(x)}{{ \eps_q^{\frac{l}{2}}}}\;\;\text{for}\;\; l= 1,2\\\\
|\frac{\partial^3 w^L_i}{\partial x^3}|&\leq& C \sum_{q=1}^{n} \frac{B_q^L(x)}{{ \eps_q^{\frac{3}{2}}}}\\\\
\text{and}\;\;\;\; |\eps_i\frac{\partial^4 w^L_i}{\partial x^4}|&\leq& C \sum_{q=1}^{n} \frac{B_q^L(x)}{\eps_q}. \end{array}\] Recalling the bounds on the
derivatives of $w_n^L$ completes the proof of the lemma for the system of $n$ equations.\\ A similar proof of the analogous results for the right boundary layer
functions holds.\eop\end{proof}
\section{The Shishkin mesh}
A piecewise
uniform Shishkin mesh with $M \times N$ mesh-intervals is now
constructed. Let $\Omega^M_t=\{t_k \}_{k=1}^{M},\;\;\Omega^N_x=\{x_j
\}_{j=1}^{N-1},\;\;\overline{\Omega}^M_t=\{t_k
\}_{k=0}^{M},\;\;\overline{\Omega}^N_x=\{x_j
\}_{j=0}^{N},\;\;\Omega^{M,N}=\Omega^M_t \times \Omega^N_x,\;\;
\overline{\Omega}^{M,N}=\overline{\Omega}^M_t \times \overline{\Omega}^N_x \;\;\mathrm{and}\;\;\Gamma^{M,N}=\Gamma \cap \overline{\Omega}^{M,N}.$
The mesh $\overline{\Omega}^M_t$ is chosen to be a uniform mesh with $M$
mesh-intervals on $[0,T]$. The mesh $\overline{\Omega}^N_x$ is a
piecewise-uniform mesh on $[0,1]$ obtained by dividing $[0,1]$ into
$2n+1$ mesh-intervals as follows
\[[0,\sigma_1]\cup\dots\cup(\sigma_{n-1},\sigma_n]\cup(\sigma_n,1-\sigma_n]\cup(1-\sigma_n,1-\sigma_{n-1}]\cup\dots\cup(1-\sigma_1,1].\]
The $n$ parameters $\sigma_r$, which determine the points separating the uniform meshes, are defined by
\begin{equation}\label{tau1}\sigma_{n}=
\min\displaystyle\left\{\frac{1}{4},2\sqrt{\frac{\eps_n}{\alpha}}\ln N\right\}\end{equation} and for $\;r=1,\;\dots \; ,n-1$
\begin{equation}\label{tau2}\sigma_{r}=\min\displaystyle\left\{\frac{\sigma_{r+1}}{2},2\sqrt{\frac{\eps_r}{\alpha}}\ln
N\right\}.\end{equation} Clearly \[
0\;<\;\sigma_1\;<\;\dots\;<\;\sigma_n\;\le\;\frac{1}{4}, \qquad
\frac{3}{4}\leq 1-\sigma_n < \; \dots \;< 1-\sigma_1 <1.\] Then, on
the sub-interval $\;(\sigma_n,1-\sigma_n]\;$ a uniform mesh with
$\;\frac{N}{2}\;$ mesh-intervals is placed, on each of the
sub-intervals
$\;(\sigma_r,\sigma_{r+1}]\;\tx{and}\;(1-\sigma_{r+1},1-\sigma_r],\;\;r=1,\dots,n-1,\;$
a uniform mesh of $\;\frac{N}{2^{n-r+2}}\;$ mesh-intervals is placed
and on both of the sub-intervals $\;[0,\sigma_1]\;$ and
$\;(1-\sigma_1,1]\;$ a uniform mesh of $\;\frac{N}{2^{n+1}}\;$
mesh-intervals is placed. In practice it is convenient to take
\begin{equation}\label{meshpts2} N=2^{n+p+1} \end{equation} for some
natural number $p$. It follows that in the sub-interval
$[\sigma_{r-1},\sigma_{r}]$ there are $N/2^{n-r+3}=2^{r+p-2}$
mesh-intervals. This construction leads to a class of $2^n$
piecewise uniform Shishkin meshes $\Omega^{M,N}$. Note that these meshes are not the same as those
constructed in \cite{GLOR}\\
The following notation is introduced: $h_j=x_{j}-x_{j-1},\; J= \{x_j: D^+ h_j=h_{j+1}-h_j \neq 0\}$. Clearly, $J$ is the set of points at which the mesh-size
changes. Let $R=\{r:\sigma_r \in J \}$. From the above construction it follows that $J$ is a subset of the set of transition points $\{\sigma_r \}_{r=1}^n
\cup\{1-\sigma_r\}_{r=1}^n$. It is not hard to see that for each point $x_j$ in the mesh-interval $(\sigma_{r-1} ,\sigma_r ]$,
\begin{equation}\label{geom7} h_j
=2^{n-r+3}N^{-1}(\sigma_r-\sigma_{r-1})\end{equation} and so the change in the mesh-size at the point $\sigma_r$ is
\begin{equation}\label{geom8} D^+ h_r=2^{n-r+3}(d_r
-d_{r-1}), \end{equation} where $d_r =\frac{\sigma_{r+1}}{2}-\sigma_r$ for
$1 \leq r \leq n$, with the conventions $d_0 =0,\; \sigma_{n+1}=1/2.$ Notice that $d_r
\ge 0$, that $\Omega^{M,N}$ is a classical uniform mesh when $d_r = 0$ for all $ r=1 \; \dots \; n$ and, from \eqref{geom8},
that \begin{equation}\label{geom5}
D^+h_r <0 \;\mathrm{if}\; d_r=0.\end{equation}
\\
Furthermore
\begin{equation}\label{geom2}\sigma_r \leq C \sqrt{\eps_r} \ln N,
\; \;\; 1 \leq r \leq n, \end{equation}
and, using (\ref{geom7}), (\ref{geom2}),
\begin{equation}\label{geom4}h_{r}+h_{r+1} \leq
CN^{-1}\ln N \left\{ \begin{array}{l}\;\; \sqrt{\eps_{r+1}}, \;\;\mathrm{if}\;\; D^+ h_r >0, \\
\;\; \sqrt{\eps_{r}}, \;\;\mathrm{if}\;\; D^+ h_r <0.\end{array}\right .\end{equation}\\
Also
\begin{equation}\label{geom-1}
\sigma_r=2^{-(s-r+1)}\sigma_{s+1}\;\mathrm{when}\; d_r=\dots =d_s
=0, \; 1 \leq r \leq s \leq n.
\end{equation}
The geometrical results in the following lemma are used later.
\begin{lemma} \label{s1} Assume that $d_r>0$ and let $0<s \leq 2$. Then the following inequalities hold
\begin{equation}\label{geom0}
B^L_r(\sigma_r)=N^{-2}.
\end{equation}
\begin{equation}\label{geom9} x^{(s)}_{r-1,r}\;\leq\;\sigma_r-h_r \;\mathrm{for} \;
1<r\leq n.\end{equation}
\begin{equation}\label{geom10}
\frac{B_{q}^{L}(\sigma_{r})}{\eps_q^s}\leq \frac{1}{\eps_r^s} \;\;
\mathrm{for} \;\; 1 \leq q \leq n, \;\;1 \leq r \leq n.
\end{equation}
\begin{equation}\label{geom3} B_q^L(\sigma_r-h_r)\leq
CB_q^L(\sigma_r)\;\;\mathrm{for}\;\; 1 \leq r \leq q \leq n.
\end{equation}
\end{lemma}
\begin{proof} The proof of \eqref{geom0} follows immediately from
the definition of $\sigma_r$ and the assumption that $d_r>0$.\\ To verify (\ref{geom9}) note that, by Lemma \ref{layers} and \eqref{meshpts2},
\[x^{(s)}_{r-1,r} < 2s\sqrt{\frac{\eps_r}{\alpha}} =
\frac{s\sigma_r}{\ln N} = \frac{s\sigma_r}{(n+p+1)\ln 2} \leq
\frac{\sigma_r}{2}.
\]
Also, by \eqref{meshpts2} and \eqref{geom7}, \[h_r
=2^{n-r+3}N^{-1}(\sigma_{r}-\sigma_{r-1})=2^{2-r-p}(\sigma_{r}-\sigma_{r-1})\leq
\frac{\sigma_{r}-\sigma_{r-1}}{2} < \frac{\sigma_r}{2}.\] It follows
that $x^{(s)}_{r-1,r}+h_r \leq \sigma_r$ as required.
\\
To verify (\ref{geom10}) note that if $q \ge r$ the result is
trivial. On the other hand, if $q<r$, by (\ref{geom9}) and Lemma
\ref{layers},
\[\frac{B_{q}^{L}(\sigma_{r})}{\eps_q^s}\leq
\frac{B_{q}^{L}(x^{(s)}_{q,r})}{\eps_q^s}=
\frac{B_{r}^{L}(x^{(s)}_{q,r})}{\eps_r^s}\leq \frac{1}{\eps_r^s}.\]
Finally, to verify (\ref{geom3}) note, from \eqref{geom7}, that
\begin{equation*}
h_r=2^{n-r+3}N^{-1}(\sigma_{r}-\sigma_{r-1}) \leq
2^{n-r+3}N^{-1}\sigma_r=2^{n-r+4}\sqrt{\frac{\eps_r}{\alpha}}
N^{-1}\ln N.
\end{equation*}
But
\begin{equation*}
e^{2^{n-r+4} N^{-1}\ln N}=(N^{\frac{1}{N}})^{2^{n-r+4}} \leq C,
\end{equation*}
so
\begin{equation*}
\sqrt{\frac{\alpha}{\eps_q }}h_r \leq
\sqrt{\frac{\eps_r}{\eps_q}}2^{n-r+4} N^{-1}\ln N \leq 2^{n-r+4}
N^{-1}\ln N \leq C,
\end{equation*} since $r\leq q$.
It follows that
\begin{equation*}
B^L _q (\sigma_r -h_r )=B^L _q
(\sigma_r)e^{\sqrt{\frac{\alpha}{\eps_q }}h_r } \leq CB^L _q
(\sigma_r)
\end{equation*} as required. \eop \end{proof}
\section{The discrete problem}
In this section a classical finite difference operator with an
appropriate Shishkin mesh is used to construct a numerical method
for (\ref{BVP}), which is shown later to be essentially second order
parameter-uniform. It is assumed henceforth that the problem data
satisfy whatever smoothness conditions are required.\\
The discrete initial-boundary value problem is now defined on any mesh by the finite difference method
\begin{equation}\label{discreteBVP}
D^-_t\vec{U}-E\delta^2_x\vec{U} +A\vec{U}=\vec{f}\;\; \mathrm{on}
\;\; \Omega^{M,N},\;\; \vec{U}=\vec{u} \;\; \mathrm{on} \;\;
\Gamma^{M,N}.
\end{equation}
This is used to compute numerical approximations to the exact
solution of (\ref{BVP}). Note that (\ref{discreteBVP}),
can also be written in the operator form
\[\vec{L}^{M,N} \vec{U}\;=\;\vec{f} \;\; \mathrm{on}
\;\; \Omega^{M,N},\;\; \vec{U}=\vec{u} \;\; \mathrm{on} \;\;
\Gamma^{M,N},\] where \[\vec{L}^{M,N} \;=\; D^-_t-E\delta^2_x+A\]
and $D^-_t,\; \delta^2_x,\; D^+_x \; \tx{and} \; D^{-}_x$ are the
difference operators
\[D^-_t\vec{U}(x_j,t_k)\;=\;\dfrac{\vec{U}(x_j,t_k)-\vec{U}(x_{j},t_{k-1})}{t_k-t_{k-1}},\]
\[\delta^2_x\vec{U}(x_j,t_k)\;=\;\dfrac{D^+_x\vec{U}(x_j,t_k)-D^-_x\vec{U}(x_j,t_k)}{(x_{j+1}-x_{j-1})/2},\]
\[D^+_x\vec{U}(x_j,t_k)\;=\;\dfrac{\vec{U}(x_{j+1},t_k)-\vec{U}(x_j,t_k)}{x_{j+1}-x_j},\]
\[D^-_x\vec{U}(x_j,t_k)\;=\;\dfrac{\vec{U}(x_j,t_k)-\vec{U}(x_{j-1},t_k)}{x_j-x_{j-1}}.\]
The following discrete results are analogous to those for the
continuous case.
\begin{lemma}\label{dmax} Let $A(x,t)$ satisfy (\ref{a1}) and (\ref{a2}).
Then, for any mesh function $\vec{\Psi}$, the inequalities $\vec
{\Psi}\;\ge\;\vec 0 \;\mathrm{on}\; \Gamma^{M,N}$ and $\vec{L}^{M,N}
\vec{\Psi}\;\ge\;\vec 0\;$ on $\Omega^{M,N}$ imply that $\;\vec
{\Psi}\ge \vec 0\;$ on $\overline{\Omega}^{M,N}.$
\end{lemma}
\begin{proof} Let $i^*, j^*, k^*$ be such that
$\Psi_{i^*}(x_{j^{*}}, t_{k^{*}})=\min_i\min_{j,k}\Psi_i(x_j,t_k)$
and assume that the lemma is false. Then $\Psi_{i^*}(x_{j^{*}},
t_{k^{*}})<0$ . From the hypotheses we have $j^*\neq 0, \;N$ and
$\Psi_{i^*} (x_{j^*},t_{k^{*}})-\Psi_{i^*}(x_{j^*-1},t_{k^{*}})\leq
0, \; \Psi_{i^*}
(x_{j^*+1},t_{k^{*}})-\Psi_{i^*}(x_{j^*},t_{k^{*}})\geq 0,$ so
$\;\delta^2_x\Psi_{i*}(x_{j*},t_{k^{*}})\;>\;0.\;$ It follows that
\[\ds\left(\vec{L}^N\vec{\Psi}(x_{j*},t_{k^{*}})\right)_{i*}\;=
\;-\eps_{i*}\delta^2_x\Psi_{i*}(x_{j*},t_{k^{*}})+\ds{\sum_{q=1}^n}
a_{i*,\;q}(x_{j*},t_{k^{*}})\Psi_{q}(x_{j*},t_{k^{*}})\;<\;0,\]
which is a contradiction, as required. \eop \end{proof}
An immediate consequence of this is the following discrete stability
result.
\begin{lemma}\label{dstab} Let $A(x,t)$ satisfy (\ref{a1}) and (\ref{a2}).
Then, for any mesh function $\vec{\Psi} $ on $\Omega$,
\[\parallel\vec{\Psi}(x_j,t_k)\parallel\;\le\;\max\left\{||\vec{\Psi}||_{\Gamma^{M,N}}, \frac{1}{\alpha}||
\vec{L}^{M,N}\vec{\Psi}||\right\}. \]
\end{lemma}
\begin{proof} Define the two functions
\[\vec{\Theta}^{\pm}(x_j,t_k)=\max\{||\vec{\Psi}||_{\Gamma^{M,N}},
\frac{1}{\alpha}||\vec{L^{M,N}}\vec{\Psi}||\}\vec{e}\pm
\vec{\Psi}(x_j,t_k)\]where $\vec{e}=(1,\;\dots \;,1)$ is the unit
vector. Using the properties of $A$ it is not hard to verify that
$\vec{\Theta}^{\pm}\geq \vec{0}$ on $\Gamma^{M,N}$ and
$\vec{L}^{M,N}\vec{\Theta}^{\pm}\geq \vec{0}$ on $\Omega^{M,N}$. It follows from Lemma \ref{dmax} that $\vec{\Theta}^{\pm}\geq \vec{0}$ on
$\overline{\Omega}^{M,N}$.\eop
\end{proof}
The following comparison result will be used in the proof of the
error estimate.
\begin{lemma}\label{comparison}(Comparison Principle) Assume that,
for each $i=1,\; \dots \;,n$,
the mesh functions $\vec{\Phi}$ and $\vec{Z}$ satisfy
\[|Z_i| \leq \Phi_i,\;\; \mathrm{on}\;\;\Gamma^{M,N}\;\; \mathrm{and}\;\;
|(\vec{L}^{M,N}\vec{Z})_i| \leq (\vec{L}^{M,N} \vec{\Phi})_i\;\; \mathrm{on}\;\;\Omega^{M,N}.\] Then, for each
$i=1,\; \dots \;,n$,
\[|Z_i| \leq \Phi_i.\]
\end{lemma}
\begin{proof} Define the two mesh functions $\vec{\Psi}^{\pm}$ by
\[\vec{\Psi}^{\pm}=\vec{\Phi} \pm \vec{Z}.\] Then, for each $i=1,\; \dots \;,n$,
satisfies
\[(\Psi^{\pm})_i\ge 0, \;\; \mathrm{on}\;\;\Gamma^{M,N}\;\; \mathrm{and}\;\;
|(\vec{L}^{M,N}\vec{Z})_i| \leq (\vec{L}^{M,N} \vec{\Phi})_i\;\; \mathrm{on}\;\;\Omega^{M,N}.\] The result follows
from an application of Lemma \ref{dmax}.
\eop \end{proof}
\section{The local truncation error}
From Lemma \ref{dstab}, it is seen that in order to bound the error
$||\vec{U}-\vec{u}||$ it suffices to bound
$\vec{L}^{M,N}(\vec{U}-\vec{u})$. But this expression satisfies
\begin{equation*}\begin{array}{l}
\vec{L}^{M,N}(\vec{U}-\vec{u})=\vec{L}^{M,N}(\vec{U})-\vec{L}^{M,N}(\vec{u})=\\
\vec{f}-\vec{L}^{M,N}(\vec{u})
=\vec{L}(\vec{u})-\vec{L}^{M,N}(\vec{u})
=(\vec{L}-\vec{L}^{M,N})\vec{u}.\end{array}\end{equation*} It
follows that
\begin{equation*}
\vec{L}^{M,N}(\vec{U}-\vec{u})=(\frac{\partial}{\partial
t}-D^-_t)\vec{u}-E(\frac{\partial^2}{\partial
x^2}-\delta^2_x)\vec{u}.
\end{equation*}
Let $\vec{V}, \vec{W}^L, \vec{W}^R$ be the discrete analogues of
$\vec{v}, \vec{w}^L, \vec{w}^R$ respectively. Then, similarly,
\begin{equation*}\vec{L}^{M,N}(\vec{V}-\vec{v})=(\frac{\partial}{\partial
t}-D^-_t)\vec{v}-E(\frac{\partial^2}{\partial x^2}-\delta^2_x)\vec{v},\end{equation*}
\begin{equation*}\vec{L}^{M,N}(\vec{W}^L-\vec{w}^L)=(\frac{\partial}{\partial
t}-D^-_t)\vec{w}^L-E(\frac{\partial^2}{\partial
x^2}-\delta^2_x)\vec{w}^L,\end{equation*}
\begin{equation*}\vec{L}^{M,N}(\vec{W}^R-\vec{w}^R)=(\frac{\partial}{\partial
t}-D^-_t)\vec{w}^R-E(\frac{\partial^2}{\partial
x^2}-\delta^2_x)\vec{w}^R,\end{equation*} and so, for each
$i=1,\;\dots \;,n$,
\begin{equation}\label{LTEV}
|(\vec{L}^{M,N}(\vec{V}-\vec{v}))_i|\leq |(\frac{\partial}{\partial t}-D^-_t)v_i|+
|\eps_i(\frac{\partial^2}{\partial x^2}-\delta^2_x)v_i|,
\end{equation}
\begin{equation}\label{LTEWL}
|(\vec{L}^{M,N}(\vec{W^L}-\vec{w^L}))_i|\leq
|(\frac{\partial}{\partial
t}-D^-_t)w^L_i|+|\eps_i(\frac{\partial^2}{\partial
x^2}-\delta^2_x)w^L_i|,
\end{equation}
\begin{equation}\label{LTEWR}
|(\vec{L}^{M,N}(\vec{W^R}-\vec{w^R}))_i|\leq
|(\frac{\partial}{\partial
t}-D^-_t)w^R_i|+|\eps_i(\frac{\partial^2}{\partial
x^2}-\delta^2_x)w^R_i|.
\end{equation}
Thus, the smooth and singular components of the local truncation
error can be treated separately. Note that, for any smooth
function $ \psi $, the following
distinct estimates of the local truncation error hold:\\
for each$(x_j,t_k)\in \Omega^{L,M}$
\begin{equation}\label{lte0}
|(\frac{\partial}{\partial t}-D^-_t)\psi(x_j,t_k)|\;\le\;
C(t_k-t_{k-1})\max_{s\;\in\;[t_{k-1},t_k]}|\frac{\partial^2\psi}{\partial
t^2}(x_j,s)|,
\end{equation}
\begin{equation}\label{lte1}
|(\frac{\partial^2}{\partial x^2}-\delta^2_x)\psi(x_j,t_k)|\;\le\;
C\max_{s\;\in\;I_j}|\frac{\partial^2\psi}{\partial x^2}(s,t_k)|,
\end{equation}
and
\begin{equation}\label{lte2}
|(\frac{\partial^2}{\partial
x^2}-\delta^2_x)\psi(x_j,t_k)|\;\le\;C(x_{j+1}-x_{j-1}) \max_{s\in
I_j}|\frac{\partial^3\psi}{\partial x^3}(s,t_k)|.
\end{equation}
Assuming, furthermore, that $x_j \notin J$, then
\begin{equation}\label{lte3}
|(\frac{\partial^2}{\partial
x^2}-\delta^2_x)\psi(x_j,t_k)|\;\le\;C(x_{j+1}-x_{j-1})^2
\max_{s\in I_j}|\frac{\partial^4\psi}{\partial x^4}(s,t_k)|.
\end{equation}
Here $I_j=[x_{j-1}, x_{j+1}]$.
\section{Error estimate}
The proof of the error estimate is broken into two parts. In the
first a theorem concerning the smooth part of the error is proved.
Then the singular part of the error is considered. A barrier
function is now constructed, which is used in both parts of the
proof.\\
For each $r \in R$, introduce a piecewise linear polynomial
$\theta_r$ on $\overline{\Omega}$, defined by
\begin{equation*} \theta_r(x)=
\left\{ \begin{array}{l}\;\; \dfrac{x}{\sigma_r}, \;\; 0 \leq x \leq \sigma_r. \\
\;\; 1, \;\; \sigma_r < x < 1-\sigma_r.
\\ \;\; \dfrac{1-x}{\sigma_r}, \;\; 1-\sigma_r \leq x \leq 1. \end{array}\right .\end{equation*}\\
It is not hard to verify that
\begin{equation}\label{theta} L^{M,N}(\theta_r(x_j)\vec{e})_i \ge
\left\{ \begin{array}{l}\;\;
\alpha \theta_{r}(x_j), \;\; \mathrm{if} \;\;x_j \notin J \\
\;\; \alpha+\dfrac{2\eps_i}{ \sigma_r
(h_r+h_{r+1})}, \;\; \mathrm{if} \;\;x_j=\sigma_r \in J. \end{array}\right .\end{equation}\\
On the Shishkin mesh $\Omega^{M,N}$ define the barrier function
$\vec{\Phi}$ by
\begin{equation}\label{barrier}\vec{\Phi}(x_j,t_k)=C[M^{-1}+N^{-2}+
N^{-2}(\ln N)^3\ds\sum_{r\in
R}\theta_r (x_j)]\vec{e},\end{equation} where $C$ is any sufficiently large constant.\\
Then, on $\Omega^{M,N}$, $\vec{\Phi}$ satisfies
\begin{equation}\label{barrierbound1}0 \leq \Phi_{i}(x_j,t_k) \leq
C(M^{-1}+N^{-2}(\ln N)^{3}),\;\; 1 \leq i \leq n.\end{equation}
Also, for $x_j \notin J$,
\begin{equation}\label{barrierbound3}
(L^{M,N}\vec{\Phi})_i(x_j,t_k) \ge C(M^{-1}+N^{-2}(\ln
N)^{3})\end{equation} and, for $x_j \in J$, using
(\ref{geom4}),(\ref{theta}),
\begin{equation}\label{barrierbound4}
(L^{M,N}\vec{\Phi}(x_j,t_k))_i \ge
\left\{\begin{array}{l}C(M^{-1}+N^{-2}+
\frac{\eps_i}{\sqrt{\eps_r \eps_{r+1}}} N^{-1} \ln N),\;\;\mathrm{if}\;\;D^+ h_r>0, \\
C(M^{-1}+N^{-2}+\frac{\eps_i}{\eps_r} N^{-1} \ln N),\;\;
\mathrm{if}\;\;D^+ h_r<0.\end{array} \right .\end{equation} The
following theorem gives the required error estimate for the smooth
component.
\begin{theorem}\label{smootherrorthm} Let $A(x,t)$ satisfy (\ref{a1}) and
(\ref{a2}). Let $\vec v$ denote the smooth component of the exact
solution from (\ref{BVP}) and $\vec V$ the smooth component of the
discrete solution from (\ref{discreteBVP}). Then
\begin{equation}\;\; ||\vec{V}-\vec{v}|| \leq C(M^{-1}+N^{-2}(\ln N)^3). \end{equation}
\end{theorem}
\begin{proof}
It suffices to show that
\begin{equation}\label{ratiobound} \frac{|(L^{M,N}(\vec{V}-\vec{v}))_i (x_j,t_k)|}{|(L^{M,N}\vec{\Phi})_i
(x_j,t_k)|} \leq C,
\end{equation}
for each $i=1,\;\dots \;,n,$ because an application of the
Comparison
Principle then yields the required result.\\ For each mesh point $x_j$ either $x_j \notin J$ or $x_j \in J$.\\
Suppose first that $x_j \notin J$. Then, from \eqref{barrierbound3},
\begin{equation}\label{smoothdenom}
(L^{M,N}\vec{\Phi}(x_j,t_k))_i \ge C(M^{-1}+N^{-2})\end{equation}
and from \eqref{lte0}, \eqref{lte3} and Lemma \ref{lsmooth1}
\begin{equation}\label{smoothnumer}
\begin{array}{lcl}|(L^{M,N}(\vec{V}-\vec{v}))_i(x_j,t_k)|&\leq &C(t_k-t_{k-1}+(x_{j+1}-x_{j-1})^2)\\ &\leq &C(M^{-1}+(h_j+h_{j+1})^2)\\
&\leq & C(M^{-1}+N^{-2}).\end{array}\end{equation} Then
\eqref{ratiobound} follows from \eqref{smoothdenom} and
\eqref{smoothnumer} as required.\\
On the other hand, when $x_j \in J$, by \eqref{lte0}, \eqref{lte2}
and Lemma \ref{lsmooth2}
\begin{equation}\label{s2}|(L^{M,N}(\vec{V}-\vec{v}))_i(x_j,t_k)|
\leq C[M^{-1}+
\eps_i(h_r+h_{r+1})(1+\sum^n_{q=i}\frac{B_q (\sigma_r
-h_r)}{\sqrt{\eps_q}})].\end{equation}
The cases $i \ge r$ and $i<r$
are treated separately.\\
Suppose first that $i \ge r$, then it is not hard to see that
\begin{equation}\label{jumpnumer1}\begin{array}{l}
|(L^{M,N}(\vec{V}-\vec{v}))_i(x_j,t_k)| \\
\leq C[M^{-1}+\eps_i(h_r+h_{r+1})(1+\frac{1}{\sqrt{\eps_i}})]
\\ \leq C[M^{-1}+(h_r+h_{r+1})\sqrt{\eps_i}] .\end{array}\end{equation}
Combining \eqref{barrierbound4} and \eqref{jumpnumer1},
\eqref{ratiobound} follows using \eqref{geom4} and the ordering of
the $\eps_i$.\\
On the other hand, if $i<r$ then $\eps_i \leq \eps_{r-1} < \eps_r$.
Also, either $d_r>0$ or $d_r=0.$\\
First, suppose that $d_r>0$. Then, by Lemma \ref{s1}, \[\sigma_r-h_r
\ge x^{(\frac{1}{2})}_{q,r} ;\; \mathrm{for} \;\; i \leq q \leq
r-1\] and so, by Lemma \ref{layers}\[\sum^{r-1}_{q=i}\frac{B_q
(\sigma_r
-h_r)}{\sqrt{\eps_q}} \leq C\frac{B_r (\sigma_r
-h_r)}{\sqrt{\eps_r}}.\] Combining this with \eqref{s2} gives
\begin{equation}\label{jumpnumer2}
|(L^{M,N}(\vec{V}-\vec{v}))_i(x_j,t_k)| \\
\leq C[M^{-1}+\frac{\eps_i}{\sqrt{\eps_r}}(h_r+h_{r+1})]
.\end{equation} Combining \eqref{barrierbound4} and
\eqref{jumpnumer2}, \eqref{ratiobound} follows using \eqref{geom4}
and the ordering of
the $\eps_i$.\\
Secondly, suppose that $d_r=0$. Then $d_{r-1}>0$ and $D^+ h_r<0$.
Then, by Lemma \ref{s1} with $r-1$ instead of $r$, \[\sigma_r-h_r
\ge \sigma_{r-1}>\sigma_{r-1}-h_{r-1} \ge
x^{(\frac{1}{2})}_{q,{r-1}}\;\; \mathrm{for}\;\; i \leq q \leq r-2\]
and so, by Lemma
\ref{layers}\[\sum^{r-2}_{q=i}\frac{B_q (\sigma_r
-h_r)}{\sqrt{\eps_q}} \leq C\frac{B_{r-1} (\sigma_r
-h_r)}{\sqrt{\eps_{r-1}}}\leq
C\frac{B_{r-1}(\sigma_{r-1})}{\sqrt{\eps_{r-1}}}=C\frac{N^{-2}}{\sqrt{\eps_{r-1}}}.\]
Combining this with \eqref{s2} and \eqref{geom4} gives
\begin{equation}\label{jumpnumer3}\begin{array}{l}
|(L^{M,N}(\vec{V}-\vec{v}))_i(x_j,t_k)| \\
\leq C[M^{-1}+\eps_i(h_r+h_{r+1})(\frac{N^{-2}}{\sqrt{\eps_{r-1}}}+
\frac{1}{\sqrt{\eps_{r}}})]\\
\leq C[M^{-1}+\sqrt{\eps_i\eps_r}N^{-3}\ln N+\eps_i N^{-1}\ln N].
\end{array}\end{equation}
Combining \eqref{barrierbound4} and \eqref{jumpnumer3},
\eqref{ratiobound} follows using the ordering of
the $\eps_i$ and noting that in this case
the middle term in the denominator is used to bound the
middle term in the numerator.\\
\eop\end{proof} Before the singular part of the error is estimated
the following lemmas are established.
\begin{lemma}\label{est1} Assume that $x_j \notin J$. Let $A(x,t)$ satisfy
(\ref{a1}) and (\ref{a2}). Then,
on $\Omega^{M,N}$, for each $1 \leq i \leq n$, the following
estimates hold
\begin{equation} |(L^{M,N}(\vec{W^L}-\vec{w^L}))_i(x_j,t_k)|\leq C(M^{-1}+\frac{(x_{j+1}-x_{j-1})^2}{\eps_1}).\end{equation} An analogous result holds for the $w^R_i$.
\end{lemma}
\begin{proof} Since $x_j \notin J$, from (\ref{lte3}) and Lemma \ref{lsingular},
it follows that
\begin{equation*}\begin{array}{lcl}|(L^{M,N}(\vec{W^L}-\vec{w^L}))_i(x_j,t_k)|&=&
|((\frac{\partial}{\partial t}-D^-_t)-E(\frac{\partial^2}{\partial
x^2}-\delta^2_x))\vec{w}^L _i(x_j,t_k)|\\
& \leq &
C(M^{-1}+(x_{j+1}-x_{j-1})^2\;\ds\max_{s\;\in\;I_j}\ds\sum_{q\;=\;1}^n
\dfrac{B^{L}_{q}(s)}{\eps_q}) \\ & \leq &
C(M^{-1}+\frac{(x_{j+1}-x_{j-1})^2}{\eps_1})\end{array}\end{equation*}
as required.\eop\end{proof}
The following decompositions are introduced
\[w^L_i=\sum_{m=1}^{r+1}w_{i,m},\] where the components
are defined by
\[w_{i,r+1}=\left\{ \begin{array}{ll} p^{(s)}_{i} & {\rm on}\;\;[0,x^{(s)}_{r,r+1})\\
w^L_i & {\rm otherwise} \end{array}\right. \]
and for each $m$, $r \ge m \ge 2$,
\[w_{i,m}=\left\{ \begin{array}{ll} p^{(s)}_{i} & \rm{on} \;\; [0,x^{(s)}_{m-1,m})\\
w^L_i-\ds\sum_{q=m+1}^{r+1} w_{i,q} & {\rm otherwise}
\end{array}\right. \]
and
\[w_{i,1}=w^L_i-\sum_{q=2}^{r+1} w_{i,q}\;\; \rm{on} \;\; [0,1]. \]
Here the polynomials $p^{(s)}_{i}$, for $s=3/2$ and $s=1$, are
defined by
\\ \[ p^{(3/2)}_{i}(x,t)=\sum_{q=0}^3
\frac{\partial^q w^{(L)}_i}{\partial
x^q}(x^{(3/2)}_{r,r+1},t)\frac{(x-x^{(3/2)})^q}{q!}\]\\ and \\
\[ p^{(1)}_{i}(x,t)=\sum_{q=0}^4
\frac{\partial^q w^{(L)}_i}{\partial
x^q}(x^{(1)}_{r,r+1},t)\frac{(x-x^{(1)})^q}{q!}.\]
\begin{lemma}\label{general} Assume that $d_r>0$. Let $A(x,t)$ satisfy (\ref{a1}) and (\ref{a2}). Then, for
each $1 \leq i \leq n$, there exists a decomposition
\[ w^L_i=\sum_{q=1}^{r+1}w_{i,q}, \] for which the following
estimates hold for each $q$ and $r$, $1 \le q \le r$,
\[|\frac{\partial^2 w_{i,q}}{\partial x^2}(x_j,t_k)| \leq C \min \{\frac{1}{\eps_q}, \frac{1}{\eps_i}\}B^L_q(x_j),\]
\[|\frac{\partial^3 w_{i,q}}{\partial x^3}(x_j,t_k)| \leq C \min \{\frac{1}{\eps_q^{3/2}}, \frac{1}{\eps_i \sqrt{\eps_q}}\}B^L_q(x_j),\]
\[|\frac{\partial^3 w_{i,r+1}}{\partial x^3}(x_j,t_k)| \leq C \min \{\sum_{q=r+1}^n \frac{B^L_q(x_j)}{\eps_q^{3/2}}, \sum_{q=r+1}^n \frac{B^L_q(x_j)}{\eps_i \sqrt{\eps_q}}\},\]
\[|\frac{\partial^4 w_{i,q}}{\partial x^4}(x_j,t_k)| \leq C \frac{B^L_q(x_j)}{\eps_i \eps_q},\]
\[|\frac{\partial^4 w_{i,r+1}}{\partial x^4}(x_j,t_k)| \leq
C\sum_{q=r+1}^n \frac{B^L_q(x_j)}{\eps_i \eps_q}.\]
Analogous results hold for the $w^R_i$ and their derivatives.
\end{lemma}
\begin{proof}
First consider the decomposition corresponding $s=3/2$.\\
From the above definitions it follows that, for each $m$, $1 \leq m
\leq r$,
$w_{i,m}=0 \;\; \rm{on} \;\; [x^{(3/2)}_{m,m+1},1]$.\\
To establish the bounds on the third derivatives it is seen that:
for $x \in [x^{(3/2)}_{r,r+1},1]$, Lemma \ref{lsingular} and $x \geq
x^{(3/2)}_{r,r+1}$ imply that
\[|\frac{\partial^3 w_{i,r+1}}{\partial x^3}(x,t)| =
|\frac{\partial^3 w^L_{i}}{\partial x^3}(x,t)| \leq
C\sum_{q=1}^n \frac{B^L_q(x)}{\eps_q^{3/2}} \leq C\sum_{q=r+1}^n
\frac{B^L_q(x)}{\eps_q^{3/2}};\]
for $x \in [0, x^{(3/2)}_{r,r+1}]$, Lemma \ref{lsingular} and $x
\leq x^{(3/2)}_{r,r+1}$ imply that
\[|\frac{\partial^3
w_{i,r+1}}{\partial x^3}(x,t)| = |\frac{\partial^3 w^L_{i}}{\partial
x^3}(x^{(3/2)}_{r,r+1},t)|\] \[\leq \sum_{q=1}^{n}
\frac{B^L_q(x^{(3/2)}_{r,r+1})}{\eps_q^{3/2}} \leq \sum_{q=r+1}^{n}
\frac{B^L_q(x^{(3/2)}_{r,r+1})}{\eps_q^{3/2}} \leq \sum_{q=r+1}^{n}
\frac{B^L_q(x)}{\eps_q^{3/2}};\]
and for each $m=r, \;\; \dots \;\;,2$, it follows that\\
for $x \in [x^{(3/2)}_{m,m+1},1]$, $w_{i,m}^{(3)}=0;$
for $x \in [x^{(3/2)}_{m-1,m},x^{(3/2)}_{m,m+1}]$, Lemma
\ref{lsingular} implies that
\[|\frac{\partial^3
w_{i,m}}{\partial x^3}(x,t)| \leq |\frac{\partial^3
w^L_{i}}{\partial x^3}(x,t)|+\sum_{q=m+1}^{r+1}|\frac{\partial^3
w_{i,q}}{\partial x^3}(x,t)|\] \[ \leq C\sum_{q=1}^n
\frac{B^L_q(x)}{\eps_q^{3/2}} \leq C\frac{B^L_m(x)}{\eps_m^{3/2}};\]
for $x \in [0, x^{(3/2)}_{m-1,m}]$, Lemma \ref{lsingular} and $x
\leq x^{(3/2)}_{m-1,m}$ imply that
\[|\frac{\partial^3
w_{i,m}}{\partial x^3}(x,t)| =|\frac{\partial^3 w^L_{i}}{\partial
x^3}(x^{(3/2)}_{m-1,m},t)|\] \[ \leq C\sum_{q=1}^n
\frac{B^L_q(x^{(3/2)}_{m-1,m})}{\eps_q^{3/2}} \leq
C\frac{B^L_m(x^{(3/2)}_{m-1,m})}{\eps_m^{3/2}} \leq
C\frac{B^L_m(x)}{\eps_m^{3/2}};
\]
for $x \in [x^{(3/2)}_{1,2},1],\;\; \frac{\partial^3
w_{i,1}}{\partial x^3}=0;$
for $x \in [0, x^{(3/2)}_{1,2}]$, Lemma \ref{lsingular} implies that
\[|\frac{\partial^3
w_{i,1}}{\partial x^3}(x,t)| \leq |\frac{\partial^3
w^L_{i}}{\partial x^3}(x,t)|+\sum_{q=2}^{r+1}|\frac{\partial^3
w_{i,q}}{\partial x^3}(x,t)| \leq C\sum_{q=1}^n
\frac{B^L_q(x)}{\eps_q^{3/2}} \leq C\frac{B^L_1(x)}{\eps_1^{3/2}}.\]
For the bounds on the second derivatives note that, for each $m$, $1
\leq m \leq r $ :
for $x \in [x^{(3/2)}_{m,m+1},1],\;\; \frac{\partial^2
w_{i,m}}{\partial x^2}=0;$
for $x \in [0, x^{(3/2)}_{m,m+1}],\;\; \int_x^{x^{(3/2)}_{m,m+1}}
\frac{\partial^3 w_{i,m}}{\partial x^3}(s,t)ds =\\ \frac{\partial^2
w_{i,m}}{\partial x^2}(x^{(3/2)}_{m-1,m},t)- \frac{\partial^2
w_{i,m}}{\partial x^2}(x,t)= -\frac{\partial^2
w_{i,m}}{\partial x^2}(x,t)$ \\
and so
\[|\frac{\partial^2
w_{i,m}}{\partial x^2}(x,t)| \leq
\int_x^{x^{(3/2)}_{m,m+1}}|\frac{\partial^3 w_{i,m}}{\partial
x^3}(s,t)|ds \leq \frac{C}{\eps_m^{3/2}}\int_{x}^{x^{(3/2)}_{m,m+1}}
B^L_m(s)ds \leq C\frac{B^L_m(x)}{\eps_m}.\] This completes the proof
of the estimates for $s=3/2$.\\
For the estimates in the case $s=1$ consider the decomposition
\[w^L_i=\sum_{m=1}^{r+1}w_{i,m}.\]
From the above definitions it follows that, for each $m$, $1 \leq m
\leq r$,
$w_{i,m}=0 \;\; \rm{on} \;\; [x^{(1)}_{m,m+1},1]$.\\
To establish the bounds on the fourth derivatives it is seen that:
for $x \in [x^{(1)}_{r,r+1},1]$, Lemma \ref{lsingular} and $x \geq
x^{(1)}_{r,r+1}$ imply that
\[|\eps_i \frac{\partial^4 w_{i,r+1}}{\partial x^4}(x,t)| =|\eps_i \frac{\partial^4 w^L_{i}}{\partial x^4}(x,t)| \leq
C\sum_{q=1}^n \frac{B^L_q(x)}{\eps_q} \leq C\sum_{q=r+1}^n
\frac{B^L_q(x)}{\eps_q};\]
for $x \in [0, x^{(1)}_{r,r+1}]$, Lemma \ref{lsingular} and $x \leq
x^{(1)}_{r,r+1}$ imply that
\[|\eps_i \frac{\partial^4 w_{i,r+1}}{\partial x^4}(x,t)| =|\eps_i \frac{\partial^4 w^L_{i}}{\partial x^4}(x^{(1)}_{r,r+1},t)|
\] \[\leq \sum_{q=1}^{n} \frac{B^L_q(x^{(1)}_{r,r+1})}{\eps_q} \leq
C\sum_{q=r+1}^{n} \frac{B^L_q(x^{(1)}_{r,r+1})}{\eps_q} \leq
C\sum_{q=r+1}^{n} \frac{B^L_q(x)}{\eps_q};\]
and for each $m=r, \;\; \dots \;\;,2$, it follows that\\
for $x \in [x^{(1)}_{m,m+1},1]$,\;\; $\frac{\partial^4
w_{i,m}}{\partial x^4}=0;$
for $x \in [x^{(1)}_{m-1,m},x^{(1)}_{m,m+1}]$, Lemma \ref{lsingular}
implies that
\[|\eps_i \frac{\partial^4 w_{i,m}}{\partial x^4}(x,t)| \leq |\eps_i \frac{\partial^4 w^L_{i}}{\partial x^4}(x,t)|+\sum_{q=m+1}^{r+1}|\eps_i \frac{\partial^4 w_{i,q}}{\partial x^4}(x,t)|
\] \[\leq C\sum_{q=1}^n \frac{B^L_q(x)}{\eps_q} \leq
C\frac{B^L_m(x)}{\eps_m};\]
for $x \in [0, x^{(1)}_{m-1,m}]$, Lemma \ref{lsingular} and $x \leq
x^{(1)}_{m-1,m}$ imply that
\[|\eps_i \frac{\partial^4 w_{i,m}}{\partial x^4}(x,t)| =
|\eps_i \frac{\partial^4 w^L_{i}}{\partial x^4}(x^{(1)}_{m-1,m},t)|
\] \[\leq C\sum_{q=1}^n \frac{B^L_q(x^{(1)}_{m-1,m})}{\eps_q} \leq
C\frac{B^L_m(x^{(1)}_{m-1,m})}{\eps_m} \leq
C\frac{B^L_m(x)}{\eps_m};
\]
for $x \in [x^{(1)}_{1,2},1],\;\; \frac{\partial^4 w_{i,1}}{\partial
x^4}=0;$
for $x \in [0, x^{(1)}_{1,2}]$, Lemma \ref{lsingular} implies that
\[|\eps_i \frac{\partial^4 w_{i,1}}{\partial
x^4}(x,t)| \leq |\eps_i \frac{\partial^4 w^L_{i}}{\partial
x^4}(x,t)|+\sum_{q=2}^{r+1}|\eps_i \frac{\partial^4
w_{i,q}}{\partial x^4}(x,t)|\] \[\leq C\sum_{q=1}^n
\frac{B^L_q(x)}{\eps_q} \leq C\frac{B^L_1(x)}{\eps_1}.\]
For the bounds on the second and third derivatives note that, for
each $m$, $1 \leq m \leq r $ :
for $x \in [x^{(1)}_{m,m+1},1],\;\; \frac{\partial^2
w_{i,m}}{\partial x^2}=0=\frac{\partial^3 w_{i,m}}{\partial x^3};$
for $x \in [0, x^{(1)}_{m,m+1}],\;\;
\ds\int_x^{x^{(1)}_{m,m+1}}\eps_i \frac{\partial^4 w_{i,m}}{\partial
x^4}(s,t)ds \\= \eps_i \frac{\partial^3 w_{i,m}}{\partial
x^3}(x^{(1)}_{m,m+1},t)- \eps_i \frac{\partial^3 w_{i,m}}{\partial
x^3}(x,t)= -\eps_i \frac{\partial^3 w_{i,m}}{\partial
x^3}(x,t)$ \\
and so
\[|\eps_i \frac{\partial^3 w_{i,m}}{\partial
x^3}(x,t)| \leq \int_x^{x^{(1)}_{m,m+1}}|\eps_i \frac{\partial^4
w_{i,1}}{\partial x^4}(s,t)|ds \\ \leq
\frac{C}{\eps_m}\int_{x}^{x^{(1)}_{m,m+1}} B^L_m(s)ds \leq
C\frac{B^L_m(x)}{\sqrt\eps_m}.\] In a similar way, it can be shown
that
\[|\eps_i \frac{\partial^2 w_{i,m}}{\partial
x^2}(x,t)| \leq C B^L_m(x).\]
The proof for the $w^R_i$ and their derivatives is similar. \eop
\end{proof}
\begin{lemma}\label{general1} Assume that $d_r>0$. Let $A(x)$ satisfy (\ref{a1}) and (\ref{a2}).
Then, for each $i$, $1 \leq i \leq n$, and each $(x_j,t_k) \in
\Omega^{M,N}$
\begin{equation}|\label{gen1}(L^{M,N}(\vec{W^L}-\vec{w}^L)_i(x_j,t_k))| \leq C
[M^{-1}+B^L_r(x_{j-1})+\frac{x_{j+1}-x_{j-1}}{\sqrt{\eps_{r+1}}}]
\end{equation}
and
\begin{equation}\label{gen2}|(L^{M,N}(\vec{W^L}-\vec{w}^L)_i(x_j,t_k))| \leq C[M^{-1}+\eps_i
\sum^r_{q=1}\frac{B^L_{q}(x_{j-1})}{\eps_q}+\frac{\eps_i}{\eps_{r+1}}
\frac{x_{j+1}-x_{j-1}}{\sqrt{\eps_{r+1}}}].\end{equation}
Assuming, furthermore, that $x_j \notin J,$ then
\begin{equation}\label{gen3}|(L^{M,N}(\vec{W^L}-\vec{w}^L)_i(x_j,t_k))| \leq C
[M^{-1}+B^L_r(x_{j-1})+\frac{(x_{j+1}-x_{j-1})^2}{\eps_{r+1}}].
\end{equation}
Analogous results hold for the $W^R-w^R_i$ and their derivatives.
\end{lemma}
\begin{proof} Using \eqref{LTEWL}, \eqref{lte0} and the bound in
Lemma \ref{lsingular}, for any $(x_j,t_k) \in \Omega^{M,N}$,
\begin{equation} \label{ls1}|(L^{M,N}(\vec{W^L}-\vec{w}^L)_i(x_j,t_k))| \leq
C[(t_k-t_{k-1})+|\eps_i(\delta^2_x-\frac{\partial^2}{\partial
x^2})w^L_i(x_j, t_k)| ].
\end{equation}
From the decompositions and bounds in Lemma \ref{general}, with
\eqref{lte1} and \eqref{lte2}, it follows from \eqref{ls1} that
\begin{equation} \begin{array}{l} |\eps_i(\delta^2_x-\frac{\partial^2}{\partial
x^2})w^L_i(x_j, t_k)| \\ \\\leq
C[\sum_{q=1}^{r}|\eps_i(D^2_x-\frac{\partial^2}{\partial
x^2})w_{i,q}(x_j, t_k)|+ |\eps_i(D^2_x-\frac{\partial^2}{\partial
x^2})w_{i,r+1}(x_j, t_k)] \\ \\
\leq C[\sum_{q=1}^{r}\max_{s \in I_j}|\eps_iw_{i,q}^{(2)}(s,t_k)|
+(x_{j+1}-x_{j-1})\max_{s \in
I_j}|\eps_iw_{i,r+1}^{(3)}(s,t_k)|)]\\ \\ \leq
C[\sum_{q=1}^{r}\min\{\frac{\eps_i}{\eps_q},1\}B^L_q
(x_{j-1})+(x_{j+1}-x_{j-1})\min\{\frac{\eps_i}{\eps_{r+1}},1\}\frac{B^L_{r+1}
(x_{j-1})}{\sqrt{\eps_{r+1}}}].
\end{array}\end{equation}
Substituting $1$ for each of the $\min$ expressions gives
\eqref{gen1} and \eqref{gen2} is obtained by substituting the
appropriate ratio $\frac{\eps_i}{\eps_q}$ in each such
expression.\\
In the remaining case when $x_j \notin J$, \eqref{lte3} can be
used instead of \eqref{lte2}, and it follows by a similar argument
to the above that
\begin{equation} |\eps_i(\delta^2_x-\frac{\partial^2}{\partial
x^2})w^L_i(x_j, t_k)| \leq
C[\sum_{q=1}^{r}\min\{\frac{\eps_i}{\eps_q},1\}B^L_q
(x_{j-1})+\frac{(x_{j+1}-x_{j-1})^2}{\eps_{r+1}}].
\end{equation}
Substituting $1$ for the $\min$ expression, as before, gives
\eqref{gen3}.\\
The proof for the $w^R_i$ and their derivatives is similar.\eop
\end{proof}
\begin{lemma}\label{est3} Let $A(x,t)$ satisfy (\ref{a1}) and (\ref{a2}).
Then, on $\Omega^{M,N}$, for each $1 \leq i \leq n$, the following
estimates hold
\begin{equation} |(L^{M,N}(\vec{W^L}-\vec{w^L}))_i(x_j,t_k)|\leq
C(M^{-1}+B^L_n(x_{j-1})).\end{equation} An analogous result holds for the $w^R_i$.
\end{lemma}
\begin{proof}
From $\;\eqref{lte1}\;$ and Lemma \ref{lsingular}, for each
$\;i=1,\dots,n\;$, it follows that on $\Omega^{M,N},$
\begin{equation*}\begin{array}{lcl}|(L^{M,N}(\vec{W^L}-\vec{w^L}))_i(x_j,t_k)|&=&
|((\frac{\partial}{\partial t}-D^-_t)-E(\frac{\partial^2}{\partial
x^2}-\delta^2_x))\vec{w}^L _i(x_j,t_k)|\\
& \leq &
C(M^{-1}+\eps_i\ds\sum_{q=i}^{n}\dfrac{B^L_q(x_{j-1})}{\eps_q})\\ &
\le & C(M^{-1}+B^L_n(x_{j-1})).\end{array}\end{equation*} The proof
for the $w^R_i$ and their derivatives is similar.\eop
\end{proof}
The following theorem provides the error estimate for the singular
component.
\begin{theorem} \label{singularerrorthm}Let $A(x,t)$ satisfy (\ref{a1}) and
(\ref{a2}). Let $\vec w$ denote the singular component of the exact
solution from (\ref{BVP}) and $\vec W$ the singular component of the
discrete solution from (\ref{discreteBVP}). Then
\begin{equation}\;\; ||\vec{W}-\vec{w}|| \leq C(M^{-1}+N^{-2}(\ln N)^{3}). \end{equation}
\end{theorem}
\begin{proof}
Since $\vec{w}=\vec{w}^L+\vec{w}^R$, it suffices to prove the result
for $\vec{w}^L$ and $\vec{w}^R$ separately. Here it is proved for
$\vec{w}^L$ by an application of Lemma \ref{comparison}. A similar proof holds for $\vec{w}^R$.\\
The proof is in two parts: $x_j$ is such that either $x_j \notin J$ or $x_j=\sigma_r \in J$. \\
First assume that $x_j \notin J$.
Each open subinterval $(\sigma_k,\sigma_{k+1})$ is treated separately.\\
First, consider $x_j \in (0,\sigma_1)$. Then, on each mesh $M$,
$x_{j+1}-x_{j-1} \leq CN^{-1}\sigma_1$ and the result follows
from (\ref{geom2}) and Lemma \ref{est1}.\\
Secondly, consider $x_j \in (\sigma_1,\sigma_2)$, then $\sigma_1
\leq x_{j-1}$ and $x_{j+1}-x_{j-1} \leq CN^{-1}\sigma_2$. The
$2^{n}$ possible meshes are divided into subclasses of two types.
On the meshes $\overline{\Omega}^{M,N}$ with $b_1=0$ the result
follows from (\ref{geom2}), (\ref{geom-1}) and Lemma \ref{est1}.
On the meshes $\overline{\Omega}^{M,N}$ with
$b_1=1$ the result follows from (\ref{geom2}), (\ref{geom0}) and Lemma \ref{general}.\\
Thirdly, in the general case $x_j \in (\sigma_m,\sigma_{m+1})$ for
$2 \leq m \leq n-1$, it follows that $\sigma_m \leq x_{j-1}$ and
$x_{j+1}-x_{j-1} \leq CN^{-1}\sigma_{m+1}$. Then
$\overline{\Omega}^{M,N}$ is divided into subclasses of three types:
$\overline{\Omega}^{M,N}_0=\{\overline{\Omega}^{M,N}: b_1= \dots
=b_m =0\},\; \overline{\Omega}^{M,N}_{r}=\{\overline{\Omega}^{M,N}:
b_r=1, \; b_{r+1}= \dots =b_m =0 \; \mathrm{for \; some}\; 1 \leq r
\leq m-1\}$ and
$\overline{\Omega}^{M,N}_m=\{\overline{\Omega}^{M,N}: b_m=1\}.$ On
$\overline {\Omega}^{M,N}_0$ the result follows from (\ref{geom2}),
(\ref{geom-1}) and Lemma \ref{est1}; on $\overline{\Omega}^{M,N}_r$
from (\ref{geom2}), (\ref{geom-1}), (\ref{geom0}) and Lemma
\ref{general}; on $\overline{\Omega}^{M,N}_m$ from (\ref{geom2}),
(\ref{geom0})
and Lemma \ref{general}. \\
Finally, for $x_j \in (\sigma_n,1)$, $\sigma_n \leq x_{j-1}$ and
$x_{j+1}-x_{j-1} \leq CN^{-1}$. Then $\overline{\Omega}^{M,N}$ is
divided into subclasses of three types:
$\overline{\Omega}^{M,N}_0=\{\overline{\Omega}^{M,N}: b_1= \dots
=b_n =0\},\; \overline{\Omega}^{M,N}_{r}=\{\overline{\Omega}^{M,N}:
b_r=1, \; b_{r+1}= \dots =b_n =0 \; \mathrm{for \; some}\; 1 \leq r
\leq n-1\}$ and
$\overline{\Omega}^{M,N}_n=\{\overline{\Omega}^{M,N}: b_n=1\}.$ On
$\overline{\Omega}^{M,N}_0$ the result follows from (\ref{geom2}),
(\ref{geom-1}) and Lemma \ref{est1}; on $\overline{\Omega}^{M,N}_r$
from (\ref{geom2}), (\ref{geom-1}), (\ref{geom0}) and Lemma
\ref{general}; on $\overline{\Omega}^{M,N}_n$ from (\ref{geom0}) and
Lemma
\ref{est3}.\\\\
Now assume that $x_j \in J$. Then $x_j=\sigma_r$, some $r$. It
suffices to show that
\begin{equation}\label{ratiobound2} \frac{|(L^{M,N}(\vec{W}^L-\vec{w}^L))_i (x_j,t_k)|}{|(L^{M,N}\vec{\Phi})_i
(x_j,t_k)|} \leq C,
\end{equation}
for each $i=1,\;\dots \;,n,$ because an application of the
Comparison Principle then yields the required result.\\
The bounds on the denominator are given in \eqref{barrierbound4}.
To bound the numerator note that either $d_r>0$ or $d_r=0$.\\
Suppose first that $d_r>0$. Then the cases $i > r$ and $i \leq r$
are treated separately.\\
If $i > r$, then, by \eqref{gen1} in Lemma \ref{general1},
\begin{equation}
|(L^{M,N}(\vec{W}^L-\vec{w}^L))_i(x_j,t_k)| \leq
C[M^{-1}+B^L_r(x_{j-1})+\frac{x_{j+1}-x_{j-1}}{\sqrt{\eps_{r+1}}}]
\end{equation}
Since $d_r>0$, by Lemma \ref{s1}, $B^L_r (x_{j-1}) = B^L_
r(\sigma_r -h_r) \leq CB^L_r(\sigma_r)= CN^{-2},$ and so
\begin{equation}
|(L^{M,N}(\vec{W}^L-\vec{w}^L))_i(x_j,t_k)| \leq
C[M^{-1}+N^{-2}+\frac{h_r+h_{r+1}}{\sqrt{\eps_{r+1}}}].
\end{equation}
Using \eqref{geom4} and the ordering of the $\eps_i$, these bounds
on the numerator and denominator lead
to \eqref{ratiobound2}.\\
If $i \leq r$, then, by \eqref{gen2} in Lemma \ref{general1},
\begin{equation}
|(L^{M,N}(\vec{W}^L-\vec{w}^L))_i(x_j,t_k)| \leq C[M^{-1}+\eps_i
\sum^r_{q=1}\frac{B^L_q(x_{j-1})}{\eps_q}+\frac{\eps_i}{\eps_{r+1}}\frac{x_{j+1}-x_{j-1}}{\sqrt{\eps_{r+1}}}].
\end{equation}
Since $d_r>0$, by Lemma \ref{s1}, $x_{j-1}=\sigma_r-h_r \ge
x^{(s)}_{q,r}$ for $1 \leq q \leq r-1$ and
\[\frac{B^L_q (x_{j-1})}{\eps_q} \leq \frac{B^L_r (x_{j-1})}{\eps_r} \leq \frac{B^L_r
(\sigma_r)}{\eps_r} = C\frac{N^{-2}}{\eps_r}.\] Thus
\begin{equation}
|(L^{M,N}(\vec{W}^L-\vec{w}^L))_i(x_j,t_k)| \leq
C[M^{-1}+\frac{\eps_i}{\eps_{r}}N^{-2}+\frac{\eps_i}{\eps_{r+1}}\frac{x_{j+1}-x_{j-1}}{\sqrt{\eps_{r+1}}}].
\end{equation}
Using \eqref{geom4} and the ordering of the $\eps_i$, these bounds
on the numerator and denominator lead
to \eqref{ratiobound2}.\\
Now suppose that $d_r=0$. Then $d_{r-1}>0$ and $D^+ h_r<0$,
because otherwise $x_j \notin J$. The cases $i \ge r$ and $i < r$
are now treated separately.\\
If $i \ge r$, then, by \eqref{gen1} in Lemma \ref{general1} with
$r$ replaced by $r-1$,
\begin{equation}
|(L^{M,N}(\vec{W}^L-\vec{w}^L))_i(x_j,t_k)| \leq
C[M^{-1}+B^L_{r-1}(x_{j-1})+\frac{x_{j+1}-x_{j-1}}{\sqrt{\eps_{r}}}]
\end{equation}
Since $d_{r-1}>0$, by Lemma \ref{s1} with $r$ replaced by $r-1$,
\[B^L_{r-1} (x_{j-1})=B^L_ {r-1}(\sigma_r -h_r) \leq
CB^L_{r-1}(\sigma_{r-1})= CN^{-2},\] and so
\begin{equation}
|(L^{M,N}(\vec{W}^L-\vec{w}^L))_i(x_j,t_k)| \leq
C[M^{-1}+N^{-2}+\frac{h_r+h_{r+1}}{\sqrt{\eps_{r}}}].
\end{equation}
Using \eqref{geom4} and the ordering of the $\eps_i$, these bounds
on the numerator and denominator lead
to \eqref{ratiobound2}.\\
If $i < r$, then by \eqref{gen2} in Lemma \ref{general1} with
$r$ replaced by $r-1$,
\begin{equation}
|(L^{M,N}(\vec{W}^L-\vec{w}^L))_i(x_j,t_k)| \leq C[M^{-1}+\eps_i
\sum^{r-1}_{q=1}\frac{B^L_q(x_{j-1})}{\eps_q}+\frac{\eps_i}{\eps_{r}}\frac{x_{j+1}-x_{j-1}}{\sqrt{\eps_{r}}}].
\end{equation}
Since $d_{r-1}>0$, by Lemma \ref{s1}with $r$ replaced by $r-1$,
$x_{j-1}=\sigma_r-h_r \ge x^{(s)}_{q,r}$ for $1 \leq q \leq r-1$
and
\[\frac{B^L_q (x_{j-1})}{\eps_q} \leq \frac{B^L_{r-1} (x_{j-1})}{\eps_{r-1}} \leq \frac{B^L_{r-1}
(\sigma_{r-1})}{\eps_{r-1}} = C\frac{N^{-2}}{\eps_{r-1}}.\] Thus
\begin{equation}
|(L^{M,N}(\vec{W}^L-\vec{w}^L))_i(x_j,t_k)| \leq
C[M^{-1}+\frac{\eps_i}{\eps_{r-1}}N^{-2}+\frac{\eps_i}{\eps_{r}}\frac{x_{j+1}-x_{j-1}}{\sqrt{\eps_{r}}}].
\end{equation}
Using \eqref{geom4} and the ordering of the $\eps_i$, these bounds
on the numerator and denominator lead to \eqref{ratiobound2}. This
completes the proof.\eop \end{proof}
The following theorem gives the required first order in time and
essentially second order in space parameter-uniform error estimate.
\begin{theorem}Let $A(x,t)$ satisfy (\ref{a1}) and
(\ref{a2}). Let $\vec u$ denote the exact solution of (\ref{BVP})
and $\vec U$ the discrete solution of (\ref{discreteBVP}). Then
\begin{equation}\;\; ||\vec{U}-\vec{u}|| \leq C\,N^{-2}(\ln N)^3. \end{equation}
\end{theorem}
\begin{proof}
An application of the triangle inequality and the results of
Theorems \ref{smootherrorthm} and \ref{singularerrorthm} leads
immediately to the required result.\eop
\end{proof}
|
1,116,691,497,566 | arxiv | \section{Introduction}
\label{sec:introduction}}
\begin{CJK}{UTF8}{gbsn}
\IEEEPARstart{R}{ecent} years have witnessed a tremendous progress in computer-aided detection/diagnosis (CAD) in medical imaging and diagnostic radiology, primarily thanks to the advancement of deep learning techniques. Having achieved great success in computer vision tasks, various deep learning models, mainly convolutional neural networks (CNNs), soon be applied to CAD. Among the applications are the early detection and diagnosis of breast cancer, lung cancer, glaucoma, and skin cancer \cite{esteva2017dermatologist,8471199,zhou2017fine-tuning,zhu2018deeplung}.
However, the small size of medical datasets continues to be an issue in obtaining
satisfactory deep learning model for CAD; in general,
bigger datasets result in better deep learning models \cite{halevy2009unreasonable}. In traditional computer vision tasks, there are many large-scale and well-annotated datasets, such as ImageNet
\footnote{http://www.image-net.org/}
(over 14M labeled images from 20k categories) and COCO
\footnote{http://mscoco.org/}
(with more than 200k annotated images across 80 categories). In contrast, some popular publicly available medical datasets are much smaller (see Table \ref{tab:medical_dataset}). For example, among the datasets for different tasks, only ChestX-ray14 and DeepLesion, contain more than 100k labeled medical images, while most datasets only have a few thousands or even hundreds of medical images.
\begin{table*}[]
\center
\small
\caption{Examples of popular datasets in the medical domain}
\begin{tabular}{|l|l|l|l|l|}
\hline
Name & Purpose & Type & Imaging & Number of Images \\ \hline
\begin{tabular}[c]{@{}c@{}}ADNI \cite{WEINER2017561}\end{tabular} &Classification & Brain &Multiple & 1921 patients \\ \hline
\multirow{1}{*}{\begin{tabular}[c]{@{}c@{}}ABIDE \cite{di2014autism}\end{tabular}} & \multirow{1}{*}{Classification} & \multirow{1}{*}{Brain} & \multirow{1}{*}{MRI} & 539 patients and 573 controls \\ \hline
\begin{tabular}[c]{@{}c@{}}ACDC \cite{acdc_web}\end{tabular} &Classification & Cardiac &MRI & 150 patients \\ \hline
\multirow{1}{*}{\begin{tabular}[c]{@{}c@{}}ChestX-ray14 \cite{article}\end{tabular}} &\multirow{1}{*}{Detection} & \multirow{1}{*}{Chest} &\multirow{1}{*}{X-ray} & 112,120 images from 30,805 patients \\ \hline
\begin{tabular}[c]{@{}c@{}}LIDC-IDRI \cite{armato2011lung}\end{tabular} &Detection & Lung &CT, X-ray & 1,018 patients \\ \hline
\begin{tabular}[c]{@{}c@{}}LUNA16 \cite{luna2016}\end{tabular} &Detection & Lung &CT & 888 images \\ \hline
\multirow{1}{*}{\begin{tabular}[c]{@{}c@{}} MURA \cite{rajpurkar2017mura}\end{tabular}} &\multirow{1}{*}{Detection} & \multirow{1}{*}{\begin{tabular}[c]{@{}c@{}}Musculo-skeletal\end{tabular}} &\multirow{1}{*}{X-ray} & \begin{tabular}[c]{@{}c@{}}40,895 images from 14,982 patients\end{tabular} \\ \hline
\begin{tabular}[c]{@{}c@{}}BraTS2018 \cite{bakas2018identifying}\end{tabular} &Segmentation & Brain &MRI & 542 images
\\ \hline
\begin{tabular}[c]{@{}c@{}}STARE \cite{hoover2000locating}\end{tabular} &Segmentation & Eye &SLO & 400 images \\ \hline
\begin{tabular}[c]{@{}c@{}}DDSM \cite{ddsm}\end{tabular} &\begin{tabular}[c]{@{}c@{}}Classification\\Detection\end{tabular} & Breast & Mammography & 2,500 patient
\\ \hline
\begin{tabular}[c]{@{}c@{}}DeepLesion \cite{yan2018deeplesion}\end{tabular} &\begin{tabular}[c]{@{}c@{}}Classification\\Detection\end{tabular} & Multiple &CT & 32,735 images from 4,427 patients \\ \hline
\begin{tabular}[c]{@{}c@{}}Cardiac MRI \cite{cardiacMRI}\end{tabular} &\begin{tabular}[c]{@{}c@{}}Classification\\Segmentation\end{tabular} & Cardiac &MRI & 7,980 images from 33 cases \\ \hline
\begin{tabular}[c]{@{}c@{}}ISIC 2018 \cite{codella2019skin}\end{tabular} &\begin{tabular}[c]{@{}c@{}}Classification\\Detection\\Segmentation \end{tabular} & Skin & Dermoscopic & 13,000 images \\ \hline
\end{tabular}
\label{tab:medical_dataset}
\end{table*}
The lack of medical datasets is represented in three aspects. First, the number of medical images in datasets is usually small. This problem is mainly due to the high cost associated with the data collection. Medical images are collected from computerized tomography (CT), Ultrasonic imaging (US), magnetic resonance imaging (MRI) scans, positron emission tomography (PET), all of which are expensive and labor-intensive. Second, only a small portion of medical images are annotated. These annotations including classification labels (e.g., benign or malignant), the segmentation annotations of lesion areas, etc., require efforts from experienced doctors. Third, it is difficult to collect enough positive cases for some rare diseases to obtain the balanced datasets.
One direct consequence of the lack of well annotated medical data is that the trained deep learning models can easily suffer from the overfitting problem \cite{chen2016deep}. As a result, the models perform very well on training datasets, but fail when dealing with new data from the problem domain. Correspondingly, many existing studies on medical image analysis adopt techniques from computer vision to address overfitting, such as reducing the complexity of the network \cite{ciompi2015automatic,shin2016deep}, adopting some regularization techniques \cite{erickson2017machine}, or using data augmentation strategies \cite{zhu2017unpaired}.
However, in essence, both decreasing model complexity and leveraging data augmentation techniques only focus on the target task on the given datasets, but \emph{do not introduce any new information into deep learning models}. Nowdays, introducing more information, beyond the given medical datasets has become a more promising approach to address the problem of small-sized medical datasets.
The idea of introducing external information to improve the performance of deep learning models for CAD is not new. For example, it is common practice to first train a deep learning model on some natural image datasets like ImageNet, and then fine tune them on target medical datasets \cite{huynh2016mo}. This process, called transfer learning \cite{pan2009survey}, implicitly introduces information from natural images. Besides natural images, multi-modal medical datasets or medical images from different but related diseases can also be used to improve the performance of deep learning models \cite{samala2018breast,yu2019annotation}.
Moreover, as experienced medical doctors (e.g., radiologists, ophthalmologists, and dermatologists) can generally give fairly accurate results, it is not surprising that their knowledge may help deep learning models to better accomplish the designated tasks. The domain knowledge of medical doctors includes the way they browse images, the particular areas they usually focus on, the features they give special attention to, and the anatomical prior knowledge they used. These types of knowledge are accumulated, summarized, and validated by a large number of practitioners over many years based on a huge amount of cases.
Note that in this survey any network that incorporate one of these types of knowledge in their training or designing process should be regarded as the one incorporated medical domain knowledge.
\begin{figure*}[htp!] \begin{centering}
\includegraphics[width=0.9\linewidth]{overview.pdf}
\centering
\caption{Methods of information categorization and incorporating methods in disease diagnosis; lesion, organ, and abnormality detection; lesion and organ segmentation.
}
\label{fig:1}
\end{centering}
\end{figure*}
In this survey, we focus on the three main tasks of medical image analysis: (1) disease diagnosis, (2) lesion, organ and abnormality detection, and (3) lesion and organ segmentation. We also include other related tasks such as the image reconstruction, image retrieval and report generation. This survey demonstrates that, for almost all tasks, identifying and carefully integrating one or more types of domain knowledge related to the designated task will improve the performance of deep learning models.
We organize existing works according to the following three aspects: the types of tasks, the types of domain knowledge that are introduced, and the ways of introducing the domain knowledge.
More specifically, in terms of the types of domain knowledge,
some of them are of high-level such as training pattern \cite{maicas2018training,tang2018attention,wang2018deep} and diagnostic pattern. Some domain knowledge are low-level, such as particular features and special areas where medical doctors pay more attention to \cite{hsu2019breast}. In particular, in disease diagnosis tasks, high-level domain knowledge is widely utilized. For an object detection task, the low-level domain knowledge, such as detection patterns and specific features where medical doctors give special attention is more commonly adopted. For lesion or organ segmentation tasks, anatomical priors and the knowledge from different modalities seem to be more useful \cite{chen2019learning,larrazabal2019anatomical,valverde2019one-shot}.
\begin{figure*}[hbp!] \begin{centering}
\includegraphics[width=0.9\linewidth]{statistical_chart.pdf}
\centering
\caption{(a) Number of papers arranged chronically (2016-2020). (b) The distribution of selected papers in different applications of medical image analysis.
}
\label{fig:statis}
\end{centering}
\end{figure*}
In terms of the integrating methods, various approaches have been designed to incorporate different types of domain knowledge into networks \cite{ji2020infant}. For example, a simple approach is to concatenate hand-crafted features with the ones extracted from deep learning models \cite{xie2016lung}. In some works, network architectures are revised to simulate the pattern of radiologists when they read images \cite{Diagnose}. Attention mechanism, which allows a network to pay more attention to a certain region of an image, is a powerful technique to incorporate domain knowledge of radiologists \cite{li2019attention}. In addition, multi-task learning and meta learning are also widely used to introduce medical domain knowledge into deep learning models \cite{chen2016automatic,yue2019cardiac}.
Although there are a number of reviews on deep learning for medical image analysis, including \cite{debelee2019survey,litjens2017survey,shen2017deep,suzuki2017survey}, they all describe the existing works from the application point of view, i.e., how deep learning techniques are applied to various medical applications. To the best of our knowledge, there is no review that gives systematic introduction on \emph{how medical domain knowledge can help deep learning models}. This aspect, we believe, is the unique feature that distinguishes deep learning models for CAD from those for general computer vision tasks
Fig. \ref{fig:1} gives the overview on how we organize the related researches. At the top level,
existing studies are classified into three main categories according to their purposes: (1) disease diagnosis, (2) lesion, organ and abnormality detection, and (3) lesion and organ segmentation.
In each category, we organize them into several groups based on the types of extra knowledge have been incorporated. At the bottom level, they are further categorized according to the different integrating approaches of the domain knowledge.
This survey contains more than 200 papers (163 are with domain knowledge), most of which are published recently (2016-2020), on a wide variety of applications of deep learning techniques for medical image analysis.
In addition, most of the corresponding works are from the conference proceedings for MICCAI, EMBC, ISBI and some journals such as TMI, Medical Image Analysis, JBHI and so on.
\begin{figure*}[htp!] \begin{centering}
\includegraphics[width=0.65\linewidth]{paper_structure3.pdf}
\centering
\caption{The organizational structure of this survey.
}
\label{fig:structure}
\end{centering}
\end{figure*}
The distribution of these papers are shown in Fig. \ref{fig:statis}(a). It can be seen that the number of papers increases rapidly from 2016 to 2020. With respect to the applications, most of them are related to disease diagnosis and lesion/organ segmentation (shown in Fig. \ref{fig:statis}(b)). To sum up, with this survey we aim to:
\begin{itemize}
\item summarize and classify different types of domain knowledge in medical areas that are utilized to improve the performance of deep learning models in various applications;
\item summarize and classify different ways of introducing medical domain knowledge into deep learning models;
\item give the outlook of challenges and future directions in integrating medical domain knowledge into deep learning models
\end{itemize}
The rest of the survey is organized as follows. Sections \ref{sec:diagnosis}, \ref{sec:detection} and \ref{sec:segmentation} introduce the existing works for the major three tasks in medical image analysis. Besides these three major tasks, other tasks in medical image analysis are described in Section \ref{sec:otherappl}. In each section, we first introduce the general architectures of deep learning models for a task, and then categorize related works according to the types of the domain knowledge to be integrated. Various incorporating methods for each type of domain knowledge are then described. Section \ref{sec:future_work} discusses research challenges, and gives the outlook of future directions. Lastly, Section \ref{sec:conclusion} concludes this survey. The structure of this survey is shown in Fig. \ref{fig:structure}.
\section{Disease Diagnosis}
\label{sec:diagnosis}
Disease diagnosis refers to the task of determining the type and condition of possible diseases based on the medical images provided. In this section, we give an overview of the deep learning models that generally used for disease diagnosis. Concretely, subsection \ref{sec:diagmodel} outlines the general structures of deep learning models used for disease diagnosis. Subsection \ref{sec:diag_natural} introduces the works that utilize knowledge from natural images or other medical datasets. Deep learning models that leverage knowledge from medical doctors are introduced in Subsection \ref{sec:diag_rediologists} in detail. Lastly, Subsection \ref{sec:diag_overview} summarizes the research of disease diagnosis.
\subsection{General Structures of Deep Learning Models Used for Disease Diagnosis}
\label{sec:diagmodel}
In the last decades, deep learning techniques, especially CNNs, have achieved a great success in disease diagnosis.
Fig. \ref{fig:cnn} shows the structure of a typical CNN that used for disease diagnosis in chest X-ray image. The CNN employs alternating convolutional and pooling layers, and contains trainable filter banks per layer. Each individual filter in a filter bank is able to generate a feature map. This process is alternated and the CNN can learn increasingly more and more abstract features that will later be used by the fully connected layers to accomplish the classification task.
\begin{figure}[htp!] \begin{centering}
\includegraphics[width=1\linewidth]{cnn.pdf}
\centering
\caption{A typical CNN architecture for medical disease diagnosis
}
\label{fig:cnn}
\end{centering}
\end{figure}
Different types of CNN architectures, from AlexNet \cite{krizhevsky2012imagenet}, GoogLeNet \cite{szegedy2015going}, VGGNet \cite{simonyan2015very}, ResNet \cite{he2016deep} to DenseNet \cite{huang2017densely}, have achieved a great success in the diagnosis of various diseases. For example,
GoogLeNet, ResNet, and VGGNet models are used in the diagnosis of canine ulcerative keratitis \cite{kim2019cnn}, and most of them achieve accuracies of over 90\% when classifying superficial and deep corneal ulcers. DenseNet is adopted to diagnose lung nodules on chest X-ray radiograph \cite{li2019multi}, and experimental results show that more than 99\% of lung nodules can be detected. In addition, it is found that VGGNet and ResNet are more effective than other network structures for many medical diagnostic tasks \cite{Diagnose,liu2018integrate,mitsuhara2019embedding,xie2019knowledge-based}.
However, the above works generally directly apply CNNs to medical image analysis or slightly modified CNNs (e.g., by modifying the number of kernals, the number of channels or the size of filters), and no medical knowledge is incorporated. In addition, these methods generally require large medical datasets to achieve a satisfactory performance
In the following subsections, we systematically review on the research that utilizes medical domain knowledge for the disease diagnosis. The types of knowledge and the incorporating methods are summarized in Table \ref{tab:diagnosis_knowledge}.
\begin{table*}[]
\small
\center
\caption{A compilation of the knowledge and corresponding incorporating methods used in disease diagnosis}
\begin{tabular}{|c|l|l|l|}
\hline
\begin{tabular}[c]{@{}l@{}}Knowledge Source\end{tabular} & \begin{tabular}[c]{@{}l@{}}Knowledge Type\end{tabular} & \begin{tabular}[c]{@{}l@{}}Incorporating Method\end{tabular} & References \\ \hline
\begin{tabular}[c]{@{}l@{}}Natural datasets\end{tabular} & \begin{tabular}[c]{@{}l@{}}Natural images\end{tabular} & \begin{tabular}[c]{@{}l@{}}Transfer learning\end{tabular} &
{\begin{tabular}[c]{@{}l@{}}\cite{bar2015deep}\cite{esteva2017dermatologist} \cite{wimmer2016convolutional}\cite{cao2018improve}\cite{wang2017chestx} \cite{hussein2017risk} \cite{li2018path}\cite{huynh2016digital} \end{tabular}} \\ \hline
\multirow{2}{*}{\begin{tabular}[c]{@{}l@{}}Medical datasets\end{tabular}} & \begin{tabular}[c]{@{}l@{}}Multi-modal images\end{tabular} & \begin{tabular}[c]{@{}l@{}}Transfer learning\end{tabular} & \begin{tabular}[c]{@{}l@{}}\cite{samala2018breast}\cite{hadad2017classification} \cite{samala2017multi}\cite{azizi2017transfer}\cite{li2020digital}\cite{han2020deep}\end{tabular}
\\ \cline{2-4}
& \begin{tabular}[c]{@{}l@{}}Datasets from other diseases\end{tabular} & \begin{tabular}[c]{@{}l@{}}Multi-task learning\end{tabular} & \begin{tabular}[c]{@{}l@{}}\cite{li2019canet}\cite{liao2019multi} \end{tabular} \\ \hline
\multirow{9}{*}{\begin{tabular}[c]{@{}l@{}}Medical doctors\end{tabular}} & \begin{tabular}[c]{@{}l@{}}Training pattern\end{tabular} & \begin{tabular}[c]{@{}l@{}}Curriculum learning\end{tabular} &\begin{tabular}[c]{@{}l@{}}\cite{maicas2018training}\cite{tang2018attention} \cite{jimenez2019medical}\cite{haarburger2019multi}\cite{zhao2020egdcl}
\cite{jimenez2020curriculum}\cite{wei2020learn}\cite{qi2020curriculum}\end{tabular} \\ \cline{2-4}
& \begin{tabular}[c]{@{}l@{}}Diagnostic patterns\end{tabular} & \begin{tabular}[c]{@{}l@{}}Network design\end{tabular} & \begin{tabular}[c]{@{}l@{}}\cite{Diagnose}\cite{gonzalez2018dermaknet}
\cite{wang2020learning}\cite{huang2020dual}
\cite{yang2020momminet}\cite{liu2020semi}\end{tabular} \\ \cline{2-4}
& \begin{tabular}[c]{@{}l@{}}Areas doctors focus on\end{tabular} & \begin{tabular}[c]{@{}l@{}}Attention mechanism
\end{tabular} & \begin{tabular}[c]{@{}l@{}}\cite{li2019attention} \cite{mitsuhara2019embedding} \cite{8637959}\cite{cui2020collaborative}
\cite{xiedg}\cite{zhang2020short}\end{tabular} \\ \cline{2-4}
& \multirow{4}{*}{\begin{tabular}[c]{@{}l@{}}Features doctors focus on\end{tabular}} & \begin{tabular}[c]{@{}l@{}}Decision level fusion\end{tabular} & \begin{tabular}[c]{@{}l@{}}\cite{huynh2016digital} \cite{moradi2016hybrid} \cite{majtner2016combining}\cite{xie2018fusing} \cite{antropova2017deep}\cite{xia2020comparison} \end{tabular} \\ \cline{3-4}
& &\begin{tabular}[c]{@{}l@{}}Feature level fusion\end{tabular} &
{\begin{tabular}[c]{@{}l@{}}\cite{xie2016lung}\cite{hagerty2019deep} \cite{cao2018improve} \cite{chai2018glaucoma} \cite{buty2016characterization}\cite{saba2020brain}\end{tabular}}
\\ \cline{3-4}
& & \begin{tabular}[c]{@{}l@{}}Input level fusion\end{tabular} &
{\begin{tabular}[c]{@{}l@{}}\cite{yang2019dscgans} \cite{xie2019knowledge-based} \cite{tan2019expert}\cite{liu2019automated}
\cite{feng2020knowledge}\end{tabular}}
\\ \cline{3-4}
& &\begin{tabular}[c]{@{}l@{}}As labels of CNNs\end{tabular} &
{\begin{tabular}[c]{@{}l@{}}\cite{chen2016automatic}\cite{hussein2017risk} \cite{murthy2017center}\end{tabular}} \\ \cline{2-4}
& {\begin{tabular}[c]{@{}l@{}}Other related information
\end{tabular}}& {\begin{tabular}[c]{@{}l@{}}Multi-task learning \\ /network design\end{tabular}} &\begin{tabular}[c]{@{}l@{}}\cite{liu2018integrate} \cite{wang2018tienet} \cite{zhang2017tandemnet}\cite{wu2019deep}
\cite{yu2020difficulty}\end{tabular}
\\ \hline
\end{tabular}
\label{tab:diagnosis_knowledge}
\end{table*}
\subsection{Incorporating Knowledge from Natural Datasets or Other Medical Datasets}
\label{sec:diag_natural}
Despite the disparity between natural and medical images, it has been demonstrated that CNNs comprehensively trained on the large-scale well-annotated natural image datasets can still be helpful for disease diagnosis tasks \cite{wimmer2016convolutional}. Intrinsically speaking, this transfer learning process introduces knowledge from natural images into the network for medical image diagnosis.
According to \cite{litjens2017survey}, the networks pre-trained on natural images can be leveraged via two different ways: by utilizing them as fixed feature extractors, and as an initialization which will then be fine-tuned on target medical datasets. These two strategies are illustrated in Fig. \ref{fig:transfer_learning}(a) and Fig. \ref{fig:transfer_learning}(b), respectively.
\begin{figure}[htp!] \begin{centering}
\includegraphics[width=1\linewidth]{transfer_learning.pdf}
\centering
\caption{Two strategies to utilize the pre-trained network on natural images: (a) as a feature extractor and (b) as an initialization which will be fine-tuned on the target dataset. }
\label{fig:transfer_learning}
\end{centering}
\end{figure}
The first strategy takes a pre-trained network, removes its last fully-connected layer, and treats the rest of the network as a fixed feature extractor. Extracted features are then fed into a linear classifier (e.g., support vector machine (SVM)), which is trained on the target medical datasets. Applications in this category include mammography mass lesion classification \cite{huynh2016digital} and chest pathology identification \cite{bar2015deep}
The success of leveraging information from natural images for disease diagnosis can be attributed to the fact that a network pre-trained on natural images, especially in the earlier layers, contain more generic features (e.g., edge detectors and color blob detectors) \cite{yosinski2014transferable}.
In the second strategy, the weights of the pre-trained network are fine-tuned based on the medical datasets. It is possible to fine-tune the weights of all layers in the network, or to keep some of the earlier layers fixed and only fine-tune some higher-level portion of the network. This can be applied to the classification of skin cancer \cite{esteva2017dermatologist}, breast cancer \cite{cao2018improve}, thorax diseases \cite{wang2017chestx}, prostate cancer \cite{li2018path} and interstitial lung diseases \cite{hussein2017risk}
Besides the information from natural images, using images from other medical datasets is also quite popular.
Medical datasets containing images of the same or similar modality as target images have similar distribution and therefore can be helpful.
For example, to classify malignant and benign breast masses in digitized screen-film mammograms (SFMs), a multi-task transfer learning DCNN is proposed to incorporate the information from digital mammograms (DMs) \cite{samala2017multi}. It is found to have significantly higher performance compared to the single-task transfer learning DCNN which only utilizes SFMs.
In addition, even medical images with different modalities can provide complementary information. For example, \cite{samala2018breast} uses a model pre-trained on a mammography dataset to show that it could obtain better results than models trained solely on the target dataset comprising digital breast tomosynthesis (DBT) images.
Another example is in prostate cancer classification, where the radiofrequency ultrasound images are first used to train the DCNN, then the model is fine-tuned on B-mode ultrasound images \cite{azizi2017transfer}. Other examples of using the images from different modalities
can be found in \cite{hadad2017classification,li2020digital,han2020deep}.
Furthermore,
as datasets of different classes can help each other in classification tasks \cite{xiao2020pk}, medical datasets featuring images of a variety of diseases can also have similar morphological structures or distribution,
which may be beneficial for other tasks.
For example, a multi-task deep learning (MTDL) method is proposed in \cite{liao2019multi}.
MTDL can simultaneously utilize multiple cancer datasets so that hidden representations among these datasets can provide more information to small-scale cancer datasets, and enhance the classification performance.
Another example is a cross-disease attention network (CANet) proposed in \cite{li2019canet}. CANet characterizes and leverages the relationship between diabetic retinopathy (DR) and diabetic macular edema (DME) in fundus images using a special designed disease-dependent attention module. Experimental results on two public datasets show that CANet outperforms other methods on diagnosing both of the two diseases.
\subsection{Incorporating Knowledge from Medical Doctors}
\label{sec:diag_rediologists}
Experienced medical doctors can give fairly accurate conclusion on the given medical images, mainly thanks to the training they have received and the expertise they have accumulated over many years. In general, they often follow some certain patterns or take some procedures when reading medical images. Incorporating these types of knowledge can improve the diagnostic performance of deep learning models.
The types of medical domain knowledge utilized in deep learning models for disease diagnosis can be summarized into the following five categories:
\begin{enumerate}
\item the training pattern,
\item the general diagnostic patterns they view images,
\item the areas on which they usually focus,
\item the features (e.g., characteristics, structures, shapes) they give special attention to, and
\item other related information for diagnosis.
\end{enumerate}
The research works for each category will be described in the following sections.
\subsubsection{Training Pattern of Medical Doctors}
\label{sec:diag_trainingprocess}
The training process of medical students has a character: they are trained by tasks with increasing difficulty.
For example, students begin with some easier tasks, such as deciding whether an image contains lesions, later are required to accomplish more challenging tasks, such as determining whether the lesions are benign or malignant. Over time, they will advance to more difficult tasks, such as determining the subtypes of lesions in images.
This training pattern can be introduced in the training process of deep neural networks via curriculum learning \cite{bengio2009curriculum}. Curriculum determines a sequence of training samples ranked in ascending order of learning difficulty. Curriculum learning has been an active research topic in computer vision and has been recently utilized for medical image diagnosis.
For example, a teacher-student curriculum learning strategy is proposed for breast screening classification from DCE-MRI \cite{maicas2018training}. The deep learning model is trained on simpler tasks before introducing the hard problem of malignancy detection. This strategy shows the better performance when compared with the other methods.
Similarly, \cite{tang2018attention} presents a CNN based attention-guided curriculum learning framework by leveraging the severity-level attributes mined from radiology reports. Images in order of difficulty (grouped by different severity-levels) are fed into the CNN to boost the learning process gradually.
In \cite{jimenez2019medical}, the curriculum learning is adopted to support the classification of proximal femur fracture from X-ray images. The approach assigns a degree of difficulty to each training sample. By first learning `easy' examples and then `hard' ones, the model can reach a better performance even with fewer data. Other examples of using curriculum learning for disease diagnosis can be found in \cite{haarburger2019multi,zhao2020egdcl,jimenez2020curriculum,wei2020learn,qi2020curriculum}.
\subsubsection{General Diagnostic Patterns of Medical Doctors}
\label{sec:diag_diagnosticpattern}
Experienced medical doctors generally follow some patterns when they read medical images. These patterns can be integrated into deep learning models with appropriately modified architectures.
For example, radiologists generally follow a three-staged approach when they read chest X-ray images: first browsing the whole image, then concentrating on the local lesion areas, and finally combining the global and local information to make decisions. This pattern is incorporated in the architecture design of the network for thorax disease classification \cite{Diagnose} (see Fig. \ref{fig:diagnosepattern}). The proposed network has three branches, one is used to view the whole image, the second for viewing the local areas, and the third one for combining the global and local information together. The network yields state-of-the-art accuracy on the ChestX-ray14 dataset. In addition, besides the information from the whole image and local lesion area, the information from lung area is also leveraged in \cite{wang2020learning}. In particular, a segmentation subnetwork is first used to locate the lung area from the whole image, and then lesion areas are generated by using an attention heatmap. Finally, the most discriminative features are fused for final disease prediction. Another example is a Dual-Ray Net proposed to deal with the front and lateral chest radiography simultaneously \cite{huang2020dual}, which also mimics the reading pattern of radiologists.
\begin{figure}[htp!] \begin{centering}
\includegraphics[width=1.0\linewidth]{diagnosis_generalpattern_new.pdf}
\centering
\caption{The example of leveraging the diagnostic pattern of radiologists for thorax disease diagnosis \cite{Diagnose}, where three branches are used to extract and combine the global and local features.}
\label{fig:diagnosepattern}
\end{centering}
\end{figure}
In the diagnosis of skin lesions, experienced dermatologists generally first locate lesions, then identify dermoscopic features from the lesion areas, and finally make diagnosis based on the features.
This pattern is mimicked in the design of the network for the diagnosis of skin lesions \cite{gonzalez2018dermaknet}. The proposed network, DermaKNet, comprised several subnetworks with dedicated tasks: lesion-skin segmentation, detection of dermoscopic features, and global lesion diagnosis. The DermaKNet achieves higher performance compared to the traditional CNN models.
In addition, in mass identification in mammogram, radiologists generally analyze the bilateral and ipsilateral views simultaneously. To emulate this reading practice, \cite{yang2020momminet} proposes MommiNet to perform end-to-end bilateral and ipsilateral analysis of mammogram images. In addition, symmetry and geometry constraints are also aggregated from these views. Experiments show the state-of-the-art mass identification accuracy on DDSM. Another example of leveraging this diagnostic pattern of medical doctors can be found in skin lesion diagnosis and thorax disease classification \cite{liu2020semi}.
\subsubsection{The Areas Medical Doctors Usually Focus on}
\label{sec:diag_attentionmap}
When experienced medical doctors read images, they generally focus on a few specific areas, as these areas are more informative than other places for the purpose of disease diagnosis. Therefore, the information about where medical doctors focus may help deep learning models yield better results.
The knowledge above is generally represented as `attention maps', which are annotations given by medical doctors indicating the areas they focus on when reading images. For example, a CNN named AG-CNN explicitly incorporates the `attention maps' for glaucoma diagnosis \cite{li2019attention}. The attention maps of ophthalmologists are collected through a simulated eye-tracking experiment, which are used to indicate where they focus when reading images. An example of capturing the attention maps of an ophthalmologist in glaucoma diagnosis is shown in Fig. \ref{fig:attentionmap}. To incorporate the attention maps, an attention prediction subnet in AG-CNN is designed, and the attention prediction loss measuring the difference between the generated and ground truth attention maps (provided by ophthalmologists) is utilized to supervise the training process. Experimental results show that AG-CNN significantly outperforms
the state-of-the-art glaucoma detection methods.
\begin{figure}[!htp] \begin{centering}
\includegraphics[width=0.8\linewidth]{diagnosis_attentionmap_new.pdf}
\centering
\caption{Example of capturing the attention maps of an ophthalmologist in glaucoma diagnosis \cite{li2019attention}. I, II, III and IV are the original blurred fundus image, the fixations of ophthalmologists with cleared regions, the order of clearing the blurred regions, and the generated attention map based on the captured fixations, respectively. V and VII represent the original fundus images. VI and VIII are the corresponding attention maps of V and VII generated by using the method in I-IV.}
\label{fig:attentionmap}
\end{centering}
\end{figure}
Another example in this category is the lesion-aware CNN (LACNN) for the classification of retinal optical coherence tomography (OCT) images \cite{8637959}. The LACNN simulates the pattern of ophthalmologists' diagnosis by focusing on local lesion-related regions. Concretely, the `attention maps' are firstly represented as the annotated OCT images delineating the lesion regions using bounding polygons. To incorporate the information, the LACNN proposes a lesion-attention module to enhance the features from local lesion-related regions while still preserving the meaningful structures in global OCT images. Experimental results on two clinically acquired OCT datasets demonstrate the effectiveness of introducing attention maps for retinal OCT image classification, with 8.3\% performance gain when compared with the baseline method.
Furthermore, \cite{mitsuhara2019embedding} proposes an Attention
Branch Network (ABN) to incorporate the knowledge given by the radiologists in diabetic retinopathy. ABN introduces a branch structure which generates attention maps that highlight the attention regions of the network. During the training process, ABN allows the attention maps to be modified with semantic segmentation labels of disease regions. The semantic labels are also annotated by radiologists as the ground truth attention maps. Experimental results on the disease grade recognition of retina images show that ABN achieves 93.73\% classification accuracy and its interpretability is clearer than conventional approaches.
Other examples of incorporating attention maps of medical doctors can be found in the diagnosis of radiotherapy-related esophageal fistula \cite{cui2020collaborative}, breast cancer diagnosis \cite{xiedg}, and short-term lesion change detection in melanoma screening \cite{zhang2020short}.
\subsubsection{Features That Medical Doctors Give Special Attention to}
\label{sec:diag_diseases}
In the last decades, many guidelines and rules have gradually developed in various medical fields to point out some important features for diagnosis. These features are called \emph{`hand-crafted features'} as they are designated by medical doctors. For example, the popular ABCD rule \cite{nachbar1994abcd} is widely used by dermatologists to classify melanocytic tumors. The ABCD rule points out four distinguishing features, namely asymmetry, border, color and differential structures, to determine whether a melanocytic skin lesion under the investigation is benign or malignant.
\begin{table}[htp!]
\footnotesize
\center
\caption{Features in the BI-RADS guideline to classify benign and malignant breast tumors in ultrasound images \cite{birads}}
\begin{tabular}{|c|c|c|}
\hline
& Benign & Malignant\\
\hline
Margin & smooth, thin, regular & irregular, thick \\
\hline
Shape & round or oval &irregular \\
\hline
Microcalcification & no & yes \\
\hline
Echo Pattern & clear & unclear \\
\hline
Acoustic Attenuation & not obvious & obvious \\
\hline
Side Acoustic Shadow & obvious & not obvious \\
\hline
\end{tabular}
\label{tab:1}
\end{table}
Another example is in the field of breast cancer diagnosis. Radiologists use the BI-RADS (Breast Imaging Reporting and Data System) score \cite{birads} to place abnormal findings into different categories, with a score of 1 indicating negative findings and a score of 6 indicating breast cancer. More importantly, for each imaging modality, BI-RADS indicates some features closely related to its scores, including margin, shape, micro-calcification, and echo pattern. For example, for breast ultrasound images, tumors with smooth, thin and regular margins are more likely to be benign, while tumors with irregular and thick margins are highly suspected to be malignant. Other features that can help to classify benign and malignant breast tumors are shown in Table \ref{tab:1}.
Similarly, for the benign-malignant risk assessment of lung nodules in \cite{hussein2017risk}, six high-level nodule features, including calcification, sphericity, margin, lobulation, spiculation and texture, have shown a tightly connection with malignancy scores (see Fig. \ref{fig:lung_nodule}).
\begin{figure}[htp!]
\begin{centering}
\includegraphics[width=0.9\linewidth]{lung_nodule_new.pdf}
\centering
\caption{Lung nodule attributes with different malignancy scores \cite{hussein2017risk}.(a) From top to the bottom, six different nodule features attribute from missing to highest prominence. (b) The number of nodules with different malignancy scores.}
\label{fig:lung_nodule}
\end{centering}
\end{figure}
These different kinds of hand-crafted features have been widely used in many traditional CAD systems. These systems generally first extract these features from medical images, and then feed them into some classifiers like SVM or Random Forest \cite{breiman2001random,cortes1995support}. For example, for the lung nodule classification on CT images, many CAD systems utilize features including the size, shape, morphology, and texture from the suspected lesion areas \cite{alilou2017intra,xie2019knowledge-based}. Similarly, in the CAD systems for the diagnosis of breast ultrasound images, features such as intensity, texture and shape are used \cite{hsu2019breast}.
When using deep learning models like CNNs, which have the ability to automatically extract representative features, there are four approaches to combining `hand-crafted features' with features extracted from CNNs.
\begin{itemize}
\item Decision-level fusion: the two types of features are fed into two classifiers separately, then the decisions from the two classifiers are combined.
\item Feature-level fusion: the two types of features are directly combined via techniques such as concatenation.
\item Input-level fusion: the hand-crafted features are represented as image patches, which are then taken as inputs to the CNNs.
\item
Usage of features as labels of CNNs: the hand-crafted features are first annotated and then utilized as labels for CNNs during the training process.
\end{itemize}
\textbf{Decision-level fusion}: The structure of this approach is illustrated in Fig. \ref{fig:decision_level}. In this approach, the hand-crafted features and the features extracted from CNNs are separately fed into two classifiers. Then, the classification results from both classifiers are combined using some decision fusion techniques to produce final classification results.
\begin{figure}[htp!]
\begin{centering}
\includegraphics[width=1\linewidth]{decision_level_fusion.pdf}
\centering
\caption{Decision-level fusion: the decisions from two classifiers, one based on hand-crafted features, and the other on the CNNs, are combined.}
\label{fig:decision_level}
\end{centering}
\end{figure}
For example, a skin lesion classification model proposed in \cite{majtner2016combining} combines the results from two SVM classifiers. The first one uses hand-crafted features (i.e., RSurf features and local binary patterns (LBP)) as input and the second employs features derived from a CNN. Both of the classifiers predict the category for each tested image with a classification score. These scores are subsequently used to determine the final classification result.
Similarly, a mammographic tumor classification method also combines features in decision-level fusion \cite{huynh2016digital}. After individually performing classification with CNN features and analytically extracted features (e.g., contrast, texture, and margin spiculation), the method adopts the soft voting to combine the outputs from both individual classifiers. Experimental results show that the performance of the ensemble classifier was significantly better than the individual ones. Other examples that utilize this approach include lung nodule diagnosis \cite{xie2018fusing}, breast cancer diagnosis \cite{antropova2017deep}, the classification of cardiac slices \cite{moradi2016hybrid} and the prediction of the invasiveness risk of stage-I lung adenocarcinomas \cite{xia2020comparison}.
\textbf{Feature-level fusion}:
In this approach, hand-crafted features and features extracted from CNNs are concatenated, and the combined features are fed into a classifier for diagnosis. The structure of this approach is illustrated in Fig. \ref{fig:feature_level_fusion}.
\begin{figure}[htp!]
\begin{centering}
\vspace{-0.2cm}
\includegraphics[width=0.9\linewidth]{feature_level_fusion.pdf}
\centering
\caption{Feature-level fusion: the hand-crafted features are combined with the features extracted from CNNs as the new feature vectors.}
\label{fig:feature_level_fusion}
\end{centering}
\end{figure}
For example, a combined-feature based classification approach called CFBC is proposed for lung nodule classification by \cite{xie2016lung}. In CFBC, the hand-crafted features (including texture and shape descriptors) and the features learned by a nine-layer CNN are combined and fed into a back-propagation neural network. Experimental results derived from a publicly available dataset show that compared with a purely CNN model, incorporating hand-crafted features improves the accuracy, sensitivity, and specificity by 3.87\%, 6.41\%, and 3.21\%, respectively.
Another example in this category is the breast cancer classification in histology images \cite{cao2018improve}. More specifically, two hand-crafted features, namely the parameter-free threshold adjacency statistics and the gray-level co-occurrence matrix, are fused with the five groups of deep learning features extracted from five different deep models. The results show that after incorporating hand-crafted features, the accuracy of the deep learning model can be significantly improved.
Other examples of employing feature-level fusion can be found in glaucoma diagnosis \cite{chai2018glaucoma}, skin lesion classification \cite{hagerty2019deep}, lung nodule classification \cite{buty2016characterization} and brain tumor diagnosis \cite{saba2020brain}.
\textbf{Input-level fusion}:
In this approach, hand-crafted features are first represented as patches highlighting the corresponding features. Then, these patches are taken as input to CNNs to make the final conclusion. Using this approach, the CNNs are expected to rely more on the hand-crafted features. It should be noted that generally speaking, one CNN is required for each type of hand-crafted feature, and information obtained from these CNNs need to be combined in some manner to make a final decision. The structure of this approach is illustrated in Fig. \ref{fig:input_level_fusion}.
\begin{figure}[htp!]
\begin{centering}
\includegraphics[width=0.9\linewidth]{input_level_fusion_new.pdf}
\centering
\caption{Input-level fusion: the hand-crafted features are represented as image patches that are taken as inputs to the CNNs.}
\label{fig:input_level_fusion}
\end{centering}
\end{figure}
For example, in \cite{tan2019expert}, three types of hand-crafted features, namely the contrast information of the initial nodule candidates (INCs) and the outer environments, the histogram of oriented gradients (HOG) feature, and the LBP feature, are transformed into the corresponding patches and are taken as inputs of multiple CNNs. The results show that this approach outperforms both conventional CNN-based approaches and traditional
machine-learning approaches based on hand-crafted features.
Another example using input-level fusion approach is the MV-KBC (multi-view knowledge-based collaborative) algorithm proposed for lung nodule classification \cite{xie2019knowledge-based}. The MV-KBC first extracts patches corresponding to three features: the overall appearance (OA), nodule's heterogeneity in voxel values (HVV) and heterogeneity in shapes (HS). Each type of patch is fed into a ResNet. Then, the outputs of these ResNets are combined to generate the final classification results.
Moreover, \cite{yang2019dscgans} proposes the dual-path semi-supervised conditional generative adversarial networks (DScGAN) for thyroid classification. More specifically, the features from the ultrasound B-mode images and the ultrasound elastography images are first extracted as the OB patches (indicating the region of interest (ROI) in B-mode images), OS patches (characterizing the shape of the nodules), and OE patches (indicating the elasticity ROI according to the B-mode ROI position). Then, these patches are used as the input of the DScGAN. Using the information from these patches is demonstrated to be effective to improve the classification performance. Other examples employ input-level fusion can be found in thyroid nodule diagnosis \cite{liu2019automated} and breast cancer diagnosis on multi-sequence MRI \cite{feng2020knowledge}.
\textbf{Usage of features as labels of CNNs:} In this approach, besides the original classification labels of images, medical doctors also provide labels for some hand-crafted features. This extra information is generally incorporated into deep learning models via the multi-task learning structure.
For example, in \cite{chen2016automatic}, the nodules in lung images were quantitatively rated by radiologists with regard to nine hand-crafted features (e.g., spiculation, texture, and margin). The multi-task learning framework is used to incorporate the above information into the main task of lung nodule classification.
In addition, for the benign-malignant risk assessment of lung nodules in low-dose CT scans \cite{hussein2017risk}, the binary labels about the presence of six high-level nodule attributes, namely the calcification, sphericity, margin, lobulation, spiculation and texture, are utilized. Six CNNs are designed and each one aims at learning one attribute. The discriminative features automatically learned by CNNs for these attributes are fused in a multi-task learning framework to obtain the final risk assessment scores.
Similarly, in \cite{murthy2017center}, each glioma nuclear image is exclusively labeled in terms of the shapes and attributes for the centermost nuclei of the image. These features are then learned by a multi-task CNN. Experiments demonstrate that the proposed method outperforms the baseline CNN
\subsubsection{Other Types of Information Related to Diagnosis}
\label{sec:diag_extraannotations}
In this section, we summarize other types of information that can help deep learning models to improve their diagnostic performance. These types of information include extra category labels and clinical diagnostic reports
\textbf{Extra category labels}
For medical images, besides a classification label (i.e., normal, malignant or benign), radiologists may give some extra category labels. For example, in the ultrasonic diagnosis of breast cancer, an image usually has a BI-RADS label that classifies the image into 0$\sim$6 \cite{birads}---category 0 suggests re-examination, categories 1 and 2 indicate that it is prone to be a benign lesion, category 3 suggests probably benign findings, categories 4 and 5 are suspected to be malignant, category 6 definitely suggests malignant).
These labels also contain information about the condition of lesions. In \cite{liu2018integrate}, the BI-RADS label for each medical image is incorporated in a multi-task learning structure as the label of an auxiliary task. Experimental results show that incorporating these BI-RADS labels can improve the diagnostic performance of major classification task. Another example of using the information from BI-RADS labels can be found in \cite{wu2019deep}.
In addition, during the process of image annotation, consensus or disagreement among experts regarding images reflects the gradeability and difficulty levels of the image, which is also a representation of medical domain knowledge. To incorporate this information,\cite{yu2020difficulty} proposes a multi-branch model structure to predict the most sensitive, most specifical and a balanced fused result for glaucoma images. Meanwhile, the consensus loss is also used to encourage the sensitivity and specificity branch to generate consistent and contradictory predictions for images with consensus and disagreement labels, respectively.
\textbf{Extra clinical diagnostic reports}
A clinical report is a summary of all the clinical findings and impressions determined during examination of a radiography study.
It usually contains rich information and reflects the findings and descriptions of medical doctors. Incorporating clinical reports into CNNs designed for disease diagnosis is typically beneficial. As medical reports are generally handled by recurrent neural networks (RNNs), to incorporate information from medical reports, generally hybrid networks containing both CNNs and RNNs are needed.
For example, the Text-Image embedding network (TieNet) is designed to classify the common thorax disease in chest X-rays \cite{wang2018tienet}. TieNet has an end-to-end CNN-RNN architecture enabling it to integrate information of radiological reports. The classification results are significantly improved (with about a 6\% increase on average in AUCs) compared with the baseline CNN purely based on medical images.
In addition, using semantic information from diagnostic reports is explored in \cite{zhang2017tandemnet} for understanding pathological bladder cancer images. A dual-attention model is designed to facilitate the high-level interaction of semantic information and visual information. Experiments demonstrate that incorporating information from diagnostic reports significantly improves the cancer diagnostic performance over the baseline method.
\subsection{Summary}
\label{sec:diag_overview}
In the previous sections, we introduced different kinds of domain knowledge and the corresponding integrating methods into the deep learning models for disease diagnosis. Generally, almost all types of domain knowledge are proven to be effective in boosting the diagnostic performance, especially using the metrics of accuracy and AUC, some examples and their quantitative improvements are listed in Table \ref{tab:diag_quan}.
\begin{table*}[]
\small
\caption{The comparison of the quantitative metrics for some disease diagnosis methods after integrating domain knowledge}
\center
\begin{tabular}{|l|l|l|l|}
\cline{1-4}
\multicolumn{1}{|c|}{Reference} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}l@{}}Baseline Model/With Domain Knowledge\end{tabular}} & \multicolumn{1}{c|}{Accuracy} & \multicolumn{1}{c|}{AUC} \\ \hline
\multirow{1}{*}{\cite{Diagnose}} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}l@{}}AG-CNN only with\\ global branch/AG-CNN\end{tabular}} & \multicolumn{1}{c|}{--/--} & \multicolumn{1}{c|}{0.840/0.871} \\ \hlin
\multirow{1}{*}{\cite{gonzalez2018dermaknet}} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}l@{}}ResNet-50/DermaKNet\end{tabular}} & \multicolumn{1}{c|}{--/--} & \multicolumn{1}{c|}{0.874/0.908} \\ \hline
\multirow{1}{*}{\cite{hadad2017classification}} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}l@{}}Fine-tuned VGG-Net/\\Fine-tuned MG-Net\end{tabular}} & \multicolumn{1}{c|}{0.900/0.930} & \multicolumn{1}{c|}{0.950/0.970} \\ \hlin
\multirow{1}{*}{\cite{hagerty2019deep}} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}l@{}}ResNet-50/ResNet-50 \\with handcrafted features\end{tabular}} &\multicolumn{1}{c|}{--/--} & \multicolumn{1}{c|}{0.830/0.940} \\ \hlin
\multirow{1}{*}{\cite{li2019attention}} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}l@{}}CNN without using \\attention map/AG-CNN\end{tabular}} & \multicolumn{1}{c|}{0.908/0.953} & \multicolumn{1}{c|}{0.966/0.975} \\ \hlin
\multirow{1}{*}{\cite{li2019canet}} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}l@{}}ResNet50/CANet\end{tabular}} & \multicolumn{1}{c|}{0.828/0.851} & \multicolumn{1}{c|}{--/--} \\ \hlin
\multirow{1}{*}{\cite{liu2018integrate}} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}l@{}}VGG16/Multi-task VGG16\end{tabular}} & 0.829 /0.833 & \multicolumn{1}{c|}{--/--} \\ \hlin
\multirow{1}{*}{\cite{maicas2018training}} & \multicolumn{1}{c|}{DenseNet/BMSL} & \multicolumn{1}{c|}{--/--} & \multicolumn{1}{c|}{0.850/0.900} \\ \hlin
\multirow{1}{*}{\cite{samala2018breast}} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}l@{}}CNN with single-stage/\\multi-stage transfer learning\end{tabular}} & \multicolumn{1}{c|}{--/--} & \multicolumn{1}{c|}{0.850/0.910} \\ \hlin
\multirow{1}{*}{\cite{tang2018attention}} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}l@{}}CNN/AGCL\end{tabular}} & \multicolumn{1}{c|}{--/--} & \multicolumn{1}{c|}{0.771/0.803} \\ \hlin
\multirow{1}{*}{\cite{yang2019dscgans}} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}l@{}}DScGAN without/with \\ domain knowledge\end{tabular}} & \multicolumn{1}{c|}{0.816/0.905} & 0.812/0.914 \\ \hlin
\end{tabular}
\label{tab:diag_quan}
\end{table*}
With respect to type of domain knowledge for disease diagnosis, the knowledge from natural images is widely incorporated in deep learning models. However, considering the gap between natural images and medical ones, using information from external medical datasets of the same diseases with similar modalities (e.g., SFM and DM) \cite{samala2017multi}, with different modalities (DBT and MM, Ultrasound) \cite{samala2018breast}, and even with different diseases \cite{liao2019multi} may be more effective. To incorporate the above information is relatively easy, and both transfer learning and multi-task learning have been widely adopted. However, there are still no comparative studies on how different extra datasets can improve the performance of deep learning models.
For the domain knowledge of medical doctors, the high-level domain knowledge (e.g., the training pattern and the diagnostic pattern) is generally common for different diseases. On the other hand, the low-level domain knowledge, such as the areas in images and features of diseases that medical doctors usually pay attention to, is generally different for different diseases. There is generally a trade-off between the versatility and the utility of domain knowledge. To diagnose a certain disease, the benefit of incorporating a versatile type of domain knowledge suitable for many diseases may not be as significant as using the one that is specific for the disease. However, identifying such specific domain knowledge may not be easy, and generally requires more efforts from medical doctors (e.g., to identify hand-crafted features or attention maps).
We believe that more kinds of medical domain knowledge can be explored and utilized for disease diagnosis. In addition, comparative studies on some benchmark datasets should be carried out with respect to different types of domain knowledge and different incorporating methods for disease diagnosis. The results can provide further insights about the utility of medical domain knowledge for deep learning models.
\section{Lesion, Organ, and Abnormality Detection}
\label{sec:detection}
Detecting lesions in medical images is an important task and a key part for disease diagnosis in many conditions. Similarly, organ detection is an essential preprocessing step for image registration, organ segmentation, and lesion detection. Detecting abnormalities in medical images, such as cerebral microbleeds in brain MRI images and hard exudates in retinal images, is also required in many applications.
In this section, the deep learning models widely used for object detection in medical images are first described in Subsection \ref{sec:detmodel}. Then, the existing works on utilizing domain knowledge from natural and medical datasets, and from medical doctors are introduced in detail in Subsection \ref{sec:det_natural} and Subsection \ref{sec:det_radiologists}, respectively. Lastly, we summarize and discuss these different types of domain knowledge and the associated incorporating methods in Subsection \ref{sec:det_overview}.
\subsection{General Structures of Deep Learning Models for Object Detection in Medical Images}
\label{sec:detmodel}
Deep learning models designed for object detection in natural images have been directly applied to detect objects in medical images. These applications include pulmonary nodule detection in CT images \cite{setio2016pulmonary}, retinal diseases detection in retinal fundus \cite{gulshan2016development} and so on.
\begin{figure}[!htp] \begin{centering}
\includegraphics[width=0.9\linewidth]{det_structure2_new.pdf}
\centering
\caption
The workflow of colitis detection method by using the Faster R-CNN structure \cite{liu2017detection}.
}
\label{fig:det_structure2}
\end{centering}
\end{figure}
One popular type of model is the two-stage detectors like the Faster R-CNN \cite{ren2015faster} and the Mask R-CNN \cite{he2017mask}. They generally consist of a region proposal network (RPN) that hypothesizes candidate object locations and a detection network that refines region proposals. Applications in this category include colitis detection in CT images \cite{liu2017detection}, intervertebral disc detection in X-ray images \cite{sa2017intervertebral} and the detection of architectural distortions in mammograms \cite{ben2017domain}. Fig. \ref{fig:det_structure2} shows an example of using Faster R-CNN structure for colitis detection \cite{liu2017detection}.
Another category is one-stage models like YOLO (You Only Look Once) \cite{redmon2016you}, SSD (Single Shot MultiBox Detector) \cite{liu2016ssd} and RetinaNet \cite{lin2017focal}. These networks skip the region proposal stage and run detection directly by considering the probability that the object appears at each point in image. Compared with the two-stage models, models in this approach are generally faster and simpler. Examples in this category can be found in breast tumor detection in mammograms \cite{platania2017automated}, pulmonary lung nodule detection in CT \cite{li2017detection}, and the detection of different lesions (e.g., liver lesion, lung lesion, bone lesion, abdomen lesion) in CT images \cite{cai2019one}. An example of using one-stage structure for lesion detection is shown in Fig. \ref{fig:det_structure31}.
\begin{figure}[!htp] \begin{centering}
\includegraphics[width=1.0\linewidth]{det_structure31_new.pdf}
\centering
\caption
An example of using one-stage structure for lesion detection in CT images \cite{cai2019one}.
}
\label{fig:det_structure31}
\end{centering}
\end{figure}
In the following subsections, we will introduce related works that incorporate external knowledge into deep learning models for object detection in medical images. A summary of these works is shown in Table \ref{tab:detection_knowledge}.
\begin{table*}[]
\small
\center
\caption{List of studies on lesion, organ, and abnormality detection and the knowledge they incorporated}
\begin{tabular}{|c|l|l|l|}
\hline
\begin{tabular}[c]{@{}l@{}}Knowledge Source\end{tabular} & \begin{tabular}[c]{@{}l@{}}Knowledge Type\end{tabular} & \begin{tabular}[c]{@{}l@{}}Incorporating Method\end{tabular} & References \\ \hline
\begin{tabular}[c]{@{}l@{}}Natural datasets\end{tabular} & \begin{tabular}[c]{@{}l@{}}Natural images\end{tabular} & \begin{tabular}[c]{@{}l@{}}Transfer learning\end{tabular} &
{\begin{tabular}[c]{@{}l@{}}\cite{shin2016deep}\cite{yap2018automated}
\cite{nappi2016deep}\cite{zhang2016automatic} \cite{tajbakhsh2016convolutional} \\ \end{tabular}} \\ \hline
\begin{tabular}[c]{@{}l@{}}Medical datasets\end{tabular} & \begin{tabular}[c]{@{}l@{}}Multi-modal images\end{tabular} & \begin{tabular}[c]{@{}l@{}}Transfer learning\end{tabular} & \begin{tabular}[c]{@{}l@{}}\cite{zhang2018breast}\cite{ben2019cross}\cite{zhao2020tripartite}\end{tabular} \\ \hline
\multirow{8}{*}{\begin{tabular}[c]{@{}l@{}}Medical doctors\end{tabular}} & \begin{tabular}[c]{@{}l@{}}Training pattern\end{tabular} & \begin{tabular}[c]{@{}l@{}}Curriculum learning\end{tabular} &\begin{tabular}[c]{@{}l@{}}\cite{tang2018attention}\cite{jesson2017cased}
\cite{astudillo2020curriculum}\end{tabular} \\ \cline{2-4}
& \multirow{5}{*}{\begin{tabular}[c]{@{}l@{}}Detection patterns\end{tabular}} & \begin{tabular}[c]{@{}l@{}}Using images collected \\under different settings \end{tabular} & \cite{li2019mvp-net}\cite{ni2020deep} \\ \cline{3-4}
& & \begin{tabular}[c]{@{}l@{}}Comparing bilateral or \\cross-view images \end{tabular}& \begin{tabular}[c]{@{}l@{}}\cite{liu2019from}\cite{2020Cross} \cite{lisowska2017thrombus,lisowska2017context}
\cite{li2020deep}\end{tabular}
\\ \cline{3-4}
& & \begin{tabular}[c]{@{}l@{}}Analyzing adjacent slices\end{tabular} & \cite{li2017detection}
\\ \cline{2-4}
& \begin{tabular}[c]{@{}l@{}}Features doctors
focus on\end{tabular} & \begin{tabular}[c]{@{}l@{}}Feature level fusion\end{tabular}&
\begin{tabular}[c]{@{}l@{}}\cite{fu2017automatic}\cite{kooi2017large} \cite{ghatwary2019esophageal}\cite{liu2019automated}
\cite{chao2020lymph}\cite{sonora2020evaluating}\end{tabular}
\\ \cline{2-4}
& \begin{tabular}[c]{@{}l@{}}Other related information
\end{tabular} & \begin{tabular}[c]{@{}l@{}}Multi-task learning \\/training pattern design\end{tabular} &\begin{tabular}[c]{@{}l@{}}\cite{tang2018attention}\cite{hwang2016self}
\cite{bakalo2019classification}\cite{liang2020weakly}\end{tabular} \\ \hline
\end{tabular}
\label{tab:detection_knowledge}
\end{table*}
\subsection{Incorporating Knowledge from Natural Datasets or Other Medical Datasets}
\label{sec:det_natural}
Similar to disease diagnosis, it is quite popular to pre-train a large natural image dataset (generally ImageNet) to introduce information for object detection in medical domain. Examples can be found in lymph node detection \cite{shin2016deep}, polyp and pulmonary embolism detection \cite{tajbakhsh2016convolutional}, breast tumor detection \cite{yap2018automated}, colorectal polyps detection \cite{nappi2016deep,zhang2016automatic} and so on.
In addition, utilizing other medical datasets (i.e., multi-modal images) is also quite common. For example, PET images are used to help the lesion detection in CT scans of livers \cite{ben2019cross}. Specifically, PET images are first generated from CT scans using a combined structure of FCN and GAN, then the synthesized PET images are used in a false-positive reduction layer for detecting malignant lesions. Quantitative results show a 28\% reduction in the average false positive per case. Besides, \cite{zhang2018breast} develops a strategy to detect breast masses from digital tomosynthesis by fine-tuning the model pre-trained on mammography datasets. Another example of using multi-modal medical images can be found in liver tumor detection \cite{zhao2020tripartite}.
\subsection{Incorporating Knowledge from Medical Doctors}
\label{sec:det_radiologists}
In this subsection, we summarize the existing works on incorporating the knowledge of medical doctors to more effectively detect objects in medical images. The types of domain knowledge from medical doctors can be grouped into the following four categories:
\begin{enumerate}
\item the training pattern,
\item the general detection patterns they view images,
\item the features (e.g., locations, structures, shapes) they give special attention to, and
\item other related information regarding detection.
\end{enumerate}
\subsubsection{Training Pattern of Medical Doctors}
\label{sec:det_trainingpattern}
The training pattern of medical doctors, which is generally characterized as being given tasks with increasing difficulty, is also used for object detection in medical images. Similar to the disease diagnosis, most works utilize the curriculum learning to introduce this pattern. For example, an attention-guided curriculum learning (AGCL) framework is presented to locate the lesion in chest X-rays \cite{tang2018attention}. During this process, images in order of difficulty (grouped by different severity-levels) are fed into the CNN gradually, and the disease heatmaps generated from the CNN are used to locate the lesion areas.
Another work is called as CASED proposed for lung nodule detection in chest CT \cite{jesson2017cased}. CASED adopts a curriculum adaptive sampling technique to address the problem of extreme data imbalance. In particular, CASED lets the network to first learn how to distinguish nodules from their immediate surroundings, and then it adds a greater proportion of difficult-to-classify global context, until uniformly samples from the empirical data distribution. In this way, CASED tops the LUNA16 challenge leader-board with an average sensitivity score of 88.35\%. Another example of using curriculum learning can be found in cardiac landmark detection \cite{astudillo2020curriculum}.
\subsubsection{General Detection Patterns of Medical Doctors}
\label{sec:det_diagnosticpattern}
When experienced medical doctors are locating possible lesions in medical images, they also display particular patterns, and these patterns can be incorporated into deep learning models for object detection.
Experienced medical doctors generally have the following patterns: (1) they usually take into account images collected under different settings (e.g., brightness and contrast), (2) they often compare bilateral images or use cross-view images, and (3) they generally read adjacent slices.
For example, to locate possible lesions during visual inspection of the CT images, radiologists would combine images collected under different settings (e.g., brightness and contrast). To imitate the above process, a multi-view feature pyramid network (FPN) is proposed in \cite{li2019mvp-net}, where multi-view features are extracted from images rendered with varied brightness and contrast. The multi-view information is then combined using a position-aware attention module. Experiments show that the proposed model achieves an absolute gain of 5.65\% over the previous state-of-the-art method
on the NIH DeepLesion dataset. Another example of using images under different settings can be found in the COVID-19 pneumonia-based lung lesion detection \cite{ni2020deep}.
In addition, the bilateral information is widely used by radiologists. For example, in standard mammographic screening, images are captured from both two breasts, and experienced radiologists generally compare bilateral mammogram images to find masses. To incorporate this pattern, a contrasted bilateral network (CBN) is proposed in \cite{liu2019from}, in which the bilateral images are first coarsely aligned and then fed into a pair of networks to extract features for the following detection steps (shown in Fig. \ref{fig:bilateral_image}). Experimental results demonstrate the effectiveness of incorporating the bilateral information.
\begin{figure}[!htp] \begin{centering}
\includegraphics[width=0.75\linewidth]{bilateral_images_new.pdf}
\centering
\caption{The workflow of mammogram mass detection by integrating the bilateral information \cite{liu2019from}, where the aligned images are fed into two networks seperately to extract features for further detection.}
\label{fig:bilateral_image}
\end{centering}
\end{figure}
Similarly, to detect acute stroke signs in non-contrast CT images, experienced neuroradiologists routinely compare the appearance and Hounsfield Unit intensities of the left and right hemispheres, and then find the regions most commonly affected in stroke episodes. This pattern is mimicked by \cite{lisowska2017context} for the detection of dense vessels and ischaemia. The experimental results show that introducing the pattern greatly improves the performance for detecting ischaemia. Other examples of integrating the bilateral feature comparison or the symmetry constrains can be found in thrombus detection \cite{lisowska2017thrombus} and hemorrhagic lesion detection \cite{li2020deep} in brain CT images.
Besides the bilateral images, the information from cross views (i.e., mediolateral-oblique and cranio-caudal) is highly related and complementary, and hence is also used for mammogram mass detection. In \cite{2020Cross}, a bipartite graph convolutional network is introduced to endow the existing methods with cross-view reasoning ability of radiologists. Concretely, the bipartite node sets are constructed to represent the relatively consistent regions, and the bipartite edge are used to model both inherent cross-view geometric constraints and appearance similarities between correspondences. This process can enables spatial visual features equipped with cross-view reasoning ability. Experimental results on DDSM dataset achieve the state-of-the-art performance (with a recall of 79.5 at 0.5 false positives per image).
When looking for small nodules in CT images , radiologists often observe each slice together with adjacent slices, similar to detecting an object in a video. This workflow is imitated in \cite{li2017detection} to detect pulmonary lung nodule in CT images, where the state-of-the-art object detector SSD is applied in this process. This method obtains state-of-the-art result with the FROC score of 0.892 in LUNA16 dataset.
\subsubsection{Features That Medical Doctors Give Special Attention to}
\label{sec:det_feature}
Similar to disease diagnosis, medical doctors also use many `hand-crafted features' to help them to find target objects (e.g., nodules or lesions) in medical images.
For example, in \cite{kooi2017large}, to detect mammographic lesions, different types of hand-crafted features (e.g., contrast features, geometrical features, and location features) are first extracted, and then concatenated with those learned by a CNN.
The results show that these hand-crafted features, particularly the location and context features
, can complement the network generating a higher specificity over the CNN alone.
Similarly, \cite{ghatwary2019esophageal} presents a deep learning model based on Faster R-CNN to detect abnormalities in the esophagus from endoscopic images. In particular, to enhance texture details, the proposed detection system incorporates the Gabor handcrafted features
with the CNN features through concatenation in the detection stage.
The experimental results on two datasets (Kvasir and MICCAI 2015) show that the model is able to surpass the state-of-the-art performance.
Another example can be found in \cite{fu2017automatic} for the detection of lung nodules, where 88 hand-crafted features, including intensity, shape, texture are extracted and combined with features extracted by a CNN and then fed into a classifier. Experimental results
demonstrate the effectiveness of the combination of handcrafted features and
CNN features.
In the automated detection of thyroid nodules, the size and shape attribute of nodules are considered in \cite{liu2019automated}. To incorporate the information above, the generating process of region proposals is constrained, and the detection results on two different datasets show high accuracy.
Furthermore, in lymph node gross tumor volume detection (GTV$_{LN}$) in oncology imaging, some attributes of lymph nodes (LNs) are also utilized \cite{chao2020lymph}. Motivated by the prior clinical knowledge that LNs from a connected lymphatic system, and the spread of cancer cells among LNs often follows certain pathways, a LN appearance and inter-LN relationship learning framework is proposed for GTV$_{LN}$ detection. More specifically, the instance-wise appearance features are first extracted by a 3D CNN, then a graph neural network (GNN) is used to model the inter-LN relationships, and the global LN-tumor spatial priors are included in this process. This method
significantly improves over state-of-the-art method. Another example of combining handcrafted features and deep features can be found in lung lesion detection \cite{sonora2020evaluating}.
\subsubsection{Other Types of Information Related to Detection}
\label{sec:det_extraannotations}
Similar with that in disease diagnosis, there are other information (e.g., radiological reports, extra labels) can also be integrated into the lesion detection process.
For example in \cite{tang2018attention}, to locate thoracic diseases on chest radiographs, the difficulty of each sample, represented as the severity level of the thoracic disease, is first extracted from radiology reports. Then, the curriculum learning technique is adopted, in which the training samples are presented to the network in order of increasing difficulty. Experiments on the ChestXray14 database validate the effectiveness on significant performance improvement over baseline methods.
Example of using extra labels can be found in \cite{hwang2016self}. In this method, the information of the classification labels is incorporated to help the lesion localization in chest X-rays and mammograms. In particular, a framework named as self-transfer learning (STL) is proposed, which jointly optimizes both classification and localization networks to help the localization network focus on correct lesions. Experimental results show that STL can achieve significantly better localization performance compared to previous weakly supervised localization approaches. More examples of using extra labels can be found in detection in mammograms \cite{bakalo2019classification,liang2020weakly}.
\subsection{Summary}
\label{sec:det_overview}
In the previous sections, we introduced different kinds of domain knowledge and the corresponding integrating methods into the deep learning models for object detection in medical images. Table \ref{tab:detquan} illustrates the quantitative improvements, in terms of sensitivity and recall, of some typical work over the baseline methods for object detection in medical images. From the results we can see that in general, integrating domain knowledge can be beneficial for detection tasks.
Similar to disease diagnosis, the high-level training pattern of medical doctors is generic and can be utilized for detecting different diseases or organs. In contrast, the low-level domain knowledge, like the detection patterns that medical doctors adopt and some hand-crafted features they give more attention when searching lesions, are generally different for different diseases. For example, the pattern of comparing bilateral images can only be utilized for detecting organs with symmetrical structures \cite{lisowska2017context,liu2019from}. In addition, we can see from Table \ref{tab:detquan} that leveraging pattern of medical doctors on average shows better performance when compared with integrating hand-crafted features. This may indicate that there is still large room to explore more effective features for object detection in medical images.
\begin{table*}[]
\caption{The comparison of the quantitative metrics for some medical object detection methods after incorporating domain knowledge}
\small
\center
\begin{threeparttable}
\begin{tabular}{|l|l|l|l|}
\cline{1-4}
\multicolumn{1}{|c|}{Reference} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}l@{}}Baseline model/With domain knowledge\end{tabular}} & \multicolumn{1}{c|}{Sensitivity} & \multicolumn{1}{c|}{Recall} \\ \hline
\multirow{1}{*}{\cite{fu2017automatic}} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}l@{}}CNN/CNN with hand-crafted features\end{tabular}} & \multicolumn{1}{c|}{0.890/0.909} &\multicolumn{1}{c|}{--/--} \\ \hlin
\multirow{1}{*}{\cite{ghatwary2019esophageal}} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}l@{}}ResNet/CNN with handcrafted features\end{tabular}} & \multicolumn{1}{c|}{--/--} &\multicolumn{1}{c|}{0.940/0.950} \\ \hlin
\multirow{1}{*}{\cite{li2017detection}} & \multicolumn{1}{c|}{SSD/MSSD} & \multicolumn{1}{c|}{0.927/0.976}&\multicolumn{1}{c|}{--/--} \\ \hlin
\multirow{1}{*}{\cite{liu2019from}} & \multicolumn{1}{c|}{Mask R-CNN/CBD} & \multicolumn{1}{c|}{--/--} &\multicolumn{1}{c|}{0.869/0.890} \\ \hlin
\multirow{1}{*}{\cite{2020Cross}} & \multicolumn{1}{c|}{Mask R-CNN/BG-RCNN} & \multicolumn{1}{c|}{--/--} &\multicolumn{1}{c|}{0.918/0.945} \\ \hlin
\multirow{1}{*}{\cite{tang2018attention}} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}l@{}}CNN/AGCL\end{tabular}}& \multicolumn{1}{c|}{--/--} &\multicolumn{1}{c|}{0.660/0.730} \\ \hlin
\end{tabular}
\label{tab:detquan}
\begin{tablenotes}
\footnotesize
\item[1] The performance is evaluated at 4 false positives per image in \cite{fu2017automatic,liu2019from,2020Cross,samala2018breast}.
\end{tablenotes}
\end{threeparttable}
\end{table*}
\section{Lesion and Organ Segmentation}
\label{sec:segmentation}
Medical image segmentation devotes to identifying pixels of lesions or organs from the background, and is generally regarded as a prerequisite step for the lesion assessment and disease diagnosis. Segmentation methods based on deep learning models have become the dominant technique in recent years and have been widely used for the segmentation of lesions such as brain tumors \cite{havaei2017brain}, breast tumors \cite{zhang2018hierarchical}, and organs such as livers \cite{christ2017automatic} and pancreas \cite{roth2015deep}.
In Subsection \ref{sec:seg_structures}, we describe the models that are generally used for object segmentation in the medical domain. Then in Subsection \ref{sec:segmentation_natural}, the works of utilizing domain knowledge from natural and other medical datasets are introduced. Then, the models utilizing domain knowledge from medical doctors are introduced in Subsection \ref{sec:seg_radiologists}. A summary of this section is provided in Subsection \ref{sec:seg_overview}.
\subsection{General Structures of Deep Learning Models for Object Segmentation in Medical Images}
\label{sec:seg_structures}
The deep learning models utilized for medical image segmentation are generally divided into three categories: the FCN (fully convolutional network) \cite{long2015fully} based models, the U-Net \cite{ronneberger2015u} based models, and the GAN \cite{goodfellow2014generative} based models.
\begin{figure}[!htp] \begin{centering}
\includegraphics[width=0.85\linewidth]{seg_structure_FCN_new.pdf}
\centering
\caption{The schematic diagram of using FCN structure for cardiac segmentation \cite{tran2016fully}.}
\label{fig:seg_FCN}
\end{centering}
\end{figure}
In particular, the FCN has been proven to perform well in various medical image segmentation tasks \cite{chen2016iterative,gibson2017towards}. Some variants of FCN, such as cascaded FCN \cite{christ2016automatic}, parallel FCN \cite{kamnitsas2017efficient} and recurrent FCN \cite{yang2017fine} are also widely used for segmentation tasks in medical images. Fig. \ref{fig:seg_FCN} illustrates an example of using FCN based model for cardiac segmentation.
\begin{figure}[!htb] \begin{centering}
\includegraphics[width=0.9\linewidth]{seg_structure_UNet_new.pdf}
\centering
\caption{The network structure of U-Net \cite{ronneberger2015u}.
}
\label{fig:seg_structure_UNet}
\end{centering}
\end{figure}
In addition, the U-Net \cite{ronneberger2015u} (shown in Fig. \ref{fig:seg_structure_UNet}) and its variants are also widely utilized for medical image segmentation. U-Net builds upon FCN structure, mainly consists of a series of convolutional and deconvolutional layers, and with the short connections between the layers of equal resolution. U-Net and its variants like UNet++\cite{zhou2018unet++} and recurrent U-Net \cite{alom2018recurrent} perform well in many medical image segmentation task
\cite{gordienko2018deep}.
In the GAN-based models \cite{yang2017automatic,zhao2018craniomaxillofacial}, the generator is used to predict the mask of the target based on some encoder-decoder structures (such as FCN or U-Net).
The discriminator serves as a shape regulator that helps the generator to obtain satisfactory segmentation results. Applications of using GAN-based models in medical image segmentation include brain segmentation \cite{kamnitsas2017unsupervised}, skin lesion segmentation \cite{izadi2018generative}, vessel segmentation \cite{lahiri2017generative} and anomaly segmentation in retinal fundus images \cite{schlegl2017unsupervised}. Fig. \ref{fig:seg_GAN} is an example of using GAN-based model for vessel segmentation in miscroscopy images
\begin{figure}[!htp] \begin{centering}
\includegraphics[width=0.85\linewidth]{seg_structure_GAN_new.pdf}
\centering
\caption{An example of using a GAN-based model for vessel segmentation \cite{lahiri2017generative}.}
\label{fig:seg_GAN}
\end{centering}
\end{figure}
\begin{table*}[]
\small
\center
\caption{The list of researches of lesion, organ segmentation and the knowledge they incorporated}
\begin{tabular}{|c|l|l|l|}
\hline
\begin{tabular}[c]{@{}l@{}}Knowledge Source\end{tabular} & \begin{tabular}[c]{@{}l@{}}Knowledge Type\end{tabular} & \begin{tabular}[c]{@{}l@{}}Incorporating\\ Method\end{tabular} & References \\ \hline
\begin{tabular}[c]{@{}l@{}}Natural datasets\end{tabular} & \begin{tabular}[c]{@{}l@{}}Natural images\end{tabular} & \begin{tabular}[c]{@{}l@{}}Transfer learning\end{tabular} &
{\begin{tabular}[c]{@{}l@{}} \cite{chen2016iterative}\cite{chen2017dcan}\cite{wu2017cascaded}\cite{zeng20173d}
\cite{tajbakhsh2016convolutional}
\end{tabular}} \\ \hline
\multirow{6}{*}{\begin{tabular}[c]{@{}l@{}}Medical datasets\end{tabular}} & \multirow{4}{*}{\begin{tabular}[c]{@{}l@{}}Multi-modal images\end{tabular}} & \begin{tabular}[c]{@{}l@{}}Transfer learning\end{tabular} & \begin{tabular}[c]{@{}l@{}}\cite{ghafoorian2017transfer}\cite{valverde2019one-shot}\end{tabular} \\ \cline{3-4}
& & \begin{tabular}[c]{@{}l@{}}Multi-task/-modal learning\end{tabular} &
{\begin{tabular}[c]{@{}l@{}}\cite{moeskops2016deep}\cite{valindria2018multi}\end{tabular}} \\ \cline{3-4}
& & \begin{tabular}[c]{@{}l@{}}Modality transformation /synthesis
\end{tabular} &
{\begin{tabular}[c]{@{}l@{}}\cite{chen2018semantic}\cite{chen2020unsupervised} \cite{jiang2018tumor} \cite{kamnitsas2017unsupervised} \cite{yan2019domain}\\\cite{yang2019unsupervised}{\cite{li2020towards}\cite{li2020dual}\cite{hu2020knowledge}}
\cite{zhang2018translating}\cite{chartsias2020disentangle}
\end{tabular}} \\ \cline{2-4}
& \multirow{2}{*}{\begin{tabular}[c]{@{}l@{}}Datasets of other diseases\end{tabular}} & Transfer learning & \cite{chen2019med3d} \\ \cline{3-4}
& & \begin{tabular}[c]{@{}l@{}}Disease domain transformation
\end{tabular}
{\cite{yu2019annotation}}
\\ \hline
\multirow{11}{*}{\begin{tabular}[c]{@{}l@{}}Medical doctors\end{tabular}} & \begin{tabular}[c]{@{}l@{}}Training pattern\end{tabular} & \begin{tabular}[c]{@{}l@{}}Curriculum learning\end{tabular} &\begin{tabular}[c]{@{}l@{}}\cite{berger2018adaptive} \cite{wang2018deep}
\cite{li2020new}\cite{kervadec2019curriculum}
\cite{zhao2019semi}\cite{zhang2020weakly}\end{tabular} \\ \cline{2-4}
& \multirow{3}{*}{\begin{tabular}[c]{@{}l@{}}Diagnostic patterns\end{tabular}} & \begin{tabular}[c]{@{}l@{}}Using different views as input\end{tabular} & \begin{tabular}[c]{@{}l@{}}\cite{wu2018joint}\cite{chen2019learning}\end{tabular} \\ \cline{3-4}
& & \begin{tabular}[c]{@{}l@{}}Attention mechanism\end{tabular} & \cite{hatamizadeh2019end}
\\ \cline{3-4}
& & \begin{tabular}[c]{@{}l@{}}Network design\end{tabular} & \cite{zhu2020lymph}\cite{jin2020deeptarget}
\\ \cline{2-4}
& \multirow{5}{*}{\begin{tabular}[c]{@{}l@{}}Anatomical priors\end{tabular}} & \begin{tabular}[c]{@{}l@{}}Incorporated in post-processing
\end{tabular} &
\begin{tabular}[c]{@{}l@{}}\cite{huang2018medical} \cite{painchaud2019cardiac,painchaud2020cardiac}
\cite{larrazabal2019anatomical}\end{tabular}
\\ \cline{3-4}
& & \begin{tabular}[c]{@{}l@{}}Incorporated in loss function\end{tabular} &
\begin{tabular}[c]{@{}l@{}}\cite{oktay2017anatomically}\cite{ravishankar2017learning} \cite{bentaieb2016topology}\cite{mirikharaji2018star} \cite{yue2019cardiac}\\\cite{zheng2019semi-supervised}
\cite{oktay2017anatomically} \cite{zotti2019convolutional}
\end{tabular}
\\ \cline{3-4}
& & \begin{tabular}[c]{@{}l@{}}Generative models \end{tabular}&
\begin{tabular}[c]{@{}l@{}}\cite{chen2019learning}\cite{dalca2018anatomical} \cite{he2019dpa-densebiasnet}\cite{luo2020shape}
\cite{song2020shape}\\\cite{boutillon2020combining}
\cite{pham2020liver}\cite{engin2020agan}
\cite{gao2020focusnetv2}\end{tabular}
\\ \cline{2-4}
& \multirow{2}{*}{\begin{tabular}[c]{@{}l@{}}Hand-crafted features\end{tabular}} & \begin{tabular}[c]{@{}l@{}}Feature level fusion\end{tabular} &\begin{tabular}[c]{@{}l@{}}\cite{kushibar2018automated}\cite{rezaei2019gland}\end{tabular}
\\ \cline{3-4}
& & \begin{tabular}[c]{@{}l@{}}Input level fusion\end{tabular} &\begin{tabular}[c]{@{}l@{}}\cite{khan2020cascading}\cite{narotamo2020combining} \end{tabular} \\ \hline
\end{tabular}
\label{tab:segmentation_knowledge}
\end{table*}
In the following sections, we will introduce research studies that incorporate domain knowledge into deep segmentation models. The summary of these works is shown in Table \ref{tab:segmentation_knowledge}.
\subsection{Incorporating Knowledge from Natural Datasets or Other Medical Datasets}
\label{sec:segmentation_natural}
It is also quite common that deep learning segmentation models are firstly trained on a large-scale natural image dataset (e.g., ImageNet) and then fine-tuned on the target datasets. Using the above transfer learning strategy to introduce knowledge from natural images has demonstrated to achieve a better performance in medical image segmentation. Examples can be found in intima-media boundary segmentation \cite{tajbakhsh2016convolutional} and prenatal ultrasound image segmentation\cite{wu2017cascaded}. Besides ImageNet, \cite{chen2016iterative} adopts the off-the-shelf DeepLab model trained on the PASCAL VOC dataset for anatomical structure segmentation in ultrasound images. This pre-trained model is also used in the deep contour-aware network (DCAN), which is designed for the gland segmentation in histopathological images \cite{chen2017dcan}.
Besides using models pre-trained on `static' datasets like ImageNet and PASCAL VOC, many deep neural networks, especially those designed for the segmentation of 3D medial images, leverage models pre-trained on large-scale video datasets. For example, in the automatic segmentation of proximal femur in 3D MRI, the C3D pre-trained model is adopted as the encoder of the proposed 3D U-Net \cite{zeng20173d}. Notably, the C3D model is trained on the Sports-1M dataset, which is the largest video classification benchmark with 1.1 million sports videos in 487 categories \cite{tran2015learning}.
In addition to natural images, using knowledge from external medical datasets with different modalities and with different diseases is also quite popular.
For example, \cite{ghafoorian2017transfer} investigates the transferability of the acquired knowledge of a CNN model initially trained for WM hyper-intensity segmentation on legacy low-resolution data to new data from the same scanner but with higher image resolution. Likewise, the images with different MRI scanners and protocols are used in \cite{valverde2019one-shot} to help the multi sclerosis segmentation process via transfer learning.
In \cite{moeskops2016deep}, the multi-task learning is adopted,
where the data of brain MRI, breast MRI and cardiac CT angiography (CTA) are used
simultaneously as multiple tasks. On the other hand, \cite{valindria2018multi} adopts
a multi-modal learning structure for organ segmentation. A dual-stream encoder-decoder
architecture is proposed to learn modality-independent, and thus, generalisable and
robust features shared among medical datasets with different modalities (MRI and CT images).
Experimental results prove the effectiveness of this multi-modal learning structure.
Moreover, many works adopt GAN-based models to achieve the domain transformation among
datasets with different modalities. For example, a model named SeUDA (unsupervised domain
adaptation) is proposed for the left/right lung segmentation process \cite{chen2018semantic}.
It leverages the semantic-aware GAN to transfer the knowledge from one chest dataset to
another. In particular, target images are first mapped towards the source data space via the
constraint of a semantic-aware GAN loss. Then the segmentation results are obtained
from the segmentation DNN learned on the source domain. Experimental results show that
the segmentation performance of SeUDA is highly competitive.
More examples of using the knowledge from images with other modalities can be found in brain MRI segmentation \cite{kamnitsas2017unsupervised,hu2020knowledge}, cardiac segmentation
\cite{yan2019domain,zhang2018translating,li2020dual,li2020towards,chartsias2020disentangle}, liver segmentation \cite{yang2019unsupervised}, lung tumor segmentation \cite{jiang2018tumor},
cardiac substructure and abdominal multi-organ segmentation \cite{chen2020unsupervised}.
There are also a few works utilize the datasets of other diseases. For instance, \cite{chen2019med3d} first builds a union dataset (3DSeg-8) by aggregating eight different 3D medical segmentation datasets, and designs the Med3D network to co-train based on 3DSeg-8. Then the pre-trained models obtained from Med3D are transferred into lung and liver segmentation tasks. Experiments show that this method not only improves the accuracy, but also accelerates the training convergence speed.
In addition, the annotated retinal images are used to help the cardiac vessel segmentation without annotations \cite{yu2019annotation}. In particular,
a shape-consistent generative adversarial network (SC-GAN) is used to generate the synthetic images and the corresponding labels. Then the synthetic images are used to train the segmentor. Experiments demonstrate the supreme accuracy of coronary artery segmentation.
\subsection{Incorporating Knowledge from Medical Doctors}
\label{sec:seg_radiologists}
The domain knowledge of medical doctors is also widely adopted when designing deep learning models for segmentation tasks in medical images. The types of domain knowledge from medical doctors utilized in deep segmentation models can be divided into four categories:
\begin{enumerate}
\item the training pattern,
\item the general diagnostic patterns they view images,
\item the anatomical priors (e.g., shape, location, topology) of lesions or organs, and
\item other hand-crafted features they give special attention to.
\end{enumerate}
\subsubsection{Training Pattern of Medical Doctors}
\label{sec:seg_trainingprocess}
Similar to disease diagnosis and lesion/organ detection, many research works for the object segmentation in medical images also mimic the training pattern of medical doctors, which involves assigning tasks that increase in difficulty over time.
In this process, the curriculum learning technique or its derivative methods like self-paced learning (SPL) are also utilized \cite{kumar2010self}.
For example, for the segmentation of multi-organ CT images \cite{berger2018adaptive}, each annotated image is divided into small patches. During the training process, patches producing large error by the network are selected with a high probability. In this manner, the network can focus sampling on difficult regions, resulting in improved performance.
In addition, \cite{wang2018deep} combines the SPL with the active learning for the pulmonary segmentation in 3D images. This system achieves the state-of-the-art segmentation results.
Moreover, a three-stage curriculum learning approach is proposed for liver tumor segmentation \cite{li2020new}. The first stage is performed on the whole input volume to initialize the network, then the second stage of learning focuses on tumor-specific features by training the network on the tumor patches, and finally the network is retrained on the whole input in the third stage. This approach exhibits significant improvement when compared with the commonly used cascade counterpart in MICCAI 2017 liver tumor segmentation (LiTS) challenge dataset. More examples can also be found in left ventricle segmentation \cite{kervadec2019curriculum}, finger bones segmentation \cite{zhao2019semi} and vessel segmentation \cite{zhang2020weakly}.
\subsubsection{General Diagnostic Patterns of Medical Doctors}
\label{sec:seg_diagnosticpattern}
In the lesion or organ segmentation tasks, some specific patterns that medical doctors adopted are also incorporated into the network.
For example, during the visual inspection of CT images, radiologists often change window widths and window centers to help to make decision on uncertain nodules. This pattern is mimicked in \cite{wu2018joint}. In particular, image patches of different window widths and window centers are stacked together as the input of the deep learning model to gain rich information. The evaluation implemented on the public LIDC-IDRI dataset indicates that the proposed method achieves promising performance on lung nodule segmentation compared with
the state-of-the-art methods.
In addition, experienced clinicians generally assess the cardiac morphology and function from multiple standard views, using both long-axis (LA) and short-axis (SA) images to form an understanding of the cardiac anatomy. Inspired by the above observation, a cardiac MR segmentation method is proposed which takes three LA and one SA views as the input \cite{chen2019learning}. In particular, the features are firstly extracted using a multi-view autoencoder (MAE) structure, and then are fed in a segmentation network. Experimental results show that this method has a superior segmentation accuracy over
state-of-the-art methods.
Furthermore, expert manual segmentation usually relies on the boundaries of anatomical structures of interest. For instance, radiologists segmenting a liver from CT images would usually trace liver edges first, and then deduce the internal segmentation mask. Correspondingly, boundary-aware CNNs are proposed in \cite{hatamizadeh2019end} for medical image
segmentation. The networks are designed to account for organ boundary information, both by providing a special network edge branch and edge-aware loss terms. The effectiveness of these boundary aware segmentation networks are tested on BraTS 2018 dataset for the task of brain tumor segmentation.
Recently, the diagnostic pattern named as `divide-and-conquer manner' is mimicked in the GTV$_{LN}$ detection and segmentation method \cite{zhu2020lymph}. Concretely, the GTV$_{LN}$ is first divided into two subgroups of `tumor-proximal' and `tumor-distal', by means of binary of soft distance gating. Then a multi-branch detection-by-segmentation network is trained with each branch specializing on learning one GTV$_{LN}$ category features. After fusing the outs from multi-branch, the method shows significant improvements on the mean recall from 72.5\% to 78.2\%. Another example of using the diagnostic pattern of medical doctors can be found in gross tumor and clinical target volume segmentation \cite{jin2020deeptarget}.
\subsubsection{Anatomical Priors of Lesions or Organs}
\label{sec:seg_feature}
In comparison to non-medical images, medical images have many anatomical priors such as the shape, position and topology of organs or lesions. Experienced medical doctors greatly rely on these anatomical priors when they are doing segmentation tasks on these images. Incorporating the knowledge of anatomical priors into deep learning models has been demonstrated to be an effective way for accurate medical image segmentation. Generally speaking, there are three different approaches to incorporate these anatomical priors into deep learning models: (1) incorporating anatomical priors in the post-processing stage, (2) incorporating anatomical priors as regularization terms in the loss function and (3) learning anatomical priors via generative models.
\textbf{Incorporating anatomical priors in post-processing stage}
The first approach is to incorporate the anatomical priors in the post processing stage. The result of a segmentation network is often blurry and post-processing is generally needed to refine the segmentation result.
For example, according to the pathology, most of breast tumors begin in glandular tissues and are located inside the mammary layer \cite{sharma2010various}. This position feature is utilized by \cite{huang2018medical} in its post-processing stage where a fully connected conditional random field (CRF) model is employed. In particular, the position of tumors and their relative locations with mammary layer are added as a new term in CRF energy function to obtain better segmentation results.
Besides, some research first learn the anatomical priors, and then incorporate them into the post-processing stage to help produce anatomically plausible segmentation results \cite{larrazabal2019anatomical,painchaud2019cardiac,painchaud2020cardiac}.
For instance, the latent representation of anatomically correct cardiac shape is first learned by using adversarial variational autoencoder (aVAE), then be used to convert erroneous segmentation maps into anatomically plausible ones \cite{painchaud2019cardiac}.
Experiments manifest that aVAE is able to accommodate any segmentation method, and convert its anatomically implausible results to plausible ones without affecting its overall geometric and clinical metrics.
Another example in \cite{larrazabal2019anatomical} introduces the post-processing step based on denoising autoencoders (DAE) for lung segmentation. In particular, the DAE is trained using only segmentation masks, then the learned representations of anatomical shape and topological constraints are imposed on the original segmentation results (as shown in Fig. \ref{fig:seg_shape_prior_postprocessing}). By applying the Post-DAE on the resulting masks from arbitrary segmentation methods, the lung anatomical segmentation of X-ray images shows plausible results.
\begin{figure}[!htp] \begin{centering}
\includegraphics[width=0.95\linewidth]{seg_shape_prior_postprocessing_new.pdf}
\centering
\caption{The example of integrating the shape prior in the post-process stage \cite{larrazabal2019anatomical}.
}
\label{fig:seg_shape_prior_postprocessing}
\end{centering}
\end{figure}
\textbf{Incorporating anatomical priors as regularization terms in the loss function}
The second approach is incorporating anatomical priors as regularization terms in the objective function of segmentation networks. For example, for the segmentation of cardiac MR images, a network called as SRSCN is proposed \cite{yue2019cardiac}. SRSCN comprises a shape reconstruction neural network (SRNN) and a spatial constraint network (SCN). SRNN aims to maintain a realistic shape of the resulting segmentation and the SCN is adopted to incorporate the spatial information of the 2D slices. The loss of the SRSCN comes from three parts: the segmentation loss, the shape reconstruction (SR) loss for shape regularization, and the spatial constraint (SC) loss to assist segmentation. The results using images from 45 patients demonstrate the effectiveness of the SR and SC regularization terms, and show the superiority of segmentation performance of the SRSCN over the conventional schemes.
Another example in this category is the one designed for skin lesion segmentation \cite{mirikharaji2018star}. In this work, the star shape prior is encoded as a new loss term in a FCN to improve its segmentation of skin lesions from their surrounding healthy skin. In this manner, the non-star shape segments in FCN prediction
maps are penalized to guarantee a global structure in segmentation results. The experimental results on the ISBI 2017 skin segmentation
challenge dataset demonstrate the advantage of regularizing FCN parameters by
the star shape prior.
More examples in this category can be found in gland segmentation \cite{bentaieb2016topology}, kidney segmentation \cite{ravishankar2017learning}, liver segmentation \cite{zheng2019semi-supervised} and cardiac segmentation \cite{oktay2017anatomically,zotti2019convolutional}.
\textbf{Learning anatomical priors via generative models}
In the third approach, the anatomical priors (especially the shape prior) are learned by some generative models first and then incorporated into segmentation networks.
For example, in the cardiac MR segmentation process, a shape multi-view autoencoder (MAE) is proposed to learn shape priors from MR images of multiple standard views \cite{chen2019learning}. The information encoded in the latent space of the trained shape MAE is incorporated into multi-view U-Net (MV U-Net) in the fuse block to guide the segmentation process.
Another example is shown in \cite{luo2020shape}, where the shape constrained network (SCN) is proposed to incorporate the shape prior into the eye segmentation network. More specifically, the prior information is first learned by a VAE-GAN, and then the pre-trained encoder and discriminator are leveraged to regularize the training process.
More examples can be found in brain geometry segmentation in MRI \cite{dalca2018anatomical}, 3D fine renal artery segmentation \cite{he2019dpa-densebiasnet},
overlapping cervical cytoplasms segmentation \cite{song2020shape}, scapula segmentation \cite{boutillon2020combining}, liver segmentation \cite{pham2020liver}, carotid segmentation \cite{engin2020agan}, and head and neck segmentation \cite{gao2020focusnetv2}.
\subsubsection{Other Hand-crafted Features}
Besides anatomical priors, some hand-crafted features are also utilized for segmentation tasks. Generally speaking, there are two ways to incorporate the hand-crafted features into deep learning models: the feature-level fusion and the input-level fusion.
In the feature-level fusion, the hand-crafted features and the features learned by the deep models are concatenated. For example, for the gland segmentation in histopathology images \cite{rezaei2019gland}, two handcrafted features, namely invariant LBP features as well as $H\&E$ components, are firstly calculated from images. Then these features are concatenated with the features generated from the last convolutional layer of the network for predicting the segmentation results. Similarly, in the brain structure segmentation \cite{kushibar2018automated}, the spatial atlas prior is first represented as a vector and then concatenated with the deep features.
For the input-level fusion, the hand-crafted features are transformed into the input patches. Then the original image patches and the feature-transformed patches are fed into a deep segmentation network. For example in \cite{khan2020cascading}, for automatic brain tumor segmentation in MRI images, three handcrafted features (i.e., mean intensity, LBP and HOG) are firstly extracted. Based on these features, a SVM is employed to generate confidence surface modality (CSM) patches. Then the CSM patches and the original patches from MRI images are fed into a segmentation network. This method achieves good performance on BRATS2015 dataset.
\begin{table*}[]
\small
\center
\caption{The comparison of the quantitative metrics for some medical object segmentation methods after incorporating domain knowledge}
\begin{tabular}{|l|l|l|}
\cline{1-3}
\multicolumn{1}{|c|}{Reference} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}l@{}}Baseline Model/With Domain Knowledge\end{tabular}} & Dice score \\ \hline
\multirow{1}{*}{\cite{chen2019learning}} & \multicolumn{1}{c|}{3D U-Net/MV U-Net} & 0.923/0.926 \\ \hlin
\multirow{1}{*}{\cite{he2019dpa-densebiasnet}} & \multicolumn{1}{c|}{V-Net/DPA-DenseBiasNet} & 0.787/0.861 \\ \hlin
\multirow{1}{*}{\cite{jiang2018tumor}} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}l@{}}Masked cycle-GAN/Tumore\\ aware semi-unsupervised\end{tabular}} & 0.630/0.800 \\ \hlin
\multirow{1}{*}{\cite{valindria2018multi}} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}l@{}}modality specific method/dual-stream \\ encoder-decoder multi-model method\end{tabular}} & 0.838/0.860 \\ \hlin
\multirow{1}{*}{\cite{yue2019cardiac}} & \multicolumn{1}{c|}{U-Net/SRSCN} & 0.737/0.830 \\ \hlin
\multirow{1}{*}{\cite{yu2019annotation}} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}l@{}}U-Net/SC-GAN\end{tabular} } & 0.742/0.824 \\ \hlin
\multirow{1}{*}{\cite{zhang2018translating}} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}l@{}} cycle- and shape-consistency GAN\\ trained without/with synthetic data\end{tabular}} & 0.678/0.744 \\ \hlin
\end{tabular}
\label{tab:seg_quan}
\end{table*}
In addition, using handcrafted features by input-level fusion is also adopted in cell nuclei segmentation \cite{narotamo2020combining}. In particular, as nuclei are expected to have an approximately round shape, a map of gradient convergence is computed and be used by CNN as an extra channel besides the fluorescence microscopy image. Experimental results show higher F1-score when compared with other methods. Another example in this category can be found in brain tumor segmentation \cite{khan2020cascading}.
\subsection{Summary}
\label{sec:seg_overview}
The aforementioned sections described researches of incorporating domain knowledge for object (lesion or organ) segmentation in medical images.
The segmentation performance of some methods is shown in Table \ref{tab:seg_quan}, where the Dice score is used with a higher score indicating a better performance.
We can see that similar to disease diagnosis, using information from natural images like ImageNet is quite popular for lesion and organ segmentation tasks. The reason behind it may be that segmentation can be seen as a specific classification problem. Meanwhile, besides the ImageNet, some video datasets can also be utilized for segmenting 3D medical images (e.g., \cite{zeng20173d}). Using extra medical datasets with different modalities has also been proven to be helpful, although most applications are limited in using MRI to help segmentation tasks in CT images \cite{valindria2018multi}. Leveraging domain knowledge from medical doctors is also widely used in segmentation tasks. In particular, the anatomical priors of organs are widely adopted. However, anatomical priors are only suitable for the segmentation of organs with fixed shapes like hearts \cite{chen2019learning} or lungs \cite{chen2019learning}.
\section{Other Medical Applications}
\label{sec:otherappl}
In this section, we briefly introduce the works on incorporating domain knowledge in other medical images analysis applications, like medical image reconstruction, medical image retrieval and medical report generation.
\subsection{Medical Image Reconstruction}
\label{sec:otherappl_reconstruction}
The objective of medical image reconstruction is reconstructing a diagnostic image from a number of measurements (e.g., X-ray projections in CT or the spatial frequency information in MRI). Deep learning based methods have been widely applied in this field \cite{qin2018convolutional,schlemper2017deep}. It is also quite common that external information is incorporated into deep learning models for medical image reconstruction.
Some methods incorporate hand-crafted features in the medical image reconstruction process. For example, a network model called as DAGAN is proposed for the reconstruction of compressed sensing magnetic resonance imaging (CS-MRI) \cite{yang2017dagan}. In the DAGAN, to better preserve texture and edges in the reconstruction process, the adversarial loss is coupled with a content loss. In addition, the frequency-domain information is incorporated to enforce similarity in both the image and frequency domains. Experimental results show that the DAGAN method provides superior reconstruction with preserved perceptual image details.
In \cite{yedder2019limited}, a new image
reconstruction method is proposed to solve the limited-angle and limited sources breast cancer diffuse optical tomography (DOT) image reconstruction
problem in a strong scattering medium. By adaptively focusing on important features and filtering irrelevant and
noisy ones using the Fuzzy Jaccard loss, the network is able to reduce false
positive reconstructed pixels and reconstruct more accurate images.
Similarly, a GAN-based method is proposed to recover MRI images of the target contrast \cite{dar2018synergistic}. The method simultaneously leverages the relatively low-spatial-frequency information available in the collected evidence for the target contrast and the relatively high-spatial frequency information available in the source contrast. Demonstrations on brain MRI datasets indicate the proposed method outperforms state-of-the-art reconstruction methods, with enhanced recovery of high-frequency tissue structure, and improved reliability against feature leakage or loss.
\subsection{Medical Image Retrieval}
\label{sec:otherappl_retrieval}
The hospitals are producing large amount of imaging data and the development of medical image retrieval, especially the content based image retrieval (CBIR) systems can be of great help to aid clinicians in browsing these large datasets.
Deep learning methods have been applied to CBIR and have achieved high performance due to their superior capability for extracting features automatically.
It is also quite common that these deep learning models for CBIR utilize external information beyond the given medical datasets. Some methods adopt the transfer learning to utilize the knowledge from natural images or external medical datasets \cite{ahmad2017sinc,khatami2018sequential,swati2019content}. For example, the VGG model pre-trained based on ImageNet is used in brain tumor retrieval process \cite{swati2019content}, where a block-wise fine-tuning strategy is proposed to enhance the retrieval performance on the T1-weighted CE-MRI dataset. Another example
can be found in x-ray image retrieval process \cite{khatami2018sequential}, where a model pre-trained on the large augmented dataset is fine-tuned on the target dataset to extract general features.
Besides, as features play an important role in the similarly analysis in CBIR, some methods fuse prior features with deep features. In particular, in the chest radiograph image retrieval process, the decision values of binary features and texture features are combined with the deep features in the form of decision-level fusion \cite{anavi2015comparative}. Similarly, the metadata such as patients' age and gender is combined with the image-based features extracted from deep CNN for X-ray chest pathology image retrieval \cite{anavi2016visualizing}. Furthermore, the features extracted from saliency areas can also be injected into the features extracted from the whole image for the high retrieval accuracy \cite{ahmad2017sinc}.
\subsection{Medical Report Generation}
\label{sec:otherappl_reports}
Recently, deep learning models for image captioning have been successfully applied for automatic generation of medical reports \cite{jing2017automatic,liu2019clinically}. It is also found that incorporating external knowledge can help deep learning models to generate better medical reports.
For example, some methods try to incorporate specific or general patterns that doctors adopt when writing reports. For example, radiologists generally write reports using certain templates. Therefore, some templates are used during the sentence generation process \cite{li2018hybrid,li2019knowledge}. Furthermore, as the explanation given by doctors is fairly simple and phrase changing does not change their meaning, a model-agnostic method is presented to learn the short text description to explain this decision process \cite{gale2018producing}.
In addition, radiologists follow some procedures when writing reports: they generally first check a patient's images for abnormal findings, then write reports by following certain templates, and adjust statements in the templates for each individual case when necessary \cite{hong2013content}. This process is mimicked in \cite{li2019knowledge}, which first transfers the visual features of medical images into an abnormality graph, then retrieves text templates based on the abnormalities and their attributes for chest X-ray images.
In \cite{zhang2020radiology}, a pre-constructed graph embedding module (modeled with a graph CNN) on multiple disease findings is utilized to assist the generation of reports. The incorporation of knowledge graph allows for dedicated feature learning
for each disease finding and the relationship modeling between them. Experiments on the publicly accessible dataset (IU-RR) demonstrate the superior performance of the method integrated with the proposed graph module.
\section{Research Challenges and Future Directions}
\label{sec:future_work}
The aforementioned sections reviewed research studies on deep learning models that incorporate medical domain knowledge for various tasks. Although using medical domain knowledge in deep learning models is quite popular, there are still many difficulties about the selection, representation and incorporating method of medical domain knowledge. In the following sections, we summarize challenges and future directions in this area.
\subsection{The Challenges Related to the Identification and Selection of Medical Domain Knowledge}
Identifying medical domain knowledge is not an easy task. Firstly, the experiences of medical doctors are generally subjective and fuzzy. Not all medical doctors can give accurate and objective descriptions on what kinds of experiences they have leveraged to finish a given task. In addition, experiences of medical doctors can vary significantly or even contradictory to each other. Furthermore, medical doctors generally utilize many types of domain knowledge simultaneously.
Finally, currently the medical domain knowledge is identified manually, and there is no existing work on the automatically and comprehensively identifying medical domain knowledge for a given area.
One solution to the automatic identifying medical knowledge is through text mining techniques on the guidelines, books, and medical reports related to different medical areas. Guidelines or books are more robust than individual experiences. Medical reports generally contain specific terms (usually adjectives) that describe the characteristics of tumors. These terms, containing important information to help doctors to make diagnosis, can potentially be beneficial for deep learning models.
Besides the identification of medical domain knowledge, how to select appropriate knowledge to help image analysis tasks is also challenging. It should be noted that a common practice of medical doctors may not be able to help deep learning models because \emph{the domain knowledge might be learned by the deep learning model from training data.} We believe that the knowledge that is not easily learned by a deep learning model can greatly help the model to improve its performance.
\subsection{The Challenges Related to the Representation of Medical Domain Knowledge}
The original domain knowledge of medical doctors is generally in the form of descriptive sentences like `we will focus more on the margin areas of a tumor to determine whether it is benign or malignant', or `we often compare bilateral images to make decision'. How to transform the knowledge into appropriate representations and incorporate it into deep learning models need a careful design.
There are generally four ways to represent a certain type of medical domain knowledge. One is to represent knowledge as patches or highlighted images (as in \cite{tan2019expert}). This is generally used when doctors pay more attention to specific areas. The second approach is to represent knowledge as feature vectors \cite{majtner2016combining}. This way is suitable when the selected domain knowledge can be described as certain features.
The third approach is to represent domain knowledge as extra labels \cite{hussein2017risk,liu2018integrate}, which is suitable for the knowledge in clinical reports or extra feature attributes of diseases. The last approach is to embed medical domain knowledge in network structure design, which is suitable to represent high-level domain knowledge like the training pattern and diagnostic pattern of medical doctors\cite{jimenez2019medical,liu2019from,2020Cross}.
\subsection{The Challenges Related to the Incorporating Methods of Medical Domain Knowledge}
Currently, there are four ways to incorporate medical domain knowledge. The first is to transform the knowledge into certain patches or highlighted images and put them as extra inputs \cite{xie2019knowledge-based}. The second approach is via concatenation \cite{rezaei2019gland}. The domain knowledge are generally transformed into feature vectors and concatenated with those extracted by deep learning models. The third way is the attention mechanism \cite{li2019attention}. The approach is applicable when doctors focus on certain areas of medical images or focus on certain time slots on medical videos. The last one is to learn the domain knowledge by using some specific network structures like generative models \cite{chen2019learning,luo2020shape}.
However, most of the existing works only incorporate a single type of medical domain knowledge, or a few types of medical domain knowledge of the same modality (e.g., a number of hand-crafted features). In practice, experienced medical doctors usually combine different experience in different stages.
There are some researches that simultaneously introduce high-level domain knowledge (e.g., diagnostic pattern, training pattern) and the low-level one (e.g., hand-crafted features, anatomical priors). In particular, the high-level domain knowledge is incorporated as input images, and low-level one is learned by using specific network structures \cite{chen2019learning}. In addition, besides incorporating into network directly, the information from low-level domain knowledge can also be used to design the training orders when combined with the easy-to-hard training pattern \cite{tang2018attention}. We believe that simultaneously incorporating multiple kinds of medical domain knowledge can better help deep learning models in various medical applications.
\subsection{Future Research Directions}
\label{sec:seg_disease}
Besides the above challenges, there are several directions that we feel need further investigation in the future.
\textbf{Domain adaptation}
Domain adaptation is developed to transfer the information from a source domain to a target one. Via techniques like adversarial learning \cite{goodfellow2014generative}, domain adaptation is able to narrow the domain shift between the source domain and the target one in input space \cite{hoffman2017cycada}, feature space \cite{long2016unsupervised,tzeng2017adversarial} and output space \cite{luo2019taking,tsai2018learning}.
It can be naturally adopted to transfer knowledge of one medical dataset to another \cite{li2020dual}, even when they have different imaging modes or belong to different diseases \cite{ghafoorian2017transfer,jiang2018tumor}.
In addition, unsupervised domain adaptation (UDA) is a promising avenue to enhance the performance of deep neural networks on the target domain, using labels only from the source domain. This is especially useful for medical field, as annotating the medical images is quite labor-intensive and the lack of annotations is quite common in medical datasets. Some examples have demonstrated the effectiveness of UDA in disease diagnosis and organ segmentation \cite{chen2018semantic,yang2019dscgans,zhang2020collaborative,liu2020pdam}, but further depth study needs to be implemented in the future.
\textbf{The knowledge graph}
We believe that the knowledge graph \cite{wang2014knowledge}, with the character of embedding different types of knowledge, is a generic and flexible approach to incorporate multi-modal medical domain knowledge. Although rarely used at present, it also shows advantage in medical image analysis tasks, especially in medical report generation \cite{li2019knowledge}. We believe that more types of knowledge graph can be used to represent and learn domain knowledge in medical image analysis tasks.
According to different relationships in graphs, there are three possible types of knowledge graphs can be established. The first knowledge graph reflects the relationship among different kinds of medical domain knowledge with respect to a certain disease. This knowledge graph can help us identify a few key types of knowledge that may help to improve the performance deep learning models. The second type of knowledge graph may reflects the relationship among different diseases. This knowledge graph can help us to find out the potential domain knowledge from other related diseases. The third type one can describe the relationship among medical datasets. These datasets can belong to different diseases and in different imaging modes (e.g., CT, MRI, ultrasound). This type of knowledge graph will help to identify the external datasets that may help to improve the performance of the current deep learning model.
\textbf{The generative models}
The generative models, like GAN and AE, have shown great promise to be applied to incorporate medical domain knowledge into deep learning models, especially for segmentation tasks. GAN has shown its capability to leverage information from extra datasets with different imaging modes (e.g., using a MRI dataset to help segmenting CT images \cite{chen2018semantic,jiang2018tumor}). In addition, GAN is able to learn important features contained in medical images in a weakly or fully unsupervised manner and therefore is quite suitable for medical image analysis.
AE-based models have already achieved a great success in extracting features, especially the shape priors in objects like organs or lesions in a fully unsupervised manner \cite{chen2019learning,luo2020shape}. The features learning by AE can also be easily integrated into the training process of networks.
\textbf{Network architecture search (NAS)}
At last, we mentioned in the previous section that one challenge is to find appropriate network architectures to incorporate medical domain knowledge. We believe one approach to address this problem is the technique of network architecture search (NAS). NAS has demonstrated its capability to automatically find a good network architecture in many computer vision tasks \cite{wistuba2019a} and has a great promise in the medical domain \cite{guo2020organ}. For instance, when some hand-crafted features are used as the domain knowledge, with the help of NAS, a network structure can be identified with the special connections between domain knowledge features and deep features. In addition, instead of designing the feature fusion method (feature-level fusion, decision-level fusion or input-level fusion) for these two kinds of features, the integrating phase and integrating intensity of these two kinds of features can also be determined during the searching process.
\section{Conclusion}
\label{sec:conclusion}
In this paper, we give a comprehensive survey on incorporating medical domain knowledge into deep learning models for various medical image analysis tasks ranging from disease diagnosis, lesion, organ and abnormality detection to lesion and organ segmentation. In addition, some other tasks such as medical image reconstruction, medical image retrieval and medical report generation are also included. For each task, we first introduce different types of medical domain knowledge, and then review some works of introducing domain knowledge into target tasks by using different incorporating methods. From this survey, we can see that with appropriate integrating methods, different kinds of domain knowledge can help deep learning models to better accomplish corresponding tasks.
Besides reviewing current works on incorporating domain knowledge into deep learning models, we also summarize challenges of using medical domain knowledge, and introduce the identification, selection, representation and incorporating method of medical domain knowledge. Finally, we give some future directions of incorporating domain knowledge for medical image analysis tasks.
\section*{Acknowledgments}
This work was supported by the National Natural Science Foundation of China [grant numbers 61976012, 61772060]; the National Key R\&D Program of China [grant number 2017YFB1301100]; and the CERNET Innovation Project [grant number NGII20170315].
\bibliographystyle{IEEEtran}
|
1,116,691,497,567 | arxiv | \section{Introduction}
It is well known that negative refraction takes place at an interface
between a usual isotropic medium (vacuum, for example) and a
material with negative permittivity and permeability (called Veselago medium,
double-negative material, or backward-wave medium).
Recently, a lot of efforts have been devoted to realization of
backward-wave materials, because the negative refraction effect offers a possibility to
create super-resolution imaging devices (among other potential applications).
The known realizations are based on the use of metal inclusions of
various shapes, especially split rings, needed to realize negative permeability.
Creation of strong artificial magnetic response, especially in the optical region, is
a big challenge, which we have to face if we want to realize negative
refraction and superlens for optical applications. In view of
this problem, various alternative approaches to create backward-wave media
have been considered in the literature. In particular, such effects can exists in
more complex materials --- chiral media --- and, what is the key advantage,
backward-wave regime can be in principle realized even if the medium has very weak or
no magnetic properties. Thus, it appears that using chiral media one could
realize negative refraction in the optical region without the need to
create artificial magnetic materials operational in that frequency range.
The physics of the effect of backward waves in chiral media is
very simple: The propagation constants of two eigenwaves in
isotropic chiral media equal $\beta=(n \pm \kappa)k_0$, where
$n=\sqrt{\epsilon\mu}$ is the usual refractive index, $\kappa$ is
the chirality parameter, and $k_0$ is the free-space wavenumber
(see, e.g., \cite{chibi}). Near the resonance of electric or /and
magnetic susceptibilities the refractive index $n$ can become
smaller than the chirality parameter $\kappa$. It means that one
of the two eigenwaves is a backward wave, because its phase
velocity is negative but the energy transport velocity is
positive. At an interface between a usual isotropic material and
such medium negative refraction takes place for this polarization
(waves of the other polarization refract positively). The earliest
publication where a possibility for such effects was established
was probably paper \cite{bokut}. In that paper, a spiral model for
a chiral optical molecule and the Lorentz dispersion model for the
permittivity was used, and a formula for the frequency range of
negative refraction was derived. Single-phase chiral substances
were considered, and magnetic properties of the medium were
neglected. Much more recently, backward waves and negative
refraction were studied in \cite{sergei1}, with the emphasis on
the limiting case when both $\epsilon$ and $\mu$ tend to zero
(this medium was called {\itshape chiral nihility}. The simplified
antenna model for chiral inclusions \cite{antenna} was used to
estimate the material parameters of mixtures of metal helices with
the desired parameters. Possibility for backward-wave regime in
chiral materials was also indicated in conference presentation
\cite{Dakhcha1}. In paper \cite{pendry1}, instead of chiral
nihility, a two-phase mixture was introduced. One phase is a
non-dispersive chiral material, and another phase consists of
resonant dipole particles. It has been assumed that when the
dipoles resonate, $\sqrt{\epsilon}$ becomes smaller than $\kappa$,
and the material can support backward waves. These results show
that the use of chirality is a very exciting new opportunity to
realize negative refraction and related effects in the optical
region in effectively uniform media (the characteristic dimensions
in the material can be much smaller than the wavelength).
Chiral media have been very intensively studied in the past years, see e.g.
\cite{chibi,serdyukov1,lakhtakia2,priou2,mariotte1}.
However, it is interesting, that although the possibility for this effect was
published in the former Soviet Union \cite{bokut}, it was not known
in the West until very recently. The authors of monograph \cite{chibi} thought that both
eigenwaves in chiral media should be forward waves, and formulated a corresponding
restriction for the material parameters [See Eq.~(2.176) on page
51]. Recent studies have shown, that one of the eigenwaves in a
chiral nihility is indeed a backward wave \cite{sergei1}.
In this paper we study eigenwaves propagating in single- and dual-phase
chiral mixtures accurately taking into
account resonant properties of chiral particles and of resonating
electric dipoles particles and electromagnetic interaction between phases.
We identify the effects that can lead to realization of backward-wave
regime and negative refraction using chiral composites.
\section{Backward waves in chiral media and enhancement of evanescent fields}
The bi-isotropic constitutive relations read
\begin{equation}\label{bi}
\left(\begin{array}{c} {\mathbf D} \\ {\mathbf B} \end{array}\right) = \left(\begin{array}{cc} \epsilon & \xi \\ \zeta & \mu \end{array}\right) \left(\begin{array}{c}
{\mathbf E} \\ {\mathbf H} \end{array}\right) = {\mathsf M} \cdot \left(\begin{array}{c} {\mathbf E} \\ {\mathbf H}
\end{array}\right)
\end{equation}
where the material matrix $\mathsf M$ contains the four scalars
$\epsilon,\xi,\zeta,\mu$.
Here we will consider only isotropic reciprocal media, in which
the following condition holds \cite{chibi}:
\begin{equation} \zeta=-\xi=j\kappa\sqrt{\epsilon_0\mu_0}\l{rec}\end{equation}
Such media are called chiral, and $\kappa$ is the chirality parameter. In lossless
media $\kappa$ is a real number.
Considering electromagnetic field in
homogeneous chiral regions it is very convenient to introduce new
field variables $\_E_\pm$ and $\_H_\pm$ that are the following
linear combinations of the fields:
\begin{equation} \_E_+={1\over 2}(\_E-j\eta\_H),\qquad\_E_-={1\over
2}(\_E+j\eta\_H)\end{equation}
\begin{equation} \_H_+={1\over 2}\left(\_H+{j\over \eta}\_E\right),\qquad\_H_-=
\left(\_H-{j\over \eta}\_E\right)\end{equation}
where $\eta=\sqrt{\mu/\epsilon}$. Vectors $\_E_\pm$ are called wavefield vectors.
The advantage of introducing the new variables comes from the fact that these
new vectors satisfy the Maxwell equations in equivalent isotropic
non-chiral media. This allows to use the known solutions for fields in
simple isotropic medium to construct solutions for fields in
chiral media.
In uniform regions the wavefield components "see" equivalent simple
isotropic media with the equivalent parameters
\begin{equation} \epsilon_\pm=\epsilon(1\pm \kappa_r),\qquad \mu_\pm=\mu(1\pm \kappa_r)\end{equation}
where $\kappa_r=\kappa/\sqrt{\epsilon\mu})$ \cite{chibi}.
The wavenumbers of the two eigenwaves
read $ k_\pm=k(1\pm \kappa_r)$,
or the refractive indices of the two equivalent media are $n_\pm=\sqrt{\epsilon\mu}
\pm \kappa$.
\begin{figure}[h]
\includegraphics[width=0.47\textwidth]{Veselago_lens3.eps}
\includegraphics[width=0.52\textwidth]{Plasmons3.eps}
\caption{An illustration of negative refraction and subwavelength
focusing by a chiral slab.}
\label{illustration}
\end{figure}
Suppose that in some frequency region
\begin{equation} {\rm Re\mit}\{\kappa\} > {\rm Re\mit}\left\{\sqrt{\epsilon\mu}\right\} \l{condition}\end{equation}
In this case
one of the eigenmodes is a backward wave.
Actually the two eigenmodes $\_E_+,\_H_+$ and $\_E_-,\_H_-$ are
plane right- and left-circularly polarized waves. We see that for
one of these two polarizations a slab of chiral material [when
\r{condition} is satisfied] behaves as a slab of an isotropic
medium with negative effective parameters (Veselago medium). The
known phenomena of negative refraction and subwavelength focusing
will take place for waves of this polarization, see an
illustration in Figure~\ref{illustration} (These pictures
have been drawn by S. Maslovski.). Numerical simulations of
focusing effect have been published in \cite{sailing_focusing}.
The condition for creation of a perfect image for one of the two
circular polarizations read
\begin{equation} \epsilon_-=\epsilon(1- \kappa_r)=-\epsilon_0,\qquad \mu_-=\mu(1- \kappa_r)=-\mu_0 \qquad
(n_-=-1)\end{equation}
\section{Mixture of chiral and dipole particles}
In \cite{pendry1} it was assumed that once knowing the effective
permittivity of a mixture of dipoles and the effective $\kappa$ of
a chiral mixture, the propagation constants of wave in a composite
material which contains both types of inclusions
could be calculated by a simple substitution to
\begin{equation}
k_\pm=(\sqrt{\epsilon_{\rm eff}} \sqrt{\mu_{\rm eff}}\pm
\kappa_{\rm eff})k_0 \label{beta}
\end{equation}
where $k_0$ is the wave number in free space.
However, this approach does not take into
account electromagnetic coupling between dipoles and helixes,
which is rather strong near resonant frequencies of at least one
phase. In this section, we introduce effective material parameters
which are calculated taking this coupling into account. We assume
that the mixture is built up by randomly distributed dipoles and
helices in a uniform matrix with the permittivity of air. Another
possibility is to construct a regular lattice of symmetrically
positioned inclusions, so that the overall response is
isotropic. In both cases the scattering losses from
individual inclusions are supposed to be compensated, and the material
can have rather low loss factor, determined only by dissipation in
particles. The mixture is assumed to be dilute, in other words particles
are not in each other near field, and the Clausius-Mossotti model
can be used.
In the following formulation, let us use normalized field
quantities: in terms of the ``ordinary'' electric and magnetic
fields and flux densities $\cal E,H,D,B$ with units V/m, A/m,
As/m$^2$, Vs/m$^2$, respectively, we deal with fields and fluxes
that are renormalized in order to have homogeneous units in each
of them:
\begin{equation}
{\mathbf E}=\sqrt{\epsilon_0}\,{\cal E}, \quad {\mathbf H}=\sqrt{\mu_0}\,{\cal
H}, \quad {\mathbf D}=\frac{{\cal D}}{\sqrt{\epsilon_0}}, \quad {\mathbf
B}=\frac{{\cal B}}{\sqrt{\mu_0}}
\end{equation}
with the free-space parameters $\epsilon_0,\mu_0$. This leaves the
materials parameters in Equation~(\ref{bi}) dimensionless. Also,
all four renormalized field quantities carry the dimension of
square root of energy density: $\sqrt{{\rm VAs/m}^3} = \sqrt{\rm
J/m^3}$.
\subsection{Effective parameters}
\label{mixing}
Consider a mixture where randomly oriented helices and randomly
oriented dipole resonators float in neutral matrix (air). Let the
polarizability matrix contribution of one element in the random
distribution of helices be
\begin{equation}
{\mathsf A} = \left(\begin{array}{cc} \alpha_{ee} & \alpha_{em} \\ \alpha_{me} & \alpha_{mm} \end{array}\right)
\end{equation}
where the averaging over spatial orientation has been included, in
other words,
\begin{equation}
\alpha_{ee} = \frac{1}{3} (\alpha_{ee,\rm x} +\alpha_{ee,\rm y} +\alpha_{ee,\rm
z}), \quad \alpha_{em} = \ldots
\end{equation}
Note also that we can assume reciprocity, in other words
\begin{equation}
\alpha_{me} = -\alpha_{em}
\end{equation}
[see Eq.~\r{rec}].
Of course, for lossless helices, $\alpha_{me}$ and $\alpha_{em}$ are pure
imaginary.
Let the averaged polarizability matrix of the dipole resonators be
\begin{equation}
{\mathsf B} = \left(\begin{array}{cc} \beta_{ee} & 0 \\ 0 & 0 \end{array}\right)
\end{equation}
with the similar averaging included. Here we have assumed that in
addition to the vanishing magnetoelectric contribution, the
magnetic response can be neglected. (This assumption is not
necessary but makes the point easier to demonstrate that we can
have different band behaviors for the macroscopic permittivity and
chirality responses.)
Using the
methods described in
\cite{ari1,serdyukov1}, the Maxwell Garnett formula in a matrix form for the effective
parameters of the mixture can be written in the following way:
\begin{equation}
({\mathsf M}_{\rm eff} - {\mathsf I})\cdot ({\mathsf M}_{\rm eff}
+ 2 {\mathsf I})^{-1} = \frac{1}{3} (n_1{\mathsf A} + n_2 {\mathsf
B})
\end{equation}
where $n_1$ and $n_2$ are the number densities of the helices and
dipoles, respectively. The explicit formula for the effective
parameter matrix
\begin{equation}
{\mathsf M}_{\rm eff} = \left(\begin{array}{cc} \epsilon_{\rm eff} & \xi_{\rm eff} \\
\zeta_{\rm eff} & \mu_{\rm eff} \end{array}\right)
\end{equation}
reads
\begin{equation}
{\mathsf M}_{\rm eff} = {\mathsf I } + \left( {\mathsf I} -
\frac{1}{3}(n_1{\mathsf A} + n_2 {\mathsf B})\right)^{-1} \cdot
(n_1{\mathsf A} + n_2 {\mathsf B})
\end{equation}
The resulting parameters read
\begin{eqnarray}
\epsilon_{\rm eff} & = & 1 + \frac{3n_1\alpha_{ee} + 3n_2\beta_{ee}
- n_1^2 (\alpha_{ee}\alpha_{mm}+\alpha_{em}^2) - n_1\alpha_{mm} n_2\beta_{ee}}{3\cdot {\rm DEN}} \label{eps} \\
\xi_{\rm eff} & = & \frac{n_1\alpha_{em}}{{\rm DEN}} \label{xi}\\
\zeta_{\rm eff} & = & -\frac{n_1\alpha_{em}}{{\rm DEN}} \label{zeta}\\
\mu_{\rm eff} & = & 1 + \frac{3n_1\alpha_{mm} - n_1^2
(\alpha_{ee}\alpha_{mm}+\alpha_{em}^2) - n_1\alpha_{mm} n_2\beta_{ee}}{3 \cdot{\rm
DEN}} \label{mu}
\end{eqnarray}
where the common denominator is
\begin{equation}
{\rm DEN} = 1 - \frac{n_1\alpha_{ee}}{3} - \frac{n_1\alpha_{mm}}{3} +
\frac{n_1^2(\alpha_{ee}\alpha_{mm}+\alpha_{em}^2)}{9} -
\frac{n_2\beta_{ee}}{3} + \frac{n_1\alpha_{mm} n_2\beta_{ee}}{9}
\end{equation}
Obviously the material parameters depends on polarizabilities in
a very complicate way, such that every material parameter depends
on all the polarizabilies of both types of inclusions.
\subsection{Polarizabilities of chiral inclusions}
Analytical models of polarizabilities of chiral inclusions are
well known in the literature, see e.g. \cite{antenna,serdyukov1,priou2,mariotte1}.
All the polarizabilies (electric, magnetic, and magneto-electric) have
the resonant behaviour with the same resonant frequency
which corresponds to the resonance of the whole particle.
At low frequencies (well below the resonance), the electric polarizability tends to
a constant, chirality parameter tends to zero linearly with the frequency, and
the magnetic polarizability is proportional to frequency squared:
\begin{equation} \alpha_{em}\sim j(ka) \alpha_{ee},\qquad \alpha_{mm}\sim -j(ka) \alpha_{em}\sim (ka)^2
\alpha_{ee}\l{hie}\end{equation}
where $a$ is the characteristic particle size.
This brings us to the following generic model of the
frequency dispersion of the polarizabilities:
\begin{equation}
\alpha_{em}={jAx\over{1-x^2+j\alpha x}},\qquad
\alpha_{ee}={B\over{1-x^2+j\alpha x}},\qquad
\alpha_{mm}={C x^2\over{1-x^2+j\alpha x}}
\end{equation}
where $x=\omega/\omega_0$ is the frequency
normalized to the resonant frequency if the inclusion $\omega_0$
and $A,B$, and $C$ are constants.
In the optical literature, so called ``hierarchy of
polarizabilities" has been established for polarizabilities of
optical molecules \cite{raab}. According to that,
the strongest response is the electric dipole response (corresponding
to the electric polarizability $\alpha_{ee}$, the chirality parameter is
weaker, and the magnetic dipole response is the weakest of the
three considered here. This rule holds for usual molecules in the
optical region, when the molecules are small as compared with the
wavelength and spiral shapes are not very pronounced.
\begin{figure}[h]
\centering
\includegraphics[width=0.2\textwidth]{chiral_particle.eps}
\caption{The shape of the canonical helix, an illustration
for the antenna model of chiral particles \cite{antenna}.}
\label{helix}
\end{figure}
For our purpose we need a more general and quantitive model, that would describe
particles of arbitrary relations between the length of the
spiral and its diameter. To determine the relations between the
polarizabilities we make use of the analytical model \cite{antenna}
and calculate the ratios between the polarizabilities of small
canonical helices shown in Figure~\ref{helix}. The result is
\begin{equation} \alpha_{em}=j\pi {a\over l}ka \, \alpha_{ee} ,\qquad \alpha_{mm}=
\left(\pi {a\over l}ka\right)^2\, \alpha_{ee} \l{good}\end{equation}
Here, $a$ is the loop radius and $l$ is the length of the dipole arm.
If $ka\ll 1$ and $\pi a/l\approx 1$, we come again to the simple
relations \r{hie}. However, for artificial chiral materials where
the proportions between the particle dimensions can be chosen at will,
and the loop radius is not always very small compared to the wavelength,
we should use the more general relations \r{good}.
Thus, we come to the following model for chiral particle polarizabilities:
\begin{equation}
\alpha_{em}={jx\over{1-x^2+j\alpha x}},\qquad
\alpha_{ee}={\nu^{-1}\over{1-x^2+j\alpha x}},\qquad
\alpha_{mm}={\nu x^2\over{1-x^2+j\alpha x}}
\end{equation}
where we have denoted by $\nu=\pi (a/l) (ka)|_{\omega=\omega_0}$ the
coefficient which depends on the electrical diameter of the
spiral at resonance and on the ``form-factor'' (ratio of the
diameter and the length
of the helix). For natural optically active materials $\nu\ll 1$, but
for artificial chiral materials its value can be of the order of unity.
For the electric dipole inclusions we adopt the conventional
Lorentz dispersion formula, writing
\begin{equation}
\beta_{ee}={1\over{x_0^2-x^2+j\alpha' x}}
\end{equation}
Here $x_0$ is the ratio between the resonant
frequency of the dipole particles and the resonant frequency of the helices.
In all expressions for the polarizabilities we omit constant (independent
from the frequency) amplitude coefficients, since they can be incorporated
into the number densities of chiral and dipole inclusions.
These models apply to inclusions whose dimensions are considerably
smaller than the wavelength. If for example the helix radius becomes
comparable to the wavelength, more complicated analytical and
numerical models should be
used (e.g. \cite{antenna,mariotte1}), but for the present purpose
we will not need more advanced models.
\section{Numerical examples}
\subsection{The role of resonant permittivity background}
\begin{figure}[h]
\centering
\includegraphics[width=0.50\linewidth]{independent_fractions}
\includegraphics[width=0.48\linewidth]{parameters_independent_fractions}
\caption{Dispersion curves and effective material parameters
for the Pendry model of independent fractions.
$\kappa=0.3\omega/\omega_0$. $n_2=0.025, \alpha_2=0.001$, resonant
frequency of the dielectric phase is $0.3\omega_0$.}
\label{independent_fractions}
\end{figure}
Let us first calculate some numerical examples for using the simple model used in
paper \cite{pendry1}, where the chiral material was assumed to have no
frequency dispersion of the chirality parameter and the interaction between
two fractions was neglected. This approximation
corresponds to the frequency region well below the resonant frequency of
chiral inclusions (chirality parameter is still proportional to the frequency,
which we assume in our calculations,
but this dependence is not of principal importance for our problem).
Figure~\ref{independent_fractions} illustrates typical dispersion in this frequency range
for a material model of paper \cite{pendry1}.
In this example, we have taken a very high chirality parameter,
to highlight the effect of chirality (in real composites, the effect is usually
pretty weak at frequencies well below the resonance). The chiral fraction
has been assumed to be lossless.
We observe that near the resonant frequency of dipole inclusions there
is a very narrow backward-wave band. This effect happens when the permittivity
is very small but still positive. Closer to the resonant frequency of dipoles there is a
wide stop band.
However, the model used in this calculation is not realistic, because
near the resonant frequencies all the material parameters resonate due to
strong field interactions between particles of two phases (see Section~\ref{mixing}).
Next, we calculate an example using the model of the mixture and
the particles polarizabilities introduced above.
\begin{figure}[h]
\centering
\includegraphics[width=0.50\linewidth]{mixture}
\includegraphics[width=0.48\linewidth]{parameters_mixture}
\caption{Dispersion curves for a chiral medium with resonant electric dipole inclusions
and its effective material parameters.
The frequency is normalized to the
resonant frequency $\omega_0$.}
\label{mixture}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=0.6\linewidth]{no_backward}
\caption{Negative and imaginary parts of the refraction indices of
two eigenmodes in the same medium as in Figure~\ref{mixture}.}
\label{indices}
\end{figure}
The resonant frequency of the electric dipole fraction is considerably lower than the resonant
frequency of the helices of the chiral mixture ($0.3\omega_0$).
The following parameters of the medium have been assumed:
$n_1=0.1$, loss factor $\alpha=0.001$, $n_2=0.05$, loss factor of the electric fraction
particles $\alpha'=0$. Parameter $\nu=0.1$. In order to
achieve a backward-wave effect, we assumed a very high
concentration of chiral particles (in naturally available
optical chiral materials the chirality parameter is usually orders of
magnitude smaller) and assume that the electric dipole particles are
lossless. The results are shown in Figure~\ref{mixture}.
We see that even under the above assumptions, there is actually no backward-wave band.
This is apparent from the plot of the refractive indices of the
two eigenmodes. The narrow frequency band where the real part of one of the indices
becomes negative is already inside the stop band.
Near the resonance of the electric phase all the material parameters
also resonate. The resonant increase of the chirality parameter looks like a
possibility to satisfy the backward-wave condition \r{condition},
because in the frequency band where the chirality parameter increases, the permittivity
takes small
values. However, this does not lead to a backward-wave regime.
The problem is that in these two-phase composites
with resonant electric dipoles embedded in a natural optically
active material the effective magnetic permeability
resonates very weakly. In the resonant band of the permittivity
the effective permeability still stays near unity,
and there is a wide stop band near the resonance of the
electric phase. We have also studied the situation when the
magnetic response of helical fraction is stronger, taking the value of
parameter $\nu$ to be equal unity. This does not change the
above conclusion, because the magnetic response considerably increases
only near the resonance of the helix.
\subsection{Interplay of the resonances of helices and dipoles}
To achieve strong chiral response in practice, the working
frequency should be close to the resonant frequency of chiral inclusions.
Interplay of two phases becomes quite strong in this situation, and the
mixture shows rather complicated frequency response. This is illustrated by
numerical examples in this section.
\begin{figure}[h]
\centering
\includegraphics[width=0.48\linewidth]{09_pendry_parameters}
\includegraphics[width=0.50\linewidth]{09_pendry}
\caption{Effective material parameters and refraction indices for a mixture of dipoles and
helixes with the parameters $n_1=0.02$, $n_2=0.05$,
$\alpha=\alpha'=0.001$, and $\nu =1$. The resonant frequency of
electric dipoles equals $0.9\omega_0$.}
\label{fig_param}
\end{figure}
An example of frequency dependence of effective material parameters and refraction
indices is shown in
Figure~\ref{fig_param}. The two phases interact strongly and all the
material parameters show resonant response near the resonant
frequencies of helices and dipoles. Although we assumed that helices
have strong magnetic properties ($\nu=1$), we see that predicted in
\cite{pendry1} backward-wave frequency band
near the resonance of dipoles falls into a stop band of the mixture.
Close
to the resonance frequency of the effective permittivity,
$\mu_{\rm eff}$ is nearly unity and $\kappa_{\rm eff}$ also has a
resonance because of coupling between dipoles and helices.
It is difficult to design a backward-wave material
to operate near the resonance of background permittivity
because the resonances of the effective permittivity and
permeability can not be tailored separately.
On the other hand, we observe that there indeed exists predicted in
\cite{bokut,sergei1} a backward-propagation band
near the resonant frequency of helices.
\subsection{Negative refraction in resonant chiral composites}
\begin{figure}[h]
\includegraphics[width=0.49\linewidth]{1_res_hierar_betas_n.eps}
\includegraphics[width=0.49\linewidth]{1_res_hierar_eps_kappa.eps}
\caption{Refractive indices and effective parameters of
a chiral material with the parameters $n_1=0.02$, $\alpha=0.001$, $\nu =0.1$.}
\label{chiral_resonant}
\end{figure}
The results of the previous section indicate that the most
appropriate approach to realize backward-wave materials and superlenses
with the use of chiral materials is the use of only chiral inclusions.
In this section we give two examples of such mixtures to
illustrate the design criteria for these materials.
Figure~\ref{chiral_resonant} gives an example of a resonant chiral
material with the parameters that are probably possible to
achieve using natural materials (parameter $\nu$ is much smaller than unity
and magnetic properties are weak, even close to the particle resonance).
This is the case first considered in \cite{bokut}. The results
shown in
Figure~\ref{chiral_resonant} lead to the conclusion that also here
there is no propagating backward-wave regime --- when the
real part of the refraction
index becomes negative, its imaginary part is very large.
\begin{figure}[h]
\includegraphics[width=0.49\linewidth]{1_res_equal_betas_n.eps}
\includegraphics[width=0.49\linewidth]{1_res_equal_eps_kappa.eps}
\caption{Refractive indices and effective parameters of
a ``chiral nihility" composite material with the parameters $n_1=0.02$, $\alpha=0.001$, $\nu=1$.}
\label{nihility}
\end{figure}
The last Figure \ref{nihility} illustrates the situation considered first in
\cite{sergei1}. In this case we chose a large value of parameter $\nu=1$.
The effective permeability is now much stronger due to a larger ratio of the
helix diameter to its length $a/l$ and (or) the electric diameter of the
helix $ka$. As a result, the permittivity and permeability
follow nearly the same dispersion laws, and the real part of their product
remains positive even when the permittivity goes into negative.
In the previous example there is a stop band in this frequency area,
but in the present situation there is a propagating backward-wave regime like
in the usual double-negative materials based on for example split rings and
wire lattices. The role of chirality parameter here is two-fold. First,
is the chirality parameter is non-zero, this backward-wave band is wider,
since its limits are given by a weaker condition \r{condition} than the
usual ${\rm Re}\sqrt{\epsilon\mu}<0$. Second, the possible physical effects in
this new material are reacher than in double-negative materials (see \cite{sergei1}).
However, such materials do not exist in nature and have to be manufactured
artificially as {\em metamaterials}. This is reasonably easy for
microwave frequencies, and there are many reports in the literature
about artificial chiral media with strong resonant response (although
we are not aware about any realization of the backward-wave regime).
For the optical frequency range, the small required particle size
makes the task very challenging, but we hope that the quickly developing
nano-technologies will make it reality.
\section{Conclusions}
Possibilities to achieve negative refraction and enhancement of
evanescent fields in a ``perfect lens'' using chiral materials
have been theoretically explored. The two known variants have been considered:
the use of a mixture of
helixes and resonant dipoles \cite{pendry1} and the use of a composite of
only helices \cite{bokut,sergei1}.
Mixing equations for the effective material parameters, which take
into account the coupling between dipoles and helixes has been given.
Numerical examples have been calculated with the use of an introduced
general dispersion law for the polarizabilities of helical particles.
This model gives a possibility to study particles with different
shapes and electrical sizes.
It has been shown that once the coupling between helices and dipoles
is taken into account, there
is a stop band in the frequency region where negative refraction
was expected to occur. However, the negative refraction can still occur in
a ``chiral nihility'' materials as has been suggested in \cite{sergei1}.
Negative refraction could exist at frequencies higher than the resonant frequency of
chiral particles. The role of chirality is seen in widening the
backward-wave frequency band and in opening a way to realize new
physical effects and possibly create new microwave and optical devices.
Realization of these new media for optical applications requires
manufacturing of chiral inclusions with controllable shape
that would exhibit resonant response in the optical regime.
\section{Introduction}
This is the author's guide to \revtex~4, the preferred submission
format for all APS journals. This guide is intended to be a concise
introduction to \revtex~4. The documentation has been separated out
into smaller units to make it easier to locate essential
information.
The following documentation is also part of the APS \revtex~4
distribution. Updated versions of these will be maintained at
the \revtex~4 homepage located at \url{http://publish.aps.org/revtex4/}.
\begin{itemize}
\item \textit{APS Compuscript Guide for \revtex~4}
\item \textit{\revtex~4 Command and Options Summary}
\item \textit{\revtex~4 Bib\TeX\ Guide}
\item \textit{Differences between \revtex~4 and \revtex~3}
\end{itemize}
This guide assumes a working \revtex~4
installation. Please see the installation guide included with the
distribution.
The \revtex\ system for \LaTeX\ began its development in 1986 and has
gone through three major revisions since then. All versions prior to
\revtex~4 were based on \LaTeX2.09 and, until now, \revtex\ did not
keep pace with the advances of the \LaTeX\ community and thus became
inconvenient to work with. \revtex~4 is designed to remedy this by
incorporating the following design goals:
\begin{itemize}
\item
Make \revtex\ fully compatible with \LaTeXe; it is now a \LaTeXe\
document class, similar in function to the standard
\classname{article} class.
\item
Rely on standard \LaTeXe\ packages for common tasks, e.g,
\classname{graphicx},
\classname{color}, and
\classname{hyperref}.
\item
Add or improve macros to support translation to tagged formats such as
XML and SGML. This added markup will be key to enhancing the
peer-review process and lowering production costs.
\item
Provide a closer approximation to the typesetting style used in
\emph{Physical Review}.
\item
Incorporate new features, such as hypertext, to make \revtex\ a
convenient and desirable e-print format.
\item
Relax the restrictions in \revtex\ that had only been necessary for
typesetting journal camera-ready copy.
\end{itemize}
To meet these goals, \revtex~4 is a complete rewrite with an emphasis
on maintainability so that it will be easier to provide enhancements.
The \revtex~4 distribution includes both a template
(\file{template.aps}) and a sample document (\file{apssamp.tex}).
The template is a good starting point for a manuscript. In the
following sections are instructions that should be sufficient for
creating a paper using \revtex~4.
\subsection{Submitting to APS Journals}
Authors using \revtex~4 to prepare a manuscript for submission to
\textit{Physical Review} or \textit{Reviews of Modern Physics}
must also read the companion document \textit{APS Compuscript Guide
for \revtex~4}
distributed with \revtex\ and follow the guidelines detailed there.
Further information about the compuscript program of the American
Physical Society may be found at \url{http://publish.aps.org/ESUB/}.
\subsection{Contact Information}\label{sec:resources}%
Any bugs, problems, or inconsistencies should reported to
\revtex\ support at \[email protected]+.
Reports should include information on the error and a \textit{small}
sample document that manifests the problem if possible (please don't
send large files!).
\section{Some \LaTeXe\ Basics}
A primary design goal of \revtex~4 was to make it as compatible with
standard \LaTeXe\ as possible so that authors may take advantage of all
that \LaTeXe\ offers. In keeping with this goal, much of the special
formatting that was built in to earlier versions of \revtex\ is now
accomplished through standard \LaTeXe\ macros or packages. The books
in the bibliography provide extensive coverage of all topics
pertaining to preparing documents under \LaTeXe. They are highly recommended.
To accomplish its goals, \revtex~4 must sometimes patch the underlying
\LaTeX\ kernel. This means that \revtex~4 requires a fairly recent version of
\LaTeXe. Versions prior to 1996/12/01 may not work
correctly. \revtex~4 will be maintained to be compatible with future
versions of \LaTeXe.
\subsection{Useful \LaTeXe\ Markup}
\LaTeXe\ markup is the preferred way to accomplish many basic tasks.
\subsubsection{Fonts}
Because \revtex~4 is based upon \LaTeXe, it inherits all of the
macros used for controlling fonts. Of particular importance are the
\LaTeXe\ macros \cmd{\textit}, \cmd{\textbf}, \cmd{\texttt} for changing to
an italic, bold, or typewriter font respectively. One should always
use these macros rather than the lower-level \TeX\ macros \cmd{\it},
\cmd{\bf}, and \cmd{\tt}. The \LaTeXe\ macros offer
improvements such as better italic correction and scaling in super-
and subscripts for example. Table~\ref{tab:fonts}
summarizes the font selection commands in \LaTeXe.
\begin{table}
\caption{\label{tab:fonts}\LaTeXe\ font commands}
\begin{ruledtabular}
\begin{tabular}{ll}
\multicolumn{2}{c}{\textbf{Text Fonts}}\\
\textbf{Font command} & \textbf{Explanation} \\
\cmd\textit\marg{text} & Italics\\
\cmd\textbf\marg{text} & Boldface\\
\cmd\texttt\marg{text} & Typewriter\\
\cmd\textrm\marg{text} & Roman\\
\cmd\textsl\marg{text} & Slanted\\
\cmd\textsf\marg{text} & Sans Serif\\
\cmd\textsc\marg{text} & Small Caps\\
\cmd\textmd\marg{text} & Medium Series\\
\cmd\textnormal\marg{text} & Normal Series\\
\cmd\textup\marg{text} & Upright Series\\
&\\
\multicolumn{2}{c}{\textbf{Math Fonts}}\\
\cmd\mathit\marg{text} & Math Italics\\
\cmd\mathbf\marg{text} & Math Boldface\\
\cmd\mathtt\marg{text} & Math Typewriter\\
\cmd\mathsf\marg{text} & Math Sans Serif\\
\cmd\mathcal\marg{text} & Calligraphic\\
\cmd\mathnormal\marg{text} & Math Normal\\
\cmd\bm\marg{text}& Bold math for Greek letters\\
& and other symbols\\
\cmd\mathfrak\marg{text}\footnotemark[1] & Fraktur\\
\cmd\mathbb\marg{text}\footnotemark[1] & Blackboard Bold\\
\end{tabular}
\end{ruledtabular}
\footnotetext[1]{Requires \classname{amsfonts} or \classname{amssymb} class option}
\end{table}
\subsubsection{User-defined macros}
\LaTeXe\ provides several macros that enable users to easily create new
macros for use in their manuscripts:
\begin{itemize}
\footnotesize
\item \cmd\newcommand\marg{\\command}\oarg{narg}\oarg{opt}\marg{def}
\item \cmd\newcommand\verb+*+\marg{\\command}\oarg{narg}\oarg{opt}\marg{def}
\item \cmd\renewcommand\marg{\\command}\oarg{narg}\oarg{opt}\marg{def}
\item \cmd\renewcommand\verb+*+\marg{\\command}\oarg{narg}\oarg{opt}\marg{def}
\item \cmd\providecommand\marg{\\command}\oarg{narg}\oarg{opt}\marg{def}
\item \cmd\providecommand\verb+*+\marg{\\command}\oarg{narg}\oarg{opt}\marg{def}
\end{itemize}
Here \meta{\\command} is the name of the macro being defined,
\meta{narg} is the number of arguments the macro takes,
\meta{opt} are optional default values for the arguments, and
\meta{def} is the actually macro definiton. \cmd\newcommand\ creates a
new macro, \cmd\renewcommand\ redefines a previously defined macro,
and \cmd\providecommand\ will define a macro only if it hasn't
been defined previously. The *-ed versions are an optimization that
indicates that the macro arguments will always be ``short'' arguments. This is
almost always the case, so the *-ed versions should be used whenver
possible.
The use of these macros is preferred over using plain \TeX's low-level
macros such as
\cmd\def{},\cmd\edef{}, and \cmd\gdef{}. APS authors must follow the
\textit{APS Compuscript Guide for \revtex~4} when defining macros.
\subsubsection{Symbols}
\LaTeXe\ has added some convenient commands for some special symbols
and effects. These are summarized in Table~\ref{tab:special}. See
\cite{Guide} for details.
\begin{table}
\caption{\label{tab:special}\LaTeXe\ commands for special symbols and effects}
\begin{ruledtabular}
\begin{tabular}{lc}
Command & Symbol/Effect\\
\cmd\textemdash & \textemdash\\
\cmd\textendash & \textendash\\
\cmd\textexclamdown & \textexclamdown\\
\cmd\textquestiondown & \textquestiondown\\
\cmd\textquotedblleft & \textquotedblleft\\
\cmd\textquotedblright & \textquotedblright\\
\cmd\textquoteleft & \textquoteleft\\
\cmd\textquoteright & \textquoteright\\
\cmd\textbullet & \textbullet\\
\cmd\textperiodcentered & \textperiodcentered\\
\cmd\textvisiblespace & \textvisiblespace\\
\cmd\textcompworkmark & Break a ligature\\
\cmd\textcircled\marg{char} & Circle a character\\
\end{tabular}
\end{ruledtabular}
\end{table}
\LaTeXe\ also removed some symbols that were previously automatically
available in \LaTeX 2.09. These symbols are now contained in a
separate package \classname{latexsym}. To use these symbols, include
the package using:
\begin{verbatim}
\usepackage{latexsym}
\end{verbatim}
\subsection{Using \LaTeXe\ packages with \revtex}\label{sec:usepackage}%
Many \LaTeXe\ packages are available, for instance, on CTAN at
\url{ftp://ctan.tug.org/tex-archive/macros/latex/required/}
and at
\url{ftp://ctan.tug.org/tex-archive/macros/latex/contrib/}
or may be available on other distribution media, such as the \TeX\
Live CD-ROM \url{http://www.tug.org/texlive/}. Some of these packages
are automatically loaded by \revtex~4 when certain class options are
invoked and are, thus, ``required''. They will either be distributed
with \revtex\ or are already included with a standard \LaTeXe\
distribution.
Required packages are automatically loaded by \revtex\ on an as-needed
basis. Other packages should be loaded using the
\cmd\usepackage\ command. To load the
\classname{hyperref} package, the document preamble might look like:
\begin{verbatim}
\documentclass{revtex}
\usepackage{hyperref}
\end{verbatim}
Some common (and very useful) \LaTeXe\ packages are \textit{a priori}
important enough that \revtex~4 has been designed to be specifically
compatible with them.
A bug stemming from the use of one of these packages in
conjunction with any of the APS journals may be reported by contacting
\revtex\ support.
\begin{description}
\item[\textbf{AMS packages}] \revtex~4 is compatible with and depends
upon the AMS packages
\classname{amsfonts},
\classname{amssymb}, and
\classname{amsmath}. In fact, \revtex~4 requires use of these packages
to accomplish some common tasks. See Section~\ref{sec:math} for more.
\revtex~4 requires version 2.0 or higher of the AMS-\LaTeX\ package.
\item[\textbf{array and dcolumn}]
The \classname{array} and \classname{dcolumn} packages are part of
\LaTeX's required suite of packages. \classname{dcolumn} is required
to align table columns on decimal points (and it in turn depends upon
the \classname{array} package).
\item[\textbf{longtable}]
\file{longtable.sty} may be used for large tables that will span more than one
page. \revtex~4 dynamically applies patches to longtable.sty so that
it will work in two-column mode.
\item[\textbf{hyperref}] \file{hyperref.sty} is a package by Sebastian Rahtz that is
used for putting hypertext links into \LaTeXe\ documents.
\revtex~4 has hooks to allow e-mail addresses and URL's to become
hyperlinks if \classname{hyperref} is loaded.
\end{description}
Other packages will conflict with \revtex~4 and should be
avoided. Usually such a conflict arises because the package adds
enhancements that \revtex~4 already includes. Here are some common
packages that clash with \revtex~4:
\begin{description}
\item[\textbf{multicol}] \file{multicol.sty} is a package by Frank Mittelbach
that adds support for multiple columns. In fact, early versions of
\revtex~4 used \file{multicol.sty} for precisely this. However, to
improve the handling of floats, \revtex~4 now has its own macros for
two-column layout. Thus, it is not necessary to use \file{multicol.sty}.
\item[\textbf{cite}] Donald Arseneau's \file{cite.sty} is often used to provide
support for sorting a \cmd\cite\ command's arguments into numerical
order and to collapse consecutive runs of reference numbers. \revtex~4
has this functionality built-in already via the \classname{natbib} package.
\item[\textbf{endfloat}] The same functionality can be accomplished
using the \classoption{endfloats} class option.
\item[\textbf{float}] \revtex~4 already contains a lot of this
functionality.
\end{description}
\section{The Document Preamble}
The preamble of a \LaTeX\ document is the set of commands that precede
the \envb{document} line. It contains a
\cmd\documentclass\ line to load the \revtex~4 class (\textit{i.~e.},
all of the \revtex~4 macro definitions), \cmd\usepackage\ macros to
load other macro packages, and other macro definitions.
\subsection{The \emph{documentclass} line}
The basic formatting of the manuscript is controlled by setting
\emph{class options} using
\cmd\documentclass\oarg{options}\aarg{\classname{revtex4}}.
The macro \cmd\documentclass\
replaces the \cmd\documentstyle\ macro of \LaTeX2.09. The optional
arguments that appear in the square brackets control the layout of the
document. At this point, one only needs to choose a journal style
(\classoption{pra}, \classoption{prb},
\classoption{prc}, \classoption{prd},
\classoption{pre}, \classoption{prl}, \classoption{prstab},
and \classoption{rmp}) and either \classoption{preprint} or
\classoption{twocolumn}. Usually, one would want to use
\classoption{preprint} for draft papers. \classoption{twocolumn} gives
the \emph{Physical Review} look and feel. Paper size options are also
available as well. In particular, \classoption{a4paper} is available
as well as the rest of the standard \LaTeX\ paper sizes. A
full list of class options is given in the \textit{\revtex~4 Command
and Options Summary}.
\subsection{Loading other packages}
Other packages may be loaded into a \revtex~4 document by using the
standard \LaTeXe\ \cmd\usepackage\ command. For instance, to load
the \classoption{graphics} package, one would use
\verb+\usepackage{graphics}+.
\section{The Front Matter}\label{sec:front}
After choosing the basic look and feel of the document by selecting
the appropriate class options and loading in whatever other macros are
needed, one is ready to move on to creating a new manuscript. After
the preamble, be sure to put in a \envb{document} line (and put
in an \enve{document} as well). This section describes the macros
\revtex~4 provides for formatting the front matter of the
article. The behavior and usage of these macros can be quite
different from those provided in either \revtex~3 or \LaTeXe. See the
included document \textit{Differences between \revtex~4 and \revtex~3} for an
overview of these differences.
\subsection{Setting the title}
The title of the manuscript is simply specified by using the
\cmd\title\aarg{title} macro. A \verb+\\+ may be used to put a line
break in a long title.
\subsection{Specifying a date}%
The \cmd\date\marg{date} command outputs the date on the
manuscript. Using \cmd\today\ will cause \LaTeX{} to insert the
current date whenever the file is run:
\begin{verbatim}
\date{\today}
\end{verbatim}
\subsection{Specifying authors and affiliations}
The macros for specifying authors and their affiliations have
changed significantly for \revtex~4. They have been improved to save
labor for authors and in production. Authors and affiliations are
arranged into groupings called, appropriately enough, \emph{author
groups}. Each author group is a set of authors who share the same set
of affiliations. Author names are specified with the \cmd\author\
macro while affiliations (or addresses) are specified with the
\cmd\affiliation\ macro. Author groups are specified by sequences of
\cmd\author\ macros followed by \cmd\affiliation\ macros. An
\cmd\affiliation\ macro applies to all previously specified
\cmd\author\ macros which don't already have an affiliation supplied.
For example, if Bugs Bunny and Roger Rabbit are both at Looney Tune
Studios, while Mickey Mouse is at Disney World, the markup would be:
\begin{verbatim}
\author{Bugs Bunny}
\author{Roger Rabbit}
\affiliation{Looney Tune Studios}
\author{Mickey Mouse}
\affiliation{Disney World}
\end{verbatim}
The default is to display this as
\begin{center}
Bugs Bunny and Roger Rabbit\\
\emph{Looney Tune Studios}\\
Mickey Mouse\\
\emph{Disney World}\\
\end{center}
This layout style for displaying authors and their affiliations is
chosen by selecting the class option
\classoption{groupedaddress}. This option is the default for all APS
journal styles, so it does not need to be specified explicitly.
The other major way of displaying this
information is to use superscripts on the authors and
affiliations. This can be accomplished by selecting the class option
\classoption{superscriptaddress}. To achieve the display
\begin{center}
Bugs Bunny,$^{1}$ Roger Rabbit,$^{1,2}$ and Mickey Mouse$^{2}$\\
\emph{$^{1}$Looney Tune Studios}\\
\emph{$^{2}$Disney World}\\
\end{center}
one would use the markup
\begin{verbatim}
\author{Bugs Bunny}
\affiliation{Looney Tune Studios}
\author{Roger Rabbit}
\affiliation{Looney Tune Studios}
\affiliation{Disney World}
\author{Mickey Mouse}
\affiliation{Disney World}
\end{verbatim}
Note that \revtex~4 takes care of any commas and \emph{and}'s that join
the author names together and font selection, as well as any
superscript numbering. Only the author names and affiliations should
be given within their respective macros.
There is a third class option, \classoption{unsortedaddress}, for
controlling author/affiliation display. The default
\classoption{groupedaddress} will actually sort authors into the
approriate author groups if one chooses to specify an affiliation for
each author. The markup:
\begin{verbatim}
\author{Bugs Bunny}
\affiliation{Looney Tune Studios}
\author{Mickey Mouse}
\affiliation{Disney World}
\author{Roger Rabbit}
\affiliation{Looney Tune Studios}
\end{verbatim}
will result in the same display as for the first case given
above even though Roger Rabbit is specified after Mickey Mouse. To
avoid Roger Rabbit being moved into the same author group as Bugs
Bunny, use the
\classoption{unsortedaddress} option instead. In general, it is safest
to list authors in the order they should appear and specify
affiliations for multiple authors rather than one at a time. This will
afford the most independence for choosing the display option. Finally,
it should be mentioned that the affiliations for the
\classoption{superscriptaddress} are presented and numbered
in the order that they are encountered. These means that the order
will usually follow the order of the authors. An alternative ordering
can be forced by including a list of \cmd\affiliation\ commands before
the first \cmd{\author} in the desired order. Then use the exact same
text for each affilation when specifying them for each author.
If an author doesn't have an affiliation, the \cmd\noaffiliation\
macro may be used in the place of an \cmd\affiliation\ macro.
\subsubsection{Collaborations}
A collaboration name can be specified with the \cmd\collaboration\
macro. This is very similar to the \cmd\author\ macro, but it can only
be used with the class option \classoption{superscriptaddress}. The
\cmd\collaboration\ macro should appear at the end of the list of
authors. The collaboration name will be appear centered in parentheses
between the list of authors and the list of
affiliations. Because collaborations
don't normally have affiliations, one needs to follow the
\cmd\collaboration\ with \cmd\noaffiliation.
\subsubsection{Footnotes for authors, collaborations, affiliations or title}\label{sec:footau}
Often one wants to specify additional information associated with an
author, collaboration, or affiliation such an e-mail address, an
alternate affiliation, or some other anicillary information.
\revtex~4 introduces several new macros just for this purpose. They
are:
\begin{itemize}
\item\cmd\email\oarg{optional text}\aarg{e-mail address}
\item\cmd\homepage\oarg{optional text}\aarg{URL}
\item\cmd\altaffiliation\oarg{optional text}\aarg{affiliation}
\item\cmd\thanks\aarg{miscellaneous text}
\end{itemize}
In the first three, the \emph{optional text} will be prepended before the
actual information specified in the required argument. \cmd\email\ and
\cmd\homepage\ each have a default text for their optional arguments
(`Electronic address:' and `URL:' respectively). The \cmd\thanks\
macro should only be used if one of the other three do not apply. Any
author name can have multiple occurences of these four macros. Note
that unlike the
\cmd\affiliation\ macro, these macros only apply to the \cmd\author\
that directly precedes it. Any \cmd\affiliation\ \emph{must} follow
the other author-specific macros. A typical usage might be as follows:
\begin{verbatim}
\author{Bugs Bunny}
\email[E-mail me at: ]{[email protected]}
\homepage[Visit: ]{http://looney.com/}
\altaffiliation[Permanent address: ]
{Warner Brothers}
\affiliation{Looney Tunes}
\end{verbatim}
This would result in the footnote ``E-mail me at: \texttt{[email protected]},
Visit: \texttt{http://looney.com/}, Permanent address: Warner
Brothers'' being attached to Bugs Bunny. Note that:
\begin{itemize}
\item Only an e-mail address, URL, or affiliation should go in the
required argument in the curly braces.
\item The font is automatically taken care of.
\item An explicit space is needed at the end of the optional text if one is
desired in the output.
\item Use the optional arguments to provide customized
text only if there is a good reason to.
\end{itemize}
The \cmd\collaboration\ , \cmd\affiliation\ , or even \cmd\title\ can
also have footnotes attached via these commands. If any ancillary data
(\cmd\thanks, \cmd\email, \cmd\homepage, or
\cmd\altaffiliation) are given in the wrong context (e.g., before any
\cmd\title, \cmd\author, \cmd\collaboration, or \cmd\affiliation\
command has been given), then a warning is given in the \TeX\ log, and
the command is ignored.
Duplicate sets of ancillary data are merged, giving rise to a single
shared footnote. However, this only applies if the ancillary data are
identical: even the order of the commands specifying the data must be
identical. Thus, for example, two authors can share a single footnote
indicating a group e-mail address.
Duplicate \cmd\affiliation\ commands may be given in the course of the
front matter, without the danger of producing extraneous affiliations
on the title page. However, ancillary data should be specified for
only the first instance of any particular institution's
\cmd\affiliation\ command; a later instance with different ancillary
data will result in a warning in the \TeX\ log.
It is preferable to arrange authors into
sets. Within each set all the authors share the same group of
affiliations. For each author, give the \cmd\author\ (and appropriate
ancillary data), then follow this author group with the needed group
of \cmd\affiliation\ commands.
If affiliations have been listed before the first
\cmd\author\ macro to ensure a particular ordering, be sure
that any later \cmd\affiliation\ command for the given institution is
an exact copy of the first, and also ensure that no ancillary data is
given in these later instances.
Each APS journal has a default behavior for the placement of these
ancillary information footnotes. The \classoption{prb} option puts all
such footnotes at the start of the bibliography while the other
journal styles display them on the first page. One can override a
journal style's default behavior by specifying explicitly the class
option
\classoption{bibnotes} (puts the footnotes at the start of the
bibliography) or \classoption{nobibnotes} (puts them on the first page).
\subsubsection{Specifying first names and surnames}
Many APS authors have names in which either the surname appears first
or in which the surname is made up of more than one name. To ensure
that such names are accurately captured for indexing and other
purposes, the \cmd\surname\ macro should be used to indicate which portion
of a name is the surname. Similarly, there is a \cmd\firstname\ macro
as well, although usage of \cmd\surname\ should be sufficient. If an
author's surname is a single name and written last, it is not
necessary to use these macros. These macros do nothing but indicate
how a name should be indexed. Here are some examples;
\begin{verbatim}
\author{Andrew \surname{Lloyd Weber}}
\author{\surname{Mao} Tse-Tung}
\end{verbatim}
\subsection{The abstract}
An abstract for a paper is specified by using the \env{abstract}
environment:
\begin{verbatim}
\begin{abstract}
Text of abstract
\end{abstract}
\end{verbatim}
Note that in \revtex~4 the abstract must be specified before the
\cmd\maketitle\ command and there is no need to embed it in an explicit
minipage environment.
\subsection{PACS codes}
APS authors are asked to supply suggested PACS codes with their
submissions. The \cmd\pacs\ macro is provided as a way to do this:
\begin{verbatim}
\pacs{23.23.+x, 56.65.Dy}
\end{verbatim}
The actual display of the PACS numbers below the abstract is
controlled by two class options: \classoption{showpacs} and
\classoption{noshowpacs}. In particular, this is now independent of
the \classoption{preprint} option. \classoption{showpacs} must be
explicitly included in the class options to display the PACS codes.
\subsection{Keywords}
A \cmd\keywords\ macro may also be used to indicate keywords for the
article.
\begin{verbatim}
\keywords{nuclear form; yrast level}
\end{verbatim}
This will be displayed below the abstract and PACS (if supplied). Like
PACS codes, the actual display of the the keywords is controlled by
two classoptions: \classoption{showkeys} and
\classoption{noshowkeys}. An explicit \classoption{showkeys} must be
included in the \cmd\documentclass\ line to display the keywords.
\subsection{Institutional report numbers}
Institutional report numbers can be specified using the \cmd\preprint\
macro. These will be displayed in the upper lefthand corner of the
first page. Multiple \cmd\preprint\ macros maybe supplied (space is
limited though, so only three or less may actually fit).
\subsection{maketitle}
After specifying the title, authors, affiliations, abstract, PACS
codes, and report numbers, the final step for formatting the front
matter of the manuscript is to execute the \cmd\maketitle\ macro by
simply including it:
\begin{verbatim}
\maketitle
\end{verbatim}
The \cmd\maketitle\ macro must follow all of the macros listed
above. The macro will format the front matter in accordance with the various
class options that were specified in the
\cmd\documentclass\ line (either implicitly through defaults or
explicitly).
\section{The body of the paper}
For typesetting the body of a paper, \revtex~4 relies heavily on
standard \LaTeXe\ and other packages (particulary those that are part
of AMS-\LaTeX). Users unfamiliar with these packages should read the
following sections carefully.
\subsection{Section headings}
Section headings are input as in \LaTeX.
The output is similar, with a few extra features.
Four levels of headings are available in \revtex{}:
\begin{quote}
\cmd\section\marg{title text}\\
\cmd\subsection\marg{title text}\\
\cmd\subsubsection\marg{title text}\\
\cmd\paragraph\marg{title text}
\end{quote}
Use the starred form of the command to suppress the automatic numbering; e.g.,
\begin{verbatim}
\section*{Introduction}
\end{verbatim}
To label a section heading for cross referencing, best practice is to
place the \cmd\label\marg{key} within the argument specifying the heading:
\begin{verbatim}
\section{\label{sec:intro}Introduction}
\end{verbatim}
In the some journal substyles, such as those of the APS,
all text in the \cmd\section\ command is automatically set uppercase.
If a lowercase letter is needed, use \cmd\lowercase\aarg{x}.
For example, to use ``He'' for helium in a \cmd\section\marg{title text} command, type
\verb+H+\cmd\lowercase\aarg{e} in \marg{title text}.
Use \cmd\protect\verb+\\+ to force a line break in a section heading.
(Fragile commands must be protected in section headings, captions, and
footnotes and \verb+\\+ is a fragile command.)
\subsection{Paragraphs and General Text}
Paragraphs always end with a blank input line. Because \TeX\
automatically calculates linebreaks and word hyphenation in a
paragraph, it is not necessary to force linebreaks or hyphenation. Of
course, compound words should still be explicitly hyphenated, e.g.,
``author-prepared copy.''
Use directional quotes for quotation marks around quoted text
(\texttt{``xxx''}), not straight double quotes (\texttt{"xxx"}).
For opening quotes, use one or two backquotes; for closing quotes,
use one or two forward quotes (apostrophes).
\subsection{One-column vs. two-column}\label{sec:widetext}
One of the hallmarks of \textit{Physical Review} is its two-column
formatting and so one of the \revtex~4 design goals is to make it easier to
acheive the \textit{Physical Review} look and feel. In particular, the
\classoption{twocolumn} option will take care of formatting the front matter
(including the abstract) as a single column. \revtex~4 has its own
built-in two-column formatting macros to provide well-balanced columns
as well as reasonable control over the placement of floats in either
one- or two-column modes.
Occasionally it is necessary to change the formatting from two-column to
one-column to better accomodate very long equations that are more
easily read when typeset to the full width of the page. This is
accomplished using the \env{widetext} environment:
\begin{verbatim}
\begin{widetext}
long equation goes here
\end{widetext}
\end{verbatim}
In two-column mode, this will temporarily return to one-column mode,
balancing the text before the environment into two short columns, and
returning to two-column mode after the environment has
finished. \revtex~4 will also add horizontal rules to guide the
reader's eye through what may otherwise be a confusing break in the
flow of text. The
\env{widetext} environment has no effect on the output under the
\classoption{preprint} class option because this already uses
one-column formatting.
Use of the \env{widetext} environment should be restricted to the bare
minimum of text that needs to be typeset this way. However short pieces
of paragraph text and/or math between nearly contiguous wide equations
should be incorporated into the surrounding wide sections.
Low-level control over the column grid can be accomplished with the
\cmd\onecolumngrid\ and \cmd\twocolumngrid\ commands. Using these, one
can avoid the horizontal rules added by \env{widetext}. These commands
should only be used if absolutely necessary. Wide figures and tables
should be accomodated using the proper \verb+*+ environments.
\subsection{Cross-referencing}\label{sec:xrefs}
\revtex{} inherits the \LaTeXe\ features for labeling and cross-referencing
section headings, equations, tables, and figures. This section
contains a simplified explanation of these cross-referencing features.
The proper usage in the context of section headings, equations,
tables, and figures is discussed in the appropriate sections.
Cross-referencing depends upon the use of ``tags,'' which are defined by
the user. The \cmd\label\marg{key} command is used to identify tags for
\revtex. Tags are strings of characters that serve to label section
headings, equations, tables, and figures that replace explicit,
by-hand numbering.
Files that use cross-referencing (and almost all manuscripts do)
need to be processed through \revtex\ at least twice to
ensure that the tags have been properly linked to appropriate numbers.
If any tags are added in subsequent editing sessions,
\LaTeX{} will display a warning message in the log file that ends with
\texttt{... Rerun to get cross-references right}.
Running the file through \revtex\ again (possibly more than once) will
resolve the cross-references. If the error message persists, check
the labels; the same \marg{key} may have been used to label more than one
object.
Another \LaTeX\ warning is \texttt{There were undefined references},
which indicates the use of a key in a \cmd\ref\ without ever
using it in a \cmd\label\ statement.
\revtex{} performs autonumbering exactly as in standard \LaTeX.
When the file is processed for the first time,
\LaTeX\ creates an auxiliary file (with the \file{.aux} extension) that
records the value of each \meta{key}. Each subsequent run retrieves
the proper number from the auxiliary file and updates the auxiliary
file. At the end of each run, any change in the value of a \meta{key}
produces a \LaTeX\ warning message.
Note that with footnotes appearing in the bibliography, extra passes
of \LaTeX\ may be needed to resolve all cross-references. For
instance, putting a \cmd\cite\ inside a \cmd\footnote\ will require at
least three passes.
Using the \classname{hyperref} package to create hyperlinked PDF files
will cause reference ranges to be expanded to list every
reference in the range. This behavior can be avoided by using the
\classname{hypernat} package available from \url{www.ctan.org}.
\subsection{Acknowledgments}
Use the \env{acknowledgments} environment for an acknowledgments
section. Depending on the journal substyle, this element may be
formatted as an unnumbered section title \textit{Acknowledgments} or
simply as a paragraph. Please note the spelling of
``acknowledgments''.
\begin{verbatim}
\begin{acknowlegments}
The authors would like to thank...
\end{acknowlegments}
\end{verbatim}
\subsection{Appendices}
The \cmd
\section{Introduction}
This document gives a brief summary of how \revtex~4 is different from
what authors may already be familiar with. The two primary design
goals for \revtex~4 are to 1) move to \LaTeXe\ and 2) improve the
markup so that infomation can be more reliably extracted for the
editorial and production processes. Both of these goals require that
authors comfortable with earlier versions of \revtex\ change their
habits. In addition, authors may already be familiar with the standard
\classname{article.cls} in \LaTeXe. \revtex~4 differs in some
important ways from this class as well. For more complete
documentation on \revtex~4, see the main \textit{\revtex~4 Author's
Guide}. The most important changes are in the markup of the front
matter (title, authors, affiliations, abstract, etc.). Please see
Sec.~\ref{sec:front}.
\section{Version of \LaTeX}
The most obvious difference between \revtex~4 and \revtex~3 is that
\revtex~4 works solely with \LaTeXe; it is not useable as a \LaTeX2.09 package.
Furthermore, \revtex~4 requires an up-to-date \LaTeX\ installation
(1996/06/01 or later); its use under older versions is not supported.
\section{Class Options and Defaults}
Many of the class options in \revtex~3 have been retained in
\revtex~4. However, the default behavior for these options can be
different than in \revtex~3. Currently, there is only one society
option, \classoption{aps}, and this is the default. Furthermore, the
selection of a journal (such as \classoption{prl}) will automatically
set the society as well (this will be true even after other societies
are added).
In \revtex~3, it was necessary to invoke the \classoption{floats}, but
this is the default for \classoption{aps} journal in
\revtex~4. \revtex~4 introduces two new class options,
\classoption{endfloats} and \classoption{endfloats*} for moving floats
to the end of the paper.
The preamble commands \cmd{\draft} and \cmd{\tighten} have been replaced
with new class options \classoption{draft} and
\classoption{tightenlines}, respectively. The \cmd{\preprint} command
is now used only for specifying institutional report numbers (typeset
in the upper-righthand corner of the first page); it no longer
influences whether PACS numbers are displayed below the abstract. PACS
display is controlled by the \classoption{showpacs} and
\classoption{noshowpacs} (default) class options.
Paper size options (\classoption{letter}, \classoption{a4paper}, etc.)
work in \revtex~4. The text ``Typeset by \revtex'' no longer appears
by default - the option \classoption{byrevtex} will place this text in
the lower-lefthand corner of the first page.
\section{One- and Two-column formatting}
\revtex~4 has excellent support for achieving the two-column
formatting in the \textit{Physical~Review} and \textit{Reviews of
Modern Physics} styles. It will balance the columns
automatically. Whereas \revtex~3 had the \cmd{\widetext} and
\cmd{\narrowtext} commands for switching between one- and two-cloumn
modes, \revtex~4 simply has a \env{widetext} environment,
\envb{widetext} \dots \enve{widetext}. One-column formatting can be
specified by choosing either the \classoption{onecolumn} or
\classoption{preprint} class option (the \revtex~3 option
\classoption{manuscript} no longer exists). Two-column formatting is
the default for most journal styles, but can be specified with the
\classoption{twocolumn} option. Note that the spacing for
\classoption{preprint} is now set to 1.5, rather than full
double-spacing. The \classoption{tightenlines} option can be used to
reduce this to single spacing.
\section{Front Matter Markup}
\label{sec:front}
\revtex~4 has substantially changed how the front matter for an article
is marked up. These are the most significant differences between
\revtex~4 and other systems for typesetting manuscripts. It is
essential that authors new to \revtex~4 be familiar with these changes.
\subsection{Authors, Affiliations, and Author Notes}
\revtex~4 has substantially changed the markup of author names,
affiliations, and author notes (footnotes giving additional
information about the author such as a permanent address or an email
address).
\begin{itemize}
\item Each author name should appear separately in
individual \cmd\author\ macros.
\item Email addresses should be marked up using the \cmd\email\ macro.
\item Alternative affiliation information should be marked up using
the \cmd\altaffiliation\ macro.
\item URLs for author home pages can be specified with a
\cmd\homepage\ macro.
\item The \cmd\thanks\ macro should only be used if one of the above
don't apply.
\item \cmd{\email}, \cmd{\homepage}, \cmd{\altaffiliation}, and
\cmd{\thanks} commands are grouped together under a single footnote for
each author. These footnotes can either appear at the bottom of the
first page of the article or as the first entries in the
bibliography. The journal style controls this placement, but it may be
overridden by using the class options \classoption{bibnotes} and
\classoption{nobibnotes}. Note that these footnotes are treated
differently than the other footnotes in the article.
\item The grouping of authors by affiliations is accomplished
automatically. Each affiliation should be in its own
\cmd{\affiliation} command. Multiple \cmd{\affiliation},
\cmd{\email}, \cmd{\homepage}, \cmd{\altaffiliation}, and \cmd{\thanks}
commands can be applied to each author. The macro \cmd\and\ has been
eliminated.
\item \cmd\affiliation\ commmands apply to all previous authors that
don't have an affiliation already declared. Furthermore, for any
particular author, the \cmd\affilation\ must follow any \cmd{\email},
\cmd{\homepage}, \cmd{\altaffiliation}, or \cmd{\thanks} commands for
that author.
\item Footnote-style associations of authors with affilitations should
not be done via explicit superscripts; rather, the class option
\classoption{superscriptaddress} should be used to accomplish this
automatically.
\item A collaboration for a group of authors can be given using the
\cmd\collaboration\ command.
\end{itemize}
Table~\ref{tab:front} summarizes some common pitfalls in moving from
\revtex~3 to \revtex~4.
\begin{table*}
\begin{ruledtabular}
\begin{tabular}{lll}
\textbf{\revtex~3 Markup} & \textbf{\revtex~4 Markup} & \textbf{Explanation}\\
& & \\
\verb+\author{Author One and Author Two}+ & \verb+\author{Author One}+ & One name per\\
& \verb+\author{Author Two}+ & \verb+\author+ \\
& & \\
\verb+\author{Author One$^{1}$}+ & \verb+\author{Author One}+& Use \classoption{superscriptaddress}\\
\dots &\dots & class option \\
\verb+\address{$^{1}$APS}+ &\verb+\affiliation{APS}+ & \\
& & \\
\verb+\thanks{Permanent address...}+ & \verb+\altaffiliation{}+& Use most
specific macro \\
\verb+\thanks{Electronic address: [email protected]}+ &
\verb+\email{[email protected]}+& available\\
\verb+\thanks{http://publish.aps.org/}+ &
\verb+\homepage{http://publish.aps.org/}+& \\
\end{tabular}
\end{ruledtabular}
\caption{Common mistakes in marking up the front matter}
\label{tab:front}
\end{table*}
\subsection{Abstracts}
\revtex~4, like \revtex~3, uses the \env{abstract} environment
\envb{abstract} \dots \enve{abstract} for the abstract. The
\env{abstract} environment must appear before the \cmd{\maketitle}
command in \revtex~4. The abstract will be formatted
appropriately for either one-column (preprint) or two-column
formatting. In particular, in the two-column case, the abstract will
automatically be placed in a single column that spans the width of the
page. It is unnecessary to use a \cmd{\minipage} or any other macro to
achieve this result.
\section{Citations and References}
\revtex~4 uses the same \cmd{\cite},\cmd{\ref}, and \cmd{\bibitem}
commmands as standard \LaTeX\ and \revtex~3. Citation handling is
based upon Patick Daly's \classname{natbib} package. The
\env{references} environment is no longer used. Instead, use the
standard \LaTeXe\ environment \env{thebibliography}.
Two new \BibTeX\ files have been included with \revtex~4,
\file{apsrev.bst} and \file{apsrmp.bst}. These will format references
in the style of \textit{Physical Review} and \textit{Reviews of Modern
Physics} respectively. In addition, these \BibTeX\ styles
automatically apply a special macro \cmd{\bibinfo} to each element of the
bibliography to make it easier to extract information for use in the
editorial and production processes. Authors are strongly urged to use
\BibTeX\ to manage their bibliographies so that the \cmd{\bibinfo}
directives will be automatically included. Other bibliography styles
can be specified by using the \cmd\bibliographystyle\ command, but
unlike standard \LaTeXe, you must give this command \emph{before} the
\envb{document} statement.
Please note that the package \classname{cite.sty} is not needed with
\revtex~4 and is incompatible.
\section{Footnotes and Tablenotes}
\label{sec:foot}
\revtex~4 uses the standard \cmd{\footnote} macro for
footnotes. Footnotes can either appear on the bottom of the page on
which they occur or they can appear as entries at the end of the
bibliography. As with author notes, the journal style option controls
the placement; however, this can be overridden with the class options
\classoption{footinbib} and \classoption{nofootinbib}.
Within a table, the \cmd{\footnote} command behaves differently. Footnotes
appear at the bottom of the table. \cmd{\footnotemark} and
\cmd{\footnotetext} are also available within the table environment so
that multiple table entries can share the same footnote text. There
is no longer a need to use a \cmd{\tablenote}, \cmd{\tablenotemark},
and \cmd{\tablenotetext} macros.
\section{Section Commands}
The title in a \cmd\section\marg{title} command will be automatically
uppercased in \revtex~4. To prevent a particular letter from being
uppercased, enclose it in curly braces.
\section{Figures}
Figures should be enclosed within either a \env{figure} or \env{figure*}
environment (the latter will cause the figure to span the full width
of the page in two-column mode). \LaTeXe\ has two convenient packages
for including the figure file itself: \classname{graphics} and
\classname{graphicx}. These two packages both define a macro
\cmd{\includegraphics} which calls in the figure. They differ in how
arguments for rotation, translation, and scaling are specified. The
package \classname{epsfig} has been re-implemented to use these
\classname{graphicx} package. The package \classname{epsfig} provides
an interface similar to that under the \revtex~3 \classoption{epsf}
class option. Authors should use these standard
\LaTeXe\ packages rather than some other alternative.
\section{Tables}
Short tables should be enclosed within either a \env{table} or \env{table*}
environmnent (the latter will cause the table to span the full width
of the page in two-column mode). The heart of the table is the
\env{tabular} environment. This will behave for the most part as in
standard \LaTeXe. Note that \revtex~4 no longer automatically adds
double (Scotch) rules around tables. Nor does the \env{tabular}
environment set various table parameters as before. Instead, a new
environment \env{ruledtabular} provides this functionality. This
environment should surround the \env{tabular} environment:
\begin{verbatim}
\begin{table}
\caption{...}
\label{tab:...}
\begin{ruledtabular}
\begin{tabular}
...
\end{tabular}
\end{ruledtabular}
\end{table}
\end{verbatim}
Under \revtex~3, tables automatically break across pages. \revtex~4
provides some of this functionality. However, this requires adding the
table a float placement option of [H] (meaning put the table
``here'') to the \envb{table} command.
Long tables are more robustly handled by using the
\classname{longtable.sty} package included with the standard \LaTeXe\
distribution (put \verb+\usepackage{longtable}+ in the preamble). This
package gives precise control over the layout of the table. \revtex~4
goes out of its way to provide patches so that the \env{longtable}
environment will work within a two-column format. A new
\env{longtable*} environment is also provided for long tables that are
too wide for a narrow column. (Note that the \env{table*} and
\env{longtable*} environments should always be used rather than
attempting to use the \env{widetext} environment.)
To create tables with columns of numbers aligned on decimal points,
load the standard \LaTeXe\ \classname{dcolumn} package and use the
\verb+d+ column specifier. The content of each cell in the column is
implicitly in math mode: Use of math delimiters (\verb+$+) is unnecessary
in a \verb+d+ column.
Footnotes within a table can be specified with the
\cmd{\footnote} command (see Sec.~\ref{sec:foot}).
\section{Font selection}
The largest difference between \revtex~3 and \revtex~4 with respect to
fonts is that \revtex~4 allows one use the \LaTeXe\ font commands such
as \cmd{\textit}, \cmd{\texttt}, \cmd{\textbf} etc. These commands
should be used in place of the basic \TeX/\LaTeX\ 2.09 font commands
such as \cmd{\it}, \cmd{\tt}, \cmd{\bf}, etc. The new font commands
better handle subtleties such as italic correction and scaling in
super- and subscripts.
\section{Math and Symbols}
\revtex~4 depends more heavily on packages from the standard \LaTeXe\
distribution and AMS-\LaTeX\ than \revtex~3 did. Thus, \revtex~4 users
should make sure their \LaTeXe\ distributions are up to date and they
should install AMS-\LaTeX\ 2.0 as well. In general, if any fine control of
equation layout, special math symbols, or other specialized math
constructs are needed, users should look to the \classname{amsmath}
package (see the AMS-\LaTeX\ documentation).
\revtex~4 provides a small number of additional diacritics, symbols,
and bold parentheses. Table~\ref{tab:revsymb} summarizes this.
\begin{table}
\caption{Special \revtex~4 symbols, accents, and boldfaced parentheses
defined in \file{revsymb.sty}}
\label{tab:revsymb}
\begin{ruledtabular}
\begin{tabular}{ll|ll}
\cmd\lambdabar & $\lambdabar$ &\cmd\openone & $\openone$\\
\cmd\altsuccsim & $\altsuccsim$ & \cmd\altprecsim & $\altprecsim$ \\
\cmd\alt & $\alt$ & \cmd\agt & $\agt$ \\
\cmd\tensor\ x & $\tensor x$ & \cmd\overstar\ x & $\overstar x$ \\
\cmd\loarrow\ x & $\loarrow x$ & \cmd\roarrow\ x & $\roarrow x$ \\
\cmd\biglb\ ( \cmd\bigrb ) & $\biglb( \bigrb)$ &
\cmd\Biglb\ ( \cmd\Bigrb )& $\Biglb( \Bigrb)$ \\
& & \\
\cmd\bigglb\ ( \cmd\biggrb ) & $\bigglb( \biggrb)$ &
\cmd\Bigglb\ ( \cmd\Biggrb\ ) & $\Bigglb( \Biggrb)$ \\
\end{tabular}
\end{ruledtabular}
\end{table}
Here is a partial list of the more notable changes between \revtex~3
and \revtex~4 math:
\begin{itemize}
\item Bold math characters should now be handle via the standard
\LaTeXe\ \classname{bm} package (use \cmd{\bm} instead of \cmd{\bbox}).
\cmd{\bm} will handle Greek letters and other symbols.
\item Use the class options \classoption{amsmath},
\classoption{amsfonts} and \classoption{amssymb} to get even more math
fonts and symbols. \cmd{\mathfrak} and \cmd{\mathbb} will, for instance, give
Fraktur and Blackboard Bold symbols.
\item Use the \classoption{fleqn} class option for making equation
flush left or right. \cmd{\FL} and \cmd{\FR} are no longer provided.
\item In place of \cmd{\eqnum}, load the \classname{amsmath} package
[\verb+\usepackage{amsmath}+] and use \cmd{\tag}.
\item In place of \cmd{\case}, use \cmd{\textstyle}\cmd{\frac}.
\item In place of the \env{mathletters} environment, load the
\classname{amsmath} package and use \env{subequations} environment.
\item In place of \cmd{\slantfrac}, use \cmd{\frac}.
\item The macros \cmd{\corresponds}, \cmd{\overdots}, and
\cmd{\overcirc} have been removed. See Table~\ref{tab:obsolete}.
\end{itemize}
\section{Obsolete \revtex~3.1 commands}
Table~\ref{tab:obsolete} summarizes more differences between \revtex~4
and \revtex~3, particularly which \revtex~3 commands are now obsolete.
\begin{table*}
\caption{Differences between \revtex~3.1 and \revtex~4
markup}\label{tab:diff31}
\label{tab:obsolete}
\begin{ruledtabular}
\begin{tabular}{lp{330pt}}
\textbf{\revtex~3.1 command}&\textbf{\revtex~4 replacement}
\lrstrut\\
\cmd\documentstyle\oarg{options}\aarg{\classname{revtex}}&\cmd\documentclass\oarg{options}\aarg{\classname{revtex4}}
\\
option \classoption{manuscript}& \classoption{preprint}
\\
\cmd\tighten\ preamble command & \classoption{tightenlines} class option
\\
\cmd\draft\ preamble command & \classoption{draft} class option
\\
\cmd\author & \cmd\author\marg{name} may appear
multiple times; each signifies a new author name.\\
& \cmd\collaboration\marg{name}:
Collaboration name (should appear after last \cmd\author)\\
& \cmd\homepage\marg{URL}: URL for preceding author\\
& \cmd\email\marg{email}: email
address for preceding author\\
& \cmd{\altaffiliation}: alternate
affiliation for preceding \cmd\author\\
\cmd\thanks & \cmd\thanks, but use only for
information not covered by \cmd{\email}, \cmd{\homepage}, or \cmd{\altaffilitiation}\\
\cmd\and & obsolete, remove this command\\
\cmd\address & \cmd\affiliation\marg{institution}\ gives the affiliation for the group of authors above\\
& \cmd\affiliation\oarg{note} lets you specify a footnote to this institution\\
& \cmd\noaffiliation\ signifies that the above authors have no affiliation\\
\cmd\preprint & \cmd\preprint\marg{number} can appear multiple times, and must precede \cmd\maketitle\\
\cmd\pacs & \cmd\pacs\ must precede \cmd\maketitle\\
\env{abstract} environment & \env{abstract} environment must precede \cmd\maketitle\\
\cmd\wideabs & obsolete, remove this command\\
\cmd\maketitle & \cmd\maketitle\ must follow
\emph{all} front matter data commands\\
\cmd\narrowtext & obsolete, remove this command\\
\cmd\mediumtext & obsolete, remove this command\\
\cmd\widetext & obsolete, replace with \env{widetext} environment\\
\cmd\FL & obsolete, remove this command\\
\cmd\FR & obsolete, remove this command\\
\cmd\eqnum & replace with \cmd\tag, load \classname{amsmath}\\
\env{mathletters} & replace with \env{subequations}, load
\classname{amsmath}\\
\env{tabular} environment & No longer puts in doubled-rules. Enclose \env{tabular} in \env{ruledtabular} to get old behavior.\\
\env{quasitable} environment & obsolete, \env{tabular} environment no longer
puts in rules\\
\env{references} environment & replace with \env{thebibliography}\verb+{}+\\
\cmd\case & replace with \cmd\textstyle\cmd\frac\\
\cmd\slantfrac & replace with \cmd\frac\\
\cmd\tablenote & replace with \cmd\footnote\\
\cmd\tablenotemark & replace with \cmd\footnotemark\\
\cmd\tablenotetext & replace with \cmd\footnotetext\lrstrut\\
\cmd\overcirc & Use standard \LaTeXe\ \cmd\mathring\ \\
\cmd\overdots & Use \cmd\dddot\ with \classoption{amsmath}\\
\cmd\corresponds & Use \cmd\triangleq\ with \classoption{amssymb}\\
\classoption{epsf} class option & \verb+\usepackage{epsfig}+\\
\end{tabular}
\end{ruledtabular}
\end{table*}
\section{Converting a \revtex~3.1 Document to \revtex~4}\label{sec:conv31}%
\revtex~3 documents can be converted to \revtex~4 rather
straightforwardly. The following checklist covers most of the major
steps involved.
\begin{itemize}
\item Change \cmd\documentstyle\verb+{revtex}+ to
\cmd\documentclass\verb+{revtex4}+, and run the document under
\LaTeXe\ instead of \LaTeX2.09.
\item
Replace the \cmd\draft\ command with the \classoption{draft} class option.
\item
Replace the \cmd\tighten\ command with the \classoption{tightenlines}
class option.
\item
For each \cmd\author\ command, split the multiple authors into
individual \cmd\author\ commands. Remove any instances of \cmd\and.
\item For superscript-style associations between authors and
affiliations, remove explicit superscripts and use the
\classoption{superscriptaddress} class option.
\item
Use \cmd\affiliation\ instead of \cmd\address.
\item
Put \cmd\maketitle\ after the \env{abstract} environment and any
\cmd\pacs\ commands.
\item If double-ruled table borders are desired, enclose \env{tabular}
enviroments in \env{ruledtabular} environments.
\item
Convert long tables to \env{longtable}, and load the
\classname{longtable} package. Alternatively, give the \env{table}
an [H] float placement parameter so that the table will break automatically.
\item
Replace any instances of the \cmd\widetext\ and \cmd\narrowtext\
commands with the \env{widetext} environment.
Usually, the \envb{widetext} statement will replace the \cmd\widetext\
command, and the \enve{widetext} statement replaces the matching
\cmd\narrowtext\ command.
Note in this connection that due to a curious feature of \LaTeX\
itself, \revtex~4 having a \env{widetext} environment means that it
also has a definition for the \cmd\widetext\ command, even though the
latter cammand is not intended to be used in your document.
Therefore, it is particularly important to remove
all \cmd\widetext\ commands when converting to \revtex~4.
\item
Remove all obsolete commands: \cmd\FL, \cmd\FR, \cmd\narrowtext, and
\cmd\mediumtext\ (see Table~\ref{tab:diff31}).
\item
Replace \cmd\case\ with \cmd\frac. If a fraction needs to be set
in text style despite being in a display equation, use the
construction \cmd\textstyle\cmd\frac. Note that \cmd\frac\ does not
support the syntax \cmd\case\verb+1/2+.
\item
Replace \cmd\slantfrac\ with \cmd\frac.
\item
Change \cmd\frak\ to \cmd\mathfrak\marg{char}\index{Fraktur} and
\cmd\Bbb\ to \cmd\mathbb\marg{char}\index{Blackboard Bold}, and invoke
one of the class options \classoption{amsfonts} or
\classoption{amssymb}.
\item
Replace environment \env{mathletters} with environment
\env{subequations} and load the \classname{amsmath} package.
\item
Replace \cmd\eqnum\ with \cmd\tag\ and load the \classname{amsmath} package.
\item
Replace \cmd\bbox\ with \cmd\bm\ and load the \classname{bm} package.
\item
If using the \cmd\text\ command, load the \classname{amsmath} package.
\item
If using the \verb+d+ column specifier in \env{tabular} environments,
load the \classname{dcolumn} package. Under \classname{dcolumn}, the
content of each \verb+d+ column cell is implicitly in math mode:
remove any \verb+$+ math delimiters appearing in cells in a \verb+d+
column.
\item
Replace \cmd\tablenote\ with \cmd\footnote, \cmd\tablenotemark\ with
\cmd\footnotemark, and \cmd\tablenotetext\ with \cmd\footnotetext.
\item
Replace \envb{references} with \envb{thebibliography}\verb+{}+;
\enve{references} with \enve{thebibliography}.
\end{itemize}
\end{document}
\section{<heading>}+, \verb+\subsection{<heading>}+,
\verb+\subsubsection{<heading>}+ & Start a new section or
subsection.\\
\verb+\section*{<heading>}+ & Start a new section without a number.\\
\verb+ |
1,116,691,497,568 | arxiv | \section{\bf Introduction}
Higher spin fields in AdS space-time are known to be a crucial ingredient of $AdS/CFT$ correspondence,
as most of the composite operators on the conforml field theory (CFT) side are holographically dual to
the higher spin modes. Perhaps the best known example of such a correspondence is the one between
the higher derivative gauge invariant operators in the $O(N)$ vector model and
the symmetric higher spin frame-like fields in $AdS_4$ \cite{pol, kp, ss, mvf, mvs, mvt}. But
generically, any operator on the CFT side
carrying multiple space-time tensor indices, is expected to be dual to some higher-spin field with
mixed symmetry.
On the other hand, the higher spin holography implies that any correlator in boundary CFT is
reproduced by the worldsheet
correlators of vertex operators in string theory, i.e. any gauge-invariant observable on the
CFT side has its dual vertex operator in
AdS string theory. For simple some simple CFT/gauge theory observables such a correspondence is
straightforward.
For example, the
string counterpart of $Tr(F^2)$ in super Yang-Mills theory is the vertex operator $V_\phi$ of
a dilaton in string theory,
while the stress-energy tensor $T_{mn}$ corresponds to the graviton's vertex operator $V_{mn}$,
polarized along the $AdS$ boundary.
One problem with checking this conjecture on the string theory side is that we know little
about $AdS$ string dynamics beyond
semiclassical approximation.
This is related to the fact that the first-quantized string theory is background-dependent.
In addition, the higher spin interactions (at least beyond the quartic order) are known to be highly
nonlocal,
while the standard low-energy effective actions, stemming from vanishing $\beta$-function constraints
in the
first-quantized theories are typically local. This altogether suggests that the second-quantized
formalism of
the string field theory \cite{witsft}
may be a more adequate formalism to approach the higher spin holography from the string theory side,
especially given
the formal similarity of background-independent string field theory (SFT) equations of motion
\cite{witsft},
and Vasiliev's equations in the unfolding formalism \cite{mvuf, mvus, mvut, mvuft} .
This naturally poses a question of how the vertex operator to CFT observable correspondence may be
extended off-shell,
in particular involving the higher spin modes.
This does not seem to be obvious.
For example, consider a composite operator given by the $N$'th power of the stress-energy tensor in CFT:
\begin{equation}
T^n\sim{T_{m_1n_1}}...T_{m_Nn_N}
\end{equation}
In the gravity limit, this operator must be a dual of a certain field of spin $2N$ in $AdS$ with mixed
symmetries.
But what is the vertex operator description of such an object in string theory?
To answer this question, one has to take the colliding limit
of $N$ graviton vertex operators in string theory. Taking such a limit does not look simple and
must not be confused with the normal ordering. Instead, in order to reproduce the correlation
functions correctly in such a
limit, one has to retain $all$ the terms, up to $all$ orders of the operator product expansion (OPE),
as the operators
are colliding at the common point.
Such a limit is well-known in the matrix model formulations of Liouville and Toda theories and plays an
important
role in extending the $AGT$ conjecture to Argyres-Douglas type of supersymmetric gauge theories with
asymptotic freedom.
The result is given by rank $N-1$ irregular Gaiotto-BMT (Bonelli-Maruyoshi-Tanzini) states
\cite{gaiof, agts, agtt, chaihos, chaihot, chaiho, gaios}
These states extend the context of the primary operators in CFT and lead to special representations of
Virasoro algebra, being the simultaneous eigenstates of $N+1$ Virasoro generators:
\begin{eqnarray}
L_n |U_N>=\rho_n |U_N>
(N\leq{n}\leq{2N})
\nonumber \\
L_n |U_N>=0 (n>2N)
\end{eqnarray}
In the previous work \cite{chaiho}
(see also \cite{nagoya, lefloch} with the related issues addressed)
it was shown that the irregular states admit the following
irregular vertex operator representation in terms of Liouville or Toda fields
:
\begin{eqnarray}
|U_N>=U_N|0>
\nonumber \\
U_N=e^{{\vec{\alpha_0}}{\vec{\phi}}+\sum_{k=1}^N{\vec{\alpha}}_k\partial^k{\vec{\phi}}}
\end{eqnarray}
where ${\vec{\phi}}=\phi_1,...\phi_D$ is either $D$-component Toda field or (in the context of the
present paper)
parametrize the coordinates of $D$-dimensional target space in bosonic string theory.
The ${\vec{\alpha}}_k$ parameters are related to the Virasoro eigenvalues (1.2) according to \cite{chaiho}
\begin{eqnarray}
\rho_n=-{{1}\over{2}}\sum_{k_1,k_2;k_1+k_2=n}{\vec{\alpha}}_{k_1}{\vec{\alpha}}_{k_2}
\end{eqnarray}
In case of $D{\geq}2$, the irregular states, apart from being eigenvalues of positive Virasoro generators,
are also the eigenstates of positive modes $W_n^{(p)}$ of the $W_n$-algebra currents
($3\leq{n}\leq{D+1}$)
where
\begin{eqnarray}
W_n^{(p)}=\oint{{dz}\over{2i\pi}}z^{p+n-1}W_n(z)
\end{eqnarray}
where $W_n$ are the spin $n$ primaries and $(n-1)N\leq{p}\leq{nN}$.
so that
\begin{eqnarray}
W_n^{(p)}U_N=\rho_n^pU_N
\end{eqnarray}
and $\rho_n^p$ are degree $n$ polynomials in the components of ${\vec{\alpha}}$.
Note that, while the maximal possible rank $n$ is always
at least $D+1$, for higher dimensions ($D>5$) it is also possible to have the higher ranks
$n>D+1$ as well. In general case, the upper bound on $n$ is in fact related to a rather complex problem
in the partition
theory. Namely, the maximal rank is given by the maximal number $n_{max}$ for which the inequality
\begin{eqnarray}
\sum_{k=1}^{n_{max}}{{(k+D-1)!\kappa(n_{max}|k)}\over{k!}}-\sum_{q=1}^{{n_{max}}-1}\sum_{k=1}^q
{{(k+D-1)!\kappa(n_{max}|k)}\over{k!}}-(D-1)!\geq{0}
\end{eqnarray}
where $\kappa(n|k)$ is the number of ordered partitions of $n$ with the length $k$:
$n=p_1+....+p_k;0<p_1\leq{p_2}...\leq{p_k}$.
The objects (1.3) are obviously not in the BRST cohomology and are off-shell (except for the regular case
${\vec{\alpha}}_k=0;k\neq{0}$ but make a complete sense in background-independent open string field theory.
On the other hand, the $U_N$-vertices are related to the onshell vertex operators for the
higher-spin fields.
That is, $U_N$ is the generating vertex for the higher-spin operators through
\begin{eqnarray}
V_{h.s.}=\sum_{s,\lbrace{k_1},...,k_s\rbrace}{H^{\mu_1...\mu_s}}({\vec{\alpha}}_0)
{{\partial^s(cU_N)}\over{\partial{\alpha_{k_1}^{\mu_1}}...\partial\alpha_{k_s}^{\mu_s}}}
|_{{\vec{\alpha}}_k=0;k\neq{0}}
\end{eqnarray}
where $c$ is the $c$-ghost, $H^{\mu_1...\mu_s}({\vec{\alpha}}_0)$ are the higher spin $s$ fields in the
target space with masses
$m={\sqrt{2(k_1+...+k_s-1)}}$, at the momentum ${\vec{\alpha}}_0$
with all the due on-shell constraints on $H$ to ensure the BRST-invariance.
Thus the correlation functions (irregular conformal blocks) of the $U_N$-vertices particularly encode the
information about the higher-spin
interactions in string theory.
At nonzero ${\vec{\alpha}}_k$ the $U_N$ vertices generate the off-shell extensions of higher-spin
wavefunctions, which can be studied using the string field theory techniques.
A question of particular interest, studied in this work, is to find the higher spin wavefunction
configurations
in terms of irregular vertex operators, solving
SFT equations of motion analytically. In case if all ${\vec{\alpha}}_k=0$, except for ${\vec{\alpha}}_0$,
the $U_N$ vertex becomes a tachyonic primary.
Then , multiplied by the space-time tachyon's wavefunction $T({\vec{\alpha}}_0)$
$\Psi=c\int{d^D\alpha_0}T({\vec{\alpha}}_0)U_N$ is an elementary solution of of string field theory
equations:
$Q\Psi+\Psi\star\Psi=0$ provided that $T$ satisfies the vanishing tachyon's $\beta$-function constraints
$\beta_T=({1\over2}\alpha_0^2-1)T+const\times{T^2}=0$ in the leading order of string perturbation theory.
This solution is elementary as it describes the $perturbative$ background change by a tachyon.
In the case of ${\vec{\alpha}}_k{\neq}0$ things become far more interesting.
The wavefunction in the string field
$\Psi=c\int{d^D{\vec{\alpha}}_0}...{d^D{\vec{\alpha}}_N}T({\vec{\alpha}}_0,...,{\vec{\alpha}}_N)U_N$
can now be regarded as a generating wavefunction for higher spin excitations in string field
theory
with the SFT solution constraints on $T$ now related to nonperturbative background change due
to higher spin excitations and the effective action on $T$ holding the keys to higher spin
interactions
at all orders, just like the well-known Schnabl's analytic solution \cite{schnabl}
describes the
physics around the minimum of nonperturbative tachyon potential
(that would be calculated up to all orders, from the string perturbation theory point of view)
The rest of this letter is organized as follows.
In the Section 2 we explore the CFT properties of irregular vertex operators, including the
behavior under finite conformal transformations, relevant to the SFT equations of motion.
In the present work, we particularly concentrate on the rank 1 and search for the SFT
analytic solutions
in the form:
\begin{eqnarray}
\Psi=c\int{d^D}{\vec{\alpha}}\int{d^D}{\vec{\beta}}T({\vec{\alpha}},{\vec{\beta}})e^{{\vec{\alpha}}
{\vec{\phi}}+
{\vec{\beta}}\partial{\vec{\phi}}}
\end{eqnarray}
We find that, in the leading order, $\Psi$ is an analytic solution if $T$ satisfies the
constraints, given by
equations of motion
described by the nonlocal effective action for generalized rolling tachyons. The nonlocality
structures are
controlled by the SFT worldsheet correlators and by the conformal transformations of the
irregular blocks.
The solution in particular provides a nice example of how the star product in string field theory
translates into the Moyal product in
the analytic SFT solutions.
Although we explicitly concentrate on rank one case in this letter, the same structure appears to
persist for higher irregular ranks as well.
In the discussion section, we investigate the physical meaning and significance of the solution,
found in this work.
In particular, we relate the nonlocalities in the noncummutative rolling tachyon structures to those of
interacting higher-spin theories,
as in the context of our calculation $T({\vec{\alpha}},{\vec{\beta}})$ is the simplest generating function
for collective higher-spin excitations and the nonlocalities in the effective action are naturally
translated into those for
the interactions of higher spins.
The effective equations of motion for $T({\vec{\alpha}},{\vec{\beta}})$ constitute the
nonperturbative generalization
of $\beta$-function equations in perturbative string theory, and possibly
can be related to nonlocal field redefinitions for Vasiliev's
invariant functionals \cite{mvs}.
We also comment on connections between the ${\vec{\beta}}$-coordinate of the solution to
T-duality and the
doubling of the space-time and comment on possible relations to double field theory formalism.
\begin{center}
\section{\bf Irregular Vertices as String Field Theory Solutions: Rank 1 Case}
\end{center}
We start with the transformation properties of the irregular vertices under the conformal transformations
$z\rightarrow{f(z)}$, necessary to
compute the correlators in string field theory.
Straightforward application of the stress tensor to the rank one irregular vertex gives infinitezimal
conformal transformation:
\begin{eqnarray}
\delta_\epsilon{U_1}({\vec{\alpha}},{\vec{\beta}})
=\lbrack\oint{{dw}\over{2i\pi}}\epsilon(w)T(w);{U_1}({\vec{\alpha}},{\vec{\beta}},z)\rbrack
\nonumber \\
=\lbrace{1\over{12}}\partial^3\epsilon\beta^2+{1\over2}\partial^2\epsilon({\vec{\alpha}}{\vec{\beta}})
+\partial\epsilon({1\over2}\alpha^2+{\vec{\beta}}{{\partial}\over{\partial{\vec{\beta}}}})
+\epsilon\partial_z\rbrace
{U_1}({\vec{\alpha}},{\vec{\beta}},z)
\end{eqnarray}
It is not difficult to obtain the finite transformations for $U_1$, by
integrating the infinitezimal transformations (2.1).
\begin{eqnarray}
{U_1}({\vec{\alpha}},{\vec{\beta}},z)=e^{{\vec{\alpha}}{\vec{\phi}}+{\vec{\beta}}\partial{\vec{\phi}}}
\rightarrow
({{df}\over{dz}})^{{\alpha^2}\over2}
e^{{\vec{\alpha}}{\vec{\phi}}+{{df}\over{dz}}{\vec{\beta}}\partial{\vec{\phi}}
+({\vec{\alpha}}{\vec{\beta}}){{d}\over{dz}}log({{df}\over{dz}})+{1\over{12}}S(f;z)}
\end{eqnarray}
where
$S(f;z)$ is the Schwarzian derivative.
It is straightforward to generalize this result to transformation laws for the irregular vertices of
arbitrary ranks.
For the arbitrary rank N the BRST and finite conformal transformations for the irregular vertices have
the form:
\begin{eqnarray}
\lbrace{Q, cU_N}\rbrace=
\lbrace\oint{{dz}\over{2i\pi}}(cT-bc\partial{c}); ce^{i\sum_{q=1}^N{\vec{\alpha_q}}\partial^q{\vec{\phi}}}
\rbrace
\nonumber \\
={1\over2}\sum_{q_1=0}^N\sum_{q_2=0}^N{{q_1!q_2!}\over{{(q_1+q_2+1)}!}}
({\vec{\alpha_{q_1}}}{\vec{\alpha_{q_2}}}):\partial^{q_1+q_2+1}ccU_N
\nonumber \\
+i\sum_{q=1}^N\sum_{p=1}^{q-1}{{q!}\over{p!(q-p)!}}\partial^{q-p}cc({\vec{\alpha}}_q
{{\partial\over{\partial{\vec{\alpha}}_{p+1}}}})U_N
\end{eqnarray}
and
\begin{eqnarray}
U_N\rightarrow({{df}\over{dz}})^{{\alpha_0^2}\over2}e^{-\sum_{q_1,q_2=1}^NS_{q_1|q_2}(f;z)
+i\sum_{q=2}^N\sum_{k=1}^{q-1}\sum_{l=1}^k{{(q-1)!}\over{k!(q-1-k)!}}
{{d^{q-k}f}\over{dz^{q-k}}}B_{k|l}(\partial{f}...\partial^{k-l+1}f)
({\vec{\alpha}}_q\partial^{l+1}{\vec{\phi}})}
\nonumber \\
\times
e^{\partial^nf({\vec{\alpha}}_q\partial{\vec{\phi}})}
\end{eqnarray}
where
$S_{q_1|q_2}(f;z)$ are the generalized Schwarzian derivatives of the rank $q_1+q_2$,
given by
\begin{eqnarray}
S_{q_1|q_2}(f;z)={{1}\over{(q_1+q_2)!}}B^{(q_1+q_2)}
({{d\over{dz}}}log{{df}\over{dz}};...{{d^{q_1+q_2}}
\over{dz^{q_1+q_2}}}log{{df}\over{dz}}
\nonumber \\
-{{q_1+q_2+1}\over{(q_1+1)!(q_2+1)!}}
B^{(q_1)}({{d\over{dz}}}log{{df}\over{dz}};...{{d^{q_1}}\over{dz^{q_1}}}log{{df}\over{dz}})
B^{(q_2)}({{d\over{dz}}}log{{df}\over{dz}};...{{d^{q_2}}\over{dz^{q_2}}}log{{df}\over{dz}})
\end{eqnarray}
where
$B^{(q)}(\partial{f}...\partial^q{f})$ are the Bell polynomials in the derivatives of ${f(z)}$
defined according to
\begin{eqnarray}
B^{(q)}(\partial{f}...\partial^q{f})=q!\sum_{k=1}^q\sum_{q|p_1...p_k}
{{\partial^{p_1}f...\partial^{p_k}f}\over{p_1!...p_k!\lambda_{p_1}!...\lambda_{p_k}!}}
\end{eqnarray}
where $q=p_1+...p_k;0<p_1\leq{p_2}...\leq{p_k}$ are the length $k$ ordered partitions
of $q$ and $\lambda_p$ is the multiplicity of an element $p$ of the partition.
Given the transformation rules for the irregular vertices, it is now
straightforward to compute the correlators relevant to
the open string field theory equations of motion.
The evaluation of the kinetic term with the string field of the form (1.9) leads to
\begin{eqnarray}
<<Q\Psi(0)\star\Psi(0)>>=<Q\Psi(0)I\circ\Psi(0)>
\nonumber \\
=lim_{w\rightarrow\infty}
\int{d^D\alpha_1}\int{d^D\beta_1}\int{d^D\alpha_2}\int{d^D\beta_2}T({\vec{\alpha}}_1,{\vec{\beta}}_1)
yT({\vec{\alpha}}_2,{\vec{\beta}}_2)w^{\alpha_2^2-1}e^{-{{{\vec{\alpha_2}}{\vec{\beta_2}}}\over{z}}}
\nonumber \\
\times\lbrack
({1\over2}\alpha_1^2-1+{\vec{\beta}}{{\partial}\over{\partial{\vec{\beta}}}})<\partial{c}
ce^{i{\vec{\alpha}}_1{\vec{\phi}}+i{\vec{\beta}}_1\partial{\vec{\phi}}}
(0)ce^{i{\vec{\alpha}}_1{\vec{\phi}}+iw^2{\vec{\beta}}_1\partial{\vec{\phi}}}(w)>
\nonumber \\
+
{\vec{\alpha_1}}{\vec{\beta_1}}
<\partial^2{c}ce^{i{\vec{\alpha}}_1{\vec{\phi}}+i{\vec{\beta}}_1\partial{\vec{\phi}}}
(0)ce^{i{\vec{\alpha}}_1{\vec{\phi}}+iw^2{\vec{\beta}}_1\partial{\vec{\phi}}}(w)>
\nonumber \\
+{{\beta^2}\over{12}}
<\partial^3{c}ce^{i{\vec{\alpha}}_1{\vec{\phi}}+i{\vec{\beta}}_1\partial{\vec{\phi}}}
(0)ce^{i{\vec{\alpha}}_1{\vec{\phi}}+iw^2{\vec{\beta}}_1\partial{\vec{\phi}}}(w)>
\rbrack
\end{eqnarray}
First of all, as it is clear from (2.7) this correlator is only
well-defined in case if the constraint
\begin{equation}
{\vec{\alpha}}{\vec{\beta}}=0
\end{equation}
is imposed.
Since for regular vertex operators $\alpha$ has a meaning of the momentum,
the orthogonality constraint (2.8) particularly implies that the $\beta$ parameter may be related to the
Fourier image of the extra coordinates in space-time in the context of double field theory and $T$-duality
(see the discussion section).
Furthermore, note that since the ghost correlator $<\partial^n{c}c(z_1)c(z_2)>=0$ for $n>2$, combined
with the constraint
(2.8) the only surviving terms in the correlator (2.7) are those proportional to $\sim{\partial{c}}c$ in
$Q\Psi$.
In addition, in the on-shell limit $\alpha_0^2\rightarrow{2}$ the correlators involving the terms
$\sim\partial{c}c{\vec{\beta}}{{\partial}\over{\partial{\vec{\beta}}}}$
and $\sim\partial^2{c}c$ are of the order of $\sim{1\over{w}}$ and vanish.
Thus the only contributing correlator in the kinetic term gives
\begin{eqnarray}
<<Q\Psi(0)\star\Psi(0)>>
\nonumber \\
=
lim_{w\rightarrow\infty}
\int{d^D\alpha_1}{d^D\beta_1}\int{d^D\alpha_2}{d^D\beta_2}T({\vec{\alpha}}_1,{\vec{\beta}}_1)
T({\vec{\alpha}}_2,{\vec{\beta}}_2)w^{\alpha_2^2-1}e^{-{{{\vec{\alpha_2}}{\vec{\beta_2}}}\over{z}}}
\nonumber \\
\times\lbrace
({1\over2}\alpha_1^2-1+{\vec{\beta}}
{{\partial}\over{\partial{\vec{\beta}}}})<\partial{c}
ce^{i{\vec{\alpha}}_1{\vec{\phi}}+i{\vec{\beta}}_1\partial{\vec{\phi}}}
(0)ce^{i{\vec{\alpha}}_1{\vec{\phi}}
+iw^2{\vec{\beta}}_1\partial{\vec{\phi}}}(w)>\rbrace
\nonumber \\
=
\int{d^D\alpha}{d^D\beta}{1\over2}(\alpha^2-1)e^{\beta^2}T({\vec{\alpha}},{\vec{\beta}})
T(-{\vec{\alpha}},-{\vec{\beta}})
\end{eqnarray}
where we used the orthogonality constraint (2.8).
This concludes the computation of the rank 1 contribution to the kinetic term
in the SFT equations of motion .
Note that, in the regularity limit $\beta^2\rightarrow{0}$ (coinciding with the on-shell limit for the rank
1 irregular operator), one can expand the exponent so that the kinetic term in the Lagrangian
becomes
\begin{eqnarray}
\sim{\int}d\alpha{d}\beta{T}(-{\vec{\alpha}},-{\vec{\beta}})({1\over2}\alpha^2+\beta^2-1)T({\vec{\alpha}},
{\vec{\beta}})+...
\nonumber \\
\sim\int{d}xdyT(x,y)({1\over2}\Box_x+\Box_y+1)T(x,y)+...
\end{eqnarray}
where we skipped the higher derivative terms.
The next step is to calculate the cubic terms in the SFT equations.
We have:
\begin{eqnarray}
<<\Psi\star\Psi\star\Psi>>=\int\prod_{j=1}^3d\alpha_jd\beta_jT({\vec{\alpha}}_j,{\vec{\beta}}_j)
<g_j^3\circ{cqe^{i{\vec{\alpha_j}}{\vec{\phi}}+i{\vec{\beta_j}}{\vec{\partial{\phi}}}}}(0)>
\end{eqnarray}
Evaluating the values of $g_j^3$ and their Schwarzian derivatives at $0$ and substituting the
transformation laws
for $\Psi$ under $g_j^3$, as well as the on-shell constraints on ${\vec{\alpha}}$,
it is straifghtforward to calculate:
\begin{eqnarray}
<<\Psi\star\Psi\star\Psi>>=\int\prod_{j=1}^3d\alpha_jd\beta_jT({\vec{\alpha}}_j,{\vec{\beta}}_j)
=
e^{{{5}\over{54}}(\beta_1^2+\beta_2^2+\beta_3^2)}(-{2\over3})^{{1\over2}\alpha_1^2-1}
(-{8\over3})^{{1\over2}\alpha_2^2+{1\over2}\alpha_3^2-2}
\nonumber \\
\times
<e^{{i{\vec{\alpha_j}}{\vec{\phi}}-{{2i}\over3}{\vec{\beta_j}}{\vec{\partial{\phi}}}}}(0)
e^{{i{\vec{\alpha_j}}{\vec{\phi}}-{{8i}\over3}{\vec{\beta_j}}{\vec{\partial{\phi}}}}}({\sqrt{3}})
e^{{i{\vec{\alpha_j}}{\vec{\phi}}-{{8i}\over3}{\vec{\beta_j}}{\vec{\partial{\phi}}}}}(-{\sqrt{3}})>
>
\nonumber \\
=
\int\prod_{j=1}^3d\alpha_jd\beta_jT({\vec{\alpha}}_j,{\vec{\beta}}_j)
\nonumber \\
exp\lbrace
{{{5}\over{54}}}(\beta_1^2+\beta_2^2+\beta_3^2)
+{{16}\over9}({\vec{\beta}}_1{\vec{\beta}}_2+{\vec{\beta}}_1{\vec{\beta}}_3+{\vec{\beta}}_2{\vec{\beta}}_3)
+{4\over{3{\sqrt{3}}}}({\vec{\alpha}}_2{\vec{\beta}}_3
\nonumber \\
-{\vec{\alpha}}_3{\vec{\beta}}_2)
+{2\over{3{\sqrt{3}}}}(4{\vec{\alpha}}_1({\vec{\beta}}_3-{\vec{\beta}}_2)+{\vec{\beta}}_1({\vec{\alpha}}_3
-{\vec{\alpha}}_2)
)\rbrace\delta(\sum_j\beta_j)\delta(\sum_j\alpha_j)
\nonumber \\
=
\int\prod_{j=1}^2d\alpha_jd\beta_jT({\vec{\alpha}}_1,{\vec{\beta}}_1)
T({\vec{\alpha}}_2,{\vec{\beta}}_2)T(-{\vec{\alpha}}_1-{\vec{\alpha}}_2,-{\vec{\beta}}_1-{\vec{\beta}}_2)
\nonumber \\
exp\lbrace
{-{{43}\over{27}}}({\vec{\beta}}_2{\vec{\beta}}_3+\beta_2^2+\beta_3^2)
+{2\over{{\sqrt{3}}}}({\vec{\alpha}}_2{\vec{\beta}}_3-{\vec{\alpha}}_3{\vec{\beta}}_2)\rbrace
\end{eqnarray}
Comparing the two-point and the three-point correlators, the irregular ansatz (1.9) solves the OSFT
equation
of motion provided that the wavefunction $T_({{\vec{\alpha}},{\vec{\beta}}})$ satisfy the Euler-Lagrange
equation following
from the cubic nonlocal effective action:
\begin{eqnarray}
S=-\int{d^Dx}{d^Dy}{\lbrace}T(x,y)e^{-\Box_y}(-{1\over2}{\Box_x}-1)T(x,y)
+{\tilde{\star}}\lbrace\tau^3(x,y)\rbrace
\end{eqnarray}
where
\begin{eqnarray}
\tau(x,y)=e^{-{{43}\over{27}}\Box_y}T(x,y)
\end{eqnarray}
is a new (nonlocal) field variable, familiar from rolling tachyon cosmology
and the star product with the tilde is defined according to
\begin{eqnarray}
{\tilde{\star}}\lbrace{T_1(x,y)...T_N(x,y)}\rbrace
=lim_{y_1,...,y_N\rightarrow{\lbrace}
{y}}e^{\sum_{i,j=1;i<j}^N{{43}\over{27}}\partial_{y_i}\partial_{y_j}}
T(x,y_1)...T(x,y_N)\rbrace
\end{eqnarray}
This defines the analytic open string field theory solution in terms of rank one irregular
vertex operators,
generating the higher-spin vertices on the leading Regge trajectory.
The generating wavefunction for higher spins is thus described, in the leading order, by the nonlocal
action (2.13).
The actions of the type (2.13) are well known, as they describe extensions of
rolling tachyon dynamics \cite{senf, sens}, relevant to cosmological models with phantom fields
.
The nonlocality coefficients appearing in the analytic solution (1.9), (2.13) must be related to
cosmological parameters
of these models, such as dark energy state parameter and the vacuum expectation values
of the rolling tachyon in the equilibrium limit (with the
SFT solution interpolating between two vacua, describing the one dressed tachyon's value
$\tau\sim{e^{const\Box}}T$ evolving into the vacuum state satisfying the Sen's conjecture
constraints
\cite{senc, sencs}.
The solution (1.9), (2.13) also defines the deformations of the BRST charge; solving the OSFT equations
with the deformed charge would then result in quartic and higher order corrections in
$\tau$. In the commutative level, nonlocal cosmological models of that type
have been considered in a number of works (e.g. see \cite{koshf, koshs, cosht, jouk}).
In the case ${\vec{\beta}}=0$ (the regular case with the higher spins decoupled) the
solution (1.9), (2.13) simplifies
and is described by the local cubic action, which is just the leading order low-energy effective
action for a tachyon in string perturbation theory. The solution (1.9), (2.13) is then the elementary
one, describing the perturbative background deformation of flat target space
in the leading order of the tachyon's $\beta$-function.
With ${\beta}$-parameter switched on, the higher-spin dynamics enters the game and the effective
action becomes non-local, describing $non-perturbative$ background deformation in open string field theory.
The rank one solution, considered so far, can be understood as the one describing generating wavefunction
for higher-spin operators on the leading trajectory.
It is then straightforward to extend this computation to describe the SFT solutions involving the
irregular
blocks of higher ranks, generating the higher-spin vertices on arbitrary Regge trajectories.
The the effective action describing the generating higher-spin wavefunction essentially remains the same:
in the leading order, it is cubic in $\tau=e^{\sum_j=1^qa_j\Box_{\beta_j}}T({\vec{\alpha}},
{\vec{\beta}}_1,...{\vec{\beta}}_q)$
( $a_j$ are the constants defining the OSFT solution)
with the structure
\begin{eqnarray}
\sim\int{d\alpha}\prod_j{d{\beta}_j}
e^{\sum_j=1^qb_j\Box_{\beta_j}}T({1\over2}\alpha^2-1)T+{\tilde{\star}}(\tau^3)
\end{eqnarray}
where $\tau$ is again related to $T$ through nonlocal field redefinition.
All the family of the effective actions for collective
higher-spin wavefunctions is essentially nonlocal.
Clearly, they must be related to the nonlocalities and the star products
appearing
in higher-spin theories and Vasiliev's equations.
It is also remarkable that the generating wavefunction
for higher spin fields thus emerges in the context of the
rolling tachyon cosmology. Since the solutions of the type (1.9), (2.13) generally
describe the nonperturbative deformation of the flat background to collective
higher-spin vacuum, it is a profound question whether such a deformation,
related to cosmological evolution of generalized rolling tachyon type objects,
is subject to constraints set up by the Sen's conjecture \cite{senc, sencs}.
\section{\bf Conclusion and Discussion}
In this work we have described simple analytic solution in open string
field theory,expressed in terms of vertex operators for irregular
conformal blocks in the free limit of Toda theory, or bosonic string theory.
The wavefunctions $T({\vec{\alpha}};{\lbrace}{\vec{\beta}}\rbrace)$ for
these vertex operators are naturally related to those for the higher-spin vertex operators in
open string theory
according to
\begin{eqnarray}
H_{\mu_1...\mu_s}({\vec{\alpha}})
\sim(-1)^s\prod_{j=1}^s{{\delta^{n_j}}\over{\delta\beta^{\mu_j}_{n_j}}}
T({\vec{\alpha}};{\lbrace}{\vec{\beta}}\rbrace)\delta({\lbrace}{\vec{\beta}}\rbrace)
\end{eqnarray}
We found that the irregular vertex operators analytically solve the equations of open string field theory,
provided that the generating wavefunctions satisfy the constraints following from the cubic effective
action
for generalized rolling tachyons. This implies that the ``engineering'' of nonperturbative
higher-spin vacuum
(the nonperturbative deformation of the background from flat to the one described by the minimum of
collective
higher-spin action, computed to all orders) can be mimicked by the evolution of a rolling tachyon
interpolating between inequivalent vacua in cosmological models for the dark energy, with the nonlocality
introduced in order to approach the ``Big Rip'' problem that occurs when the equation of state parameter
$\sigma$
in the equation of state $p=\sigma\rho$ (with $p$ and $\rho$ being the pressure and
the energy densities) is less then $-1$.
To understand the interplays between higher spin dynamics and cosmological models, it shall be
important to
establish the explicit form of the OSFT solution for higher irregular ranks, in order to include the
higher spins on subleading trajectories,
and to analyze the equations of motions for the generating wavefunctions $T({\vec{\alpha}},
\lbrace{\vec{\beta}}\rbrace)$.
We hope to elaborate on this in the future work, currently in progress.
As the $\beta$-parameters entering the generating wavefunctions are related to the higher-spin couplings,
it is
also worth commenting their physical meaning in the context of global space-time symmetries.
The $\alpha$-parameter is clearly related to the momentum of the regular part of the vertex operators.
Since $L_0U_N={1\over2}(\alpha^2+...)U_N$, the Virasoro generator
$L_0=-{1\over2}\oint{dz}zP_m^2$ where $P_m=\partial{X}_m$ is the current for space-time translations,
is obviously a quadratic Casimir. As the number of the Toda field (target space) components
increases,
so does the number of Casimir operators of the space-time symmetry algebra, as well as the highest
possible ranks of
$W_n$ currents with the irregular vertices being their eigenvectors.
Roughly, the highest $W_n$ rank grows with the number Casimirs, but this correspondence is
in not an exact match for $n>4$ and rather subtle, related to an unsolved problem in number theory
regarding the calculation of the number of ordered partitions of a given length - in general,
the rank of $W_n$ grows
faster than the rank of space-time symmetry algebra. Does this imply the presence of hidden dimensions
in the theory?
In case of the rank one, the single ${\vec{\beta}}$-parameter also seems to have an interesting
relation to the
$T$-duality transformations in the double field theory context, suggested by orthogonality relation (2.8).
Indeed, suppose $\phi$ is a target space coordinate, compactified on a circle. Then $\partial\phi$ is
an operator
for an infinitezimal the radius change, while $e^{\beta\partial\phi}$ would define the finite deformations
of the compactified dimension. This would in turn define the $T$-duality transformation with the
compactification radius
$R\sim\beta^{-1}+{\sqrt{4+\beta^{-2}}}$. The explicit construction of vertex operators in double
string theories
could then be realized in terms of operator algebras involving irregular operators acting on
regular states.
We hope to address this issue, as well as those outlined above, in the future works.
\begin{center}
{\bf Acknowledgements}
\end{center}
The author acknowledges the support of this work by the National Natural
Science Foundation of China under grant 11575119.
|
1,116,691,497,569 | arxiv | \section{Introduction}
\label{I}
We consider the \emph{Euler system} describing the time evolution of the mass density $\varrho = \varrho(t,x)$ and the momentum $\vc{v}_m = \vc{v}_m(t,x)$ of
a barotropic inviscid fluid:
\begin{equation} \label{i1}
\begin{split}
\partial_t \varrho + {\rm div}_x \vc{v}_m &= 0,\\
\partial_t \vc{v}_m + {\rm div}_x \left( \frac{\vc{v}_m \otimes \vc{v}_m}{\varrho} \right) + \nabla_x p(\varrho) &= 0,
\end{split}
\end{equation}
where
$t \in (0,T)$, and $\Omega \subset \mathbb{R}^d$, $d=2,3$ is a bounded domain. The problem is supplemented by the impermeability condition
\begin{equation} \label{i2}
\vc{v}_m \cdot \vc{n}|_{\partial \Omega} = 0,
\end{equation}
and the initial conditions
\begin{equation} \label{i3}
\varrho(0, \cdot) = \varrho_0, \ \vc{v}_m(0, \cdot) = \vc{v}_m_0.
\end{equation}
As is well known, problem \eqref{i1}--\eqref{i3} is locally well posed for sufficiently regular initial data, however, the smooth
solutions blow up in a finite time. The weak solution exists globally in time, however, the problem is essentially ill--posed
even in the class of \emph{admissible weak solutions} satisfying the energy inequality
\begin{equation} \label{i4}
\intO{ \left[ \frac{1}{2} \frac{ |\vc{v}_m|^2}{\varrho} + P(\varrho) \right] (t) } \leq
\intO{ \left[ \frac{1}{2} \frac{ |\vc{v}_m|^2}{\varrho} + P(\varrho) \right] (s) } ,\
P'(\varrho) \varrho - P(\varrho) = p(\varrho)
\end{equation}
for a.a. $s$, including $s = 0$, and any $t$,\ $0 \leq s \leq t \leq T$. First examples of non--uniqueness were obtained in the
seminal paper by DeLellis and Sz\' ekelyhidi \cite{DelSze3}, and later extended by Chiodaroli \cite{Chiod} and \cite{Fei2016},
Luo, Xie, and Xin \cite{LuXiXi} to a rather general class of initial data.
The key tool for using the convex integration machinery of \cite{DelSze3}, developed originally for the incompressible fluids, is
a suitable adaptation of the so-called \emph{Oscillatory Lemma}, proved originally in \cite{DelSze3} and extended to ``variable coefficients'' in \cite{Chiod}. Probably the most general version including ``non--local coefficients'' can be found
in \cite{Fei2016}. The limitation of this approach is due to the fact that certain quantities, in particular the initial density
and the desired energy profile, must enjoy some degree of smoothness to transform the problem to its basic form handled
in \cite{DelSze3}. The largest possible class used so far is that of \emph{piecewise continuous} functions, cf. \cite{Fei2016},
\cite{LuXiXi}.
A closer inspection of the problem reveals apparent similarity between the regularity properties required
for the coefficients in Oscillatory Lemma and their \emph{integrability} in the Riemann sense. Our goal is to extend validity
of Oscillatory Lemma to the case of Riemann integrable coefficients, specifically belonging to the class:
\[
\mathcal{R}(Q) \equiv \left\{ \ v: Q \to \mathbb{R} \ \Big|\
{\rm meas} \left\{ y \in Q \ \Big| \ v \ \mbox{is not continuous at} \ y \right\} = 0 \right\}
\]
where the symbol ``meas" stands for the Lebesgue measure.
Such an extension allows us to show the existence of weak solutions to the Euler system with a given total energy profile belonging
to $\mathcal{R}$. In particular, as the weak solutions $[\varrho, \vc{v}_m]: t \mapsto L^1(\Omega) \times L^1(\Omega; \mathbb{R}^d)$
are strongly continuous at a time $t$ if and only if the total energy is continuous at $t$, we obtain the existence of
an admissible weak solution
that is not strongly continuous
at an arbitrary given countable dense set of times $\mathcal{S} = \{ \tau_n \}_{n=1}^\infty \subset (0,T)$.
The paper is organized as follows. In Section \ref{M}, we collect the preliminary material and state our main results. In Section
\ref{O}, we show a version of Oscillatory Lemma with coefficients belonging to $\mathcal{R}$. Applications, including the proofs of the
main results, are discussed in Section \ref{A}.
\section{Preliminaries, main results}
\label{M}
We say that the functions
\[
\varrho \in C_{{\rm weak}}([0,T]; L^2(\Omega)),\ \vc{v}_m \in C_{{\rm weak}}([0,T]; L^2(\Omega; \mathbb{R}^d))
\]
represent \emph{weak solution} to the Euler problem \eqref{i1}--\ref{i3} if:
\begin{itemize}
\item $\varrho \geq 0$, $p(\varrho) \in L^1((0,T) \times \Omega)$;
\item
the equation of continuity
\begin{equation} \label{M1}
\int_0^T \intO{ \left[ \varrho \partial_t \varphi + \vc{v}_m \cdot \nabla_x \varphi \right] } \,{\rm d} t = - \intO{ \varrho_0 \varphi (0, \cdot) }
\end{equation}
holds
for any $\varphi \in C^1_{{\rm loc}}([0,T) \times \Ov{\Omega})$;
\item
the momentum equation
\begin{equation} \label{M2}
\int_0^T \intO{ \left[ \vc{v}_m \cdot \partial_t \boldsymbol{\varphi} + \left( \frac{\vc{v}_m \otimes \vc{v}_m}{\varrho} \right) : \nabla_x \boldsymbol{\varphi}
+ p(\varrho) {\rm div}_x \boldsymbol{\varphi} \right] } \,{\rm d} t = - \intO{ \vc{v}_m_0 \boldsymbol{\varphi} (0, \cdot) }
\end{equation}
holds
for any $\boldsymbol{\varphi} \in C^1_{{\rm loc}}([0,T) \times \Ov{\Omega}; \mathbb{R}^d)$, $\boldsymbol{\varphi} \cdot \vc{n} |_{\partial \Omega} = 0$.
\end{itemize}
A weak solution $[\varrho, \vc{v}_m]$ is \emph{admissible} if it satisfies the energy inequality \eqref{i4} for any $t \in (0,T)$ and a.a.
$s \in (0,T)$, $0 \leq s < t$.
\subsection{Main results, solutions with arbitrary energy profile}
We are ready to state our first result.
\begin{Theorem} \label{MT1}
Let $\Omega \subset \mathbb{R}^d$, $d = 2,3$, be a bounded domain with $C^2$ boundary. Let the initial data $\varrho_0$, $\vc{v}_m_0$ be given,
\[
0 < \underline{\varrho} \leq \varrho_0(x) \leq \Ov{\varrho} \ \mbox{for all}\ x \in \Ov{\Omega}, \ \varrho_0 \in \mathcal{R}(\Ov{\Omega}),
\]
\[
\vc{v}_m_0 \in \mathcal{R}(\Ov{\Omega}; R^d),\ {\rm div}_x \vc{v}_m_0 \in \mathcal{R}(\Ov{\Omega}), \ \vc{v}_m_0 \cdot \vc{n}|_{\partial \Omega} = 0.
\]
Let $E(t)$ be an arbitrary function satisfying
\[
0 \leq E(t) \leq \Ov{E} \ \mbox{for all}\ t \in [0,T], \ E \in \mathcal{R}[0,T].
\]
Then there exist $E_0 \geq 0$ such that the Euler system
\eqref{M1}, \eqref{M2} admits infinitely many solutions $[\varrho, \vc{v}_m]$ in $(0,T) \times \Omega$ satisfying
\[
\frac{1}{2}\underline{\varrho} \leq \varrho(t,x) \leq 2 \Ov{\varrho}\ \mbox{for all } (t,x) \in (0,T) \times \Omega ,\ \vc{v}_m \in L^\infty((0,T) \times \Omega; \mathbb{R}^d)),
\]
\[
\intO{ \left[ \frac{1}{2} \frac{|\vc{v}_m|^2}{\varrho} + P(\varrho) \right](\tau, \cdot) } = E_0 + E(\tau) \ \mbox{for a.a.}\ \tau \in (0,T).
\]
\end{Theorem}
\begin{Remark} \label{RR1}
It will be clear from the proof given below that
the density profile can be taken $\varrho = \varrho_0(x)$ as soon as ${\rm div}_x \vc{v}_m_0 = 0$. In such a case, we may consider
$\varrho_0 \equiv 1$ obtaining the same conclusion for the incompressible Euler system. Moreover, the result holds for
any bounded domain, no smoothness of the boundary is necessary.
\end{Remark}
Solutions satisfying strict energy inequality cannot be regular, cf. e.g. Constantin, E, and Titi \cite{ConETit}
or \cite{FeGwGwWi}.
Similarly to other ``wild'' solutions
produced by the method of convex integration, the solutions may experience the initial energy jump, meaning
the energy inequality \eqref{i4} may not hold for $s=0$.
However, as there is definitely
a sequence of times $\tau_n \searrow 0$ for which
\[
\intO{ \left[ \frac{1}{2} \frac{|\vc{v}_m|^2}{\varrho} + P(\varrho) \right](\tau_n, \cdot) } = E_0 + E(\tau_n).
\]
One could also deduce the existence of infinitely many solutions with the energy continuous at the initial time\color[rgb]{0,0,0}, performing the procedure
described e.g. in \cite{DelSze3}. We leave the details to the interested reader.
\subsection{Strong continuity in time}
We say that a weak solution $[\varrho, \vc{v}_m]$ of the Euler system is \emph{strongly continuous} at a time $\tau \in (0,T)$ if
\[
\varrho(t, \cdot) \to \varrho(\tau, \cdot)\ \mbox{in}\ L^1(\Omega),\
\vc{v}_m(t, \cdot) \to \vc{v}_m(\tau, \cdot)\ \mbox{in}\ L^1(\Omega; \mathbb{R}^d) \ \mbox{for}\ t \to \tau.
\]
\begin{Theorem} \label{MT2}
Let $\Omega \subset \mathbb{R}^d$, $d = 2,3$, be a bounded domain with $C^2$ boundary.
Let the initial data $\varrho_0$, $\vc{v}_m_0$ be given,
\[
0 < \underline{\varrho} \leq \varrho_0(x) \leq \Ov{\varrho} \ \mbox{for all}\ x \in \Ov{\Omega}, \ \varrho_0 \in \mathcal{R}(\Ov{\Omega}),
\]
\[
\vc{v}_m_0 \in \mathcal{R}(\Ov{\Omega}; R^d),\ {\rm div}_x \vc{v}_m_0 \in \mathcal{R}(\Ov{\Omega}), \ \vc{v}_m_0 \cdot \vc{n}|_{\partial \Omega} = 0.
\]
Let $\mathcal{S} = \{ \tau_n \}_{n=1}^\infty \subset (0,T)$ be an arbitrary (countable) set of times.
Then the Euler system admits infinitely many admissible weak solutions
that are not strongly continuous at any $\tau_n$, $n=1,2,\dots$
\end{Theorem}
Here again \emph{admissible} means the total energy is equal to a non--increasing function
for a.a. time. In particular, the solutions need not be strongly continuous at $t = 0$.
\section{Oscillatory lemma}
\label{O}
The proof of our main results depends on a generalized version of Oscillatory Lemma of De Lellis and Sz\' ekelyhidi \cite{DelSze3}.
Our starting point is its most elementary version showed in \cite[Proposition 3]{DelSze3}:
\begin{Lemma}[Oscillatory Lemma, basic form] \label{OL1}
Let $Q = (0,1) \times (0,1)^d$, $d=2,3$. Suppose that $\vc{v} \in \mathbb{R}^d$, $\mathbb{U} \in \mathbb{R}^{d \times d}_{0,{\rm sym}}$,
$e \leq \Ov{e}$ are given
constant quantities satisfying\footnote{$\mathbb{R}^{d \times d}_{0,{\rm sym}}$ denotes the space of real symmetric matrices with zero trace, while
$\lambda_{\rm max}[\cdot]$ is the maximum eigenvalue.}
\[
\frac{d}{2} \lambda_{\rm max} \left[ \vc{v} \otimes \vc{v} - \mathbb{U} \right] < e.
\]
Then there is a constant $c = c(d, \Ov{e})$ and sequences of vector functions $\{ \vc{w}_n \}_{n=1}^\infty$,
$\{ \mathbb{V}_n \}_{n=1}^\infty$,
\[
\vc{w}_n \in C^\infty_c(Q; \mathbb{R}^d),\ \mathbb{V}_n \in C^\infty_c(Q; \mathbb{R}^{d \times d}_{0, {\rm sym}})
\]
satisfying
\[ \partial_t \vc{w}_n + {\rm div}_x \mathbb{V}_n = 0,\ {\rm div}_x \vc{w}_n = 0 \mbox{ in } Q,\]
\[
\frac{d}{2} \lambda_{\rm max} \left[ (\vc{v} + \vc{w}_n) \otimes (\vc{v} + \vc{w}_n) - (\mathbb{U} +
\mathbb{V}_n ) \right] < e \ \mbox{in}\ Q \ \mbox{for all}\ n = 1,2, \dots,
\]
\[
\vc{w}_n \to 0 \ \mbox{in}\ C_{\rm weak}([0,1]; L^2((0,1)^d; \mathbb{R}^d))\ \mbox{as}\ n \to \infty,
\]
\[
\liminf_{n \to \infty} \int_Q |\vc{w}_n|^2 \dx \dt \geq c(d, \Ov{e}) \int_Q \left( e - \frac{1}{2} |\vc{v}|^2 \right)^2 \dx \dt.
\]
\end{Lemma}
\subsection{Extension by scaling}
We say that $Q \subset [0,T] \times \mathbb{R}^d$ is a \emph{block}, if
\[
Q = (t_1, t_2) \times \Pi_{i=1}^d (a_i, b_i),\ t_1 < t_2,\ a_i < b_i, \ i=1,\dots,d.
\]
The following can be easily deduced from Lemma \ref{OL1} by a scaling argument, see e.g. Chiodaroli \cite[Section 6, formula (6.9)]{Chiod}.
\begin{Lemma}[Oscillatory Lemma, scaled form] \label{OL2}
Let
\[
Q = (t_1, t_2) \times \Pi_{i=1}^d (a_i, b_i),\ t_1 < t_2,\ a_i < b_i, \ i=1,\dots,d,
\]
be a block.
Suppose that $\vc{v} \in \mathbb{R}^d$, $\mathbb{U} \in \mathbb{R}^{d \times d}_{0,{\rm sym}}$,
$e \leq \Ov{e}$, and $r > 0$ are given
constant quantities satisfying
\[
\frac{d}{2} \lambda_{\rm max} \left[ \frac{\vc{v} \otimes \vc{v}}{r} - \mathbb{U} \right] < e.
\]
Then there is a constant $c = c(d, \Ov{e})$ and sequences of vector functions $\{ \vc{w}_n \}_{n=1}^\infty$,
$\{ \mathbb{V}_n \}_{n=1}^\infty$,
\[
\vc{w}_n \in C^\infty_c(Q; \mathbb{R}^d),\ \mathbb{V}_n \in C^\infty_c(Q; \mathbb{R}^{d \times d}_{0, {\rm sym}})
\]
satisfying
\[ \partial_t \vc{w}_n + {\rm div}_x \mathbb{V}_n = 0,\ {\rm div}_x \vc{w}_n = 0 \mbox{ in } Q,\]
\[
\frac{d}{2} \lambda_{\rm max} \left[ \frac{(\vc{v} + \vc{w}_n) \otimes (\vc{v} + \vc{w}_n)}{r} - (\mathbb{U} +
\mathbb{V}_n ) \right] < e\ \mbox{in}\ Q\ \mbox{for all}\ n = 1,2, \dots,
\]
\[
\vc{w}_n \to 0 \ \mbox{in}\ C_{\rm weak}([t_1, t_2]; L^2(\Pi_{i=1}^d (a_i, b_i); \mathbb{R}^d))\ \mbox{as}\ n \to \infty,
\]
\[
\liminf_{n \to \infty} \int_Q \frac{|\vc{w}_n|^2}{r} \dx \dt \geq c(d, \Ov{e}) \int_Q \left( e - \frac{1}{2} \frac{|\vc{v}|^2}{r} \right)^2 \dx \dt.
\]
\end{Lemma}
\subsection{Oscillatory Lemma for Riemann integrable coefficients}
Our main goal is to show the following extension of Oscillatory Lemma.
\begin{Lemma}[Oscillatory Lemma, general coefficients] \label{OL3}
Let
\[
Q = (t_1, t_2) \times \Pi_{i=1}^d (a_i, b_i),\ t_1 < t_2,\ a_i < b_i, \ i=1,\dots,d,
\]
be a block.
Suppose that
\[
\vc{v} \in \mathcal{R}(\Ov{Q}; \mathbb{R}^d),\ \mathbb{U} \in \mathcal{R}(\Ov{Q}; \mathbb{R}^{d \times d}_{0,{\rm sym}}),
\ e \in \mathcal{R}(\Ov{Q}),\ r \in \mathcal{R}(\Ov{Q})
\]
be given such that
\begin{equation} \label{HYP1}
0 < \underline{r} \leq r(t,x) \leq \Ov{r},\ e(t,x) \leq \Ov{e} \ \mbox{for all}\ (t,x) \in \Ov{Q},
\end{equation}
\begin{equation} \label{HYP2}
\frac{d}{2} \sup_{\Ov{Q}}
\lambda_{\rm max} \left[ \frac{\vc{v} \otimes \vc{v}}{r} - \mathbb{U} \right] < \inf_{\Ov{Q}} e.
\end{equation}
Then there is a constant $c = c(d, \Ov{e})$ and sequences of vector functions $\{ \vc{w}_n \}_{n=1}^\infty$,
$\{ \mathbb{V}_n \}_{n=1}^\infty$,
\[
\vc{w}_n \in C^\infty_c(Q; \mathbb{R}^d),\ \mathbb{V}_n \in C^\infty_c(Q; \mathbb{R}^{d \times d}_{0, {\rm sym}})
\]
satisfying
\[ \partial_t \vc{w}_n + {\rm div}_x \mathbb{V}_n = 0,\ {\rm div}_x \vc{w}_n = 0 \mbox{ in } Q,\]
\begin{equation} \label{CONCL1}
\frac{d}{2} \sup_{\Ov{Q}} \lambda_{\rm max} \left[ \frac{(\vc{v} + \vc{w}_n) \otimes (\vc{v} + \vc{w}_n)}{r} - (\mathbb{U} +
\mathbb{V}_n ) \right] < \inf_{\Ov{Q}} {e}\ \mbox{for all}\ n = 1,2, \dots,
\end{equation}
\[
\vc{w}_n \to 0 \ \mbox{in}\ C_{\rm weak}([t_1, t_2]; L^2(\Pi_{i=1}^d (a_i, b_i); \mathbb{R}^d))\ \mbox{as}\ n \to \infty,
\]
\[
\liminf_{n \to \infty} \int_Q \frac{|\vc{w}_n|^2}{r} \dx \dt \geq c(d, \Ov{e}) \int_Q \left( e - \frac{1}{2} \frac{|\vc{v}|^2}{r} \right)^2 \dx \dt.
\]
\end{Lemma}
The remaining part of this section will be devoted to the proof of Lemma \ref{OL3}.
\subsubsection{Basic properties of Riemann integrable functions}
The leading idea is to approximate the coefficients $\vc{v}$, $\mathbb{U}$, $e$, and $r$ by piecewise constant functions
and use Lemma \ref{OL2}. The following is standard and may be found e.g. in the textbook by Zorich \cite[Chapter 11]{Zor}.
For a real valued function $v : \Ov{Q} \to \mathbb{R}$ we introduce:
\[
{\rm osc} [v](t,x) = \lim_{s \searrow 0} \left[ \sup_{ B((t,x), s) \cap \Ov{Q} } v - \inf_{ B((t,x), s) \cap \Ov{Q} } v \right],
\]
where $B((t,x), s)$ denotes the ball of radius $s$ centered at $(t,x)$.
It holds:
\begin{itemize}
\item
\begin{equation} \label{S2}
A_\eta = \left\{ (t,x) \in \Ov{Q}\ \Big|\ {\rm osc} [v] (t,x) \geq \eta \right\} \ \mbox{is closed}
\end{equation}
\item for any $v \in \mathcal{R}(\Ov{Q})$ and $\eta > 0$, the set $A_\eta$ is of zero content, meaning for any $\delta > 0$, there exists a
\emph{finite} number of (open) boxes $Q_i$ such that
\[
A_{\eta} \subset \cup_i Q_i,\ \sum_i |Q_i| < \delta.
\]
\end{itemize}
\subsubsection{Continuity of eigenvalues}
We recall the algebraic inequalities (see e.g. \cite{DelSze3})
\begin{equation} \label{O1}
\frac{1}{2} \frac{|\vc{v}|^2} {r} \leq \frac{d}{2} \lambda_{\rm max} \left[ \frac{\vc{v} \otimes \vc{v} }{r} - \mathbb{U} \right],
\ \|\mathbb{U}\|_\infty\leq \frac{d}{2} \lambda_{\rm max} \left[ \frac{\vc{v} \otimes \vc{v} }{r} - \mathbb{U} \right]
\end{equation}
for any $\vc{v} \in \mathbb{R}^d,$ $r > 0,$ $\mathbb{U} \in \mathbb{R}^{d \times d}_{0,{\rm sym}}$, where $\|\mathbb{U}\|_\infty$ denotes the operator norm of the matrix.\\
Consider the set
\[
K = \left\{ r \in (0, \infty), \vc{v} \in \mathbb{R}^d, \mathbb{U} \in \mathbb{R}^{d \times d}_{0,{\rm sym}} \
\Big|\ \underline{r} \leq r \leq \Ov{r},\ \frac{d}{2} \lambda_{\rm max} \left[ \frac{\vc{v} \otimes \vc{v} }{r} - \mathbb{U} \right]
\leq \Ov{e} \right\}.
\]
In view of \eqref{O1}, $K$ is a compact subset of $(0, \infty) \times \mathbb{R}^d \times \mathbb{R}^{d \times d}_{0,{\rm sym}}$. Moreover, as shown in
\cite{DelSze3}, the function
\[
[\vc{w}, \mathbb{U}] \mapsto \frac{d}{2} \lambda_{\rm max} \left[ {\vc{w} \otimes \vc{w} } - \mathbb{U} \right]
\]
is convex. As convex functions are Lipschitz continuous on compact subsets of their domain, we deduce there is a constant $L$ such that
\begin{equation} \label{O2}
\begin{split}
\left| \lambda_{\rm max} \left[ \frac{\vc{v}_1 \otimes \vc{v}_1 }{r_1} - \mathbb{U}_1 \right]
- \lambda_{\rm max} \left[ \frac{\vc{v}_2 \otimes \vc{v}_2 }{r_2} - \mathbb{U}_2 \right] \right| &\leq L
\Big( |r_1 - r_2| + |\vc{v}_1 - \vc{v}_2| + |\mathbb{U}_1 - \mathbb{U}_2 | \Big)\\
&\mbox{for any}\ (r_i, \vc{v}_i, \mathbb{U}_i) \in K,\ i=1,2.
\end{split}
\end{equation}
\subsubsection{Domain decomposition}
Suppose $\vc{v}$, $r$, $\mathbb{U}$, $r$, and $e$ satisfy \eqref{HYP1}, \eqref{HYP2}. It follows from \eqref{HYP2}
that there exists $\varepsilon_0 > 0$ such that
\[
\frac{d}{2} \lambda_{\rm max} \left\{ \frac{\vc{v} \otimes \vc{v}}{r} - \mathbb{U} \right\} < e - \varepsilon_0 \leq \Ov{e} \ \mbox{in}\ \Ov{Q}.
\]
In particular $(r, \vc{v}, \mathbb{U})(t,x) \in K$ for any $(t,x) \in \Ov{Q}$.
Thus for any $0 < \varepsilon \leq \varepsilon_0$
\begin{equation} \label{S3}
\frac{d}{2} \lambda_{\rm max} \left\{ \frac{\vc{v} \otimes \vc{v}}{r} - \mathbb{U} \right\} < e - \varepsilon \leq
\Ov{e}\ \mbox{in}\ \Ov{Q}.
\end{equation}
For $\eta > 0$ consider the set
\[
A_\eta \equiv
A_{\eta}[\vc{v}] \cup A_{\eta} [r] \cup A_{\eta}[\mathbb{U}] \cup A_{\eta} [e],
\]
cf. \eqref{S2}.
In accordance with our hypotheses, this is a set of zero content, meaning
there is a finite number of (open) boxes $Q^s_i(\eta)$ such that
\[
A_{\eta} \subset \cup_i Q^s_i(\eta),\
\sum_i |Q^s_i(\eta)| < \varepsilon \ \mbox{for any given}\ \eta > 0.
\]
The complement $\Ov{Q} \setminus \cup_i Q^s_i(\eta)$ is compact. Moreover, each point $y \in \Ov{Q} \setminus \cup_i Q^s_i$
has an open neighborhood $U(y)$ such that
\begin{equation} \label{S5}
|r(y_1) - r(y_2)| < 2\eta,\ |\vc{v}(y_1) - \vc{v}(y_2)| < 2\eta,\
|\mathbb{U}(y_1) - \mathbb{U}(y_2)| < 2\eta, \ |e(y_1) - e(y_2)| <2 \eta
\end{equation}
whenever $y_1, y_2 \in U(y) \cap \Ov{Q}$.
As the set $\Ov{Q} \setminus \cup_i Q^s_i(\eta)$ is compact and there is a finite number of
$Q^s_i$, we may infer that for any
given $\varepsilon > 0$, $\eta > 0$, there exists a decomposition of $\Ov{Q}$ into a finite number of blocks:
\begin{equation} \label{S6}
\begin{split}
\Ov{Q} = (\cup_{i} \Ov{Q}^s_i(\eta)) &\cup (\cup_j \Ov{Q}^r_j(\eta) ),\ Q^r_j \cap Q^r_k = \emptyset
\ \mbox{for}\ j \ne k,\\
\mbox{such that} &\\
\sum_{i} |Q^s_i(\eta)| &< \varepsilon\\
\mbox{and}&\\
|r(y_1) - r(y_2)| &<2 \eta,\ |\vc{v}(y_1) - \vc{v}(y_2)| <2 \eta,\
|\mathbb{U}(y_1) - \mathbb{U}(y_2)| < 2\eta,\ |e(y_1) - e(y_2)| <2 \eta\\
&\mbox{for any}\ y_1, y_2 \in \Ov{Q}^r_j,\ j = 1,2,\dots
\end{split}
\end{equation}
\subsubsection{Localization}
Given $0 < \varepsilon \leq \varepsilon_0$, $\eta > 0$, consider the decomposition of $\Ov{Q}$ given by \eqref{S6}. Choosing $y_j \in Q^r_j$ we fix
\[
\widetilde{r} = r(y_j), \ \widetilde{\vc{v}} = \vc{v}(y_j),\ \widetilde{\mathbb{U}} = \mathbb{U} (y_j), \ \mbox{and}\
\widetilde{e} = e(y_j).
\]
Applying the constant coefficient version of Oscillatory Lemma (Lemma \ref{OL2}) on each $Q^r_j$
we get a sequence of functions $\vc{w}^j_n$, $\mathbb{V}^j_n$, smooth and compactly supported in $Q^r_j \equiv (s_1, s_2)\times O^r_j$, such that
\begin{equation} \label{S8}
\partial_t \vc{w}^j_n + {\rm div}_x \mathbb{V}^j_n = 0,\ {\rm div}_x \vc{w}^j_n = 0 \mbox{ in } Q^r_j,
\end{equation}
\begin{equation} \label{S9}
\vc{w}^j_n \to 0 \ \mbox{in}\ C_{\rm weak}([s_1,s_2]; L^2(O^r_j; \mathbb{R}^d)),
\end{equation}
\begin{equation} \label{S10}
\frac{d}{2} \lambda_{\rm max} \left\{ \frac{(\widetilde{\vc{v}} + \vc{w}^j_n) \otimes (\widetilde{\vc{v}} + \vc{w}^j_n)}{
\widetilde{r}} -
(\widetilde{\mathbb{U}} + \mathbb{V}^j_n ) \right\} + \varepsilon < \widetilde{e} \mbox{ in } Q^r_j,
\end{equation}
and
\begin{equation} \label{S11}
\liminf_{n \to \infty} \int_{Q^r_j} \frac{|\vc{w}^j_n|^2}{\widetilde{r}} \ \dx \dt \geq c(d, \Ov{e}) \int_{Q^r_j} \left( \widetilde{e} - \frac{1}{2} \frac{|\widetilde{\vc{v}}|^2}{\widetilde{r}} \right)^2 \dx \dt.
\end{equation}
In view of the Lipschitz continuity of the eigenvalues established in \eqref{O2}, and in accordance with \eqref{S6}, we may choose
$\eta = \eta(\varepsilon)$ small enough so that
\begin{equation} \label{S12}
\frac{d}{2} \lambda_{\rm max} \left\{ \frac{({\vc{v}} + \vc{w}^j_n) \otimes ({\vc{v}} + \vc{w}^j_n)}{
{r}} -
({\mathbb{U}} + \mathbb{V}^j_n ) \right\} + \frac{\varepsilon}{2} < {e} \ \mbox{in}\ Q^r_j.
\end{equation}
By the same token, we get
\begin{equation} \label{S13}
\liminf_{n \to \infty} \int_{Q^r_j} \frac{|\vc{w}^j_n|^2}{{r}} \ \dx \dt \geq c(d, \Ov{e}) \int_{Q^r_j} \left( {e} - \frac{1}{2} \frac{|{\vc{v}}|^2}{{r}} \right)^2 \ \dx \dt - \varepsilon| Q^r_j |.
\end{equation}
Finally, setting $\vc{w}^i_n = \mathbb{V}^i_n = 0$ on $Q^s_i$ and summing up over all boxes, we obtain
sequences defined on $\Ov{Q}$ satisfying
\begin{equation} \label{S14}
\begin{split}
\liminf_{n \to \infty} \int_{Q} \frac{|\vc{w}_n|^2}{{r}} \ \dx \dt &\geq c(d, \Ov{e})
\sum_{j} \int_{Q^r_j} \left( {e} - \frac{1}{2} \frac{|{\vc{v}}|^2}{{r}} \right)^2 \ \dx \dt - \varepsilon |Q|\\
&\geq c(d, \Ov{e})
\int_{Q} \left( {e} - \frac{1}{2} \frac{|{\vc{v}}|^2}{{r}} \right)^2 \ \dx \dt - \varepsilon \left(|Q| + \Ov{e}^2 \right),
\end{split}
\end{equation}
and
\begin{equation}\label{ann}
\vc{w}_n \to 0 \ \mbox{in}\ C_{\rm weak}([t_1, t_2]; L^2(\Pi_{i=1}^d (a_i, b_i); \mathbb{R}^d)).
\end{equation}
As pointed out, the oscillatory perturbations can be constructed for any $0 < \varepsilon < \varepsilon_0$.
\subsubsection{Diagonalization argument}
To complete the proof of Lemma \ref{OL3}, it remains to get rid of the $\varepsilon-$dependent term in \eqref{S14}.
This can be achieved by a simple diagonalization argument.
By the previous subsection, for any $\varepsilon>0$ there exists $\{\vc{w}_n^\varepsilon\}_{n\in \mathbb{N}}$ such that \eqref{S14} and \eqref{ann} hold.
Combining \eqref{S14} and a basic property of the liminf, we get that there exists $n_{0,\varepsilon}$ such that for all $n\geq n_{0,\varepsilon}$ it holds
\begin{equation}\label{liminf0}
\int_{Q} \frac{|\vc{w}_n^\varepsilon|^2}{{r}} +\varepsilon (|Q| + \Ov{e}^2)\ \dx \dt \geq c(d, \Ov{e}) \int_{Q} \left( {e} - \frac{1}{2} \frac{|{\vc{v}}|^2}{{r}} \right)^2 \ \dx \dt.
\end{equation}
In addition we can fix $n_{0,\varepsilon}$ in such a way that
\begin{equation}\label{lim} d(\vc{w}_n^\varepsilon, 0) < \varepsilon \mbox{ for all } n\geq n_{0,\varepsilon}\end{equation}
where $d(\cdot,\cdot)$ is the metric defined as
$$d(\cdot,\cdot)= \sup_{t\in [0,T]} d_B (\cdot,\cdot)$$
and $d_B (\cdot,\cdot)$ is the metric induced by the weak topology on bounded sets of the Hilbert space $L^2(\Pi_{i=1}^d (a_i, b_i); \mathbb{R}^d)$.
For any $k\in \mathbb{N}$, let us choose $\varepsilon=\frac{\varepsilon_0}{k}$ then there exists a sequence $ \{\vc{w}_n^{\frac{1}{k}}\}_{n\in \mathbb{N}}$, which fulfills \eqref{liminf0} and \eqref{lim} definitely. We do not relabel such subsequence. Thus we get an infinite matrix
$$\begin{pmatrix}
\vc{w}_1^1 & \vc{w}_2^1 & \cdots & \vc{w}_k^1 &\cdots & \vc{w}_n^1&\cdots \\
\vc{w}_1^{1/2} & \vc{w}_2^{1/2} & \cdots & \vc{w}_k^{1/2} & \cdots & \vc{w}_n^{1/2} &\cdots\\
\vdots & \vdots & \ddots & \vdots &\ddots &\vdots&\cdots\\
\vc{w}_1^{1/k} & \vc{w}_2^{1/k} & \cdots & \vc{w}_k^{1/k} &\cdots & \vc{w}_n^{1/k}&\cdots\\
\vdots & \vdots & \ddots & \vdots& \ddots &\vdots&\cdots
\end{pmatrix}$$
where the $k$-th row corresponds to a sequence fulfilling \eqref{liminf0} and \eqref{lim} with $\varepsilon=\frac{\varepsilon_0}{k}$. Consider the sequence $\{\vc{w}_k^{1/k}\}_k$, which corresponds to the diagonal of the matrix above, it enjoys
\begin{equation}\label{liminf1}
\int_{Q} \frac{|\vc{w}_k^{1/k}|^2}{{r}} +\frac{1}{k} (|Q| + \Ov{e}^2)\ \dx \dt \geq c(d, \Ov{e}) \int_{Q} \left( {e} - \frac{1}{2} \frac{|{\vc{v}}|^2}{{r}} \right)^2 \ \dx \dt
\end{equation}
and
\begin{equation}
d(\vc{w}_k^{1/k}, 0)<\frac{\varepsilon_0}{k}.
\end{equation}
Taking respectively the liminf and the limit as $k\rightarrow+\infty$, we conclude that
\begin{equation}\label{liminf2}
\liminf_{k\to+\infty} \int_{Q} \frac{|\vc{w}_k^{1/k}|^2}{{r}} \dx \dt \geq c(d, \Ov{e}) \int_{Q} \left( {e} - \frac{1}{2} \frac{|{\vc{v}}|^2}{{r}} \right)^2 \ \dx \dt
\end{equation}
and
\begin{equation}
\vc{w}_k^{1/k} \to 0 \mbox{ in } C_{\rm weak}([t_1, t_2]; L^2(\Pi_{i=1}^d (a_i, b_i); \mathbb{R}^d)).
\end{equation}
\begin{Remark} \label{OR1}
The conclusion of Lemma \ref{OL3} holds if $Q$ is a bounded open set. Indeed $Q$ can be covered by a countable number of blocks on each of which we may apply the previous arguments.
The relevant result is provided by Whitney decomposition lemma (Stein \cite{STEIN1}),
see \cite[Section 4.4]{DoFeMa} for details.
\end{Remark}
\section{Applications}
\label{A}
Our ultimate goal is to apply the general version of Oscillatory Lemma to show existence of weak solutions to the compressible Euler system with given energy.
\subsection{Rewriting the Euler system as an abstract problem}
Following \cite{Fei2016}, we write the initial momentum in the form of its Helmholtz decomposition,
\[
\vc{v}_m_0 = \vc{v}_0 + \nabla_x \Phi_0,
\]
where
\[
\Delta_x \Phi_0 = {\rm div}_x \vc{v}_m_0 \ \mbox{in}\ \Omega, \ \nabla_x \Phi_0 \cdot \vc{n}|_{\partial \Omega} = 0.
\]
As the boundary $\partial \Omega$ is of class $C^2$, the standard elliptic estimates imply
$\nabla_x \Phi_0 \in W^{1,p}(\Omega; R^d)$, in particular $\nabla_x \Phi_0 \in C(\Ov{\Omega}; R^d)$, see e.g. Agmon, Douglis, and Nirenberg
\cite{ADN}.
Next, we fix the density profile
\[
\varrho(t,x) = \varrho_0 + h(t) \Delta_x \Phi_0, \ h \in C^\infty[0, \infty), \ h(0) = 0, \ h'(0) = -1.
\]
We look for solutions in the form
\[
\vc{m} = \vc{v} - h'(t) \nabla_x \Phi_0, \ {\rm div}_x \vc{v} = 0, \ \vc{v} \cdot \vc{n}|_{\partial \Omega} = 0.
\]
Seeing that
\[
\partial_t \varrho = h'(t) \Delta_x \Phi_0 = - {\rm div}_x \vc{m},
\]
we can adjust $h$ in such a way that
\[
0 < \frac{1}{2} \underline{\varrho} \leq \varrho(t,x) \leq 2 \Ov{\varrho} \ \mbox{for all}\ t \geq 0,\ x \in \Ov{\Omega}
\]
provided the initial density is uniformly bounded below and above. In addition, for $\varrho_0 \in \mathcal{R}(\Ov{\Omega})$,
we have
\[
\varrho \in \mathcal{R}([0,T] \times \Ov{\Omega}).
\]
Accordingly, we look for a vector field $\vc{v}$ solving the following problem:
\begin{equation} \label{S15}
\begin{split}
&{\rm div}_x \vc{v} = 0, \ \vc{v} \cdot \vc{n}|_{\partial \Omega} = 0,\ \vc{v}(0, \cdot) = \vc{v}_0,
\\
&\partial_t \vc{v} - h''(t) \nabla_x \Phi_0 + {\rm div}_x \left( \frac{(\vc{v} - h'(t) \nabla_x \Phi_0) \otimes (\vc{v} -
h'(t) \nabla_x \Phi_0) }{\varrho} - \frac{1}{d} \frac{|\vc{v} - h'(t) \nabla_x \Phi_0 |^2}{\varrho} \mathbb{I} \right) = 0,
\end{split}
\end{equation}
with prescribed kinetic energy
\begin{equation} \label{S16}
\frac{1}{2} \frac{|\vc{v} - h'(t)\nabla_x \Phi_0|^2}{\varrho} = \Lambda(t) - \frac{d}{2} p(\varrho) + \frac{d}{2} h''(t) \Phi_0
\end{equation}
where $\Lambda = \Lambda(t)$ is a spatially homogeneous function to be chosen below.
Obviously (cf. Chiodaroli \cite{Chiod} and \cite{Fei2016}), any \emph{weak} solution $\vc{v}$ of \eqref{S15}, \eqref{S16} gives rise to a weak solution $[\varrho, \vc{m} = \vc{v} - h'(t) \nabla_x \Phi_0]$ of the Euler system \eqref{M1}, \eqref{M2}, with the total energy
\begin{equation} \label{S17}
\intO{ \left[ \frac{1}{2} \frac{|\vc{v}_m |^2}{\varrho } + P(\varrho) \right](\tau, \cdot)} =
\Lambda (\tau) |\Omega| + \intO{ \left[ P(\varrho) - \frac{d}{2} p(\varrho) + \frac{d}{2} h''(\tau) \Phi_0 \right] }\ \mbox{for a.a.}\ \tau \in (0,T).
\end{equation}
Evoking the notation of Theorem \ref{MT1}, we set
\[
\Lambda(\tau) = \frac{E(\tau)}{|\Omega|} + \Lambda_0 (\tau) ,\ E_0 =
\Lambda_0(\tau) |\Omega| + \intO{ \left[ P(\varrho) - \frac{d}{2} p(\varrho) + h''(\tau) \Phi_0 \right] }.
\]
Thus the proof of Theorem \ref{MT1} consists in showing that for given $\varrho_0$ and $E$, there exists $E_0$ large enough
so that the problem \eqref{S15}, \eqref{S16} admits (infinitely many) weak solutions.
\subsection{Subsolutions}
We start by fixing the energy profile
\[
e = e(t,x) = \frac{E(t)}{|\Omega|} + \Lambda_0(t) - \frac{d}{2} p(\varrho) + \frac{d}{2}
h''(t) \Phi_0,\ e \in \mathcal{R}([0,T] \times \Ov{\Omega}).
\]
Similarly to \cite{DelSze3}, we introduce the space of \emph{subsolutions},
\[
\begin{split}
X_0 = &\left\{ (\vc{v} - \vc{v}_0) \in C^1([0,T] \times \Ov{\Omega}) \ \Big| \
\ \vc{v}(0, \cdot) = \vc{v}_0, \ \vc{v} \cdot \vc{n}|_{\partial \Omega} = 0, \right. \\
&\ \ {\rm div}_x \vc{v} = 0, \ \partial_t \vc{v} + {\rm div}_x \mathbb{U} = 0 \ \mbox{for some}\
\mathbb{U} \in C^1([0,T] \times \Ov{\Omega}; \mathbb{R}^{d \times d}_{0, {\rm sym}})\\&\left.
\ \frac{d}{2} \sup_{[0,T] \times \Ov{\Omega}} \lambda_{\rm max} \left[ \frac{(\vc{v} - h'(t) \nabla_x \Phi_0 ) \otimes
(\vc{v} - h'(t) \nabla_x \Phi_0 )}{\varrho}
- \mathbb{U} \right] < \inf_{[0,T] \times \Ov{\Omega}} e \right\}.
\end{split}
\]
The functions $E$ and $\vc{m}_0$ given, we fix $\Lambda_0$, together with the constant $E_0$, so that the set
$X_0$ is non--empty. This can be achieved by considering $\vc{v} = \vc{v}_0$, $\mathbb{U} = 0$ and fixing $\Lambda_0$ appropriately.
Finally, we set
\[
\Ov{e} = \sup_{[0,T] \times \Ov{\Omega}} e(t,x) < \infty.
\]
Thus, by virtue of \eqref{O1}, the set $X_0$ is bounded in $L^\infty((0,T) \times \Omega; \mathbb{R}^d)$; whence metrizable in the topology
of $C_{{\rm weak}}([0,T]; L^2(\Omega; \mathbb{R}^d))$. We denote by $X$ its closure in the corresponding metric $d$.
\subsection{Critical points of the energy functional}
Following \cite{DelSze3}, we introduce the functional
\[
I[\vc{v}] = \int_0^T \intO{ \left( \frac{1}{2} \frac{|\vc{v} - h'(t) \nabla_x \Phi_0|^2}{\varrho} - e \right) } \,{\rm d} t \ \mbox{for}\ \vc{v} \in X.
\]
The functional $I$ is convex lower--semicontinuous on the complete metric space $X$. By Baire category argument we conclude that the points of continuity must form a dense set in $X$.
The second observation is that
\[
I[\vc{v}] = 0 \ \Rightarrow \ \vc{v} \ \mbox{is a weak solution of the problem \eqref{S15}, \eqref{S16}.}
\]
Indeed, from convexity of the function
\[
[\vc{v}; \mathbb{U}] \mapsto
\frac{d}{2} \lambda_{\rm max} \left[ \frac{(\vc{v} - h'(t) \nabla_x \Phi_0) \otimes (\vc{v} - h'(t)\nabla_x \Phi_0) }{\varrho} - \mathbb{U} \right],
\]
we deduce that for any $\vc{v} \in X$ there is $\mathbb{U} \in L^\infty((0,T) \times \Omega; \mathbb{R}^{d \times d}_{0,{\rm sym}})$
\[
\begin{split}
\partial_t \vc{v} + {\rm div}_x \mathbb{U} &= 0 \ \mbox{in}\ \mathcal{D}'((0,T) \times \Omega),\\
\frac{1}{2} \frac{|\vc{v} - h'(t) \nabla_x \Phi_0 |^2}{\varrho} &\leq \frac{d}{2} \lambda_{\rm max} \left[ \frac{(\vc{v} - h'(t) \nabla_x \Phi_0) \otimes (\vc{v} - h'(t) \nabla_x \Phi_0) }{\varrho} - \mathbb{U} \right]
\leq e \ \mbox{a.e. in}\ (0,T) \times \Omega.
\end{split}
\]
Consequently, $I \leq 0$ on $X$; while $I[\vc{v}] = 0$ implies the desired relations (cf. \cite{DelSze3})
\[
\begin{split}
\frac{1}{2} &\frac{|\vc{v} - h'(t) \nabla_x \Phi_0|^2}{\varrho} = e,\\
\mathbb{U} &= \frac{(\vc{v} - h'(t) \nabla_x \Phi_0) \otimes (\vc{v} - h'(t) \nabla_x \Phi_0)}{\varrho} - \frac{d}{2} \frac{|\vc{v} - h'(t) \nabla_x \Phi_0|^2}{\varrho} \mathbb{I}
\ \mbox{ a.e. in}\ (0,T) \times \Omega.
\end{split}
\]
Thus, similarly to the arguments used in \cite{DelSze3}, it remains to observe:
\begin{equation} \label{S18}
\vc{v} \ \mbox{-- a point of continuity of}\ I \ \mbox{on}\ X \ \Rightarrow
I[\vc{v}] = 0.
\end{equation}
To show \eqref{S18}, we argue by contradiction. Assuming
\[
I[\vc{v}] = \underline{I} < 0
\]
we construct a sequence of functions
\[
\vc{v}_m \in X_0 \ \mbox{with the corresponding fields}\ \mathbb{U}_m \in C^1([0,T] \times \Ov{\Omega}; \mathbb{R}^{d\times d}_{0, {\rm sym}})
\]
such that
\[
\vc{v}_m \to \vc{v} \ \mbox{in}\ X, \ I[\vc{v}_m] \to \underline{I} < 0\ \mbox{as}\ m \to \infty.
\]
For fixed $m$, we apply Oscillatory Lemma (Lemma \ref{OL3}) for $\vc{v} = \vc{v}_m - h'(t) \nabla_x \Phi_0$, $\mathbb{U} = \mathbb{U}_m$,
$r = \varrho_0$, and $e$. We obtain sequences $\{ \vc{w}_{m,n} \}_{n=1}^\infty$, $\{ \mathbb{V}_{m,n} \}_{n=1}^\infty$
satisfying:
\begin{itemize}
\item
\[
\vc{v}_m + \vc{w}_{m,n} \in X_0 \ \mbox{with the associated fields}\ \mathbb{U}_m + \mathbb{V}_{m,n}
\ \mbox{for any}\ m,n;
\]
\item
\begin{equation} \label{S19}
\vc{v}_m + \vc{w}_{m,n} \to \vc{v}_m \ \mbox{in}\ X \ \mbox{as}\ n \to \infty
\ \mbox{for any fixed}\ m;
\end{equation}
\item
\begin{equation} \label{S20}
\begin{split}
\liminf_{n \to \infty} \int_0^T\! \!&\intO{ \frac{1}{2} \frac{|\vc{v}_{m} + \vc{w}_{m,n} - h'(t) \nabla_x \Phi_0 |^2}{\varrho} }\,{\rm d} t \\ &= \!\!
\int_0^T \!\!\intO{ \frac{1}{2} \frac{|\vc{v}_m - h'(t) \nabla_x \Phi_0|^2}{\varrho} } +
\liminf_{n \to \infty} \int_0^T\!\! \intO{ \frac{1}{2} \frac{|\vc{w}_{m,n}|^2}{\varrho} }\,{\rm d} t \\
&\geq \int_0^T \intO{ \frac{1}{2} \frac{|\vc{v}_m - h'(t) \nabla_x \Phi_0 |^2}{\varrho} } + c(d, \Ov{e})
\int_0^T \intO{ \left( \frac{1}{2} \frac{|\vc{v}_m - h'(t) \nabla_x \Phi_0|^2}{\varrho} - e \right)^2 } \,{\rm d} t \\
&\geq \int_0^T \intO{ \frac{1}{2} \frac{|\vc{v}_m - h'(t) \nabla_x \Phi_0|^2}{\varrho} } +
c(d, \Ov{e}) T^{-1} |\Omega|^{-1} (I[\vc{v}_m])^2;
\end{split}
\end{equation}
\end{itemize}
where we have used Jensen's inequality in \eqref{S20}. Relation \eqref{S20} rewritten as
\[
\liminf_{n \to \infty} I [ \vc{v}_m + \vc{w}_{m,n}] \geq I[\vc{v}_m] +
\frac{c(d, \Ov{e})}{T |\Omega|} \left( I[\vc{v}_m] \right)^2 \ \mbox{for any}\ m
\]
implies that $\vc{v}$ cannot be a point of continuity of $I$ unless $I[\vc{v}] = 0$.
We have proved Theorem \ref{MT1}.
\subsection{Points of strong continuity}
We show how Theorem \ref{MT2} follows from Theorem \ref{MT1}. Given the set $\{ \tau_n \}_{n=1}^\infty$ it is a routine matter to construct a function $E: [0,T] \to \infty$,
\[
\begin{split}
0 \leq E(t) &\leq \Ov{E} \ \mbox{for all}\ t \in [0,T],\
E \ \mbox{strictly decreasing in}\ [0,T],\\
&\lim_{t \to \tau_n -} E(t) > \lim_{t \to \tau_n +} E(t) \ \mbox{for any}\ \tau_n, \ n=1,2,\dots
\end{split}
\]
Consider the solutions $[\varrho, \vc{v}_m]$, the existence of which is guaranteed by Theorem \ref{MT1} with the energy profile
\[
\intO{ \left( \frac{1}{2} \frac{|\vc{v}_m|^2}{\varrho} + P(\varrho) \right)(\tau, \cdot) } = E_0 + E(\tau)
\ \mbox{for a.a.}\ \tau \in (0,T).
\]
As $\varrho$, $\vc{v}_m$ is uniformly bounded and $\varrho$ bounded below away from zero, the energy
\[
\tau \mapsto \intO{ \left( \frac{1}{2} \frac{|\vc{v}_m|^2}{\varrho} + P(\varrho) \right)(\tau, \cdot) }
\]
must be continuous at any point of strong continuity of $[\varrho, \vc{v}_m]$. Consequently, $\tau_n$ cannot be points of strong continuity
of $[\varrho, \vc{v}_m]$.
We have shown Theorem \ref{MT2}.
\def\cprime{$'$} \def\ocirc#1{\ifmmode\setbox0=\hbox{$#1$}\dimen0=\ht0
\advance\dimen0 by1pt\rlap{\hbox to\wd0{\hss\raise\dimen0
\hbox{\hskip.2em$\scriptscriptstyle\circ$}\hss}}#1\else {\accent"17 #1}\fi}
|
1,116,691,497,570 | arxiv | \section{Introduction}\label{sec:1
Pulsars are rapidly rotating neutron stars that emit radio wave pulses via their strong magnetic fields. They serve as extraordinary astrophysical laboratories for studying both nuclear physics and strong gravitation. Thus far, less than 2,700 pulsars have been discovered throughout the galaxy, and the majority of these pulsars have been discovered through pulsar surveys using radio telescopes. A large number of pulsars have been discovered in modern pulsar surveys, including the
Parkes multi-beam pulsar survey (PMPS; \cite{manchester2001parkes}), the high time resolution universe (HTRU) survey \cite{burke2011high}, the pulsar Arecibo L-band feed array (PALFA) survey \cite{deneva2009arecibo}, the Green Bank Telescope (GBT) drift-scan pulsar survey \cite{Boyles2012The}, the Green Bank North Celestial Cap (GBNCC) survey \cite{GBNCC}, and the low-frequency array (LOFAR) tied-array all-sky survey (LOTAAS; \cite{LOTAAS}). Modern pulsar surveys often produce a large number of potential candidates; however, only a small fraction of these candidates are actual pulsars. For example, the HTRU survey produced several million pulsar candidates, and the GBT drift-scan pulsar observation project produced more than 1.2 million pulsar candidates \cite{Boyles2012The, Lynch2012The}. The Five-hundred-meter Aperture Spherical radio Telescope (FAST) was installed on September 25, 2016, and is currently being commissioned \cite{jyg+19}. The FAST science team used a single--pixel feed and conducted an initial drift-scan pulsar survey, producing more than 10 million candidates before May 2018. Since then, a FAST 19--beam receiver has been installed, generating more than one million pulsar candidates per night. It is infeasible for such a large number of candidates to be individually examined by several experts; therefore, we propose a deep-learning--based pulsar candidate selection system. This system has been modified based on the pulsar image-based classification system (PICS) and customized for conducting the FAST drift-scan survey. We have designed a ResNet model comprising 15 layers to replace the convolutional neural networks (CNNs) in PICS. The resulting model can run on both the CPU and GPU platforms and surpasses PICS in terms of both effectiveness and runtime.
The Commensal Radio Astronomy FasT Survey (CRAFTS; \cite{CRAFTS}) is a multi-objective drift-scan survey that has been conducted based on the FAST. We intend to use the FAST 19--beam L-band receiver in drift mode to observe the visible sky of the FAST for HI emission and new pulsars. The drift-scan mode survey produces a large number of pulsar candidates because of the large survey time and small integration time ($\sim$12s); additionally, it generates pulsar candidates that emerge and subsequently disappear as the pulsars drift away. These candidates may differ from the canonical pulsar candidates that are often used for training previous pulsar classification systems. The radio frequency interference (RFI) environment of the FAST also differs from that of other telescopes. Therefore, it is crucial to train a new pulsar classifier for conducting this new large-scale survey. This study utilizes the pulsar and RFI data obtained from a previous work and other telescopes and combines them with the new data obtained from the FAST survey. This serves to produce a novel system that is intend to be more generalized than the PICS, which is its predecessor.
In Section 2, we describe the pipeline of the pulsar candidates and review various types of automatic pulsar candidate selection. We further introduce the ResNet model, which is our method. In Section 3, we introduce the FAST, PALFA, and HTRU pulsar datasets and describe the manner in which we conducted the experiments using the data obtained from the GBNCC and FAST. Our experimental results denote that the proposed pulsar candidate selection method runs efficiently and exhibits high recognition accuracy. Finally, Section 4 presents the discussions and conclusions.
\section{Machine learning method for pulsar candidate selection \label{sec:2}}
In this section, we explain the generation of pulsar candidates, summarize a previous study related to automatic pulsar candidate selection, and discuss the design of our 15-layer ResNet model.
\subsection{Pulsar candidates and selection methods}
In their study, Lyon et al. (2016) \cite{lyon2016fifty} described a general procedure for detecting pulsars. A telescope system collects radio signals over a large number of frequency channels at a high time resolution (usually < 1 $ms$ of the sample time). The RFI--contaminated frequency channels are eliminated, and the signals are further de-dispersion to form the time series for several different dispersion trails. Dispersion refers to the transmission of radio signals having different frequencies through the interstellar medium while encountering various delays. These delays, if not accounted for, can reduce the signal--to--noise ratio (S/N) of the pulsar signal.
For an unknown pulsar, the exact dispersion value is unknown \cite{M2010The}. Therefore, several different dispersion measures (DMs) should be considered. The subsequent step involves searching for the periodic signals in the time series of different DMs using Fourier analysis. Further, a fast Fourier transform (FFT) of the time series data can be obtained, and the results can be examined for the presence of significant power peaks.
Some pulsars have considerably narrow pulses. The FFT of narrow pulses tends to spread to multiple harmonic frequencies, and this problem can be addressed by harmonic summing, which is the subsequent step during pulsar search. A maximum of 32 harmonics are often summed to produce the highest signal peaks. The signals that are detected with sufficiently large Fourier power are considered to be the candidate signals. Majority of these candidates are related to an RFI or known periodicity and can be referred to as birdies. The signals that are related to these birdies or that have RFI-like distributions are discarded during this stage. Finally, a large number of remaining candidates are retained for performing further analysis. Further, the diagnostic plots and summary statistics are calculated for these candidate signals by folding the original data using each signal's DM and period. The diagnostic plots contain a small set of features that include the signal's S/N, DM, period, pulse width, integrated pulse profile, and two-dimensional (2D) plots that display the signals' variance with respect to time, frequency, and DM.
We used the PulsaR Exploration and Search TOolkit (\textsc{presto}) pipeline software for implementing the aforementioned steps. This software generates candidate diagnostic plots. Figure~\ref{fig:pulsar} presents an example of a pulsar candidate plot based on the FAST drift-scan survey.
Further, we can examine the candidates using the following four important subplots: the summed profile plot, the phase versus time plot, the phase versus frequency plot, and the DM curve plot. These plots are the four fundamental features that are considered by the human experts to classify the candidates. The positive (pulsar, Figure~\ref{fig:pulsar}) and negative (non-pulsar, Figure~\ref{fig:non_pulsar}) candidates exhibit different characteristics.
\begin{figure}[H]
\centering
\includegraphics[width=0.9\linewidth]{FAST_pulsar.eps}
\caption{Real pulsar candidate obtained from the FAST drift-scan survey. The time domain displays an intermittent signal caused by the pulsar drifting in and out of the beam. The phase versus time plot reveals a significant subpulse drift in the pulsar. }
\label{fig:pulsar}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=0.9\linewidth]{FAST_special_nonpulsar.eps}
\caption{The example of a non-pulsar candidate obtained from the FAST data. The frequency versus phase plot contains features that resemble the pulsar signals. However, upon close examination, these features become entirely vertical when the plot is refolded with DM set to zero, demonstrating that the signal is not dispersed. Thus, this candidate is most likely to have originated from the ground interference.}
\label{fig:non_pulsar}
\end{figure}
Recently, there has been an increase in the usage of computational intelligence and machine learning technology in various fields of astronomy\cite{wang2018cis}. Several researchers have successfully applied machine learning, which has significantly improved the efficiency of the pulsar searches, to perform pulsar candidate selection. Eatough et al. (2010) \cite{eatough2010selection} extracted 12 features of pulsar candidates and trained a single-hidden-layer artificial neural network, discovering a new pulsar in their PMPS data. Lee et al. (2013) \cite{lee2013peace} proposed a pulsar scoring method that can be referred to as the pulsar evaluation algorithm for candidate extraction (PEACE), which focuses on the quantitative evaluation of six features of a candidate and determines whether this candidate is a pulsar. Bates et al. (2012) \cite{Bates2012The} adopted 22 features and trained a single-hidden-layer artificial neural network. Morello et al. (2014) \cite{morello2014spinn} proposed straightforward pulsar identification using neural networks (SPINN), whose design contained six artificial features in the input layer, eight neurons in the hidden layers, and one neuron in the output layer. Lyon et al. (2016)\cite{lyon2016fifty} proposed a pulsar candidate selection method based on a decision tree, which can be referred to as the Gaussian Hellinger very fast decision tree (GH-VFDT) and contains eight features obtained from the diagnostic histograms to describe a pulsar candidate. This model has been successfully applied in LOTAAS. For improving the pulsar recognition accuracy, Guo et al. (2017) \cite{2017arXiv171110339G} proposed DCGAN + L2-SVM to address the class imbalance problem. They used DCGAN to learn features and trained an L2-support vector machine (L2-SVM) model to predict the results. The FRB search method is similar to the pulsar search method; Connor et al. (2018) \cite{Connor(2018)} applied a tree-like deep neural network (DNN) to the FRB searches. Zhang et al. (2018) \cite{2018ApJzhang} were the first to utilize the ResNet architecture for detecting fast radio bursts, and they detected 72 new pulses from FRB 121102 that had not been identified by the S/N-based methods in previous searches \cite{Gajjar2018}.
Artificial neural networks, especially CNNs, have played an important role in earlier pulsar candidate selection. For instance, Zhu et al. (2014)\cite{zhu2014searching} proposed a pulsar image-based classification system. In accordance with the four important features of a pulsar candidate (the summed profile, time versus phase plot, frequency versus phase plot, and DM curve), they developed the design of an ensemble network combined with a single-hidden-layer network, an SVM, and a CNN. Among these, CNNs have demonstrated good classification performance for 2D diagnostic plots. For example, a CNN is a typical end-to-end model when compared to other methods. In constrast, other methods require hand-crafted features for performing automatic pulsar candidate selection. Therefore, it is necessary for a pulsar expert to design artificial features, which may decrease the efficiency of pulsar candidate selection. However, the objective of this study is to replace the CNN with ResNet in PICS, which is developed using a TensorFlow framework for classifying the efficiency.
\subsection{Residual network design}
A CNN is an excellent feature extractor because it performs end-to-end learning of the raw image data to be classified. In PICS, the CNN model is relatively slow in training and classification because it is not optimized to run on either a CPU or a GPU. In addition, for legitimate pulsar candidates, there are one or more vertical lines in the 2D subplot (time versus phase and frequency versus phase); these features are usually captured in the early layers of the neural network. However, CRAFTS generates a wide range of pulsar candidates, especially those exhibiting sub-pulse drift. Deep networks must be designed to recognize these complex features. Therefore, we have adopted a residual network model \cite{He2016Deep} based on GPU acceleration for performing 2D subplot classification.
\begin{figure}[H]
\centering
\includegraphics[width=0.7\linewidth]{Residual.eps}
\caption{Illustration of a residual block. Residual learning: a building block.}
\label{fig:residual_block}
\end{figure}
A residual network can be considered to be a typical CNN. Deep networks tend to have a low back propagation error owing to gradient degradation; this problem arises during the deep network training process. The error term gradually decreases during the process of backward propagation, reducing the learning effectiveness in the top layers of the network. He et al.\cite{He2016Deep} proposed a residual learning framework (Figure ~\ref{fig:residual_block}) to address this degradation problem. The principle of this approach involves an "identity-connection" that connects the blocks of the convolutional layers and computes the shallow and deep features. The identity connections are shortcuts preventing the gradients from vanishing too rapidly. Thus, multiple residual networks can be stacked together to form deep networks without suffering from gradient degradation. In practice, this type of deep residual network can improve the classification accuracy. Hardt et al. (2016) \cite{2016arXiv161104231H}, Du et al. (2018) \cite{2018arXiv181103804D} have also proposed theories for ensuring the effectiveness of the ResNet method.
We designed the ResNet architecture according to the characteristics of the 2D subplots of pulsar candidates and the experimentation performed using various combinations of hyperparameters. We varied the network depth, feature map numbers, and kernel size and observed the changes in classification accuracy. Further, we determined the network structure and hyperparameters based on multiple factors, including the computation time, classification accuracy, and performance convergence. We conducted an experiment using networks of different depths and discovered that both a shallow 9-layer ResNet and a deep 39-layer ResNet exhibited lower accuracy than that exhibited by a 15-layer ResNet. Thus, our ResNet model comprises 15 layers (Figure~\ref{fig:ResNet model}).
The network layers include one input convolutional layer, two output layers, and 12 convolutional layers. The 12 convolutional layers are divided into three groups, each group containing two residual blocks and each residual block containing two convolutional layers.
Figure~\ref{fig:residual_block} displays the structure of a residual block; in each residual block, the parameters of the two convolutional layers are identical, and two residual blocks in the same group are observed to be identical to each other. After the residual blocks is the 14~th layer, which is a global average pooling layer that is connected to the 15~th layer, a fully connected softmax output layer.
While using this model to classify the pulsar candidates, we preprocessed the feature data before feeding them to the neural network. This procedure involved re-scaling the data to the zero mean and unit variance and shifting the peak to a phase of 0.5 for both the pulsar profile and 2D plots; however, we did not shift the peak of the DM curve.
\begin{figure*}
\centering
\includegraphics[width=\textwidth,scale=0.8]{ResNet_15.eps}
\caption{Diagram of a 15 layered ResNet model. Conv refers to convolutional operation. The "2 Residual Blocks" component contains two building blocks that possess an identical architecture. The input image is $64 \times 64 \times 1$, and the output sizes are $H \times W \times N$, where H and W denote the height and width of the tensor, and N denotes the number of features.}
\label{fig:ResNet model}
\end{figure*}
Figure~\ref{fig:ResNet model} presents the design of the ResNet model used for classifying the 2D images. We extracted four main features plots of a candidate from the pfd files; these plots included one-dimensional (1D) data arrays (summed profile and DM curve) and 2D data arrays (time versus phase and frequency versus phase subplots). Further, the original size of the feature data varied among various candidates. However, because our machine learning models functioned only using input data of identical size, we downsampled or interpolated the data to a uniform size before feeding them to our model. For a 1D subplot, we applied an SVM model in which the size of the input array was $64 \times 1$. In contrast, for the 2D subplot, we used the ResNet model in which the size of the input array was $64 \times 64$.
\begin{figure*
\centering
\includegraphics[width=\textwidth,scale=0.5]{PICS-ResNet.eps}
\caption{Diagram of the PICS--ResNet model. The first layer classifies the individual features (the pulse profile, time versus phase plot, frequency versus phase plot, and DM curve), whereas the second layer classifies the candidates based on the results of the first layer. The SVM components represent the support vector machine model, while the ResNet components represent the residual network model. See Figure~\ref{fig:ResNet model} for a schematic of the ResNet model.}
\label{fig:our method}
\end{figure*}
\section{Results}\label{sec:3}
In this section, we present the training and validation of PICS--ResNet, which is our new model and compare it with its predecessor, which is a generic PICS model. The generic PICS model used in this study exhibits the same architecture as that exhibited by the model described by Zhu et al. (2014) \cite{zhu2014searching} but with several differences. The original PICS model used a Bayesian prior to reject RFIs based on the frequency population of all the candidates (Lee et al. 2013 \cite{lee2013peace}; Zhu et al. 2014 \cite{zhu2014searching}). The PICS model in this study is trained with a larger amount of data than that used to train the original model; however, it does not reject RFIs based on their periods. Further, the results of the following three experiments designed to test robustness of the models are presented: 1. training both the models using an old dataset and testing them using the GBNCC dataset; 2. training both the models using a new dataset, including the FAST data, and testing them using the GBNCC dataset; 3. testing the models trained with the FAST data using a small set of reserved FAST test data. The results demonstrate that the PICS--ResNet model converges better in training and outperforms the PICS model in testing. Both the models display improvement after being trained with new data obtained from the FAST, exhibiting a recall rate of more than 90\% .
\subsection{Datasets and evaluation metrics}
The pulsar candidate datasets that were used included the PALFA, HTRU, GBNCC, and FAST datasets (Table \ref{tab:table0}). Some of the datasets, such as the PALFA and GBNCC datasets, were used and presented in a previous study \cite{zhu2014searching}. The HTRU dataset was obtained via private communication with C. Ng, while the FAST data was collected in this study.
Similar to Zhu et al. (2014) \cite{zhu2014searching}, we used the GBNCC dataset as an independent test dataset to verify the generalization of the models.
Table \ref{tab:table0} displays the number of positive and negative examples in each dataset. In the GBNCC dataset, the positive candidates contained two types of labels, among which one indicated the fundamental signals of the pulsars, whereas the other indicated the harmonic signals of the pulsars.
\begin{tablehere}
\caption{Number of positive and negative examples in the datasets.}\label{tab:table0}
\vspace{-1mm}\footnotesize
\begin{center} \doublerulesep 0.1pt \tabcolsep 15pt
\begin{tabular}{lccccccc}
\hline
Dataset Names & Positive & Negative & Total \\
\hline
PALFA & 3,951 & 6,672 & 10,623 \\
HTRU & 903 & 271 & 1,174 \\
FAST & 837 & 998 & 1,835 \\
GBNCC & 277 & 89,731 & 90,008 \\
\hline
\end{tabular}
\end{center}
\end{tablehere}
\begin{tablehere}
\caption{Binary classification confusion matrix, which defines all the outcomes of predictions, including true negative (TN), false negative (FN), false positive (FP), and true positive (TP).}
\label{tab:table1}
\vspace{-1mm}\footnotesize
\begin{center} \doublerulesep 0.1pt \tabcolsep 15pt
\begin{tabular}{lccccc}
\hline
Outcomes & Negative Prediction & Positive Prediction \\
\hline
RFI & True Negative & False Positive \\
True pulsar & False Negative & True Positive \\
\hline
\end{tabular}
\end{center}
\end{tablehere}
We generally consider pulsar candidate selection to be a binary classification problem despite the fact that datasets occasionally contain more than two labels.
In our dataset, pulsars and their harmonic signals considered to be positive examples, while all the RFIs are considered to be negative examples.
The evaluation metrics that are adopted for performing pulsar candidate classification are precision, recall and \textit{F$_1$} score. After defining a binary classification confusion matrix (Table \ref{tab:table1}), the metrics can be defined as follows:
\begin{equation}\rm{Precision} = \frac{\rm{TP}}{\rm{TP} + \rm{FP}},\end{equation}
\begin{equation}\rm{Recall} = \frac{\rm{TP}}{\rm{TP} + \rm{FN}},\end{equation}
\begin{equation}\textnormal{\rm{F$_1$ score}} = \frac{2 \times \rm{Precision} \times \rm{Recall}}{\rm{Precision} + \rm{Recall}}.\end{equation}
\subsection{Model training}
\begin{table*
\caption{The number of training samples was continuously increased from 2,000 to 12,000. Five validation tests were performed for each data point, and the means of the \textit{F}$_1$ score were obtained.}\label{tab:table_curve
\begin{center}\vspace{-2mm}\footnotesize \doublerulesep 0.2pt \tabcolsep 16pt
\begin{tabular*}{\textwidth}
{cccccccc}\toprule[0.65pt]
&\multicolumn{6}{c} {Training dataset}
\\
\cline{2-7\ }
&$N$=2000&$N$=4000& $N$=6000&
$N$=8000& $N$=10000& $N$=12000&
\\ \hline
PICS Training \textit{F}$_1$ score&0.98&0.98&0.99&0.99&0.99&0.99\\
PICS--ResNet Training \textit{F}$_1$ score&0.98&0.98&0.98&0.98&0.98&0.98\\
PICS validation \textit{F}$_1$ score&0.86&0.90&0.91&0.91&0.92&0.92\\
PICS--ResNet validation \textit{F}$_1$ score&0.89&0.91&0.92&0.92&0.91&0.92\\
\bottomrule[0.65pt]
\end{tabular*}
\end{center}
\end{table*}
In this subsection, we describe the training process of the 15-layer ResNet and PICS--ResNet models and observe the manner in which the models can converge. We used the same dataset, which included the PALFA, HTRU, and FAST dataset (Table \ref{tab:table0}) for training. There were a total of 13,632 labeled candidates in this dataset, among which 5,692 were pulsars (and their harmonic signals) and 7,940 were non-pulsars.
We used grid search to fine-tune the hyperparameters of the ResNet and selected the set of parameters leading to the optimal cross-validation \textit{F}$_1$ score and as well as a well-converging model. For the cross-validation tests, we randomly select 80\% of the entire dataset for training and the remaining 20\% for performing validation. Figure~\ref{fig:resnet_learning} illustrates the convergence of the 15 layers of the ResNet over the training process. The hyperparameters of the ResNet model were set to an initial learning rate of 0.01, a batch size of 32, a weight decay rate of 0.002, a ReLU leakiness of 0.1, and a training epoch of 80.
\begin{figure}[H]
\centering
\includegraphics[scale=0.4]{resnet_learning_curve.eps}
\caption{Convergence of the 15 layers of ResNet over the training process. The x-axis represents the training epochs, whereas the y-axis represents the \textit{F}$_1$ score.}
\label{fig:resnet_learning}
\end{figure}
The training and validation tests were performed to determine the manner in which the models converged and to identify the signs of overfitting. In the training process, we gradually increased the size of the training data and visualized the convergence of the training and validation \textit{F}$_1$ scores. We first used 2,000 samples to train the model, followed by the usage of 1,363 samples for validation. Next, we increased the training set size to 12,000. In each step, we randomly selected 1,363 samples as validation data; these data did not contain the training data. We performed five independent cross-validation tests to evaluate the reliability of the training process, with each test exhibiting a different random split of the training and validation data. The resulting \textit{F}$_1$ scores are presented in Table \ref{tab:table_curve} and the learning curves are plotted in Figure~\ref{fig:learning}.
Figure~\ref{fig:learning} demonstrates that the training and validation scores of both the PICS and PICS--ResNet models improve as the size of the training set increases and that the PICS--ResNet model converges better than the PICS model.
The results indicate that the training size remains insufficient for the training and validation curve to completely converge and that some degree of overfitting still exists in the models. However, the PICS--ResNet demonstrates slightly superior performance and convergence when compared to those exhibited by the PICS.
The mean validation \textit{F}$_1$ score is 0.92 for PICS and 0.92 for PICS--ResNet.
\begin{figure}[H]
\centering
\includegraphics[scale=0.4]{PICS-ResNet_learning_curve.eps}
\caption{The PICS--ResNet and PICS learning curves. The x-axis represents the training sample set from 2,000 to 12,000, whereas the y-axis represents \textit{F}$_1$ score.}
\label{fig:learning}
\end{figure}
\subsection{Generalization test using the GBNCC dataset \label{sec:gbncc}}
To independently validate our model, we applied it to a large set of GBNCC candidates that were excluded while training the model. Similar to the FAST, PALFA, and HTRU, the GBNCC survey used the \textsc{presto} search pipeline to generate candidates in the pfd file format; however, it used a low observing frequency ($\sim$300 MHz). Thus, the RFI environments are unique to the Green Bank telescope. All the candidates in the GBNCC dataset were verified and labeled by unprejudiced human experts; therefore, this dataset represents a realistic survey. However, similar to any genuine survey candidate pool, the dataset is considerably imbalanced. There are 56 pulsars, 221 harmonics, and 89,731 RFI candidates (Table \ref{tab:table0}). The result of the GBNCC dataset serve to measure the performance of our model in a realistic pulsar survey and to test its generalizability with respect to a different telescope, different observing frequency, and different RFI environment.
To determine whether the inclusion of new data improves the performance and generalization of the models, we conducted experiments using the GBNCC test data using the following two steps: 1. we trained our models using an old dataset that excluded the FAST data, and 2. we further trained the models using a large dataset that included FAST data. In training the PICS--ResNet model, we collected 11,797 training samples from the PALFA and the HTRU South survey (Table \ref{tab:table0}). The model ran for 80 epochs, usually with an iterative process and a checkpoint being saved after each epoch. In addition, we used a standard momentum optimizer as the learning algorithm for training in which the initial learning rate was set to 0.01, the weight decay was set to 0.002, and the batch size set to 32.
We used the same training set to train the PICS model; the results are presented in Table \ref{tab:table2}. PICS sorted 52 pulsar fundamental signals and 194 harmonics into the top $1\%$ of 90,008 candidates, while PICS--ResNet sorted 51 fundamental signals and 190 harmonics into the top $1\%$. The recall curves of the two models are denoted in Figure~\ref{fig:recall_curve}. The total number of pulsar fundamental signals is 56, and the number of harmonics is 221.
Table \ref{tab:table2} reveals that when trained with the old dataset, both PICS and PICS--ResNet reach a $\sim$90\% recall rate for the GBNCC test by considering only the top 1\% of the candidates.
\begin{tablehere}
\caption{Classification results of the PICS and PICS--ResNet models using the GBNCC dataset when the models were trained with the PALFA and HTRU data but not the FAST data. Only the candidates who were ranked in the top $1\%$ of the dataset were selected, and the recall rates of the pulsars and their harmonics were calculated. }
\label{tab:table2}
\vspace{-1mm}\footnotesize
\begin{center} \doublerulesep 0.1pt \tabcolsep 2pt
\begin{tabular}{lccccccc}
\hline
Method & Fundamental (recall) & Harmonic (recall) & PSR (recall) \\
\hline
PICS (Top $1\%$) & 52 ($93\%$) & 194 ($88\%$) & 246 ($89\%$)\\
PICS--ResNet (Top $1\%$) & 51 ($91\%$) & 190 ($86\%$) & 241 ($87\%$)\\
\hline
\end{tabular}
\end{center}
\end{tablehere}
To dtermine the influence of FAST data on the performance of the models, we trained the models using additional data collected from the FAST drift-scan pulsar survey.
A total of 13,632 samples (see Table \ref{tab:table0}) were used from the PALFA, HTRU, and FAST. The test results are presented in Table \ref{tab:table3}.
PICS--Resnet sorted 54 fundamental pulsars and 211 harmonics. Among the results, the top 210 candidates were all pulsars, with the first non-pulsar being the 211$^{th}$ candidate. There were 254 pulsars among the top 290 candidates. We used the same training data to train the PICS, and the resulting model sorted 52 fundamental pulsars and 201 of their harmonics into the top 1\% of all the candidates.
Table \ref{tab:table3} demonstrate that when trained with the FAST data, both PICS and PICS--ResNet exhibited superior performance, and PICS--ResNet reached a $\sim$96\% recall rate in the GBNCC test by considering only the top 1\% of all the candidates. Figure~\ref{fig:recall_curve} displays the recall curves for the two models trained using FAST data.
The results reveal that the inclusion of new data obtained from the FAST significantly improved the models' recall rates with respect to the harmonic signals of pulsars. We conclude that increasing the size and diversity of the training data can increase the generalization of the models.
\begin{tablehere}
\caption{Classification results of the PICS and PICS--ResNet models using the GBNCC dataset. The models were trained with datasets from the PALFA, HTRU, and FAST. Only the candidates ranked in the top $1\%$ of the dataset were selected, and the recall rates of the pulsars and their harmonics were calculated. }
\label{tab:table3}
\vspace{-1mm}\footnotesize
\begin{center} \doublerulesep 0.1pt \tabcolsep 2pt
\begin{tabular}{lccccccc}
\hline
Method & Fundamental (recall) & Harmonic (recall) & PSR (recall) \\
\hline
PICS (Top $1\%$) & 52 ($93\%$) & 201 ($91\%$) & 253 ($91\%$)\\
PICS--ResNet (Top $1\%$) & 54 ($96\%$) & 211 ($96\%$) & 265 ($96\%$)\\
\hline
\end{tabular}
\end{center}
\end{tablehere}
\begin{figure}[H]
\centering
\includegraphics[scale=0.4]{recall_candidate.eps}
\caption{PICS--ResNet and PICS recall curves. The x-axis represents the fraction of the candidates examined, whereas the y-axis represents the recall rate of the examined candidates. Here, the recall rate is calculated based on the fraction of pulsar signals (including 56 fundamental signals and 221 harmonic signals) ranked in the top $X$ fraction of the candidates.}
\label{fig:recall_curve}
\end{figure}
While testing using the GBNCC dataset, we also evaluated the runtime of the models.
It took the ResNet model $\sim$0.22 ms to process one 2D plot using a cluster of 24 2.7--GHz CPUs and two 1080Ti GPUs. In contrast, it took the CNN model of the PICS system $\sim$7.2 ms to process one 2D plot. It should be noted that our method ran efficiently because the CPU and GPU were running concurrently. It took PICS--ResNet 61 s and PICS 95 s to predict 10,000 candidates. Overall, it took $\sim$79 minutes for PICS--ResNet and $\sim$90 minutes for PICS to complete the classification of 90,008 candidates. The PICS--ResNet model exhibits a slightly faster performance than that exhibited by the PICS model.
\subsection{Testing using the FAST data}
We used new data collected from the FAST drift--scan survey to further verify the robustness of our model. We collected 317,018 candidates from the survey; 15,542 were labeled by human experts, among which 1,163 were pulsars or their harmonics (hereafter simply referred to as pulsars). Both the PICS and PICS--ResNet models were trained with 13,632 samples (Table \ref{tab:table0}), including 1,835 samples obtained from the FAST data. An additional 326 pulsar samples and 13,321 RFI samples from the FAST data were reserved for testing. Table \ref{tab:table4} reveals that PICS--ResNet identified 320 pulsars and missed six, while PICS identified 310 pulsars and missed 16. This demonstrates that when the FAST data is used, PICS can achieve a recall rate of 95\%, whereas PICS--ResNet can achieve a recall rate of 98\%. In this experiment, the recall rate was calculated using a score threshold of 0.5. This was because the FAST test data was not as imbalanced as the GBNCC data, and there would have been considerably few candidates in the top 1\% to include all the true positive samples. Therefore, the recall numbers in Tables \ref{tab:table4} and \ref{tab:table3} should not be directly compared.
\begin{tablehere}
\caption{Number of pulsars identified by each model when tested using the FAST data. The test set comprised a total of 326 pulsar candidates.}
\label{tab:table4}
\vspace{-1mm}\footnotesize
\begin{center} \doublerulesep 0.1pt \tabcolsep 8pt
\begin{tabular}{lccccccc}
\hline
Method & Recognition pulsar & Missing pulsar & Recall \\
\hline
PICS & 310 & 16 & $95\%$\\
PICS--ResNet & 320 & 6 & $98\%$\\
\hline
\end{tabular}
\end{center}
\end{tablehere}
\section{Discussion and conclusion}\label{sec:4}
In this study, we propose PICS--ResNet, which is a new ensemble model for pulsar candidate selection. This model inherits a two-layer structure from PICS \cite{zhu2014searching} and replaces the lower-layer CNN classifier with a ResNet-based model for performing 2D data classification. The PICS--ResNet model uses 15 layers of a deep neural network to identify 2D subplots and utilizes SVM to identify a 1D subplot. Our experimental results demonstrate that the PICS--ResNet model converges better than the PICS model during training and performs better than the PICS model when trained with the FAST data. In addition, when the FAST data is considered in training, the resulting models exhibit improved preformance in a test using the GBNCC data and a test using the FAST data. When trained with the FAST data and tested with the GBNCC data, PICS--ResNet sorts 96\% pulsars into the top 1\% of all the candidates. In addition, an experiment using FAST pulsar survey data demonstrates that PICS--ResNet can identify 98\% of pulsars with a score threshold of 0.5.
Our experiments demonstrate that the ResNet model itself does not display significant signs of overfitting while classifying the 2D image features (Figure \ref{fig:resnet_learning}); however, the ensemble model appears to display slight overfitting, as illustrated by the learning curve (Figure \ref{fig:learning}). This remains a caveat because it is helpful in practice to exhibit a considerable separation in final scores while applying this model to data. In addition, our experimental results demonstrate that the inclusion of the data obtained from the FAST enhances the recall rate of the models with respect to the pulsar (harmonic) signals. Therefore, we can conclude that the training data exhibit insufficient diversity and that gathering additional data is an effective method for improving the pulsar candidate selection accuracy. Further, the simulated data can also be used as a supplement to the real data obtained from various surveys.
The PICS--ResNet model was developed using the TensorFlow platform, and our experiments revealed that it can classify more than 1.6 million candidates per day using a dual GPU 24-core desktop computer (see Section \ref{sec:gbncc} for more details). The development code of the PICS--ResNet model can be found at https://github.com/dzuwhf/PICS-ResNet. In addition, our labeled FAST data has been made public on https://github.com/dzuwhf/FAST\_label\_data.
\vspace*{2mm} \Acknowledgements{\bahao The authors thank the referee's constructive comments and suggestions. The research work is supported by National Key R$\&$D Program of China No. 2017YFA0402600 and Natural Science Foundation of Shandong (No.ZR2015FL006). This work supported by the CAS International Partnership Program No.114A11KYSB20160008, the Strategic Priority Research Program of the Chinese Academy of Sciences Grant No. XDB23000000. This project is also supported by National Natural Science Foundation of China Grant No. 61472043, 11743002, 11873067, 11690024, 11725313, the Joint Research Fund in Astronomy (U1531242) under cooperative agreement between the NSFC and CAS and National Natural Science Foundation of China (grant No.11673005). WWZ is supported by the Chinese Academy of Science Pioneer Hundred Talents Program. The authors also thank Chavonne Bowen and Alan Ho for labeling FAST pulsar candidates, the PALFA, GBNCC team and the Arecibo Remote Command Center (ARCC) students, Cherry Ng, Meng Yu, et al. for labeling and sharing their data. }
|
1,116,691,497,571 | arxiv | \section{Introduction}
Buildings account for a quarter of total energy consumption globally and up to $40 \%$ in developed countries; the majority of which is used for heating (including hot water), cooling and ventilation. Electric heating with heat pumps is slowly displacing the use of oil and natural gas \cite{perez-lombardReviewBuildingsEnergy2008, urge-vorsatzHeatingCoolingEnergy2015}. Meanwhile, the total energy consumption for cooling is expected to grow substantially, especially in developing countries \cite{isaacModelingGlobalResidential2009, FutureCoolingAnalysis}.
Model Predictive Control (MPC) has been successfully applied to buildings in simulations and experiments many times \cite{seraleModelPredictiveControl2018,aframTheoryApplicationsHVAC2014,mirakhorliOccupancyBehaviorBased2016,drgonaAllYouNeed2020}. While the most common objectives in these studies are to reduce the energy consumption or to increase thermal comfort, there is an increasing focus on harnessing buildings for electric demand response. Considering the ongoing electrification of heating and the increase in cooling on one side, as well as the increasing share of non-dispatchable sources of electricity like wind and solar on the other side, this trend can be expected to continue. Unlike the first two objectives, which can be achieved through predictive and non-predictive methods, demand response usually requires a reasonably accurate prediction of the electricity demand several hours into the future \cite{qureshiModelPredictiveControl2014}.
However, MPC is yet to find widespread commercial application in buildings. A commonly cited reason is the high upfront engineering cost of identifying a sufficiently accurate model of the building dynamics \cite{seraleModelPredictiveControl2018,killianTenQuestionsConcerning2016,woliszSelflearningModelPredictive2020,sturzeneggerModelPredictiveClimate2016}.
Buildings have a number of properties that inform the selection of an appropriate model structure: Unlike most industrial products, they are generally unique, which means a different model must be built for each building. There is also a great variety in size and complexity, ranging from a single thermal zone to hundreds. While we usually measure and control the air temperature in the occupied zones, which have time constants in the range of minutes, most of the thermal energy is stored in the walls, floors and furniture, which can have time constants in the range of hours.
Furthermore, buildings show slowly time-varying behavior, including cyclical changes between seasons, but also long-term changes due to degrading insulation or altered furnishing. However, the instantaneous dynamics of a building can be modeled as an LTI system with sufficient accuracy \cite{seraleModelPredictiveControl2018, drgonaAllYouNeed2020}.
The execution of system identification experiments with high excitation of the heating and cooling systems is constrained by a desire to maintain occupant comfort during occupied hours and to minimize strain on the HVAC components.
Despite these challenges, we desire to obtain reasonably accurate predictions over a $24 h$ horizon, with the goal to participate in demand response.
The intention to eventually deploy our work at scale outside of academic settings adds additional challenges: Sensors in buildings are often inaccurate and biased, and a replacement or calibration may not be an option. In addition, excessive computational requirements should be avoided to keep the system hardware costs as low as possible.
To address these issues, we propose the use of adaptive, linear black-box models \cite{maddalenaDatadrivenMethodsBuilding2020,hilliardModelPredictiveControl2016}. In particular, we investigate the performance of a recently popularized approach based on behavioral systems theory, further detailed in sections \ref{SecDeePC} and \ref{DeePCdetail}. These approaches omit the step of model identification altogether. However, they have a theoretical equivalence to linear time-invariant models. Therefore, we choose autoregressive models with exogenous inputs (ARX) as a well-established reference method. Furthermore, we evaluate non-adaptive variants of both methods to assess the benefits of adaptivity.
Other methods have been used to model buildings in the existing literature, but were disregarded for the scope of this paper. Neural networks and Gaussian process models can account for nonlinearities, but they require large amounts of data to train and a computationally expensive nonlinear optimization problem to be solved. This contradicts our goal to identify a mostly linear system with limited computational effort. A combination of a state-space model and a Kalman filter for state estimation was considered as an alternative to ARX, but not chosen due to the higher complexity with no obvious advantages in this context.
This study evaluates the ability of the selected modeling approaches to predict the zone temperature with sufficient accuracy on two long-term measurements from real occupied residential buildings, detailed in section \ref{SecData}. The data stems from regular operation, meaning that there is no active excitation for the purpose of system identification. Furthermore, the data contains gaps in the measurements and unknown levels of process and measurement noise. Testing under these challenging conditions is essential to draw conclusions for a future deployment in real-world buildings at scale.
\section{Theoretical background}
\subsection{Data-enabled predictive control}\label{SecDeePC}
Data-enabled predictive control (DeePC) is a formulation of MPC based on behavioral systems theory. It was first published and named by Coulson in 2019 \cite{coulsonDataEnabledPredictiveControl2019}. Unlike classical MPC, no model of the plant is identified. The solution to the predictive control problem is generated directly by a linear combination of measured trajectories of the plant. This section briefly explains the method:
Assume a linear, time-invariant system $\mathcal{B}$ of order $n_{sys}$ with discrete-time input-output samples $z_k = $
$\begin{bmatrix}
u_k & y_k
\end{bmatrix}
^\top$
and a measurement of $T$ samples $z_{[1:T]}$. Willems' fundamental lemma \cite{willemsNotePersistencyExcitation2005} shows that any possible trajectory of system $\mathcal{B}$ of length $T_f$ can be constructed as a linear combination of the columns of the Hankel matrix of depth $L=n_{sys} + T_f$:
$\mathcal{H}_L(z_{[1:T]}) = \begin{bmatrix}
z_1 & \dots & z_{T-L+1}\\
\vdots & \ddots & \vdots\\
z_L & \dots & z_T
\end{bmatrix}$
if $\mathcal{H}_L(u_{[1:T]})$ has full row rank.
Now consider the Hankel matrices $\mathcal{H}_L(u_{[1:T]}) = \begin{bmatrix}
\mathcal{H}_{T_{ini}}^u\\
\mathcal{H}_{T_{f}}^u
\end{bmatrix}$
and $\mathcal{H}_L(y_{[1:T]}) = \begin{bmatrix}
\mathcal{H}_{T_{ini}}^y\\
\mathcal{H}_{T_{f}}^y
\end{bmatrix}$
each split into two components for initialization and prediction and combined to $\mathcal{H}_L(z_{[1:T]}) = \begin{bmatrix}
\mathcal{H}_{T_{ini}}^u\\
\mathcal{H}_{T_{ini}}^y\\
\mathcal{H}_{T_{f}}^u\\
\mathcal{H}_{T_{f}}^y
\end{bmatrix}$.
As well as the right-hand vector $v = \begin{pmatrix} u_{ini} \\ y_{ini} \\ u \\ y \end{pmatrix}$, consisting of the $T_{ini}$ most recent samples $u_{ini}$ and $y_{ini}$, for initialization and the optimization variables $u$ and $y$.
With $\mathcal{H}_L(z_{[1:T]})$ simplified to $\mathcal{H}$ and the introduction of the optimization variable $g$, which represents the linear combination of the columns of $\mathcal{H}$, the predictive controller is formulated:
\begin{align*}
&\min_{g, u, y} J(u, y) \\
&\text{subject to} \nonumber \\
&\mathcal{H} g = v \\
&u \in \mathcal{U} , y \in \mathcal{Y}
\end{align*}
Many studies employing this method have been published since it was first proposed. A recent literature review is given in \cite{markovskyBehavioralSystemsTheory2021a}.
\subsection{Persistency of excitation for different trajectory matrices}
Persistency of excitation (PE) is a condition on a measured data set for the underlying system to be identifiable. Willems' fundamental lemma formulates this condition for data in Hankel matrix form. Subsequent publications have extended this to mosaic-Hankel matrices and Page matrices, shown in Tab. \ref{tab:PEconditions}. A mosaic-Hankel matrix is a horizontal concatenation of multiple Hankel matrices. A Page matrix is similar to a Hankel matrix, but does not have repeat entries. It instead contains a continuous string of samples, similar to words on a page, hence the name. We note that the very strict condition for the Page matrix is sufficient but not necessary. In fact, a quadcopter is successfully controlled with a Page matrix in \cite{coulsonDistributionallyRobustChance2021}, despite grossly violating the stated PE condition derived in that paper. This serves as an example of the importance of experimental studies complementing the theoretical work recently published in this field.
In our study, we use Hankel-like matrices comprised of individual, disjointed, but overlapping trajectories. To the best of our knowledge, no PE condition has been formalized for such an unstructured matrix. Considering the aforementioned findings, we conduct a study on real data without an adapted PE condition available.
\begin{table}[]
\begin{tabular}{l | l | l}
Matrix & PE condition & Minimal $T$ \\ \hline
Hankel \cite{coulsonDistributionallyRobustChance2021} & $\mathcal{H}_L(u_{[1,T]})$ has full row rank & \begin{tabular}[c]{@{}l@{}}$T \geq L(m+1)-1$\\ is of order $L$\end{tabular} \\ \hline
mosaic-Hankel \cite{vanwaardeWillemsFundamentalLemma2020}& \begin{tabular}[c]{@{}l@{}}$\begin{bmatrix}
\mathcal{H}_k(u_{[0,T_1-1]}^1) & \mathcal{H}_k(u_{[0,T_2-1]}^2) & \dots & \mathcal{H}_k(u_{[0,T_q-1]}^q)
\end{bmatrix}$\\ has full row rank\end{tabular} & \begin{tabular}[c]{@{}l@{}}$\sum\limits_{i=1}^q T_i \geq k(m+q)-q$\\ is of order k\\ $len(T_i) \geq k$\\ Necessary condition\end{tabular} \\ \hline
Page \cite{coulsonDistributionallyRobustChance2021} & \begin{tabular}[c]{@{}l@{}}$\begin{pmatrix}
\mathcal{P}_L(u_{[1, T-(M-1)L]}) \\
\mathcal{P}_L(u_{[L+1, T-(M-2)L]}) \\
\vdots \\
\mathcal{P}_L(u_{[L(M-1)+1, T]}) \\
\end{pmatrix}$\\ has full row rank\end{tabular} & \begin{tabular}[c]{@{}l@{}}$T \geq L((mL+1)M-1)$\\ is of order k\\ Sufficient condition\end{tabular}
\end{tabular}
\caption{Persistent excitation condition and minimal number of sampling points for different trajectory matrix structures}
\label{tab:PEconditions}
\end{table}
\subsection{Persistent excitation in dual control} \label{PeDualControl}
In case of an adaptive scheme, where the Hankel matrix (or any other data structure) is continuously updated, continuously ensuring persistent excitation is a challenge. The published methods to increase the level of excitation for adaptive data-driven controllers can be grouped into three categories: The first is selecting suitable data from a long data log. The second is adding a perturbation to the solved predictive control trajectory. The third is including the level of excitation in the cost function \cite{berberichLinearTrackingMPC2021}.
However, buildings are significantly perturbed by the weather, which cannot be controlled. This favors the first method. Furthermore, artificially increasing the level of excitation of the control signal may increase wear on the actuators as well as constraint violations. In the case of buildings, the resulting temperature fluctuations may also decrease occupant comfort.
\subsection{DeePC in building control}
DeePC is derived for deterministic, linear, time-invariant systems. Since none of these conditions perfectly apply to buildings, experiments and high-fidelity simulations are necessary to evaluate the practically achievable level of performance.
The existing literature is sparse. Schwarz \cite{schwarzDataDrivenControlBuildings2020}, O’Dwyer \cite{odwyerDatadrivenPredictiveControl2021} and Chinde \cite{chindeDataEnabledPredictiveControl} have conducted simulation studies on DeePC in buildings. However, these are not adaptive. Kerkhof \cite{kerkhofOptimalControlAutonomous2020} has conducted a simulation study on adaptive DeePC in a greenhouse. However, a very short prediction horizon of one hour is used. Also, the dynamics of a greenhouse may be very different from those of a conventional building. Lian \cite{lianAdaptiveRobustDatadriven2021} has conducted an experimental study on adaptive robust bi-level DeePC in an educational building. Kerkhof and Lian both add artificial noise to the control signal to ensure persistent excitation.
\section{Methodology}
Since DeePC refers to a control method and we focus on system identification, the methods based on behavioral systems theory are referenced with the label ''BST'' hereinafter, rather than ''DeePC''. For each building, a total of 69 modeling methods are evaluated: One non-adaptive ARX, three adaptive ARX with different forgetting factors, five non-adaptive BST with different regularization weights, and 60 adaptive BST with different trajectory matrix widths, regularization weights and data selection methods. Non-adaptive variants are identified before the evaluation phase. Adaptive variants are updated at every step, starting with a 30 day initialization phase before the evaluation phase. For all methods, 12 steps are used for initialization and 96 steps for prediction, corresponding to $3 h$ and $24 h$.
The weather forecasts for the prediction are ideal. However, a manipulated forecast is used for the data selection for the adaptive BST variants, detailed in section \ref{WeaManMet}.
As is typical for long-term field measurements in buildings, there are many gaps in the data where sensors returned no measurement or physically impossible values. They are identified in the pre-processing. Uninterrupted trajectories of length $T_{ini}+T_f$ or greater are marked as admissible for system identification and validation. Inadmissible segments are skipped. For the adaptive ARX, the model update is halted when encountering a gap. For the non-adaptive ARX, a multi-batch identification method is used in case of gaps.
\subsection{BST variants in detail}\label{DeePCdetail}
A prediction of the zone temperature, based on the current state and a weather forecast, is performed at every time step. The control variables are the heating and cooling powers $u = \begin{bmatrix} P_{heat} & P_{cool} \end{bmatrix}$ (heating only for Basel building). The disturbance variables are the ambient temperature and solar radiation $w = \begin{bmatrix} T_{amb} & I_{sol} \end{bmatrix}$. The output variable is the zone temperature $y = T_z$. With the disturbance variable, the DeePC formulation from section \ref{SecDeePC} changes to:
$\mathcal{H}_L(z_{[1:T]}) = \begin{bmatrix}
\mathcal{H}_{T_{ini}}^u\\
\mathcal{H}_{T_{ini}}^w\\
\mathcal{H}_{T_{ini}}^y\\
\mathcal{H}_{T_{f}}^u\\
\mathcal{H}_{T_{f}}^w\\
\mathcal{H}_{T_{f}}^y
\end{bmatrix}$ and $v = \begin{pmatrix} u_{ini} \\ w_{ini} \\ y_{ini} \\ u \\ w_{forecast} \\ y \end{pmatrix}$
We further define $\hat{\mathcal{H}} = \begin{bmatrix}
\mathcal{H}_{T_{ini}}^u\\
\mathcal{H}_{T_{ini}}^w\\
\mathcal{H}_{T_{ini}}^y\\
\mathcal{H}_{T_{f}}^u\\
\mathcal{H}_{T_{f}}^w
\end{bmatrix}$ the first five elements of $\mathcal{H}_L(z_{[1:T]})$ and $\hat{v} = \begin{pmatrix} u_{ini} \\ w_{ini} \\ y_{ini} \\ u \\ w_{forecast} \end{pmatrix}$ the first five elements of $v$.
The following cost function comprised of a least-squares fit and a regularization on $g$ is minimized:
\begin{equation}
J = (\hat{\mathcal{H}} g - \hat{v})^\top W_{I} (\hat{\mathcal{H}} g - \hat{v}) + \lambda g^\top g
\label{eq:CostFunction}
\end{equation}
Which is solved analytically to:
\begin{equation}
g^* = (\hat{\mathcal{H}}^\top W_{I} \hat{\mathcal{H}} + \lambda I)^{-1} \hat{\mathcal{H}}^\top W_{I} \hat{v}
\end{equation}
\begin{equation}
y^* = \mathcal{H}_{T_{f}}^y g^*
\end{equation}
$W_{I}$ is a weighting matrix that gives the initialization steps 100 times the weight of the prediction steps. This is done to improve the accuracy of the early prediction steps, which are more critical for predictive control, since usually only the first element of the optimized trajectory is applied, before the entire optimization is repeated.
The widths of the trajectory matrices, denoted $W_H$, for the adaptive variants are $181$, $373$ and $661$. In a proper Hankel matrix, this would correspond to $3$, $5$ and $8$ days of data $(W_H = T - T_{ini} - T_f + 1)$. For the non-adaptive BST, all data from the identification phase is used to build one matrix several 1000 columns wide.
The regularization weights $\lambda$ are $10^0$, $10^1$, $10^2$, $10^3$ and $10^4$.
\subsubsection{Data selection methods in detail}\label{DatSelDetail}
We investigate the impact of different ways of composing the trajectory matrix. One method that is as close as possible to a proper Hankel matrix and three alternative approaches. Following, the four different methods used are explained. A simplified visualization is given in Fig. \ref{fig:DatSelVar}.
\textbf{\textit{Most recent:}} The $W_H$ most recent admissible trajectories are selected. If there are no gaps in the data, this yields a proper Hankel matrix. If there are gaps, we get a mosaic-Hankel matrix.
\textbf{\textit{Most correlated:}} The $W_H$ admissible trajectories whose weather is most correlated with the forecast are selected. This method serves the purpose of investigating if a selection of trajectories with a shape similar to the target trajectory is advantageous, compared to the aforementioned approach.
\textbf{\textit{Smallest RMSE:}} The $W_H$ admissible trajectories whose weather has the smallest RMSE relative to the forecast are selected. This method is similar to the correlation one, but is not invariant to scaling and offset. It is similar to a parameter-variant model, with the weather as the parameter in question. Due to the multi-step nature of the BST approach, a possible parameter variance would be captured implicitly, rather than explicitly modeled. Since ambient temperature and solar radiation generally follow a daily pattern, it also includes an element of time-variant modeling.
\textbf{\textit{Closest mean:}} The $W_H$ admissible trajectories whose averaged weather most closely match the corresponding values of the forecast are selected. This method seeks to strike a balance between selecting recent data and selecting similar data. Because average ambient conditions typically change on the scale of weeks and months, it is expected to be similar to selecting the most recent data.
Data selection is limited to the past 365 days for all methods. Since the calculation is initialized with roughly 30 days of data, the amount of available data increases with time, until the one year window is full. While a constant amount of data over the calculation period would be preferable, it is not possible due to the limited data sets.
The weather data consists of the normalized ambient temperature and solar radiation. The sum of the two correlation coefficients, RMSE and differences is used. The ambient temperature is normalized to $\hat{T}_{amb} = \frac{T_{amb} - 10 [K]}{20 [K]}$. This compresses the range $[- 10 ^{\circ} C, 30 ^{\circ} C]$ to $[-1, 1]$. For the solar radiation, the normalization is adapted to the specific sensor situations detailed in section \ref{SecData}. $\hat{I}_{sol} = I_{sol}/(500 [W/m^2])$ for Zürich and $\hat{P}_{PV} = P_{PV}/(3 [kW])$ for Basel. These values are chosen to rescale the values to roughly $[0, 1]$. The different normalization ranges counteract the fact that the ambient temperature usually stays within a narrow segment of the normalization range during $24 h$. The solar radiation, on the other hand, regularly spans much of its normalization range during one summer day.
Despite these methods not yielding actual Hankel matrices, we denote them all with the symbol $\mathcal{H}$ for simplicity.
\begin{figure}[htp]
\centering
\includegraphics[width=8cm]{Plots/DataSelectVariant.pdf}
\caption{Simple visualization of the data selection variants. Four past days are compared to one forecast reference day and categorized into closest mean, most correlated, smallest RMSE and most recent.}
\label{fig:DatSelVar}
\end{figure}
\subsubsection{Manipulated weather forecast for data selection}\label{WeaManMet}
There is a key difference between the variants based on selecting similar weather and the others: They are impacted by the inaccuracy of the weather forecast twice. Once in the selection step and once in the prediction step. While we use an ideal forecast for all variants in the prediction, which creates an equal comparison, we manipulate the forecast for the data selection step to simulate the impact of this double-dependency.
To the best of our knowledge, there is no readily available tool to generate realistic artificial forecasts based on measured data. Therefore, we defined our own method. While a real (or at least realistic) forecast would have been preferred, we consider this to reasonably serve the purpose of simulating the inaccuracy of a real forecast for the scope of this study. For the same reason, \cite{scharnhorstEnergymBuildingModel2021} relies on a self-made method for skewing of the weather data as well.
A randomized sine function is generated, that grows linearly over the prediction horizon of $24 h$. For the ambient temperature, it is used in an additive way and grows from $0 K$ to a maximum of $\pm 4 K$. For the solar radiation, it is used in a multiplicative way and grows from $1$ to a maximum $\pm 15 \%$. Fig. \ref{fig:distortWeather} shows some example trajectories for the multiplicative case.
\begin{align*}
& f_{scale} = 1 + 0.15 \frac{t}{24} sin(a t + b) \\
& \text{with} \nonumber \\
& a = \mathcal{U}(0.5 \frac{\pi}{12}, 1.5 \frac{\pi}{12}) \text{(Uniform distribution)} \\
& b = \mathcal{U}(-\pi, \pi) \\
& t = (0, 24) \text{(Hour of day)}
\end{align*}
\begin{figure}[htp]
\centering
\includegraphics[width=8cm]{Plots/distortWeather.pdf}
\caption{Five example trajectories of the distortion of the weather forecast}
\label{fig:distortWeather}
\end{figure}
\subsection{ARX variants in detail}
An ARX model has the following structure \cite{EstimateParametersARX}:
\begin{equation}
y(k) = a_1 y(k-1) + ... + a_{n_a} y(k - n_a) = b_1 u(k - n_k) + ... + b_{n_b} u(k - n_b - n_k + 1)
\end{equation}
With $n_a$ the number of poles, $n_b$ the number of zeros and $n_k$ the number of input samples that occur before the input affects the output. The parameters are estimated by a least-squares fit. The prediction over the whole horizon is iterative by chaining the one-step prediction model $T_f$ times.
For the adaptive ARX, a forgetting factor method is used \cite{MathworksRecursiveArx}:
\begin{equation}
\hat{\Theta}(k) = \hat{\Theta}(k - 1) + K(k)(y(k) - \hat{y}(k))
\end{equation}
Where $\hat{\Theta}(k)$ is the parameter estimate at time k, $y(k)$ is the measurement and $\hat{y}(k)$ is the prediction thereof. The gain has the form:
\begin{equation}
K(k) = Q(k) \Psi(k)
\end{equation}
Where $\Psi(k)$ is the gradient of the predicted output $\hat{y}(k)$ with respect to $\Theta$. We further define:
\begin{equation}
Q(k) = \frac{P(k - 1)}{\alpha + \Psi^\top(k) P(k - 1) \Psi(k)}
\end{equation}
and
\begin{equation}
P(k) = \frac{1}{\alpha} \left( P(k - 1) - \frac{P(k - 1) \Psi(k) \Psi(k)^\top P(k - 1)}{\alpha + \Psi(k)^\top P(k - 1) \Psi(k)} \right)
\end{equation}
\begin{equation}
P(0) = 10000
\end{equation}
With the forgetting factor $\alpha$. The specific values of $\alpha$ used are chosen to have time constants of $3$, $5$ and $8$ days, based on the relationship $\alpha = 1 - 1/T$. $P(0)$ is the default value of the used Matlab function.
\subsection{Analysis}
The prediction quality is quantified by the root mean square error (RMSE) between the prediction and the actual values. For the error trajectories, the mean and standard deviation (STD) of the absolute error at each prediction step is used.
\section{Data sets} \label{SecData}
Two data sets are used. The first one is an apartment near Zürich, completed in 2018 \cite{dinatalePhysicallyConsistentNeural2021}. It is located on the second floor of a small vertically stacked research neighborhood. Its construction emphasizes the use of wood and recycled materials, as well as experimental construction materials derived from fungi. It consists of a living room, two bedrooms, two bathrooms and a small entrance space. For the study, only the living room and the bedrooms, which have a total area of $93.8 m^2$, are considered. The temperatures in these rooms are measured and averaged. Heating and cooling are provided by a radiant slab with a central supply of hot and cold water. The thermal powers are measured at the connection points to the apartment. The ambient temperature and the solar radiation are measured locally. The pyranometers to measure the solar radiation are aligned with the orientation of the windows. The apartment has one large window front with a north-east orientation, sandwiched between overhanging concrete slabs. Therefore, the solar gains strongly depend on the position of the sun and can differ substantially from the global horizontal radiation.
The second location is a terraced apartment building in Basel, completed in 2015 \cite{MonitoringMinergieAEcoMFH}. It consists of seven units with 2.5 to 3.5 rooms. It is a low-energy building with a measured thermal heating energy demand of $17 kWh/(m^2a)$. It is heated by radiant slabs, supplied by a ground-source heat pump. Since only the total electric power consumption of the heat pump is measured, a lumped internal temperature of the whole building, measured in the collective ventilation exhaust before the heat recovery unit is used, marked as $T_z$ in Fig. \ref{fig:BaselPlan}. The ambient temperature is locally measured. Because there is no local measurement of the solar radiation, the power of the roof-top photovoltaic installation with a north-west/south-east orientation is used as a substitute. Due to its size, split orientation and mostly unobstructed location, it is assumed to be a good approximation of the global horizontal radiation. The windows are moderately sized and well-shaded to minimize overheating in the summer.
Both data sets span one year and are sampled with a $15$ min time step. The raw data is measured at a higher frequency and resampled. This allows to eliminate sensor noise in the pre-processing. This method could also be used in real-time.
Roughly $365$ days of data are selected for the evaluation, plus roughly $30$ days of initialization for the adaptive variants and $30$ days each of heating (and cooling) operation for the system identification of the non-adaptive variants. Depending on the presence of heating and cooling, and to compensate for gaps in the data, these numbers are slightly adjusted, shown in Tab. \ref{tab:ExpDat}. The missing data refers to the evaluation period. We note that one missing data point prevents multiple time steps from being evaluated, based on the lengths of the initialization and horizon of the prediction.
\begin{table}[]
\begin{tabular}{l|c|c}
& Zürich building & Basel building \\ \hline
Heating identification & 17.02.2020 - 17.03.2020 & 02.10.2016 - 22.11.2016 \\ \hline
Cooling identification & 21.08.2020 - 19.09.2020 &
no cooling \\ \hline
Initialization & 21.08.2020 - 20.09.2020 & 02.10.2016 - 22.11.2016 \\ \hline
Evaluation & 21.09.2020 - 21.09.2021 & 23.11.2016 - 23.11.2017 \\ \hline
Missing data & $\sim 10 \%$ & $\sim 2.5 \%$
\end{tabular}
\caption{Data selection and gaps}
\label{tab:ExpDat}
\end{table}
\begin{figure}[htp]
\centering
\includegraphics[width=8cm]{Buildings/ZurichPhoto.png}
\caption{Zürich building}
\label{fig:ZurichPhoto}
\end{figure}
\begin{figure}[htp]
\centering
\includegraphics[width=8cm]{Buildings/ZurichPlan02.png}
\caption{Zürich building plan}
\label{fig:ZurichPlan}
\end{figure}
\begin{figure}[htp]
\centering
\includegraphics[width=8cm]{Buildings/BaselPhoto.png}
\caption{Basel building}
\label{fig:BaselPhoto}
\end{figure}
\begin{figure}[htp]
\centering
\includegraphics[width=8cm]{Buildings/BaselPlan02.png}
\caption{Basel building plan}
\label{fig:BaselPlan}
\end{figure}
\section{Results}
Fig. \ref{fig:ReaTraZurich} and Fig. \ref{fig:ReaTraBasel} show a selection of specific zone temperature predictions for the Zürich and Basel buildings. We note the low temperature resolution in the latter. The following sections show a number of statistical analyses of all the data. Because of the different baselines, the two buildings are treated separately.
\begin{figure}[htp]
\centering
\includegraphics[width=12cm]{Plots/RealTrajZurich.pdf}
\caption{Selected zone temperature predictions at stated times over the horizon for Zürich building}
\label{fig:ReaTraZurich}
\end{figure}
\begin{figure}[htp]
\centering
\includegraphics[width=12cm]{Plots/RealTrajBasel.pdf}
\caption{Selected zone temperature predictions at stated times over the horizon for Basel building}
\label{fig:ReaTraBasel}
\end{figure}
\subsection{Zürich building}
Fig. \ref{fig:ZurichAllDeePC} compares the mean prediction error during the evaluation phase for all variants of the adatpive BST. A larger matrix consistently leads to better performance. The \textit{Most recent} and \textit{Closest mean} variants perform better than the \textit{Most correlated} and \textit{Smallest RMSE} variants.
\begin{figure}[htp]
\centering
\includegraphics[width=12cm]{Plots/AdaDeePcZurich.pdf}
\caption{Comparison of mean temperature prediction error for adaptive BST, Zürich}
\label{fig:ZurichAllDeePC}
\end{figure}
Fig. \ref{fig:ZurichAllRef} shows all reference methods. The adaptive ARX with higher forgetting factors (i.e. a longer trace) perform best. The BST variants perform similarly to the ARX variants, depending on the regularization weight.
For further comparison, the best variant of each of the four categories (ARX vs BST; adaptive vs non-adaptive) is selected. For the adaptive BST, the best sub-variant is \textit{Most recent} with the widest trajectory matrix and the medium regularization weight.
\begin{figure}[htp]
\centering
\includegraphics[width=12cm]{Plots/AllReferenceZurich.pdf}
\caption{Comparison of mean temperature prediction error for reference controllers, Zürich}
\label{fig:ZurichAllRef}
\end{figure}
Fig. \ref{fig:ZurichTrajMean} and Fig. \ref{fig:ZurichTrajStd} show the mean prediction errors over the horizon and the corresponding standard deviations. All variants perform similarly.
\begin{figure}[htp]
\centering
\includegraphics[width=8cm]{Plots/TrajectoryZurichMean.pdf}
\caption{Mean prediction error over time, Zürich}
\label{fig:ZurichTrajMean}
\end{figure}
\begin{figure}[htp]
\centering
\includegraphics[width=8cm]{Plots/TrajectoryZurichStd.pdf}
\caption{Standard deviation of prediction error over time, Zürich}
\label{fig:ZurichTrajStd}
\end{figure}
Fig. \ref{fig:ErrAllZurich} shows the prediction errors over the year and a cubic fit. Fig \ref{fig:ErrCompZurich} compares all fits to the ambient temperature. No clear pattern is recognizable.
\begin{figure}[htp]
\centering
\includegraphics[width=13cm]{Plots/ErrorFitAllZurich.pdf}
\caption{Seasonal prediction errors, Zürich}
\label{fig:ErrAllZurich}
\end{figure}
\begin{figure}[htp]
\centering
\includegraphics[width=12cm]{Plots/ErrorFitCompareZurich.pdf}
\caption{Seasonal prediction error fit comparison and ambient temperature, Zürich}
\label{fig:ErrCompZurich}
\end{figure}
\subsection{Basel building}
Fig. \ref{fig:BaselAllDeePC} compares the mean prediction error during the evaluation phase for all variants of the adatpive BST. As before, a wider trajectory matrix leads to better performance. The \textit{Most recent} variant with the medium regularization weight performs best. All other variants follow closely grouped.
\begin{figure}[htp]
\centering
\includegraphics[width=12cm]{Plots/AdaDeePcBasel.pdf}
\caption{Comparison of mean temperature prediction error for adaptive BST, Basel}
\label{fig:BaselAllDeePC}
\end{figure}
Fig. \ref{fig:BaselAllRef} shows all reference methods. Two of the adaptive ARX variants perform significantly better than the non-adaptive one. The BST variants perform similarly to the ARX, depending on the regularization weight.
As before, the best variant of each of the four principal categories is selected for comparison.
\begin{figure}[htp]
\centering
\includegraphics[width=12cm]{Plots/AllReferenceBasel.pdf}
\caption{Comparison of mean temperature prediction error for reference controllers, Basel}
\label{fig:BaselAllRef}
\end{figure}
Fig. \ref{fig:BaselTrajMean} and Fig. \ref{fig:BaselTrajStd} show the mean prediction errors over the horizon and the corresponding standard deviations. In general, the adaptive variants outperform the non-adaptive ones in this case, with a slight advantage for the ARX methods.
\begin{figure}[htp]
\centering
\includegraphics[width=8cm]{Plots/TrajectoryBaselMean.pdf}
\caption{Mean prediction error over time, Basel}
\label{fig:BaselTrajMean}
\end{figure}
\begin{figure}[htp]
\centering
\includegraphics[width=8cm]{Plots/TrajectoryBaselStd.pdf}
\caption{Standard deviation of prediction error over time, Basel}
\label{fig:BaselTrajStd}
\end{figure}
Fig. \ref{fig:ErrAllBasel} shows the prediction errors over the year and a cubic fit. Fig. \ref{fig:ErrCompBasel} compares all fits to the ambient temperature. There are noticeable spikes in the errors around the middle of the simulation period for both adaptive and non-adaptive variants. The similarity between ARX and BST indicates an (unknown) physical cause, rather than an issue with the methods. The adaptive variants show significantly better performance during the summer months in Fig. \ref{fig:ErrCompBasel}.
\begin{figure}[htp]
\centering
\includegraphics[width=13cm]{Plots/ErrorFitAllBasel.pdf}
\caption{Seasonal prediction errors, Basel}
\label{fig:ErrAllBasel}
\end{figure}
\begin{figure}[htp]
\centering
\includegraphics[width=12cm]{Plots/ErrorFitCompareBasel.pdf}
\caption{Seasonal prediction error fit comparison and ambient temperature, Basel}
\label{fig:ErrCompBasel}
\end{figure}
\subsection{A note on singular values of the trajectory matrices}
In dual control, the cost function often includes a term to maximize the level of excitation in the resulting trajectory \cite{klenskeApproximateDualControl2016, ebadatApplicationorientedInputDesign2017, hernandezvicenteStabilizingPredictiveControl2019, luRobustAdaptiveModel2020, zacekovaZoneMPCGuaranteed2020, bruggemannForwardlookingPersistentExcitation2022}. According to Willems' lemma, a loss of full row rank means a loss of persistent excitation in a Hankel matrix. The smallest singular value of a matrix can be thought of as a measure of how close it is to this. Therefore, we hypothesized a correlation between the smallest singular value of the trajectory matrix and the resulting prediction accuracy. However, no such correlation was found.
\section{Discussion and Conclusions}
One limitation of the study is the small data set, consisting of two similar buildings in similar climate. A greater variety of building types and climates would have been desirable, but corresponding data sets were not available.
The prediction error for the Basel building is noticeably smaller than for the Zurich building. This is likely due to the larger size and the smaller window-to-wall area fraction. Thus giving the Basel building a much higher inertia-to-disturbance ratio.
The average prediction accuracy of the BST and ARX methods is shown to be similar, with a slight advantage for the ARX. However, the computation of ARX is faster by orders of magnitude. \cite{schwarzDataDrivenControlBuildings2020} and \cite{kerkhofOptimalControlAutonomous2020} report similar ratios for the computation time comparing DeePC to conventional MPC. Furthermore, ARX has fewer tuning parameters. Excluding the worst choices of the regularization weight $\lambda$ for BST, all methods are sufficiently accurate, from an application point of view, with prediction errors well below $1 K$ over a $24 h$ horizon. In more general terms, linear time-invariant methods are sufficient to predict the zone temperatures in the studied buildings. We note that this accuracy is achieved without intentional excitation of the system for the purpose of identification.
ARX has a clear advantage in the early prediction steps, which fades toward the later prediction steps. Accuracy in the early prediction steps is more important for most applications of predictive control. However, a more consistent accuracy over a long horizon could be desirable for specific applications. For example, the day-ahead prediction of electricity consumption required for some forms of demand response \cite{qureshiModelPredictiveControl2014}.
Comparing the different methods to select data for trajectory matrices, using the most recent data is equally or more accurate than selecting data based on similar weather. It is also the simplest method to implement and the fastest to compute.
The adaptive and non-adaptive methods perform similarly, despite the data sets including seasonal transitions. However, one year is still a short span, relative to the time scales at which building materials significantly degrade \cite{AcceleratedAgeingDurability2016, ImpactBuildingEnvelope2017}. Furthermore, the data sets do not include any substantial changes in the occupancy, use or surroundings of the buildings. Accounting for such long-term changes does not require the continuous model updating methods used in this study. A periodic update as used in \cite{woliszSelflearningModelPredictive2020} should be sufficient.
Regarding the gaps in the data, the simplest possible methods of simply omitting them for the BST case and skipping them for the ARX case are found to work well.
\section{Declaration of competing interest}
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
\section{Data availability}
The data used in this study is confidential.
\section{Acknowledgements}
We would like to thank the Interreg programme for their support. We would further like to thank Loris Di Natale from EMPA and Gregor Steinke from FHNW for providing the data sets used in this study, as well as Ralph Eismann from FHNW for his advisory contributions.
|
1,116,691,497,572 | arxiv | \section*{Introduction}
In this small note we introduce a notion of self-stresses on the set
functions in two variables with generic critical points.
The notion naturally comes from a rather exotic representation of classical Maxwell frameworks
in terms of differential forms.
For the sake of clarity we work in the two-dimensional case only.
However all the definitions for the higher dimensional case are straightforward.
\section{Preliminaries}
\subsection{Classical definition of tensegrity}
For completeness of the story we start with the classical approach introduced
in~\cite{Max} by J.~C.~Maxwell in 1864. We refer any interested in rigidity
and flexibility questions to~\cite{Con,SJS2018}.
We use the following slightly modified definition of tensegrity from~\cite{Guz1}.
\begin{definition}
Let $G=(V,E)$ be an arbitrary graph on $n$ vertices.
\itemize
\item{A {\it framework} $G(P)$ in the plane is a map $f=(f_v,f_e)$:
$$
f_v: V \to \r^2, \qquad f_e: E \to S^{1},
$$
such that for every edge $v_iv_j$ the vector
$f_v(v_i)f_v(v_j)$ is a multiple of $f_e(v_iv_j)$.
}
\item{A {\it stress} $w$ on a framework is an assignment of real scalars
$w_{i,j}=w_{j,i}$ (called {\it tensions}) to its edges.}
\item{A stress $w$ is called a {\it self-stress} if, in addition, the
following equilibrium condition is fulfilled at every vertex
$p_i$:
$$
\sum\limits_{\{j|j\ne i\}} w_{i,j}e_{ij}=0.
$$
}
\item{A pair $(G(P),w)$ is a {\it tensegrity} if $w$ is a self-stress for the framework $G(P)$.}
\end{definition}
\subsection{Tensegrities and exterior forms}
In this section we recall a rather exotic interpretation of two-dimensional tensegrities
as a collection of 2-forms in $\r^3$ with certain relations.
It is rather in common with projective approach discussed by I.~Izmestiev in~\cite{Izm2017}.
\vspace{2mm}
Consider a tensegrity $(G(P),w)$ with $P=(P_1,\ldots, P_n)$.
For every point $P_i=(x_i,y_i)$ we associate a 1-form in $\r^3$:
$$
dP_i:=x_idx+y_idy+dz.
$$
(one can say that $dz$ is a normalization factor to extract tensions.)
For every edge $E_iE_j$ we consider a 2-form:
$$
dP_i\wedge dP_j.
$$
It turns out that self-stressability conditions is precisely equivalent to
$$
\sum\limits_{\{j|j\ne i\}} w_{i,j}dP_j\wedge dP_i=0.
$$
So any framework in tensegrity can be defined simply by a collection of
decomposable 2-forms in $\r^3$ (which we denote as $G(dP)$) and a self-stress $w$ as before,
We denote it by $(G(dP),w)$.
\begin{remark}
This definition perfectly suits the ``meet'' and ``join'' Cayley algebra expressions arising with
the description of existence conditions of tensegrities
(see, e.g., in~\cite{WW,DKS2010,Kar2018}).
It also rather straightforwardly provides projective invariance of tensegrity existence.
\end{remark}
\section{Case of generic functions}
One of the mysterious questions related the notion of $(G(dP),w)$ is as follows: {\it what is a natural generalizations
of the tensegrity to the case of decomposable differentiable 2-forms $($not-necessarily with constant coefficients$)$?}
\vspace{1mm}
The aim of this section is to give a partial answer to this question for differential forms whose factors are of type
$$
df+dz,
$$
where $f=f(x,y)$ is a function of two variables.
\vspace{2mm}
Let us first give a definition of tensegrity.
Secondly we show a geometric interpretation and link it to the classical case.
\subsection{Main definitions}
Let $F=(f_1,\ldots, f_n)$. Denote by $dF$ the collection of forms
$$
dF_1=df_1+dz, \quad dF_2=df_2+dz, \quad\ldots,\quad dF_n=df_n+dz.
$$
\begin{definition}
Let $F$ be a collection of functions $F=(f_1,\ldots, f_n)$ with finitely many critical points.
A {\it tensegrity} $(G(dF),w)$ is a triple: a graph $(G,F,w)$,
where functions $f_i$ are associated with vertices of a graph and edges are associated with stresses $w_{i,j}$.
\end{definition}
For a function $f$ denote the set of its critical points by $\cri(f)$;
the index of a critical point $P$ is denoted by $\ind(P)$.
\begin{definition}
A self-stress condition on $(G(dF),w)$ at a function $F_i$
$$
\sum\limits_{P_{i,k}\in \cri(f_i)} (-1)^{\ind(P_{i,k})}
\bigg(
\sum\limits_{\{j|j\ne i\}} w_{i,j}dF_j(P_{i,k})\wedge dF_i(P_{i,k})
\bigg)=0.
$$
\end{definition}
\begin{remark}
It might be also useful to consider critical points separately (say if this increases durability for certain overconstrained system).
\end{remark}
\begin{remark}
Recall that at a critical point of $F_i$
$$
dF_i\wedge dF_j=dz\wedge df_j.
$$
So one can replace every 2-form $dF_i\wedge dF_j$ in the equilibrium condition simply by $df_j$.
\end{remark}
\subsection{Geometric discussions}
{\it Lines of forces} are precisely the points when the total force $dF_i\wedge dF_j=0$
(see Figure~\ref{f.1}).
Lines of forces are defined by the equation
$$
df_i\wedge df_j=0.
$$
\begin{figure}
\epsfbox{f.1}
\caption{Level sets of two functions (gray and black), and the line where their gradients have the same direction.}
\label{f.1}
\end{figure}
It is clear that lines of forces are connecting critical points of $F_i$ (including critical points at infinity) to critical points $F_j$
usually by a graph rather than by a line.
A possible picture for a splitting of a force line is as follows:
$$
\epsfbox{f.2}
$$
\begin{remark}
One might consider the classical theory of tensegrities as follows:
For each point $P_i=(x_i,y_i)$ consider a function
functions
$$
f_i=(x-x_i)^2+(y-y_i)^2.
$$
Then (possible after a simple rescaling of stresses) one has a classical tensegrity.
This works also for case of point-hyperplane frameworks introduced recently in~\cite{EJN},
where hyperplanes are defined as linear functions.
\end{remark}
In some sense the proposed techniques can be considered as a deformation of a classical tensegrity.
\begin{example}
First we start with a graph $G$ on 5 vertices:
$$
\epsfbox{f.4}
$$
Let us consider the following 5 functions corresponding to vertices of graphs:
$$
\begin{array}{l}
a:\quad f_1=(x-3)^2+(y-3)^2;\\
b:\quad f_2=(x+3)^2+(y-3)^2;\\
c:\quad f_3=(x+3)^2+(y+3)^2;\\
d:\quad f_4=(x-3)^2+(y+3)^2;\\
o:\quad f_5=x^2+5y^2.
\end{array}
$$
Then the line of forces and the corresponding stresses are as on Figure~\ref{f.3}.
They are defined up to a choice of a real parameter $\lambda$.
Here grey curves are level sets; black curves are compact lines of forces between critical points;
the numbers indicate the stresses on edges.
The critical points of functions are marked by the corresponding capital letters.
\end{example}
\begin{figure}
\epsfbox{f.3}
\caption{Level sets, compact lines of forces, and stresses.}
\label{f.3}
\end{figure}
\begin{remark}
Finally we would like to admit that
the situation in three and higher dimensional cases repeats the two-dimensional case discussed above.
\end{remark}
\begin{remark}
In three dimensional case consider the following functions (potentials):
$$
f_{a,b,c}=\frac{k_e}{(x-a)^2+(y-b)^2+(z-c)^2}
$$
(here $k_e$ is the Coulomb constant) and take the unit stresses.
Then we arrive to classical Coulomb situation for points with unit charges in three-space.
\end{remark}
|
1,116,691,497,573 | arxiv | \section{Introduction}
Astronomical and cosmological probes of dark matter not only exist, but indicate that dark matter is five
times as prevalent in the universe than the conventional forms of matter described by the Standard Model \cite{Ade:2013zuv}.
Despite this abundance however, knowledge of dark matter remains perplexingly incomplete. Principle among
these unknowns are the mass of the dark matter (DM) particle and the nature of its interactions with the
Standard Model (SM), both of which are unconstrained over many orders of magnitude.
A diversity of theoretical models has grown to accompany the diversity of allowed phenomenology \cite{Feng:2010gw}. Extremely light and
weakly-coupled axions \cite{Kim:2008hd,Rosenberg:2000wb} are a canonical scenario of DM with phenomenology that differs drastically from that of
the more usually discussed WIMPs \cite{Bertone:2004pz,Jungman:1995df,Chung:2003fi,Servant:2002aq,Cheng:2002ej,Hooper:2007qk},
though essentially arbitrary phenomenology can be obtained from hidden sector
models \cite{Pospelov:2007mp,Pospelov:2008jd,Feng:2008ya,Feng:2008mu}, which may be designed to solve problems unrelated
to dark matter (${\it e.g.}$, generation of cosmic baryon number \cite{Kaplan:2009ag,Falkowski:2011xh,Davoudiasl:2012uw}).
Given this diversity, the experimental effort to measure such interactions has become increasingly creative. In addition to the
traditional three-pronged experimental program consisting of direct detection, which seeks to measure DM-nucleon scattering,
colliders searches for DM production and indirect detection searches for the energetic products of DM annihilation in
astroparticle experiments, are studies of even more diverse effects, ${\it e.g.}$, observed and simulated shapes of DM halos \cite{Lin:2011gj}, the
detailed nature of the CMB \cite{Galli:2011rz,Hutsi:2011vx,Finkbeiner:2011dx} and primordial element abundances \cite{Cyburt:2004yc} and cooling of astrophysical objects \cite{Raffelt:1999tx}.
Recently, the CDMS collaboration has made the interesting observation of an excess of 3 events over an expected background of
$0.4$ events, that can be interpreted as a signal detection with $\sim 2\sigma$ significance. Such a result is clearly inconclusive
on its own and should be subjected to the utmost scrutiny, especially as the favored mass $m_{DM}\simeq 8.6\,{\rm GeV}$ coincides with
the sensitivity threshold of the experiment. Despite these considerations, the result is very interesting in light of
similar anomalous results, such as from CoGeNT \cite{Aalseth:2010vx}, and in the favored-region's proximity to the predictions of some
well-motivated theoretical models \cite{Fitzpatrick:2010em}.
Describing a light DM particle with such (relatively) large interactions with the SM and that wouldn't have already been seen
elsewhere is a phenomenological challenge. There exist several ``portals'' (in effective operator language: SM-singlet operators
built only out of SM fields) by which such DM may easily communicate with the SM, each of which may naturally suggest vector,
scalar or fermionic mediators and have been studied in some detail in the context of light DM
\cite{Hooper:2012cw,Andreas:2010dz,Fitzpatrick:2010em,Falkowski:2009yz,Okada:2013cba,Choi:2013fva}. In this work we will consider a
generic model of Dirac fermionic DM interacting with the Standard Model
via a relatively light scalar mediator particle. For such a model to avoid being ruled out from the outset we consider our mediator
to be coupled to SM fermions in minimal-flavor-violating (MFV \cite{D'Ambrosio:2002ex}) fashion, suggesting a natural connection
between the physics that
generates our DM and messenger to the physics of the Higgs sector and electroweak symmetry breaking. We will describe regions
of parameter space for which our model obtains scattering in the range of the CDMS result, where the annihilations in our model
are sufficient for equalling the cosmological DM relic density and regions that are already ruled out by collider and low-energy
experiments.
The rest of this paper is divided into four sections. In Section \ref{model} we describe and discuss our simplified
model framework, in Section \ref{dmpheno} we describe our model's DM phenomenology, in Section \ref{colliders}
we describe collider and low-energy bounds that can be placed on the parameter space of our model and in Section \ref{discuss}
we present a concluding discussion.
\section{Simplified Model Framework}
\label{model}
We work in the framework of a simplified model consisting of the Standard Model supplemented by
a Dirac DM particle $\chi$ and a CP-even scalar messenger $\phi$. Since the CDMS signal is suggestive of
a WIMP whose mass is well below $M_Z / 2$, we restrict ourselves to considering dark matter which is an electroweak
singlet in order to avoid large contributions to the invisible width of the $Z$ boson \cite{Beringer:1900zz}.
Fitting the CDMS signal region will imply ${\mathcal O}(0.1 - 1)$ coupling between $\phi$ and $\bar{\chi}\chi$,
suggesting that $\phi$ should also be an electroweak singlet.
The mass of the $\chi$ is fixed by the CDMS
signal to $m_\chi \simeq 8.5$~GeV. In the discussion below, we fix the dark matter mass to this value and comment
where appropriate as to how our results would change for different masses.
In order to evade very strong bounds from
flavor-violating observables, we invoke minimal flavor violation \cite{D'Ambrosio:2002ex}
with regard to the $\phi$ coupling to quarks,
\begin{eqnarray}
\mathcal{L}_{int} = g_{\chi} \phi \bar{\chi} \chi + \sum_i g_d \lambda^d_i \phi \bar{d}_i d_i
+ \sum_i g_u \lambda^u_i \phi \bar{u}_i u_i
\end{eqnarray}
where $\lambda^d_i$ and $\lambda^u_i$ are the down-type and up-type Yukawa interactions. In addition to the
masses $m_\chi$ and $m_\phi$, the model is specified by the dimensionless
couplings to dark matter $g_\chi$, to down-type quarks (scaled by the appropriate Yukawa interaction) $g_d$, and
similarly defined coupling to up-type quarks $g_u$.
In what follows we will work primarily in the 3-dimensional space ($m_{\phi}$, $g_{\chi}$, $g_d$). We consider
two distinct cases for $g_u$:
\begin{itemize}
\item $g_u\sim 1.8\; g_d$, leading to iso-spin preserving (IP) elastic scattering in direct detection experiments; or
\item $g_u\sim -1.015\; g_d$, leading to isospin-violating (IV) scattering with $f_n/f_p\sim -0.7$, designed
to maximally weaken the sensitivity of Xenon-based searches \cite{Feng:2011vu}.
\end{itemize}
It is worth noting that even for $g_u \sim \; g_d$, the elastic scattering cross section will be similar
for protons and neutrons, owing to the relatively small contribution of the up and down quarks because of
their small Yukawa interactions. One could also write down (and put bounds on)
a coupling between $\phi$ and leptons, but such
an interaction is largely orthogonal to a discussion of the CDMS signal. Where relevant, we
will comment on the bounds on such a coupling below.
There are also potentially renormalizable interactions between $\phi$ and the Standard Model Higgs doublet, $H$.
In general, the details of the scalar potential are not very important for the phenomena of interest here, and we leave
a detailed analysis for future work. However, it is worth noting that mixing between $\phi$ and the Higgs
boson allows for $\phi$ to be produced via typical Higgs production modes, including $\phi Z$ at LEP II.
For masses less than about 110 GeV, null results of Higgs searches at LEP generically imply
that the mixing is no larger than ${\mathcal O}(10\%)$ \cite{Searches:2001ad}, although there are windows of mass
where bounds are weaker, and might even be interpreted as not very significant hints for a
positive signal \cite{Dermisek:2007yt}.
While we remain agnostic as to the origin of the simplified model framework, it is worth noting that one can imagine a
simple UV-completion of the scalar sector
based on a two Higgs doublet model augmented by a gauge singlet scalar. The two Higgs
doublets provide sufficient freedom in the Yukawa couplings so as to realize $g_u$ and $g_d$ in the desired ranges,
with the (mostly singlet) $\phi$ inheriting the couplings through modest mixing with a combination of the physical
CP even Higgs bosons.
Perhaps the most studied model containing these ingredients is the NMSSM \cite{Ellwanger:2009dp,Balazs:2007pf}.
It has been pointed out that one can find limits in the NMSSM parameter space that attain large scattering cross-sections
with a low DM mass \cite{Draper:2010ew,Carena:2011jy,Cao:2011re} although there may be some tension with other constraints
as, in supersymmetric models like these,
large cross-sections tend to come hand-in-hand with sizable couplings to $W^{\pm}/Z^0$ \cite{Das:2010ww}. Variations of supersymmetric
models consisting of the MSSM plus a singlet super-field can realize suitable cross sections
\cite{Hooper:2009gm,Belikov:2010yi,Buckley:2010ve}. For an example of a non-supersymmetric UV completion see \cite{He:2013suk}.
\section{DM Observables}
\label{dmpheno}
In this section we focus on finding regions of our parameter space that are attractive from a DM standpoint: light DM with large
elastic scattering cross-sections. Although we are particularly interested in scattering, we also calculate relic density and
discuss current annihilation cross-sections for our $\chi$ to give a sense of the cosmological history necessary in such a scenario. We
consider messenger masses in a wide range, $1\,{\rm GeV}\mathrel{\mathpalette\atversim<} m_{\phi} \mathrel{\mathpalette\atversim<} 100\,{\rm GeV}$, anticipating (as is confirmed below) that mediator masses
above $\sim 100\,{\rm GeV}$
will be non-trivially constrained by collider monojet searches\footnote{For mediator masses heavier than typical LHC center-of-mass
energies the limit should be essentially the same as the stringent EFT bounds derived in \cite{Rajaraman:2011wf}}. We use \textit{MicrOMEGAs v2.4} \cite{Belanger:2010gh}
for all elastic scattering and annihilation cross section calculations.
For our direct detection calculation we use a local DM density $\rho_0=0.3\,{\rm GeV} / \mathrm{cm}^3$ and nuclear form factors:
\begin{eqnarray*}
f^p_u=0.023,\hspace{10 mm} f^p_d&=&0.033,\hspace{10 mm} f^p_s=0.05,\\
f^n_u=0.018,\hspace{10 mm} f^n_d&=&0.042,\hspace{10 mm} f^n_s=0.05.
\label{formfactors}
\end{eqnarray*}
Appropriate values for the strange-flavored scalar form factors are hotly-debated at current
\cite{Ellis:2008hf,Gasser:1990ap,Bernard:1993nj,Pavan:2001wz,Young:2009zb,Thomas:2011cg,Junnarkar:2013ac,Alarcon:2012nr}, the choice
$f^N_s\approx 0.05$ is on the low side of proposed values, making it a conservative choice for our purposes. The
uncertainty coming from the strange quark is anyway not critical for our purposes: we consider a wide range of elastic
scattering cross-sections\footnote{This range corresponds to the lower-most and upper-most values on the $2\sigma$
ellipse of the result \cite{Agnese:2013rvf}.},
\begin{eqnarray}
10^{-6} \mathrm{pb}\mathrel{\mathpalette\atversim<}\sigma_{\mathrm{SI,N}}\mathrel{\mathpalette\atversim<} 3*10^{-4} \mathrm{pb},
\label{ddrange}
\end{eqnarray}
as interesting for our purposes. The scattering cross section depends on the couplings only through the product, $g_{\chi}g_d$.
We calculate the thermal relic density of our $\chi$ assuming that the only relevant processes at freezeout are those in
our simplified model. As always, this is a fairly heavy-handed assumption and may or may not be relevant in any particular
completion of our model. Despite this, our thermal relic calculation remains useful for denoting regions of parameter space where extra
theoretical structure\footnote{${\it e.g.}$, non-thermal evolution or dark sector states that participate in the thermal calculation} may be
necessary to increase or decrease the relic density with respect to our minimal scenario and where our model saturates the
\textit{Planck} collaboration's measurement \cite{Ade:2013zuv}, $\Omega_{CDM}h^2\approx 0.1146$, on its own. Annihilations proceed through t-channel
$\chi\bar{\chi}\rightarrow \phi\phi$ (when kinematically
available) and through s-channel $\chi\bar{\chi}\rightarrow f\bar{f}$, the former depending on the couplings only as $g_{\chi}^2$
and the latter as $g_{\chi}g_d$. Both of these processes are actually p-wave processes at leading order (suppressed by
$\upsilon^2\sim10^{-6}$) so current annihilations from our simplified model are predicted to be much below the
canonical $\sigv\sim 3*10^{-26} \mathrm{cm}^3/\mathrm{s}$. Similarly low rates are calculated in the resonant region
$2m_{\chi}\approx m_{\phi}$, although \textit{Planck}-level relic densities are achieved for much lower coupling values.
If our model were to also include a pseudo-scalar state, $a$, then there would be available s-wave processes giving current
annihilations close to the canonical value\footnote{As may be desired given the current (inconclusive, but interesting)
hints of $\sim 10\,{\rm GeV}$ DM particles annihilating to $b$'s or $\tau$'s conributing to the $\gamma$-ray spectrum at the
Galactic Center \cite{Abazajian:2012pn,Hooper:2011ti}.}. Such pseudo-scalars are not hard to come by theoretically
(${\it e.g.}$, in approximately SUSY-preserving multiplets) and would have no effect on scattering rates (momentum suppressed)
but potentially sizable effects on the other observables, such as collider production.
In Figure \ref{ddrd} we map out the combinations of $g_{\chi}$ and $m_{\phi}$ for which scattering cross sections are within the
range Eqn.\ \ref{ddrange} and for which the relic density matches the \textit{Planck} value for both IP and IV cases and for several values of $g_d$.
The features of the relic density band are easy to understand: there is a sharp upturn where the $\chi\chi\rightarrow\phi\phi$ channel becomes
phase space suppressed ($m_{\phi}\approx m_{\chi}$) and a sharp downturn in the resonant annihilation region ($m_{\phi}\approx 2m_{\chi}$).
Annihilation cross-sections (not shown) are $\sigv\mathrel{\mathpalette\atversim<} 3*10^{-30} \mathrm{cm}^3/\mathrm{s}$ on the \textit{Planck} band.
In the IV case, scattering cross-sections are reduced by destructive interference and we observe a shift of the favored region
for scattering toward larger coupling values. We observe regions where both large scattering cross-sections and $\Omega_{\chi}\approx\Omega_{CDM}$
can be obtained simultaneously, for essentially any choice of $g_d$. While this happens both for very light mediators ($m_{\phi}< 10\,{\rm GeV}$) and
for very heavy mediators ($m_{\phi}> 20\,{\rm GeV}$), we expect these regions to be in danger either from $ \ifmmode \Upsilon \else $\Upsilon$ \fi$-decay data or from collider searches. In contrast,
regions of overlap in the $m_{ \ifmmode \Upsilon \else $\Upsilon$ \fi} < m_{\phi} < 2m_{\chi}$ range are particularly hard to constrain.
\begin{figure}[hbtp]
\centering
\includegraphics[width=1.0\textwidth]{./ddrd.png}
\caption{Spin-Independent Scattering and Relic Density. The blue band denotes SI scattering cross-sections within the range Eqn. \ref{ddrange}
(darker and lighter regions describing the extent of $1\sigma$ and $2\sigma$ ellipses in the result \cite{Agnese:2013rvf}, respectively). The red band shows
where our $\chi$'s relic density is $\Omega_{\chi}\approx\Omega_{CDM}$. In the upper panels $g_u$ and $g_d$ are related such that $f_n=f_p$ (IP),
while in the lower panels $f_n/f_p=-0.7$ (IV).}
\label{ddrd}
\end{figure}
\section{Collider \& Low-Energy Constraints}
\label{colliders}
\subsection{Mono-Objects}
Intuition garnered from DM effective theory analyses over the last few years suggests that collider searches may have the
final say on the viability of this scenario \cite{Beltran:2010ww,Goodman:2010yf,Goodman:2010ku,Bai:2010hh,Rajaraman:2011wf,Fox:2011fx,Fox:2011pm,Bai:2012xg,Cotta:2012nj,Carpenter:2012rg,Bell:2012rg}.
Such searches typically look for DM direct production by studying single objects
(monojets, monophotons, etc.) recoiling off of a missing transverse momentum vector and, unlike direct detection experiments, remain
sensitive to arbitrarily small DM masses. The caveat to these searches is the
efficacy of the EFT description, which can give either an overly-conservative or an overly-optimistic sense of the
collider reach in light-mediator scenarios. For our mediators, with the DM mass fixed at $m_{\chi}=8.5\,{\rm GeV}$, there are roughly
three regimes for collider production: \textit{(i)} the mediator is very heavy compared to typical machine center-of-mass energies,
\textit{(ii)} the mediator is light compared to collider center of mass energies but heavier than $2m_{\chi}$ and \textit{(iii)}
the mediator is lighter than $2m_{\chi}$. Scenario \textit{(i)} is the regime where the EFTs should give basically the right answer,
in scenario \textit{(ii)} the mediator can be produced on-shell so we would expect the EFT bounds to be conservative relative to
the exact bounds and in scenario \textit{(iii)} the mediator can never be put on-shell, the production cross-section is a rapidly
falling function of the mono-object's $p_T$ and the EFT bounds would suggest much tighter constraints than what one would actually
get in the full calculation. Of course these regimes bleed into each other a bit, here we seek to describe this behavior.
For studies involving light vector mediators, see Refs.~\cite{An:2012va,Frandsen:2012rk,An:2012ue,Shoemaker:2011vi}.
Here we focus on LHC monojet searches, which we expect to provide the tightest constraints in this class of experiments. Monojet
bounds from the Tevatron were checked ($c.f.$, \cite{Bai:2010hh}) as well and they are not competitive with those coming from the LHC\footnote{Monophoton
bounds from LEP are irrelevant unless our mediator were to have large couplings to the electron, which seems unlikely in our
construction.}. We mimic cuts from the ATLAS analysis \cite{ATLAS:2012ky} and use the typical \textit{MadGraph}(v5)-\textit{Pythia}(v6)-\textit{PGS}(v4)
chain \cite{Alwall:2011uj,Sjostrand:2006za,pgs} (hereafter \textit{MPP}) with default ATLAS detector card to simulate signal and background rates.
\begin{figure}[hbtp]
\centering
\includegraphics[width=0.8\textwidth]{./1jet-ttMET.pdf}
\caption{
Monojet and $t\bar{t}+\mathrm{MET}$ bounds on our model in the $g_{\chi}g_d$ vs.\ $m_{\phi}$ plane (IP left panel, IV right panel).
The blue bands gives scattering cross-sections in the desired
range, as in Fig.\ \ref{ddrd}, while the gray regions are excluded by the ATLAS monojet search \cite{ATLAS:2012ky} and the ATLAS
$t\bar{t}+\mathrm{MET}$ search \cite{ATLAS-CONF-2012-167} (both at 95\% confidence) as noted in the figure. The limits in this plot
were generated with fixed $g_d=1$.}
\label{1jet}
\end{figure}
Monojet bounds are presented in Figure \ref{1jet}. The features of these curves can be easily understood: The cross-section is highly suppressed and nearly constant
in the $m_{\phi}<2m_{\chi}$ regime where the mediator cannot be put on-shell. The kink occurs at $m_{\phi}=2m_{\chi}$ whereafter the monojet bounds become more and
more constraining until the eventual fall off above typical center-of-mass energies. We know that our couplings must increase with the mediator mass in order
to have scattering cross-sections in the range Eqn.\ \ref{ddrange}, here we see that our model will actually run into monojet constraints before reaching its
ultimate perturbativity bound at $g_{\chi}g_d\sim 4\pi$. Interestingly however, Figure \ref{1jet} shows that the monojet reach is much less than that from
the heavy-flavor $t\bar{t}+\mathrm{MET}$ search for all $m_{\phi}$, this is what we will describe next.
\subsection{Heavy-Flavor Searches}
While the MFV structure of our messenger's couplings keep direct collider production of $\phi$'s highly-suppressed, the
large couplings to top and bottom quarks suggest large rates for $\phi$'s radiated off of the final states in heavy flavor (HF) production. Since our
$\phi$'s may be made to decay either dominantly to missing transverse energy (for $g_{\chi}\gg g_{d}$) or to $b\bar{b}$ (for $g_{d}\gg g_{\chi}$),
heavy flavor searches both
with and without associated MET may be applicable. HF searches with MET are typical of the suite of SUSY searches for
third-generation squarks (${\it e.g.}$, \cite{ATLAS-CONF-2012-167}), while HF searches without MET are not nearly as common.
An example of the latter is the search for signals of Higgs production in the $t\bar{t}H\rightarrow t\bar{t} b\bar{b}$ channel
(in practice, the $t\bar{t}+$b-jet channel \cite{ATLAS-CONF-2012-135,Aad:2013tua}). Here we investigate bounds on our model's
parameter space that can be derived from these two searches. Another recent work that considered heavy-flavored final states
and dark matter is \cite{Lin:2013sca}
The ATLAS analysis \cite{ATLAS-CONF-2012-167}, uses $13~\mathrm{fb}^{-1}$ of $8\,{\rm TeV}$ data to place very stringent
constraints, $\mathcal{O}(1 \mathrm{fb})$, on $t\bar{t}+\mathrm{MET}$ from BSM sources. Here we use the full
\textit{MPP} analysis chain to simulate the SM background to this
search and to get a sense of the acceptance profile for tagging the two tops in our signal. To calculate the
signal rate we assume that the acceptance (more precisely, the part of which comes from top-tagging) for signal
events is essentially the same as that for the SM background. This allows us to do an initial calculation of the
signal at parton level, before applying the more involved $m_{T2}$ cut to accurately reproduce the MET acceptance
(the quantity that is really sensitive to the kinematics of our signal events) in reasonable computational time. The
particular MET and $p_T$ cuts that we used were those of the ``110 SR'' signal region defined in \cite{ATLAS-CONF-2012-167}.
The resulting excluded region is described in Figure \ref{1jet} and is seen to be stronger for all $m_{\phi}$ than that from
the monojet search. Our model's mediator mass is bounded to be $m_{\phi}\mathrel{\mathpalette\atversim<} 45\,{\rm GeV}$ (IP) or $m_{\phi}\mathrel{\mathpalette\atversim<} 20\,{\rm GeV}$ (IV),
in both cases far smaller than the model's ultimate perturbativity bound $g_{\chi}g_d\mathrel{\mathpalette\atversim<} 4\pi$.
In the $t\bar{t}b$ channel it is more difficult to obtain an accurate bound in our parameter space. The most relevant\footnote{
The analysis \cite{ATLAS-CONF-2012-135} uses a similar data sample but is too focused on the SM Higgs to be useful in bounding our model.}
analysis in this regard is the ATLAS measurement \cite{Aad:2013tua} of the ratio of $t\bar{t}b$ and $t\bar{t}j$ (where $b$ denotes a b-tagged
jet and $j$ denotes all jets) in $4.7~\mathrm{fb}^{-1}$ of $7~\,{\rm TeV}$ data. The result is not easy to interpret as a bound in the present context,
as the measured ratio $t\bar{t}b$/$t\bar{t}j$ is found to be in excess of the SM expectation at the $1.4\sigma$ level. Rather than
trying to interpret this as evidence for new physics, we simply suppose that the measurement is roughly consistent with the SM (including a
$125\,{\rm GeV}$ Higgs) prediction and require that our model not contribute to $t\bar{t}b$ at a level greater than that from the Higgs. We
calculate both the $t\bar{t}\phi$ and $t\bar{t}H$ cross-sections using \textit{MPP} with the ``nominal''
sample selection cuts described in \cite{Aad:2013tua} to determine the ``excluded'' regions for which the $t\bar{t}\phi$ cross-section is greater
than the $t\bar{t}H$ cross-section. The result is described in Figure \ref{hfvis}, where the excluded region is compared to the preferred
regions for scattering with two choices for $g_{\chi}$, $g_{\chi}=0.01$ and $g_{\chi}=1$ (as the $t\bar{t}\phi$ cross-section depends only on $g_d$).
\begin{figure}[hbtp]
\centering
\includegraphics[width=0.8\textwidth]{./ttb.pdf}
\caption{Heavy Flavor bounds on our model in the $t\bar{t}b$ channel (IP left panel, IV right panel). As this search depends only on
the coupling $g_d$ we display, in red and blue bands, the favored regions for scattering with $g_{\chi}=0.01$ and $g_{\chi}=1$, respectively.
The gray region denotes parameter space for which the $t\bar{t}\phi$ production cross-section is greater than that for $t\bar{t}H$ production
of the SM Higgs (our rough criterion for exclusion given the result \cite{ATLAS-CONF-2012-135}).}
\label{hfvis}
\end{figure}
\subsection{B-Factory Constraints}
For mediators with $m_{\phi}\mathrel{\mathpalette\atversim<} m_{\Upsilon}\approx 10\,{\rm GeV}$ one must consider the possible signatures of our model in $ \ifmmode \Upsilon (nS) \else $\Upsilon (nS)$ \fi$
decay processes. Since our DM has $2m_{\chi}>m_{\Upsilon}$ we do not expect signatures in $ \ifmmode \Upsilon \else $\Upsilon$ \fi$ decays with invisible products (although
these would become relevant for $m_{\chi}\mathrel{\mathpalette\atversim<} 5 \,{\rm GeV}$), instead we consider radiative $ \ifmmode \Upsilon \else $\Upsilon$ \fi$ decays, $ \ifmmode \Upsilon (nS) \else $\Upsilon (nS)$ \fi\rightarrow\gamma\phi\rightarrow\gamma X$
where\footnote{Of course, ``$\phi$'' here refers to our mediator, not the light unflavored meson.} $X$ is some visible system recoiling
off of a monochromatic $\gamma$. We consider two \emph{BaBar} collaboration analyses: \cite{Lees:2011wb}, a search for photon resonances in
$\Upsilon (3S)\rightarrow \gamma + \mathrm{hadrons}$ and \cite{Lees:2012te}, a search for photon resonances in $\Upsilon (1S)\rightarrow \gamma + \tau^+\tau^-$.
Both of these results provide a bound on $g_d$ (independent of $g_{\chi}$), the former considering only quark coupling while the latter requires the
model-dependent assumption that $g_l=g_d$. We calculate the associated rates in our model space, following closely the work \cite{Yeghiyan:2009xc}.
The resulting bounds are shown in Figure \ref{ups}. The $ \ifmmode \Upsilon \else $\Upsilon$ \fi$ data limits the
$g_d$ coupling to be generally $g_d\mathrel{\mathpalette\atversim<} 0.1$ for models with $m_{\phi}\mathrel{\mathpalette\atversim<} 10~\,{\rm GeV}$, ruling out favored parts of parameter space where
$g_{\chi}$ is small. There is a large dependence on the choice of IP or IV scattering, the latter being constrained much more tightly at a
given scattering cross-section by the $\Upsilon$ data.
\begin{figure}[hbtp]
\centering
\includegraphics[width=0.8\textwidth]{./ups.pdf}
\caption{
Bounds from the radiative $ \ifmmode \Upsilon \else $\Upsilon$ \fi$-decays to hadronic and di-tau final states. Gray regions are excluded by \emph{BaBar}
analyses \cite{Lees:2011wb} and \cite{Lees:2012te} (as noted in the figure) at 90\% confidence. The red and blue bands
give direct detection favored regions for $g_{\chi}=0.01$ and $g_{\chi}=1$ as in Fig.\ \ref{hfvis}. Favored regions are
calculated for both IP (left) and IV (right) cases.}
\label{ups}
\end{figure}
\subsection{Exotic Higgs Decays}
Given the necessarily small mixing between our messenger and the SM Higgs, we expect that the
current constraint on the Higgs invisible width (about $40\%$, per \cite{Bai:2011wz}) is not tight enough to constrain our model. If
our mediator is light, $m_{\phi}\ll m_{H}$, then, as in many NMSSM discussions, we may imagine producing a
pair of boosted $\phi$'s and searching for pairs of boosted objects from their decays.
While the rate of such events depends on the details of the UV physics that give rise to our simplified model, the resultant
striking signature may be the first place in which such a model can be discovered.
An example of such an analysis is the ``ditau-jet'' search strategy, outlined in \cite{Englert:2011iz}, wherein one tries to discern
``jets'' composed of a pair of boosted $\tau$'s (${\it e.g.}$, coming from the $\phi$ decays) from generic QCD jets. In this work it was
demonstrated that (with consideration of a jet's $p_T/m_j$ ratio and application of jet-substructure techniques) one can tag ditau-jets
with high-efficiency and low-mistag rates. It was argued that, for a light-scalar model with nearly exactly the same kinematics as ours,
an appropriate series of cuts would yield effective signal and background cross-sections of $\sigma_s=0.5~\mathrm{fb}$ and $\sigma_b=0.12~\mathrm{fb}$,
and thus a $S/\sqrt{B}=5$ discovery for $\mathcal{L}=12~\ifmmode {\rm fb}^{-1}\else ${\rm fb}^{-1}$\fi$ of $14\,{\rm TeV}$ LHC data. In our model, if we assume that down-type quarks \emph{and}
down-type leptons are both normalized with the parameter $g_d$, then $BR(\phi\rightarrow\tau\tau)\sim\mathcal{O}(10\%)$. Given this then, even assuming a
scalar trilinear coupling $g_{h\phi\phi}=\sqrt{4\pi}$, our model would be far from detectable in such a search. If, however, the lepton couplings
are normalized independently of $g_d$ then, with $g_l$ such that $BR(\phi\rightarrow\tau\tau)\sim\mathcal{O}(100\%)$ , our model would also be observable in
$\mathcal{L}=12~\ifmmode {\rm fb}^{-1}\else ${\rm fb}^{-1}$\fi$ of $14\,{\rm TeV}$ LHC data.
\section{Discussion}
\label{discuss}
We have investigated diverse bounds on the parameter space of a simplified model of DM whose phenomenology could plausibly explain the
low-mass and high-cross-section signal of DM scattering in the CDMS Silicon data. Our model is typical of some extensions of the SM Higgs
sector that give light scalars coupling to SM fermions in an MFV pattern (${\it e.g.}$, coupling like a Higgs). We have shown that such models
can easily attain the necessary large scattering cross-sections for couplings of $\mathcal{O}(0.1-1)$, while also attaining the correct relic
density, in many regions of this subspace. If such a model were to be supplemented with a pseudoscalar of similar mass to our messenger $\phi$,
essentially none of the above story would change qualitatively, except that one would have the kind of canonical s-wave annihilation rates
that we may already be seeing in the Galactic Center.
We have discussed collider and low-energy B-factory bounds on our parameter space and the complementarity of these bounds. A round-up of these
results is described in Figures \ref{combo1}-\ref{combo2}, where all bounds are collected and plotted in the $g_{\chi}g_{d}$ vs.\ $m_{\phi}$ plane.
Results are given for two different choices of $g_{\chi}=1$ and $g_{\chi}=0.1$. In Fig.\ \ref{combo1} we find that, for large $g_{\chi}=1$, the
combination of $t\bar{t}+\rm{MET}$ and $\Upsilon(nS)$ data require $g_d\mathrel{\mathpalette\atversim<} 0.1$ except in the difficult region
$m_{\Upsilon(3S)}<m_{\phi}<2m_{\chi}$ where $g_d\mathrel{\mathpalette\atversim<} 1$. For smaller $g_{\chi}=0.1$ we see that the $t\bar{t}+$b-jet bound (depending only on $g_d$)
supplants the $t\bar{t}+\rm{MET}$ bound (depending on $g_{\chi}g_{d}$) to require $g_d\mathrel{\mathpalette\atversim<} 1$ for all $m_{\phi}$. In Figure \ref{combo2} we overlay
the favored regions for scattering and relic density in our parameter space. We see that the isospin-violating case is more highly constrained
than the isospin-preserving case, owing to the generally larger product $g_{\chi}g_{d}$ required to produce scattering signals at the CDMS level.
The fact that a light DM particle and scalar messenger coupling \textbf{so strongly} to SM fermions is even phenomenologically viable at this point
is very interesting. It is completely plausible that a model like ours could be discovered first in
direct detection experiments (as it may already have been!), especially
for mediator masses in the difficult range $m_{\Upsilon(3S)}<m_{\phi}<2m_{\chi}$. From what we have shown it is also plausible that such a discovery could be
corroborated (or such a model ruled out) by LHC searches for anomalous heavy flavor final states, strongly motivating a more careful look
at such signatures under more generic (${\it i.e.}$, than SM Higgs or MSSM sparticle) expectations.
\begin{figure}[hbtp]
\centering
\includegraphics[width=0.9\textwidth]{./combo1.pdf}
\caption{Combined bounds in the $g_{\chi}g_d$ vs.\ $m_{\phi}$ plane. Bounds from $t\bar{t}+\mathrm{MET}$, $t\bar{t}+$b-jet and radiative $\Upsilon$ decays
(in both hadronic and $\tau$ channels) are labelled accordingly. Monojet bounds are irrelevant, given the axes ranges plotted. We choose $g_{\chi}=1$
($g_{\chi}=0.1$) in the upper (lower) panels to translate bounds that only depend on $g_d$ onto this plane. Left and right panels correspond
to IP and IV scenarios, respectively. We use a dashed line to remind the reader that the $t\bar{t}+$b-jet bound is particularly rough (as described
in the text).}
\label{combo1}
\end{figure}
\begin{figure}[hbtp]
\centering
\includegraphics[width=0.9\textwidth]{./combo2.pdf}
\caption{As in Figure \ref{combo1}, but with the inclusion of direct detection and \textit{Planck} favored bands in blue and red, respectively.}
\label{combo2}
\end{figure}
\newpage
\section*{Acknowledgments}
The authors would like to acknowledge helpful discussions with J.~Shelton, J.~Zupan, C. Wagner, and L. Tao. The research of R.C.C. and A.R. is
supported by the National Science Foundation under grant PHY-0970173. The research of T.M.P.T. is supported in part by NSF
grant PHY-0970171 and by the University of California, Irvine through a Chancellor's fellowship.
\newpage
\bibliographystyle{JHEP}
|
1,116,691,497,574 | arxiv | \section{introduction}
Let $\mathcal{X}$ be a Banach space and $r>0$ a real. A subset $\mathcal{Y}\subseteq \mathcal{X}$ is called $r$-equilateral if
$\|x-y\|=r$ for any two distinct $x, y\in \mathcal{Y}$; it is called equilateral if it is $r$-equilateral
for some real $r>0$.
As shown by Brass and by Dekster (\cite{brass, dekster}) for each $k\in \mathbb{N}\setminus\{0,1\}$
there is $d(k)\in \mathbb{N}$
such that every normed space of dimension $d(k)$ admits a $k$ element $r$-equilateral set.
However, the smallest value of $d(k)$ is unknown and it is an open conjecture that $d(k)$
can take the value $k-1$ for each $k\in \mathbb{N}\setminus\{0,1\}$ (\cite{thompson}).
The above results
of \cite{brass, dekster} imply that any infinite dimensional Banach space contains arbitrarily
large finite equilateral sets. In fact,
by a result of Shkarin (\cite{shkarin}) every finite
ultrametric space (a metric space where the distance $d$ satisfies $d(x, z)\leq\max(d(x,y), d(y,z))$
for any points $x, y, z$) isometrically embeds in any infinite dimensional
Banach space. A surprising result was obtained by Terenzi who proved
in \cite{terenzi} that there are
infinite dimensional (separable) Banach
spaces with no infinite equilateral sets (for other spaces with this
property see \cite{glau-inf, terenzi2}). On the other
hand Mercourakis and Vassiliadis proved that
any Banach space containing
an isomorphic copy of $c_0$ admits an
infinite equilateral set (\cite{mer-pams}) and Freeman, Odell, Sari and Schlumprecht
proved that every uniformly smooth Banach space admits an infinite equilateral set (\cite{smooth}).
The difference between the example of Terenzi and the latter Banach spaces should be seen not
only in the context of the geometry of Banach spaces but also in the context of
infinite combinatorics, in particular the applicability of Ramsey methods in Banach spaces.
A natural problem if every nonseparable Banach space admits an uncountable
equilateral set has been considered in \cite{mer-pams, mer-c, equi}.
The first named author constructed in \cite{equi}
a consistent example of a nonseparable Banach space
which does not admit an uncountable equilateral set. It is of the form
$C(K)$, where $K$ is Hausdorff and compact. However, it was also
proved in \cite{equi} that it is consistent that no such Banach space
of the form $C(K)$ exists. This showed that
the problem whether a nonseparable Banach space of the form $C(K)$
admits an uncountable equilateral set is undecidable (\cite{equi}).
The main result of this paper is that there are absolute (under no extra set-theoretic assumption)
examples of nonseparable Banach spaces with no uncountable equilateral sets (necessarily not of
the form $C(K)$).
Moreover, they do not even admit an infinite equilateral sets and have density
continuum\footnote{We do not know if the density continuum is the maximal possible. The only result bounding the densities of Banach spaces with no uncountable equilateral sets was obtained by Terenzi in \cite{terenzi} using essentially an Erd\"os-Rado type argument which is an uncountable version
of the Ramsey theorem:
if the density of a Banach space $\mathcal{X}$ is bigger than $2^{(2^\omega)}$, then
$\mathcal{X}$ admits an uncountable equilateral set.}.
Our approach is to transfer some parts of the Terenzi arguments from \cite{terenzi} to the nonseparable
setting. He considered a renorming of $\ell_1$, where the norm is given for any $x\in \ell_1$ by
$$\|x\|=\|x\|_1+\sqrt{\sum_{i\in \mathbb{N}}{x(i)^2\over 2^i}}.$$
We consider renormings of $\ell_1([0,1])$ where the norm is
defined for any $x\in \ell_1([0,1])$ by
$$\|x\|_T=\|x\|_1+\|T(x)\|_\mathcal{X}, \leqno (\odot)$$
where $T:\ell_1([0,1])\rightarrow\mathcal{X}$ is an injective operator into a Banach space $\mathcal{X}$.
Renormings of $\ell_1([0,1])$ similar but different to ours were already employed in e.g., \cite{godun}
to obtain Banach spaces not admitting certain subsets. The foundation of our main result is the following:
\begin{theorem}\label{infinite} Suppose that $\mathcal{X}$ is a Banach space with a strictly convex norm and
that $T:\ell_1([0,1])\rightarrow \mathcal{X}$ is a compact bounded injective operator.
Then the equivalent renorming $(\ell_1([0,1]), \|\ \|_T)$
of $(\ell_1([0,1]), \|\ \|_1)$
admits no infinite equilateral set.
\end{theorem}
Since there exist operators as in the hypothesis of the above theorem
(Lemma \ref{operators}) we obtain:
\begin{corollary}\label{main} There is a Banach space of density continuum which does not admit
an infinite equilateral set.
\end{corollary}
In particular, this solves the question of whether there is a nonseparable
Banach space with no uncountable equilateral set (\cite{mer-c, equi}, Problem 293 of \cite{guirao}).
Another absolute construction of a nonseparable Banach space with no uncountable
equilateral set is being presented at the same time in a paper by the first named author
\cite{kottman}. However, that is a renorming of
a space $C_0(K)$ for $K$ locally compact and scattered, so it is $c_0$-saturated (by \cite{pel}).
Since a result in \cite{mer-pams} says that any Banach space which contains an isomorphic copy
of $c_0$ admits an infinite equilateral set, we conclude that
spaces of \cite{kottman} admit such infinite sets.
By an argument of Terenzi from \cite{terenzi} given any equilateral set $\mathcal{Y}$ in
a Banach space $\mathcal{X}$ we may assume that it is a $1$-equilateral set by scaling it.
Considering $\{y-y_0: y\in \mathcal{Y}\setminus\{y_0\}\}$ for any $y_0\in \mathcal{Y}$
we may assume that it is a $1$-equilateral set included in the unit sphere of $\mathcal{X}$.
Thus equilateral sets are related to the questions concerning separation of points
in the spheres of Banach spaces (see e.g. \cite{hajek-tams} for references).
Recall that a subset $\mathcal{Y}$ of a Banach space $\mathcal{X}$ is called
$\delta$-separated if $\|y-y'\|\geq \delta$ for all distinct $y, y'\in \mathcal{Y}$.
It is called $(\delta+)$-separated if $\|y-y'\|> \delta$ for all distinct $y, y'\in \mathcal{Y}$.
By Remark 3.16 \cite{hajek-tams} the unit sphere of every renorming of $\ell_1([0,1])$
contains a subset $\mathcal{Y}$ of cardinality continuum such that $\|y-y'\|\geq1+\varepsilon$
some $\varepsilon>0$ and for every two distinct $y, y'\in \mathcal{Y}$.
After proving Theorem \ref{infinite} in Section 3 we consider renormings
$\|\ \|_T$ of $\ell_1([0,1])$ as in ($\odot$) for any injective $T$ with separable range.
Some of such renormings admit many infinite equilateral sets (see Remark \ref{plus-id}). We obtain:
\begin{theorem}\label{uncountable}
Suppose that $\mathcal{X}$ is a Banach space and $T:\ell_1([0,1])\rightarrow \mathcal{X}$ is injective
and has separable range and $r>0$.
Then $(\ell_1([0,1]), \|\ \|_T)$
has the following property: Any $r$-separated subset $\mathcal{Y}\subseteq \ell_1([0,1])$ of regular uncountable cardinality has a subset $\mathcal{Z}\subseteq \mathcal{Y}$ of the same cardinality which is $(r+)$-separated.
In particular $(\ell_1([0,1]), \|\ \|_T)$ does not admit any uncountable equilateral set.
\end{theorem}
The above property for renormings induced by
$T$ injective compact and with strictly convex range is a consequence of Theorem \ref{infinite} and
some partition calculus results
(see Corollary \ref{dushnik}). Also this property is much stronger
than not having uncountable equilateral sets (see Remark \ref{sierpinski}).
A close link
between separated subsets in the sphere and Auerbach bases
was demonstrated in \cite{hajek-tams}. In fact Godun's renorming of $\ell_1([0,1])$
was designed to prove that the space has no fundamental Auerbach system (\cite{godun}).
Nevertheless, we do not know if our spaces admits an uncountable Auerbach system.
Let us also remark that considering Banach spaces without large equilateral sets which have renormings
admitting large equilateral sets (obviously $\ell_1([0,1])$ admits equilateral set of
cardinality continuum in the standard norm) is sometimes necessary to obtain
examples of the former kind. For
example, by a result of Swanepoel (\cite{swan}) any infinite dimensional
Banach space has an equivalent renorming which admits an infinite equilateral set (see also
\cite{mer-pams}). Moreover by the results of \cite{mer-pams} the existence
of a biorthogonal system of cardinality $\kappa$ in a Banach space $\mathcal{X}$ implies
the existence of an equivalent renorming of $\mathcal{X}$ which admits equilateral set of cardinality $\kappa$.
This means by a result of Todorcevic (\cite{stevo-biorth}) that it is consistent
that every nonseparable Banach space has an equivalent renorming which admits an uncountable
equilateral set. We do not know however if it is consistent that there is
a nonseparable Banach space without an equivalent renorming which admits uncountable
equilateral sets. The densities of such an example could not exceed continuum
(By a result of W. Johnson that any Banach space of density
bigger than continuum admits an uncountable biorthogonal system cf. Theorem 2.1 of \cite{sur}).
If at all possible, the construction for density equal to any consistent value
of the continuum would not be easy, as the examples in the literature of Banach spaces
which do not admit uncountable biorthogonal sets have reached only the density $\omega_2$ so far
(\cite{christina}).
Note also that it remains open if there are (even consistent) Banach spaces
(or even renormings of $\ell_1(\kappa)$) of densities in the interval $(2^\omega, 2^{(2^\omega)}]$ which do not admit
infinite or uncountable equilateral sets.
\section{Preliminaries and notation}\label{prelim}
The notation and terminology are standard. The notation $A^B$ represents the set of all functions form
a set $B$ into a set $A$. Given a set $A$ by $[A]^2$ we mean the collection of all two-element
subsets of $A$. When $f:[A]^2\rightarrow B$, we say that $A'\subseteq A$ is $b$-monochromatic
for $b\in B$ if $f[[A']^2]=\{b\}$; a set is called monochromatic if it is $b$-monochromatic for some
$b\in B$. The symbols $\omega_1$, $\omega_2$
denote the first and the second uncountable cardinals respectively. The
set of all natural numbers and the set of all rational numbers are
denoted by $\mathbb{N}$ and $\mathbb{Q}$ respectively.
All Banach spaces considered here are over the reals.
Whenever $(\mathcal{X}, \|\ \|_\mathcal{X})$ is a Banach space
we will refer only to $\mathcal{X}$ if $\|\ \|_\mathcal{X}$ is clear from the context.
We will consider norms
$\|\ \|_1$ and $\|\ \|_2$ defined as usual for a real sequence $(x(i))_{i\in \mathbb{N}}$ by
$\|x\|_1=\sum_{i\in \mathbb{N}} |x(i)|$ and $\|x\|_2=\sqrt{\sum_{i\in \mathbb{N}} x(i)^2}$.
By $\ell_1(A)$ for a set $A$ we mean all functions $x\in\mathbb{R}^{A}$ such that
$\|x\|_1<\infty$ with $\| \ \|_1$ norm and by $\ell_2$ as all functions
$x\in\mathbb{R}^\mathbb{N}$ such that $\|x\|_2<\infty$ with $\|\ \|_2$ norm.
The dual space to $\ell_1([0,1])$ is $\ell_\infty([0,1])$
together with the action
$$\langle\phi, x\rangle=\sum_{t\in [0,1]}\phi(t)x(t)$$
for any $\phi\in \ell_\infty([0,1]$ and $x\in \ell_1([0,1])$.
By a support of $x\in \ell_1(A)$ we mean $\{a\in A: x(a)\not=0\}$; it is denoted $supp(x)$.
If $x\in \ell_1(A)$ and $B\subseteq A$, then by $x|B$ we mean the coordinatewise product
of $x$ and the characteristic function of $B$.
Recall that a Banach space $(\mathcal{X}, \|\ \|_\mathcal{X})$ is strictly convex
if $\|x+y\|=\|x\|+\|y\|$ implies that there is $\lambda>0$ such that $x=\lambda y$
for any $x\not=0\not=y$. It is well known that the norm on $\ell_2$ is strictly convex.
We say that two norms $\|\ \|$ and $\|\ \|'$ on a Banach space $\mathcal{X}$
are equivalent if there are constants $c, C>0$ such that
$c\|x\|\leq\|\ x\|'\leq C\|\ x\|$ for every $x\in \mathcal{X}$. This is equivalent to the fact that
the identity is an isomorphism between $(\mathcal{X}, \|\ \|)$ and $(\mathcal{X}, \|\ \|')$.
\begin{lemma} Suppose that $\mathcal{X}, \mathcal{Y}$ are Banach spaces and $T: \mathcal{X}\rightarrow \mathcal{Y}$
is a bounded linear operator. Then the norm $\|\ \|_T$ on $\mathcal{X}$ given by
$\|x \|_\mathcal{X}+\|T(x)\|_\mathcal{Y}$ for $x\in \mathcal{X}$ is equivalent to the norm $\|\ \|_\mathcal{X}$.
If $T$ is injective and $\mathcal{Y}$ is strictly convex, then $\|\ \|_T$ is strictly convex.
\end{lemma}
\begin{proof} We have
$$\|x\|_\mathcal{X}\leq \|x\|_\mathcal{X}+\|T(x)\|_\mathcal{Y}\leq (1+\|T\|)\|x\|.$$
For strict convexity suppose that
$\|x+y\|_T=\|x\|_T+\|y\|_T$ for some nonzero $x, y\in \mathcal{X}$. So
$\|x+y\|_\mathcal{X}+\|T(x+y)\|_\mathcal{Y}=\|x\|_\mathcal{X}+\|y\|_\mathcal{X}+\|T(x)\|_\mathcal{Y}+\|T(y)\|_\mathcal{Y}$. By the triangle
inequality this means that $\|x+y\|_\mathcal{X}=\|x\|_\mathcal{X}+\|y\|_\mathcal{X}$
and $\|T(x+y)\|_\mathcal{Y}=\|T(x)\|_\mathcal{Y}+\|T(y)\|_\mathcal{Y}$. The injectivity of $T$ yields $T(x)\not=0\not=T(y)$.
The strict convexity of $\mathcal{Y}$ yields
$\lambda>0$ such that $T(x)=\lambda T(y)$. The injectivity of $T$ gives
$x=\lambda y$.
\end{proof}
Let $(I_i)_{i\in \mathbb{N}}$ be an enumeration of all subintervals
of $[0,1]$ with rational end-points. We define $x^*_i=\chi_{I_i}\in \ell_\infty([0,1])$,
where $ \chi_{I_i}$ is the characteristic function of $I_i$.
Given a nonzero $x\in \ell_1([0,1])$ we find $t\in [0,1]$ such that
$x(t)\not=0$ and an open interval $I_i\ni t$ such that
$\sum\{|x(t')|: t'\in I_i\setminus \{t\}\}<|x(t)|$. Then $\langle x^*_i, x\rangle\not=0$.
This shows that $\{x^*_i: i\in \mathbb{N}\}$ is total for $\ell_1([0,1])$ i.e.,
$x_i^*(x)=0$ for each $i\in I$ implies that $x=0$.
Observe that $\|x^*_i\|=1$ for each $i\in \mathbb{N}$.
\begin{lemma}\label{operators} There is a bounded compact injective operator
$T:\ell_1([0,1])\rightarrow \ell_2$.
\end{lemma}
\begin{proof}
Define $T$ by
$$T(x)=\Big({{x^*_i(x)}\over{2^i}}\Big)_{n\in \mathbb{N}}$$
for any $x\in \ell_1([0,1])$. As $x^*_i$s form a total set,
the operator is injective. It is also clear that the values of $T$ are in $\ell_2$
and the operator is bounded with its norm $\sqrt2$, as $\|x_i^*\|=1$ or each $i\in \mathbb{N}$.
For the compactness, use again the fact that the norms of $x_i^*$s are $1$
and so $T$ can be approximated in the operator norm by finite rank operators
which are $T$ up to the $k$-th coordinate and later $0$ for $k\in \mathbb{N}$.
As compact operators form a closed ideal this proves the compactness of $T$.
\end{proof}
\section{Proof of the main result}
\noindent{\bf Theorem 1.} {\it Suppose that $\mathcal{X}$ is a Banach space with a strictly convex norm and
that $T:\ell_1([0,1])\rightarrow \mathcal{X}$ is a compact bounded injective operator.
Then the equivalent renorming $(\ell_1([0,1]), \|\ \|_T)$
of $(\ell_1([0,1]), \|\ \|_1)$
admits no infinite equilateral set.}
\begin{proof}
Suppose that $\{x_n: n\in \mathbb{N}\}$ is equilateral in $\ell_1([0,1])$
with the norm $\|\ \|_T$. We will derive a contradiction.
By scaling it, we may assume that it is $1$-equilateral.
As the supports of $x_n$'s are countable, they are all included in some countable
$A\subseteq [0,1]$. So we need to prove that the corresponding renorming of the
separable $\ell_1(A)$ does not admit an infinite equilateral set.
By the compactness of $T$, passing to a subsequence we may assume that
$\{T(x_n): n\in \mathbb{N}\}$ converges in the norm $\|\ \|_\mathcal{X}$ to $z\in \mathcal{X}$.
As the range of $T$ is not closed, $z$ does not need to belong to it.
Since we work now with
separable $\ell_1(A)$ and $(x_n)_{n\in \mathbb{N}}$ is bounded,
by passing to a subsequence we may assume that $(x_n)_{n\in \mathbb{N}}$
converges pointwise to $y\in \ell_1(A)$, that is for every $t\in A$
the sequence $(x_n(t))_{n\in \mathbb{N}}$ converges to $y(t)$.
Let $x_n'=x_n-y$ for every $n\in \mathbb{N}$.
Then $\|x_n'-x_m'\|_T=\|x_n-x_m\|_T=1$ for all distinct $n, m\in \mathbb{N}$ and for every $t\in A$
the sequence $(x_n'(t))_{x\in \mathbb{N}}$ converges to $0$.
Moreover
$(T(x_n'))_{n\in \mathbb{N}}$ converges in the norm $\|\ \|_\mathcal{X}$ to $z'=z-T(y)$.
Fix $m\in \mathbb{N}$ and $\varepsilon>0$. Choose a finite $F\subseteq A$ such
that $\|x_m'|F-x_m'\|_1<\varepsilon/4$. As $(x_n'(t))_{n\in \mathbb{N}}$ converges to $0$,
for each $t\in F$, for sufficiently large $n\in \mathbb{N}$ we have $\|x_n'|F\|_1\leq\varepsilon/4$ and so
we have
\begin{itemize}
\item $\|x_n'\|_1\leq \|x_n'-x_n'|F\|_1+\varepsilon/4$,
\item $|x_m'\|_1\leq \|x_m'|F\|_1 +\varepsilon/4$,
\item $\|x_n'-x_m'\|_1\geq \|(x_n'-x_n'|F)- x_m'|F\|-\varepsilon/2$
\item $\|(x_n'-x_n'|F)-x_m'|F\|_1=\|(x_n'-x_n'|F)\|_1+\|x_m'|F\|_1$ as these vectors
have disjoint supports,
\end{itemize}
so we obtain
$$\|x_n'\|_1+\|x_m'\|_1-\|x_n'-x_m'\|_1\leq$$
$$\leq \|x_n'-x_n'|F\|_1+\|x_m'|F\|_1-\|(x_n'-x_n'|F)- x_m'|F\|+\varepsilon=\varepsilon.$$
So for every $m\in \mathbb{N}$ we have
$$lim_{n\rightarrow \infty}\big(\|x_n'\|_1+\|x_m'\|_1-\|x_n'-x_m'\|_1\big)=0.\leqno (*)$$
For $n\in \mathbb{N}$ define
$$c_n=1/2-\|x_n'\|_1.$$
\begin{claim}\label{one-half}
$\lim_{n\rightarrow\infty}c_n=0$.
\end{claim}
\noindent{\it Proof of the Claim:}
Since the sequence of $c_n$s is bounded (as the $x_n'$s form an equilateral set),
by passing to a subsequence we may assume that it is
converging to $c$. First suppose that $c>0$. Let $k\in \mathbb{N}$ and $\varepsilon>0$
be such that $\|x_n'\|<1/2-\varepsilon$ for all $n>k$. Then by passing to a subsequence
we may assume that $\|T(x_n')-T(x_m')\|_\mathcal{X}\leq\varepsilon$ for all $n, m\in \mathbb{N}$ as
$(T(x_n'))_{n\in \mathbb{N}}$ converges to $z'$ in $\mathcal{X}$. Fixing $m>k$
by the triangle inequality we obtain
$$\|x_n'-x_m'\|_T=\|x_n'-x_m'\|_1+\|T(x_n'-x_m')\|_\mathcal{X} \leq\|x_n'\|_1+\|x_m'\|_1+ \|T(x_n')-T(x_m')\|_\mathcal{X}<$$
$$2(1/2-\varepsilon)+\varepsilon\leq 1$$
contradicting the fact that $x_n'$s form a $1$-equilateral set.
Now suppose that $c<0$. Let $k\in \mathbb{N}$ and $\varepsilon>0$
be such that $\|x_n'\|>1/2+\varepsilon$ for all $n>k$. Fixing $m>k$
by (*) we may find $n\in \mathbb{N}$ such that $\|x_n'\|_1+\|x_m'\|_1-\|x_n'-x_m'\|_1\leq\varepsilon$.
So
$$\|x_n'-x_m'\|_T>\|x_n'-x_m'\|_1=(\|x_n'-x_m'\|_1-\|x_n'\|_1-\|x_m'\|_1)+(\|x_n'\|_1+\|x_m'\|_1)> $$
$$-\varepsilon+2(1/2+\varepsilon)\geq 1$$
contradicting the fact that $x_n'$s form a $1$-equilateral set. This completes the proof of the claim.
For distinct $m, n\in \mathbb{N}$ we have
$$\|x_n'-x_m'\|_T=\|x_n'-x_m'\|_1+\|T(x_n')-T(x_m')\|_\mathcal{X}=1,$$
so subtracting $1=1/2+1/2=(\|x_n'\|_1+c_n)+(\|x_m'\|_1+c_m)$ from both sides of the second equality, we get
$$-c_n-c_m-\|x_n'\|_1-\|x_m'\|_1+\|x_n'-x_m'\|_1+\|T(x_n')-T(x_m')\|_\mathcal{X}=0.\leqno (**)$$
Fixing any $m\in\mathbb{N}$, by going to infinity with $n\in\mathbb{N}$ by (*) and Claim \ref{one-half} we obtain
$$\|z'-T(x_m')\|_\mathcal{X}=c_m.\leqno (***)$$
Defining $u_m=T(x_m')-z'$ and combining (**) and (***) we obtain for distinct $n,m\in \mathbb{N}$
$$\|x_n'-x_m'\|_1-\|x_n'\|_1-\|x_m'\|_1=\|u_n\|_\mathcal{X}+\|u_m\|_\mathcal{X}-\|u_n-u_m\|_\mathcal{X}.$$
By the triangle inequality the right hand side of the above is non-negative while the left hand side
is non-positive which implies that both of the expressions are equal to zero. In
particular, the left hand side is zero.
Note that $u_n\not=u_m$ for distinct $n, m\in \mathbb{N}$ because otherwise we would have
$T(x_n')=T(x_m')$, which implies $x_n'=x_m'$ by the injectivity of $T$ and this contradicts
the fact that $x_n'$s form a $1$-equilateral set. So at most one $u_n$ can be zero.
By passing to an infinite subset we may assume that all are nonzero, so that we can
apply the definition of the strict convexity.
By the strict convexity of the norm in $\mathcal{X}$ we obtain $\lambda_{m, n}>0$
such that $u_m=\lambda_{m, n} u_n$. If $\lambda_{m, n}=1$ for some distinct $m, n\in \mathbb{N}$,
then $T(x_n')=T(x_m')$ which contradicts the injectivity of $T$.
Otherwise $(z'-T(x_m'))=\lambda_{m, n}(z'-T(x_n'))$ which gives
$z'(1-\lambda_{m, n})=-\lambda_{m, n}T(x_n')+T(x_m')$ and so
$$z'=T\Big( {1\over{1-\lambda_{m, n}}}\big( x_m'-\lambda_{m, n}x_n'\big)\Big)$$
By the injectivity of $T$ this means for any distinct $k, l\in \mathbb{N}$
$${1\over{1-\lambda_{m, n}}}\big( x_m'-\lambda_{m, n}x_n'\big)={1\over{1-\lambda_{k, l}}}\big( x_k'-\lambda_{m, n}x_l'\big),$$
in particular that
$$x_m'={{1-\lambda_{m, n}}\over{1-\lambda_{k, l}}}\big( x_k'-\lambda_{m, n}x_l'\big)+\lambda_{m, n}x_n'.$$
That means that $\{x_n': n\in \mathbb{N}\}$ spans a two dimensional space. However
such a space cannot admit an infinite equilateral set as its ball is compact and so any bounded sequence
has a convergent subsequence. This is the required contradiction.
\end{proof}
\begin{corollary}\label{dushnik}
Suppose that $\mathcal{X}$ is a Banach space with a strictly convex norm and
that $T:\ell_1([0,1])\rightarrow \mathcal{X}$ is a compact bounded injective operator.
Then the equivalent renorming $(\ell_1([0,1]), \|\ \|_T)$
of $(\ell_1([0,1]), \|\ \|_1)$ has the following property: Any infinite $r$-separated subset
$\mathcal{Y}\subseteq \ell_1([0,1]$ has a subset $\mathcal{Z}\subseteq \mathcal{Y}$ of the same cardinality which is $(r+)$-separated.
\end{corollary}
\begin{proof} Define a function $c: [\mathcal{Y}]^2\rightarrow\{0,1\}$
by putting $c(\{y, y'\})=0$ if and only if $\|y-y'\|_T=r$.
First consider the case when $\mathcal{Y}$ is countable.
By Ramsey's (Problem 24.1 of \cite{komjath})
theorem there is an infinite monochromatic subset of $\mathcal{Y}$. It cannot be
$0$-monochromatic as $(\ell_1([0,1]), \|\ \|_T)$ has no infinite equilateral
sets by Theorem \ref{infinite}. A $1$-monochromatic infinite subset is the required one.
Now consider an uncountable regular cardinality of $\mathcal{Y}$.
A version of Dushnik Miller theorem says that for any uncountable cardinal
$\kappa$ for any $f: [\kappa]^2\rightarrow\{0,1\}$ there is
either an infinite $0$-monochromatic set or a $1$-monochromatic subset
of cardinality $\kappa$ (Problem 24.13 of \cite{komjath}). So apply this to $c$ and use the fact that
$(\ell_1([0,1]), \|\ \|_T)$ has no infinite equilateral sets by Theorem \ref{infinite}.
A $1$-monochromatic infinite subset is the required one.
\end{proof}
\section{Renormings induced by injective separable range operators}
In this section we prove that a substantial part of the property of Corollary \ref{dushnik} holds
for much bigger class of renormings of $\ell_1([0,1])$ than
those considered in Theorem \ref{infinite}.
\vskip 6pt
\noindent{\bf Theorem 3.}
{\it Suppose that $\mathcal{X}$ is a Banach space and $T:\ell_1([0,1])\rightarrow \mathcal{X}$ is injective
and has separable range and $r>0$.
Then $(\ell_1([0,1]), \|\ \|_T)$
has the following property: Any $r$-separated subset $\mathcal{Y}\subseteq \ell_1([0,1])$ of regular uncountable cardinality has a subset $\mathcal{Z}\subseteq \mathcal{Y}$ of the same cardinality which is $(r+)$-separated.
In particular $(\ell_1([0,1]), \|\ \|_T)$ does not admit any uncountable equilateral set..}
\begin{proof}
Let $\{d_n: n\in \mathbb{N}\}$ be a dense subset of the range of $T$.
Suppose that $\kappa$ is a regular uncountable cardinal
and $\mathcal{Y}=\{x_\alpha: \alpha<\kappa\}\subseteq \ell_1([0,1])$ is $r$-separated in the norm
$\|\ \|_T$.
As the supports of $x_\alpha$s are countable, their union has cardinality at most $\kappa$.
So we may assume that $\mathcal{Y}\subseteq \ell_1(A)$ for some $A\subseteq[0,1]$ of cardinality $\kappa$.
Let $A=\{t_\xi: \xi<\kappa\}$.
Now we need to define certain function $M$. The domain of $M$ will consist of $5$-tuples
of the form
$(\varepsilon, q, F, s, n)$, where $\varepsilon, q>0$ are rationals, $F$ is a finite subset
of $[0,1]$, $s\in \mathbb{Q}^F$ and $n\in\mathbb{N}$.
Note that there are only countably many such $\varepsilon, q, n$ and given a finite
$F\subseteq [0,1]$
there are only countably many choices for $s\in \mathbb{Q}^F$.
The function $M$ will assume values in $\kappa$. It is defined as follows: if there is $\alpha<\kappa$
such that
\begin{enumerate}[(a)]
\item
$\|d_n-T(x_{\alpha})\|_\mathcal{X}<\varepsilon/10,$
\item
$|\|x_{\alpha}|([0,1]\setminus F)\|_1-q|<2\varepsilon/10$,
\item
$\|s-(x_{\alpha}|F)\|_1<\varepsilon/10$,
\end{enumerate}
then we choose minimal such $\alpha$ and define $M(\varepsilon, q, F, s, n)$ as the minimal
ordinal less than $\kappa$ such that $supp(x_\alpha)\subseteq
\{t_\xi: \xi<M(\varepsilon, q, F, s, n)\}$.
If there is no such $\alpha$, then we define $M(\varepsilon, q, F, s, n)$ anyhow.
\begin{claim}
$$C=\{\delta<\kappa: \forall (\varepsilon, q, F, s, n)\in dom(M)
[F\subseteq\{t_\xi:\xi<\delta\}\ \Rightarrow \ M(\varepsilon, q, F, s, n)<\delta]\}$$
is unbounded in $\kappa$
\end{claim}
\noindent{\it Proof of the Claim.}
Fix $\delta_0<\kappa$. By recursion define a strictly increasing $(\delta_n)_{n\in \mathbb{N}}$
in $\kappa$ such that $M(\varepsilon, q, F, s, n)<\delta_{n+1}$ whenever
$(\varepsilon, q, F, s, n)$ is in the domain of $M$ and $F\subseteq\delta_n$.
Given $\delta_n$ there are less than $\kappa$ many elements in the domain of $M$
such that $F\subseteq\delta_n$ as there are only less than $\kappa$ such $F$s, so
by the regularity of $\kappa$ the next
$\delta_{n+1}$ can be taken as the supremum of all the values under $M$ of such elements.
One sees that $\delta=\sup\{\delta_n: n\in \mathbb{N}\}$ is in $C$. As $\delta_0$ was arbitrary,
this completes the proof of the Claim.
For any $\delta\in C$ and $\alpha<\kappa$ we define
$$x_{\alpha, \delta}=x_{\alpha}|\{t_\xi: \delta\leq \xi<\kappa\}$$
\begin{claim}\label{model} If $x_{\alpha, \delta}\not=0$, then $\|x_{\alpha, \delta}\|_1\geq r/2$ for each $\alpha<\kappa$
and $\delta\in C$.
\end{claim}
\noindent{\it Proof of the Claim.}
Fix $\delta\in C$ and $\alpha<\kappa$
such that $x_{\alpha, \delta}\not=0$.
Fix a rational $\varepsilon>0$. We will show that there is $\beta<\alpha$
such that $|\|x_\beta-x_{\alpha}\|_T-2\|x_{\alpha, \delta}\|_1|<\varepsilon$. As
$\{x_\alpha: \alpha<\kappa\}$ is an $r$-separated set and $\varepsilon>0$ is arbitrary, this
is sufficient.
Find
\begin{enumerate}
\item $q\in \mathbb{Q}$ such that $|q-\|x_{\alpha, \delta}\|_1|<\varepsilon/10$,
\item $n\in \mathbb{N}$ such that
$$\|d_n-T(x_{\alpha})\|_\mathcal{X}<\varepsilon/10,$$
\item finite $F\subseteq\{t_\xi: \xi<\delta\}$ such that
$|\|x_{\alpha}|([0,1]\setminus F)\|_1-q|<2\varepsilon/10$,
\item $s\in \mathbb{Q}^F$ such that
$\|s-(x_{\alpha}|F)\|_1<\varepsilon/10$,
\end{enumerate}
So $x_{\alpha}$ satisfies the following formulas when substituted in place of $x$;
\begin{enumerate}
\item[(5)]
$\|d_n-T(x)\|_\mathcal{X}<\varepsilon/10,$
\item[(6)]
$|\|x|([0,1]\setminus F)\|_1-q|<2\varepsilon/10$,
\item [(7)]
$\|s-(x|F)\|_1<\varepsilon/10$.
\end{enumerate}
As $F\subseteq\{t_\xi: \xi<\delta\}$ since $\delta\in C$, we have $M(\varepsilon, q, F, s, n)<\delta$.
As there is $x_{\gamma}$ (for $\gamma=\alpha$) which satisfies (5) - (7) when substituted for $x$,
by the definition of $M$
there is $\beta$ such that $x_\beta$ satisfies (5) -(7) when substituted for $x$ and $x_\beta$
has its support included in $\{t_\xi: \xi<M(\varepsilon, q, F, s, n)\}$ and in particular in
$\{t_\xi: \xi<\delta\}$.
Now we estimate $\|x_\beta-x_{\alpha}\|_T$:
$$\|T(x_\beta-x_{\alpha})\|_\mathcal{X}=\|T(x_\beta)-T(x_{\alpha})\|_\mathcal{X}\leq \|T(x_\beta)-d_n\|_\mathcal{X}
+\|d_n-T(x_{\alpha})\|_\mathcal{X}
\leq 2\varepsilon/10\leqno(8)$$
by (2) and (5) for $x_\beta$ in place of $x$.
$$\|x_\beta-x_{\alpha}\|_1=\|x_\beta|F-x_{\alpha}|F\|_1+
\|x_\beta|(\{t_\xi: \xi<\delta\}\setminus F)-x_{\alpha}|(\{t_\xi: \xi<\delta\}\setminus F)\|_1
+\|x_{\alpha, \delta}\|_1\leqno(9)$$
since the support of $x_\beta$ is included in $\{t_\xi: \xi<\delta\}$.
Conditions (4) and (7) for $x_\beta$ in place of $x$ imply that
$$\|x_\beta|F-x_{\alpha}|F\|_1\leq 2\varepsilon/10.\leqno (10)$$
Conditions (1) and (3) imply that
$$\|x_{\alpha}|(\{t_\xi: \xi<\delta\}\setminus F)\|_1< 3\varepsilon/10$$
and so by (6) for $x_\beta$ in place of $x$ and the fact that the support of $x_\beta$ is included in $\{t_\xi: \xi<\delta\}$
we conclude that
$$q-5\varepsilon/10\leq \|x_\beta|(\{t_\xi: \xi<\delta\}\setminus F)
-x_{\alpha}|(\{t_\xi: \xi<\delta\}\setminus F)\|_1\leq q+5\varepsilon/10,\leqno(11)$$
so by (1) and (8) - (11) we conclude that
$$2\|x_{\alpha, \delta}\|_1-6\varepsilon/10\leq \|x_\beta
-x_{\alpha}\|_T\leq 2\|x_{\alpha, \delta}\|_1+\varepsilon,$$
which completes the proof of the Claim.
Now note that
as $\{x_\alpha: \alpha<\kappa\}$ is discrete, it cannot be contained
in any subspace of the form $\ell_1(\{t_\xi: \xi<\delta\})$ for $\delta<\kappa$ which has
density less than $\kappa$. So for $\delta\in C$ we can find $\alpha_\delta<\kappa$
such that $x_{\alpha_\delta, \delta}\not=0$ and moreover we may make sure that
$\alpha_\delta\not=\alpha_{\delta'}$ for any $\delta<\delta'$ in $C$.
Next we find $C'\subseteq C$ of cardinality $\kappa$ such that
$$supp(x_{\alpha_{\delta'}, {\delta'}})\subseteq\{t_\xi: \delta'\leq \xi<\delta\}\leqno (12)$$
for any $\delta'<\delta$ and $\delta, \delta'\in C'$.
This can be done by recursion taking at the inductive step the next $\delta\in C$ such that
the supports of the previous $x_{\alpha_{\delta'}, \delta'}$s are included
in $\{t_\xi: \xi<\delta\}$.
Now we will consider two cases, the first when there is $\theta\in C'$ such
that for any $\delta, \delta'\in C'$ such that
$\theta<\delta'<\delta$ we have
$$supp(x_{\alpha_{\delta'},\delta'})\cap supp(x_{\alpha_{\delta}})=\emptyset.$$
Then by Claim \ref{model} we have
$$\|x_{\alpha_{\delta}}-x_{\alpha_{\delta'}}\|_1\geq
\|x_{\alpha_{\delta'},\delta'}\|_1+\|x_{\alpha_{\delta},\delta}\|_1\geq r$$
for every $\theta<\delta'<\delta<\kappa$. Since $T$ is injective
$\|x_{\alpha_{\delta}}-x_{\alpha_{\delta'}}\|_T>\|x_{\alpha_{\delta}}-x_{\alpha_{\delta'}}\|_1\geq r$,
so we obtain that $\mathcal{Z}=\{x_{\alpha_\delta}: \delta\in C', \ \delta>\theta\}$.
The second case is when there is no $\theta\in C'$ as in the first case. Then
by recursion we can construct $C''\subseteq C'$ and $(\theta_\delta: \delta\in C'')$
such that
$$sup\{\xi: t_\xi\in supp(x_{\alpha_{\delta'}})\}< \theta_\delta<\delta, \ \ x_{\alpha_{\delta}}(t_{\theta_\delta})\not=0\leqno (13)$$
for every $\delta'<\delta$ with $\delta, \delta'\in C''$.
Indeed, having constructed less then $\kappa$ elements $\delta'\in C''$ whose supremum is $\theta< \kappa$
we consider all $\delta\in C'$ which are above $\theta$. Since there is no $\theta\in C'$ as
in the first case we find $\theta<\delta''<\delta$ with $\delta'', \delta\in C'$ such that
$supp(x_{\alpha_{\delta''},\delta''})\cap supp(x_{\alpha_{\delta}})\not=\emptyset$.
To complete the recursion take $\delta$ as the next element of
$C''$ and take as $t_{\theta_\delta}$ any element of the above intersection.
Since
$\kappa$ is a cardinal of uncountable cofinality, by passing to
a subset of cardinality $\kappa$ we may assume that there is $\varepsilon>0$
such that $|x_{\alpha_{\delta}}(t_{\theta_\delta})|\geq\varepsilon$ for every $\delta\in C''$.
Now we will use the following version of the Dushnik-Miller theorem: if
$\kappa$ is a regular uncountable cardinal and $c:[\kappa]^2\rightarrow\{0,1\}$,
then either there is a $0$-monochromatic set for $c$ which has its order type equal to $\omega+1$
or there a $1$-monochromatic subset of cardinality $\kappa$ for $c$ (Problem 24.32 of \cite{komjath}).
We define $c: [C'']^2\rightarrow\{0,1\}$ by $c(\delta', \delta)=0$ for $\delta'<\delta$
if and only if
$$\|x_{\alpha_{\delta}}-x_{\alpha_{\delta'}}\|_T=r.$$
So it is enough to prove that there is no $0$-monochromatic set of order type $\omega+1$.
Suppose $\{\delta_n: n\in \mathbb{N}\}\cup\{\delta_\omega\}$ forms such a set, where
$\delta_\xi<\delta_\eta$ if $\xi<\eta$ for all $\xi, \eta< \omega+1$.
Since the supports of $x_{\alpha_{\delta},\delta}$s for $\delta\in C''$
are pairwise disjoint by (12) there is $n\in \mathbb{N}$ such that
$$\|x_{\alpha_{\delta_\omega}}| supp(x_{\alpha_{\delta_n},\delta_n})\|_1\leq \varepsilon/2.$$
Also by (13) we have $t_{\theta_{\delta_\omega}}\not\in supp(x_{\alpha_{\delta_n}, \delta_n})\cup
supp(x_{\alpha_{\delta_\omega}, \delta_\omega})$, where the union is disjoint by (12).
So by Claim \ref{model} we have
$$\|x_{\alpha_{\delta_\omega}}-x_{\alpha_{\delta_n}}\|_T>
\|x_{\alpha_{\delta_\omega}}-x_{\alpha_{\delta_n}}\|_1\geq
\|x_{\alpha_{\delta_\omega}, \delta_\omega}\|_1+|x_{\alpha_{\delta_\omega}}
(t_{\theta_{\delta_\omega}})|+\|x_{\alpha_{\delta_n}, \delta_n}\|_1-\varepsilon/2
\geq r$$
which contradicts the choice of $\{\delta_n: n\in \mathbb{N}\}\cup\{\delta_\omega\}$ as $0$-monochromatic.
So the set of vectors $x_{\alpha_\delta}$, for $\delta$ in the $1$-monochromatic set of cardinality
$\kappa$, is the desired $\mathcal{Z}$.
\end{proof}
\begin{remark}\label{plus-id}
Some of the renormings considered in Theorem \ref{uncountable} admit many infinite
equilateral sets. For example we can identify $\ell_1([0,1])$ with
$\ell_1\oplus\ell_1([0,1])$ and define $T': \ell_1\oplus\ell_1([0,1])\rightarrow \ell_1\oplus\ell_2$
by $T'(x, y)=(x, T(y))$, where $T$ is as in Lemma \ref{operators}.
\end{remark}
\begin{remark}\label{sierpinski} Let us remark that the property of the spaces from Theorem \ref{uncountable} is much stronger
than not admitting uncountable equilateral sets. To see this consider
$\ell_\infty([[0,1]]^2)$ with the usual $\|\ \|_\infty$ norm. Let
$c:[[0,1]]^2\rightarrow \{0, 1\}$ be Sierpi\'nski's colouring with no
uncountable monochromatic subset (Problem 24.23 of \cite{komjath}).
So $c\in \ell_\infty([[0,1]]^2)$ and consider $f_t\in \ell_\infty([[0,1]]^2)$
defined by
$$
f_t(\{r,s\}) =
\begin{cases}
c(\{r, s\}) & \text{if $t=\min(\{r,s\})$} \\
-c(\{r, s\}) & \text{if $t=\max(\{r,s\})$} \\
0 & \text{otherwise.}
\end{cases}
$$
For distinct $t, t'\in [0,1]$ the intersection of supports of $f_t$ and $f_{t'}$
is included in $\{\{t, t'\}\}$. For $t<t'$ the value of $f_t-f_{t'}$ at $\{t, t'\}$
is $1-(-1)=2$ if $c(\{t, t'\})=1$ and it is $0$ otherwise, so
$$
\|f_t-f_{t'}\| =
\begin{cases}
1 & \text{if $c(\{t,t'\})=0$} \\
2 & \text{if $c(\{t,t'\})=1$.}
\end{cases}
$$
This means that $\{f_t: t\in [0,1]\}$ is a $1$-separated subset of
the unit sphere which does not admit any uncountable
equilateral subset but also there is no uncountable subset
which is $(1+)$-separated. Note that any countable
$1$-separated set $\{x_n: n\in \mathbb{N}\}$ in any Banach space,
contains either an infinite $1$-equilateral set or an infinite
$(1+)$-separated set by Ramsey's theorem (Problem 24.1 of \cite{komjath}).
\end{remark}
\bibliographystyle{amsplain}
|
1,116,691,497,575 | arxiv | \section{Introduction}
The presence or absence of broad optical emission lines has been historically used to separate Seyfert galaxies into two classes: Seyfert 1 galaxies
with broad permitted and narrow forbidden lines and Seyfert 2 galaxies with only narrow permitted and forbidden line emission \citep{1974ApJ...192..581K}.
Using spectropolarimetry, \cite{1985ApJ...297..621A} found broad permitted line emission in the Seyfert 2 \objectname{NGC 1068} galaxy, characteristic of a Seyfert 1 spectrum.
These observational results provided the first evidence in favor of a unified model. In this model
\citep{1993ARA&A..31..473A} both types of Seyfert galaxies are intrinsically the same with the differences being the
visibility of the central engine. A geometrically and optically-thick dusty molecular torus-like structure surrounds the central source, as well as the broad line region (BLR). Therefore the visibility of the nuclear engine depends on the viewing angles.
Following the unified model, our line of sight to Seyfert 2 galaxies is obstructed by optically thick material corresponding to
column densities of $N_H > 10^{22}$ ${\rm cm^{-2}}$ \citep{1999ApJ...522..157R}. For column densities $N_H \leq$ $10^{24}$ ${\rm cm^{-2}}$, photons above few keV can penetrate the torus creating
an unobstructed view of the nuclear source. One refers to such cases as ``Compton thin". For values of a few times $10^{24}$ ${\rm cm^{-2}}$, only high energy X-ray
emission (tens of keV) can pass through the obscuring material \citep{1997ApJ...488..164T}. For $N_H > 10^{25}$ ${\rm cm^{-2}}$, even high energy X-rays, above a few tens of keV, are Compton scattered and the nuclear source
is completely hidden from our direct view. Therefore, in order to calculate the intrinsic luminosity for an absorbed object with $N_H > 10^{22}$ ${\rm cm^{-2}}$,
we need to find an indirect method \citep{1994ApJ...436..586M}. Many authors have used purportedly isotropic indicators, such as the [\ion{O}{3}] $\lambda$5007 line
(hereafter [\ion{O}{3}]), the infrared continuum, and the 2-10 keV hard X-ray band \citep{2005ApJ...634..161H,2006A&A...453..525N,1999AJ....118.1169X,1997MNRAS.288..977A,2006A&A...457L..17H}. \cite{1999ApJS..121..473B} presented a three-dimensional diagram
for Seyfert 2 galaxies suitable to identify Compton-thick sources, using $K_\alpha$ iron emission line equivalent width and the 2-10 keV hard X-ray flux normalized to the [\ion{O}{3}] line flux, with the latter corrected for extinction and assumed to be a true indicator of the intrinsic luminosity of the source.
In this paper, we present and discuss the use of [\ion{O}{4}] $\lambda$25.89$\mu {\rm m}$ (hereafter [\ion{O}{4}]) as an isotropic quantity avoiding the limitations of previous methods. Since it has a relatively high ionization potential
(54.9 eV), it is less affected by star formation and is significantly less affected by extinction than [\ion{O}{3}] ($A_{v}\sim 5-45$ corresponds to
$A_{{\rm 25.89}\mu{\rm m}} \sim 0.1-0.9$) \citep{1998ApJ...498..579G}. Also, [\ion{O}{4}] represents an improvement
over the use of infrared continuum given the difficulty in isolating the AGN continuum from the host galaxy emission \citep{2004A&A...418..465L}. For the X-ray, we choose the
{\it Swift} Burst Alert Telescope (BAT) high Galactic latitude ($|b| > 19^o$) survey in the 14-195 keV band \citep{2005ApJ...633L..77M}. The survey covers the whole sky
at $(1 - 3) \times 10^{-11} {\rm ergs}$ ${\rm cm^{-2}}$ $s^{-1}$ and represents a complete sample including Compton thin AGNs that were missed
from previous X-ray surveys in the 2-10 keV band because of their high column densities ($N_H \sim 10^{24}$ ${\rm cm^{-2}}$). Furthermore, Compton-thick
(i.e., $N_H$ $\sim$ few $10^{24}$ ${\rm cm^{-2}}$) sources, which cannot be detected in the 2-10 keV band by the
{\it Advanced Satellite for Cosmology and Astrophysics} ({\it ASCA}), {\it Chandra} or {\it XMM-Newton}, have been detected by {\it Swift}/BAT
\citep{2005ApJ...633L..77M,bat}.
In the present work we compare the $L_{[{\rm O ~ IV}]}$ $L_{14-195 {\rm keV}}$ to $L_{[{\rm O ~ III}]}$ $L_{14-195 {\rm keV}}$ and $L_{[{\rm O ~ IV}]}$ $L_{2-10 {\rm keV}}$
to $L_{[{\rm O ~ III}]}$ $L_{2-10 {\rm keV}}$ relations for a sample of X-ray selected
nearby Seyfert Galaxies. We then use the obtained relations in combination with photoionization modeling to place constraints on the
physical properties in these emitting regions. We also discuss the different mechanisms behind these correlations and the
possibilities of using such correlations as a way to find the intrinsic luminosity of AGNs.
\section{Sample and data analysis}
Using the preliminary results from the first 3 months of the
{\it Swift}/BAT high Galactic latitude survey in the 14-195 keV band \citep{2005ApJ...633L..77M} and
the new results from the 9 month survey \citep{bat} we have compiled a sample of
40 nearby Seyfert galaxies (z $<$ 0.08, except \objectname{3C273} with z=0.16) for which the 2-10 keV, [\ion{O}{4}] $\lambda$25.89$\mu {\rm m}$ and [\ion{O}{3}] $\lambda$5007 fluxes have been measured.
The 2-10 keV fluxes are mainly from {\it ASCA} observations and were
retrieved from the TARTARUS data base \citep{2001AIPC..599..991T}, except where noted
in Tables 1 and 2. All the fluxes have been corrected for Galactic absorption and in the case of multiple observations, the mean flux was taken.
The luminosities in the [\ion{O}{3}] line
were compiled from the literature and are presented here without reddening corrections, in Tables~1 and 2. We present
[\ion{O}{4}] fluxes found in the literature and from our analysis of unpublished
archival spectra observed with the Infrared Spectrograph (IRS) \cite[see][]{2004ApJS..154...18H}
on board the {\it Spitzer Space Telescope} in the ${\rm 1^{st}}$ Long-Low
(LL1, $\lambda$ = 19.5 - 38.0 $\mu$m, 10.7$"$ $\times$ 168$"$) IRS order in Staring mode. The [\ion{O}{4}] luminosities
are presented without reddening corrections.
For the analysis of the archival {\it Spitzer} data we used the basic calibrated data (BCD) files preprocessed using the S15.3 IRS pipeline. This includes ramp fitting, dark sky subtraction,
drop correction, linearity correction and wavelength and flux calibrations. The one-dimensional (1D) spectra were extracted from the
IRS data using the SMART v6.2.4 data processing and analysis package \citep{2004PASP..116..975H}. For the extraction we used
the ``Automatic Tapered Column Point Source extraction method" for LL observations of point sources.
For the LL staring mode data we performed sky subtraction by subtracting
the BCDs between the two nodes after we created median BCDs from each node position. After that, the spectra from each node
position for LL1 were averaged to obtain the final spectrum for that order. We performed the line fit with SMART
using a polynomial to fit the continuum and a Gaussian for the line profile.
\objectname{NGC 6300} is the only source, within our sample, observed in high resolution with the Infrared Spectrograph (IRS)
in the Long-High order (LH, $\lambda$ = 18.7 - 37.2 $\mu$m, 11.1$"$ $\times$ 22.3$"$) in Staring mode. For the extraction we used
the ``Full" extraction method for LH observations of point sources. We created median BCDs from each node. Then the spectra from each node
position for LH were averaged to obtain the final spectrum. Since background observations are not available for this galaxy and we required only the
[\ion{O}{4}] line flux, we did not perform any background subtraction for this object.
We note the difficulty in deconvolving the adjacent [\ion{Fe}{2}] $\lambda$25.99$\mu {\rm m}$ line from the [\ion{O}{4}] in
low-resolution IRS spectra. We assumed that the fluxes measured in low-resolution mode are mostly from [\ion{O}{4}] given the fact that
the [\ion{Fe}{2}] is due primarily to star formation \citep{2004ApJ...614L..69H,2005ApJ...633..706W,2006ApJ...641..795O}, and we have a
hard X-ray BAT selected sample which is pre-selected to be sources in which the X-ray emission is dominated by an AGN.
In the next section we will discuss the possible contribution of [\ion{Fe}{2}] to the [\ion{O}{4}] fluxes, based on two cases within our sample:
the edge-on Seyfert 2 galaxy \objectname{NGC 3079} with a nuclear starburst having a low AGN contribution \citep{2007ApJ...655L..73G}
and another Seyfert 2, \objectname{Mrk 3}, with little or no starburst contribution \citep{2007ApJ...671..124D}.
All the flux errors are within 10\% as obtained from the reduction package {\it SMART} and
from the literature.
\section{Results}
\subsection{Reddening in the NLR}
\begin{figure}
\epsscale{.80}
\plotone{f1.eps}
\caption{ Comparison between [\ion{O}{4}] and [\ion{O}{3}] luminosities. The solid line represents the linear regression calculated for Seyfert 1 galaxies; the
dotted, dashed and point-dashed lines represents the linear regression for the sample with the Seyfert 2 galaxies corrected by extinction in the visible with
$A_v=0,1$ and $3$, respectively. For comparison purposes we group Seyfert 1.8's and 1.9's with the Seyfert 2's.
\label{fig1}}
\end{figure}
In Figure ~\ref{fig1} we compare the [\ion{O}{3}] and the [\ion{O}{4}] luminosities.
For our purposes, we group Seyfert 1.2's and 1.5's with Seyfert 1's.
From the plot we notice that Seyfert 2, Seyfert 1.9 and Seyfert 1.8 galaxies show lower [\ion{O}{3}] luminosities than Seyfert 1 galaxies. The mean
value of the luminosities our sample are:
for Seyfert 1's $\log (L_{[{\rm O~ III}]})=41.4 \pm 0.9$ and $\log (L_{[{\rm O~ IV}]})=41.2 \pm 0.8$, for Seyfert 1.9's $\log (L_{[{\rm O~ III}]})=40.5 \pm 0.8$ and
$\log (L_{[{\rm O~ IV}]})=40.9 \pm 0.4$; and for Seyfert 2's $\log (L_{[{\rm O~ III}]})=40.0 \pm 1.0$ and $\log (L_{[{\rm O~ IV}]})= 41.0 \pm 0.6$.
A linear regression
for each individual group yields a relation of the type: for Seyfert 1's $L_{[{\rm O~ III}]} \propto L_{[{\rm O~ IV}]}^{0.9 \pm 0.1}$;
Seyfert 1.9's $L_{[{\rm O~ III}]} \propto L_{[{\rm O~ IV}]}^{2 \pm 1}$ and for Seyfert 2's $L_{[{\rm O~ III}]} \propto L_{[{\rm O~ IV}]}^{1.8 \pm 0.5}$. These results are consistent with previous reports of additional reddening in the narrow line region (NLR) of Seyfert 2's \cite[e.g.,][]{1991MNRAS.250..422J,1994ApJ...436..586M,1994A&A...283..791K,2005ApJ...620..151R,2006A&A...453..525N}.
Using the extinction law derived by \cite{1989ApJ...345..245C}, the
estimated extinction in the visible is $A_{v}\sim 1-6$ mag, calculated from
the mean values for the [\ion{O}{3}] luminosity required to ``correct" the Seyfert 2's (Sy1.8/1.9/2's)
[\ion{O}{3}] to those from Seyfert 1's, assuming a ratio of total to
selective extinction of $R_{v}=3.1$ (which represents a typical value for the diffuse interstellar medium).
In Figure~\ref{fig1} we present these findings by comparing the linear regression obtained for Seyfert 1 galaxies (solid line), with those obtained for all the sample with the [\ion{O}{3}] luminosities corrected for extinction in Seyfert 2 galaxies.
Therefore, we interpreted the higher $\log (L_{[{\rm O~IV}]}/L_{[{\rm O~III}]} \ga 1$ found in Seyfert 2 galaxies to be a result of reddening.
There is a relative absence of Seyfert 1's in nearly edge-on host galaxies \citep[see][]{1980AJ.....85..198K,1995ApJ...454...95M,2001ApJ...555..663S}. Our sample
is consistent with this trend, in that we find that Seyfert 2's have, on average, more inclined host galaxies with a ratio of b/a$=0.5\pm0.3$\footnote{The values for the major and minor diameter of the host galaxy, a and b respectively, were taken from NED} contrasting with b/a$=0.7\pm0.2$ for Seyfert 1 galaxies. These results are also supported by the observed correlation between continuum reddening and inclination of the host galaxy \citep{2001ApJ...562L..29C}. Thus,
host galaxy-related obscuration may contribute to the inclination dependence of the [\ion{O}{3}] emission, although our sample is too small to draw strong conclusions.
Assuming a gas-to-dust ratio for the host galaxy of $5.2 \times 10^{21}\rm{cm^{-2}mag^{-1}}$ \citep{1985ApJ...294..599S} the additional absorbing column calculated from
the values derived for extinction ($A_v$) is in the range of $N_H \sim (2 - 10) \times 10^{21} {\rm cm^{-2}}$ for the Seyfert 2 galaxies. The median values
for the X-ray hydrogen column densities for our Seyfert 2 and Seyfert 1 samples are $N_H = 2.1 \times 10^{23} \rm{cm^{-2}}$ and $N_H = 1.2 \times 10^{21} \rm{cm^{-2}}$,
respectively \citep{bat}. The observed discrepancy between the X-ray column density and those required for the extinction of the [\ion{O}{3}] emission lines is consistent with
previous results \citep[e.g.,][]{1982ApJ...257...47M,1985ApJ...296...69R,2001A&A...365...28M}. The X-ray column density
is measured along the line of sight to the nucleus and the derived reddening measures the column density towards the NLR. Thus while different gas-to-dust ratios
may contribute, it is also likely that there is an additional attenuating gas component closer to the X-ray source which is not affecting the [\ion{O}{3}] emission, as first suggested by \cite{1982ApJ...257...47M}.
\subsection{The Correlation Between [\ion{O}{3}],[\ion{O}{4}] and the 2-10 keV Band}
\begin{figure*}
\epsscale{.80}
\plotone{f2.eps}
\caption{Correlation between [\ion{O}{4}] and [\ion{O}{3}] luminosities with hard X-ray (2-10 keV) band and BAT
(14-195 keV) luminosities. The solid line represents the linear regression calculated for the Seyfert 1 galaxies and the dashed line represents the linear regression for the Seyfert 2 galaxies. The circles in the lower right panel represent radio loud (RL) objects.
\label{fig2}}
\end{figure*}
In Figure ~\ref{fig2} we present comparisons between [\ion{O}{4}] and [\ion{O}{3}] luminosities and X-ray (2-10 keV) and BAT
(14-195 keV) luminosities. Linear fits to these relations, and statistical analysis are presented in Table ~\ref{table0}.
From the different relations, the lowest level of significance for a correlation is obtained for the [\ion{O}{4}]
versus the 2-10 keV hard X-ray. We propose that comparing the heavily-absorbed hard X-ray band in Seyfert 2's with the
``unabsorbed" [\ion{O}{4}] may have weakened any correlation. In order to corroborate this statement we
used the relation, $L_{[{\rm O~ IV}]} \propto L_{2-10 {\rm keV}}^{0.7\pm 0.1}$, derived from the Seyfert 1's to estimate the hard X-ray 2-10 keV fluxes for
the Seyfert 2 galaxies using their observed [\ion{O}{4}]. To investigate the effect of ionized absorption on the 2-10 keV X-ray continuum, we created a multiplicative table (mtable) model using the ``grid" and ``punch XSPEC" options \citep{2006PASP..118..920P} in the photoionization code CLOUDY, version 07.02.01, last described by \cite{1998PASP..110..761F}. A single zone and simple power law incident continuum model was assumed, with photon index $\Gamma = {\rm 1.8}$, and low- and high-energy cutoffs at 1 micron and 100 keV, respectively. Hydrogen column density was then varied, with hydrogen number density fixed at $n_H = 10^8~{\rm cm^{-3}}$, and ionization parameter $\log (U) = -1$, where the ionization parameter $U$ is defined as \cite[see][]{2006agna.book.....O}:
\begin{equation}
U=\frac{1}{4\pi R^2cn_H}\int^\infty_{\nu_o}\frac{L_\nu}{h\nu}d\nu=\frac{Q(H)}{4\pi R^2cn_H},
\label{u}
\end{equation}
where R is the distance to the cloud
, c is the speed of light and $Q(H)$ is the flux of ionizing photons.
The resulting table was then fed into the X-ray spectral fitting package XSPEC (version 12.3.1). We reproduced the incident power law in XSPEC using the POWERLAW additive model, attenuated by absorption from the Cloudy-produced mtable. Using the XSPEC ``flux" command, we determined the effect of various hydrogen column densities on the emergent 2-10 keV X-ray flux. Finally, we obtained the column densities needed in order to correct the observed hard X-ray
fluxes with their intrinsic counterpart derived from the [\ion{O}{4}].
In Figure ~\ref{fig3} we compare the predicted absorbing column densities with
values found in the literature (Table ~\ref{table_column}). Even though we used a simple X-ray model, the predicted column densities are in good agreement
with those from the literature. \objectname{NGC 2992} is the evident outlier from our sample. The X-ray flux has been seen to vary dramatically throughout the history of \objectname{NGC 2992} (Trippe et al., in preparation). As we mentioned before, the 2-10 keV fluxes are mainly from {\it ASCA} observations (in order to maintain consistency within our sample). {\it ASCA} observations
for this object were taken in 1994 and {\it Spitzer} IRS staring observations in 2005. Using the hard X-ray flux,
$f_{2-10keV} = 8.88 \times 10^{-11}{\rm ergs~cm^{-2}~s^{-1}}$, observed by \cite{2007ApJ...666...96M} with the {\it Rossi X-ray Timing Explorer} (RXTE) in 2005,
we obtained better agreement with the absorbing column density
predicted from the [\ion{O}{4}] (see Figure~\ref{fig3} and Table~\ref{table_column}). Overall, these results confirm the heavy absorption present in Seyfert 2 galaxies and the effectiveness of [\ion{O}{4}] as a true indicator of the intrinsic X-ray luminosity of the galaxy.
\begin{figure}
\epsscale{.80}
\plotone{f3.eps}
\caption{ Comparison between the predicted absorbing column densities and values
found in the literature for our Seyfert 2 population. The predicted values were derived using the [\ion{O}{4}] and 2-10 keV luminosity correlation found
for Seyfert 1's. The square represents the column density prediction based on
ASCA observations for the
X-ray flux in 2004 for NGC~2992 (see text for details). The dashed line represents a 1:1 correspondence.\label{fig3}}
\end{figure}
The good empirical correlation between the [\ion{O}{3}] emission line and the hard X-ray in the 2-10 keV band has been discussed extensively
\citep[e.g.,][]{1997MNRAS.288..977A,1999AJ....118.1169X,2005ApJ...634..161H,2006A&A...453..525N}. This correlation is often used to determine the
intrinsic X-ray luminosity, both in absorbed AGN where the observed X-ray flux is affected by absorption. We corroborate this in our sample, in that
we find a strong linear correlation between [\ion{O}{3}] and 2-10 keV luminosities. However, caution must be taken when using this correlation since
[\ion{O}{3}] and hard X-ray luminosities are essentially absorbed quantities, especially in Seyfert 2 sources. We suggest that the
combined effect of extinction in the [\ion{O}{3}] luminosities ($A_{v}\sim 1-6$ mag) with correspondingly heavily absorbed 2-10 keV fluxes
for Seyfert 2 galaxies results in a false correlation, thereby underestimating the true intrinsic luminosity of the AGN, particularly
for nearly X-ray Compton thick sources.
\subsection{The Correlation Between [\ion{O}{3}], [\ion{O}{4}] and the BAT Band}
Based on the statistical analysis we found equally good correlations for the [\ion{O}{3}] and [\ion{O}{4}] relations with the 14-195 keV band. As we mentioned before, the [\ion{O}{4}] represents a relatively uncontaminated
quantity and therefore the relation with an unabsorbed quantity such as the BAT fluxes should give the best correlation, assuming
that the [\ion{O}{4}] is a good indicator of the power of the AGN. However, from
the statistical analysis, this does not seem to be the case.
To explore this behavior we applied linear regression fits to the individual Seyfert 1 and Seyfert 2 galaxies in our sample (see Figure~\ref{fig2}).
The highest level of correlation was obtained for the Seyfert 1 galaxies: [\ion{O}{3}] versus the 14-195 keV ($r_s=0.88$, $P_r=1.2 \times 10^{-7}$) and
[\ion{O}{4}] versus the 14-195 keV ($r_s=0.84$, $P_r=3.6 \times 10^{-7}$). As we found previously, [\ion{O}{3}] in Seyfert 2 galaxies is
more affected by extinction than in Seyfert 1's. This could create a well-correlated linear distribution when comparing with the 14-195 keV BAT fluxes only if there is similar attenuation present in the 14-195 keV band for Seyfert 2's. Following the individual statistical analyses, Seyfert 2 objects have less correlated [\ion{O}{3}]
versus 14-195 keV hard X-ray ($r_s=0.73$, $P_r=1.6 \times 10^{-2}$) and [\ion{O}{4}] versus 14-195 keV hard X-ray luminosities
($r_s=0.64$, $P_r=3.4 \times 10^{-2}$) than Seyfert 1 galaxies. Partial absorption of the hard X-ray fluxes, especially in Seyfert 2 galaxies, with the relatively
unobscured [\ion{O}{4}] is suggested as the cause for the low level of correlation. We found that, using $L_{[{\rm O~ III}]} \propto L_{14-195 {\rm keV}}^{0.9 \pm 0.1}$ derived from the Seyfert 1 population, we underpredicted the BAT luminosities for
the Seyfert 2 galaxies. This is an expected result considering the extinction of the [\ion{O}{3}] emission. On the other hand,
using the $L_{[{\rm O~ IV}]} \propto L_{14-195 {\rm keV}}^{0.7\pm 0.1}$ derived from the Seyfert 1 population, we overpredicted the BAT luminosities for most of the
Seyfert 2 galaxies. There are two different scenarios to explain the over-prediction: an overestimation of [\ion{O}{4}] due to contamination from
[\ion{Fe}{2}] (see Section 2) and absorption in the 14-195 keV BAT band.
Although we cannot dismiss the former, we expect minimal starburst
contribution in our X-ray selected sample. However, we are aware of the importance of [\ion{Fe}{2}] emission especially in weak X-ray sources.
In order to estimate the [\ion{Fe}{2}] contamination coming from the starburst contribution we calculated the predicted [\ion{O}{4}] from the relation with the 14-195 keV luminosities
for Seyfert 1 galaxies and compared with the observed [\ion{O}{4}] luminosities for two extreme Seyfert 2 objects (see Figure ~\ref{fig2}): \objectname{NGC 3079}, with a major contribution from the starburst \citep{2007ApJ...655L..73G},
and \objectname{Mrk 3}, with no starburst \citep{2007ApJ...671..124D}. Using these luminosities we estimated an upper limit for the [\ion{Fe}{2}] starburst contribution
to be $\sim 1.5$ times the uncontaminated [\ion{O}{4}] luminosity for the low-resolution spectra of \objectname{NGC 3079}.
Comparing the full low-resolution with the full high-resolution IRS spectra obtained by \cite{2005ApJ...633..706W} we clearly see [\ion{Fe}{2}] as the dominant
component (and the impossibility of resolving the [\ion{O}{4}] in the low-resolution spectra for \objectname{NGC 3079}). We also noticed that the strong [\ion{O}{4}] emission line in \objectname{Mrk 3} shows no contribution from [\ion{Fe}{2}]. However, the nearly X-ray Compton-thick \objectname{Mrk 3} \citep{2005MNRAS.360..380B}, with uncontaminated and unabsorbed [\ion{O}{4}], still overpredicts the measured
BAT flux and the observed [\ion{O}{4}] of \objectname{NGC 3079} is consistent with the X-ray flux, if the source is nearly X-ray Compton-thick \citep{2006A&A...446..459C}, as we showed with the XSPEC simulation.
Therefore, the most likely scenario to explain the hard X-ray overprediction in Seyfert 2's is partial absorption in the 14-195 keV BAT band. In order to investigate this last statement further we used, as before, the X-ray spectral fitting package XSPEC. For energies above 10 keV, we employed the standard XSPEC model PLCABS, which simulates attenuation of a power law
continuum by dense, cold matter \citep{1997ApJ...479..184Y}. The model grid has a high energy flux calculation limit of 50 keV, so our flux range was limited to 14-50 keV. To accommodate large column densities, we set the maximum number of scatterings considered by the model to a value of 12, as suggested by \cite{1997ApJ...479..184Y}. As before, the power law photon index was set to $\Gamma={\rm 1.8}$. Other variable model parameters were set to default values, and the column density was varied to determine relative changes in the predicted 14-50 keV X-ray luminosities using the [\ion{O}{4}]- BAT correlation for Seyfert 1's, and the observed BAT 14-50 keV X-ray luminosities of the Seyfert 2 galaxies. From these results, we derived a mean column density of $N_H \approx (3.2 \pm 0.8) \times 10^{24} {\rm cm^{-2}}$ for our Seyfert 2 population.
Given the energy limitation of the previous method, this result is in agreement with a derived value of
$N_H \approx (2.4 \pm 0.8) \times 10^{24} {\rm cm^{-2}}$, calculated using the average values for the predicted and the observed BAT X-ray luminosities, and assuming a purely Thomson cross-section. From this value of column density the BAT 14-195 keV X-ray band appear to have been partially absorbed. This result is in good agreement with the large column densities observed for \objectname{Mrk 3} and \objectname{NGC 3079} (see Table~\ref{table_column}).
\subsection{Radio Loudness and [\ion{O}{4}]}
There is a well known linear correlation between the radio and the [\ion{O}{3}] luminosities, and that this correlation is similar for both radio quiet and
radio loud AGNs \citep[e.g.,][]{1999AJ....118.1169X,2001ApJ...555..650H}. We noticed that some of the objects in
our sample, which show more more scattered with respect to the linear correlation between [\ion{O}{4}] and BAT luminosities,
are radio loud sources (see Figure~\ref{fig2}, open circles). In order to investigate whether or not this is an actual trend or a sample selection effect, we expanded our original sample to include more radio loud objects. We used the sample and radio loudness classification from \cite{1999AJ....118.1169X} and selected
the objects that have been observed with {\it Spitzer} and are classified
as Seyfert galaxies (following NED and SIMBAD classification).
These new objects do not have 14-195 keV fluxes from the 9 month BAT survey and therefore are not included in any of the previous calculations
regarding the correlations of the [\ion{O}{4}] and the hard X-ray.
We obtained the [\ion{O}{4}] fluxes from the literature and {\it Spitzer} IRS staring mode archival (unpublished) spectra in low resolution. The extended sample is presented in Table~2 (the labels are the same as in Table~1).
The relationship between the [\ion{O}{3}] and [\ion{O}{4}] has been studied before in powerful Fanaroff-Riley class II radio galaxies \citep{2005A&A...442L..39H}.
\cite{2005A&A...442L..39H} found [\ion{O}{3}] not to be a true indicator of the AGN, because the optical emission from the NLR suffers extinction. In Figure~\ref{fig4} we compare the [\ion{O}{4}] and [\ion{O}{3}] luminosities for the combined sample, i.e. the original and extended sample. From the comparison
it become clear that radio loud sources exhibit higher emission line luminosities than those of other types of active galaxies. This result is in agreement
with previous findings and could be explained by a proposed bowshock model where part of the NLR emission is
being powered by a radio-emitting jets, present in radio loud AGNs \citep[eg.][]{1992MNRAS.255..351T,1995ApJ...455..468D,1997ApJ...485..112B,1998ApJ...495..680B}.
Most of the radio loud objects included in the extended sample were chosen to be powerful radio sources \citep{2005A&A...442L..39H,2006ApJ...647..161O}.
Therefore, our results are hampered by selection effects. Furthermore, our small sample does not allow us to extract any
statistically significant results to determine a clear trend that may distinguish between Type 1 and Type 2 radio loud AGNs. For example, there may be evidence
for a high [\ion{O}{4}]/[\ion{O}{3}] in radio loud Seyfert 1's, as suggested by \cite{2005A&A...442L..39H}, however the same trend is not
seen in the RL Seyfert 2 galaxies. This remains as an open question for further analysis.
\begin{figure}
\epsscale{.80}
\plotone{f4.eps}
\caption{Correlation between [\ion{O}{4}] and [\ion{O}{3}] luminosities for the combined sample, i.e. the original and extended sample.
The circles represent radio loud (RL) objects. For the purpose of comparison we group Seyfert's 1.8 and 1.9 with the Seyfert 2 galaxies. \label{fig4}}
\end{figure}
\section{Photoionization modeling}
To investigate the physical conditions in the emission line regions for both the [\ion{O}{4}] and [\ion{O}{3}], we generated
single-zone, constant-density models.
We used a set of roughly solar abundances \cite[e.g.,][]{1989AIPC..183....1G}. The logs of the abundances relative to H by number are: He: -1; C: -3.46; N: -3.92; O: -3.19; Ne: -3.96; Na: -5.69; Mg: -4.48; Al: -5.53;
Si: -4.50; P: -6.43; S: -4.82; Ar: -5.40; Ca: -5.64; Fe: -4.40 and Ni: -5.75. We assumed a column density of $10^{21}{\rm cm^{-2}}$, which is typical
of the narrow line region \citep[e.g.,][]{2000ApJ...531..278K}. Although it is likely that there is dust mixed in with the emission-line gas in the
NLRs of Seyfert galaxies \citep[e.g.,][]{1986ApJ...307..478K,1993ApJ...404L..51N}, more recent studies using optical and UV long-slit spectra obtained with the Space Telescope
Imaging Spectrograph aboard the {\it Hubble Space Telescope} indicate that the dust/gas ratios are significantly lower than in the Galactic ISM
\citep{2000ApJ...544..763K,2000ApJ...532..256K,2000ApJ...531..278K}. Therefore, we have not included dust in our models.
For the Spectral Energy Distribution (SED) we assume a broken power law as used by \cite{2004ApJ...607..794K} and similar
to that suggested for \objectname{NGC 5548} \citep{1998ApJ...499..719K} and \objectname{NGC 4151} \citep{2000ApJ...531..278K}
of the form $F_\nu \propto \nu^{-\alpha}$, with $\alpha {\rm =0.5}$ below 13.6 eV, $\alpha {\rm =1.5}$ from 13.6 eV to 1 keV and 0.8 at higher energies.
We generated a grid of photoionization models varying the ionization parameter $U$ and total hydrogen number density ($n_H$). The average BAT luminosity (14-195 keV) of the sample, 2.67$\times 10^{43}$ ergs ${\rm s^{-1}}$,
yields a flux of ionizing photons of $Q (H) \sim$1.2$\times 10^{54}$ photons ${\rm s^{-1}}$.
In order to study the physical conditions present in the emission-line regions, we need to
use intrinsic quantities (free of viewing angle effects). As discussed in the previous section, the [\ion{O}{3}] emission line seems to be more affected by
reddening in Seyfert 2 than Seyfert 1 galaxies. Consequently, we used the values derived from the Seyfert 1 galaxies in
our sample to constrain the photoionization parameters. We suggested that, given the isotropic properties for [\ion{O}{4}], the physical
conditions derived from our Seyfert 1 population could be extended to Seyfert 2 as well.
From the Seyfert 1's we find that the average [\ion{O}{4}] and [\ion{O}{3}] ratio is $\sim 1.0 \pm 0.2$.
Assuming that the [\ion{O}{3}] and [\ion{O}{4}] emission comes from the same gas, we used the generated grid of models to find the range in $U$ and $n_H$ for
which we can obtain a ratio of approximately unity for different values of extinction ($A_v$). We obtained
a range for the ionization parameter $-1.50 < \log (U) < -1.30$ and for the
hydrogen density, $ 2.0 < \log (n_H) < 4.25~{\rm (cm^{-3})}$, assuming no extinction ($A_v=0$). The
values for $n_H$ lie around the [\ion{O}{4}] critical density, $\log(n_c)\sim 3.7$, where the line intensity peaks. The critical density for the $^2{\rm P}^o_2$ level
of ${\rm O^{3+}}$, which is the upper level of the [\ion{O}{4}] line, was calculated using the formalism described in \cite{2006agna.book.....O} using atomic data from \cite{1992ApJS...80..425B} and \cite{1998A&AS..131..499G}.
\cite{2005MNRAS.358.1043B} found from their single-zone approximation a relatively small range for the values of $n_H$
($\log(n_e)\sim 5.85 \pm 0.7 ~{\rm cm^{-3}}$)\footnote{ The photoionization code CLOUDY uses the total hydrogen density ($n_H$) instead of the
electron density ($n_e$) as used by \cite{2005MNRAS.358.1043B}. We assumed, in our simple model, both densities to be roughly the same.}
and a relatively small range of $U$ ($\log (U)= -3 \pm 0.5$) where both the [\ion{O}{3}] and H$\beta$ are emitted efficiently.
For their two-zone model assuming $n_e = 10^{3} {\rm cm^{-3}}$ for the outer zone and $n_e = 10^{7} {\rm cm^{-3}}$ and $\log (U) = -1$ for the inner zone, they found
higher $U$ values for the outer zone than in the single-zone approximation ($-3.5 < \log (U) < -2$). These
are typical values for the conditions in the NLR \citep[e.g.,][]{2000ApJ...531..278K}. While our models predicted higher ionization parameters than
\cite{2005MNRAS.358.1043B}, they did not attempt to include [\ion{O}{4}].
\begin{figure}
\epsscale{.80}
\plotone{f5.eps}
\caption{ Comparison between the parameter space required to obtain a ratio of approximately unity for [\ion{O}{4}]/[\ion{O}{3}] for different extinction magnitudes for the [\ion{O}{3}].
For comparison we shown the parameter space obtained by \cite{2005MNRAS.358.1043B}.\label{fig5}}
\end{figure}
In Figure \ref{fig5}, we present the required parameter space to obtain a [\ion{O}{4}]/[\ion{O}{3}] of unity for different values of $A_v$ and compared
with that found by \cite{2005MNRAS.358.1043B} in their single-zone model. This comparison shows the
dependence of this ratio in terms of the ionization parameter, hydrogen density and extinction. For a higher extinction, the parameter space required is closer
to that found by \cite{2005MNRAS.358.1043B}, meaning that the intrinsic ratio is smaller than unity and therefore representing a region where the [\ion{O}{3}] is
dominant, relative to the [\ion{O}{4}] emission. In the absence of such large extinction ($A_v \ga 6 {\rm mag}$), it is likely that
the [\ion{O}{4}] is coming from a more highly ionized and lower density gas. Our comparison of [\ion{O}{4}] and [\ion{O}{3}] show that the single-zone model cannot reproduce the physical conditions present in the NLR gas since $U$ would be too low to produce the observed [\ion{O}{4}] and the line would be collisionally suppressed.
On the other hand, our results are in better agreement with the higher ionization state found in their constant density outer zone. However, this
higher ionization and lower density cloud population cannot fully reproduce the observed [\ion{O}{4}].
Since $U$ depends on the flux of ionizing photons ($Q(H)$), the distance to the source ($R$) and the
hydrogen density of the gas ($n_H$), we used the model grid predictions to calculate a range for the distances to the source. Then, we determined a range for
the covering factors ($Cf$) by adjusting to the observed [\ion{O}{4}] luminosity of the sample. We found a distance of the emitting gas from the central source of
$250~{\rm pc} > R > 30~{\rm pc}$ and a range of covering factors of $ 0.04 < Cf < 0.15$. The range in parameter space in our grid was dominated by matter bounded models. In the matter bounded case, all the material is ionized and some fraction of the ionizing radiation passes through the gas unabsorbed. Therefore, the covering factor must be higher than that from a radiation bounded model (in which all the ionizing radiation is absorbed in the NLR), to account for the observed strength of the emission lines. However, the inclusion of dust in the models will decrease the size of the ionized zone, which would further increase the predicted covering factors \citep[e.g.,][]{1993ApJ...404L..51N,2005MNRAS.358.1043B}. If the [\ion{O}{4}] arose in such dusty gas, it would imply higher covering factors than our models predict.
Nevertheless, the results from our dust-free models are in good agreement with the $Cf$ ($\sim 0.02 -0.2$) obtained by \cite{2005MNRAS.358.1043B} in their
radiation bounded dusty models. There is evidence for a component of high ionization matter bounded gas
in the NLR in which optical lines such as [\ion{Ne}{5}]$\lambda$3426 and [\ion{Fe}{7}] $\lambda$6087 form \citep[e.g.,][]{1996A&A...312..365B,2000ApJ...532..256K,2000ApJ...531..278K}, so it is likely that the [\ion{O}{4}] originates in the same gas, while most of the [\ion{O}{3}]
arises in conditions similar to those found by \cite{2005MNRAS.358.1043B}.
\begin{figure}
\epsscale{.80}
\plotone{f6.eps}
\caption{ Comparison between the correlation of [\ion{O}{4}] luminosities with BAT X-ray luminosities
and a grid of photoionization models for different covering factors (Cf) for a distance of $R=130~{\rm pc}$. The dashed line represents the linear regression obtained for the observed luminosities for all the sample and the open circles represent the observed Seyfert 1's luminosities (see Figure~\ref{fig2}).
$n_H$ refers to the $\log$ of the hydrogen nucleus density and $U$ to the $\log$ of the ionization parameter. The vertical dashed line indicates the critical
density for the [\ion{O}{4}] line \label{fig6}}
\end{figure}
In Figure~\ref{fig6} we present the photoionization model grid overlaid on the Seyfert 1 luminosities
and the linear regression for all the sample. Each case corresponds to a different set of covering factors ($Cf$) for a distance from the ionizing source to the gas cloud of $R=130~{\rm pc}$. The model X-ray luminosities are calculated using U, the fixed radial distance, the range in densities shown in Figure~\ref{fig6}, and our assumed
SED (see Equation~1).
As presented in Figure ~\ref{fig6}, our simple photoionization model can reproduce the observed $L_{[{\rm O~IV]}}$ to $L_{14-195 {\rm keV}}$ relationship.
In order to fit the higher luminosity sources, the models predict that
more distant or higher density gas is required for the NLR. We dismiss the latter because, at high densities (i.e. higher than the critical density for
[\ion{O}{4}]), the [\ion{O}{4}] is collisionally de-excited and the line intensity decreases. Therefore, the most likely scenario is a more distant
NLR for high luminosity sources. On the other hand, Figure ~\ref{fig6} shows that our simple photoionization model underpredicts [\ion{O}{4}] at $R=130~{\rm pc}$, even at low densities. Consequently, in the low luminosity sources the NLR must be fairly small ($R < 150$ pc) with a bigger covering factor ($Cf \ga 0.07$), since gas at or below the critical density of the [\ion{O}{4}] would not be sufficiently ionized at large distances from the source. These results suggests that the size of the NLR scales with the luminosity of the source, in agreement with the observed good correlation between the NLR size and the luminosity of the AGN \citep{2003ApJ...597..768S}.
\section{Conclusions}
We have explored the relationship between the $L_{[{\rm O~ IV}]}$-$L_{14-195~keV}$ and related correlations for a sample of X-ray selected
nearby Seyfert Galaxies. For the X-ray selected sample the [\ion{O}{4}] and the [\ion{O}{3}] luminosities are
well-correlated with the hard X-ray luminosity in the 14-195 keV band over a range of $\sim$ 3 and 5 orders of magnitude, respectively.
From the [\ion{O}{4}] and [\ion{O}{3}] comparison we derived an absorbing column density from the reddening toward the NLR for Seyfert 2 galaxies,
$N_H \sim (2 - 10) \times 10^{21} {\rm cm^{-2}}$, which is smaller than the median value for the X-ray column densities, $N_H = 2.1 \times 10^{23} \rm{cm^{-2}}$. The differences in the line of sight for the different absorbing columns are consistent with an additional attenuating gas component
close to the X-ray source.
We also suggested that part of the obscuration in the NLR of Seyfert 2 galaxies may occur in the disk of the host galaxy, given the fact that, in our sample,
Seyfert 2 galaxies are on average more inclined.
We obtained a correlation between the $L_{[{\rm O~ IV}]}$-$L_{2-10~keV}$ that is not as strong as the one between $L_{[{\rm O~ III}]}$-$L_{2-10~keV}$,
which results from comparing the heavily absorbed 2-10 keV band,
especially in Seyfert 2 galaxies, with the intrinsic (unabsorbed) [\ion{O}{4}] fluxes. As pointed out in previous works, there is a
good correlation between [\ion{O}{3}] and hard X-ray 2-10 keV luminosities. However, we conclude that the
combined effects of reddening in the [\ion{O}{3}] and absorption in the 2-10 keV band may result in a false correlation.
For the $L_{[{\rm O~ III}]}$ and $L_{[{\rm O~ IV}]}$ relations with the $L_{14-195{\rm keV} }$ BAT
band we found equally good correlations. Using the linear regression found for Seyfert 1 galaxies we
underpredict the hard X-ray 14-195 keV BAT luminosities of Seyfert 2's obtained from the $L_{[{\rm O~ III}]}$ versus hard
X-ray $L_{14-195 {\rm keV} }$ relation and overpredict the hard X-ray 14-195 keV BAT luminosities using
the $L_{[{\rm O~ IV}]}$ relation. We explain the former case due to the extinction of the [\ion{O}{3}] ($A_{v}\sim 1-6$ mag) and the latter with
absorption in the BAT band. We found that for a column density of $ N_H \sim 3 \times 10^{24} \rm{cm^{-2}}$ the BAT luminosities
could be smaller, by a factor of $\sim 5 \pm 3$, than the intrinsic luminosities derived from the Seyfert 1's relation.
Assuming that the [\ion{O}{3}] and [\ion{O}{4}] come from the same gas, and using the mean fluxes of the sample, we conclude that the emitting region for [\ion{O}{4}]
is in a higher ionization state and lower density than the components modeled for the [\ion{O}{3}] emitting region by \cite{2005MNRAS.358.1043B} and has a
mean covering factor of $Cf \sim 0.07$. The physical conditions
derived from the photoionization models indicate that [\ion{O}{4}] originates in the inner $\sim$ 150 pc of the NLR.
In conclusion, we propose the [\ion{O}{4}] as a truly isotropic property of AGNs given its high ionization potential and that it is
unaffected by reddening, which makes this line easy to identify and extract. This is true at least in galaxies with
minimal or no star formation as for our X-ray selected sample. Where star formation activity has an
important contribution this may not be the case, especially with contamination from [\ion{Fe}{2}] $\lambda$25.99$\mu {\rm m}$ in low-resolution IRS spectra.
In a future work we will expand our sample to include the complete 22 month {\it SWIFT}/BAT survey and high resolution {\it Spitzer} IRS observations to
continue to explore these open issues.
\acknowledgments
We would like to thank our anonymous referee for her/his insightful comments and suggestions that improved the paper.
This research has 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. The IRS was a collaborative venture between
Cornell University and Ball Aerospace Corporation funded by NASA through the Jet Propulsion Laboratory and Ames Research Center.
SMART was developed by the IRS Team at Cornell University and is available through the Spitzer Science Center at Caltech. Basic research in astronomy at the NRL is supported by 6.1 base funding.
\clearpage
\bibliographystyle{apj}
|
1,116,691,497,576 | arxiv | \section{Introduction and statement of the result}
A real random variable $X$ is said to be unimodal if there exists $a\in{\mathbb{R}}$ such that its distribution function ${\mathbb{P}}[X\le x]$ is convex on $(-\infty, a)$ and concave on $(a, +\infty)$. When $X$ is absolutely continuous, this means that its density is non-decreasing on $(-\infty, a]$ and non-increasing on $[a, +\infty)$. The number $a$ is called a mode of $X$ and might not be unique. A random variable with a single mode is called strictly unimodal. The problem of unimodality has been intensively studied for infinitely divisible random variables and we refer to Chapter 10 in \cite{S} for details. This problem has also been settled in the framework of hitting times of processes and R\"osler - see Theorem 1.2 in \cite{R} - showed that hitting times for points of real-valued diffusions are always unimodal. However, much less is known when the underlying process has jumps, for example when it is a L\'evy process.
In this paper we consider a real strictly $\alpha-$stable process ($1<\alpha\le 2$), which is a L\'evy process $\{X_t, \; t\ge 0\}$ starting from zero and having characteristic exponent
$$\log[\EE[e^{{\rm i} \lambda X_1}]]\; =\; -({\rm i} \lambda)^\alpha e^{-{\rm i}\pi\alpha\rho\, {\rm sgn}(\lambda)}, \quad \lambda\in{\mathbb{R}},$$
where $\rho\in[1-1/\alpha, 1/\alpha]$ is the positivity parameter of $\{X_t, \; t\ge 0\}$ that is $\rho ={\mathbb{P}}[X_1 \ge 0].$ We refer to \cite{Z} and to Chapter 3 in \cite{S} for an account on stable laws and processes. In particular, comparing the parametrisations (B) and (C) in the introduction of \cite{Z} shows that the characteristic exponent of $\{X_t, \; t\ge 0\}$ takes the more familiar form
$$c\,\vert\lambda\vert^\alpha (1 - {\rm i}\theta\tan(\frac{\pi\alpha}{2})\,{\rm sgn}(\lambda))$$
with $\rho = 1/2 + (1/\pi\alpha) \tan^{-1}(\theta\tan(\pi\alpha/2))$ and $c = \cos(\pi\alpha(\rho -1/2)).$ The constant $c$ is a scaling parameter which could take any arbitrary positive value without changing our purposes below. We are interested in the hitting times for points of $\{X_t, \; t\ge 0\}\!\!:$
$$\tau_x\; =\; \inf\{t>0, \; X_t = x\}, \quad x\in{\mathbb{R}}.$$
It is known \cite{M} that $\tau_x$ is a proper random variable which is also absolutely continuous. Recall also - see e.g. Example 43.22 in \cite{S} - that points are polar for strictly $\alpha-$stable L\'evy process with $\alpha\le 1,$ so that $\tau_x = +\infty$ a.s. in this situation. In the following we will focus on the random variable $\tau =\tau_1.$ Again, this does not cause any loss of generality since by self-similarity one has
$$\tau_x \,\stackrel{d}{=} \,x^\alpha \tau_1\qquad\mbox{and}\qquad \tau_{-x}\,\stackrel{d}{=}\, x^\alpha \tau_{-1}$$
for any $x \ge 0,$ and because the law of $\tau_{-1}$ can be deduced from that of $\tau_1$ in considering the dual process $\{-X_t, \; t\ge 0\}.$ We show the
\begin{THE} The random variable $\tau$ is unimodal.
\end{THE}
In the spectrally negative case $\rho = 1/\alpha,$ the result is plain because $\tau$ is then a positive stable random variable of order $1/\alpha,$ which is known to be unimodal - see e.g. Theorem 53.1 in \cite{S}. We will implicitly exclude this situation in the sequel and focus on the case with positive jumps. To proceed with this non-trivial situation we use several facts from the recent literature, in order to show that $\tau$ factorizes into the product of a certain unimodal random variable and a product of powers of Gamma random variables. The crucial property that the latter product preserves unimodality by independent multiplication \cite{CT} allows to conclude. Our argument follows that of \cite{TS2}, where a new proof of Yamazato's theorem for the unimodality of stable densities was established, but it is more involved. In passing we obtain a self-decomposability property for the Kanter random variable, which is interesting in itself and extends the main result of \cite{PSS}.
For the sake of clarity we divide the proof into three parts. We first consider the symmetric case $\rho =1/2$, appealing to a factorization of $\tau$ in terms of generalized Rayleigh and Beta random variables which was discovered in \cite{Y3}. Second, we deal with the spectrally positive case $\rho = 1-1/\alpha$, with the help of a multiplicative identity in law for $\tau$ involving positive stable and shifted Cauchy random variables which was obtained in \cite{TS1}. In the third part, we observe the remarkable fact that this latter identity extends to the general case $\rho \in(1-1/\alpha, 1/\alpha)$, thanks to the evaluation of the Mellin transform of $\tau$ which was performed in \cite{KKMW}, and which can actually be obtained very easily - see section 2.4. This identity allows also to show that the density of $\tau$ is real-analytic and that $\tau$ is strictly unimodal - see the final remark (a).
\section{Proof of the Theorem}
\subsection{The symmetric case} Formula (5.12) and Lemma 2.17 in \cite{Y3} yield together with the normalization (4.1) therein, which is the same as ours, the following independent factorization
\begin{equation}
\label{3Y}
\tau\; \stackrel{d}{=}\;2^{-\alpha} {\bf L}^{-\frac{\alpha}{2}}\;\times\; \left({\bf Z}_{\frac{\alpha}{2}}^{(-\frac{1}{2})}\right)^{-\frac{\alpha}{2}}\,\times\,\B_{1-\frac{1}{\alpha}, \frac{1}{\alpha}}^{-1}
\end{equation}
with the following notation, which will be used throughout the text:\\
\begin{itemize}
\item ${\bf L}$ is the unit exponential random variable.
\item $\B_{a,b}$ is the Beta random variable ($a,b >0$) with density
$$\frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)} \, x^{a-1} (1-x)^{b-1} {\bf 1}_{(0,1)}(x)$$
\item ${\bf Z}_c$ is the positive $c-$stable random variable ($0<c\le 1$), normalized such that
$$\EE\left[ e^{-\lambda {\bf Z}_c}\right] \; =\; e^{-\lambda^c}, \qquad \lambda \ge 0.$$
\item For every $t\in{\mathbb{R}}$ and every positive random variable $X$ such that $\EE[X^t]<+\infty,$ the random variable $X^{(t)}$ is the size-biased sampling of $X$ at order $t$, which is defined by
$$\EE\left[ f(X^{(t)})\right]\; =\; \frac{\EE\left[ X^t f(X)\right]}{\EE\left[ X^t\right]}$$
for every $f : {\mathbb{R}}^+\to{\mathbb{R}}$ bounded continuous.
\end{itemize}
\medskip
Notice that the above random variable ${\bf Z}_{\frac{\alpha}{2}}^{(-\frac{1}{2})}$ makes sense from the closed expression of the fractional moments of ${\bf Z}_c$:
\begin{equation}
\label{MomZ}
\EE[{\bf Z}_c^s]\; =\; \frac{\Gamma(1-s/c)}{\Gamma(1-s)}, \qquad s < c,
\end{equation}
which ensures $\EE[{\bf Z}_{\frac{\alpha}{2}}^{-\frac{1}{2}} ]< +\infty.$ Observe also that for every $t > 0$ one has ${\bf L}^{(t-1)}\stackrel{d}{=} \G_t$ where $\G_t$ is the Gamma random variable with density
$$\frac{x^{t-1}e^{-x}}{\Gamma(t)} \, {\bf 1}_{(0,+\infty)}(x).$$
It is easy to see that
\begin{equation}
\label{Size}
X^{(t)}\times\,Y^{(t)}\; \stackrel{d}{=}\; (X\times Y)^{(t)}\qquad\mbox{and}\qquad \left( X^{(t)}\right)^p\; \stackrel{d}{=}\; \left( X^p\right)^{(\frac{t}{p})}
\end{equation}
for every $t,p\in{\mathbb{R}}$ such that the involved random variables exist, and where the products in the first identity are supposed to be independent. In particular, one has
$$(\kappa X)^{(t)}\; \stackrel{d}{=}\; \kappa X^{(t)}$$
for every positive constant $\kappa.$ Combined with (\ref{3Y}) the second identity in (\ref{Size}) entails
$$\tau\; \stackrel{d}{=}\;2^{-\alpha} {\bf L}^{-\frac{\alpha}{2}}\;\times\; \left({\bf Z}_{\frac{\alpha}{2}}^{-\frac{\alpha}{2}}\right)^{(\frac{1}{\alpha})}\,\times\,\B_{1-\frac{1}{\alpha}, \frac{1}{\alpha}}^{-1}.$$
On the other hand, Kanter's factorization - see Corollary 4.1 in \cite{K} - reads
\begin{equation}
\label{Kant}
{\bf Z}_{\frac{\alpha}{2}}^{-\frac{\alpha}{2}}\; \stackrel{d}{=}\; {\bf L}^{1-\frac{\alpha}{2}}\,\times\; b_{\frac{\alpha}{2}}({\bf U})
\end{equation}
where ${\bf U}$ is uniform on $(0,1)$ and
$$b_c (u)\; = \;\frac{\sin (\pi u)}{\sin^c (\pi c u)\sin^{1-c} (\pi(1-c) u)}$$
for all $u,c\in (0,1).$ Since $b_c$ is a decreasing function from $\kappa_c = c^{-c}(1-c)^{c-1}$ to $0$ - see the proof of Theorem 4.1 in \cite{K} for this fact, let us finally notice that the support of the random variable
$${\bf K}_{c}\; =\; \kappa_{c}^{-1} b_{c}({\bf U})$$
is $[0,1].$ Putting everything together shows that
$$\tau\; \stackrel{d}{=}\;2^{-\alpha} \kappa_{\frac{\alpha}{2}}^{}\, {\bf L}^{-\frac{\alpha}{2}}\times\; \left({\bf L}^{1-\frac{\alpha}{2}}\right)^{(\frac{1}{\alpha})}\,\times\, {\bf K}_{\frac{\alpha}{2}}^{(\frac{1}{\alpha})}\,\times\,\B_{1-\frac{1}{\alpha}, \frac{1}{\alpha}}^{-1}.$$
By Theorem 3.7. and Corollary 3.14. in \cite{CT}, the random variable
$${\bf X}_\alpha\; =\; 2^{-\alpha} \kappa_{\frac{\alpha}{2}}^{}\, {\bf L}^{-\frac{\alpha}{2}}\times\; \left({\bf L}^{1-\frac{\alpha}{2}}\right)^{(\frac{1}{\alpha})}\; \stackrel{d}{=}\; 2^{-\alpha} \kappa_{\frac{\alpha}{2}}^{}\, {\bf L}^{-\frac{\alpha}{2}}\times\; \left({\bf L}^{(\frac{1}{\alpha}-\frac{1}{2})}\right)^{1-\frac{\alpha}{2}}\; \stackrel{d}{=}\; 2^{-\alpha} \kappa_{\frac{\alpha}{2}}^{}\, {\bf L}^{-\frac{\alpha}{2}}\times\; \G_{\frac{1}{\alpha}+\frac{1}{2}}^{1-\frac{\alpha}{2}}$$
is multiplicatively strongly unimodal, that is its independent product with any unimodal random variable remains unimodal. Indeed, a straightforward computation shows that the random variables $\log ({\bf L})$ and $\log ( \G_{\frac{1}{\alpha}+\frac{1}{2}})$ have a log-concave density, and the same is true for $\log({\bf X}_\alpha)$ by Pr\'ekopa's theorem.
All in all, we are reduced to show the
\begin{PROP}
\label{Ksym}
With the above notation, the random variable ${\bf K}_{\frac{\alpha}{2}}^{(\frac{1}{\alpha})}\times\B_{1-\frac{1}{\alpha}, \frac{1}{\alpha}}^{-1}$ is unimodal.
\end{PROP}
\proof A computation shows that the density of $\B_{1-\frac{1}{\alpha}, \frac{1}{\alpha}}^{-1}$ decreases on $(1,+\infty).$ Hence, there exists $F_\alpha : (0,1)\mapsto (1,+\infty)$ increasing and convex such that
$$\B_{1-\frac{1}{\alpha}, \frac{1}{\alpha}}^{-1}\;\stackrel{d}{=}\; F_\alpha({\bf U}).$$
On the other hand, up to normalization the density of ${\bf K}_{\frac{\alpha}{2}}^{(\frac{1}{\alpha})}$ writes
$$g_\alpha(x)\; =\;x^{\frac{1}{\alpha}} f_\alpha(x)$$
on $(0,1),$ where $f_\alpha$ is the density of ${\bf K}_{\frac{\alpha}{2}}.$ It follows from Lemma 2.1 in \cite{TS3} that $f_\alpha$ increases on $(0,1),$ so that the density of $g_\alpha$ also increases on $(0,1)$ and that there exists $G_\alpha : (0,1)\mapsto (0,1)$ increasing and concave such that
$${\bf K}_{\frac{\alpha}{2}}^{(\frac{1}{\alpha})}\;\stackrel{d}{=}\; G_\alpha({\bf U}).$$
We can now conclude by the lemma in \cite{TS2}.
\endproof
\begin{REMS} {\em The lemma in \cite{TS2} shows that the mode of ${\bf K}_{\frac{\alpha}{2}}^{(\frac{1}{\alpha})}\times\B_{1-\frac{1}{\alpha}, \frac{1}{\alpha}}^{-1}$ is actually 1. However, this does not give any information on the mode of $\tau.$}
\end{REMS}
\subsection{The spectrally positive case} This situation corresponds to the value $\rho = 1-1/\alpha$ of the positivity parameter. The characteristic exponent of $\{X_t, \; t\ge 0\}$ can be extended to the negative half-plane, taking the simple form
$$\log[\EE[e^{-\lambda X_1}]]\; =\; \lambda^\alpha, \qquad \lambda \ge 0.$$
With this normalization, we will use the following independent factorization which was obtained in \cite{TS1}:
$$\tau\; \stackrel{d}{=}\; {\bf U}_\alpha\;\times\; {\bf Z}_{\frac{1}{\alpha}}$$
where ${\bf U}_\alpha$ is a random variable with density
$$f_{{\bf U}_\alpha}(t)\; =\;\frac{-(\sin \pi\alpha) t^{1/\alpha}}{\pi(t^2 - 2t \cos \pi\alpha +1)}\cdot$$
It is easy to see that ${\bf U}_\alpha$ is multiplicatively strongly unimodal for $\alpha\le 3/2$ and this was used in Proposition 8 of \cite{TS1} to deduce the unimodality of $\tau$ in this situation. To deal with the general case $\alpha\in (1,2)$ we proceed via a different method. First, it is well-known and easy to see - solve e.g. Exercise 4.21 (3) in \cite{CY} - that the independent quotient
$$\left(\frac{{\bf Z}_{\alpha-1}}{{\bf Z}_{\alpha-1}}\right)^{\alpha-1}$$
has the density
$$\frac{-\sin (\pi \alpha)}{\pi (t^2 -2 t\cos \pi \alpha + 1)}$$
over ${\mathbb{R}}^+,$ whence we deduce
\begin{eqnarray*}
\tau & \stackrel{d}{=} & \left(\frac{{\bf Z}_{\alpha-1}^{\alpha-1}}{{\bf Z}_{\alpha-1}^{\alpha-1}}\right)^{(\frac{1}{\alpha})}\times\; {\bf Z}_{\frac{1}{\alpha}}\\
& \stackrel{d}{=} & \kappa_{\frac{1}{\alpha}}^{-\alpha}\left(\frac{{\bf L}^{2-\alpha}}{{\bf L}^{2-\alpha}}\right)^{(\frac{1}{\alpha})}\times\; {\bf L}^{1-\alpha}\times\;{\bf K}_{\alpha-1}^{(\frac{1}{\alpha})}\times\; ({\bf K}_{\alpha-1}^{-1})^{(\frac{1}{\alpha})}\times{\bf K}_{\frac{1}{\alpha}}^{-\alpha}
\end{eqnarray*}
with the above notation. Similarly as above, the first product with the three exponential random variables is multiplicatively strongly unimodal, whereas the random variable ${\bf K}_{\alpha-1}^{(\frac{1}{\alpha})}$ has an increasing density on $(0,1).$ Hence, reasoning as in Proposition \ref{Ksym} it is enough to show that the random variable $({\bf K}_{\alpha-1}^{-1})^{(\frac{1}{\alpha})}\times{\bf K}_{\frac{1}{\alpha}}^{-\alpha}$ has a decreasing density on $(1, +\infty).$ We show the more general
\begin{PROP}
\label{Kasym}
With the above notation, the random variable
$$({\bf K}_{\beta}^{-r})^{(t)}\,\times\,{\bf K}_{\g}^{-s}$$
has a decreasing density on $(1, +\infty)$ for every $r, s >0$ and $\beta, \g, t$ in $(0,1).$
\end{PROP}
The proof of the proposition uses the notion of self-decomposability - see Chapter 3 in \cite{S} for an account. Recall that a positive random variable $X$ is self-decomposable if its Laplace transform reads
$$\EE[e^{-\lambda X}]\; =\; \exp - \left[ a_X^{}\lambda \; +\; \int_0^\infty (1- e^{-\lambda x}) \frac{\varphi_{X}^{}(x)}{x} dx\right], \quad \lambda\ge 0,$$
for some $a_X^{}\ge 0$ which is called the drift coefficient of $X,$ and some non-increasing function $\varphi_{X}^{} : (0,+\infty)\to {\mathbb{R}}^+$ which will be henceforth referred to as the spectral function of $X.$ Introduce the following random variable
$${\bf W}_\beta\; =\; -\log({\bf K}_\beta)$$
and notice that its support is ${\mathbb{R}}^+,$ thanks to our normalization for ${\bf K}_\beta.$ A key-observation is the following
\newpage
\begin{LEM}
\label{Main}
The random variable ${\bf W}_\beta$ is self-decomposable, without drift and with a spectral function taking the value $1/2$ at $0\!+\!.$
\end{LEM}
\proof Combining (\ref{MomZ}), (\ref{Kant}) and the classical formula for the Gamma function
$$\Gamma(1-u)\; =\; \exp\left[ \gamma u \, +\, \int_0^\infty (e^{ux}-1-u x)\frac{dx}{x(e^x-1)}\right], \quad u<1,$$
(where $\gamma$ is Euler's constant) yields the following expression for the Laplace transform of ${\bf W}_\beta$ - see (3.5) in \cite{PSS}:
$$\EE[e^{-\lambda {\bf W}_\beta}]\; =\; \EE[{\bf K}_\beta^{\lambda}]\; =\; \exp - \left[ \int_0^\infty (1- e^{-\lambda x}) \frac{\varphi_{\beta}^{}(x)}{x} dx\right], \quad \lambda\ge 0,$$
with
$$\varphi_\beta(x)\; =\; \frac{e^{-x}}{1-e^{-x}}\; -\; \frac{e^{-x/\beta}}{1-e^{-x/\beta}}\; -\; \frac{e^{-x/(1-\beta)}}{1-e^{-x/(1-\beta)}}, \quad x > 0.$$
It was shown in Lemma 3 of \cite{PSS} that the function $\varphi_{\beta}$ is non-negative and an asymptotic expansion at order 2 yields $\varphi_{\beta}(0+) = 1/2.$
We finally show that $\varphi_{\beta}$ is non-increasing on $(0,+\infty).$ Following the proof and the notation of Lemma 3 in \cite{PSS} this amounts to the fact that the function $x \mapsto x\psi_\beta(x)$ therein is non-decreasing on $(0,1),$ which is a clear consequence of the following claim
\begin{equation}
\label{Clay}
t\; \mapsto\; \log(1-e^t) - \log (1-e^{rt})\quad \mbox{is convex on ${\mathbb{R}}^-$ for every $r\in (0,1).$}
\end{equation}
Let us show the claim. Differentiating twice, we see that we are reduced to prove that
$$\frac{r^2x^{r-1}(1-x)^2}{(1-x^r)^2}\; \ge \; 1, \qquad 0<x,r< 1.$$
The limit of the quantity on the left-hand side is 1 when $x\to1-,$ whereas its logarithmic derivative equals
$$\frac{(r-1)(1-x^{r+1}) + (r+1)(x^r -x)}{x(1-x)(1-x^r)}\cdot$$
The latter fraction is negative for all $0<x,r<1$ because its numerator is concave as a function of $x\in(0,1)$ which vanishes together with its derivative at $x=1\!-\!.$ This shows (\ref{Clay}) and finishes the proof of the lemma.
\endproof
\noindent
{\bf Proof of Proposition \ref{Kasym}.} Set $f_{\beta, r}$ resp. $f_{\g,s}$ for the density of $r{\bf W}_\beta$ resp. $s{\bf W}_\g.$ By multiplicative convolution, the density of $({\bf K}_{\beta}^{-r})^{(t)}\,\times\,{\bf K}_{\g}^{-s}$ writes
$$\int_1^x (xy^{-1})^t f^{}_{{\bf K}_{\beta}^{-r}} (xy^{-1}) f^{}_{{\bf K}_{\g}^{-s}} (y) \frac{dy}{y}$$
on $(1, +\infty),$ up to some normalization constant. This transforms into
$$x^{t-1}\int_1^x f_{\beta, r}(\log (x) -\log(y)) f_{\g,s}(\log(y))\frac{dy}{y^{t+1}}\; =\; x^{t-1}\int_0^{\log(x)} f_{\beta, r}(\log (x) - u) e^{-tu} f_{\g,s}(u)\, du$$
and since $t\in(0,1)$ it is enough to prove that the function
\begin{equation}
\label{Yam}
v\;\mapsto\; \int_0^v f_{\beta, r}(v-u) e^{-tu} f_{\g,s}(u)\, du
\end{equation}
is non-increasing on $(0,+\infty).$ Lemma \ref{Main} and a change of variable show that $f_{\beta, r}(u)$ resp. $e^{-tu} f_{\g,s}(u)$ is up to normalization the density of a positive self-decomposable random variable without drift and with spectral function $\varphi_{\beta}(xr^{-1})$ resp. $e^{-tx} \varphi_{\g}(xs^{-1}),$ with the notation of the proof of Lemma \ref{Main}. By additive convolution this entails that the function in (\ref{Yam}) is the constant multiple of the density of a positive self-decomposable random variable without drift and with spectral function
$$\varphi_{\beta}(xr^{-1})\; +\; e^{-tx} \varphi_{\g}(xs^{-1}).$$
By Lemma \ref{Main} this latter function takes the value 1 at $0+,$ and we can conclude by Theorem 53.4 (ii) in \cite{S}.
\hfill$\square$
\begin{REMS} {\em By Theorem 4 in \cite{JS} we know that ${\bf W}_\beta$ has also a completely monotone density, in other words - see Theorem 51.12 in \cite{S} - that its spectral function writes
$$\varphi_\beta (x)\; =\; x\int_0^\infty\!\! e^{-tx} \theta_\alpha (t)dt, \quad x\ge 0,$$
for some function $\theta_\alpha (t)$ valued in $[0,1]$ and such that
$t^{-1}\theta_\alpha (t)$ is integrable at $0+\!.$ This entails that ${\bf K}_{\beta}^{-r}$ has a completely monotone density as well for every $r>0$ - see Corollary 3 in \cite{JS}. However, this latter property does not seem true in general for $({\bf K}_{\beta}^{-r})^{(t)}\,\times\,{\bf K}_{\g}^{-s}.$}
\end{REMS}
\subsection{The general case} We now suppose $\rho\in (1-1/\alpha, 1/\alpha),$ which means that our stable L\'evy process has jumps of both signs. The symmetric case was dealt with previously but it can also be handled with the present argument. Theorem 3.10 in \cite{KKMW} computes the fractional moments of $\tau$ in closed form:
\begin{equation}
\label{closed}
\EE[\tau^s]\; =\; \frac{\sin(\frac{\pi}{\alpha})\sin(\pi\rho\alpha(s+\frac{1}{\alpha})) }{\sin(\pi\rho)\sin(\pi(s+\frac{1}{\alpha}))}\; \times\;\frac{\Gamma(1-\alpha s)}{\Gamma(1-s)}, \quad -1-\frac{1}{\alpha} < s < 1-\frac{1}{\alpha}
\end{equation}
(the initial normalization of \cite{KKMW} is the same as ours - see the introduction therein - but beware that with their notation our $\tau$ has the law of $T_0$ under ${\rm P}_{\!-1}$). On the other hand, it is easy to see from (\ref{MomZ}) and the complement formula for the Gamma function that
$$\EE\left[ \left(\frac{{\bf Z}_{\rho\alpha}^{\rho \alpha}}{{\bf Z}_{\rho\alpha}^{\rho \alpha}}\right)^{s}\right]\; =\; \frac{\sin(\pi\rho\alpha s)}{\rho\alpha\sin(\pi s)},\quad -1< s< 1.$$
Hence, a fractional moment identification entails
\begin{equation}
\label{Rk}
\tau\; \stackrel{d}{=}\; \left(\frac{{\bf Z}_{\rho\alpha}^{\rho \alpha}}{{\bf Z}_{\rho\alpha}^{\rho \alpha}}\right)^{(\frac{1}{\alpha})}\times\; {\bf Z}_{\frac{1}{\alpha}}\cdot
\end{equation}
Making the same manipulations as in the spectrally positive case, we obtain
\begin{equation}
\label{Final}
\tau \;\stackrel{d}{=}\;\kappa_{\frac{1}{\alpha}}^{-\alpha}\left(\frac{{\bf L}^{1-\rho\alpha}}{{\bf L}^{1-\rho\alpha}}\right)^{(\frac{1}{\alpha})}\times\; {\bf L}^{1-\alpha}\times\;{\bf K}_{\rho\alpha}^{(\frac{1}{\alpha})}\times\; ({\bf K}_{\rho\alpha}^{-1})^{(\frac{1}{\alpha})}\times{\bf K}_{\frac{1}{\alpha}}^{-\alpha}
\end{equation}
and we can conclude because Proposition \ref{Kasym} applies here as well.
\hfill$\square$
\subsection{A short proof of (\ref{closed})} In this paragraph we give an independent proof of the fractional moment evaluation (\ref{closed}), which is short and standard. This method was suggested to us by L.~Chaumont and P.~Patie and we would like to thank them for reminding us this classical argument. Setting $f_{X_t}$ for the density of $X_t,$ by Theorem 43.3 in \cite{S} one has
$$\EE[e^{-q\tau}]\; =\; \frac{u^q(1)}{u^q(0)}$$
where by self-similarity
$$u^q(1)\; =\; \int_0^\infty e^{-qt} f_{X_t}(1)\, dt \; =\; \int_0^\infty e^{-qt} t^{-1/\alpha}f_{X_1}(t^{-1/\alpha})\, dt,$$
and
$$u^q(0)\; =\; \int_0^\infty e^{-qt} f_{X_t}(0)\, dt \; =\; \frac{1}{\varphi(q)}$$
with $\varphi$ the Laplace exponent of the inverse local time at zero of $\{X_t, \, t\ge 0\}.$ It is well-known and easy to see by self-similarity - see e.g. Theorem 2 in \cite{St} - that
$$\varphi(q) \; =\; \kappa\, q^{\frac{\alpha-1}{\alpha}},$$
where $\kappa > 0$ is a normalizing constant to be determined later. Making a change of variable, we deduce
$$\EE[e^{-q\tau}]\; =\; {\tilde \kappa}\, q^{\frac{\alpha-1}{\alpha}} \EE[Z_1 e^{-q Z_1}]$$
with ${\tilde \kappa} =\alpha\rho\kappa$ and $Z_1 = (X_1\vert X_1\ge 0)^{-\alpha}.$ For every $s\in (0,1)$ one has
\begin{eqnarray*}
\EE[\tau^{-s}] & = & \frac{1}{\Gamma(s)} \int_0^\infty \EE[e^{-q\tau}] q^{s-1} dq\\
& = & \frac{ {\tilde \kappa}}{\Gamma(s)} \EE\left[ Z_1 \int_0^\infty e^{-q Z_1} q^{\frac{\alpha-1}{\alpha} +s-1} dq\right]\\
& = & {\tilde \kappa}\, \frac{\Gamma(1-1/\alpha+s)}{\Gamma(s)}\, \EE\!\left[ Z_1^{\frac{1}{\alpha} -s} \right] \; =\; {\tilde \kappa}\, \frac{\Gamma(1-1/\alpha+s)}{\Gamma(s)}\, \EE\!\left[ (X_1\vert X_1\ge 0)^{\alpha s -1} \right].\\
\end{eqnarray*}
On the other hand, by the formula (2.6.20) in \cite{Z} one has
$$\EE\!\left[ (X_1\vert X_1\ge 0)^{\alpha s -1} \right]\; =\; \frac{\Gamma(\alpha s ) \Gamma(1+1/\alpha-s)}{\Gamma(1-\rho +\rho\alpha s ) \Gamma(1+\rho -\rho\alpha s)}$$
and putting everything together entails
\begin{eqnarray*}
\EE[\tau^{-s}] & = & \frac{{\tilde \kappa} \Gamma(\alpha s ) \Gamma(1-1/\alpha+s) \Gamma(1+1/\alpha-s)}{\Gamma(s)\Gamma(1-\rho +\rho\alpha s ) \Gamma(1+\rho -\rho\alpha s)}
\; = \;\frac{\sin(\frac{\pi}{\alpha})\sin(\pi\rho\alpha(-s+\frac{1}{\alpha})) \Gamma(1+\alpha s)}{\sin(\pi\rho)\sin(\pi(-s+\frac{1}{\alpha}))\Gamma(1+s)}
\end{eqnarray*}
for every $s\in (0,1),$ where we used standard properties of the Gamma function and the identification of the constant comes from $\EE[\tau^{0}] =1.$ This completes the proof of (\ref{closed}) for $s\in (-1,0),$ and hence for $s\in (-1-1/\alpha, 1- 1/\alpha)$ by analytic continuation.
\hfill$\square$
\begin{REMS}{\em The above computation shows that the normalizing constant for the inverse local time at zero reads
$$\kappa\; =\; \frac{\alpha\sin(\frac{\pi}{\alpha})}{\sin(\pi\rho)}\cdot$$
One can check that this constant is the same as the one computed by Fourier inversion in \cite{St} p. 636.}
\end{REMS}
\subsection{Final remarks} (a) The identity in law (\ref{Rk}) shows that the density of $\tau$ is the multiplicative convolution of two densities which are real-analytic on $(0,+\infty).$ Indeed, it is well-known - see e.g. \cite{Z} Theorem 2.4.1 - that the density of ${\bf Z}_{\frac{1}{\alpha}}$ is real-analytic on $(0,+\infty),$ whereas the density of the first factor in (\ref{Rk}) reads
$$\frac{\sin (\pi \rho\alpha)\sin(\frac{\pi}{\alpha})t^{\frac{1}{\alpha}}}{\pi \sin(\pi\rho)(t^2 +2 t\cos(\pi \rho\alpha) + 1)}\cdot$$
Hence, the density of $\tau$ is itself real-analytic on $(0,+\infty)$ and a combination of our main result and the principle of isolated zeroes entails that $\tau$ is strictly unimodal. Besides, its mode is positive since we know from Theorem 3.15 (iii) in \cite{KKMW} in the spectrally two-sided case, and from Proposition 2 in \cite{TS1} in the spectrally positive case, that the density of $\tau$ always vanishes at $0+$ (with an infinite first derivative). The strict unimodality of $\tau$ can also be obtained in analyzing more sharply the factors in (\ref{Final}) and using Step 6 p. 212 in \cite{CT}.\\
(b) The identity in law (\ref{Rk}) can be extended in order to encompass the whole set of admissible parameters $\{\alpha\in (1, 2],\,1-1/\alpha \le \rho\le 1/\alpha\}$ of strictly $\alpha-$stable L\'evy processes that hit points in finite time a.s. Using the Legendre-Gauss multiplication formula and a fractional moment identification, one can also deduce from (\ref{Rk}) with $\rho =1/2$ the formula (5.12) of \cite{Y3}. It is possible to derive a factorization of $\tau$ with the same inverse Beta factor for $\alpha \le n/(n-1)$ and $\rho =1/n,$ but this kind of identity in law does not seem to be true in general.\\
(c) When $\rho\alpha \ge 1/2,$ the formula (\ref{Rk}) shows that the law of $\tau$ is closely related to that of the positive branch of a real stable random variable with scaling parameter $1/\alpha$ and positivity parameter $\rho\alpha.$ Indeed, Bochner's subordination - see e.g. Chapter 6 in \cite{S} or Section 3.2 in \cite{Z} for details - shows that the latter random variable decomposes into the independent product
$$\left(\frac{{\bf Z}_{\rho\alpha}^{\rho \alpha}}{{\bf Z}_{\rho\alpha}^{\rho \alpha}}\right)\times\, {\bf Z}_{\frac{1}{\alpha}}\cdot$$
(d) The identity (\ref{Rk}) is attractive in its simplicity. Compare with the distribution of first passage times of stable L\'evy processes, whose fractional moments can be computed in certain situations - see Theorem 3 in \cite{Kz} and the references therein for other recent results in the same vein - but with complicated formul\ae\, apparently not leading to tractable multiplicative identities in law. In the framework of hitting times, it is natural to ask whether (\ref{Rk}) could not help to investigate further distributional properties of $\tau,$ in the spirit of \cite{Y}. This will be the matter of further research.
\medskip
\noindent
{\bf Acknowledgement.} Ce travail a b\'en\'efici\'e d'une aide de l'Agence Nationale de la Recherche portant la r\'ef\'erence ANR-09-BLAN-0084-01.
|
1,116,691,497,577 | arxiv | \section{\bf Introduction}
\subsection{(Hereditary) cotorsion pairs in Morita rings} For a ring $R$, let $R$-Mod be the category of left $R$-modules.
For a class $\mathcal C$ of objects in abelian category $\mathcal A$, and $X\in \mathcal A$, by \ $\Ext^1_\mathcal A(X, \mathcal C)=0$ we mean
\ $\Ext^1_\mathcal A(X, C)=0$ for all $C\in \mathcal C$. Let $^\perp\mathcal C$ be the full subcategory of objects $X$ with \ $\Ext^1_\mathcal A(X, \mathcal C)=0$.
Similarly for $\mathcal C^\perp$.
\vskip5pt
Given a class \ $\mathcal X$ \ of \ $A$-modules and a class \ $\mathcal Y$ \ of \ $B$-modules, three classes
$$\left(\begin{smallmatrix}\mathcal X\\ \mathcal Y\end{smallmatrix}\right), \ \ \ \Delta(\mathcal X, \ \mathcal Y), \
\ \ \nabla(\mathcal X, \ \mathcal Y)$$ of modules over Morita ring \ $\Lambda$ \ are defined. See Subsection 3.1.
In particular, one has the monomorphism category \ ${\rm Mon}(\Lambda) = \Delta(A\mbox{\rm-Mod}, \ B\mbox{\rm-Mod})$, and the epimorphism category \
${\rm Epi}(\Lambda)= \nabla(A\mbox{\rm-Mod}, \ B\mbox{\rm-Mod})$.
\vskip5pt
Main constructions of (hereditary) cotorsion pairs in $\Lambda$-Mod are given as follows. By \ $\Tor^A_1(M, \ \mathcal U)$ $=0$ we mean \ $\Tor^A_1(M, \ U)=0$ for all $U\in\mathcal U$.
\vskip5pt
\begin{thm}\label{mainin31} \ {\rm (Theorem \ref{ctp1})} \ Let \ $\Lambda = \left(\begin{smallmatrix} A & N \\ M & B\end{smallmatrix}\right)$ be a Morita ring with $\phi = 0=\psi$, \ $(\mathcal U, \ \mathcal X)$ and \ $(\mathcal V, \ \mathcal Y)$ cotorsion pairs in $A\mbox{-}{\rm Mod}$ and $B$\mbox{\rm-Mod}, respectively.
\vskip5pt
$(1)$ \ If \ $\Tor^A_1(M, \ \mathcal U)=0 = \Tor^B_1(N, \ \mathcal V)$,
then \ $({}^\perp\left(\begin{smallmatrix}\mathcal X\\ \mathcal Y\end{smallmatrix}\right), \ \left(\begin{smallmatrix}\mathcal X\\ \mathcal Y\end{smallmatrix}\right))$ is a cotorsion pair in $\Lambda${\rm-Mod}$;$ and it is hereditary if and only if \ so are \ $(\mathcal U, \ \mathcal X)$ and \ $(\mathcal V, \ \mathcal Y)$.
\vskip5pt
$(2)$ \ If \ $\Ext_A^1(N, \ \mathcal X) =0 = \Ext_B^1(M, \ \mathcal Y)$,
then \ $(\left(\begin{smallmatrix}\mathcal U\\ \mathcal V\end{smallmatrix}\right), \ \left(\begin{smallmatrix}\mathcal U\\ \mathcal V\end{smallmatrix}\right)^\perp)$ is a cotorsion pair in $\Lambda${\rm-Mod}$;$ and
it is hereditary if and only if so are
\ $(\mathcal U, \ \mathcal X)$ and $(\mathcal V, \ \mathcal Y)$.
\end{thm}
\vskip5pt
\begin{thm}\label{mainin32} \ \ {\rm (Theorem \ref{ctp6})} \ Let $\Lambda = \left(\begin{smallmatrix} A & N \\ M & B\end{smallmatrix}\right)$ a Morita ring with $M\otimes_A N = 0 = N\otimes_BM$, \ $(\mathcal U, \ \mathcal X)$ and $(\mathcal V, \ \mathcal Y)$ cotorsion pairs in $A\mbox{-}{\rm Mod}$ and $B$\mbox{\rm-Mod}, respectively. Then
\vskip5pt
$(1)$ \ \ $(\Delta(\mathcal U, \ \mathcal V), \ \Delta(\mathcal U, \ \mathcal V)^\perp)$ is a cotorsion pair in $\Lambda${\rm-Mod}$;$ and if \ $M_A$ and \ $N_B$ are flat, then it is hereditary if and only if so are \ $(\mathcal U, \ \mathcal X)$ and \ $(\mathcal V, \ \mathcal Y)$.
\vskip5pt
$(2)$ \ \ $(^{\perp}\nabla(\mathcal X, \ \mathcal Y), \ \nabla(\mathcal X, \ \mathcal Y))$ is a cotorsion pair in $\Lambda${\rm-Mod}$;$ and if \ $_BM$ and \ $_AN$ are projective, then
it is hereditary if and only if so are \ $(\mathcal U, \ \mathcal X)$ and $(\mathcal V, \ \mathcal Y)$.
\end{thm}
We stress that, the condition \ ``$M\otimes_A N = 0 = N\otimes_BM$" in {\rm Theorem \ref{mainin32}}, can not be weakened as \ ``$\phi = 0 = \psi$" in general, as Example \ref{irem1} shows.
\vskip5pt
The cotorsion pairs $$({}^\perp\left(\begin{smallmatrix}\mathcal X\\ \mathcal Y\end{smallmatrix}\right), \ \left(\begin{smallmatrix}\mathcal X\\ \mathcal Y\end{smallmatrix}\right)),
\ \ \ \ \ \ \ \ (\Delta(\mathcal U, \ \mathcal V), \ \Delta(\mathcal U, \ \mathcal V)^\perp)$$
and
$$(\left(\begin{smallmatrix}\mathcal U\\ \mathcal V\end{smallmatrix}\right), \ \left(\begin{smallmatrix}\mathcal U\\ \mathcal V\end{smallmatrix}\right)^\perp),
\ \ \ \ \ \ \ \ (^{\perp}\nabla(\mathcal X, \ \mathcal Y), \ \nabla(\mathcal X, \ \mathcal Y))$$
given in Theorems \ref{mainin31} and \ref{mainin32} have the following relations:
$$\Delta(\mathcal U, \ \mathcal V)^{\bot} \subseteq \left(\begin{smallmatrix}\mathcal X\\ \mathcal Y\end{smallmatrix}\right); \ \ \ \ \ \ \ ^{\bot}\nabla(\mathcal X, \ \mathcal Y) \subseteq \left(\begin{smallmatrix}\mathcal U\\ \mathcal V\end{smallmatrix}\right).$$ See Theorem \ref{compare} for details. But, what important and interesting are that, {\bf they are not equal}, in general. In fact, there even exists an algebra $\Lambda$, such that the four cotorsion pairs above are pairwise different. Such an example has been given in Example \ref{ie}.
\subsection{Identifications} If \ $M=0$ or $N = 0$, \ Theorems \ref{mainin31} and \ref{mainin32} have been obtained by {\rm R. M. Zhu}, {\rm Y. Y. Peng} and {\rm N. Q. Ding} [ZPD, 3.4, 3.6];
and moreover, there hold (see {\rm [ZPD, 3.7]}) $$({}^\perp\left(\begin{smallmatrix}\mathcal X\\ \mathcal Y\end{smallmatrix}\right), \ \left(\begin{smallmatrix}\mathcal X\\ \mathcal Y\end{smallmatrix}\right)) = (\Delta(\mathcal U, \ \mathcal V), \ \Delta(\mathcal U, \ \mathcal V)^\perp)$$ and
$$(\left(\begin{smallmatrix} \mathcal U\\ \mathcal V\end{smallmatrix}\right), \ \left(\begin{smallmatrix} \mathcal U\\ \mathcal V\end{smallmatrix}\right)^\perp) =(^{\perp}\nabla(\mathcal X, \ \mathcal Y), \ \nabla(\mathcal X, \ \mathcal Y)).$$
As pointed out above, in general, they are not correct! We study the problem of identifications, i.e., when the two equalities hold true. If they are equal, then one has the cotorsion pairs
$$(\Delta(\mathcal U, \ \mathcal V), \ \left(\begin{smallmatrix}\mathcal X\\ \mathcal Y\end{smallmatrix}\right))
\ \ \ \ \mbox{and} \ \ \ \ (\left(\begin{smallmatrix} \mathcal U\\ \mathcal V\end{smallmatrix}\right), \ \nabla(\mathcal X, \ \mathcal Y)),$$
both are explicitly given. Since \ $^\perp\left(\begin{smallmatrix}\mathcal X\\ \mathcal Y\end{smallmatrix}\right)$ and \
$\Delta(\mathcal U, \ \mathcal V)^\perp$ are usually difficult to determine,
this identification is of significance, in explicitly finding abelian model structures in Morita rings.
\vskip5pt
In the rest of this section, \ $$\Lambda = \left(\begin{smallmatrix} A & N \\ M & B \end{smallmatrix}\right)$$ is a Morita ring with \ $M\otimes_A N = 0 = N\otimes_BM$. We will not state this each time.
For functors \ ${\rm T}_A: A\mbox{-Mod} \longrightarrow \Lambda\mbox{-Mod}$,
\ ${\rm T}_B: B\mbox{-Mod} \longrightarrow \Lambda\mbox{-Mod}$,
\ ${\rm H}_A: A\mbox{-Mod} \longrightarrow \Lambda\mbox{-Mod},$
and \ ${\rm H}_B: B\mbox{-Mod} \longrightarrow \Lambda\mbox{-Mod}$, see Subsection 2.4.
\vskip5pt
\begin{thm}\label{mainin41} \ {\rm (Theorem \ref{identify1})} \ Let \ $(\mathcal U, \ \mathcal X)$ and $(\mathcal V, \ \mathcal Y)$ be cotorsion pairs in $A\mbox{-}{\rm Mod}$ and in $B$\mbox{\rm-Mod}, respectively.
\vskip5pt
$(1)$ \ Assume that \ $\Tor^A_1(M, \ \mathcal U) =0 = \Tor^B_1(N, \ \mathcal V)$. If \
$M\otimes_A\mathcal U \subseteq \mathcal Y$ \ and \ $N\otimes_B\mathcal V \subseteq \mathcal X$, then
\ $\Delta(\mathcal U, \ \mathcal V) =
\ ^\perp\left(\begin{smallmatrix} \mathcal X\\ \mathcal Y\end{smallmatrix}\right) = {\rm T}_A(\mathcal U)\oplus {\rm T}_B(\mathcal V)$, and thus \ $({\rm T}_A(\mathcal U)\oplus {\rm T}_B(\mathcal V), \ \left(\begin{smallmatrix} \mathcal X\\ \mathcal{Y}\end{smallmatrix}\right))$ \ is a cotorsion pair.
\vskip10pt
$(2)$ Assume that $\Ext_B^1(M, \mathcal Y) =0 = \Ext_A^1(N, \mathcal X)$. If $\Hom_B(M, \ \mathcal Y)\subseteq \mathcal U$ and $\Hom_A(N, \mathcal X) \subseteq \mathcal V$, then
$\nabla(\mathcal X, \mathcal Y) =
\left(\begin{smallmatrix} \mathcal U \\ \mathcal V\end{smallmatrix}\right)^\perp = {\rm H}_A(\mathcal X)\oplus {\rm H}_B(\mathcal Y)$, and thus $(\left(\begin{smallmatrix} \mathcal U \\ \mathcal V\end{smallmatrix}\right), {\rm H}_A(\mathcal X)\oplus {\rm H}_B(\mathcal Y))$ is a cotorsion pair.
\end{thm}
Even if the two cotorsion pairs are not equal in general,
there are possibilities that they can be equal for some special $A$, $B$, $M$ and $N$. The following result provide such important cases: cotorsion pairs \ $(\left(\begin{smallmatrix}_A\mathcal P\\ _B\mathcal P\end{smallmatrix}\right), \ \left(\begin{smallmatrix}_A\mathcal P\\ _B\mathcal P\end{smallmatrix}\right)^\perp)$ and $(^\bot{\rm Epi}(\Lambda), \ {\rm Epi}(\Lambda)) = (^\perp\nabla (A\mbox{-}{\rm Mod}, \ B\mbox{-}{\rm Mod}), \ \nabla (A\mbox{-}{\rm Mod}, \ B\mbox{-}{\rm Mod}))$
are not equal in general (cf. Example \ref{ie}); but the following result claims that they can be the same in some special cases.
\vskip5pt
For a ring $R$, let $_R\mathcal P$ (respectively, $_R\mathcal I$) be the full subcategory of $R$-Mod of projective (respectively, injective) modules, $_R \mathcal P^{\le 1}$ (respectively, $_R\mathcal I^{\le 1}$) the full subcategory of modules
with projective (respectively, injective) dimension $\le 1$.
\begin{thm}\label{mainin42} \ {\rm (Theorem \ref{ctp4})} \ Assume that $A$ and $B$ are quasi-Frobenius rings, \ $_AN$ and $_BM$ are projective, and that $M_A$ and $N_B$ are flat. Then
\vskip5pt
${\rm (1)}$ \ \ $\Lambda$ is a Gorenstein ring with \ ${\rm inj.dim} _\Lambda\Lambda\le 1$.
\vskip5pt
${\rm (2)}$ \ \ $(^\perp\binom{_A\mathcal I}{_B\mathcal I}, \ \binom{_A\mathcal I}{_B\mathcal I}) = ({\rm Mon}(\Lambda), \ {\rm Mon}(\Lambda)^\bot);$ and it is
exactly the Gorenstein-projective cotorsion pair $({\rm GP}(\Lambda), \ _\Lambda\mathcal P^{\le 1})$. So, it is complete and hereditary, and
$${\rm GP}(\Lambda) = {\rm Mon}(\Lambda) = \ {}^\perp \ _\Lambda\mathcal P, \ \ \ \ {\rm Mon}(\Lambda)^\bot = \ _\Lambda \mathcal P^{\le 1}.$$
\vskip5pt
${\rm (2)'}$ \ \ $(\binom{_A\mathcal P}{_B\mathcal P}, \ \binom{_A\mathcal P}{_B\mathcal P}^\perp) = (^\bot{\rm Epi}(\Lambda), \ {\rm Epi}(\Lambda));$ and it
is exactly the Gorenstein-injective cotorsion pair \ $(_\Lambda \mathcal P^{\le 1}, \ {\rm GI}(\Lambda))$. So, it is complete and hereditary, and
$$ {\rm GI}(\Lambda) = {\rm Epi}(\Lambda) = \ _\Lambda\mathcal I{}^\perp, \ \ \ \ ^\bot{\rm Epi}(\Lambda) = \ _\Lambda \mathcal P^{\le 1}.$$
\end{thm}
\vskip5pt
The conditions of {\rm Theorem \ref{mainin42}} really and often occur. See Example \ref{examctp4}.
Note that \ ${\rm GP}(\Lambda) = {\rm Mon}(\Lambda)$ is a new result: it could be regarded as an application of
cotorsion theory and monomorphism category to Gorenstein-projective modules. See Remark \ref{remctp4}.
\subsection{Completeness} \ Completeness of a cotorsion pair is important, not only in the theory itself, but also in abelian model structures via Hovey correspondence ([H2]. See Theorem \ref {hoveycorrespondence}).
In view of identifications, we only discuss the completeness of cotorsion pairs in Theorem \ref{ctp1}.
\vskip5pt If cotorsion pairs \ $(\mathcal U, \mathcal X)$ and \ $(\mathcal V, \mathcal Y)$ are generated by sets $S_1$ and $S_2$, respectively,
then the cotorsion pair \ $(^\perp\left(\begin{smallmatrix}\mathcal X \\ \mathcal Y\end{smallmatrix}\right), \ \left(\begin{smallmatrix}\mathcal X \\ \mathcal Y\end{smallmatrix}\right))$
is generated by the set \ ${\rm T}_A(S_1) \cup {\rm T}_B(S_2)$, and hence complete, by a well-known theorem of P. C. Eklof and J. Trlifaj. See Proposition \ref{generatingcomplete}.
However, since the theorem of Eklof and Trlifaj has no dual versions, there is no information on the completeness of \ $(\left(\begin{smallmatrix}\mathcal U\\ \mathcal V\end{smallmatrix}\right), \ \left(\begin{smallmatrix}\mathcal U\\ \mathcal V\end{smallmatrix}\right)^\perp)$.
Also, it is more natural to start from the completeness of $(\mathcal U, \mathcal X)$ and \ $(\mathcal V, \mathcal Y)$.
So, we need module-theoretical methods to the completeness of cotorsion pairs in Morita rings.
\vskip5pt
Take \ $(\mathcal V, \ \mathcal Y)$ to be an arbitrary complete cotorsion pair in $B$-Mod.
For the cotorsion pair \ $({}^\perp\left(\begin{smallmatrix}\mathcal X\\ \mathcal Y\end{smallmatrix}\right), \ \left(\begin{smallmatrix}\mathcal X\\ \mathcal Y\end{smallmatrix}\right))$ in Theorem \ref{ctp1}(1),
taking \ $(\mathcal U, \ \mathcal X) = (_A\mathcal P, \ A\mbox{-}{\rm Mod})$, we have Theorem \ref{mainin51}(1) below;
for the cotorsion pair \ $(\left(\begin{smallmatrix}\mathcal U\\ \mathcal V\end{smallmatrix}\right), \ \left(\begin{smallmatrix}\mathcal U\\ \mathcal V\end{smallmatrix}\right)^\perp)$ in Theorem \ref{ctp1}(2),
taking \ $(\mathcal U, \ \mathcal X) = (A\mbox{-}{\rm Mod}, \ _A\mathcal I)$, we have Theorem \ref{mainin51}(2) below.
\begin{thm} \label{mainin51} \ {\rm (Theorem \ref{ctp2})} \ Assume that \ $N_B$ is flat and \ $_BM$ is projective. Let \ $(\mathcal V, \ \mathcal Y)$ be a complete cotorsion pair in $B\mbox{-}{\rm Mod}$.
\vskip5pt
$(1)$ \ If \ $M\otimes_A\mathcal P\subseteq \mathcal Y$,
then \ $({\rm T}_A(_A\mathcal P)\oplus {\rm T}_B(\mathcal V), \ \left (\begin{smallmatrix} A\mbox{-}{\rm Mod} \\ \mathcal Y\end{smallmatrix}\right))$ \ is a complete cotorsion pair.
\vskip10pt
$(2)$ \ If \ $\Hom_A(N, \ _A\mathcal I) \subseteq \mathcal V$, then \ $(\left(\begin{smallmatrix} A\mbox{-}{\rm Mod} \\ \mathcal V\end{smallmatrix}\right), \ {\rm H}_A(_A\mathcal I)\oplus {\rm H}_B(\mathcal Y))$ \ is a complete cotorsion pair.
\end{thm}
\vskip5pt
We stress that
${\rm(i)}$ \ \ If $B$ is left noetherian and \ $_BM$ is injective, then \ $M\otimes_A\mathcal P\subseteq \mathcal Y$ always
holds;
${\rm(ii)}$ \ If \ $B$ is quasi-Frobenius and \ $N_B$ is flat, then \ $\Hom_A(N, \ _A\mathcal I) \subseteq \mathcal V$ always holds.
\vskip10pt
Similarly, let \ $(\mathcal U, \ \mathcal X)$ be an arbitrary complete cotorsion pair in $A$-Mod.
For the cotorsion pair \ $({}^\perp\left(\begin{smallmatrix}\mathcal X\\ \mathcal Y\end{smallmatrix}\right), \ \left(\begin{smallmatrix}\mathcal X\\ \mathcal Y\end{smallmatrix}\right))$ in Theorem \ref{ctp1}(1),
taking $(\mathcal V, \ \mathcal Y) = (_B\mathcal P, \ B\mbox{-}{\rm Mod})$;
and for the cotorsion pair \ $(\left(\begin{smallmatrix}\mathcal U\\ \mathcal V\end{smallmatrix}\right), \ \left(\begin{smallmatrix}\mathcal U\\ \mathcal V\end{smallmatrix}\right)^\perp)$ in Theorem \ref{ctp1}(2),
taking $(\mathcal V, \ \mathcal Y) = (B\mbox{-}{\rm Mod}, \ _B\mathcal I)$, we then get Theorem \ref{ctp3}.
\subsection{Realizations} \ It turns out that Morita rings are rich in producing cotorsion pairs. In Theorem \ref{mainin31} (respectively, Theorem \ref{mainin32}),
even if we start form the projective or the injective cotorsion pair in $A$-Mod and $B$-Mod,
what we get in $\Lambda$-Mod are already pairwise``generally different" (see Definition \ref{difference}) and ``new" cotorsion pairs. Here, by a ``new" cotorsion pair we mean that it is ``generally different from"
the projective and the injective cotorsion pair, the Gorenstein-projective and the Gorenstein-projective cotorsion pair, and the flat cotorsion pair ([EJ]). For details please see Definition \ref{new}, Propositions \ref{different}, \ref{newI}, \ref{different2} and \ref{newII}.
\subsection{Abelian model structures on Morita rings}
A natural method of getting abelian model structures on Morita rings, is to see how abelian model structures on $A$-Mod and $B$-Mod induce the ones on $\Lambda$-Mod.
\vskip5pt
A Hovey triple is {\it cofibrantly generated}, if the
corresponding model category is cofibrantly generated ([H1, 2.1.17]).
For a Grothendieck category $\mathcal A$ with enough projective objects, a Hovey triple $(\mathcal C, \mathcal F, \mathcal W)$ in $\mathcal A$
is cofibrantly generated if and only if both the cotorsion pairs $(\mathcal C\cap \mathcal W, \mathcal F)$ and $(\mathcal C, \mathcal F\cap\mathcal W)$ are cogenerated by sets. See Proposition \ref {cofibrantly}.
\vskip5pt
By set argument on the completeness of cotorsion pairs, one can show that cofibrantly generated Hovey triples in $A\mbox{-}{\rm Mod}$ and $B\mbox{-}{\rm Mod}$ induce cofibrantly generated Hovey triples in $\Lambda\mbox{-}{\rm Mod}$.
\vskip5pt
\begin{thm}\label{main71} \ {\rm (Theorem \ref{cofibrantlygenHtriple})} \ Let \ $(\mathcal U', \ \mathcal X, \ \mathcal W_1)$ and \ $(\mathcal V', \ \mathcal Y, \ \mathcal W_2)$ be cofibrantly generated
Hovey triples in $A\mbox{-}{\rm Mod}$ and $B\mbox{-}{\rm Mod}$, respectively.
\vskip5pt
$(1)$ \ Assume that \ ${\rm Tor}^A_1(M, \ \mathcal U') = 0 = {\rm Tor}^B_1(N, \ \mathcal V')$, \ $M\otimes_A\mathcal U' \subseteq \mathcal Y\cap \mathcal W_2$ and \ $N\otimes_B\mathcal V' \subseteq \mathcal X\cap \mathcal W_1.$ Then
$$({\rm T}_A(\mathcal U')\oplus {\rm T}_B(\mathcal V'), \ \left(\begin{smallmatrix} \mathcal X \\ \mathcal Y\end{smallmatrix}\right), \ \left(\begin{smallmatrix} \mathcal W_1 \\ \mathcal W_2\end{smallmatrix}\right))$$
is a cofibrantly generated Hovey triple in $\Lambda\mbox{-}{\rm Mod};$ and it is hereditary with \ ${\rm Ho}(\Lambda) \cong {\rm Ho}(A)\oplus {\rm Ho}(B),$ if \ $(\mathcal U', \mathcal X, \mathcal W_1)$ and \ $(\mathcal V', \mathcal Y, \mathcal W_2)$ are hereditary.
\vskip5pt
$(2)$ \ Assume that ${\rm Ext}_B^1(M, \mathcal Y) = 0 = {\rm Ext}_A^1(N, \mathcal X)$, $\Hom_B(M, \mathcal Y) \subseteq \mathcal U'\cap \mathcal W_1$ and $\Hom_A(N, \mathcal X) \subseteq \mathcal V'\cap \mathcal W_2$. Then
$$(\left(\begin{smallmatrix} \mathcal U' \\ \mathcal V'\end{smallmatrix}\right), \ {\rm H}_A(\mathcal X)\oplus {\rm H}_B(\mathcal Y), \ \left(\begin{smallmatrix} \mathcal W_1 \\ \mathcal W_2\end{smallmatrix}\right))$$
is a cofibrantly generated
Hovey triple$;$ and it is hereditary with ${\rm Ho}(\Lambda) \cong {\rm Ho}(A)\oplus {\rm Ho}(B)$, if $(\mathcal U', \mathcal X, \mathcal W_1)$ and $(\mathcal V', \mathcal Y, \mathcal W_2)$ are hereditary.
\end{thm}
For general Hovey triples (not assumed to be cofibrantly generated), we need other arguments on the completeness of cotorsion pairs.
Using Theorem \ref{mainin51} we have
\begin{thm} \label{main72} \ {\rm (Theorem \ref{Htriple1})} \ Let \ $N_B$ be flat, $_BM$ projective, and \ $(\mathcal V', \mathcal Y, \mathcal W)$ a Hovey triple in $B$\mbox{-}{\rm Mod}.
\vskip5pt
$(1)$ \ If \ $M\otimes_A\mathcal P \subseteq \mathcal Y\cap \mathcal W$, then
$$({\rm T}_A(_A\mathcal P)\oplus {\rm T}_B(\mathcal V'), \ \left(\begin{smallmatrix} A\text{\rm\rm-Mod} \\ \mathcal Y\end{smallmatrix}\right), \ \left(\begin{smallmatrix} A\text{\rm\rm-Mod} \\ \mathcal W\end{smallmatrix}\right))$$
is a Hovey triple in $\Lambda$\mbox{-}{\rm Mod}$;$ and it is hereditary with \ ${\rm Ho}(\Lambda) \cong {\rm Ho}(B)$, if
\ $(\mathcal V', \mathcal Y, \mathcal W)$ is hereditary.
\vskip5pt
$(2)$ \ If \ $\Hom_A(N, \ _A\mathcal I)\subseteq \mathcal V'\cap \mathcal W$, then
$$(\left(\begin{smallmatrix} A\mbox{-}{\rm Mod} \\ \mathcal V'\end{smallmatrix}\right), \ \ {\rm H}_A(_A\mathcal I)\oplus {\rm H}_B(\mathcal Y), \ \ \left(\begin{smallmatrix}A\text{\rm\rm-Mod}\\ \mathcal W\end{smallmatrix}\right))$$
is a Hovey triple in $\Lambda$\mbox{-}{\rm Mod}$;$ and it is hereditary with \ ${\rm Ho}(\Lambda) \cong {\rm Ho}(B)$, if
\ $(\mathcal V', \mathcal Y, \mathcal W)$ is hereditary.
\end{thm}
Theorem \ref{main72} is not a corollary of Theorem \ref{main71}, since it needs
module-theoretical argument on the completeness of cotorsion pairs in Morita rings.
\vskip5pt
Similarly, starting from a Hovey triple in $A$-{\rm Mod} and using Theorem \ref{ctp3}, we get Theorem \ref{Htriple2}, which is also not a corollary of Theorem \ref{main71}.
Thus, by Theorems \ref{main72} and \ref{Htriple2}, we in fact get four kinds of abelian model structures on $\Lambda$\mbox{-}{\rm Mod}.
\subsection{Gpctps and projective models} \ A complete cotorsion pair \ $(\mathcal U, \ \mathcal X)$ in $A\mbox{-}{\rm Mod}$ is {\it generalized projective}, if \ $\mathcal U\cap \mathcal X = \ _A\mathcal P$ and \ $\mathcal X$ is thick (cf. Subsection 2.8).
A generalized projective cotorsion pair (or in short, a gpctp) is always hereditary and not necessarily the projective cotorsion pair $(_A\mathcal P, \ A\mbox{-}{\rm Mod})$. Following [H2] and [Gil4],
an abelian model structure is {\it projective}, if
each object is fibrant, i.e., the corresponding Hovey triple is of the form $(\mathcal U, \ A\mbox{-}{\rm Mod}, \ \mathcal X)$. Note that gpctps and projective models are in one-one correspondence, i.e.,
\ $(\mathcal U, \ \mathcal X)$ is a gpctp
in $A\mbox{-}{\rm Mod}$ if and only if $(\mathcal U, \ A\mbox{-}{\rm Mod}, \ \mathcal X)$ is a Hovey triple.
\vskip5pt
Dually, one has the notion of {\it a generalized injective cotorsion pair}, or in short, {\it gictp}, and {\it an injective model}.
\vskip5pt
The following result deals the special case of gpcpts (gictps) in Theorem \ref{main72}.
However, it is an important case with stronger result, and without extra conditions, i.e., the conditions
``$M\otimes_A\mathcal P \subseteq \mathcal Y\cap \mathcal W$" and ``$\Hom_A(N, \ _A\mathcal I)\subseteq \mathcal V'\cap \mathcal W$" in Theorem \ref{main72} can be dropped.
\vskip5pt
\begin{thm}\label{main73} {\rm(Theorem \ref{GHtriplegpgiB})} \ Assume that \ $N_B$ is flat and $_BM$ is projective.
Let \ $(\mathcal V, \ \mathcal Y)$ and $(\mathcal V', \ \mathcal Y')$ be compatible complete hereditary cotorsion pairs in $B$-{\rm Mod}, with Gillespie-Hovey triple
\ $(\mathcal V', \ \mathcal Y, \ \mathcal W)$.
\vskip5pt
$(1)$ \ If \ $(\mathcal V, \ \mathcal Y)$ and $(\mathcal V', \ \mathcal Y')$ are gpctps, then $$({\rm T}_A(_A\mathcal P)\oplus {\rm T}_B(\mathcal V), \ \left(\begin{smallmatrix} A\text{\rm\rm-Mod} \\ \mathcal Y\end{smallmatrix}\right))
\ \ \ \mbox{and} \ \ \
({\rm T}_A(_A\mathcal P)\oplus {\rm T}_B(\mathcal V'), \ \left(\begin{smallmatrix} A\text{\rm\rm-Mod} \\ \mathcal Y'\end{smallmatrix}\right))$$
are compatible gpctps in \ $\Lambda\mbox{-}{\rm Mod}$, with Gillespie-Hovey triple
$$({\rm T}_A(_A\mathcal P)\oplus {\rm T}_B(\mathcal V'), \ \left(\begin{smallmatrix} A\text{\rm\rm-Mod} \\ \mathcal Y\end{smallmatrix}\right), \ \left(\begin{smallmatrix} A\text{\rm\rm-Mod} \\ \mathcal W\end{smallmatrix}\right))$$
and \ ${\rm Ho}(\Lambda)\cong (\mathcal V'\cap \mathcal Y)/_B\mathcal P\cong {\rm Ho}(B)$.
\vskip5pt
$(2)$ \ If \ $(\mathcal V, \mathcal Y)$ and \ $(\mathcal V', \mathcal Y')$ are gictps, then
$$(\left(\begin{smallmatrix} A\mbox{-}{\rm Mod} \\ \mathcal V\end{smallmatrix}\right), \ {\rm H}_A(_A\mathcal I)\oplus {\rm H}_B(\mathcal Y)) \ \ \ \mbox{and} \ \ \ (\left(\begin{smallmatrix} A\mbox{-}{\rm Mod} \\ \mathcal V'\end{smallmatrix}\right), \ {\rm H}_A(_A\mathcal I)\oplus {\rm H}_B(\mathcal Y'))$$
are compatible gictps in $\Lambda$-{\rm Mod}, with Gillespie-Hovey triple
$$(\left(\begin{smallmatrix} A\mbox{-}{\rm Mod} \\ \mathcal V'\end{smallmatrix}\right), \ \ {\rm H}_A(_A\mathcal I)\oplus {\rm H}_B(\mathcal Y), \ \ \left(\begin{smallmatrix}A\text{\rm\rm-Mod}\\ \mathcal W\end{smallmatrix}\right))$$
and \ ${\rm Ho}(\Lambda) \cong (\mathcal V'\cap \mathcal Y)/ _B\mathcal I\cong {\rm Ho}(B)$.
\end{thm}
Similarly, starting from compatible gpctps (gictps) in $A$-{\rm Mod}, one has Theorem \ref{GHtriplegpgiA}.
\vskip5pt
\begin{cor}\label{main74} \ {\rm (Corollaries \ref{projinjtripleB}, \ref{frobB})} \ Suppose that \ $N_B$ is flat and \ $_BM$ is projective.
\vskip5pt
$(1)$ \ Let \ $(\mathcal V, \mathcal Y)$ be a gpctp in $B$-{\rm Mod}. Then
$$({\rm T}_A(_A\mathcal P)\oplus {\rm T}_B(\mathcal V), \ \ \Lambda\text{\rm\rm-Mod}, \ \ \left(\begin{smallmatrix} A\text{\rm\rm-Mod} \\ \mathcal Y\end{smallmatrix}\right))$$
is a hereditary Hovey triple, with \ ${\rm Ho}(\Lambda)\cong\mathcal V/_B\mathcal P$.
\vskip5pt
In particular, if $B$ is quasi-Frobenius, then
\ $({\rm T}_A(_A\mathcal P)\oplus {\rm T}_B(B\text{\rm\rm-Mod}), \ \ \Lambda\text{\rm\rm-Mod}, \ \ \left(\begin{smallmatrix} A\text{\rm\rm-Mod} \\ _B\mathcal I\end{smallmatrix}\right))$ \
is a hereditary {\rm Hovey} triple with \ ${\rm Ho}(\Lambda) \cong B\mbox{-}\underline{{\rm Mod}}.$
\vskip5pt
$(2)$ \ Let \ $(\mathcal V, \mathcal Y)$ be a gictp in $B$-{\rm Mod}. Then
$$(\Lambda\text{\rm\rm-Mod}, \ \ {\rm H}_A(_A\mathcal I)\oplus {\rm H}_B(\mathcal Y), \ \ \left(\begin{smallmatrix}A\text{\rm\rm-Mod}\\ \mathcal V\end{smallmatrix}\right))$$
is a hereditary Hovey triple, with \ ${\rm Ho}(\Lambda)\cong \mathcal Y/_B\mathcal I$.
\vskip5pt
In particular, if $B$ is quasi-Frobenius, then \ $(\Lambda\text{\rm\rm-Mod}, \ \ {\rm H}_A(_A\mathcal I)\oplus {\rm H}_B(B\text{\rm\rm-Mod}), \ \ \left(\begin{smallmatrix}A\text{\rm\rm-Mod}\\ _B\mathcal P\end{smallmatrix}\right))$
is a hereditary Hovey triple with \ ${\rm Ho}(\Lambda)\cong B\mbox{-}\underline{{\rm Mod}}.$
\end{cor}
Similarly, starting from a gpctp or a gictp in $A$-Mod, one get {\rm Corollaries \ref{projinjtripleA} and \ref{frobA}}.
\vskip5pt
Even if the four abelian model structures, as in Corollaries \ref{main74} and \ref{projinjtripleA},
are pairwise generally different, and they are different from the ones induced by
the projective (injective) cotorsion pairs, the Forbenius model, the Gorenstein-projective (Gorenstein-injective) model (if $\Lambda$ is Gorenstein), and the flat-cotorsion model.
See Proposition \ref{newmodel} for details.
\section{\bf Preliminaries}
\subsection{Notations}
For a ring $R$, let $R$-Mod be the category of left $R$-modules,
$_R\mathcal P$ (respectively, $_R\mathcal I$) the full subcategory of $R$-Mod of projective (respectively, injective) modules;
$_R\mathcal P^{<\infty}$ (respectively, $_R\mathcal I^{<\infty}$) the full subcategory of $R$-Mod modules of finite projective (respectively, injective) dimension. Denote by ${\rm GP}(R)$ (respectively, ${\rm GI}(R)$) the full subcategory of $R$-Mod of Gorenstein-projective (respectively, Gorenstein-injective) modules.
\vskip5pt
For a class $\mathcal C$ of objects in abelian category $\mathcal A$, let
\begin{align*}^\perp\mathcal C &= \{X\in \mathcal A \ \mid \ \Ext^1_\mathcal A(X,\mathcal C)=0\}, \ \ \ \ \ \ \ \ \ \ ^{\bot_{\ge 1}}\mathcal C = \{ X\in \mathcal A \ | \Ext^i_\mathcal A(X, \ \mathcal C) = 0, \ \forall \ i\ge 1\}, \\ \mathcal C^{\perp} & = \{X\in \mathcal A\ \mid \ \Ext^1_\mathcal A(\mathcal C, X)=0\}, \ \ \ \ \ \ \ \ \ \
\mathcal C^{\bot_{\ge 1}} = \{X\in \mathcal A | \mid \Ext^i_\mathcal A(\mathcal C, X)=0, \ \forall \ i\ge 1\}.\end{align*}
For classes $\mathcal C$ and $\mathcal D$ of objects in $\mathcal A$, by $\Hom_\mathcal A(\mathcal C, \mathcal D) = 0$ we mean
$\Hom_\mathcal A(C, D) = 0$ for all $C\in\mathcal C$ and for all $D\in\mathcal D$. Similarly for $\Ext^1_\mathcal A(\mathcal C, \mathcal D) = 0.$
\subsection{Morita rings} \ Let $A$ and $B$ be rings, $_BM_A$ a $B$-$A$-bimodule, \ $_AN_B$ an $A$-$B$-bimodule, \ $\phi: M\otimes_AN\longrightarrow B$ a $B$-bimodule map,
and $\psi: N\otimes_BM\longrightarrow A$ an $A$-bimodule map, such that
$$m'\psi(n\otimes_Bm) = \phi(m'\otimes_An)m, \ \ n'\phi(m\otimes_An) = \psi(n'\otimes_Bm)n, \ \ \forall \ m, m'\in M, \ \ \forall \ n, n'\in N. \eqno(*)$$
A {\it Morita ring} is \ $\Lambda = \Lambda_{(\phi, \psi)}:=\left(\begin{smallmatrix} A & {}_AN_B \\
{}_BM_A & B\end{smallmatrix}\right)$, with componentwise addition, and multiplication
$$\left(\begin{smallmatrix} a & n \\ m & b \end{smallmatrix}\right) \left(\begin{smallmatrix} a' & n' \\ m' & b' \end{smallmatrix}\right)
=\left(\begin{smallmatrix} aa'+\psi(n\otimes_Bm') & an'+nb' \\ ma'+bm' & \phi(m\otimes_An')+bb'\end{smallmatrix}\right).$$
The assumptions $(*)$ guarantee the associativity of the multiplication (the converse is also true).
This construction is finally formulated in [Bas]. Throughout this paper, we will assume $\phi = 0 =\psi$. This contains triangular matrix rings (i.e., $M =0$ or $N = 0$).
\vskip5pt
\subsection{Two expressions of modules over Morita rings} \ Let $\mathcal M(\Lambda)$ be the category with objects $\left(\begin{smallmatrix} X \\ Y \end{smallmatrix}\right)_{f,g}$, where $X\in A\mbox{-Mod}$, \ $Y\in B\mbox{-Mod}$, \ $f\in \Hom_B(M\otimes_AX,Y)$ and $g\in \Hom_A(N\otimes_BY, X)$,
satisfy the conditions
$$g(n\otimes_Bf(m\otimes_Ax)) = \psi(n\otimes_Bm)x, \ \ \
f(m\otimes_Ag(n\otimes_Ay)) = \phi(m\otimes_An)y, \ \ \forall \ m\in M, \ n\in N, \ x\in X, \ y\in Y.$$
The maps $f$ and $g$ are called {\it the structure maps} of $\left(\begin{smallmatrix} X \\ Y \end{smallmatrix}\right)_{f,g}$.
\vskip5pt
For $\phi = 0 = \psi,$ the conditions are just \ $g(1_N\otimes f) = 0 = f(1_M\otimes g).$
\vskip5pt
A morphism in $\mathcal M(\Lambda)$ is $\left(\begin{smallmatrix} a \\ b \end{smallmatrix}\right): \left(\begin{smallmatrix} X \\ Y\end{smallmatrix}\right)_{f,g}\longrightarrow \left(\begin{smallmatrix} X' \\ Y'\end{smallmatrix}\right)_{f',g'}$, where \ $a: X\rightarrow X'$ and $b: Y\rightarrow Y'$ are respectively an $A$-map and a $B$-map, so that the following diagrams commute:
$$\xymatrix@R= 0.7cm{M\otimes_AX\ar[d]_-f\ar[r]^-{1_M\otimes a} & M\otimes_AX' \ar[d]^-{f'} \\
Y \ar[r]^-b & Y'}\qquad \qquad \qquad
\xymatrix@R= 0.7cm{N\otimes_BY\ar[d]_-g\ar[r]^-{1_N\otimes b} & N\otimes Y'\ar[d]^-{g'} \\
X\ar[r]^-a & X'.}$$
Let $\eta_{X, Y}: \Hom_B(M\otimes_AX, Y)\cong \Hom_A(X, \ \Hom_B(M, Y))$
and $\eta'_{Y, X}: \Hom_A(N\otimes_BY, X)\cong \Hom_B(Y, \ \Hom_A(N, X))$ be the adjunction isomorphisms.
For $f\in \Hom_B(M\otimes_AX, Y)$ and
$g\in \Hom_A(N\otimes_BY, X)$, put $\widetilde{f} = \eta_{X, Y}(f)$ and $\widetilde{g} = \eta'_{X, Y}(g)$. Thus
$$\widetilde{f}(x) = ``m\mapsto f(m\otimes_Ax)", \ \forall \ x\in X; \ \ \ \ \ \widetilde{g}(y) = ``n\mapsto g(n\otimes_By)", \ \forall \ y\in Y.$$
Using the bi-functorial property of the adjunction isomorphisms one knows that
$$fb = f'(1_M\otimes_Aa) \ \ \mbox{if and only if} \ \ (M, b)\widetilde{f} = \widetilde{f'}a$$ and
$$ ag = g'(1_N\otimes_Bb) \ \ \mbox{if and only if} \ \ (N, a)\widetilde{g} = \widetilde{g'}b.$$
Let $\mathcal M'(\Lambda)$ be the category with objects $\left(\begin{smallmatrix} X \\ Y \end{smallmatrix}\right)_{\widetilde{f}, \ \widetilde{g}}$, where $X\in A\mbox{-Mod}$, \ $Y\in B\mbox{-Mod}$, \ $\widetilde{f}\in \Hom_A(X, \ \Hom_B(M, Y))$ and $\widetilde{g}\in \Hom_B(Y, \ \Hom_A(N, X))$, such that the following diagrams commute:
$$\xymatrix@R= 0.7cm {X \ar[d]_-{\widetilde{f}}\ar[r]^-{(\psi, X)h_{A, X}} & \Hom_A(N\otimes_BM, X)\ar[d]_-\cong^-{\eta'_{M, X}} \\
\Hom_B(M, Y) \ar[r]^-{(M, \widetilde{g})} & \Hom_A(M, \Hom_B(N, X))} \qquad
\xymatrix@R= 0.7cm{Y\ar[d]_-{\widetilde{g}}\ar[r]^-{(\phi, Y)h_{B, Y}} & \Hom_B(M\otimes_AN, Y)\ar[d]_-\cong
^-{\eta_{N, Y}} \\
\Hom_A(N, X) \ar[r]^-{(N, \widetilde{f})} & \Hom_A(N, \Hom_B(M, Y))}$$
where $h_{A, X}: X\rightarrow \Hom_A(A, X)$ and $h_{B, Y}: Y \rightarrow \Hom_B(B, Y)$ are the canonical isomorphisms.
The maps $\widetilde{f}$ and $\widetilde{g}$ are also called {\it the structure maps} of $\left(\begin{smallmatrix} X \\ Y \end{smallmatrix}\right)_{\widetilde{f}, \widetilde{g}}$.
\vskip5pt
For $\phi = 0 = \psi,$ the conditions are just \ $(M, \widetilde{g}) \widetilde{f} = 0 = (N, \widetilde{f}) \widetilde{g}.$
\vskip5pt
A morphism in $\mathcal M'(\Lambda)$ is $\left(\begin{smallmatrix} a \\ b \end{smallmatrix}\right): \left(\begin{smallmatrix} X \\ Y\end{smallmatrix}\right)_{\widetilde{f}, \widetilde{g}}\longrightarrow \left(\begin{smallmatrix} X' \\ Y'\end{smallmatrix}\right)_{\widetilde{f'}, \widetilde{g'}}$, where \ $a: X\rightarrow X'$ and $b: Y\rightarrow Y'$ are respectively an $A$-map and a $B$-map, so that diagrams
$$\xymatrix@R= 0.7cm{X\ar[d]_-{\widetilde{f}}\ar[r]^-{a} & X' \ar[d]^-{\widetilde{f'}} \\
\Hom_B(M, Y) \ar[r]^-{(M, b)} & \Hom_B(M, Y')}\qquad \qquad \qquad
\xymatrix@R= 0.7cm{Y\ar[d]_-{\widetilde{g}}\ar[r]^-b & Y'\ar[d]^-{\widetilde{g'}} \\
\Hom_A(N, X)\ar[r]^-{(N, a)} & \Hom_A(N, X')}$$
commute. Then
$$\left(\begin{smallmatrix} X \\ Y\end{smallmatrix}\right)_{f, g}\mapsto \left(\begin{smallmatrix} X \\ Y\end{smallmatrix}\right)_{\widetilde{f}, \widetilde{g}}, \ \ \ \ \ \ \ ``\left(\begin{smallmatrix} X \\ Y\end{smallmatrix}\right)_{f, g}\stackrel{\left(\begin{smallmatrix} a \\ b \end{smallmatrix}\right)}\longrightarrow \left(\begin{smallmatrix} X' \\ Y'\end{smallmatrix}\right)_{f', g'}"\mapsto ``\left(\begin{smallmatrix} X \\ Y\end{smallmatrix}\right)_{\widetilde{f}, \widetilde{g}}\stackrel{\left(\begin{smallmatrix} a \\ b \end{smallmatrix}\right)}\longrightarrow \left(\begin{smallmatrix} X' \\ Y'\end{smallmatrix}\right)_{\widetilde{f'}, \widetilde{g'}}"$$ gives an isomorphism $\mathcal M(\Lambda)\cong \mathcal M'(\Lambda)$ of categories.
\begin{thm}\label{modovermorita} {\rm (E. L. Green [G, 1.5])} \ Let $\Lambda =\left(\begin{smallmatrix} A & N \\
M & B\end{smallmatrix}\right)$ be a Morita ring. Then $\Lambda\mbox{-}{\rm Mod}\cong \mathcal M(\Lambda)\cong \mathcal M'(\Lambda)$ as categories.
\end{thm}
Throughout we will identify a $\Lambda$-module with $\left(\begin{smallmatrix} X \\ Y \end{smallmatrix}\right)_{f,g}$.
We will also use the expression $\left(\begin{smallmatrix} X \\ Y \end{smallmatrix}\right)_{\widetilde{f}, \widetilde{g}},$ when it is more convenient.
For convenience we will call $\left(\begin{smallmatrix} X \\ Y \end{smallmatrix}\right)_{\widetilde{f}, \widetilde{g}}$
the second expression of a $\Lambda$-module. A sequence of $\Lambda$-maps
$$\left(\begin{smallmatrix} X_1 \\ Y_1\end{smallmatrix}\right)_{f_1,g_1}\stackrel{\binom{a_1}{b_1}}\longrightarrow \left(\begin{smallmatrix} X_2 \\ Y_2 \end{smallmatrix}\right)_{f_2,g_2}\stackrel{\binom{a_2}{b_2}}\longrightarrow
\left(\begin{smallmatrix} X_3 \\ Y_3 \end{smallmatrix}\right)_{f_3,g_3}$$
is exact if and only if both the sequences
\ $X_1\stackrel{a_1}\longrightarrow X_2\stackrel{a_2}\longrightarrow X_3$ and
\ $Y_1\stackrel{b_1}\longrightarrow Y_2\stackrel{b_2}\longrightarrow Y_3$
are exact. Also, in the second expressions of $\Lambda$-modules, a sequence of $\Lambda$-maps
$$\left(\begin{smallmatrix} X_1 \\ Y_1\end{smallmatrix}\right)_{\widetilde{f_1},\widetilde{g_1}}\stackrel{\binom{a_1}{b_1}}\longrightarrow \left(\begin{smallmatrix} X_2 \\ Y_2 \end{smallmatrix}\right)_{\widetilde{f_2},\widetilde{g_2}}\stackrel{\binom{a_2}{b_2}}\longrightarrow
\left(\begin{smallmatrix} X_3 \\ Y_3 \end{smallmatrix}\right)_{\widetilde{f_3},\widetilde{g_3}}$$
is exact if and only if both the sequences
\ $X_1\stackrel{a_1}\longrightarrow X_2\stackrel{a_2}\longrightarrow X_3$ and
\ $Y_1\stackrel{b_1}\longrightarrow Y_2\stackrel{b_2}\longrightarrow Y_3$
are exact.
\subsection{Twelve functors and two recollements}
Denote by $\Psi_X$ the composition \ $N\otimes_BM\otimes_AX \stackrel {\psi\otimes 1_X} \longrightarrow A\otimes_AX\stackrel {\cong} \rightarrow X$, and denote by $\Phi_Y$
the composition \ $M\otimes_AN\otimes_B Y\stackrel{1_M\otimes g} \longrightarrow B\otimes_BY \stackrel{\cong}\rightarrow Y$.
\vskip5pt
Let $\epsilon: M\otimes_A\Hom_B(M, -)\longrightarrow {\rm Id}_{B\text{-Mod}}$ be the counit,
and $\delta: {\rm Id}_{A\text{-Mod}}\longrightarrow \Hom_B(M, M\otimes_A -)$ the unit, of the adjoint pair $(M\otimes_A-, \ \Hom_A(M, -))$. Let
\ $\epsilon': N\otimes_B\Hom_A(N, -)\longrightarrow {\rm Id}_{A\text{-Mod}}$ be the counit,
and \ $\delta': {\rm Id}_{B\text{-Mod}}\longrightarrow \Hom_A(N, N\otimes_B-)$ the unit, of the adjoint pair $(N\otimes_B-, \ \Hom_A(N, -))$.
\vskip5pt
Recall twelve functors involving $\Lambda$-Mod.
\vskip5pt
$\bullet$ \ ${\rm T}_A: A\mbox{-Mod} \longrightarrow \Lambda_{(\phi,\psi)}\mbox{-Mod}$, \quad $X \longmapsto\left(\begin{smallmatrix} X \\ M\otimes_A X\end{smallmatrix}\right)_{1_{M\otimes_A X}, \Psi_X}$.
\vskip5pt
$\bullet$ \ ${\rm T}_B: B\mbox{-Mod} \longrightarrow \Lambda_{(\phi,\psi)}\mbox{-Mod}$, \quad $Y \longmapsto\left(\begin{smallmatrix} N\otimes_BY \\ Y\end{smallmatrix}\right)_{\Phi_Y,1_{N\otimes_BY}}$.
\vskip5pt
If $\phi = \psi=0$, then \ ${\rm T}_AX= \left(\begin{smallmatrix} X \\ M\otimes_A X\end{smallmatrix}\right)_{1, 0}$ \ and \
${\rm T}_BY= \left(\begin{smallmatrix} N\otimes_BY \\ Y\end{smallmatrix}\right)_{0, 1}.$
\vskip5pt
$\bullet$ \ ${\rm U}_A: \Lambda_{(\phi,\psi)}\mbox{-Mod} \longrightarrow A\mbox{-Mod}, \quad \left(\begin{smallmatrix} X \\ Y \end{smallmatrix}\right)_{f,g} \longmapsto X$.
\vskip5pt
$\bullet$ \ ${\rm U}_B: \Lambda_{(\phi,\psi)}\mbox{-Mod} \longrightarrow B\mbox{-Mod}, \quad \left(\begin{smallmatrix} X \\ Y \end{smallmatrix}\right)_{f,g} \longmapsto Y$.
\vskip5pt
$\bullet$ \ ${\rm H}_A: A\mbox{-Mod} \longrightarrow \Lambda_{(\phi,\psi)}\mbox{-Mod},
\quad X \longmapsto \left(\begin{smallmatrix} X \\ \Hom_A(N, X)\end{smallmatrix}\right)_{\widetilde{\Psi_X}, \ \epsilon'_X}$.
Note that \ $\widetilde{\Psi_X} = \Hom_A(N, \ \Psi_X)\circ\delta'_{M\otimes_AX}$; and
${\rm H}_AX =
\left(\begin{smallmatrix} X \\ \Hom_A(N, X)\end{smallmatrix}\right)_{\widetilde{\widetilde{\Psi_X}}, \ 1}$ in the second expression.
\vskip5pt
$\bullet$ \ ${\rm H}_B: B\mbox{-Mod} \longrightarrow \Lambda_{(\phi,\psi)}\mbox{-Mod},
\quad Y \longmapsto \left(\begin{smallmatrix} \Hom_B(M,Y) \\ Y\end{smallmatrix}\right)_{\epsilon_Y, \widetilde{\Phi_Y}}$.
Note that $\widetilde{\Phi_Y} = \Hom_B(M, \ \Phi_Y)\circ \delta_{N\otimes_BY};$ and ${\rm H}_BY =
\left(\begin{smallmatrix} \Hom_B(M,Y) \\ Y\end{smallmatrix}\right)_{1, \widetilde{\widetilde{\Phi_Y}}}$ in the second expression.
\vskip5pt
If $\phi = \psi=0$, then \ ${\rm H}_A X = \left(\begin{smallmatrix} X \\ \Hom_A(N, X)\end{smallmatrix}\right)_{0, \ \epsilon'_X}, \
{\rm H}_BY = \left(\begin{smallmatrix} \Hom_B(M, Y) \\ Y\end{smallmatrix}\right)_{\epsilon_Y, 0};$
and it is convenient to use the second expression:
${\rm H}_A X = \left(\begin{smallmatrix} X \\ \Hom_A(N, X)\end{smallmatrix}\right)_{0, 1}, \ \ \
{\rm H}_BY = \left(\begin{smallmatrix} \Hom_B(M, Y) \\ Y\end{smallmatrix}\right)_{1, 0}.$
\vskip5pt
$\bullet$ \ ${\rm C}_A: \Lambda_{(\phi, \psi)}\text{\rm-Mod}\longrightarrow A\text{\rm-Mod}$, \quad $\left(\begin{smallmatrix} X\\Y\end{smallmatrix}\right)_{f,g}\longmapsto \Coker g$.
\vskip5pt
$\bullet$ \ ${\rm C}_B: \Lambda_{(\phi, \psi)}\text{\rm-Mod}\longrightarrow B\text{\rm-Mod}$, \quad $\left(\begin{smallmatrix} X\\Y\end{smallmatrix}\right)_{f,g}\longmapsto \Coker f$.
\vskip5pt
$\bullet$ \ ${\rm K}_A: \Lambda_{(\phi, \psi)}\text{\rm-Mod}\longrightarrow A\text{\rm-Mod}$, \quad $\left(\begin{smallmatrix} X\\Y\end{smallmatrix}\right)_{f,g}\longmapsto \Ker \widetilde{f}$.
\vskip5pt
$\bullet$ \ ${\rm K}_B: \Lambda_{(\phi, \psi)}\text{\rm-Mod}\longrightarrow B\text{\rm-Mod}$, \quad $\left(\begin{smallmatrix} X\\Y\end{smallmatrix}\right)_{f,g}\longmapsto \Ker \widetilde{g}$.
\vskip10pt
If $\phi = \psi=0$, then $\left(\begin{smallmatrix} X \\ 0 \end{smallmatrix}\right)_{0,0}$ is a left $\Lambda$-module for any $A$-module $_AX$,
and $\left(\begin{smallmatrix} 0 \\ Y \end{smallmatrix}\right)_{0,0}$ is a left $\Lambda$-module for any $B$-module $_BY$. (In general, they are not left $\Lambda$-modules.) In this case,
one has extra functors:
\vskip5pt
$\bullet$ \ ${\rm Z}_A: A\mbox{-Mod}\longrightarrow \Lambda_{(0,0)}\mbox{-Mod}$, \quad $X\longmapsto \left(\begin{smallmatrix} X \\ 0 \end{smallmatrix}\right)_{0,0}$.
\vskip5pt
$\bullet$ \ ${\rm Z}_B: B\mbox{-Mod}\longrightarrow \Lambda_{(0,0)}\mbox{-Mod}$, \quad $Y\longmapsto \left(\begin{smallmatrix} 0 \\ Y \end{smallmatrix}\right)_{0,0}$.
\begin{thm} \label{recollments} {\rm ([GrP, 2.4])} \ There are recollements of abelian categories $($in the sense of {\rm [FP]}$)$$:$
$$\xymatrix{A\text{\rm-Mod}\ar[rr]^-{{\rm Z}_A} && \Lambda_{(0,0)}\text{\rm-Mod} \ar[rr]^-{{\rm U}_B}\ar@<-12pt>[ll]_-{{\rm C}_A}\ar@<12pt>[ll]_-{{\rm K}_A}
&& B\text{\rm-Mod} \ar@<-12pt>[ll]_-{{\rm T}_B} \ar@<12pt>[ll]_-{{\rm H}_B}}$$
\noindent and
$$\xymatrix{B\text{\rm-Mod}\ar[rr]^-{{\rm Z}_B} && \Lambda_{(0,0)}\text{\rm-Mod} \ar[rr]^-{{\rm U}_A}\ar@<-13pt>[ll]_-{{\rm C}_B}\ar@<13pt>[ll]_-{{\rm K}_B}
&& A\text{\rm-Mod}. \ar@<-13pt>[ll]_-{{\rm T}_A} \ar@<13pt>[ll]_-{{\rm H}_A}}$$
\end{thm}
\vskip10pt
\subsection{Projective (injective) modules}
A left $\Lambda_{(\phi, \psi)}$-module $\left(\begin{smallmatrix} L_1 \\ L_2 \end{smallmatrix}\right)_{f,g}$ is projective if and only if
\ $\left(\begin{smallmatrix} L_1 \\ L_2\end{smallmatrix}\right)_{f,g}\cong {\rm T}_AP \oplus {\rm T}_BQ$
\ for some $P\in \ _A\mathcal P$ \ and $Q\in \ _B\mathcal P$; and it is injective if and only if
\ $\left(\begin{smallmatrix} L_1 \\ L_2\end{smallmatrix}\right)_{f,g}\cong {\rm H}_AI \oplus {\rm H}_BJ$
\ for some $I\in \ _A\mathcal I$ \ and $J\in \ _B\mathcal I$.
\vskip5pt
Thus, if $\phi = 0 = \phi$, a left $\Lambda_{(0, 0)}$-module $\left(\begin{smallmatrix} L_1 \\ L_2 \end{smallmatrix}\right)_{f,g}$ is projective if and only if
$$\left(\begin{smallmatrix} L_1 \\ L_2\end{smallmatrix}\right)_{f,g}\cong \left(\begin{smallmatrix} P \\ M\otimes_AP\end{smallmatrix}\right)_{1,0}\oplus \left(\begin{smallmatrix} N\otimes_BQ \\ Q \end{smallmatrix}\right)_{0,1}= \left(\begin{smallmatrix} P\oplus (N\otimes_BQ) \\ (M\otimes_AP)\oplus Q\end{smallmatrix}\right)_{\left(\begin{smallmatrix} 1 & 0 \\ 0 & 0 \end{smallmatrix}\right), \left(\begin{smallmatrix} 0 & 0 \\ 0 & 1 \end{smallmatrix}\right)}$$
for some $P\in \ _A\mathcal P$ and $Q\in \ _B\mathcal P$; and it is injective if and only if
$$\left(\begin{smallmatrix} L_1 \\ L_2\end{smallmatrix}\right)_{f,g}\cong \left(\begin{smallmatrix} I \\ \Hom_A(N,I)\end{smallmatrix}\right)_{0, \epsilon'_I}\oplus \left(\begin{smallmatrix} \Hom_B(M,J) \\ J \end{smallmatrix}\right)_{\epsilon_{_J},0}\cong \left(\begin{smallmatrix} I\oplus \Hom_B(M,J) \\ \Hom_A(N,I)\oplus J\end{smallmatrix}\right)_{\left(\begin{smallmatrix} 0 & 0 \\ 0 & \epsilon_{_J} \end{smallmatrix}\right), \left(\begin{smallmatrix} \epsilon'_{_I} & 0 \\ 0 & 0 \end{smallmatrix}\right)}$$
for some $I\in \ _A\mathcal I$ and $J\in \ _B\mathcal I$. Using the second expression of $\Lambda$-modules, a left
$\Lambda_{(0, 0)}$-module $\left(\begin{smallmatrix} L_1 \\ L_2 \end{smallmatrix}\right)_{\widetilde{f},\widetilde{g}}$ is injective if and only if
$$\left(\begin{smallmatrix} L_1 \\ L_2\end{smallmatrix}\right)_{\widetilde{f},\widetilde{g}}\cong \left(\begin{smallmatrix} I \\ \Hom_A(N,I)\end{smallmatrix}\right)_{0, 1}\oplus \left(\begin{smallmatrix} \Hom_B(M,J) \\ J \end{smallmatrix}\right)_{1,0}\cong \left(\begin{smallmatrix} I\oplus \Hom_B(M,J) \\ \Hom_A(N,I)\oplus J\end{smallmatrix}\right)_{\left(\begin{smallmatrix} 0 & 0 \\ 0 & 1 \end{smallmatrix}\right), \left(\begin{smallmatrix} 1 & 0 \\ 0 & 0 \end{smallmatrix}\right).}$$
See [GrP, 3.1].
\vskip10pt
\subsection{Cotorsion Pairs} \
Let $\mathcal A$ be an abelian category. A pair \ $(\mathcal C, \ \mathcal F)$ of classes of objects of $\mathcal A$ is
a {\it cotorsion pair} (see [S]), if \ $\mathcal C={}^\perp\mathcal F \ \ \mbox{and} \ \ \mathcal F = \mathcal C^{\perp}.$
\vskip5pt
A cotorsion pair $(\mathcal C, \mathcal F)$ is {\it complete}, if for any object $X\in \mathcal A$, there are exact sequences
$$0\longrightarrow F\longrightarrow C\longrightarrow X\longrightarrow 0, \quad \text{and}\quad
0\longrightarrow X\longrightarrow F'\longrightarrow C'\longrightarrow 0,$$
with $C, \ C'\in \mathcal C$, and \ $F, \ F'\in \mathcal F$.
\begin{prop} \label{completenessandheredity} \ {\rm ([EJ, 7.17])} \ Let \ $\mathcal A$ be an abelian category with enough projective objects and enough injective objects, and \ $(\mathcal C, \mathcal F)$ a cotorsion pair in
$\mathcal A$. Then the following are equivalent$:$
\vskip5pt
${\rm(i)}$ \ $(\mathcal C, \mathcal F)$ is complete$;$
\vskip5pt
${\rm(ii)}$ \ For any object $X\in \mathcal A$, there is an exact sequence
$0\rightarrow F\rightarrow C\rightarrow X\rightarrow 0$ with $C\in \mathcal C$ and \ $F\in \mathcal F;$
\vskip5pt
${\rm(iii)}$ \ For an any object $X\in \mathcal A$, there is exact sequence
$0\rightarrow X\rightarrow F'\rightarrow C'\rightarrow 0$ with $C'\in \mathcal C$ and \ $F'\in \mathcal F$. \end{prop}
A cotorsion pair $(\mathcal C, \mathcal F)$ is {\it cogenerated by} a set $\mathcal S$,
if $\mathcal F = \mathcal S^\perp$. One should be careful with this terminology:
in some reference, e.g., in [GT, p.99], it is also called ``{\it generated by}".
\begin{prop} \label{cogenerated} \ Let \ $\mathcal A$ be a Grothendieck category with enough projective objects.
Then any cotorsion pair in $\mathcal A$ cogenerated by a set is complete.
\end{prop}
This result is given in [ET, Theorem 10] for the module category of a ring,
and has the generality by [SS] or [Bec, 1.2.2]. It does not have a dual version.
\vskip5pt
A cotorsion pair $(\mathcal C, \mathcal F)$ is {\it hereditary}, if $\mathcal C$ is closed under the kernel of epimorphisms, and $\mathcal F$ is closed under the cokernel of monomorphisms.
\begin{prop} \label{heredity} \ {\rm ([GR, 1.2.10])} \ Let \ $\mathcal A$ be an abelian category with enough projective objects and enough injective objects, and \ $(\mathcal C, \mathcal F)$ a cotorsion pair in
$\mathcal A$. Then the following is equivalent
\vskip5pt
${\rm(i)}$ \ $(\mathcal C, \mathcal F)$ is hereditary$;$
\vskip5pt
${\rm(ii)}$ \ $\mathcal C$ is closed under the kernel of epimorphisms$;$
\vskip5pt
${\rm(iii)}$ \ $\mathcal F$ is closed under the cokernel of monomorphisms$;$
\vskip5pt
${\rm(iv)}$ \ $\Ext^2_\mathcal A(\mathcal C, \mathcal F) = 0;$
\vskip5pt
${\rm(v)}$ \ $\Ext^i_\mathcal A(\mathcal C, \mathcal F) = 0$ for $i\ge 1$.
\end{prop}
The proof of Proposition \ref{heredity} needs the assumption that abelian category $\mathcal A$ has enough projective objects and enough injective objects.
\subsection{\bf Model structures} A closed model structure on a category and a model category are introduced by D. Quillen \ [Q1] (see also [Q2]). {\it A closed model structure} on a category $\mathcal M$ is a triple
\ $(\Cofib(\mathcal M)$, \ $\Fib(\mathcal M)$, \ $\Weq(\mathcal M))$ of classes of morphisms,
where the morphisms in the three classes are respectively called cofibrations, fibrations, and weak equivalences, satisfying ${\rm (CM1) - (CM4)}$:
\vskip5pt
(CM1) \ Let $X\xlongrightarrow{f}Y\xlongrightarrow{g}Z$ be morphisms in $\mathcal M$. If two of the morphisms $f, \ g, \ gf$ are weak equivalences, then so is the third.
\vskip5pt
(CM2) \ If $f$ is a retract of $g$, and $g$ is a cofibration (fibration, weak equivalence), then so is $f$.
\vskip5pt
(CM3)=(M1) \ Given a commutative square
$$\xymatrix{A\ar[r]^-a \ar[d]_-i & X \ar[d]^-p \\
B\ar[r]^-b \ar@{.>}[ru]^-s & Y }$$
where $i\in \Cofib(\mathcal M)$ and $p\in \Fib(\mathcal M)$, if either $i\in \Weq(\mathcal M)$ or $p\in \Weq(\mathcal M)$,
then there exists a morphism $s: B\longrightarrow X$ such that $a = si, \ \ b = ps$.
\vskip5pt
(CM4) \ Any morphism $f: X\longrightarrow Y$ has a factorizations \ $f=pi$ \ with $i\in \Cofib(\mathcal M)\cap \Weq(\mathcal M)$ and $p\in \Fib(\mathcal M)$; and also \ $f=p'i'$ with $i'\in \Cofib(\mathcal M)$ and $p'\in \Fib(\mathcal M)\cap \Weq(\mathcal M)$.
\vskip5pt
Following [H1] (also [Hir]), we will call a closed model structure just as
{\it a model structure}.
\vskip5pt
A category is {\it bicomplete} if it has an arbitrary small limits and colimits.
{\it A model category} is a bicomplete category equipped with a model structure (M. Hovey [H1, 1.1.4]).
\vskip5pt
For a model structure $(\Cofib(\mathcal M)$, \ $\Fib(\mathcal M)$, \ $\Weq(\mathcal M))$ on category $\mathcal M$ with zero object,
an object $X$ is {\it trivial} if $0 \longrightarrow X $ is a weak equivalence, or, equivalently, $X\longrightarrow 0$ is a weak equivalence. It is {\it cofibrant} if $0\longrightarrow X$ is a cofibration, and it is {\it fibrant} if $X\longrightarrow 0$ is a fibration.
For a model structure on category $\mathcal M$ with zero object ($\mathcal M$ is not necessarily a model category), Quillen's homotopy category is the localization $\mathcal M[\Weq(\mathcal M)^{-1}]$, and is denoted by ${\rm Ho}(\mathcal M)$.
\vskip5pt
A model structure on an abelian category is {\it an abelian model structure}, provided that
cofibrations are exactly monomorphisms with cofibrant cokernel, and that fibrations are exactly
epimorphisms with fibrant kernel. This is equivalent to the original definition ([H2, 2.1, 4.2]), see also [Bec, 1.1.3].
{\it An abelian model category} is a bicomplete abelian category equipped with an abelian model structure.
\subsection{\bf Hovey triples} \ Let $\mathcal A$ be an abelian category.
A triple \ $(\mathcal C, \mathcal F, \mathcal W)$ \ of classes of objects is {\it a Hovey triple} in $\mathcal A$ (see [H2]), if it satisfies the conditions:
\vskip5pt
(i) \ The class \ $\mathcal W$ is {\it thick}, i.e., $\mathcal W$ is closed under direct summands,
and if two out of three terms in a short exact sequence are in $\mathcal W$, then so is the third;
\vskip5pt
(ii) \ $(\mathcal C \cap \mathcal W, \ \mathcal F)$ and \ $(\mathcal C, \ \mathcal F \cap \mathcal W)$ \ are complete cotorsion pairs.
\begin{thm} \label{hoveycorrespondence} {\rm (Hovey correspondence) \ ([H2, Theorem 2.2]; also [BR, VIII 3.5, 3.6])} \ Let $\mathcal A$ be an abelian category.
Then there is a one-to-one correspondence between abelian model structures and Hovey triples in \ $\mathcal A$, given by
$$({\rm Cofib}(\mathcal{A}), \ {\rm Fib}(\mathcal{A}), \ {\rm Weq}(\mathcal{A}))\mapsto (\mathcal{C}, \ \mathcal{F}, \ \mathcal W)$$
where \ $\mathcal C = \{\mbox{cofibrant objects}\}, \ \
\mathcal F = \{\mbox{fibrant objects}\}, \ \
\mathcal W = \{\mbox {trivial objects}\}$, with inverse $$(\mathcal{C}, \ \mathcal{F}, \ \mathcal W) \mapsto ({\rm Cofib}(\mathcal{A}), \ {\rm Fib}(\mathcal{A}), \ {\rm Weq}(\mathcal{A}))$$ where
\begin{align*} &{\rm Cofib}(\mathcal{A}) = \{\mbox{monomorphisms with cokernel in} \ \mathcal{C}\}, \ \ \
{\rm Fib}(\mathcal{A}) = \{\mbox{epimorphisms with kernel in} \ \mathcal{F} \}, \\
& {\rm Weq}(\mathcal{A}) = \{pi \ \mid \ i \ \mbox{is monic,} \ \Coker i\in \mathcal{C}\cap \mathcal W, \ p \ \mbox{is epic,} \ \Ker p\in \mathcal{F}\cap \mathcal W\}.\end{align*}
\end{thm}
We stress that, in Theorem \ref{hoveycorrespondence}, \ $\mathcal A$ is not necessarily to be bicomplete: although this is assumed in [H2, Theorem 2.2],
however, the proof given there does not use the assumption ``bicomplete".
(In fact, one can also read this from lines of [Gil2] and [Gil3].)
\vskip5pt
A cofibrantly generated model category has been introduced in [H1, 2.1.17].
Let $\mathcal A$ be a Grothendieck category with enough projective objects.
A Hovey triple $(\mathcal C, \mathcal F, \mathcal W)$ in $\mathcal A$
will be called {\it cofibrantly generated}, if cotorsion pairs $(\mathcal C\cap \mathcal W, \mathcal F)$ and $(\mathcal C, \mathcal F\cap\mathcal W)$ are cogenerated by sets.
Note that a Grothendieck category is always bicomplete (see e.g. [KS, 8.3.27]).
\begin{prop} \label{cofibrantly} {\rm ([Bec, 1.2.7; 1.2.2])} \ Let $\mathcal A$ be a Grothendieck category with enough projective objects.
Then a Hovey triple $(\mathcal C, \mathcal F, \mathcal W)$ in $\mathcal A$
is cofibrantly generated if and only if the corresponding abelian model category $\mathcal A$ is cofibrantly generated. \end{prop}
\subsection{Hereditary Hovey triples} \ A Hovey triple \ $(\mathcal C, \mathcal F, \mathcal W)$ \ is {\it hereditary}, if both
\ $(\mathcal C \cap \mathcal W, \mathcal F)$ and \ $(\mathcal C, \mathcal F \cap \mathcal W)$ \ are hereditary cotorsion pairs.
Hereditary Hovey triples enjoy the following pleasant property.
\vskip5pt
\begin{thm} \label{Ho} {\rm ([Bec, 1.1.14]; [BR, VIII 4.2]; [Gil4, 4.3])} \ Let \ $(\mathcal C, \mathcal F, \mathcal W)$ be a hereditary Hovey triple in abelian category \ $\mathcal A$. Then \ $\mathcal C \cap \mathcal F$ is a Frobenius category $($with the canonical exact structure$)$, with $\mathcal C \cap \mathcal F\cap \mathcal W$ as
the class of projective-injective objects. The composition $\mathcal C \cap \mathcal F \hookrightarrow \mathcal A \longrightarrow {\rm Ho}(\mathcal A)$ induces a triangle equivalence \ ${\rm Ho}(\mathcal A)\cong (\mathcal C \cap \mathcal F)/(\mathcal C \cap \mathcal F\cap \mathcal W),$
where $(\mathcal C \cap \mathcal F)/(\mathcal C \cap \mathcal F\cap \mathcal W)$ is the stable category of \ $\mathcal C \cap \mathcal F$ modulo \ $\mathcal C \cap \mathcal F\cap \mathcal W$.
\end{thm}
Note that the definition of ${\rm Ho}(\mathcal A)$ does not need that $\mathcal A$ is bicomplete.
By this result, hereditary Hovey triples $(\mathcal{C}, \ \mathcal{F}, \ \mathcal W)$ with $(\mathcal C \cap \mathcal F) \nsubseteqq \mathcal W$ are of special interest.
\vskip5pt
Two cotorsion pairs \ $(\Theta, \ \Theta^\perp)$ and \ $(^\perp\Upsilon, \ \Upsilon)$ are {\it compatible}
(see [Gil3]), if \ $\Ext_\mathcal A^1(\Theta, \ \Upsilon)$ $=0$ and \
$\Theta\cap \Theta^\perp ={}^\perp\Upsilon\cap \Upsilon$. The compatibility depends on the order of two cotorsion pairs. This terminology of
compatible is taken from [HJ].
\vskip5pt
J. Gillespie gives the following approach to construct all the hereditary Hovey triples.
\vskip5pt
\begin{thm} \label{GHtriple} \ {\rm (Gillespie Theorem) \ ([Gil3, 1.1])} \ Let $\mathcal A$ be an abelian category, and \ $(\Theta, \ \Theta^\perp)$ and \ $(^\perp\Upsilon, \ \Upsilon)$ complete hereditary cotorsion pairs in $\mathcal A$.
If \ $(\Theta, \ \Theta^\perp)$ and $({}^\perp\Upsilon, \ \Upsilon)$ are compatible, then
\ $(^\perp\Upsilon, \ \Theta^\perp, \ \mathcal W)$ is a hereditary Hovey triple, where
\begin{align*} \mathcal W & =
\{W\in \mathcal A \ \mid \ \exists \ \text{ an exact sequence} \ 0\rightarrow P\rightarrow F\rightarrow W\rightarrow 0 \ \mbox{with} \ F\in \Theta, \ P\in \Upsilon\} \\ & =
\{W\in \mathcal A \ \mid \ \exists \ \text{an exact sequence} \ 0\rightarrow W\rightarrow P'\rightarrow F'\rightarrow 0 \ \mbox{with} \ P'\in \Upsilon, \ F'\in \Theta\}.
\end{align*}
Conversely, any hereditary Hovey triple in $\mathcal A$ arises in this way. \end{thm}
For later applications, we will call the hereditary Hovey triple \ $(^\perp\Upsilon, \ \Theta^\perp, \ \mathcal W)$ in Theorem \ref{GHtriple}
{\it the Gillespie-Hovey triple},
induced by compatible complete hereditary cotorsion pairs \ $(\Theta, \ \Theta^\perp)$ and \ $(^\perp\Upsilon, \ \Upsilon)$. Thus,
the Gillespie-Hovey triples are exactly hereditary Hovey triples.
\subsection{Gorenstein rings} A noetherian ring $R$ is {\it a Iwanaga-Gorenstein ring}, or {\it a Gorenstein ring}, if \ ${\rm inj.dim} _RR < \infty$ and \ ${\rm inj.dim} R_R < \infty$.
In this case, it is well-known that
\vskip5pt
$\bullet$ \ ${\rm inj.dim} _RR = {\rm inj.dim} R_R$ \ and \ $_R \mathcal P^{<\infty} = \ _R \mathcal I^{<\infty}$;
\vskip5pt
$\bullet$ \ ([EJ, p. 211]) \ If \ ${\rm inj.dim} _RR \le n$, then \ $_R \mathcal P^{< \infty} = \ _R \mathcal P^{\le n} = \ _R \mathcal I^{\le n} = \ _R \mathcal I^{<\infty}$, where \ $_R \mathcal P^{\le n}$ \ ($_R \mathcal I^{\le n}$, respectively) is the full subcategory of $R$-Mod consisting of modules $X$ with ${\rm proj.dim} X \le n$ (${\rm inj.dim} X \le n$, respectively).
\vskip5pt
$\bullet$ \ ([EJ, 11.5.3]) \ ${\rm GP}(R) = \ ^{\bot_{\ge 1}} \ _R \mathcal P = \ ^{\bot_{\ge 1}} \ _R \mathcal P^{< \infty}$, and \ ${\rm GI}(R) = {}_R \mathcal I^{\bot_{\ge 1}} ={}(_R \mathcal I^{< \infty})^{\bot_{\ge 1}}$;
\vskip5pt
$\bullet$ \ ([H2, 8.6]) \ $({\rm GP}(R), \ R\mbox{-Mod}, \ _R\mathcal P^{<\infty})$ and $(R\mbox{-Mod}, \ {\rm GI}(R), \ _R\mathcal P^{<\infty})$ are hereditary Hovey triples in $R$-Mod. In particular,
\ $({\rm GP}(R), \ _R\mathcal P^{<\infty})$ and $({\rm GI}(R), \ _R\mathcal P^{<\infty})$ are complete hereditary cotorsion pairs.
\section{\bf (Hereditary) cotorsion pairs in Morita rings}
\subsection{Three classes of modules over a Morita ring} \ Let $\Lambda = \left(\begin{smallmatrix} A & N \\ M & B\end{smallmatrix}\right)$ be a Morita ring. For a class $\mathcal X$ of $A$-modules and a class $\mathcal Y$ of $B$-modules, define
\begin{align*}\left(\begin{smallmatrix}\mathcal X\\ \mathcal Y\end{smallmatrix}\right): & = \ \{\left(\begin{smallmatrix} X \\ Y\end{smallmatrix}\right)_{f, g}\in \Lambda\mbox{-}{\rm Mod} \ \mid \ X\in \mathcal X, \ \ Y\in\mathcal Y\};\\[7pt]
\Delta(\mathcal X, \ \mathcal Y): & = \ \{\left(\begin{smallmatrix} L_1 \\ L_2\end{smallmatrix}\right)_{f,g}\in \Lambda\mbox{-}{\rm Mod} \ \mid \ f: M\otimes_AL_1 \longrightarrow L_2 \ \ \text{and}
\ \ g: N\otimes_B L_2 \longrightarrow L_1
\\ & \ \ \ \ \ \ \ \ \text{are monomorphisms}, \ \ \Coker f\in \mathcal Y, \ \ \Coker g\in \mathcal X \}; \\[7pt] \nabla(\mathcal X,\ \mathcal Y): & = \ \{\left(\begin{smallmatrix} L_1 \\ L_2\end{smallmatrix}\right)_{f,g}\in \Lambda\mbox{-}{\rm Mod} \ \mid \ \widetilde{f}: L_1\longrightarrow \Hom_B(M, L_2) \ \text{and} \ \widetilde{g}: L_2\longrightarrow \Hom_A(N, L_1) \\ & \ \ \ \ \ \ \ \ \text{are epimorphisms}, \ \Ker\widetilde{f}\in \mathcal X, \ \Ker \widetilde{g}\in \mathcal Y\}.
\end{align*}
In particular, we put \begin{align*}{\rm Mon}(\Lambda): & = \Delta(A\mbox{\rm-Mod}, \ B\mbox{\rm-Mod}) = \ \{\left(\begin{smallmatrix} L_1 \\ L_2\end{smallmatrix}\right)_{f,g}\in \Lambda\mbox{-}{\rm Mod} \ \mid \ f \ \mbox{and} \ g \ \text{are monomorphisms}\};
\\[7pt] {\rm Epi}(\Lambda): & = \nabla(A\mbox{\rm-Mod}, \ B\mbox{\rm-Mod}) = \ \{\left(\begin{smallmatrix} L_1 \\ L_2\end{smallmatrix}\right)_{f,g}\in \Lambda\mbox{-}{\rm Mod} \ \mid \ \widetilde{f} \ \mbox{and} \ \widetilde{g} \ \text{are epimorphisms}\}.
\end{align*}
They will be called {\it the monomorphism category} and {\it the epimorphism category} of $\Lambda$, respectively.
\vskip5pt
It is clear that if $M\otimes_A N = 0 = N\otimes_BM$, then
$$\Delta(_A\mathcal P, \ _B\mathcal P) = \ _\Lambda\mathcal P \ \ \ \ \mbox{and} \ \ \ \ \nabla(_A\mathcal I, \ _B\mathcal I) = \ _\Lambda\mathcal I.$$
\subsection{Constructions on (hereditary) cotorsion pairs}
\begin{thm}\label{ctp1} \ Let \ $\Lambda = \left(\begin{smallmatrix} A & N \\ M & B\end{smallmatrix}\right)$ be a Morita ring with $\phi = 0=\psi$. Let \ $(\mathcal U, \ \mathcal X)$ and \ $(\mathcal V, \ \mathcal Y)$ be cotorsion pairs in $A\mbox{-}{\rm Mod}$ and $B$\mbox{\rm-Mod}, respectively.
\vskip5pt
$(1)$ \ If \ $\Tor^A_1(M, \ \mathcal U)=0 = \Tor^B_1(N, \ \mathcal V)$,
then \ $({}^\perp\left(\begin{smallmatrix}\mathcal X\\ \mathcal Y\end{smallmatrix}\right), \ \left(\begin{smallmatrix}\mathcal X\\ \mathcal Y\end{smallmatrix}\right))$ is a cotorsion pair in $\Lambda${\rm-Mod}$;$
and moreover, it is hereditary if and only if so are \ $(\mathcal U, \ \mathcal X)$ and $(\mathcal V, \ \mathcal Y)$.
\vskip10pt
$(2)$ \ If \ $\Ext_A^1(N, \ \mathcal X) =0 = \Ext_B^1(M, \ \mathcal Y)$,
then \ $(\left(\begin{smallmatrix}\mathcal U\\ \mathcal V\end{smallmatrix}\right), \ \left(\begin{smallmatrix}\mathcal U\\ \mathcal V\end{smallmatrix}\right)^\perp)$ is a cotorsion pair in $\Lambda${\rm-Mod}$;$ and moreover, it is hereditary if and only if so are \ $(\mathcal U, \ \mathcal X)$ and $(\mathcal V, \ \mathcal Y)$.
\end{thm}
\vskip5pt
\begin{thm}\label{ctp6} \ Let $\Lambda = \left(\begin{smallmatrix} A & N \\ M & B\end{smallmatrix}\right)$ a Morita ring with $M\otimes_A N = 0 = N\otimes_BM$. Let \ $(\mathcal U, \ \mathcal X)$ and $(\mathcal V, \ \mathcal Y)$ be cotorsion pairs in $A\mbox{-}{\rm Mod}$ and $B$\mbox{\rm-Mod}, respectively. Then
\vskip5pt
$(1)$ \ $(\Delta(\mathcal U, \ \mathcal V), \ \Delta(\mathcal U, \ \mathcal V)^\perp)$ is a cotorsion pair in $\Lambda${\rm-Mod}.
\vskip5pt
Moreover, if \ $M_A$ and \ $N_B$ are flat, then \ $(\Delta(\mathcal U, \ \mathcal V),$ \ $\Delta(\mathcal U, \ \mathcal V)^\perp)$ is hereditary if and only if so are \ $(\mathcal U, \ \mathcal X)$ and \ $(\mathcal V, \ \mathcal Y)$.
\vskip10pt
$(2)$ \ $(^{\perp}\nabla(\mathcal X, \ \mathcal Y), \ \nabla(\mathcal X, \ \mathcal Y))$ is a cotorsion pair in $\Lambda${\rm-Mod}.
\vskip5pt
Moreover, if \ $_BM$ and \ $_AN$ are projective, then
\ $(^{\perp}\nabla(\mathcal X, \ \mathcal Y),$ \ $\nabla(\mathcal X, \ \mathcal Y))$ is hereditary if and only if
so are \ $(\mathcal U, \ \mathcal X)$ and $(\mathcal V, \ \mathcal Y)$.
\end{thm}
\begin{Notation} \label{not} \ For convenience, we will call the cotorsion pairs in Theorem {\rm\ref{ctp1}}
{\it the cotorsion pairs in Series} {\rm I}$;$
and the ones in Theorem {\rm\ref{ctp6}} {\it the cotorsion pairs in Series} {\rm II}.\end{Notation}
\begin{exm}\label{irem1} {\it The condition \ ``$M\otimes_A N = 0 = N\otimes_BM$" in {\rm Theorem \ref{ctp6}} can not be weakened as \ ``$\phi = 0 = \psi$", in general.
\vskip5pt
For example, taking \ $\Lambda = \left(\begin{smallmatrix} A & A \\A & A\end{smallmatrix}\right)$ with $A\ne 0$ and $\phi = 0=\psi$. Then for any class \ $\mathcal U \subseteq A\mbox{-}{\rm Mod}$ and any class \ $\mathcal V\subseteq B\mbox{-}{\rm Mod}$, one has \ $\Delta(\mathcal U, \mathcal V)= \{0\}$. In fact, if \ $\left(\begin{smallmatrix} L_1 \\ L_2\end{smallmatrix}\right)_{f, g}\in \Delta(\mathcal U, \mathcal V),$ then $fg = 0=gf$. However,
$f: L_1 \longrightarrow L_2$ and $g: L_2 \longrightarrow L_1$ are monomorphisms.
Thus $L_1 = 0 = L_2$.
\vskip5pt
But $\{0\}$ can not occur in any cotorsion pair $($since $\Lambda\ne 0)$.}
\end{exm}
We will compare the cotorsion pairs in Series I with the corresponding cotorsion pairs in Series II. Comparing cotorsion pair \ $({}^\perp\left(\begin{smallmatrix}\mathcal X\\ \mathcal Y\end{smallmatrix}\right), \ \left(\begin{smallmatrix}\mathcal X\\ \mathcal Y\end{smallmatrix}\right))$ in Theorem \ref{ctp1}(1)
with cotorsion pair \ $(\Delta(\mathcal U, \ \mathcal V), \ \Delta(\mathcal U, \ \mathcal V)^\perp)$ in Theorem \ref{ctp6}(1), we get the assertion (1) below;
comparing cotorsion pair \ $(\left(\begin{smallmatrix} \mathcal U\\ \mathcal V\end{smallmatrix}\right), \ \left(\begin{smallmatrix} \mathcal U\\ \mathcal V\end{smallmatrix}\right)^\perp)$ in Theorem \ref{ctp1}(2) with
cotorsion pair \ $(^{\perp}\nabla(\mathcal X, \ \mathcal Y), \ \nabla(\mathcal X, \ \mathcal Y))$ in Theorem \ref{ctp6}(2),
we get the assertion (2) below.
\begin{thm}\label{compare} \ Let $\Lambda = \left(\begin{smallmatrix} A & N \\
M & B\end{smallmatrix}\right)$ be a Morita ring with $M\otimes_A N = 0 = N\otimes_BM$. Suppose that $(\mathcal U, \ \mathcal X)$ and $(\mathcal V, \ \mathcal Y)$ are cotorsion pairs in $A\mbox{-}{\rm Mod}$ and $B$\mbox{\rm-Mod}, respectively.
\vskip5pt
$(1)$ \ If \ $\Tor^A_1(M, \ \mathcal U) =0 = \Tor^B_1(N, \ \mathcal V)$, then the cotorsion pairs \ $$({}^\perp\left(\begin{smallmatrix}\mathcal X\\ \mathcal Y\end{smallmatrix}\right), \ \left(\begin{smallmatrix}\mathcal X\\ \mathcal Y\end{smallmatrix}\right)) \ \ \mbox{and} \ \ (\Delta(\mathcal U, \ \mathcal V), \ \Delta(\mathcal U, \ \mathcal V)^\perp)$$ in $\Lambda$-mod have a relation \ $\Delta(\mathcal U, \ \mathcal V)^{\bot} \subseteq \left(\begin{smallmatrix}\mathcal X\\ \mathcal Y\end{smallmatrix}\right),$ \ or equivalently,
\ \ $^\bot\left(\begin{smallmatrix}\mathcal X\\ \mathcal Y\end{smallmatrix}\right)\subseteq \Delta(\mathcal U, \ \mathcal V)$.
\vskip10pt
$(2)$ \ If \ $\Ext_A^1(N, \ \mathcal X)=0 = \Ext_B^1(M, \ \mathcal Y)$, then the cotorsion pairs $$(\left(\begin{smallmatrix} \mathcal U\\ \mathcal V\end{smallmatrix}\right), \ \left(\begin{smallmatrix} \mathcal U\\ \mathcal V\end{smallmatrix}\right)^\perp) \ \ \mbox{and} \ (^{\perp}\nabla(\mathcal X, \ \mathcal Y), \ \nabla(\mathcal X, \ \mathcal Y))$$ in $\Lambda$-mod have a relation \ $^{\bot}\nabla(\mathcal X, \ \mathcal Y) \subseteq \left(\begin{smallmatrix}\mathcal U\\ \mathcal V\end{smallmatrix}\right)$, \ or equivalently,
\ \ $\left(\begin{smallmatrix}\mathcal U\\ \mathcal V\end{smallmatrix}\right)^\bot \subseteq \nabla(\mathcal X, \ \mathcal Y).$
\end{thm}
\begin{rem}\label{irem2} \ {\it If $M=0$ or $N = 0$, then {\rm Theorems \ref{ctp1}, \ref{ctp6} and \ref{compare}} have been obtained by {\rm R. M. Zhu}, {\rm Y. Y. Peng} and {\rm N. Q. Ding} {\rm [ZPD]}. In particular, in that case one has $$({}^\perp\left(\begin{smallmatrix}\mathcal X\\ \mathcal Y\end{smallmatrix}\right), \ \left(\begin{smallmatrix}\mathcal X\\ \mathcal Y\end{smallmatrix}\right)) = (\Delta(\mathcal U, \ \mathcal V), \ \Delta(\mathcal U, \ \mathcal V)^\perp)$$ and $$(\left(\begin{smallmatrix} \mathcal U\\ \mathcal V\end{smallmatrix}\right), \ \left(\begin{smallmatrix} \mathcal U\\ \mathcal V\end{smallmatrix}\right)^\perp) =(^{\perp}\nabla(\mathcal X, \ \mathcal Y), \ \nabla(\mathcal X, \ \mathcal Y)).$$ See {\rm [ZPD, Proposition 3.7]}. But, in general,
{\bf they are not true!} See {\rm Example \ref{ie}}.} \end{rem}
\subsection{Induced isomorphisms between $\Ext^1$} To prove Theorems \ref{ctp1}, we need some preparations.
In the following lemma, functors ${\rm F}$ and ${\rm G}$ are not required to be exact. This is important for applications.
\begin{lem} \label{adj} \ Let $R$ and $S$ be rings, $({\rm F}, \ {\rm G})$ an adjoint pair with ${\rm F}: R\mbox{\rm -Mod}\longrightarrow S\mbox{\rm -Mod}$.
\vskip5pt
$(1)$ \ For an $X\in R$-{\rm Mod}, if \ $0 \rightarrow K \rightarrow P \rightarrow X \rightarrow 0$ is exact with $P$ projective,
such that \ $0\rightarrow {\rm F}K\rightarrow {\rm F}P\rightarrow {\rm F}X\rightarrow 0$ is exact with ${\rm F}P$ projective, then \
$\Ext_S^1({\rm F}X, \ Y)\cong \Ext_R^1(X, \ {\rm G}Y), \ \ \forall \ Y\in S\text{\rm-Mod}.$
\vskip5pt
$(2)$ \ For a $Y\in S$-{\rm Mod}, if \ $0 \rightarrow Y \rightarrow I \rightarrow C \rightarrow 0$ is exact with $I$ injective,
such that \ $0\rightarrow {\rm G}Y\rightarrow {\rm G}I\rightarrow {\rm G}C\rightarrow 0$ is exact with ${\rm G}I$ injective, then \ $\Ext_S^1({\rm F}X, \ Y)\cong \Ext_R^1(X, \ {\rm G}Y), \ \ \forall \ X\in R\text{\rm-Mod}.$
\end{lem}
\begin{proof} \ (1) \ Applying $\Hom_R(-, {\rm G}Y)$ to \ $0 \rightarrow K \rightarrow P \rightarrow X \rightarrow 0$
and applying $\Hom_S(-, Y)$ to \ $0\rightarrow {\rm F}K\rightarrow {\rm F}P\rightarrow {\rm F}X\rightarrow 0$, one gets a commutative diagram with exact rows
$$\xymatrix@C=.5cm{\Hom_S({\rm F}P, \ Y) \ar[r]\ar[d]^-\cong & \Hom_S({\rm F}K, \ Y)\ar[r]\ar[d]^-\cong & \Ext_S^1({\rm F}X, \ Y)\ar[r]\ar@{-->}[d] & 0 \\
\Hom_R(P, \ {\rm G}Y)\ar[r] & \Hom_R(K, \ {\rm G}Y)\ar[r] & \Ext_R^1(X, \ {\rm G}Y)\ar[r] & 0.}$$
Then the assertion follows from Five Lemma.
\vskip5pt
The assertion $(2)$ is the dual of (1).
\end{proof}
\begin{lem}\label{extadj1} \ Let $\Lambda = \left(\begin{smallmatrix} A & N \\
M & B\end{smallmatrix}\right)$ be a Morita ring with $\phi = 0=\psi$, \ $X\in A\mbox{-}{\rm Mod}$ and $Y\in B\mbox{-}{\rm Mod}$. Then for any $L =\left(\begin{smallmatrix} L_1\\ L_2\end{smallmatrix}\right)_{f,g}\in \Lambda\mbox{-}{\rm Mod}$ one has
\vskip5pt
$(1)$ \ If \ $\Tor_1^A(M, \ X)=0$, then $\Ext_\Lambda^1({\rm T}_AX, \ L)\cong \Ext_A^1(X, \ {\rm U}_AL).$
\vskip5pt
$(2)$ \ If \ $\Tor_1^B(N, \ Y)=0$, then \ $\Ext_\Lambda^1({\rm T}_BY, \ L)\cong \Ext_B^1(Y, \ {\rm U}_BL).$
\vskip5pt
$(3)$ \ If \ $\Ext_A^1({}N, \ X)=0$, then \ $\Ext_A^1({\rm U}_AL, \ X)\cong \Ext_\Lambda^1(L, \ {\rm H}_AX).$
\vskip5pt
$(4)$ \ If \ $\Ext_B^1(M, \ Y)=0$, then \ $\Ext_B^1({\rm U}_BL, \ Y)\cong \Ext_\Lambda^1(L, \ {\rm H}_BY).$
\end{lem}
\begin{proof} We only justify (1) and (3). The assertions (2) and (4) can be similarly proved.
\vskip5pt
(1) \ Take an exact sequence \ $0\rightarrow K\rightarrow P\rightarrow X\rightarrow 0$ with $P$ projective.
Since by assumption \ $\Tor_1^A(M, \ X)=0$, one has an exact sequence of $B$-modules
$$0\longrightarrow M\otimes_AK \longrightarrow M\otimes_AP \longrightarrow M\otimes_AX \longrightarrow 0.$$
Applying ${\rm T}_A$ (note that \ ${\rm T}_A$ is not an exact functor), one gets an exact sequence of $\Lambda$-modules
$$0\longrightarrow \left(\begin{smallmatrix} K \\ M\otimes_AK\end{smallmatrix}\right)_{1, 0}\longrightarrow \left(\begin{smallmatrix} P \\ M\otimes_AP\end{smallmatrix}\right)_{1, 0}
\longrightarrow \left(\begin{smallmatrix} X \\ M\otimes_AX\end{smallmatrix}\right)_{1, 0}\longrightarrow 0$$
where $\left(\begin{smallmatrix} P \\ M\otimes_AP\end{smallmatrix}\right)_{1, 0}$ \ is a projective $\Lambda$-module.
Consider adjoint pair $({\rm T}_A, \ {\rm U}_A)$ between $A$-Mod and $\Lambda$-Mod.
Applying Lemma \ref{adj}$(1)$ to $X$, one gets $\Ext_\Lambda^1({\rm T}_AX, \ L)\cong \Ext_A^1(X, \ {\rm U}_AL).$
\vskip5pt
(3) \ Take an exact sequence \ $0\rightarrow X\rightarrow I\rightarrow C\rightarrow 0$ with $I$ injective.
Since by assumption \ $\Ext_A^1(N, \ X)=0$, one has an exact sequence of $B$-modules
$$0\longrightarrow \Hom_A(N, X) \longrightarrow \Hom_A(N, I) \longrightarrow \Hom_A(N, C) \longrightarrow 0.$$
Applying ${\rm H}_A$ (note that \ ${\rm H}_A$ is also not an exact functor) one gets an exact sequence of $\Lambda$-modules
$$0\longrightarrow \left(\begin{smallmatrix} X \\ \Hom_A(N, X)\end{smallmatrix}\right)_{0, \epsilon_X} \longrightarrow \left(\begin{smallmatrix} I \\ \Hom_A(N, I)\end{smallmatrix}\right)_{0, \epsilon_I}
\longrightarrow \left(\begin{smallmatrix} C \\ \Hom_A(N, C) \end{smallmatrix}\right)_{0, \epsilon_C}\longrightarrow 0$$
where $\left(\begin{smallmatrix} I \\ \Hom_A(N, I)\end{smallmatrix}\right)_{0, \epsilon_I}$ is an injective $\Lambda$-module.
Consider adjoint pair $({\rm U}_A, \ {\rm H}_A)$ between $\Lambda$-Mod and $A$-Mod.
Applying Lemma \ref{adj}$(2)$ to $X$, one gets
$\Ext_A^1({\rm U}_AL, \ X)\cong \Ext_\Lambda^1(L, \ {\rm H}_AX).$
\end{proof}
\begin{lem}\label{destheta} \ Let $\Lambda = \left(\begin{smallmatrix} A & N \\
M & B\end{smallmatrix}\right)$ be a Morita ring with $\phi = 0=\psi$, \ $\mathcal X \subseteq A\mbox{-}{\rm Mod}$, and $\mathcal Y\subseteq B\mbox{-}{\rm Mod}$.
\vskip5pt
$(1)$ \ If \ $\Tor_1^A(M, \ \mathcal X)=0 = \Tor_1^B(N, \ \mathcal Y)$, then
\ $\binom{\mathcal X^\perp}{\mathcal Y^\perp} = {\rm T}_A(\mathcal X)^\perp \ \cap \ {\rm T}_B(\mathcal Y)^\perp.$
\vskip5pt
$(2)$ \ If \ $\Ext_A^1(N, \ \mathcal X)=0=\Ext_B^1(M, \ \mathcal Y)$, then \ $\binom{^\perp\mathcal X}{^\perp\mathcal Y} = \ ^\perp{\rm H}_A(\mathcal X) \ \cap \ ^\perp{\rm H}_B(\mathcal Y).$
\end{lem}
\begin{proof}
(1) \ By definition \ $L=\left(\begin{smallmatrix} L_1 \\ L_2\end{smallmatrix}\right)_{f,g}\in \binom{\mathcal X^\perp}{\mathcal Y^\perp}$ if and only if
$L_1\in \mathcal X^\perp$ and $L_2\in \mathcal Y^\perp$, or equivalently,
$\Ext_A^1(\mathcal X, \ L_1)=0 =\Ext_B^1(\mathcal Y, \ L_2)$.
Since by assumption $\Tor_1^A(M, \ \mathcal X)=0 =\Tor_1^B(N, \ \mathcal Y)$, it follows from Lemma \ref{extadj1}(1) and (2) that
$\Ext_A^1(\mathcal X, \ L_1)\cong \Ext_\Lambda^1({\rm T}_A(\mathcal X), \ L)$ and \ $\Ext_B^1(\mathcal Y, \ L_2) \cong \Ext_\Lambda^1({\rm T}_B(\mathcal Y), \ L).$
Thus,
$L=\left(\begin{smallmatrix} L_1 \\ L_2\end{smallmatrix}\right)_{f,g}\in \binom{\mathcal X^\perp}{\mathcal Y^\perp}$ if and only if
$$\Ext_\Lambda^1({\rm T}_A(\mathcal X), \ L) = 0 = \Ext_\Lambda^1({\rm T}_B(\mathcal Y), \ L)$$
i.e., $L\in {\rm T}_A(\mathcal X)^\perp \ \cap \ {\rm T}_B(\mathcal Y)^\perp$.
\vskip5pt
(2) \ Similarly, $L=\left(\begin{smallmatrix} L_1 \\ L_2\end{smallmatrix}\right)_{f,g}\in \binom{{}^\perp\mathcal X}{^\perp\mathcal Y}$
if and only if $\Ext_A^1(L_1, \ \mathcal X)=0$ and $\Ext_B^1(L_2, \ \mathcal Y)=0$.
Since \ $\Ext_A^1(N, \ \mathcal X)=0$ and \ $\Ext_B^1(M, \ \mathcal Y)=0$, by Lemma \ref{extadj1}(3) and (4),
$\Ext_A^1(L_1, \ \mathcal X)\cong \Ext_\Lambda^1(L, \ {\rm H}_A(\mathcal X))$ and \ $\Ext_B^1(L_2, \ \mathcal Y) \cong \Ext_\Lambda^1(L, \ {\rm H}_B(\mathcal Y)).$
Thus,
$L\in \left(\begin{smallmatrix}^\perp\mathcal X\\ ^\perp\mathcal Y\end{smallmatrix}\right)$ if and only if $L\in \ ^\perp{\rm H}_A(\mathcal X) \ \cap \ ^\perp{\rm H}_B(\mathcal Y)$.
\end{proof}
\subsection{Proof of Theorem \ref{ctp1}} $(1)$ \ To prove that \ $({}^\perp\binom{\mathcal X}{\mathcal Y}, \ \left(\begin{smallmatrix}\mathcal X\\ \mathcal Y\end{smallmatrix}\right))$ is a cotorsion pair,
it suffices to show \ $\left(\begin{smallmatrix}\mathcal X\\ \mathcal Y\end{smallmatrix}\right) = (^\perp\left(\begin{smallmatrix}\mathcal X\\ \mathcal Y\end{smallmatrix}\right))^\perp.$
\vskip5pt
In fact, since \ $(\mathcal U, \ \mathcal X)$ and \ $(\mathcal V, \ \mathcal Y)$ are cotorsion pairs,
it follows that \ $\left(\begin{smallmatrix}\mathcal X\\ \mathcal Y\end{smallmatrix}\right) = \binom{\mathcal U^\perp}{\mathcal V^\perp}$.
Since by assumption \ $\Tor^A_1(M, \ \mathcal U)=0 = \Tor^B_1(N, \ \mathcal V)$, it follows from
Lemma \ref{destheta}(1) that
$$\left(\begin{smallmatrix}\mathcal U^\perp\\\mathcal V^\perp\end{smallmatrix}\right) = {\rm T}_A(\mathcal U)^\perp \ \cap \ {\rm T}_B(\mathcal V)^\perp =
({\rm T}_A(\mathcal U) \ \cup \ {\rm T}_B(\mathcal V))^\perp.$$
Thus
\begin{align*} (^\perp\left(\begin{smallmatrix}\mathcal X\\ \mathcal Y\end{smallmatrix}\right))^\perp & = (^\perp\left(\begin{smallmatrix}\mathcal U^\perp\\ \mathcal V^\perp\end{smallmatrix}\right))^\perp
= \{^\perp [({\rm T}_A(\mathcal U) \ \cup \ {\rm T}_B(\mathcal V))^\perp]\}^\perp \\ &
= ({\rm T}_A(\mathcal U) \ \cup \ {\rm T}_B(\mathcal V))^\perp = \left(\begin{smallmatrix}\mathcal X\\ \mathcal Y\end{smallmatrix}\right)
\end{align*}
here one uses the fact \ $(^\bot(\mathcal S^\bot))^\bot = \mathcal S^\bot,$ for any class $\mathcal S$ of modules.
\vskip5pt
If $(\mathcal U, \ \mathcal X)$ and $(\mathcal V, \ \mathcal Y)$ are hereditary, then $\mathcal X$ and $\mathcal Y$ are closed under taking the cokernels of monomorphisms.
By the construction of $\left(\begin{smallmatrix}\mathcal X\\ \mathcal Y\end{smallmatrix}\right)$, it is clear that $\left(\begin{smallmatrix}\mathcal X\\ \mathcal Y\end{smallmatrix}\right)$ is also closed under taking the cokernels of monomorphisms, i.e.,
$({}^\perp\left(\begin{smallmatrix}\mathcal X\\ \mathcal Y\end{smallmatrix}\right), \ \left(\begin{smallmatrix}\mathcal X\\ \mathcal Y\end{smallmatrix}\right))$ is hereditary.
\vskip5pt
Conversely, let $({}^\perp\left(\begin{smallmatrix}\mathcal X\\ \mathcal Y\end{smallmatrix}\right), \ \left(\begin{smallmatrix}\mathcal X\\ \mathcal Y\end{smallmatrix}\right))$ be hereditary.
Using functors
\ ${\rm Z}_A$ and \ ${\rm Z}_B$, one sees that $\mathcal X$ and $\mathcal Y$ are closed under taking the cokernels of monomorphisms, i.e., $(\mathcal U, \ \mathcal X)$ and $(\mathcal V, \ \mathcal Y)$ are hereditary.
\vskip5pt
$(2)$ \ Similarly, it suffices to show \ $\left(\begin{smallmatrix}\mathcal U\\ \mathcal V\end{smallmatrix}\right) = \ ^\perp(\left(\begin{smallmatrix}\mathcal U\\ \mathcal V\end{smallmatrix}\right)^\perp).$
In fact, by Lemma \ref{destheta}(2) one has
$$\left(\begin{smallmatrix}\mathcal U\\ \mathcal V\end{smallmatrix}\right) = \left(\begin{smallmatrix}^\perp\mathcal X\\ ^\perp\mathcal Y\end{smallmatrix}\right) = \ ^\perp{\rm H}_A(\mathcal X) \ \cap \ ^\perp{\rm H}_B(\mathcal Y) =
\ ^\perp ({\rm H}_A(\mathcal X) \ \cup \ {\rm H}_B(\mathcal Y)).$$
Thus
\begin{align*} \ ^\perp(\left(\begin{smallmatrix}\mathcal U\\ \mathcal V\end{smallmatrix}\right)^\perp) & = \ ^\perp(\left(\begin{smallmatrix}^\perp\mathcal X\\ ^\perp\mathcal Y\end{smallmatrix}\right)^\perp)
= \ ^\perp\{ [^\perp ({\rm H}_A(\mathcal X) \ \cup \ {\rm H}_B(\mathcal Y))]^\perp\} \\ &
= \ ^\perp ({\rm H}_A(\mathcal X) \ \cup \ {\rm H}_B(\mathcal Y)) =\left(\begin{smallmatrix}\mathcal U\\ \mathcal V\end{smallmatrix}\right)
\end{align*}
here one uses the fact \ $^\bot((^\bot\mathcal S)^\bot) = \ ^\bot\mathcal S,$ for any class $\mathcal S$ of modules.
\vskip5pt
If $(\mathcal U, \ \mathcal X)$ and $(\mathcal V, \ \mathcal Y)$ are hereditary, then $\mathcal U$ and $\mathcal V$ are closed under taking the kernels of epimorphisms.
By construction, $\left(\begin{smallmatrix}\mathcal U\\ \mathcal V\end{smallmatrix}\right)$ is also closed under taking the kernels of epimorphisms, i.e.,
\ $(\left(\begin{smallmatrix}\mathcal U\\ \mathcal V\end{smallmatrix}\right), \ \left(\begin{smallmatrix}\mathcal U\\\mathcal V\end{smallmatrix}\right)^\perp)$ is hereditary. One can see the converse, by using functors
\ ${\rm Z}_A$ and \ ${\rm Z}_B$. \hfill $\square$
\subsection{Induced isomorphisms between $\Ext^1$ (continued)}
\begin{lem}\label{extadj2} \ Let $\Lambda = \left(\begin{smallmatrix} A & N \\
M & B\end{smallmatrix}\right)$ be a Morita ring with $\phi = 0=\psi$, \ $L =\left(\begin{smallmatrix} L_1\\ L_2\end{smallmatrix}\right)_{f,g}$ a $\Lambda$-module.
\vskip5pt
$(1)$ \ If $g$ is a monomorphism, then \ $\Ext_A^1({\rm C}_AL, \ X)\cong \Ext_\Lambda^1(L, \ {\rm Z}_AX)$, \ $\forall \ X\in A\mbox{\rm -Mod}$.
\vskip5pt
$(2)$ \ If $f$ is a monomorphism, then \ $\Ext_B^1({\rm C}_BL, \ Y)\cong \Ext_\Lambda^1(L, \ {\rm Z}_BY)$, \ $\forall \ Y\in B\mbox{\rm -Mod}$.
\vskip5pt
$(3)$ \ If $\widetilde{f}$ is an epimorphism, then \ $\Ext_\Lambda^1({\rm Z}_AX, \ L)\cong \Ext_A^1(X, \ {\rm K}_AL)$, \ $\forall \ X\in A\mbox{\rm -Mod}$.
\vskip5pt
$(4)$ \ If $\widetilde{g}$ is an epimorphism, then \ $\Ext_\Lambda^1({\rm Z}_BY, \ L)\cong \Ext_B^1(Y, \ {\rm K}_BL)$, \ $\forall \ Y\in B\mbox{\rm -Mod}$.
\end{lem}
\begin{proof} We only prove (1) and (3). The assertions (2) and (4) can be similarly proved.
\vskip5pt
(1) \ Taking an exact sequence
$$0\longrightarrow \left(\begin{smallmatrix} K_1 \\ K_2\end{smallmatrix}\right)_{s,t}\xlongrightarrow{\left(\begin{smallmatrix} i_1 \\ i_2 \end{smallmatrix}\right)} \left(\begin{smallmatrix} P_1 \\ P_2\end{smallmatrix}\right)_{u,v}\xlongrightarrow{\left(\begin{smallmatrix}p_1 \\ p_2\end{smallmatrix}\right)} \left(\begin{smallmatrix} L_1 \\ L_2\end{smallmatrix}\right)_{f,g}\longrightarrow 0$$
with $\left(\begin{smallmatrix} P_1 \\ P_2\end{smallmatrix}\right)_{u,v}$ a projective module, one gets a commutative diagram with exact rows:
$$\xymatrix{& N\otimes_B K_2 \ar[r]^-{1\otimes i_2}\ar[d]_-t & N\otimes_B P_2\ar[d]_-v\ar[r]^-{1\otimes p_2} & N\otimes_B L_2\ar[d]^-g \ar[r] & 0 \\
0\ar[r] & K_1\ar[r]^-{i_1} & P_1\ar[r]^-{p_1} & L_1\ar[r] & 0}$$
Since $g$ is a monomorphism, by Snake Lemma
$$0\longrightarrow \Coker t\longrightarrow \Coker v\longrightarrow \Coker g\longrightarrow 0$$
is exact. Since $P = \left(\begin{smallmatrix} P_1 \\ P_2\end{smallmatrix}\right)_{u,v}$ is projective,
$\Coker v = {\rm C}_A P$ is a projective $A$-module.
\vskip5pt
Consider adjoint pair $({\rm C}_A, \ {\rm Z}_A)$ between $\Lambda$-Mod and $A$-Mod. (Note that ${\rm C}_A$ is not exact.)
Applying Lemma \ref{adj}(1) to $L =\left(\begin{smallmatrix} L_1\\ L_2\end{smallmatrix}\right)_{f,g}$, one gets
$$\Ext_A^1({\rm C}_AL, \ X) \cong \Ext_\Lambda^1(L, \ {\rm Z}_AX), \ \forall \ X\in A\mbox{\rm -Mod}.$$
\vskip5pt
(3) \ Similarly, taking an exact sequence of $\Lambda$-modules
$$0\longrightarrow \left(\begin{smallmatrix} L_1 \\ L_2\end{smallmatrix}\right)_{f,g}\xlongrightarrow{\left(\begin{smallmatrix} \sigma_1 \\ \sigma_2 \end{smallmatrix}\right)}
\left(\begin{smallmatrix} I_1 \\ I_2 \end{smallmatrix}\right)_{u,v}\xlongrightarrow{\left(\begin{smallmatrix}\pi_1 \\ \pi_2\end{smallmatrix}\right)}
\left(\begin{smallmatrix} C_1 \\ C_2 \end{smallmatrix}\right)_{s,t}\longrightarrow 0$$
with $\left(\begin{smallmatrix} I_1 \\ I_2 \end{smallmatrix}\right)_{u,v}$ an injective module, one gets a commutative diagram with exact rows:
$$\xymatrix{& M\otimes_A L_1\ar[r]^-{1\otimes \sigma_1}\ar[d]_-{f} & M\otimes_A I_1 \ar[r]^-{1\otimes \pi_1}\ar[d]_-{u} & M\otimes_A C_1\ar[r]\ar[d]^-{s} & 0 \\
0\ar[r] & L_2\ar[r]^-{\sigma_2} & I_2 \ar[r]^-{\pi_2} & C_2\ar[r] & 0}$$
Using adjoint isomorphism, one gets a commutative diagram with exact rows:
$$\xymatrix{0\ar[r] & L_1\ar[r]^-{\sigma_1}\ar[d]_-{\widetilde{f}} & I_1 \ar[r]^-{\pi_1}\ar[d]_-{\widetilde{u}} & C_1\ar[r]\ar[d]^-{\widetilde{s}} & 0 \\
0\ar[r] & \Hom_B(M, L_2)\ar[r]^-{(M, \sigma_2)} & \Hom_B(M, I_2) \ar[r]^-{(M, \pi_2)} & \Hom_B(M, C_2)}$$
Since $\widetilde{f}$ is an epimorphism, by Snake Lemma that
$$0\longrightarrow \Ker\widetilde{f}\longrightarrow \Ker\widetilde{u} \longrightarrow \Ker\widetilde{s}\longrightarrow 0$$
is exact. Since $I = \left(\begin{smallmatrix} I_1 \\ I_2\end{smallmatrix}\right)_{u, v}$ is injective,
$\Ker \widetilde{u} = {\rm K}_A I$ is an injective $A$-module.
\vskip5pt
Consider adjoint pair $({\rm Z}_A, \ {\rm K}_A)$ between $A$-Mod and $\Lambda$-Mod. Applying Lemma \ref{adj}$(2)$ to $L =\left(\begin{smallmatrix} L_1\\ L_2\end{smallmatrix}\right)_{f,g}$, one gets
$\Ext_\Lambda^1({\rm Z}_AX, \ L)\cong \Ext_A^1(X, \ {\rm K}_AL), \ \forall \ X\in A\mbox{\rm -Mod}.$
\end{proof}
\subsection{Key lemmas for Theorem \ref{ctp6}} \ The following lemma will play an important role in the proof of Theorem \ref{ctp6}.
\begin{lem}\label{desdelta} \ Let $\Lambda = \left(\begin{smallmatrix} A & N \\
M & B\end{smallmatrix}\right)$ be a Morita ring with $M\otimes_A N = 0 = N\otimes_BM$, \ \ $\mathcal X \subseteq A\mbox{-}{\rm Mod}$, and \ $\mathcal Y\subseteq B\mbox{-}{\rm Mod}$.
\vskip5pt
$(1)$ \ If \ $\mathcal X \supseteq \ _A \mathcal I$ \ and \ $\mathcal Y\supseteq \ _A\mathcal I $, \ then
\ $\Delta({}^\perp\mathcal X, \ {}^\perp\mathcal Y) = \ ^\perp{\rm Z}_A(\mathcal X) \ \cap \ ^\perp{\rm Z}_B(\mathcal Y).$
\vskip5pt
$(2)$ \ If \ $\mathcal X \supseteq \ _A \mathcal P$ \ and \ $\mathcal Y\supseteq \ _B \mathcal P$, \ then \ $\nabla(\mathcal X^\perp, \ \mathcal Y^\perp) = {\rm Z}_A(\mathcal X)^\perp \ \cap \ {\rm Z}_B(\mathcal Y)^\perp.$
\end{lem}
\begin{proof} (1) \ Let $L=\left(\begin{smallmatrix} L_1 \\ L_2\end{smallmatrix}\right)_{f,g}\in \Delta({}^\perp\mathcal X, \ {}^\perp\mathcal Y)$.
By definition $f$ and $g$ are monomorphisms, and $\Coker f\in {}^\perp\mathcal Y$ and $\Coker g\in {}^\perp\mathcal X$.
Since $g$ is a monomorphism and \ $\Ext_B^1({\rm C}_AL, \ \mathcal X) = \Ext_A^1(\Coker g, \ \mathcal X) = 0$, it follows from Lemma \ref{extadj2}(1) that
$\Ext_\Lambda^1(L, \ {\rm Z}_A(\mathcal X)) = 0$, i.e., $L\in \ ^\perp{\rm Z}_A(\mathcal X)$.
Similarly, since $f$ is a monomorphism and \ $\Ext_B^1({\rm C}_BL, \ \mathcal Y) = \Ext_B^1(\Coker f, \ \mathcal Y) = 0$, by Lemma \ref{extadj2}(2),
$\Ext_\Lambda^1(L, \ {\rm Z}_B(\mathcal Y)) = 0$, i.e., $L\in \ ^\perp{\rm Z}_B(\mathcal Y)$. Thus, $L\in \ ^\perp{\rm Z}_A(\mathcal X) \ \cap \ ^\perp{\rm Z}_B(\mathcal Y)$.
\vskip5pt
Conversely, let $L=\left(\begin{smallmatrix} L_1 \\ L_2\end{smallmatrix}\right)_{f,g}\in \ ^\perp{\rm Z}_A(\mathcal X) \ \cap \ ^\perp{\rm Z}_B(\mathcal Y)$, i.e.,
$\Ext_\Lambda^1(L, \ {\rm Z}_A(\mathcal X)) = 0 = \Ext_\Lambda^1(L, \ {\rm Z}_B(\mathcal Y))$.
\vskip5pt
{\bf Claim 1:} \ $\Hom_A(g, \ X): \Hom_A(L_1, \ X)\longrightarrow \Hom_A(N\otimes_BL_2, \ X)$ is an epimorphism,
for any module $X\in \mathcal X$. In fact, for any $A$-map $u: N\otimes_B L_2\longrightarrow X$,
consider $A$-map $g' =\left(\begin{smallmatrix}u\\ g\end{smallmatrix}\right): N\otimes_BL_2\longrightarrow X\oplus L_1$ and
the exact sequence of $A$-modules
$$0\longrightarrow X\xlongrightarrow{\left(\begin{smallmatrix} 1\\ 0\end{smallmatrix}\right)}X\oplus L_1\xlongrightarrow{(0,1)}L_1\longrightarrow 0.$$
Put \ $f' = (0, f): M\otimes_A(X\oplus L_1)\longrightarrow L_2.$ Then $\left(\begin{smallmatrix} X\oplus L_1 \\ L_2\end{smallmatrix}\right)_{f',g'}$ is indeed a $\Lambda$-module.
We stress that this is a place where one needs the assumption \ $M\otimes_A N = 0 = N\otimes_BM$:
Given any $U\in A\mbox{-Mod}$ and $V\in B\mbox{-Mod}$, for arbitrary $u\in \Hom_B(M\otimes_AU, V)$ and $v\in \Hom_A(N\otimes_BV, U)$,
\ $\left(\begin{smallmatrix} U \\ V \end{smallmatrix}\right)_{u, v}$ is always a left $\Lambda$-module, since the conditions
\ $v(1_N\otimes u) = 0$ and $u(1_M\otimes v) = 0$ automatically hold.
\vskip5pt
Then one can check that
$$
0\longrightarrow\left(\begin{smallmatrix} X\\ 0\end{smallmatrix}\right)_{0,0}\xlongrightarrow{\left(\begin{smallmatrix}\left(\begin{smallmatrix} 1 \\ 0 \end{smallmatrix}\right) \\ 0 \end{smallmatrix}\right)}\left(\begin{smallmatrix} X\oplus L_1 \\ L_2\end{smallmatrix}\right)_{f',g'}\xlongrightarrow{\left(\begin{smallmatrix} (0,1) \\ 1 \end{smallmatrix}\right)} \left(\begin{smallmatrix} L_1\\ L_2\end{smallmatrix}\right)_{f,g}\longrightarrow 0$$
is an exact sequence of $\Lambda$-modules. Since $\left(\begin{smallmatrix} X\\ 0\end{smallmatrix}\right)_{0,0} = {\rm Z}_A X\in {\rm Z}_A(\mathcal X)$ and $L\in \ ^\perp{\rm Z}_A(\mathcal X)\cap \ ^\perp{\rm Z}_B(\mathcal Y)$,
this exact sequence splits. Thus there is a $\Lambda$-map $$\left(\begin{smallmatrix}\left(\begin{smallmatrix} a \\ b\end{smallmatrix}\right) \\ \beta\end{smallmatrix}\right):\left(\begin{smallmatrix} L_1 \\ L_2\end{smallmatrix}\right)_{f,g}\longrightarrow \left(\begin{smallmatrix} X\oplus L_1 \\ L_2\end{smallmatrix}\right)_{f',g'}$$ such that
$\left(\begin{smallmatrix} (0,1) \\ 1 \end{smallmatrix}\right)\left(\begin{smallmatrix}\left(\begin{smallmatrix} a \\ b\end{smallmatrix}\right) \\ \beta\end{smallmatrix}\right)={\rm Id}_L$.
So $b={\rm Id}_{L_1}$ and $\beta ={\rm Id}_{L_2}$. Thus one gets a commutative diagram
$$\xymatrix@C=2cm{N\otimes_BL_2\ar[d]_-g\ar@{=}[r] & N\otimes_BL_2\ar[d]^-{g'=\left(\begin{smallmatrix} u \\ g \end{smallmatrix}\right)} \\ L_1\ar[r]^-{\left(\begin{smallmatrix} a \\ 1 \end{smallmatrix}\right)} & X\oplus L_1}$$
and hence $u=ag$. This proves {\bf Claim 1}.
\vskip5pt
{\bf Claim 2:} \ $g$ is a monomorphism. In fact, embedding $N\otimes_BL_2$ into an injective $A$-module one has a monomorphism $i: N\otimes_BL_2\hookrightarrow I$. By assumption $I\in \mathcal X$, hence
$\Hom_A(g, \ I): \Hom_A(L_1, \ I)\longrightarrow \Hom_A(N\otimes_BL_2, \ I)$ is an epimorphism, by {\bf Claim 1}. Hence there is an $A$-map $v: L_1\longrightarrow I$ such that $vg = i.$
Thus, $g$ is a monomorphism.
\vskip5pt
Similar as {\bf Claim 1}, one has
\vskip5pt
{\bf Claim 3:} \ $\Hom_A(f, \ Y): \Hom_B(L_2, \ Y)\longrightarrow \Hom_B(M\otimes_AL_1, \ Y)$ is an epimorphism, for any module $Y\in \mathcal Y$.
\vskip5pt
Similar as {\bf Claim 2}, one has
\vskip5pt
{\bf Claim 4:} \ $f$ is a monomorphism.
\vskip5pt
We omit the similar proof of {\bf Claim 3} and {\bf Claim 4}.
\vskip5pt
Now, since $g$ and $f$ are monic, by Lemma \ref{extadj2}(1) and (2) one has
$$\Ext_A^1(\Coker g, \ \mathcal X) = \Ext_A^1({\rm C}_AL, \ \mathcal X) \cong \Ext_\Lambda^1(L, \ {\rm Z}_A(\mathcal X)) = 0,$$
$$\Ext_B^1(\Coker f, \ \mathcal Y) = \Ext_B^1({\rm C}_BL, \ \mathcal Y)\cong \Ext_\Lambda^1(L, \ {\rm Z}_B(\mathcal Y)) = 0$$
\noindent By definition, $L=\left(\begin{smallmatrix} L_1 \\ L_2\end{smallmatrix}\right)_{f,g}\in \Delta({}^\perp\mathcal X, \ {}^\perp\mathcal Y)$. This completes the proof of $(1)$.
\vskip5pt
(2) \ This can be similarly proved, however, it is difficult to say that it is the dual of (1), thus we include a justification. It will be much convenient to use the second expression of a
$\Lambda$-module.
\vskip5pt
Let $L=\left(\begin{smallmatrix} L_1 \\ L_2\end{smallmatrix}\right)_{\widetilde{f}, \widetilde{g}}\in \nabla(\mathcal X^\perp, \ \mathcal Y^\perp)$, i.e., $\widetilde{f}$ and $\widetilde{g}$ are epimorphisms, and $\Ker \widetilde{f}\in \mathcal X^\perp$ and $\Ker \widetilde{g}\in \mathcal Y^\perp$.
Since $\widetilde{f}$ is an epimorphism and \ $\Ext_B^1(\mathcal X, \ {\rm K}_AL) = \Ext_A^1(\mathcal X, \ \Ker\widetilde{f}) = 0$, by Lemma \ref{extadj2}(3), $L\in {\rm Z}_A(\mathcal X)^\perp$.
Similarly, since $\widetilde{g}$ is an epimorphism and \ $\Ext_B^1(\mathcal Y, \ {\rm K}_BL) = \Ext_B^1(\mathcal Y, \ \Ker\widetilde{g}) = 0$, by Lemma \ref{extadj2}(4),
$L\in {\rm Z}_B(\mathcal Y)^\perp$. Thus, $L\in {\rm Z}_A(\mathcal X)^\perp \ \cap \ {\rm Z}_B(\mathcal Y)^\perp$.
\vskip5pt
Conversely, let $L=\left(\begin{smallmatrix} L_1 \\ L_2\end{smallmatrix}\right)_{\widetilde{f}, \widetilde{g}}\in \ {\rm Z}_A(\mathcal X)^\perp \ \cap \ {\rm Z}_B(\mathcal Y)^\perp$.
\vskip5pt
{\bf Claim 1:} $\Hom_B(Y, \ \widetilde{g}): \Hom_B(Y, \ L_2)\longrightarrow \Hom_B(Y, \ \Hom_A(N, L_1))$ is an epimorphism, for any module $Y\in \mathcal Y$.
In fact, $\forall \ u\in \Hom_B(Y, \ \Hom_A(N,L_1))$,
consider $B$-map $\widetilde{g'}:=(u,\widetilde{g}): Y\oplus L_2\longrightarrow \Hom_A(N, \ L_1)$. Thus \ $g'\in \Hom_A(N\otimes_B(Y\oplus L_2), \ L_1)$.
Put \ $\widetilde{f'}= \binom{0}{\widetilde{f}}: L_1\longrightarrow \Hom_B(M,Y)\oplus \Hom_B(M,L_2).$ Thus \ $f'\in \Hom_B(M\otimes_AL_1, \ Y\oplus L_2)$.
Since \ $M\otimes_A N = 0 = N\otimes_BM$, \ $\left(\begin{smallmatrix} L_1 \\ Y\oplus L_2\end{smallmatrix}\right)_{\widetilde{f'}, \widetilde{g'}}$ is indeed a $\Lambda$-module.
\vskip5pt
Then one has the exact sequence
$$
0\longrightarrow\left(\begin{smallmatrix} L_1\\ L_2\end{smallmatrix}\right)_{\widetilde{f}, \widetilde{g}}\xlongrightarrow{\binom{1}{\binom{0}{1}}}
\left(\begin{smallmatrix} L_1 \\ Y\oplus L_2\end{smallmatrix}\right)_{\widetilde{f'}, \widetilde{g'}}\xlongrightarrow{\left(\begin{smallmatrix} 0 \\ (1,0) \end{smallmatrix}\right)} \left(\begin{smallmatrix} 0\\ Y\end{smallmatrix}\right)_{0,0}\longrightarrow 0.$$
(We stress that it is much convenient to use the second expression of
$\Lambda$-modules. Otherwise, say, it is not direct to see that ${\binom{1}{\binom{0}{1}}}$ is a $\Lambda$-map.)
\vskip5pt
\noindent Since $\left(\begin{smallmatrix} 0\\ Y\end{smallmatrix}\right)_{0,0} = {\rm Z}_B Y\in {\rm Z}_B(\mathcal Y)$ and $L\in \ {\rm Z}_A(\mathcal X)^\perp\cap \ {\rm Z}_B(\mathcal Y)^\perp$,
this exact sequence splits, i.e., there is a $\Lambda$-map $$\left(\begin{smallmatrix}\alpha \\ (a,b)\end{smallmatrix}\right):\left(\begin{smallmatrix} L_1 \\ Y\oplus L_2\end{smallmatrix}\right)_{\widetilde{f'},\widetilde{g'}}\longrightarrow \left(\begin{smallmatrix} L_1 \\ L_2\end{smallmatrix}\right)_{\widetilde{f},\widetilde{g}}$$ such that
$\left(\begin{smallmatrix} \alpha \\ (a,b) \end{smallmatrix}\right)\left(\begin{smallmatrix} 1 \\ \binom{0}{1}\end{smallmatrix}\right)={\rm Id}_L$.
So $\alpha=\Id_{L_1}$ and $b={\rm Id}_{L_2}$.
This gives the commutative diagram
$$\xymatrix@C=2cm{Y\oplus L_2\ar[d]_-{\widetilde{g'}=(u,\widetilde{g})}\ar[r]^-{(a,1)} & L_2\ar[d]^-{\widetilde{g}} \\
\Hom_A(N, L_1)\ar@{=}[r] & \Hom_A(N,L_1)}$$
\noindent commutes. Hence $u=\widetilde{g}a$. This proves {\bf Claim 1}.
\vskip5pt
{\bf Claim 2:} \ $\widetilde{g}$ is an epimorphism. In fact, taking a $B$-epimorphism $q: Q\longrightarrow \Hom_A(N, L_1)$ with $Q$ projective. Then $Q\in \mathcal Y$, hence
$\Hom_B(Q, \ \widetilde{g}): \Hom_B(Q, \ L_2)\longrightarrow \Hom_B(Q, \ \Hom_A(N, \ L_1))$ is an epimorphism. So there is a $B$-map $v: Q\longrightarrow L_2$ with $q = \widetilde{g}v.$ This proves {\bf Claim 2}.
\vskip5pt
Similarly, \ $\Hom_A(X, \ \widetilde{f}): \Hom_A(X, \ L_1)\longrightarrow \Hom_A(X, \ \Hom_B(M,L_2))$ is an epimorphism for any $X\in \mathcal X$; and
\ $\widetilde{f}$ is an epimorphism.
\vskip5pt
It follows from Lemma \ref{extadj2}(3) and (4) that \ $$\Ext_A^1(\mathcal X, \ \Ker\widetilde{f}) = \Ext_A^1(\mathcal X, \ {\rm K}_AL)\cong \Ext_\Lambda^1({\rm Z}_A(\mathcal X), \ L) = 0$$
and that
$$ \Ext_B^1(\mathcal Y, \ \Ker\widetilde{g}) = \Ext_B^1(\mathcal Y, \ {\rm K}_BL) \cong \Ext_\Lambda^1({\rm Z}_B(\mathcal Y), \ L) = 0.$$
By definition, $L=\left(\begin{smallmatrix} L_1 \\ L_2\end{smallmatrix}\right)_{\widetilde{f},\widetilde{g}}\in \nabla(\mathcal X^\perp, \ \mathcal Y^\perp)$. This completes the proof.
\end{proof}
\vskip5pt
\begin{lem}\label{deltaher} \ \ Let $\Lambda = \left(\begin{smallmatrix} A & N \\
M & B\end{smallmatrix}\right)$ be a Morita ring with $M\otimes_A N = 0 = N\otimes_BM$.
\vskip5pt
$(1)$ \ Assume that $M_A$ and $N_B$ are flat modules.
Then \ $\Delta(\mathcal U, \ \mathcal V)$ is closed under the kernels of epimorphisms if and only if \ $\mathcal U$ and \ $\mathcal V$ are closed under the kernels of epimorphisms.
\vskip5pt
$(2)$ \ Assume that $_BM$ and $_AN$ are projective.
Then $\nabla(\mathcal X, \ \mathcal Y)$ is closed under the cokernels of monomorphisms if and only if \ $\mathcal X$ and \ $\mathcal Y$ are closed under the cokernels of monomorphisms.
\end{lem}
\begin{proof} \ $(1)$ \ Assume that \ $\mathcal U$ and \ $\mathcal V$ are closed under the kernels of epimorphisms.
Let \ $0\longrightarrow \left(\begin{smallmatrix} L_1 \\ L_2\end{smallmatrix}\right)_{f, g}
\longrightarrow \left(\begin{smallmatrix} M_1 \\ M_2 \end{smallmatrix}\right)_{u, v}\longrightarrow
\left(\begin{smallmatrix} N_1 \\ N_2 \end{smallmatrix}\right)_{s, t}\longrightarrow 0$ \
be an exact sequence with $\left(\begin{smallmatrix} M_1 \\ M_2 \end{smallmatrix}\right)_{u, v}, \ \left(\begin{smallmatrix} N_1 \\ N_2 \end{smallmatrix}\right)_{s, t}\in \Delta(\mathcal U, \ \mathcal V)$.
Thus $u, v, s, t$ are monomorphisms, $\Coker u \in \mathcal V, \ \Coker v\in \mathcal U, \ \Coker s\in \mathcal V$, and \ $\Coker t\in \mathcal U.$
Since $M_A$ is flat, one has the commutative diagram with exact rows:
$$\xymatrix@R=0.5cm{0\ar[r] & M\otimes_A L_1\ar[r]^-{1\otimes \alpha}\ar[d]_-{f} & M\otimes_A M_1 \ar[r]\ar[d]_-{u} & M\otimes_A N_1\ar[r]\ar[d]^-{s} & 0 \\
0\ar[r] & L_2\ar[r] & M_2 \ar[r]& N_2\ar[r] & 0.}$$
Since $1\otimes \alpha$ and $u$ are monomorphisms, so is $f$. By Snake Lemma and the assumption
that $\mathcal V$ is closed under the kernels of epimorphisms,
one knows that $\Coker f \in \mathcal V$. Similarly, $g$ is a monomorphism and $\Coker g \in \ \mathcal U$. By definition $\left(\begin{smallmatrix} L_1 \\ L_2 \end{smallmatrix}\right)_{f, g}\in \Delta(\mathcal U, \ \mathcal V)$.
This proves that \ $\Delta(\mathcal U, \ \mathcal V)$ is closed under the kernels of epimorphisms.
\vskip5pt
Conversely, using functors \ ${\rm T}_A$ and \ ${\rm T}_B$, one sees that
\ $\mathcal U$ and \ $\mathcal V$ are closed under the kernels of epimorphisms.
\vskip5pt
$(2)$ \ Assume that \ $\mathcal X$ and \ $\mathcal Y$ are closed under the cokernels of monomorphisms. Let \ $0\longrightarrow \left(\begin{smallmatrix} L_1 \\ L_2\end{smallmatrix}\right)_{f, g}
\longrightarrow \left(\begin{smallmatrix} M_1 \\ M_2 \end{smallmatrix}\right)_{u, v}\longrightarrow
\left(\begin{smallmatrix} N_1 \\ N_2 \end{smallmatrix}\right)_{s, t}\longrightarrow 0$
be an exact sequence of $\Lambda$-modules with $\left(\begin{smallmatrix} L_1 \\ L_2 \end{smallmatrix}\right)_{f, g}\in \nabla(\mathcal X, \ \mathcal Y)$ and \ $\left(\begin{smallmatrix} M_1 \\ M_2 \end{smallmatrix}\right)_{u, v}\in \nabla(\mathcal X, \ \mathcal Y)$.
Thus \ $\widetilde{f}, \ \widetilde{g}, \ \widetilde{u}, \ \widetilde{v}$ are epimorphisms, $\Ker \widetilde{f} \in \mathcal X, \ \Ker \widetilde{g}\in \mathcal Y, \ \Ker \widetilde{u} \in \mathcal X$, and \ $\Ker \widetilde{v}\in \mathcal Y.$
Since \ $_BM$ is projective, one has the commutative diagram with exact rows
$$\xymatrix@R=0.5cm{0\ar[r] & L_1\ar[r]\ar[d]_-{\widetilde{f}} & M_1 \ar[r]\ar[d]_-{\widetilde{u}} & N_1\ar[r]\ar[d]^-{\widetilde{s}} & 0 \\
0\ar[r] & \Hom_B(M, L_2)\ar[r] & \Hom_B(M, M_2) \ar[r]^-{(M, \beta)} & \Hom_B(M, N_2)\ar[r] & 0.}$$
Since $\widetilde{u}$ and $(M, \beta)$ are epimorphisms, so is $\widetilde{s}$. By Snake Lemma and the assumption that $\mathcal X$ is closed under taking the cokernels of monomorphisms,
one knows that $\Ker \widetilde{s} \in \mathcal X$. Similarly, $\widetilde{t}$ is an epimorphism and \ $\Ker \widetilde{t} \in \mathcal Y$. By definition $\left(\begin{smallmatrix} N_1 \\ N_2 \end{smallmatrix}\right)_{s, t}
\in \nabla(\mathcal X, \ \mathcal Y)$. This proves that \ $\nabla(\mathcal X, \ \mathcal Y)$ is closed under the cokernels of monomorphisms.
\vskip5pt
Conversely, using functors \ ${\rm H}_A$ and \ ${\rm H}_B$, one sees that
\ $\mathcal X$ and \ $\mathcal Y$ are closed under the cokernels of monomorphisms.
\end{proof}
\subsection{Proof of Theorem \ref{ctp6}} $(1)$ \ It suffices to prove \ $\Delta(\mathcal U, \ \mathcal V) = \ ^\perp(\Delta(\mathcal U, \ \mathcal V)^\perp).$
In fact, $\Delta(\mathcal U, \ \mathcal V) = \Delta(^\perp\mathcal X, \ ^\perp\mathcal Y)$.
Since \ $\mathcal X$ contains all the injective $A$-modules and \ $\mathcal Y$ contains all the injective $B$-modules, it follows from
Lemma \ref{desdelta}(1) that
$$\Delta(^\perp\mathcal X, \ ^\perp\mathcal Y) = \ ^\perp{\rm Z}_A(\mathcal X) \ \cap \ ^\perp{\rm Z}_B(\mathcal Y) =
\ ^\perp({\rm Z}_A(\mathcal X) \ \cup \ {\rm Z}_B(\mathcal Y)).$$
Thus
\begin{align*} ^\perp(\Delta(\mathcal U, \ \mathcal V)^\perp) & = \ ^\perp(\Delta(^\perp\mathcal X, \ ^\perp\mathcal Y)^\perp)
= \ ^\perp \{[^\perp({\rm Z}_A(\mathcal X) \ \cup \ {\rm Z}_B(\mathcal Y))]^\perp\} \\ &
= \ ^\perp ({\rm Z}_A(\mathcal X) \ \cup \ {\rm Z}_B(\mathcal Y)) = \Delta(\mathcal U, \ \mathcal V).
\end{align*}
\vskip5pt
By Lemma \ref{deltaher}(1), $\Delta(\mathcal U, \ \mathcal V)$ is closed under the kernels of epimorphisms if and only if
$\mathcal U$ and $\mathcal V$ are closed under the kernels of epimorphisms.
That is, \ $(\Delta(\mathcal U, \ \mathcal V), \ \Delta(\mathcal U, \ \mathcal V)^\perp)$ is hereditary if and only if $(\mathcal U, \ \mathcal X)$ and $(\mathcal V, \ \mathcal Y)$ are hereditary.
\vskip5pt
$(2)$ \ Similarly, it suffices to show \ $\nabla(\mathcal X, \ \mathcal Y) = \ (^{\perp}\nabla(\mathcal X, \ \mathcal Y))^\perp.$
In fact, $\nabla(\mathcal X, \ \mathcal Y) = \nabla(\mathcal U^\perp, \ \mathcal V^\perp)$.
Since \ $\mathcal U$ contains all the projective $A$-modules and \ $\mathcal V$ contains all the projective $B$-modules, it follows from
Lemma \ref{desdelta}(2) that
$$\nabla(\mathcal U^\perp, \ \mathcal V^\perp) = \ {\rm Z}_A(\mathcal U)^\perp \ \cap \ {\rm Z}_B(\mathcal V)^\perp =
({\rm Z}_A(\mathcal U) \ \cup \ {\rm Z}_B(\mathcal V))^\perp.$$
Thus
\begin{align*} (^{\perp}\nabla(\mathcal X, \ \mathcal Y))^\perp & = (^{\perp}\nabla(\mathcal U^\perp, \ \mathcal V^\perp))^\perp
= \{^\perp[({\rm Z}_A(\mathcal U) \ \cup \ {\rm Z}_B(\mathcal V))^\perp]\}^\perp \\ &
= ({\rm Z}_A(\mathcal U) \ \cup \ {\rm Z}_B(\mathcal V))^\perp = \nabla(\mathcal X, \ \mathcal Y).
\end{align*}
\vskip5pt
By Lemma \ref{deltaher}(2), \ $\nabla(\mathcal X, \ \mathcal Y)$ is
closed under the cokernels of monomorphisms if and only if $\mathcal X$ and $\mathcal Y$ are closed under the cokernels of monomorphisms. That is, \ $(^{\perp}\nabla(\mathcal X, \ \mathcal Y), \ \nabla(\mathcal X, \ \mathcal Y))$ is hereditary if and only if \ $(\mathcal U, \ \mathcal X)$ and $(\mathcal V, \ \mathcal Y)$ are hereditary. \hfill $\square$
\subsection{Proof of Theorem \ref{compare}}
$(1)$ \ By Theorem \ref{ctp1}(1), one has cotorsion pair \ $({}^\perp\left(\begin{smallmatrix}\mathcal X\\ \mathcal Y\end{smallmatrix}\right), \ \left(\begin{smallmatrix}\mathcal X\\ \mathcal Y\end{smallmatrix}\right))$;
and by Theorem \ref{ctp6}(1), one has cotorsion pair \ $(\Delta(\mathcal U, \ \mathcal V), \ \Delta(\mathcal U, \ \mathcal V)^\perp)$. We will prove $\Delta(\mathcal U, \ \mathcal V)^{\bot} \subseteq \left(\begin{smallmatrix}\mathcal X\\ \mathcal Y\end{smallmatrix}\right).$ By Lemma \ref{desdelta}(1) one has
$$\Delta(\mathcal U, \ \mathcal V)^{\bot} = [\Delta({}^\perp\mathcal X, \ {}^\perp\mathcal Y)]^{\bot} = [^\perp{\rm Z}_A(\mathcal X) \ \cap \ ^\perp{\rm Z}_B(\mathcal Y)]^{\bot}.$$
Since by assumption \ $\Tor^A_1(M, \ \mathcal U) =0 = \Tor^B_1(N, \ \mathcal V)$, it follows from Lemma \ref{destheta}(1) that
$$\left(\begin{smallmatrix} \mathcal X\\ \mathcal Y\end{smallmatrix}\right) = \left(\begin{smallmatrix} \mathcal U^\perp \\ \mathcal V^\perp \end{smallmatrix}\right)
= {\rm T}_A(\mathcal U)^\perp \ \cap \ {\rm T}_B(\mathcal V)^\perp = ({\rm T}_A(\mathcal U) \ \cup \ {\rm T}_B(\mathcal V))^\perp.$$
Thus, to show \ $\Delta(\mathcal U, \ \mathcal V)^{\bot} \subseteq \left(\begin{smallmatrix}\mathcal X\\ \mathcal Y\end{smallmatrix}\right),$ it suffices to show
$${\rm T}_A(\mathcal U) \ \cup \ {\rm T}_B(\mathcal V) \subseteq \ ^\perp{\rm Z}_A(\mathcal X)\ \cap \ ^\perp{\rm Z}_B(\mathcal Y).$$
In fact, since $N\otimes_BM = 0$, the structure map $g = 0$ of any $\Lambda$-module in ${\rm T}_A(\mathcal U)$ is a monomorphism,
it follows from Lemma \ref{extadj2}(1) that
$$\Ext^1_{\Lambda}({\rm T}_A(\mathcal U), \ {\rm Z}_A(\mathcal X)) \cong \Ext^1_A({\rm C}_A{\rm T}_A(\mathcal U), \ \mathcal X) = \Ext^1_A(\mathcal U, \ \mathcal X) = 0.$$
By Lemma \ref{extadj2}(2) one has
$$\Ext^1_{\Lambda}({\rm T}_A(\mathcal U), \ {\rm Z}_B(\mathcal Y)) \cong \Ext^1_A({\rm C}_B{\rm T}_A(\mathcal U), \ \mathcal Y) = 0$$
since ${\rm C}_B{\rm T}_A = 0$.
So ${\rm T}_A(\mathcal U)\subseteq \ ^\perp{\rm Z}_A(\mathcal X)\ \cap \ ^\perp{\rm Z}_B(\mathcal Y)$.
\vskip5pt
Similarly, by Lemma \ref{extadj2}(1) one has
$$\Ext^1_{\Lambda}({\rm T}_B(\mathcal V), \ {\rm Z}_A(\mathcal X)) \cong \Ext^1_A({\rm C}_A{\rm T}_B(\mathcal V), \ \mathcal X) = 0$$
since ${\rm C}_A{\rm T}_B = 0.$ Since $M\otimes_AN = 0$, the structure map $f=0$ of any $\Lambda$-module in ${\rm T}_B(\mathcal V)$ is a monomorphism,
it follows from Lemma \ref{extadj2}(2) that
$$\Ext^1_{\Lambda}({\rm T}_B(\mathcal V), \ {\rm Z}_B(\mathcal Y)) \cong \Ext^1_A({\rm C}_B{\rm T}_B(\mathcal V), \ \mathcal Y) = \Ext^1_A(\mathcal V, \ \mathcal Y) = 0.$$
So ${\rm T}_B(\mathcal V)\subseteq \ ^\perp{\rm Z}_A(\mathcal X)\ \cap \ ^\perp{\rm Z}_B(\mathcal Y)$. This completes the proof of (1).
\vskip5pt
(2) \ Comparing cotorsion pair \ $(\left(\begin{smallmatrix} \mathcal U\\ \mathcal V\end{smallmatrix}\right), \ \left(\begin{smallmatrix} \mathcal U\\ \mathcal V\end{smallmatrix}\right)^\perp)$ in Theorem \ref{ctp1}(2)
with \ $(^{\perp}\nabla(\mathcal X, \ \mathcal Y), \ \nabla(\mathcal X, \ \mathcal Y))$ in Theorem \ref{ctp6}(2), we will prove \ $^{\bot}\nabla(\mathcal X, \ \mathcal Y) \subseteq \left(\begin{smallmatrix}\mathcal U\\ \mathcal V\end{smallmatrix}\right).$ This can be similarly done as (1). For convenience we include a brief justification. By Lemma \ref{desdelta}(2) one has
$$^{\bot}\nabla(\mathcal X, \ \mathcal Y) = \ ^{\bot}\nabla(\mathcal U^\perp, \ \mathcal V^\perp) = \ ^{\bot}[{\rm Z}_A(\mathcal U)^\perp \ \cap \ {\rm Z}_B(\mathcal V)^\perp].$$
By Lemma \ref{destheta}(2), one has
$$\left(\begin{smallmatrix} \mathcal U\\ \mathcal V\end{smallmatrix}\right) = \left(\begin{smallmatrix} ^{\bot}\mathcal X \\ ^{\bot} \mathcal Y\end{smallmatrix}\right) = \ ^{\bot}[{\rm H}_A(\mathcal X) \ \cup \ {\rm H}_B(\mathcal Y)].$$
So, it suffices to show \ ${\rm H}_A(\mathcal X) \ \cup \ {\rm H}_B(\mathcal Y) \subseteq {\rm Z}_A(\mathcal U)^\perp \ \cap \ {\rm Z}_B(\mathcal V)^\perp.$
\vskip5pt
In fact, since \ $\Ext_A^1(N, \ \mathcal X)=0$, it follows from Lemma \ref{extadj1}(3) that
$$\Ext^1_{\Lambda}({\rm Z}_A(\mathcal U), \ {\rm H}_A(\mathcal X)) \cong \Ext^1_A({\rm U}_A{\rm Z}_A(\mathcal U), \ \mathcal X) = \Ext^1_A(\mathcal U, \ \mathcal X) = 0$$
and
$$\Ext^1_{\Lambda}({\rm Z}_B(\mathcal V), \ {\rm H}_A(\mathcal X)) \cong \Ext^1_A({\rm U}_A{\rm Z}_B(\mathcal V), \ \mathcal X) = 0.$$
Thus ${\rm H}_A(\mathcal X)\subseteq {\rm Z}_A(\mathcal U)^\perp \ \cap \ {\rm Z}_B(\mathcal V)^\perp$.
\vskip5pt
Since \ $\Ext_B^1(M, \ \mathcal Y)=0$, it follows from Lemma \ref{extadj1}(4) that
$$\Ext^1_{\Lambda}({\rm Z}_A(\mathcal U), \ {\rm H}_B(\mathcal Y)) \cong \Ext^1_A({\rm U}_B{\rm Z}_A(\mathcal U), \ \mathcal Y) = 0$$
and
$$\Ext^1_{\Lambda}({\rm Z}_B(\mathcal V), \ {\rm H}_B(\mathcal Y)) \cong \Ext^1_A({\rm U}_B{\rm Z}_B(\mathcal V), \ \mathcal Y) = \Ext^1_A(\mathcal V, \ \mathcal Y) = 0,$$
which show ${\rm H}_B(\mathcal Y)\subseteq {\rm Z}_A(\mathcal U)^\perp \ \cap \ {\rm Z}_B(\mathcal V)^\perp$.
This completes the proof. \hfill $\square$
\section{\bf Identifications}
The aim of this section is, on one hand, to prove that the four constructions of cotorsion pairs, given in Theorem \ref{ctp1} and Theorem \ref{ctp6},
are pairwise generally different; and on the other hand, to study the problem of identifications, i.e.,
we will show that, in many important cases, the cotorsion pairs in Series I coincide with the corresponding ones in Series II. Then we will get cotorsion pairs
$$(\Delta(\mathcal U, \ \mathcal V), \ \left(\begin{smallmatrix}\mathcal X\\ \mathcal Y\end{smallmatrix}\right))
\ \ \ \ \mbox{and} \ \ \ \ (\left(\begin{smallmatrix} \mathcal U\\ \mathcal V\end{smallmatrix}\right), \ \nabla(\mathcal X, \ \mathcal Y)).$$
Since the both cotorsion pairs are explicitly given,
they will be greatly helpful in finding Hovey triples, i.e., the abelian model structures on Morita rings.
\subsection{Generally different cotorsion pairs}
Mainly for the convenience in Section 6, we introduce the following notion.
\begin{defn} \label{difference} \ Let \ $\Omega$ be a class of Morita rings, $(\mathcal X, \ \mathcal Y)$ and \ $(\mathcal X', \ \mathcal Y')$ cotorsion pairs defined in $\Lambda\mbox{-}{\rm Mod}$, for arbitrary
Morita rings $\Lambda\in \Omega$.
We say that \ $(\mathcal X, \ \mathcal Y)$ and \ $(\mathcal X', \ \mathcal Y')$ are generally different, provided that there exist $\Lambda\in \Omega$, such that \ $(\mathcal X, \ \mathcal Y) \ne (\mathcal X', \ \mathcal Y')$ in $\Lambda\mbox{-}{\rm Mod}$.
\end{defn}
\begin{exm} \label{gdsame} Generally different cotorsion pairs could be the same for some special Morita rings.
\vskip5pt
For example,
$(_\Lambda\mathcal P, \ \Lambda\mbox{\rm-Mod})$ and $(\binom{_A\mathcal P}{_B\mathcal P}, \ \binom{_A\mathcal P}{_B\mathcal P}^\perp)$ are cotorsion pairs in $\Lambda\mbox{-}{\rm Mod}$, \ $\forall \ \Lambda\in \Omega$, where
\ $\Omega = \{ \mbox{Morita ring} \ \Lambda = \left(\begin{smallmatrix}A & N \\ M & B\end{smallmatrix}\right) \ | \ \phi= 0 = \psi, \ _BN \ \mbox{and} \ _AM \ \mbox{are projective}\}.$
If $M \ne 0$, then \ $\binom{A}{0}_{0, 0}\notin \ _\Lambda\mathcal P$. Thus \ $_\Lambda\mathcal P\ne \binom{_A\mathcal P}{_B\mathcal P}$ for
$\Lambda\in \Omega$ with $M\ne 0$. Hence
$(_\Lambda\mathcal P, \ \Lambda\mbox{\rm-Mod})$ and $(\binom{_A\mathcal P}{_B\mathcal P}, \ \binom{_A\mathcal P}{_B\mathcal P}^\perp)$ are generally different cotorsion pairs.
But they are the same for $\Lambda\in \Omega$ with $M = 0 = N$.
\end{exm}
\subsection{The four cotorsion pairs are pairwise generally different} \ By Theorems \ref{ctp1} and \ref{ctp6}, the four kinds of cotorsion pairs
$$(^\perp\left(\begin{smallmatrix}\mathcal X\\ \mathcal Y\end{smallmatrix}\right), \ \left(\begin{smallmatrix}\mathcal X\\ \mathcal Y\end{smallmatrix}\right)), \ \ \ \
(\Delta(\mathcal U, \ \mathcal V), \ \Delta(\mathcal U, \ \mathcal V)^\perp)$$
and
$$(\left(\begin{smallmatrix} \mathcal U\\ \mathcal V\end{smallmatrix}\right), \ \left(\begin{smallmatrix} \mathcal U\\ \mathcal V\end{smallmatrix}\right)^\perp), \ \ \ \ (^{\perp}\nabla(\mathcal X, \ \mathcal Y), \ \nabla(\mathcal X, \ \mathcal Y))$$
are defined in $\Lambda$-Mod, \ $\forall \ \Lambda\in \Omega$, where $$\Omega = \{\Lambda = \left(\begin{smallmatrix} A & N \\
M & A\end{smallmatrix}\right) \ | \ M\otimes_AN = 0 = N\otimes_BM, \ M_A \ \mbox{and} \ N_B \ \mbox{are flat}, \ _BM \ \mbox{and} \ _AN \ \mbox{are projective}\}.$$
We will show that they are pairwise generally different. For convenience, we will call the cotorsion pairs above the first, the second, the third, and the fourth cotorsion pairs.
\begin{exm} \label{ie} Let $A = B$ be the path algebra $k(1 \longrightarrow 2)$, where ${\rm char} \ k\ne 2$. Write the conjunction of paths from right to left. Thus $e_1Ae_2 = 0$ and $e_2Ae_1 \cong k$. Take $M = N = Ae_2\otimes_ke_1A$. Then $M\otimes_AN = 0 = N\otimes_AM$. Let $\Lambda$ be the Morita ring $\left(\begin{smallmatrix} A & N \\
M & A\end{smallmatrix}\right).$ Then $\Lambda\in \Omega.$
\vskip5pt
Note that \ $_AM = \ _AN$ is isomorphic to the simple projective left $A$-module $Ae_2 = S_2$, and that \ $M_A = N_A$ is isomorphic to the simple projective right $A$-module $e_1A$. Then \ $M\otimes_AAe_1\cong Ae_2\otimes_k (e_1A\otimes_AAe_1) \cong S_2$. To see the left $A$-module structure on $\Hom_A(M, Ae_1)$, note that
$\Hom_A(M, Ae_1)\cong \Hom_A(Ae_2, Ae_1) \cong e_2A e_1 \cong k$ as $k$-spaces. For $f\in \Hom_A(M, Ae_1)$ given by $f(e_2\otimes_k e_1) = e_1$, one has $e_1f = f$. Thus $\Hom_A(M, Ae_1)\cong S_1$ as left $A$-modules. The Auslander-Reiten quiver of $A$ is
$$\xymatrix@R=0.3cm@C=0.6cm{& Ae_1\ar[dr]^-\pi
\\ S_2\ar[ur]^-\sigma & & S_1}$$
\vskip5pt
Take \ $(\mathcal U, \mathcal X) = (A\mbox{-}{\rm Mod}, \ _A\mathcal I) = (\mathcal V, \ \mathcal Y)$. Note that $M\otimes_A\mathcal U \nsubseteq \mathcal Y, \ N\otimes_B\mathcal V \nsubseteq \mathcal X.$
Take \ $L = \left(\begin{smallmatrix}Ae_1\\ Ae_1\end{smallmatrix}\right)_{\sigma, \sigma}$.
Then $L\in {\rm Mon(\Lambda)} = \Delta(A\mbox{-}{\rm Mod}, \ A\mbox{-}{\rm Mod}) = \Delta(\mathcal U, \ \mathcal V)$ and $L\in \left(\begin{smallmatrix}_A\mathcal I\\ _A\mathcal I\end{smallmatrix}\right) =
\left(\begin{smallmatrix}\mathcal X\\ \mathcal Y\end{smallmatrix}\right).$
Consider the exact sequence of $\Lambda$-modules
$$0\longrightarrow \left(\begin{smallmatrix}Ae_1\\ Ae_1\end{smallmatrix}\right)_{\sigma, \sigma}
\stackrel{\left(\begin{smallmatrix} \binom{1}{0}\\ \binom{1}{0}\end{smallmatrix}\right)}\longrightarrow \left(\begin{smallmatrix} Ae_1\oplus Ae_1\\ Ae_1\oplus Ae_1\end{smallmatrix}\right)_{\left(\begin{smallmatrix}\sigma & \sigma\\ 0 & \sigma\end{smallmatrix}\right), \left(\begin{smallmatrix}\sigma & \sigma\\ 0 & \sigma\end{smallmatrix}\right)}
\stackrel{\left(\begin{smallmatrix} (0, 1) \\ (0,1)\end{smallmatrix}\right)}
\longrightarrow \left(\begin{smallmatrix}Ae_1\\ Ae_1\end{smallmatrix}\right)_{\sigma, \sigma}\longrightarrow 0.$$
This exact sequence does not split. In fact, if it splits, then
there is a $\Lambda$-map $\left(\begin{smallmatrix} (a,b) \\ (c,d)\end{smallmatrix}\right): \left(\begin{smallmatrix} Ae_1\oplus Ae_1\\ Ae_1\oplus Ae_1\end{smallmatrix}\right)_{\left(\begin{smallmatrix}\sigma & \sigma\\ 0 & \sigma\end{smallmatrix}\right), \left(\begin{smallmatrix}\sigma & \sigma\\ 0 & \sigma\end{smallmatrix}\right)}\longrightarrow \left(\begin{smallmatrix}Ae_1\\ Ae_1\end{smallmatrix}\right)_{\sigma, \sigma}$ such that
$\left(\begin{smallmatrix} (a,b) \\ (c,d)\end{smallmatrix}\right) \left(\begin{smallmatrix} \binom{1}{0}\\ \binom{1}{0}\end{smallmatrix}\right) = \left(\begin{smallmatrix} 1 \\ 1\end{smallmatrix}\right),$
i.e., $a = 1=c.$ Since the following diagrams
$$\xymatrix@R= 0.7cm{S_2\oplus S_2\ar[d]_-{\left(\begin{smallmatrix}\sigma & \sigma\\ 0 & \sigma\end{smallmatrix}\right)}\ar[r]^-{(1, b)} & S_2 \ar[d]^-{\sigma} \\
Ae_1\oplus Ae_1 \ar[r]^-{(1, d)} & Ae_1}\qquad \qquad \qquad
\xymatrix@R= 0.7cm{S_2\oplus S_2\ar[d]_-{\left(\begin{smallmatrix}\sigma & \sigma\\ 0 & \sigma\end{smallmatrix}\right)}\ar[r]^-{(1, d)} & S_2 \ar[d]^-{\sigma} \\
Ae_1\oplus Ae_1 \ar[r]^-{(1, b)} & Ae_1}$$
commute, \ $d+1 = b$ and $b+1 = d$, which is a contradiction, since ${\rm char} \ k\ne 2$.
\vskip5pt
Thus \ ${\rm Ext}^1_\Lambda(L, L)\ne 0$. This means \ $L\notin \ ^\perp\left(\begin{smallmatrix}_A\mathcal I\\ _A\mathcal I\end{smallmatrix}\right)$. Since $L\in {\rm Mon(\Lambda)}$,
\ $^\perp\left(\begin{smallmatrix}_A\mathcal I\\ _A\mathcal I\end{smallmatrix}\right) \ne {\rm Mon(\Lambda)}$.
Thus, the first cotorsion pair
is not equal to the second one, i.e.,
$$(^\perp\left(\begin{smallmatrix}\mathcal X\\ \mathcal Y\end{smallmatrix}\right), \ \left(\begin{smallmatrix}\mathcal X\\ \mathcal Y\end{smallmatrix}\right)) = ({}^\perp\left(\begin{smallmatrix}_A\mathcal I\\ _A\mathcal I\end{smallmatrix}\right), \ \left(\begin{smallmatrix}_A\mathcal I\\ _A\mathcal I\end{smallmatrix}\right)) \ne
({\rm Mon}(\Lambda), \ {\rm Mon}(\Lambda)^\bot) = (\Delta(\mathcal U, \ \mathcal V), \ \Delta(\mathcal U, \ \mathcal V)^\perp).$$
\vskip5pt
Since $\left(\begin{smallmatrix} \mathcal U\\ \mathcal V\end{smallmatrix}\right) = \left(\begin{smallmatrix} A\mbox{-}{\rm Mod}\\ A\mbox{-}{\rm Mod}\end{smallmatrix}\right) = \Lambda\mbox{-}{\rm Mod}$,
it follows that $(\left(\begin{smallmatrix} \mathcal U\\ \mathcal V\end{smallmatrix}\right), \ \left(\begin{smallmatrix} \mathcal U\\ \mathcal V\end{smallmatrix}\right)^\perp) = (\Lambda\mbox{-}{\rm Mod}, \ _\Lambda\mathcal I).$
Since \ $L\notin \ ^\perp\left(\begin{smallmatrix}_A\mathcal I\\ _A\mathcal I\end{smallmatrix}\right)$, the first cotorsion pair
is not equal to the third one:
$$(^\perp\left(\begin{smallmatrix}\mathcal X\\ \mathcal Y\end{smallmatrix}\right), \ \left(\begin{smallmatrix}\mathcal X\\ \mathcal Y\end{smallmatrix}\right)) = ({}^\perp\left(\begin{smallmatrix}_A\mathcal I\\ _A\mathcal I\end{smallmatrix}\right), \ \left(\begin{smallmatrix}_A\mathcal I\\ _A\mathcal I\end{smallmatrix}\right)) \ne
(\Lambda\mbox{-}{\rm Mod}, \ _\Lambda\mathcal I) = (\left(\begin{smallmatrix} \mathcal U\\ \mathcal V\end{smallmatrix}\right), \ \left(\begin{smallmatrix} \mathcal U\\ \mathcal V\end{smallmatrix}\right)^\perp).$$
\vskip5pt
Since $\left(\begin{smallmatrix} Ae_1\\ 0\end{smallmatrix}\right)\notin {\rm Mon}(\Lambda) = \Delta(\mathcal U, \ \mathcal V)$, the second cotorsion pair
is not equal to the third one:
$$(\Delta(\mathcal U, \ \mathcal V), \ \Delta(\mathcal U, \ \mathcal V)^\perp)
\ne
(\Lambda\mbox{-}{\rm Mod}, \ _\Lambda\mathcal I) = (\left(\begin{smallmatrix} \mathcal U\\ \mathcal V\end{smallmatrix}\right), \ \left(\begin{smallmatrix} \mathcal U\\ \mathcal V\end{smallmatrix}\right)^\perp).$$
\vskip5pt
By definition $\nabla(\mathcal X, \ \mathcal Y) = \nabla(_A\mathcal I, \ _A\mathcal I) = \ _\Lambda \mathcal I$. Thus
$(^{\perp}\nabla(\mathcal X, \ \mathcal Y), \ \nabla(\mathcal X, \ \mathcal Y)) = (\Lambda\mbox{-}{\rm Mod}, \ _\Lambda \mathcal I)$, i.e., the fourth cotorsion pair is exactly
third cotorsion pair. Therefore, the first cotorsion pair is not equal to the fourth one, and the second cotorsion pair
is not equal to the fourth one.
\vskip5pt
Finally, to see the third cotorsion pair is not equal to the fourth one, namely,
$$(\left(\begin{smallmatrix} \mathcal U\\ \mathcal V\end{smallmatrix}\right), \ \left(\begin{smallmatrix} \mathcal U\\ \mathcal V\end{smallmatrix}\right)^\perp)\ne (^{\perp}\nabla(\mathcal X, \ \mathcal Y), \ \nabla(\mathcal X, \ \mathcal Y))$$
we take $(\mathcal U, \mathcal X) = (_A\mathcal P, \ A\mbox{-}{\rm Mod}) = (\mathcal V, \ \mathcal Y)$.
Note that \
$\Hom_B(M, \ \mathcal Y)\nsubseteq \mathcal U, \ \Hom_A(N, \ \mathcal X) \nsubseteq \mathcal V.$
\vskip5pt
Take $L = \left(\begin{smallmatrix}Ae_1\\ Ae_1\end{smallmatrix}\right)_{\sigma, \sigma}$ as above.
Then $L\in \left(\begin{smallmatrix}_A\mathcal P\\ _A\mathcal P\end{smallmatrix}\right) =
\left(\begin{smallmatrix}\mathcal U\\ \mathcal V\end{smallmatrix}\right).$
Since $\widetilde{\sigma}: Ae_1 \longrightarrow \Hom_A(M, Ae_1)\cong S_1$ is exactly the epimorphism $\pi: Ae_1\longrightarrow S_1$, by definition
$L\in {\rm Eip(\Lambda)} = \nabla (A\mbox{-}{\rm Mod}, \ A\mbox{-}{\rm Mod}) = \nabla(\mathcal X, \ \mathcal Y)$.
Since \ ${\rm Ext}^1_\Lambda(L, L)\ne 0$, \ $L\notin \left(\begin{smallmatrix}_A\mathcal P\\ _A\mathcal P\end{smallmatrix}\right)^\perp$. Thus
$\nabla(\mathcal X, \ \mathcal Y)\ne \left(\begin{smallmatrix}_A\mathcal P\\ _A\mathcal P\end{smallmatrix}\right)^\perp$, and hence the third cotorsion pair is not equal to the fourth one:
$$(\left(\begin{smallmatrix} \mathcal U\\ \mathcal V\end{smallmatrix}\right), \ \left(\begin{smallmatrix} \mathcal U\\ \mathcal V\end{smallmatrix}\right)^\perp) = (\left(\begin{smallmatrix}_A\mathcal P\\ _A\mathcal P\end{smallmatrix}\right), \ \left(\begin{smallmatrix}_A\mathcal P\\ _A\mathcal P\end{smallmatrix}\right)^\perp)\ne (^{\perp}\nabla(\mathcal X, \ \mathcal Y), \ \nabla(\mathcal X, \ \mathcal Y))$$
All together, we have proved that the four cotorsion pairs are pairwise generally different.
In fact, we have found an example $\Lambda$, such that the four constructions of cotorsion pairs in $\Lambda$-Mod
are pairwise different.
\end{exm}
\subsection{Main results on identification}
\begin{thm}\label{identify1} Let \ $\Lambda = \left(\begin{smallmatrix} A & N \\ M & B \end{smallmatrix}\right)$ be a Morita ring with \ $M\otimes_A N = 0 = N\otimes_BM$, \ $(\mathcal U, \ \mathcal X)$ and $(\mathcal V, \ \mathcal Y)$ be cotorsion pairs in $A\mbox{-}{\rm Mod}$ and in $B$\mbox{\rm-Mod}, respectively.
\vskip5pt
$(1)$ \ Assume that \ $\Tor^A_1(M, \ \mathcal U) =0 = \Tor^B_1(N, \ \mathcal V)$. If
$M\otimes_A\mathcal U \subseteq \mathcal Y$ or \ $N\otimes_B\mathcal V \subseteq \mathcal X$, then
$$(\Delta(\mathcal U, \ \mathcal V), \ \Delta(\mathcal U, \ \mathcal V)^\bot) =
(^\perp\left(\begin{smallmatrix} \mathcal X\\ \mathcal Y\end{smallmatrix}\right), \ \left(\begin{smallmatrix} \mathcal X \\ \mathcal Y\end{smallmatrix}\right)).$$
Thus \ \ $(\Delta(\mathcal U, \ \mathcal{V}), \ \left(\begin{smallmatrix} \mathcal X\\ \mathcal{Y}\end{smallmatrix}\right))$ \ is a cotorsion pair in $\Lambda\mbox{-}{\rm Mod}$.
\vskip5pt
Moreover, if $M\otimes_A\mathcal U \subseteq \mathcal Y$ and \ $N\otimes_B\mathcal V \subseteq \mathcal X$, then
\ $\Delta(\mathcal U, \ \mathcal V) = {\rm T}_A(\mathcal U)\oplus {\rm T}_B(\mathcal V).$
\vskip10pt
$(2)$ \ Assume that \ $\Ext_B^1(M, \ \mathcal Y) =0 = \Ext_A^1(N, \ \mathcal X)$. If \
$\Hom_B(M, \ \mathcal Y)\subseteq \mathcal U$ or \ $\Hom_A(N, \ \mathcal X) \subseteq \mathcal V$, then
$$(^\perp\nabla(\mathcal X, \ \mathcal Y), \ \nabla(\mathcal X, \ \mathcal Y)) =
(\left(\begin{smallmatrix} \mathcal U\\ \mathcal V\end{smallmatrix}\right), \ \left(\begin{smallmatrix} \mathcal U \\ \mathcal V\end{smallmatrix}\right)^\perp).$$
Thus \ \ $(\left(\begin{smallmatrix} \mathcal U \\ \mathcal V\end{smallmatrix}\right), \ \nabla(\mathcal X, \ \mathcal Y))$ \ is a cotorsion pair in $\Lambda\mbox{-}{\rm Mod}$.
\vskip5pt
Moreover, if \ $\Hom_B(M, \ \mathcal Y)\subseteq \mathcal U$ and \ $\Hom_A(N, \ \mathcal X) \subseteq \mathcal V$, then
\ $\nabla(\mathcal X, \ \mathcal Y)
={\rm H}_A(\mathcal X)\oplus {\rm H}_B(\mathcal Y).$
\end{thm}
\subsection{Applications} In Theorem \ref{identify1}, taking one of $(\mathcal U, \ \mathcal X)$ and
$(\mathcal V, \ \mathcal Y)$ being the projective cotorsion pair or the injective cotorsion pair,
and another being an arbitrary cotorsion pair, one has
\begin{cor}\label{identification1} \ Let \ $\Lambda = \left(\begin{smallmatrix} A & N \\ M & B \end{smallmatrix}\right)$ be a Morita ring with \ $M\otimes_A N = 0 = N\otimes_BM$.
\vskip10pt
$(1)$ \ If \ $N_B$ is flat, then for any
cotorsion pair \ $(\mathcal V, \ \mathcal Y)$ in $B\mbox{-}{\rm Mod}$
one has
$$(\Delta(_A\mathcal P, \ \mathcal V), \ \Delta(_A\mathcal P, \ \mathcal V)^\bot) =
(^\perp\left(\begin{smallmatrix} A\text{\rm\rm-Mod}\\ \mathcal Y\end{smallmatrix}\right), \ \left(\begin{smallmatrix} A\text{\rm-Mod} \\ \mathcal Y\end{smallmatrix}\right)).$$
Thus \ \ $(\Delta(_A\mathcal P, \ \mathcal{V}), \ \left(\begin{smallmatrix} A\text{\rm-Mod}\\ \mathcal{Y}\end{smallmatrix}\right))$ \ is a cotorsion pair in $\Lambda\mbox{-}{\rm Mod}$.
\vskip5pt
Moreover, if \ $M\otimes_A \mathcal P \subseteq \mathcal Y$ $($e.g., this is the case if $B$ is left noetherian and $_BM$ is injective$)$,
then
\ $\Delta(_A\mathcal P, \ \mathcal V) = {\rm T}_A(_A\mathcal P)\oplus {\rm T}_B(\mathcal V).$
\vskip5pt
$(2)$ \ If $M_A$ is flat, then for any
cotorsion pair \ $(\mathcal U, \ \mathcal X)$ in $A\mbox{-}{\rm Mod}$ one has
$$(\Delta(\mathcal U, \ _B\mathcal P), \ \Delta(\mathcal U, \ _B\mathcal P)^\bot) =
({}^\perp\left(\begin{smallmatrix}\mathcal X\\ B\mbox{-}{\rm Mod}\end{smallmatrix}\right), \ \left(\begin{smallmatrix}\mathcal X\\ B\mbox{-}{\rm Mod}\end{smallmatrix}\right)).$$
Thus \ $(\Delta(\mathcal{U}, \ _B\mathcal P), \ \binom{\mathcal{X}}{B\text{\rm-Mod}})$ is a cotorsion pair in $\Lambda\mbox{-}{\rm Mod}$.
\vskip5pt
Moreover, if \ $N\otimes_B \mathcal P\subseteq \mathcal X$ $($e.g., this is the case if \ $A$ is left noetherian and $_AN$ is injective$)$, then
\ $\Delta(\mathcal U, \ _B\mathcal P)= {\rm T}_A(\mathcal U)\oplus {\rm T}_B(_B\mathcal P).$
\vskip5pt
$(3)$ \ If \ $_BM$ is projective, then for any cotorsion pair $(\mathcal V, \ \mathcal Y)$ in \ $B\mbox{-}{\rm Mod}$ one has
$$(^\bot\nabla(_A\mathcal I, \ \mathcal Y), \ \nabla(_A\mathcal I, \ \mathcal Y))
= (\left(\begin{smallmatrix} A\mbox{-}{\rm Mod}\\ \mathcal V\end{smallmatrix}\right), \ \left(\begin{smallmatrix}A\mbox{-}{\rm Mod}\\ \mathcal V\end{smallmatrix}\right)^\perp).$$
Thus \ $(\binom{A\text{\rm-Mod}}{_B\mathcal V}, \ \nabla(_A\mathcal I, \ \mathcal Y))$ is a cotorsion pair in $\Lambda\mbox{-}{\rm Mod}$.
\vskip5pt
Moreover, if \ $\Hom_A(N, \ _A\mathcal I)\subseteq \mathcal V$ \ $($e.g., this is the case if \ $B$ is quasi-Frobenius and $N_B$ is flat$)$, then
\ $\nabla(_A\mathcal I, \ \mathcal Y) ={\rm H}_A(_A\mathcal I)\oplus {\rm H}_B(\mathcal Y).$
\vskip5pt
$(4)$ \ If $_AN$ is projective, then for any cotorsion pair
$(\mathcal U, \ \mathcal X)$ in $A\mbox{-}{\rm Mod}$ one has
$$(^\bot\nabla(\mathcal X, \ _B\mathcal I), \ \nabla(\mathcal X, \ _B\mathcal I)) = (\left(\begin{smallmatrix}\mathcal U\\ B\mbox{-}{\rm Mod}\end{smallmatrix}\right), \ \left(\begin{smallmatrix}\mathcal U\\ B\mbox{-}{\rm Mod}\end{smallmatrix}\right)^\perp).$$
Thus $(\binom{\mathcal U}{B\text{\rm-Mod}}, \ \nabla(_A\mathcal X, \ _B\mathcal I))$ is a cotorsion pair in $\Lambda\mbox{-}{\rm Mod}$.
\vskip5pt
Moreover, if \ $\Hom_B(M, \ _B\mathcal I)\subseteq \mathcal U$ \ $($e.g., this is the case if \ $A$ is quasi-Frobenius and $M_A$ is flat$)$, then \ $\nabla(\mathcal X, \ _B\mathcal I) = {\rm H}_A(\mathcal X)\oplus {\rm H}_B(_B\mathcal I).$
\end{cor}
\begin{proof}
(1) \ Taking \ $(\mathcal U, \ \mathcal X) = (_A\mathcal P, \ A\mbox{-}{\rm Mod})$ in Theorem \ref{identify1}(1). Then
\ $N\otimes_B\mathcal V \subseteq A\mbox{-}{\rm Mod} = \mathcal X$. By Theorem \ref{identify1}(1) one has
\ $(\Delta(_A\mathcal P, \ \mathcal V), \ \Delta(_A\mathcal P, \ \mathcal V)^\bot) =
(^\perp\left(\begin{smallmatrix} A\text{\rm\rm-Mod}\\ \mathcal Y\end{smallmatrix}\right), \ \left(\begin{smallmatrix} A\text{\rm-Mod} \\ \mathcal Y\end{smallmatrix}\right)).$
\vskip5pt
If $M\otimes_A \mathcal P \subseteq \mathcal Y$, i.e.,
$M\otimes_A\mathcal U = M\otimes_A\mathcal P \subseteq \mathcal Y$,
then by Theorem \ref{identify1}(1),
$\Delta(_A\mathcal P, \ \mathcal V) = {\rm T}_A(_A\mathcal P)\oplus {\rm T}_B(\mathcal V).$
\vskip5pt
Assume that \ $B$ is left noetherian and $_BM$ is injective. For any $P\in \ _A\mathcal P$,
as a left $B$-module, \ $M\otimes_AP$ is a direct summand of
a direct sum of copies of $_BM$. Since $_BM$ is injective and $B$ is left noetherian,
$M\otimes_AP$ is an injective left $B$-module, and hence it is in \ $\mathcal Y$. Thus
$M\otimes_A\mathcal P \subseteq \mathcal Y$, and hence
$\Delta(_A\mathcal P, \ \mathcal V) = {\rm T}_A(_A\mathcal P)\oplus {\rm T}_B(\mathcal V),$ by Theorem \ref{identify1}(1).
\vskip10pt
(2) \ Taking \ $(\mathcal V, \ \mathcal Y) = (_B\mathcal P, \ B\mbox{-}{\rm Mod})$ in Theorem \ref{identify1}(1). Then
\ $M\otimes_A\mathcal U \subseteq B\mbox{-}{\rm Mod} = \mathcal Y$. By Theorem \ref{identify1}(1) one has
\ $(\Delta(\mathcal U, \ _B\mathcal P), \ \Delta(\mathcal U, \ _B\mathcal P)^\bot) =
({}^\perp\left(\begin{smallmatrix}\mathcal X\\ B\mbox{-}{\rm Mod}\end{smallmatrix}\right), \ \left(\begin{smallmatrix}\mathcal X\\ B\mbox{-}{\rm Mod}\end{smallmatrix}\right)).$
\vskip5pt
If $N\otimes_B\mathcal P\subseteq \mathcal X$, then by Theorem \ref{identify1}(1),
$\Delta(\mathcal U, \ _B\mathcal P)= {\rm T}_A(\mathcal U)\oplus {\rm T}_B(_B\mathcal P).$
\vskip5pt
Assume that \ $A$ is left noetherian and $_AN$ is injective. Then
\ $N\otimes_B\mathcal P \subseteq \ _A\mathcal I \subseteq \mathcal X,$ and hence
$\Delta(\mathcal U, \ _B\mathcal P)= {\rm T}_A(\mathcal U)\oplus {\rm T}_B(_B\mathcal P).$
\vskip10pt
(3) \ Taking \ $(\mathcal U, \ \mathcal X) = (A\mbox{-}{\rm Mod}, \ _A\mathcal I)$ in Theorem \ref{identify1}(2). Then
\ $\Hom_B(M, \ \mathcal Y) \subseteq A\mbox{-}{\rm Mod} = \mathcal U.$
By Theorem \ref{identify1}(2) one has
\ $(^\bot\nabla(_A\mathcal I, \ \mathcal Y), \ \nabla(_A\mathcal I, \ \mathcal Y))
= (\left(\begin{smallmatrix} A\mbox{-}{\rm Mod}\\ \mathcal V\end{smallmatrix}\right), \ \left(\begin{smallmatrix}A\mbox{-}{\rm Mod}\\ \mathcal V\end{smallmatrix}\right)^\perp).$
\vskip5pt If \ $\Hom_A(N, \ _A\mathcal I)\subseteq \mathcal V$,
then by Theorem \ref{identify1}(2), $\nabla(_A\mathcal I, \ \mathcal Y) ={\rm H}_A(_A\mathcal I)\oplus {\rm H}_B(\mathcal Y)$.
\vskip5pt
Assume that $B$ is quasi-Frobenius and $N_B$ is flat. Then
\ $\Hom_A(N, \ _A\mathcal I)\subseteq \ _B\mathcal I = \ _B\mathcal P\subseteq \mathcal V$, and thus
\ $\nabla(_A\mathcal I, \ \mathcal Y)
={\rm H}_A(_A\mathcal I)\oplus {\rm H}_B(\mathcal Y)$, by Theorem \ref{identify1}(2).
\vskip10pt
(4) \ Taking $(\mathcal V, \ \mathcal Y) = (B\mbox{-}{\rm Mod}, \ _B\mathcal I)$ in Theorem \ref{identify1}(2). Then
\ $\Hom_A(N, \ \mathcal X) \subseteq B \mbox{-}{\rm Mod} = \mathcal V.$
By Theorem \ref{identify1}(2) one has
\ $(^\bot\nabla(\mathcal X, \ _B\mathcal I), \ \nabla(\mathcal X, \ _B\mathcal I)) = (\left(\begin{smallmatrix}\mathcal U\\ B\mbox{-}{\rm Mod}\end{smallmatrix}\right), \ \left(\begin{smallmatrix}\mathcal U\\ B\mbox{-}{\rm Mod}\end{smallmatrix}\right)^\perp).$
\vskip5pt
If \ $\Hom_B(M, \ _B\mathcal I)\subseteq \mathcal U$,
then by Theorem \ref{identify1}(2), \ $\nabla(\mathcal X, \ _B\mathcal I)={\rm H}_A(\mathcal X)\oplus {\rm H}_B(_B\mathcal I)$.
\vskip5pt
Assume that $A$ is quasi-Frobenius and $M_A$ is flat.
Then \ $\Hom_B(M, \ _B\mathcal I)\subseteq \ _A\mathcal I = \ _A\mathcal P\subseteq \mathcal U,$
and hence \ $\nabla(\mathcal X, \ _B\mathcal I)={\rm H}_A(\mathcal X)\oplus {\rm H}_B(_B\mathcal I)$.
\end{proof}
\vskip5pt
\subsection{Proof of Theorem \ref{identify1}} \ (1) \ By Theorem \ref{compare}(1), one has cotorsion pairs
$$(^\perp\left(\begin{smallmatrix} \mathcal X\\ \mathcal Y\end{smallmatrix}\right), \ \left(\begin{smallmatrix} \mathcal X \\ \mathcal Y\end{smallmatrix}\right)), \ \ \ \ \ (\Delta(\mathcal U, \ \mathcal V), \ \Delta(\mathcal U, \ \mathcal V)^\perp)$$
with \ $^\bot\left(\begin{smallmatrix}\mathcal X \\ \mathcal Y\end{smallmatrix}\right)\subseteq \Delta(\mathcal U, \ \mathcal V)$.
To see that they are equal, it remains to prove $\Delta(\mathcal U, \ \mathcal V)\subseteq \ ^\perp\left(\begin{smallmatrix} \mathcal X \\ \mathcal Y\end{smallmatrix}\right)$.
\vskip5pt
Let \ $L = \left(\begin{smallmatrix} L_1 \\ L_2\end{smallmatrix}\right)_{f, g}\in \Delta(\mathcal U, \ \mathcal V)$. By definition there are exact sequences
$$0\rightarrow M\otimes_AL_1\xlongrightarrow{f}L_2\xlongrightarrow{p_1}\Coker f\rightarrow 0 \ \ \ \mbox{and} \ \ \
0\rightarrow N\otimes_BL_2\xlongrightarrow{g}L_1\xlongrightarrow {p_2}\Coker g\rightarrow 0$$
with $\Coker f\in \ \mathcal V$ and $\Coker g\in \ \mathcal U$. Since $N\otimes_BM = 0 = M\otimes_AN$, it follows that
$$1\otimes p_1: N\otimes_B L_2 \longrightarrow N\otimes_B \Coker f \ \ \ \ \mbox{and} \ \ \
1\otimes p_2: M\otimes_A L_1 \longrightarrow M\otimes_A \Coker g$$ are isomorphisms.
\vskip5pt
{\bf Case I:} \ Assume that \ $M\otimes_A\mathcal U \subseteq \mathcal Y$. Then
$$M\otimes_A L_1\cong M\otimes_A\Coker g\in M\otimes_A\mathcal U \subseteq \mathcal Y.$$
Since $(\mathcal V, \ \mathcal Y)$ is a cotorsion pair, the exact sequence
$$0\longrightarrow M\otimes_AL_1\xlongrightarrow{f}L_2\xlongrightarrow{p_1}\Coker f\longrightarrow 0$$
splits. Thus there are $B$-maps $f': L_2\longrightarrow M\otimes_AL_1$ and $\sigma_1: \Coker f \longrightarrow L_2$
such that
$$f'f = 1_{M\otimes_AL_1}, \ \ \ p_1\sigma_1 = 1_{\Coker f}, \ \ \ ff' + \sigma_1p_1 = 1_{L_2}, \ \ f'\sigma_1 = 0.$$
Thus \ $\left(\begin{smallmatrix} f' \\ p_1\end{smallmatrix}\right): L_2 \longrightarrow (M\otimes_AL_1)\oplus \Coker f$ is a $B$-isomorphism, and $$\left(\begin{smallmatrix} 1 \\ \left(\begin{smallmatrix} f' \\ p_1\end{smallmatrix}\right)\end{smallmatrix}\right): \ L = \left(\begin{smallmatrix} L_1 \\ L_2\end{smallmatrix}\right)_{f, g} \cong \left(\begin{smallmatrix} L_1 \\ (M\otimes_AL_1)\oplus \Coker f \end{smallmatrix}\right)_{\binom{1}{0}, \ g(1\otimes p_1)^{-1}}$$
is a $\Lambda$-isomorphism, and
$$0\rightarrow \left(\begin{smallmatrix} N\otimes_B\Coker f \\ \Coker f\end{smallmatrix}\right)_{0,1}\xlongrightarrow{\left(\begin{smallmatrix} g(1\otimes p_1)^{-1} \\ \binom{0}{1}\end{smallmatrix}\right)}
\left(\begin{smallmatrix} L_1 \\ (M\otimes_AL_1)\oplus \Coker f \end{smallmatrix}\right)_{\binom{1}{0}, \ g(1\otimes p_1)^{-1}}\xlongrightarrow{\left(\begin{smallmatrix} p_2 \\ (1\otimes p_2,0)\end{smallmatrix}\right)}
\left(\begin{smallmatrix} \Coker g \\ M\otimes_A\Coker g\end{smallmatrix}\right)_{1,0}\rightarrow 0$$
is an exact sequence of $\Lambda$-modules, i.e.,
$$0\longrightarrow {\rm T}_B\Coker f\longrightarrow \left(\begin{smallmatrix} L_1 \\ (M\otimes_AL_1)\oplus \Coker f \end{smallmatrix}\right)_{\binom{1}{0}, \ g(1\otimes p_1)^{-1}}\longrightarrow {\rm T}_A\Coker g\longrightarrow 0$$
is exact.
\vskip5pt
Since $\Coker f\in \ \mathcal V$ and $(\mathcal V, \ \mathcal Y)$ is a cotorsion pair,
by Lemma \ref{extadj1}(2), ${\rm T}_B\Coker f\in \ ^\perp \binom{\mathcal{X}}{\mathcal Y}$.
Since $\Coker g \in \mathcal U$ and $(\mathcal U, \ \mathcal X)$ is a cotorsion pair, by Lemma \ref{extadj1}(1), ${\rm T}_A\Coker g\in \ ^\perp \binom{\mathcal{X}}{\mathcal Y}$.
Thus $L\cong \left(\begin{smallmatrix} L_1 \\ (M\otimes_AL_1)\oplus \Coker f \end{smallmatrix}\right)_{\binom{1}{0}, \ g(1\otimes p_1)^{-1}}\in \ ^\perp \binom{\mathcal{X}}{\mathcal Y}$.
\vskip5pt
{\bf Case II:} \ Assume that \ $N\otimes_B\mathcal V \subseteq \mathcal X$. \ This is similar to {\bf Case I}.
We include the main step. Since $N\otimes_B L_2\cong N\otimes_B\Coker f\in N\otimes_B\mathcal V \subseteq \mathcal X,$
the exact sequence
$$0\longrightarrow N\otimes_BL_2\xlongrightarrow{g}L_1\xlongrightarrow {p_2}\Coker g\longrightarrow 0$$
splits. Then $L = \left(\begin{smallmatrix} L_1 \\ L_2\end{smallmatrix}\right)_{f, g} \cong \left(\begin{smallmatrix}(N\otimes_BL_2)\oplus \Coker g\\ L_2 \end{smallmatrix}\right)_{f(1\otimes p_2)^{-1}, \binom{1}{0}}$
and
$$0\rightarrow \left(\begin{smallmatrix} \Coker g \\ M\otimes_A\Coker g\end{smallmatrix}\right)_{1,0}\xlongrightarrow{\left(\begin{smallmatrix} \binom{0}{1} \\ f(1\otimes p_2)^{-1} \end{smallmatrix}\right)}
\left(\begin{smallmatrix}(N\otimes_BL_2)\oplus \Coker g\\ L_2 \end{smallmatrix}\right)_{f(1\otimes p_2)^{-1}, \binom{1}{0}} \xlongrightarrow{\left(\begin{smallmatrix} (1\otimes p_1,0) \\ p_1 \end{smallmatrix}\right)}
\left(\begin{smallmatrix} N\otimes_B\Coker f \\ \Coker f\end{smallmatrix}\right)_{0,1}\rightarrow 0$$
is an exact sequence of $\Lambda$-modules, i.e.,
$$0\longrightarrow {\rm T}_A\Coker g \longrightarrow \left(\begin{smallmatrix}(N\otimes_BL_2)\oplus \Coker g\\ L_2 \end{smallmatrix}\right)_{f(1\otimes p_2)^{-1}, \binom{1}{0}}\longrightarrow {\rm T}_B\Coker f\longrightarrow 0$$
is exact.
By Lemma \ref{extadj1}(1), ${\rm T}_A\Coker g \in \ ^\perp\binom{\mathcal{X}}{\mathcal{Y}}$; and
by Lemma \ref{extadj1}(2), ${\rm T}_B\Coker f \in \ ^\perp\binom{\mathcal{Y}}{\mathcal{Y}}.$
Thus $L\in \ ^\perp\binom{\mathcal{Y}}{_B\mathcal{Y}}$.
\vskip5pt
Finally, assume that $M\otimes_A\mathcal U \subseteq \mathcal Y$ and \ $N\otimes_B\mathcal V \subseteq \mathcal X$.
Then from the proof above one sees that both \ $0\rightarrow M\otimes_AL_1\xlongrightarrow{f}L_2\xlongrightarrow{p_1}\Coker f\rightarrow 0$ \ and
\ $0\rightarrow N\otimes_BL_2\xlongrightarrow{g}L_1\xlongrightarrow {p_2}\Coker g\rightarrow 0$ split, and
$$L\cong\left(\begin{smallmatrix} \Coker g \\ M\otimes_A\Coker g\end{smallmatrix}\right)_{1, 0}
\oplus \left(\begin{smallmatrix} N\otimes_B \Coker f \\ \Coker f\end{smallmatrix}\right)_{0, 1} = {\rm T}_A \Coker g\oplus {\rm T}_B\Coker f.$$
Thus \ $\Delta(\mathcal U, \ \mathcal V) \subseteq {\rm T}_A(\mathcal U)\oplus {\rm T}_B(\mathcal V).$ The inclusion
\ $ {\rm T}_A(\mathcal U)\oplus {\rm T}_B(\mathcal V)\subseteq \Delta(\mathcal U, \ \mathcal V)$ is clear. Thus shows
$\Delta(\mathcal U, \ \mathcal V) = {\rm T}_A(\mathcal U)\oplus {\rm T}_B(\mathcal V).$
\vskip10pt
(2) \ By Theorem \ref{compare}(2), one has cotorsion pairs
$$(\left(\begin{smallmatrix} \mathcal U\\ \mathcal V\end{smallmatrix}\right), \ \left(\begin{smallmatrix} \mathcal U \\ \mathcal V\end{smallmatrix}\right)^\perp), \ \ \ \ \ (^\perp\nabla(\mathcal X, \ \mathcal Y), \ \nabla(\mathcal X, \ \mathcal Y))$$
with \ $\left(\begin{smallmatrix}\mathcal U \\ \mathcal V\end{smallmatrix}\right)^\bot \subseteq \nabla(\mathcal X, \ \mathcal Y)$.
To see that they are equal, it remains to prove $\nabla(\mathcal X, \ \mathcal Y)\subseteq \left(\begin{smallmatrix} \mathcal U \\ \mathcal V\end{smallmatrix}\right)^\bot$.
\vskip5pt
Here it is much more convenient to use the second expression of $\Lambda$-modules. Thus, let \ $L = \left(\begin{smallmatrix} L_1 \\ L_2\end{smallmatrix}\right)_{\widetilde{f}, \widetilde{g}}\in \nabla(\mathcal X, \ \mathcal Y)$, where $\widetilde{f}\in \Hom_A(X, \ \Hom_B(M, Y))$ and
$\widetilde{g}\in \Hom_B(Y, \ \Hom_A(N, X))$. By definition there are exact sequences
$$0\rightarrow \Ker\widetilde{f}\xlongrightarrow{i_1}L_1\xlongrightarrow{\widetilde{f}}\Hom_B(M, L_2)\rightarrow 0 \ \ \ \mbox{and} \ \ \ 0\rightarrow \Ker\widetilde{g}\xlongrightarrow{i_2}L_2\xlongrightarrow{\widetilde{g}}\Hom_A(N, L_1)\rightarrow 0$$
with $\Ker \widetilde{f}\in \ \mathcal X$ and $\Ker \widetilde{g}\in \ \mathcal Y$. Since $M\otimes_AN = 0 = N\otimes_BM$, it follows that
$$(N, \ i_1): \Hom_A(N, \Ker\widetilde{f})\cong \Hom_A(N, L_1) \ \ \ \mbox{and} \ \ \ (M, \ i_2): \Hom_B(M, \Ker\widetilde{g})\cong \Hom_B(M, L_2).$$
{\bf Case I:} \ Assume that \ $\Hom_B(M, \mathcal Y)\subseteq \mathcal U$. Then
$$\Hom_B(M, \ L_2)\cong \Hom_B(M, \ \Ker\widetilde{g})\in \Hom_B(M, \mathcal Y)\subseteq \mathcal U.$$
Since $(\mathcal U, \ \mathcal X)$ is a cotorsion pair, the exact sequence
$$0\longrightarrow \Ker\widetilde{f}\xlongrightarrow{i_1}L_1\xlongrightarrow{\widetilde{f}}\Hom_B(M, L_2)\longrightarrow 0$$
splits. Thus there are $A$-maps $\alpha: \Hom_B(M,\ L_2)\longrightarrow L_1$ and $\pi_1: L_1 \longrightarrow \Ker\widetilde{f}$
such that
$$\pi_1i_1 = 1_{\Ker\widetilde{f}}, \ \ \ \widetilde{f}\alpha = 1_{\Hom_B(M,L_2)}, \ \ \ \alpha\widetilde{f} + i_1\pi_1 = 1_{L_1}, \ \ \ \pi_1\alpha = 0.$$
Hence \ $\left(\begin{smallmatrix} \pi_1 \\ \widetilde{f}\end{smallmatrix}\right): L_1 \cong \Ker\widetilde{f}\oplus \Hom_B(M, L_2)$ and
$$\left(\begin{smallmatrix} \binom{\pi_1}{\widetilde{f}} \\ 1 \end{smallmatrix}\right): \ L = \left(\begin{smallmatrix} L_1 \\ L_2\end{smallmatrix}\right)_{\widetilde{f}, \widetilde{g}} \cong \left(\begin{smallmatrix} \Ker\widetilde{f}\oplus \Hom_B(M, \ L_2) \\ L_2\end{smallmatrix}\right)_{(0, \ 1), \ \binom{(N, \pi_1)\widetilde{g}}{0}}.$$
Moreover,
$$0 \rightarrow \left(\begin{smallmatrix} (M, \ \Ker\widetilde{g}) \\ \Ker\widetilde{g}\end{smallmatrix}\right)_{1, 0}
\xrightarrow{\left(\begin{smallmatrix} \binom{0}{(M, i_2)} \\ i_2\end{smallmatrix}\right)}
\left(\begin{smallmatrix} \Ker\widetilde{f}\oplus (M, \ L_2) \\ L_2\end{smallmatrix}\right)_{(0, 1), \ \binom{(N, \pi_1)\widetilde{g}}{0}}\xlongrightarrow{\left(\begin{smallmatrix} (1,0) \\ (N, \pi_1)\widetilde{g}\end{smallmatrix}\right)}
\left(\begin{smallmatrix} \Ker\widetilde{f} \\ (N, \ \Ker\widetilde{f}) \end{smallmatrix}\right)_{0, 1}\rightarrow 0$$
is an exact sequence of $\Lambda$-modules, i.e.,
$$0\longrightarrow {\rm H}_B\Ker \widetilde{g}\longrightarrow
\left(\begin{smallmatrix} \Ker\widetilde{f}\oplus \Hom_B(M, \ L_2) \\ L_2\end{smallmatrix}\right)_{(0, \ 1), \ \binom{(N, \pi_1)\widetilde{g}}{0}}
\longrightarrow {\rm H}_A\Ker \widetilde{f}\longrightarrow 0$$
is exact. (We stress that all the $\Lambda$-modules are in the second expression.)
\vskip5pt
Since $\Ker\widetilde{g}\in \ \mathcal Y$ and $(\mathcal V, \ \mathcal Y)$ is a cotorsion pair,
by Lemma \ref{extadj1}(4), ${\rm H}_B\Ker\widetilde{g}\in \binom{\mathcal{U}}{\mathcal V}^\bot$.
Since $\Ker\widetilde{f} \in \mathcal X$ and $(\mathcal U, \ \mathcal X)$ is a cotorsion pair, by Lemma \ref{extadj1}(3), ${\rm H}_A\Ker\widetilde{f}\in \binom{\mathcal{U}}{\mathcal V}^\perp$.
Thus $L\cong \ \left(\begin{smallmatrix} \Ker\widetilde{f}\oplus \Hom_B(M,L_2) \\ L_2\end{smallmatrix}\right)_{(0, \ 1), \ \binom{(N, \pi_1)\widetilde{g}}{0}}\in \binom{\mathcal{U}}{\mathcal V}^\perp$.
\vskip5pt
{\bf Case II:} \ Assume that \ $\Hom_A(N, \ \mathcal X)\subseteq \mathcal V$. \ This is similar to {\bf Case I}.
Since $\Hom_A(N, \ L_1)\cong \Hom_A(N, \ \Ker\widetilde{f})\in \Hom_A(N, \ \mathcal X)\subseteq \mathcal V,$
the exact sequence
$$0\longrightarrow \Ker\widetilde{g}\xlongrightarrow{i_2}L_2\xlongrightarrow{\widetilde{g}}\Hom_A(N, L_1)\longrightarrow 0$$
splits. Thus $L = \left(\begin{smallmatrix} L_1 \\ L_2\end{smallmatrix}\right)_{\widetilde{f}, \widetilde{g}} \cong \left(\begin{smallmatrix}L_1 \\ \Ker\widetilde{g}\oplus \Hom_A(N, L_1) \end{smallmatrix}\right)_{\binom{(M, i_2)^{-1}\widetilde{f}}{0}, (0, 1)}$
and
$$0 \rightarrow \left(\begin{smallmatrix} \Ker\widetilde{f} \\ (N, \Ker\widetilde{f})\end{smallmatrix}\right)_{0,1}
\xlongrightarrow{\left(\begin{smallmatrix} i_1 \\ \binom{0}{(N, i_1)}\end{smallmatrix}\right)}
\left(\begin{smallmatrix} L_1 \\ \Ker\widetilde{g}\oplus (N, L_1) \end{smallmatrix}\right)_{\binom{(M, i_2)^{-1}\widetilde{f}}{0}, (0, 1)}
\xlongrightarrow{\left(\begin{smallmatrix} (M, i_2)^{-1}\widetilde{f} \\ (1,0) \end{smallmatrix}\right)}
\left(\begin{smallmatrix} (M, \Ker\widetilde{g}) \\ \Ker\widetilde{g} \end{smallmatrix}\right)_{1, 0}\rightarrow 0$$
is exact, i.e.,
$$0\longrightarrow {\rm H}_A\Ker \widetilde{f}\longrightarrow \left(\begin{smallmatrix} L_1 \\ \Ker\widetilde{g}\oplus \Hom_A(N, L_1) \end{smallmatrix}\right)_{\binom{(M, i_2)^{-1}\widetilde{f}}{0}, (0, 1)}
\longrightarrow {\rm H}_B\Ker \widetilde{g}\longrightarrow 0$$
is exact. By Lemma \ref{extadj1}(3), ${\rm H}_A\Ker\widetilde{f} \in \binom{\mathcal{U}}{\mathcal{V}}^\perp$; and by Lemma \ref{extadj1}(4), ${\rm H}_B\Ker\widetilde{g} \in \binom{\mathcal{U}}{\mathcal{V}}^\perp.$
Thus $L\in \binom{\mathcal{U}}{\mathcal{V}}^\perp$.
\vskip10pt
Finally, assume that \ $\Hom_B(M, \ \mathcal Y)\subseteq \mathcal U$ and \ $\Hom_A(N, \ \mathcal X) \subseteq \mathcal V$. Then both
\ $0\rightarrow \Ker\widetilde{f}\xlongrightarrow{i_1}L_1\xlongrightarrow{\widetilde{f}}\Hom_B(M, L_2)\rightarrow 0$ \ and \ $0\rightarrow \Ker\widetilde{g}\xlongrightarrow{i_2}L_2\xlongrightarrow{\widetilde{g}}\Hom_A(N, L_1)\rightarrow 0$ splits, and
$$L=\left(\begin{smallmatrix}L_1\\ L_2\end{smallmatrix}\right)_{\widetilde{f},\widetilde{g}}\cong \left(\begin{smallmatrix} \Ker\widetilde{f}\\ \Hom_A(N, \Ker\widetilde{f})\end{smallmatrix}\right)_{0, 1}\oplus
\left(\begin{smallmatrix} \Hom_B(M, \Ker \widetilde{g}) \\ \Ker \widetilde{g}\end{smallmatrix}\right)_{1,0}= {\rm H}_A\Ker\widetilde{f}\oplus {\rm H}_B\Ker \widetilde{g}
\in {\rm H}_A(\mathcal X)\oplus {\rm H}_B(\mathcal Y).$$
Conversely, it is clear that
${\rm H}_A(\mathcal X)\oplus {\rm H}_B(\mathcal Y)\subseteq \nabla(\mathcal X, \ \mathcal Y)$.
Thus \ $\nabla(\mathcal X, \ \mathcal Y)
={\rm H}_A(\mathcal X)\oplus {\rm H}_B(\mathcal Y)$.
\hfill $\square$
\subsection{Remark} In Theorem \ref{compare}, taking one of $(\mathcal U, \ \mathcal X)$ and
$(\mathcal V, \ \mathcal Y)$ being the projective cotorsion pair or the injective cotorsion pair,
and another being an arbitrary cotorsion pair, we conclude as follows: where ``$=$" follows from Corollary \ref{identification1}, and
``$\ne$" follows from Example \ref{ie}.
\vskip5pt
$(1)$ \ If \ $(\mathcal U, \ \mathcal X) = (_A\mathcal P, \ A\mbox{-}{\rm Mod})$, and \ $(\mathcal V, \ \mathcal Y)$
is an arbitrary cotorsion pair in $B\mbox{-}{\rm Mod}$, then
$$(\Delta(_A\mathcal P, \ \mathcal V), \ \Delta(_A\mathcal P, \ \mathcal V)^\bot) = ({}^\perp\left(\begin{smallmatrix}A\mbox{-}{\rm Mod}\\ \mathcal Y\end{smallmatrix}\right), \ \left(\begin{smallmatrix}A\mbox{-}{\rm Mod}\\ \mathcal Y\end{smallmatrix}\right)).$$
But, in general \ $(^\bot\nabla(A\mbox{-}{\rm Mod}, \ \mathcal Y), \ \nabla(A\mbox{-}{\rm Mod}, \ \mathcal Y))\ne (\left(\begin{smallmatrix}_A\mathcal P\\ \mathcal V\end{smallmatrix}\right), \ \left(\begin{smallmatrix}_A\mathcal P\\ \mathcal V\end{smallmatrix}\right)^\perp).$
\vskip5pt
$(2)$ \ If \ $(\mathcal U, \ \mathcal X) = (A\mbox{-}{\rm Mod}, \ _A\mathcal I)$, and \ $(\mathcal V, \ \mathcal Y)$ an arbitrary cotorsion pair in $B\mbox{-}{\rm Mod}$,
then in general \ $(\Delta(A\mbox{-}{\rm Mod}, \ \mathcal V), \ \Delta(A\mbox{-}{\rm Mod}, \ \mathcal V)^\bot)\ne ({}^\perp\left(\begin{smallmatrix}_A\mathcal I\\ \mathcal Y\end{smallmatrix}\right), \ \left(\begin{smallmatrix}_A\mathcal I\\ \mathcal Y\end{smallmatrix}\right)).$
However, one has
$$(^\bot\nabla(_A\mathcal I, \ \mathcal Y), \ \nabla(_A\mathcal I, \ \mathcal Y)) = (\left(\begin{smallmatrix}A\mbox{-}{\rm Mod}\\ \mathcal V\end{smallmatrix}\right), \ \left(\begin{smallmatrix}A\mbox{-}{\rm Mod}\\ \mathcal V\end{smallmatrix}\right)^\perp).$$
$(3)$ \ If \ $(\mathcal U, \ \mathcal X)$ is an arbitrary cotorsion pair in $A\mbox{-}{\rm Mod}$, and
\ $(\mathcal V, \ \mathcal Y) = (_B\mathcal P, \ B\mbox{-}{\rm Mod})$, then
$$(\Delta(\mathcal U, \ _B\mathcal P), \ \Delta(\mathcal U, \ _B\mathcal P)^\bot)= (^\perp\left(\begin{smallmatrix}\mathcal X\\ B\mbox{-}{\rm Mod}\end{smallmatrix}\right), \ \left(\begin{smallmatrix}\mathcal X\\ B\mbox{-}{\rm Mod}\end{smallmatrix}\right)).$$
But, in general \ $(^\bot\nabla(\mathcal X, \ B\mbox{-}{\rm Mod}), \ \nabla(\mathcal X, \ B\mbox{-}{\rm Mod}))\ne (\left(\begin{smallmatrix}\mathcal U\\ _B\mathcal P\end{smallmatrix}\right), \ \left(\begin{smallmatrix}\mathcal U\\ _B\mathcal P\end{smallmatrix}\right)^\perp).$
\vskip5pt
$(4)$ \ If \ $(\mathcal U, \ \mathcal X)$ is an arbitrary cotorsion pair in $A\mbox{-}{\rm Mod}$, and
\ $(\mathcal V, \ \mathcal Y) = (B\mbox{-}{\rm Mod}, \ _B\mathcal I)$, then in general
\ $(\Delta(\mathcal U, \ B\mbox{-}{\rm Mod}), \ \Delta(\mathcal U, \ B\mbox{-}{\rm Mod})^\bot)\ne ({}^\perp\left(\begin{smallmatrix}\mathcal X\\ _B\mathcal I\end{smallmatrix}\right), \ \left(\begin{smallmatrix}\mathcal X\\ _B\mathcal I\end{smallmatrix}\right)).$
However, one has
$$(^\perp\nabla(\mathcal X, \ _B\mathcal I), \ \nabla(\mathcal X, \ _B\mathcal I)) = (\left(\begin{smallmatrix}\mathcal U\\ B\mbox{-}{\rm Mod}\end{smallmatrix}\right), \ \left(\begin{smallmatrix}\mathcal U\\ B\mbox{-}{\rm Mod}\end{smallmatrix}\right)^\perp).$$
\vskip5pt
The above information is listed in Table 1 below, where
$$\mathcal A: = A\mbox{-}{\rm Mod}, \ \ \ \ \mathcal B: = B\mbox{-}{\rm Mod}, \ \ \ \ \mbox{proj.}: = \mbox{projective}.$$
\vskip10pt
\centerline{\bf Table 1: \ Cotorsion pairs in $\Lambda$-Mod}
\vspace{-10pt}
$${\tiny\begin{tabular}{|c|c|c|c|c|}
\hline
\phantom{\LARGE 0} & \multicolumn{2}{c|} {\tabincell{c}{Cotorsion pairs in Series I\\[3pt] $\varphi=0=\psi$}}
& \multicolumn{2}{c|} {\tabincell{c}{Cotorsion pairs in Series II\\[3pt] $M\otimes_AN=0 = N\otimes_BM$}}
\\[8pt]\hline
\tabincell{c}{$(_A\mathcal U, \ _A\mathcal X)$ \\ $(_B\mathcal V, \ _B\mathcal Y)$}
& \tabincell{c}{$\Tor_1(M,\mathcal U)=0$ \\ [3pt] $\Tor_1(N,\mathcal V)=0$: \\[3pt] $(^\perp\binom{\mathcal X}{\mathcal Y},\binom{\mathcal X}{\mathcal Y})$}
& \tabincell{c}{$\Ext^1(N, \mathcal X)=0$ \\ [3pt]$\Ext^1(M, \mathcal Y)=0$: \\[3pt] $(\binom{\mathcal U}{\mathcal V}, \binom{\mathcal U}{\mathcal V}^\perp)$}
& \tabincell{c}{$(\Delta(\mathcal U, \mathcal V), \ \Delta(\mathcal U, \mathcal V)^\bot)$}
& \tabincell{c}{$(^\bot\nabla(\mathcal X, \mathcal Y), \ \nabla(\mathcal X, \mathcal Y))$}
\\[15pt] \hline
\tabincell{c}{$(\mathcal P, \mathcal A$) \\ $(\mathcal V, \mathcal Y)$}
&\tabincell{c}{$N_B$ flat \\ [3pt] $(^\perp\binom{\mathcal A}{\mathcal Y}, \binom{\mathcal A}{\mathcal Y})$}
& \tabincell{c}{$_AN$, $_BM$ proj.: \\[3pt] $(\binom{\mathcal P}{\mathcal V}, \ \binom{\mathcal P}{\mathcal V}^\perp)$}
& \tabincell{c}{$(\Delta(\mathcal P, \mathcal V), \ \Delta(\mathcal P, \mathcal V)^\bot)$. \\[3pt] If $N_B$ flat then it is \\ [3pt]$(^\perp\binom{\mathcal A}{\mathcal Y}, \binom{\mathcal A}{\mathcal Y})$ \\[3pt] thus it is \\ [3pt]$(\Delta(\mathcal P, \mathcal V), \binom{\mathcal A}{\mathcal Y})$}
& \tabincell{c}{$(^\bot\nabla(\mathcal A, \mathcal Y), \ \nabla(\mathcal A, \mathcal Y))$. \\[3pt] Even if $_AN$, $_BM$ proj., \\[3pt]it $\ne (\binom{\mathcal P}{\mathcal V}, \ \binom{\mathcal P}{\mathcal V}^\perp)$\\[3pt] in general.}
\\[30pt] \hline
\tabincell{c}{$(\mathcal A, \mathcal I$) \\ $(\mathcal V, \mathcal Y)$}
& \tabincell{c}{$M_A$, $N_B$ flat: \\ [3pt] $(^\perp\binom{\mathcal I}{\mathcal Y}, \ \binom{\mathcal I}{\mathcal Y})$}
& \tabincell{c}{$_BM$ proj.: \\[3pt] $(\binom{\mathcal A}{\mathcal V}, \ \binom{\mathcal A}{\mathcal V}^\perp)$}
& \tabincell{c}{$(\Delta(\mathcal A, \mathcal V), (\Delta(\mathcal A, \mathcal V)^\bot).$ \\ [3pt] Even if $M_A$, $N_B$ flat \\ [3pt] it $\ne (^\perp\binom{\mathcal I}{\mathcal Y}, \ \binom{\mathcal I}{\mathcal Y})$\\ [3pt] in general}
& \tabincell{c}{$(^\bot\nabla(\mathcal I, \mathcal Y), \ \nabla(\mathcal I, \mathcal Y)).$ \\ [3pt] If $_BM$ proj. then it is \\ [3pt]$(\binom{\mathcal A}{\mathcal V}, \ \binom{\mathcal A}{\mathcal V}^\perp)$\\[3pt]thus it is \\[3pt]$(\binom{\mathcal A}{\mathcal V}, \ \nabla(\mathcal I, \mathcal Y))$}
\\[30pt] \hline
\tabincell{c}{$(\mathcal U, \mathcal X$) \\ $(\mathcal P, \mathcal B)$}
& \tabincell{c}{$M_A$ flat: \\[3pt] $(^\perp\binom{\mathcal X}{\mathcal B}, \ \binom{\mathcal X}{\mathcal B})$}
& \tabincell{c}{$_BM, \ _AN$ proj.: \\[3pt] $(\binom{\mathcal U}{\mathcal P}, \ \binom{\mathcal U}{\mathcal P}^\perp)$}
& \tabincell{c}{$(\Delta(\mathcal U, \mathcal P), \Delta(\mathcal U, \mathcal P)^\bot).$ \\[3pt] If $M_A$ flat then it is \\[3pt]$(^\perp\binom{\mathcal X}{\mathcal B}, \binom{\mathcal X}{\mathcal B})$\\[3pt] thus it is \\ [3pt]$((\Delta(\mathcal U, \mathcal P), \ \binom{\mathcal X}{\mathcal B})$}
& \tabincell{c}{$(^\bot\nabla(\mathcal X, \mathcal B), \nabla(\mathcal X, \mathcal B))$. \\[3pt] Even if $_BM, \ _AN$ proj., \\[3pt] it $\ne (\binom{\mathcal U}{\mathcal P}, \ \binom{\mathcal U}{\mathcal P}^\perp)$\\[3pt]in general }
\\[30pt] \hline
\tabincell{c}{$(\mathcal U, \mathcal X)$ \\ $(\mathcal B, \mathcal I)$}
& \tabincell{c}{$M_A$, $N_B$ flat: \\[3pt] $(^\perp\binom{\mathcal X}{\mathcal I}, \ \binom{\mathcal X}{\mathcal I})$}
& \tabincell{c}{$_AN$ proj.: \\[3pt] $(\binom{\mathcal U}{\mathcal B}, \ \binom{\mathcal U}{\mathcal B}^\bot)$}
& \tabincell{c}{$(\Delta(\mathcal U, \mathcal B), \ \Delta(\mathcal U, \mathcal B)^\bot)$.\\[3pt]Even if $M_A$, $N_B$ flat,\\ [3pt]it $\ne (^\perp\binom{\mathcal X}{\mathcal I}, \ \binom{\mathcal X}{\mathcal I})$\\ [3pt]in general}
& \tabincell{c}{$(^\bot\nabla(\mathcal X, \mathcal I), \ \nabla(\mathcal X, \mathcal I)).$ \\ [3pt] If $_AN$ proj. then it is \\ [3pt]$(\binom{\mathcal U}{\mathcal B}, \ \binom{\mathcal U}{\mathcal B}^\perp)$\\[3pt]thus it is \\[3pt]$(\binom{\mathcal U}{\mathcal B}, \ \nabla(\mathcal X, \mathcal I))$}
\\[30pt] \hline
\end{tabular}}$$
\vskip5pt
\subsection{Monomorphism categories and epimorphism categories} \ Even if in the case of ``$\ne$" in general, the two cotorsion pairs can be the same, in some special cases.
\vskip5pt
If $_AN$ and $_BM$ are projective, then cotorsion pairs \ $(\left(\begin{smallmatrix}_A\mathcal P\\ _B\mathcal P\end{smallmatrix}\right), \ \left(\begin{smallmatrix}_A\mathcal P\\ _B\mathcal P\end{smallmatrix}\right)^\perp)$ and $(^\bot{\rm Epi}(\Lambda), \ {\rm Epi}(\Lambda)) = (^\perp\nabla (A\mbox{-}{\rm Mod}, \ B\mbox{-}{\rm Mod}), \ \nabla (A\mbox{-}{\rm Mod}, \ B\mbox{-}{\rm Mod}))$
are not equal in general (cf. Example \ref{ie}); but the following result claims that they can be the same in some special cases.
\vskip5pt
Also, if $M_A$ and $N_B$ are flat, then \ $({}^\perp\left(\begin{smallmatrix}_A\mathcal I\\ _B\mathcal I\end{smallmatrix}\right), \ \left(\begin{smallmatrix}_A\mathcal I\\ _B\mathcal I\end{smallmatrix}\right))
\ne ({\rm Mon}(\Lambda), \ {\rm Mon}(\Lambda)^\bot)$ in general
(cf. Example \ref{ie}); but the following result claims that they can be the same in some special cases.
\vskip5pt
\begin{thm}\label{ctp4} \ Let \ $\Lambda = \left(\begin{smallmatrix} A & N \\ M & B \end{smallmatrix}\right)$ be a noetherian Morita ring with $M\otimes_A N = 0 = N\otimes_BM$.
Assume that $A$ and $B$ are quasi-Frobenius rings, that $_AN$ and $_BM$ are projective, and that $M_A$ and $N_B$ are flat. Then
\vskip5pt
${\rm (1)}$ \ \ $\Lambda$ is a Gorenstein ring with ${\rm inj.dim} _\Lambda\Lambda \le 1$, and
$_\Lambda \mathcal P^{<\infty} = \ _\Lambda \mathcal P^{\le 1} = \binom{_A\mathcal P}{_B\mathcal P} = \binom{_A\mathcal I}{_B\mathcal I} = \ _\Lambda \mathcal I^{\le 1} = \ _\Lambda \mathcal I^{<\infty}.$
\vskip5pt
${\rm (2)}$ \ \ The cotorsion pair $(^\perp\binom{_A\mathcal I}{_B\mathcal I}, \ \binom{_A\mathcal I}{_B\mathcal I})$ coincides with \ $({\rm Mon}(\Lambda), \ {\rm Mon}(\Lambda)^\bot);$ and it is
exactly the Gorenstein-projective cotorsion pair $({\rm GP}(\Lambda), \ _\Lambda\mathcal P^{\le 1})$. So, it is complete and hereditary, and
$${\rm GP}(\Lambda) = {\rm Mon}(\Lambda) = {}^\perp \ _\Lambda\mathcal P, \ \ \ \ {\rm Mon}(\Lambda)^\bot = \ _\Lambda \mathcal P^{\le 1}.$$
\vskip5pt
${\rm (2)'}$ \ \ The cotorsion pair $(\binom{_A\mathcal P}{_B\mathcal P}, \ \binom{_A\mathcal P}{_B\mathcal P}^\perp)$ coincides with \ $(^\bot{\rm Epi}(\Lambda), \ {\rm Epi}(\Lambda));$ and it
is exactly the Gorenstein-injective cotorsion pair \ $(_\Lambda \mathcal P^{\le 1}, \ {\rm GI}(\Lambda))$. So, it is complete and hereditary, and
$$ {\rm GI}(\Lambda) = {\rm Epi}(\Lambda) = \ _\Lambda\mathcal I{}^\perp, \ \ \ \ ^\bot{\rm Epi}(\Lambda) = \ _\Lambda \mathcal P^{\le 1}.$$
\end{thm}
\vskip5pt
\begin{exm}\label{examctp4} $(1)$ \ {\it We give an example to justify the existence of the assumptions in {\rm Theorem \ref{ctp4}}.
Let $Q$ be the quiver
$$\xymatrix{1\ar[r] & 2\ar[r] & 3\ar[r] & \cdots \ar[r] & n \ar@/_20pt/[llll]}$$
\vskip5pt
\noindent and $A=kQ/J^h$, where $J$ is the ideal of path algebra $kQ$ generated by all the arrows, and $2\le h\le n$. Then $A$ is a self-injective algebra, in particular, a quasi-Frobenius ring. Let $e=e_i$, $e'=e_j$, where $1\le i < j\le n$, satisfying $j-i\ge h$. Then \ $e'Ae=e_jAe_i=0$.
Put $M: = Ae\otimes_k e'A$. Then $_AM$ and $M_A$ are projective, and $M\otimes_AM = (Ae\otimes_k e'A)\otimes_A (Ae\otimes_k e'A) = Ae\otimes_k(e'A\otimes_A Ae)\otimes_k e'A = 0.$
\vskip5pt
Take \ $\Lambda = \left(\begin{smallmatrix} A & M \\ M & A \end{smallmatrix}\right)$. Then $\Lambda$ satisfies
all the conditions in {\rm Theorem \ref{ctp4}}.}\end{exm}
\begin{rem}\label{remctp4} {\rm $(1)$} \ {\it Non-zero Morita rings $\Lambda$ in {\rm Theorem \ref{ctp4}} do not satisfy the sufficient condition for self-injective algebras in {\rm [GrP, Proposition 3.7]}. In fact,
$\Lambda$ can not be quasi-Frobenius$:$ otherwise ${\rm Mon}(\Lambda) = {\rm GP}(\Lambda) = \Lambda$-{\rm Mod}, which is absurd $!$
\vskip5pt
{\rm $(2)$} \ Although {\rm Theorem \ref{ctp4}} does not give new cotorsion pairs, in the sense that
they are just the Gorenstein-projective $($respectively, Gorenstein-injective$)$ cotorsion pairs,
however, \ ${\rm GP}(\Lambda) = {\rm Mon}(\Lambda)$ is a new result. In the special case of triangular matrix rings, this is known, by
{\rm [LiZ, Thm. 1.1], [XZ, Cor.1.5], [Z2, Thm.1.4], [LuoZ1, Thm.4.1], [ECIT, Thm.3.5]}. For more
relations between monomorphism categories and the Gorenstein-projective modules, we refer to {\rm [Z1], [GrP], [LuoZ2], [GaP], [ZX], [HLXZ]}.}
\end{rem}
\subsection{Modules $\binom{_A\mathcal P}{_B\mathcal P}$ and $\binom{_A\mathcal I}{_B\mathcal I}$} To prove Theorem \ref{ctp4}, we need the following fact, which is of independent interest.
\vskip5pt
\begin{lem}\label{proj-injdim} \ Let \ $\Lambda = \left(\begin{smallmatrix} A & N \\ M & B \end{smallmatrix}\right)$ be a Morita ring with $M\otimes_A N = 0 = N\otimes_BM.$
\vskip5pt
$(1)$ \ Assume that $_AN$ and $_BM$ are projective modules. Let $\binom{P}{Q}_{f, g}\in \binom{_A\mathcal P}{_B\mathcal P}$. Then
$$0\rightarrow
\left(\begin{smallmatrix} N\otimes_BQ\\M\otimes_AP \end{smallmatrix}\right)_{0,0}
\stackrel{\left(\begin{smallmatrix} -\binom{g}{1} \\ \binom{1}{f}\end{smallmatrix}\right)} \longrightarrow
\left(\begin{smallmatrix} P\\ M\otimes_AP\end{smallmatrix}\right)_{1, 0}
\oplus \left(\begin{smallmatrix} N\otimes_BQ \\ Q\end{smallmatrix}\right)
_{0, 1}\stackrel{\left(\begin{smallmatrix} \binom{1}{f}, & \binom{g}{1}\end{smallmatrix}\right)}\longrightarrow
\left(\begin{smallmatrix} P \\ Q\end{smallmatrix}\right)_{f,g}\rightarrow 0$$
is a projective resolution of $\binom{P}{Q}_{f, g}$. In particular, ${\rm proj.dim} \binom{P}{Q}_{f, g} \le 1,
\ \forall \ \binom{P}{Q}_{f, g}\in \binom{_A\mathcal P}{_B\mathcal P}$.
\vskip5pt
$(2)$ \ Assume that $M_A$ and $N_B$ are flat modules. Let $\binom{I}{J}_{f, g}\in \binom{_A\mathcal I}{_B\mathcal I}$. Then
$$0\rightarrow
\left(\begin{smallmatrix}I\\ J\end{smallmatrix}\right)_{f, g}\stackrel {\left(\begin{smallmatrix} \binom {1} {\widetilde{g}} \\ \binom {\widetilde{f}}{1}\end{smallmatrix}\right)} \longrightarrow
\left(\begin{smallmatrix} I\\ \Hom_A(N, I)\end{smallmatrix}\right)_{0, \epsilon'_{_I}}\oplus \left(\begin{smallmatrix} \Hom_B(M, J)\\ J\end{smallmatrix}\right)_{\epsilon_{_J}, 0}
\stackrel {\left(\begin{smallmatrix}\binom{\widetilde{f}}{1}, & -\binom{1}{\widetilde{g}}\end{smallmatrix}\right)} \longrightarrow
\left(\begin{smallmatrix}\Hom_B(M, J)\\ \Hom_A(N, I)\end{smallmatrix}\right)_{0, 0}\rightarrow 0$$
is an injective resolution of $\binom{I}{J}_{f, g}$. In particular, ${\rm inj.dim} \binom{I}{J}_{f, g} \le 1, \ \forall \ \binom{I}{J}_{f, g}\in \binom{_A\mathcal I}{_B\mathcal I}$.\end{lem}
\begin{rem} {\it The condition $M\otimes_A N = 0 = N\otimes_BM$ can not be relaxed to $\phi = 0 = \psi$. Otherwise, for example in {\rm (1)}, $\Ker \left(\begin{smallmatrix} \binom{1}{f}, & \binom{g}{1}\end{smallmatrix}\right)=\left(\begin{smallmatrix} N\otimes_BQ\\M\otimes_AP \end{smallmatrix}\right)_{-(1_M\otimes_Ag),-(1_N\otimes_Bf)}$, which is no longer a projective left $\Lambda$-module.}
The similar remark for (2).
\end{rem}
\vskip5pt \noindent {\bf Proof of Lemma \ref{proj-injdim}} (1) \ Thanks to the assumption $M\otimes_A N = 0 = N\otimes_BM,$ the given maps are $\Lambda$-maps (otherwise $\left(\begin{smallmatrix} -\binom{g}{1} \\ \binom{1}{f}\end{smallmatrix}\right)$ is not necessarily a $\Lambda$-map in general, even if $\phi = 0 = \psi$). We omit the details.
The given sequence of $\Lambda$-modules is exact, since
$$\xymatrix {0\ar[r] & N\otimes_BQ \ar[r]^-{\binom{-g}{1}} & P \oplus (N\otimes_BQ) \ar[rr]^-{(1, g)} && P \ar[r] & 0}$$
and
$$\xymatrix {0\ar[r] & M\otimes_AP \ar[r]^-{\binom{-1}{f}} & (M\otimes_AP) \oplus Q \ar[rr]^-{(f, 1)} && Q \ar[r] & 0}$$
are exact.
\vskip5pt
We claim that $\left(\begin{smallmatrix} N\otimes_BQ\\M\otimes_AP \end{smallmatrix}\right)_{0,0}$ is a projective left $\Lambda$-module.
In fact, since $_AN$ and $_BQ$ are projective, $N\otimes_BQ$ is a projective left $A$-module. Since $M\otimes_AN = 0$, it follows that
$$\left(\begin{smallmatrix} N\otimes_BQ \\ 0\end{smallmatrix}\right)_{0, 0} = \left(\begin{smallmatrix} N\otimes_BQ\\M\otimes_AN\otimes_BQ\end{smallmatrix}\right)_{0, 0}$$
is a projective left $\Lambda$-module. Similarly, $\left(\begin{smallmatrix} 0 \\ M\otimes_AP\end{smallmatrix}\right)_{0, 0}$ is a projective left $\Lambda$-module. Thus, $\left(\begin{smallmatrix} N\otimes_BQ\\M\otimes_AP \end{smallmatrix}\right)_{0,0} = \binom{N\otimes_BQ}{0}_{0, 0}\oplus \binom{0}{M\otimes_AP}_{0, 0}$ is a projective left $\Lambda$-module.
\vskip10pt
(2) \ This can be similarly proved as (1). Since \ $N\otimes_BM = 0= M\otimes_A N,$ the given sequence is an exact sequence of $\Lambda$-maps.
Since $M_A$ is flat and $_BJ$ is injective, \ $\Hom_B(M, J)$ is an injective left $A$-module, and hence
$$\left(\begin{smallmatrix}\Hom_B(M, J)\\ 0\end{smallmatrix}\right)_{0, 0} = \left(\begin{smallmatrix}\Hom_B(M, J)\\ \Hom_A(N, \ \Hom_B(M, J))\end{smallmatrix}\right)_{0, 0}$$
is an injective left $\Lambda$-module. Similarly, $\binom{0}{\Hom_A(N, I)}_{0, 0}$ is an injective left $\Lambda$-module.
Thus, $\binom{\Hom_B(M, J)}{\Hom_A(N, I)}_{0, 0} = \binom{\Hom_B(M, J)}{0}_{0, 0}\oplus \binom{0}{\Hom_A(N, I)}_{0, 0}$ is an injective left $\Lambda$-module.
\hfill $\square$
\subsection {Proof of Theorem \ref{ctp4}.} \ (1) \ Since $A$ is quasi-Frobenius, $_AA\in \ _A\mathcal I$.
Since $B$ is quasi-Frobenius and $_BM$ is projective, \ $_BM\in \ _B\mathcal I$.
Since \ $M_A$ and $N_B$ are flat, it follows from Lemma \ref{proj-injdim}(2) that \ ${\rm inj.dim} _\Lambda \binom{A}{M}_{1, 0} \le 1$.
\vskip5pt
Similarly, since $A$ is quasi-Frobenius and $_AN$ is projective, $_AN\in \ _A\mathcal I$.
Since $B$ is quasi-Frobenius, \ $_BB\in \ _B\mathcal I$.
Since \ $M_A$ and $N_B$ are flat, ${\rm inj.dim} _\Lambda \binom{N}{B}_{0, 1} \le 1,$
by Lemma \ref{proj-injdim}(2).
\vskip5pt
Thus, ${\rm inj.dim} _\Lambda\Lambda \le 1$.
By the right module version of Lemma \ref{proj-injdim}(2) one knows \ ${\rm inj.dim} \Lambda_\Lambda \le 1.$
Thus $\Lambda$ is a Gorenstein ring.
\vskip5pt
Since $\Lambda$ is Gorenstein with ${\rm inj.dim} _\Lambda\Lambda \le 1$, it is well-known that $_\Lambda \mathcal P^{<\infty} = \ _\Lambda \mathcal P^{\le 1} = \ _\Lambda \mathcal I^{\le 1} = \ _\Lambda \mathcal I^{<\infty}.$
\vskip5pt
Since $_AN$ and $_BM$ are projective modules, it follows from Lemma \ref{proj-injdim}(1) that \ $\binom{_A\mathcal P}{_A\mathcal P} \subseteq \ _\Lambda \mathcal P^{\le 1}$. On the other hand, for any $\left(\begin{smallmatrix}X\\ Y\end{smallmatrix}\right)_{f, g}\in \ _\Lambda \mathcal P^{\le 1},$
let $0 \longrightarrow \binom{P_{11}}{P_{12}} \longrightarrow \binom{P_{01}}{P_{02}} \longrightarrow \left(\begin{smallmatrix}X\\ Y\end{smallmatrix}\right)_{f, g} \longrightarrow 0$ be a projective resolution of $\left(\begin{smallmatrix}X\\ Y\end{smallmatrix}\right)_{f, g}$. Then
one has exact sequence $0\longrightarrow P_{11} \longrightarrow P_{01} \longrightarrow X \longrightarrow 0$. Since $_AN$ is projective, $P_{11}$ and $P_{01}$ are projective (cf. Subsection 2.5), and hence injective. Thus the exact sequence splits and hence $X$ is projective. Similarly, $Y$ is projective. This shows $\left(\begin{smallmatrix}X\\ Y\end{smallmatrix}\right)_{f, g}\in \binom{_A\mathcal P}{_A\mathcal P}$. Hence $\ _\Lambda \mathcal P^{\le 1} = \binom{_A\mathcal P}{_A\mathcal P} = \binom{_A\mathcal I}{_A\mathcal I}$.
\vskip5pt
(2) \ \ By (1), $\Lambda$ is Gorenstein and $\binom{_A\mathcal I}{_B\mathcal I} = \ _\Lambda \mathcal P^{<\infty}$. Thus, ${}^\perp\binom{_A\mathcal I}{_B\mathcal I} = {\rm GP}(\Lambda)$
and $({}^\perp\binom{_A\mathcal I}{_B\mathcal I}, \ \binom{_A\mathcal I}{_B\mathcal I})$ is just the Gorenstein-projective cotorsion pair, so it is complete and hereditary.
\vskip5pt
By Theorem \ref{compare}(1), ${\rm Mon}(\Lambda)^\bot \subseteq \binom{_A\mathcal I}{_B\mathcal I} = \ _\Lambda \mathcal P^{<\infty}$.
Thus, to see $({\rm Mon}(\Lambda), \ {\rm Mon}(\Lambda)^\bot) = ({\rm GP}(\Lambda), \ _\Lambda \mathcal P^{<\infty}),$ it suffices to show \ ${\rm Mon}(\Lambda)\subseteq {\rm GP}(\Lambda)$.
Since $\Lambda$ is Gorenstein, ${\rm GP}(\Lambda) = \ ^{\bot_{\ge 1}} \ _\Lambda \mathcal P$. See Subsection 2.9.
While ${\rm inj.dim} _\Lambda\Lambda \le 1$, each projective $\Lambda$-module is of injective dimension $\le 1$.
It follows that \ $^\bot \ _\Lambda \mathcal P = \ ^{\bot_{\ge 1}} \ _\Lambda \mathcal P.$
Thus, it suffices to show \ ${\rm Mon}(\Lambda)\subseteq \ ^\bot \ _\Lambda \mathcal P$, namely,
it suffices to show
$$\Ext_\Lambda^1({\rm Mon}(\Lambda), \ {\rm T}_A(_A\mathcal P)\oplus {\rm T}_B(_B\mathcal P))=0.$$
This is indeed true. In fact, let \ $\left(\begin{smallmatrix} X \\ Y\end{smallmatrix}\right)_{f,g}\in {\rm Mon}(\Lambda).$ For any $P\in \ _A\mathcal P$, there is an exact sequence
$$0\longrightarrow \left(\begin{smallmatrix} 0 \\ M\otimes_AP\end{smallmatrix}\right)_{0,0}\xlongrightarrow{\binom{0}{1}}
{\rm T}_A P = \left(\begin{smallmatrix} P \\ M\otimes_AP\end{smallmatrix}\right)_{1,0}\xlongrightarrow{\binom{1}{0}}
\left(\begin{smallmatrix} P \\ 0 \end{smallmatrix}\right)\longrightarrow 0.$$
By Lemma \ref{extadj2}(2) one has
$$\Ext_\Lambda^1(\left(\begin{smallmatrix} X \\ Y\end{smallmatrix}\right)_{f,g}, \ \left(\begin{smallmatrix} 0 \\ M\otimes_AP\end{smallmatrix}\right)_{0,0})=
\Ext_B^1(\Coker f, \ M\otimes_AP) = 0$$
since $M\otimes_AP$ is projective as a left $B$-module (and hence injective). By Lemma \ref{extadj2}(1), one has
$$\Ext_\Lambda^1(\left(\begin{smallmatrix} X \\ Y\end{smallmatrix}\right)_{f,g}, \ \left(\begin{smallmatrix} P \\ 0 \end{smallmatrix}\right)_{0,0})=
\Ext_A^1(\Coker g, \ P)=0.$$
Thus $\Ext_\Lambda^1(\left(\begin{smallmatrix} X \\ Y\end{smallmatrix}\right)_{f,g}, \ {\rm T}_A P )= 0$. This shows $\Ext_\Lambda^1({\rm Mon}(\Lambda), \ {\rm T}_A(_A\mathcal P))=0.$
\vskip5pt
Similarly, $\Ext_\Lambda^1({\rm Mon}(\Lambda), \ {\rm T}_B(_B\mathcal P))=0.$ Thus \ $({\rm Mon}(\Lambda), \ {\rm Mon}(\Lambda)^\bot) = ({\rm GP}(\Lambda), \ _\Lambda \mathcal P^{<\infty}),$
in particular,
\ ${\rm Mon}(\Lambda) = \mathcal {GP}(\Lambda) = \ {}^\perp \ _\Lambda\mathcal P, \ \ \ \ {\rm Mon}(\Lambda)^\bot = \ _\Lambda \mathcal P^{\le 1}.$
\vskip5pt
The assertion \ ${\rm (2)'}$ \ is the dual of (2). \hfill $\square$
\section{\bf Completeness}
To study abelian model structures on Morita rings, a key step is to know the completeness of
cotorsion pairs in Morita rings.
\subsection{Completeness via cogenerations by sets}
First, by [ET, Theorem 10], one has
\vskip5pt
\begin{prop}\label{generatingcomplete} \ Let $\Lambda = \left(\begin{smallmatrix} A & N \\ M & B\end{smallmatrix}\right)$ be a Morita ring with $\phi = 0=\psi$, $(\mathcal U, \mathcal X)$ and $(\mathcal V, \mathcal Y)$ cotorsion pairs in $A\mbox{-}{\rm Mod}$ and $B\mbox{-}{\rm Mod}$, cogenerated by sets $S_1$ and $S_2$, respectively.
\vskip5pt
$(1)$ \ If $\Tor^A_1(M, \mathcal U)=0 = \Tor^B_1(N, \mathcal V)$, then cotorsion pair $(^\perp\binom{\mathcal X}{\mathcal Y}, \binom{\mathcal X}{\mathcal Y})$ is cogenerated by ${\rm T}_A(S_1) \cup {\rm T}_B(S_2),$ and hence complete.
\vskip5pt
{\rm (2)} \ If \ $M\otimes_A N = 0 = N\otimes_BM$, then cotorsion pair \ $(^\perp\nabla(\mathcal X, \ \mathcal Y), \ \nabla(\mathcal X, \ \mathcal Y))$ is generated by ${\rm Z}_A(S_1) \cup {\rm Z}_B(S_2)$, and hence complete.
\end{prop}
\begin{proof} \ (1) \ By Theorem \ref{ctp1}(1), \ $(^\perp\binom{\mathcal X}{\mathcal Y}, \ \binom{\mathcal X}{\mathcal Y})$ is a cotorsion pair in $\Lambda$-Mod. By Lemma \ref{destheta}(1),
\ $\left (\begin{smallmatrix} \mathcal X \\ \mathcal Y\end{smallmatrix}\right) = \left (\begin{smallmatrix} S_1^\perp \\ S_2^\perp\end{smallmatrix}\right) = ({\rm T}_A(S_1) \cup {\rm T}_B(S_2))^\perp.$ Thus,
\ $(^\perp\binom{\mathcal X}{\mathcal Y}, \ \binom{\mathcal X}{\mathcal Y})$ is complete, by Proposition \ref{cogenerated}.
\vskip5pt
(2) \ Without loss of generality, one may assume that $S_1 \supseteq \ _A\mathcal P$ and
$S_2 \supseteq \ _B\mathcal P$. Then by Lemma \ref{desdelta}(2) one has \
$\nabla(\mathcal X, \ \mathcal Y) = \nabla(S_1^\perp, \ S_2^\perp) = ({\rm Z}_A(S_1) \cup {\rm Z}_B(S_2))^\perp.$
\end{proof}
\vskip5pt
Proposition \ref{generatingcomplete} gives some information on the completeness of the cotorsion pairs in Morita rings.
However, since Proposition \ref{cogenerated} has no dual versions, there are no results on the completeness of $(\left(\begin{smallmatrix}\mathcal U\\ \mathcal V\end{smallmatrix}\right), \left(\begin{smallmatrix}\mathcal U\\ \mathcal V\end{smallmatrix}\right)^\perp)$ and $(\Delta(\mathcal U, \mathcal V)$, $\Delta(\mathcal U, \ \mathcal V)^{\bot})$; moreover, it is more natural to study the completeness of the cotosion pairs given in
Theorems \ref{ctp1} and \ref{ctp6}, directly from the completeness of $(\mathcal U, \mathcal X)$ and $(\mathcal V, \mathcal Y)$, rather than requiring that
they are cogenerated by sets.
Thus, we need module-theoretical methods to the completeness of the cotorsion pairs in Morita rings.
\vskip5pt
Such a general investigation is difficult.
We will deal with this question, by assuming that one of \ $(\mathcal U, \ \mathcal X)$ and \ $(\mathcal V, \ \mathcal Y)$
is arbitrary, and that another is the projective or injective cotorsion pair. In view of Section 4, we only consider cotorsion pairs in Theorem \ref{ctp1}.
\subsection{Main results on completeness} Take $(\mathcal V, \mathcal Y)$ to be an arbitrary complete cotorsion pair in $B$-Mod.
For cotorsion pair $({}^\perp\left(\begin{smallmatrix}\mathcal X\\ \mathcal Y\end{smallmatrix}\right), \left(\begin{smallmatrix}\mathcal X\\ \mathcal Y\end{smallmatrix}\right))$ in Theorem \ref{ctp1}(1),
taking $(\mathcal U, \ \mathcal X) = (_A\mathcal P, A\mbox{-}{\rm Mod})$, we have assertion (1) below;
for cotorsion pair $(\left(\begin{smallmatrix}\mathcal U\\ \mathcal V\end{smallmatrix}\right), \left(\begin{smallmatrix}\mathcal U\\ \mathcal V\end{smallmatrix}\right)^\perp)$ in Theorem \ref{ctp1}(2),
taking $(\mathcal U, \mathcal X) = (A\mbox{-}{\rm Mod}, \ _A\mathcal I)$, we have assertion (2) below.
\begin{thm} \label{ctp2} \ Let \ $\Lambda = \left(\begin{smallmatrix} A & N \\ M & B \end{smallmatrix}\right)$ be a Morita ring with $\phi = 0= \psi$,
and \ $(\mathcal V, \ \mathcal Y)$ a complete cotorsion pair in $B\mbox{-}{\rm Mod}$. Suppose that \ $N_B$ is flat and \ $_BM$ is projective.
\vskip5pt
$(1)$ \ If \ $M\otimes_A\mathcal P\subseteq \mathcal Y$,
then \ $({}^\perp\binom{A\mbox{-}{\rm Mod}}{\mathcal Y}, \ \binom{A\mbox{-}{\rm Mod}}{\mathcal Y})$ is a complete cotorsion pair in $\Lambda\mbox{-}{\rm Mod};$
and it is hereditary if \ $(\mathcal V, \ \mathcal Y)$ is hereditary.
\vskip5pt
Moreover, if \ $M\otimes_AN = 0= N\otimes_BM$, then \ \ ${}^\perp\left (\begin{smallmatrix} A\mbox{-}{\rm Mod} \\ \mathcal Y\end{smallmatrix}\right) = {\rm T}_A(_A\mathcal P)\oplus {\rm T}_B(\mathcal V),$ \ and hence
$$({\rm T}_A(_A\mathcal P)\oplus {\rm T}_B(\mathcal V), \ \left (\begin{smallmatrix} A\mbox{-}{\rm Mod} \\ \mathcal Y\end{smallmatrix}\right))$$ is a complete cotorsion pair$;$ and it is hereditary if \ $(\mathcal V, \ \mathcal Y)$ is hereditary.
\vskip10pt
$(2)$ \ If \ $\Hom_A(N, \ _A\mathcal I) \subseteq \mathcal V$, then \ $(\binom{A\mbox{-}{\rm Mod}}{\mathcal V}, \ \binom{A\mbox{-}{\rm Mod}}{\mathcal V}^\perp)$
is a complete cotorsion pair in $\Lambda\mbox{-}{\rm Mod};$ and it is hereditary if \ $(\mathcal V, \ \mathcal Y)$ is hereditary.
\vskip5pt
Moreover, if \ $M\otimes_AN = 0= N\otimes_BM$, then \ \ $\binom{A\mbox{-}{\rm Mod}}{\mathcal V}^\perp = {\rm H}_A(_A\mathcal I)\oplus {\rm H}_B(\mathcal Y)$, and hence
$$(\left(\begin{smallmatrix} A\mbox{-}{\rm Mod} \\ \mathcal V\end{smallmatrix}\right), \ {\rm H}_A(_A\mathcal I)\oplus {\rm H}_B(\mathcal Y))$$ is a complete cotorsion pair$;$
and it is hereditary if \ $(\mathcal V, \ \mathcal Y)$ is hereditary. \end{thm}
\begin{rem}\label{remcomplete1} (1) \ {\it If $B$ is left noetherian and \ $_BM$ is injective, then \ $M\otimes_A\mathcal P\subseteq \mathcal Y$ always
holds.}
\vskip5pt
$(2)$ \ {\it If \ $B$ is quasi-Frobenius and \ $N_B$ is flat, then \ $\Hom_A(N, \ _A\mathcal I) \subseteq \mathcal V$ always holds.}
\end{rem}
\vskip5pt
Take $(\mathcal U, \mathcal X)$ to be an arbitrary complete cotorsion pair in $A$-Mod.
For cotorsion pair $({}^\perp\left(\begin{smallmatrix}\mathcal X\\ \mathcal Y\end{smallmatrix}\right), \left(\begin{smallmatrix}\mathcal X\\ \mathcal Y\end{smallmatrix}\right))$ in Theorem \ref{ctp1}(1),
taking $(\mathcal V, \mathcal Y) = (_B\mathcal P, B\mbox{-}{\rm Mod})$, we have assertion (1) below;
for cotorsion pair $(\left(\begin{smallmatrix}\mathcal U\\ \mathcal V\end{smallmatrix}\right), \left(\begin{smallmatrix}\mathcal U\\ \mathcal V\end{smallmatrix}\right)^\perp)$ in Theorem \ref{ctp1}(2),
taking $(\mathcal V, \mathcal Y) = (B\mbox{-}{\rm Mod}, \ _B\mathcal I)$, we have assertion (2) below.
\begin{thm} \label{ctp3} \ Let \ $\Lambda = \left(\begin{smallmatrix} A & N \\ M & B \end{smallmatrix}\right)$ be a Morita ring with $\phi = 0= \psi$, and \ $(\mathcal U, \ \mathcal X)$ a complete cotorsion pair in $A\mbox{-}{\rm Mod}$.
Suppose that \ $M_A$ is flat and \ $_AN$ is projective.
\vskip5pt
$(1)$ \ If \ $N\otimes_B \mathcal P\subseteq \mathcal X$,
then \ $({}^\perp\binom{\mathcal X}{B\mbox{-}{\rm Mod}}, \ \binom{\mathcal X}{B\mbox{-}{\rm Mod}})$ is a complete cotorsion pair in $\Lambda\mbox{-}{\rm Mod};$ and it is hereditary if \ $(\mathcal U, \ \mathcal X)$ is hereditary.
\vskip5pt
Moreover, if $M\otimes_AN = 0= N\otimes_BM$, then \ \ ${}^\perp\left(\begin{smallmatrix}\mathcal X\\ B\mbox{-}{\rm Mod}\end{smallmatrix}\right) = {\rm T}_A(\mathcal U)\oplus {\rm T}_B(_B\mathcal P)$, \ and hence
$$ ({\rm T}_A(\mathcal U)\oplus {\rm T}_B(_B\mathcal P),\ \left(\begin{smallmatrix}\mathcal X\\ B\mbox{-}{\rm Mod}\end{smallmatrix}\right))$$ is a complete cotorsion pair$;$ and it is hereditary if \ $(\mathcal U, \ \mathcal X)$ is hereditary.
\vskip10pt
$(2)$ \ If \ $\Hom_B(M, \ _B\mathcal I) \subseteq \mathcal U$, then
\ $(\binom{\mathcal U}{B\mbox{-}{\rm Mod}}, \ \binom{\mathcal U}{B\mbox{-}{\rm Mod}}^\perp)$ is a complete cotorsion pair in $\Lambda\mbox{-}{\rm Mod};$ and it is hereditary if \ $(\mathcal U, \ \mathcal X)$ is hereditary.
\vskip5pt
Moreover, if $M\otimes_AN = 0= N\otimes_BM$, then \ $\left(\begin{smallmatrix}\mathcal U\\ B\mbox{-}{\rm Mod}\end{smallmatrix}\right)^\perp = {\rm H}_A(\mathcal X)\oplus {\rm H}_B(_B\mathcal I)$, \ and hence
$$(\left(\begin{smallmatrix}\mathcal U\\ B\mbox{-}{\rm Mod}\end{smallmatrix}\right), \ {\rm H}_A(\mathcal X)\oplus {\rm H}_B(_B\mathcal I))$$ is a complete cotorsion pair$;$ and it is hereditary if \ $(\mathcal U, \ \mathcal X)$ is hereditary.
\end{thm}
\begin{rem}\label{remcomplete2} $(1)$ \ {\it If \ $A$ is left noetherian and \ $_AN$ is injective, then \ $N\otimes_B\mathcal P\subseteq \mathcal X$ always holds.}
\vskip5pt
(2) \ {\it If \ $A$ is quasi-Frobenius and \ $M_A$ is flat, then \ $\Hom_B(M, \ _B\mathcal I) \subseteq \mathcal U$ always holds.}
\end{rem}
\subsection{Lemmas for Theorem \ref{ctp2}} \ To prove Theorem \ref{ctp2}(1), we need
\vskip5pt
\begin{lem}\label{completeness1} \ Let \ $\Lambda = \left(\begin{smallmatrix} A & N \\ M & B \end{smallmatrix}\right)$ be a Morita ring with $\phi = 0= \psi$.
Suppose $_BM$ is projective. For a $\Lambda$-module $\left(\begin{smallmatrix}L_1\\ L_2\end{smallmatrix}\right)_{f, g}$, let $\pi: P \longrightarrow L_1$ be an epimorphism with $_AP$ projective,
and \ $0\longrightarrow Y\stackrel \sigma \longrightarrow V\stackrel {\pi'}\longrightarrow L_2\longrightarrow 0$ an exact sequence.
Then there is an exact sequence of the form$:$
$$0\rightarrow {\left(\begin{smallmatrix}K \\(M\otimes P)\oplus Y \end{smallmatrix}\right)_{\alpha,\beta}} \longrightarrow
{\left(\begin{smallmatrix} P \\ M\otimes P\end{smallmatrix}\right)_{1, 0}}\oplus
{\left(\begin{smallmatrix} N\otimes V \\ V\end{smallmatrix}\right)_{0, 1}}
\xlongrightarrow {(\left(\begin{smallmatrix} \pi \\ f(1\otimes \pi)\end{smallmatrix}\right), \left(\begin{smallmatrix} g(1\otimes \pi')\\ \pi'\end{smallmatrix}\right))}
{\left(\begin{smallmatrix} L_1 \\ L_2\end{smallmatrix}\right)_{f,g}} \rightarrow 0.$$
\end{lem}
\begin{proof} \ For convenience, rewrite the sequence as
$$0\rightarrow \left(\begin{smallmatrix}K\\(M\otimes_AP)\oplus Y \end{smallmatrix}\right)_{\alpha, \beta} \longrightarrow\left(\begin{smallmatrix} P\oplus (N\otimes_BV) \\ (M\otimes_AP)\oplus V\end{smallmatrix}\right)_{\left(\begin{smallmatrix} 1_{M\otimes P} & 0 \\ 0 & 0 \end{smallmatrix}\right), \left(\begin{smallmatrix} 0 & 0 \\ 0 & 1_{N\otimes V} \end{smallmatrix}\right)}\xlongrightarrow{\left(\begin{smallmatrix}
(\pi, \ g(1_N\otimes \pi')) \\ (f(1_M\otimes \pi), \ \pi')\end{smallmatrix}\right)}
\left(\begin{smallmatrix} L_1 \\ L_2\end{smallmatrix}\right)_{f,g}\rightarrow 0.$$
\vskip5pt
We claim that \ $\left(\begin{smallmatrix}
(\pi, g(1_N\otimes \pi')) \\ (f(1_M\otimes \pi), \ \pi')\end{smallmatrix}\right)$ is a $\Lambda$-epimorphism.
In fact, by $\phi = 0 = \psi$, \ $f(1_M\otimes g) = 0 = g(1_N\otimes f)$. Hence
$$f(1_M\otimes g(1_N\otimes \pi')) = 0: \ M\otimes_AN\otimes_B V\longrightarrow L_2$$
and $$g(1_N\otimes f(1_M\otimes \pi)) = 0: \ N\otimes_B M\otimes P\longrightarrow L_1.$$
Thus, the following diagrams commute:
$$\xymatrix@R=0.7cm@C=1.2cm{(M\otimes_AP)\oplus (M\otimes_AN\otimes_B V)
\ar[d]_-{\left(\begin{smallmatrix} 1_{M\otimes P} &0 \\ 0&0 \end{smallmatrix}\right)}
\ar[rr]^-{(1_M \otimes \pi, 1_M\otimes g(1_N\otimes \pi'))} && M\otimes_AL_1\ar[d]^-f \\
(M\otimes_AP)\oplus V \ar[rr]^-{(f(1_M\otimes \pi), \ \pi')} && L_2}$$
$$\xymatrix@R=0.7cm@C=1.2cm{(N\otimes_BM\otimes_A P)\oplus (N\otimes_BV) \ar[d]_-{\left(\begin{smallmatrix} 0& 0 \\ 0& 1_{N\otimes V} \end{smallmatrix}\right)}
\ar[rr]^-{(1_N\otimes f(1_M\otimes \pi), 1_N\otimes \pi')}&& N\otimes_BL_2 \ar[d]^-g \\
P\oplus (N\otimes_BV) \ar[rr]^-{(\pi, \ g(1_N\otimes \pi'))} && L_1}$$
i.e., $\left(\begin{smallmatrix}
(\pi, g(1_N\otimes \pi') \\ (f(1_M\otimes \pi), \ \pi')\end{smallmatrix}\right)$ is a $\Lambda$-map. Clearly, it is an epimorphism.
\vskip5pt
It remains to see that \ $\Ker \left(\begin{smallmatrix}
(\pi, g(1_N\otimes \pi') \\ (f(1_M\otimes \pi), \ \pi')\end{smallmatrix}\right)$ is of the form $\left(\begin{smallmatrix}K\\ (M\otimes_A P)\oplus Y\end{smallmatrix}\right)_{\alpha, \beta}$.
\vskip5pt
In fact, as a $\Lambda$-module, $\Ker \left(\begin{smallmatrix}
(\pi, g(1_N\otimes \pi') \\ (f(1_M\otimes \pi), \ \pi')\end{smallmatrix}\right)$ is of the form
$\left(\begin{smallmatrix} K\\ K'\end{smallmatrix}\right)_{\alpha,\beta}$,
where $K' = \Ker (f(1_M\otimes \pi), \ \pi')$.
Thus, it suffices to show \ $\Ker (f(1_M\otimes \pi), \ \pi') \cong (M\otimes_AP)\oplus Y.$
\vskip5pt
Since \ $_AP$ is projective, \ $M\otimes_AP$ is a direct summand of copies of \ $_BM$, as a left $B$-module.
While by assumption \ $_BM$ is projective, it follows that
\ $M\otimes_AP$ is a projective left $B$-module. Thus, there is a $B$-map $h$ such that the following diagram commutes:
$$\xymatrix{& & M\otimes_AP\ar@{-->}[d]_{h} \ar[r]^-{1_M\otimes \pi} & M\otimes L_1\ar[d]^f \ar[r] & 0 \\
0\ar [r] & Y \ar[r]^-\sigma & V \ar[r]^{\pi'} & L_2 \ar[r] & 0.}$$
Then it is clear that
$$\xymatrix{0 \ar[r] & (M\otimes_AP)\oplus Y \ar[rr]^-{\left(\begin{smallmatrix} 1 & 0\\ -h & \sigma\end{smallmatrix}\right)} && (M\otimes_AP)\oplus V\ar[rr]^-{(f(1_M\otimes \pi), \ \pi')}
&& L_2 \ar[r] & 0}$$
is exact. This completes the proof.
\end{proof}
\vskip5pt
To prove Theorem \ref{ctp2}(2), we need the following lemma, in which it is more convenient to write
a $\Lambda$-module in the second expression.
\begin{lem}\label{completeness3} \ Let \ $\Lambda = \left(\begin{smallmatrix} A & N \\ M & B \end{smallmatrix}\right)$ be a Morita ring with $\phi = 0= \psi$. Suppose $N_B$ is flat. For $\Lambda$-module $\left(\begin{smallmatrix}L_1\\ L_2\end{smallmatrix}\right)_{\widetilde{f}, \widetilde{g}}$, let $\sigma: L_1 \longrightarrow I$ be a monomorphism with $_AI$ injective,
and \ $0\longrightarrow L_2 \stackrel {\sigma'} \longrightarrow Y\stackrel {\pi}\longrightarrow V\longrightarrow 0$ an exact sequence.
Then there is an exact sequence of the form$:$
$$0\rightarrow \left(\begin{smallmatrix} L_1 \\ L_2\end{smallmatrix}\right)_{\widetilde{f},\widetilde{g}}
\xlongrightarrow{({\left(\begin{smallmatrix} \sigma \\ (N, \sigma)\widetilde{g} \end{smallmatrix}\right)} \\ {\left(\begin{smallmatrix} (M, \sigma')\widetilde{f} \\ \sigma' \end{smallmatrix}\right)})}
\left(\begin{smallmatrix} I \\ \Hom_A(N, I)\end{smallmatrix}\right)_{0, 1} \oplus \left(\begin{smallmatrix} \Hom_B(M, Y) \\ Y\end{smallmatrix}\right)_{1, 0}
\longrightarrow
\left(\begin{smallmatrix} C \\ \Hom_A(N, I)\oplus V\end{smallmatrix}\right)_{\widetilde{\alpha},\widetilde{\beta}}\rightarrow 0.$$
\end{lem}
\begin{proof} Rewrite the sequence as
$$0\rightarrow \left(\begin{smallmatrix} L_1 \\ L_2\end{smallmatrix}\right)_{\widetilde{f},\widetilde{g}}
\xlongrightarrow{\left(\begin{smallmatrix} {\left(\begin{smallmatrix} \sigma \\ (M, \sigma')\widetilde{f} \end{smallmatrix}\right)} \\ {\left(\begin{smallmatrix} (N, \sigma)\widetilde{g} \\ \sigma' \end{smallmatrix}\right)} \end{smallmatrix}\right)} \left(\begin{smallmatrix} I\oplus \Hom_B(M, Y) \\ \Hom_A(N, I)\oplus Y\end{smallmatrix}\right)_{{\left(\begin{smallmatrix} 0 & 0 \\ 0 & 1\end{smallmatrix}\right)}, {\left(\begin{smallmatrix} 1 & 0 \\ 0 & 0 \end{smallmatrix}\right)}}
\longrightarrow
\left(\begin{smallmatrix} C \\ \Hom_A(N, I)\oplus V\end{smallmatrix}\right)_{\widetilde{\alpha}, \widetilde{\beta}}\rightarrow 0.$$
Since $\phi = 0 = \psi,$ \ $(M, \widetilde{g}) \widetilde{f} = 0 = (N, \widetilde{f}) \widetilde{g},$ and hence
$(M, (N, \sigma)\widetilde{g}) \widetilde{f} = 0 = (N, (M, \sigma')\widetilde{f}) \widetilde{g}.$ Thus
the following diagrams commute:
$$\xymatrix@R=1cm{L_1 \ar[d]_-{\widetilde{f}}\ar[rr]^-{\left(\begin{smallmatrix} \sigma \\ (M, \sigma')\widetilde{f} \end{smallmatrix}\right)} && I\oplus \Hom_A(M,Y)\ar[d]^-{{\left(\begin{smallmatrix} 0 & 0 \\ 0 & 1\end{smallmatrix}\right)}} \\
\Hom_B(M, L_2)\ar[rr]^-{\left(\begin{smallmatrix} (M, (N, \sigma)\widetilde{g}) \\ (M, \sigma')\end{smallmatrix}\right)}&& \Hom_B(M, \Hom_A(N, I))\oplus \Hom_A(M,Y)}$$
$$\xymatrix@R=1cm{L_2 \ar[d]_{\widetilde{g}}\ar[rr]^-{\left(\begin{smallmatrix} (N, \sigma)\widetilde{g} \\ \sigma'\end{smallmatrix}\right)}
&& \Hom_A(N, I)\oplus Y\ar[d]^-{\left(\begin{smallmatrix} 1 & 0 \\ 0 & 0 \end{smallmatrix}\right)} \\
\Hom_A(N, L_1)\ar[rr]^-{\left(\begin{smallmatrix} (N, \sigma)\\ (N, (M, \sigma')\widetilde{f})\end{smallmatrix}\right)} && \Hom_A(N, I)\oplus \Hom_A(N, \Hom_B(M, Y))}.$$
Therefore the map $\left(\begin{smallmatrix} {\left(\begin{smallmatrix} \sigma \\ (M, \sigma')\widetilde{f} \end{smallmatrix}\right)} \\ {\left(\begin{smallmatrix} (N, \sigma)\widetilde{g} \\ \sigma' \end{smallmatrix}\right)} \end{smallmatrix}\right)$ is a $\Lambda$-map. Clearly, it is a monomorphism.
\vskip10pt
Write $\Coker \left(\begin{smallmatrix} {\left(\begin{smallmatrix} \sigma \\ (M, \sigma')\widetilde{f} \end{smallmatrix}\right)} \\ {\left(\begin{smallmatrix} (N, \sigma)\widetilde{g} \\ \sigma' \end{smallmatrix}\right)} \end{smallmatrix}\right)$
as
$\left(\begin{smallmatrix} C\\ C'\end{smallmatrix}\right)_{\widetilde{\alpha},\widetilde{\beta}}$.
Then $C'$ is the cokernel of $B$-monomorphism $\left(\begin{smallmatrix} (N, \sigma)\widetilde{g} \\ \sigma' \end{smallmatrix}\right)$.
Since $N_B$ is flat and $_AI$ is injective, it follows that $\Hom_A(N, I)$ is an injective left $B$-module. Thus there is a $B$-map $h$ such that the diagram
$$\xymatrix{0\ar[r] & L_2\ar[d]_-{\widetilde{g}} \ar[r]^-{\sigma'} & Y\ar@{-->}[d]^-h \ar[r]^-{\pi}& V\ar[r] & 0 \\
0\ar[r] & \Hom_A(N, L_1) \ar[r]^-{(N, \sigma)} & \Hom_A(N, I) & }$$
commutes. Therefore
$$\xymatrix{0 \ar[r] & L_2\ar[rr]^-{\left(\begin{smallmatrix} (N, \sigma)\widetilde{g} \\ \sigma' \end{smallmatrix}\right)} &&
\Hom_A(N, I)\oplus Y\ar[rrr]^-{\left(\begin{smallmatrix} 1_{(N, I)} & -h \\ 0 & \pi \end{smallmatrix}\right)}
&&& \Hom_A(N, I)\oplus V \ar[r] & 0}$$
is an exact sequence. It follows that \ $C' \cong \Hom_A(N, I)\oplus V.$ This completes the proof. \end{proof}
\subsection{Proof of Theorem \ref{ctp2}}
\vskip5pt
(1) \ Since \ $N_B$ is flat, by Theorem \ref{ctp1}(1), \ \ $(^\perp\binom{A\mbox{-}{\rm Mod}}{\mathcal Y}, \ \binom{A\mbox{-}{\rm Mod}}{\mathcal Y})$ is a cotorsion pair;
and it is hereditary if \ $(\mathcal V, \ \mathcal Y)$ is hereditary.
\vskip5pt
Since \ $(\mathcal V, \ \mathcal Y)$ is complete, for any $\Lambda$-module $\left(\begin{smallmatrix}L_1\\ L_2\end{smallmatrix}\right)_{f, g}$,
there is an exact sequence $$0\longrightarrow Y \longrightarrow V\longrightarrow L_2\longrightarrow 0$$
with \ $V\in \mathcal V$ and \ $Y\in\mathcal Y$. Since \ $_BM$ is projective, by Lemma \ref{completeness1},
there is an exact sequence of $\Lambda$-modules of the form$:$
$$0\longrightarrow \left(\begin{smallmatrix}K\\ (M\otimes_AP)\oplus Y\end{smallmatrix}\right)_{\alpha,\beta} \longrightarrow \left(\begin{smallmatrix} P \\ M\otimes_AP\end{smallmatrix}\right)_{1, 0}
\oplus
\left(\begin{smallmatrix} N\otimes_BV \\ V\end{smallmatrix}\right)_{0, \ 1} \longrightarrow
\left(\begin{smallmatrix} L_1 \\ L_2\end{smallmatrix}\right)_{f,g}\longrightarrow 0$$
where $_AP$ is projective. Since by assumption \ $M\otimes_A\mathcal P\subseteq \mathcal Y$,
$(M\otimes_AP)\oplus Y\in \mathcal Y$, and hence $\left(\begin{smallmatrix}K\\ (M\otimes_AP)\oplus Y \end{smallmatrix}\right)_{\alpha, \beta}\in
\binom{A\mbox{-}{\rm Mod}}{\mathcal Y}.$
\vskip5pt
On the other hand, $\left(\begin{smallmatrix} P \\ M\otimes_AP\end{smallmatrix}\right)_{1, 0}
= {\rm T}_AP$ is a projective $\Lambda$-module, so it is in $\ ^\perp\binom{A\mbox{-}{\rm Mod}}{\ _B\mathcal Y}.$
Also, $\left(\begin{smallmatrix} N\otimes_BV \\ V\end{smallmatrix}\right)_{0, 1} = {\rm T}_BV\in {\rm T}_B(\mathcal V)$.
Since $N_B$ is flat and \ $\Ext^1_B(\mathcal V, \ \mathcal Y) = 0$, by Lemma \ref{extadj1}(2), $\left(\begin{smallmatrix} N\otimes_BV \\ V\end{smallmatrix}\right)_{0, 1} = {\rm T}_BV\in
\ ^\perp\binom{A\mbox{-}{\rm Mod}}{\mathcal Y}.$ This shows the completeness of \ $(^\perp\binom{A\mbox{-}{\rm Mod}}{\mathcal Y}, \ \binom{A\mbox{-}{\rm Mod}}{\mathcal Y})$.
\vskip5pt
Finally, if $M\otimes_AN = 0= N\otimes_BM$, then by Corollary \ref{identification1}(1) one has
${}^\perp\left (\begin{smallmatrix} A\mbox{-}{\rm Mod} \\ \mathcal Y\end{smallmatrix}\right) = \Delta(_A\mathcal P, \mathcal V) = {\rm T}_A(_A\mathcal P)\oplus {\rm T}_B(\mathcal V)$.
\vskip10pt
(2) \ Since \ $_BM$ is projective, by Theorem \ref{ctp1}(2), $(\binom{A\mbox{-}{\rm Mod}}{\mathcal V}, \ \binom{A\mbox{-}{\rm Mod}}{\mathcal V}^\perp)$ is a cotorsion pair; and it is hereditary if \ $(\mathcal V, \ \mathcal Y)$ is hereditary.
\vskip5pt
Since \ $(\mathcal V, \ \mathcal Y)$ is complete, for any $\Lambda$-module $\left(\begin{smallmatrix}L_1\\ L_2\end{smallmatrix}\right)_{\widetilde{f}, \widetilde{g}}$,
there is an exact sequence \ $0\longrightarrow L_2 \longrightarrow Y\longrightarrow V\longrightarrow 0$
with \ $Y\in \mathcal Y$ and \ $V\in\mathcal V$. Since \ $N_B$ is flat, by Lemma \ref{completeness3}, there is an exact sequence of $\Lambda$-modules of the form$:$
$$0\rightarrow \left(\begin{smallmatrix} L_1 \\ L_2\end{smallmatrix}\right)_{\widetilde{f},\widetilde{g}}
\longrightarrow
\left(\begin{smallmatrix} I \\ \Hom_A(N, I)\end{smallmatrix}\right)_{0, 1} \oplus \left(\begin{smallmatrix} \Hom_B(M, Y) \\ Y\end{smallmatrix}\right)_{1, 0}
\longrightarrow
\left(\begin{smallmatrix} C \\ \Hom_A(N, I)\oplus V\end{smallmatrix}\right)_{\widetilde{\alpha},\widetilde{\beta}}\rightarrow 0$$
where $_AI$ is injective. Since $\left(\begin{smallmatrix} I \\ \Hom_A(N, I)\end{smallmatrix}\right)_{0, 1}$ is an injective $\Lambda$-module, it is in $\binom{A\mbox{-}{\rm Mod}}{\mathcal V}^\perp$.
Since \ $_BM$ is projective and $\Ext^1_B(\mathcal V, \mathcal Y) = 0$, it follows from
Lemma \ref{extadj1}(4) that \ $\left(\begin{smallmatrix} \Hom_B(M, Y) \\ Y\end{smallmatrix}\right)_{1, 0} = {\rm H}_BY\in
\ \binom{A\mbox{-}{\rm Mod}}{\mathcal V}^\perp.$
\vskip5pt
Since by assumption \ $\Hom_A(N, \ _A\mathcal I)\subseteq \mathcal V$, \ $\Hom_A(N, I) \in \mathcal V$, and hence $\left(\begin{smallmatrix} C \\ \Hom_A(N, I)\oplus V\end{smallmatrix}\right)_{\widetilde{\alpha},\widetilde{\beta}}\in
\binom{A\mbox{-}{\rm Mod}}{\mathcal V}.$ This proves the completeness of \ $(\binom{A\mbox{-}{\rm Mod}}{\mathcal V}, \ \binom{A\mbox{-}{\rm Mod}}{\mathcal V}^\perp)$.
\vskip5pt
Finally, if $M\otimes_AN = 0= N\otimes_BM$, then by Corollary \ref{identification1}(3) one has \ $\binom{A\mbox{-}{\rm Mod}}{\mathcal V}^\perp = \nabla(_A\mathcal I, \ \mathcal Y) ={\rm H}_A(_A\mathcal I)\oplus {\rm H}_B(\mathcal Y)$. \hfill $\square$
\vskip5pt
\subsection{Lemmas for Theorem \ref{ctp3}} To see Theorem \ref{ctp3}(1), we need
\begin{lem}\label{completeness2} \ Let \ $\Lambda = \left(\begin{smallmatrix} A & N \\ M & B \end{smallmatrix}\right)$ be a Morita ring with $\phi = 0= \psi$.
Suppose $_AN$ is projective. For a $\Lambda$-module $\left(\begin{smallmatrix}L_1\\ L_2\end{smallmatrix}\right)_{f, g}$,
let $\pi: Q \longrightarrow L_2$ be an epimorphism with $_BQ$ projective, and
\ $0\longrightarrow X\stackrel \sigma \longrightarrow U\stackrel {\pi'}\longrightarrow L_1\longrightarrow 0$ an exact sequence.
Then there is an exact sequence of the form$:$
$$0\rightarrow \left(\begin{smallmatrix}X\oplus (N\otimes Q)\\ K\end{smallmatrix}\right)_{\alpha,\beta}
\longrightarrow
\left(\begin{smallmatrix} U\\ M\otimes U\end{smallmatrix}\right)_{1, 0}
\oplus \left(\begin{smallmatrix} N\otimes Q\\ Q\end{smallmatrix}\right)_{0, 1}
\xlongrightarrow{(\left(\begin{smallmatrix} \pi' \\ f(1\otimes \pi')\end{smallmatrix}\right), \left(\begin{smallmatrix}
g(1\otimes \pi) \\ \pi \end{smallmatrix}\right))}
\left(\begin{smallmatrix} L_1 \\ L_2\end{smallmatrix}\right)_{f,g}\rightarrow 0.$$
\end{lem}
\begin{proof} The proof is similar to Lemma \ref{completeness1}. We include the points.
Rewrite the sequence as
$$0\rightarrow \left(\begin{smallmatrix}X\oplus (N\otimes_BQ)\\ K\end{smallmatrix}\right)_{\alpha,\beta}
\longrightarrow\left(\begin{smallmatrix} U\oplus (N\otimes_BQ) \\ (M\otimes_AU)\oplus Q\end{smallmatrix}\right)_{\left(\begin{smallmatrix} 1 & 0 \\ 0 & 0 \end{smallmatrix}\right), \left(\begin{smallmatrix} 0 & 0 \\ 0 & 1 \end{smallmatrix}\right)}\xlongrightarrow{\left(\begin{smallmatrix}
(\pi', g(1_N\otimes \pi)) \\ (f(1_M\otimes \pi'), \pi)\end{smallmatrix}\right)}
\left(\begin{smallmatrix} L_1 \\ L_2\end{smallmatrix}\right)_{f,g}\rightarrow 0.$$
The map \ $\left(\begin{smallmatrix}
(\pi', g(1_N\otimes \pi)) \\ (f(1_M\otimes \pi'), \pi)\end{smallmatrix}\right)$ is a $\Lambda$-epimorphism, since the diagrams commute:
$$\xymatrix@R=0.6cm@C=1.2cm{(M\otimes_AU)\oplus (M\otimes_AN\otimes_B Q) \ar[d]_-{\left(\begin{smallmatrix} 1 &0 \\ 0&0 \end{smallmatrix}\right)}
\ar[rr]^-{(1\otimes \pi', 1_M\otimes g(1_N\otimes \pi))} && M\otimes_AL_1 \ar[d]^-f \\
(M\otimes_AU)\oplus Q \ar[rr]^-{(f(1\otimes \pi'), \pi)} && L_2}$$
$$\xymatrix@R=0.6cm@C=1.2cm{(N\otimes_BM\otimes_A U)\oplus (N\otimes_BQ) \ar[d]_-{\left(\begin{smallmatrix} 0& 0 \\ 0& 1 \end{smallmatrix}\right)}
\ar[rr]^-{(1_N\otimes f(1\otimes \pi'), 1_N\otimes \pi)}&& N\otimes_BL_2
\ar[d]^-g \\
U\oplus (N\otimes_BQ)\ar[rr]^-{(\pi', \ g(1_N\otimes \pi))} && L_1.}$$
\vskip5pt
\noindent It remains to prove \ $\Ker (\pi', \ g(1_N\otimes \pi))\cong X\oplus (N\otimes_BQ)$.
Since \ $_AN$ is projective, \ $N\otimes_BQ$ is a projective left $A$-module. Thus, there is an $A$-map $h$ such that the diagram
$$\xymatrix{& & N\otimes_BQ\ar@{-->}[d]_{h} \ar[r]^-{1_N\otimes \pi} & N\otimes L_2\ar[d]^g \ar[r] & 0 \\
0\ar [r] & X \ar[r]^-\sigma & U \ar[r]^{\pi'} & L_1 \ar[r] & 0.}$$
commutes. Then
$$\xymatrix{0 \ar[r] & X\oplus(N\otimes_BQ)\ar[rrr]^-{\left(\begin{smallmatrix} \sigma & -h \\ 0 & 1\end{smallmatrix}\right)} &&& U\oplus (N\otimes_BQ)\ar[rr]^-{(\pi', \ g(1_N\otimes \pi))}
&& L_1 \ar[r] & 0}$$
is exact. This completes the proof.\end{proof}
\vskip5pt
To prove Theorem \ref{ctp3}(2), we need the following lemma, in which
the second expression of a $\Lambda$-module is more convenient.
\begin{lem}\label{completeness4} \ Let \ $\Lambda = \left(\begin{smallmatrix} A & N \\ M & B \end{smallmatrix}\right)$ be a Morita ring with $\phi = 0= \psi$. Suppose $M_A$ is flat.
For $\Lambda$-module $\left(\begin{smallmatrix}L_1\\ L_2\end{smallmatrix}\right)_{\widetilde{f}, \widetilde{g}}$, let $\sigma: L_2 \longrightarrow J$ be a monomorphism with $_BJ$ injective, and
\ $0\longrightarrow L_1 \stackrel {\sigma'} \longrightarrow X\stackrel {\pi}\longrightarrow U\longrightarrow 0$ an exact sequence.
Then there is an exact sequence of the form$:$
$$0\rightarrow \left(\begin{smallmatrix} L_1 \\ L_2\end{smallmatrix}\right)_{\widetilde{f},\widetilde{g}}
\xlongrightarrow{({\left(\begin{smallmatrix} \sigma' \\ (N, \sigma')\widetilde{g} \end{smallmatrix}\right)} \\ {\left(\begin{smallmatrix} (M, \sigma)\widetilde{f} \\ \sigma \end{smallmatrix}\right)})}
\left(\begin{smallmatrix} X \\ \Hom_A(N, X)\end{smallmatrix}\right)_{0, 1} \oplus \left(\begin{smallmatrix} \Hom_B(M, J) \\ J\end{smallmatrix}\right)_{1, 0}
\longrightarrow
\left(\begin{smallmatrix} U\oplus \Hom_B(M, J) \\ C\end{smallmatrix}\right)_{\widetilde{\alpha},\widetilde{\beta}}\rightarrow 0.$$
\end{lem}
\begin{proof} The proof is similar to Lemma \ref{completeness3}. We include the points.
First, as in the proof of Lemma \ref{completeness3}, one can show that
the map $$\left(\begin{smallmatrix} {\left(\begin{smallmatrix} \sigma' \\ (M, \sigma)\widetilde{f} \end{smallmatrix}\right)} \\ {\left(\begin{smallmatrix} (N, \sigma')\widetilde{g} \\ \sigma \end{smallmatrix}\right)} \end{smallmatrix}\right): \left(\begin{smallmatrix} L_1 \\ L_2\end{smallmatrix}\right)_{\widetilde{f},\widetilde{g}}
\longrightarrow \left(\begin{smallmatrix} X\oplus \Hom_B(M, J) \\ \Hom_A(N, X)\oplus J\end{smallmatrix}\right)_{{\left(\begin{smallmatrix} 0 & 0 \\ 0 & 1\end{smallmatrix}\right)}, {\left(\begin{smallmatrix} 1 & 0 \\ 0 & 0 \end{smallmatrix}\right)}}$$ is a $\Lambda$-monomorphism. We omit the details.
\vskip5pt
Write $\Coker \left(\begin{smallmatrix} {\left(\begin{smallmatrix} \sigma' \\ (M, \sigma)\widetilde{f} \end{smallmatrix}\right)} \\ {\left(\begin{smallmatrix} (N, \sigma')\widetilde{g} \\ \sigma \end{smallmatrix}\right)} \end{smallmatrix}\right)$
as
$\left(\begin{smallmatrix} C'\\ C\end{smallmatrix}\right)_{\widetilde{\alpha},\widetilde{\beta}}$.
Then \ $C' \cong \Coker\left(\begin{smallmatrix} \sigma' \\ (M, \sigma)\widetilde{f} \end{smallmatrix}\right)$.
Since $M_A$ is flat and $_BJ$ is injective, it follows that $\Hom_B(M, J)$ is an injective left $A$-module. Thus there is an $A$-map $h$ such that the diagram
$$\xymatrix{0\ar[r] & L_1\ar[d]_-{\widetilde{f}} \ar[r]^-{\sigma'} & X\ar@{-->}[d]^-h \ar[r]^-{\pi}& U\ar[r] & 0 \\
0\ar[r] & \Hom_B(M, L_2) \ar[r]^-{(M, \sigma')} & \Hom_B(M, J) & }$$
commutes. Therefore $$\xymatrix{0 \ar[r] & L_1\ar[rr]^-{\left(\begin{smallmatrix} \sigma' \\ (M, \sigma)\widetilde{f} \end{smallmatrix}\right)} && X\oplus\Hom_B(M, J)\ar[rrr]^-
{\left(\begin{smallmatrix} \pi & 0 \\ -h & 1_{(M, J)} \end{smallmatrix}\right)}
&&& U\oplus \Hom_B(M, J) \ar[r] & 0}$$
is exact, and hence \ $C' \cong U\oplus \Hom_B(M, J).$ \end{proof}
\subsection{Proof of Theorem \ref{ctp3}} \ The proof is similar to Theorem \ref{ctp2}.
\vskip5pt
(1) \ Since \ $M_A$ is flat, by Theorem \ref{ctp1}(1), \ $({}^\perp\binom{\mathcal X}{B\mbox{-}{\rm Mod}}, \ \binom{\mathcal X}{B\mbox{-}{\rm Mod}})$ is a cotorsion pair;
and it is hereditary if \ $(\mathcal U, \ \mathcal X)$ is hereditary.
\vskip5pt
For any $\Lambda$-module $\left(\begin{smallmatrix}L_1\\ L_2\end{smallmatrix}\right)_{f, g}$, \ since \ $(\mathcal U, \ \mathcal X)$ is complete,
there is an exact sequence $$0\longrightarrow X \longrightarrow U\longrightarrow L_1\longrightarrow 0$$
with \ $U\in \mathcal U$, $X\in\mathcal X$.
Since \ $_AN$ is projective, by Lemma \ref{completeness2}, there is an exact sequence of $\Lambda$-modules of the form$:$
$$0\longrightarrow \left(\begin{smallmatrix}X\oplus(N\otimes Q)\\ K\end{smallmatrix}\right)_{\alpha,\beta} \longrightarrow \left(\begin{smallmatrix} U \\ M\otimes U\end{smallmatrix}\right)_{1, 0}
\oplus \left(\begin{smallmatrix} N\otimes Q \\ Q\end{smallmatrix}\right)_{0, 1} \longrightarrow
\left(\begin{smallmatrix} L_1 \\ L_2\end{smallmatrix}\right)_{f,g}\longrightarrow 0$$
where $_BQ$ is projective. Since by assumption $N\otimes_B\mathcal P\subseteq \mathcal X$, it follows that $X\oplus(N\otimes_BQ)\in \mathcal X$, and hence $\left(\begin{smallmatrix}X\oplus(N\otimes Q)\\ K\end{smallmatrix}\right)_{\alpha, \beta}\in
\binom{\mathcal X}{B\mbox{-}{\rm Mod}}.$
\vskip5pt
Since $\left(\begin{smallmatrix} N\otimes Q \\ Q\end{smallmatrix}\right)_{0, 1}\in \ _\Lambda\mathcal P$, it is in \ $^\perp\binom{_A\mathcal X}{B\mbox{-}{\rm Mod}}.$
Since \ $M_A$ is flat and \ $\Ext^1_A(\mathcal U, \mathcal X) = 0$, by Lemma \ref{extadj1}(1), \ $\left(\begin{smallmatrix} U \\ M\otimes U\end{smallmatrix}\right)_{1, 0}
= {\rm T}_AU\in
\ ^\perp\binom{\mathcal X}{B\mbox{-}{\rm Mod}}.$ Thus, $({}^\perp\binom{\mathcal X}{B\mbox{-}{\rm Mod}}, \ \binom{\mathcal X}{B\mbox{-}{\rm Mod}})$ is complete.
\vskip5pt
Finally if $M\otimes_AN = 0= N\otimes_BM$, then by Corollary \ref{identification1}(2) one has
\ \ ${}^\perp\left (\begin{smallmatrix} A\mbox{-}{\rm Mod} \\ \mathcal Y\end{smallmatrix}\right) = \Delta(_A\mathcal P, \mathcal V)= {\rm T}_A(_A\mathcal P)\oplus {\rm T}_B(\mathcal V)$.
\vskip10pt
(2) \ Since \ $_AN$ is projective, by Theorem \ref{ctp1}(2), $(\binom{_A\mathcal U}{B\mbox{-}{\rm Mod}}, \ \binom{_A\mathcal U}{B\mbox{-}{\rm Mod}}^\perp)$ is a cotorsion pair;
and it is hereditary if \ $(\mathcal U, \ \mathcal X)$ is hereditary.
\vskip5pt For any $\Lambda$-module $\left(\begin{smallmatrix}X\\ Y\end{smallmatrix}\right)_{\widetilde{f}, \widetilde{g}}$, \ since \ $(\mathcal U, \ \mathcal X)$ is complete,
there is an exact sequence \ $0\longrightarrow L_1 \longrightarrow X \longrightarrow U\longrightarrow 0$
with \ $X\in\mathcal X$, \ $U\in \mathcal U$. Since \ $M_A$ is flat,
by Lemma \ref{completeness4}, there is an exact sequence$:$
$$0\rightarrow \left(\begin{smallmatrix} L_1 \\ L_2\end{smallmatrix}\right)_{\widetilde{f},\widetilde{g}}
\longrightarrow \left(\begin{smallmatrix} X \\ \Hom_A(N, X)\end{smallmatrix}\right)_{0, 1} \oplus \left(\begin{smallmatrix} \Hom_B(M, J) \\ J\end{smallmatrix}\right)_{1, 0}
\longrightarrow
\left(\begin{smallmatrix} U\oplus \Hom_B(M, J) \\ C\end{smallmatrix}\right)_{\widetilde{\alpha}, \widetilde{\beta}}\rightarrow 0$$
where $_BJ$ is injective. Since \ $\left(\begin{smallmatrix} \Hom_B(M, J) \\ J\end{smallmatrix}\right)_{1, 0}$ is an injective $\Lambda$-module, it is in
$\binom{\ _A\mathcal U}{B\mbox{-}{\rm Mod}}^\perp.$ Since \ $_AN$ is projective and $\Ext^1_A(\mathcal U, \mathcal X) = 0$, by Lemma \ref{extadj1}(3), $\left(\begin{smallmatrix} X \\ \Hom_A(N, X)\end{smallmatrix}\right)_{0, 1}
= {\rm H}_AX\in
\ \binom{\ _A\mathcal U}{B\mbox{-}{\rm Mod}}^\perp.$
\vskip5pt
Since by assumption \ $\Hom_B(M, \ _B\mathcal I)\subseteq \mathcal U$, \ $\Hom_B(M, J)\in\mathcal U$, and hence \ $\left(\begin{smallmatrix} U\oplus \Hom_B(M, J) \\ C \end{smallmatrix}\right)_{\widetilde{\alpha},\widetilde{\beta}}\in
\binom{\ _A\mathcal U}{B\mbox{-}{\rm Mod}}.$ So \
$(\binom{_A\mathcal U}{B\mbox{-}{\rm Mod}}, \ \binom{_A\mathcal U}{B\mbox{-}{\rm Mod}}^\perp)$ is complete.
\vskip5pt
Finally if $M\otimes_AN = 0= N\otimes_BM$, then by Corollary \ref{identification1}(4) one has
$\binom{_A\mathcal U}{B\mbox{-}{\rm Mod}}^\perp = \nabla(\mathcal X, \ _B\mathcal I) = {\rm H}_A(\mathcal X)\oplus {\rm H}_B(_B\mathcal I).$ \hfill $\square$
\vskip10pt
\subsection{Remark} Under the framework of one of $(\mathcal U, \ \mathcal X)$ and
$(\mathcal V, \ \mathcal Y)$ being an arbitrary complete cotorsion pair, and another being the projective or the injective one, the careful reader will find that the completeness of the following cotorsion pairs
\begin{align*}&({}^\perp\left(\begin{smallmatrix}_A\mathcal I\\ \mathcal Y\end{smallmatrix}\right), \ \left(\begin{smallmatrix}_A\mathcal I\\ \mathcal Y\end{smallmatrix}\right)),
\ \ \ \ \ \ ({}^\perp\left(\begin{smallmatrix}\mathcal X\\ _B\mathcal I\end{smallmatrix}\right), \ \left(\begin{smallmatrix}\mathcal X\\ _B\mathcal I\end{smallmatrix}\right)) \ \ \ \ \ \
(\mbox{if} \ M_A \ \mbox{and} \ N_B \ \mbox{are flat})\\ &
(\left(\begin{smallmatrix}_A\mathcal P\\ \mathcal V\end{smallmatrix}\right), \ \left(\begin{smallmatrix}_A\mathcal P\\ \mathcal V\end{smallmatrix}\right)^\perp),
\ \ \ \ \ \ (\left(\begin{smallmatrix}\mathcal U\\ _B\mathcal P\end{smallmatrix}\right), \ \left(\begin{smallmatrix}\mathcal U\\ _B\mathcal P\end{smallmatrix}\right)^\perp) \ \ \ \ \ \
(\mbox{if} \ _BM \ \mbox{and} \ _AN \ \mbox{are projective})\end{align*}
have not been discussed (also they will be not used in constructing Hovey triples in Section 7).
An interesting special cases of $(\left(\begin{smallmatrix}_A\mathcal P\\ _B\mathcal P\end{smallmatrix}\right), \ \left(\begin{smallmatrix}_A\mathcal P\\ _B\mathcal P\end{smallmatrix}\right)^\perp)$ and
$({}^\perp\left(\begin{smallmatrix}_A\mathcal I\\ _B\mathcal I\end{smallmatrix}\right), \ \left(\begin{smallmatrix}_A\mathcal I\\ _B\mathcal I\end{smallmatrix}\right))$
have been treated in Theorem \ref{ctp4}.
\subsection{Triangular matrix rings} \ For the case of $M = 0$ one has
\begin{prop}\label{triangular}
Let $\Lambda=\left(\begin{smallmatrix} A & N \\ 0 & B \end{smallmatrix}\right)$ be an upper triangular matrix ring.
Suppose that \ $(\mathcal U, \ \mathcal X)$ and $(\mathcal V, \ \mathcal Y)$ are complete cotorsion pairs
in $A$-{\rm Mod} and $B$-{\rm Mod}, respectively.
\vskip5pt
$(1)$ \ Assume that $\Tor_1^B(N, \ \mathcal V)=0$. If
\ $N\otimes_B\mathcal V\subseteq \mathcal X$, then the cotorsion pair
$$(\Delta(\mathcal U, \ \mathcal V), \ \left(\begin{smallmatrix} \mathcal X \\ \mathcal Y\end{smallmatrix}\right))
=({\rm T}_A(\mathcal U)\oplus {\rm T}_B(\mathcal V), \ \left(\begin{smallmatrix} \mathcal X \\ \mathcal Y\end{smallmatrix}\right))$$
is complete.
\vskip5pt
$(2)$ \ Assume that $\Ext_A^1(N, \ \mathcal X)=0$. If \ $\Hom_A(N, \ \mathcal X) \subseteq \mathcal V$, then the cotorsion pair
$$(\left(\begin{smallmatrix} \mathcal U \\ \mathcal V\end{smallmatrix}\right), \ \nabla(\mathcal X, \ \mathcal Y))
=(\left(\begin{smallmatrix} \mathcal U \\ \mathcal V\end{smallmatrix}\right), \ {\rm H}_A(\mathcal X)\oplus {\rm H}_B(\mathcal Y))$$
is complete.
\end{prop}
For lower triangle matrix rings (i.e., $N = 0$) one has the similar results. We omit the details. For proof of Proposition \ref{triangular} we need
\begin{lem}\label{horseshoe} {\rm ([AA, 3.1])}
Let $\mathcal C$ be an abelian category with enough projectives and injectives.
Assume that $(\mathcal A, \mathcal B)$ be a hereditary cotorsion pair in $\mathcal C$,
and $0\longrightarrow X\xlongrightarrow{f}Y\xlongrightarrow{g}Z\longrightarrow 0$
be an exact sequence in $\mathcal C$.
\vskip5pt
$(1)$ \ Assume that $X$ and $Z$ have special right $\mathcal A$-approximation, i.e., there are exact sequences:
$$0\longrightarrow B_1\longrightarrow A_1\longrightarrow X\longrightarrow 0, \qquad
0\longrightarrow B_2\longrightarrow A_2\longrightarrow Z\longrightarrow 0,$$
with $A_i\in \mathcal A$, $B_i\in \mathcal B$, $i=1, 2$. Then $Y$ has a special right $\mathcal A$-approximation.
\vskip5pt
$(2)$ \ Assume that $X$ and $Z$ have special left $\mathcal B$-approximation, i.e., there are exact sequences
$$0\longrightarrow X\longrightarrow B_1\longrightarrow A_1\longrightarrow 0, \qquad
0\longrightarrow Z\longrightarrow B_2\longrightarrow A_2\longrightarrow 0$$
with $B_i\in \mathcal B$, $A_i\in \mathcal A$, $i=1, 2$.
Then $Y$ has a special left $\mathcal B$-approximation.
\end{lem}
\vskip5pt
\noindent {\bf Proof of Proposition \ref{triangular}.}
(1) \ By Theorem \ref{identify1}(1), $(\Delta(\mathcal U, \ \mathcal V), \ \left(\begin{smallmatrix} \mathcal X \\ \mathcal Y\end{smallmatrix}\right))
=({\rm T}_A(\mathcal U)\oplus {\rm T}_B(\mathcal V), \ \left(\begin{smallmatrix} \mathcal X \\ \mathcal Y\end{smallmatrix}\right))$ is a cotorsion pair. Let $\binom{L_1}{L_2}_g$ be a $\Lambda$-module. By the completeness of $(\mathcal U, \ \mathcal X)$ and $(\mathcal V, \ \mathcal Y)$, one has the exact sequences
$$0\longrightarrow X\xlongrightarrow{\sigma_1}U\xlongrightarrow{\pi_1}L_1\longrightarrow 0, \qquad
0\longrightarrow Y\xlongrightarrow{\sigma_2}V\xlongrightarrow{\pi_2}L_2\longrightarrow 0$$
in $A$-Mod and $B$-Mod respectively, with $U\in \mathcal U$, \ $X\in \mathcal X$, \ $V\in \mathcal V$, and \ $Y\in \mathcal Y$.
Then
$$0\longrightarrow \left(\begin{smallmatrix} X \\ 0 \end{smallmatrix}\right)_0\xlongrightarrow{\binom{\sigma_1}{0}}
\left(\begin{smallmatrix} U \\ 0 \end{smallmatrix}\right)_0\xlongrightarrow{\binom{\pi_1}{0}}
\left(\begin{smallmatrix} L_1 \\ 0 \end{smallmatrix}\right)_0\longrightarrow 0$$
is the special right $\Delta(\mathcal U, \ \mathcal V)$-approximation of $\binom{L_1}{0}_0$. Also, since
$N\otimes_B\mathcal V\subseteq \mathcal X$,
$$0\longrightarrow \left(\begin{smallmatrix} N\otimes_BV \\ Y \end{smallmatrix}\right)_{1\otimes\sigma_2} \xlongrightarrow{\binom{1}{\sigma_2}}
\left(\begin{smallmatrix} N\otimes_BV \\ V \end{smallmatrix}\right)_1\xlongrightarrow{\binom{0}{\pi_2}}
\left(\begin{smallmatrix} 0 \\ L_2 \end{smallmatrix}\right)_0\longrightarrow 0
$$
is the special right $\Delta(\mathcal U, \ \mathcal V)$-approximation of $\binom{0}{L_2}_0$.
Since $$0\longrightarrow \left(\begin{smallmatrix} L_1 \\ 0 \end{smallmatrix}\right) \longrightarrow
\left(\begin{smallmatrix} L_1 \\ L_2 \end{smallmatrix}\right)_g \longrightarrow
\left(\begin{smallmatrix} 0 \\ L_2 \end{smallmatrix}\right) \longrightarrow 0$$
is exact, it follows from Lemma \ref{horseshoe}(1) that
$\Lambda$-module $\binom{L_1}{L_2}_g$ has a special right $\Delta(\mathcal U, \ \mathcal V)$-approximation
$$0\longrightarrow \left(\begin{smallmatrix} X' \\ Y' \end{smallmatrix}\right)_s \longrightarrow
\left(\begin{smallmatrix} U' \\ V' \end{smallmatrix}\right)_h \longrightarrow
\left(\begin{smallmatrix} L_1 \\ L_2 \end{smallmatrix}\right)_g \longrightarrow 0$$
with $\binom{U'}{V'}_h\in \Delta(\mathcal U, \ \mathcal V)$ and $\binom{X'}{Y'}_s\in \binom{\mathcal X}{\mathcal Y}$.
This proves the completeness.
\vskip5pt
The assertion (2) can be similarly proved.
\hfill $\square$
\vskip5pt
Theorems \ref{ctp2}, \ref{ctp3}, and Proposition \ref{triangular} are new, even when $M = 0$ or $N =0$.
\section{\bf Realizations}
In Table 1, taking \ $(\mathcal U, \ \mathcal X)$ and \ $(\mathcal V, \ \mathcal Y)$ to be the projective cotorsion pair or the injective cotorsion pair, we get Table 2 below.
This section is to show that these cotorsion pairs in Table 2 are pairwise generally different and ``new" in some sense. For details see Definitions \ref{difference} and \ref{new}, Propositions \ref{different}, \ref{newI}, \ref{different2} and \ref{newII}.
All these results are new, even for $M= 0$ or $N = 0$. Thus, it turns out that Morita rings are rich in producing ``new" cotorsion pairs.
\subsection{Cotorsion pairs in Series I in Table 2 are pairwise generally different} To save the space, in Table 2 we use
\ $\mathcal A: = A\mbox{-}{\rm Mod}, \ \ \mathcal B: = B\mbox{-}{\rm Mod}, \ \ \mbox{proj.}: = \mbox{projective},$ \
$\mathcal M: = {\rm Mon}(\Lambda) = \Delta(A\mbox{\rm-Mod}, \ B\mbox{\rm-Mod})$ \ and \ $\mathcal E: = {\rm Epi}(\Lambda) = \nabla(A\mbox{\rm-Mod}, \ B\mbox{\rm-Mod}).$
\vskip5pt
{\bf About Table 2:} (i) \ It is clear that (see also Subsection 3.1)
$$(\Delta(_A\mathcal P, \ _B\mathcal P), \ \Delta(_A\mathcal P, \ _B\mathcal P)^\bot) = (_\Lambda\mathcal P, \ \Lambda\mbox{\rm-Mod}); \ \ \ (^\bot\nabla(_A\mathcal I, \ _B\mathcal I), \ \nabla(_A\mathcal I, \ _B\mathcal I)) = (\Lambda\mbox{-}{\rm Mod}, \ _\Lambda\mathcal I).$$
\vskip5pt
(ii) \ Denote by $R_{\mathcal X, \ \mathcal Y}$ the cotorsion pair where \
$\left(\begin{smallmatrix}\mathcal X\\ \mathcal Y\end{smallmatrix}\right)$ is at the right hand side, i.e.,
$R_{\mathcal X, \ \mathcal Y} = ({}^\perp\left(\begin{smallmatrix}\mathcal X\\ \mathcal Y\end{smallmatrix}\right), \ \left(\begin{smallmatrix}\mathcal X\\ \mathcal Y\end{smallmatrix}\right))$.
Similarly, \ $L_{\mathcal U, \ \mathcal V} = (\left(\begin{smallmatrix}\mathcal U\\ \mathcal V\end{smallmatrix}\right), \ \left(\begin{smallmatrix}\mathcal U\\ \mathcal V\end{smallmatrix}\right)^\perp)$.
\vskip5pt
(iii) \ The cotorsion pairs in columns 2 and 3 in Table 2 are
cotorsion pairs in Series I,
and the ones in columns 4 and 5 are the cotorsion pairs in Series II. See Notation \ref{not}.
\vskip5pt
\vskip10pt
\centerline{\bf Table 2: \ Cotorsion pairs in $\Lambda$-Mod}
\vspace{-10pt}
$${\tiny\begin{tabular}{|c|c|c|c|c|}
\hline
\phantom{\LARGE 0} & \multicolumn{2}{c|} {\tabincell{c}{Hereditary cotorsion pairs in Series I\\[3pt] $\varphi=0=\psi$}}
& \multicolumn{2}{c|} {\tabincell{c}{Cotorsion pairs in Series II\\[3pt] $M\otimes_AN=0 = N\otimes_BM$}}
\\[8pt]\hline
\tabincell{c}{$(_A\mathcal U, \ _A\mathcal X)$ \\ $(_B\mathcal V, \ _B\mathcal Y)$}
& \tabincell{c}{$\Tor_1(M,\mathcal U)=0$ \\ [3pt] $\Tor_1(N,\mathcal V)=0$: \\[3pt] $(^\perp\binom{\mathcal X}{\mathcal Y},\binom{\mathcal X}{\mathcal Y})$}
& \tabincell{c}{$\Ext^1(N,\mathcal X)=0$ \\ [3pt]$\Ext^1(M,\mathcal Y)=0$: \\[3pt] $(\binom{\mathcal U}{\mathcal V},\binom{\mathcal U}{\mathcal V}^\perp)$}
& \tabincell{c}{$(\Delta(\mathcal U,\mathcal V), \ \Delta(\mathcal U, \mathcal V)^\bot)$}
& \tabincell{c}{$(^\bot\nabla(\mathcal X,\mathcal Y), \ \nabla(\mathcal X,\mathcal Y))$}
\\[15pt] \hline
\tabincell{c}{$(\mathcal P, \mathcal A$) \\ $(\mathcal P, \mathcal B)$}
&\tabincell{c}{$(_\Lambda\mathcal P, \Lambda\mbox{\rm-Mod})$}
& \tabincell{c}{$_AN$, $_BM$ proj.: \\[3pt] $(\binom{\mathcal P}{\mathcal P}, \ \binom{\mathcal P}{\mathcal P}^\perp)$}
& \tabincell{c}{$(_\Lambda\mathcal P, \Lambda\mbox{\rm-Mod})$}
& \tabincell{c}{$(^\bot{\rm Epi}(\Lambda), \ {\rm Epi}(\Lambda)).$ \\[3pt] Even if $_AN$, $_BM$ proj., \\[3pt]$(^\bot\mathcal E, \mathcal E)\ne (\binom{\mathcal P}{\mathcal P}, \ \binom{\mathcal P}{\mathcal P}^\perp)$\\[3pt] in general.}
\\[20pt] \hline
\tabincell{c}{$(\mathcal P, \mathcal A$) \\ $(\mathcal B, \mathcal I)$}
& \tabincell{c}{$N_B$ flat: \\ [3pt] $(^\perp\binom{\mathcal A}{\mathcal I}, \ \binom{\mathcal A}{\mathcal I})$}
& \tabincell{c}{$_AN$ proj.: \\[3pt] $(\binom{\mathcal P}{\mathcal B}, \ \binom{\mathcal P}{\mathcal B}^\perp)$}
& \tabincell{c}{$(\Delta(\mathcal P, \mathcal B), (\Delta(\mathcal P, \mathcal B)^\bot).$ \\ [3pt] If $N_B$ flat then it is \\ [3pt] $(^\perp\binom{\mathcal A}{\mathcal I}, \ \binom{\mathcal A}{\mathcal I})$\\ [3pt] thus it is \\ [3pt] $(\Delta(\mathcal P, \mathcal B), \ \binom{\mathcal A}{\mathcal I})$}
& \tabincell{c}{$(^\perp\nabla(\mathcal A, \mathcal I), \nabla(\mathcal A, \mathcal I)).$ \\ [3pt] If $_AN$ proj. then it is \\ [3pt]$(\binom{\mathcal P}{\mathcal B}, \ \binom{\mathcal P}{\mathcal B}^\perp)$\\[3pt]thus it is \\[3pt]$(\binom{\mathcal P}{\mathcal B}, \ \nabla(\mathcal A, \mathcal I))$}
\\[30pt] \hline
\tabincell{c}{$(\mathcal A, \mathcal I$) \\ $(\mathcal P, \mathcal B)$}
& \tabincell{c}{$M_A$ flat: \\[3pt] $(^\perp\binom{\mathcal I}{\mathcal B}, \ \binom{\mathcal I}{\mathcal B})$}
& \tabincell{c}{$_BM$ proj.: \\[3pt] $(\binom{\mathcal A}{\mathcal P}, \ \binom{\mathcal A}{\mathcal P}^\perp)$}
& \tabincell{c}{$(\Delta(\mathcal A, \mathcal P), \Delta(\mathcal A, \mathcal P)^\bot).$ \\[3pt] If $M_A$ flat then it is \\[3pt]$(^\perp\binom{\mathcal I}{\mathcal B}, \binom{\mathcal I}{\mathcal B})$\\[3pt] thus it is \\ [3pt]$((\Delta(\mathcal A, \mathcal P), \ \binom{\mathcal I}{\mathcal B})$}
& \tabincell{c}{$(^\bot\nabla(\mathcal I, \mathcal B), \nabla(\mathcal I, \mathcal B))$. \\[3pt] If $_BM$ proj. then it is\\[3pt]$(\binom{\mathcal A}{\mathcal P}, \ \binom{\mathcal A}{\mathcal P}^\perp)$\\[3pt]thus it is \\[3pt] $(\binom{\mathcal A}{\mathcal P}, \ \nabla(\mathcal I, \mathcal B))$}
\\[30pt] \hline
\tabincell{c}{$(\mathcal A, \mathcal I)$ \\ $(\mathcal B, \mathcal I)$}
& \tabincell{c}{$M_A$, $N_B$ flat: \\[3pt] $(^\perp\binom{\mathcal I}{\mathcal I}, \ \binom{\mathcal I}{\mathcal I})$}
& \tabincell{c}{$(\Lambda\mbox{-}{\rm Mod}, \ _\Lambda\mathcal I)$}
& \tabincell{c}{$({\rm Mon}(\Lambda), \ {\rm Mon}(\Lambda)^\bot)$.\\[3pt]Even if $M_A$, $N_B$ flat,\\ [3pt] $(\mathcal M, \ \mathcal M^\bot)\ne (^\perp\binom{\mathcal I}{\mathcal I}, \ \binom{\mathcal I}{\mathcal I})$\\ [3pt]in general}
& \tabincell{c} {$(\Lambda\mbox{-}{\rm Mod}, \ _\Lambda\mathcal I)$}
\\[20pt]\hline
\end{tabular}}$$
\vskip10pt
From the proof of Proposition \ref{different} we will see that, in the most cases,
the eight hereditary cotorsion pairs in Series I in Table 2 are pairwise different.
\vskip5pt
\begin{prop}\label{different} \ Let $\Lambda = \left(\begin{smallmatrix} A & N \\
M & B\end{smallmatrix}\right)$ be a Morita ring with $\phi = 0=\psi$.
Then the eight hereditary cotorsion pairs in {\rm Series I} in {\rm Table 2} are pairwise generally different, in the sense of {\rm Definition \ref{difference}}.\end{prop}
\begin{proof} \ All together there are $\binom{8}{2} = 28$ situations.
\vskip5pt {\bf Step 1.} \ If \ $A$ and $B$ are not semisimple,
then the cotorsion pairs in {\rm Series I} in the same columns are pairwise different. This occupies $2\binom{4}{2} = 12$ situations.
\vskip5pt
For example, since $A$ is not semisimple, $A\mbox{-}{\rm Mod}\ne \ _A\mathcal I.$
Thus \ $\left(\begin{smallmatrix} A\mbox{-}{\rm Mod}\\ _B\mathcal I\end{smallmatrix}\right)
\ne \left(\begin{smallmatrix} _A\mathcal I\\ _B\mathcal I\end{smallmatrix}\right),$
and hence
$$(^\perp\left(\begin{smallmatrix} A\mbox{-}{\rm Mod}\\ _B\mathcal I\end{smallmatrix}\right), \ \left(\begin{smallmatrix} A\mbox{-}{\rm Mod}\\ _B\mathcal I\end{smallmatrix}\right))
\ne (^\perp\left(\begin{smallmatrix} _A\mathcal I\\ _B\mathcal I\end{smallmatrix}\right), \ \left(\begin{smallmatrix} _A\mathcal I\\ _B\mathcal I\end{smallmatrix}\right)).$$
{\bf Step 2.} \ The projective cotorsion pair $(_\Lambda\mathcal P, \ \Lambda\mbox{\rm-Mod})$
is generally different from all other seven cotorsion pairs in Series I. This occupies $4$ situations.
\vskip5pt
In fact, taking \ $_AN = A = B = \ _BM \ne 0$, then \
$$\left(\begin{smallmatrix} N\\ 0\end{smallmatrix}\right)_{0, 0}\in \left(\begin{smallmatrix}_A\mathcal P\\ _B\mathcal P\end{smallmatrix}\right), \ \ \ \
\left(\begin{smallmatrix} N\\ 0\end{smallmatrix}\right)_{0, 0}\in \left(\begin{smallmatrix}_A\mathcal P\\ B\mbox{-}{\rm Mod}\end{smallmatrix}\right), \ \ \ \
\left(\begin{smallmatrix} N\\ 0\end{smallmatrix}\right)_{0, 0}\in \left(\begin{smallmatrix} A\mbox{-}{\rm Mod}\\ _B\mathcal P\end{smallmatrix}\right), \ \ \ \ \left(\begin{smallmatrix} N\\ 0\end{smallmatrix}\right)_{0, 0}\in \Lambda\mbox{-}{\rm Mod}$$
but \ $\binom{N}{0}_{0, 0}\notin \ _\Lambda\mathcal P$.
Thus \ $(_\Lambda\mathcal P, \ \Lambda\mbox{\rm-Mod})$ is generally different from
$$(\left(\begin{smallmatrix}_A\mathcal P\\ _B\mathcal P\end{smallmatrix}\right), \ \left(\begin{smallmatrix}_A\mathcal P\\ _B\mathcal P\end{smallmatrix}\right)^\perp), \ \ \ (\left(\begin{smallmatrix}_A\mathcal P\\ B\mbox{-}{\rm Mod}\end{smallmatrix}\right), \ \left(\begin{smallmatrix}_A\mathcal P\\ B\mbox{-}{\rm Mod}\end{smallmatrix}\right)^\perp), \ \ \ (\left(\begin{smallmatrix}A\mbox{-}{\rm Mod}\\ _B\mathcal P\end{smallmatrix}\right), \ \left(\begin{smallmatrix}A\mbox{-}{\rm Mod}\\ _B\mathcal P\end{smallmatrix}\right)^\perp), \ \ \ (\Lambda\mbox{\rm-Mod}, \ _\Lambda\mathcal I).$$
{\bf Step 3.} \ Similarly, the injective cotorsion pair $(\Lambda\mbox{\rm-Mod}, \ _\Lambda\mathcal I)$
is generally different from all other seven cotorsion pairs in Series I. This occupies $3$ situations.
\vskip5pt
{\bf Step 4.} \ Assume that \ $A$ and $B$ are not semisimple. Under some extra conditions we will show the following remaining $9$ cases (listed in the order of comparing each cotorsion pair with the ones after):
\begin{align*}& L_{_A\mathcal P, \ _B\mathcal P} \ne R_{A\mbox{-}{\rm Mod}, \ _B\mathcal I}; \ \ \ \ \ \ \ \ L_{_A\mathcal P, \ _B\mathcal P}\ne R_{_A\mathcal I, \ B\mbox{-}{\rm Mod}};
\ \ \ \ \ \ \ \ \ \ L_{_A\mathcal P, \ _B\mathcal P}\ne R_{_A\mathcal I, \ _B\mathcal I};
\\ &
R_{A\mbox{-}{\rm Mod}, \ _B\mathcal I} \ne L_{_A\mathcal P, \ B\mbox{-}{\rm Mod}}; \ \ \ \ R_{A\mbox{-}{\rm Mod}, \ _B\mathcal I} \ne L_{A\mbox{-}{\rm Mod}, \ _B\mathcal P};
\ \ \ \ \ \ L_{_A\mathcal P, B\mbox{-}{\rm Mod}}\ne R_{_A\mathcal I, \ B\mbox{-}{\rm Mod}};
\\ &
L_{_A\mathcal P, B\mbox{-}{\rm Mod}}\ne R_{_A\mathcal I, \ _B\mathcal I}; \ \ \ \ \ \ \ \ \
R_{_A\mathcal I, \ B\mbox{-}{\rm Mod}}\ne L_{A\mbox{-}{\rm Mod}, \ _B\mathcal P}; \ \ \ \ \ \
L_{A\mbox{-}{\rm Mod}, \ _B\mathcal P}\ne R_{_A\mathcal I, \ _B\mathcal I}.
\end{align*}
To see the inequalities involving $L_{_A\mathcal P, \ _B\mathcal P}= (\binom{_A\mathcal P}{_B\mathcal P}, \ \binom{_A\mathcal P}{_B\mathcal P}^\perp)$, it suffices to show
$$\left(\begin{smallmatrix}_A\mathcal P\\ _B\mathcal P\end {smallmatrix}\right) \ne \ ^\perp\left(\begin{smallmatrix}A\mbox{-}{\rm Mod} \\ _B\mathcal I\end {smallmatrix}\right), \ \ \ \
\left(\begin{smallmatrix}_A\mathcal P\\ _B\mathcal P\end {smallmatrix}\right) \ne \ ^\perp\left(\begin{smallmatrix}_A\mathcal I\\ B\mbox{-}{\rm Mod}\end {smallmatrix}\right), \ \ \ \
\left(\begin{smallmatrix}_A\mathcal P\\ _B\mathcal P\end {smallmatrix}\right) \ne \ ^\perp\left(\begin{smallmatrix}_A\mathcal I\\_B\mathcal I\end {smallmatrix}\right).$$
Since $B$ is not semisimple, there is a non-projective $B$-module $Y$. By Lemma \ref{extadj1}(2),
\ $ {\rm T}_BY = \left(\begin{smallmatrix}N\otimes_BY\\ Y\end {smallmatrix}\right)_{0, 1}\in \ ^\perp\left(\begin{smallmatrix}A\mbox{-}{\rm Mod}\\ _B\mathcal I\end {smallmatrix}\right),$ but
$\left(\begin{smallmatrix}N\otimes_BY\\ Y\end {smallmatrix}\right)_{0,1}\notin \left(\begin{smallmatrix}_A\mathcal P\\ _B\mathcal P\end {smallmatrix}\right)$.
This shows \ $\left(\begin{smallmatrix}_A\mathcal P\\ _B\mathcal P\end {smallmatrix}\right) \ne \ ^\perp\left(\begin{smallmatrix}A\mbox{-}{\rm Mod}\\ _B\mathcal I\end {smallmatrix}\right)$.
Since $A$ is not semisimple, there is a non-projective $A$-module $X$. By Lemma \ref{extadj1}(1),
\ ${\rm T}_A X= \left(\begin{smallmatrix}X\\ M\otimes_AX\end {smallmatrix}\right)_{1,0}\in \ ^\perp\left(\begin{smallmatrix}_A\mathcal I\\ B\mbox{-}{\rm Mod}\end {smallmatrix}\right)\cap
\ \ ^\perp\left(\begin{smallmatrix}_A\mathcal I\\ _B\mathcal I\end {smallmatrix}\right)$. But $\left(\begin{smallmatrix}X\\ M\otimes_AX\end {smallmatrix}\right)_{1,0}
\notin \left(\begin{smallmatrix}_A\mathcal P\\ _B\mathcal P\end {smallmatrix}\right)$.
This shows \ $\left(\begin{smallmatrix}_A\mathcal P\\ _B\mathcal P\end {smallmatrix}\right) \ne \ ^\perp\left(\begin{smallmatrix}_A\mathcal I\\ B\mbox{-}{\rm Mod}\end {smallmatrix}\right)$ and
\ $\binom{_A\mathcal P}{_B\mathcal P} \ne
\ ^\perp\left(\begin{smallmatrix}_A\mathcal I\\ _B\mathcal I\end {smallmatrix}\right).$
\vskip10pt
For the next inequalities involving \ $R_{A\mbox{-}{\rm Mod}, \ _B\mathcal I}= (^\perp\binom{A\mbox{-}{\rm Mod}}{_B\mathcal I}, \ \binom{A\mbox{-}{\rm Mod}}{_B\mathcal I}),$
we need to find conditions such that
$$^\perp\left(\begin{smallmatrix}A\mbox{-}{\rm Mod}\\ _B\mathcal I\end {smallmatrix}\right) \ne \left(\begin{smallmatrix}_A\mathcal P\\ B\mbox{-}{\rm Mod}\end {smallmatrix}\right), \ \ \ \ \ \ \ ^\perp\left(\begin{smallmatrix}A\mbox{-}{\rm Mod}\\_B\mathcal I\end {smallmatrix}\right) \ne \left(\begin{smallmatrix}A\mbox{-}{\rm Mod}\\ _B\mathcal P\end {smallmatrix}\right).$$
Taking $A = B = M = N\ne 0$ and a non-projective $B$-module $Y$,
then ${\rm T}_BY = \left(\begin{smallmatrix}N\otimes_BY\\ Y\end {smallmatrix}\right)_{0, 1} = \left(\begin{smallmatrix}Y\\ Y\end {smallmatrix}\right)_{0, 1} \in \ ^\perp\left(\begin{smallmatrix}A\mbox{-}{\rm Mod}\\_B\mathcal I\end {smallmatrix}\right)$ by Lemma \ref{extadj1}(2),
but \ $\left(\begin{smallmatrix}Y\\ Y\end {smallmatrix}\right)_{0,1}\notin \left(\begin{smallmatrix}_A\mathcal P\\ B\mbox{-}{\rm Mod}\end {smallmatrix}\right)$ \ and \ $\left(\begin{smallmatrix}Y\\ Y\end {smallmatrix}\right)_{0,1}\notin \left(\begin{smallmatrix}A\mbox{-}{\rm Mod}\\ _B\mathcal P\end {smallmatrix}\right)$.
\vskip10pt
To see $L_{_A\mathcal P, \ B\mbox{-}{\rm Mod}}\ne R_{_A\mathcal I, \ B\mbox{-}{\rm Mod}}$ and \ $L_{_A\mathcal P, \ B\mbox{-}{\rm Mod}}\ne R_{_A\mathcal I, \ _B\mathcal I}$,
it suffices to show
$$\left(\begin{smallmatrix}_A\mathcal P\\ B\mbox{-}{\rm Mod}\end {smallmatrix}\right) \ne \ ^\perp\left(\begin{smallmatrix}_A\mathcal I\\ B\mbox{-}{\rm Mod}\end {smallmatrix}\right),
\ \ \ \ \ \ \ \left(\begin{smallmatrix}_A\mathcal P\\ B\mbox{-}{\rm Mod}\end {smallmatrix}\right) \ne \ ^\perp\left(\begin{smallmatrix}_A\mathcal I\\ _B\mathcal I\end {smallmatrix}\right).$$
For a non-projective $A$-module $X$, \ $ {\rm T}_AX = \binom{X}{M\otimes_AX}_{1, 0}\in \ ^\perp\left(\begin{smallmatrix}_A\mathcal I\\ B\mbox{-}{\rm Mod}\end {smallmatrix}\right)\cap
\ ^\perp\binom{_A\mathcal I}{_B\mathcal I}$,
but $\left(\begin{smallmatrix}X\\ M\otimes_AX \end {smallmatrix}\right)_{1, 0}\notin \binom{_A\mathcal P}{B\mbox{-}{\rm Mod}}$.
\vskip10pt
Finally, we show that \ $L_{A\mbox{-}{\rm Mod}, \ _B\mathcal P}$ is generally different from \ $R_{_A\mathcal I, \ B\mbox{-}{\rm Mod}}$ \ and \ $R_{_A\mathcal I, \ _A\mathcal I}$.
Taking $A = B = M = N\ne 0$ and a non-projective $A$-module $X$, it suffices to see
$${}^\perp\left(\begin{smallmatrix}_A\mathcal I\\ B\mbox{-}{\rm Mod}\end {smallmatrix}\right)\ne \left(\begin{smallmatrix} A\mbox{-}{\rm Mod}\\_B\mathcal P\end {smallmatrix}\right), \ \ \ \ \ \
\left(\begin{smallmatrix} A\mbox{-}{\rm Mod}\\ _B\mathcal P\end {smallmatrix}\right) \ne \ ^\perp\left(\begin{smallmatrix}_A\mathcal I\\_B\mathcal I\end {smallmatrix}\right).$$
In fact, by Lemma \ref{extadj1}(2), \ ${\rm T}_AX = \left(\begin{smallmatrix}X\\ M\otimes_AX\end {smallmatrix}\right)_{1, 0}= \left(\begin{smallmatrix}X\\ X\end {smallmatrix}\right)_{1, 0}\in \ ^\perp\left(\begin{smallmatrix}_A\mathcal I\\ B\mbox{-}{\rm Mod}\end {smallmatrix}\right)\cap \ ^\perp\left(\begin{smallmatrix}_A\mathcal I\\ _B\mathcal I\end {smallmatrix}\right)$;
but $\left(\begin{smallmatrix}X\\ X\end {smallmatrix}\right)_{1, 0}\notin \left(\begin{smallmatrix}A\mbox{-}{\rm Mod}\\ _B\mathcal P\end {smallmatrix}\right)$.
\vskip5pt
This completes the proof. \end{proof}
\subsection{``New" cotorsion pairs in Series I in Table 2}
Taking off the projective cotorsion pair and
the injective one from Series I of Table 2,
the remaining six hereditary cotorsion pairs
\begin{align*}& R_{_A\mathcal I, \ _B\mathcal I} =
(^\perp\left(\begin{smallmatrix}_A\mathcal I\\ _B\mathcal I\end {smallmatrix}\right), \ \left(\begin{smallmatrix}_A\mathcal I\\ _B\mathcal I\end {smallmatrix}\right)),
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ L_{_A\mathcal P, \ _B\mathcal P} =
(\left(\begin{smallmatrix}_A\mathcal P\\ _B\mathcal P\end {smallmatrix}\right), \ \left(\begin{smallmatrix}_A\mathcal P\\ _B\mathcal P\end {smallmatrix}\right)^\perp),
\\ & R_{A\mbox{-}{\rm Mod}, \ _B\mathcal I}= (^\perp\left(\begin{smallmatrix}A\mbox{-}{\rm Mod}\\ _B\mathcal I\end {smallmatrix}\right), \ \left(\begin{smallmatrix}A\mbox{-}{\rm Mod}\\ _B\mathcal I\end {smallmatrix}\right)),
\ \ \ \ \ L_{_A\mathcal P, \ B\mbox{-}{\rm Mod}} = (\left(\begin{smallmatrix}_A\mathcal P\\ B\mbox{-}{\rm Mod}\end {smallmatrix}\right), \
\left(\begin{smallmatrix}_A\mathcal P\\ B\mbox{-}{\rm Mod}\end {smallmatrix}\right)^\perp),
\\ & R_{_A\mathcal I, \ B\mbox{-}{\rm Mod}} = (^\perp\left(\begin{smallmatrix}_A\mathcal I\\ B\mbox{-}{\rm Mod}\end {smallmatrix}\right), \ \left(\begin{smallmatrix}_A\mathcal I\\B\mbox{-}{\rm Mod}\end {smallmatrix}\right)),
\ \ \ \ \ L_{A\mbox{-}{\rm Mod}, \ _B\mathcal P} = (\left(\begin{smallmatrix}A\mbox{-}{\rm Mod}\\ _B\mathcal P\end {smallmatrix}\right), \ (\left(\begin{smallmatrix}A\mbox{-}{\rm Mod}\\ _B\mathcal P\end {smallmatrix}\right)^\perp)\end{align*}
are ``new", in the following sense.
\begin{defn} \label{new} \ A cotorsion pair in $\Lambda\mbox{-}{\rm Mod}$ is said to be ``new", provided that it is generally different from all of the following cotorsion pairs:
\vskip5pt
$\bullet$ \ the projective cotorsion pair \ $(_\Lambda\mathcal P, \ \Lambda\mbox{-}{\rm Mod});$
$\bullet$ \ the injective cotorsion pair \ $(\Lambda\mbox{-}{\rm Mod}, \ _\Lambda\mathcal I);$
$\bullet$ \ the Gorenstein-projective cotorsion pair \ $({\rm GP}(\Lambda), \ _\Lambda\mathcal P^{<\infty})$, if \ $\Lambda$ is a Gorenstein ring;
$\bullet$ \ the Gorenstein-projective cotorsion pair \ $(_\Lambda\mathcal P^{<\infty}, \ {\rm GI}(\Lambda))$, if \ $\Lambda$ is a Gorenstein ring;
$\bullet$ \ the flat cotorsion pair \ $(_\Lambda{\rm F}, \ _\Lambda{\rm C})$.
\end{defn}
\begin{prop}\label{newI} \ Let $\Lambda = \left(\begin{smallmatrix} A & N \\ M & B \end{smallmatrix}\right)$ be a Morita ring with $\phi = 0= \phi$. Then the following six cotorsion pairs
$$R_{_A\mathcal I, \ _B\mathcal I}, \ \ \ L_{_A\mathcal P, \ _B\mathcal P}, \ \ \ R_{A\mbox{-}{\rm Mod}, \ _B\mathcal I}, \ \ \ L_{_A\mathcal P, \ B\mbox{-}{\rm Mod}}, \ \ \ R_{_A\mathcal I, \ B\mbox{-}{\rm Mod}}, \ \ \ L_{A\mbox{-}{\rm Mod}, \ _B\mathcal P}$$ are ``new", in the sense of {\rm Definition \ref{new}}.
\end{prop}
To prove Proposition \ref{newI} we need some preparations.
\begin{lem} \label{agoralg} \ {\rm ([GaP, 4.15])} \ Let $\Lambda = \left(\begin{smallmatrix} A & N \\ N & A \end{smallmatrix}\right)$ be a Morita ring with $N\otimes_AN = 0$.
Assume that \ $_AN$ and $N_A$ are projective. If $A$ is a Gorenstein ring, then so is $\Lambda$.
\end{lem}
\begin{lem}\label{notgor1} \ Let $\Lambda = \left(\begin{smallmatrix} A & N \\ M & B \end{smallmatrix}\right)$ be a Morita ring with $\phi = 0= \phi$. Then the cotorsion pairs
\ $R_{_A\mathcal I, \ _B\mathcal I}$ and $L_{_A\mathcal P, \ _B\mathcal P}$ are generally different from the Gorenstein-projective cotorsion pair and
the Gorenstein-injective cotorsion pair.
\end{lem}
\begin{proof} \ Take \ $\Lambda$ to be the Morita ring $\Lambda = \left(\begin{smallmatrix} A & N \\ N & A \end{smallmatrix}\right)$, constructed in Example \ref{ie}. Thus
$A$ is the path algebra $k(1 \longrightarrow 2)$ with ${\rm char} \ k\ne 2$, $N = Ae_2\otimes_ke_1A$, and $N\otimes_AN = 0$. By Lemma \ref{agoralg}, $\Lambda$ is a Gorenstein algebra.
\vskip5pt
{\bf Claim 1.} \ $R_{_A\mathcal I, \ _B\mathcal I} = (^\perp\left(\begin{smallmatrix}_A\mathcal I\\ _B\mathcal I\end {smallmatrix}\right), \ \left(\begin{smallmatrix}_A\mathcal I\\ _B\mathcal I\end {smallmatrix}\right))$
is generally different from $({\rm GP}(\Lambda), \ _\Lambda\mathcal P^{<\infty})$.
\vskip5pt
In fact, since \ $N\otimes_A S_2 = Ae_2\otimes_k(e_1Ae_2) =0$, \ $\left(\begin{smallmatrix}S_2\\ 0\end{smallmatrix}\right)_{0, 0} = {\rm T}_AS_2$ is a projective $\Lambda$-module, thus
$\left(\begin{smallmatrix}S_2\\ 0\end{smallmatrix}\right)_{0, 0}\in \ _\Lambda\mathcal P^{<\infty}$, but \ $\left(\begin{smallmatrix}S_2\\ 0\end{smallmatrix}\right)_{0, 0}\notin \left(\begin{smallmatrix}_A\mathcal I\\ _A\mathcal I\end{smallmatrix}\right)$. Thus $\left(\begin{smallmatrix}_A\mathcal I\\ _A\mathcal I\end{smallmatrix}\right)\ne \ _\Lambda\mathcal P^{<\infty}$, and hence
$$R_{_A\mathcal I, \ _B\mathcal I} = (^\perp\left(\begin{smallmatrix}_A\mathcal I\\ _B\mathcal I\end {smallmatrix}\right), \ \left(\begin{smallmatrix}_A\mathcal I\\ _B\mathcal I\end {smallmatrix}\right))
\ne ({\rm GP}(\Lambda), \ _\Lambda\mathcal P^{<\infty}).$$
\vskip5pt
{\bf Claim 2.} \ $R_{_A\mathcal I, \ _B\mathcal I} = (^\perp\left(\begin{smallmatrix}_A\mathcal I\\ _B\mathcal I\end {smallmatrix}\right), \ \left(\begin{smallmatrix}_A\mathcal I\\ _B\mathcal I\end {smallmatrix}\right))$
is generally different from $(_\Lambda\mathcal P^{<\infty}, \ {\rm GI}(\Lambda))$.
\vskip5pt
In fact, by Example \ref{ie} one knows
$L = \left(\begin{smallmatrix}Ae_1\\ Ae_1\end{smallmatrix}\right)_{\sigma, \sigma}\notin \ ^\perp\left(\begin{smallmatrix}_A\mathcal I\\ _A\mathcal I\end{smallmatrix}\right).$
The following $\Lambda$-projective resolution of $_\Lambda L$
$$0\longrightarrow \left(\begin{smallmatrix}S_2\\ S_2\end{smallmatrix}\right)_{0, 0}
\stackrel{\left(\begin{smallmatrix} \binom{\sigma}{-1}\\ \binom{1}{-\sigma}\end{smallmatrix}\right)}\longrightarrow \left(\begin{smallmatrix} Ae_1\oplus S_2\\ S_2 \oplus Ae_1\end{smallmatrix}\right)_{\left(\begin{smallmatrix}1 & 0 \\ 0 & 0\end{smallmatrix}\right), \left(\begin{smallmatrix}0 & 0\\ 0 & 1\end{smallmatrix}\right)}
\stackrel{\left(\begin{smallmatrix} (1, \sigma) \\ (\sigma,1)\end{smallmatrix}\right)}
\longrightarrow \left(\begin{smallmatrix}Ae_1\\ Ae_1\end{smallmatrix}\right)_{\sigma, \sigma}\longrightarrow 0$$
shows that ${\rm proj.dim} _\Lambda L = 1$.
So $L\in \ _\Lambda\mathcal P^{<\infty}$, and hence $^\perp\left(\begin{smallmatrix}_A\mathcal I\\ _B\mathcal I\end {smallmatrix}\right)\ne \ _\Lambda\mathcal P^{<\infty}$. Thus
$$R_{_A\mathcal I, \ _B\mathcal I} = (^\perp\left(\begin{smallmatrix}_A\mathcal I\\ _B\mathcal I\end {smallmatrix}\right), \ \left(\begin{smallmatrix}_A\mathcal I\\ _B\mathcal I\end {smallmatrix}\right))\ne (_\Lambda\mathcal P^{<\infty}, \ {\rm GI}(\Lambda)).$$
\vskip5pt
{\bf Claim 3.} \ $L_{_A\mathcal P, \ _B\mathcal P} =
(\left(\begin{smallmatrix}_A\mathcal P\\ _B\mathcal P\end {smallmatrix}\right), \ \left(\begin{smallmatrix}_A\mathcal P\\ _B\mathcal P\end {smallmatrix}\right)^\perp)$
is generally different from $({\rm GP}(\Lambda), \ _\Lambda\mathcal P^{<\infty})$.
\vskip5pt
In fact, by Example \ref{ie} one knows
$L = \left(\begin{smallmatrix}Ae_1\\ Ae_1\end{smallmatrix}\right)_{\sigma, \sigma}\notin \left(\begin{smallmatrix}_A\mathcal P\\ _A\mathcal P\end{smallmatrix}\right)^\perp.$
By {\bf Claim 2}, $L\in \ _\Lambda\mathcal P^{<\infty}$. Thus
$\left(\begin{smallmatrix}_A\mathcal P\\ _A\mathcal P\end{smallmatrix}\right)^\perp \ne \ _\Lambda\mathcal P^{<\infty}$, and hence
$$L_{_A\mathcal P, \ _B\mathcal P} =
(\left(\begin{smallmatrix}_A\mathcal P\\ _B\mathcal P\end {smallmatrix}\right), \ \left(\begin{smallmatrix}_A\mathcal P\\ _B\mathcal P\end {smallmatrix}\right)^\perp)
\ne ({\rm GP}(\Lambda), \ _\Lambda\mathcal P^{<\infty}).$$
\vskip5pt
{\bf Claim 4.} \ $L_{_A\mathcal P, \ _B\mathcal P} =
(\left(\begin{smallmatrix}_A\mathcal P\\ _B\mathcal P\end {smallmatrix}\right), \ \left(\begin{smallmatrix}_A\mathcal P\\ _B\mathcal P\end {smallmatrix}\right)^\perp)$
is generally different from $(_\Lambda\mathcal P^{<\infty}, \ {\rm GI}(\Lambda))$.
\vskip5pt
In fact, since \ $\Hom_A(N, S_1) = 0$, \ $\left(\begin{smallmatrix} 0\\ S_1\end{smallmatrix}\right)_{0, 0} = {\rm H}_BS_1$ is an injective $\Lambda$-module, thus
$\left(\begin{smallmatrix}0\\ S_1\end{smallmatrix}\right)_{0, 0}\in \ _\Lambda\mathcal I^{<\infty} = \ _\Lambda\mathcal P^{<\infty}$,
but \ $\left(\begin{smallmatrix}0\\ S_1\end{smallmatrix}\right)_{0, 0}\notin \left(\begin{smallmatrix}_A\mathcal P\\ _A\mathcal P\end{smallmatrix}\right)$.
Thus $\left(\begin{smallmatrix}_A\mathcal P\\ _A\mathcal P\end{smallmatrix}\right)\ne \ _\Lambda\mathcal P^{<\infty}$, and hence
$$L_{_A\mathcal P, \ _B\mathcal P} =
(\left(\begin{smallmatrix}_A\mathcal P\\ _B\mathcal P\end {smallmatrix}\right), \ \left(\begin{smallmatrix}_A\mathcal P\\ _B\mathcal P\end {smallmatrix}\right)^\perp)
\ne (_\Lambda\mathcal P^{<\infty}, \ {\rm GI}(\Lambda)).$$
This completes the proof. \end{proof}
\begin{lem}\label{notgor2} \ Let $\Lambda = \left(\begin{smallmatrix} A & N \\ M & B \end{smallmatrix}\right)$ be a Morita ring with $\phi = 0= \phi$. Then the cotorsion pairs
$$R_{A\mbox{-}{\rm Mod}, \ _B\mathcal I}, \ \ \ L_{_A\mathcal P, \ B\mbox{-}{\rm Mod}}, \ \ \ R_{_A\mathcal I, \ B\mbox{-}{\rm Mod}}, \ \ \ L_{A\mbox{-}{\rm Mod}, \ _B\mathcal P}$$ are generally different from the Gorenstein-projective cotorsion pair and
the Gorenstein-injective cotorsion pair.
\end{lem}
\begin{proof} \ Choose quasi-Frobenius rings $A$ and $B$, bimodules \ $_BM_A$ and $_AN_B$, satisfying the following conditions (i), (ii), (iii), (iv):
\vskip5pt
(i) \ \ \ $A$ and $B$ are quasi-Frobenius and not semisimple;
(ii) \ \ $_AN$ and $_BM$ are non-zero projective modules, and $M_A$ and $N_B$ are flat;
(iii) \ \ $\Lambda = \left(\begin{smallmatrix} A & N \\ M & B \end{smallmatrix}\right)$ is a Morita ring with $M\otimes_A N = 0 = N\otimes_BM;$
(iv) \ \ $\Lambda$ is a noetherian ring.
\vskip5pt
By Remark \ref{examctp4}, such $\Lambda$'s exist! By Theorem \ref{ctp4}, \ $\Lambda$ is a Gorenstein ring with ${\rm inj.dim} \Lambda \le 1$, the Gorenstein-projective cotorsion pair \ $({\rm GP}(\Lambda), \ \mathcal P^{\le 1})$ is exactly
\ $({}^\perp\binom{_A\mathcal I}{_B\mathcal I}, \ \binom{_A\mathcal I}{_B\mathcal I})$, and the Gorenstein-injective cotorsion pair \ $(_\Lambda \mathcal P^{\le 1}, \ {\rm GI}(\Lambda))$ is exactly \
$(\left(\begin{smallmatrix}_A\mathcal P\\ _B\mathcal P\end {smallmatrix}\right), \ \left(\begin{smallmatrix}_A\mathcal P\\ _B\mathcal P\end {smallmatrix}\right)^\perp)$.
\vskip5pt
{\bf Claim 1.} \ $R_{A\mbox{-}{\rm Mod}, \ _B\mathcal I}$ and \ $R_{_A\mathcal I, \ B\mbox{-}{\rm Mod}}$
are generally different from the Gorenstein-projective cotorsion pair.
Since $A$ and $B$ are not semisimple, $A\mbox{-}{\rm Mod}\ne \ _A\mathcal I$ \ and \ $B\mbox{-}{\rm Mod}\ne \ _B\mathcal I$. Thus \
$\left(\begin{smallmatrix}A\mbox{-}{\rm Mod}\\ _B\mathcal I\end {smallmatrix}\right)\ne \left(\begin{smallmatrix}_A\mathcal I\\ _B\mathcal I\end {smallmatrix}\right)$ and \ $\left(\begin{smallmatrix}_A\mathcal I\\B\mbox{-}{\rm Mod}\end {smallmatrix}\right)\ne \left(\begin{smallmatrix}_A\mathcal I\\ _B\mathcal I\end {smallmatrix}\right)$, and hence
$$R_{A\mbox{-}{\rm Mod}, \ _B\mathcal I}
= (^\perp\left(\begin{smallmatrix}A\mbox{-}{\rm Mod}\\ _B\mathcal I\end {smallmatrix}\right), \ \left(\begin{smallmatrix}A\mbox{-}{\rm Mod}\\ _B\mathcal I\end {smallmatrix}\right))\ne ({}^\perp\left(\begin{smallmatrix}_A\mathcal I\\ _B\mathcal I\end {smallmatrix}\right), \ \left(\begin{smallmatrix}_A\mathcal I\\ _B\mathcal I\end {smallmatrix}\right)) = ({\rm GP}(\Lambda), \ _\Lambda\mathcal P^{\le 1})$$
and
$$R_{_A\mathcal I, \ B\mbox{-}{\rm Mod}} = (^\perp\left(\begin{smallmatrix}_A\mathcal I\\ B\mbox{-}{\rm Mod}\end {smallmatrix}\right), \ \left(\begin{smallmatrix}_A\mathcal I\\B\mbox{-}{\rm Mod}\end {smallmatrix}\right))\ne ({}^\perp\left(\begin{smallmatrix}_A\mathcal I\\ _B\mathcal I\end {smallmatrix}\right), \ \left(\begin{smallmatrix}_A\mathcal I\\ _B\mathcal I\end {smallmatrix}\right)) = ({\rm GP}(\Lambda), \ _\Lambda\mathcal P^{\le 1}).$$
\vskip5pt
{\bf Claim 2.} \ $L_{_A\mathcal P, \ B\mbox{-}{\rm Mod}}$ and \ $L_{A\mbox{-}{\rm Mod}, \ _B\mathcal P}$
are generally different from the Gorenstein-projective cotorsion pair.
Since $A$ is not semisimple, there is a non-projective $A$-module $X$. By Lemma \ref{extadj1}(1),
\ ${\rm T}_AX = \binom{X}{M\otimes_AX}_{1, 0}\in \ ^\perp\binom{_A\mathcal I}{_B\mathcal I}$,
but $\left(\begin{smallmatrix}X\\ M\otimes_AX \end {smallmatrix}\right)_{1, 0} \notin \binom{_A\mathcal P}{B\mbox{-}{\rm Mod}}$,
which shows \ $\binom{_A\mathcal P}{B\mbox{-}{\rm Mod}} \ne \ ^\perp\binom{_A\mathcal I}{_B\mathcal I}.$
Hence
$$L_{_A\mathcal P, \ B\mbox{-}{\rm Mod}} = (\left(\begin{smallmatrix}_A\mathcal P\\ B\mbox{-}{\rm Mod}\end {smallmatrix}\right), \ \left(\begin{smallmatrix}_A\mathcal P\\ B\mbox{-}{\rm Mod}\end {smallmatrix}\right)^\perp)\ne
({}^\perp\left(\begin{smallmatrix}_A\mathcal I\\ _B\mathcal I\end {smallmatrix}\right), \ \left(\begin{smallmatrix}_A\mathcal I\\ _B\mathcal I\end {smallmatrix}\right))= ({\rm GP}(\Lambda), \ _\Lambda\mathcal P^{\le 1}).$$
Similarly, \ $L_{A\mbox{-}{\rm Mod}, \ _B\mathcal P} = (\left(\begin{smallmatrix}A\mbox{-}{\rm Mod}\\ _B\mathcal P\end {smallmatrix}\right), \ (\left(\begin{smallmatrix}A\mbox{-}{\rm Mod}\\ _B\mathcal P\end {smallmatrix}\right)^\perp)\ne
({}^\perp\left(\begin{smallmatrix}_A\mathcal I\\ _B\mathcal I\end {smallmatrix}\right), \ \left(\begin{smallmatrix}_A\mathcal I\\ _B\mathcal I\end {smallmatrix}\right))= ({\rm GP}(\Lambda), \ _\Lambda\mathcal P^{\le 1})
.$
\vskip5pt
{\bf Claim 3.} \ $R_{A\mbox{-}{\rm Mod}, \ _B\mathcal I}$ and \ $R_{_A\mathcal I, \ B\mbox{-}{\rm Mod}}$ are generally different from the Gorenstein-injective cotorsion pair.
Since $B$ is not semisimple, there is a non-projective $B$-module $Y$. Then
\ $ {\rm T}_BY = \left(\begin{smallmatrix}N\otimes_BY\\ Y\end {smallmatrix}\right)_{0, 1}\in \ ^\perp\left(\begin{smallmatrix}A\mbox{-}{\rm Mod}\\ _B\mathcal I\end {smallmatrix}\right)$ by Lemma \ref{extadj1}(2), but
$\left(\begin{smallmatrix}N\otimes_BY\\ Y\end {smallmatrix}\right)_{0,1}\notin \left(\begin{smallmatrix}_A\mathcal P\\ _B\mathcal P\end {smallmatrix}\right)$.
This shows \ $^\perp\left(\begin{smallmatrix}A\mbox{-}{\rm Mod}\\ _B\mathcal I\end {smallmatrix}\right)\ne \left(\begin{smallmatrix}_A\mathcal P\\ _B\mathcal P\end {smallmatrix}\right)$.
Thus
$$R_{A\mbox{-}{\rm Mod}, \ _B\mathcal I}
= (^\perp\left(\begin{smallmatrix}A\mbox{-}{\rm Mod}\\ _B\mathcal I\end {smallmatrix}\right), \ \left(\begin{smallmatrix}A\mbox{-}{\rm Mod}\\ _B\mathcal I\end {smallmatrix}\right))\ne (\left(\begin{smallmatrix}_A\mathcal P\\ _B\mathcal P\end {smallmatrix}\right), \ \left(\begin{smallmatrix}_A\mathcal P\\ _B\mathcal P\end {smallmatrix}\right)^\perp)= (_\Lambda \mathcal P^{\le 1}, \ {\rm GI}(\Lambda)).$$
Similarly, \ $R_{_A\mathcal I, \ B\mbox{-}{\rm Mod}}= (^\perp\left(\begin{smallmatrix}_A\mathcal I\\ B\mbox{-}{\rm Mod}\end {smallmatrix}\right), \ \left(\begin{smallmatrix}_A\mathcal I\\ B\mbox{-}{\rm Mod}\end {smallmatrix}\right))\ne (\left(\begin{smallmatrix}_A\mathcal P\\ _B\mathcal P\end {smallmatrix}\right), \ \left(\begin{smallmatrix}_A\mathcal P\\ _B\mathcal P\end {smallmatrix}\right)^\perp)= (_\Lambda \mathcal P^{\le 1}, \ {\rm GI}(\Lambda)).$
\vskip5pt
{\bf Claim 4.} \ $L_{_A\mathcal P, \ B\mbox{-}{\rm Mod}}$ and \ $L_{A\mbox{-}{\rm Mod}, \ _B\mathcal P}$
are generally different from the Gorenstein-injective cotorsion pair.
Since $B$ is not semisimple, $B\mbox{-}{\rm Mod}\ne \ _B\mathcal P$. Thus \
$\left(\begin{smallmatrix}_A\mathcal P \\ B\mbox{-}{\rm Mod}\end {smallmatrix}\right)\ne \left(\begin{smallmatrix}_A\mathcal P\\ _B\mathcal P\end {smallmatrix}\right)$, and hence $$L_{_A\mathcal P, \ B\mbox{-}{\rm Mod}} = (\left(\begin{smallmatrix}_A\mathcal P\\ B\mbox{-}{\rm Mod}\end {smallmatrix}\right), \
\left(\begin{smallmatrix}_A\mathcal P\\ B\mbox{-}{\rm Mod}\end {smallmatrix}\right)^\perp)\ne (\left(\begin{smallmatrix}_A\mathcal P\\ _B\mathcal P\end {smallmatrix}\right), \ \left(\begin{smallmatrix}_A\mathcal P\\ _B\mathcal P\end {smallmatrix}\right)^\perp) = (_\Lambda \mathcal P^{\le 1}, \ {\rm GI}(\Lambda)).$$
Similarly, \ $L_{A\mbox{-}{\rm Mod}, \ _B\mathcal P} = (\left(\begin{smallmatrix}A\mbox{-}{\rm Mod}\\ _B\mathcal P\end {smallmatrix}\right), \ (\left(\begin{smallmatrix}A\mbox{-}{\rm Mod}\\ _B\mathcal P\end {smallmatrix}\right)^\perp)\ne (\left(\begin{smallmatrix}_A\mathcal P\\ _B\mathcal P\end {smallmatrix}\right), \ \left(\begin{smallmatrix}_A\mathcal P\\ _B\mathcal P\end {smallmatrix}\right)^\perp) = (_\Lambda \mathcal P^{\le 1}, \ {\rm GI}(\Lambda)).$
\vskip5pt
This completes the proof. \end{proof}
\vskip5pt
We also need the following result due to P. A. Krylov and E. Yu. Yardykov [KY].
\begin{lem}\label{flat} {\rm ([KY, Corollary 2.5])}
Let $L=\left(\begin{smallmatrix}X\\ Y\end{smallmatrix}\right)_{f,g}$ be a flat $\Lambda$-module. Then $\Coker g$ is a flat $A$-module and $\Coker f$ is a flat $B$-module.
\end{lem}
\vskip5pt
\noindent {\bf Proof of Proposition \ref{newI}.} \ By Proposition \ref{different}, these six cotorsion pairs
are generally different from the projective cotorsion pair and the injective one; and
they are generally different from the Gorenstein-projective cotorsion pair and the Gorenstein-injective one, by Lemmas \ref{notgor1} and \ref{notgor2}. It remains to show that they are generally different from the flat cotorsion pair.
\vskip5pt
In fact, choose rings $A$ and $B$ such that they admit non flat modules (such a ring $A$ of course exists! For example, just take a finite-dimensional algebra $A$ which is not semi-simple. Then
$A$ has a finitely generated module $M$ which is not projective, and $M$ is not flat). Taking non-flat modules \ $_AX$ and $_BY$, by Lemma \ref{flat}, all the following $\Lambda$-modules are not flat:
$$\left(\begin{smallmatrix} X\\ 0\end{smallmatrix}\right)_{0, 0}, \ \ \left(\begin{smallmatrix} 0\\ Y\end{smallmatrix}\right)_{0,0},
\ \ {\rm T}_A X = \left(\begin{smallmatrix}X\\ M\otimes_AX\end{smallmatrix}\right)_{1,0}, \ \ {\rm T}_BY = \left(\begin{smallmatrix} N\otimes_BY\\ Y\end{smallmatrix}\right)_{0,1}.
$$
\vskip5pt
\noindent However,
\vskip5pt
$\bullet$ \ For the cotorsion pair \ $R_{_A\mathcal I, \ _B\mathcal I} = (^\perp\left(\begin{smallmatrix}_A\mathcal I\\ _B\mathcal I\end {smallmatrix}\right), \ \left(\begin{smallmatrix}_A\mathcal I\\ _B\mathcal I\end {smallmatrix}\right))$, one has \
${\rm T}_A X = \binom{X}{M\otimes_AX}_{1,0}\in {}^\perp\binom{_A\mathcal I}{_B\mathcal I}$, by Lemma \ref{extadj1}(1).
\vskip5pt
$\bullet$ \ For the cotorsion pair \ $R_{A\mbox{-}{\rm Mod}, \ _B\mathcal I} = (^\perp\binom{A\mbox{-}{\rm Mod}}{_B\mathcal I}, \ \binom{A\mbox{-}{\rm Mod}}{_B\mathcal I})$,
one has \ ${\rm T}_BY = \binom{N\otimes_BY}{Y}_{0,1}\in \ ^\perp\binom{A\mbox{-}{\rm Mod}}{_B\mathcal I}$, by Lemma \ref{extadj1}(2).
\vskip5pt
$\bullet$ \ For the cotorsion pair \ $L_{_A\mathcal P, \ B\mbox{-}{\rm Mod}} = (\binom{_A\mathcal P}{B\mbox{-}{\rm Mod}}, \ \binom{_A\mathcal P}{B\mbox{-}{\rm Mod}}^\perp),$ one has
\ $\binom{0}{Y}_{0,0}\in \binom{_A\mathcal P}{B\mbox{-}{\rm Mod}}$.
\vskip5pt
$\bullet$ \ For the cotorsion pair \ $R_{_A\mathcal I, \ B\mbox{-}{\rm Mod}} = ({}^\perp\binom{_A\mathcal I}{B\mbox{-}{\rm Mod}}, \ \binom{_A\mathcal I}{B\mbox{-}{\rm Mod}})$, one has \
${\rm T}_A X = \binom{X}{M\otimes_AX}_{1,0}\in {}^\perp\binom{_A\mathcal I}{B\mbox{-}{\rm Mod}}$, by Lemma \ref{extadj1}(1).
\vskip5pt
$\bullet$ \ For the cotorsion pair \ $L_{A\mbox{-}{\rm Mod}, \ _B\mathcal P} = (\binom{A\mbox{-}{\rm Mod}}{_B\mathcal P}, \ \binom{A\mbox{-}{\rm Mod}}{_B\mathcal P}^\perp)$, one has
\ $\binom{X}{0}_{0, 0}\in \binom{A\mbox{-}{\rm Mod}}{_B\mathcal P}$.
\vskip5pt
\noindent
In conclusion, the five cotorsion pairs $R_{_A\mathcal I, \ _B\mathcal I}, \ \ R_{A\mbox{-}{\rm Mod}, \ _B\mathcal I}, \ \ L_{_A\mathcal P, \ B\mbox{-}{\rm Mod}}, \ \ R_{_A\mathcal I, \ B\mbox{-}{\rm Mod}}, \ \ L_{A\mbox{-}{\rm Mod}, \ _B\mathcal P}$ are generally different from the flat cotorsion pair.
\vskip5pt
Finally, \ for the cotorsion pair \ $L_{_A\mathcal P, \ _B\mathcal P} =
(\left(\begin{smallmatrix}_A\mathcal P\\ _B\mathcal P\end {smallmatrix}\right), \ \left(\begin{smallmatrix}_A\mathcal P\\ _B\mathcal P\end {smallmatrix}\right)^\perp)$,
we take $\Lambda$ to be the Morita ring and $L = \binom{Ae_1}{Ae_1}_{\sigma, \sigma}$, as given in Example \ref{ie}. Then $L\in \left(\begin{smallmatrix}_A\mathcal P\\ _B\mathcal P\end {smallmatrix}\right)$.
But $L$ is not a flat $\Lambda$-module (otherwise, since $L$ is finitely generated, $L$ is projective, which is absurd).
\vskip5pt
This completes the proof. \hfill $\square$
\subsection{Cotorsion pairs in Series II in Table 2 are pairwise generally different} Also, in the most cases, the eight cotorsion pairs in Series II in Table 2 are pairwise different.
\vskip5pt
\begin{lem}\label{different1} \ Let $\Lambda = \left(\begin{smallmatrix} A & N \\
M & B\end{smallmatrix}\right)$ be a Morita ring with $M\otimes_A N=0=N\otimes_BM$, \ $(\mathcal U, \ \mathcal X)$ and \ $(\mathcal U', \ \mathcal X')$ cotorsion pairs in $A\mbox{-}{\rm Mod}$, and \ $(\mathcal V, \ \mathcal Y)$ and \ $(\mathcal V', \ \mathcal Y')$ cotorsion pairs in $B\mbox{-}{\rm Mod}$.
Then
\vskip5pt
$(1)$ \ $\Delta(\mathcal U, \ \mathcal V)=\Delta(\mathcal U', \ \mathcal V')$ if and only if \ $\mathcal U=\mathcal U'$ and \ $\mathcal V=\mathcal V'.$
\vskip5pt
$(2)$ \ $\nabla(\mathcal X, \ \mathcal Y) = \nabla(\mathcal X', \ \mathcal Y')$ if and only if \
$\mathcal X = \mathcal X'$ and \ $\mathcal Y = \mathcal Y'$.
\end{lem}
\begin{proof} \ $(1)$ \ This follows from the fact $${\rm T}_A U = \left(\begin{smallmatrix}U\\ M\otimes_AU\end {smallmatrix}\right)_{1,0} \in
\Delta(\mathcal U, \ \mathcal V), \ \ \forall \ U\in \mathcal U; \ \
\ \ \ {\rm T}_B V = \left(\begin{smallmatrix}N\otimes_BV \\ V\end {smallmatrix}\right)_{0,1} \in
\Delta(\mathcal U, \ \mathcal V), \ \ \forall \ V\in \mathcal V.$$
$(2)$ \ This follows from the fact \ ${\rm H}_A X = \left(\begin{smallmatrix} X \\ \Hom_A(N, X)\end{smallmatrix}\right)_{0, 1}
\in \nabla(\mathcal X, \ \mathcal Y), \ \ \forall \ X\in \mathcal X,$ and \ $
{\rm H}_BY = \left(\begin{smallmatrix} \Hom_B(M, Y) \\ Y\end{smallmatrix}\right)_{1, 0} \in
\nabla(\mathcal X, \ \mathcal Y), \ \ \forall \ Y\in \mathcal Y,$ here we use the second expression of $\Lambda$-modules. \end{proof}
\begin{prop}\label{different2}
Let $\Lambda = \left(\begin{smallmatrix} A & N \\
M & B\end{smallmatrix}\right)$ be a Morita ring with $M\otimes_A N=0=N\otimes_BM$.
Then the eight cotorsion pairs in {\rm Series II} in {\rm Table 2} are pairwise generally different.\end{prop}
\begin{proof} \ All together there are $\binom{8}{2} = 28$ situations.
\vskip5pt
{\bf Step 1.} \ By Lemma \ref{different1}, the cotorsion pairs in {\rm Series II} in the same columns of Table 2 are pairwise different. This occupies $2\binom{4}{2} = 12$ situations.
\vskip5pt
{\bf Step 2.} \ $(_\Lambda\mathcal P, \ \Lambda\mbox{\rm-Mod})$
is generally different from all other cotorsion pairs in Series II in Table 2. This occupies $4$ situations.
\vskip5pt
In fact, taking $A$ to be a non semisimple ring and $N$ a non injective $A$-module. Then $\binom{N}{0}_{0,0}\in \Lambda$-Mod.
Since the map $0\longrightarrow \Hom_A(N,N)$ is not epic, it follows that
$$\left(\begin{smallmatrix}N\\ 0\end{smallmatrix}\right)_{0,0}\notin {\rm Epi}(\Lambda), \ \ \ \left(\begin{smallmatrix}N\\ 0\end{smallmatrix}\right)_{0,0}\notin \nabla(A\mbox{\rm-Mod}, \ _B\mathcal I), \ \ \
\left(\begin{smallmatrix}N\\ 0\end{smallmatrix}\right)_{0,0}\notin\nabla(_A\mathcal I, \ B\mbox{\rm-Mod}), \ \ \ \left(\begin{smallmatrix}N\\ 0\end{smallmatrix}\right)_{0,0}\notin \ _\Lambda\mathcal I.$$
So $(_\Lambda\mathcal P, \ \Lambda\mbox{\rm-Mod})$ is generally different from \ \
$(^\perp{\rm Epi}(\Lambda), \ {\rm Epi}(\Lambda))$, \ \ \ $(^\perp\nabla(A\mbox{\rm-Mod}, \ _B\mathcal I), \ \nabla(A\mbox{\rm-Mod}, \ _B\mathcal I))$, \ \ \ $(^\perp\nabla(\mathcal I, \ B\mbox{\rm-Mod}), \ \nabla(\mathcal I, \ B\mbox{\rm-Mod}))$, \ and
\ $(\Lambda\mbox{\rm-Mod}, \ _\Lambda\mathcal I)$.
\vskip5pt
{\bf Step 3.} \ Similarly, \ $(\Lambda\mbox{\rm-Mod}, \ _\Lambda\mathcal I)$
is generally different from other cotorsion pairs in Series II. This occupies $3$ situations.
\vskip5pt
{\bf Step 4.} \ Assume that $M \neq 0\ne N$. It remains to show the following $9$ cases:
\begin{align*}& {\rm Epi}(\Lambda)\ne \Delta(_A\mathcal P, \ B\mbox{-}{\rm Mod})^\perp; \ \ \ \ \ \ {\rm Epi}(\Lambda)\ne \Delta (A\mbox{-}{\rm Mod}, \ _B\mathcal P)^\perp ;
\ \ \ \ \ \ \ \ {\rm Epi}(\Lambda)\ne {\rm Mon}(\Lambda)^\perp;
\\ &
\Delta(_A\mathcal P, \ B\mbox{-}{\rm Mod})^\perp \ne \nabla(A\mbox{-}{\rm Mod}, \ _B\mathcal I); \ \ \ \ \ \ \ \ \ \ \ \ \ \Delta(_A\mathcal P, \ B\mbox{-}{\rm Mod})^\perp \ne \nabla(_A\mathcal I, \ B\mbox{-}{\rm Mod});
\\ &\nabla(A\mbox{-}{\rm Mod}, \ _B\mathcal I) \ne \Delta(A\mbox{-}{\rm Mod}, \ _B\mathcal P)^\perp;
\ \ \ \ \ \ \ \ \ \ \ \ \
\nabla(A\mbox{-}{\rm Mod}, \ _B\mathcal I)\ne {\rm Mon}(\Lambda)^\perp; \\ &
\Delta(A\mbox{-}{\rm Mod}, \ _B\mathcal P)^\perp\ne \nabla(_A\mathcal I, \ B\mbox{-}{\rm Mod}); \ \ \ \ \ \ \ \ \ \ \ \ \
\nabla(_A\mathcal I, \ B\mbox{-}{\rm Mod})\ne {\rm Mon}(\Lambda)^\perp.
\end{align*}
First, we see the inequalities involving ${\rm Epi}(\Lambda) = \nabla(A\mbox{-}{\rm Mod}, \ B\mbox{-}{\rm Mod})$.
Let $_AI$ be the injective envelope of $_AN$. By Lemma \ref{extadj2}(1) one has
$${\rm Z}_AI= \left(\begin{smallmatrix}I\\ 0\end{smallmatrix}\right )_{0,0}\in \Delta(_A\mathcal P, \ B\mbox{-}{\rm Mod})^\perp\cap \Delta(A\mbox{-}{\rm Mod}, \ _B\mathcal P)^\perp \cap \Delta(A\mbox{-}{\rm Mod}, \ B\mbox{-}{\rm Mod})^\perp .$$
But $\widetilde{g}: 0\longrightarrow \Hom_A(N, I)$ is not epic, so $\binom{I}{0}_{0,0}\notin \nabla(A\mbox{-}{\rm Mod}, \ B\mbox{-}{\rm Mod}) = {\rm Epi}(\Lambda)$.
This shows ${\rm Epi}(\Lambda)\ne \Delta(_A\mathcal P, \ B\mbox{-}{\rm Mod})^\perp$, \ \ ${\rm Epi}(\Lambda)\ne \Delta(A\mbox{-}{\rm Mod}, \ _B\mathcal P)^\perp$ and \ ${\rm Epi}(\Lambda)\ne {\rm Mon}(\Lambda)^\perp$.
\vskip10pt
Next, we see the two inequalities involving \ $\Delta(_A\mathcal P, \ B\mbox{-}{\rm Mod})^\perp.$
By Lemma \ref{extadj2}(1),
\ ${\rm Z}_AN=\binom{N}{0}_{0,0}\in \Delta(_A\mathcal P, \ B\mbox{-}{\rm Mod})^\perp$. But $\widetilde{g}: 0\longrightarrow \Hom_A(N,N)\ne 0$ is not epic, so $\binom{N}{0}_{0,0}$ is not in $\nabla(A\mbox{-}{\rm Mod}, \ _B\mathcal I)$ and $\nabla(_A\mathcal I, \ B\mbox{-}{\rm Mod})$.
This shows the two inequalities.
\vskip10pt
Next, to see \ $\nabla(A\mbox{-}{\rm Mod}, \ _B\mathcal I) \ne \Delta(A\mbox{-}{\rm Mod}, \ _B\mathcal P)^\perp$ and $\nabla(A\mbox{-}{\rm Mod}, \ _B\mathcal I)\ne \Delta (A\mbox{-}{\rm Mod}, \ B\mbox{-}{\rm Mod})^\perp$,
Let $_BJ$ be the injective envelope of $_BM$. By Lemma \ref{extadj2}(2),
\ ${\rm Z}_BJ=\binom{0}{J}_{0,0}$ is in $\Delta(A\mbox{-}{\rm Mod}, \ _B\mathcal P)^\perp$ and \ $\Delta(A\mbox{-}{\rm Mod}, \ B\mbox{-}{\rm Mod})^\perp$.
But $\widetilde{f}: 0\longrightarrow \Hom_B(M, J)$ is not epic, so $\binom{0}{J}_{0,0}\notin \nabla(A\mbox{-}{\rm Mod}, \ _B\mathcal I)$.
\vskip10pt
Finally, to see the two inequalities involving \ $\nabla(_A\mathcal I, \ B\mbox{-}{\rm Mod})$.
Let $_AI$ be the injective envelope of $_AN$. By Lemma \ref{extadj2}(1),
\ ${\rm Z}_AI=\binom{I}{0}_{0,0}$ is in \ $\Delta(A\mbox{-}{\rm Mod}, \ _B\mathcal P)^\perp$ \ and \ $\Delta(A\mbox{-}{\rm Mod}, \ B\mbox{-}{\rm Mod})^\perp$.
But $\widetilde{g}: 0\longrightarrow \Hom_A(N, I)$ is not epic, so $\binom{I}{0}_{0,0}\notin \nabla(_A\mathcal I, \ B\mbox{-}{\rm Mod})$.
This shows \ $\Delta(A\mbox{-}{\rm Mod}, \ _B\mathcal P)^\perp\ne \nabla(_A\mathcal I, \ B\mbox{-}{\rm Mod})$
and \ $\nabla(_A\mathcal I, \ B\mbox{-}{\rm Mod})\ne {\rm Mon}(\Lambda)^\perp.$
\end{proof}
\subsection{``New" cotorsion pairs in Series II in Table 2}
In Series II of Table 2, taking off the projective cotorsion pair and
the injective one, the remaining six cotorsion pairs
are ``new".
\vskip5pt
\begin{prop}\label{newII} \ Let $\Lambda = \left(\begin{smallmatrix} A & N \\ M & B \end{smallmatrix}\right)$ be a Morita ring with $M\otimes_AN=0 = N\otimes_BM$.
Then all the six cotorsion pairs
\begin{align*} & ({\rm Mon}(\Lambda), \ {\rm Mon}(\Lambda)^\bot), \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (^\bot{\rm Epi}(\Lambda), \ {\rm Epi}(\Lambda))
\\ &(\Delta(_A\mathcal P, \ B\mbox{-}{\rm Mod}), \ \Delta(_A\mathcal P, \ B\mbox{-}{\rm Mod})^\bot),
\ \ \ \ \ \ (^\perp\nabla(A\mbox{-}{\rm Mod}, \ _B\mathcal I), \ \nabla(A\mbox{-}{\rm Mod}, \ _B\mathcal I))
\\ &
(\Delta(A\mbox{-}{\rm Mod}, \ _B\mathcal P), \ \Delta(A\mbox{-}{\rm Mod}, \ _B\mathcal P)^\bot),
\ \ \ \ \ \ (^\bot\nabla(_A\mathcal I, \ B\mbox{-}{\rm Mod}), \ \nabla(_A\mathcal I, \ B\mbox{-}{\rm Mod}))\end{align*} are ``new", in the sense of \ {\rm Definition \ref{new}}.
\end{prop}
\vskip5pt
To prove Proposition \ref{newII}, we first show
\vskip5pt
\begin{lem}\label{nongor3} \ Let $\Lambda = \left(\begin{smallmatrix} A & N \\ M & B \end{smallmatrix}\right)$ be a Morita ring with $M\otimes_AN=0 = N\otimes_BM$.
Then the cotorsion pairs
\ $({\rm Mon}(\Lambda), \ {\rm Mon}(\Lambda)^\bot)$ and $(^\bot{\rm Epi}(\Lambda), \ {\rm Epi}(\Lambda))$ are generally different from the Gorenstein-projective cotorsion pair
and the Gorenstein-injective one.
\end{lem}
\begin{proof} \ Take \ $\Lambda$ to be the Morita ring $\Lambda = \left(\begin{smallmatrix} A & N \\ N & A \end{smallmatrix}\right)$, constructed in Example \ref{ie}. Thus
$A$ is the path algebra $k(1 \longrightarrow 2)$ with ${\rm char} \ k\ne 2$, $N = Ae_2\otimes_ke_1A$, and $N\otimes_AN = 0$. By Lemma \ref{agoralg}, $\Lambda$ is a Gorenstein algebra.
\vskip5pt
{\bf Claim 1.} \ $({\rm Mon}(\Lambda), \ {\rm Mon}(\Lambda)^\bot)$
is generally different from $({\rm GP}(\Lambda), \ _\Lambda\mathcal P^{<\infty})$.
\vskip5pt
In fact, $L = \left(\begin{smallmatrix}Ae_1\\ Ae_1\end{smallmatrix}\right)_{\sigma, \sigma}\in {\rm Mon}(\Lambda).$
By {\bf Claim 2} in the proof of Lemma \ref{notgor1}, ${\rm proj.dim}_\Lambda L = 1.$ Thus $L$ is not Gorenstein-projective (otherwise $L$ is projective, which is absurd. Note that a Gorenstein-projective module of finite projective dimension is projective. See [EJ, 10.2.3]).
So $L\notin {\rm GP}(\Lambda)$. Thus ${\rm Mon}(\Lambda)\ne {\rm GP}(\Lambda)$, and hence
$$({\rm Mon}(\Lambda), \ {\rm Mon}(\Lambda)^\bot)
\ne ({\rm GP}(\Lambda), \ _\Lambda\mathcal P^{<\infty}).$$
\vskip5pt
{\bf Claim 2.} \ $({\rm Mon}(\Lambda), \ {\rm Mon}(\Lambda)^\bot)$
is generally different from $(_\Lambda\mathcal P^{<\infty}, \ {\rm GI}(\Lambda))$.
\vskip5pt
In fact, the following $\Lambda$-projective resolution of $\left(\begin{smallmatrix}Ae_1\\ 0\end{smallmatrix}\right)_{0, 0}$
$$0\longrightarrow {\rm T}_BS_2 = \left(\begin{smallmatrix}0\\ S_2\end{smallmatrix}\right)_{0, 0}
\stackrel{\binom{0}{1}}\longrightarrow {\rm T}_A(Ae_1) = \left(\begin{smallmatrix} Ae_1\\ S_2\end{smallmatrix}\right)_{1, 0}
\stackrel{\left(\begin{smallmatrix} 1 \\ 0\end{smallmatrix}\right)}
\longrightarrow \left(\begin{smallmatrix}Ae_1\\ 0\end{smallmatrix}\right)_{0, 0}\longrightarrow 0$$
shows that $\left(\begin{smallmatrix}Ae_1\\ 0\end{smallmatrix}\right)_{0, 0}\in \ _\Lambda\mathcal P^{<\infty}$.
Since $N\otimes_AAe_1\cong S_2$, \ $\left(\begin{smallmatrix}Ae_1\\ 0\end{smallmatrix}\right)_{0, 0}\notin {\rm Mon}(\Lambda)$.
Thus \ ${\rm Mon}(\Lambda)\ne \ _\Lambda\mathcal P^{<\infty}$, and hence
$$({\rm Mon}(\Lambda), \ {\rm Mon}(\Lambda)^\bot)\ne (_\Lambda\mathcal P^{<\infty}, \ {\rm GI}(\Lambda)).$$
\vskip5pt
{\bf Claim 3.} \ $(^\bot{\rm Epi}(\Lambda), \ {\rm Epi}(\Lambda))$
is generally different from $({\rm GP}(\Lambda), \ _\Lambda\mathcal P^{<\infty})$.
\vskip5pt
In fact, by {\bf Claim 2}, $\left(\begin{smallmatrix}Ae_1\\ 0\end{smallmatrix}\right)_{0, 0}\in \ _\Lambda\mathcal P^{<\infty}$. Since $\Hom_A(N, Ae_1)\cong S_1 \ne 0$ (cf. Example\ref{ie}),
$\left(\begin{smallmatrix}Ae_1\\ 0\end{smallmatrix}\right)_{0, 0}\notin {\rm Epi}(\Lambda)$.
Thus
${\rm Epi}(\Lambda) \ne \ _\Lambda\mathcal P^{<\infty}$, and hence
$$(^\bot{\rm Epi}(\Lambda), \ {\rm Epi}(\Lambda))
\ne ({\rm GP}(\Lambda), \ _\Lambda\mathcal P^{<\infty}).$$
\vskip5pt
{\bf Claim 4.} \ $(^\bot{\rm Epi}(\Lambda), \ {\rm Epi}(\Lambda))$
is generally different from $(_\Lambda\mathcal P^{<\infty}, \ {\rm GI}(\Lambda))$.
\vskip5pt
In fact, $L = \left(\begin{smallmatrix}Ae_1\\ Ae_1\end{smallmatrix}\right)_{\sigma, \sigma}\in {\rm Epic}(\Lambda)$ and $\Ext^1_\Lambda(L, L)\ne 0$ \ (cf. Example \ref{ie}).
So \ $L\notin \ ^\bot{\rm Epi}(\Lambda)$. However, $L\in \ _\Lambda\mathcal P^{<\infty}$.
Thus \ $^\bot{\rm Epi}(\Lambda)\ne \ _\Lambda\mathcal P^{<\infty}$, and hence
$$(^\bot{\rm Epi}(\Lambda), \ {\rm Epi}(\Lambda))
\ne (_\Lambda\mathcal P^{<\infty}, \ {\rm GI}(\Lambda)).$$
This completes the proof. \end{proof}
\vskip5pt
\begin{lem}\label{nongor4} \ Let $\Lambda = \left(\begin{smallmatrix} A & N \\ M & B \end{smallmatrix}\right)$ be a Morita ring with $M\otimes_AN=0 = N\otimes_BM$.
Then the cotorsion pairs
\begin{align*} & (\Delta(_A\mathcal P, \ B\mbox{-}{\rm Mod}), \ \Delta(_A\mathcal P, \ B\mbox{-}{\rm Mod})^\bot),
\ \ \ \ (^\perp\nabla(A\mbox{-}{\rm Mod}, \ _B\mathcal I), \ \nabla(A\mbox{-}{\rm Mod}, \ _B\mathcal I))
\\ &
(\Delta(A\mbox{-}{\rm Mod}, \ _B\mathcal P), \ \Delta(A\mbox{-}{\rm Mod}, \ _B\mathcal P)^\bot),
\ \ \ \ (^\bot\nabla(_A\mathcal I, \ B\mbox{-}{\rm Mod}), \ \nabla(_A\mathcal I, \ B\mbox{-}{\rm Mod}))\end{align*} are generally different from the Gorenstein-projective cotorsion pair
and the Gorenstein-injective one.
\end{lem}
\begin{proof} \ Choose rings $A$ and $B$, bimodules \ $_BM_A$ and $_AN_B$, such that
\vskip5pt
(i) \ \ \ $A$ and $B$ are quasi-Frobenius and not semisimple;
(ii) \ \ $_AN$ and $_BM$ are non-zero projective modules, and $M_A$ and $N_B$ are flat;
(iii) \ \ $M\otimes_A N = 0 = N\otimes_BM;$
(iv) \ \ $\Lambda$ is noetherian.
\vskip5pt
By Remark \ref{examctp4}, such $\Lambda$'s always exist! By Theorem \ref{ctp4}, \ $\Lambda$ is a Gorenstein ring with ${\rm inj.dim} \Lambda \le 1$, \ $({\rm GP}(\Lambda), \ \mathcal P^{\le 1}) = ({\rm Mon}(\Lambda), \ {\rm Mon}(\Lambda)^\bot)$,
\ and \ $(_\Lambda \mathcal P^{\le 1}, \ {\rm GI}(\Lambda)) = (^\bot{\rm Epi}(\Lambda), \ {\rm Epi}(\Lambda))$.
\vskip5pt
{\bf Claim 1.} \ $(\Delta(_A\mathcal P, \ B\mbox{-}{\rm Mod}), \ \Delta(_A\mathcal P, \ B\mbox{-}{\rm Mod})^\bot)$ and \ $(\Delta(A\mbox{-}{\rm Mod}, \ _B\mathcal P), \ \Delta(A\mbox{-}{\rm Mod}, \ _B\mathcal P)^\bot)$ are generally different from the Gorenstein-projective cotorsion pair.
\vskip5pt
In fact, since \ $A$ and $B$ are not semisimple, \ $A\mbox{-}{\rm Mod}\ne \ _A\mathcal P$ and
\ $B\mbox{-}{\rm Mod}\ne \ _B\mathcal P$. By Lemma \ref{different1},
\ $\Delta(_A\mathcal P, \ B\mbox{-}{\rm Mod})\ne \Delta(A\mbox{-}{\rm Mod}, \ B\mbox{-}{\rm Mod}) = {\rm Mon}(\Lambda)$, and
$\Delta(A\mbox{-}{\rm Mod}, \ _B\mathcal P) \ne {\rm Mon}(\Lambda)$.
Thus
$$(\Delta(_A\mathcal P, \ B\mbox{-}{\rm Mod}), \ \Delta(_A\mathcal P, \ B\mbox{-}{\rm Mod})^\bot)
\ne ({\rm Mon}(\Lambda), \ {\rm Mon}(\Lambda)^\bot) = ({\rm GP}(\Lambda), \ \mathcal P^{\le 1})$$
and
$$(\Delta(A\mbox{-}{\rm Mod}, \ _B\mathcal P), \ \Delta(A\mbox{-}{\rm Mod}, \ _B\mathcal P)^\bot)\ne ({\rm Mon}(\Lambda), \ {\rm Mon}(\Lambda)^\bot) = ({\rm GP}(\Lambda), \ \mathcal P^{\le 1}).$$
\vskip5pt
{\bf Claim 2.} \ $(^\perp\nabla(A\mbox{-}{\rm Mod}, \ _B\mathcal I), \ \nabla(A\mbox{-}{\rm Mod}, \ _B\mathcal I))$
and \ $(^\bot\nabla(_A\mathcal I, \ B\mbox{-}{\rm Mod}), \ \nabla(_A\mathcal I, \ B\mbox{-}{\rm Mod}))$ are generally different from the Gorenstein-projective cotorsion pair.
\vskip5pt
In fact, by Lemma \ref{extadj2}(3),
\ ${\rm Z}_AA=\binom{A}{0}_{0,0}\in \ ^\perp\nabla(A\mbox{-}{\rm Mod}, \ _B\mathcal I)\cap \ ^\bot\nabla(_A\mathcal I, \ B\mbox{-}{\rm Mod})$.
But $f: M\otimes_AA\longrightarrow 0$ is not monic, so $\binom{A}{0}_{0,0}\notin \Delta(A\mbox{-}{\rm Mod}, \ B\mbox{-}{\rm Mod}) = {\rm Mon}(\Lambda)$.
This shows \ $^\perp\nabla(A\mbox{-}{\rm Mod}, \ _B\mathcal I)\ne {\rm Mon}(\Lambda)$ and \ $^\bot\nabla(_A\mathcal I, \ B\mbox{-}{\rm Mod})\ne {\rm Mon}(\Lambda)$.
Thus
$$(^\perp\nabla(A\mbox{-}{\rm Mod}, \ _B\mathcal I), \ \nabla(A\mbox{-}{\rm Mod}, \ _B\mathcal I))
\ne ({\rm Mon}(\Lambda), \ {\rm Mon}(\Lambda)^\bot) = ({\rm GP}(\Lambda), \ \mathcal P^{\le 1})$$
and
$$(^\bot\nabla(_A\mathcal I, \ B\mbox{-}{\rm Mod}), \ \nabla(_A\mathcal I, \ B\mbox{-}{\rm Mod}))\ne ({\rm Mon}(\Lambda), \ {\rm Mon}(\Lambda)^\bot) = ({\rm GP}(\Lambda), \ \mathcal P^{\le 1}).$$
\vskip5pt
{\bf Claim 3.} \ $(\Delta(_A\mathcal P, \ B\mbox{-}{\rm Mod}), \ \Delta(_A\mathcal P, \ B\mbox{-}{\rm Mod})^\bot)$ and \ $(\Delta(A\mbox{-}{\rm Mod}, \ _B\mathcal P), \ \Delta(A\mbox{-}{\rm Mod}, \ _B\mathcal P)^\bot)$ are generally different from the Gorenstein-injective cotorsion pair.
\vskip5pt
In fact, let $I$ be the injective envelope of \ $_AN$. By Lemma \ref{extadj2}(1),
\ ${\rm Z}_AI=\binom{I}{0}_{0,0}\in \Delta(_A\mathcal P, \ B\mbox{-}{\rm Mod})^\perp\cap \Delta(A\mbox{-}{\rm Mod}, \ _B\mathcal P)^\perp$.
But $\widetilde{g}: 0\longrightarrow \Hom_A(N, I)$ is not epic, so $\binom{I}{0}_{0,0}\notin \nabla(A\mbox{-}{\rm Mod}, \ B\mbox{-}{\rm Mod}) = {\rm Epi}(\Lambda)$.
This shows $\Delta(_A\mathcal P, \ B\mbox{-}{\rm Mod})^\perp\ne {\rm Epi}(\Lambda)$ and \ $\Delta(_A\mathcal P, \ B\mbox{-}{\rm Mod})^\perp\ne {\rm Epi}(\Lambda)$.
Thus
$$(\Delta(_A\mathcal P, \ B\mbox{-}{\rm Mod}), \ \Delta(_A\mathcal P, \ B\mbox{-}{\rm Mod})^\bot)
\ne (^\bot{\rm Epi}(\Lambda), \ {\rm Epi}(\Lambda)) = (_\Lambda \mathcal P^{\le 1}, \ {\rm GI}(\Lambda))$$
and
$$(\Delta(_A\mathcal P, \ B\mbox{-}{\rm Mod}), \ \Delta(_A\mathcal P, \ B\mbox{-}{\rm Mod})^\bot)\ne (^\bot{\rm Epi}(\Lambda), \ {\rm Epi}(\Lambda)) = (_\Lambda \mathcal P^{\le 1}, \ {\rm GI}(\Lambda)).$$
\vskip5pt
{\bf Claim 4.} \ $(^\perp\nabla(A\mbox{-}{\rm Mod}, \ _B\mathcal I), \ \nabla(A\mbox{-}{\rm Mod}, \ _B\mathcal I))$ and \ $(^\bot\nabla(_A\mathcal I, \ B\mbox{-}{\rm Mod}), \ \nabla(_A\mathcal I, \ B\mbox{-}{\rm Mod}))$ are generally different from the Gorenstein-injective cotorsion pair.
\vskip5pt
In fact, since \ $A$ and \ $B$ are not semisimple, \ $_A\mathcal I \ne A\mbox{-}{\rm Mod}$ \ and
\ $_B\mathcal I\ne B\mbox{-}{\rm Mod}$. By Lemma \ref{different1},
\ $\nabla(_A\mathcal I, \ B\mbox{-}{\rm Mod})\ne \nabla(A\mbox{-}{\rm Mod}, \ B\mbox{-}{\rm Mod}) = {\rm Epi}(\Lambda)$
\ and \ $\nabla(A\mbox{-}{\rm Mod}, \ _B\mathcal I)\ne {\rm Epi}(\Lambda)$.
Thus
$$(^\bot\nabla(_A\mathcal I, \ B\mbox{-}{\rm Mod}), \ \nabla(_A\mathcal I, \ B\mbox{-}{\rm Mod}))\ne (^\bot{\rm Epi}(\Lambda), \ {\rm Epi}(\Lambda)) = (_\Lambda \mathcal P^{\le 1}, \ {\rm GI}(\Lambda))$$
and $$(^\perp\nabla(A\mbox{-}{\rm Mod}, \ _B\mathcal I), \ \nabla(A\mbox{-}{\rm Mod}, \ _B\mathcal I))
\ne (^\bot{\rm Epi}(\Lambda), \ {\rm Epi}(\Lambda)) = (_\Lambda \mathcal P^{\le 1}, \ {\rm GI}(\Lambda)).$$
This completes the proof. \end{proof}
\vskip5pt
\noindent {\bf Proof of Proposition \ref{newII}.} \ By Proposition \ref{different2}, these six cotorsion pairs
are generally different from the projective cotorsion pair and the injective one.
By Lemmas \ref{nongor3} and \ref{nongor4}, they are generally different from the Gorenstein-projective cotorsion pair
and the Gorenstein-injective one. It remains to show that they are generally different from the flat cotorsion pair.
\vskip5pt
In fact, choose rings $A$ and $B$ such that they admit non flat modules (such a ring of course exists! See the proof of Proposition \ref{newI}). Taking non flat modules \ $_AX$ and $_BY$, by Lemma \ref{flat}, all the following $\Lambda$-modules are not flat:
$$\left(\begin{smallmatrix} X\\ 0\end{smallmatrix}\right)_{0, 0}, \ \ \left(\begin{smallmatrix} 0\\ Y\end{smallmatrix}\right)_{0,0},
\ \ {\rm T}_A X = \left(\begin{smallmatrix}X\\ M\otimes_AX\end{smallmatrix}\right)_{1,0}, \ \ {\rm T}_BY = \left(\begin{smallmatrix} N\otimes_BY\\ Y\end{smallmatrix}\right)_{0,1}.
$$
\vskip5pt
\noindent However,
\vskip5pt
$\bullet$ \ For the cotorsion pair \ $({\rm Mon}(\Lambda), \ {\rm Mon}(\Lambda)^\bot)$ , one has \
${\rm T}_A X = \binom{X}{M\otimes_AX}_{1,0}\in {\rm Mon}(\Lambda)$.
\vskip5pt
$\bullet$ \ For the cotorsion pair \ $(\Delta(_A\mathcal P, \ B\mbox{-}{\rm Mod}), \ \Delta(_A\mathcal P, \ B\mbox{-}{\rm Mod})^\perp)$,
one has \ ${\rm T}_BY = \binom{N\otimes_BY}{Y}_{0,1}\in \Delta(_A\mathcal P, \ B\mbox{-}{\rm Mod})$.
\vskip5pt
$\bullet$ \ For the cotorsion pair \ $(^\perp\nabla(A\mbox{-}{\rm Mod}, \ _B\mathcal I), \ \nabla(A\mbox{-}{\rm Mod}, \ _B\mathcal I))$, one has
\ $\binom{0}{Y}_{0,0}\in \ ^\perp\nabla(A\mbox{-}{\rm Mod}, \ _B\mathcal I)$, by Lemma \ref{extadj2}(4).
\vskip5pt
$\bullet$ \ For the cotorsion pair \ $(\Delta(A\mbox{-}{\rm Mod}, \ _B\mathcal P), \ \Delta(A\mbox{-}{\rm Mod}, \ _B\mathcal P)^\bot)$, one has \
${\rm T}_A X = \binom{X}{M\otimes_AX}_{1,0}\in \Delta(A\mbox{-}{\rm Mod}, \ _B\mathcal P)$.
\vskip5pt
$\bullet$ \ For the cotorsion pair \ $(^\bot\nabla(_A\mathcal I, \ B\mbox{-}{\rm Mod}), \ \nabla(_A\mathcal I, \ B\mbox{-}{\rm Mod}))$, one has
\ $\binom{X}{0}_{0, 0}\in \ ^\bot\nabla(_A\mathcal I, \ B\mbox{-}{\rm Mod})$, by Lemma \ref{extadj2}(3).
\vskip5pt
In conclusion, the five cotorsion pairs are different from the flat cotorsion pair.
\vskip5pt
Finally, to see $(^\bot{\rm Epi}(\Lambda), \ {\rm Epi}(\Lambda))$ is generally different from the flat cotorsion pair, choose a ring $A$ such that $A$ admits a flat (left) module which is not projective.
(For example, the ring $\Bbb Z$ of integers has a flat module $_{\Bbb Z}\Bbb Q$, but $_{\Bbb Z}\Bbb Q$ is not projective, or equivalently, $_{\Bbb Z}\Bbb Q$ is not free.)
Let
$\Lambda = \left(\begin{smallmatrix} A & 0 \\ 0 & A \end{smallmatrix}\right) = A\times A$. Then ${\rm Epi}(\Lambda) = \Lambda\mbox{-}{\rm Mod}$, and hence
$^\bot{\rm Epi}(\Lambda) = \ _\Lambda\mathcal P$. By the choice of $A$, \ $^\bot{\rm Epi}(\Lambda) = \ _\Lambda\mathcal P$ is strictly contained in $_\Lambda{\rm F}$, the class of flat $\Lambda$-modules. It follows that
$(^\bot{\rm Epi}(\Lambda), \ {\rm Epi}(\Lambda))$ is generally different from the flat cotorsion pair. \hfill $\square$
\section{\bf Abelian model structures on Morita rings}
Based on results in the previous sections, we will see how abelian model structures on $A$-Mod and $B$-Mod induce abelian model structures on Morita rings; and
we will see that all these abelian model structures obtained on Morita rings are pairwise generally different, and they are generally different from the six well-known abelian model structures (cf. Proposition \ref{newmodel}).
\subsection{Cofibrantly generated Hovey triples in Morita rings}
Let $R$ be a ring. Recall that a Hovey triple $(\mathcal C, \mathcal F, \mathcal W)$ in $R\mbox{-}{\rm Mod}$ is cofibrantly generated,
if both the cotorsion pairs $(\mathcal C\cap \mathcal W, \mathcal F)$ and $(\mathcal C, \mathcal F\cap\mathcal W)$ are cogenerated by sets.
If a model structure on $R\mbox{-}{\rm Mod}$ is clear in context, we write Quillen's homotopy category simply as ${\rm Ho}(R)$.
\vskip5pt
\begin{thm}\label{cofibrantlygenHtriple} \ Let \ $\Lambda=\left(\begin{smallmatrix} A & N \\
M & B\end{smallmatrix}\right)$ be a Morita ring with $M\otimes_AN=0 = N\otimes_BM$, \ \ $(\mathcal U', \ \mathcal X, \ \mathcal W_1)$ and \ $(\mathcal V', \ \mathcal Y, \ \mathcal W_2)$ cofibrantly generated
Hovey triples in $A\mbox{-}{\rm Mod}$ and $B\mbox{-}{\rm Mod}$, respectively.
\vskip5pt
$(1)$ \ Suppose that \ ${\rm Tor}^A_1(M, \ \mathcal U') = 0 = {\rm Tor}^B_1(N, \ \mathcal V')$, \ $M\otimes_A\mathcal U' \subseteq \mathcal Y\cap \mathcal W_2$ and \ $N\otimes_B\mathcal V' \subseteq \mathcal X\cap \mathcal W_1.$ Then
$$({\rm T}_A(\mathcal U')\oplus {\rm T}_B(\mathcal V'), \ \left(\begin{smallmatrix} \mathcal X \\ \mathcal Y\end{smallmatrix}\right), \ \left(\begin{smallmatrix} \mathcal W_1 \\ \mathcal W_2\end{smallmatrix}\right))$$
is a cofibrantly generated Hovey triple in $\Lambda\mbox{-}{\rm Mod};$ and it is hereditary with \ ${\rm Ho}(\Lambda) \cong {\rm Ho}(A)\oplus {\rm Ho}(B),$ provided that \ $(\mathcal U', \mathcal X, \mathcal W_1)$ and \ $(\mathcal V', \mathcal Y, \mathcal W_2)$ are hereditary.
\vskip5pt
$(2)$ \ Suppose that ${\rm Ext}_B^1(M, \mathcal Y) = 0 = {\rm Ext}_A^1(N, \mathcal X)$, $\Hom_B(M, \mathcal Y) \subseteq \mathcal U'\cap \mathcal W_1$ and $\Hom_A(N, \mathcal X) \subseteq \mathcal V'\cap \mathcal W_2$. Then
$$(\left(\begin{smallmatrix} \mathcal U' \\ \mathcal V'\end{smallmatrix}\right), \ {\rm H}_A(\mathcal X)\oplus {\rm H}_B(\mathcal Y), \ \left(\begin{smallmatrix} \mathcal W_1 \\ \mathcal W_2\end{smallmatrix}\right))$$
is a cofibrantly generated
Hovey triple$;$ and it is hereditary with ${\rm Ho}(\Lambda) \cong {\rm Ho}(A)\oplus {\rm Ho}(B)$, provided that $(\mathcal U', \mathcal X, \mathcal W_1)$ and $(\mathcal V', \mathcal Y, \mathcal W_2)$ are hereditary.
\end{thm}
\begin{proof} \ Put \ $\mathcal U: = \mathcal U'\cap \mathcal W_1, \ \ \mathcal X': = \mathcal X\cap \mathcal W_1, \ \ \mathcal V: = \mathcal V'\cap \mathcal W_2, \ \ \mathcal Y':= \mathcal Y\cap \mathcal W_2.$
\vskip5pt
Since \ $(\mathcal U', \ \mathcal X, \ \mathcal W_1)$ is a cofibrantly generated Hovey triple in $A$-{\rm Mod},
\ $(\mathcal U, \ \mathcal X)$ and $(\mathcal U', \ \mathcal X')$ are cotorsion pairs in $A$-{\rm Mod}, cogenerated by, say,
set $S_1$ and set $S_1'$, respectively. \ Similarly, \ $(\mathcal V, \ \mathcal Y)$ and $(\mathcal V', \ \mathcal Y')$ are cotorsion pairs in $B$-{\rm Mod}, cogenerated by, say, set $S_2$ and set $S_2'$, respectively.
\vskip5pt
(1) \ Since \ ${\rm Tor}^A_1(M, \ \mathcal U) \subseteq {\rm Tor}^A_1(M, \ \mathcal U') = 0$
and \ ${\rm Tor}^B_1(N, \ \mathcal V) \subseteq {\rm Tor}^B_1(N, \ \mathcal V') = 0$,
it follows from Theorem \ref{ctp1}(1) that \ $(^\perp\binom{\mathcal X}{\mathcal Y}, \ \binom{\mathcal X}{\mathcal Y})$ is a cotorsion pair in $\Lambda$-Mod; and it is cogenerated by set ${\rm T}_A(S_1)\oplus {\rm T}_B(S_2)$, by Proposition \ref{generatingcomplete}(1).
\vskip5pt
Since \ $M\otimes_A\mathcal U \subseteq M\otimes_A\mathcal U' \subseteq \mathcal Y$ \ and \
$N\otimes_B\mathcal V \subseteq N\otimes_B\mathcal V' \subseteq \mathcal X,$ by Theorem \ref{identify1}(1), \ $^\perp\binom{\mathcal X}{\mathcal Y} = {\rm T}_A(\mathcal U)\oplus {\rm T}_B(\mathcal V)$.
Thus, \ $({\rm T}_A(\mathcal U)\oplus {\rm T}_B(\mathcal V), \ \left(\begin{smallmatrix} \mathcal X \\ \mathcal Y\end{smallmatrix}\right))$ is
a cotorsion pair, cogenerated by set ${\rm T}_A(S_1)\oplus {\rm T}_B(S_2)$.
\vskip5pt
Similarly, \ $({\rm T}_A(\mathcal U')\oplus {\rm T}_B(\mathcal V'), \ \left(\begin{smallmatrix} \mathcal X' \\ \mathcal Y'\end{smallmatrix}\right))$ is a cotorsion pair,
cogenerated by set ${\rm T}_A(S'_1)\oplus {\rm T}_B(S'_2)$.
\vskip5pt
Since \
$M\otimes_A\mathcal U' \subseteq \mathcal W_2$ \ and \ $N\otimes_B\mathcal V'\subseteq \mathcal W_1$, one has
$$({\rm T}_A(\mathcal U')\oplus {\rm T}_B(\mathcal V'))\cap \left(\begin{smallmatrix} \mathcal W_1 \\ \mathcal W_2\end{smallmatrix}\right)
= {\rm T}_A(\mathcal U'\cap\mathcal W_1)\oplus {\rm T}_B(\mathcal V'\cap \mathcal W_2) = {\rm T}_A(\mathcal U)\oplus {\rm T}_B(\mathcal V).$$
Also, \ $\left(\begin{smallmatrix} \mathcal X \\ \mathcal Y\end{smallmatrix}\right)\cap \left(\begin{smallmatrix} \mathcal W_1 \\ \mathcal W_2\end{smallmatrix}\right)
= \left(\begin{smallmatrix} \mathcal X\cap \mathcal W_1 \\ \mathcal Y\cap \mathcal W_2\end{smallmatrix}\right) = \left(\begin{smallmatrix} \mathcal X' \\ \mathcal Y'\end{smallmatrix}\right).$
Since \ $\mathcal W_1$ and \ $\mathcal W_2$ are thick, \ $\left(\begin{smallmatrix} \mathcal W_1 \\ \mathcal W_2\end{smallmatrix}\right)$
is thick.
Thus
$$({\rm T}_A(\mathcal U')\oplus {\rm T}_B(\mathcal V'), \ \left(\begin{smallmatrix} \mathcal X\\ \mathcal Y\end{smallmatrix}\right), \ \left(\begin{smallmatrix} \mathcal W_1 \\ \mathcal W_2\end{smallmatrix}\right))$$
is a cofibrantly generated Hovey triple.
\vskip5pt
If \ $(\mathcal U', \mathcal X, \mathcal W_1)$ and \ $(\mathcal V', \mathcal Y, \mathcal W_2)$ are hereditary Hovey triples, then so is \ $({\rm T}_A(\mathcal U')\oplus {\rm T}_B(\mathcal V'), \ \left(\begin{smallmatrix} \mathcal X \\ \mathcal Y\end{smallmatrix}\right), \ \left(\begin{smallmatrix} \mathcal W_1 \\ \mathcal W_2\end{smallmatrix}\right))$. Since \ $M\otimes_A\mathcal U'\subseteq \mathcal Y\cap \mathcal W_2$ and \ $N\otimes_B\mathcal V'\subseteq \mathcal X\cap \mathcal W_1,$
by Theorem \ref{Ho} one has
\begin{align*}{\rm Ho}(\Lambda)&
\cong (({\rm T}_A(\mathcal U')\oplus {\rm T}_B(\mathcal V'))\cap \left(\begin{smallmatrix} \mathcal X \\ \mathcal Y\end{smallmatrix}\right))/(({\rm T}_A(\mathcal U')\oplus {\rm T}_B(\mathcal V'))
\cap \left(\begin{smallmatrix} \mathcal X\cap \mathcal W_1 \\ \mathcal Y\cap\mathcal W_2\end{smallmatrix}\right))
\\ & \cong ({\rm T}_A(\mathcal U'\cap \mathcal X)\oplus {\rm T}_B(\mathcal V'\cap\mathcal Y))/(({\rm T}_A(\mathcal U\cap\mathcal X)\oplus {\rm T}_B(\mathcal V\cap \mathcal Y))
\\ & \cong [{\rm T}_A(\mathcal U'\cap\mathcal X)/ {\rm T}_A(\mathcal U\cap \mathcal X)]\oplus [{\rm T}_B(\mathcal V'\cap\mathcal Y)/ {\rm T}_B(\mathcal V\cap \mathcal Y)]
\\ & \cong [(\mathcal U'\cap \mathcal X)/(\mathcal U\cap \mathcal X)]\oplus [(\mathcal V'\cap \mathcal Y)/(\mathcal V\cap \mathcal Y)]
\\& = {\rm Ho}(A)\oplus {\rm Ho}(B).\end{align*}
\vskip5pt
$(2)$ Since \ ${\rm Ext}_A^1(M, \ \mathcal Y') \subseteq {\rm Ext}_A^1(M, \ \mathcal Y) = 0$
and \ ${\rm Ext}_B^1(N, \ \mathcal X') \subseteq {\rm Ext}_B^1(N, \ \mathcal X) = 0$, by Theorem \ref{ctp1}(2), \ $(\binom{\mathcal U'}{\mathcal V'}, \ \binom{\mathcal U'}{\mathcal V'}^\perp)$ is a cotorsion pair in $\Lambda$-Mod.
\vskip5pt
Since \ ${\rm Hom}_B(M, \mathcal Y') \subseteq {\rm Hom}_B(M, \mathcal Y) \subseteq \mathcal U'$ \ and \
${\rm Hom}_A(N, \mathcal X') \subseteq {\rm Hom}_A(N, \mathcal X) \subseteq \mathcal V',$ by Theorem \ref{identify1}(2) one has
$$(\left(\begin{smallmatrix} \mathcal U' \\ \mathcal V'\end{smallmatrix}\right), \ \left(\begin{smallmatrix} \mathcal U' \\ \mathcal V'\end{smallmatrix}\right)^\perp)= (^\perp\nabla(\mathcal X', \ \mathcal Y'), \ \nabla(\mathcal X', \ \mathcal Y'))$$
and $\binom{\mathcal U'}{\mathcal V'}^\perp = {\rm H}_A(\mathcal X')\oplus {\rm H}_B(\mathcal Y')$. By Proposition \ref{generatingcomplete}(2), $(^\perp\nabla(\mathcal X', \ \mathcal Y'), \ \nabla(\mathcal X', \ \mathcal Y'))$
is cogenerated by set ${\rm Z}_A(S'_1)\oplus {\rm Z}_B(S'_2)$.
Thus, \ $(\left(\begin{smallmatrix} \mathcal U' \\ \mathcal V'\end{smallmatrix}\right), \ \ {\rm H}_A(\mathcal X')\oplus {\rm H}_B(\mathcal Y'))$ is
a cotorsion pair, cogenerated by set ${\rm Z}_A(S'_1)\oplus {\rm Z}_B(S'_2)$.
\vskip5pt
Similarly, \ $(\left(\begin{smallmatrix} \mathcal U \\ \mathcal V\end{smallmatrix}\right), \ \ {\rm H}_A(\mathcal X)\oplus {\rm H}_B(\mathcal Y))$ is
a cotorsion pair, cogenerated by set ${\rm Z}_A(S_1)\oplus {\rm Z}_B(S_2)$.
\vskip5pt
Note that \ $\left(\begin{smallmatrix} \mathcal U' \\ \mathcal V'\end{smallmatrix}\right)\cap \left(\begin{smallmatrix} \mathcal W_1 \\ \mathcal W_2\end{smallmatrix}\right)
= \left(\begin{smallmatrix} \mathcal U'\cap \mathcal W_1 \\ \mathcal V'\cap \mathcal W_2\end{smallmatrix}\right) = \left(\begin{smallmatrix} \mathcal U \\ \mathcal V\end{smallmatrix}\right).$
Since \
${\rm Hom}_A(N, \mathcal X) \subseteq \mathcal W_2$ \ and \ ${\rm Hom}_B(M, \mathcal Y) \subseteq \mathcal W_1$, one has
$$({\rm H}_A(\mathcal X)\oplus {\rm H}_B(\mathcal Y))\cap \left(\begin{smallmatrix} \mathcal W_1 \\ \mathcal W_2\end{smallmatrix}\right)
= {\rm H}_A(\mathcal X\cap\mathcal W_1)\oplus {\rm H}_B(\mathcal Y\cap \mathcal W_2) = {\rm H}_A(\mathcal X')\oplus {\rm H}_B(\mathcal Y').$$
Since \ $\mathcal W_1$ and \ $\mathcal W_2$ are thick, \ $\left(\begin{smallmatrix} \mathcal W_1 \\ \mathcal W_2\end{smallmatrix}\right)$
is thick.
Thus
$$(\left(\begin{smallmatrix} \mathcal U'\\ \mathcal V'\end{smallmatrix}\right), \ {\rm H}_A(\mathcal X)\oplus {\rm H}_B(\mathcal Y), \ \left(\begin{smallmatrix} \mathcal W_1 \\ \mathcal W_2\end{smallmatrix}\right))$$
is a cofibrantly generated Hovey triple.
\vskip5pt
If \ $(\mathcal U', \mathcal X, \mathcal W_1)$ and \ $(\mathcal V', \mathcal Y, \mathcal W_2)$ are hereditary, then \ $(\left(\begin{smallmatrix} \mathcal U'\\ \mathcal V'\end{smallmatrix}\right), \ {\rm H}_A(\mathcal X)\oplus {\rm H}_B(\mathcal Y), \ \left(\begin{smallmatrix} \mathcal W_1 \\ \mathcal W_2\end{smallmatrix}\right))$
is hereditary. Since \ ${\rm Hom}_A(N, \mathcal X) \subseteq \mathcal V'$ and \
${\rm Hom}_B(M, \mathcal Y) \subseteq \mathcal U',$
by Theorem \ref{Ho} one has
\begin{align*}{\rm Ho}(\Lambda)&
\cong (\left(\begin{smallmatrix} \mathcal U' \\ \mathcal V'\end{smallmatrix}\right)\cap ({\rm H}_A(\mathcal X)\oplus {\rm H}_B(\mathcal Y)))/(\left(\begin{smallmatrix} \mathcal U \\ \mathcal V\end{smallmatrix}\right)\cap ({\rm H}_A(\mathcal X)\oplus {\rm H}_B(\mathcal Y)))
\\ & \cong ({\rm H}_A(\mathcal U'\cap \mathcal X)\oplus {\rm H}_B(\mathcal V'\cap\mathcal Y))/(({\rm H}_A(\mathcal U\cap\mathcal X)\oplus {\rm H}_B(\mathcal V\cap \mathcal Y))
\\ & \cong [{\rm H}_A(\mathcal U'\cap\mathcal X)/ {\rm H}_A(\mathcal U\cap \mathcal X)]\oplus [{\rm H}_B(\mathcal V'\cap\mathcal Y)/ {\rm H}_B(\mathcal V\cap \mathcal Y)]
\\ & \cong [(\mathcal U'\cap \mathcal X)/(\mathcal U\cap \mathcal X)]\oplus [(\mathcal V'\cap \mathcal Y)/(\mathcal V\cap \mathcal Y)]
\\& = {\rm Ho}(A)\oplus {\rm Ho}(B).\end{align*}
\end{proof}
From Theorem \ref{cofibrantlygenHtriple} and its proof, one easily sees the following.
\begin{cor}\label{cofibrantlygenGHtriple} \ Let $\Lambda=\left(\begin{smallmatrix} A & N \\
M & B\end{smallmatrix}\right)$ be a Morita ring with $M\otimes_AN=0 = N\otimes_BM$.
Let $(\mathcal U, \mathcal X)$ and $(\mathcal U', \mathcal X')$ be
compatible hereditary cotorsion pairs in $A$-{\rm Mod}, generated by sets $S_1$ and $S_1'$, respectively, with Gillespie-Hovey triple $(\mathcal U', \mathcal X, \mathcal W_1)$.
Let $(\mathcal V, \mathcal Y)$ and $(\mathcal V', \mathcal Y')$ be
compatible hereditary cotorsion pairs in $B$-{\rm Mod}, generated by sets $S_2$ and $S_2'$, respectively, with Gillespie-Hovey triple $(\mathcal V', \mathcal Y, \mathcal W_2)$.
\vskip5pt
$(1)$ \ Assume that ${\rm Tor}^A_1(M, \ \mathcal U') = 0 = {\rm Tor}^B_1(N, \ \mathcal V'), \ \ M\otimes_A\mathcal U' \subseteq \mathcal Y', \
N\otimes_B\mathcal V' \subseteq \mathcal X'.$ Then
$$({\rm T}_A(\mathcal U)\oplus {\rm T}_B(\mathcal V), \ \left(\begin{smallmatrix} \mathcal X \\ \mathcal Y\end{smallmatrix}\right))
\ \ \ \mbox{and} \ \ \
({\rm T}_A(\mathcal U')\oplus {\rm T}_B(\mathcal V'), \ \left(\begin{smallmatrix} \mathcal X' \\ \mathcal Y'\end{smallmatrix}\right))$$
are compatible complete hereditary cotorsion pairs in \ $\Lambda\mbox{-}{\rm Mod}$, with Gillespie-Hovey triple
$$({\rm T}_A(\mathcal U')\oplus {\rm T}_B(\mathcal V'), \ \left(\begin{smallmatrix} \mathcal X \\ \mathcal Y\end{smallmatrix}\right), \ \left(\begin{smallmatrix} \mathcal W_1 \\ \mathcal W_2\end{smallmatrix}\right))$$
and \ ${\rm Ho}(\Lambda) \cong
{\rm Ho}(A)\oplus {\rm Ho}(B).$
\vskip5pt
$(2)$ \ Assume that ${\rm Ext}_B^1(M, \mathcal Y) = 0 = {\rm Ext}_A^1(N, \mathcal X)$, $\Hom_B(M, \mathcal Y) \subseteq \mathcal U$ and $\Hom_A(N, \mathcal X) \subseteq \mathcal V$. Then
$$(\left(\begin{smallmatrix} \mathcal U \\ \mathcal V\end{smallmatrix}\right), \ {\rm H}_A(\mathcal X)\oplus {\rm H}_B(\mathcal Y))
\ \ \ \mbox{and} \ \ \
(\left(\begin{smallmatrix} \mathcal U' \\ \mathcal V'\end{smallmatrix}\right), \ {\rm H}_A(\mathcal X')\oplus {\rm H}_B(\mathcal Y'))$$
are compatible complete hereditary cotorsion pairs in \ $\Lambda\mbox{-}{\rm Mod}$, with Gillespie-Hovey triple
$$(\left(\begin{smallmatrix} \mathcal U' \\ \mathcal V'\end{smallmatrix}\right), \ {\rm H}_A(\mathcal X)\oplus {\rm H}_B(\mathcal Y), \ \left(\begin{smallmatrix} \mathcal W_1 \\ \mathcal W_2\end{smallmatrix}\right))$$
and \ ${\rm Ho}(\Lambda) \cong {\rm Ho}(A)\oplus {\rm Ho}(B).$
\end{cor}
\subsection{Hovey triples in Morita rings} We stress that, all the results in the rest of this section are not consequences of Theorem \ref{cofibrantlygenHtriple}, or Corollary \ref{cofibrantlygenGHtriple},
since they need module-theoretical arguments on the completeness of cotorsion pairs in Morita rings, developed in Section 5. Thus, all these results are new even for $M = 0$ or $N = 0$.
\begin{thm}\label{Htriple1} \ Let $\Lambda=\left(\begin{smallmatrix} A & N \\
M & B\end{smallmatrix}\right)$ be a Morita ring with $M\otimes_AN=0 = N\otimes_BM$.
Let \ $(\mathcal V', \ \mathcal Y, \ \mathcal W)$ be a Hovey triple in $B$\mbox{-}{\rm Mod}. Suppose that \ $N_B$ is flat and $_BM$ is projective.
\vskip5pt
$(1)$ \ If \ $M\otimes_A\mathcal P \subseteq \mathcal Y\cap \mathcal W$, then
$$({\rm T}_A(_A\mathcal P)\oplus {\rm T}_B(\mathcal V'), \ \left(\begin{smallmatrix} A\text{\rm\rm-Mod} \\ \mathcal Y\end{smallmatrix}\right), \ \left(\begin{smallmatrix} A\text{\rm\rm-Mod} \\ \mathcal W\end{smallmatrix}\right))$$
is a Hovey triple in $\Lambda$\mbox{-}{\rm Mod}$;$ and it is hereditary with \ ${\rm Ho}(\Lambda) \cong {\rm Ho}(B)$, provided that
\ $(\mathcal V', \mathcal Y, \mathcal W)$ is hereditary.
\vskip5pt
$(2)$ \ If \ $\Hom_A(N, \ _A\mathcal I)\subseteq \mathcal V'\cap \mathcal W$, then
$$(\left(\begin{smallmatrix} A\mbox{-}{\rm Mod} \\ \mathcal V'\end{smallmatrix}\right), \ \ {\rm H}_A(_A\mathcal I)\oplus {\rm H}_B(\mathcal Y), \ \ \left(\begin{smallmatrix}A\text{\rm\rm-Mod}\\ \mathcal W\end{smallmatrix}\right))$$
is a Hovey triple in $\Lambda$\mbox{-}{\rm Mod}$;$ and it is hereditary with \ ${\rm Ho}(\Lambda) \cong {\rm Ho}(B)$, provided that
\ $(\mathcal V', \mathcal Y, \mathcal W)$ is hereditary.
\end{thm}
\begin{proof} \ Put \ $\mathcal V: = \mathcal V'\cap \mathcal W, \ \ \mathcal Y':= \mathcal Y\cap \mathcal W.$
Since \ $(\mathcal V', \ \mathcal Y, \ \mathcal W)$ is a Hovey triple in $B$\mbox{-}{\rm Mod},
\ $(\mathcal V, \ \mathcal Y)$ and \ $(\mathcal V', \ \mathcal Y')$ are complete cotorsion pairs in $B$\mbox{-}{\rm Mod}.
\vskip5pt
$(1)$ \ Since \ $M\otimes_A \mathcal P \subseteq \mathcal Y,$
it follows from Theorem \ref{ctp2}(1) that
\ $({\rm T}_A(_A\mathcal P)\oplus {\rm T}_B(\mathcal V), \ \left(\begin{smallmatrix}A\text{\rm\rm-Mod}\\ \mathcal Y\end{smallmatrix}\right))$ is a complete cotorsion pair in $\Lambda$-Mod.
Similarly, \ $({\rm T}_A(_A\mathcal P)\oplus {\rm T}_B(\mathcal V'), \ \left(\begin{smallmatrix}A\text{\rm\rm-Mod}\\ \mathcal Y'\end{smallmatrix}\right))$ is a complete cotorsion pair.
\vskip5pt
Since \ $M\otimes_A \mathcal P \subseteq \mathcal W$, it follows that
$$({\rm T}_A(_A\mathcal P)\oplus {\rm T}_B(\mathcal V'))\cap \left(\begin{smallmatrix} A\text{\rm-Mod} \\ \mathcal W\end{smallmatrix}\right) ={\rm T}_A(_A\mathcal P)\oplus {\rm T}_B(\mathcal V'\cap \mathcal W)={\rm T}_A(_A\mathcal P)\oplus {\rm T}_B(\mathcal V).$$
Clearly, $\left(\begin{smallmatrix} A\text{\rm-Mod} \\ \mathcal Y\end{smallmatrix}\right)\cap \left(\begin{smallmatrix} A\text{\rm-Mod} \\ \mathcal W\end{smallmatrix}\right) = \left(\begin{smallmatrix} A\text{\rm-Mod} \\ \mathcal Y'\end{smallmatrix}\right)$.
Since $\mathcal W$ is a thick class of $B$\mbox{-}{\rm Mod}, \ $\left(\begin{smallmatrix} A\text{\rm-Mod} \\ \mathcal W\end{smallmatrix}\right)$ is a thick class of $\Lambda$\mbox{-}{\rm Mod}.
By definition
\ $({\rm T}_A(_A\mathcal P)\oplus {\rm T}_B(\mathcal V'), \ \left(\begin{smallmatrix} A\text{\rm\rm-Mod} \\ \mathcal Y\end{smallmatrix}\right), \ \left(\begin{smallmatrix} A\text{\rm\rm-Mod} \\ \mathcal W\end{smallmatrix}\right))$
is a Hovey triple.
\vskip5pt
If \ $(\mathcal V', \ \mathcal Y, \ \mathcal W)$ is a hereditary Hovey triple,
then by Theorem \ref{ctp2}(1), both \ $({\rm T}_A(_A\mathcal P)\oplus {\rm T}_B(\mathcal V), \ \left(\begin{smallmatrix}A\text{\rm\rm-Mod}\\ \mathcal Y\end{smallmatrix}\right))$ and \ $({\rm T}_A(_A\mathcal P)\oplus {\rm T}_B(\mathcal V'), \ \left(\begin{smallmatrix}A\text{\rm\rm-Mod}\\ \mathcal Y'\end{smallmatrix}\right))$ are hereditary cotorsion pairs, and hence \ $({\rm T}_A(_A\mathcal P)\oplus {\rm T}_B(\mathcal V'), \ \left(\begin{smallmatrix} A\text{\rm\rm-Mod} \\ \mathcal Y\end{smallmatrix}\right), \ \left(\begin{smallmatrix} A\text{\rm\rm-Mod} \\ \mathcal W\end{smallmatrix}\right))$ is a hereditary Hovey triple. By Theorem \ref{Ho} one has
\begin{align*}{\rm Ho}(\Lambda)& \cong (({\rm T}_A(_A\mathcal P)\oplus {\rm T}_B(\mathcal V'))\cap \left(\begin{smallmatrix} A\text{\rm\rm-Mod} \\ \mathcal Y\end{smallmatrix}\right)) /(({\rm T}_A(_A\mathcal P)\oplus {\rm T}_B(\mathcal V'))\cap \left(\begin{smallmatrix} A\text{\rm\rm-Mod} \\ \mathcal Y\cap\mathcal W\end{smallmatrix}\right))
\\ & \cong ({\rm T}_A(_A\mathcal P)\oplus {\rm T}_B(\mathcal V'\cap\mathcal Y))/(({\rm T}_A(_A\mathcal P)\oplus {\rm T}_B(\mathcal V'\cap \mathcal Y'))\\ & \cong {\rm T}_B(\mathcal V'\cap\mathcal Y)/ {\rm T}_B(\mathcal V'\cap \mathcal Y')
\\ & \cong (\mathcal V'\cap \mathcal Y)/(\mathcal V'\cap \mathcal Y')\cong {\rm Ho}(B).\end{align*}
\vskip5pt
$(2)$ \ The proof is similar as $(1)$. We include the main steps. Since \ $\Hom_A(N, \ _A\mathcal I) \subseteq \mathcal V$,
by Theorem \ref{ctp2}(2), $(\left(\begin{smallmatrix} A\mbox{-}{\rm Mod} \\ \mathcal V\end{smallmatrix}\right), \ {\rm H}_A(_A\mathcal I)\oplus {\rm H}_B(\mathcal Y))$ is a complete cotorsion pair.
Similarly, \ $(\left(\begin{smallmatrix} A\mbox{-}{\rm Mod} \\ \mathcal V'\end{smallmatrix}\right), \ {\rm H}_A(_A\mathcal I)\oplus {\rm H}_B(\mathcal Y'))$ is a complete cotorsion pair.
\vskip5pt
Clearly \ $\left(\begin{smallmatrix} A\text{\rm\rm-Mod} \\ \mathcal V'\end{smallmatrix}\right)\cap \left(\begin{smallmatrix} A\text{\rm\rm-Mod} \\ \mathcal W\end{smallmatrix}\right) = \left(\begin{smallmatrix} A\text{\rm\rm-Mod} \\ \mathcal V\end{smallmatrix}\right).$
Since
$\Hom_A(N, \ _A\mathcal I) \subseteq \mathcal W$, it follows that
$$({\rm H}_A(_A\mathcal I)\oplus {\rm H}_B(\mathcal Y))\cap \left(\begin{smallmatrix} A\text{\rm\rm-Mod} \\ \mathcal W\end{smallmatrix}\right)
= {\rm H}_A(_A\mathcal I)\oplus {\rm H}_B(\mathcal Y\cap \mathcal W) = {\rm H}_A(_A\mathcal I)\oplus {\rm T}_B(\mathcal Y').$$
Also, $\left(\begin{smallmatrix} A\text{\rm-Mod} \\ \mathcal W\end{smallmatrix}\right)$ is a thick class of $\Lambda$\mbox{-}{\rm Mod}. By definition
$$(\left(\begin{smallmatrix} A\mbox{-}{\rm Mod} \\ \mathcal V'\end{smallmatrix}\right), \ \ {\rm H}_A(_A\mathcal I)\oplus {\rm H}_B(\mathcal Y), \ \ \left(\begin{smallmatrix}A\text{\rm\rm-Mod}\\ \mathcal W\end{smallmatrix}\right))$$
is a Hovey triple. Moreover, it is hereditary if \ $(\mathcal V', \ \mathcal Y, \ \mathcal W)$ is hereditary. In this case,
by Theorem \ref{Ho} one has
\begin{align*}{\rm Ho}(\Lambda\mbox{-}{\rm Mod})& \cong (\left(\begin{smallmatrix} A\text{\rm\rm-Mod} \\ \mathcal V'\end{smallmatrix}\right)\cap ({\rm H}_A(_A\mathcal I)\oplus {\rm H}_B(\mathcal Y))) /(\left(\begin{smallmatrix} A\text{\rm\rm-Mod} \\ \mathcal V'\cap \mathcal W\end{smallmatrix}\right)\cap ({\rm H}_A(_A\mathcal I)\oplus {\rm H}_B(\mathcal Y)))
\\ & \cong ({\rm H}_A(_A\mathcal I)\oplus {\rm H}_B(\mathcal V'\cap\mathcal Y))/({\rm H}_A(_A\mathcal I)\oplus {\rm H}_B(\mathcal V'\cap\mathcal Y')) \\ & \cong {\rm H}_B(\mathcal V'\cap\mathcal Y)/{\rm H}_B(\mathcal V'\cap\mathcal Y')
\\ & \cong (\mathcal V'\cap \mathcal Y)/(\mathcal V'\cap\mathcal Y') \cong {\rm Ho}(B).\end{align*}
\end{proof}
From Theorem \ref{Htriple1} and its proof, one has
\begin{cor}\label{GHtriple1} \ Let $\Lambda=\left(\begin{smallmatrix} A & N \\
M & B\end{smallmatrix}\right)$ be a Morita ring with $M\otimes_AN=0 = N\otimes_BM$, $(\mathcal V, \ \mathcal Y)$ and $(\mathcal V', \ \mathcal Y')$
compatible complete hereditary cotorsion pairs in $B$-{\rm Mod}, with Gillespie-Hovey triple
\ $(\mathcal V', \ \mathcal Y, \ \mathcal W)$. Suppose that \ $N_B$ is flat and $_BM$ is projective.
\vskip5pt
$(1)$ \ If \ $M\otimes_A\mathcal P \subseteq \mathcal Y'$, then
$$({\rm T}_A(_A\mathcal P)\oplus {\rm T}_B(\mathcal V), \ \left(\begin{smallmatrix} A\text{\rm\rm-Mod} \\ \mathcal Y\end{smallmatrix}\right))
\ \ \ \mbox{and} \ \ \
({\rm T}_A(_A\mathcal P)\oplus {\rm T}_B(\mathcal V'), \ \left(\begin{smallmatrix} A\text{\rm\rm-Mod} \\ \mathcal Y'\end{smallmatrix}\right))$$
are compatible complete hereditary cotorsion pairs in \ $\Lambda\mbox{-}{\rm Mod}$, with Gillespie-Hovey triple
$$({\rm T}_A(_A\mathcal P)\oplus {\rm T}_B(\mathcal V'), \ \left(\begin{smallmatrix} A\text{\rm\rm-Mod} \\ \mathcal Y\end{smallmatrix}\right), \ \left(\begin{smallmatrix} A\text{\rm\rm-Mod} \\ \mathcal W\end{smallmatrix}\right))$$
and \ ${\rm Ho}(\Lambda)\cong {\rm Ho}(B)$.
\vskip5pt
$(2)$ If \ $\Hom_A(N, \ _A\mathcal I)\subseteq \mathcal V$, then
$$(\left(\begin{smallmatrix} A\mbox{-}{\rm Mod} \\ \mathcal V\end{smallmatrix}\right), \ {\rm H}_A(_A\mathcal I)\oplus {\rm H}_B(\mathcal Y)) \ \ \ \mbox{and} \ \ \ (\left(\begin{smallmatrix} A\mbox{-}{\rm Mod} \\ \mathcal V'\end{smallmatrix}\right), \ {\rm H}_A(_A\mathcal I)\oplus {\rm H}_B(\mathcal Y'))$$
are compatible complete hereditary cotorsion pairs in $\Lambda$-{\rm Mod}, with Gillespie-Hovey triple
$$(\left(\begin{smallmatrix} A\mbox{-}{\rm Mod} \\ \mathcal V'\end{smallmatrix}\right), \ \ {\rm H}_A(_A\mathcal I)\oplus {\rm H}_B(\mathcal Y), \ \ \left(\begin{smallmatrix}A\text{\rm\rm-Mod}\\ \mathcal W\end{smallmatrix}\right))$$
and \ ${\rm Ho}(\Lambda) \cong {\rm Ho}(B)$.
\end{cor}
Similar as Theorem \ref{Htriple1}, starting from a Hovey triple in $A$-{\rm Mod} and using Theorem \ref{ctp3}, we get
\begin{thm}\label{Htriple2} \ Let $\Lambda=\left(\begin{smallmatrix} A & N \\
M & B\end{smallmatrix}\right)$ be a Morita ring with $M\otimes_AN=0 = N\otimes_BM$.
Let \ $(\mathcal U', \ \mathcal X, \ \mathcal W)$ be a Hovey triple in $A$\mbox{-}{\rm Mod}.
Suppose that \ $M_A$ is flat and \ $_AN$ is projective.
\vskip5pt
$(1)$ \ If \ $N\otimes_B\mathcal P \subseteq \mathcal X\cap \mathcal W$, then
$$({\rm T}_A(\mathcal U')\oplus {\rm T}_B(_B\mathcal P), \ \left(\begin{smallmatrix}\mathcal X \\ B\text{\rm\rm-Mod}\end{smallmatrix}\right), \ \left(\begin{smallmatrix}\mathcal W \\ B\text{\rm\rm-Mod}\end{smallmatrix}\right))$$
is a Hovey triple$;$ and it is hereditary with \ ${\rm Ho}(\Lambda) \cong {\rm Ho}(A)$, provided that
\ $(\mathcal U', \mathcal X, \mathcal W)$ is hereditary.
\vskip5pt
$(2)$ \ If \ $\Hom_B(M, \ _B\mathcal I) \subseteq \mathcal U'\cap \mathcal W,$ then
$$(\left(\begin{smallmatrix}\mathcal U' \\ B\text{\rm\rm-Mod}\end{smallmatrix}\right), \ \ {\rm H}_A(\mathcal X)\oplus {\rm H}_B(_B\mathcal I), \ \ \left(\begin{smallmatrix}\mathcal W \\ B\text{\rm\rm-Mod}\end{smallmatrix}\right))$$
is a Hovey triple$;$ and it is hereditary with \ ${\rm Ho}(\Lambda) \cong {\rm Ho}(A)$, provided that
\ $(\mathcal U', \mathcal X, \mathcal W)$ is hereditary.
\end{thm}
\begin{cor}\label{GHtriple2} \ Let $\Lambda=\left(\begin{smallmatrix} A & N \\
M & B\end{smallmatrix}\right)$ be a Morita ring with $M\otimes_AN=0 = N\otimes_BM$, \ $(\mathcal U, \mathcal X)$ and \ $(\mathcal U', \mathcal X')$
compatible complete hereditary cotorsion pairs in $A$-{\rm Mod}, with Gillespie-Hovey triple
\ $(\mathcal U', \ \mathcal X, \ \mathcal W)$. Suppose that \ $M_A$ is flat and \ $_AN$ is projective.
\vskip5pt
$(1)$ \ If \ $N\otimes_B\mathcal P \subseteq \mathcal X'$, then
$$({\rm T}_A(\mathcal U)\oplus {\rm T}_B(_B\mathcal P), \ \left(\begin{smallmatrix}\mathcal X \\ B\text{\rm\rm-Mod}\end{smallmatrix}\right))
\ \ \ \mbox{and} \ \ \
({\rm T}_A(\mathcal U')\oplus {\rm T}_B(_B\mathcal P), \ \left(\begin{smallmatrix}\mathcal X' \\ B\text{\rm\rm-Mod}\end{smallmatrix}\right))$$
are compatible complete hereditary cotorsion pairs in \ $\Lambda\mbox{-}{\rm Mod}$, with Gillespie-Hovey triple
$$({\rm T}_A(\mathcal U')\oplus {\rm T}_B(_B\mathcal P), \ \left(\begin{smallmatrix}\mathcal X \\ B\text{\rm\rm-Mod}\end{smallmatrix}\right), \ \left(\begin{smallmatrix}\mathcal W \\ B\text{\rm\rm-Mod}\end{smallmatrix}\right))$$
and \ ${\rm Ho}(\Lambda)\cong {\rm Ho}(A)$.
\vskip5pt
$(2)$ \ If \ $\Hom_B(M, \ _B\mathcal I) \subseteq \mathcal U,$ then
$$(\left(\begin{smallmatrix}\mathcal U \\ B\text{\rm\rm-Mod}\end{smallmatrix}\right), \ {\rm H}_A(\mathcal X)\oplus {\rm H}_B(_B\mathcal I)) \ \ \ \mbox{and} \ \ \
(\left(\begin{smallmatrix}\mathcal U' \\ B\text{\rm\rm-Mod}\end{smallmatrix}\right), \ {\rm H}_A(\mathcal X')\oplus {\rm H}_B(_B\mathcal I))$$
are compatible complete hereditary cotorsion pairs in $\Lambda$-{\rm Mod}, with Gillespie-Hovey triple
$$(\left(\begin{smallmatrix}\mathcal U' \\ B\text{\rm\rm-Mod}\end{smallmatrix}\right), \ \ {\rm H}_A(\mathcal X)\oplus {\rm H}_B(_B\mathcal I), \ \ \left(\begin{smallmatrix}\mathcal W \\ B\text{\rm\rm-Mod}\end{smallmatrix}\right))$$
and \ ${\rm Ho}(\Lambda) \cong {\rm Ho}(A)$.
\end{cor}
\subsection{Gillespie-Hovey triples in Morita rings, via generalized projective (injective) cotorsion pairs}
The notion of generalized projective (injective) cotorsion pairs is essentially due to H. Becker [Bec].
\begin{defn} \label{genprojctp} $(1)$ \ \ A complete cotorsion pair \ $(\mathcal X, \ \mathcal Y)$ in an abelian category $\mathcal A$ with enough projective objects
is {\it a generalized projective cotorsion pair}, or in short, gpctp, provided that
${\rm (i)}$ \ \ $\mathcal X\cap \mathcal Y = \mathcal P$, where $\mathcal P$ is the class of projective objects of $\mathcal A;$
${\rm (ii)}$ \ \ the class \ $\mathcal Y$ is thick.
\vskip5pt
${\rm (1')}$ \ \ A complete cotorsion pair \ $(\mathcal X, \ \mathcal Y)$ in an abelian category $\mathcal A$ with enough injective objects is {\it a generalized injective cotorsion pair}, or in short, gictp, provided that
${\rm (i')}$ \ \ $\mathcal X\cap \mathcal Y = \mathcal I$, where $\mathcal I$ is the class of injective objects of $\mathcal A;$
${\rm (ii')}$ \ \ the class \ $\mathcal X$ is thick.
\end{defn}
\begin{exmrem} A {\rm gpctp} $($respectively, {\rm gictp}$)$ is not necessarily the projective $($respectively, injective$)$ cotorsion pair \ $(\mathcal P, \ \mathcal A)$ \
$($respectively, \ $(\mathcal A, \ \mathcal I))$.
\vskip5pt
$(1)$ \ {\rm ([H2])} \ For a Gorenstein ring $R$, the Gorenstein-projective cotorsion pair
\ $({\rm GP}(R), \ _R\mathcal P^{<\infty})$ is a {\rm gpctp}. Dually,
\ $(_R\mathcal P^{<\infty}, \ {\rm GI}(R))$ is a {\rm gictp}.
\vskip5pt
$(2)$ \ Let ${\rm Ch}(R)$ be the complex category of modules over ring $R$, \ $\mathcal E$ the class of
acyclic complexes, and ${\rm dg}\mathcal P$ the class of dg projective complexes $Q$
$($see {\rm [Sp], [AF]}$)$, i.e., components of $Q$ are projective and \ $\Hom^\bullet (Q, \mathcal E)$ is acyclic.
By {\rm [EJX]}, \ $({\rm dg}\mathcal P, \ \mathcal E)$ is a cotorsion pair, and
${\rm dg}\mathcal P\cap \mathcal E$ is exactly the class of projective objects of ${\rm Ch}(R)$. That is,
$${\rm dg}\mathcal P\cap \mathcal E = \{\bigoplus\limits_{i\in\Bbb Z} P^i(P) \ | \ P\in \ _R\mathcal P\}$$
where \ $P^i(P): \ \cdots \rightarrow 0 \rightarrow P \stackrel{{\rm Id}}\rightarrow P \rightarrow 0 \rightarrow \cdots $ is the complex with $i$-th and $(i+1)$-th component $P$. By {\rm [Sp]} $($also {\rm [BN]}$)$, for any complex $X$ there is
an epimorphism $Q\longrightarrow X$ which is a quasi-isomorphism. Thus, \ $({\rm dg}\mathcal P, \ \mathcal E)$ is complete, and hence generalized projective.
Dually, there is a {\rm gictp} \ $(\mathcal E, \ {\rm dg}\mathcal I)$. See {\rm [Gil1]} for an important development of this work.
\vskip5pt
$(3)$ \ A {\rm gpctp} $(\mathcal X, \mathcal Y)$ is hereditary, $\mathcal X$ is a Frobenius category $($with the canonical exact structure$)$, and $\mathcal P$ is
the class of projective-injective objects.
\vskip5pt
$(3')$ \ A {\rm gictp} $(\mathcal X, \mathcal Y)$ is hereditary, $\mathcal Y$ is a Frobenius category, and $\mathcal I$ is
the class of projective-injective objects. \end{exmrem}
\vskip5pt
Taking gpctps or gictps in Corollary \ref{GHtriple1}, we get a stronger and an improved result without extra conditions (i.e., the conditions
``$M\otimes_A\mathcal P \subseteq \mathcal Y'$" and ``$\Hom_A(N, \ _A\mathcal I)\subseteq \mathcal V$" in Corollary \ref{GHtriple1} can be dropped).
This is the reason we list it as a theorem.
\vskip5pt
\begin{thm}\label{GHtriplegpgiB} \ Let $\Lambda=\left(\begin{smallmatrix} A & N \\
M & B\end{smallmatrix}\right)$ be a Morita ring with $M\otimes_AN=0 = N\otimes_BM$. Suppose that \ $N_B$ is flat and $_BM$ is projective.
\vskip5pt
$(1)$ \ Let \ $(\mathcal V, \ \mathcal Y)$ and $(\mathcal V', \ \mathcal Y')$ be compatible gpctps in $B$-{\rm Mod}, with Gillespie-Hovey triple
\ $(\mathcal V', \ \mathcal Y, \ \mathcal W)$.
Then
$$({\rm T}_A(_A\mathcal P)\oplus {\rm T}_B(\mathcal V), \ \left(\begin{smallmatrix} A\text{\rm\rm-Mod} \\ \mathcal Y\end{smallmatrix}\right))
\ \ \ \mbox{and} \ \ \
({\rm T}_A(_A\mathcal P)\oplus {\rm T}_B(\mathcal V'), \ \left(\begin{smallmatrix} A\text{\rm\rm-Mod} \\ \mathcal Y'\end{smallmatrix}\right))$$
are compatible gpctps in \ $\Lambda\mbox{-}{\rm Mod}$, with Gillespie-Hovey triple
$$({\rm T}_A(_A\mathcal P)\oplus {\rm T}_B(\mathcal V'), \ \left(\begin{smallmatrix} A\text{\rm\rm-Mod} \\ \mathcal Y\end{smallmatrix}\right), \ \left(\begin{smallmatrix} A\text{\rm\rm-Mod} \\ \mathcal W\end{smallmatrix}\right))$$
and \ ${\rm Ho}(\Lambda)\cong (\mathcal V'\cap \mathcal Y)/_B\mathcal P\cong {\rm Ho}(B)$.
\vskip5pt
$(2)$ \ Let \ $(\mathcal V, \mathcal Y)$ and \ $(\mathcal V', \mathcal Y')$ be compatible gictps in $B$-{\rm Mod}, with Gillespie-Hovey triple
\ $(\mathcal V', \ \mathcal Y, \ \mathcal W)$. Then
$$(\left(\begin{smallmatrix} A\mbox{-}{\rm Mod} \\ \mathcal V\end{smallmatrix}\right), \ {\rm H}_A(_A\mathcal I)\oplus {\rm H}_B(\mathcal Y)) \ \ \ \mbox{and} \ \ \ (\left(\begin{smallmatrix} A\mbox{-}{\rm Mod} \\ \mathcal V'\end{smallmatrix}\right), \ {\rm H}_A(_A\mathcal I)\oplus {\rm H}_B(\mathcal Y'))$$
are compatible gictps in $\Lambda$-{\rm Mod}, with Gillespie-Hovey triple
$$(\left(\begin{smallmatrix} A\mbox{-}{\rm Mod} \\ \mathcal V'\end{smallmatrix}\right), \ \ {\rm H}_A(_A\mathcal I)\oplus {\rm H}_B(\mathcal Y), \ \ \left(\begin{smallmatrix}A\text{\rm\rm-Mod}\\ \mathcal W\end{smallmatrix}\right))$$
and \ ${\rm Ho}(\Lambda) \cong (\mathcal V'\cap \mathcal Y)/ _B\mathcal I\cong {\rm Ho}(B)$.
\end{thm}
\begin{proof} $(1)$ \ Since \ $_BM$ is projective, $M\otimes_A \mathcal P \subseteq \ _B\mathcal P.$ \
Since cotorsion pair \ $(\mathcal V', \mathcal Y')$ \ is generalized projective,
\ $M\otimes_A \mathcal P \subseteq \ _B\mathcal P = \mathcal V'\cap \mathcal Y'\subseteq \mathcal Y'\subseteq \mathcal Y.$
\vskip5pt
Thus, by Corollary \ref{GHtriple1}(1), $$({\rm T}_A(_A\mathcal P)\oplus {\rm T}_B(\mathcal V), \ \left(\begin{smallmatrix} A\text{\rm\rm-Mod} \\ \mathcal Y\end{smallmatrix}\right))
\ \ \ \mbox{and} \ \ \
({\rm T}_A(_A\mathcal P)\oplus {\rm T}_B(\mathcal V'), \ \left(\begin{smallmatrix} A\text{\rm\rm-Mod} \\ \mathcal Y'\end{smallmatrix}\right))$$
are compatible complete hereditary cotorsion pairs in \ $\Lambda\mbox{-}{\rm Mod}$, with Gillespie-Hovey triple
$$({\rm T}_A(_A\mathcal P)\oplus {\rm T}_B(\mathcal V'), \ \left(\begin{smallmatrix} A\text{\rm\rm-Mod} \\ \mathcal Y\end{smallmatrix}\right), \ \left(\begin{smallmatrix} A\text{\rm\rm-Mod} \\ \mathcal W\end{smallmatrix}\right))$$
and \ ${\rm Ho}(\Lambda)\cong {\rm Ho}(B) \cong (\mathcal V'\cap \mathcal Y)/_B\mathcal P$. Since
$$_\Lambda\mathcal P = {\rm T}_A(_A\mathcal P)\oplus {\rm T}_B(_B\mathcal P)
= \{\left(\begin{smallmatrix}P \\ M\otimes_A P\end{smallmatrix}\right)\oplus \left(\begin{smallmatrix}N\otimes_B Q \\ Q\end{smallmatrix}\right) \ | \ P\in \ _A\mathcal P,
\ Q\in \ _B\mathcal P\}$$
and \ $M\otimes_A \mathcal P \subseteq \mathcal Y$, it follows that
$$({\rm T}_A(_A\mathcal P)\oplus {\rm T}_B(\mathcal V))\cap \left(\begin{smallmatrix} A\text{\rm-Mod} \\ \mathcal Y\end{smallmatrix}\right)
={\rm T}_A(_A\mathcal P)\oplus {\rm T}_B(\mathcal V\cap \mathcal Y)={\rm T}_A(_A\mathcal P)\oplus {\rm T}_B(_B\mathcal P)= \ _\Lambda\mathcal P.$$
Since $\mathcal Y$ is thick, $\left(\begin{smallmatrix}A\text{\rm\rm-Mod}\\ \mathcal Y\end{smallmatrix}\right)$ is thick.
Thus, cotorsion pair $({\rm T}_A(_A\mathcal P)\oplus {\rm T}_B(\mathcal V), \ \left(\begin{smallmatrix}A\text{\rm\rm-Mod}\\ \mathcal Y\end{smallmatrix}\right))$ is generalized projective.
Similarly, \ $({\rm T}_A(_A\mathcal P)\oplus {\rm T}_B(\mathcal V'), \ \left(\begin{smallmatrix}A\text{\rm\rm-Mod}\\ \mathcal Y'\end{smallmatrix}\right))$ is generalized projective.
\vskip5pt
$(2)$ \ Since \ $N_B$ is flat, $\Hom_A(N, \ _A\mathcal I) \subseteq \ _B \mathcal I$.
Since \ $(\mathcal V, \mathcal Y)$ \ is generalized injective,
$\Hom_A(N, \ _A\mathcal I) \subseteq \ _B \mathcal I = \mathcal V\cap \mathcal Y \subseteq \mathcal V$.
\vskip5pt
Thus, by Corollary \ref{GHtriple1}(2), $$(\left(\begin{smallmatrix}\mathcal U \\ B\text{\rm\rm-Mod}\end{smallmatrix}\right), \ {\rm H}_A(\mathcal X)\oplus {\rm H}_B(_B\mathcal I)) \ \ \ \mbox{and} \ \ \
(\left(\begin{smallmatrix}\mathcal U' \\ B\text{\rm\rm-Mod}\end{smallmatrix}\right), \ {\rm H}_A(\mathcal X')\oplus {\rm H}_B(_B\mathcal I))$$
are compatible complete hereditary cotorsion pairs, with Gillespie-Hovey triple
$$(\left(\begin{smallmatrix}\mathcal U' \\ B\text{\rm\rm-Mod}\end{smallmatrix}\right), \ \ {\rm H}_A(\mathcal X)\oplus {\rm H}_B(_B\mathcal I), \ \ \left(\begin{smallmatrix}\mathcal W \\ B\text{\rm\rm-Mod}\end{smallmatrix}\right))$$
and \ ${\rm Ho}(\Lambda) \cong {\rm Ho}(B)\cong (\mathcal U'\cap \mathcal X)/ _A\mathcal I$.
Since
$$_\Lambda\mathcal I ={\rm H}_A(_A\mathcal I)\oplus {\rm H}_B(_B\mathcal I)= \{\left(\begin{smallmatrix} I \\ \Hom_A(N,I)\end{smallmatrix}\right)\oplus \left(\begin{smallmatrix} \Hom_B(M,J) \\ J \end{smallmatrix}\right) \ | \ I\in \ _A\mathcal I, \ J\in \ _B\mathcal I\}$$
and $\Hom_A(N, \ _A\mathcal I) \subseteq \mathcal V$,
it follows that $$\left(\begin{smallmatrix} A\text{\rm-Mod} \\ \mathcal V\end{smallmatrix}\right)\cap ({\rm H}_A(_A\mathcal I)\oplus {\rm H}_B(\mathcal Y))
={\rm H}_A(_A\mathcal I)\oplus {\rm H}_B(\mathcal V\cap \mathcal Y)={\rm H}_A(_A\mathcal I)\oplus {\rm H}_B(_B\mathcal I)= \ _\Lambda\mathcal I.$$
Since \ $\mathcal V$ is thick, \ $\binom{A\text{\rm\rm-Mod}}{\mathcal V}$ is thick.
Thus \ $(\left(\begin{smallmatrix} A\mbox{-}{\rm Mod} \\ \mathcal V\end{smallmatrix}\right), \ {\rm H}_A(_A\mathcal I)\oplus {\rm H}_B(\mathcal Y))$ is generalized injective. Similarly, \ $(\left(\begin{smallmatrix} A\mbox{-}{\rm Mod} \\ \mathcal V'\end{smallmatrix}\right), \ {\rm H}_A(_A\mathcal I)\oplus {\rm H}_B(\mathcal Y'))$ is generalized injective.
\end{proof}
\vskip5pt
Similarly, taking gpctps or gictps in Corollary \ref{GHtriple2}, we get a stronger and an improved result with weaker conditions.
\vskip5pt
\begin{thm}\label{GHtriplegpgiA} \ Let $\Lambda=\left(\begin{smallmatrix} A & N \\
M & B\end{smallmatrix}\right)$ be a Morita ring with $M\otimes_AN=0 = N\otimes_BM$. Suppose that \ $M_A$ is flat and \ $_AN$ is projective.
\vskip5pt
$(1)$ \ Let \ $(\mathcal U, \ \mathcal X)$ and $(\mathcal U', \ \mathcal X')$ be compatible gpctps in $A$-{\rm Mod}, with Gillespie-Hovey triple
\ $(\mathcal U', \ \mathcal X, \ \mathcal W)$.
Then
$$({\rm T}_A(\mathcal U)\oplus {\rm T}_B(_B\mathcal P), \ \left(\begin{smallmatrix}\mathcal X \\ B\text{\rm\rm-Mod}\end{smallmatrix}\right))
\ \ \ \mbox{and} \ \ \
({\rm T}_A(\mathcal U')\oplus {\rm T}_B(_B\mathcal P), \ \left(\begin{smallmatrix}\mathcal X' \\ B\text{\rm\rm-Mod}\end{smallmatrix}\right))$$
are compatible gpctps in \ $\Lambda\mbox{-}{\rm Mod}$, with Gillespie-Hovey triple
$$({\rm T}_A(\mathcal U')\oplus {\rm T}_B(_B\mathcal P), \ \left(\begin{smallmatrix}\mathcal X \\ B\text{\rm\rm-Mod}\end{smallmatrix}\right), \ \left(\begin{smallmatrix}\mathcal W \\ B\text{\rm\rm-Mod}\end{smallmatrix}\right))$$
and \ ${\rm Ho}(\Lambda)\cong (\mathcal U'\cap \mathcal X)/_A\mathcal P\cong {\rm Ho}(A)$.
\vskip5pt
$(2)$ \ Let \ $(\mathcal U, \mathcal X)$ and \ $(\mathcal U', \mathcal X')$ be compatible gictps in $A$-{\rm Mod}, with Gillespie-Hovey triple
\ $(\mathcal U', \ \mathcal X, \ \mathcal W)$. Then
$$(\left(\begin{smallmatrix}\mathcal U \\ B\text{\rm\rm-Mod}\end{smallmatrix}\right), \ {\rm H}_A(\mathcal X)\oplus {\rm H}_B(_B\mathcal I)) \ \ \ \mbox{and} \ \ \
(\left(\begin{smallmatrix}\mathcal U' \\ B\text{\rm\rm-Mod}\end{smallmatrix}\right), \ {\rm H}_A(\mathcal X')\oplus {\rm H}_B(_B\mathcal I))$$
are compatible gictps in $\Lambda$-{\rm Mod}, with Gillespie-Hovey triple
$$(\left(\begin{smallmatrix}\mathcal U' \\ B\text{\rm\rm-Mod}\end{smallmatrix}\right), \ \ {\rm H}_A(\mathcal X)\oplus {\rm H}_B(_B\mathcal I), \ \ \left(\begin{smallmatrix}\mathcal W \\ B\text{\rm\rm-Mod}\end{smallmatrix}\right))$$
and \ ${\rm Ho}(\Lambda)\cong (\mathcal U'\cap \mathcal X)/ _A\mathcal I\cong {\rm Ho}(A)$.
\end{thm}
\subsection{Projective (Injective) models on Morita rings}
An abelian model structure on (abelian) category $\mathcal A$ is {\it projective} (respectively, {\it injective}) if
each object is fibrant (respectively, cofibrant), i.e.,
the Hovey triple is of form $(\mathcal X, \mathcal A, \mathcal Y)$ \ (resectively, $(\mathcal A, \mathcal Y, \mathcal X)$). See [H2], [Gil2].
\vskip5pt
The following observation clarifies the relation between projective (respectively, injective) models and gpctp (respectively, a gictp).
\vskip5pt
\begin{lem}\label{Htriple} {\rm ([Bec, 1.1.9]; [Gil3, 1.1])} \ Let \ $(\mathcal X, \ \mathcal Y)$ be a complete cotorsion pair in abelian category $\mathcal A$ with enough projective objects and enough injective objects. Then
\vskip5pt
$(1)$ \ $(\mathcal X, \ \mathcal A, \ \mathcal Y)$ is a $($hereditary$)$ Hovey triple if and only if
\ $(\mathcal X, \ \mathcal Y)$ is a generalized projective cotorsion pair.
\vskip5pt
$(1')$ \ $(\mathcal A, \ \mathcal Y, \ \mathcal X)$ is a $($hereditary$)$ Hovey triple if and only if
\ $(\mathcal X, \ \mathcal Y)$ is a generalized injective cotorsion pair.
\end{lem}
\vskip5pt
Any gpctp $(\mathcal V, \ \mathcal Y)$ in $B$-{\rm Mod} gives compatible gpctps
$(_B\mathcal P, \ B\mbox{-}{\rm Mod})$ and $(\mathcal V, \ \mathcal Y)$.
Any gictp $(\mathcal V, \ \mathcal Y)$ in $B$-{\rm Mod} gives compatible gictps
$(\mathcal V, \ \mathcal Y)$ and $(B\mbox{-}{\rm Mod}, \ _B\mathcal I)$.
Thus, by Theorem \ref{GHtriplegpgiB} one gets:
\vskip5pt
\begin{cor}\label{projinjtripleB} \ Let $\Lambda=\left(\begin{smallmatrix} A & N \\
M & B\end{smallmatrix}\right)$ be a Morita ring with $M\otimes_AN=0 = N\otimes_BM$. Suppose that \ $N_B$ is flat and \ $_BM$ is projective.
\vskip5pt
$(1)$ \ Let \ $(\mathcal V, \mathcal Y)$ be a gpctp in $B$-{\rm Mod}. Then
$$({\rm T}_A(_A\mathcal P)\oplus {\rm T}_B(\mathcal V), \ \ \Lambda\text{\rm\rm-Mod}, \ \ \left(\begin{smallmatrix} A\text{\rm\rm-Mod} \\ \mathcal Y\end{smallmatrix}\right))$$
is a hereditary Hovey triple, and \ ${\rm Ho}(\Lambda)\cong\mathcal V/_B\mathcal P$.
\vskip5pt
$(2)$ \ Let \ $(\mathcal V, \mathcal Y)$ be a gictp in $B$-{\rm Mod}. Then
$$(\Lambda\text{\rm\rm-Mod}, \ \ {\rm H}_A(_A\mathcal I)\oplus {\rm H}_B(\mathcal Y), \ \ \left(\begin{smallmatrix}A\text{\rm\rm-Mod}\\ \mathcal V\end{smallmatrix}\right))$$
is a hereditary Hovey triple, and \ ${\rm Ho}(\Lambda)\cong \mathcal Y/_B\mathcal I$.
\end{cor}
\vskip5pt
If $B$ is quasi-Frobenius, then \ $(B\mbox{-}{\rm Mod}, \ _B\mathcal I)$
is a gpctp, and
\ $(_B\mathcal P, \ B\mbox{-}{\rm Mod})$ is a gictp. By Corollary \ref{projinjtripleB} one gets
\begin{cor}\label{frobB} \ Let $\Lambda=\left(\begin{smallmatrix} A & N \\
M & B\end{smallmatrix}\right)$ be a Morita ring with $M\otimes_AN=0 = N\otimes_BM$. Suppose that $B$ is quasi-Frobenius, $N_B$ is flat and \ $_BM$ is projective. Then
\vskip5pt
$(1)$ \ $({\rm T}_A(_A\mathcal P)\oplus {\rm T}_B(B\text{\rm\rm-Mod}), \ \ \Lambda\text{\rm\rm-Mod}, \ \ \left(\begin{smallmatrix} A\text{\rm\rm-Mod} \\ _B\mathcal I\end{smallmatrix}\right))$
is a hereditary Hovey triple$;$ and \ ${\rm Ho}(\Lambda)\cong B\mbox{-}\underline{{\rm Mod}}.$
\vskip5pt
$(2)$ \ $(\Lambda\text{\rm\rm-Mod}, \ \ {\rm H}_A(_A\mathcal I)\oplus {\rm H}_B(B\text{\rm\rm-Mod}), \ \ \left(\begin{smallmatrix}A\text{\rm\rm-Mod}\\ _B\mathcal P\end{smallmatrix}\right))$
is a hereditary Hovey triple$;$ and \ ${\rm Ho}(\Lambda)\cong B\mbox{-}\underline{{\rm Mod}}.$
\end{cor}
\vskip5pt
Similar as Corollary \ref{projinjtripleB}, by Theorem \ref{GHtriplegpgiA} one gets
\vskip5pt
\begin{cor}\label{projinjtripleA} \ Let $\Lambda=\left(\begin{smallmatrix} A & N \\
M & B\end{smallmatrix}\right)$ be a Morita ring with $M\otimes_AN=0 = N\otimes_BM$. Suppose that \ $M_A$ is flat and \ $_AN$ is projective.
\vskip5pt
$(1)$ \ Let \ $(\mathcal U, \mathcal X)$ be a gpctp in $A$-{\rm Mod}. Then
$$({\rm T}_A(\mathcal U)\oplus {\rm T}_B(_B\mathcal P), \ \ \Lambda\text{\rm\rm-Mod}, \ \ \left(\begin{smallmatrix}\mathcal X \\ B\text{\rm\rm-Mod}\end{smallmatrix}\right))$$
is a hereditary Hovey triple, and \ ${\rm Ho}(\Lambda)\cong\mathcal U/_A\mathcal P$.
\vskip5pt
$(2)$ \ Let \ $(\mathcal U, \mathcal X)$ be a gictp in $A$-{\rm Mod}. Then
$$(\Lambda\text{\rm\rm-Mod}, \ \ {\rm H}_A(\mathcal X)\oplus {\rm H}_B(_B\mathcal I), \ \ \left(\begin{smallmatrix}\mathcal U \\ B\text{\rm\rm-Mod}\end{smallmatrix}\right))$$
is a hereditary Hovey triple, and \ ${\rm Ho}(\Lambda)\cong \mathcal X/_A\mathcal I$.
\end{cor}
\vskip5pt
If $A$ is quasi-Frobenius, then \ $(A\mbox{-}{\rm Mod}, \ _A\mathcal I)$ is a gpctp, and
\ $(_A\mathcal P, \ A\mbox{-}{\rm Mod})$
is a gictp. By Corollary \ref{projinjtripleA} one gets
\begin{cor}\label{frobA} \ Let $\Lambda=\left(\begin{smallmatrix} A & N \\
M & B\end{smallmatrix}\right)$ be a Morita ring with $M\otimes_AN=0 = N\otimes_BM$. Suppose that $A$ is quasi-Frobenius, \ $M_A$ is flat and \ $_AN$ is projective. Then
\vskip5pt
$(1)$ \ $({\rm T}_A(A\text{\rm\rm-Mod})\oplus {\rm T}_B(_B\mathcal P), \ \ \Lambda\text{\rm\rm-Mod}, \ \ \left(\begin{smallmatrix}_A\mathcal I \\ B\text{\rm\rm-Mod}\end{smallmatrix}\right))$
is a hereditary Hovey triple$;$ and \ ${\rm Ho}(\Lambda)\cong A\mbox{-}\underline{{\rm Mod}}.$
\vskip5pt
$(2)$ \ $(\Lambda\text{\rm\rm-Mod}, \ \ {\rm H}_A(A\text{\rm\rm-Mod})\oplus {\rm H}_B(_B\mathcal I), \ \ \left(\begin{smallmatrix}_A\mathcal P \\ B\text{\rm\rm-Mod}\end{smallmatrix}\right))$
is a hereditary Hovey triple$;$ and \ ${\rm Ho}(\Lambda)\cong A\mbox{-}\underline{{\rm Mod}}.$
\end{cor}
\subsection{Generally different Hovey triples}
\begin{lem} \label{thesameHoveytriple} \ Let \ $(\mathcal C, \ \mathcal F, \ \mathcal W)$ and \ $(\mathcal C', \ \mathcal F', \ \mathcal W')$ be Hovey triples in abelian category $\mathcal A$.
If $$(\mathcal C\cap \mathcal W, \ \mathcal F) = (\mathcal C'\cap \mathcal W', \ \mathcal F'), \ \ \ \ (\mathcal C, \ \mathcal F\cap \mathcal W) = (\mathcal C', \ \mathcal F'\cap \mathcal W')$$
then \ $(\mathcal C, \ \mathcal F, \ \mathcal W) = (\mathcal C', \ \mathcal F', \ \mathcal W')$.
\end{lem}
In fact, by Theorem \ref{hoveycorrespondence}, the corresponding two abelian model structures are the same. Thus $\mathcal W = \mathcal W'$.
\begin{defn} \label{Hdifference} \ Let \ $\Omega$ be a class of Morita rings, \ $(\mathcal C, \ \mathcal F, \ \mathcal W)$ and \ $(\mathcal C', \ \mathcal F', \ \mathcal W')$ Hovey triples defined in $\Lambda\mbox{-}{\rm Mod}$, for arbitrary Morita rings $\Lambda\in \Omega$.
They are said to be generally different Hovey triples, provided that there is $\Lambda\in \Omega$, such that \ $(\mathcal C, \ \mathcal F, \ \mathcal W)\ne (\mathcal C', \ \mathcal F', \ \mathcal W')$ in $\Lambda\mbox{-}{\rm Mod}$.
\end{defn}
\begin{lem} \label{Hdifference1} \ Hovey triples \ $(\mathcal C, \ \mathcal F, \ \mathcal W)$ and \ $(\mathcal C', \ \mathcal F', \ \mathcal W')$ in $\Lambda\mbox{-}{\rm Mod}$ are generally different if and only if
\ $(\mathcal C\cap \mathcal W, \ \mathcal F)$ \ and $(\mathcal C'\cap \mathcal W', \ \mathcal F')$ are generally different, or,
\ $(\mathcal C, \ \mathcal F\cap \mathcal W)$ and \ $(\mathcal C', \ \mathcal F'\cap \mathcal W')$ are generally different, as cotorsion pairs.
\end{lem}
\begin{proof} The ``only if" part follows from Lemma \ref{thesameHoveytriple}. Conversely, without loss of generality, we may assume that
there are $A$, $B$, \ $_BM_A$ and $_AN_B$, such that \ $\mathcal F = \mathcal F'$ and $\mathcal C\cap \mathcal W\ne \mathcal C'\cap \mathcal W'$.
Then, either $\mathcal C \ne \mathcal C'$, or $\mathcal W\ne \mathcal W'$. Hence $(\mathcal C, \ \mathcal F, \ \mathcal W) \ne (\mathcal C', \ \mathcal F', \ \mathcal W')$ for the corresponding $\Lambda$. \end{proof}
\begin{exm} \label{Hgdsame} \ Generally different Hovey triples could be the same in special cases.
\vskip5pt
For example,
$(_\Lambda\mathcal P, \ \Lambda\mbox{\rm-Mod}, \ \Lambda\mbox{\rm-Mod})$ and $(\binom{_A\mathcal P}{_B\mathcal P}, \ \binom{_A\mathcal P}{_B\mathcal P}^\perp, \ \Lambda\mbox{\rm-Mod})$ are Hovey triples.
Since \ $(_\Lambda\mathcal P, \ \Lambda\mbox{\rm-Mod})$ and $(\binom{_A\mathcal P}{_B\mathcal P}, \ \binom{_A\mathcal P}{_B\mathcal P}^\perp)$ are generally different (cf. Example \ref{gdsame}), by
Lemma \ref{Hdifference1}, the two Hovey triples are
generally different. But, if $M = 0 = N$, then they are the same.
\end{exm}
\begin{prop} \label{newmodel} $(1)$ \ The two Hovey triples in {\rm Theorem \ref{cofibrantlygenHtriple}} are generally different.
\vskip5pt
$(2)$ \ The four Hovey triples in {\rm Theorems \ref{Htriple1} and \ref{Htriple2}} are pairwise generally different.
\vskip5pt
$(3)$ \ The four Hovey triples in {\rm Theorems \ref{GHtriplegpgiB} and \ref{GHtriplegpgiA}} are pairwise generally different.
\vskip5pt
$(4)$ \ The four Hovey triples in {\rm Corollaries \ref{projinjtripleB} and \ref{projinjtripleA}} are pairwise generally different.
\vskip5pt
$(5)$ \ The four Hovey triples in {\rm Corollaries \ref{frobB} and \ref{frobA}} are pairwise generally different.
\vskip5pt
$(6)$ \ All the Hovey triples in $(1)$- $(5)$ are generally different from the following Hovey triples$:$
\vskip5pt
\hskip20pt $\bullet$ \ $(_\Lambda\mathcal P, \ \Lambda\mbox{-}{\rm Mod}, \ \Lambda\mbox{-}{\rm Mod});$
\vskip5pt
\hskip20pt $\bullet$ \ $(\Lambda\mbox{-}{\rm Mod}, \ _\Lambda\mathcal I, \ \Lambda\mbox{-}{\rm Mod});$
\vskip5pt
\hskip20pt $\bullet$ \ the Frobenius model ${\rm ([Gil2])}:$ \ $(\Lambda\mbox{-}{\rm Mod}, \ \Lambda\mbox{-}{\rm Mod}, \ _\Lambda\mathcal P)$ $($if $\Lambda$ is quasi-Frobenius$);$
\vskip5pt
\hskip20pt $\bullet$ \ $({\rm GP}(\Lambda), \ \Lambda\mbox{-}{\rm Mod}, \ _\Lambda\mathcal P^{<\infty})$ $($if $\Lambda$ is Gorenstein$);$
\vskip5pt
\hskip20pt $\bullet$ \ $(\Lambda\mbox{-}{\rm Mod}, \ {\rm GI}(\Lambda), \ _\Lambda\mathcal P^{<\infty})$ $($if $\Lambda$ is Gorenstein$);$
\vskip5pt
\hskip20pt $\bullet$ \ the flat-cotorsion Hovey triple \ $({\rm F}(\Lambda), \ {\rm C}(\Lambda), \ \Lambda\mbox{-}{\rm Mod})$ $($see {\rm [BBE], [EJ, 7.4.3]}$)$.
\end{prop}
\begin{proof} \ (1) \ Let $k$ be a field. In Theorem \ref{cofibrantlygenHtriple},
taking \ $\Lambda = \left(\begin{smallmatrix} k & k \\
0 & k\end{smallmatrix}\right)$ and
\ $\mathcal U' = k\mbox{-}{\rm Mod} = \mathcal X = \mathcal W_1 = \mathcal V' = \mathcal Y = \mathcal W_2$, then all the conditions are satisfied. To see that
$({\rm T}_A(\mathcal U')\oplus {\rm T}_B(\mathcal V'), \ \left(\begin{smallmatrix} \mathcal X \\ \mathcal Y\end{smallmatrix}\right), \ \left(\begin{smallmatrix} \mathcal W_1 \\ \mathcal W_2\end{smallmatrix}\right))$
\ and
$(\left(\begin{smallmatrix} \mathcal U' \\ \mathcal V'\end{smallmatrix}\right), \ {\rm H}_A(\mathcal X)\oplus {\rm H}_B(\mathcal Y), \ \left(\begin{smallmatrix} \mathcal W_1 \\ \mathcal W_2\end{smallmatrix}\right))$
are different Hovey triples, it suffices to see that cotorsion pairs
$(_\Lambda\mathcal P, \ \Lambda\mbox{-}{\rm Mod})$ and $(\Lambda\mbox{-}{\rm Mod}, \ _\Lambda\mathcal I)$ are different. This is clear since $_\Lambda\mathcal P \subsetneqq \Lambda\mbox{-}{\rm Mod}$.
\vskip5pt
To show (2), (3), (4), (5), by the definition of generally different Hovey triples, it suffices to prove (5), since the Hovey triples in
Corollaries \ref{frobB} and \ref{frobA} are respectively the special cases of the Hovey triples in {\rm Theorems \ref{Htriple1} and \ref{Htriple2}}
(or, in {\rm Theorems \ref{GHtriplegpgiB} and \ref{GHtriplegpgiA}}; or, in {\rm Corollaries \ref{projinjtripleB} and \ref{projinjtripleA}}).
While for the four kinds of Hovey triples in Corollaries \ref{frobB} and \ref{frobA}, one can easily see that they are pairwise generally different.
\vskip5pt
(6) \ It suffices to show that the four Hovey triples in Corollaries \ref{frobB} and \ref{frobA} are generally different from the six Hovey triples listed above.
Then, all together there are 24 cases, and all these 24 cases are easy, except the following cases.
\vskip5pt
To see Hovey triple $({\rm T}_A(_A\mathcal P)\oplus {\rm T}_B(B\text{\rm\rm-Mod}), \ \ \Lambda\text{\rm\rm-Mod}, \ \ \left(\begin{smallmatrix} A\text{\rm\rm-Mod} \\ _B\mathcal I\end{smallmatrix}\right))$
in Corollary \ref{frobB}(1) is generally different from $({\rm GP}(\Lambda), \ \Lambda\mbox{-}{\rm Mod}, \ _\Lambda\mathcal P^{<\infty})$ (if $\Lambda$ is Gorenstein), we take $\Lambda$ to be the Morita rings as in Theorem \ref{ctp4}.
Then $_\Lambda \mathcal P^{<\infty} = \binom{_A\mathcal I}{_B\mathcal I}\ne \left(\begin{smallmatrix} A\text{\rm\rm-Mod} \\ _B\mathcal I\end{smallmatrix}\right)$ if $A$ is not semisimple.
\vskip5pt
To see the Hovey triple $(\Lambda\text{\rm\rm-Mod}, \ \ {\rm H}_A(_A\mathcal I)\oplus {\rm H}_B(B\text{\rm\rm-Mod}), \ \ \left(\begin{smallmatrix}A\text{\rm\rm-Mod}\\ _B\mathcal P\end{smallmatrix}\right))$
in Corollary \ref{frobB}(2) is generally different from $(\Lambda\mbox{-}{\rm Mod}, \ {\rm GI}(\Lambda), \ _\Lambda\mathcal P^{<\infty})$ \ (if $\Lambda$ is Gorenstein),
we take $\Lambda$ to be the Morita rings as in Theorem \ref{ctp4}.
Then $_\Lambda \mathcal P^{<\infty} = \binom{_A\mathcal P}{_B\mathcal P}\ne \left(\begin{smallmatrix} A\text{\rm\rm-Mod} \\ _B\mathcal P\end{smallmatrix}\right)$ if $A$ is not semisimple.
\vskip5pt
To see the Hovey triple $({\rm T}_A(A\text{\rm\rm-Mod})\oplus {\rm T}_B(_B\mathcal P), \ \ \Lambda\text{\rm\rm-Mod}, \ \ \left(\begin{smallmatrix}_A\mathcal I \\ B\text{\rm\rm-Mod}\end{smallmatrix}\right))$
in Corollary \ref{frobA}(1) is generally different from $({\rm GP}(\Lambda), \ \Lambda\mbox{-}{\rm Mod}, \ _\Lambda\mathcal P^{<\infty})$ (if $\Lambda$ is Gorenstein), we take $\Lambda$ to be the Morita rings as in Theorem \ref{ctp4}.
Then $_\Lambda \mathcal P^{<\infty} = \binom{_A\mathcal I}{_B\mathcal I}\ne \left(\begin{smallmatrix} _A\mathcal I \\ B\text{\rm\rm-Mod}\end{smallmatrix}\right)$ if $B$ is not semisimple.
\vskip5pt
To see the Hovey triple $(\Lambda\text{\rm\rm-Mod}, \ \ {\rm H}_A(A\text{\rm\rm-Mod})\oplus {\rm H}_B(_B\mathcal I), \ \ \left(\begin{smallmatrix}_A\mathcal P \\ B\text{\rm\rm-Mod}\end{smallmatrix}\right))$
in Corollary \ref{frobA}(2) is generally different from $(\Lambda\mbox{-}{\rm Mod}, \ {\rm GI}(\Lambda), \ _\Lambda\mathcal P^{<\infty})$ \ (if $\Lambda$ is Gorenstein), we take $\Lambda$ to be the Morita rings as in Theorem \ref{ctp4}.
Then $_\Lambda \mathcal P^{<\infty} = \binom{_A\mathcal P}{_B\mathcal P}\ne \left(\begin{smallmatrix} _A\mathcal P \\ B\text{\rm\rm-Mod}\end{smallmatrix}\right)$ if $B$ is not semisimple.
\end{proof}
|
1,116,691,497,578 | arxiv | \section{\bf Introduction}\label{section1}
In this paper, we study the Ricci flow on K\"ahler manifolds defined by
$$X_{n,k} = \mathbb{P}(\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}\oplus \mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-k))$$ for $k, n \in \mathbb{N}^+$. Such manifolds are holomorphic $\mathbb{C}\mathbb{P}^1$ bundle over the projective space $\mathbb{C}\mathbb{P}^{n-1}$. They are called Hirzebruch surfaces when $n=2$ and $X_{n,1}$ is exactly $\mathbb{C}\mathbb{P}^n$ blown-up at one point. The maximal compact subgroup of the automorphism group of $X_{n,k}$ is given by $G_{n,k}=U(n)/ \mathbb{Z}_k$ (\cite{Cal}).
The unnormalized Ricci flow introduced by Hamilton \cite{H1} is defined on a Riemannian manifold $M$ starting with a Remannian metric $g_0$ by
\begin{equation}\label{rf}
\ddt{g} = -Ric(g), ~~~g(0) = g_0.
\end{equation}
We apply the Ricci flow (\ref{rf}) to $X_{n,k}$ with a $G_{n,k}$-invariant initial K\"ahler metric. In \cite{SW1}, it is shown that the Ricci flow (\ref{rf}) must develop finite time singularity and it either shrinks to a point, collapses to $\mathbb{C}\mathbb{P}^{n-1}$ or contracts an exceptional divisor, in Gromov-Hausdorff topology.
When the flow shrinks to a point, $X_{n,k}$ is a Fano manifold and $1\leq k < n$. It is shown by Zhu \cite{Zhu} that the flow must develop Type I singularities and the rescaled Ricci flow converges in Cheeger-Gromov-Hamilton sense to the unique compact K\"ahler-Ricci soliton on $X_{n,k}$ constructed in \cite{Koi, Cao1, WZ}.
When the flow collapses to $\mathbb{C}\mathbb{P}^{n-1}$, it is shown by Fong \cite{Fo} that the flow must develop Type I singularities and the rescaled Ricci flow converges in Cheeger-Gromov-Hamilton sense to the ancient solution that splits isometrically as $\mathbb{C}^{n-1} \times \mathbb{C}\mathbb{P}^1$.
Our main result is to show that the flow must also develop Type I singularities when it does not collapse and the blow-up limit is a nontrivial complete shrinking gradient K\"ahler-Ricci soliton. Combined with the results of Zhu \cite{Zhu} and Fong \cite{Fo}, we have the following theorem.
\begin{theorem}\label{main} Let $X$ be $\mathbb{C}\mathbb{P}^n$ blown-up at one point. Then the Ricci flow on $X$ must develop Type I singularities for any $U(n)$-invariant initial K\"ahler metric.
\smallskip
Let $g(t)$ be the smooth solution defined on $t\in [0, T)$, where $T\in (0, \infty)$ is the singular time. For every $K_j\rightarrow \infty$, we consider the rescaled Ricci flow $(X, g_j(t'))$ defined on $[-K_j T, 0)$ by
$$g_j(t') = K_j g(T+K_j^{-1} t').$$ Then one and only one of the following must occur.
\begin{enumerate}
\item[(1)] If $\liminf_{t\rightarrow T} (T-t)^{-1}Vol(g(t)) =\infty$, then $(X, g_j(t'), p)$ subconverges in Cheeger-Gromov-Hamilton sense to a complete shrinking non-flat gradient K\"ahler-Ricci soliton on a complete K\"ahler manifold homeomorphic to $\mathbb{C}^n$ blown-up at one point, for any $p$ in the exceptional divisor.
\smallskip
\item[(2)] If $\liminf_{t\rightarrow T} (T-t)^{-1}Vol(g(t)) \in (0, \infty)$, then $(X, g_j(t'), p_j)$ subconverges in Cheeger-Gromov-Hamilton sense to $(\mathbb{C}^{n-1}\times \mathbb{C}\mathbb{P}^1, g_{\mathbb{C}^n} \oplus (-t') g_{FS})$, where $g_{\mathbb{C}^{n-1}}$ is the standard flat metric on $\mathbb{C}^{n-1}$ and $g_{FS}$ the Fubini-Study metric on $\mathbb{C}\mathbb{P}^1$ for any sequence of points $p_j$ \cite{Fo}.
\smallskip
\item[(3)] If $\liminf_{t\rightarrow T} (T-t)^{-1} Vol(g(t)) = 0$, then $(X, g_j(t'))$ converges in Cheeger-Gromov-Hamilton sense to the unique compact shrinking K\"ahler-Ricci soliton on $\mathbb{C}\mathbb{P}^n$ blown-up at one point \cite{Zhu}.
\end{enumerate}
\end{theorem}
The generalization of Theorem \ref{main} for $X_{n,k}$ is given in section 6. In order to exclude Type II singularities, we first prove a lower bound for the holomorphic bisectional curvature and then we apply Cao's splitting theorem for the K\"ahler Ricci flow with nonnegative holomorphic bisectional curvature \cite{Cao2}. Theorem \ref{main} gives evidence that the K\"ahler-Ricci flow can only develop Type I singularities for K\"ahler surfaces and if the flow does not collapse in finite time. Combined with the results of \cite{SW1, SW2}, Theorem \ref{main} verifies that the flow indeed performs a geometric canonical surgery with minimal singularities in the K\"ahler case. We also remark that the shrinking soliton as the pointed blow-up limit is trivial if the parabolic rescaling takes place at a fixed base point outside the exceptional divisor $D_0$. We believe that the blow-up limit should be the unique homothetically rotationally symmetric complete shrinking soliton on $\mathbb{C}^2$ blown-up at one point constructed by Feldman-Ilmanen-Knopf in \cite{FIK}. Unfortunately, we are unable to show that that limiting complete K\"ahler manifold is biholomorphic to $\mathbb{C}^n$ blown-up at one point, although it has the same topological structure with the unitary group $U(n)$ lying in the isometry group of the limiting soliton.
The organization of the paper is as follows. In section 2, we introduce the Calabi ansatz. In section 3, we obtain a lower bound for the holomorphic bisectional curvature. In section 4, we prove the flow must develop Type I singularities if non-collapsing. In section 5, we construct the blow-up limit. In section 6, we discuss some generalizations of Theorem \ref{main}.
We would also like to mention that we have been informed by Davi Maximo that he has a different approach to understand the singularity formation in similar settings \cite{Ma}.
\bigskip
\section{\bf Calabi symmetry}
In this section, we introduce the Calabi ansatz on $\mathbb{C}\mathbb{P}^n$ blown-up at one point introduced by Calabi \cite{Cal} (also see \cite{Cao1, FIK, SW1}). From now on, we let $X$ be $\mathbb{C}\mathbb{P}^n$ blown-up at one point and it is in fact a $\mathbb{C}\mathbb{P}^1$ bundle over $\mathbb{C}\mathbb{P}^{n-1}$ given by
$$X= \mathbb{P}( \mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}\oplus \mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-1)).$$
Let $D_0$ be the exceptional divisor of $X$ defined by the image of the section $(1,0)$ of $\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}\oplus \mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-1)$ and $D_\infty$ be the divisor of $X$ defined by the image of the section $(0,1)$ of $\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}\oplus \mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-1)$. Both the $0$-section $D_0$ and the $\infty$-section are complex hypersurfaces in $X$ isomorphic to $\mathbb{C}\mathbb{P}^{n-1}$. The K\"ahler cone on $X$ is given by
$$\mathcal{K}= \{ -a[D_0] + b [D_\infty] ~|~ 0<a<b\}.$$ In particular, when $n=2$, $D_0$ is a holomorphic $S^2$ with self-intersection number $-1$.
Let $z=(z_1, ..., z_n)$ be the standard holomorphic coordinates on $\mathbb{C}^n$. Let $\rho = \log |z|^2 = \log (|z_1|^2 + |z_2|^2 + ... + |z_n|^2)$. We consider a smooth convex function $u=u(\rho)$ for $\rho\in (-\infty, \infty)$ satisfying the following conditions.
\begin{itemize}
\item[(1)] $u'' >0$ for $\rho\in (-\infty, \infty)$.
\item[(2)] There exist $0<a<b$ and smooth function $u_0, u_\infty: [0, \infty) \rightarrow \mathbb{R}$ such that $$u_0'(0)>0, ~ u_\infty'(0)>0,$$
$$u_0(e^\rho) = u(\rho) - a \rho, ~~~u_\infty(e^{-\rho}) = u(\rho) - b\rho. $$
\end{itemize}
For any $u$ satisfying the above conditions, $\omega=\sqrt{-1}\partial\dbar u$ defines a smooth K\"ahler metric on $\mathbb{C}^n\setminus \{0\}$ and it extends to a smooth global K\"ahler metric on $\mathbb{C}\mathbb{P}^n$ blown-up at one point in the K\"ahler class $-a[D_0]+b[D_\infty]$.
On $\mathbb{C}^n\setminus \{0\}$, the K\"ahler metric $g$ induced by $u$ is given by
\begin{equation}
g_{i\bar j} = e^{-\rho} u' \delta_{i\bar j} + e^{-2\rho} \bar z_i z_j (u'' - u').
\end{equation}
Obviously, the K\"ahler metric $g$ induced by $u$ is invariant under the standard unitary $U(n)$ transformations on $\mathbb{C}^n$.
We define the Ricci potential of $\omega= \sqrt{-1}\partial\dbar u$ by
%
\begin{equation}v=- \log \det g = n\rho - (n-1) \log u'(\rho) - \log u''(\rho).
%
\end{equation}
and the Ricci tensor of $g$ is given by
$$R_{i\bar j} = e^{-\rho} v' \delta_{ij} + e^{-2\rho} \bar z_i z_j (v'' - v').$$
After applying a unitary transformation, we can assume $z=(z_1, 0, ..., 0)$ and then
$$\{ g_{i\bar j} \} = e^{-\rho} diag\{ u'', u', ..., u'\}$$
$$R_{i\bar j}= \sqrt{-1} e^{-\rho} diag\{ v'', v', ..., v' \}.$$
The Calabi symmetry is preserved by the Ricci flow, in other words, the evolving K\"ahler metric is invariant under the $U(n)$-action if the Ricci flow starts with a $U(n)$-invariant K\"ahler metric on $X$.
In \cite{SW1}, it is shown that the K\"ahler-Ricci flow on $X$ can be reduced to the following parabolic equation for $u= u(\rho, t)$ for $\rho\in \mathbf{R}$.
\begin{equation}
\ddt{} u(\rho, t) = \log u''(\rho, t) + (n-1) \log u'(\rho, t) - n\rho +c_t,
\end{equation}
where $$c_t = -\log u''(0, t) - (n-1) u'(0, t)$$ and $u'(\rho, t) = \frac{\partial}{\partial \rho} u(\rho, t).$ The evolving K\"ahler form $\omega(t)$ is then given by $$\omega(t) = \sqrt{-1}\partial\dbar u(\rho, t).$$
It is also shown in \cite{SW1} that if the initial K\"ahler class is given by $-a_0 [D_0]+b_0[D_\infty]$, the evolving K\"ahler class is given by
$$[\omega(t)] = -a_t [D_0]+b_t [D_\infty], ~~a_t= a_0 -(n-1)t, ~~ b_t = b_0 -(n+1)t.$$
In particular, we have an immediate bound for $u'(\rho, t)$
\begin{equation}
\lim_{\rho\rightarrow-\infty} u'(\rho, t) = a_t, ~~\lim_{\rho\rightarrow \infty} u'(\rho, t) = b_t.
\end{equation}
\section{\bf A lower bound for the holomorphic bisectional curvature}
In this section, we will obtain a lower bound for the holomorphic bisectional curvature. We consider the Ricci flow (\ref{rf}) on $X$ with a $U(n)$-invariant initial K\"ahler metric in the K\"ahler class $-a_0 [D_0]+b_0[D_\infty]$. For our purpose, it suffices to consider the case $$ 0< a_0 (n+1) < b_0 (n-1).$$ This assumption is shown in \cite{SW1} to be equivalent to the condition
$$\liminf_{t\rightarrow T} Vol(g(t))>0, ~~ \textnormal{or}, ~~\liminf_{t\rightarrow T} (T-t)^{-1} Vol(g(t))= \infty $$
and then the K\"ahler-Ricci flow will contract the exceptional divisor $D_0$ at the singular time $$T= \frac{a_0}{n-1}.$$
We will assume through out this section that the initial K\"ahler class lies in $-a_0[D_0] + b_0 [D_\infty]$ with $ 0< a_0 (n+1) < b_0 (n-1).$
The following theorem is proved in \cite{TZha}.
\begin{theorem}\label{loccin}For any relatively compact set $K$ of $X\setminus D_0$ and $k>0$, there exists $C_{K, k}>0$ such that for all $t\in [0, T)$,
$$|| g(t)||_{C^k(K, g_0)} \leq C_{K, k}.$$
\end{theorem}
It immediately implies that the Ricci flow converges in local $C^\infty$ topology outside the exceptional divisor $D_0$ as $t\rightarrow T$.
The evolution equations for $u', u'', u'''$ are derived in \cite{SW1} as below.
\begin{eqnarray} \label{upevolution}
\ddt{} u' & = & \frac{u'''}{u''} + \frac{(n-1) u''}{u'} -n \\ \label{udpevolution}
\ddt{} u'' & = & \frac{u^{(4)}}{u''} - \frac{(u''')^2}{(u'')^2} + \frac{(n-1)u'''}{u'} - \frac{(n-1) (u'')^2}{(u')^2} \\ \nonumber
\ddt{} u''' & = & \frac{u^{(5)}}{u''} - \frac{3u''' u^{(4)}}{(u'')^2} + \frac{2(u''')^3}{(u'')^3}+ \frac{(n-1) u^{(4)}}{u'} \\ \label{utpevolution}
&& \mbox{} - \frac{3(n-1) u'' u'''}{(u')^2} + \frac{2(n-1) (u'')^3}{(u')^3}.
\end{eqnarray}
The following lemma is proved in \cite{SW1} for the collapsing case when $a_0 (n+1) > b_0 (n-1) $ and the same proof can be applied here. We include the proof for the sake of completeness.
\begin{lemma} There exists $C>0$ such that for all $t\in [0, T)$ and $\rho \in (-\infty, \infty)$,
\begin{equation}\label{u'}
(n-1) (T-t) \leq u' \leq C
\end{equation}
and
\begin{equation}\label{u''}
0\leq \frac{u''}{u'} \leq C, ~~-C \leq \frac{u'''}{u''} \leq C.
\end{equation}
\end{lemma}
\begin{proof} The estimate (\ref{u'}) follows from the monotonicity of $u'$ with $a_t < u' \leq b_t$ and $a_t= (n-1)(T-t).$
We apply the maximum principle to prove (\ref{u''}). It is straightforward to verify that for all $t\in [0, T)$,
$$ \lim_{\rho\rightarrow -\infty} \frac{u''(\rho, t)}{u'(\rho, t)}= \lim_{\rho\rightarrow \infty} \frac{u''(\rho, t)}{u'(\rho, t)}=0 $$
$$\lim_{\rho\rightarrow -\infty} \frac{u'''(\rho, t)}{u''(\rho, t)}=1, ~~ \lim_{\rho\rightarrow \infty} \frac{u'''(\rho, t)}{u''(\rho, t)}=-1 . $$
Let $H= \frac{u''}{u'}$. $H$ is strictly positive for all $\rho\in (-\infty, \infty)$ and $t\in [0, T)$. The evolution for $H$ is given by
$$\ddt{ H}= \frac{ H''}{u''} + \frac{2H'}{u'} - \frac{2H^2 - H}{u'} .$$
Therefore $\sup_{\rho\in (-\infty, \infty), t\in [0, T)} H \leq C$ for some uniform constant $C>0$ by applying the maximum principle.
Let $G=\frac{u'''}{u''}$. Then the evolution for $G$ is given by
$$\ddt{} G = \frac{1}{u''} G'' + \left( \frac{n-1}{u'} - \frac{u'''}{(u'')^2} \right) G' - \frac{2(n-1)u''}{(u')^2} \left( G- \frac{u''}{u'} \right).$$
Therefore $\sup_{\rho\in (-\infty, \infty), t\in [0, T)} |G| \leq C$ for some uniform constant $C>0$ by combining the maximum principle and the uniform upper bound for $H$.
\end{proof}
By taking the trace, we obtain an explicit expression for the scalar curvature
\begin{equation}
R = - \frac{ \ddt{u''}}{u''} - \frac{(n-1) \ddt{u'}}{u'}= -\frac{u^{(4)}}{(u'')^2} + \frac{(u''')^2}{(u'')^3} - \frac{2(n-1) u'''}{u'u''} - \frac{(n-1)(n-2) u''}{(u')^2} + \frac{n(n-1)}{u'} .
\end{equation}
\begin{corollary}There exists $C>0$ such that for all $\rho\in (-\infty, \infty)$ and $t\in [0, T)$,
\begin{equation}\label{c4}
-\frac{ u^{(4)}}{ ( u'')^2} + \frac{ (u''')^2 }{ (u'')^3} \geq - \frac{C}{T-t}.
\end{equation}
\end{corollary}
\begin{proof} Since the scalar curvature $R$ is uniformly bounded below, there exists $C_1>0$ such for all $t\in [0, T)$ and $\rho\in (-\infty, \infty)$,
$$ -\frac{u^{(4)}}{(u'')^2} + \frac{(u''')^2}{(u'')^3} - \frac{2(n-1) u'''}{u'u''} - \frac{(n-1)(n-2) u''}{(u')^2} + \frac{n(n-1)}{u'} \geq - C_1. $$
There also exist $C_2, C_3>0$ such that
$$u' \geq C_2 (T-t)$$
and
$$\left| \frac{u''}{u'} \right| + \left| \frac{ u''' } { u'' } \right| \leq C_3.$$
The estimate (\ref{c4}) immediately follows from the above estimates.
\end{proof}
The holomorphic bisectional curvature $R_{i\bar j k \bar l}$ is computed in \cite{Cao1} and is given by
\begin{eqnarray*}
R_{i\bar j k \bar l} &=& e^{-2\rho} ( u' - u'' ) (\delta_{ij}\delta_{kl} + \delta_{il} \delta_{kj} )\\
&& + e^{-2\rho} (3u'' - 2 u' - u''') (\delta_{ij} \delta_{kl1} + \delta_{il} \delta_{kj1} + \delta_{kl} \delta_{ij1} + \delta_{kj}\delta_{il1} )\\
&& + e^{-2\rho} \left( 6u''' - 11u'' - u^{(4)} + 6 u' + \frac{(u'' - u''')^2}{u''} \right) \delta_{ijkl1}\\
&& + e^{-2\rho} \frac{ (u' - u'')^2}{ u'} (\delta_{ij\hat 1} \delta_{kl1} + \delta_{il\hat 1} \delta_{kj1} + \delta_{kl\hat 1} \delta_{ij1} + \delta_{kj \hat 1}\delta_{il1} )
\end{eqnarray*}
Here $\delta_{ij1}$ and $\delta_{ijkl1}$ vanish unless all the indices are $1$, while $\delta{ij\hat 1}$ vanishes unless $i=j\neq 1$.
For any point $p$ on $\mathbb{C}^n \setminus \{ 0\}$, we can assume the coordinates at $p$ are given by $z(p)=(z_1, ..., z_n) = (z_1, 0, ..., 0)$ after a unitary transformation.
Then all the nonvanishing terms of the holomorphic bisectional curvature are given by
\begin{eqnarray*}
&&R_{1\bar 1 1 \bar 1} = e^{-2\rho} \left( - u^{(4)} + \frac{ ( u''')^2}{u''} \right) \\
&&R_{k\bar k k \bar k} = 2e^{-2\rho} ( u ' - u''), ~k>1 \\
&&R_{1\bar 1 k \bar k} = e^{-2\rho} \left( - u''' + \frac{ ( u'')^2 }{u'} \right), ~ k>1 \\
&& R_{k \bar k l \bar l}= e^{-2\rho}( u' - u'' ) , ~k>1, l>1, ~k\neq l.
\end{eqnarray*}
\begin{lemma} There exists $C>0$ such that on for all $t\in [0, T)$, $p = (z_1, 0, ..., 0)$ and $i, j, k, l$, we have at $(p, t)$,
\begin{equation}
R_{i\bar j k \bar l} \geq - \frac{C}{T-t} (g_{i\bar j} g_{k\bar l} + g_{i\bar l} g_{k \bar j}).
\end{equation}
Furthermore,
\begin{equation}
|R_{i\bar j k \bar l}| \leq \frac{C}{T-t} (g_{i\bar j} g_{k\bar l} + g_{i\bar l} g_{k \bar j})
\end{equation}
unless $i=j=k=l=1$.
\end{lemma}
\begin{proof} Since $p = (z_1, 0, ..., 0)$, it suffices to verify the estimates for $R_{1\bar 1 1 \bar 1}$, $R_{1\bar 1 k \bar k}$ and $R_{k\bar k l \bar l}$ for $k, l = 2, ... , n$.
Let $Q_{i\bar j k \bar l} = g_{i\bar j} g_{k\bar l} + g_{i\bar l} g_{k\bar j}.$ Then
\begin{eqnarray*}
&&Q_{1\bar 1 1 \bar 1} = 2e^{-2\rho} (u'')^2 \\
&&Q_{k\bar k k \bar k} = 2e^{-2\rho} ( u')^2, ~k>1 \\
&&Q_{1\bar 1 k \bar k} = e^{-2\rho} u' u'', ~ k>1 \\
&& Q_{k \bar k l \bar l}= e^{-2\rho}( u' )^2 , ~k>1, l>1, ~k\neq l.
\end{eqnarray*}
Comparing $R_{i\bar j k\bar l}$ and $Q_{i\bar j k \bar l}$, the lemma follows immediately.
\end{proof}
\begin{proposition}\label{lowbd} The holomorphic bisectional curvature is uniformly bounded below by $-C(T-t)^{-1}$ on $X\times [0, T)$ for some fixed constant $C>0$.
\end{proposition}
\begin{proof}
It suffices to calculate the lower bound of the holomorphic bisectional curvature at a point $p = (z_1, 0, ..., 0)$ and $t\in [0, T)$. Let $V=V^i \frac{\partial}{\partial z_i}$ and $W= W^i \frac{\partial}{\partial z_i}$ be two vectors in $TX_p$.
Then there exists $C>0$ such that
\begin{eqnarray*}
&& R_{i\bar j k\bar l} V^i V^{\bar j} W^k W^{\bar l} \\
&=& R_{1\bar1 1 \bar 1} V^1V^{\bar 1} W^1 W^{\bar 1} + (1- \delta_{ijkl 1} ) R_{i\bar j k \bar l} W^k W^{\bar l} \\
&\geq & - \frac{2C}{T-t} g_{1\bar 1}g_{1\bar 1} \left|V^1 \right|^2 \left|W^1 \right|^2 - \frac{C}{T-t} ( g_{i\bar j} g_{ k\bar l} + g_{i\bar l} g_{k\bar j} ) \left|V^i \right| \left| V^{\bar j} \right| \left| W^k \right| \left| W^{\bar l} \right|\\
&\geq & - \frac{4C}{T-t} \left|V \right|^2_{g} \left|W \right|^2_g.
\end{eqnarray*}
\end{proof}
\begin{definition} Let $g$ be a K\"ahler metric on a K\"ahler manifold $M$. At each point $p\in X$, we can choose the normal coordinates at $p$ such that for $i, j =1 , ..., n$, $g_{i\bar j}(p) =\delta_{i\bar j}$ is the identity matrix and $$R_{i\bar j} (p)= \delta_{ij} \lambda_j.$$ We define the $k^{th}$ symmetric polynomial of Ricci curvature of $g$ at $p$ by
\begin{equation}
\sigma_k = \sigma_k(Ric(g)) = \sum_{j_1< j_2< ... < j_k} \lambda_{j_1} \lambda_{j_2} ... \lambda_{j_k}
\end{equation} for $1\leq k\leq n$.
\end{definition}
The next proposition gives a uniform bound for $\sigma_k$ in terms of the curvature tensor $R_{1\bar 1 1 \bar 1}$ at each point $z=(z_1, 0, ..., 0)$.
\begin{proposition} \label{sigmak} There exists $C>0$ such that for all $(p, t) \in X\times [0, \infty)$,
\begin{equation}
| \sigma_k (p, t) | \leq \frac{C|Rm(p, t)|} { (T-t)^{k -1} }.
\end{equation}
\end{proposition}
\begin{proof} For any point $p\in \mathbb{C}^n \setminus \{0\}$, we can assume that $p=(z_1, 0, ..., 0)$. Then the eigenvalues of $Ric(g)$ at $p$ with respect to $g$ are given by
\begin{eqnarray*}
&&\lambda_1 = - \frac{\ddt{u''}}{u''}= - \frac{ u^{(4)}} {(u'')^2} + \frac{ (u''')^2}{(u'')^3} - \frac{(n-1) u'''}{u'u''} + \frac{(n-1) u''}{(u')^2}\\
&& \lambda_2 = ... = \lambda_n = -\frac{\ddt{u'}}{u'} = - \frac{u'''}{u'u''} - \frac{ (n-1) u'' }{(u')^2} + \frac{ n }{u'}.
\end{eqnarray*}
Then $(T-t)|\lambda_j|$ is uniformly bounded for $j=2, ..., n$ and
$$|\sigma_k|(p, t) = \sum_{j_1< j_2< ... < j_k } |\lambda_{j_1} \lambda_{j_2} ... \lambda_{j_k}| \leq C (T-t)^{-(k-1)}|\lambda_1| \leq C (T-t)^{-(k-1)} |Rm|_g(p,t).$$
\end{proof}
\begin{lemma}\label{nonfl} For any $p \in D_0$, we have
\begin{equation}
|Ric(p, t)|_{g(t)} \geq \frac{1}{T-t}.
\end{equation}
\end{lemma}
\begin{proof} It suffices to compute $e^{-\rho} v'$ which is one of the eigenvalues in the Ricci tensors since $D_0 = \{ \rho= -\infty\}$.
\begin{eqnarray*}
\lim_{\rho\rightarrow -\infty} e^{-\rho} v'(\rho) & =& - \lim_{\rho\rightarrow -\infty} \frac{ u'''}{u' u''} - \lim_{\rho\rightarrow -\infty} \frac{(n-1) u''}{(u')^2} + \lim_{\rho\rightarrow -\infty} \frac{n}{u'}\\
&=&(n-1) \lim_{\rho\rightarrow -\infty} (u')^{-1}\\
&=&\frac{1}{T-t}.
\end{eqnarray*}
Therefore $|Ric|_g$ is uniformly bounded below by $(T-t)^{-1}$ along the exceptional divisor $D_0$.
\end{proof}
\section{\bf Type I singularities}
In this section, we prove that the Ricci flow must develop Type I singularities with the same assumptions in section 4.
Let's first recall the definition for a Type I singularity of the Ricci flow.
\begin{definition} Let $(M, g(t))$ be a smooth solution of the Ricci flow (\ref{rf}) for $t\in [0, T)$ with $T<\infty$. It is said to develope a Type I singularity at $T$ if it cannot be smoothly extended past $T$ and there exists $C>0$ such that for all $t\in [0, T)$,
\begin{equation}
\sup_{M} |Rm(g(t))|_{g(t)} \leq \frac{C}{T-t}.
\end{equation}
\end{definition}
The following splitting theorem is proved in \cite{Cao2} as a complex analogue of Hamilton's splitting theorem on Riemannian manifolds with nonnegative curvature operator \cite{H2}.
\begin{theorem}\label{splitting}
Let $g$ be a complete solution of the K\"ahler-Ricci flow
on a noncompact simply connected K\"ahler manifold $M$ of dimension $n$
for $t\in (-\infty, \infty)$ with bounded and nonnegative holomorphic
bisectioanl curvature. Then either $g$ is of positive Ricci curvature for all
$p \in M$ and all $t\in (-\infty, \infty)$, or $(M, g)$ splits holomorphically isometrically into a
product $\mathbb{C}^k \times N^{n-k}$ $(k \geq 1)$ flat in $\mathbb{C}^k$ direction and $N$ being of nonnegative holomorphic bisectioanl curvature and positive Ricci curvature.
\end{theorem}
We are now able to exclude Type II singularities.
\begin{theorem} \label{type1} Let $X$ be $\mathbb{C}\mathbb{P}^n$ blown-up at one point and $g(t)$ be the solution of the K\"ahler-Ricci flow on $X$ starting with a $U(n)$-invariant K\"ahler metric $g_0$. If $g_0$ lies in the K\"ahler class $$-a_0 [D_0] + b_0 [D_\infty]$$ for $ 0< a_0(n+1) < b_0 (n-1)$. Then the flow develops Type I singularities at $T=a_0/(n-1)$.
\end{theorem}
\begin{proof} Suppose the flow develops Type II singularities. Let $t_j$ be an increasing sequence converging to $T=(n-1)a_0>0$ and $p_j$ a sequence of points on $X$ such that
$$K_j = |Rm(p_j, t_j)|_{g(t_j)} = \sup_{X } |Rm|_{g(t_j)}$$
and
$$\lim_{j\rightarrow \infty} (T-t_j)^{-1} K_j^{-1} = 0. $$
Applying the standard parabolic rescaling, we define
$$ g_j(t) = K_j g(t_j + K_j^{-1} t).$$
After extracting a convergent subsequence, $(X, g_j(t), p_j) $ converges in pointed Cheeger-Gromov-Hamilton sense to a complete eternal solution $(X_\infty, g_\infty(t), p_\infty)$ on a complete K\"ahler manifold $X_\infty$ of dimension $n$. Furthermore, by the lower bound of the holomorphic bisectional curvature of $g(t)$ by Proposition \ref{lowbd}, the limiting K\"ahler metric $g_\infty(t)$ has nonnegative holomorphic bisectional curvature everywhere on $X_\infty$. On the other hand, the symmetric product of the Ricci curvature $g_\infty$ vanishes everywhere in $X_\infty$,
$$\sigma_k (Ric(g_\infty)) = 0$$ for $2\leq k \leq n$.
This implies that the Ricci curvature of $g_\infty$ is not positive at each point of $X_\infty$. By applying the splitting theorem \ref{splitting} for $(n-1)$ times, $(\tilde X_\infty, \tilde g_\infty, \tilde p_\infty)$, the eternal solution on the universal cover of $(X_\infty, g_\infty, p_\infty)$, splits holomophically isometrically into $\mathbb{C}^{n-1} \times N$, where $N$ is a compact or complete Riemann surface with positive scalar curvature. By the classification of eternal solutions of real dimension $2$ by Hamilton \cite{H3}, $(N, \tilde g_\infty(t)|_N )$ is a steady gradient soliton and hence it must be the cigar soliton. However, it violates Peralman's local non-collapsing \cite{P1}, so does $(X_\infty, g_\infty)$. It then leads to a contradiction.
\end{proof}
\section{\bf Blow-up limits}
In this section, we will prove that the blow-up limit of the Ricci flow near the singular time $T$ along the exceptional divisor is a nontrivial complete shrinking gradient K\"ahler-Ricci soliton.
We first prove a diameter bound of the exceptional divisor $D_0$.
\begin{lemma} \label{restcp}For all $t \in [0, T)$,
\begin{equation}
g(t)|_{S} = a_0(n-1)(T-t) g_{FS}.
\end{equation}
and so
\begin{equation}
diam (S, g(t) |_{D_s} ) = \alpha_n (a_0 (n-1)(T-t))^{1/2}
\end{equation}
where $g_{FS}$ is a Fubini-Study metric on $\mathbb{C}\mathbb{P}^{n-1}$ and $\alpha_n$ is the diameter of $(\mathbb{C}\mathbb{P}^{n-1}, g_{FS})$.
\end{lemma}
\begin{proof}
The K\"ahler metric $g(t)$ is the metric completion of the following metric on $\mathbb{C}^n \setminus \{0\}$
$$ \omega(t) = a_0 (n-1) (T-t) \sqrt{-1}\partial\dbar \rho + \sqrt{-1}\partial\dbar u_0(e^\rho, t),$$ where $u_0(\cdot, t)$ is smooth and for each $t\in [0, T)$ with $u'(0, t)>0$. Note that after extending $\sqrt{-1}\partial\dbar \rho = \sqrt{-1}\partial\dbar \log |z|^2 $ to $\mathbb{C}\mathbb{P}^n$ blown-up at one point, its restriction on $D_0$ is exactly a Fubini-Study metric. The lemma then follows immediately.
\end{proof}
Now we can complete the proof of Theorem \ref{main} by identifying the blow-up limit of the Ricci flow at the singular time.
\begin{proposition}\label{blowuplim} Fix any $p\in D_0$. Then for every $K_j \rightarrow \infty$, the rescaled Ricci flows $(X, g_j(t)), p)$ defined on $[-K_j T, 0)$ by
$$g_j (t) = K_j g(T+ K_j^{-1} t)$$
subconverges in Cheeger-Gromov-Hamilton sense to a complete shrinking gradient K\"ahler-Ricci soliton on a complete K\"ahler manifold homeomorphic to $\mathbb{C}^n$ blown-up at one point.
\end{proposition}
\begin{proof} We first show that the blow-up limit is a nontrivial complete shrinking soliton. Fix any point $p\in D_0$ in the exceptional divisor. Since $(X, g(t))$ is a Type I Ricci flow, the rescaled Ricci flow $(X, g_t(t), p)$ always subconverges to a shrinking gradient soliton $(X_\infty, g_\infty(t), p_\infty)$ in pointed Cheeger-Gromov-Hamilton sense, by the compactness result of Naber \cite{Nab}. Such a limiting soliton cannot be flat because of Lemma \ref{nonfl}. In particular, $(X_\infty, g_\infty, p_\infty)$ is a complete shrinking gradient K\"ahler-Ricci soliton on a complete K\"ahler manifold $X_\infty$.
We now show that $X_\infty$ is in fact homeomorphic to $\mathbb{C}^n$ blown-up at one point. Fix a closed interval $[a, b] \subset (-\infty, 0)$, the rescaled Ricci flow $g_j (t) $ restricted to $D_0$ is uniformly equivalent to a fixed standard Fubini-Study metric on $\mathbb{C}\mathbb{P}^{n-1}$ for all $j$ and $t\in [a, b]$ by Lemma \ref{restcp} and so there exist $d, D>0$ such that the diameter of $D_0$ with respect to $g_j(t)$ is uniformly bounded between $d$ and $D$ for all $j$ and $t\in [a, b]$.
We denote by $$B_{g}(p, R)$$ the geodesic ball with respect to $g$ centered at $p$ with radius $R$. We then consider
$$\mathcal{B}_{j, t}(D_0, R) = \cup_{p\in D_0} B_{g_j (t)}(p, R)$$
for each $t\in [a, b]$. By choosing $R$ sufficiently large, we have
$$ B_{g_j(t)} (p, R) \subset \mathcal{B}_{j, t}(D_0, R) \subset B_{g_j(t)} (p, 2R) $$ for any point $p\in D_0$ becaue $g_j(t)$ is $U(n)$-invariant. By definition, for all $t\in [a, b]$, $B_{g_j(t)}(p, R)$ subconverges to $B_{g_\infty(t)} (p_\infty, R)$ in Cheeger-Gromov-Hamilton sense and so $B_{g_\infty(t)}(p_\infty, R)$ is homeomorphic to $B_{g_j(t)}(p, R)$ for sufficiently large $j$. We then obtain an exhaustion $B_{g_\infty (t)} (p, R_k)$ with each $R_k$ sufficiently large and $R_k\rightarrow \infty$. Each of them is homeomorphic to $\mathbb{C}^n$ blown-up at one point. Therefore $X_\infty$ is homeomorphic to $\mathbb{C}^n$ blown-up at one point.
\end{proof}
We remark that the convergence in the above proof is $U(n)$-equivariant and the limiting shrinking soliton $(X_\infty, g_\infty, p_\infty)$ is invariant under a free action of the unitary group $U(n)$. We also remark that the Type I blow-up limit is a trivial shrinking soliton if one chooses a fixed base point outside the exceptional divisor $D_0$. This is because the flow converges in local $C^\infty$ topology outside $D_0$ to a smooth K\"ahler metric on $X\setminus D_0$ by Theorem \ref{loccin} \cite{TZha}.
Combing Theorem \ref{type1} and Proposition \ref{blowuplim}, we complete the proof of Theorem \ref{main}.
\section{\bf Some generalizations}
In this section, we discuss some generalizations of Theorem \ref{main}. First, Theorem \ref{main} can be easily generalized to $X_{n,k}$ defined in section 1 by the same argument in the previous sections.
\begin{theorem}\label{main2} The Ricci flow on $X_{n,k}$ must develop Type I singularities for any $G_{n,k}$-invariant initial K\"ahler metric.
\smallskip
Let $g(t)$ be the smooth solution defined on $t\in [0, T)$, where $T\in (0, \infty)$ is the singular time. For every $K_j\rightarrow \infty$, we consider the rescaled Ricci flow $(X, g_j(t'))$ defined on $[-K_j T, 0)$ by
$$g_j(t') = K_j g(T+K_j^{-1} t').$$ Then one and only one of the following must occur.
\begin{enumerate}
\item[(1)] If $\liminf_{t\rightarrow T} (T-t)^{-1} Vol(g(t)) =\infty$, then $(X, g_j(t'), p)$ subconverges in Cheeger-Gromov-Hamilton sense to a complete nontrivial shrinking gradient K\"ahler-Ricci soliton on a complete K\"ahler manifold homeomorphic to the total space of $L^{-k} =\mathcal{O}_{\mathbb{C}\mathbb{P}^{n-1}}(-k)$, for any $p$ in the exceptional divisor.
\smallskip
\item[(2)] If $\liminf_{t\rightarrow T} (T-t)^{-1}Vol(g(t)) \in (0, \infty)$, then $(X, g_j(t'), p_j)$ subconverges in Cheeger-Gromov-Hamilton sense to $(\mathbb{C}^{n-1}\times \mathbb{C}\mathbb{P}^1, g_{\mathbb{C}^{n-1}} \oplus (-t') g_{FS})$, where $g_{\mathbb{C}^{n-1}}$ is the standard flat metric on $\mathbb{C}^{n-1}$ and $g_{FS}$ the Fubini-Study metric on $\mathbb{C}\mathbb{P}^1$ for any sequence of points $p_j$ \cite{Fo}.
\smallskip
\item[(3)] If $\liminf_{t\rightarrow T} (T-t)^{-1} Vol(g(t)) = 0$, then $(X, g_j(t'))$ converges in Cheeger-Gromov-Hamilton sense to the unique compact shrinking K\"ahler-Ricci soliton on $X_{n,k}$ blown-up at one point \cite{WZ}.
\end{enumerate}
\end{theorem}
We can also consider the Calabi symmetry introduced by Calabi \cite{Cal} for projective bundles over a K\"ahler-Einstein manifold (also see \cite{Li, SY}). In particular, we can consider the Ricci flow on generalizations of $X_{n,k}$
$$X_{m,n, k}= \mathbb{P}(\mathcal{O}_{\mathbb{C}\mathbb{P}^n} \oplus \mathcal{O}_{\mathbb{C}\mathbb{P}^n}(-k)^{\oplus (m+1)}),~~~k=1, 2, ... .$$
Similar results are obtained for $X_{m, n, k}$ in \cite{SY} for global Gromov-Hausdorff convergence at the singular time, as those for $X_{n,k}$ in \cite{SW1}.
Furthermore, one can obtain the same lower bound for the holomorphic bisectional curvature as in Proposition \ref{lowbd}.
\begin{proposition} Let $g(t)$ be the solution of the Ricci flow on $X_{m,n, k}$ for an initial K\"ahler metric with Calabi symmetry. Then if $1 \leq m \leq n$ and if $$\liminf_{t\rightarrow T} Vol(g(t))>0$$ where $T>0$ is the singular time, then the holomorphic bisectional curvature of $g(t)$ is uniformly bounded below by $-\frac{C}{T-t}$ for some constant $C>0$.
\end{proposition}
Although we are unable to exclude Type II singularities, one can show by the same argument in section 4, that the universal cover of the blow-up limit is an eternal solution of the Ricci flow which splits into $\mathbb{C}^{n} \times N^{m+1}$ flat in $\mathbb{C}^n$ and $N^{m+1}$ of nonnegative holomorphic bisectional curvature, if the flow develops Type II singularities. Of course, a Type I bound for the scalar curvature suffices to prove a similar theorem as Theorem \ref{main}.
\bigskip
\bigskip
\noindent{\bf Acknowledgements} The author would like to thank Zhenlei Zhang for many stimulating discussions. He would like also like to thank Valentino Tosatti for many helpful suggestions.
\bigskip
\bigskip
|
1,116,691,497,579 | arxiv | \section{Discussion}
The current standard for the evaluation of an unsupervised method involves the use of an AlexNet architecture trained on ImageNet and tested on class-level tasks.
To understand and measure the various biases introduced by this pipeline on DeepCluster\xspace, we consider a different training set, a different architecture and an instance-level recognition task.
\begin{table}[t!]
\centering
\begin{tabular}{@{}lc cc cc c cc c cc@{}}
\toprule
&~~~~& &~~~~& \multicolumn{2}{c}{Classification} &~~& \multicolumn{2}{c}{Detection} &~~& \multicolumn{2}{c}{Segmentation} \\
\cmidrule{5-6} \cmidrule{8-9} \cmidrule{11-12}
Method && Training set && \textsc{fc6-8}& \textsc{all} && \textsc{fc6-8}& \textsc{all} && \textsc{fc6-8}& \textsc{all} \\
\midrule
Best competitor &&ImageNet&& $~63.0~$ & $~67.7~$ && $~43.4^{\dagger}$ & $~53.2~$ && $~35.8^{\dagger}$ & $~37.7~$ \\
\midrule
DeepCluster\xspace&& ImageNet && $~72.0~$ & $~73.7~$ && $~51.4~$ & $~55.4~$ && $~43.2~$ & $~45.1~$ \\
DeepCluster\xspace&& YFCC100M && $~67.3~$ & $~69.3~$ && $~45.6~$ & $~53.0~$ && $~39.2~$ & $~42.2~$ \\
\bottomrule
\end{tabular}
\vspace{\soustable}
\caption{
Impact of the training set on the performance of DeepCluster\xspace measured on the \textsc{Pascal} VOC transfer tasks as described in Sec.~\ref{sec:pascal}.
We compare ImageNet with a subset of $1$M images from YFCC100M~\cite{thomee2015new}.
Regardless of the training set, DeepCluster\xspace outperforms the best published numbers on most tasks.
Numbers for other methods produced by us are marked with a ${\dagger}$
}
\label{tab:flickr}
\end{table}
\subsection{ImageNet versus YFCC100M}
ImageNet is a dataset designed for a fine-grained object classification challenge~\cite{russakovsky2015imagenet}.
It is object oriented, manually annotated and organised into well balanced object categories.
By design, DeepCluster\xspace favors balanced clusters and, as discussed above, our number of cluster $k$ is somewhat comparable with the number of labels in ImageNet.
This may have given an unfair advantage to DeepCluster\xspace over other unsupervised approaches when trained on ImageNet.
To measure the impact of this effect, we consider a subset of randomly-selected $1$M images from the YFCC100M dataset~\cite{thomee2015new} for the pre-training.
Statistics on the hashtags used in YFCC100M suggests that the underlying ``object classes'' are severly unbalanced~\cite{joulin2016learning},
leading to a data distribution less favorable to DeepCluster\xspace.
Table~\ref{tab:flickr} shows the difference in performance on \textsc{Pascal} VOC of DeepCluster\xspace pre-trained on YFCC100M compared to ImageNet.
As noted by Doersch~\emph{et al}\onedot~\cite{doersch2015unsupervised}, this dataset is not object oriented, hence the performance are expected to drop by a few percents.
However, even when trained on uncured Flickr images, DeepCluster\xspace outperforms the current state of the art by a significant margin on most tasks (up to $+4.3\%$ on classification and $+4.5\%$ on semantic segmentation).
We report the rest of the results in the supplementary material with similar conclusions.
This experiment validates that DeepCluster\xspace is robust to a change of image distribution, leading
to state-of-the-art general-purpose visual features even if this distribution is not favorable to its design.
\subsection{AlexNet versus VGG}
In the supervised setting, deeper architectures like VGG or ResNet~\cite{he2015delving} have a much higher accuracy on ImageNet than AlexNet.
We should expect the same improvement if these architectures are used with an unsupervised approach.
Table~\ref{tab:vocVGG} compares a VGG-16 and an AlexNet trained with DeepCluster\xspace on ImageNet and tested on the \textsc{Pascal} VOC 2007 object detection task with fine-tuning.
We also report the numbers obtained with other unsupervised approaches~\cite{doersch2015unsupervised,wang2017transitive}.
Regardless of the approach, a deeper architecture leads to a significant improvement in performance on the target task.
Training the VGG-16 with DeepCluster\xspace gives a performance above the state of the art, bringing us to only $1.4$ percents below the supervised topline.
Note that the difference between unsupervised and supervised approaches remains in the same ballpark for both architectures (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot $1.4\%$).
Finally, the gap with a random baseline grows for larger architectures,
justifying the relevance of unsupervised pre-training for complex architectures when little supervised data is available.
\begin{table}[t!]
\parbox[c][16em][t]{.45\linewidth}{
\centering
\begin{tabular}{@{}l c c c c@{}}
\toprule
Method &~~~& AlexNet && VGG-16 \\
\midrule
ImageNet labels && $56.8$ && $67.3$ \\
Random && $47.8$ && $39.7$ \\
\midrule
Doersch~\emph{et al}\onedot~\cite{doersch2015unsupervised} && $51.1$ && $61.5$ \\
Wang and Gupta~\cite{wang2015unsupervised} && $47.2$ && $60.2$ \\
Wang~\emph{et al}\onedot~\cite{wang2017transitive} && -- && $63.2$ \\
\midrule
DeepCluster\xspace && $\textbf{55.4}$ && $\textbf{65.9}$ \\
\bottomrule
\end{tabular}
\vspace{\soustable}
\caption{
\textsc{Pascal} VOC 2007 object detection with AlexNet and VGG-16.
Numbers are taken from Wang~\emph{et al}\onedot~\cite{wang2017transitive}.
}
\label{tab:vocVGG}
}
\hfill
\parbox[c][16em][t]{.45\linewidth}{
\centering
\begin{tabular}{@{}lc c c c@{}}
\toprule
Method &~~~& Oxford$5$K && Paris$6$K \\
\midrule
ImageNet labels && $72.4$ && $81.5$ \\
Random && $~6.9$ && $22.0$ \\
\midrule
Doersch~\emph{et al}\onedot~\cite{doersch2015unsupervised} && $35.4$ && $53.1$ \\
Wang~\emph{et al}\onedot~\cite{wang2017transitive} && $42.3$ && $58.0$ \\
\midrule
DeepCluster\xspace && $\textbf{61.0}$ && $\textbf{72.0}$ \\
\bottomrule
\end{tabular}
\vspace{\soustable}
\caption{
mAP on instance-level image retrieval on Oxford and Paris dataset with a VGG-16.
We apply R-MAC with a resolution of $1024$ pixels and $3$ grid levels~\cite{tolias2015particular}.
}
\label{tab:retrieval}
}
\end{table}
\subsection{Evaluation on instance retrieval}
The previous benchmarks measure the capability of an unsupervised network to capture class level information.
They do not evaluate if it can differentiate images at the instance level.
To that end, we propose image retrieval as a down-stream task.
We follow the experimental protocol of Tolias~\emph{et al}\onedot~\cite{tolias2015particular} on two datasets, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, Oxford Buildings~\cite{Philbin07} and Paris~\cite{Philbin08}.
Table~\ref{tab:retrieval} reports the performance of a VGG-16 trained with different approaches obtained with Sobel filtering, except for Doersch~\emph{et al}\onedot~\cite{doersch2015unsupervised} and Wang~\emph{et al}\onedot~\cite{wang2017transitive}.
This preprocessing improves by $5.5$ points the mAP of a supervised VGG-16 on the Oxford dataset, but not on Paris.
This may translate in a similar advantage for DeepCluster\xspace, but it does not account for the average differences of $19$ points.
Interestingly, random convnets perform particularly poorly on this task compared to pre-trained models.
This suggests that image retrieval is a task where the pre-training is essential and studying it as a down-stream task could give further insights about the quality of the features produced by unsupervised approaches.
\section{Introduction}
\input{intro.tex}
\section{Related Work}
\input{related.tex}
\section{Method}
\input{method.tex}
\section{Experiments}
\input{results.tex}
\input{discussion.tex}
\section{Conclusion}
In this paper, we propose a scalable clustering approach for the unsupervised learning of convnets.
It iterates between clustering with $k$-means the features produced by the convnet and
updating its weights by predicting the cluster assignments as pseudo-labels in a discriminative loss.
If trained on large dataset like ImageNet or YFCC100M, it achieves performance that are significantly
better than the previous state-of-the-art on every standard transfer task.
Our approach makes little assumption about the inputs, and does not require much domain specific knowledge, making it a
good candidate to learn deep representations specific to domains where annotations are scarce.
\paragraph{Acknowledgement.}
We thank Alexandre Sablayrolles and the rest of the FAIR team for their feedback and fruitful discussion around this paper.
We would like to particularly thank Ishan Misra for spotting an error in our evaluation setting of Table~\ref{tab:linear}.
\clearpage
\bibliographystyle{splncs}
\subsection{Preliminaries}
Modern approaches to computer vision, based on statistical learning, require good image featurization.
In this context, convnets are a popular choice for mapping raw images to a vector space of fixed dimensionality.
When trained on enough data, they constantly achieve the best performance on standard classification benchmarks~\cite{he2015delving,krizhevsky2012imagenet}.
We denote by $f_\theta$ the convnet mapping, where $\theta$ is the set of corresponding parameters.
We refer to the vector obtained by applying this mapping to an image as feature or representation.
Given a training set $X = \{x_1, x_2, \dots , x_N\}$ of $N$ images, we want to find a parameter $\theta^*$ such that the mapping $f_{\theta^*}$ produces good general-purpose features.
These parameters are traditionally learned with supervision, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot each image $x_n$ is associated with a label $y_n$ in $\{0,1\}^k$.
This label represents the image's membership to one of $k$ possible predefined classes.
A parametrized classifier $g_W$ predicts the correct labels on top of the features $f_\theta(x_n)$.
The parameters $W$ of the classifier and the parameter $\theta$ of the mapping are then jointly learned by optimizing the following problem:
\begin{eqnarray}\label{eq:sup}
\min_{\theta, W} \frac{1}{N} \sum_{n=1}^N \ell\left(g_W\left( f_\theta(x_n) \right), y_n\right),
\end{eqnarray}
where $\ell$ is the multinomial logistic loss, also known as the negative log-softmax function.
This cost function is minimized using mini-batch stochastic gradient descent~\cite{bottou2012stochastic} and backpropagation to compute the gradient~\cite{lecun1998gradient}.
\subsection{Unsupervised learning by clustering}
When $\theta$ is sampled from a Gaussian distribution, without any learning, $f_\theta$ does not produce good features.
However the performance of such random features on standard transfer tasks, is far above the chance level.
For example, a multilayer perceptron classifier on top of the last convolutional layer of a random AlexNet achieves 12\% in accuracy on ImageNet while the chance is at $0.1\%$~\cite{noroozi2016unsupervised}.
The good performance of random convnets is intimately tied to their convolutional structure which gives a strong prior on the input signal.
The idea of this work is to exploit this weak signal to bootstrap the discriminative power of a convnet.
We cluster the output of the convnet and use the subsequent cluster assignments as ``pseudo-labels'' to optimize Eq.~(\ref{eq:sup}).
This deep clustering (DeepCluster) approach iteratively learns the features and groups them.
Clustering has been widely studied and many approaches have been developed for a variety of circumstances. In the absence of points of comparisons,
we focus on a standard clustering algorithm, $k$-means. Preliminary results with other clustering algorithms indicates that this choice is not crucial.
$k$-means takes a set of vectors as input, in our case the features $f_\theta(x_n)$ produced by the convnet, and clusters them into $k$ distinct groups based on a geometric criterion.
More precisely, it jointly learns a $d\times k$ centroid matrix $C$ and the cluster assignments $y_n$ of each image $n$ by solving the following problem:
\begin{equation}
\label{eq:kmeans}
\min_{C \in \mathbb{R}^{d\times k}}
\frac{1}{N}
\sum_{n=1}^N
\min_{y_n \in \{0,1\}^{k}}
\| f_\theta(x_n) - C y_n \|_2^2
\quad
\text{such that}
\quad
y_n^\top 1_k = 1.
\end{equation}
Solving this problem provides a set of optimal assignments $(y_n^*)_{n\le N}$ and a centroid matrix $C^*$.
These assignments are then used as pseudo-labels; we make no use of the centroid matrix.
Overall, DeepCluster\xspace alternates between clustering the features to produce pseudo-labels using Eq.~(\ref{eq:kmeans})
and updating the parameters of the convnet by predicting these pseudo-labels using Eq.~(\ref{eq:sup}).
This type of alternating procedure is prone to trivial solutions; we describe how to avoid such degenerate solutions in the next section.
\subsection{Avoiding trivial solutions}
The existence of trivial solutions is not specific to the unsupervised training of neural networks, but to any method that jointly learns a discriminative classifier and the labels.
Discriminative clustering suffers from this issue even when applied to linear models~\cite{xu2005maximum}.
Solutions are typically based on constraining or penalizing the minimal number of points per cluster~\cite{bach2008diffrac,joulin2012convex}.
These terms are computed over the whole dataset, which is not applicable to the training of convnets on large scale datasets.
In this section, we briefly describe the causes of these trivial solutions and give simple and scalable workarounds.
\\
\noindent\textbf{Empty clusters.}
A discriminative model learns decision boundaries between classes.
An optimal decision boundary is to assign all of the inputs to a single cluster~\cite{xu2005maximum}.
This issue is caused by the absence of mechanisms to prevent from empty clusters and arises in linear models as much as in convnets.
A common trick used in feature quantization~\cite{johnson2017billion} consists in automatically reassigning empty clusters during the $k$-means optimization.
More precisely, when a cluster becomes empty, we randomly select a non-empty cluster and use its centroid with a small random perturbation as the new centroid for the empty cluster.
We then reassign the points belonging to the non-empty cluster to the two resulting clusters.
\\
\noindent\textbf{Trivial parametrization.}
If the vast majority of images is assigned to a few clusters, the parameters $\theta$ will exclusively discriminate between them.
In the most dramatic scenario where all but one cluster are singleton, minimizing Eq.~(\ref{eq:sup}) leads to a
trivial parametrization where the convnet will predict the same output regardless of the input.
This issue also arises in supervised classification when the number of images per class is highly unbalanced.
For example, metadata, like hashtags, exhibits a Zipf distribution, with a few labels dominating the whole distribution~\cite{joulin2016learning}.
A strategy to circumvent this issue is to sample images based on a uniform distribution over the classes, or pseudo-labels.
This is equivalent to weight the contribution of an input to the loss function in Eq.~(\ref{eq:sup}) by the inverse of the size of its assigned cluster.
\subsection{Implementation details}
\noindent\textbf{Convnet architectures.}
For comparison with previous works, we use a standard AlexNet~\cite{krizhevsky2012imagenet} architecture.
It consists of five convolutional layers with $96$, $256$, $384$, $384$ and $256$ filters; and of three fully connected layers.
We remove the Local Response Normalization layers and use batch normalization~\cite{ioffe2015batch}.
We also consider a VGG-16~\cite{simonyan2014very} architecture with batch normalization.
Unsupervised methods often do not work directly on color and different strategies have been considered as alternatives
~\cite{doersch2015unsupervised,noroozi2016unsupervised}.
We apply a fixed linear transformation based on Sobel filters to remove color and increase local contrast~\cite{bojanowski2017unsupervised,paulin2015local}.
\\
\noindent\textbf{Training data.}
We train DeepCluster\xspace on ImageNet~\cite{deng2009imagenet} unless mentioned otherwise.
It contains~$1.3$M images uniformly distributed into~$1,000$ classes.
\\
\noindent\textbf{Optimization.}
We cluster the central cropped images features and perform data augmentation (random horizontal flips and crops of random sizes and aspect ratios) when training the network.
This enforces invariance to data augmentation which is useful for feature learning~\cite{dosovitskiy2014discriminative}.
The network is trained with dropout~\cite{srivastava2014dropout}, a constant step size, an~$\ell_2$
penalization of the weights~$\theta$ and a momentum of~$0.9$.
Each mini-batch contains $256$ images.
For the clustering, features are PCA-reduced to $256$ dimensions, whitened and $\ell_2$-normalized.
We use the $k$-means implementation of Johnson~\emph{et al}\onedot~\cite{johnson2017billion}.
Note that running k-means takes a third of the time because a forward pass on the full dataset is needed.
One could reassign the clusters every $n$ epochs, but we found out that our setup on ImageNet (updating the clustering every epoch) was nearly optimal.
On Flickr, the concept of epoch disappears: choosing the tradeoff between the parameter updates and the cluster reassignments is more subtle.
We thus kept almost the same setup as in ImageNet.
We train the models for $500$ epochs, which takes $12$ days on a Pascal P100 GPU for AlexNet.
\\
\noindent\textbf{Hyperparameter selection.}
We select hyperparameters on a down-stream task,~\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, object classification on
the validation set of \textsc{Pascal} VOC with no fine-tuning.
We use the publicly available code of Kr\"ahenb\"uhl\footnote{https://github.com/philkr/voc-classification}.
\subsection{Preliminary study}
We measure the information shared between two different assignments $A$ and $B$ of the same data by the
Normalized Mutual Information (NMI), defined as:
\begin{equation*}
\mathrm{NMI}(A;B)=\frac{\mathrm{I}(A;B)}{\sqrt{\mathrm{H}(A) \mathrm{H}(B)}}
\end{equation*}
where $\mathrm{I}$ denotes the mutual information and $\mathrm{H}$ the entropy. This measure can be applied to any assignment coming
from the clusters or the true labels.
If the two assignments $A$ and $B$ are independent, the NMI is equal to $0$.
If one of them is deterministically predictable from the other, the NMI is equal to $1$.
\\
\noindent\textbf{Relation between clusters and labels.}
Figure~\subref*{fig:edges} shows the evolution of the NMI between the cluster assignments and the ImageNet labels during training.
It measures the capability of the model to predict class level information.
Note that we only use this measure for this analysis and not in any model selection process.
The dependence between the clusters and the labels increases over time, showing that our features progressively capture information
related to object classes.
\\
\noindent\textbf{Number of reassignments between epochs.}
At each epoch, we reassign the images to a new set of clusters, with no guarantee of stability.
Measuring the NMI between the clusters at epoch $t-1$ and $t$ gives an insight on the actual
stability of our model.
Figure~\subref*{fig:reassing} shows the evolution of this measure during training.
The NMI is increasing, meaning that there are less and less reassignments and the clusters are stabilizing over time.
However, NMI saturates below $0.8$, meaning that a significant fraction of images are regularly reassigned between epochs.
In practice, this has no impact on the training and the models do not diverge.
\\
\noindent\textbf{Choosing the number of clusters.}
We measure the impact of the number $k$ of clusters used in $k$-means on the quality of the model.
We report the same down-stream task as in the hyperparameter selection process, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot mAP on the \textsc{Pascal} VOC 2007 classification validation set.
We vary $k$ on a logarithmic scale, and report results after $300$ epochs in Figure~\subref*{fig:choice-k}.
The performance after the same number of epochs for every $k$ may not be directly comparable, but it reflects the hyper-parameter selection process
used in this work.
The best performance is obtained with $k=10,000$.
Given that we train our model on ImageNet, one would expect $k = 1000$ to yield the best results, but apparently some amount of over-segmentation is beneficial.
\subsection{Visualizations}
\label{sec:viz}
\noindent\textbf{First layer filters.}
Figure~\ref{fig:filters} shows the filters from the first layer of an AlexNet trained with DeepCluster\xspace on raw RGB images and images preprocessed with a Sobel filtering.
The difficulty of learning convnets on raw images has been noted before~\cite{bojanowski2017unsupervised,doersch2015unsupervised,noroozi2016unsupervised,paulin2015local}.
As shown in the left panel of Fig.~\ref{fig:filters}, most filters capture only color information that typically plays a little role for object classification~\cite{van2011evaluating}.
Filters obtained with Sobel preprocessing act like edge detectors.
\\
\begin{figure}[t]
\centering
\begin{tabular}{cccccccc}
\multicolumn{2}{c}{\texttt{conv1}} &~~~& \multicolumn{2}{c}{\texttt{conv3}} &~~~& \multicolumn{2}{c}{\texttt{conv5}}\\
\includegraphics[width=0.15\linewidth]{images-filters-smlayer0-channel43.jpeg}&
\includegraphics[width=0.15\linewidth]{images-filters-layer0-channel43.jpeg}&&
\includegraphics[width=0.15\linewidth]{images-filters-smlayer2-channel5.jpeg}&
\includegraphics[width=0.15\linewidth]{images-filters-layer2-channel5.jpeg}&&
\includegraphics[width=0.15\linewidth]{images-filters-smlayer4-channel45.jpeg}&
\includegraphics[width=0.15\linewidth]{images-filters-layer4-channel45.jpeg}
\\
\includegraphics[width=0.15\linewidth]{images-filters-smlayer0-channel46.jpeg}&
\includegraphics[width=0.15\linewidth]{images-filters-layer0-channel46.jpeg}&&
\includegraphics[width=0.15\linewidth]{images-filters-smlayer2-channel80.jpeg}&
\includegraphics[width=0.15\linewidth]{images-filters-layer2-channel80.jpeg}&&
\includegraphics[width=0.15\linewidth]{images-filters-smlayer4-channel2.jpeg}&
\includegraphics[width=0.15\linewidth]{images-filters-layer4-channel2.jpeg}
\end{tabular}
\caption{
Filter visualization and top $9$ activated images from a subset of $1$ million images from YFCC100M for target filters in the
layers \texttt{conv1}, \texttt{conv3} and \texttt{conv5} of an AlexNet trained with DeepCluster\xspace on ImageNet.
The filter visualization is obtained by learning an input image that maximizes the response to a target filter~\cite{yosinski2015understanding}.
}
\label{fig:activ}
\end{figure}
\begin{figure}[t]
\centering
\begin{tabular}{ccccccc}
Filter $0$ && Filter $33$ && Filter $145$ && Filter $194$
\\
\includegraphics[width=0.24\linewidth]{images-filters-layer4-channel0.jpeg}&&
\includegraphics[width=0.24\linewidth]{images-filters-layer4-channel33.jpeg}&&
\includegraphics[width=0.24\linewidth]{images-filters-layer4-channel145.jpeg}&&
\includegraphics[width=0.24\linewidth]{images-filters-layer4-channel194.jpeg}
\\
Filter $97$ && Filter $116$ && Filter $119$ && Filter $182$
\\
\includegraphics[width=0.24\linewidth]{images-filters-layer4-channel97.jpeg}&&
\includegraphics[width=0.24\linewidth]{images-filters-layer4-channel116.jpeg}&&
\includegraphics[width=0.24\linewidth]{images-filters-layer4-channel119.jpeg}&&
\includegraphics[width=0.24\linewidth]{images-filters-layer4-channel182.jpeg}
\end{tabular}
\caption{
Top $9$ activated images from a random subset of $10$ millions images from YFCC100M for target filters in the last convolutional layer.
The top row corresponds to filters sensitive to activations by images containing objects.
The bottom row exhibits filters more sensitive to stylistic effects.
For instance, the filters $119$ and $182$ seem to be respectively excited by background blur and depth of field effects.
}
\label{fig:waouh}
\end{figure}
\noindent\textbf{Probing deeper layers.}
We assess the quality of a target filter by learning an input image that maximizes its activation~\cite{erhan2009visualizing,zeiler2014visualizing}.
We follow the process described by Yosinki~\emph{et al}\onedot~\cite{yosinski2015understanding} with a
cross entropy function between the target filter and the other filters of the same layer.
Figure~\ref{fig:activ} shows these synthetic images as well as the $9$ top activated images from a subset of $1$ million images
from YFCC100M.
As expected, deeper layers in the network seem to capture larger textural structures.
However, some filters in the last convolutional layers seem to be simply replicating
the texture already captured in previous layers, as shown on the second row of Fig.~\ref{fig:waouh}.
This result corroborates the observation by Zhang~\emph{et al}\onedot~\cite{zhang2016split} that features
from \texttt{conv3} or \texttt{conv4} are more discriminative than those from \texttt{conv5}.
Finally, Figure~\ref{fig:waouh} shows the top $9$ activated images of some \texttt{conv5} filters that seem to be semantically coherent.
The filters on the top row contain information about structures that highly corrolate with object classes.
The filters on the bottom row seem to trigger on style, like drawings or abstract shapes.
\subsection{Linear classification on activations}
\label{sec:linear}
Following Zhang~\emph{et al}\onedot~\cite{zhang2016split}, we train a linear classifier on top of different frozen convolutional layers.
This layer by layer comparison with supervised features exhibits where a convnet starts to be task specific, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot specialized in object classification.
We report the results of this experiment on ImageNet and the Places dataset~\cite{zhou2014learning} in Table~\ref{tab:linear}.
We choose the hyperparameters by cross-validation on the training set.
On ImageNet, DeepCluster\xspace outperforms the state of the art from \texttt{conv3} to \texttt{conv5} layers by $3-5\%$.
The largest improvement is observed in the \texttt{conv4} layer, while the \texttt{conv1} layer performs poorly,
probably because the Sobel filtering discards color.
Consistently with the filter visualizations of Sec.~\ref{sec:viz}, \texttt{conv3} works better than \texttt{conv5}.
Finally, the difference of performance between DeepCluster\xspace and a supervised AlexNet grows significantly on higher layers:
at layers \texttt{conv2-conv3} the difference is only around $6\%$,
but this difference rises to $14.4\%$ at \texttt{conv5}, marking where the AlexNet probably stores most of the class level information.
In the supplementary material, we also report the accuracy if a MLP is trained on the last layer; DeepCluster\xspace outperforms the state of the art by $8\%$.
\begin{table}[t]
\centering
\resizebox{\columnwidth}{!}{%
\begin{tabular}{@{}l c ccccc c ccccc@{}}
\toprule
&~~~& \multicolumn{5}{c}{ImageNet} &~~~& \multicolumn{5}{c}{Places} \\
\cmidrule{3-7} \cmidrule{9-13}
Method && \texttt{conv1} & \texttt{conv2} & \texttt{conv3} & \texttt{conv4} & \texttt{conv5} && \texttt{conv1} & \texttt{conv2} & \texttt{conv3} & \texttt{conv4} & \texttt{conv5} \\
\midrule
Places labels && -- & -- & -- & -- & -- && $22.1$ & $35.1$ & $40.2$ & $43.3$ & $44.6$ \\
ImageNet labels && $19.3$ & $36.3$ & $44.2$ & $48.3$ & $50.5$ && $22.7$ & $34.8$ & $38.4$ & $39.4$ & $38.7$ \\
Random && $11.6$ & $17.1$ & $16.9$ & $16.3$ & $14.1$ && $15.7$ & $20.3$ & $19.8$ & $19.1$ & $17.5$ \\
\midrule
Pathak~\emph{et al}\onedot~\cite{pathak2016context} && $14.1$ & $20.7$ & $21.0$ & $19.8$ & $15.5$ && $18.2$ & $23.2$ & $23.4$ & $21.9$ & $18.4$ \\
Doersch~\emph{et al}\onedot~\cite{doersch2015unsupervised} && $16.2$ & $23.3$ & $30.2$ & $31.7$ & $29.6$ && $19.7$ & $26.7$ & $31.9$ & $32.7$ & $30.9$ \\
Zhang~\emph{et al}\onedot~\cite{zhang2016colorful} && $12.5$ & $24.5$ & $30.4$ & $31.5$ & $30.3$ && $16.0$ & $25.7$ & $29.6$ & $30.3$ & $29.7$ \\
Donahue~\emph{et al}\onedot~\cite{donahue2016adversarial} && $17.7$ & $24.5$ & $31.0$ & $29.9$ & $28.0$ && $21.4$ & $26.2$ & $27.1$ & $26.1$ & $24.0$ \\
Noroozi and Favaro~\cite{noroozi2016unsupervised} && $\textbf{18.2}$ & $28.8$ & $34.0$ & $33.9$ & $27.1$ && $23.0$ & $32.1$ & $35.5$ & $34.8$ & $31.3$ \\
Noroozi~\emph{et al}\onedot~\cite{noroozi2017representation} && $18.0$ & $\textbf{30.6}$ & $34.3$ & $32.5$ & $25.7$ && $\textbf{23.3}$ & $\textbf{33.9}$ & $36.3$ & $34.7$ & $29.6$ \\
Zhang~\emph{et al}\onedot~\cite{zhang2016split} && $17.7$ & $29.3$ & $35.4$ & $35.2$ & $32.8$ && $21.3$ & $30.7$ & $34.0$ & $34.1$ & $32.5$ \\
\midrule
DeepCluster\xspace && $12.9$ & $29.2$ & $\textbf{38.2}$ & $\textbf{39.8}$ & $\textbf{36.1}$ && $18.6$ & $30.8$ & $\textbf{37.0}$ & $\textbf{37.5}$ & $\textbf{33.1}$ \\
\bottomrule
\end{tabular}
}
\vspace{\soustable}
\caption{
Linear classification on ImageNet and Places using activations from the convolutional layers of an AlexNet as features.
We report classification accuracy on the central crop.
Numbers for other methods are from Zhang~\emph{et al}\onedot~\cite{zhang2016split}.
}
\label{tab:linear}
\end{table}
The same experiment on the Places dataset provides some interesting insights:
like DeepCluster\xspace, a supervised model trained on ImageNet suffers from a decrease of performance for higher layers (\texttt{conv4} versus \texttt{conv5}).
Moreover, DeepCluster\xspace yields \texttt{conv3-4} features that are comparable to those trained with ImageNet labels.
This suggests that when the target task is sufficently far from the domain covered by ImageNet, labels are less important.
\subsection{\textsc{Pascal} VOC 2007}
\label{sec:pascal}
Finally, we do a quantitative evaluation of DeepCluster\xspace on image classification, object detection and semantic segmentation on \textsc{Pascal} VOC.
The relatively small size of the training sets on \textsc{Pascal} VOC ($2,500$ images) makes this setup closer to a ``real-world'' application,
where a model trained with heavy computational resources, is adapted to a task or a dataset with a small number of instances.
Detection results are obtained using \texttt{fast-rcnn}\footnote{https://github.com/rbgirshick/py-faster-rcnn}; segmentation results are obtained using the code of Shelhamer~\emph{et al}\onedot\footnote{https://github.com/shelhamer/fcn.berkeleyvision.org}.
For classification and detection, we report the performance on the test set of \textsc{Pascal} VOC 2007 and choose our hyperparameters on the validation set.
For semantic segmentation, following the related work, we report the performance on the validation set of \textsc{Pascal} VOC 2012.
\begin{table}[t]
\centering
\begin{tabular}{lc cc c cc c cc}
\toprule
&~~~~~~~~& \multicolumn{2}{c}{Classification} &~~~& \multicolumn{2}{c}{Detection} &~~~& \multicolumn{2}{c}{Segmentation} \\
\cmidrule{3-4} \cmidrule{6-7} \cmidrule{9-10}
Method && \textsc{fc6-8}& \textsc{all} && \textsc{fc6-8}& \textsc{all} && \textsc{fc6-8} & \textsc{all} \\
\midrule
ImageNet labels && $~78.9~$ & $~79.9~$ && -- & $~56.8~$ && -- & $~48.0~$ \\
Random-rgb && $~33.2~$ & $~57.0~$ && $~22.2~$ & $~44.5~$ && $~15.2~$ & $~30.1~$ \\
Random-sobel && $~29.0~$ & $~61.9~$ && $~18.9~$ & $~47.9~$ && $~13.0~$ & $~32.0~$ \\
\midrule
Pathak~\emph{et al}\onedot~\cite{pathak2016context} && $~34.6~$ & $~56.5~$ && -- & $~44.5~$ && -- & $~29.7~$ \\
Donahue~\emph{et al}\onedot~\cite{donahue2016adversarial}$^*$ && $~52.3~$ & $~60.1~$ && -- & $~46.9~$ && -- & $~35.2~$ \\
Pathak~\emph{et al}\onedot~\cite{pathak2016learning} && -- & $~61.0~$ && -- & $~52.2~$ && -- & -- \\
Owens~\emph{et al}\onedot~\cite{owens2016ambient}$^*$ && $~52.3~$ & $~61.3~$ && -- & -- && -- & -- \\
Wang and Gupta~\cite{wang2015unsupervised}$^*$ && $~55.6~$ & $~63.1~$ && $~32.8^\dagger$ & $~47.2~$ && $~26.0^\dagger$ & $~35.4^\dagger$ \\
Doersch~\emph{et al}\onedot~\cite{doersch2015unsupervised}$^*$ && $~55.1~$ & $~65.3~$ && -- & $~51.1~$ && -- & -- \\
Bojanowski and Joulin~\cite{bojanowski2017unsupervised}$^*$ && $~56.7~$ & $~65.3~$ && $~33.7^\dagger$ & $~49.4~$ && $~26.7^\dagger$ & $~37.1^\dagger$ \\
Zhang~\emph{et al}\onedot~\cite{zhang2016colorful}$^*$ && $~61.5~$ & $~65.9~$ && $~43.4^\dagger$ & $~46.9~$ && $~35.8^\dagger$ & $~35.6~$ \\
Zhang~\emph{et al}\onedot~\cite{zhang2016split}$^*$ && $~63.0~$ & $~67.1~$ && -- & $~46.7~$ && -- & $~36.0~$ \\
Noroozi and Favaro~\cite{noroozi2016unsupervised} && -- & $~67.6~$ && -- & $~53.2~$ && -- & $~37.6~$ \\
Noroozi~\emph{et al}\onedot~\cite{noroozi2017representation} && -- & $~67.7~$ && -- & $~51.4~$ && -- & $~36.6~$ \\
\midrule
DeepCluster\xspace && $\textbf{70.4}$ & $\textbf{73.7}$ && $\textbf{51.4}$ & $\textbf{55.4}$ && $\textbf{43.2}$ & $\textbf{45.1}$ \\
\bottomrule
\end{tabular}
\vspace{\soustable}
\caption{
Comparison of the proposed approach to state-of-the-art unsupervised feature learning on classification, detection and segmentation on \textsc{Pascal} VOC.
$^*$ indicates the use of the data-dependent initialization of Kr\"ahenb\"uhl~\emph{et al}\onedot~\cite{krahenbuhl2015data}.
Numbers for other methods produced by us are marked with a $\dagger$.
}
\label{tab:voc}
\end{table}
Table~\ref{tab:voc} summarized the comparisons of DeepCluster\xspace with other feature-learning approaches on the three tasks.
As for the previous experiments, we outperform previous unsupervised methods on all three tasks, in every setting.
The improvement with fine-tuning over the state of the art is the largest on semantic segmentation ($7.5\%$).
On detection, DeepCluster\xspace performs only slightly better than previously published methods.
Interestingly, a fine-tuned random network performs comparatively to many unsupervised methods, but performs poorly if only \textsc{fc6-8} are learned.
For this reason, we also report detection and segmentation with \textsc{fc6-8} for DeepCluster\xspace and a few baselines.
These tasks are closer to a real application where fine-tuning is not possible.
It is in this setting that the gap between our approach and the state of the art is the greater (up to $9\%$ on classification).
\section{Additional results}
\subsection{Classification on ImageNet}
\label{sec:mlpImageNet}
Noroozi and Favaro~\cite{noroozi2016unsupervised} suggest to evaluate networks trained in an unsupervised way by freezing the convolutional layers and retrain on ImageNet the fully connected layers using labels and reporting accuracy on the validation set.
This experiment follows the observation of Yosinki \emph{et al}\onedot~\cite{yosinski2014transferable} that general-purpose features appear in the convolutional layers of an AlexNet.
We report a comparison of DeepCluster\xspace to other AlexNet networks trained with no supervision, as well as random and supervised baselines in Table~\ref{tab:mlpImageNet}.
\begin{table}[t]
\centering
\begin{tabular}{@{}lcr@{}}
\toprule
Method & Pre-trained dataset & Acc@1 \\
\midrule
Supervised & ImageNet & 59.7 \\
Supervised Sobel & ImageNet & 57.8 \\
Random & - & 12.0 \\
\midrule
Wang \emph{et al}\onedot~\cite{wang2015unsupervised} & YouTube100K~\cite{LiangLWLLY14} & 29.8 \\
\midrule
Doersch \emph{et al}\onedot~\cite{doersch2015unsupervised} & ImageNet & 30.4 \\
Donahue \emph{et al}\onedot~\cite{donahue2016adversarial} & ImageNet & 32.2 \\
Noroozi and Favaro~\cite{noroozi2016unsupervised} & ImageNet & 34.6 \\
Zhang \emph{et al}\onedot~\cite{zhang2016colorful} & ImageNet & 35.2 \\
Bojanowski and Joulin~\cite{bojanowski2017unsupervised} & ImageNet & 36.0 \\
\midrule
DeepCluster\xspace & ImageNet & 44.0 \\
DeepCluster\xspace & YFCC100M & 39.6 \\
\bottomrule
\end{tabular}
\caption{
Comparison of DeepCluster\xspace to AlexNet features pre-trained supervisedly and unsupervisedly on different datasets.
A full multi-layer perceptron is retrained on top of the frozen pre-trained features.
We report classification accuracy (acc@1).
Expect for Noroozi and Favaro~\cite{noroozi2016unsupervised},
all the numbers are taken from Bojanowski and Joulin~\cite{bojanowski2017unsupervised}.
}
\label{tab:mlpImageNet}
\end{table}
DeepCluster\xspace outperforms state-of-the-art unsupervised methods by a significant margin, achieving $8\%$ better accuracy than the previous best performing method.
This means that DeepCluster\xspace \emph{halves the gap} with networks trained in a supervised setting.
\subsection{Stopping criterion}
We monitor how the features learned with DeepCluster\xspace evolve along the training epochs on a down-stream task: object classification on the validation set of \textsc{Pascal} VOC with no fine-tuning.
We use this measure to select the hyperparameters of our model as well as to check when the features stop improving.
In Figure~\ref{convergence}, we show the evolution of both the classification accuracy on this task and a measure of the clustering quality (NMI between the cluster assignments and the true labels) throughout the training.
Unsurprisingly, we notice that the clustering and features qualities follow the same dynamic.
The performance saturates after $400$ epochs of training.
\begin{figure}[!h]
\centering
\includegraphics[scale = 0.5]{figures-doublescale.pdf}
\caption{In red: validation mAP \textsc{Pascal} VOC classification performance.
In blue: evolution of the clustering quality.}
\label{convergence}
\end{figure}
\section{Further discussion}
In this section, we discuss some technical choices and variants of DeepCluster\xspace more specifically.
\subsection{Alternative clustering algorithm}
\noindent\textbf{Graph clustering.}
We consider Power Iteration Clustering (PIC)~\cite{lin2010power} as an alternative clustering method.
It has been shown to yield good performance for large scale collections~\cite{douze2017evaluation}.
Since PIC is a graph clustering approach, we generate a nearest neighbor graph by connecting all images to their 5 neighbors in the Euclidean space of image descriptors.
We denote by $f_\theta(x)$ the output of the network with parameters $\theta$ applied to image $x$.
We use the sparse graph matrix $G=\mathbb{R}^{n \times n}$.
We set the diagonal of $G$ to 0 and non-zero entries are defined as
\[
w_{ij} = e^{-\frac{\|f_\theta(x_i) - f_\theta(x_j)\|^2}{\sigma^2}}
\]
with $\sigma$ a bandwidth parameter.
In this work, we use a variant of PIC~\cite{douze2017evaluation,CL12} that does:
\begin{enumerate}
\item
Initialize $v \leftarrow [1/n,..., 1/n]^\top \in \mathbb{R}^n$;
\item
Iterate \[
v \leftarrow N_1(\alpha (G + G^\top) v + (1 - \alpha) v),
\]
where $\alpha = 10^{-3}$ is a regularization parameter and $N_1:v \mapsto v / \|v\|_1$ the L1-normalization function;
\item
Let $G'$ be the directed unweighted subgraph of $G$ where we keep edge $i \rightarrow j$ of $G$ such that
\[
j = \mathrm{argmax}_j w_{ij}(v_j - v_i).
\]
If $v_i$ is a local maximum (ie. $\forall j\ne i, v_j \le v_i$), then no edge starts from it.
The clusters are given by the connected components of $G'$.
Each cluster has one local maximum.
\end{enumerate}
An advantage of PIC clustering is not to require the setting beforehand of the number of clusters.
However, the parameter $\sigma$ influences the number of clusters: when it is larger, the edges become more uniform and the number of clusters decreases, and the other way round when $\sigma$ increased.
In the following, we set $\sigma=0.2$.
As our PIC implementation relies on a graph of nearest neighbors, we show in Figure~\ref{nn-retrieval} some query images and their $3$ nearest neighbors in the feature space with a network trained with the PIC version of DeepCluster and a random baseline.
A randomly initialized network performs quite well in some cases (sunset query for example), where the image has a simple low-level structure.
The top row in Fig.~\ref{nn-retrieval} seems to represent a query for which the performance of the random network is quite good whereas the bottom query is too complex for the random network to retrieve good matches.
Moreover, we notice that the nearest neighbors matches are largely improved after the network has been trained with DeepCluster\xspace.
\begin{figure}[!h]
\centering
\begin{tabular}{ccccccc}
Query &&& Random &&& DeepCluster\xspace PIC
\\
\includegraphics[scale = 0.2]{images-query_04-003-959.jpg}&&&
\includegraphics[scale = 0.2]{images-random_04-003-959.jpg}&&&
\includegraphics[scale = 0.2]{images-imnetsob_04-003-959.jpg}
\\
\includegraphics[scale = 0.2]{images-query_14-024-196.jpg}&&&
\includegraphics[scale = 0.2]{images-random_14-024-196.jpg}&&&
\includegraphics[scale = 0.2]{images-imnetsob_14-024-196.jpg}
\\
\includegraphics[scale = 0.2]{images-query_21-002-768.jpg}&&&
\includegraphics[scale = 0.2]{images-random_21-002-768.jpg}&&&
\includegraphics[scale = 0.2]{images-imnetsob_21-002-768.jpg}
\\
\includegraphics[scale = 0.2]{images-query_11-001-098.jpg}&&&
\includegraphics[scale = 0.2]{images-random_11-001-098.jpg}&&&
\includegraphics[scale = 0.2]{images-imnetsob_11-001-098.jpg}
\end{tabular}
\caption{Images and their $3$ nearest neighbors in a subset of Flickr in the feature space. The query images are shown on the left column.
The following $3$ columns correspond to a randomly initialized network and the last $3$ to the same network after training with PIC DeepCluster\xspace.
}
\label{nn-retrieval}
\end{figure}
\noindent\textbf{Comparison with $k$-means.}
First, we give an insight about the distribution of the images in the clusters.
We show in Figure~\ref{clustersize} the sizes of the clusters produced by the $k$-means and PIC versions of DeepCluster\xspace at the last epoch of training (this distribution is stable along the epochs).
We observe that $k$-means produces more balanced clusters than PIC.
Indeed, for PIC, almost one third of the clusters are of a size lower than $10$ while the biggest cluster contains roughly $3000$ examples.
In this situation of very unbalanced clusters, it is important in our method to train the convnet by sampling images based on a uniform distribution over the clusters to prevent the biggest cluster from dominating the training.
\begin{figure}[!h]
\centering
\includegraphics[scale = 0.5]{figures-clustersize.pdf}
\caption{Sizes of clusters produced by the $k$-means and PIC versions of DeepCluster\xspace at the last epoch of training.}
\label{clustersize}
\end{figure}
We report in Table~\ref{tab:pic} the results for the different \textsc{Pascal} VOC transfer tasks with a model trained with the PIC version of DeepCluster.
For this set of transfer tasks, the models trained with $k$-means and PIC versions of DeepCluster\xspace perform in comparable ranges.
\begin{table}[h!]
\centering
\begin{tabular}{@{}l c cc c cc c cc@{}}
\toprule
&Clustering algorithm& \multicolumn{2}{c}{Classification} &~~& \multicolumn{2}{c}{Detection} &~~& \multicolumn{2}{c}{Segmentation} \\
\cmidrule{3-4} \cmidrule{6-7} \cmidrule{9-10}
&& \textsc{fc6-8} & \textsc{all} && \textsc{fc6-8} & \textsc{all} && \textsc{fc6-8} & \textsc{all} \\
\midrule
DeepCluster\xspace& $k$-means & 72.0 & 73.7 && 51.4 & 55.4 && 43.2 & 45.1 \\
DeepCluster\xspace& PIC & 71.0 & 73.0 && 53.6 & 54.4 && 42.4 & 43.8 \\
\bottomrule
\end{tabular}
\caption{
Evaluation of PIC versus $k$-means on \textsc{Pascal} VOC transfer tasks.
}
\label{tab:pic}
\end{table}
\subsection{Variants of DeepCluster\xspace}
In Table~\ref{tab:flick}, we report the results of models trained with different variants of DeepCluster\xspace: a different training set, an alternative clustering method or without input preprocessing.
In particular, we notice that the performance of DeepCluster\xspace on raw RGB images degrades significantly.
\begin{table}[t]
\centering
\resizebox{\columnwidth}{!}{%
\begin{tabular}{@{}l c ccccc c ccccc@{}}
\toprule
&~~~& \multicolumn{5}{c}{ImageNet} &~~~& \multicolumn{5}{c}{Places} \\
\cmidrule{3-7} \cmidrule{9-13}
Method && \texttt{conv1} & \texttt{conv2} & \texttt{conv3} & \texttt{conv4} & \texttt{conv5} && \texttt{conv1} & \texttt{conv2} & \texttt{conv3} & \texttt{conv4} & \texttt{conv5} \\
\midrule
Places labels && -- & -- & -- & -- & -- && $22.1$ & $35.1$ & $40.2$ & $43.3$ & $44.6$ \\
ImageNet labels && $19.3$ & $36.3$ & $44.2$ & $48.3$ & $50.5$ && $22.7$ & $34.8$ & $38.4$ & $39.4$ & $38.7$ \\
Random && $11.6$ & $17.1$ & $16.9$ & $16.3$ & $14.1$ && $15.7$ & $20.3$ & $19.8$ & $19.1$ & $17.5$ \\
\midrule
Best competitor ImageNet && $\textbf{18.2}$ & $30.6$ & $35.4$ & $35.2$ & $32.8$ && $23.3$ & $\textbf{33.9}$ & $36.3$ & $34.7$ & $32.5$ \\
\midrule
DeepCluster\xspace && $13.4$ & $32.3$ & $\textbf{41.0}$ & $39.6$ & $\textbf{38.2}$ && $19.6$ & $33.2$ & $\textbf{39.2}$ & $\textbf{39.8}$ & $34.7$ \\
DeepCluster\xspace YFCC100M && $13.5$ & $30.9$ & $38.0$ & $34.5$ & $31.4$ && $19.7$ & $33.0$ & $38.4$ & $39.0$ & $35.2$\\
DeepCluster\xspace PIC && $13.5$ & $32.4$ & $40.8$ & $\textbf{40.5}$ & $37.8$ && $19.5$ & $32.9$ & $39.1$ & $39.5$ & $\textbf{35.9}$ \\
DeepCluster\xspace RGB && $18.0$ & $\textbf{32.5}$ & $39.2$ & $37.2$ & $30.6$ && $\textbf{23.8}$ & $32.8$ & $37.3$ & $36.0$ & $31.0$ \\
\bottomrule
\end{tabular}
}
\caption{
Impact of different variants of our method on the performance of DeepCluster\xspace.
We report classification accuracy of linear classifiers on ImageNet and Places using activations from the convolutional layers of an AlexNet as features.
We compare the standard version of DeepCluster\xspace to different settings: YFCC100M as pre-training set; PIC as clustering method; raw RGB inputs.
}
\label{tab:flick}
\end{table}
In Table~\ref{tab:classif}, we compare the performance of DeepCluster\xspace depending on the clustering method and the pre-training set.
We evaluate this performance on three different classification tasks: with fine-tuning on \textsc{Pascal} VOC, without, and by retraining the full MLP on ImageNet.
We report the classification accuracy on the validation set.
Overall, we notice that the regular version of DeepCluster\xspace (with $k$-means) yields better results than the PIC variant on both ImageNet and the uncured dataset YFCC100M.
\begin{table}[h!]
\centering
\begin{tabular}{@{}lcccccc@{}}
\toprule
Dataset & Clustering&& \multicolumn{2}{c}{\textsc{Pascal} VOC} &~~& ImageNet \\
\cmidrule{4-5} \cmidrule{7-7}
&&& \textsc{fc6-8} & \textsc{all} && \textsc{fc6-8} \\
\midrule
ImageNet & $k$-means && 72.0 & 73.7 && 44.0 \\
ImageNet & PIC && 71.0 & 73.0 && 45.9 \\
YFCC100M & $k$-means && 67.3 & 69.3 && 39.6 \\
YFCC100M & PIC && 66.0 & 69.0 && 38.6 \\
\bottomrule
\end{tabular}
\caption{
Performance of DeepCluster\xspace with different pre-training sets and clustering algorithms measured as classification accuracy on \textsc{Pascal} VOC and ImageNet.
}
\label{tab:classif}
\end{table}
\section{Additional visualisation}
\subsection{Visualise VGG-$16$ features}
We assess the quality of the representations learned by the VGG-$16$ convnet with DeepCluster\xspace.
To do so, we learn an input image that maximises the mean activation of a target filter in the last convolutional layer.
We also display the top $9$ images in a random subset of $1$ million images from Flickr that activate the target filter maximally.
In Figure~\ref{fig:vgg}, we show some filters for which the top $9$ images seem to be semantically or stylistically coherent.
We observe that the filters, learned without any supervision, capture quite complex structures.
In particular, in Figure~\ref{fig:human}, we display synthetic images that correspond to filters that seem to focus on human characteristics.
\begin{figure}[t]
\centering
\begin{tabular}{cccccccc}
\includegraphics[width=0.16\linewidth]{images-vgg-layer13-channel191.jpeg}
\includegraphics[width=0.16\linewidth]{images-vgg-layer12-channel191-small.jpeg}&
\includegraphics[width=0.16\linewidth]{images-vgg-layer13-channel5.jpeg}
\includegraphics[width=0.16\linewidth]{images-vgg-layer12-channel5-small.jpeg}&
\includegraphics[width=0.16\linewidth]{images-vgg-layer13-channel86.jpeg}
\includegraphics[width=0.16\linewidth]{images-vgg-layer12-channel86-small.jpeg}&
\\
\includegraphics[width=0.16\linewidth]{images-vgg-layer13-channel83.jpeg}
\includegraphics[width=0.16\linewidth]{images-vgg-layer12-channel83-small.jpeg}&
\includegraphics[width=0.16\linewidth]{images-vgg-layer13-channel63.jpeg}
\includegraphics[width=0.16\linewidth]{images-vgg-layer12-channel63-small.jpeg}&
\includegraphics[width=0.16\linewidth]{images-vgg-layer13-channel155.jpeg}
\includegraphics[width=0.16\linewidth]{images-vgg-layer12-channel155-small.jpeg}&
\\
\includegraphics[width=0.16\linewidth]{images-vgg-layer13-channel225.jpeg}
\includegraphics[width=0.16\linewidth]{images-vgg-layer12-channel225-small.jpeg}&
\includegraphics[width=0.16\linewidth]{images-vgg-layer13-channel232.jpeg}
\includegraphics[width=0.16\linewidth]{images-vgg-layer12-channel232-small.jpeg}&
\includegraphics[width=0.16\linewidth]{images-vgg-layer13-channel240.jpeg}
\includegraphics[width=0.16\linewidth]{images-vgg-layer12-channel240-small.jpeg}&
\\
\includegraphics[width=0.16\linewidth]{images-vgg-layer13-channel175.jpeg}
\includegraphics[width=0.16\linewidth]{images-vgg-layer12-channel175-small.jpeg}&
\includegraphics[width=0.16\linewidth]{images-vgg-layer13-channel3.jpeg}
\includegraphics[width=0.16\linewidth]{images-vgg-layer12-channel3-small.jpeg}&
\includegraphics[width=0.16\linewidth]{images-vgg-layer13-channel43.jpeg}
\includegraphics[width=0.16\linewidth]{images-vgg-layer12-channel43-small.jpeg}&
\end{tabular}
\caption{
Filter visualization and top 9 activated images (immediate right to the corresponding synthetic image) from a subset of 1 million images from YFCC100M for target filters in the last convolutional layer of a VGG-$16$ trained with DeepCluster\xspace.}
\label{fig:vgg}
\end{figure}
\begin{figure}[t]
\centering
\begin{tabular}{cccccccc}
\includegraphics[width=0.24\linewidth]{images-vgg-layer13-channel110.jpeg}
\includegraphics[width=0.24\linewidth]{images-vgg-layer13-channel17.jpeg}
\includegraphics[width=0.24\linewidth]{images-vgg-layer13-channel45.jpeg}
\includegraphics[width=0.24\linewidth]{images-vgg-layer13-channel143.jpeg}
\\
\includegraphics[width=0.24\linewidth]{images-vgg-layer13-channel176.jpeg}
\includegraphics[width=0.24\linewidth]{images-vgg-layer13-channel126.jpeg}
\includegraphics[width=0.24\linewidth]{images-vgg-layer13-channel125.jpeg}
\includegraphics[width=0.24\linewidth]{images-vgg-layer13-channel7.jpeg}
\end{tabular}
\caption{
Filter visualization by learning an input image that maximizes the response to a target filter~\cite{yosinski2015understanding} in the last convolutional layer of a VGG-$16$ convnet trained with DeepCluster\xspace.
Here, we manually select filters that seem to trigger on human characteristics (eyes, noses, faces, fingers, fringes, groups of people or arms).
}
\label{fig:human}
\end{figure}
\subsection{AlexNet}
It is interesting to investigate what clusters the unsupervised learning approach actually learns.
Fig.~2(a) in the paper suggests that our clusters are correlated with ImageNet classes.
In Figure~\ref{fig:clusters}, we look at the purity of the clusters with the ImageNet ontology to see which concepts are learned.
More precisely, we show the proportion of images belonging to a \emph{pure cluster} for different synsets at different depths in the ImageNet ontology (pure cluster = more than 70\% of images are from that synset).
The correlation varies significantly between categories, with the highest correlations for animals and plants.
\begin{figure}[t]
\centering
\includegraphics[width=1\linewidth]{images-treeim.pdf} \\
\caption{
{
\footnotesize
Proportion of images belonging to a \emph{pure cluster} for different synsets in the ImageNet ontology (pure cluster = more than 70\% of images are from that synset).
We show synsets at depth 6, 9 and 13 in the ontology, with ``zooms'' on the animal and mammal synsets.
}
}
\vspace{-1em}
\label{fig:clusters}
\end{figure}
In Figure~\ref{fig:waouh2}, we display the top 9 activated images from a random subset of 10 millions images from YFCC100M for the first $100$ target filters in the last convolutional layer
(we selected filters whose top activated images do not depict humans).
We observe that, even though our method is purely unsupervised, some filters trigger on images containing particular classes of objects.
Some other filters respond strongly on particular stylistic effects or textures.
\begin{figure}[t]
\centering
\begin{tabular}{ccccccc}
Filter $0$ && Filter $1$ && Filter $2$ && Filter $3$
\\
\includegraphics[width=0.23\linewidth]{images-filter-layer4-channel0-small.jpeg}&&
\includegraphics[width=0.23\linewidth]{images-filter-layer4-channel1-small.jpeg}&&
\includegraphics[width=0.23\linewidth]{images-filter-layer4-channel2-small.jpeg}&&
\includegraphics[width=0.23\linewidth]{images-filter-layer4-channel3-small.jpeg}
\\
Filter $5$ && Filter $6$ && Filter $7$ && Filter $8$
\\
\includegraphics[width=0.23\linewidth]{images-filter-layer4-channel5-small.jpeg}&&
\includegraphics[width=0.23\linewidth]{images-filter-layer4-channel6-small.jpeg}&&
\includegraphics[width=0.23\linewidth]{images-filter-layer4-channel7-small.jpeg}&&
\includegraphics[width=0.23\linewidth]{images-filter-layer4-channel8-small.jpeg}
\\
Filter $9$ && Filter $10$ && Filter $11$ && Filter $13$
\\
\includegraphics[width=0.23\linewidth]{images-filter-layer4-channel9-small.jpeg}&&
\includegraphics[width=0.23\linewidth]{images-filter-layer4-channel10-small.jpeg}&&
\includegraphics[width=0.23\linewidth]{images-filter-layer4-channel11-small.jpeg}&&
\includegraphics[width=0.23\linewidth]{images-filter-layer4-channel13-small.jpeg}
\\
Filter $14$ && Filter $16$ && Filter $18$ && Filter $24$
\\
\includegraphics[width=0.23\linewidth]{images-filter-layer4-channel14-small.jpeg}&&
\includegraphics[width=0.23\linewidth]{images-filter-layer4-channel16-small.jpeg}&&
\includegraphics[width=0.23\linewidth]{images-filter-layer4-channel18-small.jpeg}&&
\includegraphics[width=0.23\linewidth]{images-filter-layer4-channel24-small.jpeg}
\\
Filter $24$ && Filter $25$ && Filter $26$ && Filter $27$
\\
\includegraphics[width=0.23\linewidth]{images-filter-layer4-channel24-small.jpeg}&&
\includegraphics[width=0.23\linewidth]{images-filter-layer4-channel25-small.jpeg}&&
\includegraphics[width=0.23\linewidth]{images-filter-layer4-channel26-small.jpeg}&&
\includegraphics[width=0.23\linewidth]{images-filter-layer4-channel27-small.jpeg}
\\
Filter $28$ && Filter $30$ && Filter $33$ && Filter $34$
\\
\includegraphics[width=0.23\linewidth]{images-filter-layer4-channel28-small.jpeg}&&
\includegraphics[width=0.23\linewidth]{images-filter-layer4-channel30-small.jpeg}&&
\includegraphics[width=0.23\linewidth]{images-filter-layer4-channel33-small.jpeg}&&
\includegraphics[width=0.23\linewidth]{images-filter-layer4-channel34-small.jpeg}
\end{tabular}
\end{figure}
\begin{figure}[t]
\centering
\begin{tabular}{ccccccc}
Filter $36$ && Filter $41$ && Filter $42$ && Filter $43$
\\
\includegraphics[width=0.23\linewidth]{images-filter-layer4-channel36-small.jpeg}&&
\includegraphics[width=0.23\linewidth]{images-filter-layer4-channel41-small.jpeg}&&
\includegraphics[width=0.23\linewidth]{images-filter-layer4-channel42-small.jpeg}&&
\includegraphics[width=0.23\linewidth]{images-filter-layer4-channel43-small.jpeg}
\\
Filter $45$ && Filter $52$ && Filter $55$ && Filter $56$
\\
\includegraphics[width=0.23\linewidth]{images-filter-layer4-channel45-small.jpeg}&&
\includegraphics[width=0.23\linewidth]{images-filter-layer4-channel52-small.jpeg}&&
\includegraphics[width=0.23\linewidth]{images-filter-layer4-channel55-small.jpeg}&&
\includegraphics[width=0.23\linewidth]{images-filter-layer4-channel56-small.jpeg}
\\
Filter $57$ && Filter $58$ && Filter $59$ && Filter $60$
\\
\includegraphics[width=0.23\linewidth]{images-filter-layer4-channel57-small.jpeg}&&
\includegraphics[width=0.23\linewidth]{images-filter-layer4-channel58-small.jpeg}&&
\includegraphics[width=0.23\linewidth]{images-filter-layer4-channel59-small.jpeg}&&
\includegraphics[width=0.23\linewidth]{images-filter-layer4-channel60-small.jpeg}
\\
Filter $62$ && Filter $63$ && Filter $65$ && Filter $66$
\\
\includegraphics[width=0.23\linewidth]{images-filter-layer4-channel62-small.jpeg}&&
\includegraphics[width=0.23\linewidth]{images-filter-layer4-channel63-small.jpeg}&&
\includegraphics[width=0.23\linewidth]{images-filter-layer4-channel65-small.jpeg}&&
\includegraphics[width=0.23\linewidth]{images-filter-layer4-channel66-small.jpeg}
\\
Filter $67$ && Filter $68$ && Filter $69$ && Filter $70$
\\
\includegraphics[width=0.23\linewidth]{images-filter-layer4-channel67-small.jpeg}&&
\includegraphics[width=0.23\linewidth]{images-filter-layer4-channel68-small.jpeg}&&
\includegraphics[width=0.23\linewidth]{images-filter-layer4-channel69-small.jpeg}&&
\includegraphics[width=0.23\linewidth]{images-filter-layer4-channel70-small.jpeg}
\\
Filter $74$ && Filter $75$ && Filter $76$ && Filter $78$
\\
\includegraphics[width=0.23\linewidth]{images-filter-layer4-channel74-small.jpeg}&&
\includegraphics[width=0.23\linewidth]{images-filter-layer4-channel75-small.jpeg}&&
\includegraphics[width=0.23\linewidth]{images-filter-layer4-channel76-small.jpeg}&&
\includegraphics[width=0.23\linewidth]{images-filter-layer4-channel78-small.jpeg}
\end{tabular}
\end{figure}
\begin{figure}[t]
\centering
\begin{tabular}{ccccccc}
Filter $79$ && Filter $81$ && Filter $83$ && Filter $84$
\\
\includegraphics[width=0.23\linewidth]{images-filter-layer4-channel79-small.jpeg}&&
\includegraphics[width=0.23\linewidth]{images-filter-layer4-channel81-small.jpeg}&&
\includegraphics[width=0.23\linewidth]{images-filter-layer4-channel83-small.jpeg}&&
\includegraphics[width=0.23\linewidth]{images-filter-layer4-channel84-small.jpeg}
\\
Filter $85$ && Filter $88$ && Filter $90$ && Filter $91$
\\
\includegraphics[width=0.23\linewidth]{images-filter-layer4-channel85-small.jpeg}&&
\includegraphics[width=0.23\linewidth]{images-filter-layer4-channel88-small.jpeg}&&
\includegraphics[width=0.23\linewidth]{images-filter-layer4-channel90-small.jpeg}&&
\includegraphics[width=0.23\linewidth]{images-filter-layer4-channel91-small.jpeg}
\\
Filter $92$ && Filter $93$ && Filter $96$ && Filter $97$
\\
\includegraphics[width=0.23\linewidth]{images-filter-layer4-channel92-small.jpeg}&&
\includegraphics[width=0.23\linewidth]{images-filter-layer4-channel93-small.jpeg}&&
\includegraphics[width=0.23\linewidth]{images-filter-layer4-channel96-small.jpeg}&&
\includegraphics[width=0.23\linewidth]{images-filter-layer4-channel97-small.jpeg}
\end{tabular}
\caption{
Top $9$ activated images from a random subset of $10$ millions images from YFCC100M for the $100$ first filters in the last convolutional layer of an AlexNet trained with DeepCluster\xspace (we do not display humans).
}
\label{fig:waouh2}
\end{figure}
\section{Erratum [18/03/2019]}
In the original version of the paper, we report classification accuracy averaged over $10$ crops for the linear classifier experiments on Places and ImageNet datasets.
However, other methods report accuracy of the central crop so our comparison wasn't fair.
Nevertheless, it does not change the conclusion of these experiments, our approach still outperforms the state of the art from \texttt{conv3} to \texttt{conv5} layers.
In Table~\ref{tab:erratum}, we show our results both for single and $10$ crops.
\begin{table}[t]
\centering
\resizebox{\columnwidth}{!}{%
\begin{tabular}{@{}l c ccccc c ccccc@{}}
\toprule
&~~~& \multicolumn{5}{c}{ImageNet} &~~~& \multicolumn{5}{c}{Places} \\
\cmidrule{3-7} \cmidrule{9-13}
Method && \texttt{conv1} & \texttt{conv2} & \texttt{conv3} & \texttt{conv4} & \texttt{conv5} && \texttt{conv1} & \texttt{conv2} & \texttt{conv3} & \texttt{conv4} & \texttt{conv5} \\
\midrule
DeepCluster\xspace single crop && $12.9$ & $29.2$ & $38.2$ & $39.8$ & $36.1$ && $18.6$ & $30.8$ & $37.0$ & $37.5$ & $33.1$ \\
DeepCluster\xspace $10$ crops && $13.4$ & $32.3$ & $41.0$ & $39.6$ & $38.2$ && $19.6$ & $33.2$ & $39.2$ & $39.8$ & $34.7$ \\
\bottomrule
\end{tabular}
}
\vspace{\soustable}
\caption{
Linear classification on ImageNet and Places using activations from the convolutional layers of an AlexNet as features.
}
\label{tab:erratum}
\end{table}
|
1,116,691,497,580 | arxiv | \section{Introduction}
The Standard ($\Lambda$-CDM) Model in cosmology gives a phenomenological description of
the observed {\it Dark Energy} (DE) and {\it Dark Matter} (DM). It is based on the
use of a small positive cosmological constant $\Lambda$ and a {\it Cold Dark Matter} (CDM),
and is consistent with all observations coming from the existing cosmological, Solar
system and ground-based laboratory data. However, the $\Lambda$-CDM Model cannot be the
ultimate answer to DE, since it implies its time-independence. For example, the
`primordial' DE responsible for inflation in the early universe was different from $\Lambda$
and unstable. The {\it dynamical} (ie. time-dependent) models of DE can be easily
constructed by using the $f(R)$ gravity theories, defined via replacing the scalar
curvature $R$ by a function $f(R)$ in the gravitational action. The $f(R)$ gravity
provides the self-consistent non-trivial alternative to the $\Lambda$-CDM Model --- see
eg., refs.~\cite{stu,fr,odin} for a review. The use of $f(R)$ gravity in the
inflationary cosmology was pioneered by Starobinsky \cite{star}. Viable
$f(R)$-gravity-based models of the current DE are also known \cite{cde1,cde2,cde3}, and
the combined inflationary-DE models are possible too \cite{stu}.
Despite of the apparent presence of the higher derivatives, an $f(R)$ gravity theory
can be free of ghosts and tachyons. The corresponding stability conditions are well
known -- see Sec.~2 below. Under those conditions, it is always possible to prove the
classical equivalence of an $f(R)$ gravity theory to the certain scalar-tensor
theory of gravity \cite{eq,bac,ma}. Dynamics of the spin-2 part of metric in $f(R)$
gravity (compared to Einstein gravity) is not modified, but there is the extra
propagating scalar field (called scalaron) given by the spin-0 part of metric. By the
classical equivalence above we mean that both theories lead to the same inflaton scalar
potential and, therefore, the same inflationary dynamics. However, the physical nature
of inflaton in each theory is different. In the $f(R)$ gravity and $F({\cal R})$ supergravity
inflaton field is part of metric, whereas in the scalar-tensor gravity and supergravity
inflaton is a matter particle. Therefore, the inflaton interactions with other matter fields
are different in both theories. It gives rise to different inflaton decay rates and different
reheating in the post-inflationary universe.
In our recent papers \cite{our1,our2,our3,our4,our5,our6,our7,our8} we proposed the new
supergravity theory (we call it $F({\cal R})$-supergravity), and studied some of its
physical applications (see also refs.~\cite{our9,our10} for our earlier related work).
The $F({\cal R})$-supergravity can be considered as the $N=1$ locally supersymmetric
extension of $f(R)$ gravity in four space-time dimensions.~\footnote{Another
(unimodular) $F(R)$ supergravity theory was proposed in ref.~\cite{nish}.} Supergravity
is well-motivated in High-Energy Physics theory beyond the Standard Model of
elementary particles. Supergravity is also the low-energy effective action of
Superstrings. As was demonstrated in ref.~\cite{our1}, an $F({\cal R})$ supergravity is
classically equivalent to the $N=1$ Poincar\'e supergravity coupled to a dynamical
(quintessence) chiral superfield, whose (non-trivial) K\"ahler potential and superpotential
are dictated by the chiral (holomorphic) function $F$. The classical equivalence is achieved
via a non-trivial field redefinition \cite{our1} that gives rise to a non-trivial Jacobian in
the path integral formulation of those quantum field theories (below their unitarity bounds).
Hence, their classical equivalence is expected to be broken in quantum theory.~\footnote{See
ref.~\cite{kmy} for the first steps in quantizing $f(R)$ gravity theories.}
The natural embedding of the Starobinsky $(R+R^2)$-inflationary model into $F({\cal R})$
supergravity was found in ref.~\cite{our7}. It provides the very economical realization
of chaotic inflation (at early times) in supergravity, which is consistent with observations
\cite{wmap7} and gives a simple solution to the $\eta$-problem in supergravity \cite{eta}.
The natural question arises, whether $F({\cal R})$ supergravity is also capable to describe
the present DE or have a positive cosmological constant. It is non-trivial because the
standard supergravity with usual matter can only have a negative or vanishing cosmological
constant \cite{gibb}. It takes place since the usual (known) matter does not violate the
{\it Strong Energy Condition} (SEC) \cite{hawe}. A violation of SEC is required for an
accelerating universe, and it is easily achieved in $f(R)$ gravity due to the fact that the
quintessence field in $f(R)$ gravity is part of metric (ie. the unusual matter). Similarly,
the quintessence scalar superfield in $F({\cal R})$ supergravity is part of super-vielbein, and
it also gives rise to a violation of SEC. In this Letter we further extend the Ansatz used
in ref.~\cite{our7} for $F$-function, and apply it to get a positive cosmological constant
in the regime of low spacetime curvature (at late times).
Our paper is organized as follows. In sec.~2 we briefly recall the superspace
construction of $F({\cal R})$ supergravity, its relation to $f(R)$ gravity and the stability
conditions. In sec.~3 we define our model of $F({\cal R})$ supergravity, and compute its
cosmological constant. Sec.~4 is our conclusion.
Throughout the paper we use the units $c=\hbar=M_{\rm Pl}=1$ in terms of the (reduced)
Planck mass $M_{\rm Pl}$, with the spacetime signature $(+,-,-,-)$. Our basic notation
of General Relativity coincides with that of ref.~\cite{landau}. An AdS-spacetime has a
positive scalar curvature, and a dS-spacetime has a negative scalar curvature in our
notation.
\section{$F({\cal R})$ supergravity and $f(R)$ gravity}
A concise and manifestly supersymmetric description of supergravity is given
by superspace. We refer the reader to the textbooks \cite{ss1,ss2,ss3} for details of
the superspace formulation of supergravity. A construction of the $F({\cal R})$
supergravity action goes beyond the supergravity textbooks.
The most succinct formulation of $F({\cal R})$ supergravity exist in a chiral
4D, $N=1$ superspace where it is defined by the action \cite{our1}
\begin{equation} \label{act}
S_F = \int d^4x d^2\theta\, {\cal E} F({\cal R}) + {\rm H.c.}
\end{equation}
in terms of a holomorphic function $F({\cal R})$ of the covariantly-chiral scalar
curvature superfield ${\cal R}$, and the chiral superspace density ${\cal E}$. The chiral
$N=1$ superfield ${\cal R}$ has the scalar curvature $R$ as the field coefficient
at its $\theta^2$-term. The chiral superspace density ${\cal E}$ (in a WZ gauge) reads
\begin{equation} \label{cde}
{\cal E} = e \left( 1- 2i\theta\sigma_a\bar{\psi}^a +3\theta^2 X\right)
\end{equation}
where $e=\sqrt{-g}$, $\psi^a$ is gravitino, and $X=S-iP$ is the complex scalar
auxiliary field (it does not propagate in the theory (\ref{act}) despite of the
apparent presence of the higher derivatives \cite{our1}).
A bosonic $f(R)$ gravity action is given by \cite{stu,fr,odin}
\begin{equation} \label{mgrav}
S_f = \int d^4x \,\sqrt{-g}\, f(R) \end{equation}
in terms of the real function $f(R)$ of the scalar curvature $R$. The relation between
the master chiral superfield function $F({\cal R})$ in eq.~(\ref{act}) and the
corresponding bosonic function $f(R)$ in eq.~(\ref{mgrav}) can be established by
applying the standard formulae of superspace \cite{ss1,ss2,ss3} and ignoring the
fermionic contributions. For simplicity, we also ignore the complex nature of $F$ and
$X$ in what follows.
The embedding of $f(R)$ gravity into $F({\cal R})$ supergravity is given by
\cite{our1,our2,our3}
\begin{equation} \label{emb1}
f(R) = f(R,X(R))
\end{equation}
where the function $f(R,X)$ (or the gravity Lagrangian ${\cal L}$) is defined by
\begin{equation} \label{emb2}
{\cal L}= f(R,X) = 2F'(X) \left[ \fracmm{1}{3} R +4 X^2 \right] + 6XF(X)
\end{equation}
and the function $X=X(R)$ is determined by solving an algebraic equation,
\begin{equation} \label{emb3}
\fracmm{\pa f(R,X)}{\pa X} =0
\end{equation}
The primes denote the derivatives with respect to the given argument. Equation
(\ref{emb3}) arises by varying the action (\ref{act}) with respect to the
auxiliary field $X$. It cannot be explicitly solved for $X$ in a generic $F({\cal R})$
supergravity theory.
The cosmological constant in $F({\cal R})$ supergravity, in the regime of low space-time
curvature, is thus given by
\begin{equation} \label{cc1}
\Lambda = - f(0,X_0)
\end{equation}
where $X_0=X(0)$. It should be mentioned that $X_0$ represents the vacuum expectation
value of the auxiliary field $X$ that determines the scale of the supersymmetry
breaking. Both inflation and DE imply $X_0\neq 0$.
The $f(R)$-gravity stability conditions in our notation are given by \cite{stu,our6}
\begin{equation} \label{csta}
f'(R) < 0
\end{equation}
and
\begin{equation} \label{qsta}
f''(R) > 0
\end{equation}
The first (classical stability) condition (\ref{csta}) is related to the sign factor
in front of the Einstein-Hilbert term (linear in $R$) in the $f(R)$-gravity action,
and it ensures that graviton is not a ghost. The second (quantum stability) condition
(\ref{qsta}) ensures that scalaron is not a tachyon. In $F(R)$ supergravity
eq.~(\ref{csta}) is replaced by a stronger condition \cite{our6},
\begin{equation} \label{cs}
F'(X) < 0
\end{equation}
Equation (\ref{cs}) guarantees the classical stability of the $f(R)$-gravity embedding
into the full $F({\cal R})$ supergravity against small fluctuations of the axion field $P$
\cite{our6}.
To describe the early universe inflation (ie. in the regime of {\it high} spacetime
curvature $R\to -\infty$), the function $f(R)$ should have the profile
\begin{equation} \label{infp}
f(R) = - \fracmm {1}{2}R + R^2 A(R) \equiv f_{\rm EH}(R) + R^2 A(R)
\end{equation}
with the slowly varying function $A(R)$ in the sense
\begin{equation} \label{svar}
\abs{A'(R)} \ll \abs{ \fracmm{A(R)}{R} } \quad {\rm and}
\abs{A''(R)} \ll \abs{ \fracmm{A(R)}{R^2} }
\end{equation}
The simplest choice $A=const. >0$ gives rise to the Starobinsky model \cite{star}
with
\begin{equation} \label{star1}
f_S(R) = -\fracmm{1}{2}R +\fracmm{R^2}{12M^2_{\rm inf}}
\end{equation}
where the inflaton (scalaron) mass $M_{\rm inf}$ has been introduced.
To describe DE in the present universe, ie. in the regime with {\it low} spacetime
curvature $R$, the function $f(R)$ should be close to the Einstein-Hilbert (linear)
function $f_{\rm EH}(R)$ with a small positive $\Lambda$,
\begin{equation} \label{dep}
\abs{f(R)-f_{\rm EH}(R) } \ll \abs{f_{\rm EH}(R)},\quad
\abs{f'(R)-f'_{\rm EH} } \ll 1,\quad
\abs{Rf''(R)} \ll 1
\end{equation}
ie. $f(R)\approx -\fracmm{1}{2}R -\Lambda$ for small $R$ with the very small and positive
$\Lambda\approx 10^{-118}(M^4_{\rm Pl})$.
\section{Cosmological constant}
Equations (\ref{emb2}) and (\ref{cc1}) imply
\begin{equation} \label{cc2}
\Lambda =-8F'(X_0)X_0^2 -6X_0F(X_0)
\end{equation}
where $X_0$ is a solution to the algebraic equation
\begin{equation} \label{cc3}
4X_0^2F''(X_0) + 11X_0F'(X_0) + 3F(X_0) =0
\end{equation}
As is clear from eq.~(\ref{cc2}), to have $\Lambda\neq 0$, one must have $X_0\neq 0$, ie.
a (spontaneous) supersymmetry breaking. However, in order to proceed further, one
needs a reasonable Ansatz for the $F$-function in eq.~(\ref{act}).
The simplest opportunity is given by expanding the function $F({\cal R})$ in Taylor
series with respect to ${\cal R}$. Since the $N=1$ chiral superfield ${\cal R}$ has
$X$ as its leading field component (in $\theta$-expansion), one may expect that the Taylor
expansion is a good approximation as long as $\abs{X_0}\ll 1(M_{\rm Pl})$. As was
demonstrated in ref.~\cite{our7}, a viable (successful) description of inflation is
possible in $F({\cal R})$ supergravity, when keeping the {\it cubic} term ${\cal R}^3$ in the
Taylor expansion of the $F({\cal R})$ function. It is, therefore, natural to expand the
function $F$ up to the cubic term with respect to ${\cal R}$, and use it as our Ansatz
here,
\begin{equation} \label{an}
F({\cal R}) = f_0 - \fracmm{1}{2}f_1{\cal R} + \fracmm{1}{2}f_2{\cal R}^2
-\fracmm{1}{6}f_3{\cal R}^3
\end{equation}
with some real coeffieints $f_0,f_1,f_2,f_3$. The Ansatz (\ref{an}) differs from the
one used in ref.~\cite{our7} by the presence of the new parameter $f_0$ only. It is
worth emphasizing here that $f_0$ is {\it not} a cosmological constant because one
still has to eliminate the auxiliary field $X$. The stability conditions in the case
(\ref{an}) require
\begin{equation} \label{sa1}
f_1>0~~, \qquad f_2>0~~,\qquad f_3 > 0
\end{equation}
and
\begin{equation} \label{sa2}
f_2^2 < f_1 f_3
\end{equation}
Inflation requires $f_3\gg 1$ and $f_2^2\gg f_1$ \cite{our7}.~\footnote{The stronger condition
$f^2_2\ll f_1f_3$ was used in ref.~\cite{our7} for simplicity.}
As was shown in ref.~\cite{our7}, in the high-curvature regime the
effective $f(R)$-gravity action (originating from the $F({\cal R})$ supergravity defined
by eqs.~(\ref{act}) and (\ref{an}) with $f_0=0$) takes the form of eq.~(\ref{star1})
with $f_3=15M^2_{\rm inf}$. To meet the WMAP observations \cite{wmap7}, the parameter
$f_3$ should be approximately $6.5\cdot 10^{10}(N_e/50)^2$, where $N_e$ is the number
of e-foldings \cite{our7}. The cosmological constant in the high-curvature regime
does not play a significant role and may be ignored there.
In the low curvature regime, in order to recover the Einstein-Hilbert term, one has
to fix $f_1=3/2$ \cite{our7}. Then the Ansatz (\ref{an}) leads to the gravitational Lagrangian
\begin{equation}
\label{frx}
f(R,X) = -5f_3X^4 +11f_2X^3 - \fracmm{1}{3}f_3\left(R+\fracmm{63}{2f_3}\right)
X^2 + \left(6f_0 + \fracmm{2}{3}f_2R\right)X - \fracmm{1}{2}R
\end{equation}
and the auxiliary field equation
\begin{equation} \label{afe}
X^3 - \fracmm{33f_2}{20f_3}X^2 +\fracmm{1}{30}\left( R + \fracmm{63}{2f_3}\right)
X - \fracmm{1}{30f_3}\left( f_2R +9f_0\right)=0
\end{equation}
whose formal solution is available via the standard Cardano (Vi\`ete)
formulae \cite{cv}.
In the low-curvature regime we find a cubic equation for $X_0$ in the form
\begin{equation} \label{cub}
X_0^3 - \left(\fracmm{33f_2}{20f_3}\right)X_0^2
+\left(\fracmm{21}{20f_3}\right)X_0 -\left(\fracmm{3f_0}{10f_3}\right)=0
\end{equation}
`Linearizing' eq.~(\ref{cub}) with respect to $X_0$ brings the solution $X_0=2f_0/7$
whose substitution into the action (\ref{frx}) gives rise to a {\it negative} cosmological
constant, $\Lambda_0=-6f_0^2/7$. This way we recover the standard supergravity case.
Equations (\ref{frx}) and (\ref{cub}) allow us to write down the exact eq.~(\ref{cc1})
for the cosmological constant in the factorized form
\begin{equation} \label{cce}
\Lambda(X_0) = - \fracmm{11f_2}{4}X_0 ( X_0-X_-)(X_0-X_+)
\end{equation}
where $X_{\pm}$ are the roots of the quadratic equation $x^2-\fracmm{21}{11f_2}x
+\fracmm{18f_0}{11f_2}=0$, ie.
\begin{equation} \label{roots}
X_{\pm} = \fracmm{21}{22f_2}\left[ 1\pm \sqrt{ 1- \fracmm{2^3\cdot 11}{7^2}f_0f_2}\right]
\end{equation}
Since $f_0f_2$ is supposed to be very small, both roots $X_{\pm}$ are real and positive.
Equation (\ref{cce}) implies that $\Lambda>0$ when either (I) $X_0<0$, or (II) $X_0$ is inside
the interval $(X_-,X_+)$.
By using {\it Matematica} we were able to numerically confirm the existence of solutions to
eq.~(\ref{cub}) in the region (I) when $f_0<0$, but not in the region (II). So, to this end,
we continue with the region (I) only. All real roots of eq.~(\ref{cub}) are given by
\begin{equation} \label{3roots}
\eqalign{
(X_0)_1 = ~&~ 2\sqrt{-Q} \cos\left(\fracmm{\vartheta}{3}\right) +\fracmm{11f_2}{20f_3}~~,\cr
(X_0)_2 = ~&~ 2\sqrt{-Q} \cos\left(\fracmm{\vartheta+2\pi}{3}\right) +\fracmm{11f_2}{20f_3}~~,\cr
(X_0)_3 = ~&~ 2\sqrt{-Q} \cos\left(\fracmm{\vartheta+4\pi}{3}\right) +\fracmm{11f_2}{20f_3}~~,\cr }
\end{equation}
in terms of the Cardano-Vi\`ete parameters
\begin{equation} \label{cvp}
\eqalign{
Q= ~&~ -\fracmm{11f_2}{2^2\cdot 5f_3} -\fracmm{7^2}{2^4\cdot5^2 f_3^2}\approx -\fracmm{11f_2}{20f_3}~~,\cr
R= ~&~ -\fracmm{3\cdot 7\cdot 11f_2}{2^5\cdot 5^2 f^2_3} + \fracmm{3f_0}{2^2\cdot 5f_3}
+\fracmm{11^3f_2^3}{2^6\cdot 5^3f_3^3} \approx -\fracmm{1}{20f_3}\left(-\fracmm{21}{2}Q+3f_0\right)~~\cr}
\end{equation}
and the angle $\vartheta$ defined by
\begin{equation} \label{vq}
\cos\vartheta =\fracmm {R}{\sqrt{-Q^3}}
\end{equation}
The Cardano discriminant reads $D=R^2+Q^3$. All three roots are real provided that $D<0$. It is
known to be the case in the high-curvature regime \cite{our7}, and it is also the case when $f_0$
is extremely small. Under our requirements on the parameters the angle $\vartheta$ is very close to zero,
so the relevant solutions $X_0<0$ are given by the 2nd and 3rd lines of eq.~(\ref{3roots}), with
$X_0\approx f_0/10$.
\section{Conclusion}
We demonstrated that it is possible to have a {\it positive} cosmological constant (at low spacetime
curvature or late times) in the particular $F({\cal R})$ supergravity (without its coupling to
super-matter) described by the Ansatz (\ref{an}). The same Ansatz is applicable for describing
a viable chaotic inflation in supergravity (at high spacetime curvature or early times). The positive
cosmological constant was technically achieved as the {\it non-linear} effect with respect to the
superspace curvature and spacetime curvature in the relatively narrow part of the parameter space
(it is, therefore, highly constrained).
In the particular $F({\cal R})$ supergravity model we considered, the effective $f(R)$ gravity function
is essentially given by the Starobinsky function $(-{\fracmm12} R+\fracmm{1}{12M^2_{\rm inf}} R^2)$ in the
high curvature regime, and by the DE-like function $(-{\fracmm12} R-\Lambda)$ in the low curvature
regime. Therefore, our model has a cosmological solution which describes an inflationary universe
of the quasi-dS type with the Hubble function $H(t)\approx \fracmm{M^2_{inf}}{6}(t_{end}-t)$ at early
times $t< t_{end}$ and an accelerating universe of the dS-type with $H=\Lambda$ at late times.
It is similar to the known cosmological solutions unifying inflation and DE in $f(R)$ gravity
\cite{stu}.
Of course, describing the DE in the present universe requires an enormous fine-tuning
of our parameters in the $F$-function. However, it is the common feature of all known
approaches to the DE. This paper does not contribute to `explaining' the smallness of
the cosmological constant.
\newpage
|
1,116,691,497,581 | arxiv | \section{Introduction}\label{sec:intro}
A surface is one of the central subjects in condensed matter physics.
An open surface changes the spatial symmetry of the system and perturbs the bulk electronic states, which sometimes brings about novel surface states qualitatively different from the bulk one~\cite{PhysRev.115.869,PhysRevLett.43.43,PhysRevLett.81.1953}.
In particular, in topological insulators, the surface provides a stage where the nontrivial topological
nature manifests itself, in the form of topologically protected surface states~\cite{PhysRevB.25.2185,PhysRevB.48.11851,RevModPhys.82.3045}.
Such topological surface states exhibit a number of interesting properties, such as a dissipationless chiral edge current, which are of special interest for device applications as well as the fundamental science.
Among many systems with a topologically nontrivial character, magnetic Chern insulators (MCIs) have recently attracted special interest.
The MCIs are magnetically ordered insulators that possess the electronic structure characterized
by a nonzero Chern number~\cite{PhysRevB.62.R6065,PhysRevLett.87.116801}.
The topologically nontrivial nature usually originates in the so-called spin Berry phase carried by a noncoplanar magnetic long-range order.
Such noncoplanar magnetic structures are often characterized by the spin scalar chirality: $\chi_{lmn} \propto (\bm{S}_l\times\bm{S}_m)\cdot\bm{S}_n$ defined for three spins $\bm{S}_l$, $\bm{S}_m$, and $\bm{S}_n$.
A typical example of MCIs was recently reported in the Kondo lattice model on a triangular lattice near 1/4 and 3/4 fillings~\cite{PhysRevLett.101.156402, JPSJ.79.083711,PhysRevLett.105.266405}.
This MCI is accompanied by a four-sublattice noncoplanar magnetic order with a nonzero scalar chirality, as shown in Figs.~\ref{fig:schematics}(a) and \ref{fig:schematics}(b).
The instability toward such peculiar magnetic ordering was discussed from the viewpoint of the Fermi surface properties~\cite{PhysRevLett.101.156402, PhysRevLett.108.096401, PhysRevB.90.060402}.
Different types of MCIs were also reported in other lattice systems~\cite{PhysRevLett.109.166405,barros2014novel}.
MCIs exhibit the topological quantum Hall effect, associated with the nonzero Chern numbers.
In the quantum Hall insulating state, the surfaces become metallic and retain the chiral edge currents.
However, in contrast to ordinary (nonmagnetic) Chern insulators, the emergence of such topologically protected edge states is not trivial.
This is because, in general, the magnetic structure is perturbed by the spatial symmetry breaking by the surface.
Once this occurs in MCIs, the topological nature is also perturbed by the surface, which might significantly affect the edge states.
In fact, in the previous study~\cite{JPSJ.83.073706}, the authors found that, for the chiral edge states in the MCI on the triangular lattice, the four-sublattice noncoplanar magnetic order is substantially reconstructed and ferromagnetic (FM) correlations develop near the edges.
Surprisingly, the FM correlations enhance the total amount of chiral edge current up to almost twice, rather than suppress it.
It is interesting how the edge magnetic reconstruction leads to the enhancement of the chiral edge current.
However, the precise description of the reconstructed edge state is complicated: it is obtained only after the minimization of many-body free energy by optimizing spin configurations near the edges.
It is desirable to establish an effective model, which captures the essential physics of the edge reconstruction in the MCI.
In this paper, we show that the essential character of reconstructed edge states can be described by a simple junction model between the bulk four-sublattice ordered state and a FM layer.
The electronic state near the interfaces in the junction system gives a good approximation to
the edge states in the optimized spin configuration:
the total chiral current is largely enhanced by the FM junction.
Furthermore, we find that the total chiral current is strongly affected by the magnetic structure of the capping layers; for instance, it is suppressed substantially by an antiferromagnetic (AFM) junction. Through the careful analysis of electronic band structures of the edge states, we discuss the origin of the enhanced chiral edge current in terms of a variant of the double-exchange mechanism.
\begin{figure}[h]
\begin{center}
\includegraphics[width=15cm]{fig_test_ferro_conjugation_11.eps}
\caption{\label{fig:schematics}
Schematics of (a) a four-sublattice spin order on a triangular lattice,
(b) spin directions of the four-sublattice order which form a tetrahedron,
(c) a MCI with open edges in the $x$ direction,
and (d) a MCI with capping magnetic layers.
In (c), spin configurations are optimized to minimize the free energy~\cite{JPSJ.83.073706}.
See the text for details.
}
\end{center}
\end{figure}
\section{Model and method}\label{sec:model}
In this study, we focus on the MCI with the four-sublattice noncoplanar spin configuration on a triangular lattice as schematically shown in Figs.~\ref{fig:schematics}(a) and \ref{fig:schematics}(b).
This is realized in the Kondo lattice model at 1/4 filling~\cite{JPSJ.79.083711,PhysRevLett.105.266405}, whose Hamiltonian is given by
\begin{equation}
\hat{\mathcal{H}} = -t \sum_{\braket{l, m}}\sum_s(\hat{c}^\dagger_{ls}\hat{c}_{ms} + {\rm h.c.}) - J_{\rm H}\sum_l\hat{\bm{s}}_l\cdot{\bm{S}}_l.
\label{Hamiltonian}
\end{equation}
Here, the first term represents hopping of itinerant electrons, where $\hat{c}^\dagger_{ls}(\hat{c}_{ls})$ denotes the creation (annihilation) operator of an itinerant electron on site $l$ with spin $s=\uparrow, \downarrow$, $t$ is the transfer integral, and the sum $\braket{l,m}$ is taken over nearest neighbor sites on a triangular lattice.
The second term is the on-site exchange interaction between localized spins $\bm{S}_l$ and itinerant electron spins $\bm{\hat{s}}_l = \sum_{s, s'}\hat{c}^\dagger_{ls}\bm{\sigma}_{ss'}\hat{c}_{ls'}$ ($\bm{\sigma}$ is the vector representation of the Pauli matrix);
$J_{\rm H}$ denotes the coupling constant.
Hereafter, we assume $\bm{S}_l$ to be a classical vector with $|\bm{S}_l|$ = 1, and take $t=1$ as an energy unit and the lattice constant as a length unit.
Also, we denote the coordinate of site $l$ by ($i$, $j$) measured from the origin defined on the left edge or interface
[see Figs.~\ref{fig:schematics}(c) and \ref{fig:schematics}(d)].
In the following calculations, the system size is taken to be $L_x \times L_y$ sites with an open (periodic) boundary condition in the $x$ ($y$) direction.
All the following calculations are done at 1/4 filling and $J_{\rm H}=3$.
The four-sublattice noncoplanar order in the MCI shown in Figs.~\ref{fig:schematics}(a) and \ref{fig:schematics}(b) is given by the spin configurations,
$\bm{S}_{(2m, 2n )} = \frac{1}{\sqrt{3}}( 1, 1, 1)$,
$\bm{S}_{(2m, 2n+1)} = \frac{1}{\sqrt{3}}( -1, -1, 1)$,
$\bm{S}_{(2m+1, 2n )} = \frac{1}{\sqrt{3}}( 1, -1, -1)$, and
$\bm{S}_{(2m+1, 2n+1)} = \frac{1}{\sqrt{3}}( -1, 1, -1)$;
$m$ and $n$ are integers in $0\leq m \leq L_x/2-1$ and $0\leq n \leq L_y/2-1$.
We denote this perfectly ordered configuration as $\{\bm{S}_l^{\rm 4sub}\}$.
Meanwhile, the optimized spin configuration in the system with open edges in the $x$ direction [see Fig.~\ref{fig:schematics}(c)] is denoted by $\{\bm{S}_l^{\rm opt}\}$.
The optimized state was obtained in the previous study by using the Langevin simulation with
the kernel polynomial expansion method at zero temperature;
for the details, the readers are referred to Ref.~\cite{JPSJ.83.073706}.
In addition to them, we consider the ``junction-type" spin configurations, $\{\bm{S}^{\theta}_l\}$.
This is defined by replacing each edge ($i=0$ and $i=L_x-1$) of $\{\bm{S}^{\rm 4sub}_l\}$ by a magnetically ordered layer.
We call the replaced edges the capping layers.
In the capping layers, we assume a twist of spins from $\{\bm{S}^{\rm 4sub}_l\}$ by an angle $\theta$.
Namely, the configuration $\{\bm{S}^{\theta}_l\}$ is given by
\begin{equation}
\label{eq:S^theta}
\bm{S}^{\theta}_{(i,j)} =
\begin{cases}
\bm{S}_z\cos(\theta/2) + (-1)^j\bm{S}_{xy}\sin(\theta/2) & (i=0) \\
-\bm{S}_z\cos(\theta/2) + (-1)^j\bar{\bm{S}}_{xy}\sin(\theta/2) & (i=L_x-1) \\
\bm{S}^{\rm 4sub}_{(i,j)} & ({\rm otherwise}),
\end{cases}
\end{equation}
where $\bm{S}_z=(0, 0, 1)$, $\bm{S}_{xy}=\frac{1}{\sqrt{2}}(1, 1, 0)$, and $\bar{\bm{S}}_{xy}=\frac{1}{\sqrt{2}}(1 ,-1, 0)$.
Note that the spins in the capping layers are aligned ferromagnetically (antiferromagnetically) for $\theta=0(\pi)$, which are denoted by $\{\bm{S}^{\rm FM}_l\} = \{\bm{S}^{\theta=0}_l\}$ and $\{\bm{S}^{\rm AFM}_l\} = \{\bm{S}^{\theta=\pi}_l\}$, respectively.
We also note that $\{\bm{S}^{\theta}_l\}$ with $\theta=\cos^{-1}(-1/3)\sim0.6\pi$ coincides with $\{\bm{S}_l^{\rm 4sub}\}$.
In other words, the configuration $\{\bm{S}^{\theta}_l\}$ continuously interpolates the perfect four-sublattice order, FM, and AFM interfaces by changing $\theta$.
For each spin configuration, we diagonalize the fermionic part of the Hamiltonian in Eq.~(\ref{Hamiltonian}), and obtain the electronic state at zero temperature.
We calculate the electronic band structure and the local electric current density parallel to the edge, $j_{\parallel}(i)$, which is defined as
\begin{equation}
\label{eq:j}
j_\parallel(i) = \frac{1}{2{\rm i}L_y}\sum_{j=0}^{L_y-1}\sum_{s=\uparrow, \downarrow}
\langle\hat{c}^\dagger_{(i,j)s}\hat{c}_{(i,j+1)s} - {\rm h.c.}\rangle,
\end{equation}
and compare the results for $\{\bm{S}_l^\theta\}$ and $\{\bm{S}_l^{\rm opt}\}$.
\section{Results and discussions}\label{sec:results}
\begin{figure}[h]
\begin{center}
\includegraphics[width=14cm]{fig_test_band_00030_00_04.eps}
\caption{\label{fig:band}
Band structures of the MCIs with
FM capping layers, $\{\bm{S}_l^{\rm FM}\}$,
AFM capping layers, $\{\bm{S}_l^{\rm AFM}\}$,
the optimized spin configuration for the system with open edges, $\{\bm{S}_l^{\rm opt}\}$,
and the four-sublattice order, $\{\bm{S}_l^{\rm 4sub}\}$.
The data are calculated at $J_{\rm H}=3$ and for $L_x\times L_y=34\times136$.
(a) shows the overall energy spectra, and
(b) is an enlarged figure of (a) near the chemical potential for 1/4 filling.
The dispersions of the edge states in the bulk gap ($-4.0 \lesssim E \lesssim -2.8$) almost coincide with each other for $\{\bm{S}_l^{\rm FM}\}$ and $\{\bm{S}_l^{\rm opt}\}$.
}
\end{center}
\end{figure}
Figure~\ref{fig:band} shows the electronic band structures as functions of the momentum in the $y$ direction, $k_y$, for the four types of spin configurations, $\{\bm{S}_l^{\rm FM}\}$, $\{\bm{S}_l^{\rm AFM}\}$, $\{\bm{S}_l^{\rm 4sub}\}$, and $\{\bm{S}_l^{\rm opt}\}$:
Fig.~\ref{fig:band}(a) shows the overall band structure, and Fig.~\ref{fig:band}(b) is the enlarged figure near the chemical potential for 1/4 filling.
Here, we calculate the band structures for the systems with $L_x\times L_y = 34\times 136$ sites; for $\{\bm{S}_l^{\rm opt}\}$, the spin configuration obtained for $34\times 34$ sites is repeated in the $y$ direction.
As shown in Fig.~\ref{fig:band}(a), the band structure is split into four bunches.
For the perfect four-sublattice order $\{\bm{S}_l^{\rm 4sub}\}$, each bunch forms an isolated band separated by finite energy gaps, if the periodic boundary condition is assumed also for the $x$ direction.
In the presence of open edges, however, the chiral edge states appear so as to traverse the energy gaps corresponding to 1/4 and 3/4 fillings with the crossing points at $k_y=\pi/2$, as shown in Fig.~\ref{fig:band}(a).
The existence of chiral edge states results from the nontrivial topological property of the four-sublattice ordered state.
As indicated in the data for $\{\bm{S}_l^{\rm opt}\}$ in Fig.~\ref{fig:band}, albeit the edge reconstruction considerably modifies the dispersions of the chiral edge states, it does not destroy the topologically protected edge states~\cite{JPSJ.83.073706}.
The situation is similar to the cases of the capping layers for both $\{\bm{S}_l^{\rm FM}\}$ and $\{\bm{S}_l^{\rm AFM}\}$, but the form of the dispersions are distinct between the two cases:
the dispersions for $\{\bm{S}_l^{\rm FM}\}$ almost coincide with those for $\{\bm{S}_l^{\rm opt}\}$, while the results for $\{\bm{S}_l^{\rm AFM}\}$ look very different from all other cases as shown in Fig.~\ref{fig:band}(b).
The results suggest that reconstructed edge states in the system with open edges are well described by that with one layer of FM ``skin".
\begin{figure}[t]
\begin{minipage}{14pc}
\includegraphics[width=7cm]{fig_test_dirac_ene_02.eps}
\end{minipage}\hspace{2cm}
\begin{minipage}{8cm}
\vspace{-1cm}
\caption{\label{fig:dirac} Energy of the band crossing point near 1/4 filling, $E_{\rm cross}$,
as a function of the angle $\theta$ in Eq.~(\ref{eq:S^theta}).
The data are calculated at $J_{\rm H} = 3$ and for $L_x\times L_y=34\times136$.
The square, circle, upward triangle, and downward triangle indicate $E_{\rm cross}$ for
$\{\bm{S}_l^{\rm FM}\}$, $\{\bm{S}_l^{\rm AFM}\}$,
$\{\bm{S}_l^{\rm opt}\}$, and $\{\bm{S}_l^{\rm 4sub}\}$, respectively.
See the text for details.
}
\end{minipage}\hspace{20cm}
\end{figure}
In order to further examine how the edge spin configuration affects the chiral edge mode, we consider the crossing energy of chiral modes at $k_y=\pi/2$ near 1/4 filling.
Figure~\ref{fig:dirac} shows the crossing energy $E_{\rm cross}(\{\bm{S}_l^\theta\})$ as a function of $\theta$ (red curve).
Note that the data for $\{\bm{S}^{\rm FM}_l\}$, $\{\bm{S}^{\rm AFM}_l\}$, and $\{\bm{S}^{\rm 4sub}_l\}$ are on the red curve at $\theta=0$, $\pi$, and $\sim0.6\pi$, respectively.
The result indicates that $E_{\rm cross}(\{\bm{S}_l^\theta\})$ monotonically decreases as the edge spin configuration changes from FM to AFM by increasing $\theta$.
This behavior will be discussed later, in comparison with the enhancement of the chiral edge current.
In Fig.~\ref{fig:dirac}, we also plot the crossing energy for $\{\bm{S}^{\rm opt}_l\}$, $E_{\rm cross}(\{\bm{S}^{\rm opt}_l\})$.
Here, we define the twist angle $\theta$ for $\{\bm{S}^{\rm opt}_l\}$ by the relative angle between the neighboring spins in the edge layer: $\theta=\cos^{-1}\left(\bm{S}_{(0, j)}^{\rm opt}\cdot\bm{S}_{(0, j+1)}^{\rm opt}\right)$. We note that $\theta$ is insensitive to $j$.
With this definition, $E_{\rm cross}(\{\bm{S}^{\rm opt}_l\})$ is almost on the curve of $E(\{\bm{S}^{\theta}_l\})$, as shown in Fig.~\ref{fig:dirac}.
This is rather surprising because the reconstruction of the spin configuration from $\{\bm{S}^{\rm 4sub}_l\}$ is not limited to the edge layer~\cite{JPSJ.83.073706}.
Furthermore, the value of $E_{\rm cross}(\{\bm{S}^{\rm opt}_l\})$ is very close to $E_{\rm cross}(\{\bm{S}^{\rm FM}_l\})$.
The results indicate quantitatively that the reconstructed edge states for $\{\bm{S}^{\rm opt}_l\}$ are close to those for the simple junction model with $\{\bm{S}^{\rm FM}_l\}$.
\begin{figure}[h]
\begin{center}
\includegraphics[width=14cm]{fig_test_current_dis_00030_05.eps}
\caption{\label{fig:current}
(a) Electric current densities $j_{\parallel}$ as functions of the distance from the edge, $i$, under several spin configurations calculated at $J_{\rm H} = 3$ and for $L_x\times L_y=34\times 34$.
The squares, circles, upward triangles, and downward triangles indicate $j_{\parallel}$ for
$\{\bm{S}_l^{\rm FM}\}$, $\{\bm{S}_l^{\rm AFM}\}$,
$\{\bm{S}_l^{\rm opt}\}$, and $\{\bm{S}_l^{\rm 4sub}\}$, respectively.
The lines are the guides for the eye.
(b) Integrated chiral current $j_{\rm chiral}$ as a function of $\theta$.
The square, circle, upward triangle, and downward triangle indicate $j_{\rm chiral}$
for $\{\bm{S}_l^{\rm FM}\}$, $\{\bm{S}_l^{\rm AFM}\}$,
$\{\bm{S}_l^{\rm opt}\}$, and $\{\bm{S}_l^{\rm 4sub}\}$, respectively.
}
\end{center}
\end{figure}
In Fig.~\ref{fig:current}(a), we plot spatial modulations of the local current density $j_{\parallel}(i)$ in Eq.~(\ref{eq:j}) for the spin configurations
$\{\bm{S}_l^{\rm FM}\}$, $\{\bm{S}_l^{\rm AFM}\}$, $\{\bm{S}_l^{\rm opt}\}$, and $\{\bm{S}_l^{\rm 4sub}\}$.
The calculations are done for the systems with $L_x\times L_y = 34\times 34$ sites.
The result shows that the current densities for $\{\bm{S}_l^{\rm FM}\}$ and $\{\bm{S}_l^{\rm opt}\}$ are similar to each other:
$j_\parallel(0)$ is lower than that for $\{\bm{S}_l^{\rm 4sub}\}$, whereas $j_\parallel(1)$ and $j_\parallel(2)$ are much larger than those for $\{\bm{S}_l^{\rm 4sub}\}$.
The decrease of $j_\parallel(0)$ for $\{\bm{S}_l^{\rm FM}\}$ and $\{\bm{S}_l^{\rm opt}\}$ reflects
the suppression of local spin scalar chirality at the edges because of FM spin correlations (see also Ref.~\cite{JPSJ.83.073706}).
We also plot the integrated chiral current $j_{\rm chiral}$ in Fig.~\ref{fig:current}(b),
which is defined as $j_{\rm chiral} = \sum_{i=0}^{L_x/2-1} j_\parallel(i)$, as a function of $\theta$ (red curve).
As shown in the figure, $j_{\rm chiral}$ shows almost 1.5 times larger values for $\{\bm{S}_l^{\rm FM}\}$ and $\{\bm{S}_l^{\rm opt}\}$,
compared with that for $\{\bm{S}_l^{\rm 4sub}\}$.
In contrast, for $\{\bm{S}_l^{\rm AFM}\}$, $j_{\rm chiral}$ is reduced to almost half of the value for $\{\bm{S}_l^{\rm 4sub}\}$.
Comparing the Fig.~\ref{fig:current}(b) with Fig.~\ref{fig:dirac}, we find a similar tendency
between the $\theta$ dependences of $j_{\rm chiral}$ and $E_{\rm cross}$.
This is understood as follows.
The amount of the chiral current is roughly proportional to the ``bandwidth"
of filled edge states,
i.e., the difference between the chemical potential and the band bottom energy of the edge state.
This means that $E_{\rm cross}$ gives a good measure for the amount of chiral current $j_{\rm chiral}$, as the band bottom is almost unchanged for different spin configurations.
The above consideration leads us to associate the increase of $E_{\rm cross}$, or the widening of chiral ``bandwidth", with the double-exchange mechanism~\cite{PhysRev.82.403}.
In the double-exchange mechanism, the underlying spins are ferromagnetically aligned so as to gain the kinetic energy of itinerant electrons. The gain of the effective kinetic energy in the edge mode for $\{\bm{S}_l^{\rm FM}\}$ and $\{\bm{S}_l^{\rm opt}\}$ implies that a similar mechanism works in both cases;
the FM correlation suppresses the current density in the outmost layer, but it increases the total amount of edge current by optimizing the kinetic energy in the vicinity of system edges.
This double-exchange type mechanism may be generic in a wide class of MCIs, and perhaps gives a guiding principle to the general problem of edge reconstruction.
Sensitivity of chiral edge current to the edge magnetic structure provides another interesting possibility.
The magnitude of the chiral current could be controlled in several ways:
for example, by making an interface to a magnetic material as discussed above, and by applying a magnetic field to the surface of MCIs.
\section{Summary}\label{sec:summary}
We have numerically investigated the energy spectra and chiral edge current in magnetic Chern insulators with open surfaces or interfaces to magnetic capping layers.
For the magnetic Chern insulator realized in the quarter-filled Kondo lattice model on the triangular lattice,
we clarified that the edge states for the optimized spin configuration are well described by the junction model with the ferromagnetic interface.
Furthermore, we revealed the close correlation between the total amount of chiral edge current and the energy of the crossing point of the edge states.
We also found that the ferromagnetic arrangement of edge magnetic moments maximizes the total amount of the edge current.
The results suggest that the ferromagnetic edge reconstruction is driven by the optimization of kinetic energy in the edge region,
i.e., a variant of the double-exchange mechanism.
We also discussed the possibility of controlling the chiral edge current via an external magnetic force, by utilizing the sensitivity of the chiral edge current to the local magnetic structure in the edge region.
\ack
R.O. is supported by the Program for Leading Graduate Schools, MEXT, Japan, via the Advanced Leading Graduate Course for Photon Science.
Y.A. acknowledges support from OIST.
This research was supported by Grants-in-Aid for Scientific Research (Grants No. 24340076, 24740221, and 26400339),
the Strategic Programs for Innovative Research (SPIRE), MEXT, and the Computational Materials Science Initiative (CMSI), Japan.
\section*{References}
\providecommand{\newblock}{}
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1,116,691,497,582 | arxiv | \section{Introduction}
A scalable quantum computer is expected solve difficult problems that are intractable with classical computing technology. Scaling such a machine to a useful size will necessarily require fault-tolerant components that protect quantum information as the data is processed~\cite{Calderbank96, Steane96, Dennis02, Terhal15, Brown16, Campbell17}. If we are to see the realisation of a quantum computer, its design must respect the constraints of the quantum architecture that can be prepared in the laboratory. In many cases, for instance superconducting qubits~\cite{Raussendorf07, DiVincenzo09, Fowler12a, Corcoles15, Kelly15, Takita16}, this restricts us to two-dimensional architectures.
Leading candidate models for fault-tolerant quantum computation are based on the surface code~\cite{Kitaev03, Dennis02} due to its high threshold~\cite{Dennis02, Wang03, Raussendorf07, Fowler12a} and multitude of ways of performing Clifford gates~\cite{Raussendorf06, Bombin09, Hastings15, Brown17, Yoder17}. Universal quantum computation is possible if this gate set is supplemented by a non-Clifford gate. Among the most feasible approaches to realise a non-Clifford gate is by the use of magic-state distillation~\cite{Bravyi05}. However, this is somewhat prohibitive as it is estimated that most of the resources of a quantum computer will be expended by these protocols~\cite{Fowler12a, OGorman17}.
Here we provide a promising alternative to magic-state distillation with the surface code. Remarkably, we show that we can perform a fault-tolerant non-Clifford gate with three overlapping copies of the surface code that interact locally. Each of the two-dimensional arrays of live qubits replicates a copy of the three-dimensional generalisation of the surface code over a time that scales with the size of the array. We use that the full three-dimensional model is natively capable of performing a controlled-controlled-phase gate~\cite{Kubica15, Vasmer18} to realise a two-dimensional non-Clifford gate. The procedure makes essential use of just-in-time gauge fixing; a concept recently introduced by Bomb\'{i}n in Ref.~\cite{Bombin18}. This enables us to recover the three-dimensional surface-code model using parity measurements of weight no greater than four. Research into such technology is presently under intensive development~\cite{Corcoles15, Kelly15, Takita16}, as these are the minimal requirements to realise the surface-code model.
The non-Clifford gate presented here circumvents fundamental limitations of two-dimensional models~\cite{Eastin09, Bravyi13a, Pastawski15, OConnor18, Webster18} by dynamically preparing a three-dimensional system using a two-dimensional array of active qubits. In the past, there has been a significant effort to realise a non-Clifford gate with two-dimensional quantum error-correcting codes~\cite{Bravyi15, OConnor16, Jones16, Yoder16, Yoder17a}. However, these proposals are unlikely to function reliably as the size of the system diverges. It is remarked in Ref.~\cite{Bombin18} that we should understand fault-tolerant quantum operations, not in terms of quantum error-correcting codes, but instead by the processes they perform. Notably, in our scheme, error-detecting measurements are realised dynamically. This is in contrast to the more conventional approach where we make stabilizer measurements on static quantum error-correcting codes to identify errors. As we will see, the process is well characterised by connecting the surface code with the topological cluster-state model~\cite{Raussendorf05, Raussendorf06, Raussendorf07, Raussendorf07a}, a measurement-based model with a finite threshold error rate below which it will function reliably at an arbitrary size. Furthermore, as we will see, the cluster state offers a natural static language to characterise the dynamical quantum process using a time-independent entangled resource state.
We begin by defining measurement-based model, and we explain how we project the non-Clifford gate onto a two-dimensional surface. We finally discuss the just-in-time decoder that permits a two-dimensional implementation of the gate. Microscopic details of the system and proof of its threshold are deferred to appendices.
The topological cluster state model~\cite{Raussendorf05} is described in three dimensions. However, we need only maintain a two-dimensional array of its qubits at a given moment to realise the system~\cite{Raussendorf07}. Specifically, we destructively measure each qubit immediately after it has interacted with all of the other qubits that are specified by the cluster state. This method of generating the model on the fly gives rise to a time-like direction, see Fig.~\ref{Fig:Raussendorf}(a).
We use the topological cluster state to realise the three-dimensional surface code~\cite{Hamma05}. We define the surface code on a lattice with arbitrary geometry with one qubit on each edge which we index $e$. The model is specified by two types of stabilizers, star and plaquette operators, denoted $A_v$ and $B_f$, see Appendix~\ref{App:Microscopics} for details. Stabilizers specify the code states of the model, $|\psi \rangle$, such that $A_v |\psi \rangle = B_f |\psi \rangle = |\psi \rangle$ for all code states. Star operators are associated to the vertices, $v$, of the lattice such that $A_v = \prod_{\partial e \ni v} X_e$ where $\partial e$ is the set of vertices at the boundary of $e$ and $X_e $ and $Z_e$ are Pauli operators acting on $e$. Plaquettes $B_f$ lie on lattice faces $f$ such that $B_f = \prod_{e \in \partial f} Z_f$ where $\partial f$ are the edges that bound $f$.
\begin{figure}
\includegraphics{Raussendorf.pdf}
\caption{(a)~The topological cluster-state model is a three-dimensional model that propagates quantum information over time with only a two-dimensional array of live qubits at any given moment. We show a grey plane of live qubits that propagates in the direction of the time arrow. (b)~Grey loops of show the connectivity of plaquette measurements that returned the -1 outcome. An arbitrary state is initialised fault tolerantly by initialising the system with an encoded two-dimensional fixed-gauge surface code on the grey face at the left of the image. (c)~The boundary configurations of the three copies of the surface code required to perform a local transversal controlled-controlled-phase gate. The first code has rough boundaries on the top and the bottom of the lattice. The middle(right) code has rough boundaries on the left and right(front and back) sides of the lattice. The orientation of the boundaries determines the time direction in which we can move the planes of live qubits. \label{Fig:Raussendorf}}
\end{figure}
To connect the three-dimensional surface code with the topological cluster state~\cite{Raussendorf05} we consider initialising the surface code in the $+1$ eigenvalue eigenstate of the logical Pauli-X operator by measurement. We consider initialising all of the physical qubits in the $|+\rangle$ state and then measure all of the plaquette operators. Up to an error correction step, this completes initialisation. To measure a plaquette operator $B_f$ we prepare an ancilla qubit, $a$, in the $| + \rangle _a$ state and couple it to the qubits that bound $f$ with controlled-phase gates, i.e., we apply $U = \prod_{e \in \partial f} CZ_{e, a}$ with $CZ_{j,k} = (1 + Z_j + Z_k - Z_jZ_k) / 2$ the controlled-phase gate. It may be helpful to imagine placing the ancilla qubit on face $f$. Measuring the ancilla qubit in the Pauli-X basis will recover the value of the face operator. However, we observe that before the ancillas are measured we have the topological cluster-state model~\cite{Raussendorf05} where now the qubits of the surface code give the qubits of the primal lattice of the model and the ancilla qubits make up the qubits of its dual lattice.
The surface code lies in the common $+1$ eigenvalue eigenstate of all the face operators. Measuring all the dual qubits of the cluster state projects its primal qubits into a random gauge of the three-dimensional surface code where, up to certain constraints, all the face measurements take random values. Henceforth, unless there is ambiguity, we refer to the model with face operators fixed onto their $+1$ eigenvalue eigenstate as the surface code. Otherwise we call it the random-gauge surface code. It is important to realise the fixed gauge surface code to perform the controlled-controlled-phase gate~\cite{Vasmer18}.
We use error correction to recover the surface code from the random gauge model~\cite{Paetznick13, Anderson14, Bombin15, Vuillot18}. We note that the product of all the face operators that bound a cell return identity, i.e. $\prod_{f \in \partial c} B_f = 1$ where $\partial c$ at the set of faces that bound cell $c$. As such, supposing all of the measurements are made noiselessly, there must be an even parity of measurements that return the $-1$ outcome about each cell. This in turn constrains the plaquette operator measurements to respect loop-like configurations on the dual lattice, see Fig.~\ref{Fig:Raussendorf}(b). To recover the fixed-gauge surface code we apply a Pauli-X operator with a membrane like support whose boundary terminates at each component of the loop configuration.
Further, we can initialise the surface code in an arbitrary state fault tolerantly if, before face measurements are made, we replace the unentangled qubits on one side of one boundary of the lattice with an encoded surface code, for instance the grey face shown to the left of Fig.~\ref{Fig:Raussendorf}(a)~\cite{Brown18}. We refer to this face as the initial face. Imposing that the face operators of the surface code are fixed in the $+1$ eigenvalue eigenstate mean no loop configurations will terminate at this boundary. This method of initialisation is a dimension jump~\cite{Bombin16}.
We can now explain how we can embed the three-dimensional surface code that performs a non-Clifford gate in two dimensions. There are several constraints the system must satisfy if we realise a controlled-controlled-phase gate with a two-dimensional system. We first point out that the orientation of boundaries of the topological cluster state are important for the transmission of logical information~\cite{Brown18}. Moreover, they constrain the temporal directions of the model. We consider again the cluster state in terms of the three-dimensional surface code. The surface code model has two types of boundary; rough and smooth~\cite{Hamma05}. If we couple ancilla to the surface code to recover the topological cluster state as specified above then the rough(smooth) boundaries of the surface code give rise to the primal(dual) boundaries of the cluster state. If we only maintain a two-dimensional array of qubits, the plane must contain two distinct primal boundaries that are well separated by two distinct dual boundaries to support the encoded information. The grey plane in Fig.~\ref{Fig:Raussendorf}(a) is suitable, for example.
Secondly, the boundaries of the three surface codes must be correctly configured to perform a transversal controlled-controlled-phase gate~\cite{Kubica15, Vasmer18}. Fig.~\ref{Fig:Raussendorf}(c) shows the boundaries configured such that the qubit at coordinate $P = (x,y,z)$ of each code interacts with the respective qubit at the same location of the other codes via transversal controlled-controlled-phase gates. To perform the gate locally, these three lattices must overlap while maintaining these boundary conditions.
Finally, if we only maintain a two-dimensional array of the three-dimensional system, it is important that all of the qubits that need to interact with one another must be live at the same time. We show a system that satisfies all of these constraints in Fig.~\ref{Fig:Locality}. The figure shows a three-dimensional spacetime diagram of two of the codes moving orthogonally to one another. We omit the third code as it can travel in parallel to one of the codes already shown. The first code that has rough boundaries on the top and bottom of its volume and the live plane of qubits moves right across the page. The second code has rough edges on its left and right faces and moves upwards in the Figure. The controlled-controlled-phase gate is made at the cubic region where the codes intersect. We find that all of the appropriate qubits are active at the right moment by choosing two diagonal planes of live qubits for each code. We can also see that the planes we choose all have two well separated rough and smooth boundaries within their respective volume.
\begin{figure}
\includegraphics{Locality.pdf}
\caption{Two codes travelling in different temporal directions cross. The third code is omitted as it can run in parallel with one of the two shown. Live qubits of the spacetime history are shown on light grey planes. The transversal gate is applied in the cubic region in the middle. It will be applied on the qubits shown at the dark grey plane where the two-dimensional arrays of qubits are overlapping. \label{Fig:Locality}}
\end{figure}
We are now ready to consider an embedding the three-dimensional spacetime shown in Fig.~\ref{Fig:Locality} onto a two-dimensional manifold. We find that one of the codes has to move with respect to the other. This can be naturally incorporated in the procedure to generate layers of the topological cluster state, see Appendix~\ref{App:Microscopics}. We consider a point $P$ in the spacetime diagram in the region where the controlled-controlled-phase gate is performed such that a qubit of each of the two models must interact. The coordinates of the locations of the two codes change differently with time. The first code that travels upwards in the spacetime diagram has coordinates $P = (x' , t')$, the other that moves from left to right has coordinates $P= (t, y)$ with time $t = t'$. We neglect the $z$-coordinate as this is static. We imagine projecting the three-dimensional system onto a two-dimensional plane such that $y = t' = 0$, we now observe that $t = x'$. We conclude that one code must move with respect to the other to ensure all of the qubits that must interact are local at the right points in time.
We use a decoder to fix the topological cluster-state model onto the surface code using data from the ancilla qubit measurements. In the case that there are measurement errors we will necessarily introduce small Pauli errors onto the system that will translate into Clifford errors upon application of the transversal non-Clifford gate. Measurement errors in this model take the form of strings which are detected by defects that lie at their endpoints. A decoder must attempt to close these endpoints and will fix the gauge according to the error and the correction. This will necessarily lead to gauge-fixing errors. Provided the errors that are introduced during gauge fixing are small and are supported on a correctable region though, the Clifford errors the transversal gate will introduce are also correctable. We can therefore prove a fault-tolerance threshold under the gauge fixing procedure by showing the errors we introduce during gauge fixing are small in comparison to the distance of the code.
\begin{figure}
\includegraphics{JustInTimeIntuition.pdf}
\caption{\label{Fig:JIT} The spacetime diagram of an error on the dual qubits of the topological cluster-state where time travels upwards. The grey area shows the two-dimensional area of live qubits at a given moment. At the point where an error is discovered on the left diagram, it is unlikely that the defects should be paired due to their separation. We therefore defer matching the defects to a later time after more information emerges as decoding progresses, as in the middle figure. After enough time, the most likely outcome is that the defects we discovered in the left figure should be paired. The error we introduce fills the interior of the error, and the chosen correction.}
\end{figure}
We aim to fix the gauge of a three-dimensional model. However, we will only maintain a two-dimensional array of live qubits. As such the decoder has a limited amount of information available to make decisions about how to pair defects. To overcome this issue we defer correcting pairs of defects to a later time once we have more certainty that two defects should be matched. This leads the errors to spread over the time matching is deferred, see Fig.~\ref{Fig:JIT}. We propose a just-in-time decoder~\cite{Harrington04, Bravyi13b} that will defer the pairing of defects such that the spread of errors is controlled. Broadly speaking, we find that a just-in-time decoder will work if the pairing of two defects is deferred until both defects have existed for a time proportional to their separation in spacetime. We make this statement precise in Appendix~\ref{App:JIT} and prove it controls the spread of errors.
Supposing an independent and identically distributed error model that is characterised in terms of chunks~\cite{Gacs86, Grimmett99, Gray01} we can show that the just-in-time decoder will not spread a connected component of the error by more than a constant factor of the size of the component. We further find that this spread error model can be decoded by a renormalisation-group decoder. We prove a threshold against the spread error model using a renormalisation-group decoder in Appendix~\ref{App:JIT} , see Lemma~\ref{Lemma:RG}. We then prove that the just-in-time decoder will give rise to spread errors with a constant spread, see Lemma~\ref{Lemma:JIT}, thus justifying the noise model. In contrast, the threshold theorem for just-in-time decoding given by Bomb\'{i}n.~\cite{Bombin18} uses a minimum-weight perfect matching decoder~\cite{Edmonds65, Dennis02}.
One should worry that the just-in-time gauge fixing process will add errors that may significantly decrease the logical failure rate of the system. We argue that we can make this effect relatively benign in post processing. The errors introduced by the just-in-time decoder are twofold. Firstly, it may directly introduce a logical failure by incorrectly matching defects and, secondly, if the decoder does succeed, it will introduce large errors to the primal qubits of the system that need to be decoded globally once the gate is complete.
Rather than considering the protocol as a gate that can be used on the fly in some algorithm, we use it to produce high-fidelity magic states by inputting Pauli-X eigenstates that are prepared fault tolerantly. Once gauge fixing is completed, we can simulate gauge-fixing again globally with a high-performance decoder~\cite{Wang03, Raussendorf06, Wootton12}. We can then compare the output of the high-performance decoder with the just-in-time decoder. If their results do not agree, we discard the output.
We denote the failure rates of the high-threshold(just-in-time) decoder $\overline{P}_{\text{HP}}$($\overline{P}_{\text{JIT}}$). Both decay rapidly with system size below threshold, but we suppose $\overline{P}_{\text{HP}} \ll \overline{P}_{\text{JIT}}$. In the event that the decoders disagree, we discard the state. This occurs with likelihood $ \sim \overline{P}_{\text{JIT}}$. In the case that the decoders agree, the state that we output is logically incorrect with likelihood $\sim \overline{P}_{\text{HP}}\overline{P}_{\text{JIT}}$. The use of a high-threshold decoder therefore improves the fidelity of the post-selected output states. The failure rate of the just-in-time decoder then only determines the rate at which magic states should be discarded.
We can also use the output of the high-performance decoder to deal with the errors spread to the surface code with just-in-time decoding. We can compare the output of the high-performance decoder with the correction produced by the just-in-time decoder. The discrepancy in their outputs should indicate the approximate locations of the gauge-fixing errors. This information can be fed to the decoder we use to decode the errors on the primal qubits that will flag the discrepancy as qubits that are highly likely to support an error. Indeed there have been a number of results showing how to improve decoders by using knowledge of likely errors~\cite{Delfosse14a, Fowler13, Nickerson17, Criger18}. Ref.~\cite{Bombin18} treats these flagged qubits as erasure errors that are highly likely to support linking charges~\cite{Yoshida15, Bombin18a}. The proof given in Appendix~\ref{App:JIT} shows that we have a threshold without these considerations, but implementations of this protocol should use a decoder that accounts for these effects to improve their performance. After post-selection then we might expect the system to perform as though it were gauge fixed globally with some known erasure errors.
To summarise, we have shown how to perform a fault-tolerant controlled-controlled-phase gate with a two-dimensional surface-code architecture and we have proved it has a threshold. Next, it is important to compare the resource scaling of this scheme compared with more conventional two-dimensional approaches to fault-tolerant quantum computation, namely, surface-code quantum computation with magic state distillation~\cite{Bravyi05, Fowler12a}. Given that gauge-fixing errors will spread phase errors as we apply the three-qubit transversal gate, the logical error rate of this scheme is likely to decay more slowly than approaches using magic state distillation where we do not rely on gauge fixing. However, the spacetime volume of realising a fault-tolerant controlled-controlled-phase gate, $\sim 30 d^3$, is an order of magnitude smaller than a single distillation routine, as such, these schemes are clearly deserved of further comparison. It is likely that the optimal choice will depend on the error rate of the physical hardware.
It will also be interesting to compare the protocol introduced here to that presented by Bomb\'{i}n~\cite{Bombin18}. This protocol is very similar to that presented here, except it is based on the color code such that a transversal $T$ gate is performed over time via single-qubit rotations. This will make for a very interesting comparison since, even though decoding technology for the color code model~\cite{Delfosse14, Bombin15a, Brown16a, Kubica17, Kubica18, Aloshious18, Brown19} remains lacking in comparison to the surface code~\cite{Raussendorf07, Fowler12a}, the fact that the non-Clifford operation is performed using single-qubit rotations instead of a weight-three gate will mean that fewer errors will be spread during computational processes. To begin comparing these protocols fairly it will first be important to improve the decoding algorithms we have for the color code.
Finally, it is likely that there will be several ways to optimise the present scheme. Although we find transversal gates via a mapping between the color code and the surface code~\cite{Kubica15, Vasmer18} such that we arrive at quite a specific lattice, it will be surprising if we cannot find ways of performing a constant-depth locality-preserving gate with other lattices~\cite{Nickerson18} for the topological cluster-state model. Indeed, history has shown that the gates a given model is able to achieve is connected with the macroscopic properties of a system, not its microscopic details. Developing our understanding of measurement-based quantum computation by decomposing it in terms of its topological degrees of freedom~\cite{Bombin18, Brown18} is likely to be a promising route towards better models of two-dimensional fault-tolerant quantum computation.
\begin{acknowledgements}
I am grateful to A. Doherty, M. Kesselring, N. Nickerson and S. Roberts for helpful and supportive discussions, and in particular S. Bartlett, C. Chubb, C. Dawson and S. Flammia for patiently following various incarnations of these results during the preparation of this manuscript. I also thank D. Poulin for many discussions on non-Clifford operations. This work is supported by the University of Sydney Fellowship Programme and the Australian Research Council via the Centre of Excellence in Engineered Quantum Systems(EQUS) project number CE170100009.
\end{acknowledgements}
|
1,116,691,497,583 | arxiv | \section{Background}
\label{sec:background}
This section summarizes the essential aspects of blockchain technology and discusses initial research efforts at the intersection of BPM and blockchains.
\subsection{Blockchain Technology}
In its original form, Blockchain is a distributed database technology that builds on a tamper-proof list of timestamped transaction records. Among others, it is used for cryptocurrencies such as Bitcoin~\citep{nakamoto2008bitcoin}.
Its innovative power stems from allowing parties to transact with others they do not trust over a computer network in which nobody is trusted. This is enabled by a combination of peer-to-peer networks, consensus-making, cryptography, and market mechanisms.
Blockchain derives its name from the fact that its essential data structure is a chained list of blocks. This chain of blocks is distributed over a peer-to-peer network, in which every node maintains the latest version of it. Blocks can contain information about transactions. In this way, we can for instance know that a buyer has ordered 200 items of a particular type of material from a vendor at a specific time. When a new block is added to the blockchain, it is signed using cryptographic methods. In this way, it can be checked if its content and its signature match. For example, if we take the content $c=$"Buyer orders 200 items from vendor" and apply a specific hash function $h(c)$, we get a unique result $r$. Every block is associated with a hash generated from its content \emph{and} the hash value of the previous block in the list. Hash values thus uniquely represent not only the transactions within blocks but also the ordering of every block. This mechanism is at the basis of the chain. In case somebody would try to alter a transaction, this would change the hash value of its block, and therefore break the chain. Since every node can create blocks in a peer-to-peer network, there has to be consensus on the new version of the blockchain including a new block.
This is achieved with consensus algorithms that are based on concepts like proof-of-work or proof-of-stake~\citep{Bentov2016}, and more recently \textit{proof-of-elapsed-time}\footnote{Intel: Proof of elapsed time (PoET). Available from \url{http://intelledger.github.io/}}.
In proof-of-work, miners guess a value for a specific field, to fulfill the condition that $r$ must be smaller than a threshold (which is dynamically adjusted by the network based on a predefined protocol). In proof-of-stake, miner selection considers the size of their stake , i.e., amount of cryptocurrency held by them. The rationale is that a high stake is a strong motivation for not cheating: if the miners cheat (and this is detected), the respective cryptocurrency will be devalued.
The network protocols and dynamic adjustment of thresholds are designed to avoid network overload.
In summary, these foundational blockchain concepts support two important notions that are also essential for business processes: the blockchain as a (tamper-proof) data structure captures the history and the current state of the network and transactions move the system to a new state.
Blockchain offers an additional concept that is important for business processes, called \textit{smart contracts}~\citep{szabo1997formalizing}. Consider again the example of the buyer ordering 200 items from the vendor. Business processes are subject to rules on how to respond to specific conditions. If, for instance, the vendor does not deliver within two weeks, the buyer might be entitled to receive a penalty payment. Such business rules can be expressed by smart contracts. For instance, the \textit{Ethereum} blockchain supports a Turing-complete programming language for smart contracts\footnote{\url{https://www.ethereum.org/}}. The code in these languages is deterministic and relies on a closed-world assumption: only information that is stored on the blockchain is available in the runtime environment. Smart contract code is deployed with a specific type of transaction. As with any other blockchain transaction, the deployment of smart contract code to the blockchain is immutable.
Once deployed, smart contracts offer a way to execute code directly on the blockchain network, like the conditional transfer of money in our example if a certain condition is fulfilled.
By using blockchain technology,
untrusted parties can establish trust in the truthful execution of the code. Smart contracts can be used to implement business collaborations in general and inter-organizational business processes in particular. The potential of blockchain-based distributed ledgers to enable collaboration in open environments has been successfully tested in diverse fields ranging from diamonds trading to securities settlement~\citep{walport2016distributed}.
At this stage, it has to be noted that blockchain technology still faces numerous general technological challenges. A mapping study by \cite{Yli-Huumo:2016:PlosOne} found that a majority of these challenges have not been addressed by the research community, albeit we note that blockchain developer communities actively discuss some of these challenges and suggest a myriad of potential solutions\footnote{\url{http://www.the-blockchain.com/2017/01/24/adi-ben-ari-outstanding-challenges-blockchain-technology-2017/}}.
Some of them can be addressed by using private or consortium blockchain instead of a fully open network~\citep{mougayar2016business}. In general, the technological challenges include the following~\cite{swan2015blockchain}.
\begin{description}
\item{\textbf{Throughput}} in the Ethereum blockchain is limited to approx.~15 transaction inclusions per second (tps) currently. In comparison, transaction volumes for the VISA payment network are 2,000 tps on average, with a tested capacity of up to 50,000 tps. However, the experimental Red Belly Blockchain which particularly caters to private or consortium blockchains has achieved more than 400,000 tps in a lab test\footnote{\url{http://poseidon.it.usyd.edu.au/~concurrentsystems/rbbc/}}.
\item{\textbf{Latency}} is also an issue. Transaction inclusion in the absence of network congestion takes a certain amount of time. In addition, a number of confirmation blocks are typically recommended to ensure the transaction does not get removed due to accidental or malicious forking. That means that transactions can be seen as committed after 60 minutes on average in Bitcoin, or 3 to 10 minutes in Ethereum. Even with improvements of techniques like the \emph{lightning network} or \emph{side chains} spawned off from the main chain, blockchains are unlikely to achieve latencies as low as centrally-controlled systems.
\item{\textbf{Size and bandwidth}} limitations are variations of the throughput issue: if the transaction volume of VISA were to be processed by Bitcoin, the full replication of the entire blockchain data structure would pose massive problems. \cite{Yli-Huumo:2016:PlosOne} quote 214 PB per year, thus posing a challenge in data storage and bandwidth. Private and consortium chains and concepts like the lightning network or side chains all aim to address these challenges. In this context it is worth noting that most everyday users can use \emph{wallets} instead, which require only small amounts of storage.
\item{\textbf{Usability}} is limited at this point, in terms of both developer support (lack of adequate tooling) and end-user support (hard to use and understand). Recent advances on developer support include efforts by some of the authors towards model-driven development of blockchain applications~\citep{Weber:2016:BPM,Garcia:2017:BPM,Tran:2017:CAISE}.
\item{\textbf{Security}} will always pose a challenge on an open network like a public blockchain. Security is often discussed in terms of the CIA properties~\citep{dhillon2000technical}. First, \textit{confidentiality} is per se low in a distributed system that replicates all data over its network, but can be addressed by targeted encryption~\citep{7546538}. Second, \textit{integrity} is a strong suit of blockchains, albeit challenges do exist~\citep{Eyal2014,Gervais:2016:SPP:2976749.2978341}. Third, \textit{availability} can be considered high in terms of reads from blockchain due to the wide replication, but is less favorable in terms of write availability~\cite{Weber:2017:SRDS}.
New attack vectors exist around forking, e.g., through network segregation~\cite{NG:2017:DSN}. These are particularly relevant in private or consortium blockchains.
\item{\textbf{Wasted resources,}} particularly electricity, are due to the consensus mechanism, where miners constantly compete in a race to mine the next block for a high reward. In an empirical analysis, \cite{Weber:2017:SRDS} found that about 10\% of announced new blocks on the Ethereum network were uncles (forks of length 1). This can be seen as wasteful, but is just a small indication of the vast duplication of effort in \emph{proof-of-work} mechanisms. Longer forks (at most of length 3) were extremely rare, so accidental forking seems unlikely in a well-connected network like the Internet -- but could occur if larger nations were cut off temporarily or even permanently. Alternatives to the proof-of-work, like \emph{proof-of-stake}~\citep{Bentov2016}, have been discussed for a while and would be much more efficient. At the time of writing, they remain an unproven but highly interesting alternative. Proof-of-work makes very low assumptions in trusting other participants, which is well suited for an open network managing digital assets. Designing more efficient protocols without relaxing these assumptions has proven a challenge.
\item{\textbf{Hard forks}} are changes to the protocol of a blockchain which enable transactions or blocks which were previously considered invalid~\citep{conf/p2p/DeckerW13}. They essentially change the rules of the game and therefore require adoption by a vast majority of the miners to be effective~\citep{7163021}. While hard forks can be controversial in public blockchains, as demonstrated by the split of the Ethereum blockchain into a hard forked main chain and Ethereum Classic (ETC), this is less of an issue for private and consortium blockchains where such a consensus is more easily found.
\end{description}
Many of these general technological challenges of blockchains are currently the focus of the emerging body of research. As noted, our main interest is in the \textit{potential} of blockchain technology to enable a shift in BPM research. Our belief is vested both in the novel technological properties discussed above and in the already available attempts of using blockchain technology in the definition and implementation of fundamentally novel business processes. We review these attempts in the following.
\subsection{Business Processes and Blockchain Technology}
We are not the first to identify the application potential of blockchain technology to business processes. In fact, several blockchains are currently adopted in various domains to facilitate the operation of new business processes. For example, \cite{DBLP:journals/bise/NoferGHS17} list applications in the financial sector including cryptocurrency transactions, securities trading and settlement, and insurances as well as non-financial applications such as notary services, music distribution, and various services like proof of existence, authenticity, or storage. Other works describe application scenarios involving blockchain technology in logistics and supply chain processes, for instance in the agricultural sector~\citep{Risks-Blockchain-2017}.
A proposal to support inter-organizational processes through blockchain technology is described by~\cite{Weber:2016:BPM}: large parts of the control flow and business logic of inter-organizational business processes can be compiled from process models into smart contracts which ensure the joint process is correctly executed. So-called \textit{trigger} components allow connecting these inter-organizational process implementations to Web services and internal process implementations. These triggers serve as a bridge between the blockchain and enterprise applications.
The cryptocurrency concept enables the optional implementation of conditional payment and built-in escrow management at defined points within the process, where this is desired and feasible.
\begin{figure}[t]
\begin{center}
\includegraphics[width=1\textwidth]{supply.pdf}
\caption{Supply Chain Scenario from~\cite{Weber:2016:BPM}}\label{fig:collaboration}
\end{center}
\end{figure}
To illustrate these capabilities, Figure~\ref{fig:collaboration} shows a simplified supply chain scenario, where a bulk buyer orders goods from a manufacturer. The manufacturer, in turn, orders supplies through a middleman, which are sent from the supplier to the manufacturer via a special carrier.
Without global monitoring each participant has restricted visibility of the overall progress. This may very well be a basis for misunderstandings and shifting blame in cases of conflict. Model-driven approaches such as proposed by~\cite{Weber:2016:BPM,Garcia:2017:BPM} produce code of smart contracts that implement the process (see Figure~\ref{fig:code}).
If executed using smart contracts on a blockchain, typical barriers complicating the deployment of inter-organizational processes can be removed.
(i) The blockchain can serve as an immutable public ledger, so that participants can review a trustworthy history of messages to pinpoint the source of an error. This means that all state-changing messages have to be recorded in the blockchain. (ii) Smart contracts can offer independent process monitoring from a global viewpoint, such that only expected messages are accepted, and only if they are sent from the player registered for the respective role in the process instance.
(iii) Encryption can ensure that only the data that must be visible is public, while the remaining data is only readable for the process participants that require it.
These capabilities demonstrate how blockchains can help organizations to implement and execute business processes across organizational boundaries even if they cannot agree on a trusted third party.
This is a fundamental advance, because the core aspects of this technology enable support of enterprise collaborations going far beyond asset management, including the management of entire supply chains, tracking food from source to consumption to increase safety, or sharing personal health records in privacy-ensuring ways amongst medical service providers.
The technical realization of this advance is still nascent at this stage, although some early efforts can be found in the literature. For example, smart contracts that enforce a process execution in a trustworthy way can be generated from BPMN process models~\citep{Weber:2016:BPM} and from domain-specific languages~\citep{Frantz:ECAS:2016}. Further cost optimizations are proposed by~\cite{Garcia:2017:BPM}. Figure~\ref{fig:code} shows a code excerpt that was generated by this approach.
In a closely related work, \cite{DBLP:conf/icsoc/HullBCDHV16} emphasize the affinity of artifact-centric process specification~\citep{DBLP:journals/debu/CohnH09,DBLP:conf/bpm/MarinHV12} for blockchain execution.
\begin{figure}[t]
\begin{center}
\includegraphics[width=.8\textwidth]{smart-contract-snippet2.png}
\caption{Smart contract snippet illustrating how code is generated from a BPMN model. It shows the implementation of function \texttt{PlaceOrder} from the above process model. This function is to be executed by the Manufacturer, which is checked in line 6. Subsequently, we check if the function is activated in line 7. If so, any custom task logic is executed, and the activation of tasks is updated in line 9. For more details, see~\cite{Garcia:2017:BPM}.}\label{fig:code}
\end{center}
\end{figure}
Even at this stage, research on the benefits and potentials of blockchain technology is mixed with studies that highlight or examine issues and challenges. For example, \cite{Norta:2015:BIR,Norta:2016:ICACDS} discusses ways to ensure secure negotiation and creation of smart contracts for Decentralized Autonomous Organizations (DAOs), among others in order to avoid attacks like the DAO hack during which approx. US\$ 60M were stolen. This in turn was remediated
by a hard fork of the Ethereum blockchain, which was controversial among the respective mining node operators and resulted in a part of the public Ethereum network splintering off into the \emph{Ethereum Classic} (ETC) network. This split, in turn, caused major issues for the network in the medium term, allowing among others \emph{replay attacks} where transactions from Ethereum can be replayed on ETC.
A formal analysis of smart contract participants using game theory and formal methods is conducted by~\cite{Bigi:2015:Degano}. As pointed out by~\cite{Norta:2016:ICACDS}, the assumption of perfect rationality underlying the game-theoretic analysis is unlikely to hold for human participants.
These examples show that blockchain technology and its application to BPM are at an important crossroads: technical realization issues blend with promising application scenarios; early implementations mix with unanticipated challenges. It is timely, therefore, to discuss in broad and encompassing ways where open questions lie that the scholarly community should be interested in addressing. We do so in the two sections that follow.
\section{Blockchain Technology and BPM Capabilities}
\label{sec:capab}
There are also challenges and opportunities for BPM and blockchain technology beyond the classical BPM lifecycle. We refer to the BPM capability areas~\citep{rosemann2015six} beyond the methodological support we reflected above, including strategy, governance, information technology, people, and culture.
\subsection{Strategy}
Strategic alignment refers to the active management of connections between organizational priorities and business processes~\citep{rosemann2015six}, which aims at facilitating effective actions to improve business performance.
Currently, various approaches to BPM assume that the corporate strategy is defined first and business processes are aligned with the respective strategic imperatives~\citep{DBLP:books/daglib/0031128}.
Blockchain technology challenges these approaches to strategic alignment.
For many companies, blockchains define a potential threat to their core business processes. For instance, the banking industry could see a major disintermediation based on blockchain-based payment services~\citep{guo2016blockchain}. Also lock-in effects~\citep{tassey2000standardization} might deteriorate when, for example, the banking service is not the banking network itself anymore, but only the interface to it. These developments could lead to business processes and business models being under strong influence of technological innovations outside of companies.
\subsection{Governance}
BPM governance refers to appropriate and transparent accountability in terms of roles, responsibilities, and decision processes for different BPM-related programs, projects, and operations~\citep{rosemann2015six}.
Currently, BPM as a management approach builds on the explicit definition of BPM-related roles and responsibilities with a focus on the internal operations of a company.
Blockchain technology might change governance towards a more externally oriented model of self-governance based on smart contracts.
Research on corporate governance investigates agency problems and mechanisms to provide effective incentives for intended behavior~\citep{shleifer1997survey}. Smart contracts can be used to establish new governance models as exemplified by The Decentralized Autonomous Organization (The DAO)~\footnote{\url{https://daohub.org}}. It is an important question in how far this idea of The DAO can be extended towards reducing the agency problem of management discretion or eventually eliminate the need for management altogether.
Furthermore, the revolutionary change suggested by The DAO for organization shows just how disruptive this technology can be, and whether similarly radical changes could apply to BPM.
\subsection{Information Technology}
BPM-related information technology subsumes all systems that support process execution, such as process-aware information systems and business process management systems. These systems typically assume central control over the process.
Blockchain technology enables novel ways of process execution, but several challenges in terms of security and privacy have to be considered. While the visibility of encrypted data on a blockchain is restricted, it is up to the participants in the process to ensure that these mechanisms are used according to their confidentiality requirements. Some of these requirements are currently being investigated in the financial industry\footnote{\url{https://gendal.me/2016/04/05/introducing-r3-corda-a-distributed-ledger-designed-for-financial-services/}}. Further challenges can be expected with the introduction of the General Data Protection Regulation\footnote{\url{http://eur-lex.europa.eu/legal-content/EN/TXT/?uri=uriserv:OJ.L\_{}.2016.119.01.0001.01.ENG}}.
It is also not clear, which new attack scenarios on blockchain networks might emerge~\citep{hurlburt2016might}. Therefore, guidelines for using private, public, or consortium-based blockchains are required~\citep{mougayar2016business}. It also has to be decided what types of smart contract and which cryptocurrency are allowed to be used in a corporate setting.
\subsection{People}
People in this context refers to all individuals, possibly in different roles, who engage with BPM~\citep{rosemann2015six}.
Currently, these are people who work as process analyst, process manager, process owner or in other process-related roles. The roles of these individuals are shaped by skills in the area of management, business analysis and requirements engineering.
In this capability area, the use of blockchain technology requires extensions of their skill sets.
New required skills relate to partner and contract management, software enginering, and cryptography.
Also, people have to be willing to design blockchain-based collaborations within the frame of existing regulations to enable adoption. This implies that research into blockchain-specific technology acceptance is needed, extending the established technology acceptance model~\citep{venkatesh2003user}.
\subsection{Culture}
Organizational culture is defined by the collective values of a group of people in an organization~\citep{rosemann2015six}.
Currently, BPM is discussed in relation to organizational culture~\citep{brocke2011culture} from a perspective that emphasizes an affinity with clan and hierarchy culture~\citep{stemberger-2017}. These cultural types are often found in the many companies that use BPM as an approach for documentation. Blockchains are likely to influence organizational culture towards a stronger emphasis on flexibility and an outward-looking perspective. In the competing values framework by~\cite{cameron2005diagnosing}, these aspects are associated with an adhocracy organizational culture. Furthermore, not only consequences of blockchain adoption have to be studied, but also antecedants. These include organizational factors that facilitate early and successful adoption.
\section{Introduction}
\label{sec:intro}
Business process management (BPM) is concerned with the design, execution, monitoring, and improvement of business processes. Systems that support the enactment and execution of processes
have extensively been used by companies to streamline and automate \textit{intra}-organizational processes. Yet, for \textit{inter}-organizational processes, challenges of joint design and a lack of mutual trust have hampered a broader uptake.
Emerging \textit{blockchain} technology has the potential to drastically change the environment in which inter-organizational processes are able to operate. Blockchains offer a way to execute processes in a trustworthy manner even in a network without any mutual trust between nodes. Key aspects are specific algorithms that lead to consensus among the nodes and market mechanisms that motivate the nodes to progress the network. Through these capabilities, this technology has the potential to shift the discourse in BPM research about how systems might enable the enactment, execution, monitoring or improvement of business process within or across business networks.
In this paper, we describe what we believe are the main new challenges and opportunities of blockchain technology for BPM. This leads to directions for research activities to investigate both challenges and opportunities.
Section~\ref{sec:background} provides a background on fundamental concepts of blockchain technology and an illustrative example of how this technology applies to business processes.
Section~\ref{sec:bpm-lc} focuses on the impact of blockchains on the traditional \textit{BPM lifecycle phases}~\citep{DBLP:books/daglib/0031128}. Section~\ref{sec:capab} goes beyond it and asks which impact blockchains might have on core capability areas of BPM~\citep{rosemann2015six}.
Section~\ref{Discu} summarizes this discussion by emphasizing seven future research directions.
\section{Blockchain Technology and the BPM Lifecycle}\label{sec:bpm-lc}
In this section, we discuss blockchain in relation to the traditional BPM lifecycle~\citep{DBLP:books/daglib/0031128} including the following phases:
identification, discovery, analysis, redesign, implementation, execution, monitoring, and adaptation.
Using the traditional BPM lifecycle as a framework of reference allows us to discuss many incremental changes that blockchains might provide.
\subsection{Identification}
Process identification is concerned with the high-level description and evaluation of a company from a process-oriented perspective, thus connecting strategic alignment with process improvement. Currently, identification is mostly approached from an inward-looking perspective~\citep{DBLP:books/daglib/0031128}.
Blockchain technology adds another relevant perspective for evaluating high-level processes in terms of the implied strengths, weaknesses, opportunities, and threats. For example, how can a company systematically identify the most suitable processes for blockchains or the most threatened ones?
Research is needed into how this perspective can be integrated into the identification phase. Because blockchains have affinity with the support of inter-organizational processes, process identification may need to encompass not only the needs of one organization, but broader known and even unknown partners.
\subsection{Discovery}
Process discovery refers to the collection of information about the current way a process operates and its representation as an \textit{as-is} process model. Currently, methods for process discovery are largely based on interviews, walkthroughs and documentation analysis, complemented with automated process discovery techniques over non-encrypted event logs generated by process-aware information systems~\citep{DBLP:books/sp/Aalst16}.
Blockchain technology defines new challenges for process discovery techniques: the information may be fragmented and encrypted; accounts and keys can change frequently; and payload data may be stored partly on-chain and partly off-chain.
For example, how can a company discover an overall process from blockchain transactions when these might not be logically related to a process identifier?
This fragmentation might require a repeated alignment of information from all relevant parties operating on the blockchain. Work on matching could represent a promising starting point to solve this problem~\citep{DBLP:books/daglib/0032976,gal2011uncertain,DBLP:conf/bpm/CayogluDDFGHKLLLMOSSTUWW13}. There is both the risk and opportunity of conducting process mining on blockchain data. An opportunity could involve establishing trust in how a process or a prospective business partner operates, while a risk is that other parties might be able to understand operational characteristics from blockchain transactions.
There are also opportunities for reverse engineering business processes, among others, from smart contracts.
\subsection{Analysis}
Process analysis refers to obtaining insights into issues relating to the way
a business process currently operates. Currently, the analysis of processes mostly builds on data that is available inside of organizations or from perceptions shared by internal and external process stakeholders~\citep{DBLP:books/daglib/0031128}.
Records of processes executed on the blockchain yield valuable information that can help to assess the case load, durations, frequencies of paths, parties involved, and correlations between unencrypted data items. These pieces of information can be used to discover processes, detect deviations, and conduct root cause analysis~\citep{DBLP:books/sp/Aalst16},
ranging from small groups of companies to an entire industry at large.
The question is which effort is required to bring the available blockchain transaction data into a format that permits such analysis.
\subsection{Redesign}
Process redesign deals with the systematic improvement of a process. Currently, approaches like redesign heuristics build on the assumption that there are recurring patterns of how a process can be improved~\citep{DBLP:journals/bise/VanwerschSVVGPM16}. Blockchain technology offers novel ways of improving specific business processes or resolving specific problems.
For instance, instead of involving a trustee to release a payment if an agreed condition is met, a buyer and a seller of a house might agree on a smart contract instead. The question is where blockchains can be applied for optimizing existing interactions and where new interaction patterns without a trusted central party can be established, potentially drawing on insights from related research on Web service interaction~\citep{barros2005service}.
A promising direction for developing blockchain-appropriate abstractions and heuristics may come from data-aware workflows~\citep{DBLP:conf/bpm/MarinHV12} and BPMN choreography diagrams~\citep{DBLP:journals/is/DeckerW11}. Both techniques combine two primary ingredients of blockchain, namely data and process, in a holistic manner that is well-suited for top-down design of cross-organizational processes.
It might also be beneficial to formulate blockchain-specific redesign heuristics that could mimic how
Incoterms~\citep{ramberg2011icc} define standardized interactions in international trade. Specific challenges for redesign include the joint engineering of blockchain processes between all parties involved, an ongoing problem for choreography design.
\subsection{Implementation}
Process implementation refers to the procedure of transforming a \textit{to-be} model into software components executing the business process. Currently, business processes are often implemented using process-aware information systems or business process management systems inside single organizations.
In this context, the question is how can the involved parties make sure that the implementation that they deploy on the blockchain supports their process as desired.
Some of the challenges regarding the transformation of a process model to blockchain artifacts are discussed by~\cite{Weber:2016:BPM}. Several ideas from earlier work on choreography can be reused in this new setting~\citep{DBLP:conf/caise/AalstW01,mendling2008ws,2008-Weber-IJBPIM,DBLP:journals/is/DeckerW11,RE-14:Protos,AAMAS-Comma-12}.
It has to be noted that choreographies have not been adopted by industry to a large extent yet. Despite this, they are especially helpful in inter-organizational settings, where it is not possible to control and monitor a complete process in a centralized fashion because of organizational borders~\citep{breu13}. To verify that contracts between choreography stakeholders have been fulfilled, a trust basis, which is not under control of a particular party, needs to be established. Blockchains may serve to establish this kind of trust between stakeholders.
An important engineering challenge on the implementation level is the identification and definition of abstractions for the design of blockchain-based business process execution.
Libraries and operations for engines are required, accompanied by modeling primitives and language extensions of BPMN. Software patterns and anti-patterns will be of good help to engineers designing blockchain-based processes.
There is also a need for new approaches for quality assurance, correctness, and verification, as well as
for new corresponding correctness criteria.
These can build on existing notions of compliance~\citep{DBLP:journals/toit/AalstDORV08}, reliability~\cite{DBLP:conf/saint/SattanathanTNMM08}, quality of services~\citep{DBLP:journals/tse/ZengBNDKC04} or data-aware workflow verification~\citep{DBLP:conf/pods/CalvaneseGM13}, but will have to go further in terms of consistency and consideration of potential payments. Furthermore, dynamic partner binding and rebinding is a challenge that requires attention. Process participants will have to find partners, either manually or automatically on dedicated marketplaces using dedicated look-up services. The property of inhabiting a certain role in a process might itself be a tradable asset. For example, a supplier might auction off the role of shipper to the highest bidder as part of the process. Finally, as more and more companies use blockchain, there will be a proliferation of smart contract templates available for use. Tools for finding templates appropriate for a given style of collaboration will be essential. All these characteristics emphasize the need for specific testing and verification approaches.
\subsection{Execution}
Execution refers to the instantiation of individual cases and their information-technological processing. Currently, such execution is facilitated by process-aware information systems or business process management systems~\citep{DBLP:books/daglib/0031128}.
For the actual execution of a process deployed on a blockchain following the method of~\cite{Weber:2016:BPM}, several differences with the traditional ways exist.
During the execution of an instance, messages between participants need to be passed as blockchain transactions to the smart contract; resulting messages need to be observed from the blocks in the blockchain. Both of these can be achieved by integrating blockchain technology directly with existing enterprise systems or through the use of dedicated integration components, such as the triggers suggested by~\cite{Weber:2016:BPM}. First prototypes like Caterpillar as a BPMS that build on blockchains are emerging~\cite{DBLP:conf/bpm/Lopez-PintadoGD17}.
The main challenge here involves ensuring correctness and security, especially when monetary assets are transferred using this technology.
\subsection{Monitoring}
Process monitoring refers to collecting events of process executions, displaying them in an understandable way, and triggering alerts and escalation in cases where undesired behavior is observed.
Currently, such process execution data is recorded by systems that support process execution~\citep{DBLP:books/daglib/0031128}.
First, we face issues in terms of data fragmentation and encryption as in the analysis phase. For example, the data on the blockchain alone will likely not be enough to monitor the process, but require an integration with local off-chain data.
Once such tracing in place, the global view of the process can be monitored independently by each involved party.
This provides a suitable basis for continuous conformance and compliance checking and monitoring of service-level agreements.
Second, based on monitoring data exchanged via the blockchain, it is possible to verify if a process instance meets the original process model and the contractual obligations of all involved process stakeholders. For this, blockchain technology can be exploited to store the process execution data and handoffs between process participants. Notably, this is even possible without the usage of smart contracts, i.e., in a first-generation blockchain like the one operated by Bitcoin~\citep{PSHW17}.
\subsection{Adaptation and Evolution}
Runtime adaptation refers to the concept of changing the process during execution. In traditional approaches, this can for instance be achieved by allowing participants in a process to change the model during its execution~\citep{DBLP:books/daglib/0030179}.
Interacting partners might take a defensive stance in order to avoid certain types of adaptation.
As discussed by~\cite{Weber:2016:BPM}, blockchain can be used to enforce conformance with the model, so that participants can rely on the joint model being followed.
In such a setting, adaptation is by default something to be \emph{avoided:} if a participant can change the model, this could be used to gain an unfair advantage over the other participants. For instance, the rules of retrieving cryptocurrency from an escrow account could be changed or the terms of payment.
In this setting, process adaptation must strictly adhere to defined paths for it, e.g., any change to a deployed smart contract may require a transaction signed by all participants.
In contrast, the method proposed by~\citep{PSHW17} allows runtime adaptation, but assumes that relevant participants monitor the execution and react if a change is undesired.
If smart contracts enforce the process, there are also problems arising in relation to evolution: new smart contracts need to be deployed to reflect changes to a new version of the process model. Porting running instances from an old version to a new one would require effective coordination mechanisms involving all participants. Some challenges for choreographies are summarized by~\cite{DBLP:journals/is/FdhilaIRR15}.
\section{Seven Future Research Directions}\label{Discu}
Blockchains will fundamentally shift how we deal with transactions in general, and therefore how organizations manage their business processes within their network. Our discussion of challenges in relation to the BPM lifecycle and beyond points to seven major future research directions. For some of them we expect viable insights to emerge sooner, for others later. The order loosely reflects how soon such insights might appear.
\begin{enumerate}
\item Developing a diverse set of \textit{execution and monitoring systems} on blockchain. Research in this area will have to demonstrate the feasibility of using blockchains for process-aware information systems. Among others, design science and algorithm engineering will be required here. Insights from software engineering and distributed systems will be informative.
\item Devising new \textit{methods for analysis and engineering} business processes based on blockchain technology. Research in this topic area will have to investigate how blockchain-based processes can be efficiently specified and deployed. Among others, formal research methods and design science will be required to study this topic. Insights from software engineering and database research will be informative here.
\item \textit{Redesigning processes} to leverage the opportunities granted by blockchain. Research in this context will have to investigate how blockchain may allow re-imagining specific processes and the collaboration with external stakeholders. The whole area of choreographies may be re-vitalized by this technology. Among others, design science will be required here. Insights from operations management and organizational science will be informative.
\item Defining appropriate methods for \textit{evolution and adaptation}. Research in this area will have to investigate the potential guarantees that can be made for certain types of evolution and adaptation. Among others, formal research methods will be required here. Insights from theoretical computer science and verification will be informative.
\item Developing techniques for identifying, discovering, and analyzing relevant processes for the \textit{adoption} of blockchain technology. Research on this topic will have to investigate which characteristics of blockchain as a technology best meet requirements of specific processes. Among others, empirical research methods and design science will be required. Insights from management science and innovation research will be informative here.
\item Understanding the \textit{impact on strategy and governance} of blockchains, in particular regarding new business and governance models enabled by revolutionary innovation based on blockchain. Research in this topic area will have to study which processes in an enterprise setting could be onoverganized differently using blockchain and which consequences this brings. Among others, empirical research methods will be required to investigate this topic. Insights from organizational science and business research will be informative.
\item Investigating the \textit{culture shift} towards openness in the management and execution of business processes, and on hiring as well as upskilling people as needed. Research in this topic area will have to investigate how corporate culture changes with the introduction of blockchains, and in how far this differs from the adoption of other technologies. Among others, empirical methods will be required for research in this area. Insights from organizational science and business research will be informative.
\end{enumerate}
The BPM and the Information Systems community have a unique opportunity to help shape this fundamental shift towards a distributed, trustworthy infrastructure to promote inter-organizational processes. With this paper we aim to provide clarity, focus, and impetus for the research challenges that are upon us.
\bibliographystyle{ACM-Reference-Format}
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1,116,691,497,584 | arxiv |
\subsection{Analysis of Algorithm~\ref{algo:wam2}}
For a fixed scaling $s: E\to [0,1]$, let $\ell$ be the number of eigenvalues
of the normalized Laplacian $N^{s}$ of $G^{s}$ that are in the range $(0,\zeta]$.
Let $\ell_0 = \ceil{ \frac{\gamma_2 k^2}{(1-\zeta)^2} }$, which, by Theorem~\ref{thm:fewsmall},
is an upper bound
on $\ell$ when the graph $G^{s}$ is $\gamma_2$-almost regular.
Then we consider the following potential function: %
\begin{align}\label{eq:potential}
\Upsilon(s) \stackrel{\mathrm{def}}{=}
\begin{cases}
\zeta^{\ell_0 - \ell} \det\nolimits_{\ell}(N^{s}) & \ell\leq \ell_0 \\
\det_{\ell_0}(N^{s}) & \ell > \ell_0.
\end{cases}
\end{align}
Thus, $\Upsilon(s)$ is always a product of exactly $\ell_0$ numbers between $(0,\zeta]$.
It is not hard to see the following alternative form of $\Upsilon(s)$, which will be helpful for analyzing it.
\begin{proposition}\label{prop:altups}
Let $\lambda_1,\ldots,\lambda_{\ell_0}$ be the smallest $\ell_0$ nonzero eigenvalues of $N^{s}$.
Then
\begin{align*}
\Upsilon(s) = \prod_{i=1}^{\ell_0} \min\setof{\lambda_i,\zeta}.
\end{align*}
\end{proposition}
\begin{corollary}[of Proposition~\ref{prop:altups}]
\label{cor:altups}
For any $\ell' \leq \ell_0$, we have
$\Upsilon(s) \leq \det_{\ell'}(N^{s}) \zeta^{\ell_0 - \ell'}$.
\end{corollary}
Since we always maintain the $\gamma_2$-regularity,
we will also need to consider the determinant of the degree matrix,
$\det\nolimits_{+}(D)$, as a potential function.
We show that starting from an almost-regular graph,
$\det\nolimits_{+}(D)$ can only decrease at a limited speed when edges are downscaled.
The proof of the lemma below is deferred to Appendix~\ref{sec:aphared}.
\begin{lemma}\label{lem:decmost}
Let $G = (V,E)$ be a $\gamma$-almost regular graph.
Let $s: E\to (0,1]$ be a strictly positive scaling,
and let $S \stackrel{\mathrm{def}}{=} \prod_{e\in E} \frac{1}{s_e}$.
Then we have
\begin{align*}
\det\nolimits_{+}(D^s) \geq \kh{1 - \frac{2 \gamma k}{n}}^{2 \log S} \det\nolimits_{+}(D).
\end{align*}
\end{lemma}
As a result of the above lemma,
we then show that the while loop at Lines~\ref{line:while1s}-\ref{line:while1t}
of $\WTM$ will terminate after a bounded number of iterations.
The proof of the following lemma is also deferred to Appendix~\ref{sec:aphared}.
\begin{lemma}\label{lem:tt1}
Suppose the input to $\WTM$ satisfies that
$G^{s_0}$ is $\gamma_1/2$-regular
and $\gamma_2 \geq 16 \gamma_1$.
Consider the first $t$ iterations of the outermost while loop.
Then the total number of iterations executed so far by the inner while loop
at Lines~\ref{line:while1s}-\ref{line:while1t} is at most $2t$.
\end{lemma}
We now prove our key lemma, which shows that our potential
$\Upsilon$ will decrease at least as fast as a geometric series with rate bounded away from $1$.
As a result, the smallest nonzero eigenvalue of the normalized Laplacian must also
decrease at a steady rate.
This implies that eventually we will be able to find a $\rho < 1$ at Line~\ref{line:rhofind},
and therefore terminate the outermost while loop in finite time.
\begin{lemma}
Suppose the input to $\WTM$ satisfies that
$G^{s_0}$ is $\gamma_1/2$-regular,
$\gamma_2 \geq 16 \gamma_1$,
$\psi_2 \geq 1024 \psi_1$,
and
\begin{align*}
\psi_2 \geq \frac{1024 \gamma_2^2}{(1 - \zeta)^2}.
\end{align*}
Consider the first $t$ iterations of the outermost while loop,
and let $s^1$ be the scaling obtained at the end of the $t^{\mathrm{th}}$ iteration.
Then
\begin{align*}
\frac{\Upsilon(s^1)}{\Upsilon(s^0)}
\leq
\kh{ 1 - \frac{\psi_2 k^2}{16 n^2} }^t.
\end{align*}
\end{lemma}
\begin{proof}
We first show that the potential reduces by a certain amount at Line~\ref{line:wackH}.
\begin{claim}\label{lem:line14}
Let the scalings before and after one execution of Line~\ref{line:wackH} be $s$ and $s'$ respectively.
Then we have
\begin{align*}
\frac{\Upsilon(s')}{\Upsilon(s)} \leq 1 - \frac{\psi_2 k^2}{4 n^2}.
\end{align*}
\end{claim}
\begin{proof}[Proof of Claim~\ref{lem:line14}]
Let $\ell$ be the number of eigenvalues of $N^s$ that are between $(0,\zeta]$.
Since $G^{s}$ is $\gamma_2$-almost regular, we have $\ell \leq \ell_0$ by Theorem~\ref{thm:fewsmall}.
Thus $\Upsilon(s) = \det\nolimits_{\ell}(N^s) \zeta^{\ell_0 - \ell}$.
Also, by Corollary~\ref{cor:altups}, we have
$\Upsilon(s') \leq \det\nolimits_{\ell}(N^{s'}) \zeta^{\ell_0 - \ell}$.
Therefore, it suffices to show that
\begin{align}
\frac{\det\nolimits_{\ell}(N^{s'})}{\det\nolimits_{\ell}(N^{s})} \leq 1 - \frac{\psi_2 k^2}{4 n^2}.
\end{align}
We then do so by considering the variational characterization $\det\nolimits_{\ell}$
from Lemma~\ref{lem:potentialfrac}.
Let $X = (D^s)^{\dag/2} F$ and $X' = (D^{s'})^{\dag/2} F'$
be the optimal matrix that minimizes the variational characterization
of $\det\nolimits_{\ell}(N^s)$ and $\det\nolimits_{\ell}(N^{s'})$ respectively.
Here $F$ and $F'$ are both $nk\times \ell$ matrices whose columns
are bottom nonzero eigenvectors of $N^{s}$ and $N^{s'}$ respectively.
It then suffices to show that the fraction in the characterization
reduces by much after the execution of Line~\ref{line:wackH},
even if we do not switch from $X$ to $X'$,
since switching to the latter can only decrease the potential function.
Namely, since by the optimality of $X'$ we have
\begin{align*}
\frac{ \det\kh{ X^T L^{s'} X } }{ \det\kh{X^T D^{s'} X} } \geq
\frac{ \det\kh{ (X')^T L^{s'} (X') } }{ \det\kh{(X')^T D^{s'} (X')} },
\end{align*}
it suffices to show
\begin{align}\label{eq:sufficesss}
\kh{ \frac{ \det\kh{ X^T L^{s'} X } }{ \det\kh{X^T D^{s'} X} } } \Big /
\kh{ \frac{ \det\kh{ X^T L^{s} X } }{ \det\kh{X^T D^{s} X} } } \leq
1 - \frac{\psi_2 k^2}{4 n^2}.
\end{align}
For the numerator we have by matrix determinant lemma
\begin{align}
\frac{ \det\kh{ X^T L^{s'} X } }
{ \det\kh{ X^T L^{s} X } }
= & 1 - \frac{3}{4}\cdot (b_{uv}^s)^T X \kh{X^T L^s X}^{-1} X^T (b_{uv}^s) \notag \\
= & 1 - \frac{3}{4}\cdot (b_{uv}^s)^T \kh{D^{s}}^{\dag/2} F
\kh{ F^T \kh{ D^{s} }^{\dag/2} L^s \kh{D^{s}}^{\dag/2} F}^{-1} F^T \kh{D^{s}}^{\dag/2} b_{uv}^s \notag \\
= & 1 - \frac{3}{4}\cdot H_{\zeta}^{s}(u,v)
\leq 1 - \frac{3\psi_2 k^2}{4n^2},
\label{eq:detnum}
\end{align}
where the last equality follows from that
$F$'s columns are eigenvectors corresponding to all eigenvalues of $N^{s}$ that are between $(0,\zeta]$.
For the denominator, we have, again by matrix determinant lemma,
\begin{align*}
\frac{ \det\kh{ X^T D^{s'} X } }
{ \det\kh{ X^T D^{s} X } } \geq &
{ 1 - \frac{3}{4}(e_{u\la v}^{s})^T X (X^T D^s X)^{-1} X^T e_{u\la v}^{s} -
\frac{3}{4}(e_{v\la u}^{s})^T X (X^T D^s X)^{-1} X^T e_{v\la u}^{s} } \\ = &
{ 1 - \frac{3}{4}(e_{u\la v}^{s})^T X X^T e_{u\la v}^{s} -
\frac{3}{4}(e_{v\la u}^{s})^T X X^T e_{v\la u}^{s} }
\qquad
\text{(as $X^T D^{s} X = F^T F = I$)} \\ = &
1 - \frac{3}{4}(e_{u\la v}^{s})^T (D^{s})^{\dag/2} F F^T (D^{s})^{\dag/2} (e_{u\la v}^{s}) -
\frac{3}{4}(e_{v\la u}^{s})^T (D^{s})^{\dag/2} F F^T (D^{s})^{\dag/2} (e_{v\la u}^{s}) \\ = &
1 - \frac{3}{4}\norm{ F^T(u) (D^{s}_u)^{\dag/2} \phi_{uv} }^2 -
\frac{3}{4}\norm{ F^T(v) (D^{s}_v)^{\dag/2} \phi_{uv} }^2,
\end{align*}
where in the last equality we let $F(u)$ be the $u^{th}$ row block of $F$.
Since $G^{s}$ is $\gamma_2$-almost regular,
we have
$\norm{(D^s_u)^{\dag/2} \phi_{uv}}^2 \leq \frac{\gamma_2\cdot k}{n}$ and
$\norm{(D^s_v)^{\dag/2} \phi_{uv}}^2 \leq \frac{\gamma_2\cdot k}{n}$.
By Lemma~\ref{lem:fsu2}, we have
$\lambda_{\mathrm{max}}(F(u) F(u)^T) \leq \frac{\gamma_2 k}{(1 - \zeta)^2 n}$, and
$\lambda_{\mathrm{max}}(F(v) F(v)^T) \leq \frac{\gamma_2 k}{(1 - \zeta)^2 n}$, and therefore
\begin{align*}
& \norm{F^T(u) (D^s_u)^{\dag/2} \phi_{uv}}^2 \leq \frac{\gamma_2^2\cdot k^2}{(1 - \zeta)^2 n^2} \\
& \norm{F^T(v) (D^s_v)^{\dag/2} \phi_{uv}}^2 \leq \frac{\gamma_2^2\cdot k^2}{(1 - \zeta)^2 n^2}.
\end{align*}
This give us
\begin{align}\label{eq:detdom}
\frac{ \det\kh{ X^T D^{s'} X } }
{ \det\kh{ X^T D^{s} X } } \geq & 1 - \frac{3\gamma_2^2\cdot k^2}{2(1 - \zeta)^2 n^2}.
\end{align}
(\ref{eq:detnum}),(\ref{eq:detdom}) coupled with $\psi_2 \geq \frac{1024 \gamma_2^2}{(1 - \zeta)^2}$
imply~(\ref{eq:sufficesss}), which finishes the proof of the claim.
\end{proof}
We next show that during the first while loop at Lines~\ref{line:while1s}-\ref{line:while1t},
the potential function cannot increase much.
\begin{claim}\label{claim:firstwhile}
Consider a fixed iteration of the outermost while loop.
Suppose in this iteration of the outer while loop, the total number of iterations executed by the first inner while loop
at Lines~\ref{line:while1s}-\ref{line:while1t} is $t_1$.
Let $s,s'$ be the scalings
before and after the while loop respectively.
Then
\begin{align*}
\frac{\Upsilon(s')}{\Upsilon(s)}
\leq
\kh{ 1 - \frac{96 \gamma_2^2 k^2}{(1-\zeta)^2 n^2} }^{-t_1}.
\end{align*}
\end{claim}
\begin{proof}[Proof of Claim~\ref{claim:firstwhile}]
We know that before Line~\ref{line:wackH}, the graph is $\gamma_2$-almost regular.
Then at Line~\ref{line:wackH} we halve the scale of a single edge,
so the graph $G^{s}$ is $4\gamma_2$-almost regular afterwards.
We also know that after the while loop terminates at Line~\ref{line:while1t},
the graph $G^{s'}$ is $\gamma_2$-almost regular, by the termination condition of the while loop.
Now consider the following process for obtaining $s'$ from $s$.
\begin{enumerate}
\item While $s\neq s'$:
\begin{enumerate}
\item For each $(u,v)\in E$ such that $s_{uv} > s'_{uv}$,
let $s_{uv} \gets s_{uv} / 2$.
\end{enumerate}
\end{enumerate}
We first argue, by induction, that at the end of each iteration of the while loop of the above process,
$G^s$ is $4\gamma_2$-almost regular.
As noted above, initially, the graph $G^s$ is $4\gamma_2$-almost regular.
For the induction step,
consider a fixed iteration, and let $F$ be the edges $(u,v)$ for which $s_{uv} > s'_{uv}$.
Since we decrease the weights of all edges in $F$ by a same factor,
from the point view of leverage scores, it is equivalent
to increase the weights of all other edges by a same multiple.
Therefore the leverage scores of edges in $F$ can only decrease,
and thus can be at most $\frac{4\gamma_2 k}{n}$ after this iteration.
As for edges $(u,v)$ not in $F$, they satisfy $s_{uv} = s'_{uv}$.
We know that in $G^{s'}$ their leverage scores are at most $\frac{4\gamma_2 k}{n}$,
as $G^{s'}$ is $\gamma_2$-almost regular.
Since $G^{s}$'s weights always dominate those of $G^s$, their leverage scores in $G^{s}$
can only be smaller, and thus at most $\frac{4\gamma_2 k}{n}$ as well.
We then argue that at any point of the process, the graph is $16\gamma_2$-almost regular.
This follows by noting that at any point of the algorithm, $G^{s}$'s edge weights
are within a factor $4$ of those at the end of previous iteration, and those
at the end of the current iteration.
We now show that in this process, each time we let $s_{uv}\gets s_{uv}/2$,
the potential function increases by at most $(1 - \frac{400\gamma_2^2 k^2}{(1 - \zeta)^2 n^2})^{-1}$,
which implies the statement of this claim.
Let $q,q'$ be the scalings before and after one execution of $s_{uv}\gets s_{uv}/2$.
Let $\ell$ be the number of nonzero eigenvalues of $N^q$ that are between $(0,\zeta]$,
and let $\ell_1 = \min(\ell,\ell_0)$.
Then we have $\Upsilon(q) = \det\nolimits_{\ell_1}(N^{q}) \zeta^{\ell_0 - \ell_1}$,
and, by Corollary~\ref{cor:altups},
$\Upsilon(q') \leq \det\nolimits_{\ell_1}(N^{q'}) \zeta^{\ell_0 - \ell_1}$.
Thus it suffices to show
\begin{align}\label{eq:32}
\frac{\det\nolimits_{\ell_1}(N^{q'})}{\det\nolimits_{\ell_1}(N^{q})} \leq
\kh{ 1 - \frac{400\gamma_2^2 k^2}{(1 - \zeta)^2n^2} }^{-1}.
\end{align}
As in our proof of Claim~\ref{lem:line14}, we once again consider the variational characterization
in Lemma~\ref{lem:potentialfrac}.
Let $X = (D^{q})^{\dag/2} F$ where $F\in\mathbb{R}^{nk\times \ell_1}$'s columns are
nonzero bottom eigenvectors of $N^{q}$.
The numerator of the characterization can only decrease.
For the denominator, we have by matrix determinant lemma
\begin{align*}
\frac{ \det\kh{ X^T D^{q'} X } }
{ \det\kh{ X^T D^{q} X } } \geq &
{ 1 - \frac{3}{4}(e_{u\la v}^{q})^T X (X^T D^q X)^{-1} X^T e_{u\la v}^{q} -
\frac{3}{4}(e_{v\la u}^{q})^T X (X^T D^q X)^{-1} X^T e_{v\la u}^{q} } \\ = &
{ 1 - \frac{3}{4}(e_{u\la v}^{q})^T X X^T e_{u\la v}^{q} -
\frac{3}{4}(e_{v\la u}^{q})^T X X^T e_{v\la u}^{q} }
\qquad
\text{(as $X^T D^{q} X = F^T F = I$)} \\ = &
1 - \frac{3}{4}(e_{u\la v}^{q})^T (D^{q})^{\dag/2} F F^T (D^{q})^{\dag/2} (e_{u\la v}^{q}) -
\frac{3}{4}(e_{v\la u}^{q})^T (D^{q})^{\dag/2} F F^T (D^{q})^{\dag/2} (e_{v\la u}^{q}) \\ = &
1 - \frac{3}{4}\norm{ F^T(u) (D^{q}_u)^{\dag/2} \phi_{uv} }^2 -
\frac{3}{4}\norm{ F^T(v) (D^{q}_v)^{\dag/2} \phi_{uv} }^2,
\end{align*}
where in the last equality we let $F(u)$ be the $u^{th}$ row block of $F$.
Since $G^{q}$ is $16\gamma_2$-almost regular,
we have
$\norm{(D^q_u)^{\dag/2} \phi_{uv}}^2 \leq \frac{16\gamma_2\cdot k}{n}$ and
$\norm{(D^q_v)^{\dag/2} \phi_{uv}}^2 \leq \frac{16\gamma_2\cdot k}{n}$.
By Lemma~\ref{lem:fsu2}, we have
$\lambda_{\mathrm{max}}(F(u) F(u)^T) \leq \frac{16\gamma_2 k}{(1 - \zeta)^2 n}$, and
$\lambda_{\mathrm{max}}(F(v) F(v)^T) \leq \frac{16\gamma_2 k}{(1 - \zeta)^2 n}$, and therefore
\begin{align*}
& \norm{F^T(u) (D^q_u)^{\dag/2} \phi_{uv}}^2 \leq \frac{256\gamma_2^2\cdot k^2}{(1 - \zeta)^2 n^2} \\
& \norm{F^T(v) (D^q_v)^{\dag/2} \phi_{uv}}^2 \leq \frac{256\gamma_2^2\cdot k^2}{(1 - \zeta)^2 n^2}.
\end{align*}
This give us
\begin{align}\label{eq:detdom1}
\frac{ \det\kh{ X^T D^{q'} X } }
{ \det\kh{ X^T D^{q} X } } \geq & 1 - \frac{400 \gamma_2^2\cdot k^2}{(1 - \zeta)^2 n^2}.
\end{align}
This then implies~(\ref{eq:32}) and finishes the proof of the claim.
\end{proof}
We next show that during the second while loop at Lines~\ref{line:while2s}-\ref{line:while2t},
the potential function cannot increase much either.
\begin{claim}\label{claim:secondwhile}
Consider a fixed iteration of the outermost while loop.
Suppose in this iteration of the outer while loop, the total number of iterations executed by the first inner while loop
at Lines~\ref{line:while2s}-\ref{line:while2t} is $t_2$.
Let $s,s'$ be the scalings
before and after the while loop respectively.
Then
\begin{align*}
\frac{\Upsilon(s')}{\Upsilon(s)}
\leq
\kh{ 1 + \frac{3\psi_1 k^2}{n^2} }^{t_2}.
\end{align*}
\end{claim}
\begin{proof}[Proof of Claim~\ref{claim:secondwhile}]
Let $q,q'$ be the scalings before and after one execution of $s_{uv}\gets s_{uv}\cdot 2$.
Let $\ell$ be the number of nonzero eigenvalues of $N^q$ between $(0,\zeta]$,
and let $\ell_2 = \min(\ell,\ell_0)$.
Then we have $\Upsilon(q) = \det\nolimits_{\ell_2}(N^{q}) \zeta^{\ell_0 - \ell_2}$,
and, by Corollary~\ref{cor:altups},
$\Upsilon(q') \leq \det\nolimits_{\ell_2}(N^{q'}) \zeta^{\ell_0 - \ell_2}$.
Thus it suffices to show
\begin{align}\label{eq:54}
\frac{\det\nolimits_{\ell_2}(N^{q'})}{\det\nolimits_{\ell_2}(N^{q})} \leq 1 + \frac{3\psi_1 k^2}{n^2}.
\end{align}
As in our proofs of Claims~\ref{lem:line14},~\ref{claim:firstwhile},
we also consider the variational characterization
in Lemma~\ref{lem:potentialfrac}.
Let $X = (D^{q})^{\dag/2} F$ where $F\in\mathbb{R}^{nk\times \ell_2}$'s columns are
nonzero bottom eigenvectors of $N^{q}$.
The denominator of the characterization can only increase.
For the numerator we have by matrix determinant lemma
\begin{align}
\frac{ \det\kh{ X^T L^{q'} X } }
{ \det\kh{ X^T L^{q} X } }
= & 1 + 3 (b_{uv}^q)^T X \kh{X^T L^q X}^{-1} X^T (b_{uv}^q) \notag \\
= & 1 + 3 (b_{uv}^q)^T \kh{D^{q}}^{\dag/2} F
\kh{ F^T \kh{ D^{q} }^{\dag/2} L^q \kh{D^{q}}^{\dag/2} F}^{-1} F^T \kh{D^{q}}^{\dag/2} b_{uv}^q \notag \\
\leq & 1 + 3 H_{\zeta}^{q}(u,v)
\leq 1 + \frac{3 \psi_1 k^2}{n^2}.
\label{eq:det54}
\end{align}
This implies~(\ref{eq:54}) and finishes the proof the claim.
\end{proof}
For the first $t$ iterations of the outermost while loop,
let $t_1$ be the total number of iterations executed by the first inner while loop,
and $t_2$ be the total number of iterations executed by the second inner while loop.
Then we have
$t_1 \leq 2t$ by Lemma~\ref{lem:tt1},
and $t_2 \leq t + t_1 \leq 3t$.
Therefore, by Claims~\ref{lem:line14},~\ref{claim:firstwhile},~\ref{claim:secondwhile},
we have
\begin{align*}
\frac{\Upsilon(s^1)}{\Upsilon(s^{0})}
\leq \kh{1 - \frac{\psi_2 k^2}{4 n^2}}^{t}
\kh{ 1 - \frac{400 \gamma_2^2 k^2}{(1-\zeta)^2 n^2} }^{-2t}
\kh{ 1 + \frac{3\psi_1 k^2}{n^2} }^{3t}
\leq \kh{ 1 - \frac{\psi_2 k^2}{16 n^2} }^t,
\end{align*}
where the last inequality follows from
$\psi_2 \geq 1024 \psi_1$ and
$\psi_2 \geq \frac{1024 \gamma_2^2}{(1 - \zeta)^2}$.
\end{proof}
Notice that by design, after $\WTM$ terminates,
each edge $(u,v)$ with $s_{uv} < s_{uv}^0$ satisfies either
$R^s(u,v) \geq \frac{\gamma_1 k}{n}$ or
$H^s_{\zeta}(u,v) \geq \frac{\psi_1 k^2}{n^2}$.
We show that the total number of such edges is small.
\begin{lemma}\label{lem:ne1}
After $\WTM$ terminates, the number of edges $(u,v)$ with $s_{uv} < s_{uv}^0$
such that $R^s(u,v) \geq \frac{\gamma_1 k}{n}$ is at most
$\frac{ n\cdot \rank{D^{s}} }{\gamma_1 k}$.
\end{lemma}
\begin{proof}
Since $R^{s}(u,v)$'s are leverage scores, we have
\begin{align*}
\sum_{(u,v)\in E} R^{s}(u,v) = \rank{D_s}.
\end{align*}
Then the desired bound follows.
\end{proof}
\begin{lemma}\label{lem:ne2}
After $\WTM$ terminates, the number of edges $(u,v)$ with $s_{uv} < s_{uv}^0$
such that $H^s_{\zeta}(u,v) \geq \frac{\psi_1 k^2}{n^2}$ is at most
$\frac{ n^2 \gamma_2 }{\psi_1 (1 - \zeta)^2}$.
\end{lemma}
\begin{proof}
Let $\ell$ be the number of eigenvalues of $N^{s}$ between $(0,\zeta]$.
Let $\lambda_1,\ldots,\lambda_{\ell}$ be all eigenvalues between $(0,\zeta]$,
and let $f_1,\ldots,f_{\ell}$ be a set of orthonormal eigenvectors.
By Theorem~\ref{thm:fewsmall}, $\ell \leq \frac{\gamma_2 k^2}{(1 - \zeta)^2}$.
Also we have by Proposition~\ref{prop:ellavg} that
\begin{align*}
\sum_{(u,v)\in E} H^{s}_{\zeta}(u,v) = \ell.
\end{align*}
Thus our desired bound follows.
\end{proof}
\subsection{Analysis of Algorithm~\ref{algo:hed}}
Our analysis of Algorithm~\ref{algo:hed} will mostly focus on
bounding the total number of deleted edges at Line~\ref{line:delsmall}.
Since we only delete edges with $(s_{uv}/s'_{uv})^2 \leq \rho < 1$,
by the termination condition of the second inner while loop of $\WTM$ we know that
each deleted edge satisfies either $R^s(u,v)\geq \frac{\gamma_1 k}{n}$ or
$H^s_{\zeta}(u,v) \geq \frac{\psi_1 k^2}{n^2}$,
where $\gamma_1 = \gamma / 16$ and $\psi_1 = \psi/1024$.
Thus we will show that the numbers of both types of edges are small.
Consider fixing a while loop iteration of $\HED$ where we go the ``else'' branch.
Let $s'$ be the scaling we obtain after we invoke $\HARGD$ at Line~\ref{line:argd},
$s$ be the scaling returned by $\WTM$, and $\hat{s}$ be the scaling obtained
after we delete the small weight edges at Line~\ref{line:delsmall}
(but before we reset the edge weights at Line~\ref{line:resetw}).
Then by an (almost) identical proof to that of Lemma~\ref{lem:nerd},
we have:
\begin{lemma}\label{lem:sdddd}
The number of edges $(u,v)\in E$ with $(s_{uv}/s'_{uv})^2 \leq \rho$
such that $R^{s}(u,v)\geq \frac{\gamma_1\cdot k}{n}$ is at most
\begin{align*}
\frac{2n}{\gamma_1 k} \kh{ \rank{D^{s'}} - \rank{D^{\shat}} }.
\end{align*}
\end{lemma}
This means that we can charge the number of deleted edges at Line~\ref{line:delsmall} that
satisfy $R^s(u,v)\geq \frac{\gamma_1\cdot k}{n}$ to the rank change of $D$.
We will then bound the number of deleted edges that satisfy
$H^{s}_{\zeta}(u,v)\geq \frac{\psi_1 k^2}{n^2}$ by considering the number of nontrivial (Definition~\ref{def:nontrivial})
zero eigenvalues of the normalized Laplacian,
which,
by Theorem~\ref{thm:fewsmall}, is at most
$2 \gamma k^2$ when the graph is $\gamma$-almost regular.
To that end,
for a normalized Laplacian matrix $N$,
let $\eta(N)$ denote the number of nontrivial
zero eigenvalues of $N$.
Additionally, let $s''$ denote the scaling we obtain after we invoke $\HARGD$ at Line~\ref{line:argd} in the {\em next}
iteration of the while loop.
The following lemma characterizes how $\eta(N)$ changes after a while loop iteration.
\begin{lemma}\label{lem:etachange}
We have
\begin{align}\label{eq:etachange}
\eta(N^{s''}) \geq \eta(N^{s'}) + 1 -
\floor{ \frac{2 \gamma k}{n} \kh{\rank{D^{s'}} - \rank{D^{s''}}} + \sqrt{\frac{1}{16nk}}}.
\end{align}
\end{lemma}
Before proving the lemma, we first give the proof of Theorem~\ref{thm:hared}.
\begin{proof}[Proof of Theorem~\ref{thm:hared}]
Since we always maintain $\gamma$-almost regularity,
the number of nontrivial zero eigenvalues can be at most
$2\gamma k^2$. Therefore by Lemma~\ref{lem:etachange}
the total number of iterations of the while loop is at most
$6\gamma k^2$.
Thus, the algorithm terminates in finite time.
Also, by the termination condition, the resulting graph $G^s$ must be
a $(\gamma,\zeta,\psi)$-almost regular expander.
By Lemma~\ref{lem:ne2}, the total number of deleted edges
with $H^s_{\zeta}(u,v) \geq \frac{\psi_1 k^2}{n^2}$ is at most
\begin{align*}
\frac{n^2 \gamma}{\psi_1 (1 - \zeta)^2}\cdot 6\gamma k^2 \leq
\frac{10000 n^2 \gamma^2 k^2}{\psi (1 - \zeta)^2}.
\end{align*}
By Lemma~\ref{lem:sdddd} and Theorem~\ref{thm:hargd},
the total number of deleted edges with $R^{s}(u,v)\geq \frac{\gamma_1\cdot k}{n}$,
plus those deleted by $\HARGD$, is at most
\begin{align*}
\frac{8n}{(\gamma/32) k}\cdot nk = \frac{256 n^2}{\gamma}.
\end{align*}
These two bounds coupled with Lemmas~\ref{lem:ne1},~\ref{lem:ne2} imply
the desired bound on the number of rescaled edges, and thus finish the proof. %
\end{proof}
It then remains to proof Lemma~\ref{lem:etachange}, for which we need the following lemma.
\begin{lemma}\label{lem:231}
We have
\begin{align*}
\lambda_{\mathrm{max}}\kh{ \kh{D^{\dag/2} L^{\hat{s}} D^{\dag/2}}^{\dag} } \leq \frac{1}{128nk^3 \gamma \cdot \rho}.
\end{align*}
\end{lemma}
\begin{proof}
Let $\shat_{\min} = \min\setof{\shat_{uv}: \shat_{uv}\neq 0}$.
Since $\shat$ is obtained from $s$ by zeroing out the edges $(u,v)$ with $(s_{uv}/s'_{uv})^2\leq \rho$,
and no $(s_{uv}/s'_{uv})^2$ is in the range $(\rho,\rho\cdot \alpha)$,
we have
\begin{align*}
\shat_{\min}^2 \geq \rho\cdot \alpha\cdot (s'_{\min})^2 \geq 128nk^3\gamma \alpha_2 \rho,
\end{align*}
where $s'_{\min} = \min_{s'_{uv} > 0} s'_{uv}$ is defined at Line~\ref{line:smin},
and the last inequality follows from $\alpha \geq 128nk^3\gamma\cdot \alpha_2 / (s'_{\min})^2$.
Let $F = \setof{(u,v)\in E: \shat_{uv} > 0}$. Then we have
\begin{align*}
D^{\dag/2} L^{\shat} D^{\dag/2} \pgeq \shat_{\min}^2 D^{\dag/2} L^{F} D^{\dag/2}
\pgeq 128nk^3\gamma \alpha_2 \rho D^{\dag/2} L^{F} D^{\dag/2},
\end{align*}
and therefore
\begin{align}\label{eq:4nka2}
\kh{ D^{\dag/2} L^{\shat} D^{\dag/2} }^{\dag} \pleq
\frac{1}{128nk^3\gamma\alpha_2 \rho} \kh{ D^{\dag/2} L^{F} D^{\dag/2} }^{\dag}.
\end{align}
By definition,
\begin{align*}
\alpha_{2} = \max_{F'\subseteq E} \lambda_{\mathrm{max}}\kh{\kh{D^{\dag/2} L^{F'} D^{\dag/2}}^{\dag}}
\geq \lambda_{\mathrm{max}}\kh{ \kh{ D^{\dag/2} L^{F} D^{\dag/2} }^{\dag} }.
\end{align*}
This coupled with~(\ref{eq:4nka2}) implies that
\begin{align*}
\kh{ D^{\dag/2} L^{\shat} D^{\dag/2} }^{\dag} \pleq
\frac{1}{128nk^3\gamma \rho} I
\end{align*}
as desired.
\end{proof}
\begin{proof}[Proof of Lemma~\ref{lem:etachange}]
Since $s$ and $s'$ have the same support, we have
$\eta(N^{s'}) = \eta(N^s)$.
Let $z = \eta(N^s)$.
Let $0 = \lambda_1 = \ldots = \lambda_{z} < \lambda_{z+1} \leq \rho$ be the smallest $z+1$ nontrivial eigenvalues
of $N^s$, and let $f_1,\ldots,f_{z+1}$ be a corresponding set of orthonormal, nontrivial eigenvectors.
Since $G^{\hat{s}}$ is a subgraph of $G^{s}$, $f_1,\ldots,f_{z}$ are also zero eigenvectors of
$(D^s)^{\dag/2} L^{\hat{s}} (D^s)^{\dag/2}$.
We first show that $f_{z+1}$ is not in the range of $(D^s)^{\dag/2} L^{\hat{s}} (D^s)^{\dag/2}$.
In particular, we will show that the projection of $f_{z+1}$ onto the range of
$(D^s)^{\dag/2} L^{\hat{s}} (D^s)^{\dag/2}$, i.e.
\begin{align*}
\kh{ (D^s)^{\dag/2} L^{\hat{s}} (D^s)^{\dag/2} }^{\dag/2}
\kh{ (D^s)^{\dag/2} L^{\hat{s}} (D^s)^{\dag/2} }^{1/2} f_{z+1},
\end{align*}
has small norm.
First note that
\begin{align}
\norm{\kh{ (D^s)^{\dag/2} L^{\hat{s}} (D^s)^{\dag/2} }^{1/2} f_{z+1}}^2 =
& f_{z+1}^T (D^s)^{\dag/2} L^{\hat{s}} (D^s)^{\dag/2} f_{z+1} \notag \\ \leq
& f_{z+1}^T (D^s)^{\dag/2} L^{s} (D^s)^{\dag/2} f_{z+1} \notag \\ =
& \lambda_{z+1} \leq \rho.
\label{eq:232}
\end{align}
Then
\begin{align}
& \norm{ \kh{ (D^s)^{\dag/2} L^{\hat{s}} (D^s)^{\dag/2} }^{\dag/2}
\kh{ (D^s)^{\dag/2} L^{\hat{s}} (D^s)^{\dag/2} }^{1/2} f_{z+1} }^2 \notag \\ =
& f_{z+1}^T \kh{ (D^s)^{\dag/2} L^{\hat{s}} (D^s)^{\dag/2} }^{1/2}
\kh{ (D^s)^{\dag/2} L^{\hat{s}} (D^s)^{\dag/2} }^{\dag}
\kh{ (D^s)^{\dag/2} L^{\hat{s}} (D^s)^{\dag/2} }^{1/2} f_{z+1} \notag \\
\leq &
\lambda_{\mathrm{max}}\kh{ \kh{ (D^s)^{\dag/2} L^{\hat{s}} (D^s)^{\dag/2} }^{\dag} }
\norm{ \kh{ (D^s)^{\dag/2} L^{\hat{s}} (D^s)^{\dag/2} }^{1/2} f_{z+1} }^2 \notag \\
\leq &
\lambda_{\mathrm{max}}\kh{ \kh{ D^{\dag/2} L^{\hat{s}} D^{\dag/2} }^{\dag} }
\norm{ \kh{ (D^s)^{\dag/2} L^{\hat{s}} (D^s)^{\dag/2} }^{1/2} f_{z+1} }^2 \notag \\
\leq &
\frac{1}{128nk^3\gamma\rho} \cdot \rho = \frac{1}{128nk^3\gamma},
\label{eq:firstproj}
\end{align}
where the last inequality follows from Lemma~\ref{lem:231} and~(\ref{eq:232}).
Define a matrix $\mathcal{F}\in \mathbb{R}^{(z+1)\times nk}$ by
\begin{align*}
\mathcal{F} :=
\begin{pmatrix}
f_1^T \\
\vdots \\
f_{z+1}^T
\end{pmatrix} \in \mathbb{R}^{(z+1)\times nk}
\end{align*}
and a function $F: V\to \mathbb{R}^{(z+1)\times k}$ by
\begin{align*}
F(u) =
\begin{pmatrix}
(f_1)_u^T \\
\vdots \\
(f_{z+1})_u^T
\end{pmatrix} \in \mathbb{R}^{(z+1)\times nk}
\end{align*}
Since $G^{s}$ is $\gamma$-almost regular,
we have by Lemma~\ref{lem:fsu2} that $\lambda_{\mathrm{max}}(F(u) F(u)^T)\leq\frac{2\gamma k}{n}$
and by Theorem~\ref{thm:fewsmall} that $z+1 \leq 2\gamma k^2$.
Now consider projecting the rows of $\mathcal{F}$ twice: (i) project each row onto
the null space of $(D^{s})^{\dag/2} L^{\shat} (D^{s})^{\dag/2}$, and then (ii)
project each row onto the range of $(D^{s})^{\dag/2} D^{s''}(D^{s})^{\dag/2}$.
Specifically, let $\Pi_1,\Pi_2\in\mathbb{R}^{nk\times nk}$ be the projection matrix
onto the range of $(D^{s})^{\dag/2} L^{\shat} (D^{s})^{\dag/2}$ and $(D^{s})^{\dag/2} D^{s''}(D^{s})^{\dag/2}$
respectively,
and let $\Pi_1^\bot,\Pi_2^\bot$ be the projection matrix onto the maximal subspace orthogonal to
the range of $\Pi_1$ and $\Pi_2$ respectively.
Then we consider the matrix $\mathcal{F}':= \mathcal{F} \Pi_1^{\bot} \Pi_2$.
In order to prove the lemma, it suffices to show that
the rank of $\mathcal{F}'$ is at least
\begin{align}\label{eq:rankp}
z + 1 -
\floor{ \frac{2 \gamma k}{n} \kh{\rank{D^{s'}} - \rank{D^{s''}}} + \sqrt{\frac{1}{16nk}}},
\end{align}
because this will imply that the number of zero eigenvalues
of $(D^{s})^{\dag/2} L^{s''} (D^{s})^{\dag/2}$ (and hence of $N^{s''}$) is at least
$nk - \rank{D^{s''}} + (\ref{eq:rankp})$.
Since
$$\mathcal{F}' (\mathcal{F}')^{T} = \mathcal{F} \Pi_1^{\bot} \Pi_2 \Pi_1^{\bot} \mathcal{F}^T \pleq
\mathcal{F} \Pi_1^{\bot} \Pi_1^{\bot} \mathcal{F}^T
\pleq \mathcal{F} \mathcal{F}^T = I_{(z+1)\times (z+1)},$$
we know that all singular values of $\mathcal{F}'$ is at most $1$.
Thus, it suffices to show that
\begin{align*}
\trace{\mathcal{F}' (\mathcal{F}')^T} \geq
z + 1 -
\kh{ \frac{2 \gamma k}{n} \kh{\rank{D^{s'}} - \rank{D^{s''}}} + \sqrt{\frac{1}{16nk}} },
\end{align*}
which will imply that the number of nonzero singular values of $\mathcal{F}'$
is at least~(\ref{eq:rankp}).
To this end, let us first write $\trace{\mathcal{F} \Pi_2 \mathcal{F}^T}$ as
\begin{align}
& \trace{\mathcal{F} \Pi_2 \mathcal{F}^T} \notag \\ =
& \trace{\mathcal{F} (\Pi_1^{\bot} + \Pi_1) \Pi_2 (\Pi_1^{\bot} + \Pi_1)\mathcal{F}^T} \notag \\
= & \trace{\mathcal{F} \Pi_1^{\bot} \Pi_2 \Pi_1^{\bot} \mathcal{F}^T}
+ \trace{\mathcal{F} \Pi_1 \Pi_2 \mathcal{F}^T} +
\trace{\mathcal{F}\Pi_1^{\bot} \Pi_2 \Pi_1 \mathcal{F}^T} \notag \\
\leq & \trace{\mathcal{F} \Pi_1^{\bot} \Pi_2 \Pi_1^{\bot} \mathcal{F}^T} \notag\\
+ & \sqrt{ \trace{\mathcal{F} \Pi_1 \mathcal{F}^T} \trace{\mathcal{F} \Pi_2 \mathcal{F}^T} }
+ \sqrt{\trace{\mathcal{F}\Pi_1^{\bot} \Pi_2 \Pi_1^{\bot} \mathcal{F}^T} \trace{\mathcal{F} \Pi_1 \mathcal{F}^T}},
\label{eq:trineq}
\end{align}
where the last equality follows from expanding,
and the last inequality follows from Cauchy-Schwarz.
This combined with~(\ref{eq:firstproj}) gives that
\begin{align}
\trace{\mathcal{F}' (\mathcal{F}')^T} =
& \trace{\mathcal{F} \Pi_1^{\bot} \Pi_2 \Pi_1^{\bot} \mathcal{F}^T} \notag \\ \geq
& \trace{\mathcal{F} \Pi_2 \mathcal{F}^T} - 2 \sqrt{\frac{z+1}{128nk^3\gamma}} \notag\\ \geq
& \trace{\mathcal{F} \Pi_2 \mathcal{F}^T} - 2 \sqrt{\frac{2\gamma k^2}{128nk^3\gamma}} \notag \\ =
& \trace{\mathcal{F} \Pi_2 \mathcal{F}^T} - \sqrt{\frac{1}{16nk}}. \label{eq:pi1}
\end{align}
Using the fact that $\lambda_{\mathrm{max}}(F(u) F(u)^T) \leq \frac{2\gamma k}{n}$,
we have
\begin{align*}
\trace{\mathcal{F} \Pi_2^{\bot} \mathcal{F}^T}
\leq \frac{2 \gamma k}{n} \kh{\rank{D^{s'}} - \rank{D^{s''}}},
\end{align*}
and hence
\begin{align}
\trace{\mathcal{F} \Pi_2 \mathcal{F}^T} =
& \trace{\mathcal{F} \mathcal{F}^{T}} - \trace{\mathcal{F} \Pi_2^{\bot} \mathcal{F}^T} \notag\\ \geq
& z+1 - \frac{2 \gamma k}{n} \kh{\rank{D^{s'}} - \rank{D^{s''}}}. \label{eq:pi2}
\end{align}
Combining~(\ref{eq:pi1}) and~(\ref{eq:pi2}) finishes the proof of the lemma.
\end{proof}
\subsection{Our results}
In this paper, we initiate the study of {\em weighted} graph sparsification by linear sketching.
To state our results, we need to introduce some notation first.
Let $w\in\mathbb{R}_{\geq 0}^{\binom{n}{2}}$ denote the weights of the edges
of the input graph, where $w_e = 0$ means that there is no edge in the edge slot $e$.
A {\em linear sketch of $N = N(n)$ measurements} consists of a (random) {\em sketching matrix}
$\Phi \in \mathbb{R}^{N\times \binom{n}{2}}$, and a (randomized) {\em recovery algorithm} $\mathcal{A}$
that takes as input $\Phi w\in\mathbb{R}^{N}$ and outputs a sparsifier of the graph with edge weights $w$.
Note that, by definition, the linear sketch is {\em non-adaptive}.
We focus on a natural class of linear sketches that we call {\em incidence sketches},
in which each row of the sketching matrix $\Phi$ is supported\footnote{Recall that
the support of a vector is the set of indices at which it is non-zero.}
on edges incident on a {\em single} vertex
(which could be different for different rows).
This class captures all linear sketches
that are implementable in a
distributed computing setting,
where the edges are stored across $n$ machines such that machine $i$ has all edges incident on the $i^{\mathrm{th}}$ vertex
(a.k.a.{\em simultaneous communication model}).
Moreover, it also covers {\em all} aforementioned linear sketches used in previous works
for unweighted cut sparsification~\cite{AhnGM12},
spectral sparsification~\cite{KapralovLMMS17,KapralovMMMNST20}, and spanner computation~\cite{FiltserKN21}.
We now present our results for computing these three kinds of sparsifiers in weighted graphs.
When describing our results, we use $w_{\max}$ and $w_{\min}$ to denote the largest
and the smallest non-zero edge weights, respectively, and always assume $w_{\max} \geq 1 \geq w_{\min}$.
We also write $\Otil(\cdot)$ to hide $\mathrm{polylog}(n, \epsilon^{-1}, \frac{w_{\max}}{w_{\min}})$ factors.
\paragraph{Weighted cut sparsification.}{
We design an incidence sketch with a near-linear number of measurements
for computing a $(1 + \epsilon)$-cut sparsifier of a weighted graph.
\begin{theorem}[Algorithm for weighted cut sparsification]
\label{thm:cutalg}
For any $\epsilon\in(0,1)$,
there exists an incidence sketch
with random sketching matrix $\Phi_1\in\mathbb{R}^{N_1\times \binom{n}{2}}$ satisfying
$N_1\leq \Otil(n \epsilon^{-3})$ and
a recovery algorithm $\mathcal{A}_1$, such that for any $w\in\mathbb{R}_{\geq 0}^{\binom{n}{2}}$,
$\mathcal{A}_1(\Phi_1 w)$ returns, with probability $1 - \frac{1}{\mathrm{poly}(n)}$,
a $(1+\epsilon)$-cut sparsifier of the graph with edge weights $w$.
\end{theorem}
Thus, we achieve a similar performance as the linear sketch for unweighted graphs given in~\cite{AhnGM12},
which uses $O(n \epsilon^{-2} \mathrm{poly} (\log n))$ measurements.
It is well known that even to detect the connectivity of a graph, $\Omega(n)$ linear measurements
are needed. Therefore, our upper bound in Theorem~\ref{thm:cutalg} is nearly optimal
in $n$. %
Similar to~\cite{Indyk06,AhnGM12,KapralovLMMS14,KapralovMMMNST20,FiltserKN21},
by using Nisan's well known pseudorandom number generator~\cite{Nisan92},
we can turn our linear sketching algorithm to a low space streaming algorithm.
\begin{corollary}[of Theorem~\ref{thm:cutalg}]
There is a single pass turnstile streaming algorithm with $\Otil(n \epsilon^{-3})$ space
that, at any given point of the stream, recovers a $(1+\epsilon)$-cut sparsifier of the current graph
with high probability.
\end{corollary}
Note that in the turnstile model, the stream consists of arbitrary edge weight updates.
}
\paragraph{Weighted spectral sparsification.}{
We design an incidence sketch with about $n^{6/5}$ measurements for computing
a $(1 + \epsilon)$-spectral sparsifier of a weighted graph.
\begin{theorem}[Algorithm for weighted spectral sparsification]
\label{thm:spectralalg}
For any $\epsilon\in(0,1)$,
there exists an incidence sketch
with random sketching matrix $\Phi_2\in\mathbb{R}^{N_2\times \binom{n}{2}}$ satisfying
$N_2\leq \Otil(n^{6/5} \epsilon^{-4})$ and
a recovery algorithm $\mathcal{A}_2$, such that for any $w\in\mathbb{R}_{\geq 0}^{\binom{n}{2}}$,
$\mathcal{A}_2(\Phi_2 w)$ returns, with probability $1 - \frac{1}{\mathrm{poly}(n)}$,
a $(1+\epsilon)$-spectral sparsifier of the graph with edge weights $w$.
\end{theorem}
Similar to the cut sparsification case, we have the following corollary:
\begin{corollary}[of Theorem~\ref{thm:spectralalg}]
There is a single pass turnstile streaming algorithm with $\Otil(n^{6/5} \epsilon^{-4})$ space
that, at any given point of the stream, recovers a $(1+\epsilon)$-spectral sparsifier of the current graph
with high probability.
\end{corollary}
We complement this result by showing that a superlinear number of measurements are indeed necessary
for any incidence sketch to recover some $O(1)$-spectral sparsifier.
\begin{restatable}[Lower bound for weighted spectral sparsification]{theorem}{spectrallb}
\label{thm:spectrallb}
There exist constants $\epsilon,\delta \in (0,1)$
such that
any incidence sketch of $N$ measurements that computes a $(1+\epsilon)$-spectral sparsifier
with probability $\geq 1 - \delta$ on any $w$
must satisfy $N \geq n^{21/20-o(1)}$.
\end{restatable}
Note that this is in sharp contrast to the unweighted case, where a near-linear number of incidence sketch measurements
are sufficient for computing an $O(1)$-spectral sparsifier~\cite{KapralovLMMS17,KapralovMMMNST20}.
Theorem~\ref{thm:spectrallb} also draws a distinction between
spectral sparsification and cut sparsification, %
as for the latter a near-linear number of measurements are enough even in the weighted case
(by Theorem~\ref{thm:cutalg}).
}
\paragraph{Weighted spanner computation.}{
We first show that any $o(n^2)$ linear measurements can only recover
a spanner of stretch that in general depends {\em linearly} on $\frac{w_{\max}}{w_{\min}}$.
This differs fundamentally from the case of cut or spectral sparsification,
where we can recover an $O(1)$-sparsifier, whose error is completely independent of $\frac{w_{\max}}{w_{\min}}$,
using a sublinear-in-$n^2$ number of measurements.
Specifically, we prove the following proposition, whose proof appears in Appendix~\ref{sec:appendspanner}.
\begin{proposition}\label{prop:spanner}
Any linear sketch (not necessarily an incidence sketch) of $N$ measurements that
computes an $o(\frac{w_{\max}}{w_{\min}})$-spanner with probability $\geq .9$ on any $w$
must satisfy $N \geq \Omega(n^2)$.
\end{proposition}
This proposition is a consequence of that edge weights are {\em proportional} to the edge lengths.
More specifically,
consider a complete graph where we weight a uniformly random edge by $w_{\min}$
and all other $\binom{n}{2} - 1$ edges by $w_{\max}$.
Then, while we can ignore the $w_{\min}$-weight edge in an $O(1)$-cut or spectral sparsifier,
we have to include it in any $o(\frac{w_{\max}}{w_{\min}})$-spanner,
since otherwise the shortest path length between its two endpoints would have been blown up by
at least a factor of $\frac{2w_{\max}}{w_{\min}}$.
Now note that, as the weight of this edge is smaller than all other edges, in order to find it
we have to essentially recover all entries of the edge weight
vector $w$, which inevitably requires
$\Omega(n^2)$ linear measurements\footnote{In our actual proof of the proposition,
we have to add some random Gaussian noise to each edge's weight for the lower bound to carry out.}.
In light of the above proposition, we turn our focus to graphs with $\frac{w_{\max}}{w_{\min}} = O(1)$,
and study the stretch's optimal dependence on $n$.
On such graphs, the approach in~\cite{FiltserKN21} is able to obtain
the following tradeoff between the stretch of the spanner and the number of linear measurements needed.
\begin{theorem}[Algorithm for weighted spanner computation~\cite{FiltserKN21}]
\label{thm:spectralspanner}
For any constant $\alpha\in[0,1]$,
there exists an incidence sketch
with random sketching matrix $\Phi_3\in\mathbb{R}^{N_3\times \binom{n}{2}}$ satisfying
$N_3\leq \Otil(n^{1+\alpha})$ and
a recovery algorithm $\mathcal{A}_3$, such that for any
$w\in\mathbb{R}_{\geq 0}^{\binom{n}{2}}$ with
$\frac{w_{\max}}{w_{\min}} \leq O(1)$,
$\mathcal{A}_3(\Phi_3 w)$ returns, with probability $1 - \frac{1}{\mathrm{poly}(n)}$,
an $\Otil(n^{\frac{2}{3}(1 - \alpha)})$-spanner of the graph with edge weights $w$.
\end{theorem}
\cite{FiltserKN21} also conjectured that on unweighted graphs, to obtain a spanner of stretch
$O(n^{\frac{2}{3}-\epsilon})$ for any constant $\epsilon > 0$, a superlinear number of measurements are needed
for any linear sketch (in other words, the tradeoff is optimal at $\alpha = 0$).
We make progress on this question by showing that this is indeed true
for a natural class of linear sketches (i.e. incidence sketches)
on ``almost'' unweighted graphs (i.e. those with $\frac{w_{\max}}{w_{\min}} = O(1)$).
In fact, we show that in such a setting, the tradeoff obtained in the above theorem is
{\em optimal for all} $\alpha < 1/10$.
\begin{restatable}[Lower bound for weighted spanner computation]{theorem}{spannerlb}
\label{thm:spannerlb}
For any constant $\alpha \in (0,1/10)$,
there exist constants $C\geq 1, \delta \in (0,1)$
such that
any incidence sketch of $N$ measurements that computes
an $o(n^{\frac{2}{3}(1 - \alpha)})$-spanner with probability $\geq 1 - \delta$
on any $w$ with $\frac{w_{\max}}{w_{\min}} \leq C$ must satisfy $N\geq n^{1+\alpha-o(1)}$.
\end{restatable}
}
\subsection{Roadmap}
The rest of this paper is structured as follows.
We start by giving an overview of the techniques used in proving our main results.
Specifically, Section~\ref{sec:ovcutalg} gives an overview
of our algorithms for weighted cut and spectral sparsification.
Then in Section~\ref{sec:ovspectrallb}, we give an overview
of our lower bound for weighted spectral sparsification.
Note that we do {\em not} give a separate overview
of our lower bound for weighted spanner computation,
because the ideas are similar
to the ones described in Section~\ref{sec:ovspectrallb}. %
Section~\ref{sec:preli} contains some
preliminaries that we will rely on throughout. Then Sections~\ref{sec:algocut} and~\ref{sec:algospectral} present
in detail our algorithms and their analysis for
weighted cut and spectral sparsification, respectively.
The remaining Sections~\ref{sec:hdg}-\ref{sec:spanner} are
devoted to proving our lower bounds for weighted spectral
sparsification and spanner computation,
both of which rely on a number of new tools
for analyzing certain {\em matrix-weighted} graphs.
Section~\ref{sec:hdg} first defines such matrix-weighted graphs and sets up some notation.
Then Sections~\ref{sec:fewsmall},\ref{sec:hargd},\ref{sec:hared},\ref{sec:hdevs} present the new tools we develop
for analyzing them.
Finally, Sections~\ref{sec:lbspectral},\ref{sec:spanner}
prove our lower bounds for weighted spectral sparsification
and spanner computation, respectively.
\section{Introduction}\label{sec:intro}
\input{intro}
\section{Overview of weighted cut and spectral sparsification algorithms}\label{sec:ovcutalg}
\input{ovalg}
\section{Overview of lower bound for weighted spectral sparsification}\label{sec:ovspectrallb}
\input{ovsplb}
\section{Preliminaries}\label{sec:preli}
\input{pre}
\section{A linear sketching algorithm for weighted cut sparsification}\label{sec:algocut}
\input{techniques.tex}
\section{A linear sketching algorithm for weighted spectral sparsification}\label{sec:algospectral}
\input{weight-spectral.tex}
\section{Preliminaries on matrix-weighted graphs}\label{sec:hdg}
\input{hdg}
\section{Almost regular graphs have only few small eigenvalues}\label{sec:fewsmall}
\input{fewsmall2}
\section{Almost regular graph decomposition}\label{sec:hargd}
\input{hargd}
\section{Almost regular expander decomposition}\label{sec:hared}
\input{hared}
\section{Expanders are preserved under vertex sampling}\label{sec:hdevs}
\subsection{Warm-up: ordinary expanders are preserved under vertex sampling}\label{sec:expwarmup}
\input{ordexp}
\subsection{Matrix-weighted expanders are preserved under vertex sampling}\label{sec:mtexpsamp}
\input{exp}
\section{A lower bound for weighted spectral sparsification}\label{sec:lbspectral}
\input{lbspectral}
\section{A lower bound for weighted spanner computation}\label{sec:spanner}
\input{spanner}
\subsection{Overview of the algorithm for weighted cut sparsification}\label{sec:cutoverview}
\paragraph{Recap of the unweighted cut sparsification algorithm in~\cite{AhnGM12}.}{
At a high-level, the approach taken is to reduce cut sparsification to (repeatedly) recovering a spanning forest
of a subgraph of the input graph, obtained by sampling edges
{\em uniformly} at some rate $p\in(0,1)$ known beforehand. This task is then further reduced to the task of sampling an edge connecting $S$ to $\bar{S}$ for an arbitrary subset $S$ of vertices, as this can be used to create a spanning forest by growing connected components.
Now to implement this latter task, in the sketching phase, we apply an $\ell_0$-sampler sketch to the incidence vector
of each vertex $u$ (i.e. each column of the edge-vertex incidence matrix) in the {\em sub-sampled graph}.
Then in the recovery phase, in order to recover an edge going out of a vertex set $S$,
we add up the sketches of the vertices inside $S$. By linearity, this summed sketch is taken
over the sum of the incidence vectors of vertices inside $S$, and the latter contains
exactly the edges going out of $S$, since the edges inside cancel out.
As a result, we can recover an edge going out of $S$, and create a spanning forest of the sub-sampled graph.
Note that this approach crucially utilizes the fact that the edges are sampled uniformly at a rate
that is known {\em beforehand}.
This means that we can sample all $\binom{n}{2}$ edge slots beforehand, and apply the linear sketch only
to the sampled edge slots.
}
\paragraph{Our approach for weighted cut sparsification.}{
We also reduce the task to recovering a spanning forest in a sub-sampled graph.
However, the latter graph is now obtained by sampling edges {\em non-uniformly}.
Specifically, we need to recover a spanning forest in a subgraph obtained by
sampling each edge $e$ with probability $\min\setof{w_e p,1}$ for some
parameter $p\in(0,1)$ that is known beforehand.
Therefore, in order to apply the idea as in the unweighted case, we will now need to design
a variant of $\ell_0$-sampler that, given a vector $x\in\mathbb{R}_{\geq 0}^{N}$
and a parameter $p\in(0,1)$, recovers a nonzero entry of $x$ after each entry $i\in[N]$
is sampled with probability $\min\setof{x_i p, 1}$. %
We call such a sampler ``weighted edge sampler''.
Note that the edge weights are {\em not} known to us beforehand, so we cannot
sample the edge slots with our desired probabilities as in the unweighted graph case.
We instead build such a weighted edge sampler using
a {\em rejection sampling} process, in which we sample edges uniformly at $\Otil(1)$ geometric rates,
but use {\em $\ell_1$-samplers}
to try recovering edges at each rate, and only output a recovered edge $e$ if the sampling rate $\approx w_e p$.
We then show that with high probability, we can efficiently find a desired edge.
Roughly,
our analysis involves proving that
there exists a geometric rate $q$ such that,
after uniformly sampling edges at rate $q$,
the total weight of edges $e$ satisfying $w_e p\approx q$ accounts for a large portion
of that of all sampled edges. As a result, by using a few independent $\ell_1$-samplers,
we can find one such edge with high probability.
}
\subsection{Overview of the algorithm for weighted spectral sparsification}\label{sec:spectraloverview}
\newcommand\sq{\mathrm{sq}}
As in previous linear sketches for unweighted graphs~\cite{KapralovLMMS14,KapralovMMMNST20},
the key task is to recover edges with $\Omegatil(1)$ effective resistances (or in weighted case,
$\Omegatil(1)$-leverage scores), which we refer to as {\em heavy edges}.
The high-level idea used in previous works is to (i) compute, for each vertex pair $s,t$,
a set of vertex potentials $x_{s,t}\in\mathbb{R}^{n}$ induced by an electrical flow from $s$ to $t$,
and then (ii) apply an $\ell_2$-heavy hitter to $B_G x_{s,t} \in\mathbb{R}^{\binom{n}{2}}$ to try
recovering the edge $(s,t)$, where $B_G$ is the edge-vertex incidence matrix of $G$
(see Section~\ref{sec:gmat}).
They achieve (i) by simulating an iterative refinement process in~\cite{LiMP13}.
To achieve (ii),
they make a key observation that
\begin{align*}
\norm{ B_G x_{s,t} }_2^2 = x_{s,t}^T B_G^T B_G x_{s,t} = x_{s,t}^T L_G x_{s,t}
\end{align*}
is the energy of $x_{s,t}$,
and the entry of $B_G x_{s,t}$ indexed by edge $(s,t)$
is $(B_G x_{s,t})_{(s,t)} = b_{s,t}^T x = x_s - x_t$.
Therefore by the energy minimization characterization of effective resistances
(Fact~\ref{fact:energymin}),
whenever the effective resistance between $s,t$ is $b_{s,t}^T L_G^{\dag} b_{s,t} \geq \Omegatil(1)$,
we have
$$ (B_G x_{s,t})_{(s,t)}^2 \geq \Omegatil(1) \norm{ B_G x_{s,t} }_2^2, $$
and hence the entry $(s,t)$ is an $\ell_2$-heavy hitter.
However, when the graph is weighted,
we are only allowed to access the graph through linear measurements on its weight vector $w_G$.
As a result, we can only apply $\ell_2$-heavy hitters
to $W_G B_G x_{s,t}$, whose squared $\ell_2$-norm is $x_{s,t}^T B_G^T W_G^2 B_G x_{s,t}$.
Now notice that $B_G^T W_G^2 B_G$ is the Laplacian matrix of a ``squared'' graph (call it $G^{\sq}$),
which has the same edges as $G$, but whose edges are weighted by $w_e^2$ as opposed to $w_e$.
Therefore, we will be recovering
edges that are heavy in $G^{\sq}$ instead of in $G$ if we apply the same approach as in previous works.
Unfortunately, a heavy edge in $G$ is not in general heavy in $G^{\sq}$, since the energy on the edges
with very large weights will blow up when we square the edge weights
(i.e. $w_e^2(x_u - x_v)^2 \gg w_e(x_u - x_v)^2$),
and hence make the total energy
grow unboundedly.
\begin{figure}[ht!]
\centering
\begin{tikzpicture}
[scale = 0.5]%
\node (1)[style={circle, fill=black, draw=black,}] at (1.757, 1.757) {};
\node (3)[style={circle, fill=black, draw=black,}] at (1.757, 10.243){};
\node (5)[style={circle, fill=black, draw=black,}] at (10.243, 10.243){};
\node (7)[style={circle, fill=black, draw=black,}] at (10.243, 1.757){};
\node (2)[style={circle, fill=black, draw=black,}] at (0, 6){};
\node (4)[style={circle, fill=black, draw=black,}] at (6, 12){};
\node (6)[style={circle, fill=black, draw=black,}] at (12, 6){};
\node (8)[style={circle, fill=black, draw=black,}] at (6, 0){};
\node (9) at (1,1) {$S_{n^{4/5}-1}$};
\node (10) at (-1.2,6) {$S_0$};
\node (11) at (1,11) {$S_1$};
\node (12) at (6,13.2) {$S_2$};
\node (13) at (11.2,11.2) {$S_{\frac{n^{4/5}}{2}-1}$};
\node (14) at (13.8,6) {$S_{\frac{n^{4/5}}{2}}$};
\node (15) at (11.5,0.5) {$S_{\frac{n^{4/5}}{2}+1}$};
\node (16) at (6,-1.2) {$S_{\frac{n^{4/5}}{2}+2}$};
\node (17) at (8.3,11.3) {$\dots$};
\node (18) at (3.7,0.7) {$\dots$};
\draw[-,decoration={zigzag,segment length=4}] (1) [out=135,in=270] edge[decorate] (2);
\draw[-,decoration={zigzag,segment length=4}] (2) [out=90,in=225] edge[decorate] (3);
\draw[-,decoration={zigzag,segment length=4}] (3) [out=45,in=180] edge[decorate] (4);
\draw[-,decoration={zigzag,segment length=4}] (5) [out=315,in=90] edge[decorate] (6);
\draw[-,decoration={zigzag,segment length=4}] (6) [out=270,in=45] edge[decorate] (7);
\draw[-,decoration={zigzag,segment length=4}] (7) [out=225,in=0] edge[decorate] (8);
\draw[-,draw=red] (2) [out=0,in=180] to (6);
\end{tikzpicture}
\caption{A block cycle graph on $n$ vertices.
Each $S_i$ represents a block of $n^{1/5}$ vertices connected by a clique
and each zigzag represents the edges of a complete bipartite graph between adjacent blocks.
The red edge represents the crossing edge.
All edges along the cycle have weights $n^{2/5}$, and the crossing edge has weight $1$.}
\label{fig:block}
\end{figure}
To see an intuitive example, suppose $G$ is a ``block cycle graph'' on $n$ vertices whose edges are
generated as follows (see also Figure~\ref{fig:block}):
\begin{enumerate}
\item Partition the vertices into $n^{4/5}$ blocks $S_0,\ldots,S_{n^{4/5}-1}$,
each with $n^{1/5}$ vertices.
\item For each $0\leq i< n^{4/5}$, add on $S_i$
a complete graph of $n^{1/5}$ vertices with edge weights $n^{2/5}$, i.e. $n^{2/5} K_{n^{1/5}}$.
\item For each $0\leq i< n^{4/5}$, add on $(S_i,S_{i+1})$
a complete bipartite graph of $2 n^{1/5}$ vertices with edge weights $n^{2/5}$
and bipartition $(S_i,S_{i+1})$\footnote{Note that we consider $i+1$ as $0$ when $i=n^{4/5}-1$.},
i.e. $n^{2/5} K_{n^{1/5},n^{1/5}}$.
\item Finally, add a ``crossing edge'' $e^*$ of weight $1$ between a randomly chosen vertex pair $s,t$.
\end{enumerate}
We note that, in this construction, the crossing edge $e^*$ spans $\Omega(n^{4/5})$ consecutive blocks,
Note that, typically, the crossing edge $e^*$ spans $\Omega(n^{4/5})$ consecutive blocks,
and therefore has effective resistance (and also leverage score) $\Omega(1)$.
\begin{proposition}\label{prop:largeer}
If $s\in S_i, t\in S_j$ such that $\min\setof{|i-j|,n^{4/5}-|i-j|} \geq \Omega(n^{4/5})$,
then the effective resistance of $e^*$ satisfies $r_{e^*} \geq \Omega(1)$.
\end{proposition}
\begin{proof}%
Let $s,t$ be the endpoints of the crossing edge $e^*$.
By the energy minimization characterization of effective resistances
(Fact~\ref{fact:energymin}), it suffices to show that there is a set of vertex potentials
whose normalized energy with respect to $s,t$ is $O(1)$. %
Specifically, consider the set of potentials $x\in\mathbb{R}^n$ such that
$x_u = \frac{i}{n^{4/5}}$ for all $u\in S_i$.
Then we have $x_s - x_t = \Theta(1)$,
and the total energy is $\sum_{e=(u,v)} w_e(x_u - x_v)^2 = n^{6/5}\cdot n^{2/5} (\Theta(n^{-4/5}))^2 + \Theta(1)
= \Theta(1)$, as desired.
\end{proof}
However, in the squared graph $G^{\sq}$, all edge weights along the cycle
are blown up by a factor of $n^{2/5}$, and thus $e^*$ only has
leverage score $O(n^{-2/5})$ in $G^{\sq}$. To recover in a vector $x$ every entry with $\ell_2$-contribution
$\geq n^{-2/5} \norm{x}_2^2$, one will need an $\Omega(n^{-2/5})$ factor
blowup in the number of linear measurements, resulting in a total of $n^{7/5}$ measurements needed to
recover $e^*$.
We can in fact improve the number of linear measurements needed for recovering $e^*$
to $\Otil(n^{6/5})$ using the {\em vertex sampling} trick, an idea first used in~\cite{FiltserKN21} for
sketching spanners.
Namely, consider sampling a vertex set $C\subset V$ by including each vertex with probability $n^{-1/5}/100$,
and looking at the vertex-induced subgraph $G^{\sq}[C]$.
Then one can show that, conditioned on $e^*\in G^{\sq}[C]$,
with constant probability, the two endpoints of $e^*$ will be disconnected in $G^{\sq}[C]\setminus e^*$.
As a result, the leverage score of $e^*$ becomes $1$ in $G^{\sq}[C]$,
and we can recover $e^*$ by recovering heavy edges in $G^{\sq}[C]$, which, as will show,
can be done using $\Otil(|C|) \approx \Otil(n^{4/5})$ measurements.
Since $e^*\in G[C]$ with probability $\approx \frac{1}{n^{2/5}}$,
repeating this sampling process independently
for $\Otil(n^{2/5})$ times allows us to recover $e^*$ in at least one vertex induced subgraph.
This results in a linear sketch of $\Otil(n^{6/5})$ measurements.
What if we slightly increase each block's size to $n^{1/5+\delta}$
and decrease the edge weights along the cycle to $n^{2/5-3\delta}$?
While one can still verify that the crossing edge $e^*$
has leverage score $\Omega(1)$, applying the same vertex sampling process as above
will not disconnect the endpoints of $e^*$ with $\Omega(1)$ probability.
However, one can alternatively show that,
with constant probability,
the number of edges along the cycle reduces by a factor of $n^{2/5}$.
Since now the energy of each edge only blows up by a factor smaller than $n^{2/5}$ in $G^{\sq}$,
this will also make the leverage score of $e^*$ become $\Omega(1)$ in $G^{\sq}[C]$,
and thus we can apply the same linear sketch of $\Otil(n^{6/5})$ measurements.
The above warm-up seems to suggest that the sampling rate of $\approx n^{-1/5}$ is a sweet spot
for recovering heavy edges in any graphs with the block cycle structure.
Indeed, we prove a key vertex sampling lemma showing that in {\em any} weighted graph $G$,
a heavy edge $e$ in $G$ is also likely heavy in a vertex-induced subgraph of $G^{\sq}$ obtained by sampling vertices
at rate $\approx n^{-1/5}$.
This is proved by carefully analyzing the structures of the edges of different weights after vertex sampling,
and then explicitly constructing a set of vertex potentials with small total energy in the induced subgraph.
Finally, by integrating this lemma into an iterative refinement process in~\cite{LiMP13}
(as the authors did in~\cite{KapralovLMMS14,KapralovMMMNST20})
and a spectral sparsification algorithm in~\cite{Koutis14},
we are able to recover a spectral sparsifier of $G$ using $\Otil(n^{6/5})$
linear measurements.
We note that the latter step of using heavy edge recovery to build a spectral sparsifier
is also more involved than in the unweighted case and requires a few extra techniques;
we refer the reader to the overview at the beginning of Section~\ref{sec:algmainspectral}
and the discussion therein for more details.
We note that this method of recovering heavy edges by vertex sampling is inspired by the one
used in~\cite{KapralovLMMS14} for spanners.
However, for spectral sparsification,
the correctness of such a method follows from fairly different reasoning,
and the proof is arguably more involved.
\subsection{The hard distribution}~\label{sec:harddist}
We first state how we generate the input weighted graph $G = (V,E,w)$.
Let $n$ be the number of vertices
and define $s \stackrel{\mathrm{def}}{=} n^{1/5}$ and
$\ell \stackrel{\mathrm{def}}{=} n^{4/5}$.
We choose a random permutation $\pi:1..n\to 1..n$
and construct a block cycle graph as follows.
The $i$-th block (where $0\leq i < \ell$) consists
of vertices $\pi(s i + 1),\ldots,\pi(s i + s)$.
For simplicity we denote
the $a$-th vertex in the $i$-th block
(i.e. $\pi(s i + a)$) as $u_{i,a}$.
The block index $i$ will always be modulo $\ell$ implicitly.
We then add a complete graph to each block, and a complete bipartite graph
between each pair of adjacent blocks.
Namely, for each $0\leq i < \ell$, we add a graph $G_{i}$ with edges connecting
$u_{i,a},u_{i,b}$ for all $a < b \in \setof{1,\ldots,s}$,
and add another bipartite graph $G_{i,i+1}$ with edges connecting
$u_{i,a},u_{i+1,b}$ for all $a,b \in \setof{1,\ldots,s}$.
Finally, with probability $1/2$,
we add an edge between
vertices $\pi(1)$ and $\pi(n/2+1)$ (assume $n$ is even).
We refer to this edge as the {\em crossing edge} with respect to $\pi$
and any other edge in $G$ as a {\em non-crossing edge} with respect to $\pi$.
We will omit ``with respect to $\pi$'' when the underlying permutation $\pi$ is clear.
We next describe how the edge weights are determined.
The weights of all non-crossing edges
are drawn independently from $\mathcal{N}(8n^{2/5}, n^{4/5} \log^{-1} n)$
(the Gaussian distribution with mean $8n^{2/5}$ and
variance $n^{4/5} \log^{-1} n$). The weight of the crossing edge
is drawn from the standard Gaussian $\mathcal{N}(0,1)$.
If the crossing edge has negative weight, we say the input is {\em invalid}, and
accept any sketch as a valid sketch.
Our goal will be to detect the presence/absence of the crossing edge with high probability.
In the following, we will call the conditional distribution on the presence of the crossing edge
the {\em {\sf Yes} distribution}, and call the conditional distribution on the absence of the crossing edge
the {\em {\sf No} distribution}.
We then show that with high probability, the effective resistance of the crossing edge is large,
and therefore any linear sketch for computing spectral sparsifiers must distinguish between
the two distributions with good probability.
\begin{proposition}\label{prop:erbig}
With probability at least $1 - 1/n$,
all non-crossing edges have weights in the range $[4 n^{2/5}, 12 n^{2/5}]$,
and as a result the effective resistance
between vertices $\pi(1)$ and $\pi(n/2+1)$ is at least $1/48$
in a {\sf No} instance.
\end{proposition}
\begin{proposition}\label{prop:0.6}
Any linear sketch that can compute a $1.0001$-spectral sparsifier with probability $0.9$
can distinguish between the {\sf Yes} and {\sf No} distributions with probability $0.6$.
\end{proposition}
The first proposition follows from an application of the Chernoff bound (Theorem~\ref{thm:GaussianChernoff}).
The proof of the second proposition is deferred to Appendix~\ref{sec:apovlb}.
In the following, we will assume, for ease of our analysis, that the sketch will be given the permutation
$\pi$ {\em after} computing the linear sketch.
That is, the recovery algorithm $\mathcal{A}$ takes as input both $\Phi w$ and $\pi$.
We will show that even with this extra piece of information,
any incidence sketch with $n^{21/20-\epsilon}$ measurements for constant $\epsilon>0$ cannot distinguish between the {\sf Yes}
and {\sf No} distributions with high probability.
\subsection{A bound on the success probability via effective resistance}\label{sec:tver}
We first show that for our lower bound instance, any incidence sketch can be reduced to a more restricted class of linear sketches
by only increasing the number of measurements by an $O(\log n)$ factor.
Specifically,
let us fix an arbitrary orientation of the edges,
and consider sketches taken over the weighted signed edge-vertex incidence matrix $B^{w}\in\mathbb{R}^{\binom{n}{2}\times n}$,
where the latter is given by
\begin{align*}
B_{eu}^w =
\begin{cases}
w_e & \text{$e\in E$ and $u$ is $e$'s head} \\
-w_e & \text{$e\in E$ and $u$ is $e$'s tail} \\
0 & \text{otherwise.} \\
\end{cases}
\end{align*}
That is, the algorithm must choose a (random) sketching matrix $\Phi\in \mathbb{R}^{k\times \binom{n}{2}}$ with
the $e^{\mathrm{th}}$ column $\phi_e\in\mathbb{R}^{k}$ corresponding to the edge slot $e$.
The sketch obtained is then $\Phi B^w \in \mathbb{R}^{k\times n}$.
Notice that the total number of measurements in $\Phi B^w$ is $k n$,
as each vertex applies the sketching matrix $\Phi$ to its incident edges.
Let us call this class of sketch {\em signed sketches}.
By Yao's minimax principle~\cite{Yao77}, to prove a lower bound for distinguishing the {\sf Yes} and {\sf No} distributions,
it suffices to focus on deterministic sketches.
The proof of the proposition below appears in Appendix~\ref{sec:apovlb}.
\begin{proposition}[Reduction to signed sketches]
\label{prop:B}
Consider any incidence sketch of
$N$ measurements
with a deterministic sketching matrix $\Phi\in\mathbb{R}^{N\times \binom{n}{2}}$ and
a recovery algorithm $\mathcal{A}$ that, given $\Phi w$ and $\pi$,
distinguishes between the {\sf Yes} and {\sf No} distributions
with probability at least $0.6$.
Then there exists a signed sketch with a sketching matrix $\Phi'\in\mathbb{R}^{k\times \binom{n}{2}}$,
where $k = O(1)\cdot \max\{1, \frac{N\log n}{n}\}$,
and a recovery algorithm $\mathcal{A}'$ that, given $\Phi' B^w$ and $\pi$,
distinguishes between the {\sf Yes} and {\sf No} distributions with probability at least $0.55$.
\end{proposition}
Let us now fix a sketching matrix $\Phi\in\mathbb{R}^{k\times \binom{n}{2}}$
and aim to obtain an upper bound on the success probability of any signed sketch using $\Phi$.
For notational convenience, let us write $(\Phi B^w)_{\mathrm{yes}}$ to denote $\Phi B^w$ conditioned on
the presence of the crossing edge %
and $(\Phi B^w)_{\mathrm{no}}$ to denote $\Phi B^w$ conditioned on
the absence of the crossing edge. %
We will also write $(\Phi B^w)_{\pi,\mathrm{yes}}$ or $(\Phi B^w)_{\pi,\mathrm{no}}$
to denote an extra conditioning on the permutation being $\pi$ in addition
to the presence/absence of the crossing edge.
Then to bound the success probability of any signed sketch using $\Phi$ ,
it suffices to show that the total variation distance (TV-distance) between
$(\Phi B^w)_{\mathrm{yes}}$ and $(\Phi B^w)_{\mathrm{no}}$ is small.
To state our upper bound on the TV-distance, %
we need to first introduce some notation.
For an edge $(u,v)$, define $b_{uv}\in\mathbb{R}^{nk}$ by writing it as a block vector (with block size $k$)
as follows:
\begin{align}\label{eq:defbb}
b_{uv} =
\begin{blockarray}{c@{}c@{}cl}
\begin{block}{(c@{}c@{}c)l}
& 0 \\
& \vdots \\
& \phi_{uv} & & \text{$u^{\mathrm{th}}$ block} \\
& 0 \\
& \vdots \\
& -\phi_{uv} & & \text{$v^{\mathrm{th}}$ block} \\
& 0 \\
& \vdots \\
\end{block}
\end{blockarray}\in \mathbb{R}^{nk},
\end{align}
where $\phi_{uv}\in\mathbb{R}^{k}$ is the column of $\Phi$ corresponding to the edge slot $(u,v)$.
For a permutation $\pi$,
we then define $L_{\pi} = \sum_{\text{non-crossing $(u,v)$}} n^{4/5}\log^{-1} n\,b_{uv} b_{uv}^T$.
The following proposition is essentially a consequence of Theorem~\ref{thm:tvd}~\cite{DevroyeMR18},
which bounds the TV-distance between multivariate Gaussians with the same mean.
We give its proof in Appendix~\ref{sec:apovlb}.
Note that we use $\dag$ to denote taking the Moore-Penrose pseudoinverse of a matrix.
\begin{proposition}\label{prop:tver}
For any permutation $\pi$ such that
$b_{\pi(1)\pi(n/2+1)}$ is in the range\footnote{Recall that the range
of a symmetric matrix is the linear span of its columns.}
of $L_{\pi}$,
\begin{align*}
d_{\tv}((\Phi B^w)_{\pi,\mathrm{yes}},(\Phi B^w)_{\pi,\mathrm{no}}) \leq O(1)\cdot
\min\setof{1, b_{\pi(1)\pi(n/2+1)} L_{\pi}^{\dag} b_{\pi(1)\pi(n/2+1)}}.
\end{align*}
\end{proposition}
Our plan is then to show that $b_{\pi(1)\pi(n/2+1)} L_{\pi}^{\dag} b_{\pi(1)\pi(n/2+1)}$
is small on average for every choice of a signed sketch with $k = n^{1/20-\epsilon}$
for constant $\epsilon > 0$.
Note that if $k=1$ and each $\phi_{uv}\in \setof{0,1}$,
then $L_{\pi}$ is exactly the Laplacian matrix of the graph
(call it $\mathcal{H}_{\pi}$) that is formed by the non-crossing edges
$(u,v)$ such that $\phi_{uv} = 1$,
where
each edge is weighted by $n^{4/5} \log^{-1} n$.
Therefore, $b_{\pi(1)\pi(n/2+1)} L_{\pi}^{\dag} b_{\pi(1)\pi(n/2+1)}$ is the effective resistance
between $\pi(1)$ and $\pi(n/2+1)$ in $\mathcal{H}_{\pi}$, if $\phi_{\pi(1)\pi(n/2+1)} \neq 0$
(otherwise $b_{\pi(1)\pi(n/2+1)}$ is the zero vector).
In fact,
to get a quick intuition as to why we should expect the effective resistance
between $\pi(1)$ and $\pi(n/2+1)$
to be small,
let us assume $\phi_{uv} = 1$ for all edge slots $(u,v)$.
Then, for any permutation $\pi$,
$\mathcal{H}_{\pi}$ is the graph formed by {\em all} non-crossing edges,
each weighted by $n^{4/5} \log^{-1} n$.
Note that these weights are about $n^{2/5}$ times larger than
the weights $\Theta(n^{2/5})$ in the actual input graph (Proposition~\ref{prop:erbig}).
As a result, the effective resistance between $\pi(1)$ and $\pi(n/2+1)$ is about $n^{2/5}$ times smaller
than the effective resistance between them in the input graph (the former roughly equals $n^{-2/5}$).
When $k > 1$, we can view $L$ as the Laplacian of a {\em matrix-weighted} graph (again, call it $\mathcal{H}_{\pi}$)
formed by the non-crossing edges,
where each edge $(u,v)$ has a $k\times k$ matrix weight $n^{4/5}\log^{-1} n\, \phi_{uv} \phi_{uv}^T$.
Now $b_{\pi(1)\pi(n/2+1)} L_{\pi}^{\dag} b_{\pi(1)\pi(n/2+1)}$ can be seen as
the (generalized) effective resistance
between $\pi(1)$ and $\pi(n/2+1)$ in $\mathcal{H}_{\pi}$.
\subsection{Warm-up: one-row signed sketches have small TV-distance}\label{sec:onerow}
As a warm-up, we show that for any signed sketch,
in the case that $k=1$ and the sketching matrix $\Phi$ has $0/1$ entries,
we have, for any constant $\epsilon > 0$,
\begin{align}\label{eq:dtvonerow}
\expec{\pi}{d_{\tv}\kh{ (\Phi B^w)_{\pi,\yes}, (\Phi B^w)_{\pi,\no} }} \leq \frac{1}{n^{1/5 - O(\epsilon)}}.
\end{align}
By Proposition~\ref{prop:tver}, we know that
$d_{\tv}\kh{ (\Phi B^w)_{\pi,\yes}, (\Phi B^w)_{\pi,\no} }$
can be bounded by the effective resistance
between $\pi(1),\pi(n/2+1)$ in $\mathcal{H}_{\pi}$
if $\phi_{\pi(1)\pi(n/2+1)} = 1$, and is zero otherwise.
Here $\mathcal{H}_{\pi}$ is formed by the non-crossing edges whose $\phi_{uv} = 1$,
where each edge $(u,v)$ has scalar weight $n^{4/5}\log^{-1} n$.
We can focus on the $\Phi$'s whose number of nonzero entries is at least $n^{9/5 + \epsilon}$,
since otherwise
\begin{align*}
\prob{\pi}{\phi_{\pi(1)\pi(n/2+1)} = 1} = \frac{\nnz(\Phi)}{\binom{n}{2}} \leq
\frac{1}{n^{1/5-O(\epsilon)}},
\end{align*}
and we would already have our desired result~(\ref{eq:dtvonerow}).
Our proof of~(\ref{eq:dtvonerow}) will rely on
decomposing $\mathcal{H}_{\pi}$ into expanders with large minimum degree.
Since $\mathcal{H}_{\pi}$'s edges all have the same weight $n^{4/5}\log^{-1} n$,
it is more convenient to work with the {\em unweighted version of $\mathcal{H}_{\pi}$},
which we denote by $H_{\pi}$.
We now briefly review the definition of unweighted expanders,
as well as state a known expander decomposition lemma that we will utilize.
\begin{definition}[Expander]
An unweighted graph $H = (V,E)$ is a $\zeta$-expander for some $\zeta \in [0,1]$
if its conductance is at least $\zeta$, namely,
for every nonempty $S\subset V$, we have
\begin{align*}
|E(S,V-S)| \geq \zeta \cdot \min\setof{ \vol(S), \vol(V-S) },
\end{align*}
where $\vol(S)$ is the total degree of vertices in $S$.
\end{definition}
Note that in the lemma below, we slightly abuse the notion of ``regular graphs''.
Specifically, we will say a graph is regular if
its minimum vertex degree $d_{\min}$ is not much smaller than the average degree $d$.
\begin{restatable}[Almost regular expander decomposition, see e.g.~\cite{KapralovKTY21}]{lemma}{lemared}
\label{lem:ared}
Given an unweighted graph $H = (V,E)$ with average degree $d\geq 16$, there exists
a subgraph $I = (U,F)$ where $U\subseteq V$ and $F\subseteq E$
such that $I$ is a $\frac{1}{16\log n}$-expander with
minimum degree $d_{\min} \geq \frac{d}{16}$.
\end{restatable}
We will also need the following lemma, which shows that
a random vertex-induced subgraph of an expander with large minimum degree is almost
certainly an expander.
We give the proof of this lemma in Section~\ref{sec:expwarmup}.
To the best of our knowledge, even this result was not known before.
\begin{restatable}[Expanders are preserved under vertex sampling]{lemma}{lemedvs}
\label{lem:edvs}
There exists a $\theta = \theta(n) = n^{o(1)}$ with the following property.
Consider an unweighted $\frac{1}{16\log n}$-expander $H=(V,E)$ %
with minimum degree $d_{\min} \geq 4\cdot 10^6 \cdot \theta(n)$.
For any
$s \geq \frac{4\cdot 10^6}{d_{\min}}\cdot \theta(n)\cdot n$,
let $C\subseteq V$ be a uniformly random vertex subset of size $s$.
Then with probability at least $1 - 1/n^7$,
the vertex-induced subgraph $H[C]$ is a $\frac{1}{n^{o(1)}}$-expander
with minimum degree at least $\frac{s}{2n}\cdot d_{\min}$.
\end{restatable}
\begin{proof}[Proof of~(\ref{eq:dtvonerow}) using Lemmas~\ref{lem:ared},\ref{lem:edvs}]
As argued above we can assume w.l.o.g. that $\nnz(\Phi) \geq n^{9/5 + \epsilon}$.
We want to obtain, for each edge slot $e$ satisfying $\phi_e = 1$,
conditioned on $e$ being the crossing edge w.r.t. $\pi$,
an upper bound (call it $u_e$) on the typical effective resistance
between the endpoints of $e$ in the graph $\mathcal{H}_{\pi}$.
In other words, conditioned on $e$ being the crossing edge,
$u_e$ should be an upper bound on the effective resistance between the endpoints of $e$
in $\mathcal{H}_{\pi}$ with high probability over $\pi$.
Then the total variation distance between $(\Phi B^w)_{\yes}$ and $(\Phi B^w)_{\no}$
can be bounded by
\begin{align}\label{eq:dtvue}
\expec{\pi}{d_{\tv}\kh{ (\Phi B^w)_{\pi,\yes}, (\Phi B^w)_{\pi,\no} } } \leq
O(1)\cdot \frac{1}{\binom{n}{2}} \sum_{e: \phi_e = 1} u_e.
\end{align}
To obtain the $u_e$'s, let us define the unweighted graph
$H_{\phi} = (V,E_{\phi})$
where $E_{\phi}$ contains {\em all} edges $e$ whose $\phi_e = 1$
(including the ones not present in the input graph, i.e. $|E_{\phi}| = \nnz(\Phi))$.
Now consider the following process, %
where we repeatedly delete an expander subgraph from $H_{\phi}$ and obtain
$u_e$'s for the edges in the expander.
\begin{enumerate}
\item While $|E_{\phi}| \geq 10^9 n^{9/5+\epsilon}$:
\begin{enumerate}
\item Find a subgraph $I = (U,F)$ of $H_{\phi} = (V,E_{\phi})$ that is a
$\frac{1}{16\log n}$-expander with minimum degree
$d_{\min} \geq \frac{|E_{\phi}|}{8n}$
(existence is guaranteed by Lemma~\ref{lem:ared}).
\item For each edge $f\in F$,
let $u_f\gets \kh{\frac{10^9 n^{9/5+\epsilon}}{|E_{\phi}|}}^2$.
\label{step:2b}
\item Delete the edges in $F$ from $H_{\phi}$ by letting $E_{\phi} \gets E_{\phi} \setminus F$.
\end{enumerate}
\item Let $u_f \gets 1$ for all $f$ in the remaining $E_{\phi}$.
\end{enumerate}
To show that $u_e$'s are valid upper bounds, let us consider a fixed iteration of the while loop.
For $i=0,\ldots,n^{4/5}-1$, let $U_i$ denote the vertices in $I$ that are
in the $i^{\mathrm{th}}$ block of the input block cycle graph:
\begin{align*}
U_i \stackrel{\mathrm{def}}{=} U\cap \setof{\pi(n^{1/5} i + 1), \ldots, \pi(n^{1/5} i + n^{1/5})}.
\end{align*}
Then by Chernoff bounds, with probability at least $1 - 1/n^5$ over the random choice of $\pi$,
we have $|U_i| \geq \frac{|U|}{2 n^{4/5}} \geq \frac{4\cdot 10^6}{d_{\min}}\cdot |U|^{1+\epsilon}$.
Then by invoking Lemma~\ref{lem:edvs}, %
with probability at least $1 - 1/n^4$ over $\pi$,
all vertex-induced subgraphs $I[U_i\cup U_{i+1}]$ are
$\frac{1}{n^{o(1)}}$-expanders with minimum degree at least $\frac{|E_{\phi}|}{16 n^{9/5}}$.
Using this fact, we obtain the following claim, whose proof appears in Appendix~\ref{sec:apovlb}.
\begin{claim}\label{claim:ue}
For each edge $f\in F$,
conditioned on $f$ being the crossing edge,
with probability at least $1-1/n^2$ over $\pi$,
the effective resistance between the endpoints of $f$ in $\mathcal{H}_{\pi}$
is at most $u_f$.
\end{claim}
Now let us divide the above process for obtaining $u_e$'s into $O(\log n)$ phases,
where in phase $i \in \setof{1,\ldots,O(\log n)}$, we have $|E_{\phi}| \in (\binom{n}{2} / 2^i, \binom{n}{2}/2^{i-1}]$.
Then we have
\begin{align*}
\sum_{e: \phi_e = 1} u_e =
& \sum_{e: \phi_e = 1} \kh{\frac{10^9 n^{9/5+\epsilon}}{|E_{\phi}|}}^2
\leq
n^{O(\epsilon)}\cdot
\sum_{i=1}^{O(\log n)} \frac{\binom{n}{2}}{2^i} \cdot \kh{\frac{2^i}{n^{1/5}}}^2 \\
\leq & n^{8/5 + O(\epsilon)} \sum_{1}^{O(\log n)} 2^i \leq
n^{9/5 + O(\epsilon)}
\end{align*}
where in the first line we have used $n^{O(\epsilon)}$ to hide the constant factors,
and the last inequality holds since in the last phase we have $2^i\leq n^{1/5}$.
Plugging this into~(\ref{eq:dtvue}) finishes the proof.
\end{proof}
\subsection{The general case: proof of Theorem~\ref{thm:spectrallb}}\label{sec:multirow}
Note that even though for $k = 1$, the TV-distance is $\Otil(n^{-1/5})$, this does not imply
that $k$ must be large for the TV-distance to become $\Omega(1)$.
By Proposition~\ref{prop:B},
in order to prove Theorem~\ref{thm:spectrallb},
it suffices to prove the following:
\begin{restatable}{theorem}{tvdsmall}
\label{thm:tvdsmall}
For any fixed sketching matrix
$\Phi\in\mathbb{R}^{k\times \binom{n}{2}}$ where
$k \leq n^{1/20 - \epsilon}$ for some constant $\epsilon > 0$,
we have
\begin{align*}
\expec{\pi}{d_{\tv}\kh{ (\Phi B^w)_{\pi,\yes}, (\Phi B^w)_{\pi,\no} } } \leq o(1).
\end{align*}
\end{restatable}
By Proposition~\ref{prop:tver}, our goal is to bound the ``effective resistance''
$b_{\pi(1)\pi(n/2+1)}L_{\pi}^{\dag} b_{\pi(1)\pi(n/2+1)}$
between
vertices $\pi(1),\pi(n/2+1)$
in the matrix-weighted graph $\mathcal{H}_{\pi}$ consisting of the non-crossing edges,
where edge $(u,v)$
has matrix weight $n^{4/5} \log^{-1} n\, \phi_{uv}\phi_{uv}^T\in\mathbb{R}^{k\times k}$.
We will do so by (significantly) generalizing our previous approach based on expander decomposition
for ordinary graphs in Section~\ref{sec:onerow}.
Our approach for the $k=1$ case essentially consists of two steps:
(i) decomposing the graph $H_{\phi}$ into large expander subgraphs and
(ii) proving that a random vertex induced subgraph of an expander is still an expander.
First note that there does not appear to be a combinatorial analog of conductance in matrix-weighted graphs,
which suggests that we should define expanders in an algebraic way.
Let us first recall the algebraic characterization of expanders for ordinary, unweighted graphs.
The definition is based on eigenvalues of the {\em normalized Laplacian} of the graph,
which is given by $N = D^{-1/2} L D^{-1/2}$,
where $D$ is a diagonal matrix with $D_{uu}$ equal to the degree $d_u$ of $u$.
\begin{definition}[Algebraic definition of ordinary, unweighted expanders]
\label{def:uwexp}
An unweighted graph $H$ is a $\zeta$-expander for some $\zeta\in[0,1]$ if
the smallest nonzero eigenvalue of its normalized Laplacian matrix $N$ is at least $\zeta$.
\end{definition}
By Cheeger's inequality~\cite{AlonM85},
for $\zeta \geq \Omegatil(1)$,
this definition translates to that
the graph $H$ is a union of vertex-disjoint combinatorial expanders, each with conductance $\Omegatil(1)$.
To come up with an analogous definition for matrix-weighted graphs,
let us first define their associated matrices formally.
\paragraph{Matrices associated with matrix-weighted graphs.}{
We consider a $k\times k$ matrix-weighted graph $H = (V,E)$ with $|V| = n$.
For each edge $(u,v)\in E$,
there is a vector $\phi_{uv}\in\mathbb{R}^{k}$, indicating that
$(u,v)$ is weighted by the $k\times k$ rank-$1$ matrix $\phi_{uv} \phi_{uv}^T$.
\begin{definition}[Degree matrices]
For a vertex $u$, its generalized degree is given by
\begin{align*}
D_u = \sum_{u\sim v} \phi_{uv} \phi_{uv}^T\in\mathbb{R}^{k\times k}.
\end{align*}
We then define the $nk\times nk$ degree matrix $D$ as a block diagonal matrix
(with block size $k\times k$),
with the $u^{\mathrm{th}}$ block on the diagonal being $D_{uu} = D_u\in\mathbb{R}^{k\times k}$.
\end{definition}
\begin{definition}[Laplacian matrices]
The Laplacian matrix is given by $L = \sum_{(u,v)\in E} b_{uv} b_{uv}^T$,
where $b_{uv}$'s are defined in~(\ref{eq:defbb}).
\end{definition}
We will call $b_{uv}$ the {\em incidence vector} of edge $(u,v)$.
Note that the Laplacian matrix here differs from the connection Laplacian matrix~\cite{KyngLPSS16} which is also defined
to be a block matrix. In particular, the Laplacian matrix of a matrix-weighted graph is not necessarily block diagonally dominant
(bDD) (Definition 1.1 of~\cite{KyngLPSS16}).
\begin{definition}[Normalized Laplacian matrices]
The normalized Laplacian matrix is given by $N\stackrel{\mathrm{def}}{=} D^{\dag/2} L D^{\dag/2}$.
Equivalently, we have $N = \sum_{(u,v)\in E} D^{\dag/2} b_{uv} b_{uv}^T D^{\dag/2}$.
\end{definition}
We will call $D^{\dag/2} b_{uv}$ the {\em normalized incidence vector} of edge $(u,v)$.
The following proposition says that similar to scalar-weighted graphs,
the eigenvalues of the normalized Laplacian of a matrix-weighted graph are also between $[0,2]$.
The proof this proposition appears in Appendix~\ref{sec:aphdg} (in ``Proof of Proposition~\ref{prop:eigvalofN}'').
\begin{proposition}
The eigenvalues of $N$ are between $[0,2]$.
\end{proposition}
}
Now, a first attempt might be to define matrix-weighted expanders to also be graphs
whose normalized Laplacians' nonzero eigenvalues are large,
and then try to decompose any matrix-weighted graph into large
expander subgraphs. However, we show that the latter goal may not be achievable in general, by presenting
in Appendix~\ref{sec:aphd}
a hard instance, for which any large subgraph has a small nonzero eigenvalue.
\paragraph{Our approach.}{
In light of the hard instance, we loosen the requirement of being an expander
by allowing small eigenvalues, but requiring instead that
each edge, compared to the average, does not have too large ``contribution'' to the small eigenvectors.
Formally, we want that every edge's normalized incidence vector has small
(weighted) projection onto the bottom eigenspace.
We will also need an analog of ``almost regularity'', which for ordinary, unweighted graphs says that
the minimum degree is large.
We give the formal definition of an almost regular matrix-weighted expander below.
\begin{definition}[Almost regular matrix-weighted expanders]
\label{def:hde1}
For a $k\times k$ matrix-weighted graph $H$,
let $\lambda_1\leq \ldots\leq\lambda_{nk}$ be the eigenvalues of its normalized Laplacian $N$, and
let $f_1,\ldots,f_{nk} \in \mathbb{R}^{nk}$ be a set of corresponding orthonormal eigenvectors.
We say $H$ is a $(\gamma,\zeta,\psi)$-almost regular expander if
\begin{enumerate}
\item ($\gamma$-almost regularity) For every vertex $u$ and every incident edge $(u,v)\in E$, we have
\begin{align}\label{eq:arty}
\phi_{uv}^T D_u^{\dag} \phi_{uv} \leq \frac{\gamma \cdot k}{n}.
\end{align}
\item ($(\zeta,\psi)$-expander) For every edge $(u,v)\in E$ we have
\begin{align}\label{eq:expdef}
\kh{D^{\dag/2} b_{uv} }^T
\kh{ \sum_{i: \lambda_i\in (0,\zeta]} \frac{1}{\lambda_i} f_i f_i^T } D^{\dag/2} b_{uv} \leq
\frac{\psi\cdot k^2}{n^2}.
\end{align}
\end{enumerate}
\end{definition}
The LHS of~(\ref{eq:arty}) is the so-called {\em leverage score} of $\phi_{uv}$ w.r.t. $D_u$
(Definition~\ref{def:deflvg}).
It is known that the sum of leverage scores equals the rank of the matrix:
\begin{proposition}
For any fixed vertex $u$,
$\sum_{(u,v)\in E} \phi_{uv}^T D_{u}^{\dag} \phi_{uv} = \rank{D_u}$.
\end{proposition}
Since $D_u$ is a $k\times k$ matrix, we have $\rank{D_u}\leq k$.
Therefore, in the case that $u$ has $\Omega(n)$ incident edges,~(\ref{eq:arty})
is essentially saying that no incident edge's leverage score exceeds the average by too much.
To get intuition for condition~(\ref{eq:expdef}),
we need the following two results.
The first theorem is proved in Section~\ref{sec:fewsmall}.
The second proposition is proved in Appendix~\ref{sec:apovlb}.
\begin{theorem}\label{thm:fewsmallintro}
Let $H$ be a $k\times $k-matrix weighted graph that is $\gamma$-almost regular
(in the sense of~(\ref{eq:arty})).
Then for any $\zeta \in (0,1)$,
the number of eigenvalues of its normalized Laplacian that are between $(0,\zeta]$
is at most $\frac{\gamma \cdot k^2}{(1 - \zeta)^2}$.
\end{theorem}
\begin{proposition}\label{prop:ellavg}
Let $\ell$ be the number of $\lambda_i$'s
that are between $(0,\zeta]$. Then
\begin{align}
\sum_{(u,v)\in E}
\kh{D^{\dag/2} b_{uv} }^T
\kh{ \sum_{i: \lambda_i\in (0,\zeta]} \frac{1}{\lambda_i} f_i f_i^T } D^{\dag/2} b_{uv}
= \ell.
\end{align}
\end{proposition}
Therefore,
in the case that $|E| = \Omega(n^2)$,~(\ref{eq:expdef}) is essentially saying that
the LHS for every edge $(u,v)$ does not exceed the average by too much.
We then show that every dense enough matrix-weighted graph can indeed be made into an expander
by downscaling a small number of edges.
To this end, let us define, for a {\em scaling} $s: E\to[0,1]$,
the rescaled graph $H^s$, which is obtained from $H$
by rescaling each edge $(u,v)$'s weight to $s_{uv}^2 \phi_{uv} \phi_{uv}^T$.
The proof of the following theorem appears in Sections~\ref{sec:hargd} and~\ref{sec:hared}.
\begin{theorem}\label{thm:edintro}
There is an algorithm that,
given any $k\times k$ matrix-weighted graph $H = (V,E)$ with $|E| \geq \Omega(n^2)$,
outputs a scaling $s: E \to [0,1]$ such that
\begin{enumerate}
\item The rescaled graph $H^s$ is a $(\gamma,\zeta,\psi)$-almost regular expander
for
\begin{align*}
\gamma = 8 \log n,\ \zeta = \frac{1}{\log n},\ \psi = 16 k^2 \log^3 n.
\end{align*}
\item The number of edges $(u,v)\in E$ with $s_{uv} < 1$ is $o(n^2)$.
\end{enumerate}
\end{theorem}
We next show that almost-regular expanders are preserved under vertex sampling.
However, we will now use a different notion of ``preservation''.
To state our specific result, let us define some additional notations.
For a vertex subset $C\subseteq V$, we write $L_{G[C]}$ to denote the Laplacian
of the vertex-induced subgraph $G[C]$.
We also let $D_{CC}$ be the submatrix of $D$ (the degree matrix of the original graph $H$)
with rows and columns restricted to vertices in $C$,
and let $(f_i)_{C}$ denote, for an eigenvector $f_i$, the vector $f_i$ with indices restricted to $C$.
We then have the following theorem, whose proof appears in Section~\ref{sec:mtexpsamp}.
\begin{theorem}\label{thm:expsampintro}
There exists a $\theta = \theta(n) \leq n^{o(1)}$ with the following property.
Let $H = (V,E)$ be a $k\times k$ matrix-weighted,
$(\gamma,\zeta,\psi)$-almost regular expander where $\zeta\leq 1/\log n$.
For an
$s \geq 2\cdot 10^6 \gamma \psi \zeta^{-1} k^2 \theta(n)$,
let $C\subseteq V$ be a uniformly random vertex subset of size $s$.
Then with probability at least $1 - 1/n^5$, we have
that
\begin{enumerate}
\item The null space of $D_{CC}^{\dag/2} L_{G[C]} D_{CC}^{\dag/2}$ is exactly
the linear span of $\setof{(f_i)_C : \lambda_i = 0}$.
\item For all vectors $x\in\mathbb{R}^{|C|k}$ such that
$x^T (f_i)_C = 0,\forall i: \lambda_i = 0$,
\begin{align}\label{eq:preservance}
x^T \kh{ D_{CC}^{\dag/2} L_{G[C]} D_{CC}^{\dag/2} }^{\dag} x \leq
n^{o(1)}\cdot
x^T \kh{ \frac{n^2}{s^2} \sum_{i: \lambda_i\in(0,\zeta]} \frac{1}{\lambda_i} (f_i)_C (f_i)_C^T +
\frac{n}{s}\cdot \frac{1}{\zeta} I } x.
\end{align}
\end{enumerate}
\end{theorem}
We argue that~(\ref{eq:preservance}) is roughly saying that
the pseudoinverse of the subgraph $G[C]$ can be bounded by the pseudoinverse of the original
graph that is (i) restricted to indices in $C$ and (ii) rescaled in a certain way.
For technical reasons, on the LHS of~(\ref{eq:preservance}) we
normalize the Laplacian of the vertex-induced subgraph using the degree matrix
of the {\em original graph $H$}.
As for the RHS,
we can see it as a rescaled version of the pseudoinverse of $N$ restricted to $C$,
by noting that
\begin{align*}
\kh{ N^{\dag} }_{CC} = \sum_{\lambda_i > 0} \frac{1}{\lambda_i} (f_i)_C (f_i)_C^T.
\end{align*}
Thus, on the RHS of~(\ref{eq:preservance}) we blow up the small eigenvalues
quadratically in $1/(\text{sampling\ rate})$,
but blow up the large eigenvalues linearly in $1/(\text{sampling\ rate})$.
With these tools, we are finally able to prove Theorem~\ref{thm:spectrallb}.
We present the proof in Section~\ref{sec:lbspectral}.
}
\subsubsection{Techniques for proving Theorems~\ref{thm:fewsmallintro},~\ref{thm:edintro},~\ref{thm:expsampintro}}
We now explain, at a very high level,
the techniques that we use to prove these three key theorems,
as well as their connections to previous works.
More details can be found in the subsequent sections.
\paragraph{Proof of Theorem~\ref{thm:fewsmallintro}.}{
We consider the ``spectral embedding'' %
induced by the bottom eigenvectors of the normalized Laplacian,
which maps each vertex to a rectangular matrix.
Such a spectral embedding may be seen as the matrix-weighted counterpart of the ones for scalar-weighted graphs,
which map each vertex to a {\em vector}.
The latter embeddings were previously used to prove higher-order Cheeger inequalities~\cite{LouisRTV12,LeeGT14}.
We show that, for matrix-weighted graphs that are almost regular (in the sense of~(\ref{eq:arty})),
the spectral embedding has vertex-wise bounded spectral norm (Lemma~\ref{lem:fsu2}),
and as a result the number of bottom eigenvectors must be small (Theorem~\ref{thm:fewsmall}).
}
\paragraph{Proof of Theorem~\ref{thm:edintro}.}{
Our proof consists of two steps:
(i) decomposing the graph into an almost regular graph (Theorem~\ref{thm:hargd}), and
(ii) decomposing the graph into an almost regular expander (Theorem~\ref{thm:hared}).
In achieving (ii), we actually invoke (i) repeatedly to maintain the almost regularity of the graph.
As noted above, the almost regularity condition~(\ref{eq:arty}) is essentially saying that
no incident edge has leverage score too large comparing to the average.
A similar task to (i) has in fact been investigated by a previous work~\cite{CohenLMMPS15},
where the authors showed that given a set of vectors, one can,
by downscaling a small number of them,
make every vector have small leverage score comparing to the average.
This result is achieved by an algorithm that iteratively downscales vectors
with large leverage scores, while analyzing how each vector's leverage score changes in the process.
While it is possible to directly invoke the result from~\cite{CohenLMMPS15} to get
a large almost regular graph, its guarantee does not suffice for our purpose of smoothly incorporating (i) into (ii).
In particular, since we will repeatedly invoke (i) in (ii), we need,
in addition to that the number of rescaled edges is small,
an extra bound on the number of completely deleted edges (i.e. those rescaled to $0$) that
is proportional to the rank change of the degree matrix $D$.
As a result, we design a more involved algorithm for obtaining the scaling
as well as carry out a more careful analysis of the algorithm.
Achieving (ii) turns out to be much more challenging.
Although the LHS of~(\ref{eq:expdef}) may be seen as a leverage score,
there is the intrinsic difficulty that whenever the edge weights change,
so do the eigenvalues and eigenvectors of the normalized Laplacian,
as well as the degree matrix itself (hence also the normalized incidence vectors).
Thus it is not clear how the LHS of~(\ref{eq:expdef}) will change. %
As a result, when trying to obtain a desired scaling,
we have to use some global measure of progress.
This is in contrast to (i), where we can track the leverage score change of each edge {\em locally}.
We resolve this issue
by considering, as a potential function, the determinant of the normalized Laplacian restricted to the bottom eigenspace.
In other words, our potential function is the product of the eigenvalues of $N$ that are between
$(0,\zeta]$\footnote{Due to technical reasons, the actual potential function slightly differs from the one stated here.}.
We show that, by a delicate global analysis of such a potential function, we are able to
make the graph an expander by only downscaling a small number of edges.
}
\paragraph{Proof of Theorem~\ref{thm:expsampintro}.}{
Our proof is motivated by the {\em approximate Gaussian elimination} of
the Laplacian matrices of scalar-weighted graphs, which was previously used as an algorithmic tool for solving
graph structured linear systems~\cite{KyngLPSS16,KyngS16}
and building data structures for dynamically maintaining effective resistances~\cite{DurfeeKPRS17,LiZ18,LiPYZ20}.
Our approach also relies
on analyzing matrix-valued martingales which played a key role in~\cite{KyngS16}.
which have played key roles in constructing vertex/subspace sparsifiers~\cite{KyngS16,LiS18,ForsterGLPSY21}.
Let us first briefly review the Gaussian elimination of the Laplacian matrix of a scalar-weighted graph.
Roughly speaking, by eliminating the row and column of $L$ corresponding to a vertex $u$,
we can obtain another Laplacian matrix $L'$ supported on $V\setminus\setof{u}$ whose pseudoinverse {\em equals}
the pseudoinverse of the original $L$ restricted to $V\setminus\setof{u}$
(i.e. $(L')^{\dag} = (L^{\dag})_{V\setminus\setof{u}V\setminus\setof{u}}$).
Given a vertex subset $C\subseteq V$,
one can also eliminate the vertices outside of $C$ one by one
and get a Laplacian matrix $L''$ supported
on $C$ with the same property that $(L'')^{\dag} = (L^{\dag})_{CC}$.
The matrix $L''$ is referred to as the {\em Schur complement} of $L$ onto $C$.
However, the graphs associated with $L'$ and $L''$ could be dense,
which are inefficient for algorithm design.
Therefore~\cite{KyngS16} showed that one can perform an {\em approximate} Gaussian elimination
by, upon each elimination, implicitly sub-sampling the edges in $L'$.
They then showed that we eventually get a good approximation to $L''$
by analyzing a matrix-valued martingale induced by this process.
We now explain how to apply this idea to prove Theorem~\ref{thm:expsampintro}.
Since we are considering an induced subgraph $G[C]$ where $C$ is a uniformly random
subset of size $s$, we can also view the process for choosing $C$
as deleting a sequence of $n-s$ vertices from $V$ uniformly at random.
Our goal is to compare the pseudoinverse of $G[C]$ with that of the original graph,
therefore it suffices to compare it with the Schur complement of $L$ onto $C$.
We will in fact do such a comparison upon the elimination of every vertex.
That is, if we let $C_i$ be the set of remaining vertices at the $i^{th}$ step,
then we want to compare the Laplacian of $G[C_i]$ with the Schur complement of $L$ onto $C_i$.
At a high level,
we do so by setting up a matrix-valued martingale,
and show that it has good concentration
when $G$ is a matrix-weighted expander (in the sense of Definition~\ref{def:hde1}).
}
\subsection{Matrices}
\begin{definition}[Pseudoinverse]
Let $A$ be an $n\times n$ symmetric matrix.
Let $\lambda_1,\lambda_2,\ldots,\lambda_n$
be the eigenvalues of $A$ and let $v_1,v_2,\ldots,v_n$
be orthonormal (i.e. unit and orthogonal)
eigenvectors of $A$,
then by the spectral theorem
$A = \sum_{i=1}^{n} \lambda_i v_i v_i^T$.
The pseudoinverse of $A$ is then defined as
\begin{align}
A^{\dag} \stackrel{\mathrm{def}}{=} \sum_{\lambda_i\neq 0}
\frac{1}{\lambda_i} v_i v_i^{T}.
\end{align}
\end{definition}
\begin{definition}[Matrix partial ordering]
\label{def:matrixorder}
For two $n\times n$ symmetric matrices $A,B$,
we write $A\pleq B$ if for any vector $x\in\mathbb{R}^{n}$ we have
$x^{T} A x \leq x^T B x$
(i.e. $B - A$ is positive semi-definite).
\end{definition}
\begin{fact}[Properties of the matrix partial ordering]\label{fact:inverseordering}
\quad
\begin{enumerate}
\item If $A \pleq B$ and $C \pleq D$, then $A + C \pleq B + D$.
\item If $A \pleq B$ where $A,B\in\mathbb{R}^{n\times n}$,
then for any $W$ with $n$ rows, $W^T A W \pleq W^T \BB W$.
\item For two positive semidefinite matrices $A,B$
that have the same null space,
$A\pleq B$ implies that $B^{\dag} \pleq A^{\dag}$.
\end{enumerate}
\end{fact}
\begin{theorem}[Matrix chernoff bound~\cite{Tropp12}]\label{thm:matrixchernoff}
Let $X_1,\ldots,X_m\in\mathbb{R}^{n\times n}$ be independent random positive
semidefinite matrices such that $\lambda_{\mathrm{max}}(X_i)\leq R, \forall i$
and $\ex{\sum_{i=1}^{m} X_i} = I_{n\times n}$ where $I_{n\times n}$ is the
$n\times n$ identity matrix. Then with probability at least
$1 - 2n \exp\setof{-\frac{\epsilon^2}{3R}}$
\begin{align}
(1 - \epsilon) I \pleq \sum_{i=1}^{m} X_i \pleq (1 + \epsilon) I.
\end{align}
\end{theorem}
\begin{definition}[Leverage scores]
\label{def:deflvg}
Let $a_1,\ldots,a_m \in \mathbb{R}^{n}$ and
$A = \sum_{i=1}^{m} a_i a_i^T \in \mathbb{R}^{n\times n}$.
The leverage score of $a_i$ w.r.t. $A$ is defined to be
$\tau_i(A) = a_i^T A^{\dag} a_i$.
\end{definition}
\begin{fact}\label{thm:dual}
Let $a_1,\ldots,a_m \in \mathbb{R}^{n}$ and
$A = \sum_{i=1}^{m} a_i a_i^T \in \mathbb{R}^{n\times n}$.
Also let $B = (a_1 ,\ldots , a_m )\in\mathbb{R}^{n\times m}$,
so we have $A = B B^T$.
Let $b\in\mathbb{R}^{n}$
be a vector in the span of $a_1,\ldots,a_m$.
Then we have
\begin{align*}
\min_{Bx = b} \norm{x}_2^2 = b^{T} (B B^T)^{\dag} b.
\end{align*}
\end{fact}
\subsection{Multivariate Gaussian distributions}
\begin{definition}\label{def:multivargaussian}
Let $\mu\in \mathbb{R}^{d}$ be a vector
and $\Sigma\in\mathbb{R}^{d\times d}$ be a matrix.
We say a random vector $x\in\mathbb{R}^d$ follows a multivariate Gaussian distribution
with mean $\mu$ and covariance matrix $\Sigma$, denoted by $\mathcal{N}(\mu,\Sigma)$, if
\begin{enumerate}
\item Each $x_i$ distributes as $\mathcal{N}(\mu_i,\Sigma_{ii})$,
a univariate Gaussian with mean $\mu_i$ and variance $\Sigma_{ii}$. \label{cond:1}
\item $\ex{(x_i - \mu_i)(x_j - \mu_j)} = \Sigma_{ij}$ for all pairs $i,j$.
Or equivalently,
\begin{align}
\ex{(x - \mu) (x - \mu)^{T}} = \Sigma.
\end{align}
\label{cond:2}
\end{enumerate}
\end{definition}
\begin{fact}
The covariance matrix $\Sigma$ is symmetric and positive semi-definite.
\end{fact}
\begin{fact}\label{thm:sumgaussian}
Let $x_1,x_2 \in \mathbb{R}^d$ be independent random vectors
such that $x_1\sim \mathcal{N}(\mu_1,\Sigma_1)$ and $x_2\sim \mathcal{N}(\mu_2,\Sigma_2)$.
Then $x_1 + x_2 \sim \mathcal{N}(\mu_1 + \mu_2, \Sigma_1 + \Sigma_2)$.
\end{fact}
\begin{theorem}[$\ell_1$-distance between multivariate Gaussians with the same mean~\cite{DevroyeMR18}]
\label{thm:tvd}
Let $\mu\in\mathbb{R}^{d}$ and let
$\Sigma_1,\Sigma_2\in\mathbb{R}^{d\times d}$ be positive semidefinite
such that
\begin{enumerate}
\item $\Sigma_1$ and $\Sigma_2$ have the same null space.
\item $\Sigma_2 = \Sigma_1 + w w^T$ for some $w\in\mathbb{R}^{d}$.
\end{enumerate}
Then we have
\begin{align*}
d_{TV} \kh{ \mathcal{N}(\mu,\Sigma_1), \mathcal{N}(\mu,\Sigma_2) }
= \Theta\kh{ \min\setof{1, w^T \Sigma_1^{\dag} w} }. %
\end{align*}
\end{theorem}
\begin{theorem}[Chernoff bound for univariate Gaussians, Theorem 9.3 of~\cite{MitzenmacherU2017}]
\label{thm:GaussianChernoff}
Let $X$ be a univariate Gaussian with mean $\mu$ and variance $\sigma^2 > 0$:
$X\sim \mathcal{N}(\mu,\sigma^2)$. Then for any $a > 0$
\begin{align}
\pr{ \sizeof{ X - \mu } \geq a \sigma } \leq 2 e^{-a^2/2}.
\end{align}
\end{theorem}
\subsection{$\ell_2$-heavy hitter, $\ell_p$-sampler and $\ell_2$ estimation}
\begin{proposition}[$\ell_2$-heavy hitters~\cite{KapralovLMMS14}]\label{prop:heavyhitter}
For any $\eta > 0$, there is a decoding algorithm $D$ and a distribution on
matrices $A \in \mathbb{R}^{O(\eta^{-2} \mathrm{polylog}(N))\times N}$ such that,
for any $x\in \mathbb{R}^{N}$, given $A x$, the algorithm $D$ returns a vector $\xtil$
such that $\xtil$ has $O(\eta^{-2}\mathrm{polylog}(N))$ non-zeros and satisfies
$$\norm{x - \xtil}_{\infty} \leq \eta \norm{x}_2$$ with high probability over the choice of $A$.
The sketch $A x$ can be maintained and decoded in
$O(\eta^{-2} \mathrm{polylog}(N))$ space.
\end{proposition}
Given a vector $x$ of size $N$ and a number $\delta > 0$, an $\ell_p$ sampler is an algorithm that output an index $i$ with probability
$$
p_i \in (1 \pm N^{-c}) \frac{\sizeof{x_i}^p}{\norm{x}^p_p} \pm O(N^{-c})
$$
where $c$ is arbitrary constant. The algorithm may also output $\textsf{Fail}$ with probability at most $\delta$.
For any $0 \le p \le 2$, there is a polylog size linear sketch for $\ell_p$-sampler.
\begin{proposition}[\cite{JowhariST11,CormodeF14}] \label{l0-sampler}
For any constant $c$ and $0<\delta<1$, there is a linear sketch for $\ell_0$-sampling with measurement size $O(\log^2 n \log 1/\delta)$.
\end{proposition}
\begin{proposition}[\cite{JayaramW21}] \label{l1-sampler}
For any $0 < p < 2$, any $\epsilon,\delta_1,\delta_2>0$ and any constant $c>0$, there is a linear sketch with measurement size $O(\log^2 N (\log \log N)^2 \log (1/\delta_1) + \epsilon^{-p} \log N \log^2(1/\delta_2)\log (1/\delta_1))$ such that given an $N$-dimensional vector $x$, with probability $(1-\delta_1)$, it can recover an index $i$, such that the probablity of outputting $i$ is
$$
p_i \in (1 \pm N^{-c}) \frac{\sizeof{x_i}^p}{\norm{x}^p_p} \pm O(N^{-c})
$$
Moreover, if the sketch does output an index $i$, then it also recovers a value $x'_i$ such that $\sizeof{x_i}\le x'_i \le (1 + \epsilon)\sizeof{x_i}$, with probability $1-\delta_2$.
\end{proposition}
\begin{proposition}[\cite{KapralovLMMS14}] \label{l2-heavy-hitter}
For any $\epsilon>0$ and any constant $c>0$, there is a linear sketch of size $O(\epsilon^{-2} \mathrm{polylog}(N))$ such that given an $N$-dimensional vecotr $x$, with probability $N^{-c}$, we can recover a vector $x'$ such that $\norm{x-x'}_{\infty} \le \epsilon \norm{x}_2$.
\end{proposition}
\begin{proposition}[\cite{johnson1984extensions}] \label{l2-norm}
For any $0<\delta<1$ and any constant $c>0$, there is a linear sketch of size $O(\delta^{-2} \log N)$ such that given an $N$-dimensional vector $x$, with probability $N^{-c}$, we can recover a vector $x'$ such that $\norm{x'}_2 \in (1 \pm \delta) \norm{x}_2$.
\end{proposition}
\subsection{Edge strengths and cut sparsifiers}
Given a graph $G$, a {\em $k$-strongly connected component} is a maximal vertex induced subgraph whose minimum cut size is at least $k$. Thus, all $k$-strongly connected components form a partition of the entire vertex set.
The following fact gives an equivalent way of obtaining the $k$-strongly connected components.
\begin{fact}\label{fact:removecut}
Given any graph and integer $k$,
consider a process
where we iteratively remove (the edges across) an arbitrary cut of size strictly smaller than $k$,
until there is no such cut left.
Then the connected components in the resulting
graph are the $k$-strongly connected components of the original graph.
\end{fact}
The {\em strength} of an edge $e$ in the graph, denoted $k_e$, is the maximum value of $k$ such that a $k$-strong component of $G$ contains both endpoints of $e$. The weighted sum of the inverse of the strength of every edge in a graph is at most $n-1$.
\begin{claim} [\cite{BenczurK15}] \label{clm:sum_n-1} For any weighted graph $G=(V,F,w)$ on $n$ vertices, $\sum_{f\in F}\frac{w_f}{k_f}\leq n-1$.
\end{claim}
Given a graph $G$, an $(1 \pm \epsilon)$-cut sparsifier $G'$ is a subgraph of $G$ such that any cut in $G$ is preserved in $G'$ to within a factor of $(1 \pm \epsilon)$. The seminal result~\cite{BenczurK15} shows that, for any graph $G$, if we construct a graph $G'$ as follows: we include each edge $e$ in $G'$ with probability $p_e \ge \Omega(\frac{w_e \log n}{\epsilon^2 k_e})$, and give weight $\frac{w_e}{p_e}$ if it gets chosen, then $G'$ is a $(1 \pm \epsilon)$ cut sparsifier of $G$ with high probability.
\subsection{Graph matrices, leverage scores, and spectral sparsifiers}\label{sec:gmat}
Fix an arbitrary orientation of all possible $\binom{n}{2}$ edge slots of the graph.
Let $B \in \mathbb{R}^{\binom{n}{2} \times n}$ be the {\em edge-vertex incidence matrix} of an undirected, unweighted complete graph over $n$ vertices. That is, for every edge $e=(u,v)$ oriented from $u\to v$,
there is a row $b_e\in\mathbb{R}^{n}$ in $B$ corresponding to $e$ such that the column $u$ has value $1$,
the column $v$ has value $-1$, and all other columns have value $0$.
We also write $b_e$ as $b_{u,v}$.
For a graph $G$,
we write $B_G\in\mathbb{R}^{\binom{n}{2}\times n}$ to denote the matrix
obtained from $B$ by zeroing out rows corresponding to absent edges in $G$.
Given any weighted graph $G$, let $W_G \in \mathbb{R}^{\binom{n}{2} \times \binom{n}{2}}$ be the diagonal matrix whose diagonal entries are the weights of the edges corresponding to them, i.e. $(W_{G})_{ee} = w_e$. If an edge $e$ is not present in the graph, then
$(W_G)_{ee} = 0$. The {\em Laplacian matrix} of $G$ is given by $L_G = B_G^{\top} W_G B_G = (W_G^{1/2} B_G)^{\top} (W_G^{1/2} B_G)$.
Notice that for unweighted graphs, we have $L_G = B_G^T B_G$.
For any edge $e$, %
its {\em leverage score} $\tau_e$ is given by $\tau_e = (\sqrt{w_e} b_e)^{\top} L^{\dagger}_G (\sqrt{w_e} b_e) = w_e b_e^{\top} L^{\dagger}_G b_e$, where $L^{\dagger}_G$ is the Moore-Penrose pseudoinverse of the Laplacian matrix $L_G$.
\begin{fact}
The sum of the leverage scores of all edges $\sum_e \tau_e = \mathrm{rank}(W_G^{1/2} B_G) \le n-1$.
\end{fact}
In general, for {\em any} matrix $C\in\mathbb{R}^{m\times n}$ whose $i^{\mathrm{th}}$ row is denoted by
$c_i\in\mathbb{R}^{n}$,
we define the leverage score of the $i^{th}$ row by $\tau_i = c_i^T (C^T C)^{\dag} c_i$,
and we once again have $\sum_{i=1}^{m} \tau_i = \rank{C}$.
If we view the graph $G$ as an electrical network where each edge $e$ has resistance $1/w_e$,
then the {\em effective resistance} between the two vertices $s,t$ is given by $r_{s,t} = b_{s,t}^{\top} L^{\dagger}_G b_{s,t}$.
For an edge $e = (s,t)$, we also define its effective resistance as $r_e = r_{s,t}$.
Thus, the leverage score $\tau_e = w_e r_e$.
If we inject $f$ units of electrical flow into a vertex $s$ and extract $1$ unit from a vertex $t$,
then $f L^{\dagger} b_{s,t} \in \mathbb{R}^{n}$ is referred to as {\em the set of vertex potentials induced by the electrical flow}.
The relation between vertex potentials of the electrical flow is characterized by Ohm's Law.
\begin{fact}[Ohm's Law]
Let $x = f L^{\dag} b_{s,t}\in\mathbb{R}^{n}$ be the set of vertex potentials when we send
$f$ units of electrical flow
from $s$ to $t$. Then
we have $x_s - x_t = f (b_{s,t} L^{\dag} b_{s,t})$.
Moreover, for any edge $e = (u,v)$ with $x_u \geq x_v$,
the flow on this edge is in the direction of $u\to v$ and has amount exactly
$w_e (x_u - x_v)$.
\end{fact}
It is also known that the vertex potentials induced by an electrical flow minimizes the total energy.
Specifically, for an arbitrary set of vertex potentials $x\in \mathbb{R}^{n}$,
we define its {\em energy} to be $x^T L_G x = \sum_{e = (u,v)\in G} w_e (x_u - x_v)^2$,
and define its {\em normalized energy with respect to vertices $s,t$} to be $\frac{x^T L_G x}{(x^T b_{s,t})^2}$
(i.e. the energy divided by $(x_s - x_t)^2$).
Then we have:
\begin{fact}\label{fact:energymin}
The vertex potentials induced by an electrical flow from $s$ to $t$ minimizes
the normalized energy with respect to $s,t$.
That is, for any $f > 0$, we have
$$ f L^{\dag} b_{s,t} \in \argmin_{x\in\mathbb{R}^{n}} \frac{x^T L_G x}{(x^T b_{s,t})^2}, $$
and thus, by plugging in $x = L^{\dag} b_{s,t}$,
the smallest normalized energy w.r.t. $s,t$ is
\begin{align*}
\min_{x\in\mathbb{R}^{n}} \frac{x^T L_G x}{(x^T b_{s,t})^2} =
\frac{ b_{s,t} L_G^{\dag} L_G L_G^{\dag} b_{s,t}}{(b_{s,t}^T L_G^{\dag} b_{s,t})^2} =
\frac{b_{s,t} L_G^{\dag} b_{s,t}}{(b_{s,t}^T L_G^{\dag} b_{s,t})^2} =
\frac{1}{{b_{s,t}^T L_G^{\dag} b_{s,t}}},
\end{align*}
exactly $1$ over the effective resistance between $s,t$.
\end{fact}
Given a graph $G$ with Laplacian matrix $L_G$, a $(1 \pm \epsilon)$-spectral sparsifier $G'$ is a graph with Laplacian matrix $L_{G'}$ such that for any $x \in \mathbb{R}^{n}$, $x^{\top} L_G' x \in (1 \pm \epsilon) x^{\top} L_G x$. In other words, $\frac{1}{1+\epsilon} L_G \preceq L_{G'} \preceq (1+\epsilon) L_G$ (recall Definition~\ref{def:matrixorder}). %
We also use $L_G \approx_{1+\epsilon} L_{G'}$ to denote the same relation between $L_G$ and $L_{G'}$.
For two scalars $a,b\geq 0$, we also write $a\approx_{1+\epsilon} b$ to denote $\frac{a}{1+\epsilon} \leq b \leq (1 + \epsilon) a$.
If we sample each edge of a graph with probability proportional to its leverage score or larger,
and reweight it accordingly,
then with high probability we get a spectral sparsifier.
In fact, this sampling process gives a good spectral approximation for any $C\in\mathbb{R}^{m\times n}$
\begin{theorem} [\cite{SpielmanS11,Tropp12}] \label{clm:lev-sample}
Let $\epsilon > 0$.
Given a matrix $C \in \mathbb{R}^{m \times n}$, let $p_1,\ldots,p_m\in\mathbb{R}$ be such that
$1 \geq p_i \geq \min\setof{1, 100 \tau_i \epsilon^{-2}\log n}$ for all $i \in [m]$.
Let $\tilde{W}\in\mathbb{R}^{m\times m}$ be a diagonal matrix such that
$\tilde{W}_{ii} = 1/p_i$ with probability $p_i$ and $\tilde{W}_{ii} = 0$ otherwise.
Then with high probability,
$$
C^T \tilde{W} C \approx_{\epsilon} C^T C.
$$
\end{theorem}
Recall that when $C = W_G^{1/2} B_G$, we have $L_G = C^T C$.
Thus if we sample each edge $e$ with probability $p_e = \min\setof{1, 100 \tau_e \epsilon^{-2} \log n}$
and reweight it to $w_e / p_e$ if sampled, we get, by the claim above
and that $\sum_{e} \tau_e \leq n - 1$, a $(1+\epsilon)$-spectral sparsifier of $O(n\epsilon^{-2} \log n)$ edges.
We could also do an oversampling, where we sample each edge $e$ with some probability
$p_e \geq \min\setof{1, 100 \tau_e \epsilon^{-2} \log n}$ and also reweight it to $w_e/p_e$ if sampled,
and then we get a $(1 + \epsilon)$-spectral sparsifier of $O(\sum_{e} p_e)$ edges.
\subsection{A model oblivious algorithm for weighted cut sparsification}
We present a model-oblivious algorithm $\WGM$ for weighted cut sparsification in Figure~\ref{fig:wcutalg},
which generalizes the idea presented in~\cite{RubinsteinSW18} for unweighted graphs.
Here, we assume the edge weights of the input graph are between $[1,U]$ for some $U\geq 1$ that is known
to us.
We characterize the performance of the algorithm in the lemma below.
\begin{lemma}\label{prop:cutsparsifier}
Fix $\alpha = 800,\beta = 400,\gamma = 100$ at Line~\ref{line:abc} of the algorithm $\WGM$.
Let $\epsilon$ be an arbitrary number in $(0,1)$.
Then,
the algorithm $\WGM$ outputs, with high probability, a $(1+\epsilon)$-cut sparsifier of the input graph $G$
with $O(n \epsilon^{-2} \log n)$ edges.
\end{lemma}
To prove the proposition, we will need the following
lemma.
\begin{lemma}\label{lem:strengthl}
In each iteration $\ell$, with high probability:
\begin{enumerate}
\item The edges in $\Gtil$ that survive the contractions have strength at most $2^{\ell-1}$. \label{item:strengthupw}
\item The edges in $\Gtil$ that are within the components $C_1,\ldots,C_r$ have strength
$[2^{\ell-4},2^{\ell}]$. \label{item:strengthlbw}
\end{enumerate}
\end{lemma}
\begin{proof}
We show that at the end of each iteration $\ell$,
the edges within $C_1,\ldots,C_r$ have strength $\geq 2^{\ell-4}$,
and the edges between them have strength at most $2^{\ell-1}$.
Then by noting that all edges in the graph have strength at most $n U$,
the lemma follows by an induction on $\ell$.
Consider iteration $\ell$ of the for loop.
Let $V_1,\ldots,V_{t}$ be the partition of $\Gtil$ into
maximal $2^{\ell-4}$-strongly connected components.
Then by Fact~\ref{fact:removecut} there exists a way to arrive at these components by starting from the entire graph $\Gtil$
and iteratively removing a cut with size $< 2^{\ell-4}$.
By applying a Chernoff bound and a union bound over the sequence of (at most $n-1$) cuts
that we remove in this process,
we have that after sampling, each of these cuts has size at most $50 \log n$.
As a result, the partition $C_1,\ldots,C_r$ of $G_{\ell}$ into maximal $100 \log n$-strongly connected components is a refinement
of $V_1,\ldots,V_{t}$.
This implies that all edges within $C_1,\ldots,C_r$ have strength $\geq 2^{\ell-4}$.
Now consider the partition $V_1,\ldots,V_{s}$ of $\Gtil$ into maximal $2^{\ell-1}$-strongly connected components.
After sampling at Line~\ref{line:sample3}, these components still have min cut $\geq 100 \log n$ with high probability.
Thus, the partition $C_1,\ldots,C_r$ of $G_{\ell}$ into maximal $100\log n$-strongly connected components
is a coarsening of $V_1,\ldots,V_{s}$. This implies that all edges going across different $C_i$'s
have strength at most $2^{\ell-1}$. This finishes the proof of the lemma.
\end{proof}
\begin{proof}[Proof of Lemma~\ref{prop:cutsparsifier}]
By Lemma~\ref{lem:strengthl}, whenever we sample an edge $e$ of the graph, we sample it
with probability $p_e \geq \min\setof{1, 50 w_e k_e^{-1} \epsilon^{-2} \log n }$ and re-weight it to $w_e / p$.
Moreover, in the last iteration $\ell = 0$, all edges are within the components $C_1,\ldots,C_r$
(since each edge's strength is at least $1$). Therefore ultimately all edges get sampled in our algorithm.
As a result, we get with high probability a $(1+\epsilon)$-cut sparsifier. %
On the other hand, the probability $p_e$ with which we sample $e$ also satisfies
$p_e \leq 800 w_e k_e^{-1} \epsilon^{-2} \log n$, and thus we get a sparsifier
with $O(n\epsilon^{-2}\log n)$ edges.
\end{proof}
\begin{figure}
\begin{algbox}
$H\gets \WGM(G, \epsilon)$\\
\begin{enumerate}
\item Initially, let the sparsifier $H$ be an empty graph, and let $\Gtil \gets G$.
\item Fix some sufficiently large constants $\alpha > \beta > \gamma > 0$. \label{line:abc}
\item For $\ell \in\setof {\log (nU), \log (nU) - 1, \ldots, 1,0}$:
\begin{enumerate}
\item Obtain $G_{\ell}$ from $\Gtil$ by keeping each edge w.p.
{\color{blue} $p_e := \min\setof{\beta \cdot w_e\cdot 2^{-\ell} \log n,1}$} and with
{\color{blue} weight $(\beta \cdot w_e\cdot 2^{-\ell} \log n)/p_e$}. \label{line:sample3}
\item In each connected component of $G_{\ell}$:
\begin{enumerate}
\item While there exists a cut of weight $\leq \gamma \cdot \log n$,
remove the edges in that cut and recurse on both sides; repeat until there is no such cut.
\item Let $C_1,\ldots, C_r$ be the connected components induced by the remaining edges.
\item For each edge $e$ in $\Gtil$ with endpoints in the same $C_i$,
add it to $H$ with probability
{\color{blue} $p_e:= \min\setof{\alpha \cdot w_e\cdot \epsilon^{-2} 2^{-\ell} \log n,1}$}
and with {\color{blue} weight $w_e/p_e$}. \label{line:sample4}
\item Update $\Gtil$ by contracting each of $C_1,\ldots,C_r$. \label{line:contraction2}
\end{enumerate}
\end{enumerate}
\end{enumerate}
\end{algbox}
\caption{Model oblivious algorithm for weighted cut sparsification.}
\label{fig:wcutalg}
\end{figure}
\subsection{Implementation by linear sketching}
Now we show how to implement the algorithm by linear sketching. The implementation is motivated by the techniques first used in~\cite{AhnGM12a}. Note that it suffices to implement the two edge sampling processes at Lines~\ref{line:sample3},~\ref{line:sample4} and the contraction operations at Line~\ref{line:contraction2}.
We note that the implementation of both sampling processes can be seen as the following task:
we first independently generate a uniformly random real number $R_e \in [0,1)$ for each edge slot $e$,
and then recover all edges satisfying $R_e < w_e p$ for some given $p$ (which in iteration $\ell$ equals $\beta 2^{-\ell} \log n$
for the first process and $\alpha \epsilon^{-2} 2^{-\ell} \log n$ for the second process).
Here we can generate the $R_e$'s offline, but have to recover the sampled edges using linear sketching.
We achieve the latter by repeatedly finding a spanning forest formed by the sampled edges.
We will show that the sampled edges can be found by performing $\Otil(1)$ iterations of spanning forest recovery.
We shall first show how to recover a spanning forest formed by the sampled edges via linear sketching.
To this end, we need a linear sketching subroutine that we call {\em weighted edge sampler},
with the following guarantee.
\begin{lemma} \label{clm:wei_l0}
Let $R_1,\ldots,R_N$ be $N$ numbers independently and uniformly at random
generated from $[0,1)$, let $c > 0$ be an arbitrary constant,
and let $p \in (0,1)$ be a parameter.
There exists a linear sketch of $\mathrm{polylog}(N,1/p)$ measurements with the following guarantee.
For any vector $w\in\mathbb{R}^{N}$, %
if there exists an entry $e$ such that $R_e \le w_e p$ and $w_e > N^{-c}$,
then with high probability,
the sketch recovers an index $e'$ such that $R_{e'} \leq (1 + \epsilon) w_{e'} p$
along with a $(1+\epsilon)$-approximate estimate of $w_{e'}$.
\end{lemma}
To find a spanning forest of the sampled edges (those with $R_e < w_e p$),
we apply the weighted edge sampler sketch to the incidence vectors of the vertices in $G$. Specifically, we fix an arbitrary orientation of each of the $\binom{n}{2}$ potential edges. Then for a vertex $u\in G$, we consider its incidence vector $b_u \in \mathbb{R}^{\binom{n}{2}}$ given by \begin{align}
(b_u)_e=
\begin{cases}
w_e & \text{$e\in G$, $u$ is $e$'s head} \\
-w_e & \text{$e\in G$, $v$ is $e$'s tail} \\
0 & \text{$e\notin G$ or $e$ does not touch $u$.}
\end{cases}
\end{align}
Let $t = \mathrm{polylog}(n)$. For any $i \in [t]$,
let $A_i$ be an independently generated weighted edge sampler sketching matrix.
For each $i\in [t]$ and vertex $u\in G$, we compute the sketch $A_i b_u$.
Thus we make $\Otil(n)$ measurements in total.
Now using the sketches we have taken,
we recover a spanning forest of sampled edges via a $\Otil(1)$-round process as follows.
In the first round, for each vertex $u$, we find an arbitrary outgoing edge using the weighted edge sampler sketch $A_1 b_u$. We then find all connected components induced by these edges, and add up the sketches $A_2 b_u$ of vertices within the same component. Note that the edges within the same component cancel out in the summation, so the resulting sketches are in fact taken over the outgoing edges of each component. As a result, in the next round we are able to find an outgoing edge of each component.
We then proceed similarly in the $i^{\mathrm{th}}$ round using sketches $A_{i} b_u$'s.
Since in each round, the number of components is at least reduced by a factor of $2$, we can find a spanning forest in $O(\log n)$ rounds of this process.
In order to iteratively find $\Otil(1)$ edge-disjoint spanning forests,
each time we find one,
we ``delete'' the found edges from the other linear sketches, and restart the $O(\log n)$-round process above.
Note that however, since we do not have the exact weights of the edges (Lemma~\ref{clm:wei_l0} only gives approximation of them),
we do not delete the found edges completely, bur rather decrease each of their weights by an $\Omega(1)$ factor.
Finally, to implement the contraction operations at Line~\ref{line:contraction2}, we once again add the sketches of the vertices within each contracted component, just as we did in finding spanning forests. Then starting from the next iteration, the sketches work for the contracted graph $\Gtil$.
We conclude this subsection by proving that the sampled edges can be recovered
by $\Otil(1)$ edge-disjoint spanning forests.
\begin{lemma}
The edges in $G_{\ell}$ can be found by $O(\log^2 n)$ edge-disjoint spanning forests.
\end{lemma}
\begin{proof}
We first show that the edges in $G_{\ell}$ all have low strength.
\begin{claim}
Every edge in the graph $G_{\ell}$ has strength $O(\log n)$.
\end{claim}
\begin{proof}
By Lemma~\ref{lem:strengthl}, we know that at the beginning of iteration $\ell$,
all edges in $\Gtil$ have strength at most $2^{\ell}$.
This means that there is a way of removing all edges in the graph
by iteratively removing a cut of size $\leq 2^{\ell}$.
Then after sampling at Line~\ref{line:sample3}, these cuts all have size $O(\log n)$.
Thus it follows that all edges in $G_{\ell}$ have strength $O(\log n)$.
\end{proof}
Let $C > 0$ be a constant such that all edges in $G_{\ell}$ have strength $\leq C \log n$.
Then this claim implies that $G_{\ell}$ is {\em uniformly sparse}, in the sense that
for any vertex induced subgraph $G[S]$ where $S\subseteq V$,
the number of edges in $G[S]$ is $(|S| - 1) C \log n$.
Indeed, each edge in $G[S]$ has strength only smaller than in $G$,
and thus all edges in $G[S]$ can be removed by iteratively (for at most $|S|-1$ times) removing a cut
of size $C \log n$.
This in particular means that at any point,
a spanning forest contains an $1 / (C \log n)$ fraction of the total remaining edges.
Thus $O(\log^2 n)$ edge-disjoint spanning forests recover all edges in $G_{\ell}$.
\end{proof}
\subsection{Proof of Lemma~\ref{clm:wei_l0}}
Roughly, we will simulate the non-uniform sampling process, where we want to sample
each $e$ with probability $w_e p$, by sampling
the elements uniformly, but at different geometric rates.
We will then essentially implement a {\em rejection sampling} process.
Specifically, when subsampling all elements at some uniform rate $q$, we use $\ell_1$-samplers to recover
a few elements that are sampled. We will then check if any one of the sampled elements $e$
satisfies $w_e p \approx q$. If so, we will output this element; otherwise, we go the next sampling
rate and repeat this step. We will show that with high probability, we will successfully recover
a desired element.
\let\oldell\ell
\renewcommand*{\ell}{\gamma}
\begin{proof}[Proof of Lemma~\ref{clm:wei_l0}]
We will analyze the linear sketch given in Figure~\ref{fig:wl0}.
The basic idea of the algorithm is as follows: for each $0 \le j \le \ell$, we maintain $\mathrm{polylog}(N,\frac{1}{p})$ number of $\ell_1$-sketches given by Proposition~\ref{l1-sampler} with failure probability $\delta_1,\delta_2 = N^{-100}$ that work for $w^j$ where $w^j$ is the vector generated by sampling each entry of $w$ with probability $2^{-j}$ (i.e. $q$ in the overview above). For each sampled element $e$, we check if $w_e p$ is indeed larger than $2^{-j}$, if so, $e$ indeed gets sampled and we can output $e$, otherwise $e$ might not get sampled, and we discard $e$. We prove that whenever there are elements that get sampled, we will find one of them with high probability.
\begin{figure}
\begin{algbox}
$e\gets \WL(w\in \mathbb{R}^{N},p, R_1,\ldots,R_N)$\\
\begin{itemize}
\item Let $t = 10^4 c \ell \log N$
where $\ell$ is the minimum integer such that $2^{-\ell}<p N^{-c}$.
\item For each $1 \le i \le t$ and each $0 \le j \le \ell$, independently generate an $\ell_1$-sampling
matrix $A_i^j$.
\item For each $0 \le j \le \ell$, check if $R_e > 2^{-j}$. If so, for each $1 \le i \le t$, replace the column of $A_i^j$ that corresponds to element $e$ by the $0$ vector.
\item Compute $A_i^j w$ for each pair of $i$ and $j$, and recover a sampled element $e_i^j$ and a weight $w'_{e_i^j}$, which is a $(1+\epsilon)$-factor approximation to $\sizeof{w_{e_i^j}}$.
\item For each recovered pair $e_i^j,w'_{e_i^j}$, check if $R_{e_i^j} \le w'_{e_i^j} p $. If there exists such an element $e_i^j$, output an arbitrary one of them, otherwise output $\mathsf{No Element}$.
\end{itemize}
\end{algbox}
\caption{Weighted edge sampler.}
\label{fig:wl0}
\end{figure}
The analysis is conditioned on the event that none of the $\ell_1$-samplers fails, which is a high probability event. Since the algorithm only uses $\ell_1$ samplers, we can without loss of generality assume that each element has positive weight. Note that the algorithm uses $t \ell$ number of $\ell_1$ sampling matrices in total, so the total number of measurements of this sketch is $\mathrm{polylog}(N,U,\frac{1}{p})$. Moreover, by Proposition~\ref{l1-sampler}, if the algorithm outputs an element $e$, we have $R_e \le w'_e \le (1+\epsilon) w_e p$. %
it is sufficient to prove that if there exists $e$ such that $R_e \le w_e p$, with high probability, we will not output $\textsf{No Element}$.
For any $1 \le k \le \ell$, let $S_k$ be the set of elements $e$ such that $2^{-k} \le w_e p < 2^{-k+1}$, and let $S_0$ be the set of elements $e$ such that $w_e p \ge 1$. For any $0 \le j \le \ell$, let $S_k^j$ be the set of elements $e \in S_k$ such that $R_e \le 2^{-j}$. For any $j$ and $k$, let $n_k = \sizeof{S_k}$, $n_k^j=\sizeof{S_k^j}$ and $W_k^j$ be the total weight of elements in $S_k^j$. The following claim follows from Chernoff bound.
\begin{claim} \label{clm:sam_con}
With high probability, for any $0 \le j,k \le \ell$, if $n_k > 1000 \cdot 2^j \log N$, then $\sizeof{2^j n_k^j - n_k} < n_k/2$, otherwise $n_k^j < 1500 \log N$.
\end{claim}
\begin{proof}
For each element $e \in n_k$, the probability that $e \in n_k^j = 2^{-j}$, so the size of $n_k^j$ is the sum of $\sizeof{n_k}$ independent random $0/1$ variables each with expectation $2^{-j}$. The expected size of $n_k^j$ is $2^{-j} n_k$. If $\sizeof{n_k} > 1000 \cdot 2^j \log N$, by Chernoff bound, the probability that $\sizeof{2^j n_k^j - n_k} > n_k/2$ is at most $< 2 e^{\frac{n_k}{2^{j+4}}} < N^{-50}$.
If $\sizeof{n_k} \le 1000 \cdot 2^j \log N$, the probability that $\sizeof{n_k^j} \ge 1500 \log N$ is at most the probability of the case when $n_k = 1000 \cdot 2^j \log N$. So the probability is less than $N^{-50}$.
\end{proof}
Let $k^*$ be the index that maximizes $\frac{n_{k^*}}{2^{k^*}}$. If $n_{k^*} > 1000 \cdot 2^{k^*} \log N$, by Claim~\ref{clm:sam_con}, $n_{k^*}^{k^*}> \frac{n_{k^*}}{2^{k^*+1}}$. Since each element in $S_{k^*}$ has weight at least $2^{-k^*}/p$, $W_{k^*}^{k^*} > \frac{n_{k^*}}{2^{2k^*+1}p}$. On the other hand, for any $k > k^*$, we have $n_k^{k^*} < \max \{ 1500 \log N, \frac{3n_k}{2^{k^*+1}}\}$ by Claim~\ref{clm:sam_con}. Since each element in $S_k$ has weight at most $2^{-k+1}/p$, we have $W_k^{k^*} < \max\{ \frac{3000 \log N}{2^k p} , \frac{3n_k}{2^{k+k^*+1}}\}$. As $n_{k^*} > 1000 \cdot 2^{k^*} \log N$, $\frac{3000 \log N}{2^k p} < \frac{3 n_{k^*}}{2^{k+k^*} p} < 3 W_{k^*}^{k^*}$. Also, by definition of $k^*$, $\frac{n_{k^*}}{2^{k^*}} \ge \frac{n_k}{2^k}$, which means $\frac{3n_k}{2^{k+k^*+1}} \le \frac{3n_{k^*}}{2^{2k^*+1}} < 3 W_{k^*}^{k^*}$. Thus, we have $W_k^{k^*} < 3W_{k^*}^{k^*}$. Note that for any $1 \le i \le t$, $e_i^{k^*}$ is obtained by an $\ell_1$ sampler from $\cup_{k=0}^{\ell} S_k^{k^*}$, so with probability at least $\frac{1}{3\ell + 3}$, $e_i^{k^*}$ is an element in $S_k$ such that $k \le k^*$. In this case, $R_e \le 2^{-k^*} \le w_e p$, and so the algorithm will output an element that gets sampled. Since $t=10^4 \ell \log N$, there exists such an $i$ with high probability.
If $n_{k^*} \le 1000 \cdot 2^{k^*} \log N$, then for any $k$, $\frac{n_k}{2^k} \le 1000 \log N$. Let $e$ be a maximum weight element such that $R_e < w_e p$. Suppose $e \in S_k$, then we have $R_e < w_e p < 2^{-k+1}$. Let $k'$ be the largest index such that $e \in S_k^{k'}$, we have $k' \ge k-1$. For any $e'$ such that $e' \in S_{k''}$ with $k''< k$, by definition of $e$ and $k$, $R_{e'} > w_{e'} p \ge 2 ^ {-k''} \ge 2^{-k+1} \ge 2^{k'}$, which means $e' \notin S_{k''}^{k'}$. So $S_{k''}^k = \emptyset$ for any $k''<k$. On the other hand, for any $k'' \ge k$, $n_{k''}^{k'} < \max\{ 1500 \log N , \frac{3n_{k''}}{2^{k'}+1}\}$ by Claim~\ref{clm:sam_con}. Since any element in $S_{k''}$ has weight at most $2^{-k''+1}/p$, $W_{k''}^{k'} < \{ \frac{3000 \log N}{2^{k''} p} , \frac{3n_{k''}}{2^{k''+k'+1}}\}$. Since $w_e \ge \frac{1}{2^k p}$ and $k'' \ge k$, $\frac{3000 \log N}{2^{k''} p} \le 3000 \log N w_e$. Moreover, since $\frac{n_{k''}}{2^{k''}} \le 1000 \log N$, $\frac{3n_{k''}}{2^{k''+k'+1}} \le \frac{3000 \log N}{2^{k'+1} p} \le 3000 \log N w_e$. So $W_{k''}^{k'} < 3000 \log N w_e$ for any $k'' \ge k$, which means for any $1 \le i \le t$, $e_i^{k'} = e$ with probability at least $\frac{1}{3000 (\ell+1) \log N}$. Since $t = 10^4 \ell \log N$, with high probability, there exists one $i$ such that $e_i^{k'} = e$ and the algorithm will output an element that gets sampled.
So in both cases, the algorithm succeeds with high probability.
\end{proof}
\let\ell\oldell
\subsection{A vertex sampling lemma}\label{sec:vertexsample}
In this subsection we prove a key vertex sampling lemma.
This lemma will enable us to recover heavy edges in $G$ by subsampling vertices at rate $\approx n^{-1/5}$
and then recovering edges in the vertex-induced subgraph of $G^{\sq}$.
\begin{lemma}[Vertex sampling lemma]\label{lem:vertex-sample}
Let $\omega\in (0,1)$.
Given a weighted graph $G$,
let $C$ be a vertex set obtained by including each vertex in $G$ with probability $\frac{\omega}{100n^{1/5}}$
independently.
For any edge $e$ in $G$ with leverage score $w_e b_e^T L_G^{\dag} b_e \geq \omega$,
conditioned on $e\in G[C]$,
with probability at least $.1$,
its leverage score in $G^{\sq}[C]$ satisfies
$w_e^2 b_e^T L_{G^{\sq}[C]}^{\dag} b_e \geq 1/1000$.
\end{lemma}
Roughly, our proof of the lemma proceeds as follows: (i) group vertices according to their potentials
induced by an electrical flow between the endpoints of $e$ in $G$;
(ii) analyze the structure of the edges in the vertex-induced subgraph based on their weights
and the potential difference between their endpoints; (iii)
explicitly construct a set of vertex potentials in $G^{\sq}$ that certifies
the heaviness of the edge $e$.
\begin{proof}[Proof of Lemma~\ref{lem:vertex-sample}]
Let $e=(s,t)$ and without loss of generality, assume $w_e=1$
(since we could always scale all edge weights simultaneously without changing any leverage scores).
Since the leverage score $\tau_e \geq \omega$,
we also have that the effective resistance $b_e^{T} L_G^{\dagger} b_e \geq \omega$.
We use the electrical network view of the graph $G$, and let $x = \frac{L_G^{\dag} b_e}{b_e^T L_G^{\dag} b_e} \in\mathbb{R}^{n}$
be the set of vertex potentials induced by an
electrical flow from $s$ to $t$
of $\frac{1}{b_e^{T} L_G^{\dagger} b_e}\leq \frac{1}{\omega}$ units.
We also assume without loss of generality $x_t=0$ (since we could always shift all vertex potentials by $x_t$ otherwise).
Since $b_e^{T} L_G^{\dagger} b_e$ is the effective resistance between $s$ and $t$, we have $x_s=1$ by Ohm's law.
Moreover, by Fact~\ref{fact:energymin},
the normalized energy of $x$ w.r.t. $s,t$ satisfies that
\begin{align*}
x^T L_G x =
\frac{1}{{b_e^T L_G^{\dag} b_e}} \leq 1/\omega.
\end{align*}
We now partition the vertices other than $s$ and $t$
into $n^{4/5}$ groups based on their potentials.
Specifically, the $i^{\mathrm{th}}$ group $S_i$ contains all vertices satisfying
$(i-1)\cdot n^{-4/5}\leq x_u \leq i\cdot n^{-4/5}$,
where we break ties arbitrarily.
For an edge $f = (u,v)$, we say $f$ passes through $S_i$ if
$x_u \leq (i-1)\cdot n^{-4/5} < i\cdot n^{-4/5} \leq x_v$ or
$x_v \leq (i-1)\cdot n^{-4/5} < i\cdot n^{-4/5} \leq x_u$.
\begin{claim}
For any $i\in [n^{4/5}]$,
the total weight of edges that pass through $S_i$ is at most
$n^{4/5} / \omega$.
\label{claim:totalw}
\end{claim}
\begin{proof}
Consider an edge $f = (u,v)$ that passes through $S_i$,
and assume without loss of generality $x_u \geq x_v$.
By Ohm's law, the flow on edge $f$ is in the direction $u\to v$ and has amount
$w_f (x_u - x_v) \geq w_f n^{-4/5}$. Since the total amount of flow across the cut
\begin{align*}
\kh{\setof{t}\cup S_1\cup\ldots \cup S_{i},\ S_{i+1}\cup\ldots \cup S_{n^{4/5}}\cup\setof{s}}
\end{align*}
is at most $1/\omega$, we have
\begin{align*}
\sum_{\text{$f$: $f$ passes through $S_i$}} w_f n^{-4/5} \leq 1/\omega,
\end{align*}
which means that the total weight of such edges is at most $n^{4/5}/\omega$.
\end{proof}
We now consider what happens when we look at a vertex-induced subgraph $G[C]$
where $C$ is obtained by including each vertex (other than $s,t$)
with probability $\frac{\omega}{100 n^{1/5}}$, and then also including $s,t$.
We say an edge $f = (u,v)$ is {\em intermediate} if $\setof{u,v}\cap\setof{s,t} = \emptyset$.
We say a group $S_i$ is {\em good} if (i)
none of the vertices in $S_i$ gets sampled in $C$, and (ii) all intermediate edges that pass through $S_i$ and have
both endpoints in $C$ have weight at most $n^{2/5}$.
We say $S_i$ is {\em bad} otherwise.
\begin{claim}\label{claim:allgood}
With probability at least $2/3$, the number of good $S_i$'s with $i\in(\frac{1}{4}n^{4/5},\frac{3}{4}n^{4/5}]$
is at least $n^{4/5}/20$.
\end{claim}
\begin{proof}
First, by Markov's inequality,
at least $.8$ fraction of the $S_i$'s with $i\in(\frac{1}{4}n^{4/5},\frac{3}{4}n^{4/5}]$
have size at most $10 n^{1/5}$.
For any fixed $S_i$ with $|S_i|\leq 10n^{1/5}$,
we have
\begin{align*}
\ex{\sizeof{S_i\cap C}} \leq \frac{|S_i| {\omega}}{100 n^{1/5}} \leq \frac{{\omega}}{10}
\leq \frac{1}{10}.
\end{align*}
Therefore, once again by Markov's inequality,
the probability that (i) happens for any fixed $S_i$ with $|S_i|\leq 10n^{1/5}$ is at least $.9$.
On the other hand, by Claim~\ref{claim:totalw},
the total number of edges with weight $> n^{2/5}$ that pass through $S_i$
is at most $n^{2/5}/\omega$.
The probability of any intermediate edge belonging to $G[S]$ is $\kh{\frac{{\omega}}{100 n^{1/5}}}^2 =
\frac{\omega^2}{10000 n^{2/5}}$.
These combined give us that the expected number of intermediate edges with weight $> n^{2/5}$ that pass through $S_i$
and have both endpoints in $C$ is at most $1/10000$.
Now an application of Markov's inequality gives that
(ii) happens for $S_i$ with probability at least $1 - 1/10000$.
Therefore, by a union bound, each $S_i$ with $|S_i|\leq 10n^{1/5}$
is good with probability at least $.89$.
Thus the expected number of bad $S_i$'s with $|S_i|\leq 10 n^{1/5}$
is at most $.11 n^{4/5}$.
By Markov's inequality, the number of
bad $S_i$'s with $|S_i|\leq 10 n^{1/5}$
is at most $.33 n^{4/5}$ with probability $\geq 2/3$.
Thus, with probability at least $2/3$,
the number of good $S_i$'s with $i\in(\frac{1}{4}n^{4/5},\frac{3}{4}n^{4/5}]$
is at least $.4 n^{4/5} - .33 n^{4/5} \geq .05 n^{4/5}$, as desired.
\end{proof}
We will now construct a set of vertex potentials (call it $y\in\mathbb{R}^{|C|}$) in $G[C]$ from $x$.
We will show that the energy of the new set of potentials is small in $G^{\sq}[C]$ (i.e. even with edge weights squared),
but the potential difference between $s,t$ is still large, which result in
a small normalized energy w.r.t. $s,t$, and thus certify the ``heaviness'' of edge $(s,t)$ in $G^{\sq}[C]$.
Specifically, we obtain $y$ by ``collapsing'' the vertex potentials within each bad $S_i$,
so that the intermediate edges that do not pass through any good $S_i$'s will have both endpoints getting the same potential.
To take care of the edges incident on $s$ or $t$ that span less than $1/4$ fraction
of the groups, we will also
collapse the vertex potentials in the range $[0,1/4]$ and $[3/4,1]$.
Precisely, $y_u$ is given as follows for each $u\in C$:
\begin{enumerate}
\item If $x_u \geq 3/4$, then set $y_u \gets 1$.
\item Otherwise, suppose $x_u\in S_i$ for some $i\leq \frac{3}{4} n^{4/5}$.
Count the number of good $S_j$'s with $j\in (\max\setof{i,\frac{1}{4}n^{4/5}}, \frac{3}{4} n^{4/5}]$ and let
$k$ denote that number. Then set $y_u \gets 1 - k n^{-4/5}$.
\end{enumerate}
\begin{claim}\label{claim:yx}
$y$ satisfies the following properties:
\begin{enumerate}
\item For any $u,v$, $|y_u - y_v| \leq |x_u - x_v|$. \label{item:uv}
\item With probability at least $2/3$,
$y_s - y_t \geq 1/20$. \label{item:st}
\item With probability at least $.8$,
$y^T L_{G^{\sq}[C]} y \leq 2$. \label{item:yty}
\end{enumerate}
\end{claim}
\begin{proof}
Note that for any $i$, $C\cap S_i \neq \emptyset$ implies that $S_i$ is bad.
Therefore, by our construction, all vertices within the same bad $S_i$ will end up having
the same potentials in $y$. As a result, for any $u,v$,
$|y_u - y_v|$ equals $n^{-4/5}$ times the number of good groups $S_i$ between
$x_u$ and $x_v$ such that $i\in(\frac{1}{4} n^{4/5}, \frac{3}{4} n^{4/5}]$, which implies~\ref{item:uv}.
By Claim~\ref{claim:allgood}, with probability $2/3$, the number of good $S_i$'s
with $i\in (\frac{1}{4} n^{4/5}, \frac{3}{4} n^{4/5}]$ is at least $n^{4/5}/20$.
Thus we have~\ref{item:st}.
We then prove~\ref{item:yty}.
First note that for edges that do not pass through any good $S_i$, both their endpoints
have the same potential in $y$. So the total energy contributed by these edges is zero.
Now the remaining edges can be divided into two types: (A)
edges that are incident on $s$ or $t$; (B) intermediate edges
that pass through some good $S_i$.
By the definition of good $S_i$'s, edges of type (B) have weight at most $n^{2/5}$ each.
For edges of type (A), they are of the form $(s,u)$ or $(v,t)$. If $x_u \geq 3/4$ or
$x_v \leq 1/4$, then once again both endpoints of the edge have the same potential in $y$,
and the energy contribution from such edges is zero.
We thus focus on the edges of type (A) such that $x_u < 3/4$ or $x_v > 1/4$,
and refer to those edges as type (A').
For any type (A') edge $f = (a,b)$,
we have $|x_a - x_b|\geq 1/4$,
and thus by Ohm's law the amount of flow on $f$ is at least $w_f/4$.
Since the total amount of flow going out of $s$ or going into $t$
is upper bounded by $1/\omega$,
the total weight of type (A') edges is at most $8/\omega$.
This means that the number of type (A') edges $f$ with $w_f > 1$ is at most $8/\omega$.
Thus the expected number of such edges in $G[C]$ is at most
$(8/\omega) \frac{\omega}{100 n^{1/5}} \leq \frac{1}{10}$.
By Markov's inequality, none of such edges is in $G[C]$ with probability at least $.9$
(call this event $\mathcal{E}_1$).
As for any type (A') edge $f = (a,b)$ with $w_f \leq 1$, we have
$w_f^2 (y_a - y_b)^2 \leq w_f (x_a - x_b)^2$.
This combined with the fact that each $f$ belongs to $G[C]$ with probability $\frac{\omega}{100 n^{1/5}}$,
the expected contribution of type (A') edges with weight at most $1$ to $y^T L_{G^{\sq}[C]} y$ is at most
$\frac{\omega}{100 n^{1/5}}\cdot x^T L_G x \leq \frac{1}{100 n^{1/5}} \leq \frac{1}{100}$.
Thus this contribution does not exceed $1$ with probability $.99$ (call this event $\mathcal{E}_2$).
Finally we consider type (B) edges. Since their weights are at most $n^{2/5}$ each, squaring the edge weights
blows up the energy on them by at most a factor of $n^{2/5}$.
On the other hand, since these are intermediate edges,
the probability that any such edge belongs to $G[C]$ is
$\kh{ \frac{\omega}{100 n^{1/5}} }^2 = \frac{\omega^2}{10000n^{2/5}}$.
Therefore, the expected total contribution of type (B) edges to $y^T L_{G^{\sq}[C]} y$ is at most
$\frac{\omega^2}{10000 n^{2/5}}\cdot n^{2/5} x^T L_G x \leq \frac{\omega}{10000} \leq \frac{1}{10000}$.
Thus this contribution does not exceed $1$ with probability $1 - 1/10000$
(call this event $\mathcal{E}_3$).
By a union bound, $\mathcal{E}_1,\mathcal{E}_2,\mathcal{E}_3$ simultaneously happen with probability at least $.8$,
in which case we have $y^T L_{G^{\sq}[C]} y \leq 2$.
\end{proof}
By Claim~\ref{claim:yx} and a union bound over the events in the claim,
we have with probability at least $.1$ that
the normalized energy of $y$ with respect to $s,t$ in $G^{\sq}[C]$
is $\frac{y^T L_{G^{\sq}[C]} y}{(y_s - y_t)^2} \leq 800$.
\end{proof}
\subsection{Recovery of heavy edges}\label{sec:heavyedge}
Armed with the vertex sampling lemma,
we are now ready to design a linear sketch to recover all heavy edges in $G$.
In doing so, we will also need to invoke two other linear sketches
for sparsifying and recovering heavy edges in $G^{\sq}$, respectively.
We summarize their performance in the two lemmas below,
and prove them later in Section~\ref{sec:sparsifyG^2}.
We note that the sketch designed in Lemma~\ref{lem:shv} is basically a direct application of $\ell_2$-heavy hitters.
The sketch designed in Lemma~\ref{lem:sparsifyG^2} is essentially a reduction from sparsification to heavy edge recovery,
which will be very similar to our main linear sketching algorithm for weighted spectral sparsification
in the next subsection (Section~\ref{sec:algmainspectral}).
\begin{restatable}{lemma}{sqsparsify}
For any parameter $\epsilon_2 > 0$ and any integer $n$,
there exists a linear sketch with sketching matrix
$\SSQ\in \mathbb{R}^{n \epsilon_2^{-4} \mathrm{polylog}(n,\epsilon_2^{-1},\frac{w_{\max}}{w_{\min}})\times \binom{n}{2}}$
and recovery algorithm $\SQR$
such that, given an input graph $G$ of $n$ vertices with weight vector $w_G$,
$\SQR(\SSQ w_G)$ returns
a $(1+\epsilon_2)$-spectral sparsifier of $G^{\sq}$ with high probability.
\label{lem:sparsifyG^2}
\end{restatable}
\begin{restatable}{lemma}{lemshv}
For any parameters $\omega_3,\epsilon_3\in(0,1)$ and any integer $n$,
there exists a linear sketch with sketching matrix
$\SSV\in\mathbb{R}^{n \omega_3^{-1} \epsilon_3^{-2} \mathrm{polylog}(n)\times \binom{n}{2}}$ and recovery algorithm
$\SVR$ such that, for an input graph $G$ of $n$ vertices with weight vector $w_G$ and
another graph $\Gtil$,
$\SVR(\SSV w_G,\Gtil)$ recovers a set of edges $F$ in $G$ along with estimates of their weights $\tilde{w}_f$'s
such that with high probability
\begin{enumerate}
\item $F$ contains all edges $e$ satisfying
\begin{align*}
\frac{((w_G)_e b_e^T L_{\Gtil}^{\dag} b_e)^2}{ b_e^T L_{\Gtil}^{\dag} L_{G^{\sq}} L_{\Gtil}^{\dag} b_e }
\geq \omega_3.
\end{align*}
\item All edges $f\in F$ satisfy $\frac{1}{1 + \epsilon_3} (w_G)_f \leq \tilde{w}_f \leq (1 + \epsilon_3) (w_G)_f$.
\end{enumerate}
\label{lem:shv}
\end{restatable}
We remark that to understand the first guarantee of the above lemma, one should think of $\Gtil$
as a good spectral sparsifier of $G^{\sq}$, so that the numerator $((w_G)_e b_e^T L_{\Gtil}^{\dag} b_e)^2 \approx
((w_G)_e b_e^T L_{G^{\sq}}^{\dag} b_e)^2$
and the denominator
$b_e^T L_{\Gtil}^{\dag} L_{G^{\sq}} L_{\Gtil}^{\dag} b_e \approx
b_e^T L_{G^{\sq}}^{\dag} b_e$, and thus the LHS
$\approx (w_G)_e^2 b_e^T L_{G^{\sq}}^{\dag} b_e$, the leverage score of $e$ in $G^{\sq}$.
That is, this guarantee is essentially saying that all heavy edges in $G^{\sq}$ will be recovered.
We now describe the linear sketch for recovering heavy edges in $G$ in Figure~\ref{fig:hvs},
and characterize its performance in the lemma below.
\begin{figure}[!htbp]
\begin{algbox}
$\HVS(G,\omega_1,\epsilon_1)$
\quad
\begin{enumerate}
\item Let $t = \ceil{10000 \omega_1^{-2} n^{2/5}\log n}$ and
let $V_1,\ldots,V_t$ be vertex subsets, each obtained by
including each vertex in $G$ with probability $\frac{\omega_1}{100 n^{1/5}}$ independently.
\hfill \textcolor{blue}{(Subsample the vertices sufficiently many times to cover every edge)}
\item For each $i\in [t]$, let
$\SSQ_i \in \mathbb{R}^{|V_i| \epsilon_2^{-4} \mathrm{polylog}(|V_i|,\epsilon_2^{-1}, \frac{w_{\max}}{w_{\min}})\times \binom{|V_i|}{2}}$
be a sketching matrix with error $\epsilon_2 = .01$ (Lemma~\ref{lem:sparsifyG^2}),
and let $\SSV_i\in\mathbb{R}^{|V_i|\omega_3^{-1}\epsilon_3^{-2}\mathrm{polylog}(|V_i|)\times \binom{|V_i|}{2}}$
with $\omega_3 = 1/2000$ and $\epsilon_3 = \epsilon_1/100$ (Lemma~\ref{lem:shv}).
\item Concatenate the following sketches as $\SHV w_G$:
\begin{enumerate}
\item $\SSQ_1 w_{G[V_1]}, \ldots, \SSQ_t w_{G[V_t]}$, where
$w_{G[V_i]}\in\mathbb{R}^{\binom{|V_i|}{2}}$ is the weight vector of the vertex induced subgraph $G[V_i]$.
\hfill \textcolor{blue}{(Create a sparsification sketch for each vertex-induced subgraph)}
\item $\SSV_1 w_{G[V_1]}, \ldots, \SSV_t w_{G[V_t]}$.
\hfill \textcolor{blue}{(Create a heavy edge sketch for each vertex-induced subgraph)}
\end{enumerate}
\end{enumerate}
\quad
$\HVR(\SHV w_G)$ %
\begin{enumerate}
\item For each $i\in[t]$:
\begin{enumerate}
\item Use Lemma~\ref{lem:sparsifyG^2} to
recover from $\SSQ_i w_{G[V_i]}$ a sparsifier $\Gtil_i$ of $G^{\sq}[V_i]$.
\hfill \textcolor{blue}{($\Gtil_i$ is a $1.01$-spectral sparsifier of $G^{\sq}[V_i]$)}
\item Use the recovery algorithm from Lemma~\ref{lem:shv} to recover from
$\SSV_i w_G[V_i]$ and $\Gtil_i$ a set $F$ of edges along with their estimated weights $\tilde{w}_f$'s,
and mark all edges in $F$ as heavy.
\end{enumerate}
\hfill \textcolor{blue}{(Feed the sparsifier $\Gtil_i$ to the recovery algorithm
to get all heavy edges)}
\item Return all edges marked heavy along with the estimates
of their weights (if for some edge there are multiple estimates of its weight, pick
an arbitrary one).
\end{enumerate}
\end{algbox}
\caption{Linear sketch for recovering heavy edges in $G$.}
\label{fig:hvs}
\end{figure}
\begin{restatable}%
{lemma}{heavyedge}
For any parameters $\omega_1 \in (n^{-4/5}\log n,1)$, $\epsilon_1\in(0,1)$ and integer $n$,
there exists a linear sketch with sketching matrix
$\SHV\in\mathbb{R}^{n^{6/5}\omega_1^{-1} \epsilon_1^{-2}
\mathrm{polylog} (n, \frac{w_{\max}}{w_{\min}})
\times \binom{n}{2}}$
and recovery algorithm $\HVR$
such that,
for an input graph $G$ of $n$ vertices with weight vector $w_G$,
$\HVR(\SHV w_G)$ recovers a set $F$ of edges in $G$ along with estimates of their weights
$\tilde{w}_f$'s
such that with high probability
\begin{enumerate}
\item All edges $e$ whose leverage score in $G$ satisfy $(w_G)_e b_e^T L_{G}^{\dag} b_e \geq \omega_1$
belong to $F$. \label{item:g1}
\item All edges $f\in F$ satisfy $\frac{1}{1 + \epsilon_1} (w_G)_f \leq \tilde{w}_f \leq (1 + \epsilon_1) (w_G)_f$.
\label{item:g2}
\end{enumerate}
\label{lem:heavyedge}
\end{restatable}
We now prove Lemma~\ref{lem:heavyedge}
using Lemmas~\ref{lem:vertex-sample},\ref{lem:sparsifyG^2},\ref{lem:shv}.
\begin{proof}[Proof of Lemma~\ref{lem:heavyedge}
using Lemmas~\ref{lem:vertex-sample},\ref{lem:sparsifyG^2},\ref{lem:shv}]
\textbf{Number of linear measurements.}{
We observe that with high probability, $|V_i|\leq 100 \omega_1 n^{4/5}$ for all $i\in[t]$, and
thus each $\SSQ_i w_G[V_i] \in \mathbb{R}^{\omega_1 n^{4/5}\mathrm{polylog}(n,\frac{w_{\max}}{w_{\min}})}$
and each $\SSV_i w_G[V_i] \in \mathbb{R}^{\omega_1 n^{4/5} \epsilon_1^{-2}\mathrm{polylog}(n)}$.
Therefore, the total number of linear measurements is bounded by
\begin{align*}
t\cdot \omega_1 n^{4/5} \epsilon_1^{-2}\mathrm{polylog}(n,\frac{w_{\max}}{w_{\min}}) \leq
n^{6/5} \omega_1^{-1} \epsilon_1^{-2} \mathrm{polylog}(n,\frac{w_{\max}}{w_{\min}}).
\end{align*}
}
\textbf{Guarantee~\ref{item:g1}.}{
Consider fixing any edge $e$ with $w_e b_e^T L_G^{\dag} b_e \geq \omega_1$.
By Lemma~\ref{lem:vertex-sample},
for each $i\in [t]$, with probability $\frac{\omega_1^2}{1000 n^{2/5}}$,
we have $e\in G[V_i]$ and $w_e^2 b_e^T L_{G^{\sq}[V_i]}^{\dag} b_e\geq 1/1000$ (call this event $\mathcal{E}_i$).
Therefore at least one of $\mathcal{E}_1,\ldots,\mathcal{E}_t$ happens with probability
$\geq 1 - (1 - \frac{\omega_1^2}{1000 n^{2/5}})^{t} \geq 1 - 1/n^{-10}$.
Whenever an $\mathcal{E}_i$ happens, using the fact that $\Gtil_i$ is a $1.01$-spectral sparsifier
of $G^{\sq}[V_i]$ (by Lemma~\ref{lem:sparsifyG^2}),
we have
\begin{align*}
(w_G)_e b_e^T L^{\dag}_{\Gtil_i} b_e \geq \frac{1}{1.01}
(w_G)_e b_e^T L^{\dag}_{G^{\sq}[V_i]} b_e
\end{align*}
and
\begin{align*}
b_e^T L_{\Gtil_i}^{\dag} L_{G^{\sq}[V_i]} L^{\dag}_{\Gtil_i} b_e \leq 1.01
b_e^T L_{\Gtil_i}^{\dag} L_{\Gtil_i} L^{\dag}_{\Gtil_i} b_e =
b_e^T L^{\dag}_{\Gtil_i} b_e \leq 1.01^2
b_e^T L^{\dag}_{G^{\sq}[V_i]} b_e,
\end{align*}
and thus
\begin{align*}
\frac{((w_G)_e b_e^T L^{\dag}_{\Gtil_i} b_e)^2}{b_e^T L_{\Gtil_i}^{\dag} L_{G^{\sq}[V_i]} L^{\dag}_{\Gtil_i} b_e}
\geq
1.01^{-3} (w_G)_e^2 b_e^T L_{G^{\sq}[V_i]}^{\dag} b_e \geq 1.01^{-3}/1000 \geq 1/2000.
\end{align*}
By Lemmas~\ref{lem:shv},
with high probability, $e$ is among the recovered edges. Therefore by a union bound
over all such edges $e$, we have the desired result.
}
\textbf{Guarantee~\ref{item:g2}.}{
This follows directly from Lemma~\ref{lem:shv}.
}
\end{proof}
\subsection{Main algorithm for weighted spectral sparsification}\label{sec:algmainspectral}
We now show how to use the heavy edge recovery sketch in the previous section to obtain a
spectral sparsifier of $G$.
We first briefly summarize the main ideas.
The first idea is to use the iterative refinement process in~\cite{LiMP13}
as in the previous works on unweighted graphs~\cite{KapralovLMMS14,KapralovMMMNST20}.
That is, we consider, for some large $\alpha = \mathrm{poly}(w_{\max},n),\ t = \mathrm{polylog}(\epsilon^{-1},\frac{w_{\max}}{w_{\min}},n)$
and a constant $\beta\in(0,1)$,
the sequence of graphs
\begin{align*}
G + \alpha K_n, G + \beta \alpha K_n, G + \beta^2 \alpha K_n, \ldots,
G + \beta^t \alpha K_n,
\end{align*}
which have the properties that
\begin{enumerate}
\item $\alpha K_n$ is an $O(1)$-spectral sparsifier of $G + \alpha K_n$.
\item $G + \beta^k \alpha K_n$ is an $O(1)$-spectral sparsifier of $G + \beta^{k+1} \alpha K_n$ for all $k\geq 0$.
\item $G + \beta^t \alpha K_n$ is a $(1 + \epsilon)$-spectral sparsifier of $G$.
\end{enumerate}
The idea is then to iteratively obtain a sparsifier of each of these graphs,
where we use the sparsifier of $G + \beta^k \alpha K_n$ to guide the sparsification
of $G + \beta^{k+1} \alpha K_n$ (in particular, we use the sparsifier of the former
to estimate the effective resistances and leverage scores in the latter).
Thus it boils down to how to sparsify $G^{\supi{k+1}} := G + \beta^{k+1} \alpha K_n$ using heavy edge recovery.
\begin{remark}
We remark that, when given access to a heavy edge recovery sketch,
the sparsification of $G^{\supi{k+1}}$
is relatively easy to achieve in the unweighted case, for the following reason.
Consider, in an unweighted graph,
an edge $e = (s,t)$ with effective resistance (thus also leverage score)
$r_e$, and let $x_{s,t}\in\mathbb{R}^{n}$ be the set of vertex potentials induced by an electrical flow
from $s$ to $t$ and assume w.l.o.g. $x_s - x_t = 1$. By Fact~\ref{fact:energymin},
we have $x^T L_G x = 1/r_e$.
Now notice that we can also assume $x_u \in [0,1]$ for all $u$,
since letting $x_u\gets 1$ for all $x_u > 1$ and $x_v \gets 0$ for all $x_v < 0$
can only decrease the total energy.
This means that the energy $(x_s - x_t)^2 = 1$ on edge $e=(s,t)$ is the largest among all edges,
and thus if we sample all edges uniformly at rate $\approx r_e$, the total energy will be
$\Otil(1)$ with high probability by Chernoff bounds. This implies that in the latter subsampled graph,
$e$ (if sampled) becomes heavy with high probability.
Since $r_e$ is exactly (up to an $O(\log n)$ factor) the probability with which we want to sample $e$,
we can apply the heavy edge sketch to subgraphs of $G$ obtained by sampling edges
at geometrically decreasing rates, and then try to recover each edge $e$
from the subgraph with sampling rate $\approx r_e$.
However, for weighted graphs, the energy on some other edges of very large weights can be unboundedly big.
Thus concentration bounds no longer give us high success probability of recovering an edge $e$
when sampling all edges uniformly at rate $\approx \tau_e = w_e r_e$, even though
the energy reduces significantly in expectation.
\end{remark}
To sparsify $G^{\supi{k+1}}$,
we will utilize the spectral sparsification framework in~\cite{Koutis14},
which is itself model oblivious.
The framework works as follows:
\begin{enumerate}
\item Fix some constant $p\in (0,1)$.
\item While the number of edges in the graph is $> n \epsilon^{-2} \mathrm{polylog}(n)$:
\begin{enumerate}
\item Find all edges whose leverage score $\geq \omega := \epsilon^2/\mathrm{polylog}(n)$,
and call these edges $F$.
\item Sample each edge not in $F$ with probability
$p$, and multiply its weight by $1/p$ if sampled.
\end{enumerate}
\end{enumerate}
Notice that since the leverage scores of all edges sum up to at most $n-1$,
the total number of edges in $F$ is at most $n\epsilon^{-2} \mathrm{polylog}(n)$.
Thus, in each while loop iteration, the number of edges decreases by a constant factor,
and as a result there can be at most $O(\log n)$ iterations.
Then using Theorem~\ref{clm:lev-sample}, we have that the final graph
is a $(1 + \epsilon)$-spectral sparsifier of $G^{\supi{k+1}}$.
Notice that here, the first step in the while loop is exactly the recovery of heavy edges.
We now describe the difficulty that arises in implementing the above process
using linear sketching {\em non-adaptively}, and our way around it.
First, let $E_0 \supseteq E_1\supseteq\ldots\supseteq E_{O(\log n)}$ be such that
$E_0 = \binom{V}{2}$ and $E_{i+1}$ is obtained by subsampling each edge slot in $E_i$
with probability $p$ (the constant fixed at the first step of the above process).
We apply the heavy edge recovery sketch to each $G^{\supi{k+1}}[E_i]$.
We then implement each iteration of the while loop in the above process.%
At first, we recover all heavy edges in $G^{\supi{k+1}}$ using the sketch of $G^{\supi{k+1}}[E_0] = G^{\supi{k+1}}$,
and call these edges $F_0$.
We would like to sample each edge in $G^{\supi{k+1}}[\binom{V}{2}\setminus F_0]$ with probability $p$,
and multiply its weight by $1/p$ if sampled.
Then in the next iteration, we would want to recover heavy edges in the latter subsampled graph.
That is, we would like to have a sketch
of the graph $(1/p) G^{\supi{k+1}}[E_1\setminus F_0] + G^{\supi{k+1}}[F_0]$. However, we only have a sketch
of $G^{\supi{k+1}}[E_1]$. By linearity, we can multiply it by $1/p$
and add to it $G^{\supi{k+1}}[F_0\setminus E_1]$ and get a sketch
of $(1/p)G^{\supi{k+1}}[E_1] + G^{\supi{k+1}}[F_0\setminus E_1]$.
Nonetheless, this sketch is still not taken on our desired graph
$(1/p) G^{\supi{k+1}}[E_1\setminus F_0] + G^{\supi{k+1}}[F_0]$,
since the weights of the edges in $E_1\cap F_0$ in the former graph
are larger than in the latter by a factor of $1/p$. We say these edges are {\em overweighted} by $1/p$,
and call the former graph {\em overweighted} graph.
One might hope to further subtract from the sketch
$(1/p - 1)G^{\supi{k+1}}[E_1\cap F_0]$ to bring down the weights of the overweighted edges by a factor of $(1/p)$.
However, notice that we do {\em not} have the exact weights of the edges in $F_0$
from our heavy edge recovery sketch.
Rather, we only have some estimates of their weights.
Moreover,
while in the second iteration we are only looking to subtract
edges that are overweighted by $1/p$,
in subsequent iterations, we might need to subtract edges that
are overweighted by $\mathrm{poly}(n)$,
which means that our weight estimates for such edges must have inverse polynomial accuracy for the subtraction to work.
Our way around this issue is to repeatedly {\em re-estimate} the weights of the overweighted edges.
Specifically, we show that the edges that are overweighted the most must be heavy in the overweighted graph.
Thus we can apply the heavy edge recovery sketch to the overweighted graph, get estimates
of the weights of these edges, and bring their weights down by a factor of $(1/p)$.
We then repeatedly apply this step $O(\log n)$ times (where we re-estimate the weights each time)
until there are no overweighted edges. Since we only bring down the edge weights by a constant factor each time
and always re-estimate the weights once changed,
we will never have too large an error.
We now present in Figure~\ref{fig:wss} our main algorithm for weighted spectral sparsification,
which invokes the heavy edge recovery sketch in Section~\ref{sec:heavyedge}.
Specifically, for each graph $G^{(k)}$ in the iterative refinement process,
we apply independent heavy edge recovery sketches to subgraphs of $G^{(k)}$
obtained by sampling edges at geometrically decreasing rates.
Then in the recovery step, we first simulate the iterative refinement process
using an outer for loop of $k$, and then implement the framework from~\cite{Koutis14}
in an inner for loop of $i$.
Inside each iteration of the inner for loop, we start with the sketch of an overweighted graph $Z$,
and then gradually bring down the weights of the overweighted edges by repeatedly recovering heavy
edges in the current (overweighted) graph and subtracting a constant fraction of their weights.
Finally when there are no overweighted edges left, we recover the heavy edges in the resulting graph
and then go to the next sampling rate.
\begin{figure}[!htbp]
\vspace{-30pt}
\begin{algbox}
$\WSS(G,\epsilon)$
\quad
\begin{enumerate}
\item Let $t = \ceil{10000 \log (10^6 \epsilon^{-1} \frac{w_{\max}}{w_{\min}}\cdot n^{10})}$,
and then for each $k = 0,1,2,\ldots,t$:
\begin{enumerate}
\item %
Let $E_0^{\supi{k}}\supseteq E_1^{\supi{k}}\supseteq\ldots\supseteq E_t^{\supi{k}}$
be edge subsets %
where $E_0^{\supi{k}} = \binom{V}{2}$ and $E_i^{\supi{k}}$
is obtained by sub-sampling each edge slot in $E_{i-1}^{\supi{k}}$
with probability $(1+1/1000)^{-1}$.
\item For each pair $0\leq i,j\leq t$, use Lemma~\ref{lem:heavyedge} to generate a sketching matrix
$(\SHV)_{i,j}^{\supi{k}} \in \mathbb{R}^{O(n^{6/5}\omega_1^{-1} \epsilon_1^{-2}
\mathrm{polylog}(n,\frac{w_{\max}}{w_{\min}}))\times \binom{n}{2}}$
with $\omega_1 = \epsilon^2/(10^{12} t^2 \log n)$,
$\epsilon_1 = \epsilon / (10^6 t)$. %
\end{enumerate}
\item For each $0\leq k,i,j \leq t$,
let $G^{\supi{k}} \gets G + (1+1/10^4)^{-k} 10^6 w_{\max} n^5 K_n$,
and compute the sketch
$(\SHV)_{i,j}^{\supi{k}} w_{G^{\supi{k}}[E^{\supi{k}}_{i}]}$, where $w_{G^{\supi{k}}[E^{\supi{k}}_{i}]}\in\mathbb{R}^{\binom{n}{2}}$ is the weight vector
of $G^{\supi{k}}[E^{\supi{k}}_i]$. %
\hfill \textcolor{blue}{(Take sufficiently many independent heavy edge sketches on
each subsampled graph)}
\item Concatenate these sketches as $\SSS w_G$.
\end{enumerate}
\quad
$H^{\supi{t}} = \WSR(\SSS w_G)$
\begin{myEnumerate}
\item Initially, let $H^{\supi{0}} \gets 10^6 w_{\max} n^5 K_n$.
\hfill \textcolor{blue}{($H^{\supi{0}}$ is a $1.001$-spectral sparsifier of $G^{\supi{0}}$)}
\item For $k = 1,2,\ldots,t$:
\begin{myEnumerate}
\item Let $H^{\supi{k}} \gets \emptyset$.
\hfill \textcolor{blue}{($H^{\supi{k}}$ will be a $(1+\epsilon/1000)$-spectral sparsifier of $G^{\supi{k}}$)}
\item Set $c_e\gets 0$ for all $e\in \binom{V}{2}$.
\hfill \textcolor{blue}{($c_e$ will be s.t. $e$ is added to $H^{\supi{k}}$ when $i = c_e$)}
\item For $i = 0,1,\ldots,t$:
\begin{myEnumerate}
\item Let $Z\gets (1 + 1/1000)^i G^{\supi{k}}[E_i^{\supi{k}}] + H^{\supi{k}}[\binom{V}{2}\setminus E^{\supi{k}}_i]$.
\hfill \textcolor{blue}{($Z$ records the graph on which our linear sketches are currently taken)}
\item Compute sketches $s_j := (\SHV)^{\supi{k}}_{i,j} w_Z, j\in[0,t]$,
where $w_Z$ is $Z$'s weight vector.
\item For each $f\in H^{\supi{k}}\cap E_i^{\supi{k}}$, let
$\delta_f \gets i - c_f$, and let $\delta_f \gets 0$ for all other edges.
\hfill \textcolor{blue}{($(w_Z)_f$ needs to be brought down by a factor of $(1+1/1000)^{\delta_f}$)}
\item Let $j\gets 0$. Then while $\exists f: \delta_f > 0$, do the following:
\begin{itemize}
\item Use Lemma~\ref{lem:heavyedge} to recover from $s_j$ a set $F$ of edges and then let $j\gets j+1$.
\item For each $f\in F$ such that $\delta_f > 0$, let $\tilde{w}_f$ be the estimate of its weight:
\begin{itemize}
\item $Z\gets Z - (1 - (1+1/1000)^{-1}) \tilde{w}_f f$.
\item $s_{j'}\gets s_{j'} - (1 - (1+1/1000)^{-1}) (\SHV)^{\supi{k}}_{i,j'} (\tilde{w}_f \chi_f)$
for all $j'\in [0,t]$.
\item $\delta_f \gets \delta_f - 1$.
\hfill \textcolor{blue}{(Bring down $(w_Z)_f$ by $(1+1/1000)$ and update all sketches accordingly)}
\end{itemize}
\end{itemize}
\item Use Lemma~\ref{lem:heavyedge} to recover from $s_j$ a set $F^*$ of edges.
\item For each edge $f\in F^*$ with estimated weight $\tilde{w}_f$ such that
$\tilde{w}_f b_f^T L_{H^{\supi{k-1}}}^{\dag} b_f \geq 8 \omega_1$
and $f$ is not already in $H^{\supi{k}}$,
add $f$ to $H^{\supi{k}}$ with weight $\tilde{w}_f$,
and let $c_f\gets i$.
\hfill \textcolor{blue}{(Add recovered heavy edges to $H^{\supi{k}}$,
then go to the next sampling rate)}
\end{myEnumerate}
\end{myEnumerate}
\end{myEnumerate}
\end{algbox}
\caption{Linear sketch for weighted spectral sparsification.}
\label{fig:wss}
\end{figure}
The performance of our main linear sketching algorithm for weighted graph sparsification
is characterized in Theorem~\ref{thm:sparsifyG}.
\begin{restatable}{theorem}{sparsifyG}
For any parameter $\epsilon > 0$ and any integer $n$,
there exists a linear sketch with sketching matrix
$\SSS\in \mathbb{R}^{n^{6/5} \epsilon^{-4} \mathrm{polylog}(n,\epsilon^{-1},\frac{w_{\max}}{w_{\min}})\times \binom{n}{2}}$
and recovery algorithm $\WSR$
such that, given an input graph $G$ of $n$ vertices with weight vector $w_G$,
$\WSR(\SSS w_G)$ returns
a $(1+\epsilon)$-spectral sparsifier of $G$ with high probability.
\label{thm:sparsifyG}
\end{restatable}
Before proving the theorem, we shall first give some useful intermediate lemmas.
The following proposition directly follows from the definition of spectral sparsifiers.
\begin{proposition}
For any two graphs $G_1,G_2$ whose weight vectors satisfy
that $(w_{G_1})_e \approx_{1+\epsilon} (w_{G_2})_e$
for all $e\in \binom{V}{2}$,
$G_1$ is a $(1 + \epsilon)$-spectral sparsifier of $G_2$.
\end{proposition}
This proposition then immediately implies the following two lemmas.
\begin{lemma}\label{label:basecaseG}
$10^6 w_{\max} n^5 K_n$ is a $1.001$-spectral sparsifier of $G^{\supi{0}}$.
\end{lemma}
\begin{lemma}
For all $k\geq 1$, $G^{\supi{k-1}}$ is a $1.001$-spectral sparsifier of $G^{\supi{k}}$.
\end{lemma}
\begin{lemma}
$G^{\supi{t}}$ is a $(1 + \epsilon/2)$-spectral sparsifier of $G$.
\end{lemma}
\begin{proof}
By definition
\begin{align*}
L_{G^{\supi{t}}} = & L_G + L_{(1+10^{-4})^{-t} 10^6 w_{\max} n^5 K_n} \\
\pleq & L_G + .1 \epsilon {w_{\min}} n^{-5} L_{K_n},
\end{align*}
where the last line follows from our choice of $t$.
Thus, the largest eigenvalue of the second term is bounded by $.1\epsilon w_{\min} n^{-4}$.
By standard lower bounds on the second smallest eigenvalue, %
the second smallest eigenvalue of $L_{G}$ is at least $w_{\min} / n^2$.
Therefore we have
\begin{align*}
- .1\epsilon L_{G} \pleq L_{G^{\supi{t}}} - L_{G} \pleq .1 \epsilon L_{G},
\end{align*}
which implies that $G^{\supi{t}}$ is a $(1 + \epsilon/2)$-spectral sparsifier of $G$.
\end{proof}
Fix an iteration of the outer for loop of $k$.
Then for an iteration of the inner for loop of $i$,
let $H_i^{\supi{k}}$ be the $H^{\supi{k}}$ at the {\em beginning} of the iteration,
and let $F_i^{\supi{k}}$ be the edges in $H_i^{\supi{k}}$.
Define graph $$J_i^{\supi{k}} := (1 + 1/1000)^{{i}} G^{\supi{k}}[E_i^{\supi{k}} \setminus F_i^{\supi{k}}] +
\sum_{\ell=0}^{{i-1}} (1+1/1000)^{\ell} G^{\supi{k}}[F_{\ell+1}^{\supi{k}}\setminus F_{\ell}^{\supi{k}}].$$
Notice that $F^{\supi{k}}_0 = \emptyset$ and $J_0^{\supi{k}} = G^{\supi{k}}$.
Also by the way we are assigning values to $c_f$ in the algorithm,
we have, at the beginning of the for loop (of $i$) iteration, $f \in F^{\supi{k}}_{c_f + 1}\setminus F^{\supi{k}}_{c_f}$
for all $f\in F^{\supi{k}}_i$.
Thus we also have
\begin{align}\label{eq:Jdef2G}
J_i^{\supi{k}} := (1 + 1/1000)^i G^{\supi{k}}[E_i^{\supi{k}} \setminus F_i^{\supi{k}}] +
\sum_{f\in F_i^{\supi{k}}} (1 + 1/1000)^{c_f} (w_{G^{\supi{k}}})_f f.
\end{align}
\begin{lemma}\label{lem:inductionkG}
Suppose $H^{\supi{k-1}}$ is a $1.001$-spectral sparsifier of $G^{\supi{k-1}}$.
Then with high probability, for all $0\leq i < t$,
\begin{enumerate}
\item After the while loop inside the $i^{\mathrm{th}}$ iteration terminates,
for all $f$,
$$\frac{1}{1+\epsilon/10000} (w_{J_{i}^{\supi{k}}})_f \leq (w_{Z})_f \leq (1 + \epsilon/10000) (w_{J_{i}^{\supi{k}}})_f.$$
\label{item:whileG}
\item For all $f\in F^{\supi{k}}_{i+1}\setminus F^{\supi{k}}_i$, %
$$\frac{1}{1+\epsilon/(10^6 t)} (w_{J_{i+1}^{\supi{k}}})_f \leq
(w_{H^{\supi{k}}_{i+1}})_f
\leq (1 + \epsilon / (10^6 t)) (w_{J_{i+1}^{\supi{k}}})_f.$$
\label{item:wtili+1G}
\item All edges in $F^{\supi{k}}_{i+1}$ have leverage scores in $J^{\supi{k}}_{i}$ at least $4\omega_1$.
\label{item:o4G}
\item $J^{\supi{k}}_{i+1}$ is a $(1 + \epsilon/(10^4 t))$-spectral sparsifier of $J^{\supi{k}}_i$.
\label{item:jkG}
\end{enumerate}
\end{lemma}
\begin{proof}
We prove all statements of this lemma by induction on $i$.
For $i = 0$, since $\delta_f = 0$ for all $f$,
the while loop will not execute.
Thus throughout this iteration we have $Z = J^{\supi{k}}_0 = G^{\supi{k}}$. %
This immediately gives~\ref{item:whileG}.
By Lemma~\ref{lem:heavyedge}, the $F^*$ we recover in this iteration
contains all edges whose leverage score in $G^{\supi{k}}$ is at least $\omega_1$,
and all edges in $F^*$ have weight estimates satisfying~\ref{item:wtili+1G}.
Since $H^{\supi{k-1}}$ is a $1.001$-spectral sparsifier of $G^{\supi{k-1}}$,
and $G^{\supi{k-1}}$ is in turn a $1.001$-spectral sparsifier of of $G^{\supi{k}}$,
we have that $H^{\supi{k-1}}$ is a $1.003$-spectral sparsifier of $G^{\supi{k}}$.
As a result, we know that,
at the last step of the for loop iteration,
all edges with leverage score at least $\geq 10\omega_1$ in $G^{\supi{k}}$ will be added to $H^{\supi{k}}$,
and all edges added to $H^{\supi{k}}$ have leverage score at least $\geq 4\omega_1$ in $G^{\supi{k}}$,
so we have~\ref{item:o4G}.
This means that
$J^{\supi{k}}_{1}$ is obtained by sampling a set of edges in $J^{\supi{k}}_0$ whose leverage scores in $J^{\supi{k}}_0$ are
at most $10 \omega_1$ with probability $(1+1/1000)^{-1}$,
and multiply their weights by $(1+1/1000)$
if sampled. Using Theorem~\ref{clm:lev-sample}, we have~\ref{item:jkG}.
We now do an inductive step. Suppose all four statements hold
for iterations $0,1,\ldots,i-1$ where $1 < i < t$.
We show that they also hold for iteration $i$.
We first need to analyze the while loop inside iteration $i$.
Let us number a while loop iteration by the value of $j$ at the {\em end} of the iteration.
\begin{claim}\label{claim:whileapxG}
At the end of while loop iteration $j$ where $j \leq t$, we have
for all $f\in E_i^{\supi{k}}\cap F_i^{\supi{k}}$
$$\frac{1}{(1+2\epsilon_1)^j} \cdot (1+1/1000)^{\delta_f} (w_{J_{i}^{\supi{k}}})_f \leq (w_{Z})_f \leq
(1 + 2\epsilon_1)^{j} (1+1/1000)^{\delta_f} (w_{J_{i}^{\supi{k}}})_f.$$
\label{item:weightZG}
\end{claim}
\begin{proof}
We prove this claim by an induction on $j$.
First we show that the statement is true for $j=0$ at the beginning of while loop iteration $1$.
Here all $f\in E_i^{\supi{k}}\cap F_i^{\supi{k}}$ satisfy
that $(w_Z)_f = (1+1/1000)^{i} (w_{G^{\supi{k}}})_f$.
Since we set $\delta_f \gets i - c_f$ before the while loop,
and by~(\ref{eq:Jdef2G}) $(w_{J^{\supi{k}}_i})_f = (1 + 1/1000)^{c_f} (w_{G^{\supi{k}}})_f$,
we have
$(w_Z)_f = (1 + 1/1000)^{\delta_f} (w_{J^{\supi{k}}_i})_f$, as desired.
Now suppose the statement is true
at the end of iteration $j-1$ where $1 < j \leq t$.
We then show that the statement is also true at the end of iteration $j$.
Let $Z_0$ be the $Z$ before our updates to $Z$ in iteration $j$ and let $Z_1$ be the $Z$ after our updates.
By Lemma~\ref{lem:heavyedge}, all edges recovered $f\in F$ have their estimated edge weights
$\tilde{w}_f \in [\frac{1}{1+\epsilon_1} (w_{Z_0})_f, (1 + \epsilon_1) (w_{Z_0})_f]$.
Therefore after our updates, we have for any $f\in F$ such that $\delta_f > 0$ that
$(w_{Z_1})_f \in [\frac{1}{1+2\epsilon_1} (1+1/1000)^{-1} (w_{Z_0})_f, (1 + 2\epsilon_1) (1+1/1000)^{-1} (w_{Z_0})_f]$,
and $(w_{Z_1})_f = (w_{Z_0})_f$ for other edges $f$.
Since we let $\delta_f\gets \delta_f - 1$ for such edges, and do not change the $\delta_f$'s of other edges,
we have our desired statement for $j$.
\end{proof}
\begin{claim}
The while loop terminates after at most $t$ iterations.
\end{claim}
\begin{proof}
It suffices to show that $\max_f {\delta_f}$ decreases by $1$ in each while loop iteration.
Since $\delta_f \leq t$ for any $f$, this will imply that there can be at most $t$ iterations.
Then it boils down to showing that for all $f^*$ with $\delta_{f^*} = \max_f \delta_f$,
$f^*$ belongs to the recovered edge set $F$.
Since $f^*\in F^{\supi{k}}_i$, by~\ref{item:o4G} of our induction hypothesis,
the leverage score of $f^*$ in $J_{i-1}^{\supi{k}}$ is at least $4\omega_3$.
Notice that by~\ref{item:jkG} of our induction hypothesis,
$J^{\supi{k}}_{i-1}$ is a $(1+1/1000)$-spectral sparsifier of $G^{\supi{k}}$.
Then using the fact that $H^{\supi{k-1}}$ is a $1.003$-spectral sparsifier of $G^{\supi{k}}$
(which we proved at the beginning of the proof of this lemma), we have that
$H^{\supi{k-1}}$ is a $1.005$-spectral sparsifier of $J_{i-1}^{\supi{k}}$.
By Claim~\ref{claim:whileapxG}, we have at the beginning of each while loop that, for all $f$,
\begin{align}\label{eq:wzf1G}
(w_Z)_f \in &
[\frac{1}{(1+2\epsilon_1)^t}(1+1/1000)^{\delta_f} (w_{J^{\supi{k}}_i})_f,(1 + 2\epsilon_1)^t (1 + 1/1000)^{\delta_f} (w_{J^{\supi{k}}_i})_f] \notag\\
\subseteq &
[\frac{1}{1.01}(1+1/1000)^{\delta_f} (w_{J^{\supi{k}}_i})_f,1.01 (1 + 1/1000)^{\delta_f} (w_{J^{\supi{k}}_i})_f].
\end{align}
Since $\delta_{f^*} \geq \delta_f$ for all $f$,
the above implies
\begin{align}
L_{Z} \pleq {1.03} (1 + 1/1000)^{\delta_{f^*}} L_{J_{i-1}^{\supi{k}}}.
\label{eq:ZHG}
\end{align}
By inverting both sides, we then get
$$L_{Z}^{\dag} \pgeq 1.03^{-1} (1 + 1/1000)^{-\delta_{f^*}} L_{J_{i-1}^{\supi{k}}}^{\dag}.$$
Combining this with~(\ref{eq:wzf1G}),
the leverage score of $f^*$ in $Z$ satisfies
\begin{align*}
(w_Z)_{f^*} b_{f^*} L_{Z}^{\dag} b_{f^*} \geq
\frac{1}{1.01\cdot 1.03} (w_{J_i^{\supi{k}}})_{f^*} b_{f^*}^T L_{J^{\supi{k}}_{i-1}}^{\dag} b_{f^*} \geq
\frac{4\omega_3}{1.01\cdot 1.03} \geq \omega_3,
\end{align*}
as desired.
\end{proof}
By Claim~\ref{claim:whileapxG}, after the while loop terminates,
we have that for all $f$,
\begin{align*}
(w_Z)_f \in & [\frac{1}{(1+2\epsilon_1)^t} (w_{J^{\supi{k}}_i})_f, (1 + 2\epsilon_1)^t (w_{J^{\supi{k}}_i})_f] \\
\subseteq & [\frac{1}{(1+\epsilon/10000)} (w_{J^{\supi{k}}_i})_f, (1 + \epsilon/10000) (w_{J^{\supi{k}}_i})_f],
\end{align*}
and thus we have~\ref{item:whileG}.
This also implies that $Z$ is a $(1 + \epsilon/10000)$-spectral sparsifier of $J^{\supi{k}}_i$,
and as result, for each edge $f$, its leverage scores in $Z$ and $J^{\supi{k}}_i$ are within
a $(1+\epsilon/10000)^2 < 1.01$ factor of each other.
For all edges in $E^{\supi{k}}_i\setminus F^{\supi{k}}_i$, their weights in $Z$ equal exactly
their weights in $J^{\supi{k}}_i$,
therefore by Lemma~\ref{lem:heavyedge},
all edges recovered in $F^*$ not in $F_i^{\supi{k}}$ have weight estimates satisfying~\ref{item:wtili+1G}.
Notice that by~\ref{item:jkG} of our induction hypothesis,
$J^{\supi{k}}_i$ is a $(1+1/1000)$-spectral sparsifier of $G^{\supi{k}}$.
Then using the fact that $H^{\supi{k-1}}$ is a $1.003$-spectral sparsifier of $G^{\supi{k}}$
(which we proved at the beginning of this proof),
we have that $H^{\supi{k-1}}$ is a $1.01$-spectral sparsifier of $Z$.
Thus, at the last step of the for iteration,
all edges added to $H^{\supi{k}}$ have leverage score at least $\geq 5\omega_1$ in $Z$,
and all edges with leverage score $\geq 9\omega_1$ in $Z$ will be added to $H^{\supi{k}}$.
Thus we also know that all edges added to $H^{\supi{k}}$ have leverage score $\geq 4\omega_1$ in $J^{\supi{k}}_i$
(which gives~\ref{item:o4G}),
and all edges with leverage score $\geq 10\omega_1$ in $J^{\supi{k}}_i$ will be added to $H^{\supi{k}}$.
The above reasoning also implies that
$J^{\supi{k}}_{i+1}$ is obtained by sampling a set of edges in $J^{\supi{k}}_i$ whose leverage score
is at most $10 \omega_1$ with probability $(1+1/1000)^{-1}$,
and multiply their weights by $(1+1/1000)$
if sampled. Using Theorem~\ref{clm:lev-sample}, we have~\ref{item:jkG}.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:sparsifyG}]
\textbf{Number of linear measurements.}{
Notice that each $(\SHV)_{i,j}^{\supi{k}} w_{G^{\supi{k}}[E^{\supi{k}}_{i}]}\in\mathbb{R}^{n^{6/5}
\omega_1^{-1}\epsilon_1^{-2}\mathrm{polylog}(n,\frac{w_{\max}}{w_{\min}})}$,
so the total number of linear measurements is bounded by
\begin{align*}
t^3 n^{6/5} \omega_1^{-1}\epsilon_1^{-2}\mathrm{polylog}(n,\frac{w_{\max}}{w_{\min}}) \leq
n^{6/5} \epsilon^{-4} \mathrm{polylog}(n,\frac{w_{\max}}{w_{\min}},\epsilon^{-1}).
\end{align*}
}
\textbf{Spectral sparsifier guarantee.}
By Lemma~\ref{label:basecaseG},
$H^{\supi{0}}$ is a $1.001$-spectral sparsifier of $G^{\supi{0}}$.
We then show that whenever $H^{\supi{k-1}}$ is a $1.001$-spectral sparsifier of $G^{\supi{k-1}}$,
$H^{\supi{k}}$ is a $(1 + \epsilon/1000)$-spectral sparsifier of $G^{\supi{k}}$ with high probability.
Notice that inside each iteration of the outermost for loop of $k$,
for $i = t$, we have that with high probability
$E^{\supi{k}}_t = \emptyset$. This means that $J^{\supi{k}}_t$ consists of solely edges
in $F^{\supi{k}}_t$. Thus by Lemma~\ref{lem:inductionkG}, $H^{\supi{k}}_t$ is a
$(1+\epsilon/(10^6t))$-spectral sparsifier of $J^{\supi{k}}_t$.
Also by Lemma~\ref{lem:inductionkG}, $J^{\supi{k}}_t$ is a $(1+\epsilon/(10^4 t))^t$-spectral sparsifier
of $G^{\supi{k}}$. These combined imply that
$H^{\supi{k}}$ is a $(1 + \epsilon/1000)$-spectral sparsifier of $G^{\supi{k}}$.
Now applying an induction on $k$,
we have that $H^{\supi{t}}$ is a $(1 + \epsilon/1000)$-spectral sparsifier of $G^{\supi{t}}$.
Since $G^{\supi{t}}$ is a $(1 + \epsilon/2)$-spectral sparsifier of $G$,
$H^{\supi{t}}$ is a $(1 + \epsilon)$-spectral sparsifier of $G$, as desired.
\end{proof}
\subsection{Sparsification of \texorpdfstring{$G^{\sq}$}{}}\label{sec:sparsifyG^2}
\subsubsection{Sparsification of \texorpdfstring{$G^{\sq}$}{} by heavy edge recovery}\label{sec:similar}
We first give in Figure~\ref{fig:sqs} the linear sketch for sparsifying $G^{\sq}$ using the recovery of heavy edges
in Lemma~\ref{lem:shv}.
We will then prove Lemma~\ref{lem:shv} later in Section~\ref{sec:shv}.
The ideas for the former linear sketch are the same as the ones we used in Section~\ref{sec:algmainspectral},
since both are about how to sparsify a graph by repeatedly recovering heavy edges.
\begin{figure}[!htbp]
\begin{algbox}
$\SQS(G,\epsilon_2)$ %
\quad
\begin{enumerate}
\item Let $t = \ceil{10000 \log (10^6 \epsilon_2^{-1} \frac{w_{\max}}{w_{\min}}\cdot n^{10})}$,
and then for each $k = 0,1,2,\ldots,t$:
\begin{enumerate}
\item %
Let $E_0^{\supi{k}}\supseteq E_1^{\supi{k}}\supseteq\ldots\supseteq E_t^{\supi{k}}$ be subsets of edge slots
where $E_0^{\supi{k}} = \binom{V}{2}$ and $E_i^{\supi{k}}$ is obtained by sub-sampling each edge slot in $E_{i-1}^{\supi{k}}$ with probability
$(1+1/1000)^{-2}$.
\item For each pair $0\leq i,j\leq t$,
let $(\SSV)^{\supi{k}}_{i,j}\in \mathbb{R}^{n \omega_3^{-1}\epsilon_3^{-2}\mathrm{polylog}(n)\times \binom{n}{2}}$ be a sketching matrix
in Lemma~\ref{lem:shv} with $\omega_3 = \epsilon_2^2/(10^{12} t^2\log n)$, $\epsilon_3 = \epsilon_2/(10^6t)$.
\end{enumerate}
\item For each $k,i,j \in [0,t]$,
let $G^{\supi{k}} \gets G + {(1+10^{-4})^{-k} 10^6 w_{\max} n^5} K_n$
and compute the sketch
$(\SSV)_{i,j}^{\supi{k}} w_{G^{\supi{k}}[E^{\supi{k}}_{i}]}$
where $w_{G^{\supi{k}}[E^{\supi{k}}_{i}]}\in\mathbb{R}^{\binom{n}{2}}$ is
weight vector of the subgraph $G^{\supi{k}}[E^{\supi{k}}_i]$. %
\item Concatenate these sketches as $\SSQ w_G$.
\end{enumerate}
\quad
$\SQR(\SSQ w_G)$
\begin{myEnumerate}
\item Initially, let $H^{\supi{0}} \gets 10^6 w_{\max} n^5 K_n$ (where $(H^{\supi{0}})^{\sq}$ is a $1.001$-spectral sparsifier of $(G^{\supi{0}})^{\sq}$).
\item For $k = 1,2,\ldots,t$:
\begin{myEnumerate}
\item Let $H^{\supi{k}} \gets \emptyset$ (where $(H^{\supi{k}})^{\sq}$ will be a $(1+\epsilon_2/1000)$-spectral sparsifier of $(G^{\supi{k}})^{\sq}$).
\item Set $c_e\gets 0$ for all $e\in \binom{V}{2}$
(where $c_e$ will be such that $e$ is added to $H^{\supi{k}}$ when $i = c_e$).
\item For $i = 0,1,\ldots,t$:
\begin{myEnumerate}
\item Let $Z\gets (1 + 1/1000)^{i} G^{\supi{k}}[E_i^{\supi{k}}] + H^{\supi{k}}[\binom{V}{2}\setminus E_i^{\supi{k}}]$
($Z$ is to record
what graph our linear sketches are taken over).
\item Compute sketches $s_j:=(\SSV)^{\supi{k}}_{i,j} w_Z, j\in[0,t]$, where $w_Z$ is the weight vector of $Z$.
\item For each $f\in H^{\supi{k}}\cap E_i^{\supi{k}}$, let
$\delta_f \gets i - c_f$, and let $\delta_f \gets 0$ for all other edges.
\item Let $j\gets 0$. Then while $\exists f: \delta_f > 0$, do the following:
\begin{myEnumerate}
\item Use Lemma~\ref{lem:shv} to recover from $s_j$ and $(H^{\supi{k-1}})^{\sq}$ an edge set $F$, and $j\gets j+1$.
\item For each $f\in F$ such that $\delta_f > 0$, let $\tilde{w}_f$ be the estimate of its weight:
\begin{myEnumerate}
\item $Z\gets Z - (1 - (1 + 1/1000)^{-1}) \tilde{w}_f f$.
\item $s_{j'} \gets s_{j'} - (1 - (1 + 1/1000)^{-1})(\SSV)^{\supi{k}}_{i,j'} (\tilde{w}_f \chi_f)$
for all $j\leq j' \leq t$
\item $\delta_f \gets \delta_f - 1$.
\end{myEnumerate}
\end{myEnumerate}
\item Use Lemma~\ref{lem:shv} to recover from $s_t$ and $(H^{\supi{k-1}})^{\sq}$ a set $F^*$ of edges.
\item For each edge $f\in F^*$ with estimated weight $\tilde{w}_f$ such that
$\tilde{w}_f^2 b_f^T L_{(H^{\supi{k-1}})^{\sq}}^{\dag} b_f \geq 8 \omega_3$
and $f$ is not already in $H^{\supi{k}}$,
add $f$ to $H^{\supi{k}}$ with weight $\tilde{w}_f$,
and let $c_f\gets i$.
\end{myEnumerate}
\end{myEnumerate}
\item Return $(H^{\supi{t}})^{\sq}$.
\end{myEnumerate}
\end{algbox}
\caption{Linear sketch for sparsifying $G^{\sq}$.}
\label{fig:sqs}
\end{figure}
The performance of the linear sketch is characterized by Lemma~\ref{lem:sparsifyG^2}.
\sqsparsify*
The proof of Lemma~\ref{lem:sparsifyG^2} will also be largely similar to that
of Theorem~\ref{thm:sparsifyG} in Section~\ref{sec:algmainspectral}, However, we still include the proof here for completeness.
As in Section~\ref{sec:algmainspectral}, we first prove some useful intermediate lemmas.
\begin{lemma}\label{label:basecase}
$(10^6 w_{\max} n^5 K_n)^{\sq}$ is a $1.001$-spectral sparsifier of $(G^{\supi{0}})^{\sq}$.
\end{lemma}
\begin{proof}
Let $\Pi\in\mathbb{R}^{n\times n}$ be the projection matrix on the $n-1$-dimensional subspace
orthogonal to the all-one vector. Then $\Pi$ has $n-1$ eigenvalues of $1$ and one eigenvalue of $0$
(with eigenvector being the all-one vector).
It is known that $L_{K_n} = n \Pi$, so $L_{(10^6w_{\max} n^5 K_n)^{\sq}} = 10^{12} w_{\max}^2 n^{11} \Pi$,
and has all $n-1$ nonzero eigenvalues equal to $10^{12} w_{\max}^2 n^{11}$.
On the other hand, notice that by expanding
\begin{align*}
L_{(G^{\supi{0}})^{\sq}} = & L_{(G + 10^6 w_{\max} n^5 K_n)^{\sq}} \\ =
& L_{G^{\sq}} + 2\cdot 10^6 w_{\max} n^5 L_G + 10^{12} w_{\max}^2 n^{10} L_{K_n}.
\end{align*}
By standard bounds on the largest eigenvalue of the Laplacian matrix of a weighted graph,
the largest eigenvalue of an $n$-vertex graph with maximum weight $w_{\max}$
is at most $nw_{\max}$.
Therefore,
the largest eigenvalue of $L_{(G^{\supi{0}})^{\sq} - (10^6 w_{\max} n^5 K_n)^{\sq}}$ is at most $3\cdot 10^6 w_{\max}^2 n^6$.
This combined with $L_{(G^{\supi{0}})^{\sq}} \geq L_{(10^6 w_{\max} n^5 K_n)^{\sq}}$
shows that
\begin{align*}
- \frac{1}{10^5 n^5} L_{(G^{\supi{0}})^{\sq}} \pleq
L_{(G^{\supi{0}})^{\sq}} - L_{(10^6 w_{\max} n^5 K_n)^{\sq}} \pleq
\frac{1}{10^5 n^5} L_{(G^{\supi{0}})^{\sq}}.
\end{align*}
This implies that
\begin{align*}
\frac{1}{1.001} L_{(G^{\supi{0}})^{\sq}} \pleq
L_{(10^6 w_{\max} n^5 K_n)^{\sq}} \pleq 1.001 L_{(G^{\supi{0}})^{\sq}}
\end{align*}
as desired.
\end{proof}
\begin{lemma}
For all $k\geq 1$, $(G^{\supi{k-1}})^{\sq}$ is a $1.001$-spectral sparsifier of $(G^{\supi{k}})^{\sq}$.
\end{lemma}
\begin{proof}
By definition
\begin{align*}
L_{(G^{\supi{k-1}})^{\sq}} = & L_{(G + (1+10^{-4})^{-(k-1)} 10^6 w_{\max} n^5 K_n)^{\sq}} \\
= & L_{G^{\sq}} + 2(1 + 10^{-4})^{-(k-1)} 10^6 w_{\max} n^5 L_G + (1+10^{-4})^{-2(k-1)} 10^{12} w_{\max}^2 n^{10} L_{K_n}
\end{align*}
and
\begin{align*}
L_{(G^{\supi{k}})^{\sq}} = & L_{(G + (1+10^{-4})^{-k} 10^6 w_{\max} n^5 K_n)^{\sq}} \\
= & L_{G^{\sq}} + 2(1 + 10^{-4})^{-k} 10^6 w_{\max} n^5 L_G + (1+10^{-4})^{-2k} 10^{12} w_{\max}^2 n^{10} L_{K_n}.
\end{align*}
The second terms of the above two expressions are $(1+10^{-4})$-sparsifiers of each other,
and the third terms of the above two expressions are $(1+10^{-4})^2$-sparsifiers of each other.
Therefore, $L_{(G^{\supi{k-1}})^{\sq}}$ is a $1.001$-spectral sparsifier of $L_{(G^{\supi{k}})^{\sq}}$.
\end{proof}
\begin{lemma}
$(G^{\supi{t}})^{\sq}$ is a $(1 + \epsilon_2/2)$-spectral sparsifier of $G^{\sq}$.
\end{lemma}
\begin{proof}
By definition
\begin{align*}
L_{(G^{\supi{t}})^{\sq}} = & L_{(G + (1+10^{-4})^{-t} 10^6 w_{\max} n^5 K_n)^{\sq}} \\
= & L_{G^{\sq}} + 2(1 + 10^{-4})^{-t} 10^6 w_{\max} n^5 L_G + (1+10^{-4})^{-2t} 10^{12} w_{\max}^2 n^{10} L_{K_n} \\
\pleq & L_{G^{\sq}} + .1 \epsilon_2 \frac{w_{\min}^2}{w_{\max}} n^{-5} L_G +
.001 \epsilon_2^2 \frac{w_{\min}^4}{w_{\max}^2} n^{-10} L_{K_n},
\end{align*}
where the last line follows from our choice of $t$.
Thus, the largest eigenvalue of the sum of the last two terms is bounded by $.2\epsilon_2 w_{\min}^2 n^{-4}$.
By standard lower bounds on the second smallest eigenvalue, %
the second smallest eigenvalue of $L_{G^{\sq}}$ is at least $w_{\min}^2 / n^2$.
Therefore we have
\begin{align*}
- .2\epsilon_2 L_{G^{\sq}} \pleq L_{(G^{\supi{t}})^{\sq}} - L_{G^{\sq}} \pleq .2 \epsilon_2 L_{G^{\sq}},
\end{align*}
which implies that $(G^{\supi{t}})^{\sq}$ is a $(1 + \epsilon_2/2)$-spectral sparsifier of $G^{\sq}$.
\end{proof}
Fix an iteration of the outer for loop of $k$.
Then for an iteration of the inner for loop of $i$,
let $H_i^{\supi{k}}$ be the $H^{\supi{k}}$ at the {\em beginning} of the iteration,
and let $F_i^{\supi{k}}$ be the edges in $H_i^{\supi{k}}$.
Define graph $$J_i^{\supi{k}} := (1 + 1/1000)^i G^{\supi{k}}[E_i^{\supi{k}} \setminus F_i^{\supi{k}}] +
\sum_{\ell=0}^{i-1} (1+1/1000)^{\ell} G^{\supi{k}}[F_{\ell+1}^{\supi{k}}\setminus F_{\ell}^{\supi{k}}].$$
Notice that $F^{\supi{k}}_0 = \emptyset$ and $J_0^{\supi{k}} = G^{\supi{k}}$.
Also by the way we are assigning values to $c_f$ in the algorithm,
we have, at the beginning of the for loop (of $i$) iteration, $f \in F^{\supi{k}}_{c_f + 1}\setminus F^{\supi{k}}_{c_f}$
for all $f\in F^{\supi{k}}_i$.
Thus we also have
\begin{align}\label{eq:Jdef2}
J_i^{\supi{k}} := (1 + 1/1000)^i G^{\supi{k}}[E_i^{\supi{k}} \setminus F_i^{\supi{k}}] +
\sum_{f\in F_i^{\supi{k}}} (1 + 1/1000)^{c_f} (w_{G^{\supi{k}}})_f f.
\end{align}
\begin{lemma}\label{lem:inductionk}
Suppose $(H^{\supi{k-1}})^{\sq}$ is a $1.001$-spectral sparsifier of $(G^{\supi{k-1}})^{\sq}$.
Then with high probability, for all $0\leq i < t$,
\begin{enumerate}
\item After the while loop inside the $i^{\mathrm{th}}$ iteration terminates,
for all $f$,
$$\frac{1}{1+\epsilon_2/10000} (w_{J_{i}^{\supi{k}}})_f \leq (w_{Z})_f \leq (1 + \epsilon_2/10000) (w_{J_{i}^{\supi{k}}})_f.$$
\label{item:while}
\item For all $f\in F^{\supi{k}}_{i+1}\setminus F^{\supi{k}}_i$, %
$$\frac{1}{1+\epsilon_2/(10^6 t)} (w_{J_{i+1}^{\supi{k}}})_f \leq
(w_{H^{\supi{k}}_{i+1}})_f
\leq (1 + \epsilon_2 / (10^6 t)) (w_{J_{i+1}^{\supi{k}}})_f.$$
\label{item:wtili+1}
\item All edges in $F^{\supi{k}}_{i+1}$ have leverage scores in $(J^{\supi{k}}_{i})^{\sq}$ at least $4\omega_3$.
\label{item:o4}
\item $(J^{\supi{k}}_{i+1})^{\sq}$ is a $(1 + \epsilon_2/(10^4 t))$-spectral sparsifier of $(J^{\supi{k}}_i)^{\sq}$.
\label{item:jk}
\end{enumerate}
\end{lemma}
\begin{proof}
We prove all statements of this lemma by induction on $i$.
For $i = 0$, since $\delta_f = 0$ for all $f$,
the while loop will not execute.
Thus throughout this iteration we have $Z = J^{\supi{k}}_0 = G^{\supi{k}}$. %
This immediately gives~\ref{item:while}.
Since $(H^{\supi{k-1}})^{\sq}$ is a $1.001$-spectral sparsifier of $(G^{\supi{k-1}})^{\sq}$,
and $(G^{\supi{k-1}})^{\sq}$ is in turn a $1.001$-spectral sparsifier of of $(G^{\supi{k}})^{\sq}$,
we have that $(H^{\supi{k-1}})^{\sq}$ is a $1.003$-spectral sparsifier of $(G^{\supi{k}})^{\sq}$.
Therefore by letting $\Gtil = (H^{\supi{k-1}})^{\sq}$, we have
\begin{align}\label{eq:bygtil}
& \frac{((w_{G^{\supi{k}}})_e b_e^T L_{\Gtil}^{\dag} b_e)^2}{ b_e^T L_{\Gtil}^{\dag} L_{(G^{\supi{k}})^{\sq}} L_{\Gtil}^{\dag} b_e }
\approx_{1.003^4}
\frac{((w_{G^{\supi{k}}})_e b_e^T L_{(G^{\supi{k}})^{\sq}}^{\dag} b_e)^2}{ b_e^T L_{(G^{\supi{k}})^{\sq}}^{\dag} b_e }
= (w_{G^{\supi{k}}})_e^2 b_e^T L_{(G^{\supi{k}})^{\sq}}^{\dag} b_e,
\end{align}
where in the first step we have used
\begin{align*}
(w_{G^{\supi{k}}})_e b_e^T L_{\Gtil}^{\dag} b_e \approx_{1.003} (w_{G^{\supi{k}}})_e b_e^T L_{(G^{\supi{k}})^{\sq}}^{\dag} b_e
\end{align*}
and
\begin{align*}
b_e^T L_{\Gtil}^{\dag} L_{(G^{\supi{k}})^{\sq}} L_{\Gtil}^{\dag} b_e \approx_{1.003}
b_e^T L_{\Gtil}^{\dag} L_{\Gtil} L_{\Gtil}^{\dag} b_e =
b_e^T L_{\Gtil}^{\dag} b_e \approx_{1.003}
b_e^T L_{(G^{\supi{k}})^{\sq}}^{\dag} b_e.
\end{align*}
Therefore by Lemma~\ref{lem:shv}, the $F^*$ we recover in this iteration
contains all edges whose leverage score in $(G^{\supi{k}})^{\sq}$ is at least $1.1\omega_3$,
and all edges in $F^*$ have weight estimates satisfying~\ref{item:wtili+1}.
Also, at the final step of this for loop iteration,
since $(H^{\supi{k-1}})^{\sq}$ is a $1.003$-spectral sparsifier of $(G^{\supi{k}})^{\sq}$,
all edges with leverage score $\geq 10 \omega_3$ in $(G^{\supi{k}})^{\sq}$ will be added to $H^{\supi{k}}$,
and all edges added to $H^{\supi{k}}$ have leverage score at least $\geq 4\omega_3$ in $(G^{\supi{k}})^{\sq}$,
so we have~\ref{item:o4}.
This means that
$(J^{\supi{k}}_{1})^{\sq}$ is obtained by sampling a set of edges in $(J^{\supi{k}}_0)^{\sq}$ whose leverage scores in $(J^{\supi{k}}_0)^{\sq}$ are
at most $10 \omega_3$ with probability $(1+1/1000)^{-2}$,
and multiply their weights by $(1+1/1000)^2$
if sampled. Using Theorem~\ref{clm:lev-sample}, we have~\ref{item:jk}.
We now do an inductive step. Suppose all four statements hold
for iterations $0,1,\ldots,i-1$ where $1 < i < t$.
We show that they also hold for iteration $i$.
We first need to analyze the while loop inside iteration $i$.
Let us number a while loop iteration by the value of $j$ at the {\em end} of the iteration.
\begin{claim}\label{claim:whileapx}
At the end of while loop iteration $j$ where $j \leq t$, we have
for all $f\in E_i^{\supi{k}}\cap F_i^{\supi{k}}$
$$\frac{1}{(1+2\epsilon_3)^j} \cdot (1+1/1000)^{\delta_f} (w_{J_{i}^{\supi{k}}})_f \leq (w_{Z})_f \leq
(1 + 2\epsilon_3)^{j} (1+1/1000)^{\delta_f} (w_{J_{i}^{\supi{k}}})_f.$$
\label{item:weightZ}
\end{claim}
\begin{proof}
We prove this claim by an induction on $j$.
First we show that the statement is true for $j=0$ at the beginning of while loop iteration $1$.
Here all $f\in E_i^{\supi{k}}\cap F_i^{\supi{k}}$ satisfy
that $(w_Z)_f = (1+1/1000)^{i} (w_{G^{\supi{k}}})_f$.
Since we set $\delta_f \gets i - c_f$ before the while loop,
and by~(\ref{eq:Jdef2}) $(w_{J^{\supi{k}}_i})_f = (1 + 1/1000)^{c_f} (w_{G^{\supi{k}}})_f$,
we have
$(w_Z)_f = (1 + 1/1000)^{\delta_f} (w_{J^{\supi{k}}_i})_f$, as desired.
Now suppose the statement is true
at the end of iteration $j-1$ where $1 < j \leq t$.
We then show that the statement is also true at the end of iteration $j$.
Let $Z_0$ be the $Z$ before our updates to $Z$ in iteration $j$ and let $Z_1$ be the $Z$ after our updates.
By Lemma~\ref{lem:shv}, all edges recovered $f\in F$ have their estimated edge weights
$\tilde{w}_f \in [\frac{1}{1+\epsilon_3} (w_{Z_0})_f, (1 + \epsilon_3) (w_{Z_0})_f]$.
Therefore after our updates, we have for any $f\in F$ such that $\delta_f > 0$ that
$(w_{Z_1})_f \in [\frac{1}{1+2\epsilon_3} (1+1/1000)^{-1} (w_{Z_0})_f, (1 + 2\epsilon_3) (1+1/1000)^{-1} (w_{Z_0})_f]$,
and $(w_{Z_1})_f = (w_{Z_0})_f$ for other edges $f$.
Since we let $\delta_f\gets \delta_f - 1$ for such edges (those with $\delta_f > 0$),
and do not change the $\delta_f$'s of other edges,
we have our desired statement for $j$.
\end{proof}
\begin{claim}
The while loop terminates after at most $t$ iterations.
\end{claim}
\begin{proof}
It suffices to show that $\max_f {\delta_f}$ decreases by $1$ in each while loop iteration.
Since $\delta_f \leq t$ for any $f$, this will imply that there can be at most $t$ iterations.
Then it boils down to showing that for all $f^*$ with $\delta_{f^*} = \max_f \delta_f$,
$f^*$ belongs to the recovered edge set $F$.
Since $f^*\in F^{\supi{k}}_i$, by~\ref{item:o4} of our induction hypothesis,
the leverage score of $f^*$ in $(J_{i-1}^{\supi{k}})^{\sq}$ is at least $4\omega_3$.
Notice that by~\ref{item:jk} of our induction hypothesis,
$(J^{\supi{k}}_{i-1})^{\sq}$ is a $(1+1/1000)$-spectral sparsifier of $(G^{\supi{k}})^{\sq}$.
Then using the fact that $(H^{\supi{k-1}})^{\sq}$ is a $1.003$-spectral sparsifier of $(G^{\supi{k}})^{\sq}$
(which we proved at the beginning of the proof of this lemma), we have that
$(H^{\supi{k-1}})^{\sq}$ is a $1.005$-spectral sparsifier of $(J_{i-1}^{\supi{k}})^{\sq}$.
By Claim~\ref{claim:whileapx}, we have at the beginning of each while loop that, for all $f$,
\begin{align}\label{eq:wzf1}
(w_Z)_f \in &
[\frac{1}{(1+2\epsilon_3)^t}(1+1/1000)^{\delta_f} (w_{J^{\supi{k}}_i})_f,(1 + 2\epsilon_3)^t (1 + 1/1000)^{\delta_f} (w_{J^{\supi{k}}_i})_f] \notag\\
\subseteq &
[\frac{1}{1.01}(1+1/1000)^{\delta_f} (w_{J^{\supi{k}}_i})_f,1.01 (1 + 1/1000)^{\delta_f} (w_{J^{\supi{k}}_i})_f].
\end{align}
Since $\delta_{f^*} \geq \delta_f$ for all $f$,
the above implies
\begin{align}
L_{Z^{\sq}} \pleq {1.03} (1 + 1/1000)^{2 \delta_{f^*}} L_{(J_{i-1}^{\supi{k}})^{\sq}}
\pleq {1.04} (1 + 1/1000)^{2 \delta_{f^*}} L_{(H^{\supi{k-1}})^{\sq}}, \label{eq:ZH}
\end{align}
where the second inequality follows from that $(H^{\supi{k-1}})^{\sq}$ is a
$1.005$-spectral sparsifier of $(J^{\supi{k}}_{i-1})^{\sq}$.
Let $\Gtil = (H^{\supi{k-1}})^{\sq}$.
Now in order to show that $f^*$ will be recovered as an edge in $F^*$, by Lemma~\ref{lem:shv}, it suffices
to show
\begin{align}\label{eq:show}
\frac{ ( (w_{Z})_{f^*} b_{f^*}^T L_{\Gtil}^{\dag} b_{f^*} )^2 }
{b_{f^*}^T L_{\Gtil}^{\dag} L_{Z^{\sq}} L_{\Gtil}^{\dag} b_{f^*} } \geq \omega_3.
\end{align}
By~(\ref{eq:ZH}), the denominator satisfies
\begin{align*}
b_{f^*}^T L_{\Gtil}^{\dag} L_{Z^{\sq}} L_{\Gtil}^{\dag} b_{f^*}
\leq {1.04} (1 + 1/1000)^{2 \delta_{f^*}} b_{f^*}^T L_{\Gtil}^{\dag} b_{f^*}.
\end{align*}
Therefore, the LHS of~(\ref{eq:show}) is at least
$$(w_Z)_{f^*}^2(1 + 1/1000)^{-2 \delta_{f^*}} b_{f^*}^T L_{\Gtil}^{\dag} b_{f^*}\geq
1.005^{-1}(w_Z)_{f^*}^2 (1 + 1/1000)^{-2 \delta_{f^*}} b_{f^*}^T L_{(J^{\supi{k}}_{i-1})^{\sq}}^{\dag} b_{f^*},
$$
where the inequality follows from that $(H^{\supi{k-1}})^{\sq}$ is a $1.005$-spectral sparsifier
of $(J^{\supi{k}}_{i-1})^{\sq}$.
Finally, using~(\ref{eq:wzf1}) and that the leverage score of $f^*$ is at least $2\omega_3$ in $(J^{\supi{k}}_{i-1})^{\sq}$,
we have that the above is at least $\omega_3$, proving~(\ref{eq:show}).
\end{proof}
By Claim~\ref{claim:whileapx}, after the while loop terminates,
we have that for all $f$,
\begin{align*}
(w_Z)_f \in & [\frac{1}{(1+2\epsilon_3)^t} (w_{J^{\supi{k}}_i})_f, (1 + 2\epsilon_3)^t (w_{J^{\supi{k}}_i})_f] \\
\subseteq & [\frac{1}{(1+\epsilon_2/10000)} (w_{J^{\supi{k}}_i})_f, (1 + \epsilon_2/10000) (w_{J^{\supi{k}}_i})_f],
\end{align*}
and thus we have~\ref{item:while}.
This also implies that $Z^{\sq}$ is a $(1 + \epsilon_2/10000)^2$-spectral sparsifier of $(J^{\supi{k}}_i)^{\sq}$,
and as result, for each edge $f$, its leverage scores in $Z^{\sq}$ and $(J^{\supi{k}}_i)^{\sq}$ are within
a $(1+\epsilon_2/10000)^3 < 1.01$ factor of each other.
For all edges in $E^{\supi{k}}_i\setminus F^{\supi{k}}_i$, their weights in $Z$ equal exactly
their weights in $J^{\supi{k}}_i$,
therefore by Lemma~\ref{lem:shv},
all edges recovered in $F^*$ not in $F_i^{\supi{k}}$ have weight estimates satisfying~\ref{item:wtili+1}.
Notice that by~\ref{item:jk} of our induction hypothesis,
$(J^{\supi{k}}_i)^{\sq}$ is a $(1+1/1000)$-spectral sparsifier of $(G^{\supi{k}})^{\sq}$.
Then using the fact that $(H^{\supi{k-1}})^{\sq}$ is a $1.003$-spectral sparsifier of $(G^{\supi{k}})^{\sq}$
(which we proved at the beginning of this proof),
we have that $(H^{\supi{k-1}})^{\sq}$ is a $1.01$-spectral sparsifier of $Z^{\sq}$.
By letting $\Gtil = (H^{\supi{k-1}})^{\sq}$
we have, similar to~(\ref{eq:bygtil}),
\begin{align}
& \frac{((w_{Z})_e b_e^T L_{\Gtil}^{\dag} b_e)^2}{ b_e^T L_{\Gtil}^{\dag} L_{Z^{\sq}} L_{\Gtil}^{\dag} b_e }
\approx_{1.01^4}
\frac{((w_{Z})_e b_e^T L_{Z^{\sq}}^{\dag} b_e)^2}{ b_e^T L_{Z^{\sq}}^{\dag} b_e }
= (w_{Z})_e^2 b_e^T L_{Z^{\sq}}^{\dag} b_e.
\end{align}
Therefore by Lemma~\ref{lem:shv}, the $F^*$ we recover in this iteration
contains all edges whose leverage scores in $(Z)^{\sq}$ are at least $1.5\omega_3$.
Also, at the last step of the for iteration, since $(H^{\supi{k-1}})^{\sq}$ is a $1.01$-spectral sparsifier of $Z^{\sq}$,
all edges added to $H^{\supi{k}}$ have leverage scores at least $\geq 5\omega_3$ in $(Z)^{\sq}$,
and all edges with leverage scores $\geq 9\omega_3$ in $(Z)^{\sq}$ will be added to $H^{\supi{k}}$.
Thus we also know that all edges added to $H^{\supi{k}}$ have leverage scores at least $4\omega_3$ in $(J^{\supi{k}}_i)^{\sq}$
(which gives~\ref{item:o4}),
and all edges with leverage scores $\geq 10\omega_3$ in $(J^{\supi{k}}_i)^{\sq}$ will be added to $H^{\supi{k}}$,
The above reasoning also implies that
$(J^{\supi{k}}_{i+1})^{\sq}$ is obtained by sampling a set of edges in $(J^{\supi{k}}_i)^{\sq}$ whose leverage score
is at most $10 \omega_3$ with probability $(1+1/1000)^{-2}$,
and multiply their weights by $(1+1/1000)^2$
if sampled. Using Theorem~\ref{clm:lev-sample}, we have~\ref{item:jk}.
\end{proof}
\begin{proof}[Proof of Lemma~\ref{lem:sparsifyG^2}]
\textbf{Number of linear measurements.}{
Notice that each $(\SSV)_{i,j}^{\supi{k}} w_{G^{\supi{k}}[E^{\supi{k}}_{i}]}\in\mathbb{R}^{n \omega_3^{-1}\epsilon_3^{-2}\mathrm{polylog}(n)}$,
so the total number of linear measurements is bounded by
\begin{align*}
t^3 n \omega_3^{-1}\epsilon_3^{-2}\mathrm{polylog}(n) \leq n \epsilon_2^{-4} \mathrm{polylog}(n,\frac{w_{\max}}{w_{\min}},\epsilon_2^{-1}).
\end{align*}
}
\textbf{Spectral sparsifier guarantee.}
By Lemma~\ref{label:basecase},
$(H^{\supi{0}})^{\sq}$ is a $1.001$-spectral sparsifier of $(G^{\supi{0}})^{\sq}$.
We then show that whenever $(H^{\supi{k-1}})^{\sq}$ is a $1.001$-spectral sparsifier of $(G^{\supi{k-1}})^{\sq}$,
$(H^{\supi{k}})^{\sq}$ is a $(1 + \epsilon_2/1000)$-spectral sparsifier of $(G^{\supi{k}})^{\sq}$ with high probability.
Notice that inside each iteration of the outermost for loop of $k$,
for $i = t$, we have that with high probability
$E^{\supi{k}}_t = \emptyset$. This means that $J^{\supi{k}}_t$ consists of solely edges
in $F^{\supi{k}}_t$. Thus by Lemma~\ref{lem:inductionk}, $(H^{\supi{k}}_t)^{\sq}$ is a
$(1+\epsilon_2/(10^6t))^2$-spectral sparsifier of $(J^{\supi{k}}_t)^{\sq}$.
Also by Lemma~\ref{lem:inductionk}, $(J^{\supi{k}}_t)^{\sq}$ is a $(1+\epsilon_2/(10^4 t))^t$-spectral sparsifier
of $(G^{\supi{k}})^{\sq}$. These combined imply that
$(H^{\supi{k}})^{\sq}$ is a $(1 + \epsilon_2/1000)$-spectral sparsifier of $(G^{\supi{k}})^{\sq}$.
Now applying an induction on $k$,
we have that $(H^{\supi{t}})^{\sq}$ is a $(1 + \epsilon_2/1000)$-spectral sparsifier of $(G^{\supi{t}})^{\sq}$.
Since $(G^{\supi{t}})^{\sq}$ is a $(1 + \epsilon_2/2)$-spectral sparsifier of $G^{\sq}$,
$(H^{\supi{t}})^{\sq}$ is a $(1 + \epsilon_2)$-spectral sparsifier of $G^{\sq}$, as desired.
\end{proof}
\subsubsection{Recovery of heavy edges in \texorpdfstring{$G^{\sq}$}{}}\label{sec:shv}
We now give in Figure~\ref{fig:svs} a linear sketch for recovering heavy edges in $G^{\sq}$ and its analysis.
This linear sketch is essentially a direct application of $\ell_2$-heavy hitters.
\begin{figure}[ht]
\begin{algbox}
$\SVS(G,\omega_3,\epsilon_3)$
\quad
\begin{enumerate}
\item Let $A\in\mathbb{R}^{\eta^{-2}\mathrm{polylog}(n)\times\binom{n}{2}}$ be an $\ell_2$-heavy hitter sketching
matrix with $\eta = \sqrt{\omega_3} \epsilon_3 / 10000$ (Proposition~\ref{prop:heavyhitter}).
\item Let $J\in\mathbb{R}^{O(\delta^{-2}\log n)\times \binom{n}{2}}$ be an $\ell_2$-estimation sketching matrix
with $\delta = .01$ (Proposition~\ref{l2-norm})
\item Concatenate the sketches $A W_G B_G$ and $J W_G B_G$ as $\SSV w_G$.
\end{enumerate}
\quad
$\SVR(\SSV w_G, \Gtil)$
\begin{enumerate}
\item For each vertex pair $s,t\in V$:
\begin{enumerate}
\item Let $x = L_{\Gtil}^{\dag} b_{s,t} \in \mathbb{R}^{n}$ be the set of vertex potentials induced
by a unit electrical flow from $s$ to $t$ in $\Gtil$.
\item Let $z\in \mathbb{R}^{\binom{n}{2}}$ be an approximation to $W_G B_G x$ that is recovered
from $A W_G B_G x$ using the recovery algorithm in Proposition~\ref{prop:heavyhitter}.
\item Let $\beta$ be an estimate of $\norm{W_G B_G x}_2$ recovered from $J W_G B_G x$ using
Proposition~\ref{l2-norm}.
\item If $z_{(s,t)} \geq .9\sqrt{\omega_3} \beta$, mark edge $(s,t)$ as heavy,
and estimate its weight as $\tilde{w}_{(s,t)} = \frac{z_{(s,t)}}{x^T b_{s,t}}$.
\end{enumerate}
\item Return all edges marked heavy along with the estimates of their weights.
\end{enumerate}
\end{algbox}
\caption{Linear sketch for recovering heavy edges in $G^{\sq}$.}
\label{fig:svs}
\end{figure}
The performance of the linear sketch is characterized in Lemma~\ref{lem:shv}:
\lemshv*
\begin{proof}[Proof of Lemma~\ref{lem:shv}]
\textbf{Number of linear measurements.}{
Notice that the sketches we compute satisfy
$A W_G B_G \in \mathbb{R}^{\omega_3^{-1} \epsilon_3^{-2} \mathrm{polylog}(n)\times n}$
and $J W_G B_G \in \mathbb{R}^{O(\log n)\times n}$.
Therefore by concatenating them, the total number of linear measurements
is bounded by $n \omega_3^{-1} \epsilon_3^{-2} \mathrm{polylog}(n)$.
}
\textbf{First guarantee.}{
By the guarantee of the $\ell_2$-heavy hitter sketch, we have
\begin{align}
|\kh{ W_G B_G x }_{(s,t)} - z_{(s,t)}| \leq & \eta \norm{W_G B_G x}_2 \notag \\
\leq & (\sqrt{\omega_3} \epsilon_3/10000) \norm{W_G B_G x}_2
\qquad \text{(by the value of $\eta$)} \notag \\
= & (\sqrt{\omega_3} \epsilon_3/10000) \sqrt{b_{s,t}^T L_{\Gtil}^{\dag} L_{G^{\sq}} L_{\Gtil}^{\dag} b_{s,t}}
\label{eq:errz}
\end{align}
By the guarantee of the $\ell_2$-estimation sketch
\begin{align}\label{eq:betaer}
\beta \in \left[\frac{1}{1.01} \sqrt{b_{s,t}^T L_{\Gtil}^{\dag} L_{G^{\sq}} L_{\Gtil}^{\dag} b_{s,t}},
1.01 \sqrt{b_{s,t}^T L_{\Gtil}^{\dag} L_{G^{\sq}} L_{\Gtil}^{\dag} b_{s,t}}\right].
\end{align}
Now fix any $e$ with
\begin{align}\label{eq:heavy3}
\frac{((w_e)_G b_e^T L_{\Gtil}^{\dag} b_e)^2}{ b_e^T L_{\Gtil}^{\dag} L_{G^{\sq}} L_{\Gtil}^{\dag} b_e }
\geq \omega_3.
\end{align}
We then write
\begin{align}
| z_e - \kh{ W_G B_G x }_{e} | \leq & \frac{\epsilon_3}{10000}
\sqrt{\omega_3 b_{e}^T L_{\Gtil}^{\dag} L_{G^{\sq}} L_{\Gtil}^{\dag} b_{e}}
\qquad \text{by~(\ref{eq:errz})}
\notag \\
\leq & \frac{\epsilon_3}{10000} \sqrt{\omega_3} \sqrt{1/\omega_3} w_e b_{e}^T L_{\Gtil}^{\dag} b_{e}
\qquad \text{(by~(\ref{eq:heavy3}))}
\notag \\
\leq & .001 \epsilon_3 w_e b_e^T L_{\Gtil}^{\dag} b_e \notag \\
= & .001\epsilon_3 \kh{ W_G B_G x }_{e}. \label{eq:.01}
\end{align}
This implies that
\begin{align*}
z_e \geq & .999 w_e b_e^T L_{\Gtil}^{\dag} b_e \\
\geq & .999 \sqrt{{\omega_3} b_{e}^T L_{\Gtil}^{\dag} L_{G^{\sq}} L_{\Gtil}^{\dag} b_{e}}
\qquad \text{(by~(\ref{eq:heavy3}))}
\\
\geq & .999 \sqrt{{\omega_3}} \frac{\beta}{1.01}
\qquad \text{(by~(\ref{eq:betaer}))}
\\
\geq & .9 \sqrt{\omega_3} \beta.
\end{align*}
Therefore $e$ will be marked as heavy.
}
\textbf{Second guarantee.}{
For $e$ such that $z_e \geq .9\sqrt{\omega_3}\beta$, we have
\begin{align*}
(W_G B_G x)_e \geq & z_e - \frac{\epsilon_3}{10000} \sqrt{\omega_3 b_{e}^T L_{\Gtil}^{\dag} L_{G^{\sq}} L_{\Gtil}^{\dag} b_{e}}
\qquad \text{(by~(\ref{eq:errz}))}
\\
\geq & .9\sqrt{\omega_3} \frac{1}{1.01}\sqrt{b_{e}^T L_{\Gtil}^{\dag} L_{G^{\sq}} L_{\Gtil}^{\dag} b_{e}} -
\frac{\epsilon_3}{10000}\sqrt{b_{e}^T L_{\Gtil}^{\dag} L_{G^{\sq}} L_{\Gtil}^{\dag} b_{e}}
\qquad \text{(by~(\ref{eq:betaer}))}
\\
\geq & .87 \sqrt{\omega_3 b_{e}^T L_{\Gtil}^{\dag} L_{G^{\sq}} L_{\Gtil}^{\dag} b_{e}}
\qquad \text{(by $\epsilon_3 < 1$).}
\end{align*}
Since $(W_G B_G x)_e = w_e b_e^T L_{\Gtil}^{\dag} b_e$,
we have
\begin{align}\label{eq:heavy4}
\frac{((w_e)_G b_e^T L_{\Gtil}^{\dag} b_e)^2}{ b_e^T L_{\Gtil}^{\dag} L_{G^{\sq}} L_{\Gtil}^{\dag} b_e }
\geq .7 \omega_3.
\end{align}
We then write
\begin{align}
| z_e - \kh{ W_G B_G x }_{e} | \leq & \frac{\epsilon_3}{10000}
\sqrt{\omega_3 b_{s,t}^T L_{\Gtil}^{\dag} L_{G^{\sq}} L_{\Gtil}^{\dag} b_{s,t}}
\qquad \text{(by~(\ref{eq:errz}))}
\notag \\
\leq & %
\frac{\epsilon_3}{10000} \sqrt{\omega_3} \sqrt{1/(.7\omega_3)} w_e b_{e}^T L_{\Gtil}^{\dag} b_{e}
\qquad \text{(by~(\ref{eq:heavy4}))}
\notag \\
\leq & .01 \epsilon_3 w_e b_e^T L_{\Gtil}^{\dag} b_e
\notag \\
= & .01\epsilon_3 \kh{ W_G B_G x }_{e}.
\end{align}
By dividing both sides by $x^T b_e$, we have
\begin{align*}
\sizeof{ \frac{z_e}{x^T b_e} - \frac{w_e b_e^T L_{\Gtil}^{\dag} b_e}{x^T b_e} } \leq
.01\epsilon_3 \frac{w_e b_e^T L_{\Gtil}^{\dag} b_e}{x^T b_e}.
\end{align*}
Since
${x^T b_e} =
{b_e^T L_{\Gtil}^{\dag} b_e}$,
the above is equivalent to $\sizeof{\frac{z_e}{x^T b_e} - w_e} \leq .01\epsilon_3 w_e$,
and thus we conclude
$\frac{1}{1+\epsilon_3} w_e \leq \frac{w_e b_e^T L_{\Gtil}^{\dag} b_e}{x^T b_e} \leq (1 + \epsilon_3) w_e$,
as desired.
}
\end{proof}
\let\omega\oldomega
|
1,116,691,497,585 | arxiv | \section{Introduction}
The goal of this paper is to understand diffeomorphism symmetries
in the canonical formalism at the classical level. The putative
generators of infinitesimal general coordinate transformations feature
in all canonical quantization approaches, but debate persists in the
literature as to what aspects of the diffeomorphism group are realized
at the classical level as canonical transformations
\cite{Bergmann72,Isham85,Lee90,Salisbury83}.
This issue is intimately
related to the meaning of time in quantum gravity.
In this paper we extend recent work by Pons and Shepley \cite{Pons95}
concerning constrained systems. We analyze diffeomorphism symmetries
using in a natural way the equivalence of the Hamiltonian and the
Lagrangian approaches to generally covariant systems. We show that
infinitesimal transformations which are projectable under the Legendre
map are a basis for the generators of the gauge group. This group is
much larger than the original group of spacetime diffeomorphisms
because it acts on the space of spacetime metrics, whereas the
diffeomorphism group acts on the underlying manifold. Since we retain
the full set of canonical variables, the associated infinitesimal
generators are new; they are realized on the full set of phase space
variables and must at least depend in a specific way on the lapse
function and shift vector of the spacetime metric in a given
coordinate patch. The results are contrasted and compared with
earlier work by Salisbury and Sundermeyer \cite{Salisbury83} on the
realizeability of general coordinate transformations as canonical
transformations.
The formalism we shall develop encompasses all generally covariant
Lagrangian dynamical models containing configuration variables which
are either metric components or which may be used to construct a
metric. We begin in Section 2 with a rederivation of the relation
between gauge symmetries in Lagrangian and Hamiltonian formalisms.
After introducing the notions of lapse and shift in Section 3, we show
that diffeomorphism-induced gauge transformations are projectable
under the Legendre transformation if and only if infinitesimal
variations depend on the lapse and shift but not on their time
derivatives. These projectable infinitesimal transformations thus
contain a compulsory dependence on the normal to the chosen time
foliation. We illustrate these ideas with the relativistic particle,
canonical gravity, and the relativistic string.
In Section 4 we turn our attention to the construction of canonical
generators of the metric-dependent gauge group. These objects
generate symmetry transformations on the full set of canonical
variables. We show that every generator with non-vanishing time
component acts as an evolution generator on at least one member of
every equivalence class of solutions. Section 5 contains a discussion
of gauge fixing and the elimination of redundancy in initial
conditions. In Section 6, our conclusion, we discuss the nature
of the diffeomorphism-induced gauge group.
The Appendix illustrates the projectability conditions in a model, the
Nambu-Goto string, in which the lapse and shift depend on time
derivatives of the dynamical variables.
\section{Noether Hamiltonian Symmetries}
We begin by rederiving some results of Batlle {\it et al.}
\cite{Batlle89} for first order Lagrangians $L(q, \dot q)$. We
exclude Lagrangians which explicitly depend on time $t$ since we are
interested in reparameterization covariant systems. We start with a
Noether Lagrangian symmetry,
$$\delta L = {dF}/{dt},
$$
and we will investigate the conversion of this symmetry to the
Hamiltonian formalism. Defining
\begin{equation}
G = ({\partial L}/{\partial \dot q^i}) \delta q^i - F,
\end{equation}
we can write
\begin{equation} [L]_i\delta q^i + \frac{dG}{dt} = 0,
\label{noet}
\end{equation}
where $[L]_i$ is the Euler-Lagrange functional derivative of $L$,
$$ [L]_i = \alpha_i - W_{is}\ddot q^s,
$$
where
$$ W_{ij}\equiv {\partial^2L\over\partial\dot q^i\partial\dot q^j}
\quad {\rm{}and} \quad \alpha_i\equiv
- {\partial^2L\over\partial\dot q^i\partial q^s}\dot q^s
+ {\partial L\over\partial q^i} .
$$
Here we consider the general case where the mass matrix or Hessian
${\bf{}W}=(W_{ij})$ may be a singular matrix. In this case there
exists a kernel for the pullback ${\cal F}\!L^*$ of the Legendre map
${\cal F}\!L$ from configuration-velocity space $TQ$ (the tangent
bundle $TQ$ of the configuration space $Q$) to phase space $T^*Q$ (the
cotangent bundle). This kernel is spanned by the vector fields
\begin{equation} {\bf\Gamma}_\mu
= \gamma^i_\mu {\partial\over\partial\dot q^i} ,
\label{GAMMA}
\end{equation}
where $\gamma^i_\mu$ are a basis for the null vectors of $W_{ij}$.
The Lagrangian time-evolution differential operator can therefore be
expressed as:
\begin{equation} {\bf X} = {\partial\over\partial t}
+ \dot q^{s}{\partial\over\partial q^{s}}
+a^{s}(q,\dot q){\partial\over\partial \dot q^{s}}
+\lambda^{\mu}{\bf\Gamma}_{\mu}
\equiv {\bf X}_{o} +\lambda^{\mu}{\bf\Gamma}_{\mu} ,
\label{EVOLOP}
\end{equation}
where $a^{s}$ are functions which are determined by the formalism, and
$\lambda^{\mu}$ are arbitrary functions.
It is not necessary to use the Hamiltonian technique to find the
${\bf\Gamma}_\mu$, but it does facilitate the calculation:
\begin{equation} \gamma^i_\mu
= {\cal F}\!L^*\left( \partial\phi_\mu\over\partial p_i \right) ,
\label{gam}
\end{equation}
where the $\phi_\mu$ are the Hamiltonian primary first class
constraints.
Notice that the highest derivative in (\ref{noet}), $\ddot q^i$,
appears linearly. Because $\delta L$ is a symmetry, (\ref{noet}) is
identically satisfied, and therefore the coefficient of $\ddot q^i$
vanishes:
\begin{equation}
W_{is} \delta q^s - {\partial G \over\partial\dot q^i} = 0 .
\label{w-g}
\end{equation}
We contract with a null vector $\gamma^i_\mu$ to find
that
$$ {\bf\Gamma}_\mu G = 0 .
$$
It follows that $G$ is projectable to a function
$G_{\rm H}$ in~$T^*Q$; that is, it is the pullback of
a function (not necessarily unique) in
$T^*Q$:
$$
G = {\cal F}\!L^*(G_{\rm H}) .
$$
This important property, valid for any conserved quantity associated
with a Noether symmetry, was first pointed out by Kamimura
\cite{kami82}. Observe that $G_{\rm H}$ is determined up to the
addition of linear combinations of the primary constraints.
Substitution of this result in (\ref{w-g}) gives
$$ W_{is} \left[ \delta q^s - {\cal F}\!L^*
\left({\partial G_{\rm H}\over\partial p_s}\right) \right] = 0 ,
$$
and so the brackets enclose a null vector of {\bf W}:
\begin{equation}
\delta q^i - {\cal F}\!L^*
\left({\partial G_{\rm H}\over\partial p_i}\right)
= \sum_\mu r^\mu \gamma^i_\mu ,
\label{dq}
\end{equation}
for some $r^\mu(t, q, \dot q)$.
We shall investigate the projectability of variations generated by
diffeomorphisms in the following section. Assume for now that an
infinitesimal transformation $\delta q^i$ is projectable:
$${\bf\Gamma}_\mu \delta q^i =0.
$$
Notice that if $\delta q^i$ is projectable, so must be $r^\mu$,
so that $r^\mu = {\cal F}\!L^* (r^\mu_{\rm H})$.
Then, using (\ref{gam}) and (\ref{dq}), we see that
$$\delta q^i
= {\cal F}\!L^*
\left({\partial (G_H + \sum_\mu r^\mu_{\rm H}\phi_\mu)\over\partial
{p_i}}\right).
$$
We now redefine $G_{\rm H}$ to absorb the piece
$\sum_\mu r^\mu_{\rm H} \phi_\mu$, and from now on we will have
$$ \delta q^i = {\cal F}\!L^*
\left({\partial G_{\rm H} \over\partial p_i}\right).
$$
Define
$$ \hat p_i = {\partial L\over\partial\dot q^i};
$$
after eliminating (\ref{w-g}) times $\ddot q^i$ from (\ref{noet}),
we obtain
\begin{eqnarray}
\left({\partial L\over\partial q^i}
- \dot q^s {\partial \hat p_i\over\partial q^s} \right)
{\cal F}\!L^*({\partial G_{\rm H}\over\partial p_i})
&+& \dot q^i {\partial \over\partial q^{i}}
{\cal F}\!L^* (G_{\rm H}) \nonumber\\
&+& {\cal F}\!L^* ({\partial G_{\rm H}\over\partial t}) =0,
\end{eqnarray}
which simplifies to
\begin{equation}
{\partial L\over\partial q^i} {\cal F}\!L^*(
{\partial G_{\rm H}\over\partial p_i} )
+ \dot q^i {\cal F}\!L^*({\partial G_{\rm H}\over\partial q^i})
+ {\cal F}\!L^*
({\partial G_{\rm H}\over\partial t}) =0.
\label{noet-wg}
\end{equation}
Now let us invoke two identities \cite{Batlle86} that are at the core
of the connection between the Lagrangian and the Hamiltonian
equations of motion. They are
$$ \dot q^i = {\cal F}\!L^* ({\partial H\over\partial p_i})
+ v^\mu(q, \dot q) {\cal F}\!L^*
({\partial \phi_\mu\over\partial p_i}),
$$
and
$$ {\partial L\over\partial q^i}
= - {\cal F}\!L^* ({\partial H\over\partial q^i})
- v^\mu(q, \dot q) {\cal F}\!L^*
({\partial \phi_\mu\over\partial q^i});
$$
where $H$ is any canonical Hamiltonian, so that
${\cal F}\!L^*(H)
= \dot q^i (\partial L / \partial \dot q^i) - L =\hat E$,
the Lagrangian energy, and the functions $v^\mu$ are
determined so as to render the first relation an identity.
Notice the important relation
\begin{equation} {\bf\Gamma}_\mu v^\nu = \delta_\mu^\nu,
\end{equation}
which stems from applying ${\bf\Gamma}_\mu$ to the first identity
and taking into account that
\[ {\bf\Gamma}_\mu \circ{\cal F}\!L^* = 0.
\]
Substitution of these two identities into (\ref{noet-wg}) yields
(where $\{\,,\,\}$ is the Poisson Bracket)
$$ {\cal F}\!L^*\{G_{\rm H},H\}
+ v^\mu {\cal F}\!L^*\{G_{\rm H},\phi_\mu \}
+{\cal F}\!L^* ({\partial G_{\rm H}\over\partial t})
=0.
$$
This result can be split through the action of ${\bf\Gamma}_\mu$ into
\begin{equation}
{\cal F}\!L^*\{G_{\rm H},H\}
+{\cal F}\!L^* ({\partial G_{\rm H}\over\partial t}) =0,
\nonumber
\end{equation}
and
\begin{equation} {\cal F}\!L^*\{G_{\rm H},\phi_\mu \} = 0; \nonumber
\end{equation}
or equivalently,
\begin{equation} \{G_{\rm H},H\}
+ ({\partial G_{\rm H}\over\partial t}) = pc ,
\label{dgdt}
\end{equation}
and
\begin{equation} \{G_{\rm H},\phi_\mu \} = pc,
\label{g,phi}
\end{equation}
where $pc$ stands for any linear combination of primary
constraints. We have arrived at a neat characterization for a
generator $G_{\rm H}
$ of Noether transformations in the canonical formalism.
Up to now we have considered general Noether symmetries, encompassing
rigid (global) as well as gauge (local) transformations. Let us
finally specialize to gauge transformations. For reparameterization
covariant theories, except for a small number of exceptional cases not
important for this paper \cite{pep94},
a gauge generator will be of the form
$$ G_{\rm H}(t) = \epsilon(t) G_0(q,p) + \dot\epsilon(t)G_1(q,p),
$$
where $\epsilon(t)$ is an arbitrary function.
Because of the arbitrariness
of $\epsilon(t)$, and recognizing that the Poisson Bracket of the
Hamiltonian with primary constraints yields secondary constraints, we
learn from (\ref{dgdt}) that
$$
G_1 = pc,
$$
\begin{equation} G_0 = - \{G_1,H\} + pc,
\label{ggen}
\end{equation}
and
\begin{equation} \{G_0,H\} = pc;
\label{ggen2}
\end{equation}
while from (\ref{g,phi}) we deduce that
\begin{equation} \{G_0, pc\} = pc, \label{test1}
\end{equation}
and
\begin{equation} \{G_1, pc\} = pc.
\label{test2}
\end{equation}
It can be shown from
(\ref{ggen}) that $G_0$ must contain a piece which is a secondary
constraint, while (\ref{test1}) and (\ref{test2}) show that
both $G_0$ and the primary
constraint $G_1$ are first class.
\section{Diffeomorphism-Induced Gauge Symmetries}
We specialize now to generally covariant dynamical models in which a
metric can be constructed with the configuration variables (but not
with velocity variables). We assume in addition that no further gauge
symmetry exists. We shall illustrate our results with the
relativistic particle with an auxiliary variable and with general
relativity. Our first objective is to determine the general form of
projectable variations resulting from diffeomorphisms on a coordinate
patch.
If a metric exists in a coordinate system {$\{x^\mu\}$} the line
element may always be written in the form
\begin{equation}
ds^2=-N^2 (dx^0)^2 + g_{ab}(N^adx^0+dx^a)(N^bdx^0+dx^b)
\label{3.1}
\end{equation}
with contravariant metric components given by
\begin{equation}
(g^{\mu \nu}) = \pmatrix{
-N^{-2} & N^{-2}N^a \cr
N^{-2}N^a & e^{a b}-N^{-2}N^a N^b
} , \label{3.2}
\end{equation}
with $e^{ab}g_{bc}=\delta^{a}_{c}$.
The lapse function $N$ and shift vector $N^a$ will play important
roles in our discussion. Our index conventions are
that greek indices range from 0 to $M$, where $M$ is the
dimension of the spacelike hypersurfaces of the time foliation.
Latin indices range from 1 to $M$.
Explicitly, the configuration space variables are $N^\mu$ (with
$N^0 \equiv N$) and $g_{a b}$. The unit normal $n^{\mu}$ to
the spacelike hypersurfaces is given by
\begin{equation}
n^{\mu}=\delta^{\mu}_0 N^{-1} - \delta^{\mu}_a N^{-1} N^a\
{\rm{},\ so\ that}\ n^\mu n^\nu g_{\mu\nu}=-1.
\label{3.3}
\end{equation}
Since $e^{a b}$ is the inverse of the three-metric $g_{a b}$, the
contravariant components of the spacetime metric are
\begin{equation}
g^{\mu \nu}=e^{ab}\delta^\mu_a\delta^\nu_b - n^\mu n^\nu.
\label{3.4}
\end{equation}
Diffeomorphism covariance prevents the
lapse $N$ and shift $N^a$ from being fixed by the equations
of motion in any generally covariant dynamical model.
Specifically, since the $N^{\mu}$ are arbitrary, $\ddot N^{\mu}$
are undetermined. The evolution operator (\ref{EVOLOP}) acting on
$\dot N^{\mu}$ must therefore serve only to relate the arbitrary
functions $\lambda^{\mu}$ to the $\ddot N^{\mu}$.
Consequently, ${\bf \Gamma}_{\mu}\dot N^{\nu}$ must
form a nonsingular matrix.
Further, ${\bf \Gamma}_{\mu}$ acting on any other
velocity must give zero, since we are assuming no other
gauge symmetry. It follows that the null
vectors of the Hessian {\bf W} (see \ref{GAMMA}) are spanned by
\begin{equation}
{\bf{\Gamma}}_{\mu}={\partial \over \partial {\dot N }^{\mu}}.
\label{3.5}
\end{equation}
Since there are $M+1$ of the $N^\mu$, these null vectors span the
arbitrary component of the Lagrangian evolution operator
(\ref{EVOLOP}).
Now consider infinitesimal coordinate transformations $x^\mu
\rightarrow x^\mu - \epsilon^\mu(x)$ , with $\epsilon^\mu$ arbitrary
functions of the coordinate variables $x^\nu$. The corresponding
variations of the components of the metric tensor (the Lie
derivative of the metric along $\epsilon^\mu$) are
(${}_{,\mu}\equiv\partial/\partial x^\mu$):
\begin{equation}
\delta g^{\mu\nu}
= g^{\mu\nu}_{~~,\rho} \epsilon^\rho
- g^{\mu\rho} \epsilon^\nu_{,\rho}
- g^{\rho\nu} \epsilon^\mu_{,\rho},
\label{3.6}
\end{equation}
or
\begin{equation}
\delta g_{\mu\nu} = g_{\mu\nu,\rho} \epsilon^\rho
+ g_{\mu\rho} \epsilon^\rho_{,\nu}
+ g_{\rho\nu} \epsilon^\rho_{,\mu}.
\label{3.6a}
\end{equation}
The variations of the $N^\mu$ are readily calculated:
\begin{equation}
\delta N = N_{,\mu} \epsilon^\mu
+N \epsilon^0_{,0} -N N^a \epsilon^0_{,a},
\label{3.8}
\end{equation}
\begin{eqnarray}
\delta N^a &=& N^a_{,\mu} \epsilon^\mu
+ N^a \epsilon^0_{,0} -(N^2 e^{a b}
+ N^a N^b)\epsilon^0_{,b} \nonumber\\
&&~~~~~~+\epsilon^a_{,0} -N^b \epsilon^a_{,b}.
\label{3.9}
\end{eqnarray}
Thus the variations of the $N^\mu$ do depend on ${\dot N}^\mu =
N^\mu_{,0}$ (but the variations of $g_{a b}$ do not), assuming as we
have above, that $\epsilon^\mu$ depends only on the coordinates.
Consequently the variations of $N^\mu$ are clearly not projectable;
projectability is attained only if we permit $\epsilon^\mu$ to depend
on $N^\mu$. The requirement that derivatives of $\delta N^\mu$ with
respect to ${\dot N}^\mu$ vanish implies that
\begin{equation}
\epsilon^0 + N {\partial \epsilon^0 \over \partial N} = 0,
\label{3.10}
\end{equation}
\begin{equation} {\partial \epsilon^0 \over \partial N^a }= 0,
\label{3.11}
\end{equation}
\begin{equation} N^a {\partial \epsilon^0 \over \partial N}
+ {\partial \epsilon^a \over \partial N} = 0,
\label{3.12}
\end{equation}
\begin{equation}
\epsilon^0 \delta^a_b
+ {\partial \epsilon^a \over \partial N^b}=0.
\label{3.13}
\end{equation}
These equations were first obtained in \cite{Salisbury83} using a
rather different approach. In \cite{Salisbury83}, the following
requirement was introduced for diffeomorphism-induced gauge
transformations: Consider $\delta_1 x^\mu = -\epsilon_1^\mu(x, g(x))$
and $\delta_2 x^\mu = -\epsilon_2^\mu(x, g(x))$; then ask for
conditions to be satisfied by $\epsilon_1, \epsilon_2$ such that
$[\delta_1,\delta_2] x^\mu $ has no explicit time derivatives of
$\epsilon_1$ or $\epsilon_2$. We will discuss in the next section the
reason why this latter approach gives results coincident with ours.
We feel that the requirement of projectability (independence of the
$\delta N^\mu$ on ${\dot N}^\mu$ in this case) is a more natural
approach.
The general solution of the $\epsilon^\mu$ equations
(\ref{3.10}-\ref{3.13}) is
\begin{equation} \epsilon^\mu = \delta^\mu_a \xi^a + n^\mu \xi^0,\
{\rm{}so\ that\ }
\epsilon^0={\xi^0\over N},\
\epsilon^a=\xi^a - {N^a\over N} \xi^0,
\label{3.14}
\end{equation}
where $\xi^a $ and $\xi^0$ are arbitrary functions of the spacetime
variables $x^\mu$ and $g_{ab}$ but are independent of $N^\mu$. The
dependence on the $M$-surface metric plays no role in our present
arguments but is required, as we show in Section 6, in order that the
diffeomorphism-induced transformations form a group. The result
(\ref{3.14}) is true in a more general context than we have been
treating. The Appendix will illustrate this point with an example,
the Nambu-Goto string, in which the metric is built with velocity
variables as well as configuration space variables.
\subsection{The Free Relativistic Particle with Auxiliary Variable}
We illustrate first with the unit-mass relativistic free particle
model with auxiliary variable described by the Lagrangian
\begin{equation} L = \frac{1}{2e} \dot x^\mu \dot x^\nu \eta_{\mu\nu}
- \frac{1}{2} e,
\label{partlag}
\end{equation}
where ${ x^{\mu}}$ is the vector variable in Minkowski spacetime,
with metric $(\eta_{\mu\nu})={\rm{}diag}(-1,1,1,1)$, and $e$ is an
auxiliary variable whose equation of motion gives
$e = (-{\dot x}^\mu {\dot x}_\mu)^{1/2}$.
Substituting this value of $e$ into the
Lagrangian leads to the free particle Lagrangian
$-(-{\dot x}^\mu {\dot x}_\mu)^{1/2}$.
The following Noether gauge transformation is well-known to describe
the reparameterization invariance for this Lagrangian
($\delta L = {d\over dt}(\epsilon L)$):
\begin{equation}
\delta x^\mu = \epsilon {\dot x}_{\mu}, \ \delta e
= \epsilon \dot e + {\dot \epsilon} e.
\label{3.16}
\end{equation}
Here $\epsilon$ is an infinitesimal arbitrary function of the
evolution parameter $t$. Comparing (\ref{3.16}) with (\ref{3.8}),
we observe that $e$ may be interpreted as a lapse, with
corresponding metric $g_{0 0}=-e^2$.
The kernel of the pullback map ${\cal F}\!L^*$ is defined in
(\ref{GAMMA}); here it is spanned by the vector field
${\bf\Gamma} = \partial/ \partial \dot e$.
The condition that a function $f$ in
configuration-velocity space be projectable to phase space is
$$ {\bf\Gamma} f = \frac{\partial f}{\partial \dot e} = 0.
$$
The Noether transformation (\ref{3.16}) is not projectable to phase
space, since ${\bf\Gamma} \delta e \neq 0$. Projectable
transformations are of the form (\ref{3.14}):
\begin{equation}
\epsilon(t,e) = {\xi(t)}/{e}.
\label{proj3.16}
\end{equation}
The Noether variations then become:
\begin{equation}
\delta x^{\mu} = \xi \frac{{\dot x}^\mu}{e},\ \delta e = \dot \xi.
\label{del-x}
\end{equation}
The arbitrary function describing the Noether gauge transformation is
$\xi(t)$. What we have achieved is a change of the generator of the
gauge transformations. This leads to a change of the gauge algebra
which in our case becomes Abelian. But from the point of view of the
gauge symmetry of our model we still have the same mappings of
solutions onto gauge equivalent solutions. That is, on a given
dynamical trajectory $ x^\mu_0(t), \, e_0(t)$ we can match the
transformation given by an arbitrary $\epsilon_0(t)$ with $\xi_0(t)$
defined as $\xi_0(t) \equiv e_0(t) \epsilon_0(t)$. It is in comparing
a transformation on one dynamical trajectory with that acting on
another where the change has occurred. In Section 6 we shall
elaborate further on the issue of the gauge group.
The canonical Hamiltonian is
$$ H = \frac{1}{2} e ( p^\mu p_\mu + 1),
$$
and there is a primary constraint $\pi \simeq 0$, where $\pi$ is the
variable conjugate to $e$. The evolution operator vector field
$\{-,H\}+ \lambda(t) \{-,\pi\}$ yields the secondary constraint
$\frac{1}{2} (p^\mu p_\mu + 1) \simeq 0$. Both the primary and the
secondary constraints are first class. The arbitrary function
$\lambda$ is a reflection of the gauge invariance of the model. The
solutions of the equations of motion are:
$$ x^\mu(t) = x^\mu(0) + p^\mu(0) \left(e(0) t
+ \int_0^t d\tau \int_0^\tau d\tau' \, \lambda(\tau')\right),
$$
$$ e(t) = e(0) + \int_0^t d\tau \, \lambda(\tau),
$$
$$ p^\mu(t) = p^\mu(0),
$$
$$ \pi(t) = \pi(0),
$$
with the initial conditions satisfying the constraints.
Gauge transformations relate trajectories obtained through different
choices of $\lambda(t)$. Consider an infinitesimal change
$\lambda \rightarrow \lambda + \delta\lambda$. Then the change
in the trajectories (keeping the initial conditions intact) is:
$$ \delta x^\mu(t) = p^\mu(0)
\left(\int_0^t d\tau \int_0^\tau d\tau' \,
\delta\lambda(\tau')\right),
$$
$$ \delta e(t) = \int_0^t d\tau \, \delta\lambda(\tau),
$$
$$ \delta p^\mu(t) = 0 , \, \delta \pi(t) = 0,
$$
which is nothing but a particular case of the projectable gauge
transformations displayed above with
$$\xi(t) =\int_0^t d\tau \int_0^\tau d\tau' \, \delta\lambda(\tau').
$$
\subsection{Diffeomorphisms in Canonical General Relativity}
Up to a boundary piece, the Einstein-Hilbert Lagrangian can be written
as \cite{Wald84}
\begin{equation}
{\cal L} = ({}^{3}\!g)^{\frac{1}{2}} N ({}^3\!R + K_{ab}K^{ab}
- K^2) , \label{lagr}
\end{equation}
where ${}^{3}\!g$ is the determinant of the 3-metric tensor in
(\ref{3.1}), ${}^{3}\!g = \det(g_{ab})$, ${}^3\!R$ is the scalar
curvature computed from the 3-metric, and $K_{ab}$ is the second
fundamental form (extrinsic curvature) for the constant-time
3-surfaces :
\begin{equation}
K_{ab}={1\over 2N}(\dot g_{ab} - N_{a|b} - N_{b|a}),
\end{equation}
with ${}_|$ meaning covariant differentiation with respect to the
3-metric connection. Notice that the lapse $N$ and shift $N^a$ of the
4-metric all appear, but their time-derivatives do not.
We may directly apply our general formalism, with no notational
changes, to conclude that projectable infinitesimal coordinate
transformations must be of the form
\begin{equation}
x^\mu \rightarrow x^\mu - \delta^\mu_a \xi^a - n^\mu \xi^0.
\label{sptm3.xx}
\end{equation}
Notice also that for any specific spacetime metric $g_{\mu\nu}(x)$ we
can implement {\it any} infinitesimal diffeomorphism $x^\mu
\rightarrow x^\mu - \epsilon^\mu(x)$ by taking the set $\xi^0, \xi^i$
(assuming $N \neq 0$) as:
$$
\xi^0 = N \epsilon^0 , \quad \xi^a = \epsilon^a + N^a \epsilon^0;
$$
therefore we are not restricting the (infinitesimal) diffeomorphisms
that can act on any specific metric. What we achieved is a set of
generators of the gauge group which can be projected to the phase
space.
\section{The Hamiltonian Gauge Generators}
Our objective in this section is to derive the full set of
diffeomorphism-induced gauge generators for the class of dynamical
models treated in Section 2. Since the $\xi^{\mu}$ are now the
arbitrary functions of time appearing in the variations $\delta q^i$
of Section 2, we modify the argument leading to the general form of
the symmetry generators to conclude that these generators must be of
the form
$$ G(t) = \int d^M\!x \left(\xi^\mu({\bf x},t) G^{(0)}_\mu
+ \dot\xi^\mu({\bf x},t)G^{(1)}_\mu\right).
$$
When we use $\epsilon^\mu = \delta^\mu_a \xi^a + n^\mu \xi^0$ in
(\ref{3.6}), the algebra of the infinitesimal transformations ceases
to be the standard diffeomorphism algebra. The standard algebra is
that of Lie derivatives:
$\epsilon^\mu_3 = \epsilon^\nu_2 \epsilon^\mu_{1,\nu}
- \epsilon^\nu_1 \epsilon^\mu_{2,\nu}
=({\cal L}_{\epsilon_2}\epsilon_1)^{\mu} $.
In our case the commutator of two infinitesimal transformations yields
an $\epsilon^\mu_3$ of the form of (\ref{3.14}), with
\begin{equation}
\xi_3^a = \xi^b_2\xi^a_{1,b} - \xi^b_1\xi^a_{2,b}
- e^{a b}\xi^0_1 \xi^0_{2,b}
+ e^{a b}\xi^0_2 \xi^0_{1,b},
\label{comrel-a}
\end{equation}
\begin{equation}
\xi_3^0 = \xi^a_2 \xi^0_{1,a} - \xi^a_1 \xi^0_{2,a} .
\label{comrel-0}
\end{equation}
These are the new commutation relations of the gauge algebra in
configuration-velocity space. The commutation rules of the gauge
generators in phase space coincide with the commutation relations in
configuration-velocity space as long as at least all but one of the
$M+1$ gauge generators are linear in the momenta (see
\cite{Batlle89a}). This amount of linearity holds in our models:
(\ref{3.14}) implies that time-independent $M$-space diffeomorphisms
are always projectable. This means that each associated canonical
generator must be linear in the momenta; otherwise the transformation
of configuration variables will depend on velocities. We can
conclude that we know the Poisson Bracket rules
$G[\xi_3] = \{G[\xi_1], G[\xi_2] \}$
for our gauge generators. Thus in comparing
(\ref{comrel-a}) and (\ref{comrel-0}) with $G[\xi_3]$ we deduce the
algebra for these generators. In what follows, we use the convention
that repeated indices imply both summation and M-dimensional
integration; we use primed indices where necessary to make sure that
separate integrations are clearly delineated, though we drop the
primes on the indices where no loss of clarity is involved:
\begin{equation}
\{G_\mu^{(0)},G_{\nu '}^{(0)}\}
=C_{\mu \nu '}^{\alpha ''} G_{\alpha ''}^{(0)}
+({d \over dt}C_{\mu {\nu '}}^{\alpha ''} ) G_{\alpha ''}^{(1)},
\label{5.20}
\end{equation}
\begin{equation}
\{G_\mu^{(0)},G_{\nu '}^{(1)}\}
=C_{\mu \nu '}^{\alpha ''} G_{\alpha ''}^{(1)},
\label{5.21}
\end{equation}
and
\begin{equation}
\{G_\mu^{(1)},G_{\nu '}^{(1)}\}=0,
\label{5.22}
\end{equation}
where the structure coefficients are given by
\begin{eqnarray}
C_{0 0'}^{a''}
&=& e^{a b}({\bf x}'')\bigg(\delta^M ({\bf x}-{\bf x}'')
\nonumber\\
&&~~~~+\delta^M ({\bf x}'-{\bf x}'')\bigg)
{\partial \over \partial x^b} \delta^M ({\bf x}-{\bf x}'),
\label{5.22a}
\end{eqnarray}
\begin{equation} C_{0 0'}^{0''}=0, \label{5.22b}
\end{equation}
\begin{equation} C_{a 0'}^{0''}
=\delta^M ({\bf x}-{\bf x}'')
{\partial \over \partial x^a} \delta^M ({\bf x}-{\bf x}')
= - C_{0' a}^{0''}, \label{5.22c}
\end{equation}
\begin{equation} C_{a 0'}^{b''}= 0, \label{5.22d}
\end{equation}
\begin{eqnarray} C_{a b'}^{c''}&=&
\bigg(\delta_a^c \delta^M ({\bf x}''-{\bf x}')
{\partial \over \partial x^b} \nonumber\\
&&~~~~+ \delta_b^c \delta^M ({\bf x}''-{\bf x})
{\partial \over \partial x^a}\bigg)
\delta^M ({\bf x}-{\bf x}'), \label{5.22e}
\end{eqnarray}
and
\begin{equation} C_{a b'}^{0''}=0. \label{5.22f}
\end{equation}
We now construct these generators explicitly. The canonical
Hamiltonian (such that its pullback under the Legendre transformation
gives the Lagrangian energy) for the class of models under discussion
is
\begin{equation}
H = N^\mu {\cal H}_\mu, \label{444}
\end{equation}
where the ${\cal H}_\mu$ are independent of $N^\mu$ and $P_\mu$, and
the primary constraints are $P_\mu$, the canonical variables conjugate
to $N^\mu$. The secondary constraints are
$\dot P_\mu = \{P_\mu , H \} = - {\cal H}_\mu$,
and no more constraints appear. It was shown in
\cite{Pons95} that the canonical Hamiltonian always takes the form
(\ref{444}). All constraints are required to be first class; the
reason is that $\xi^\mu$ and $\dot\xi^\mu$ appear in (\ref{3.6}), and
so all constraints are required to build the spacetime gauge
generators that our theory possesses.
The Dirac Hamiltonian $H_{\rm D}$ is constructed by the addition to
$H$ of a linear combination (with arbitrary functions
$\lambda^\mu$) of the primary constraints:
\begin{equation}
H_{\rm D} = H + \lambda^\mu P_\mu.
\label{dirach}
\end{equation}
$G^{(1)}_\mu$ must be a primary constraint, so the simplest choice is
$G^{(1)}_\mu = P_\mu$. It is now necessary to apply (\ref{ggen}):
$ G^{(0)}_\mu = -\{G^{(1)}_\mu , H \} + pc, $ implying
$$
G^{(0)}_\mu = {\cal H}_\mu + A_\mu^\nu P_\nu .
$$
From (\ref{5.20}) we deduce that
$$
\{{\cal H}_\mu, {\cal H}_\nu \}
= C^\sigma_{\mu\nu}{\cal H}_\sigma ,
$$
since $\{N^\mu,{\cal H}_\nu \}=\{P_\mu,{\cal H}_\nu \}=0$.
The $A^\nu_\mu$ are determined by applying condition (\ref{ggen2}) to
$G^{(0)}_\mu$:
\begin{eqnarray}
\nonumber pc &=& \{{\cal H}_\mu + A_\mu^\nu P_\nu , H \} \\
\nonumber &=& N^\nu \{{\cal H}_\mu, {\cal H}_\nu \}
+ A^\nu_\mu \{P_\nu , H \} \\
\nonumber &=& N^\nu C^\sigma_{\mu\nu}{\cal H}_\sigma - A^\nu_\mu
{\cal H}_\nu,
\end{eqnarray}
which implies
$$ A_\mu^\nu = N^\rho C^\nu_{\mu\rho}
$$
up to an irrelevant arbitrary linear combination of primary
constraints that would add an ineffective piece to the gauge
generator. (By ineffective we mean that the added piece is quadratic
in the constraints.) We ignore this piece and take the
simplest solutions available for $A_{\mu}^{\nu}$.
It is trivial to check the fulfillment of conditions
(\ref{test1},\ref{test2}). By use of the Jacobi identity we find
\begin{eqnarray} \nonumber
0 &\equiv& \{{\cal H}_\alpha,\{{\cal H}_\beta,{\cal H}_\gamma \}\}
+ \{{\cal H}_\beta,\{{\cal H}_\gamma,{\cal H}_\alpha \}\} \\
&~~~~&+ \{{\cal H}_\gamma,\{{\cal H}_\alpha,{\cal H}_\beta \}\}
\nonumber \\
\nonumber &=&\left(C_{\beta \gamma}^\rho C_{\alpha \rho}^\sigma
+ C_{\gamma \alpha}^\rho C_{\beta \rho}^\sigma
+ C_{\alpha \beta}^\rho C_{\gamma \rho}^\sigma\right)
{\cal H}_\sigma
+ \{{\cal H}_\alpha,C_{\beta \gamma}^\rho \} {\cal H}_\rho\\
&~~~~&
+ \{{\cal H}_\beta,C_{\gamma \alpha}^\rho \} {\cal H}_\rho
+ \{{\cal H}_\gamma,C_{\alpha \beta}^\rho \} {\cal H}_\rho,
\label{5.22g}
\end{eqnarray}
together with
\begin{equation} {d \over dt} C_{\alpha \beta}^\gamma
= N^\rho \{C_{\alpha \beta}^\gamma, {\cal H}_\rho \},
\label{5.22h}
\end{equation}
and it is straightforward to show that
the generators $G_\mu^{(0)}$ and $G_\mu^{(1)}$ do satisfy the algebra
(\ref{5.20}-\ref{5.22}).
We have therefore obtained the full set of
diffeo\-mor\-phism-induced gauge generators:
\begin{equation}
G(t) = P_\mu \dot\xi^\mu
+ ( {\cal H}_\mu + N^\rho C^\nu_{\mu\rho} P_\nu) \xi^\mu,
\label{thegen}
\end{equation}
(where the repeated index, to repeat, involves an integration).
Note that $G(t)$ generates variations in the full phase space. It is
straightforward to verify that it does generate the correct
variations of $N^{\mu}$ (\ref{3.8}-\ref{3.9}) under the
diffeomorphism-induced gauge transformations (\ref{3.14}).
The preceding discussion applies with no modification of notation to
canonical general relativity.
We continue with some general remarks on diffeomorphism generators.
Our first observation is that every generator $G(t)$ with
$\xi^0 \neq 0$ is interpretable as a global
time translation generator for at
least one member of every gauge equivalence class of solutions. To
demonstrate this property we note that for a given set of functions
$\xi^\mu$ in the expansion
\begin{equation}
\epsilon^\mu = \delta^\mu_a \xi^a + n^\mu \xi^0,
\label{diff4.xx}
\end{equation}
we can solve for the $N^\mu$ which render
$\epsilon^\mu = \delta^\mu_0$:
\begin{equation}
\epsilon^\mu = \delta^\mu_0
=\delta^\mu_a \xi^a
+ (N^{-1}\delta^\mu_0-N^{-1}N^a \delta^\mu_a)\xi^0.
\label{solv.xx}
\end{equation}
The solution is $N^\mu = \xi^\mu$, which renders $G(t)$ in
(\ref{thegen}) identical to the Dirac Hamiltonian (\ref{dirach}),
once we take into account that the equations of motion provide
$\dot{N}^\mu = \lambda^\mu$. Therefore the gauge
generator contains, for any solution of the equations of motion,
the dynamical evolution as a particular case.
At this point we are ready to understand the coincidence of the two
approaches mentioned in the previous section: The same conditions
(\ref{3.10}-\ref{3.13}) are obtained if (1) one asks for the
projectability of (\ref{3.6a}), or (2) one asks for
$[\delta_1,\delta_2] x^\mu $ not to have any explicit time
derivative of $\epsilon_1$ or $\epsilon_2$. It seems odd that
conditions imposed on {\it one} transformation (projectability)
and conditions imposed on the commutation of {\it two}
transformations should give the same results.
The reason lies in the structure of the gauge generators in
phase space: They are constructed with linear combinations of
constraints with arbitrary functions and their first time derivatives.
Let us consider two of these generators $G[\xi_1], G[\xi_2]$. Their
Poisson Bracket, an equal time commutator, is on general grounds
$G[\xi_3]$ for some $\xi_3$. It is impossible to get for $\xi_3$ the
standard diffeomorphism rule
$\xi^\mu_3 = \xi_2^\nu \xi_{1,\nu}^\mu - \xi_1^\nu \xi_{2,\nu}^\mu $:
In such a case $\dot\xi_3$, which appears in $G[\xi_3]$, will depend
on the second time derivatives of $\xi_1$ and $\xi_2$, and this
dependence {\it cannot} be generated by the equal time Poisson Bracket
$\{G[\xi_1], G[\xi_2]\}$. Nesting of Poisson Brackets would introduce
yet higher time derivatives. This is why general reparameterization
covariance cannot be implemented in this form in the Hamiltonian
formalism. The argument applies to any reparameterization covariant
theory.
This was the argument used in \cite{Salisbury83} to realize
diffeomorphisms in the canonical formalism. In fact the arena in
\cite{Salisbury83} was the reduced space defined by the variables
$(g_{a b}, K^{a b})$. In this case there are no time derivatives of
the arbitrary functions $\xi^{\mu}$ in the variations generated by
(\ref{thegen}), but the argument still applies in the same way, as
shown above. Once the obstruction to projectability is identified
through the form of $[\delta_1,\delta_2] x^\mu $, the assumption of a
metric dependence in $\epsilon^\mu$ and the requirement that
$[\delta_1,\delta_2] x^\mu $ must not have any explicit time
derivative of $\epsilon_1$ or $\epsilon_2$ leads to equations
(\ref{3.10}-\ref{3.13}), the projectability condition.
We should caution that the algebra (\ref{5.20}-\ref{5.22}) is
satisfied only under the condition that there is no other gauge
symmetry in addition to diffeomorphism-induced symmetry. Under more
general circumstances, pure diffeomorphisms (even the field dependent
variety given by \ref{3.14}) are not realizable as canonical
transformations; they must be accompanied by related internal gauge
transformations \cite{Salisbury83b}. The issue of projectability for
these models will be addressed in another paper.
Gauge theories like electromagnetism or Yang-Mills in Minkowski
spacetime share with general relativity the property that gauge
transformations (diffeomorphisms in general relativity) need to be
constructed with arbitrary functions and their spacetime first
derivatives. The gauge generators are made up of two pieces,
associated with a primary and a secondary constraint; it is therefore
mandatory that all these theories have secondary constraints. This is
the way by which the canonical formalism is able to provide us with
the right gauge transformations.
For the sake of completeness, let us now apply these ideas to our
relativistic particle (\ref{partlag}). The gauge generator is, from
(\ref{thegen}),
$$
G(t) = {\dot \xi}(t) \pi + \xi(t) \frac{1}{2} (p_\mu p^\mu + 1),
$$
with $\xi$ an arbitrary function of time. One can easily check that
$G(t)$ generates the (projectable) transformations (\ref{del-x})
introduced above. Notice also that if $\xi$ is a constant, the
secondary constraint generates a rigid (time-independent) Noether
symmetry, whereas the primary one does not. Primary constraints
generate gauge symmetries only in the case when they do not lead to
secondary constraints through the stabilization algorithm.
Finally, notice that we do not modify ``by hand'' the
Hamiltonian by adding to it the secondary constraint with a new
Lagrange multiplier. This modification, the so called Dirac
conjecture, turns out not only to be unnecessary but to break the
equivalence with the Lagrangian theory as well \cite{Gracia88}.
\section{Gauge Fixing and Reduced Formalism}
\subsection{The Gauge Fixing Procedure}
One of the methods to eliminate the superfluous degrees of freedom of
a gauge theory is through the introduction of a new set of
constraints. This is the gauge fixing procedure, which according to
\cite{Pons95}, can be performed in two different steps: the first is
to fix the dynamics, the second to fix the redundancy of the initial
conditions (though this need not be the order in which the whole set
of constraints is introduced).
First, to fix the dynamics---to determine specific values for the
functions $\lambda^\mu$ in (\ref{dirach})---we must introduce $M+1$
constraints, $\varphi_\mu \simeq 0 $, such that
$\det|\{\varphi_\mu , P_\nu \}| \neq 0$. A typical set could be
\begin{equation}
\varphi_\mu = N^\mu - f^\mu,
\label{gaugef}
\end{equation}
with $f^\mu$ ($f^0 \neq 0$) a given set of functions not depending on
$P_{\mu}$ or $N^{\nu}$ (the simplest choice could be
$f^a = 0,\, f^0 = 1$). We could also think of $f^\mu$
as a not yet determined set of functions. These gauge fixing
constraints fix $\lambda^\mu$ in $H_{\rm D}$ to be zero and then:
\begin{equation}
H_{\rm D}^{\small red}
= N^\mu {\cal H}_\mu
\cong f^\mu {\cal H}_\mu,
\label{hred}
\end{equation}
where we have used Dirac's notation of strong equality, $\cong$, to
mean an equality up to {\it quadratic} pieces in the constraints,
including the gauge fixing ones. In practice this strong equality
tells us that we can substitute $f^\mu$ for $N^\mu$ within the
Hamiltonian due to the fact that ${\cal H}_\mu$ are constraints, too.
Once this set of evolution-fixing constraints $\varphi_\mu \simeq 0 $
has been introduced, with a given set of functions $f^\mu$, the gauge
transformations---strictly speaking---have disappeared. In fact, if
we require the gauge generators to be consistent with the new
constraints, $\{\varphi_\mu ,\, G(t) \} = 0$, we get the relations
\begin{equation}
\dot \xi^\mu + \xi^\nu N^\sigma C^\mu_{\nu\sigma} = 0,
\label{nogaug}
\end{equation}
which means that the functions $\xi^\mu$ cease to be arbitrary (at
least with respect to the time dependence), and hence there are no
more gauge transformations. The transformations $G(t)$ satisfying
(\ref{nogaug}) can be called, as is usual in other contexts, residual
gauge transformations, but it must be emphasized that they are not
true gauge transformations in phase space, because the arbitrariness
that was present in the Dirac Hamiltonian has been eliminated. We
encounter a parallel case in electromagnetism, for instance, when
after introducing the Lorentz gauge $\partial_\mu A^\mu = 0$, we are
left with a residual gauge symmetry,
$A^\mu \rightarrow A^\mu + \partial_\mu \Lambda$,
provided $\Lambda$ satisfies $\Box \Lambda = 0$.
Another way to view the residual gauge transformations, which is more
interesting to us, is to consider the situation at a given initial
time, $t=0$. Let
$\alpha^\mu({\bf x}) = \xi^\mu(0,{\bf x}),\,
\beta^\mu({\bf x}) = \dot\xi^\mu(0,{\bf x})$;
they are related by (\ref{nogaug}):
$\beta^\mu + \alpha^\nu N^\sigma C^\mu_{\nu\sigma} = 0$.
We are left with the ``residual'' gauge transformation at $t=0$
$$
G_{\rm R}(0) = P_\mu \beta^\mu
+ ( {\cal H}_\mu + N^\rho C^\nu_{\mu\rho} P_\nu) \alpha^\mu
= {\cal H}_\mu \alpha^\mu ,
$$
with $\alpha^\mu$ an arbitrary function of $M$-space variables.
The role of $G_{\rm R}(0)$ is that it generates transformations on the
initial value surface that describe a redundancy that is still left in
the formalism, and we must eliminate it in order to arrive at the true
degrees of freedom. Thus we must introduce a new set of gauge fixing
constraints, $\chi_\mu \simeq 0$, with the requirements:
\begin{enumerate}
\item
The dynamical evolution, which is already fixed, must preserve these
new constraints.
\item
$\{ \chi_\mu , G_{\rm R}(0) \} = 0 $ must imply $\alpha^\mu = 0$.
\end{enumerate}
Obviously, to satisfy the second condition, we need
$\det|\{\chi_\mu , {\cal H}_\nu \}| \neq 0$,
and to satisfy the first we need:
\begin{equation}
\{\chi_\mu, H_{\rm D} \} + {\partial \chi_\mu\over\partial t}
= f^\nu \{\chi_\mu, {\cal H}_\nu \}
+ {\partial\chi_\mu\over\partial t} \simeq 0.
\end{equation}
Notice that the first and the second conditions are only compatible if
at least one of the gauge fixing constraints $\chi_\mu$, for instance
$\chi_0$, has explicit time dependence: $\partial{\chi_0}/\partial{t}
\neq 0$. This result implies that time needs to be defined
classically through a function of the canonical variables.
\subsection{The Reduced Formalism}
Notice that if we perform the partial gauge fixing defined in
(\ref{gaugef}), $N^\mu - f^\mu \simeq 0$ in the spirit of keeping
$f^\mu$ undetermined, then we can interpret
$$
H_{\rm D}^{\small red} = f^\mu {\cal H}_\mu
$$
in (\ref{hred}) as the Hamiltonian for the reduced phase space
described by all variables other than $P_{\mu}$ and $N^{\nu}$. In
this reduced space we have a dynamical theory defined by a vanishing
canonical Hamiltonian and a set of constraints, which now become
primary, ${\cal H}_\mu \simeq 0$. Then the new Dirac Hamiltonian is
$H_{\rm D}^{\small red}$ and the new gauge generator is
$$
G^{\small red} = \xi^\mu {\cal H}_\mu.
$$
Thus we see that the constraints ${\cal H}_\mu$ do generate gauge
transformations {\it in the reduced phase space}.
We identify here a frequent source of confusion in the literature when
it is claimed that all first class constraints, either primary or
secondary (or tertiary, etc.), generate gauge transformations. For
generally covariant theories with a metric, in the original phase
space, only specific combinations, as in (\ref{thegen}), of primary
and secondary constraints generate gauge transformations. But in the
reduced formalism, since the old secondary constraints take the role
of primary constraints and there are no more constraints, these new
primary constraints generate gauge transformations in the reduced
space.
As to the gauge fixing procedure in the reduced phase space, since
there are only primary constraints, there is only one step to be
undertaken: to fix the evolution. Notice that the same argument we
used previously to show that one of the gauge-fixing constraints must
be the definition of time applies here as well.
\subsection{From the Reduced to the Original Formalism}
In the case of generally covariant theories, we have seen that the
reduced formalism consists in the elimination of the primary
constraints, $P_\mu$, and their canonical conjugate variables,
$N^\mu$, through a partial gauge fixing
$N^\mu = f^\mu, \, \, (f^0 \neq 0)$, with $f^\mu$
arbitrary functions of spacetime as well as of
the reduced variables. Then we obtain a reduced
theory which has ${\cal H}_\mu \simeq 0$ as primary constraints (no
secondary constraints appear),
$H^{\small red} = f^\mu {\cal H}_\mu$
as the Dirac Hamiltonian, and
$G^{\small red} = \xi^\mu {\cal H}_\mu$
as the generator of gauge symmetries. The new
bracket for the set of the reduced variables is just the Dirac
Bracket, which in our case is trivially obtained as the old Poisson
Bracket when acting with the reduced variables.
One may wonder whether there is a way to restore the full theory from
the reduced one. In these cases where the constraints eliminated in
the process of reduction are canonical momenta we will see that there
exists such a method. This is the enlargement procedure:
Consider a theory defined by a canonical Hamiltonian $H_{\rm C}$ and a
set of primary constraints $\phi_\mu \simeq 0$. The Dirac Hamiltonian
is $H_{\rm D} = H_{\rm C} + \lambda^\mu\phi_\mu$ ($\lambda^\mu$ are
arbitrary functions). Let us suppose we have applied the
stabilization algorithm to obtain secondary, tertiary, etc.,
constraints and that we have finally obtained a set of Noether gauge
generators, described by a single $G[\xi]$, where $\xi$ stands for a
set $\xi^\alpha$ of infinitesimal arbitrary functions of spacetime.
$G[\xi]$ is assumed to be a local functional of $\xi^\alpha$ (that is,
it depends linearly on $\xi^\alpha$ and a finite number of its time
derivatives, according to the length of the stabilization algorithm).
We also assume the commutation algebra for $G[\xi]$ to be
$$
\{G[\xi_1] , \, G[\xi_2] \} = G[\xi_3],
$$
with
$\xi_3^\alpha = C_{\beta\gamma}^\alpha \xi^{\beta}_1 {\xi}^{\gamma}_2$
(this is a general property, and the $C_{\beta\gamma}^\alpha$ are not
necessarily the same as previously defined; remember that repeated
indices imply both summation and integration).
According to Section 2, there exist functionals $A^\mu_\nu[\xi]$ and
$B^\mu[\xi]$ such that
$$
\{G[\xi],H_{\rm C}\} + {\partial G[\xi]\over\partial t}
= B^\mu[\xi]\phi_\mu, \quad
\{G[\xi],\phi_\nu \}
= A^{\mu}_\nu[\xi]\phi_{\mu}.
$$
The enlargement procedure consists in promoting the arbitrary
functions $\lambda^\mu$ to the status of canonical variables; let us
call them $N^\mu$ for obvious reasons. Let us introduce canonical
momenta $P_\mu$ associated with the new variables (we thus trivially
enlarge the Poisson Bracket) and require these new momenta to be the
primary constraints of the enlarged theory.
The enlarged canonical Hamiltonian will then be
$H_{\rm C} + N^\mu\phi_\mu$, and the new Dirac Hamiltonian will be
$H_{\rm E} = H_{\rm C} + N^\mu\phi_\mu + \eta^\mu P_\mu$,
with $\eta^\mu$ new arbitrary functions. It is straightforward to
verify that the dynamics of the original theory and that of the
enlarged theory coincide as far as the evolution of the original
variables is concerned. However, note that the $\phi^\mu$ have become
secondary constraints.
Now we will show how to enlarge the corresponding gauge generators
$G[\xi]$. Since the new primary constraints $P_\mu$ must appear
within the enlarged gauge generators $G_{\rm E}[\xi]$, we will assume
the general form $G_{\rm E}[\xi] = G[\xi] + S^\mu[\xi] P_\mu$, with
$S^\mu$ to be determined through the requirements of Section 2. It
turns out that $S^\mu = B^\mu + N^\nu A^\mu_\nu P_\mu$. The enlarged
gauge generator therefore has the following form:
\begin{equation}
G_{\rm E}[\xi] = G[\xi] + B^\mu[\xi] P_\mu
+ N^{\nu} A^{\mu}_{\nu}[\xi] P_{\mu}. \label{enlargen}
\end{equation}
The commutation algebra for $G_{\rm E}[\xi]$ is:
$$
\{G_{\rm E}[\xi_1] , \, G_{\rm E}[\xi_2] \}
= G_{\rm E}[\xi_3] + {\cal O}(P^2),
$$
where ${\cal O}$ is pure quadratic in the (new) primary constraints:
\begin{eqnarray}
&&{\cal O}(P^2) \nonumber\\
&&~~ = \left\{B^\mu[\xi_1] + N^\nu A^\mu_\nu[\xi_1] , \,
B^\sigma[\xi_2] + N^\rho A^\sigma_\rho[\xi_2] \right\}
P_\mu P_\sigma.\nonumber
\end{eqnarray}
In our particular case of general covariant theories,
$B^\mu[\xi] = \dot\xi^\mu$ and
$A^\mu_\nu[\xi] = \xi^\rho C_{\rho\nu}^\mu$. Notice
that in this particular case the term ${\cal O}(P^2)$ vanishes.
The procedure of enlargement here devised is completely general, and
it is valid for any gauge theory no matter how complicated its
structure of constraints may be.
\section{The Gauge Group}
The gauge group is a subgroup of the symmetry group of the system. A
symmetry is a transformation that maps solutions of the equations of
motion into solutions. From a physical standpoint, gauge symmetry
reflects a redundancy in the description. Mathematically, a gauge
transformation is characterized by its functional dependence on
arbitrary functions. The functional dependence is expected to be
local in the sense of depending on the values of the functions and on
a finite number of derivatives. This is the definition for classical
mechanics and classical field theory. It is most convenient to define
gauge symmetries in a more restrictive way as local transformations
which leave the action invariant up to boundary terms. Our analysis
is based on the Noether identities which result from this invariance
under infinitesimal local symmetries.
Let us make some formal remarks on the nature of the
diffeomorphism-induced gauge group of the type discussed in this
paper. Let ${\it Riem}({\cal M})$ be the space
of (pseudo) Riemannian metrics of the spacetime manifold ${\cal M}$,
and let ${\bf Diff}({\cal M})$ be the group of diffeomorphisms in
${\cal M}$. An element of the gauge group
${\bf G}[{\it Riem}({\cal M})]$ is a regular map
${\it Riem}({\cal M}) \rightarrow {\it Riem}({\cal M})$
such that each $g \in {\it Riem}({\cal M})$
undergoes a diffeomorphic transformation, that is, a transformation
dictated by a specific element of ${\bf Diff}({\cal M})$
(thus keeping the action invariant). (Other fields are also affected
by this diffeomorphic transformation, but for this discussion we
devote our attention to the metric.) But this element of
${\bf Diff}({\cal M})$ may be different if we consider
the action of the same element of the gauge
group on a different $g' \in {\it Riem}({\cal M})$.
To determine an element of the gauge group we must assign to each
$g \in {\it Riem}({\cal M})$ the specific spacetime
diffeomorphism which is going to act on $g$. More precisely,
an element, $d$, of the gauge group is a map
\begin{eqnarray} \nonumber
d: {\it Riem}({\cal M}) &\longrightarrow&
{\bf Diff}({\cal M})\\
g &\longrightarrow& d[g]
\end{eqnarray}
such that we can build out of it a regular map
\begin{eqnarray} \nonumber
{\bf G}: {\it Riem}({\cal M}) &\longrightarrow&
{\it Riem}({\cal M})\\
g &\longrightarrow& (d[g])(g) .
\end{eqnarray}
Now let us consider the generators of the gauge group {\bf G}. We
use, for the sake of generality, a condensed notation where $\Phi^i$
stands for the fields that are present in the theory; the action is
denoted by ${\cal S}$, and $i$ includes continuous spacetime indices
(so that repeated indices imply both summation and integration). Let
$\epsilon^\alpha$ be arbitrary functions of spacetime variables, and
$\delta_\epsilon \Phi^i = R_\alpha^i \epsilon^\alpha$ be a complete
set of infinitesimal gauge transformations. These satisfy the Noether
identities $(\delta {\cal S} / \delta \Phi^i) R_\alpha^i = 0$.
Obviously we do not alter the Noether identities by taking a different
linear combination of variations
$R_\alpha^i \rightarrow \bar R_\alpha^i
= \Lambda_\alpha^\beta(\Phi) R_\beta^i$, even when the
$\Lambda_\alpha^\beta$ depend on $\Phi^i$. No gauge equivalent
trajectories are eliminated through this transformation, presuming
that $\bf\Lambda$ is invertible. In our case (see \ref{3.14}) the
requirement of projectability fixed
\begin{equation}
\Lambda^\mu{}_\nu (g)
= n^\mu \delta^0_\nu +\delta^\mu_a \delta^a_\nu,
\end{equation}
which is clearly invertible. The algebra corresponding to this new
choice of generators contains field-dependent structure coefficients.
One might conclude that this ``soft'' algebra structure signifies that
the symmetry transformations no longer form a group. We do have a
group, which acts not on the spacetime manifold $\cal M$ but on ${\it
Riem}({\cal M})$. Note that elements of the diffeomorphism-induced
gauge group must depend on the full metric. Dependence on the lapse
and shift is fixed by (\ref{3.14}). Perhaps more surprising is the
fact that the full group depends non-locally on the hypersurface
induced metric. This is a direct consequence of the structure
coefficients in (\ref{5.22a}): Repeated nesting of the commutator
produces spatial derivatives of $g_{a b}$ to infinite order
\cite{Bergmann72}.
To summarize, we have discussed some aspects of the canonical approach
to generally covariant theories. In particular we have emphasized the
special way in which the canonical formalism describes the
diffeomorphism covariance of these theories. The gauge group for
these theories is larger than the diffeomorphism group. The canonical
gauge generators are just one of the possible bases for the gauge
algebra, although projectability of transformations generated by the
larger group from configuration-velocity space to phase space fixes
the dependence on the lapse and shift uniquely. We have displayed the
canonical generators for the gauge symmetries of these theories on the
entire phase space. Transformations may be pulled back to the entire
configuration-velocity space.
In this paper we assumed that the only gauge symmetries are generated
by diffeomorphisms. When other gauge symmetries occur, related
internal gauge transformations must be taken into account. This topic
and the question of projectability of the gauge transformation group
onto the full constraint hypersurface will be dealt with in future
papers. It is also our intention to explore further the relationship
between our results and matters pertaining to quantization,
particularly the question of time in quantum gravity.
\section*{Acknowledgments}
JMP and DCS would like to thank the Center for Relativity of The
University of Texas at Austin for its hospitality. JMP acknowledges
support by CIRIT and by CICYT contracts AEN95-0590 and GRQ 93-1047.
DCS acknowledges support by National Science Foundation Grant
PHY94-13063.
|
1,116,691,497,586 | arxiv | \section{Introduction}
There has been spectacular technological advances in the use of
high-quality optomechanical oscillators for force sensing, enabling
ultra-sensitive force measurements of single spin, charge,
acceleration, magnetic field and mass~\cite{Rugar04_Nat,
Cleland98_Nat, Krause12_NatPhot, Forstner12_PRL,
Jensen08_NatNano}. Such advances have opened up the exciting
possibility of experimentally studying quantum light-matter
interactions in macroscopic structures~\cite{Purdy13_Sci,
Verhagen12_Nat}, hence paving the way towards new technologies for
quantum information science and metrology~\cite{Mancini03_PRL,
Brooks12_Nat,purdy2013,safavi2013}.
Thermal and measurement noises impose major limitations to the
accuracy of mechanical force sensors. Furthermore, while the
development of higher quality and lower mass mechanical oscillators
has played a central role in advancing the sensitivity of
optomechanical force sensors \cite{gavartin}; such oscillators also
have increased sensitivity to their environment. This introduces new
sources of noise that can cause fluctuations in parameters, such as
the effective oscillator temperature and mechanical resonance
frequency. As optomechanical technology continue to advance, it can be
expected that methods to characterize, monitor, and control these
additional noise sources, in conjunction with thermal and measurement
noise, will become increasingly important. In this context,
statistical signal processing techniques that are provably optimal in
a theoretical sense offer the potential to improve the actual sensing
performance significantly, beyond the heuristic curve-fitting
procedures commonly employed in the field.
In this paper, we introduce a statistical framework to study the
problem of parameter estimation from a noisy optomechanical
system. This problem is especially relevant to the recent
optomechanics experiments reported in
Refs.~\cite{gavartin,harris_bowen2013}. We derive analytic
expressions for the Cram\'er-Rao lower bound on the estimation errors
and apply various estimation techniques to experimental data to
estimate the parameters of an optomechanical system, including the
force power, mechanical resonance frequency, damping rate, and
measurement noise power.
Our analytic results provide convenient expressions of the estimation
errors as a function of system parameters and measurement time and
should be valuable to optomechanical experiment design. Another
highlight of our study is the use of the expectation-maximization (EM)
algorithm \cite{dempster, shumway_stoffer, levy}, which is generalized
here for a complex Gauss-Markov model and applied to a cavity
optomechanical system, both for the first time to our knowledge.
Among the estimators we have studied, including the one used in
Refs.~\cite{gavartin,harris_bowen2013}, we find that the
root-mean-square errors of the EM algorithm in estimating the force
noise power are the lowest, following the Cram\'er-Rao bound most
closely and beating the estimator in
Refs.~\cite{gavartin,harris_bowen2013} by more than a factor of 5 for
longer measurement times.
Our framework is also naturally applicable to quantum systems that can
be described by a homogeneous Gauss-Markov model
\cite{wiseman_milburn}, such as quantum optomechanical systems
\cite{chen2013, aspelmeyer2013,wheatley, yonezawa,iwasawa} and atomic
spin ensembles \cite{stockton,petersen}. This makes our study, and
the EM algorithm in particular, relevant not just to future precision
sensing and system identification applications, but also to
fundamental tests of quantum mechanics
\cite{chen2013,aspelmeyer2013,testing_quantum}.
\section{Experiment}
To motivate our theoretical model and numerical analysis, we first
describe the optomechanical experiment presented in
Ref.~\cite{harris_bowen2013} that was used to produce the data.
The transducer under consideration consists of a room temperature
microtoroidal resonator that simultaneously supports mechanical modes
sensitive to external forces and high quality optical modes that
permit ultra-precise readout of the mechanical displacement.
We couple shot-noise limited 1550nm laser light into a whispering
gallery mode of the microtoroid via a tapered optical fiber which is
nested inside an all fiber inteferometer. Excitation of the mechanical
mode, which has fundamental frequency, damping rate and effective mass
of $\Omega_{\rm m}\!=\!40.33~\rm MHz$, $\gamma \!=\!23~\rm kHz$ and
$m_{\rm eff}\!=\! 7~\rm ng$ respectively, induces phase fluctuations
on the transmitted light which is measured by shot-noise limited
homodyne detection.
To maintain constant coupling of optical power into the microtoroid we
use an amplitude and phase modulation technique to actively lock the
toroid-taper separation~\cite{Chow12_OptExp} and laser frequency
respectively. The relative phase of the bright local oscillator to the
signal is controlled via a piezoactuated fiber stretcher that
precisely tunes the optical path length in one arm of the
inteferometer.
To specifically demonstrate power estimation, a small incoherent
signal is applied to the mechanical oscillator in addition to the
thermal fluctuations. This is achieved by the electrostatic gradient
force applied by a nearby electrode driven with white noise from a
signal generator~\cite{Lee10_PRL}.
The measurement record is acquired from the homodyne signal by
electronic lock-in detection which involves demodulation of the
photocurrent at the mechanical resonance frequency allowing real time
measurement of the slowly evolving quadratures of motion, denoted
$I(t)$ and $Q(t)$ where $x(t)\!=\!I(t)\cos(\Omega_{m}t) +
Q(t)\sin(\Omega_{m}t)$. The room temperature thermal fluctuations of
the mechanical mode are observed with a signal-to-noise ratio of
$37$dB and calibrated via the optical response to a known reference
modulation~\cite{Schliesser08}. The resulting force sensitivity,
which can be extracted from Fourier analsis of the measurment record,
will depend on the specific protocol used.
Here we evaluate the force sensitivity of 3 parameter estimation
protocols relative to the Cram\'er-Rao lower bound.
\section{Theory}
\subsection{\label{model}Continuous-time model}
A simple linear Gaussian model for the mechanical mode can be
described by the following equation for the complex analytic signal
$z(t)$ of the mechanical-mode displacement:
\begin{eqnarray}
\diff{z(t)}{t} = -\gamma z(t)+i\Omega z(t) + \xi(t),
\label{a}
\end{eqnarray}
where $\Omega$ is the mechanical resonance frequency relative to
$\Omega_m$, $\gamma$ is the damping rate, and $\xi(t)$ is the
stochastic force as a sum of the thermal noise and the
signal. $\xi(t)$ is assumed to be a complex zero-mean white Gaussian
noise \cite{vantrees3} with power $A$ and covariance function
\begin{eqnarray}
E\Bk{\xi(t)\xi^*(t')} = A\delta(t-t'),
\qquad
E\Bk{\xi(t)\xi(t')} = 0.
\label{xi}
\end{eqnarray}
The measurements can
be modeled in continuous time as
\begin{eqnarray}
y(t) = C z(t) + \eta(t),
\label{y}
\end{eqnarray}
where $C$ is a real parameter and $\eta(t)$ is the measurement noise,
assumed to be a complex additive white Gaussian noise with power $R$:
\begin{eqnarray}
E\Bk{\eta(t)\eta^*(t')} = R\delta(t-t'),
\qquad
E\Bk{\eta(t)\eta(t')} = 0.
\label{eta}
\end{eqnarray}
We assume that the parameters
\begin{eqnarray}
\theta = (\Omega, \gamma, A, C, R)^\top
\label{theta}
\end{eqnarray}
are constant in time, such that $z(t)$, $\xi(t)$, $y(t)$, and
$\eta(t)$ are stationary stochastic processes given $\theta$. $y(t)$,
in particular, has a power spectrum given by
\begin{eqnarray}
S_y(\omega|\theta) &\equiv&
\lim_{T\to\infty} E\Bk{\frac{1}{T} \abs{\int_{-T/2}^{T/2}dt y(t)\exp(-i\omega t)}^2} \label{Sy} \\
&=& AS(\omega) + R, \\
S(\omega) &\equiv& \frac{C^2}{(\omega-\Omega)^2+\gamma^2}.
\end{eqnarray}
Although this simple model suffices to describe our experiment, it is
not difficult to generalize our entire formalism to describe more
complicated dynamics and colored noise \cite{vantrees}. This is done
by generalizing $z(t)$ to a vector of state variables for more
mechanical and optical modes, Eq.~(\ref{a}) to a vectoral equation of
motion, and the parameters $(\Omega, \gamma, A, C, R)$ to matrices
that describe the coupled-mode dynamics and the noise statistics.
\subsection{Binary hypothesis testing}
Although hypothesis testing \cite{levy,vantrees3} is not the focus of
our study, the theory is useful for the derivation of the Cram\'er-Rao
bound, so we present the topic here briefly for completeness.
Suppose that there are two hypotheses, denoted by $\mathcal H_0$ and
$\mathcal H_1$, with prior probabilities $P_0$ and $P_1 = 1-P_0$. From
a measurement record $Y$, with a probability density $P(Y|\mathcal
H_0)$ or $P(Y|\mathcal H_1)$ that depends on the hypothesis, one
wishes to decide which hypothesis is true. Given the densities and a
decision rule, one can compute $P_{jk}$, the probability that
$\mathcal H_j$ is chosen when $\mathcal H_k$ is true. The average
error probability is
\begin{eqnarray}
P_e \equiv P_{10}P_0 + P_{01} P_1.
\end{eqnarray}
$P_e$ can be minimized using a Bayes likelihood-ratio test:
\begin{eqnarray}
\Lambda \equiv \frac{P(Y|\mathcal H_1)}{P(Y|\mathcal H_0)}
\overset{\mathcal H_1}{\underset{\mathcal H_0}{\gtrless}} \frac{P_0}{P_1},
\end{eqnarray}
which means that $\mathcal H_1$ is chosen if $\Lambda \ge P_0/P_1$ and
vice versa. The resulting $P_e$ is often difficult to compute
analytically, but can be bounded by upper and lower bounds. For $P_0 =
P_1 = 1/2$ \cite{levy,kailath},
\begin{eqnarray}
\frac{1}{2}\Bk{1-\sqrt{1-F^2(0.5)}}\le \min P_e
\le
\frac{1}{2}\min_{0\le s\le 1} F(s),
\end{eqnarray}
where the upper bound is the Chernoff bound,
\begin{eqnarray}
F(s) \equiv E\Bk{\Lambda^s|\mathcal H_0} =
\int dY P(Y|\mathcal H_0)
\Bk{\frac{P(Y|\mathcal H_1)}{P(Y|\mathcal H_0)}}^s,
\label{chernoff}
\end{eqnarray}
and $F(0.5)$ is known as the Bhattacharyya distance between the two
probability densities \cite{kailath}.
Let $Y$ be a record of continuous measurements:
\begin{eqnarray}
Y = \{x(t); -T/2\le t \le T/2\}.
\end{eqnarray}
If $x(t)$ is a realization of a real zero-mean stationary process with spectrum $S_x(\omega|\mathcal H_j)$, the exponent of $F(s)$ in the case of stationary processes and long observation time (SPLOT) is known to be \cite{levy}
\begin{eqnarray}
\Gamma_F &\equiv \lim_{T\to\infty} -\frac{1}{T}\ln F(s)
\\
&= \frac{1}{2}\int_{-\infty}^{\infty} \frac{d\omega}{2\pi}
\ln \frac{s S_x(\omega|\mathcal H_0) + (1-s) S_x(\omega|\mathcal H_1)}
{S_x^{s}(\omega|\mathcal H_0)S_x^{1-s}(\omega|\mathcal H_1)}.
\label{exponent}
\end{eqnarray}
This expression means that $F(s)$ has the form of
\begin{eqnarray}
F(s) = \beta(T)\exp\bk{-\Gamma_F T},
\end{eqnarray}
where $-\ln \beta(T)$ is asymptotically smaller than $T$:
\begin{eqnarray}
\lim_{T\to\infty} -\frac{1}{T}\ln \beta(T) = 0,
\end{eqnarray}
and therefore $\beta(T)$ decays more slowly than $\exp(-\Gamma_F T)$.
For the model in Sec.~\ref{model}, $y(t)$ is a complex signal, and (\ref{exponent}) needs to be modified. This can be done by assuming that $y(t)$ is bandlimited in $[-\pi b,\pi b]$ and considering a real signal $x(t)$ given by
\begin{eqnarray}
x(t) \equiv y(t)\exp(i\omega_0 t) + y^*(t)\exp(-i\omega_0t),
\end{eqnarray}
where $\omega_0$ is a carrier frequency assumed to be $> \pi b$. We then have
\begin{eqnarray}
S_x(\omega|\mathcal H_j) = S_y(\omega-\omega_0|\mathcal H_j) +
S_y(-\omega-\omega_0|\mathcal H_j).
\end{eqnarray}
Using this expression in (\ref{exponent}) leads to another expression for the Chernoff exponent given by
\begin{eqnarray}
\Gamma_F
= \int_{-\pi b}^{\pi b} \frac{d\omega}{2\pi}
\ln \frac{s S_y(\omega|\mathcal H_0) + (1-s) S_y(\omega|\mathcal H_1)}
{S_y^s(\omega|\mathcal H_0)S_y^{1-s}(\omega|\mathcal H_1)}.
\label{exponent2}
\end{eqnarray}
Note the absence of the $1/2$ factor in (\ref{exponent2}) compared with (\ref{exponent}).
\subsection{Cram\'er-Rao Bound}
We now consider the estimation of $\theta$ from $Y$ with a probability
density given by $P(Y|\theta)$. Defining the estimate as $\hat
\theta(Y)$, the error covariance matrix is
\begin{eqnarray}
\Sigma(\theta) \equiv E\BK{\Bk{\hat \theta(Y)-\theta}
\Bk{\hat \theta(Y)-\theta}^\top\Big|\theta},
\\
E[g(Y)|\theta] \equiv \int dY P(Y|\theta) g(Y).
\end{eqnarray}
Assuming that $\hat \theta$ satisfies the unbiased condition
\begin{eqnarray}
E\Bk{\hat\theta(Y)\Big|\theta} = \theta,
\end{eqnarray}
the Cram\'er-Rao bound on $\Sigma$ is \cite{levy}
\begin{eqnarray}
\Sigma(\theta) \ge J^{-1}(\theta),
\end{eqnarray}
where $J(\theta)$ is known as the Fisher information matrix:
\begin{eqnarray}
J(\theta) \equiv E\BK{
\nabla\Bk{\ln P(Y|\theta)}
\nabla^\top\Bk{\ln P(Y|\theta)}\Big|\theta},
\\
\nabla \equiv \bk{\parti{}{\theta_1},\parti{}{\theta_2},\dots}^\top.
\end{eqnarray}
It turns out that $J(\theta)$ can be related to the Bhattacharyya
distance in a hypothesis testing problem with
\begin{eqnarray}
P(Y|\mathcal H_0) &= P(Y|\theta),
\\
P(Y|\mathcal H_1) &= P(Y|\theta').
\end{eqnarray}
$F(s)$ defined in (\ref{chernoff}) becomes a function of $\theta$ and
$\theta'$, and $J(\theta)$ can be expressed as \cite{vantrees3}
\begin{eqnarray}
J(\theta) &= -4 \nabla \nabla^\top \ln
F(0.5,\theta,\theta')\Big|_{\theta'=\theta}.
\label{J}
\end{eqnarray}
For the model in Section~\ref{model}, $y(t)$ is a realization of a
stationary process given $\theta$, so $J(\theta)$ for the estimation
of $\theta$ from $Y=\{y(t); -T/2 \le t \le T/2\}$ in the SPLOT case
can be obtained by combining (\ref{exponent2}) and (\ref{J}):
\begin{eqnarray}
\Gamma_J &\equiv \lim_{T\to\infty}\frac{J(\theta)}{T}
\\
&= 4\nabla\nabla^\top\int_{-\pi b}^{\pi b}
\frac{d\omega}{2\pi}
\ln \frac{S_y(\omega|\theta) + S_y(\omega|\theta')}
{2\sqrt{S_y(\omega|\theta)S_y(\omega|\theta')}}\Bigg|_{\theta'=\theta},
\end{eqnarray}
where $S_y(\omega|\theta)$ is given by (\ref{Sy}). This expression
means that $J(\theta)$ for any stationary-process parameter estimation
problem increases linearly with time as $T\to\infty$, in the sense of
\begin{eqnarray}
J(\theta) &= \Gamma_J T + o(T),
\end{eqnarray}
where $o(T)$ is asymptotically smaller than $T$:
\begin{eqnarray}
\lim_{T\to\infty} \frac{o(T)}{T} = 0.
\end{eqnarray}
In the asymptotic limit, maximum-likelihood (ML) estimation can attain
the Cram\'er-Rao bound \cite{shumway_stoffer}, so the bound is a
meaningful indicator of estimation error. Despite the asymptotic
assumption, the simpler analytic expressions are more convenient to
use for experimental design purposes.
Although the preceding formalism is applicable to the estimation of
any of the parameters, in the following we focus on $A$, the force
noise power. The Cram\'er-Rao bound on the mean-square estimation
error $\Sigma_A$ is
\begin{eqnarray}
\Sigma_A &\equiv E\BK{\Bk{\hat A(Y)-A}^2\Big|\theta}
\ge J_A^{-1},
\label{CRBA}
\\
\Gamma_A &\equiv \lim_{T\to\infty} \frac{J_A}{T} =
\int_{-\pi b}^{\pi b}\frac{d\omega}{2\pi}
\frac{S^2(\omega)}{[AS(\omega)+R]^2}.
\label{GammaA}
\end{eqnarray}
This bound allows us to investigate the efficiency of the parameter
estimation algorithms presented in the next section.
\section{\label{algorithms}Parameter estimation algorithms}
\subsection{Averaging}
We first consider the estimator used in
Refs.~\cite{gavartin,harris_bowen2013}:
\begin{eqnarray}
\hat A_{\rm avg} = G\int_{-T/2}^{T/2} dt |y(t)|^2,
\qquad
G = \Bk{T\int_{-\pi b}^{\pi b} \frac{d\omega}{2\pi} S(\omega)}^{-1}.
\label{avg}
\end{eqnarray}
The rationale for this simple averaging estimator is that, in the absence of measurement noise ($R = 0$), it is an unbiased estimate for $T\to\infty$:
\begin{eqnarray}
\lim_{T\to\infty} E\bk{\hat A_{\rm avg}\Big|\theta,R = 0} = A.
\end{eqnarray}
The unbiased condition breaks down, however, in the presence of
measurement noise, and we are therefore motivated to find a better
estimator.
\subsection{Radiometer}
The ``radiometer'' estimator described in Ref.~\cite{vantrees3} can be
easily generalized for complex variables. The result is
\begin{eqnarray}
\hat A_{{\rm rad}} = G\Bk{\int_{-T/2}^{T/2}dt \int_{-T/2}^{T/2} dt' y^*(t) h(t-t') y(t') - B},
\end{eqnarray}
where $h(t-t')$ filters $y(t')$ before correlating the result with $y^*(t)$, and $G$ and $B$ are parameters chosen to enforce the unbiased condition.
We see that the averaging estimator $\hat A_{{\rm avg}}$ given by (\ref{avg}) also has the radiometer form. It can be shown that, for $T\to\infty$,
\begin{eqnarray}
G = \Bk{T\int_{-\pi b}^{\pi b} \frac{d\omega}{2\pi} H(\omega) S(\omega)}^{-1},
\\
B = T \int_{-\pi b}^{\pi b} \frac{d\omega}{2\pi} H(\omega) R,
\\
H(\omega) \equiv \int_{-\infty}^{\infty} dt h(t)\exp(-i\omega t).
\end{eqnarray}
The mean-square error, on the other hand, has the asymptotic expression
\begin{eqnarray}
\lim_{T\to\infty} \Sigma_A T =
G^2\int_{-\pi b}^{\pi b} \frac{d\omega}{2\pi} H^2(\omega) S_y^2(\omega|\theta).
\end{eqnarray}
This expression coincides with the Cram\'er-Rao bound given by (\ref{CRBA}) and (\ref{GammaA}) if we set
\begin{eqnarray}
H(\omega) = \frac{S(\omega)}{[DS(\omega)+R]^2},
\label{H}
\end{eqnarray}
and $A$ happens to be equal to $D$. For any other value of $A$, the radiometer is suboptimal.
\subsection{Expectation-maximization (EM) algorithm}
A major shortcoming of the radiometer is its requirement of parameters
other than $A$ to be known exactly. Another issue is that it assumes
continuous time and relies on asymptotic arguments, when the
measurements are always discrete and finite in practice. We find that
the EM algorithm \cite{dempster,shumway_stoffer,levy}, which performs
maximum-likelihood (ML) estimation and is applicable to the linear
Gaussian model we consider here, overcomes both of these problems.
ML estimation aims to find the set of parameters $\theta$ that
maximizes the log-likelihood function $\ln P(Y|\theta)$. This task can
be significantly simplified by the EM algorithm if there exist hidden
data $Z$ that results in simplified expressions for $P(Z|Y,\theta)$
and $P(Y,Z|\theta)$. Starting with a trial $\theta = \theta^{0}$, the
algorithm considers the estimated log-likelihood function
\begin{eqnarray}
Q(\theta,\theta^{k}) \equiv \int dZ P(Z|Y,\theta^{k})
\ln P(Y,Z|\theta),
\end{eqnarray}
where the superscript $k$ is an index denoting the EM iteration, and
finds the $\theta^{k+1}$ for the next iteration by maximizing $Q$:
\begin{eqnarray}
\theta^{k+1} = \arg \max_{\theta} Q(\theta,\theta^{k}).
\end{eqnarray}
The iteration is halted when the difference betwen $\theta^{k+1}$ and
$\theta^{k}$ reaches a prescribed threshold, and the final
$\theta^{k+1}$ is taken to be the EM estimate
$\hat\theta_{{\rm EM}}$.
To apply the EM algorithm to our model in Section~\ref{model}, we
consider a complex discrete-time Gauss-Markov model:
\begin{eqnarray}
z_{j+1} = {f} z_j + w_j,
\label{z}\\
y_j = c z_j + v_j,
\quad
j = 0,1,\dots,J.
\label{Y}
\end{eqnarray}
In general, $z_j$ and $y_j$ can be column vectors, and ${f}$ and $c$
are matrices. $w_j$ and $v_j$ are complex independent zero-mean
Gaussian random variables with covariances given by
\begin{eqnarray}
E\bk{w_jw_k^\dagger} = q\delta_{jk},
\qquad
E\bk{w_jw_k^\top} = 0,
\label{w}
\\
E\bk{v_jv_k^\dagger} = r\delta_{jk},
\qquad
E\bk{v_jv_k^\top} = 0,
\label{v}
\end{eqnarray}
where $^\dagger$ denotes the conjugate transpose, $^\top$ denotes the
transpose, and $q$ and $r$ are covariance matrices. The parameters of
interest $\theta$ are the components of ${f}$, $c$, $q$, and $r$. The
EM algorithm for a real Gauss-Markov model described in
Refs.~\cite{shumway_stoffer,levy} is generalized to account for
complex variables in \ref{complexEM}. The problem may become
ill-conditioned when too many parameters are taken to be unknown and
multiple ML solutions exist \cite{shumway_stoffer,levy,guta_yamamoto},
so we choose a parameterization with known $q$:
\begin{eqnarray}
{f} &= \exp\Bk{\bk{i\Omega-\gamma} \delta t},
\\
c &= C\sqrt{A\frac{1-\exp(-2\gamma\delta t)}{2\gamma \delta t}},
\\
q &= \delta t,
\\
r &= \frac{R}{\delta t},
\end{eqnarray}
where $\delta t$ is the sampling period. With the EM estimates $\hat
f_{{\rm EM}}$, $\hat c_{{\rm EM}}$, and $\hat r_{{\rm EM}}$
and assuming that $\delta t$ and $C$ are known by independent
calibrations, we can retrieve estimates of $\Omega$, $\gamma$, $A$,
and $R$:
\begin{eqnarray}
\hat \Omega_{{\rm EM}} &= \frac{\arg\hat f_{{\rm EM}}}{\delta t},
\\
\hat\gamma_{{\rm EM}} &= -\frac{\ln |\hat f_{{\rm EM}}|}{\delta t},
\\
\hat A_{{\rm EM}} &= \frac{\hat c_{{\rm EM}}^2}{C^2}
\frac{2\hat\gamma_{{\rm EM}}\delta t}
{1-\exp(-2\hat\gamma_{{\rm EM}}\delta t)},
\\
\hat R_{{\rm EM}} &= \hat r_{{\rm EM}} \delta t.
\end{eqnarray}
It can be shown that the ML parameter estimator for the Gauss-Markov
model is asymptotically efficient \cite{shumway_stoffer}, meaning that
it attains the Cram\'er-Rao bound in the limit of $T\to\infty$.
\section{Application to experimental data}
\subsection{Procedure}
There are two records of experimental data, one with thermal noise in
$\xi(t)$ and one with additional applied white noise in $\xi(t)$,
leading to a different $A$ for each record, denoted by $A^{(0)}$ and
$A^{(1)}$. Each record contains $J_{{\rm max}}+1=3,750,001$ points
of $y_j^{(n)}$. With a sampling frequency $b = 1/\delta t = 15~$MHz,
the total time for each record is $T_{{\rm max}} =
(J_{{\rm max}} + 1)\delta t \approx 0.25$~s. From independent
calibrations, we also obtain $C = 2.61\times
10^{-2}~({\rm fN}/\sqrt{{\rm Hz}})^{-1}$. To investigate the
errors with varying $T$, we divide each record into slices of records
with various $T$, resulting in $M(T) =
{\rm floor}(T_{{\rm max}}/T)$ number of trials for each
$T$. Using a desktop computer (Intel Core i7-2600 [email protected] with
16GB RAM) and MATLAB, we apply each of the three estimators in
Section~\ref{algorithms} to each trial to produce an estimate $\hat
A_{m,l}^{(n)}(T)$, where $m$ denotes the trial and $l$ denotes the
estimator. The EM iteration is stopped when the fractional difference
between the current estimate of $A$ and the previous value is less
than $10^{-7}$. For the averaging and radiometer estimators, true
values for $\Omega$, $\gamma$, and $R$ are needed, and since we do not
know them, we estimate them by applying the EM algorithm to the whole
records. This is reasonable because $T_{{\rm max}} \gg
4~{\rm ms} \ge T$, and we expect
$\hat\theta_{{\rm EM}}^{(n)}(T_{{\rm max}})$ to be much closer
to the true values $\theta^{n}$ than the short-time estimates. The EM
algorithm for each $T$, on the other hand, does not use $\hat
\theta_{{\rm EM}}^{(n)}(T_{{\rm max}})$ at all and produces its
own estimates each time. The parameter $D$ in (\ref{H}) is taken to be
$\hat A_{{\rm EM}}^{(0)}(T_{{\rm max}})$. The estimation errors
are computed by
\begin{eqnarray}
\Sigma_l^{(n)}(T) &= \frac{1}{M(T)}\sum_{m=1}^{M(T)} \Bk{\hat A_{m,l}^{(n)}(T) -A^{(n)}}^2,
\end{eqnarray}
and compared with the SPLOT Cram\'er-Rao bound $J_A^{-1} \approx
(\Gamma_J T)^{-1}$ by assuming $\theta^{(n)} = \hat\theta_{
{\rm EM}}^{(n)} (T_{{\rm max}})$.
Note that the estimation error in general contains two components:
\begin{eqnarray}
\Sigma &= \frac{1}{M}\sum_{m=1}^{M} \bk{\hat A_{m} -\bar A}^2 +
\bk{\bar A-A}^2,
\end{eqnarray}
where
\begin{eqnarray}
\bar A &\equiv \frac{1}{M}\sum_{m=1}^{M} \hat A_{m}
\end{eqnarray}
is the sample mean of the estimate, the first component is the sample
variance, and the second component is the square of the estimate
\emph{bias} with respect to the true value $A$. Unlike
Refs.~\cite{gavartin,harris_bowen2013}, our error analysis is able to
account for the bias component more accurately by referencing with the
much more accurate long-time EM estimates.
\subsection{Results}
Applied to the two records, the EM algorithm produces the following
estimates:
\begin{eqnarray}
\hat A_{{\rm EM}}^{(0)}(T_{\rm max}) &=& 2.4748/C^2
=3.64\times 10^{3}~{\rm fN}^2\,{\rm Hz}^{-1},
\\
\hat \Omega_{{\rm EM}}^{(0)}(T_{{\rm max}}) &=& -1.8582\times 10^{4}~{\rm rad~s}^{-1},
\\
\hat \gamma_{{\rm EM}}^{(0)}(T_{{\rm max}}) &=& 5.5730\times 10^{4}~{\rm rad~s}^{-1},
\\
\hat R_{{\rm EM}}^{(0)}(T_{{\rm max}}) &=& 1.4532\times 10^{-13}~{\rm Hz}^{-1},
\\
\hat A_{{\rm EM}}^{(1)}(T_{\rm max}) &=& 2.6926/C^2
= 3.96\times 10^{3}~{\rm fN}^2\,{\rm Hz}^{-1},
\\
\hat \Omega_{{\rm EM}}^{(1)}(T_{{\rm max}}) &=& -1.8668\times 10^{4}~{\rm rad~s}^{-1},
\\
\hat \gamma_{{\rm EM}}^{(1)}(T_{{\rm max}}) &=& 5.6156\times 10^{4}~{\rm rad~s}^{-1},
\\
\hat R_{{\rm EM}}^{(1)}(T_{{\rm max}}) &=& 1.4703\times 10^{-13}~{\rm Hz}^{-1}.
\end{eqnarray}
The algorithm takes $\approx 3.3$ hours to run for each record. These
values are then used as references to analyze the estimators at
shorter times.
\begin{figure}[htbp]
\centerline{\includegraphics[width=\textwidth]{RMSE_combined}}
\caption
Root-mean-square
force-noise-power estimation errors and the asymptotic Cram\'er-Rao
bound versus time in log-log scale. Left: the force contains
thermal noise only. Right: the force contains thermal noise and an
applied noise.}
\label{MSE_combined}
\end{figure}
Figure~\ref{MSE_combined} plots the root-mean-square errors
$\sqrt{\Sigma_l^{(n)}(T)}$ and the SPLOT Cram\'er-Rao bound
$J_A^{-1/2} \to (\Gamma_JT)^{-1/2}$ versus time $T$ in log-log
scale. The two plots show very similar behavior. A few observations
can be made:
\begin{enumerate}
\item The averaging estimator is more accurate than the radiometer for
short times but becomes much worse for longer times. We cannot
explain the short-time errors because our analytic results rely on
the long-time limit, although the errors there are so high relative
to the estimate that they are irrelevant to real applications. The
large long-time errors can be attributed to the bias and
suboptimality of the estimator.
\item The radiometer beats the averaging estimator and approaches the
Cram\'er-Rao bound for longer times. This is consistent with our
SPLOT analysis, as we have chosen $D = \hat
A_{{\rm EM}}^{(0)}(T_{{\rm max}})$ and the radiometer should
be near-optimal.
\item The EM estimator beats the other estimators at all times and
follow the Cram\'er-Rao bound more closely, even though we allow the
averaging and radiometer estimators to have the unfair advantage of
accessing more accurate values of $\Omega$, $\gamma$, and $R$. This
may be explained by the fact that the EM algorithm is formulated to
perform ML estimation on discrete measurements for any finite $T$,
unlike the other estimators that rely only on asymptotic arguments.
\item The EM estimator takes a much longer time to compute
(computation time $\approx 200~$s for one trial with $J+1 = 60,000$
points and $T = 4~$ms) than the other estimators ($\approx 0.3$~ms
for the averaging estimator, $\approx 16$~ms for the radiometer). If
computation time is a concern, the radiometer estimator may be
preferable, although its performance depends heavily on the accuracy
of the other assumed parameters, and the EM method can still be
useful for estimating such parameters in offline system
identification.
\end{enumerate}
To gain further insight into the finite gap between the errors and the
Cram\'er-Rao bound, in Figure~\ref{raw_spectra} we plot the raw
spectrum of $y_j^{(n)}$, defined as
\begin{eqnarray}
s_y^{(n)}(\omega) \equiv \frac{1}{T_{{\rm max}}}
\abs{\delta t\sum_{j=0}^{J_{{\rm max}}} y_j^{(n)}
\exp(-i\omega j\delta t)}^2.
\end{eqnarray}
\begin{figure}[htbp]
\centerline{\includegraphics[width=\textwidth]{raw_spectra}}
\caption{Raw spectra $s_y^{(n)}(\omega)$ of
the measurement records $y_j^{(n)}$ in log scale.}
\label{raw_spectra}
\end{figure}
The figure shows that our model does not exactly match the experiment
in two ways:
\begin{enumerate}
\item The data show a second weaker resonance peak.
\item The noise floor of the data rolls off at higher frequencies
due to the presence of an RF notch filter in the experiment
prior to data acquisition.
\end{enumerate}
Despite the mismatch, our results are in reasonable agreement with the
theory. To improve the estimation accuracy further, the weaker
resonance can be modeled by including another mode in our linear
Gaussian model, while the noise-floor roll-off can be removed by a
whitening filter before applying the estimators.
\section{Outlook}
In this paper we have followed the paradigm of orthodox statistics to
investigate parameter estimation for an optomechanical system,
focusing on unbiased and ML estimators and the Cram\'er-Rao bound. For
detection applications \cite{ting_hero_rugar} with uncertain
parameters, the ML estimator can form the basis of more advanced
hypothesis testing techniques, such as the generalized
likelihood-ratio test \cite{levy}. The assumption of static parameters
means that the presented techniques are most suited to system
identification purposes. For sensing applications, the parameters are
often time-varying, and Bayesian estimators, such as the extended and
unscented Kalman filters for continuous variables \cite{simon}, the
generalized-pseudo-Bayesian and interacting-multiple-model algorithms
for finite-state dynamical hypotheses \cite{bar-shalom}, and particle
filtering \cite{particle}, may be more suitable.
Since the Gauss-Markov model often remains valid for quantum systems
\cite{wiseman_milburn}, a quantum extension of our study is
straightforward. This means that the presented techniques are
potentially useful for future quantum sensing and system
identification applications, such as optomechanical force sensing
\cite{chen2013,wheatley,yonezawa,iwasawa}, atomic magnetometry
\cite{stockton,petersen}, and fundamental tests of quantum mechanics
\cite{chen2013,aspelmeyer2013,testing_quantum}. We expect our
parametric methods to lead to more accurate quantum sensing and
control than robust quantum control methods \cite{stockton,james},
which may be too conservative for the highly controlled environment of
typical quantum experiments.
There also exist quantum versions of the Cram\'er-Rao bound that
impose fundamental limits to the parameter estimation accuracy for a
quantum system with any measurement \cite{helstrom,twc,tsang_open},
and it may be interesting to explore how close the classical bounds
presented here can get to the quantum limits.
The continued improvement of optomechanical devices for applications
and fundamental science requires precise engineering of the mechanical
resonance frequency, dissipation rate and effective mass. This
necessitates a deep understanding of how these mechanical properties
depend on differing materials and fabrication techniques. The
mechanical resonance frequency is easily predicted via a numerical
eigenmode analysis using the geometry of the structure and the Youngs
modulus of the material. It is much more challenging to predict the
level of mechanical dissipation, where numerical models are not as
well established and multiple decay channels usually exist. Effective
experimental characterization of such dissipation channels requires
high precision force estimation to accurately quantify the oscillators
coupling to the environment. This is critical to advancing
optomechanics in applications such as quantum memories and quantum
information~\cite{Bagheri11_NatNano, Rabl10_NatPhys}. A more
immediate application for high precision force estimation is that of
temperature sensing and bolometry where small relative changes of the
signal power are of interest, for example, in detecting submillimeter
wavelengths in radio astronomy~\cite{Griffin00_NucInst} or even to
search for low energy events in particle
physics~\cite{Alessandrello98_PRB}. Given the demonstrated success of
our statistical techniques, we envision them to be similarly useful
for all these applications.
\section*{Acknowledgments}
S.~Z.~Ang and M.~Tsang acknowledge support by the Singapore National
Research Foundation under NRF Grant.~No.~NRF-NRFF2011-07.
W.~P.~Bowen and G.~I.~Harris acknowledge funding from the Australian
Research Council Centre of Excellence CE110001013 and Discovery
Project DP0987146. Device fabrication was undertaken within the
Queensland Node of the Australian Nanofabrication Facility.
|
1,116,691,497,587 | arxiv | \section{Introduction}
In human-robot collaboration, it is crucial for a cooperative robot (``co-robot'') to have the abilities of perception of human activities and corresponding appropriate decision-making to understand and interact with human peers. In order to provide these important capabilities, an artificial cognitive model integrating perception, reasoning, and decision making modules is required by intelligent co-robots to respond to humans effectively.
Artificial cognition has its origin in cybernetics; its intention is to create a science of mind based on logic \cite{Varela_99}.
Among other mechanisms,
cognitivism is a most widely used cognitive paradigm \cite{Vernon_TEC07}.
Several cognitive architectures were developed within this paradigm,
including ACT-R \cite{Anderson_AP96}
(and its extensions ACT-R/E \cite{trafton2013act}, ACT-R$\Phi$ \cite{dancy2013act}, etc),
Soar \cite{laird2012soar},
C4 \cite{Isla_IJCAI01},
and architectures for robotics \cite{Burghart_ICHR05}.
Because an architecture represents the connection and interaction of different cognitive components,
it cannot accomplish a specific task
on its own
without specifying each component that can provide knowledge to the cognitive architecture.
The combination of the cognitive architecture and components
is usually referred to as a {cognitive model} \cite{Vernon_TEC07}.
\begin{figure}[!ht]
\centering
\includegraphics[width= 0.8 \textwidth]{ArchitectureCogSys.png}
\caption{
Overview of the SRAC model for robot response to human activities.
The novel self-reflection module allows a co-robot to reason about
when the learned knowledge no longer applies.
Decisions are made by considering both human activity category distributions
and robot action risks.
Entities in ellipses are prior knowledge to the SRAC model.
Information flows from modules with lighter colors to those with darker colors.
}\label{fig:ArchitectureCogSys}
\end{figure}
Implementing such an artificial cognitive system is challenging,
since the high-level processes (e.g., reasoning and decision making)
must be able to seamlessly work with the low-level components, e.g., perception, under significant uncertainty in a complex environment \cite{Schmid_CogSysRes12}.
In the context of human-robot collaboration,
perceiving human behaviors is a necessary component,
where uncertainty arises due to human behavior complexity,
including variations in human motions and appearances,
and challenges of machine vision,
such as lighting changes and occlusion.
This perception uncertainty is addressed in this work
using the bag-of-visual-words (BoW) representation
based on local spatio-temporal features,
which has previously shown promising performance \cite{Le_CVPR11,Alahi_CVPR12,Zhang_IROS11}.
To further process the perceptual data,
a high-level reasoning component is necessary for a co-robot to make decisions.
In recent years, topic modeling has attracted increasing attention
in human behavior discovery and recognition
due to its ability to generate a distribution over activities of interest,
and its promising performance using BoW representations in robotics applications \cite{Zhang_IROS11,Girdhar_IJRR13}.
However, previous work only aimed at human behavior understanding;
the essential task of incorporating topic modeling into cognitive decision making
(e.g., selecting a response action)
is not well analyzed.
Traditional activity recognition systems typically use accuracy as a performance metric \cite{Niebles_IJCV08}.
Because the accuracy metric ignores the distribution of activity categories,
which is richer and more informative than a single label,
it is not appropriate for decision making.
For example,
in a task of behavior understanding with two categories,
assume that two recognition systems obtain two distributions
$[0.8,0.2]$ and $[0.55, 0.45]$ on a given observation,
and the ground truth indicates the first category is correct.
Although both systems are accurate,
in the sense that the most probable category matches the ground truth,
the first model obviously performs better,
since it better separates the correct from the incorrect assignment.
Previous studies did not consider this important phenomenon.
In real-world applications, artificial cognitive models must be applied in an online fashion.
If a co-robot is unable to determine whether its knowledge is accurate, then
if it observes a new human behavior that was not presented during the training phase,
it cannot be correctly recognized,
because the learned behavior recognition model no longer applies.
Decision making based on incorrect recognition
in situations like these
can result in inappropriate or even unsafe robot action response.
Thus,
an artificial cognitive model requires the capability to self-reflect
whether the learned activity recognition system
becomes less applicable,
analogous to human self-reflection on learned knowledge,
when applied in a new unstructured environment.
This problem was not well investigated in previous works.
In this paper, we develop a novel artificial cognitive model,
based on topic models,
for robot decision making in response to human behaviors.
Our model is able to incorporate human behavior distributions
and take into account robot action risks to make more appropriate decisions (we label this ``risk-aware").
Also, our model is able to identify new scenarios when the learned recognition subsystem is less applicable (we label this ``self-reflective").
Accordingly, we call our model the \emph{self-reflective, risk-aware artificial cognitive} (SRAC) model.
Our primary contributions are twofold:
\begin{itemize}
\item
Two novel indicators are proposed.
The \emph{interpretability indicator} ($I_I$)
enables a co-robot to interpret category distributions
in a similar manner to humans.
The online \emph{generalizability indicator} ($I_G$)
measures the human behavior recognition model's generalization capacity
(i.e., how well unseen observations can be represented by the learned model).
\item
A novel artificial cognitive model (i.e., SRAC) is introduced
based on topic models and the indicators,
which is able to consider robot action risks and perform self-reflection
to improve robot decision making in response to human activities in new situations.
\end{itemize}
The rest of the paper is organized as follows.
We first overview the related work in Section \ref{sec:RelatedWork}.
Then, the artificial cognitive architecture
and its functional modules are described in Section \ref{sec:CognitiveSystem}.
Section \ref{sec:TopicModels} introduces the new indicators.
Section \ref{sec:DecisionMaking} presents self-reflective risk-aware decision making.
Experimental results are discussed in Section \ref{sec:ExperimentResults}.
Finally,
we conclude our paper in Section \ref{sec:Conclusion}.
\section{Related Work}\label{sec:RelatedWork}
In this section, we provide an overview of a variety of methods related to our proposed SRAC cognitive model for human-robot teaming, including human activity recognition, topic models,
and artificial cognitive modeling.
\subsection{Human Activity Recognition}
We focus our review on the commonly used sequential and space-time volume methods \cite{Aggarwal_CSUR11}.
A comprehensive review of different aspects of human activity recognition (HAR)
is presented in \cite{Aggarwal_CSUR11} and \cite{borges2013video}.
A popular sequential method is to use centroid trajectories to identify human activities in visual data,
in which a human is represented as a single point
indicating the human's location.
Chen and Yang represented a human with just a point to derive the gait features for the pedestrian detection \cite{chen2014extraction}.
Ge \emph{et al.} extracted pedestrian trajectories from video to automatically detect small groups of people traveling together \cite{ge2012vision}.
These methods can avoid the influence of human appearances such as dresses and carrying, but are not able to recognize activities involving various relative body-part movements.
Another sequential method relies on human shapes,
including human contours and silhouettes.
Singh \emph{et al.} extracted directional vectors from the silhouette contours and utilize the distinct data distribution of these vectors in a vector space for activity recognition \cite{Singh_2008}.
Junejo \emph{et al.} transformed silhouettes of a human from every frame into time-series, then each of these time series is converted into the symbolic vector to represent actions \cite{junejo2014silhouette}.
A third method is based on body-part models.
Zhang and Parker implemented a bio-inspired predictive orientation decomposition (BIPOD) to construct representations of people from skeleton trajectories for the activity recognition and prediction, where the human body is decomposed into five body parts \cite{zhang2015bio}.
However, techniques based on shapes and/or body-part models
rely on human and body-part detection,
which are hard-to-solve problems due to occlusions and dynamic backgrounds, among others.
Space-time volume methods use local features to represent local texture and motion variations regardless of global human appearance and activity.
A large number of HAR methods are based on SIFT features \cite{Lowe_IJCV04}
and its extensions \cite{Sun_CVPR09}.
For example, Behera \emph{et al.} proposed a random forest that unifies randomization, discriminative relationships mining and a Markov temporal structure for real-time activity recognition with SIFT features \cite{behera2014real}.
However, SIFT features only encode appearance information
and are not able to represent temporal information.
STIP features were introduced in \cite{Schuldt_ICPR04}
and SVMs were applied to classify human activities.
Dollar \emph{et al.} used separable filters in spatial and temporal dimensions to extract features for HAR \cite{Dollar_ICCV05}.
Four-dimensional features were also introduced in \cite{Zhang_IROS11} to combine depth information to classify human activities.
Previous work focused only on recognizing human activities
but did not discuss the consequent issue:
how a co-robot can make decisions based on recognition results,
especially when risks are associated with different human activities.
\subsection{Topic Models and Evaluation}
Among other machine learning techniques,
topic modeling has been widely applied to HAR.
Zhao \emph{et al.} incorporated Bayesian learning into an undirected topic model and proposed a "relevance topic model" for the unstructured social group activity recognition \cite{zhao2013relevance}.
A semi-latent topic model trained in a supervised fashion was introduced in \cite{Wang_PAMI09}
and used to classify activities in videos.
Zhang and Parker adopted topic models to classify activities in 3D point clouds from color-depth cameras on mobile robots \cite{Zhang_IROS11}.
Topic models were also widely used to discover human activities in streaming data.
The use of topic models was explored in \cite{Huynh_DAP08} to discover daily activity patterns in wearable sensor data.
An unsupervised topic model was proposed in \cite{Farrahi_TIST11} to detect daily routines from streaming location and proximity data.
Taking temporal and/or object relational information into account, Freedman \emph{et al.} explored a new method using topic models for both plan recognition and activity recognition objective \cite{freedman2014temporal}.
Although there is a significant body of work introducing and developing sophisticated topic models,
few efforts have been undertaken to evaluate them.
Existing methods are dominated by either intrinsic methods,
(e.g., computing the probability of held-out documents to evaluate generalization ability \cite{Wallach_ICML2009})
or extrinsic methods that rely on external tasks,
(e.g., information retrieval \cite{Wei_ICRDIR06}).
Some work also focused on evaluation of topic modeling's interpretability as semantically coherent concepts.
For example,
Chang \emph{et al.} demonstrated that the probability of held-out documents is
not always a good indicator of human judgment \cite{chang_nips09}.
Newman \emph{et al.} showed that metrics based on word co-occurrence statistics
are able to predict human evaluations of topic quality \cite{Newman_HLT10}.
As recently pointed out by Blei \cite{Blei_ComACM12},
topic model evaluation is an essential research topic.
Despite this, previous works use only the accuracy metric to evaluate topic modeling results in HAR tasks;
issues such as the model's interpretability and generalizability have not been studied.
In this paper, we analyze these two aspects of topic model evaluation in HAR tasks,
explore their relationship,
and show how they can be used to improve robot decision making.
\subsection{Artificial Cognitive Modeling}
Artificial cognition has its origin in cybernetics with the intention to create a science of mind based on logic \cite{Varela_99}.
Among other cognitive paradigms, cognitivism has undoubtedly been predominant to date
\cite{Vernon_TEC07}.
Within the cognitivism paradigm,
several cognitive architectures were developed,
including
Soar \cite{laird2012soar},
ACT-R \cite{Anderson_AP96}
(and its extensions ACT-R/E \cite{trafton2013act}, ACT-R$\Phi$ \cite{dancy2013act}, etc),
C4 \cite{Isla_IJCAI01},
and architectures for robotics \cite{Burghart_ICHR05},
which are relatively independent of applications \cite{Gray_HCI97}.
Because architectures represent the mechanism for cognition but lack the relevant information for using that mechanism, they cannot accomplish anything in
their own right and need to be provided with knowledge to conduct a specific task.
The combination of a cognitive architecture and a particular knowledge set
is generally referred to as a \emph{cognitive model} \cite{Vernon_TEC07}.
The knowledge incorporated in cognitive models
is typically determined by human designers \cite{Vernon_TEC07}.
The knowledge can be also learned and adapted using machine learning techniques.
Cognitive models have been widely used in human-machine interaction and robotic vision applications.
For example, cognitive modeling was adopted in \cite{Duric_IEEE02,lallee2014efaa,baxter2013cognitive} to construct intelligent human-machine interaction systems.
Cognitive perception systems were also used to recognize traffic signs \cite{Yang_TST13,halbrugge2013act},
interpret traffic behaviors \cite{Nagel_AIM04,lawitzky2013interactive},
and recognize human activities \cite{Crowley_CVS06,brdiczka2009learning}.
Over the last decade, probabilistic models of cognition,
as an alternative of deterministic cognitive models,
have attracted more attention in cognitive development \cite{Xu_Cog11}.
For example, an adaptive remote data mirroring system was proposed applying
dynamic decision networks in \cite{bencomo2013dynamic}.
Another cognitive model was introduced in \cite{kafai2012dynamic}
to apply dynamic Bayesian networks for vehicle classification.
Probabilistic models have also been widely used for learning and reasoning in cognitive modeling \cite{Chater_TCS06}.
We believe we are the first to adopt topic models for the construction of reliable artificial cognitive models and show that they are particularly suited for this task.
We demonstrate topic modeling's ability to combine risks in decision making.
In addition, we develop two evaluation metrics
and show their effectiveness in model selection and decision making.
These aspects were not addressed in previous artificial cognitive modeling research.
\section{Topic Modeling for Artificial Cognition}\label{sec:CognitiveSystem}
\subsection{Cognitive Architecture Overview}
The proposed SRAC model is inspired by
the C4 cognitive architecture \cite{Isla_IJCAI01}.
As shown in Fig. \ref{fig:ArchitectureCogSys},
our model is organized into four modules by their functionality:
\begin{itemize}
\item \emph{Sensory and perception}:
Visual cameras observe the environment.
Then, the perception system
builds a BoW representation from raw data, which can be processed by topic models.
\item \emph{Probabilistic reasoning}:
Topic models are applied to reason about human activities,
which are trained off-line and used online.
The training set is provided as a prior that encodes a history of sensory information.
This module uses the proposed indicators to select topic models that better match human's perspective,
and to discover new activities in an online fashion.
\item \emph{Decision making}:
Robot action risk based on topic models and the evaluation results is estimated and
a response robot action that minimizes this risk is selected.
The risk is provided as a prior to the module.
\item \emph{Navigation and motor system}:
The selected robot action is executed in response to human activities.
\end{itemize}
\subsection{Topic Modeling}
Latent Dirichlet Allocation (LDA) \cite{Blei_JMLR03}, which showed promising activity recognition performance in our prior work \cite{ Zhang_IROS11}, is applied in the SRAC model.
Given a set of observations $\{\boldsymbol{w}\}$,
LDA models each of $K$ activities as a multinomial distribution of all possible visual words
in the dictionary $\boldsymbol{D}$.
This distribution is parameterized by
$\boldsymbol{\varphi} \!=\! \{\varphi_{w_1}, \dots, \varphi_{w_{|\boldsymbol{D}|}}\}$,
where $\varphi_{w}$ is the probability that the word ${w}$ is generated by the activity.
LDA also represents each $\boldsymbol{w}$ as a collection of visual words,
and assumes that each word $w \in \boldsymbol{w}$ is associated with a latent activity assignment $z$.
By applying the visual words to connect observations and activities,
LDA models $\boldsymbol{w}$ as a multinomial distribution over the activities,
which is parameterized by $\boldsymbol{\theta} \!=\! \{\theta_{z_1},\dots,\theta_{z_K}\}$,
where $\theta_{z}$ is the probability that $\boldsymbol{w}$ is generated by the activity $z$.
LDA is a Bayesian model,
which places Dirichlet priors on the multinomial parameters:
$\boldsymbol{\varphi} \!\sim\! \operatorname{Dir}(\boldsymbol{\beta})$ and
$\boldsymbol{\theta} \!\sim\! \operatorname{Dir}(\boldsymbol{\alpha})$,
where $\boldsymbol{\beta} \!=\! \{\beta_{w_1},\dots,\beta_{w_{|\boldsymbol{D}|}}\}$
and $\boldsymbol{\alpha} \!=\! \{\alpha_{z_1},\dots,\alpha_{z_K}\}$ are
the concentration hyperparameters.
One of the major objectives in HAR tasks to is to estimate the parameter $\boldsymbol{\theta}$,
i.e., the per-observation activity proportion.
However, exact parameter estimation is intractable in general \cite{Blei_JMLR03}.
Our model applies Gibbs sampling \cite{Griffiths_NAS04} to compute the \emph{per-observation activity distribution} $\boldsymbol{\theta}$,
based on two considerations:
1) This sampling-based method is generally accurate,
since it asymptotically approaches the correct distribution \cite{Porteous_KDD08}, and
2) This method can be used to intrinsically evaluate topic model's performance \cite{Wallach_ICML2009},
thereby providing a consistent method to infer, learn, and evaluate topic models.
At convergence,
the element $\theta_{z_k} \!\in\! \boldsymbol{\theta}$, $k \!=\! 1,\dots,K$, is estimated by:
\begin{eqnarray}
\hat{\theta}_{z_k} = \frac{n_{z_k} + \alpha_{z_k}}{\sum_{z}{(n_z + \alpha_z)}},
\end{eqnarray}
where $n_z$ is the number of times that a visual word is assigned to activity $z_k$ in the observation.
The incorporation of topic models into cognitive modeling has several important advantages.
First, as a probabilistic reasoning approach,
it serves as a bridge to allow information to flow from the perception module to the decision making module.
Second, the ability to model per-observation activity distribution allows topic models
to take into account the risks of all robot actions in a probabilistic way
and make an appropriate decision.
Third, by introducing an extrinsic evaluation metric for topic model selection,
the constructed cognitive system is able to accurately interpret human activities.
Fourth, the unsupervised nature of topic modeling, which is explored using our new intrinsic metric,
facilitates online discovery of new knowledge (e.g., human activities).
All these advantages allow us to apply topic models to construct an artificial cognitive system
that is able to better interpret human activities,
discover new knowledge and react more appropriately and safely to humans,
which is highly desirable for real-world online human-robot interaction scenarios.
\section{Interpretability and Generalizability}\label{sec:TopicModels}
To improve artificial cognitive modeling,
we introduce two novel indicators and discuss their relationship in this section,
which are the core of the \emph{Self-Reflection} module in Fig. \ref{fig:ArchitectureCogSys}.
\subsection{Interpretability Indicator} \label{subsec:ii}
We observe that accuracy is not an appropriate assessment metric for robot decision making,
since it only considers the most probable human activity category and ignores the others.
To utilize the category distribution, which contains much richer information,
the \emph{interpretability indicator}, denoted by $I_I$, is introduced.
$I_I$ is able to encode how well topic modeling matches human common sense.
Like the accuracy metric, $I_I$ is an extrinsic metric,
meaning that it requires a ground truth to compute.
Formally, $I_I$ is defined as follows:
\begin{definition}[Interpretability indicator]\label{def:II}
Given the observation $\boldsymbol{w}$
with the ground truth $g$ and the distribution $\boldsymbol{\theta}$ over $K \geq 2$ categories,
let $\boldsymbol{\theta}_s = (\theta_1,$ $\dots, \theta_{k-1}, \theta_k,$ $\theta_{k+1}, \dots, \theta_K)$ denote
the sorted proportion satisfying
$\theta_1 \geq \cdots \geq \theta_{k-1} \geq \theta_k \geq \theta_{k+1} \geq \cdots \geq \theta_K \geq 0$
and $\sum_{i=1}^{K} \theta_i = 1$,
and let $k \in \{1,\cdots, K\}$ represent the index of the assignment in $\boldsymbol{\theta}_s$ that matches $g$.
The interpretability indicator $I_I (\boldsymbol{\theta},g) = I_I (\boldsymbol{\theta}_s,\!k)$ is defined as:
\begin{eqnarray} \label{eq:II}
\small{
I_I (\boldsymbol{\theta}_s,\!k) \triangleq
\frac{1}{a}
\left( \frac{K \!-\! k}{K \!-\! 1} + \mathds{1}(k \!=\! K) \right)
\left( \frac{\theta_k}{\theta_1} \!-\!
\frac{\theta_{k + \mathds{1}(k \neq K)}}{\theta_k} \!+\! b \right)
\!\!
}
\end{eqnarray}
where $\mathds{1}(\cdot)$ is the indicator function, and $a=2$, $b=1$ are normalizing constants
\end{definition}
The indicator $I_I$ is defined over the per-observation category proportion $\boldsymbol{\theta}$,
which takes values in the $(K\!-\!1)$-simplex \cite{Blei_JMLR03}.
The sorted proportion $\boldsymbol{\theta}_s$ is computed through sorting $\boldsymbol{\theta}$,
which is inferred by topic models.
In the definition, the ground truth is represented by its location in $\boldsymbol{\theta}_s$,
i.e., the $k$-th most probable assignment in $\boldsymbol{\theta}_s$ matches the ground truth label.
The indicator function $\mathds{1}(\cdot)$ in Eq. (\ref{eq:II}) is adopted to deal with the special case when $k = K$.
For an observation in an activity recognition task with $K$ categories,
given its ground truth index $k$ and sorted category proportion $\boldsymbol{\theta}_s$,
we summarize $I_I$'s properties as follows:
\begin{proposition}[$I_I$'s properties]\label{prop1}
The interpretability indicator $I_I(\boldsymbol{\theta},g) = I_I(\boldsymbol{\theta}_s,k)$ satisfies the following properties:
1. If $k = 1$, $\forall \boldsymbol{\theta}_s$, $I_I(\boldsymbol{\theta}_s,k) \geq 0.5$.
2. If $k = K$, $\forall \boldsymbol{\theta}_s$, $I_I(\boldsymbol{\theta}_s,k) \leq 0.5$.
3. $\forall \boldsymbol{\theta}_s$, $I_I(\boldsymbol{\theta}_s,k) \in [0,1]$.
4. $\forall k \in \{1,\dots, K\}$ and $\boldsymbol{\theta}_s$, $\boldsymbol{\theta}'_s$ such that $\theta_1 \geq\theta_1'$, $\theta_k = \theta_k'$ and $\theta_{k+\mathds{1}(k \neq K)} = \theta_{k+\mathds{1}(k \neq K)}'$,
$I_I (\boldsymbol{\theta}_s,k) \leq I_I (\boldsymbol{\theta}'_s,k)$ holds.
5. $\forall k \in \{1,\dots, K\}$ and $\boldsymbol{\theta}_s$, $\boldsymbol{\theta}'_s$ such that $\theta_{k+\mathds{1}(k \neq K)} \geq\! \theta_{k + \mathds{1}(k \neq K)}'$,
$\theta_1 \!=\! \theta_1'$ and
$\theta_k \!=\! \theta_k'$,
$I_I (\boldsymbol{\theta}_s,k) \leq I_I (\boldsymbol{\theta}'_s,k)$ holds.
6. $\forall k \in \{1,\dots, K\}$ and $\boldsymbol{\theta}_s$, $\boldsymbol{\theta}'_s$ such that
$\theta_k \geq \theta_k'$,
$\theta_1 = \theta_1'$ and
$\theta_{k+\mathds{1}(k \neq K)} = \theta_{k + \mathds{1}(k \neq K)}'$,
$I_I (\boldsymbol{\theta}_s,k) \geq I_I (\boldsymbol{\theta}'_s,k)$ holds.
7. $\forall k, k' \in \{1,\dots, K\}$ such that $k \leq k' < K$ and
$\forall \boldsymbol{\theta}_s$, $\boldsymbol{\theta}'_s$ such that
$\theta_k = \theta_k'$,
$\theta_1 = \theta_1'$ and
$\theta_{k+\mathds{1}(k \neq K)} = \theta_{k + \mathds{1}(k \neq K)}'$,
$I_I (\boldsymbol{\theta}_s,k) \geq I_I (\boldsymbol{\theta}'_s,k')$ holds.
\end{proposition}
\begin{pf}
See \ref{appendix:1}.
\end{pf}
The indicator $I_I$ is able to quantitatively measure
how well topic modeling can match human common sense,
because it captures three essential considerations to
simulate the process of how humans evaluate the category proportion $\boldsymbol{\theta}$:
\begin{itemize}
\item
A topic model performs better, in general, if it obtains a larger $\theta_k$ (Property 6).
In addition, a larger $\theta_k$ generally indicates
$\theta_k$ is closer to the beginning in $\boldsymbol{\theta}_s$
and further away from the end (Property 7).
\textit{Example}:
A topic model obtaining the sorted proportion
$[0.4, \boxed{0.35}, 0.15, 0.10]$
performs better than a model obtaining $[0.4, \boxed{0.30}, 0.15, 0.15]$,
where the ground truth is marked with a box, i.e., $k = 2$ in the example.
\item A smaller difference between $\theta_k$ and $\theta_1$ indicates better modeling performance (Properties 4 and 5), in general.
Since the resulting category proportion is sorted,
a small difference between $\theta_k$ and $\theta_1$ guarantees $\theta_k$ has an even smaller difference from $\theta_2$ to $\theta_{k-1}$.
\textit{Example}: A topic model obtaining the sorted proportion
$[0.4, \boxed{0.3}, 0.2, 0.1]$
performs better than the model with the proportion $[0.5, \boxed{0.3}, 0.2, 0]$.
\item A larger distinction between $\theta_k$ and $\theta_{k+1}$ generally indicates better modeling performance (Properties 5 and 6),
since it better separates the correct assignment from the incorrect assignments with lower probabilities.
\textit{Example}: A topic model obtaining the sorted proportion
$[0.4, \boxed{0.4}, 0.1, 0.1]$ performs better than the topic model obtaining the proportion
$[0.4, \boxed{0.4}, 0.2, 0]$.
\end{itemize}
\begin{comment}
We normalize $I_I$ to fall in the range $[0,1]$ (Property 3),
with a greater value indicating a better interpreted model.
If the most probable assignment matches the ground truth,
it is guaranteed that $I_I$ is greater or equal to 0.5 (Property 1).
Similarly, when the least probable assignment matches the ground truth (Property 2), $I_I$ is no greater than 0.5.
During the training phase,
topic modeling groups the training set $\mathcal{W}_{train}$ into clusters
$\mathcal{W}_{c_1} \cup \mathcal{W}_{c_2} \cup \dots \cup \mathcal{W}_{c_K}$.
Because topic models are unsupervised,
it is necessary to associate the resulting clusters with pre-defined categories.
We introduce a procedure called \emph{Topic Mapping}, based on $I_I$, to automatically perform topic association:
\begin{definition}[Topic mapping]
Let $\boldsymbol{g} = \{g_1,\dots, g_M\}$ be the ground truth of a set of observations
$\mathcal{W} = \{\boldsymbol{w}_1,\dots,\boldsymbol{w}_M\}$.
We denote the interpretability indicator over $\mathcal{W}$ to be:
\begin{eqnarray}
I_I(\mathcal{W},\boldsymbol{g}) = \frac{1}{M}\sum_{m = 1}^{M} I_I(\boldsymbol{w}_m, g_m).
\end{eqnarray}
Let $\boldsymbol{g} = \{g_1, \dots, g_K\}$ be the ground truth
and $\boldsymbol{c} = \{c_1, \dots,$ $ c_K\}$ be the cluster indices
of $\mathcal{W}_{c_1}, \dots, \mathcal{W}_{c_K}$.
We define topic mapping to be a bijective function $f: \boldsymbol{c} \rightarrow \boldsymbol{g}$ such that
\begin{eqnarray}
f = \operatorname*{arg\,max}_{f'} \frac{1}{K} \sum_{k = 1}^{K} I_I(\mathcal{W}_k, f'(c_k)).
\end{eqnarray}
\end{definition}
Intuitively, topic mapping is a one-to-one and onto mapping of the cluster indices to the ground truth
that maximizes topic modeling's average interpretability on the training set.
Given the training set,
the topic model whose hyper-parameters (e.g., $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$ in LDA)
maximize this average interpretability is selected as our final model to reason about human activities.
In this work, we employ the Hungarian algorithm \cite{Kuhn_NRLQ55} to solve
this topic mapping problem.
\end{comment}
The indicator $I_I$ extends the \emph{accuracy} metric $I_A$
(i.e., rate of correctly recognized data),
as described in Proposition \ref{prop2}:
\begin{proposition}[Relationship of $I_I$ and $I_A$] \label{prop2}
The accuracy measure $I_A$ is a special case of $I_I(\boldsymbol{\theta}_s, k)$, when
$\theta_1 = 1.0$, $\theta_2 =\! \dots \!= \theta_K = 0$,
and $k = 1$ or $k = K$.
\end{proposition}
\begin{pf}
See \ref{appendix:2}.
\end{pf}
\subsection{Generalizability Indicator} \label{subsec:ig}
An artificial cognitive model requires the crucial capability of detecting new situations and being aware that the learned knowledge becomes less applicable
in an online fashion.
To this end, we propose the \emph{generalizability indicator} ($I_G$),
an intrinsic metric
that does not require ground truth to compute and consequently can be used online.
The introduction of $I_G$ is inspired by the perplexity metric
(also referred to as held-out likelihood),
which evaluates a topic model's generalization ability
on a fraction of held-out instances using cross-validation \cite{Musat_IJCAI11}
or unseen observations \cite{Blei_NIPS05}.
The perplexity is defined as the log-likelihood of words in an observation \cite{Wallach_ICML2009}.
Because different observations may contain a different number of visual words,
we compute the \emph{Per-Visual-Word Perplexity ($Pvwp$)}.
Mathematically, given the trained topic model $\mathcal {M}$ and an observation $\boldsymbol{w}$, $Pvwp$ is defined as follows:
\begin{eqnarray}\label{eq:pvwp}
\small{
Pvwp(\boldsymbol{w} | \mathcal{M}) \!=\! \frac{1}{N} \log\! P(\boldsymbol{w} | \mathcal{M})
\!=\! \frac{1}{N} \log\! \prod_{n = 1}^{N} \!P(w_n| \boldsymbol{w}_{<n}, \mathcal{M})\!\!\!\!
}
\end{eqnarray}
where $N = |\boldsymbol{w}|$ is the number of visual words in $\boldsymbol{w}$,
and the subscript $<\!n$ denotes positions before $n$.
Because $P(\boldsymbol{w} | \mathcal{M})$ is a probability that satisfies $P(\boldsymbol{w} | \mathcal{M}) \!\leq\! 1$,
it is guaranteed $Pvwp(\boldsymbol{w}|\mathcal{M}) \!\leq\!0$.
The left-to-right algorithm,
presented in Algorithm \ref{alg:pvwpEstimation}, is used to estimate $Pvwp$,
which is an accurate and efficient Gibbs sampling method to estimate perplexity \cite{Wallach_ICML2009}.
The algorithm decomposes $P(\boldsymbol{w} | \mathcal{M})$ in an incremental, left-to-right fashion,
where the subscript $\neg n$ is a quantity that excludes data from the $n$th position.
Given observations $\mathcal{W} \!=\! \{\boldsymbol{w}_1,\dots,\boldsymbol{w}_M\}$,
$Pvwp(\mathcal{W}| \mathcal{M})$ is defined as the average of each observation's perplexity:
\begin{eqnarray}
Pvwp(\mathcal{W} | \mathcal{M}) = \frac{1}{M} \sum_{m = 1}^{M} Pvwp(\boldsymbol{w}_m | \mathcal{M})
\end{eqnarray}
\begin{algorithm}
\SetAlgoLined
\SetKwData{Left}{left}\SetKwData{This}{this}\SetKwData{Up}{up}
\SetKwFunction{Union}{Union}\SetKwFunction{FindCompress}{FindCompress}
\SetKwInOut{Input}{Input}\SetKwInOut{Output}{Output}
\SetNlSty{textrm}{}{:}
\SetKwComment{tcc}{/*}{*/}
\small{
\Input{
$\boldsymbol{w}$ (observation), $\mathcal{M}$ (trained topic model), and
$R$ (number of particles)
}
\Output{
$Pvwp(\boldsymbol{w}|\mathcal{M})$
}
\BlankLine
Initialize $l = 0$ and $N = |\boldsymbol{w}|$;
\For{each position $n = 1$ to $N$ in $\boldsymbol{w}$}{
Initialize $p_n = 0$;
\For{each particle $ r = 1$ to $R$}{
\For{$n' < n$}{
Sample $z_{n'}^{(r)} \sim P(z_{n'}^{(r)} | w_{n'}, \{ \boldsymbol{z}_{<n}^{(r)} \}_{\neg n'}, \mathcal{M})$;
}
Compute $p_n = p_n + \sum_t P(w_n, z_n^{(r)} = t | z_{<n}^{(r)}, \mathcal{M})$;
Sample $z_n^{(r)} \sim P(z_n^{(r)} | w_n, z_{<n}^{(r)}, \mathcal{M})$;
}
Update $p_n = \frac{p_n}{R}$ and $l = l + \log p_n$;
}
\Return{ $Pvwp(\boldsymbol{w}|\mathcal{M}) \simeq \frac{l}{N}$. }
}
\caption{Left-to-right $Pvwp$ estimation}\label{alg:pvwpEstimation}
\end{algorithm}
Based on $Pvwp$,
the generalizability indicator $I_G$,
on previously unseen observations in the testing phase or
using the held-out instances in cross-validation,
is defined as follows:
\begin{definition}[Generalizability indicator]
Let $\mathcal{M}$ denote a trained topic model,
$\mathcal{W}_{\text{valid}}$ denote the validation dataset that is used in the training phase,
and $\boldsymbol{w}$ be an previously unseen observation.
We define the generalizability indicator:
\begin{eqnarray}\label{eq:IG}
\footnotesize{
I_G(\boldsymbol{w}) \!\triangleq\!
\begin{dcases}
\frac{\exp(Pvwp(\boldsymbol{w}| \mathcal{M}))}{c \cdot \exp(Pvwp(\mathcal{W}_{valid}| \mathcal{M}))} \\
\quad\;\;
{\text{if }
\footnotesize{ \exp(Pvwp(\boldsymbol{w}|\mathcal{M})) \!<\! c \!\cdot\! \exp(Pvwp(\mathcal{W}_{valid}| \mathcal{M})) }
} \\
\; \text{$1$} \quad
\!\!{\text{if } \footnotesize{ \exp(Pvwp(\boldsymbol{w}|\mathcal{M})) \!\geq\! c\! \cdot\! \exp(Pvwp(\mathcal{W}_{valid}|\mathcal{M}))}}
\end{dcases}\!\!\!\!
}
\end{eqnarray}
where $c \in [1, \infty)$ is a constant encoding novelty levels.
\end{definition}
Besides considering the topic model's generalization ability,
$I_G$ also evaluates whether previously unseen observations are well-represented by the training set,
i.e., whether the training set used to train the topic model is exhaustive.
The training set is defined as \emph{exhaustive} when it contains instances from all categories
that can possibly be observed in the testing phase \cite{Dundar_ICML12}.
When some categories are missing and not represented by the training set,
it is defined as \emph{non-exhaustive}; in this case, novel categories emerge in the testing phase.
Since it is impractical, often impossible, to define an exhaustive training set,
mainly because some of the categories may not exist at the time of training,
the ability to discover novelty and be aware that the learned model
is less applicable is essential for safe, adaptive decision making.
The indicator $I_G$ provides this ability through evaluating
how well new observations are represented by the validation set in the training phase.
We constrain $I_G$'s value in the range $(0, 1]$, with a greater value indicating less novelty,
which means an observation can be better encoded by the training set and the topic model generalizes better on this observation.
The constant $c$ in Eq. (\ref{eq:IG}) provides the flexibility to encode the degree to which we
consider an observation to be novel.
\subsection{Indicator Relationship}
While the interpretability indicator interprets human activity distributions
in a way that is similar to human reasoning,
the generalizability indicator endows a co-robot with the self-reflection capability.
We summarize their relationship
in the cases when a training set is exhaustive
(i.e., training contains all possible categories)
and non-exhaustive
(i.e., new human behavior occurs during testing), as follows:
\begin{comment}
In human-robot interaction applications,
one major objective is to make the modeling recognition result match human common sense as closely as possible.
This is captured by the $I_I$ metric, i.e., a better evaluation result with a greater $I_I$ value indicates better recognition performance.
However, due to its extrinsic nature,
the indicator $I_I$ cannot be directly applied on previously unseen observations without knowledge of ground truth,
i.e., a topic model's interpretability cannot be evaluated online during the testing phase.
On the other hand, as an intrinsic metric,
the indicator $I_G$ can be computed to evaluate the topic model's generalization ability over new observations.
If we can understand the relationship between $I_G$ and $I_I$,
the relationship between a topic model's generalizability and interpretability,
it should be possible to apply the topic model's generalizability to indicate its interpretability in an online fashion.
\end{comment}
\emph{Observation (Relationship of $I_G$ and $I_I$)}:
Let $\mathcal{W}_{train}$ be
the training dataset used to train a topic model,
and $I_I$ and $I_G$ be the model's interpretability and generalizability indicators.
\begin{itemize}
\item If $\mathcal{W}_{train}$ is exhaustive,
then $I_G \rightarrow 1$ and $I_I$ is generally independent of $I_G$.
\item If $\mathcal{W}_{train}$ is non-exhaustive,
then $I_G$ takes values that are much smaller than $1$; $I_I$ also takes small values and is moderately to strongly correlated with $I_G$.
\end{itemize}
\begin{table}[tbh]
\caption{
Meaning and relationship of $I_I$ and $I_G$.
The gray area denotes that the situation is generally impossible.
} \label{tab:ImplicationIiIg}
\begin{center}
\vspace{-8pt}
\begin{tabular}{ r|p{3.5cm}|p{4.5cm}| }
\multicolumn{1}{r}{}
& \multicolumn{1}{c}{$I_G$: low}
& \multicolumn{1}{c}{$I_G$: high} \\
\cline{2-3}
$I_I$:$\;$ low & Category is novel \newline Model is \emph{not} applicable & Category is \emph{not} novel \newline Model is \emph{not} well interpreted \\
\cline{2-3}
$I_I$: high & \cellcolor{black!25} & Category is \emph{not} novel \newline Model is well interpreted \\
\cline{2-3}
\end{tabular}
\end{center}
\end{table}
This observation answers the critical question of
whether a better generalized topic model can lead to better recognition performance.
Intuitively, if $\mathcal{W}_{train}$ is non-exhaustive and a previously unseen observation $\boldsymbol{w}$ belongs to a novel category,
which is indicated by a small $I_G$ value,
a topic model trained on $\mathcal{W}_{train}$ cannot accurately classify $\boldsymbol{w}$.
On the other hand, if $\boldsymbol{w}$ belongs to a category that is known in $\mathcal{W}_{train}$,
then $I_G \!\rightarrow\! 1$ and the recognition performance over $\boldsymbol{w}$ only depends on
the model's performance on the validation set used in the training phase.
The meaning and relationship of the indicators $I_I$ and $I_G$ are summarized in Table \ref{tab:ImplicationIiIg},
where the gray area denotes that it is
generally impossible for a topic model
to obtain a low generalizability but a high interpretability,
as a model is never correct when presented with a novel activity.
\begin{table}[tb]
\caption{Risk levels as prior knowledge to our cognitive model.}\label{tab:RiskLevel}
\vspace{-8pt}
{
\begin{center}
\tabcolsep=0.2cm
\begin{tabular}{|l|c|l|}
\hline
\quad Levels & Values & $\qquad\qquad\qquad$ Definition\\
\hline\hline
Low risk & [1,30] & Unsatisfied with the robot's performance.\\
\hline
Medium risk & [31,60] & Annoyed or upset by the robot's actions.\\
\hline
High risk & [61,90] & Interfered with, interrupted, or obstructed.\\
\hline
Critical risk & [95,100] & Injured or worse (i.e., a safety risk).\\
\hline
\end{tabular}
\end{center}
}
\end{table}
\section{Self-Reflective Risk-Aware Decision Making} \label{sec:DecisionMaking}
Another contribution of this research is a decision making framework
that is capable of incorporating activity category distribution,
robot self-reflection (enabled by the indicators),
and co-robot action risk,
which is realized in the module of \emph{Decision Making}
in Fig. \ref{fig:ArchitectureCogSys}.
Our new self-reflective risk-aware decision making algorithm is presented in Algorithm \ref{alg:Decision}.
\begin{algorithm}
\SetAlgoLined
\SetKwData{Left}{left}\SetKwData{Name}{Name}\SetKwData{Up}{up}
\SetKwFunction{KwFn}{Fn}
\SetKwInOut{Input}{Input}
\SetKwInOut{Output}{Output}
\SetKwInOut{Func}{Func.}
\SetNlSty{textrm}{}{:}
\SetKwComment{tcc}{/*}{*/}
\SetKwFunction{Union}{Union}
{
\Input{
$\boldsymbol{w}$ (observation), $\mathcal{M}$ (trained topic model), and
$\mathcal{N}$ (decision making bipartite network)
}
\Output{
$a^{\star}$ (Selected robot action with minimum risk)
}
\BlankLine
Estimate per-observation activity proportion $\boldsymbol{\theta}$ of $\boldsymbol{w}$;
Compute generalizability indicator $I_G(\boldsymbol{w})$;
\For{each robot action $ i = 1$ to $S$}{
Estimate activity-independent risk:
$r^{in}_i \!=\! \frac{1}{K}\sum_{j = 1}^{K} \! r_{ij}$;
Calculate activity-dependent risk:
$r^{de}_i = \sum_{j = 1}^{K} (\theta_j \cdot r_{ij})$;
Combine activity-independent and dependent risks,
and assign to per-observation action risk vector:
$\boldsymbol{r}^a(i) = (1 - I_G(\boldsymbol{w})) \cdot r_i^{in} + I_G(\boldsymbol{w}) \cdot r_i^{de}$;
}
Select optimal robot action $a^{\star}$ with minimum risk in $\boldsymbol{r}^a$;
\Return{ $a^{\star}$. }
}
\caption{Self-reflective risk-aware decision making}\label{alg:Decision}
\end{algorithm}
Given the robot action set $\boldsymbol{a} = \{a_1, \dots, a_S \}$
and the human activity set $\boldsymbol{z} = \{z_1, \dots, z_K\}$,
an action-activity risk $r_{ij}$ is defined as the amount of discomfort, interference, or harm that can be expected to occur
during the time period if the robot takes a specific action $a_i, \forall i \in \{1,\dots,S\}$
in response to an observed human activity $z_j, \forall j \in \{1,\dots,K\}$.
While $\boldsymbol{\theta}$ and $I_G$ are computed online,
the risks $\boldsymbol{r} = \{r_{ij}\}_{S \times K}$, with each element $r_{ij} \in [0, 100]$, are manually estimated off-line by domain experts
and used as a prior in the decision making module.
In practice, the amount of risk is categorized into a small number of risk levels for simplicity's sake.
To assign a value to $r_{ij}$, a risk level is first selected.
Then, a risk value is determined within that risk level.
As listed in Table \ref{tab:RiskLevel},
we define four risk levels with different risk value ranges in our application.
We intentionally leave a five-point gap between critical risk and high risk
to increase the separation of critical risk from high risk actions.
\begin{figure}[h]
\centering
\includegraphics[width= 0.6 \textwidth]{RiskExample.png}
\caption{
An illustrative example of a bipartite network (left) and the per-observation activity distribution (right) in assistive robotics applications.
}\label{fig:RiskExample}
\end{figure}
A bipartite network
$\mathcal{N} \!=\! \{\boldsymbol{a}, \boldsymbol{z}, \boldsymbol{r}\}$
is proposed to graphically illustrate the risk matrix $\boldsymbol{r}$
of robot actions $\boldsymbol{a}$ associated with human activities $\boldsymbol{z}$.
In this network, vertices are divided into two disjoint sets $\boldsymbol{a}$ and $\boldsymbol{z}$,
such that every edge with a weight $r_{ij}$ connects a vertex $a_i \in \boldsymbol{a}$ to a vertex $z_j \in \boldsymbol{z}$.
An example of such a bipartite network is depicted in Fig. \ref{fig:RiskExample} for assistive robotics applications.
The bipartite network also has a tabular representation (for example, in Table \ref{tab:RiskMat}).
Given the bipartite network,
for a new observation $\boldsymbol{w}$,
after $\boldsymbol{\theta}$
and $I_G(\boldsymbol{w})$ are computed in the probabilistic reasoning module,
the robot action $a^\star \in \boldsymbol{a}$ is selected according to:
\begin{eqnarray}\label{eq:ActionSelection}
\small{
a^\star \!=\! \mathop{\arg \min}\limits_{a_i:i = 1, \dots, S}
\left(
\frac{1 \!-\! I_G(\boldsymbol{w})}{K} \!\cdot\! \sum_{j=1}^{K} r_{ij} +
I_G(\boldsymbol{w}) \!\cdot\! \sum_{j = 1}^{K}{(\theta_j \!\cdot\! r_{ij})}
\right) \!\!
}
\end{eqnarray}
The risk of taking a specific robot action is determined by two separate components:
activity-independent and activity-dependent action risks.
The activity-independent risk (that is $\frac{1}{K}\sum_{j=1}^{K} r_{ij}$) measures the inherent risk of an action,
which is independent of the human activity context information,
i.e., computing this risk does not require the category distribution.
For example, the robot action ``standing-by'' generally has a smaller risk than ``moving backward'' in most situations.
The activity-dependent risk (that is $\sum_{j = 1}^{K}{(\theta_j \!\cdot\! r_{ij})}$) is the average risk weighted by context-specific information (i.e., the human activity distribution).
The combination of these two risks is controlled by $I_G$,
which automatically encodes preference over robot actions.
When the learned model generalizes well over $\boldsymbol{w}$, i.e., $I_G(\boldsymbol{w}) \!\rightarrow\! 1$,
the decision making process prefers co-robot actions that are more appropriate to the recognized human activity.
Otherwise, if the model generalizes poorly,
indicating new human activities occur
and the learned model is less applicable,
our decision making module
would ignore the recognition results
and select co-robot actions with lower activity-independent risk.
\section{Experiments} \label{sec:ExperimentResults}
\subsection{Datasets and Visual Features}
We employ three real-world benchmark datasets to evaluate our cognitive model on HAR tasks,
which are widely used in the machine vision community:
the Weizmann activity dataset \cite{Gorelick_PAMI07}, the KTH activity dataset \cite{Laptev_IJCV05},
and the UTK 3D activity dataset \cite{Zhang_IROS11}.
Illustrative examples from each activity category in the datasets are depicted in Fig. \ref{fig:ActionDataset}.
\begin{figure}[h]
\centering
\subfigure[Weizmann dataset]{
\label{fig:Weizmann}
\begin{minipage}[b]{1\textwidth}
\centering
\includegraphics[width=1\textwidth]{WeizmannDataset.pdf}
\end{minipage}}
\subfigure[KTH dataset]{
\label{fig:KTH}
\begin{minipage}[b]{1\textwidth}
\centering
\includegraphics[width=1\textwidth]{KthDataset.pdf}
\end{minipage}}
\subfigure[UTK3D dataset (3D view)]{
\label{fig:UTK}
\begin{minipage}[b]{1\textwidth}
\centering
\includegraphics[width=1\textwidth]{Utk3dDataset.pdf}
\end{minipage}}
\caption{
Exemplary frames of actions in the datasets used in our experiments.
}
\label{fig:ActionDataset}
\end{figure}
In our experiments,
we apply different types of local visual features to encode these datasets.
For 2D datasets that contain only color videos
(i.e., the Weizmann and KTH datasets),
we use two different features:
scale-invariant feature transform (SIFT) features \cite{Lowe_IJCV04}
and
space-time interest points (STIP) features \cite{Laptev_IJCV05}.
For 3D datasets that contain both color and depth videos (i.e., the UTK dataset),
we use the 4-dimensional local spatio-temporal features (4D-LSTF) \cite{Zhang_IROS11}.
SIFT features are the most commonly applied local visual features
and have desirable characteristics including invariance to transformation,
rotation and scale, and robustness to partial occlusion \cite{Lowe_IJCV04}.
We employ the algorithm and implementation in \cite{Lowe_IJCV04} to detect and describe SIFT features.
A disadvantage of SIFT features in HAR tasks is that these features are extracted in a frame-by-frame fashion,
i.e., SIFT features do not capture any temporal information.
To encode time information,
we also apply STIP along with the histogram of oriented gradients (HOG)
and histogram of optical flow (HOF) descriptors \cite{Laptev_IJCV05}.
These two types of features are extracted using only color or intensity information.
Previous work has demonstrated that local features incorporating both depth and color information can greatly improve recognition accuracy \cite{Zhang_IROS11}.
Therefore, for the 3D UTK dataset we use 4D-LSTF \cite{Zhang_IROS11} features,
which are highly robust and distinct and are generated using both color and depth videos.
It is also noteworthy that SIFT and STIP features can be directly extracted from color or depth videos in the 3D dataset.
These feature extraction algorithms generate a collection of feature vectors for each visual observation.
Then, the feature vectors are clustered into discrete visual words using the $k$-means algorithm,
and the number of clusters is set equal to the dictionary size.
Lastly, each feature vector is indexed by a discrete word that represents cluster assignment.
At this point,
each observation is encoded by a BoW representation, which can be perceived by topic modeling.
Although we only test the most widely used features,
one should note that
our artificial cognitive model is capable of incorporating different types of local visual features,
since our reasoning and decision making process is independent of the features given their BoW representation.
\subsection{Activity Recognition} \label{sec:experiment-II}
We first evaluate the SRAC model's capability to recognize human activities using the interpretability indicator $I_I$, when the training set is exhaustive.
In this experiment, each dataset is split into disjoint training and testing sets.
We randomly select 75\% of data instances in each category as the training set,
and employ the rest of the instances for testing.
During training,
fourfold cross-validation is used to estimate model parameters.
Then, the interpretability of the topic model is computed using the testing set,
which is fully represented by the training set and does not contain novel human activities.
This training-testing process is repeated five times to obtain reliable results.
\begin{figure}[!htb]
\subfigure[Weizmann dataset]{
\label{fig:ii_weizmann}
\begin{minipage}[b]{0.5\textwidth}
\centering
\includegraphics[width=1\textwidth]{weizmann_interpretability.png}
\end{minipage}}
\subfigure[KTH dataset]{
\label{fig:ii_kth}
\begin{minipage}[b]{0.5\textwidth}
\centering
\includegraphics[width=1\textwidth]{kth_interpretability.png}
\end{minipage}}
\subfigure[UTK3D dataset]{
\label{fig:ii_utk3d}
\begin{minipage}[b]{1\textwidth}
\centering
\includegraphics[width=0.5\textwidth]{utk3d_interpretability.png}
\end{minipage}}
\caption{
Variations of model interpretability and its standard deviation versus dictionary size using different visual features over benchmark datasets.
}
\label{fig:ii}
\end{figure}
Experimental results of the interpretability and its standard deviation versus the dictionary size are illustrated in Fig. \ref{fig:ii}.
Our SRAC model obtains promising recognition performance in terms of interpretability:
0.989 is obtained using the STIP feature and a dictionary size 1800 on the Weizmann dataset, 0.952 using the STIP feature and a dictionary size 2000 on the KTH dataset,
and 0.936 using the 4D-LSTF feature and a dictionary size 1600 on the UTK3D dataset.
In general, STIP features perform better than SIFT features for color data,
and 4D-LSTF features perform the best for RGB-D visual data.
The dictionary size in the range [1500, 2000] can generally result in satisfactory human activity recognition performance.
The results are also very consistent,
as illustrated by the small error bars in Fig. \ref{fig:ii},
which demonstrates our interpretability indicator's consistency.
The model's interpretability is also evaluated over different activity categories
using the UTK3D dataset,
which includes more complex activities (i.e., sequential activities) and contains more information (i.e., depth).
It is observed that topic modeling's interpretability varies for different activities.
This performance variation is affected by three main factors:
the topic model's modeling ability, feature and BoW's representability,
and human activity complexity and similarity.
For example, since the LDA topic model and SIFT features are not capable of modeling time,
the reversal human activities including ``lifting a box'' and ``removing a box'' in the UTK3D dataset cannot be well distinguished, as illustrated in Fig. \ref{fig:ii_activity}.
Since sequential activities (e.g., ``removing a box'') are more complex
than repetitive activities (e.g., ``waving''),
they generally result in low interpretability.
Since ``pushing'' and ``walking'' are similar,
which share motions such as moving forward,
they can also reduce interpretability.
This observation provides general guidance for designing future recognition systems with the SRAC model.
\begin{figure*}
\centering
\includegraphics[width= 1 \textwidth]{utk3d_activity_interpretability.png}
\caption{
Model interpretability over the activities in the UTK3D dataset using different features and a dictionary size of 1600.
}\label{fig:ii_activity}
\end{figure*}
\begin{comment}
\begin{figure*}[!t]
\centerline{
\subfloat[Weizmann dataset + STIP features]{
\includegraphics[width=0.33\textwidth]{./figures/weizmann_pvwp.pdf}\label{fig:pvwp_weizmann}
}
\hfil
\subfloat[KTH dataset + STIP features]{
\includegraphics[width=0.33\textwidth]{./figures/kth_pvwp.pdf}\label{fig:pvwp_kth}
}
\hfil
\subfloat[UTK3D dataset + 4D-LSTF features]{
\includegraphics[width=0.33\textwidth]{./figures/utk3d_pvwp.pdf}\label{fig:pvwp_utk3d}
}
}
\caption{
Variations of topic modeling's $Pvwp$ versus dictionary size over validation set, known and unknown testing sets.
} \label{fig:pvwp}
\centerline{
\subfloat[Weizmann dataset]{
\includegraphics[width=0.33\textwidth]{./figures/weizmann_generalization.pdf}\label{fig:ig_weizmann}
}
\hfil
\subfloat[KTH dataset]{
\includegraphics[width=0.33\textwidth]{./figures/kth_generalization.pdf}\label{fig:ig_kth}
}
\hfil
\subfloat[UTK3D dataset]{
\includegraphics[width=0.33\textwidth]{./figures/utk3d_generalization.pdf}\label{fig:ig_utk3d}
}
}
\caption{
Variations of our model's generalizability versus dictionary size over known and unknown testing sets for all datasets.
} \label{fig:ig}
\end{figure*}
\end{comment}
\subsection{Knowledge Discovery} \label{sec:experiment-IG}
We evaluate the SRAC model's capability to discover new situations
using the generalizability indicator $I_G$,
when the training dataset is non-exhaustive (i.e., new human activities occur during testing).
A non-exhausted setup is created by dividing the used benchmark datasets as follows.
We place all data instances of one activity in the \emph{unknown testing set},
and randomly select 25\% of the instances from the remaining activities in the \emph{known testing set}.
The remaining instances are placed in the training set for learning,
based on fourfold cross-validation.
To evaluate the model's ability to discover each individual human activity,
given a dataset that contains $K$ activity categories,
the experiments are repeated $K$ times,
each using one category as the novel activity.
Visual features that achieve the best model interpretability over each dataset
are used in this set of experiments
i.e., STIP features for the Weizmann and KTH datasets
and 4D-LSTF features for the UTK3D dataset.
Variations of $Pvwp$ values versus the dictionary size over the validation set (in cross-validation),
known testing set, and unknown testing set
are shown in Fig. \ref{fig:pvwp}.
Several important phenomena are observed.
First,
there exists a large $Pvwp$ gap between the known and unknown testing sets,
as shown by the gray area in the figure,
indicating that topic models
generalize differently over data instances
from known and unknown activities.
A better generalization result indicates a less novel instance,
which can be better represented by the training set.
Since data instances from the known testing and validation sets are well represented by the training set,
the $Pvwp$ gap between them is small.
As shown in Fig. \ref{fig:pvwp_weizmann},
it is possible that the known testing set's $Pvwp$ value
is greater than the $Pvwp$ value of the validation set,
if its data instances can be better represented by the training set.
Second, Fig. \ref{fig:pvwp}
shows that the gap's width varies over different datasets:
the Weizmann dataset generally has the largest $Pvwp$ gap,
followed by the KTH dataset, and then the UTK3D dataset.
The gap's width mainly depends on the observation's novelty,
in terms of the novel activity's similarity to the activities in the training dataset.
This similarity is encoded by the portion of overlapping features.
A more novel activity is generally represented by a set of more distinct visual features
with less overlapping with the features existing during training,
which generally results in a larger gap.
For example, activities in the Weizamann dataset share fewer motions
and thus contain a less number of overlapping features,
which leads to a larger gap.
Third, when the dictionary size increases,
the model's $Pvwp$ values decrease at a similar rate.
This is because in this case,
the probability of a specific codeword appearing in an instance decreases,
resulting in a decreasing $Pvwp$ value.
\begin{figure*}
\subfigure[Weizmann dataset + STIP features]{
\label{fig:pvwp_weizmann}
\begin{minipage}[b]{0.5\textwidth}
\centering
\includegraphics[width=1.05\textwidth]{weizmann_pvwp.pdf}
\end{minipage}}
\subfigure[KTH dataset + STIP features]{
\label{fig:pvwp_kth}
\begin{minipage}[b]{0.5\textwidth}
\centering
\includegraphics[width=1.05\textwidth]{kth_pvwp.pdf}
\end{minipage}}
\subfigure[UTK3D dataset + 4D-LSTF features]{
\label{fig:pvwp_utk3d}
\begin{minipage}[b]{1\textwidth}
\centering
\includegraphics[width=0.5\textwidth]{utk3d_pvwp.pdf}
\end{minipage}}
\caption{
Variations of topic modeling's $Pvwp$ versus dictionary size over validation set, known and unknown testing sets.
}
\label{fig:pvwp}
\end{figure*}
The generalizability indicator $I_G$'s characteristics
are also empirically validated on the known and unknown testing sets,
as illustrated in Fig. \ref{fig:ig}.
An important characteristic of $I_G$ is its invariance to dictionary size.
Because $Pvwp$ over testing and validation sets has similar decreasing rate,
the division operation in Eq. (\ref{eq:IG}) removes the variance to dictionary size.
In addition, a more novel activity generally leads to a smaller $I_G$ value.
For example, the Weizmann dataset has a smaller $I_G$ value over the unknown testing set,
because its activities are more novel in the sense that they share less overlapping motions.
In general, we observe $I_G$ is smaller than 0.5 for unknown activities
and greater than 0.7 for activities that are included in training sets.
As indicated by the gray area in Fig. \ref{fig:ig},
similar to $Pvwp$,
there exists a large gap between the $I_G$ values over the unknown and known testing datasets.
The average $I_G$ gap across different dictionary sizes is $0.69$ for the Weizmann dataset,
$0.48$ for the KTH dataset, and $0.36$ for the UTK3D dataset.
This reasoning process, based on $I_G$, provides a co-robot with the critical self-reflection capability, and allows a robot to reason about when new situations occur as well as when the learned model becomes less applicable.
\begin{figure*}
\subfigure[Weizmann dataset]{
\label{fig:ig_weizmann}
\begin{minipage}[b]{0.5\textwidth}
\centering
\includegraphics[width=1.05\textwidth]{weizmann_generalization.pdf}
\end{minipage}}
\subfigure[KTH dataset]{
\label{fig:ig_kth}
\begin{minipage}[b]{0.5\textwidth}
\centering
\includegraphics[width=1.05\textwidth]{kth_generalization.pdf}
\end{minipage}}
\subfigure[UTK3D dataset]{
\label{fig:ig_utk3d}
\begin{minipage}[b]{1\textwidth}
\centering
\includegraphics[width=0.505\textwidth]{utk3d_generalization.pdf}
\end{minipage}}
\caption{
Variations of our model's generalizability versus dictionary size over known and unknown testing sets for all datasets.
}\label{fig:ig}
\vspace{-6pt}
\end{figure*}
We have pointed out that the indicator $I_G$ is heavily affected by the novelty of an activity
in terms of its proportion of overlapping features.
To validate this conclusion,
we generate a synthetic dataset
by manually controlling the proportion of overlapping visual words in the testing instances.
In order to make the characteristics of the synthetic dataset as close as possible to real-world datasets,
features used in the simulation are borrowed from the KTH dataset.
Instances of two activities (i.e., ``bending'' and ``waving2'')
are used to train a topic model, which is then applied as a classifier to perform recognition in this experiment.
This topic model is also applied to generate overlapping visual words for a testing instance.
Another topic model, whose parameters are learned using activities ``siding'' and ``jacking'',
is used to generate non-overlapping words in the testing instance.
A dictionary of size 1800 is adopted, which is created using the visual words of the KTH dataset.
The used four activities contain 906 unique visual words,
with each pair of activities sharing less than $1\%$ overlapping words.
We generate $50$ instances for each testing set,
with the number of words in each instance set to $112$,
which is the average number of visual words in real-world instances.
We present the results of five simulations in Fig. \ref{fig:syn},
which clearly shows that, in general, $I_G$'s value over testing instances increases linearly with
the percentage of features that overlap with the features of known activities in the training set.
\begin{figure}
\centering
\includegraphics[width= 0.8 \textwidth]{syn_experiment.pdf}
\caption{
Variations of our model generalizability versus percentage of overlapping features in synthetic data.
}\label{fig:syn}
\end{figure}
\subsection{Relationship of $I_G$ and $I_I$}
Here,
we empirically analyze the relationship
between the interpretability and generalizability indicators.
We first validate the correlation of $I_I$ and $I_G$.
In addition,
we investigate additional relationships of $I_I$ and $I_G$, such as the probability that $I_I \leq I_G$.
While we are able to employ the exhaustive experimental setup from Section \ref{sec:experiment-II} to analyze $I_G$ and $I_I$'s relationship when testing instances are fully represented by the training set,
unfortunately,
we cannot use the non-exhaustive setup in Section \ref{sec:experiment-IG} to validate this relationship in cases where $I_G$ takes small values.
This is because ground truth cannot be assigned to instances belonging to novel activities to compute $I_I$,
since these activities only exist in the testing set
and are not presented to our model during the training phase.
Inspired by the method used to generate synthetic data in Section \ref{sec:experiment-IG},
we adopt a semi-exhaustive experimental setup by replacing certain portions of words in each testing instance with visual words from novel activities.
This experimental setup is used to validate the indicators' relationship when the training set cannot fully represent testing instances.
Each experiment is performed using $F$ folds, where $F$ is the number of activities in a dataset.
In each fold,
we take all instances of one activity out from the dataset,
which is treated as a novel activity that is not presented to the topic model in the learning phase.
Then, we randomly select 75\% of the instances of the remaining activities as training set,
which is further divided into training and validation sets to perform four-fold cross-validation.
The rest of the instances are used as an ``initial'' testing set.
During the testing phase,
the novel activity's word distribution is used to generate new visual words to replace a proportion of the words in each instance in the initial testing set.
This testing is performed six times within each of the $F$ folds
using different replacement rates (i.e., $0.25$, $0.35$, $\dots$, $0.75$).
Testing results from all $F$ folds are used to investigate $I_I$ and $I_G$'s relationship.
In this experimental setup,
we use features that achieve the best interpretability over each dataset.
In addition, we set the dictionary size to 1600,
which achieves the best interpretability over all datasets in general.
This experimental setup is semi-exhaustive in the sense that,
although training data cannot fully represent testing instances due to the replaced features that are generated from unknown activities,
the remaining non-replaced features are presented to the model during the learning phase,
and the ground truth assigned to each testing instance remains the same, which is also known to the model.
It is noteworthy that we do not use very high or very low replacement rates.
A very low replacement rate makes the experimental setup equivalent to the exhaustive setup.
When using a very high replacement rate,
testing instances can be viewed as being drawn from the novel activity; in this case the ground truth associated with a testing instance would be meaningless or incorrect.
\begin{table}[tb]
\caption{
Relationship between $I_I$ and $I_G$
over exhaustive and semi-exhaustive datasets.
}
\label{tab:RelationIiIg}
\begin{center}
\begin{tabular}{|c|rc|cc|}
\hline
\multirow{2}{*}{Dataset + Features } & \multicolumn{2}{c|}{Exhaustive} & \multicolumn{2}{c|}{Semi-exhaustive} \\ \cline{2-3} \cline{4-5}
& $\rho_{I,G}$ & $P_{I_I \!\leq\! I_G}$ & $\rho_{I,G}$ & $P_{I_I \!\leq\! I_G}$ \\
\hline\hline
Weizmann + STIP & $-0.065$ & 0.456 & 0.664 & 0.912 \\
KTH + STIP & $0.036$ & 0.324 & 0.685 & 0.853 \\
UTK3D + 4D-LSTF & $0.097$ & 0.275 & 0.714 & 0.896 \\
\hline
\end{tabular}
\end{center}
\end{table}
We empirically analyze the correlation between $I_I$ and $I_G$,
using both exhaustive and semi-exhaustive datasets,
in order to determine whether better generalizability indicates better interpretability.
The Pearson correlation coefficient is used to measure the strength and direction of the linear relationship between these two indicators.
Given a dataset $\mathcal{W} = \{\boldsymbol{w}_1,\dots,\boldsymbol{w}_{\mathcal{|W|}}\}$ and its ground truth $\boldsymbol{g} = \{g_1,\dots,g_{\mathcal{|W|}}\}$,
this correlation is mathematically defined as follows:
\begin{eqnarray}
\rho_{I,G}
= {E[(\boldsymbol{I}_I
- \mu_{I_I})(\boldsymbol{I}_G -\mu_{I_G})] \over \sigma_{I_I}\sigma_{I_G}},
\end{eqnarray}
where $\boldsymbol{I}_G \!=\! \{I_G(\boldsymbol{w}_1), \dots, I_G(\boldsymbol{w}_{\mathcal{|W|}})\}$
and $\boldsymbol{I}_I \!=\! \{I_I(\boldsymbol{w}_1, g_1),$
$\dots, I_I(\boldsymbol{w}_{\mathcal{|W|}}, g_{\mathcal{|W|}})\}$
are vectors of interpretability and generalizability indicators for all of the instances in the dataset,
$\mu$ is the mean and $\sigma$ is the standard deviation of the indicators in the vector.
Our experimental results are listed in Table \ref{tab:RelationIiIg}.
For the exhaustive dataset, topic models which perform better on generalizability are not necessarily better interpreted,
which is indicated by the weak linear correlation between the indicators.
This is because, when testing on exhaustive datasets,
$I_G$ takes values closer to $1$. But the model's interpretability takes a wide range of values,
depending on the model's modeling capacity, feature representability and dataset complexity,
as explained in Section \ref{sec:experiment-II}.
For semi-exhaustive datasets,
$I_I$ and $I_G$ are moderately to strongly correlated,
which indicates that a poor generalizability usually leads to a poor interpretability.
Since $I_G$ reflects the novelty of an instance as discussed in Section \ref{sec:experiment-IG},
a low $I_G$'s value means the instance is badly represented by the training set.
Therefore, the trained model cannot obtain a good interpretability over the instance of an activity
that is not well represented during the training phase.
We also check an additional relationship, i.e., the probability that $I_I$ is smaller than or equal to $I_G$.
Given a labeled dataset $\mathcal{W} = \{\boldsymbol{w}_1,\dots,\boldsymbol{w}_M\}$
and its ground truth $\boldsymbol{g} = \{g_1,\dots,g_M\}$,
this probability is defined as follows:
\begin{eqnarray}
P_{I_I \leq I_G} = \frac{1}{M} \sum_{m = 1}^{M}
\mathds{1}(I_I(\boldsymbol{w}_m,g_m) \leq I_G(\boldsymbol{w}_m)).
\end{eqnarray}
The experimental results are presented in Table \ref{tab:RelationIiIg}.
One of the most important observations is that,
for a majority of testing instances (more than $85\%$) in the semi-exhaustive experiment,
$I_G$'s value is greater than $I_I$'s value.
This again shows that
a poor generalizability usually indicates a poor interpretability.
Using the exhaustive experimental setup,
it is more probable that $I_G$ takes smaller values than $I_I$.
This is because when the training set is exhaustive,
the topic model is well trained and can well recognize testing instances,
which leads to $I_I \!\rightarrow\! 1$ for most of testing instances.
On the other hand,
although $I_G$ also takes a large value in general,
it is usually slightly smaller than one,
because features in testing instances usually do not completely overlap with features in training instances.
\subsection{Decision Making}
We assess our SRAC model's decision making capability
using a Turtlebot 2 robot in a human following task,
which is important in many human-robot teaming applications.
In this task,
a robotic follower needs to
decide at what distance to follow the human teammate.
We are interested in three human behaviors:
``walking'' in a straight line, ``turning'', and ``falling'', shown in Fig. \ref{fig:DecisionEg}.
With perfect perception and reasoning,
i.e., a robot always perfectly interprets human activities,
we assume the ideal robot actions are to ``stay far from the human''
when he or she is walking in a straight line (to not interrupt the human),
``move close to the human'' when the subject is turning (to avoid losing the target),
and ``stop beside the human'' when he or she is falling (to provide assistance).
\begin{figure}
\subfigure[Falling]{
\label{fig:fall}
\begin{minipage}[b]{0.32\textwidth}
\centering
\includegraphics[width=1\textwidth]{fall.png}
\end{minipage}}
\subfigure[Turning]{
\label{fig:turn}
\begin{minipage}[b]{0.32\textwidth}
\centering
\includegraphics[width=1\textwidth]{turn.png}
\end{minipage}}
\subfigure[Walking]{
\label{fig:walk}
\begin{minipage}[b]{0.32\textwidth}
\centering
\includegraphics[width=1\textwidth]{walk.png}
\end{minipage}}
\caption{
Experiment setup for validating the SRAC model's decision making ability
in a human following task using a Turtlebot 2 robot.
}
\label{fig:DecisionEg}
\end{figure}
\begin{table}[t]
\centering
\caption{The risk matrix used in the robot following task.}
\label{tab:RiskMat}
\tabcolsep=0.35cm
\begin{tabular}{|c||c|c|c|}
\hline
Robot Actions & Falling & Turning & Walking \\
\hline\hline
Stay besides humans & 0 & 20 & 50 \\
\hline
Move close & 90 & 0 & 20 \\
\hline
Stay far away & 95 & 80 & 0 \\
\hline
\end{tabular}
\end{table}
In order to qualitatively assess the performance,
we collect 20 color-depth instances from each human behaviors to train the SRAC model,
using a BoW representation based on 4D-LSTF features.
The risk matrix used in this task is presented in Table \ref{tab:RiskMat}.
We evaluate our model in two circumstances.
\texttt{Case$\,$1}: exhaustive training (i.e., no unseen human behaviors occur in testing).
In this case, the subjects only perform the three activities during testing with small variations in motion speed and style.
\texttt{Case$\,$2}: non-exhaustive training (i.e., novel movements occur during testing).
In this case, the subjects not only perform the activities with large variations,
but also add additional movements (such as jumping and squatting)
which are not observed in the training phase.
During testing,
each activity is performed 40 times.
The model performance is measured using failure rate,
i.e., the percentage with which the robot fails to stop besides to help the human
or loses the target.
Experimental results
are presented in Table \ref{tab:ExpResults},
where the traditional methodology,
which selects the co-robot actions only based on the most probable human activity, is used as a baseline for comparison.
We observe that
the proposed SRAC model significantly decreases the failure rate
in both exhaustive and non-exhaustive setups.
When the training set is exhaustive and no new activities occur during testing (\texttt{Case$\,$1}),
the results demonstrate that
incorporating human activity distributions and robot action risks improves decision making performance.
When the training set is non-exhaustive and new activities occur during testing (\texttt{Case$\,$2}),
the SRAC model significantly outperforms the baseline model.
In this situation, if $I_G$ has a very small value,
according to Eq. \ref{eq:ActionSelection},
our model tends to select safer robot actions,
i.e., ``stay beside humans,'' since its average risk is the lowest,
which is similar to the human common practice ``playing it safe in uncertain times.''
The results show the importance of self-reflection for decision making especially under uncertainty.
\begin{table}[htbp]
\centering
\caption{Failure rate (\%) in exhaustive (\texttt{Case$\,$1}) and non-exhaustive (\texttt{Case$\,$2}) experimental settings.}
\label{tab:ExpResults}
\tabcolsep=0.35cm
\begin{tabular}{|c|c||c|c|}
\hline
Exp. settings & Models & Fail to assist & Fail to follow \\
\hline\hline
Exhaustive & Baseline & 10.5\% & 15\% \\
\cline{2-4}
(\texttt{Case$\,$1}) & \textbf{SRAC} &0.5\% & 5.5\% \\
\hline
Non-exhaustive & Baseline & 45.5\% & 60\% \\
\cline{2-4}
(\texttt{Case$\,$2}) & \textbf{SRAC} & 24.5\% & 35.5\% \\
\hline
\end{tabular}
\end{table}
\begin{comment}
To intuitively explain the benefit from our model, two case studies are provided as follows.
\emph{Distribution over activity categories is important for decision making}:
Case study in \texttt{Case$\,$1}:
When training is exhaustive, $I_G$ is close to 1.
Assume $\boldsymbol{\theta} = \{0.2, 0.42, 0.38\}$ and $I_G \!=\! 1$,
Let's consider the situation that the final decision is made only based on a single activity without considering the activity distribution.
Since the human activity ``moving around'' has the highest probability,
the robot action ``push wheelchair'' will be selected,
because it is the most appropriate action with no risk for this activity.
However, this decision is not optimal.
It is quite possible that the correct activity is ``cooking'',
since it also has a high probability that is similar to the activity ``moving around''.
In this case, ``push wheelchair'' will cause a high risk of value 80.
On the other hand, when considering the distribution over all possible activities,
the decision making process will select the action ``do housework'' according to Eq. (\ref{eq:ActionSelection}) with an average risk of 29.5,
which is much smaller than the average risk of 49.4 for the robot action ``push wheelchair''.
\emph{Model evaluation is important for decision making}:
Let's assume $\boldsymbol{\theta} = \{0.1, 0.8, 0.1\}$,
that is, the human activity ``moving around'' dominates the distribution.
In the case of $I_G \!=\! 0.9$,
``push wheelchair'' has the lowest average risk of 21.58 among all possible robot actions.
Since the model well generalizes over the observation,
the decision making system is confident about the distribution.
Thus, the best robot action ``push wheelchair'' can be safely selected,
even if it has high risks for the other human activities.
On the other hand, if $I_G \!=\! 0.1$, i.e., the model generates badly on the observation,
the average risk of ``push wheelchair'' increases significantly to 54.25.
Although this robot action has no risk to the most probable activity,
it is still not selected by our decision making module,
because ``stand by \& observe'' leads to the lowest average risk of 36.7,
which also has the lowest activity-independent risk (38.33) among all possible actions.
\end{comment}
\section{Conclusion}\label{sec:Conclusion}
In this paper, we construct an artificial cognitive model that provides co-robots with both
accurate perception and new information discovery capabilities, which
enables safe, reliable robot decision making for HAR tasks in human-robot interaction applications.
The proposed SRAC model exploits topic modeling,
which is unsupervised and allows for the discovery of new knowledge without presence in training.
In addition, topic modeling is also able to treat activity estimation as a distribution and incorporate risks for each action response,
which is beneficial for the system's ability to make decisions.
In order to provide the capability of accurate human activity interpretation,
we define a new interpretability indicator ($I_I$) and demonstrate
its ability to enable a robot to interpret category distribution in a similar fashion
to humans.
The indicator $I_I$ is applied to map detected clusters to known activity categories and select the best interpreted model.
In addition, to provide the ability of knowledge discovery,
we introduce a novel generalizability indicator ($I_G$). %
It measures how well an observation can be represented by the learned knowledge, which
allows for self-reflection that can enable the SRAC model to identify new scenarios.
We applied the proposed SRAC model in extensive experiments to demonstrate the effectiveness
using both synthetic and real-world datasets.
We show that our model performs extremely well in terms of interpretability;
that is, our model's recognition results closely and consistently match human common sense.
We demonstrate that, using $I_G$, our cognitive model is capable of discovering new knowledge,
i.e., observations from new activity categories that are not considered in the training phase can be automatically detected.
We also examine the relationship between $I_I$ and $I_G$ and show,
both analytically and experimentally,
that $I_G$ can also be used as an indicator for $I_I$.
The results reveal that scenarios with a low $I_G$ score for an observation will equate to a low $I_I$ score with high confidence,
i.e., a badly generalized model is likely to be inaccurate.
We further demonstrate the advantages of using distributions over activity categories,
as well as the importance of the evaluation metrics
in order to create a system capable of safe, reliable decision making.
|
1,116,691,497,588 | arxiv | \section{Introduction}\label{sec:s1}
Since the introduction in Kingsbury \cite{king1,king2} of complex wavelet\ transform s implemented by the dual-tree scheme, the complex wavelet s (DT\_CW), wavelet\ frames and wavelet packet s (WPs) have become a field of active research that appears in multiple applications (\cite{jalob1,jalob2,jalob3,xie_wang_zhao_chen,barakin,bay_sele,bhan_zhao,bhan_zhao_zhu}, to name a few). The advantages of the DT\_CWs over the standard real wavelet\ transform s stem from their approximate shift invariance and some directionality inherent to tensor-products of the DT\_CWs.
However, the direction ality of the DT\_CWs is very limited (only 6 directions) and this is a drawback for image processing\ applications. The tight tensor-product complex wavelet\ frames TP\_CTF$_{n}$ with differen t number of directions, are designed in \cite{bhan_zhao,bhan_zhao_zhu} and some of them, in particular TP\_CTF$_{6}$ and TP\_CTF$^{\downarrow}_{n}$, demonstrate excellent performance (in terms of PSNR) for image denoising and inpainting. The number of directions in both 2D TP\_CTF$_{6}$ and TP\_CTF$^{\downarrow}_{n}$ frames is 14 and remains the same for all decomposition\ levels.
Some of the disadvantages of the above 2D TP\_CTF$_{6}$ and TP\_CTF$^{\downarrow}_{n}$ frames are mentioned in
\cite{che_zhuang}. For example, ``limited and fixed number of directions is undesirable in practice especially
when the resolution of an image is very high that requires large number of directional filters in order to capture as many
features with different orientations as possible" (\cite{che_zhuang}).
In addition, ``due to the fixed number
of 1D filters, the number of free parameters
is limited which prevents the search of optimal filter bank
systems for image processing" (\cite{che_zhuang}).
According to \cite{che_zhuang}, the remedy for the above shortcomings is in the incorporation of the two-layer structure that is inherent in the TP\_CTF$_{6}$ and TP\_CTF$^{\downarrow}_{n}$ frames into directional filter bank s introduced in \cite{bhan_zhuang,zhuang}.
The complex wavelet packet s (Co\_WPs) is an alternative way to overcome the above disadvantages. The first version of complex WPs appears in \cite{jalob1} after the introduction of the complex wavelet s by Kingsbury. The complex wavelet\ transform s in \cite{jalob1,jalob2,jalob3} are extended to the Co\_WP transform s by the application of the same filter s as used in the DT\_CW transform s
to the high-frequency\ bands. Although the low- and high-frequency\ bands in DT\_CW are approximat ely analytic, this is not the case for the Co\_WPs designed in \cite{jalob1,jalob2,jalob3}.
In addition, as shown in \cite{bay_sele} (Fig. 1), much energy passes into the negative half-bands of the spectr a. Another approach to the design of Co\_WPs is described in \cite{bay_sele}. It is suggested in \cite{bay_sele} that in order to retain an approximat e analytic ity of the dual-tree WP transform s, the filter bank s for the second decomposition of the transform s should be the same for both stems of the tree.
Although the potential advantages of the Co\_WP transform s are apparent, so far, to the best of our knowledge, none of the Co\_WP schemes in the literature have the desired properties such as perfect frequency\ separation,
Hilbert transform\ relation between real and imaginary parts of the Co\_WPs,
orthonormality of shifts of real and imaginary parts of the Co\_WPs,
unlimited number of direction s in the multidimensional case,
a variety of free parameters,
and fast and easy implementation.
Our motivation in this paper is to fill this gap. We design a family of Co\_WP transform s which possess all the above properties. As a base for the design, we use the family of discrete-time\ WPs originated from periodic\ discrete spline s of different orders that are described in \cite{ANZ_book3} (Chapter 4). The wavelet packet s $\psi_{[m],l}^{2r}$, where $m$ is the decomposition\ level, $l=0,...,2^{m}-1$ is the index of the related frequency\ band and $2r$ is the order of the generating discrete spline, are symmetr ic, well localized in time domain (although are not compactly supported), their DFTs spectr a are flat, and provide a refined split of the frequency\ domain. The WP transform s are executed in the frequency\ domain using the Fast Fourier\ transform\ (FFT). By varying the order $2r$, we can supply the WPs $\psi_{[m],l}^{2r}$ with any number of local vanishing moment s without increase of the computational cost. Different combinations of the shifts in these WPs provide a variety of orthonormal\ bases of the space of $N$-periodic\ signal s.
To derive the Co\_WPs, we expand the WPs $\psi_{[m],l}^{2r}$ to periodic\ analytic\ discrete-time\ signal s $\bar{\psi}_{\pm[m],l}^{2r}=\psi_{[m],l}^{2r}\pm i\,\th_{[m],l}^{2r}$, where $\th_{[m],l}^{2r}$ is the discrete\ periodic\ Hilbert transform\ (HT) of the WP $\psi_{[m],l}^{2r}$. The waveform s $\th_{[m],l}^{2r}$ are antisymmetr ic, and for all $l\neq0, \,2^{m}-1,$ orthonormal properties similar to the properties of the WPs $\psi_{[m],l}^{2r}$ take place. To achieve orthonormal ity, the waveform s $\th_{[m],l}^{2r}, \;l=0, \,2^{m}-1$ are slightly corrected at the expense of minor deviation from antisymmetr y and we get a new orthonormal\ complimentary WP (cWP) system $\left\{\f_{[m],l}^{2r}\right\}, \;m=1,...,M,\;l=0, ...,2^{m}-1$, where for $l\neq0, \,2^{m}-1$, the WPs satisfy $\f_{[m],l}^{2r}=\th_{[m],l}^{2r}$. The magnitude spectr a of the cWPs $\f_{[m],l}^{2r}$ coincide with the spectr a of the respective WPs $\psi_{[m],l}^{2r}$ and, similarly to the WPs $\psi_{[m],l}^{2r}$, the cWPs $\f_{[m],l}^{2r}$ provide a variety of orthonormal\ bases for the space of $N$-periodic\ signal s.
Correspondingly, we define the quasi-analytic\ WP systems (qWP) as $$ \Psi^{2r}_{\pm[m],l}=\psi^{2r}_{[m],l} \pm i\f^{2r}_{[m],l}, \quad m=1,...,M,\;l=0,...,2^{m}-1,$$ where all the WPs with indices other than $l=0,2^{m}-1$ are analytic.
For the implementation of the transform s with the complex qWPs we do not use the dual-tree scheme with differen t filter bank s for real and imaginary wavelet s but use the scheme with a single complex filter bank\ in the first step of the transform, and a real filter bank\ in the additional steps.
A dual-tree structure type appears in the 2D case when two sets of qWPs are defined as the tensor products of 1D qWPs
\begin{equation}\label{psipsi}
\Psi_{++[m],j,l}^{2r}[k,n] \stackrel{\Delta}{=} \Psi_{+[m],\k}^{2r}[k]\,\Psi_{+[m],l}^{2r}[n], \quad
\Psi_{+-[m],j,l}^{2r}[k,n] \stackrel{\Delta}{=} \Psi_{+[m],\k}^{2r}[k]\,\Psi_{-[m],l}^{2r}[n]
\end{equation}
and processing\ with the qWPs $\Psi_{+\pm[m],j,l}^{2r}$ is executed separately.
The real and imaginary parts of the qWPs $\Psi_{+\pm[m],j,l}^{2r}$ are the 2D waveform s oriented in multiple direction s, specifically the $2(2^{m+1}-1)$ directions at the $m$-th decomposition\ level. Such an abundant direction ality proved to be efficient in the examples on image denoising and inpainting. It is worth mentioning that the WPs of one- and two-dimensions have a localized oscillating structure, which is useful for detection of transient oscillating events in 1D signal s and oscillating patterns in the images (for example, ``Barbara" in Fig. \ref{barb30}).
Both one- and two-dimensional transform s are implemented in a very fast ways by using FFT.
The paper is organized as follows:
Section \ref{sec:s2} outlines briefly periodic\ discrete-time\ WPs originated from discrete spline s and corresponding transform s that form a basis for the design of Co\_WPs. The analysis\ $\tilde{\mathbf{F}}$ and synthesis\ $\mathbf{F}$ filter bank s for the WP transform s are described.
Section \ref{sec:s3} outlines the construction of discrete-time\ periodic\ analytic\ signal s. This section also introduces complimentary sets of WPs (cWPs), analytic\ and quasi-analytic\ WPs (qWPs).
Section \ref{sec:s4} describes the implementation of the cWP and qWP transform s. The filter bank s for one step of analysis\ and synthesis\ transform s are introduced. It is interesting to note that subsequent application of the direct and inverse qWP filter bank s to a signal\ $\mathbf{x}$ produces the analytic\ signal\ $\bar{\mathbf{x}}=\mathbf{x}+i\,H(\mathbf{x})$. All the subsequent steps of cWP and qWP transform s are implemented with the same filter bank s $\tilde{\mathbf{F}}$ and $\mathbf{F}$ as used in the above WP transform s (section \ref{sec:s2}).
Section \ref{sec:s5} extends the design of 1D qWPs to the 2D case. The 2D qWPs are defined via tensor products as shown in Eq. \rf{psipsi}. Directionality of the 2D qWPs is discussed.
Section \ref{sec:s6} describes the implementation of the 2D qWP transform s by a dual-tree.
Section \ref{sec:s7} presents a few experimental results for images restoration degraded by either strong additive noise or by missing many of the pixels. In one example, both the degradation sources are present.
Section \ref{sec:s8} discusses the results.
The Appendix contains proof of a proposition.
\paragraph{Notations and abbreviations}
$N=2^{j}$, $M=2^{m},\,m<j$, $N_{m}=2^{j-m}$ and $\Pi[N]$ is a space of real-valued $N$-periodic\ signal s.
$\Pi[N,N]$ is the space of two-dimensional $N$-periodic\ arrays in both vertical and horizontal directions.
$\w\stackrel{\Delta}{=} e^{2\pi\,i/N}$.
A signal\ $\mathbf{x}=\left\{ x[k]\right\}\in\Pi[N]$ is represent ed by its inverse discrete\ Fourier\ transform\ (DFT)
\begin{equation}\label{fsx}
\begin{array}{lll
x[k]&=&\frac{1}{N}\sum_{n=0}^{N-1}\hat{x}[n]\,\w^{kn}=\frac{1}{N}\sum_{n=-N/2}^{N/2-1}\hat{x}[n]\,\w^{kn},\\%\nonumber
\hat{x}[n] &=&\sum_{k=0}^{N-1}{x}[k]\,\w^{-kn},\quad \hat{x}[-n]=\hat{x}[N-n]= \hat{x}[n]^{\ast},
\end{array}
\end{equation}
where $\cdot^{\ast}$ means complex conjugate. In particular,
\(
\hat{x}[0]=\sum_{k=0}^{N-1}{x}[k]\) and \( \hat{x}[N/2]=\sum_{k=0}^{N-1}(-1)^{k}\,{x}[k]\) are real numbers.
The DFT of an $N_{m}$-periodic\ signal\ is $\hat{x}[n]_{m} =\sum_{k=0}^{N_{m}-1}{x}[k]\,\w^{-kn2^{m}}$.
The abbreviation PR means perfect reconstruction.
The abbreviations 1D and 2D mean one-dimensional and two-dimensional, respectively. FFT is the fast Fourier\ transform, HT is the Hilbert transform, $H(\mathbf{x})$ is the discrete\ periodic\ HT of a signal\ $\mathbf{x}$.
The abbreviations WP, cWP and qWP mean wavelet packet s (typically spline-based wavelet packet s $\psi^{2r}_{[m],l}$), complimentary wavelet packet s $\f^{2r}_{[m],l}$ and quasi-analytic\ wavelet packet s $\Psi^{2r}_{\pm[m],l}$, respectively, in 1D case, and wavelet packet s $\psi^{2r}_{[m],j,l}$, complimentary wavelet packet s $\f^{2r}_{[m],j,l}$ and quasi-analytic\ wavelet packet s $\Psi^{2r}_{+\pm[m],l,j}$, respectively, in 2D case.
\begin{equation}\label{u2r}
U^{4r}[n]\stackrel{\Delta}{=} \frac{1}{2}\left(\cos^{4r}\frac{\pi\,n}{N} +\sin^{4r}\frac{\pi\,n}{N}\right).
\end{equation}
Peak Signal-to-Noise ratio (PSNR) in decibels (dB) is
\[
PSNR\srr10\log_{10}\left(\frac{N\,255^2}{\sum_{k=1}^N(x_{k}-\tilde
x_{k})^2}\right)\; dB.
\]
SBI stands for split Bregman iterations and p-filter\ means periodic\ filter.
\section{Outline of orthonormal WPs originated from discrete spline s: preliminaries}\label{sec:s2}
This section provides a brief outline of periodic\ discrete-time\ wavelet packet s originated from discrete spline s and corresponding transform s. For details see Chapter 4 in \cite{ANZ_book3}.
\subsection{ Periodic discrete spline s }\label{sec:ss21}
The centered span-two $N$-periodic\ discrete\ B-spline\ of order $2r$ is defined as the IDFT of the sequence\
\begin{eqnarray*}\label{bs_2}
\hat{b}^{2r}[n]&=&\cos^{2r}\frac{\pi\,n}{N},\quad {b}^{2r}[k]=\frac{1}{N}\sum_{n=-N/2}^{N/2-1}\w^{kn}\, \cos^{2r}\frac{\pi\,n}{N}.
\end{eqnarray*}
The B-spline s are non-negative symmetr ic finite-length signal s (up to periodization). Only the samples ${b}^{2r}[k], \;k=-r,...r,$ are non-zero.
The signal s $
{s}^{2r}[k]\stackrel{\Delta}{=} \sum_{l=0}^{N/2-1}q[l]\,{b}^{2r}[k-2l],
$
which are referred to as discrete spline s, form an $N/2$-dimensional subspace $^{2r}{\mathcal{S}}_{[1]}^{0}$ of the space $\Pi[N]$ whose basis consists of two-sample shifts of the B-spline\ $\mathbf{b}^{2r}$.
Here $\mathbf{q}=\left\{ q[l]\right\}, \;l=0, ... ,N/2-1,$ is a real-valued sequence. The DFT of the discrete spline\ $\mathbf{{s}}^{2r}$ is
\begin{eqnarray*}
\label{adsf1}
\hat{ {s}}^{2r}[n] =\hat{q}[n]_{1}\, \hat{ {b}}^{2r}[n] =\hat{q}[n]_{1}\,\cos^{2r}\frac{\pi\,n}{N}.
\end{eqnarray*}
A discrete spline\ $\psi_{[1],0}^{2r}\in{} ^{2r}{\mathcal{S}}_{[1]}^{0}$ is defined through its inverse DFT (iDFT):
\begin{equation*}\label{psi0}
\psi_{[1],0}^{2r}[k]\stackrel{\Delta}{=}\frac{1}{N}\sum_{n=-N/2}^{N/2-1}\w^{kn}\,\frac{\cos^{2r}\frac{\pi\,n}{N}}{\sqrt{ U^{4r}[n]}},
\end{equation*} where $ U^{4r}[n]$ is defined in Eq. \rf{u2r}.
\begin{proposition} [\cite{ANZ_book3}, Proposition 3.6]\label{pro:psi0}
Two-sample shifts $\left\{ \psi_{[1],0}^{2r}[\cdot-2l]\}\right\}, \;l=0, ... ,N/2-1,$ of the discrete spline s $ \psi_{[1],0}^{2r}$ form an orthonormal\ basis of the subspace $^{2r}{\mathcal{S}}_{[1]}^{0}\subset\Pi[N]$.
The orthogonal\ projection of a signal\ $\mathbf{x}\in \Pi[N]$ onto the space ${} ^{2r}{\mathcal{S}}_{[1]}^{0}$ is the signal\ $\mathbf{x}_{[1]}^{0}\in\Pi[N]$ such that
\begin{eqnarray}\nonumber
x_{[1]}^{0}[k]&=&\sum_{l=0}^{N/2-1}y_{[1]}^{0}[l]\, \psi_{[1],0}^{2r}[k-2l] ,\quad
y_{[1]}^{0}[l] =\left\langle \mathbf{x},\, \psi_{[1],0}^{2r}[\cdot-2l] \right\rangle
= \sum_{k=0}^{N-1}h_{[1]}^{0}[k-2l] \,x[k], \\\label{y0_del} h_{[1]}^{0}[k] &=&\psi_{[1],0}^{2r}[k],\quad
\hat{{h}}_{[1]}^{0}[n] = \hat{ \psi}_{[1],0}^{2r}[n]=\frac{\cos^{2r}\frac{\pi\,n}{N}}{\sqrt{U^{4r}[n]}}.
\end{eqnarray}
\end{proposition}
\subsection{ Orthogonal complement for subspace $ ^{2r}{\mathcal{S}}_{[1]}^{0}$ }\label{sec:ss22}
The orthogonal\ complement for ${} ^{2r}{\mathcal{S}}_{[1]}^{0}$ in the signal\ space $\Pi[N]$ is denoted by
${} ^{2r}{\mathcal{S}}_{[1]}^{1}$. Thus, $\Pi[N]={\mathcal{S}}_{[0]}={} ^{2r}{\mathcal{S}}_{[1]}^{0}\bigoplus^{2r}{\mathcal{S}}_{[1]}^{1}$. The orthonormal\ basis in
the subspace is formed by the two-sample shifts $\left\{ \psi_{[1],1}^{2r}[\cdot-2l]\}\right\}, \;l=0, ... ,N/2-1,$ of the discrete spline\ $ \psi_{[1],1}^{2r}$, which is defined through its inverse DFT (iDFT):
\begin{equation*}\label{psi1}
\psi_{[1],1}^{2r}[k]\stackrel{\Delta}{=}\frac{1}{N}\sum_{n=-N/2}^{N/2-1}\w^{kn}\,\frac{\w^{n}\,\sin^{2r}\frac{\pi\,n}{N}}{\sqrt{ U^{4r}[n]}}.
\end{equation*}
\begin{proposition} [\cite{ANZ_book3}, Proposition 4.1]\label{pro:psi1}
The orthogonal\ projection of a signal\ $\mathbf{x}\in \Pi[N]$ onto the space ${} ^{2r}{\mathcal{S}}_{[1]}^{1}$ is the signal\ $\mathbf{x}_{[1]}^{1}\in\Pi[N]$ such that
\begin{eqnarray}\nonumber
x_{[1]}^{1}[k]&=&\sum_{l=0}^{N/2-1}y_{[1]}^{1}[l]\, \psi_{[1],1}^{2r}[k-2l] ,\quad
y_{[1]}^{1}[l] =\left\langle \mathbf{x},\, \psi_{[1],1}^{2r}[\cdot-2l] \right\rangle
= \sum_{k=0}^{N-1}h_{[1]}^{1}[k-2l] \,x[k], \\\label{y1_del} h_{[1]}^{1}[k] &=&\psi_{[1],1}^{2r}[k],\quad
\hat{{h}}_{[1]}^{1}[n] = \hat{ \psi}_{[1],1}^{2r}[n]=\frac{\w^{n}\,\sin^{2r}\frac{\pi\,n}{N}}{\sqrt{ U^{4r}[n]}}.
\end{eqnarray}
\end{proposition}
The signal s $\psi_{[1],0}^{2r}$ and $\psi_{[1],1}^{2r}$ are referred to as the discrete-spline\ wavelet packet s of order $2r$ from the first decomposition\ level . They are the impulse response s of the low- and high-pass periodic\ filter s (p-filter s) $\mathbf{h}_{[1]}^{0}$ and $\mathbf{h}_{[1]}^{1}$, respectively.
\begin{rmk}\label{0_N2rem}We emphasise that the DFTs $\hat{\psi}_{[1],0}^{2r}[N/2]=0$ and $\hat{\psi}_{[1],1}^{2r}[0]=0$.\end{rmk}
Figure \ref{ds_wav1LA} displays the discrete-spline\ wavelet packet s $ \psi_{[1],0}^{2r}$ and $ \psi_{[1],1}^{2r}$ of different orders
and magnitudes of their DFT spectr a (which are the p-filter s $\mathbf{h}_{[1],0}$ and $\mathbf{h}_{[1],1}$ magnitude response s).
It is seen that the wavelet s are well localized in time domain. The spectr a are flat and their shapes tend to rectangular as their orders increase.
\begin{figure}[H]
\begin{center}
\resizebox{12cm}{5cm}{
\includegraphics{png/ds_wav1L_1A.png}
}
\end{center}
\caption{ Left: wavelet packet s $ \psi_{[1],0}^{2r}$ (red lines) and $ \psi_{[1],1}^{2r}$ (blue lines), $r=1,3,5$.
Right: magnitude spectr a of $ \psi_{[1],0}^{2r}$ (red lines) and $ \psi_{[1],1}^{2r}$ (blue lines)}
\label{ds_wav1LA}
\end{figure}
\subsection{One-level wavelet packet\ transform\ of a signal}\label{sec:ss23}
The transform\ of a signal\ $\mathbf{x}\in\Pi[N]$ into the pair $\left\{\mathbf{ y}_{[1]}^{0},\mathbf{ y}_{[1]}^{1}\right\}$ of signal s from $\Pi[N/2]$ is
referred to as the one-level wavelet packet\ transform\ (WPT) of the signal\ $\mathbf{x}$. According to Propositions \ref{pro:psi0} and \ref{pro:psi1},
the transform\ is implemented by filter ing $\mathbf{x}$ with time-reversed half-band low- and high-pass p-filter s ${\mathbf{h}}_{[1]}^{0}$
and ${\mathbf{h}}_{[1]}^{1}$, respectively, which is followed by downsampling. Thus the p-filter s ${\mathbf{h}}_{[1]}^{0}$
and ${\mathbf{h}}_{[1]}^{1}$ form a critically sampled analysis\ p-filter bank\ $\tilde{\mathbf{H}}_{[1]}^{1}$. Eqs. \rf{y0_del} and \rf{y1_del} imply that its
modulation matrix is
\begin{equation}\label{aa_modma10
\begin{array}{lll}
\tilde{\mathbf{M}}[n]&=& \left(
\begin{array}{cc}
\hat{h}_{[1]}^{0}[n] & \hat{h}_{[1]}^{0}\left[n+\frac{N}{2}\right]\\
\ \hat{h}^{1}_{[1]}[n] & \hat{h}^{1}_{[1]}\left[n+\frac{N}{2}\right]\\
\end{array}
\right)=\left(
\begin{array}{cc}
\beta[n] & \beta\left[n+\frac{N}{2}\right] \\
\alpha[n] &\alpha\left[n+\frac{N}{2}\right] \\
\end{array}
\right)=\left(
\begin{array}{cc}
\beta[n] & \w^{-n}\alpha[n] \\
\alpha[n] &-\w^{n}\beta[n] \\
\end{array}
\right),\\
\beta[n] &=& \frac{\cos^{2r}\frac{\pi\,n}{N}}{\sqrt{U^{4r}[n]}}, \quad
\alpha[n]=\w^{n}\,\beta\left[n+\frac{N}{2}\right]=\w^{n}\, \frac{\sin^{2r}\frac{\pi\,n}{N}}{\sqrt{U^{4r}[n]}}.
\end{array}
\end{equation}
The analysis\ modulation matri x $\tilde{\mathbf{M}}[n]/\sqrt{2}$, as well as the matrix $\tilde{\mathbf{M}}[-n]/\sqrt{2}$ are unitary matrices. Therefore, the synthesis\ modulation matri x
is
\begin{eqnarray}\label{sa_modma10}
\mathbf{M}[n]=\left(
\begin{array}{cc}
\beta[n] &\alpha[n] \\
\w^{-n}\alpha[n] &-\w^{n}\beta[n] \\
\end{array}
\right)= \tilde{\mathbf{M}}[n]^{T}.
\end{eqnarray}
Consequently, the synthesis\ p-filter bank\ coincides with the analysis\ p-filter bank\ and, together, they form a perfect reconstruction (PR) p-filter bank .
The one-level WP transform\ of a signal\ $\mathbf{x}$ and its inverse are represent ed in a matrix form:
\begin{equation}\label{mod_repAS1}
\left(
\begin{array}{c}
\hat{y}_{[1]}^{0}[n]_{1} \\
\hat{y}_{[1]}^{1}[n]_{1}\\
\end{array}
\right)=\frac{1}{2}\tilde{\mathbf{M}}[-n]\cdot \left(
\begin{array}{l}
\hat{x}[n] \\
\hat{x}[\vec{n}]
\end{array}
\right),\quad \left(
\begin{array}{l}
\hat{x}[n] \\
\hat{x}[\vec{n}]
\end{array}
\right)={\mathbf{M}}[n]\cdot \left(
\begin{array}{c}
\hat{y}_{[1]}^{0}[n]_{1} \\
\hat{y}_{[1]}^{1}[n]_{1}\\
\end{array}
\right)
\end{equation}
where $\vec{n}=n+{N}/{2}$.
\subsection{Extension of transform s to deeper decomposition\ levels}\label{sec:ss24}
\subsubsection{Second-level p-filter bank s}\label{sec:sss241}
The signal s $\mathbf{y}_{[1]}^{\la},\;\la=0,1,$ belong to the space $\Pi[N/2]\subset\Pi[N]$. The space $\Pi[N/2]$ can be split into mutually orthogonal\ subspaces, which we denote by $\Pi^{0}[N/2]$ and $\Pi^{1}[N/2]$,
in a way that is similar to the split of the space $\Pi[N]$. Projection of a signal\ $\mathbf{Y}\in \Pi[N/2]$ onto these subspaces and the inverse operation are done using the analysis\ and synthesis\ p-filter bank s
$\tilde{\mathbf{H}}_{[2]}=\left\{\mathbf{h}^{0}_{[2]},{\mathbf{h}}^{1}_{[2]}\right\}=\mathbf{H}_{[2]}$ (Eq. \ref{pfs2}), which operate in the space $\Pi[N/2]$. The frequency response s of the p-filter s are
\begin{eqnarray}\label{pfs2}
\hat{{h}}_{[2]}^{0}[n]_{1}= \beta[2n] \quad
\hat{{h}}_{[2]}^{1}[n]_{1} = \alpha[2n],
\end{eqnarray}
where $\beta[n]$ and $\alpha[n]$ are defined in Eq. \rf{aa_modma10}. The impulse response s of the p-filter s $\mathbf{h}^{0}_{[2]}$ and $\mathbf{h}^{1}_{[2]}$ are orthogonal\ to each other in the space $\Pi[N/2]$ and their 2-sample shifts are mutually orthogonal\
\begin{equation*}\label{or_pf2}
\sum_{k=0}^{N/2-1}{h}_{[2]}^{\la}[k-2l] \,{h}_{[2]}^{\mu}[k-2p] =\delta[\la-\mu]\,\delta[l-p],\quad \la,\mu=0,1.
\end{equation*}
The orthogonal\ projections of a signal\ $\mathbf{Y}\in \Pi[N/2]$ onto the subspaces $\Pi^{0}[N/2]$ and $\Pi^{1}[N/2]$ are
\begin{equation*}\label{ort_proj2}
Y^{\mu}[k]=\sum_{l=0}^{N/4-1}y_{[2]}^{\mu}[l]\, {h}_{[2]}^{\mu}[k-2l] , \quad
y_{[2]}^{\mu}[l] = \sum_{k=0}^{N/2-1}{h}_{[2]}^{\mu}[k-2l] Y[k],
\end{equation*}
where $\mu=0,1.$
The modulation matri ces of the p-filter bank\
${\mathbf{H}}_{[2]}$ are
\begin{eqnarray}
\tilde{\mathbf{M}}_{[2]}[n]=\tilde{\mathbf{M}}[2n],\quad
{\mathbf{M}}_{[2]}[n]={\mathbf{M}}[2n]\label{sa_modma20},
\end{eqnarray}
where the modulation matri ces $\tilde{\mathbf{M}}[n]$ and ${\mathbf{M}}[n]$ are defined in Eqs. \rf{1a_modma10} and \rf{sa_modma10}, respectively.
\subsubsection{Second-level WPTs}\label{sec:sss242}
By the application of the analysis\ p-filter bank\ $\tilde{\mathbf{H}}_{[2]}$ (section \ref{sec:sss241} and Eq. \ref{pfs2}) to the signal s $
y_{[1]}^{\la}[k] = \sum_{n=0}^{N-1}h_{[1]}^{\la}[n-2k] x[n],\;\mu, ~\la=0,1,$ that belong to $\Pi[N/2]$, we get their orthogonal\ projections $\mathbf{y}_{[1]}^{\la,0}$ and $\mathbf{y}_{[1]}^{\la,1}\in \Pi[N/2]$ onto the subspaces $\Pi^{0}[N/2]$ and $\Pi^{1}[N/2]$:
\begin{eqnarray*}\label{ort_proj2y00}
y_{[1]}^{\la,\mu}[k]&=&\sum_{l=0}^{N/4-1}y_{[2]}^{\rr}[l]\, {h}_{[2]}^{\mu}[k-2l] , \quad
y_{[2]}^{\rr}[l] = \sum_{k=0}^{N/2-1}h_{[2]}^{\mu}[k-2l] \, y_{[1]}^{\la}[k]
\\\nonumber
&=& \sum_{k=0}^{N/2-1}{h}_{[2]}^{\mu}[k-2l] \, \sum_{n=0}^{N-1}{h}_{[1]}^{\la}[n-2k] x[n]= \sum_{n=0}^{N-1}x[n]\,{\psi}_{[2],\rr}^{2r}[n-4l], \\\label{psi20}
{\psi}_{[2],\rr}^{2r}[n] &\stackrel{\Delta}{=}& \sum_{k=0}^{N/2-1}{h}_{[2]}^{\mu}[k] \,{h}_{[1]}^{\la}[n-2k]=\sum_{k=0}^{N/2-1}{h}_{[2]}^{\mu}[k] \, {\psi}_{[1],\la}^{2r}[n-2k].
\end{eqnarray*}
where
$\rr=\left\{
\begin{array}{ll}
\mu, & \hbox{if $\la=0$;} \\
3-\mu, & \hbox{if $\la=1$.}
\end{array}
\right.$
The signal\ ${\psi}_{[2],\rr}^{2r}$ is a linear combination of 2-sample shifts of the discrete-spline\ WP ${\psi}_{[1],\la}^{2r}$, therefore ${\psi}_{[2],\rr}^{2r}\in{} ^{2r}{\mathcal{S}}_{[1]}^{\la}\subset \Pi[N]$. Its DFT is
\begin{equation}\label{spec_psi20}
\hat{ {\psi}}_{[2],\rr}^{2r}[n]=\hat{{\psi}}_{[1],\la}^{2r}[n]\,\hat{h}_{[2]}^{\mu}[n]_{1}.
\end{equation}
\begin{proposition} [\cite{ANZ_book3}]\label{psi20_pro} The norms of the signal s ${\psi}_{[2],\rr}^{2r}\in \Pi[N]$ are equal to one. The 4-sample shifts $\left\{\psi_{[2],\rr}^{2r}[\cdot-4l] \right\},\;l=0,...,N/4-1,$ of this signal\ are mutually orthogonal\ and signal s with differen t indices $\rr$ are orthogonal\ to each other.\end{proposition}
Thus, the signal\ space $\Pi[N]$ splits into four mutually orthogonal\ subspaces $\Pi[N]=\bigoplus_{\rr=0}^{3} {}^{2r}\mathcal{S}_{[1]}^{\rr}$ whose orthonormal\ bases are formed by 4-sample shifts $\left\{\psi_{[2],\rr}^{2r}[\cdot-4l] \right\},\;l=0,...,N/4-1,$ of the signal s $\psi_{[2],\rr}^{2r}$, which are referred to as the second-level discrete-spline\ wavelet packet s of order $2r$.
The orthogonal\ projection of a signal\ $\mathbf{x}\in \Pi[N]$ onto the subspace ${}^{2r}\mathcal{S}_{[2]}^{\rr}$ is the signal\
\begin{equation*}\label{s20_rep}
x_{[2]}^{\rr}[k]= \sum_{l=0}^{N/4-1}\left\langle \mathbf{x}, \,\psi_{[2],\rr}^{2r}[\cdot-4l] \right\rangle\,{\psi}_{[2],\rr}^{2r}[k-4l]=\sum_{l=0}^{N/4-1}y_{[2]}^{\rr}[l]\,{\psi}_{[2],\rr}^{2r}[k-4l], ~~
k=0, \ldots , N -1.
\end{equation*}
Practically, derivation of the wavelet packet\ transform\ coefficient s $\mathbf{y}_{[1]}^{\la},\;\la=0,1,$ from $\mathbf{x}$ and the inverse operation are implemented using Eq. \rf{mod_repAS1}, while the transform\
$\mathbf{y}_{[1]}^{\la}\longleftrightarrow\mathbf{y}_{[2]}^{\rr}$ are implemented similarly using the modulation matri ces of the p-filter bank\
${\mathbf{H}}_{[2]}$ defined in Eq. \rf{sa_modma20}.
The second-level wavelet packet s $\psi_{[2],\rr}^{2r}$ are derived from the first-level wavelet packet s $\psi_{[1],\la}^{2r}$ by filter ing the latter with the p-filter s $\mathbf{h}_{[2]}^{\mu},\;\la,\mu=0,1,\;\rr=\left\{
\begin{array}{ll}
\mu, & \hbox{if $\la=0$;} \\
3-\mu, & \hbox{if $\la=1$.}
\end{array}
\right.
$.
Figure \ref{dss_wq_s2} displays the second-level wavelet packet s originating from discrete\ spline s of orders 2, 6 and 10 and their DFTs. One can observe that the wavelet packet s are symmetr ic and well localized in time domain.
Their spectr a are flat and their shapes tend to rectangular as their orders increase. They split the frequency\ domain into four quarter-bands.
\begin{figure}[H]
\begin{center}
\resizebox{14cm}{5cm}{
\includegraphics{png/dss_wq_s2_2A}
}
\end{center}
\caption{Left: second-level discrete-spline\ wavelet packet s of different orders; left to right: $\psi_{[2],0}^{2r}\to\psi_{[2],1}^{2r}\to\psi_{[2],2}^{2r}\to\psi_{[2],3}^{2r}$. Right: magnitude DFT spectr a of these wavelet packet s}
\label{dss_wq_s2}
\end{figure}
\subsubsection{Transforms to deeper levels}\label{sec:sss243}
The WPTs to deeper decomposition\ levels are implemented iteratively, while the transform\ coefficient s $\left\{\mathbf{y}_{[m+1]}^{\rr}\right\}$ are derived by filter ing the coefficient s $\left\{\mathbf{y}_{[m]}^{\la}\right\}$ with the p-filter s $\mathbf{h}^{\mu}_{[m+1]},$ where $\la=0,...,2^m-1,\;\mu=0,1$ and $\rr=\left\{
\begin{array}{ll}
2\la +\mu, & \hbox{if $\la$ is even;} \\
2\la +(1-\mu), & \hbox{if $\la$ is odd.}
\end{array}
\right.
$
The transform\ coefficient s are ${y}_{[m]}^{\la}[l]=\left\langle \mathbf{x},\psi^{2r}_{[m],\la}[\cdot, -2^{m}l] \right\rangle$, where the signal s $\psi^{2r}_{[m],\la}$ are normalized, orthogonal\ to each other in the space $\Pi[N]$, and their $2^{m}l-$sample shifts are mutually orthogonal. They are referred to as level-$m$ discrete-spline\ wavelet packet s of order $2r$. The set $\left\{\psi^{2r}_{[m],\la}[\cdot, -2^{m}l] \right\},\;\la=0,...,2^m-1,\;l=0,...N/2^m-1,$ constitutes an orthonormal\ basis of the space $\Pi[N]$ and generates its split into $2^m$ orthogonal\ subspaces. The next-level wavelet packet s $\psi^{2r}_{[m+1],\rr}$ are derived by filter ing the wavelet packet s $\psi^{2r}_{[m],\la}$ with the p-filter s $\mathbf{h}^{\mu}_{[m+1]}$ such that
\begin{equation}\label{mlev_wq}
{\psi}_{[m+1],\rr}^{2r}[n] =\sum_{k=0}^{N/2^{m}-1}{h}_{[m+1]}^{\mu}[k] \, {\psi}_{[m],\la}^{2r}[n-2^{m}k].
\end{equation}
Note that the frequency response\ of an $m-$level p-filter\ is $ \hat{h}^{\mu}_{[m]}[n]=\hat{h}^{\mu}_{[1]}[2^{m-1}n].$
A scheme of fast implementation of the discrete-spline-based WPT is described in \cite{ANZ_book3}. The transform s are executed in the spectral domain using the Fast Fourier transform (FFT) by the application of critically sampled two-channel filter bank s to the half-band spectr al components of a signal. For example, the Matlab execution of the 8-level 12-th-order WPT of a signal comprising 245760 samples, takes 0.2324 seconds.
\subsection{ 2D WPTs}\label{sec:ss25}
A standard way to extend the one-dimensional (1D) WPTs to multiple dimensions is the tensor-product extension.
The 2D one-level WPT of a signal\ $\mathbf{x}=\left\{x[k,n]\right\},\;k,n=0,...,N-1,$ which belongs to $\Pi[N,N]$, consists of the application of 1D WPT to columns of $\mathbf{x}$, which is followed by the application of the transform\ to rows of the coefficient s array. As a result of the 2D WPT of signal s from $\Pi[N,N]$, the space becomes split
into four mutually orthogonal\ subspaces
$ \Pi[N,N]=\bigoplus_{j,l=0}^{1}\,^{2r} {\mathcal{S}}^{j,l}_{[1]}.$
The subspace ${}^{2r} {\mathcal{S}}^{j,l}_{[1]}$ is a linear hull of two-sample shifts of the 2D wavelet packet s\\
$\left\{\psi_{[1],j,l}^{2r}[k-2p,n-2t]\right\} ,\;p,t,=0,...,N/2-1,$ that form an orthonormal\ basis of ${}^{2r} {\mathcal{S}}^{j,l}_{[1]}$. The orthogonal\ projection of the signal\ $\mathbf{x}\in\Pi[N,N]$ onto the subspace ${}^{2r} {\mathcal{S}}^{j,l}_{[1]}$ is the signal\ $\mathbf{x}_{[1]}^{j,l}\in\Pi[N,N]$ such that
\begin{equation*}\label{op_2d}
{x}_{[1]}^{j,l}[k,n]=\sum_{p,t=0}^{N/2-1} y_{[1]}^{j,l}[p,t] \,\psi_{[1],j,l}^{2r}[k-2p,n-2t], \quad j,l =0,1,
\end{equation*}
The 2D wavelet packet s are $\psi_{[1],j,l}^{2r}[n,m]\stackrel{\Delta}{=} \psi_{[1],j }^{2r}[n]\, \psi_{[1],l}^{2r}[m], \quad j,l=0,1.$ They are normalized and orthogonal\ to each other in the space $\Pi[N,N]$. It means that \\ $\sum_{n,m=0}^{N-1}\psi_{[1],j1 ,l1}^{2r}[n,m]\,\psi_{[1],j2 ,l2}^{2r}[n,m]=\delta[j1-j2]\,\delta[l1-l2]$. Their two-sample shifts in either direction are mutually orthogonal. The transform\ coefficient s are $$y_{[1]}^{j,l}[p,t] =\left\langle \mathbf{x},\psi_{[1],j,l}^{2r}[\cdot-2p,\cdot-2t] \right\rangle=\sum_{n,m=0}^{N-1} \psi_{[1],j,l}^{2r}[n-2p,m-2t]\: x[n,m].$$
By the application of the above transform s iteratively to blocks of the transform\ coefficient s down to $m$-th level, we get that the space $ \Pi[N,N]$ is decomposed into $4^{m}$ mutually orthogonal\ subspaces
$ \Pi[N,N]=\bigoplus_{j,l=0}^{2^{m}-1}\,^{2r} {\mathcal{S}}^{j,l}_{[m]}.$ The orthogonal\ projection of the signal\ $\mathbf{x}\in\Pi[N,N]$ onto the subspace ${}^{2r} {\mathcal{S}}^{j,l}_{[m]}$ is the signal\ $\mathbf{x}_{[m]}^{j,l}\in\Pi[N,N]$ such that
\begin{eqnarray*}\label{op_2d}
{x}_{[m]}^{j,l}[k,l]&=&\sum_{p,t=0}^{N/2^{m}-1} y_{[m]}^{j,l}[p,t] \,\psi_{[m],j ,l}^{2r}[k-2^{m}p,l-2^{m}t], \quad j,l =0,...,2^{m}-1,\\
\psi_{[m],j ,l}^{2r}[k,n]&=&\psi_{[m],j}^{2r}[k]\,\psi_{[m],l}^{2r}[n],\quad y_{[m]}^{j,l}[p,t]
=\left\langle \mathbf{x},\psi_{[m],j ,l}^{2r}[\cdot-2^{m}p,\cdot-2^{m}t] \right\rangle.
\end{eqnarray*}
The 2D tensor-product wavelet packet s $\psi_{[m],j ,l}^{2r}$ are well localized in the spatial domain, their 2D DFT spectr a are flat and provide a refined split of the frequency\ domain of signal s from $ \Pi[N,N].$\footnote{Especially it is true for WPs derived from higher-order discrete spline s.} The drawback is that the wavelet packet s are oriented in ether horizontal or vertical directions or are not oriented at all.
Figure \ref{psifpsi2_2} displays the tenth-order 2D wavelet packet s from the second decomposition\ level and their magnitude spectr a.
\begin{figure}[H]
\centering
\includegraphics[width=3.2in]{png/psi2_2.png}
\hfil
\includegraphics[width=3.2in]{png/fpsi2_2.png}%
\caption{WPs from the second decomposition\ level (left) and their magnitude spectr a (right)}
\label{psifpsi2_2}
\end{figure}
\section{(Quasi-)analytic and complementary WPs}\label{sec:s3}
In this section we define analytic\ and the so-called quasi-analytic\ WPs related to the discrete-spline-based WPs discussed in Section \ref{sec:s2} and introduce an orthonormal\ set of waveforms which are complementary to the above WPs.
\subsection{ Analytic periodic\ signal s}\label{sec:ss31}
A signal\ $\mathbf{x}\in\Pi[N]$ is represent ed by its inverse DFT. Then, Eq. \rf{fsx} can be written as follows:
\begin{eqnarray*}\label{fsx+}
x[k]&=&\frac{\hat{x}[0]+(-1)^{k}\hat{x}[N/2]}{N}+\frac{2}{N}\sum_{n=1}^{N/2-1}\frac{\hat{x}[n]\,\w^{kn}+(\hat{x}[n]\,\w^{kn})^{\ast}}{2}.
\end{eqnarray*}
Define the real-valued signal\ $\mathbf{h}\in\Pi[N]$ and two complex-valued signal s $\mathbf{\bar{x}}_{+}$ and $\mathbf{\bar{x}}_{-}$ such that
\begin{equation}
\label{yy}
\begin{array}{lll}
h[k]&\stackrel{\Delta}{=}&\frac{2}{N}\sum_{n=1}^{N/2-1}\frac{\hat{x}[n]\,\w^{kn}-\hat{x}[n]^{\ast}\,\w^{-kn}}{2i},\\
\bar{x}_{\pm}[k]&\stackrel{\Delta}{=}&x[k]\pm ih[k]=\frac{\hat{x}[0]+(-1)^{k}\hat{x}[N/2]}{N}\\&+&\frac{2}{N}\sum_{n=1}^{N/2-1}
\left\{
\begin{array}{ll}
\hat{x}[ n]\,\w^{ kn}, & \hbox{for $\bar{x}_{+}$;} \\
\hat{x}[ n]^{\ast}\,\w^{- kn}=\hat{x}[ N-n]\,\w^{- k(N-n)}, & \hbox{for $\bar{x}_{-}$.}
\end{array}
\right.
\end{array}
\end{equation}
The signal s' $\mathbf{\bar{x}}_{\pm}$ DFT spectr a are
\begin{equation}
\label{xpm_sp}
\begin{array}{lll}
\hat{\bar{x}}_{+}[n] &=& \left\{
\begin{array}{ll}
\hat{x}[n], & \hbox{if $n=0$, or $n=N/2$ ;} \\
2\hat{x}[n], & \hbox{if $0<n<N/2$;} \\
0, & \hbox{if $-N/2<n<0\Longleftrightarrow N/2<n<N$,}
\end{array}
\right. \\
\hat{\bar{x}}_{-}[n] &=& \left\{
\begin{array}{ll}
\hat{x}[n], & \hbox{if $n=0$, or $n=N/2$ ;} \\
2\hat{x}[n], & \hbox{if $-N/2<n<0\Longleftrightarrow N/2<n<N$;} \\
0, & \hbox{if $n=0$, or $n=N/2$.}
\end{array}
\right.
\end{array}
\end{equation}
The spectr um of $\mathbf{\bar{x}}_{+}$ comprises only non-negative frequencies and vice versa for $\mathbf{\bar{x}}_{-}$.
$\mathbf{x}=\mathfrak{Re}(\mathbf{\bar{x}}_{\pm})$ and $\mathfrak{Im}(\mathbf{\bar{x}}\pm)=\pm\mathbf{h}$. The signal s $\mathbf{\bar{x}}_{\pm}$ are referred to as periodic\ analytic signal s.
The signal's $\mathbf{h}$ DFT spectr um is
\begin{equation*}\label{yft}
\hat{h}[n]=\left\{
\begin{array}{ll}
-i\,\hat{x}[n], & \hbox{if $0<n<N/2$;} \\
i\, \hat{x}[n], & \hbox{if $-N/2<n<0\Longleftrightarrow N/2<n<N$;} \\
0, & \hbox{if $n=0$, or $n=N/2$.}
\end{array}
\right.
\end{equation*}
Thus, the signal\ $\mathbf{h}$ where $\mathbf{h}=H(\mathbf{x})$ can be regarded as the Hilbert transform\ (HT) of a discrete-time\ periodic\ signal\ $\mathbf{x}$, (see \cite{opp}, for example).
\begin{proposition} \label{pro:ysym}
\par\noindent
\begin{enumerate}
\item The HT $\mathbf{h}=H(\mathbf{x})$ is invariant with respect to circular shift in $\Pi[N]$. That means that $\mathbf{\tilde{h}}=\mathbf{h}[\cdot +m]$ is the HT of $\mathbf{\tilde{x}}=\mathbf{x}[\cdot +m]$.
\item If the signal\ $\mathbf{x}\in\Pi[N]$ is symmetr ic about a grid point $k=K$ than $\mathbf{h}=H(\mathbf{x})$ is antisymmetr ic about \emph{K} and $h[K]=0$.
\item Assume that a signal\ $\mathbf{x}\in\Pi[N]$ and $\hat{x}[0]=\hat{x}[N/2]=0$. Then,
\begin{enumerate}
\item The norm of its HT is $\|H(\mathbf{x}) \|=\|\mathbf{x} \|$.
\item Magnitude spectr a of the signal s $\mathbf{x}$ and $\mathbf{h}=H(\mathbf{x})$ coincide.
\end{enumerate}
\end{enumerate}\end{proposition}
\goodbreak\medskip\noindent{\small\bf Proof: }
\begin{enumerate}
\item The DFT of the signal $\mathbf{\tilde{x}}$ is $\hat{\tilde{x}}[n]=\w^{mn}\,\hat{x}[n]$. Denote by $\mathbf{\bar{\tilde{x}}}_{+}$ the analytic\ signal\ related to $\mathbf{\tilde{x}}$. Equation \rf{xpm_sp} implies that $\hat{\bar{\tilde{x}}}_{+}[n]=\w^{mn}\,\hat{\bar{x}}[n]$. Consequently, $\bar{\tilde{x}}_{+}[k]=\bar{x}_{+}[k+m]$. The same relation holds for $\mathbf{\tilde{h}}=\mathfrak{Im}(\mathbf{\bar{\tilde{x}}})$.
\item Assume that $\mathbf{x}\in\Pi[N]$ is symmetr ic about $K=0$. Then, its DFT is $$\hat{x}[n]=x[0]+(-1)^{n}x[N/2] +2\sum_{k=1}^{N/2-1}x[k] \,\cos(2\pi kn/N)=\hat{x}[-n].$$
Then, due to Eq. \rf{yy}, $h[k]=2/N\sum_{n=1}^{N/2-1}x[n]\sin(2\pi kn/N)=h[-k]$ and $h[0]=0$. Extension of the proof to $K\neq0$ is straightforward.
\item
\begin{enumerate}
\item The squared norm is
\(
\|\mathbf{h} \|^{2}=\frac{1}{N}\sum_{n=-N/2}^{N/2-1}|\hat{h}[n]|^{2}=\frac{1}{N}\sum_{n=-N/2}^{N/2-1}|\hat{x}[n]|^{2}.
\
\item The coincidence of the magnitude spectr a is straightforwared.
\end{enumerate}\end{enumerate}{\vrule height7pt width7pt depth0pt}\par\bigskip
\subsection{Analytic WPs}\label{sec:ss32}
The analytic\ spline-based WPs and their DFT spectr a are derived from the corresponding WPs $\left\{\psi^{2r}_{[m],l}\right\},\;m=1,...,M,\;l=0,...,2^{m}-1,$ in line with the scheme in Section \ref{sec:ss31}. Recall that for all $l\neq0$, the DFT $\hat{\psi}^{2r}_{[m],l}[0]=0$ and for all $l\neq2^{m}-1$, the DFT $\hat{\psi}^{2r}_{[m],l}[N/2]=0$.
Denote by $\th^{2r}_{[m],l}=H(\psi^{2r}_{[m],l})$ the discrete\ periodic\ HT of the wavelet packet\ $\psi^{2r}_{[m],l}$, such that the DFT is
\begin{equation*}\label{th_df}
\hat{\th}^{2r}_{[m],l}[n]=\left\{
\begin{array}{ll}
-i\,\hat{\psi}^{2r}_{[m],l}[n], & \hbox{if $0<n<N/2$;} \\
i\, \hat{\psi}^{2r}_{[m],l}, & \hbox{if $-N/2<n<0$;} \\
0, & \hbox{if $n=0$, or $n=N/2$ .}
\end{array}
\right.
\end{equation*}
Then, the corresponding analytic\ WPs are
\begin{equation*}\label{awq}
\bar{\psi}^{2r}_{\pm[m],l}=\psi^{2r}_{[m],l} \pm i\th^{2r}_{[m],l}.
\end{equation*}
\paragraph{Properties of the analytic\ WPs}
\begin{enumerate}
\item The DFT spectr a of the analytic\ WPs $\bar{\psi}^{2r}_{+[m],l}$ and $\bar{\psi}^{2r}_{-[m],l}$ are located within the bands $[0,N/2]$ and $[N/2,N]\Longleftrightarrow[-N/2,0]$, respectively.
\item The real component ${\psi}^{2r}_{[m],l}$ is the same for both WPs $\bar{\psi}^{2r}_{\pm[m],l}$. {It} is a symmetr ic oscillating waveform.
\item\label{prop3}
The HT WPs $\th^{2r}_{[m],l}=H({\psi}^{2r}_{[m],l})$ are antisymmetr ic oscillating waveforms.
\item For all $l\neq0, \,2^{m}-1$, the norms $\left\| \th^{2r}_{[m],l}\right\|=1$. Their magnitude spectr a $\left|\hat{\th}^{2r}_{[m],l}[n]\right|$ coincide with the magnitude spectr a of the respective WPs $\psi^{2r}_{[m],l}$.
\item\label{prop5} When $l=0$ or $l=2^{m}-1$, the magnitude spectr a of $\th^{2r}_{[m],l}$ coincide with that of ${\psi}^{2r}_{[m],l}$ everywhere except for the points $n=0$ or $N/2,$ respectively, and the waveforms' norms are no longer equal to 1.
\end{enumerate}
Properties in items \ref{prop3}--\ref{prop5} follow directly from Proposition \ref{pro:ysym}.
\begin{proposition} \label{pro:teta_oo} { For all $l\neq0, \,2^{m}-1$,
the shifts of the HT WPs $\left\{\th^{2r}_{[m],l}[\cdot-2^{m}l]\right\}$ are orthogonal\ to each other in the space $\Pi[N]$. The orthogonal ity does not take place for for $\th^{2r}_{[m],0}$ and $\th^{2r}_{[m],2^{m}-1}$.}
\end{proposition}
\goodbreak\medskip\noindent{\small\bf Proof: } The inner product is
\begin{eqnarray*}
&& \left\langle \th^{2r}_{[m],l},\th^{2r}_{[m],l}[\cdot-2^{m}l] \right\rangle =\frac{1}{N}\sum_{n=-N/2}^{N/2-1}\w^{2^{m}ln}\left|\hat{\th}^{2r}_{[m],l}[n] \right|^{2}\\
&& =\frac{1}{N}\sum_{n=-N/2}^{N/2-1}\w^{2^{m}ln}\left|\hat{\psi}^{2r}_{[m],l}[n] \right|^{2}=\left\langle \psi^{2r}_{[m],l},\psi^{2r}_{[m],l}[\cdot-2^{m}l] \right\rangle=0.
\end{eqnarray*}
{\vrule height7pt width7pt depth0pt}\par\bigskip
Figure \ref{psi_theta2} displays the wavelet packet s ${\psi}^{2r}_{[2],l}$ and ${\th}^{2r}_{[2],l},\;r=1,3,5,\; l=0,1,2,3,$ and their magnitude spectr a.
\begin{figure}[H]
\resizebox{16cm}{10cm}{
\centering
\includegraphics{png/psi_theta2.png}
}
\caption{WPs ${\psi}^{2r}_{[2],l}$ (first, third and fifth from the top left frames) and ${\th}^{2r}_{[2],l}$ (second, fourth and sixth from the top left frames) from the second decomposition\ level and their magnitude spectr a, respectively (right frames)}
\label{psi_theta2}
\end{figure}
\subsection{Complementary set of wavelet packets and quasi-analytic\ WPs}\label{sec:ss33}
\subsubsection{Complementary orthonormal\ WPs}\label{sec:sss331}
The discrete-spline-based WPs $\left\{\psi^{2r}_{[m],l}\right\}$ are normalized and their $2^{m}$-sample shifts are mutually orthogonal. Combinations of shifts of several wavelet packet s can form orthonormal\ bases for the signal\ space $\Pi[N]$. On the other hand, it is not true for the set $\left\{\th^{2r}_{[m],l}\right\},\;l=0,...2^{m}-1,$ of the antysymmetr ic waveform s, which are the HTs of the WPs $\left\{\psi^{2r}_{[m],l}\right\}$.
At the decomposition\ level \emph{m}, the waveform s $\left\{\th^{2r}_{[m],l}\right\},\;l=1,...2^{m}-2,$ are normalized and their $2^{m}$-sample shifts are mutually orthogonal, but the norms of the
waveform s $\th^{2r}_{[m],0}$ and $\th^{2r}_{[m],2^{m}-1}$ are close but not equal to 1 and their shifts are not mutually orthogonal. It happens because the values $\hat{\th}^{2r}_{[m],j}[0]$ and $\hat{\th}^{2r}_{[m],j}[N/2]$ are missing in their DFT spectr a\footnote{Recall that these values are real}. This keeping in mind, we upgrade the set $\left\{\th^{2r}_{[m],l}\right\},\;l=0,...2^{m}-1$ in the following way.
Define
a set $\left\{\f^{2r}_{[m],l}\right\},\;m=1,...,M, \;l=0,...,2^{m}-1,$ of signal s from the space $\Pi[N]$ via their DFTs:
\begin{equation}\label{phi_df}
\hat{\f}^{2r}_{[m],l}[n]=\left\{
\begin{array}{ll}
-i\,\hat{\psi}^{2r}_{[m],l}[n], & \hbox{if $0<n<N/2$;} \\
i\, \hat{\psi}^{2r}_{[m],l}[n], & \hbox{if $-N/2<n<0$ ;}\\
\hat{\psi}^{2r}_{[m],l}[n], & \hbox{if $n=0$, or $n=N/2 .$}
\end{array}
\right.
\end{equation}
For all $l\neq0, 2^{m}-1,$ the signal s $\f^{2r}_{[m],l}$ coincide with $\th^{2r}_{[m],l}=H(\psi^{2r}_{[m],l})$.
\begin{proposition} \label{pro:phi_oo}
\par\noindent
\begin{description}
\item[-] The magnitude spectr a $\left|\hat{\f}^{2r}_{[m],l}[n]\right|$ coincide with the magnitude spectr a of the respective WPs $\psi^{2r}_{[m],l}$.
\item[-] For any $m=1,...,M,$ and $l=1,...,2^{m}-2,$ the signal s \ $\f^{2r}_{[m],l}$ are antisymmetr ic oscillating waveforms. For $l=0$ and $l=2^{m}-1$, the shapes of the signal s are near antisymmetr ic.
\item[-] The orthonormal ity properties that are similar to the properties of WPs $\psi^{2r}_{[m],l}$ hold for the signal s $\f^{2r}_{[m],l}$ such that
\begin{eqnarray*}
\label{onp}
\left\langle\f^{2r}_{[m],l}[\cdot -p\,2^{m}],\f^{2r}_{[m],\la}[\cdot -s\,2^{m}] \right\rangle= \delta[\la,l]\,\delta[p,s].
\end{eqnarray*}
\end{description}
\end{proposition}
The proof of Proposition \ref{pro:phi_oo} is similar to the proof of Proposition \ref{pro:teta_oo}.
Figure \ref{teta_phi} displays the signal s ${\th}^{6}_{[2],l},\;l=0,3$ and ${\f}^{6}_{[2],l},\;l=0,3$, from the second decomposition\ level and their magnitude spectr a. Lack of the values $\hat{\th}^{2r}_{[m],j}[0]$ and $\hat{\th}^{2r}_{[m],j}[N/2]$ in the DFTs of ${\th}^{6}_{[2],l},\;l=0,3$, are seen. Addition of $\hat{\psi}^{2r}_{[m],j}[0]$ and $\hat{\psi}^{2r}_{[m],j}[N/2]$ to the above spectr a results in the antisymmetr y distortion.
\begin{figure}[H]
\resizebox{17cm}{4cm}{
\centering
\includegraphics{png/teta_phi.png}
}
\caption{Left: signal s ${\th}^{6}_{[2],l},\;l=0,3$ (top), and ${\f}^{6}_{[2],l},\;l=0,3$ (bottom). Right: their magnitude DFT spectr a, respectively}
\label{teta_phi}
\end{figure}
We call the signal s $\left\{\f^{2r}_{[m],l}\right\},\;m=1,...,M, \;l=0,...,2^{m}-1$, the \emph{complementary wavelet packets} (cWPs). Similarly to the WPs $\left\{\psi^{2r}_{[m],l}\right\},$ differen ent combinations of the cWPs can provide differen ent orthonormal\ bases for the space $\Pi[N]$. These can be, for example, the wavelet\ bases $$\left\{\bigoplus_{r=0}^{N/2^{M}}\f^{2r}_{[M],0}[\cdot-r\,2^{M}]\right\}\bigoplus_{m=1}^{M}
\left\{\bigoplus_{r=0}^{N/2^{m}}\f^{2r}_{[m],1}[\cdot-r\,2^{m}]\right\}.$$
or a Best Basis \cite{coiw1} type.
\subsubsection{Quasi-analytic\ WPs}\label{sec:sss332}
The sets of complex-valued WPs, which we refer to as the quasi-analytic\ wavelet packets (qWP), are defined as
\begin{equation*}\label{qaz}
\Psi^{2r}_{\pm[m],l}=\psi^{2r}_{[m],l} \pm i\f^{2r}_{[m],l}, \quad m=1,...,M,\;l=0,...,2^{m}-1,
\end{equation*}
where $\f^{2r}_{[m],l}$ are the cWPs defined in Eq. \rf{phi_df}. The qWPs $\Psi^{2r}_{\pm[m],l}$ differ from the analytic\ WPs $\bar{\psi}^{2r}_{\pm[m],l}$ by the addition of the two values $\pm i\,\hat{\psi}^{2r}_{[m],l}[0]$ and $\pm i\,\hat{\psi}^{2r}_{[m],l}[N/2]$ into their DFT spectr a, respectively. For a given decomposition\ level $m$, these values are zero for all $l$ except for $l_{0}=0$ and $l_{m}=2^{m}-1$. It means that for all $l$ except for $l_{0}$ and $l_{m}$, the qWPs $\Psi^{2r}_{\pm[m],l}$ are analytic.
The DFTs of qWPs are
\begin{eqnarray}\label{qa_df}
\hat{\Psi}^{2r}_{+[m],l}[n]&=&\left\{
\begin{array}{ll}
(1+i)\hat{\psi}^{2r}_{[m],l}[n], & \hbox{if $n=0$ or $n= N/2$;} \\
2\hat{\psi}^{2r}_{[m],l}[n], & \hbox{if $0< n< N/2$;} \\
0 & \hbox{if $ N/2<n<N$,}
\end{array}
\right.\\\nonumber
\hat{\Psi}^{2r}_{-[m],l}[n]&=&\left\{
\begin{array}{ll}
(1-i)\hat{\psi}^{2r}_{[m],l}[n], & \hbox{if $n=0$ or $n= N/2$;} \\
0 & \hbox{if $0< n< N/2$;} \\
2\hat{\psi}^{2r}_{[m],l}[n], & \hbox{if $ N/2< n< N$.}
\end{array}
\right.
\end{eqnarray}
\subsubsection{Design of cWPs and qWPs}\label{sec:sss333}
The DFTs of the first-level WPs are
\begin{equation*}\label{df_wq1}
\hat{\psi}^{2r}_{[1],0}[n]=\frac{\cos^{2r}\frac{\pi\,n}{N}}{\sqrt{ U^{4r}[n]}}=\beta[n],\quad \hat{\psi}^{2r}_{[1],1}[n]=\frac{\w^{n}\,\sin^{2r}\frac{\pi\,n}{N}}{\sqrt{ U^{4r}[n]}}=\alpha[n],
\end{equation*}
where the sequence\ $ U^{4r}[n]$ is defined in Eq. \rf{u2r}. Consequently, the DFTs of the first-level cWPs are
\begin{eqnarray}\label{df_cwq1}
\hat{\f}^{2r}_{[1],0}[n]=\left\{
\begin{array}{ll}
-i\,\beta[n], & \hbox{if $0<n<N/2$;} \\
i\, \beta[n], & \hbox{if $N/2<n<N$;} \\
\sqrt{2}, & \hbox{if $n=0$; }
\\
0, & \hbox{ if $n=N/2,$}
\end{array}
\right.\quad \hat{\f}^{2r}_{[1],1}[n]=\left\{
\begin{array}{ll}
-i\,\alpha[n], & \hbox{if $0<n<N/2$;} \\
i\, \alpha[n], & \hbox{if $N/2<n<N$;} \\
0, & \hbox{if $n=0$; }
\\
-\sqrt{2}, & \hbox{ if $n=N/2.$}
\end{array}
\right.
\end{eqnarray}
Due to Eq. \rf{spec_psi20}, the DFT of the second-level WPs are
\begin{eqnarray}\nonumber
\hat{ {\psi}}_{[2],\rr}^{2r}[n]&=&\hat{{\psi}}_{[1],\la}^{2r}[n]\,\hat{h}_{[2]}^{\mu}[n]_{1},\quad \la,\mu=0,1,\;\rr=2\la+\left\{
\begin{array}{ll}
\mu, & \hbox{if $\la=0$;} \\
1-\mu, & \hbox{if $\la=1$.}
\end{array}
\right.,\\\label{df_wq2}
\hat{h}_{[2]}^{0}[n]_{1}&=& \beta[2n],\quad \hat{h}_{[2]}^{1}[n]_{1}=\alpha[2n].
\end{eqnarray}
For example, assume $\la=\mu=0$ then we have
\[\hat{ \psi}_{[2],0}^{2r}[n]=\hat{{\psi}}_{[1],0}^{2r}[n]\,\hat{h}_{[2]}^{0}[n]_{1}=\frac{\cos^{2r}\frac{\pi\,n}{N}}{\sqrt{ U^{4r}[n]}}\,\frac{\cos^{2r}\frac{2\pi\,n}{N}}{\sqrt{U^{4r}[2n]}}.\]
Keeping in mind that the sequence\ $\beta[2n]=\cos^{2r}(2\pi\,n/{N})/\sqrt{U^{4r}[2n]}$ is $N/2-$periodic, we have that the DFT of the corresponding cWP is
\[\hat{\f}^{2r}_{[2],0}[n]=\widehat{H({ \psi}_{[2],0}^{2r})}[n]=\beta[2n]\,\left\{
\begin{array}{ll}
-i\,\beta[n], & \hbox{if $0<n<N/2$;} \\
i\, \beta[n], & \hbox{if $N/2<n<N$;} \\
2, & \hbox{if $n=0$; }
\\
0, & \hbox{ if $n=N/2,$}
\end{array}
\right.=\hat{{\f}}_{[1],0}^{2r}[n]\,\hat{h}_{[2]}^{0}[n]_{1}=\hat{{\f}}_{[1],0}^{2r}[n]\,\hat{h}_{[1]}^{0}[2n]_{1}.
\]
Similar relations hold for all the second-level cWPs and a general statement is true.
\begin{proposition} \label{pro:cwq_des} Assume that for a WP $\psi_{[m+1],\rr}^{2r}$ the relation in Eq. \rf{mlev_wq} holds. Then, for the cWP $\f_{[m+1],\rr}^{2r}$ we have
\begin{eqnarray*}\label{mlev_cwq}
{\f}_{[m+1],\rr}^{2r}[n] &=&\sum_{k=0}^{N/2^{m}-1}{h}_{[m+1]}^{\mu}[k] \, {\f}_{[m],\la}^{2r}[n-2^{m}k]\Longleftrightarrow\hat{\f}_{[m+1],\rr}^{2r}[\n]=
\hat{h}_{[1]}^{\mu}[2^{m}\n]_{m}\,\hat{\f}_{[m],\la}^{2r}[\n],\\\nonumber
\hat{h}_{[1]}^{0}[\n] &=&\hat{\psi}^{2r}_{[1],0}[\n]=\beta[\n],\quad \hat{h}_{[1]}^{1}[\n] = \hat{\psi}^{2r}_{[1],1}[\n]=\alpha[\n].
\end{eqnarray*}
\end{proposition}
\begin{corollary} \label{cor:cwq_des} Assume that for a WP $\psi_{[m+1],\rr}^{2r}$, the relation in Eq. \rf{mlev_wq} holds. Then, for the qWP $\Psi_{\pm[m+1],\rr}^{2r}$ we have
\begin{eqnarray}\label{mlev_qwq}
\Psi_{\pm[m+1],\rr}^{2r}[n] =\sum_{k=0}^{N/2^{m}-1}{h}_{[m+1]}^{\mu}[k] \, \Psi_{\pm[m],\la}^{2r}[n-2^{m}k]\Longleftrightarrow\hat{\Psi}_{\pm[m+1],\rr}^{2r}[\n]=
\hat{h}_{[1]}^{\mu}[2^{m}\n]_{m}\,\hat{\Psi}_{\pm[m],\la}^{2r}[\n].
\end{eqnarray}
\end{corollary}
\section{Implementation of cWP and qWP transform s}\label{sec:s4}
Implementation of transform s with WPs ${\psi}_{[m],\la}^{2r}$ was discussed in Section \ref{sec:s2}. In this section, we extent that the transform\ scheme to the transform s with cWPs ${\f}_{[m],\la}^{2r}$ and qWPs $\Psi_{[m],\la}^{2r}$.
\subsection{One-level transform s}\label{sec:ss41}
Denote by ${} ^{2r}{\mathcal{C}}_{[1]}^{0}$ the subspace of the signal\ space $\Pi[N]$, which is the linear hull of the set $\mathbf{W}_{[1]}^{0}=\left\{{\f}_{[1],0}^{2r}[\cdot-2k]\right\},\;k=0,...,N/2-1$. The signal s from the set $\mathbf{W}_{[1]}^{0}$ form an orthonormal\ basis of the subspace ${} ^{2r}{\mathcal{C}}_{[1]}^{0}$. Denote by ${} ^{2r}{\mathcal{C}}_{[1]}^{1}$ the orthogonal\ complement of the subspace ${} ^{2r}{\mathcal{C}}_{[1]}^{0}$ in the space $\Pi[N]$. The signal s from the set $\mathbf{W}_{[1]}^{1}=\left\{{\f}_{[1],1}^{2r}[\cdot-2k]\right\},\;k=0,...,N/2-1$ form an orthonormal\ basis of the subspace ${} ^{2r}{\mathcal{C}}_{[1]}^{1}$.
\begin{proposition} \label{pro:phi1}
The orthogonal\ projections of a signal\ $\mathbf{x}\in \Pi[N]$ onto the spaces ${} ^{2r}{\mathcal{C}}_{[1]}^{\mu},\;\mu=0,1$ are the signal s $\mathbf{x}_{[1]}^{\mu}\in\Pi[N]$ such that
\begin{eqnarray*}\label{c1_del1}
x_{[1]}^{\mu}[k]&=&\sum_{l=0}^{N/2-1}c_{[1]}^{\mu}[l]\, \f_{[1],\mu}^{2r}[k-2l] ,\quad
c_{[1]}^{\mu}[l] =\left\langle \mathbf{x},\, \f_{[1],\mu}^{2r}[\cdot-2l] \right\rangle
= \sum_{k=0}^{N-1}g_{[1]}^{\mu}[k-2l] \,x[k], \\\nonumber g_{[1]}^{\mu}[k] &=&\f_{[1],\mu}^{2r}[k],\quad
\hat{{g}}_{[1]}^{\mu}[n] = \hat{ \f}_{[1],\mu}^{2r}[n],\quad \mu=0,1.
\end{eqnarray*}
The DFTs $ \hat{ \f}_{[1],\mu}^{2r}[n]$ of the first-level cWPs are given in Eq. \rf{df_cwq1}.
\end{proposition}
The transform s $\mathbf{x}\rightarrow \mathbf{c}_{[1]}^{0}\bigcup\mathbf{c}_{[1]}^{1}$ and back are implemented using the analysis\ $ \tilde{\mathbf{M}}^{c}[n] $ and the synthesis\
$ \mathbf{M}^{c}[n] $ modulation matri ces:
\begin{equation}
\label{aa_modma11}
\begin{array}{lll}
\tilde{\mathbf{M}}^{c}[n]&\stackrel{\Delta}{=}& \left(
\begin{array}{cc}
\hat{g}_{[1]}^{0}[n] & \hat{g}_{[1]}^{0}\left[n+\frac{N}{2}\right]\\
\ \hat{g}^{1}_{[1]}[n] & \hat{g}^{1}_{[1]}\left[n+\frac{N}{2}\right]\\
\end{array}
\right)=\left(\begin{array}{cc}
\check{\beta}[n] & -\check{\beta}\left[n+\frac{N}{2}\right] \\
\check{\alpha}[n] &- \check{\alpha}\left[n+\frac{N}{2}\right] \\
\end{array}
\right),\\
{\mathbf{M}}^{c}[n]&\stackrel{\Delta}{=}& \left(\begin{array}{cc}
\check{\beta}[n] & \check{\alpha}[n] \\
-\check{\beta}\left[n+\frac{N}{2}\right] &- \check{\alpha}\left[n+\frac{N}{2}\right] \\
\end{array}
\right),
\\
\check{\beta}[n] &=& \left\{
\begin{array}{ll}
{\beta}[0], & \hbox{if $n=0$;} \\
-i{\beta}[n], & \hbox{otherwise,}
\end{array}
\right.
\quad
\check{\alpha}[n]= \left\{
\begin{array}{ll}
{\alpha}[N/2], & \hbox{if $n=N/2$;} \\
-i\alpha[n], & \hbox{otherwise.}
\end{array}
\right.
\end{array}
\end{equation}
The sequence s $\beta[n]$ and $\alpha[n]$ are given in Eq. \rf{aa_modma10}.
Similarly to Eq. \rf{mod_repAS1}, the one-level cWP transform\ of a signal\ $\mathbf{x}$ and its inverse are:
\begin{equation*}\label{mod_repAC1}
\left(
\begin{array}{c}
\hat{c}_{[1]}^{0}[n]_{1} \\
\hat{c}_{[1]}^{1}[n]_{1}\\
\end{array}
\right)=\frac{1}{2} \tilde{\mathbf{M}}^{c}[-n]\cdot \left(
\begin{array}{l}
\hat{x}[n] \\
\hat{x}[\vec{n}]
\end{array}
\right),\quad \left(
\begin{array}{l}
\hat{x}[n] \\
\hat{x}[\vec{n}]
\end{array}
\right)={\mathbf{M}}^{c}[n]\cdot \left(
\begin{array}{c}
\hat{c}_{[1]}^{0}[n]_{1} \\
\hat{c}_{[1]}^{1}[n]_{1}\\
\end{array}
\right),
\end{equation*}
where $\vec{n}=n+{N}/{2}$.
Define the p-filter s $$\mathbf{q}^{l}_{\pm[1]}\stackrel{\Delta}{=} \mathbf{h}^{j}_{[1]}\pm i\,\mathbf{g}^{j}_{[1]}=\psi^{2r}_{[1],l}\pm i\,\f^{2r}_{[1],l}={\Psi}^{2r}_{\pm [1],l}, \quad l=0,1.$$
Equation \rf{qa_df} implies that their frequency response s are
\begin{eqnarray*}\label{Psi1_dfp}
\hat{q}^{0}_{+[1]}[n]=\left\{
\begin{array}{ll}
(1+i)\sqrt{2}, & \hbox{if $n=0$;} \\
2\beta[n], & \hbox{if $0< n< N/2$;} \\
0 & \hbox{if $ N/2\leq n<N$,}
\end{array}
\right.\quad
\hat{q}^{1}_{+[1]}[n]=\left\{
\begin{array}{ll}
-(1+i)\sqrt{2}, & \hbox{if $n= N/2$;} \\
2\alpha[n], & \hbox{if $0< n< N/2$;} \\
0, & \hbox{if $ N/2< n\leq N$.}
\end{array}
\right.\\\label{Psi1_dfm}
\hat{q}^{0}_{-[1]}[n]=\left\{
\begin{array}{ll}
(1-i)\sqrt{2}, & \hbox{if $n=0$;} \\
2\beta[n], & \hbox{if $ N/2<n<N$,} \\
0 & \hbox{if $0< n\leq N/2$;}
\end{array}
\right.\quad
\hat{q}^{1}_{-[1]}[n]=\left\{
\begin{array}{ll}
-(1-i)\sqrt{2}, & \hbox{if $n= N/2$;} \\
2\alpha[n] & \hbox{if $ N/2< n\leq N$;} \\
0, & \hbox{if $0\leq n< N/2$.}
\end{array}
\right.
\end{eqnarray*}
Thus, the analysis\ modulation matri ces for the p-filter s $\mathbf{q}^{l}_{\pm[1]}$ are
\begin{eqnarray}\label{aa_modma10p}
\tilde{\mathbf{M}}_{+}^{q}[n]&=& \left(
\begin{array}{cc}
\hat{q}_{+[1]}^{0}[n] & 0\\
\hat{q}^{1}_{+[1]}[n] & -\sqrt{2}(1+i)\,\delta[n-N/2]\\
\end{array}
\right)= \tilde{\mathbf{M}}[n]+i\, \tilde{\mathbf{M}}^{c}[n],\\\label{aa_modma10m}
\tilde{\mathbf{M}}_{-}^{q}[n]&=& \left(
\begin{array}{cc}
(1-i)\sqrt{2}\delta[n] & \hat{q}_{-[1]}^{0}[n] \\
0 & \hat{q}^{1}_{-[1]}[n]\\
\end{array}
\right)= \tilde{\mathbf{M}}[n]-i\, \tilde{\mathbf{M}}^{c}[n],
\end{eqnarray}
where the modulation matri x $\tilde{\mathbf{M}}[n]$ is defined in Eq. \rf{aa_modma10} and $\tilde{\mathbf{M}}^{c}[n]$ is defined in Eq. \rf{aa_modma11}.
Application of the matrices $\tilde{\mathbf{M}}_{\pm}^{q}[n]$ to the vector\ $( \hat{x}[n] ,
\hat{x}[\vec{n}])^{T}$ produces the vector s
\begin{equation*}\label{mod_decAQ1}
\left(
\begin{array}{c}
\hat{z}_{\pm[1]}^{0}[n]_{1} \\
\hat{z}_{\pm[1]}^{1}[n]_{1}\\
\end{array}
\right)=\frac{1}{2}( \tilde{\mathbf{M}}_{\pm}^{q}[n])^{*}\cdot \left(
\begin{array}{l}
\hat{x}[n] \\
\hat{x}[\vec{n}]
\end{array}
\right)= \left(
\begin{array}{c}
\hat{y}_{[1]}^{0}[n]_{1} \\
\hat{y}_{[1]}^{1}[n]_{1}\\
\end{array}
\right)\mp i\, \left(
\begin{array}{c}
\hat{c}_{[1]}^{0}[n]_{1} \\
\hat{c}_{[1]}^{1}[n]_{1}\\
\end{array}
\right).
\end{equation*}
Define the matrices ${\mathbf{M}}_{\pm}^{q}[n]\stackrel{\Delta}{=}\tilde{\mathbf{M}}_{\pm}^{q}[n]={\mathbf{M}}[n]\pm i\,{\mathbf{M}}^{c}[n]$ and apply these matrices to the vector s\\ $(\hat{z}_{\pm[1]}^{0}[n]_{1} ,
\hat{z}_{\pm[1]}^{1}[n]_{1})^{T}$. Here the modulation matri x ${\mathbf{M}}[n]$ is defined in Eq. \rf{sa_modma10} and ${\mathbf{M}}^{c}[n]$ is defined in Eq. \rf{aa_modma11}.
\begin{proposition} \label{pro:Mq_z}The following relations hold
\begin{eqnarray*}\label{Mq_z1}
&& {\mathbf{M}}_{\pm}^{q}[n]\cdot \left(
\begin{array}{c}
\hat{z}_{\pm[1]}^{0}[n]_{1} \\
\hat{z}_{\pm[1]}^{1}[n]_{1}\\
\end{array}
\right)=\mathbf{M}[n]\cdot \left(
\begin{array}{c}
\hat{y}_{[1]}^{0}[n]_{1} \\
\hat{y}_{[1]}^{1}[n]_{1}\\
\end{array}
\right) +{\mathbf{M}}^{c}[n]\cdot \left(
\begin{array}{c}
\hat{c}_{[1]}^{0}[n]_{1} \\
\hat{c}_{[1]}^{1}[n]_{1}\\
\end{array}
\right) \\\nonumber&&\pm i \left( \mathbf{M}^{c}[n]\cdot \left(
\begin{array}{c}
\hat{y}_{[1]}^{0}[n]_{1} \\
\hat{y}_{[1]}^{1}[n]_{1}\\
\end{array}
\right) -\mathbf{M}[n]\cdot \left(
\begin{array}{c}
\hat{c}_{[1]}^{0}[n]_{1} \\
\hat{c}_{[1]}^{1}[n]_{1}\\
\end{array}
\right) \right)
\\\label{Mq_z2}&&=2\left(\left(
\begin{array}{l}
\hat{x}[n] \\
\hat{x}[n+N/2]
\end{array}
\right)
{\pm}i \,\left(
\begin{array}{l}
\hat{h}[n] \\
\hat{h}[n+N/2]
\end{array}
\right)\right)
=2\left(
\begin{array}{l}
\hat{\bar{x}}_{\pm}[n] \\
\hat{\bar{x}}_{\pm}[n+N/2]
\end{array}
\right)
,
\end{eqnarray*}
where $\mathbf{h}$ is the HT of the signal\ $\mathbf{x}\in \Pi[N]$ and $\mathbf{\bar{x}}_{\pm}$ are the analytic\ signal s associated with $\mathbf{x}$.
\end{proposition}
\goodbreak\medskip\noindent{\small\bf Proof: } In Appendix section \ref{sec:ap2}
\begin{definition} \label{def:a_mvs} The matrices $\tilde{\mathbf{M}}_{\pm}^{q}[n]$ and ${\mathbf{M}}_{\pm}^{q}[n]$ are called the analysis\ and synthesis\ modulation matri ces for the qWP transform, respectively.\end{definition}
\begin{corollary} \label{cor:a_mvs}Successive application of filter bank s defined by the analysis\ and synthesis\ modulation matri ces $\tilde{\mathbf{M}}_{\pm}^{q}[n]$ and ${\mathbf{M}}_{\pm}^{q}[n]$ to a signal\ $\mathbf{x}\in \Pi[N]$ produces the analytic\ signal s $\mathbf{\bar{x}}_{\pm}$ associated with $\mathbf{x}$.\end{corollary}
\begin{corollary} \label{cor:on_sys}A signal\ $\mathbf{x}\in \Pi[N]$ is represent ed by the redundant orthonormal\ system
\begin{eqnarray*}\label{on_sys}
x[k]&=&\frac{1}{2}\sum_{\mu=0}^{1}\sum_{l=0}^{N/2-1}\left(y_{[1]}^{\mu} [l]\psi_{[1],\mu}^{2r}[k-2l] +
c_{[1]}^{\mu} [l] \f_{[1],\mu}^{2r}[k-2l]\right),\\\label{on_sys_yc}
y_{[1]}^{\mu} [l]&=& \left\langle \mathbf{x},\, \psi_{[1],\mu}^{2r}[\cdot-2l] \right\rangle,\quad
c_{[1]}^{\mu} [l]= \left\langle \mathbf{x},\, \f_{[1],\mu}^{2r}[\cdot-2l] \right\rangle.
\end{eqnarray*}
Thus, the system
\begin{equation*}\label{on_sysTF}
\mathbf{F} \stackrel{\Delta}{=} \left\{\psi_{[1],0}^{2r}[\cdot-2l] \right\}\bigoplus \left\{\psi_{[1],1}^{2r}[\cdot-2l] \right\}\bigcup\left\{\f_{[1],0}^{2r}[\cdot-2l] \right\}\bigoplus \left\{\f_{[1],1}^{2r}[\cdot-2l] \right\}
\end{equation*}
form a tight frame of the space $ \Pi[N]$.
\end{corollary}
\subsection{Multi-level transform s}\label{sec:ss42}
It was explained in Section \ref{sec:sss242} that the second-level transform\ coefficient s $\mathbf{y}_{[2]}^{\rr}$ are
\begin{eqnarray*}
{y}_{[2]}^{\rr} [l]&=& \sum_{n=0}^{N-1}x[n]\,{\psi}_{[2],\rr}^{2r}[n-4l], \quad {\psi}_{[2],\rr}^{2r}[n] =\sum_{k=0}^{N/2-1}{h}_{[2]}^{\mu}[k] \, {\psi}_{[1],\la}^{2r}[n-2k]\Longrightarrow\\
{y}_{[2]}^{\rr} [l]&=& \sum_{k=0}^{N/2-1}h_{[2]}^{\mu}[k-2l] \, y_{[1]}^{\la}[k], \quad \la,\mu=0,1,\;\rr=\left\{
\begin{array}{ll}
\mu, & \hbox{if $\la=0$ ;} \\
3-\mu, & \hbox{if $\la=1$.}
\end{array}
\right.
\end{eqnarray*}
The frequency response s of the p-filter s $\mathbf{h}_{[2]}^{\mu}$ are given in Eq. \rf{pfs2} and Eq. \rf{df_wq2}. Recall that $\hat{h}_{[2]}^{\mu}[n]=\hat{h}_{[1]}^{\mu}[2n]$. The direct and inverse transform s
$\mathbf{y}_{[1]}^{\la}\longleftrightarrow\mathbf{y}_{[2]}^{2\la}\bigcup\mathbf{y}_{[2]}^{2\la+1}$ are implemented using the analysis\ and synthesis\ modulation matri ces $\tilde{\mathbf{M}}[2n]$ and $\mathbf{M}[2n]$, that are defined in Eqs. \rf{aa_modma10} and \rf{sa_modma10} respectively:
\begin{equation*}\label{mod_decA20}
\left(
\begin{array}{c}
\hat{y}_{[2]}^{\rr0}[n]_{2} \\
\hat{y}_{[2]}^{\rr1}[n]_{2}\\
\end{array}
\right)=\frac{1}{2} \tilde{\mathbf{M}}[-2n]\cdot \left(
\begin{array}{l}
\hat{y}_{[1]}^{\la}[n]_{1} \\
\hat{y}_{[1]}^{\la}[\vec{n}]_{1}
\end{array}
\right),\quad\left(
\begin{array}{l}
\hat{y}_{[1]}^{\la}[n]_{1} \\
\hat{y}_{[1]}^{\la}[\vec{n}]_{1}
\end{array}
\right)={\mathbf{M}}[2n]\cdot\left(
\begin{array}{c}
\hat{y}_{[2]}^{\rr0}[n]_{2} \\
\hat{y}_{[2]}^{\rr1}[n]_{2}\\
\end{array}
\right),
\end{equation*}
where
\begin{equation*}\label{rr_01}
\rr0=\left\{
\begin{array}{ll}
0, & \hbox{if $\la=0$;} \\
3, & \hbox{if $\la=1$,}
\end{array}
\right. \quad \rr1=\left\{
\begin{array}{ll}
1, & \hbox{if $\la=0$;} \\
2, & \hbox{if $\la=1$,}
\end{array}
\right.\quad \vec{n}=n+N/4.
\end{equation*}
The second-level transform\ coefficient s $\mathbf{c}_{[2]}^{\rr}$ are
\begin{eqnarray*}
{c}_{[2]}^{\rr} [l]&=& \sum_{n=0}^{N-1}x[n]\,{\f}_{[2],\rr}^{2r}[n-4l], \quad {\f}_{[2],\rr}^{2r}[n] =\sum_{k=0}^{N/2-1}{h}_{[2]}^{\mu}[k] \, {\f}_{[1],\la}^{2r}[n-2k]\Longrightarrow\\
{c}_{[2]}^{\rr} [l]&=& \sum_{k=0}^{N/2-1}h_{[2]}^{\mu}[k-2l] \, c_{[1]}^{\la}[k], \quad \la,\mu=0,1,\;\rr=\left\{
\begin{array}{ll}
\mu, & \hbox{if $\la=0$ ;} \\
3-\mu, & \hbox{if $\la=1$.}
\end{array}
\right.
\end{eqnarray*}
We emphasize that the p-filter s $\mathbf{h}_{[2]}^{\mu}$ for the transform\ $\mathbf{c}_{[1]}^{\la}\longleftrightarrow\mathbf{c}_{[2]}^{2\la}\bigcup\mathbf{c}_{[2]}^{2\la+1}$ are the same that the p-filter s for the transform\ $\mathbf{y}_{[1]}^{\la}\longleftrightarrow\mathbf{y}_{[2]}^{2\la}\bigcup\mathbf{y}_{[2]}^{2\la+1}$. Therefore, the direct and inverse transform s
$\mathbf{c}_{[1]}^{\la}\longleftrightarrow\mathbf{c}_{[2]}^{2\la}\bigcup\mathbf{c}_{[2]}^{2\la+1}$ are implemented using the same analysis\ and synthesis\ modulation matri ces $\tilde{\mathbf{M}}[2n]$ and $\mathbf{M}[2n]$. Apparently, it is the case also for the transform s $\mathbf{z}_{\pm[1]}^{\la}\longleftrightarrow\mathbf{z}_{\pm[2]}^{2\la}\bigcup\mathbf{z}_{\pm[2]}^{2\la+1}$. The transform s to subsequent decomposition\ levels are implemented in an iterative way:
\begin{eqnarray*}\label{mod_decA2m}
\left(
\begin{array}{c}
\hat{z}_{\pm[m+1]}^{\rr0}[n]_{m+1} \\
\hat{z}_{\pm[m+1]}^{\rr1}[n]_{m+1}\\
\end{array}
\right)&=&\frac{1}{2} \tilde{\mathbf{M}}[-2^{m}n]\cdot \left(
\begin{array}{l}
\hat{z}_{\pm[m]}^{\la}[n]_{m} \\
\hat{z}_{\pm[m]}^{\la}[\vec{n}]_{m}
\end{array}
\right),\\\nonumber\left(
\begin{array}{l}
\hat{z}_{\pm[m]}^{\la}[n]_{m} \\
\hat{z}_{\pm[m]}^{\la}[\vec{n}]_{m}
\end{array}
\right)&=&{\mathbf{M}}[2^{m}n]\cdot\left(
\begin{array}{c}
\hat{z}_{\pm[m+1]}^{\rr0}[n]_{m+1} \\
\hat{z}_{\pm[m+1]}^{\rr1}[n]_{m+1}\\
\end{array}
\right),
\end{eqnarray*}
where
\(
\rr0=\left\{
\begin{array}{ll}
2\la, & \hbox{if $\la$ is even;} \\
2\la+1, & \hbox{if $\la$ is odd,}
\end{array}
\right. \) and vice versa for $\rr1$, $\vec{n}=n+N/2^{m+1}$ and $m=1,...,M$.
By the application of the inverse DFT to the arrays $\left\{ \hat{z}_{\pm[m+1]}^{\rr}[n]_{m+1}\right\}$, we get the arrays\\ $\left\{ z_{\pm[m+1]}^{\rr}[k]=y_{[m+1]}^{\rr}[k]\pm i\,c_{[m+1]}^{\rr}[k]\right\}$ of the transform\ coefficient s with the qWPs $\Psi^{2r}_{\pm[m+1],\rr}$.
\begin{rmk}\label{rem:zyc_cyz}We stress that by operating on the transform\ coefficient s $\left\{ z_{\pm[m]}^{\rr}[k]\right\}$, we simultaneously operate on the arrays $\left\{ y_{[m]}^{\rr}[k]\right\}$ and $\left\{ c_{[m]}^{\rr}[k]\right\}$, which are the coefficient s for the transform s with the WPs $\psi^{2r}_{[m],\rr}$ and cWPs $\f^{2r}_{[m],\rr}$, respectively. The execution speed of the transform\ with the qWPs $\left\{ \Psi_{\pm[m]}^{2r}\right\}= \psi_{[m]}^{2r}\pm i \f_{[m]}^{2r}$ is the same as the speed of the transform s with either WPs $\left\{ \psi_{[m]}^{2r}\right\}$ or cWPs $\left\{ \f_{[m]}^{2r}\right\}$.\end{rmk}
The transform s are executed in the spectral domain using the FFT by the application of critically sampled two-channel filter bank s to the half-band spectr al components $(\hat{x}[n],\hat{x}[n+N/2])^{T}$ of a signal.
The diagrams in Figs. \ref{dia_wq_A} and \ref{dia_wq_S} illustrate the three-level forward and inverse qWPTs of a signal\ with quasi-analytic\ wavelet packet s, which use the analysis\ $\tilde{\mathbf{M}}^{q}[n]$ and the synthesis\ ${\mathbf{M}}^{q}[n]$ modulation matri ces, respectively, for the transform s to and from the first decomposition\ level, respectively, and the modulation matri ces $\tilde{\mathbf{M}}[2^{m}n]$ and ${\mathbf{M}}[2^{m}n]$ for the subsequent levels.
\begin{figure}[H]
\begin{center}
\resizebox{15cm}{8cm}{
\includegraphics{png/dia_wq_aA.png}
}
\end{center}
\caption{ Forward qWTP of a signal\ $\mathbf{X}$ down to the third decomposition\ level with quasi-analytic\ wavelet packet s. Here $\vec{n}$ means ${n}+N/8$ }
\label{dia_wq_A}
\end{figure}
\begin{figure}[H]
\begin{center}
\resizebox{15cm}{8cm}{
\includegraphics{png/dia_wq_SAh.png}
}
\end{center}
\caption{ Inverse qWTP from the transform\ coefficient s from the third decomposition\ level that results in restoration of the signal\ $\mathbf{X}$ and its HT $H(\mathbf{X})$ }
\label{dia_wq_S}
\end{figure}
\begin{rmk}\label{rem:recon}The decomposition\ of a signal\ $\mathbf{x}\in\Pi[N]$ down to the $M$-th level produces $2MN$ transform\ coefficient s $\left\{ y_{[m]}^{\rr}[k]\right\}\bigcup\left\{ c_{[m]}^{\rr}[k]\right\}$. Such a redundancy provides many options for the signal\ reconstruction. Some of them are listed below.
\begin{itemize}
\item A basis compiled from either WPs $\left\{ \psi_{[m]}^{2r}\right\}$ or $\left\{ \f_{[m]}^{2r}\right\}$.
\begin{itemize}
\item Wavelet basis.
\item Best bases \cite{coiw1}, Local discriminant bases \cite{sai,sai2}.
\item WPs from a single decomposition\ level.
\end{itemize}
\item Combination of bases compiled from both $\left\{ \psi_{[m]}^{2r}\right\}$ and $\left\{ \f_{[m]}^{2r}\right\}$ WPs generates a tight frame of the space $\Pi[N]$ with redundancy rate 2. The bases for $\left\{ \psi_{[m]}^{2r}\right\}$ and $\left\{ \f_{[m]}^{2r}\right\}$ can have a different structure.
\item Frames with increased redundancy rate. For example, a combined reconstruction\ from several decomposition\ levels.
\end{itemize}
\end{rmk}
The collection of WPs $\left\{ \psi_{[m]}^{2r}\right\}$ and cWPs $\left\{ \f_{[m]}^{2r}\right\}$, which originate from discrete spline s of differen t orders $2r$, provides a variety of waveform s that are (anti)symmetr ic, well localized in time domain. Their DFT spectr a are flat and the spectr a shapes tend to rectangles when the order $2r$ increases. Therefore, they can be utilized as a collection of band-pass filter s which produce a refined split of the frequency\ domain into bands of differen t widths. The (c)WPs can be used as testing waveform s for the signal\ analysis, such as a dictionary for the Matching Pursuit procedures \cite{mal,azk_MP}.
\section{Two-dimensional complex wavelet packet s}\label{sec:s5}
A standard design scheme for 2D wavelet packet s is outlined in Section \ref{sec:ss25}. The 2D wavelet packet s are defined as the tensor products of 1D WPs such that
\begin{equation*}\label{psipsi0}
\psi_{[m],j ,l}^{2r}[k,n]=\psi_{[m],j}^{2r}[k]\,\psi_{[m], l}^{2r}[n].
\end{equation*}
The $2^{m}$-sample shifts of the WPs $\left\{\psi_{[m],j ,l}^{2r}\right\},\;j , l=0,...,2^{m}-1,$ in both directions form an orthonormal\ basis for the space $\Pi[N,N]$ of arrays that are $N$-periodic\ in both directions. The DFT spectr um of such a WP is concentrated in four symmetr ic spots in the frequency\ domain as it is seen in Fig. \ref{psifpsi2_2}.
Similar properties are inherent to the 2D cWPs such that
\begin{equation*}\label{phiphi}
\f_{[m],j ,l}^{2r}[k,n]=\f_{[m],j}^{2r}[k]\,\f_{[m], l}^{2r}[n].
\end{equation*}
\subsection{Design of 2D directional WPs}\label{sec:ss51}
\subsubsection{2D complex WPs and their spectr a }\label{sec:sss511}
The WPs $\left\{\psi_{[m],j ,l}^{2r}\right\}$ as well as the cWPs $\left\{\f_{[m],j ,l}^{2r}\right\}$ lack the directionality property which is needed in many applications that process 2D data. However, real-valued 2D wavelet packet s oriented in multiple directions can be
derived from tensor products of complex quasi-analytic\ qWPs $\Psi_{\pm[m],\rr}^{2r}$.
The complex 2D qWPs are defined as follows:
\begin{eqnarray*} \label{qwp_2d}
\Psi_{++[m],j , l}^{2r}[k,n] &\stackrel{\Delta}{=}& \Psi_{+[m],j}^{2r}[k]\,\Psi_{+[m], l}^{2r}[n], \\\nonumber
\Psi_{+-[m],j ,l}^{2r}[k,n] &\stackrel{\Delta}{=}& \Psi_{+[m],j}^{2r}[k]\,\Psi_{-[m], l}^{2r}[n],
\end{eqnarray*}
where $ m=1,...,M,\;j ,l=0,...,2^{m}-1,$ and $k ,n=-N/2,...,N/2-1$.
The real and imaginary parts of these 2D qWPs are
\begin{equation}
\label{vt_pm}
\begin{array}{lll}
\vt_{+[m],j ,l}^{2r}[k,n] &\stackrel{\Delta}{=}& \mathfrak{Re}(\Psi_{++[m],j ,l}^{2r}[k,n]) = \psi_{[m],j ,l}^{2r}[k,n]-\f_{[m],j ,l}^{2r}[k,n], \\
\vt_{-[m],j ,l}^{2r}[k,n] &\stackrel{\Delta}{=}& \mathfrak{Re}(\Psi_{+-[m],j ,l}^{2r}[k,n]) = \psi_{[m],j ,l}^{2r}[k,n]+\f_{[m],j ,l}^{2r}[k,n],\\%\label{th_pm}
\end{array}
\end{equation}
\begin{equation}
\label{th_pm}
\begin{array}{lll}
\th_{+[m],j ,l}^{2r}[k,n] &\stackrel{\Delta}{=}& \mathfrak{Im}(\Psi_{++[m],j ,l}^{2r}[k,n]) = \psi_{[m],j}^{2r}[k]\,\f_{[m], l}^{2r}[n]+\f_{[m],j}^{2r}[k]\,\psi_{[m], l}^{2r}[n], \\
\th_{-[m],j ,l}^{2r}[k,n] &\stackrel{\Delta}{=}& \mathfrak{Im}(\Psi_{+-[m],j ,l}^{2r}[k,n]) = \f_{[m],j}^{2r}[k]\,\psi_{[m], l}^{2r}[n]-\psi_{[m],j}^{2r}[k]\,\f_{[m], l}^{2r}[n].
\end{array}
\end{equation}
The DFT spectr a of the 2D qWPs $\Psi_{++[m],j ,l}^{2r},\;j ,l=0,...,2^{m}-1,$ are the tensor products of the one-sided spectr a of the qWPs:
\begin{equation*}\label{fPsi_pp}
\hat{ \Psi}_{++[m],j ,l}^{2r}[p,q] =\hat{ \Psi}_{+[m],j}^{2r}[p]\,\hat{\Psi}_{+[m], l}^{2r}[q]
\end{equation*}
and, as such, they fill the quadrant $k ,n=0,...,N/2-1$ of the frequency\ domain, while the spectr a of $\Psi_{+-[m],j ,l}^{2r},\;j ,l=0,...,2^{m}-1,$ fill the quadrant $k =0,...,N/2-1,\;n=-N/2,...,-1$. Figures \ref{fpp_2} and \ref{fmp_2} display the magnitude spectr a of the tenth-order 2D qWPs $\Psi_{++[2],j ,l}^{10}$ and $\Psi_{+-[2],j ,l}^{10}$ from the second decomposition\ level, respectively.
\begin{figure}[H]
\centering
\includegraphics[width=6.2in]{png/fpp_2.png}
\caption{Magnitude spectr a of 2D qWPs $\Psi_{++[2],j ,l}^{10}$ from the second decomposition\ level}
\label{fpp_2}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=6.2in]{png/fpm_2.png}%
\caption{Magnitude spectr a of 2D qWPs $\Psi_{+-[2],j ,l}^{10}$ from the second decomposition\ level}
\label{fmp_2}
\end{figure}
\begin{rmk}\label{rem:dif_lev}The 2D qWPs $\Psi_{+\pm[m],j , l}^{2r}$ are the tensor products of 1D qWPs from the decomposition\ level $m$. However, there is no problems to design the 2D qWPs as a tensor products of 1D qWPs from differen t decomposition\ levels such as
\(\Psi_{+\pm[m,s],j , l}^{2r}[k,n] \stackrel{\Delta}{=} \Psi_{+[m],j}^{2r}[k]\,\Psi_{\pm[s], l}^{2r}[n].\)
\end{rmk}
\subsubsection{Directionality of real-valued 2D WPs}\label{sec:sss512}
It is seen in Fig. \ref{fpp_2} that the DFT spectr a of the qWPs $\Psi_{+\pm[m],j ,l}^{10}$ effectively occupy relatively small squares in the frequency\ domain. For deeper decomposition\ levels, sizes of the corresponding squares decrease on geometric progression. Such configurations of the spectr a lead to the directionality of the real-valued 2D WPs $ \vt_{\pm[m],j ,l}^{2r}$ and $ \th_{\pm[m],j ,l}^{2r}$.
Assume, for example, that $N=512, \;m=3,\; j=2,\; l=5$ and denote $\Psi[k,n]\stackrel{\Delta}{=} \Psi_{++[3],2 ,5}^{2r}[k,n]$ and $ \vt[k,n]\stackrel{\Delta}{=}\mathfrak{Re}(\Psi[k,n])$. Its magnitude spectr um $\left|\hat{\Psi}[j,l]\right|$, displayed in Fig, \ref{78_178}, effectively occupies the square of size $40\times 40$ \emph{pixels} centered around the point $\mathbf{C}=[\k_{0},\n_{0}]$, where $\k_{0}=78, \;\n_{0}=178$. Thus, the WP $\Psi$ is represent ed by
\begin{eqnarray*}
\label{psi78_178}
\Psi[k,n] &=& \frac{1}{N^{2}}\sum_{\k,\n=0}^{N/2-1}\w^{k\k+n\n}\, \hat{\Psi}[\k,\n]\approx{\w^{\k_{0}k+\n_{0}n}}\,\underline{\Psi}[k,n] \\\nonumber
\underline{\Psi}[k,n] &\stackrel{\Delta}{=}& \frac{1}{N^{2}}\sum_{\k,\n=-20}^{19}\w^{k\k+n\n} \, \hat{\Psi}[\k+\k_{0},\n+\n_{0}].
\end{eqnarray*}
Consequently, the real-valued WP $\vt $ is represent ed as follows:
\begin{eqnarray*}
\label{th78_178}
\vt[k,n] \approx{\cos\frac{2\pi(\k_{0}k+\n_{0}n)}{N}}\,\underline{\vt}[k,n] ,\quad \underline{\vt}[k,n] \stackrel{\Delta}{=}\mathfrak{Re}(\underline{\Psi}[k,n]).
\end{eqnarray*}
The spectr um of the 2D signal\ $\underline{\vt}$ comprises only f low frequencies in both directions and the signal\ $\underline{\vt}$ does not have a directionality. But the 2D signal\ $\cos\frac{2\pi(\k_{0}k+\n_{0}n)}{N}$ is oscillating in the direction of the vector $\vec{V}_{++[2],2 ,5}=178\vec{i}+78\vec{j}$. The 2D WP $\vt[k,n]$ is well localized in the spatial domain as is seen from Eq. \rf{vt_pm} and the same is true for the low-frequency\ signal\ $\underline{\vt}$. Therefore, WP $\vt[k,n]$ can be regarded as the directional cosine modulated by the localized low-frequency\ signal\ $\underline{\vt}$.
The same arguments, which to some extent are similar to the discussion in Section 6.2 of \cite{bhan_zhao}, are applicable to all four real-valued 2D WPs defined in Eqs. \rf{vt_pm} -- \rf{th_pm}. Figure \ref{AB_3_2_5} displays the low-frequency\ signal\ $\underline{\vt}$, its magnitude spectr um and the 2D WP $\vt[k,n]$.
\begin{SCfigure}
\centering
\caption{Magnitude spectr a of 2D qWP $\Psi$ (left) and $\mathfrak{Re}(\Psi)=\vt$ (right)}
\includegraphics[width=3.2in]{png/78_178.png}
\label{78_178}
\end{SCfigure}
\begin{figure}[H]
\centering
\includegraphics[width=3.2in]{png/AB3_2_5.png}
\caption{Center: Low-frequency\ signal\ $\underline{\vt}$. Left: Its magnitude spectr um. Right: 2D WP $\vt[k,n]$}
\label{AB_3_2_5}
\end{figure}
Figures \ref{pp_2_2d} and \ref{fpp_2_2d} display WPs $\vt_{+[2],j ,l}^{10},\;j,l=0,1,2,3,$ from the second decomposition\ level and their magnitude spectr a, respectively.
Figures \ref{pm_2_2d} and \ref{fpm_2_2d} display WPs $\vt_{-[2],j ,l}^{10},\;j,l=0,1,2,3,$ from the second decomposition\ level and their magnitude spectr a, respectively.
\begin{figure}[H]
\centering
\includegraphics[width=6.2in]{png/pp_2_2d.png}
\caption{WPs $\vt_{+[2],j ,l}^{10}$ from the second decomposition\ level}
\label{pp_2_2d}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=5.2in]{png/fpp_2_2d.png}
\caption{Magnitude spectr a of WPs $\vt_{+[2],j ,l}^{10}$ from the second decomposition\ level }
\label{fpp_2_2d}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=6.2in]{png/pm_2_2d.png}
\caption{WPs $\vt_{-[2],j ,l}^{10}$ from the second decomposition\ level}
\label{pm_2_2d}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=5.2in]{png/fpm_2_2d.png}
\caption{Magnitude spectr a of WPs $\vt_{-[2],j ,l}^{10}$ from the second decomposition\ level }
\label{fpm_2_2d}
\end{figure}
\begin{rmk}\label{direc_rem}Note that orientations of the vector s $\vec{V}_{++[m],j, l}$ and $\vec{V}_{++[m],j+1, l+1}$ are approximat ely the same. These vector s determine the orientations of the WPs $\vt_{+[m],j ,l}^{2r}$ and $\vt_{+[m],j+1 , l+1}^{2r}$, respectively. Thus, these WPs have approximat ely the same orientation. Consequently, the WPs from the $m$-th decomposition\ level are oriented in $2^{m+1}-1$ differen t directions. The same is true for the WPs $\vt_{-[m],j ,l}^{2r}$. Thus, altogether, at the level $m$ we have WPs oriented in $2(2^{m+1}-1)$ differen t directions. It is seen in Figs. \ref{pp_2_2d}, \ref{pm_2_2d} and in Figs. \ref{pp_3_2d}, \ref{pm_3_2d} that display the WPs $\vt_{\pm[3],j ,l}^{2r}$.\end{rmk}
\begin{figure}[H]
\centering
\includegraphics[width=6.2in]{png/pp_3_2d.png}
\caption{WPs $\vt_{+[3],j ,l}^{10}$ from the third decomposition\ level }
\label{pp_3_2d}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=6.2in]{png/pm_3_2d.png}%
\caption{WPs $\vt_{-[3],j ,l}^{10}$ (right) from the third decomposition\ level }
\label{pm_3_2d}
\end{figure}
\section{Implementation of 2D qWP transform s}\label{sec:s6}
The spectr a of 1D qWPs $\left\{\Psi_{+[m],j }^{2r}\right\},\;j =0,...,2^{m}-1$, fill the non-negative half-band $[0,N/2]$, and vice versa for the qWPs $\left\{\Psi_{-[m],j }^{2r}\right\},\;j =0,...,2^{m}-1$ . Therefore, the spectr a of 2D qWPs $\left\{\Psi_{++[m],j ,l}^{2r}\right\},\;j ,l=0,...,2^{m}-1$ fill the quadrant $[0,N/2-1]\times[0,N/2-1]$ of the frequency\ domain, while the spectr a of 2D qWPs $\left\{\Psi_{+-[m],j ,l}^{2r}\right\}$ fill the quadrant $[0,N/2-1]\times[-N/2,-1]$. It is clearly seen in Fig. \ref{fpm_1_2d}.
\begin{figure}[H]
\centering
\includegraphics[width=3.0in]{png/fp_1_2d.png}
\hfil
\includegraphics[width=3.0in]{png/fpm_1_2d.png}%
\caption{Magnitude spectr a of qWPs $\Psi_{++[1],j ,l}^{10}$ (left) and $\Psi_{+-[1],j ,l}^{10}$ (right) from the first decomposition\ level }
\label{fpm_1_2d}
\end{figure}
Consequently, the spectr a of the real-valued 2D WPs $\left\{\vt_{+[m],j ,l}^{2r}\right\},\;j ,l=0,...,2^{m}-1$, and $\left\{\vt_{-[m],j ,l}^{2r}\right\}$ fill the pairs of quadrant $\mathbf{Q}_{+}\stackrel{\Delta}{=}[0,N/2-1]\times[0,N/2-1]\bigcup[-N/2,-1]\times[-N/2,-1]$ and $\mathbf{Q}_{-}\stackrel{\Delta}{=}[0,N/2-1]\times[-N/2,-1]\bigcup[-N/2,-1]\times[0,N/2-1]$, respectively (Figs. \ref{fpp_2_2d} and \ref{fpm_2_2d}).
By this reason, none linear combination of the WPs $\left\{\vt_{+[m],j ,l}^{2r}\right\}$ and their shifts can serve as a basis in the signal\ space $\Pi[N,N]$. The same is true for WPs $\left\{\vt_{-[m],j ,l}^{2r}\right\}$. However, combinations of the WPs $\left\{\vt_{\pm[m],j ,l}^{2r}\right\}$ provide frames of the space $\Pi[N,N]$.
\subsection{One-level 2D transform s}\label{sec:ss61}
The one-level 2D qWP transform s of a signal\ $\mathbf{X}=\left\{X[k,n] \right\}\in\Pi[N,N]$ are implemented by a tensor-product scheme mentioned in Section \ref{sec:ss25}.
\subsubsection{Direct transforms with qWPs $\Psi^{2r}_{+\pm[1]}$}\label{sec:sss611}
Denote by $\tilde{\mathbf{T}}_{\pm}^{h}$ the 1D transform s of row signal s from $\Pi[N]$ with the analysis\ modulation matri ces $\tilde{\mathbf{M}}_{\pm}^{q}$ which are defined in Eq. \rf{aa_modma10p}. Application of these transform s to rows of a signal\ \textbf{X} produces the coefficient\ arrays
\begin{eqnarray*}\label{tTh+x}
\tilde{\mathbf{T}}_{+}^{h}\mathbf{\cdot}\mathbf{X} &=& \left(\za_{+}^{0},\za_{+}^{1}\right),\quad \za_{+}^{j}[k,n]=\eta^{j}[k,n]-i\,\xi^{j}[k,n],
\\\nonumber \tilde{\mathbf{T}}_{-}^{h}\mathbf{\cdot}\mathbf{X} &=& \left(\za_{-}^{0},\za_{-}^{1}\right),\quad \za_{-}^{j}[k,n]=\eta^{j}[k,n]+i\,\xi^{j}[k,n]=(\za_{+}^{j}[k,n])^{*},
\\\nonumber
{\eta}^{j}[k,n] &=&\left\langle \mathbf{X}[k,\cdot],{\psi}^{2r}_{[1],j}[\cdot -2n]\right\rangle,\quad {\xi}^{j}[k,n]=\left\langle \mathbf{X}[k,\cdot],{\f}^{2r}_{[1],j}[\cdot -2n]\right\rangle,\;j=0,1.
\end{eqnarray*}
Denote by ${\mathbf{T}}_{\pm}^{h}$ the 1D inverse transform s with the synthesis\ modulation matri ces ${\mathbf{M}}_{\pm}^{q}$. Due to Proposition \ref{pro:Mq_z}, application of these transform s to rows of the coefficient\ arrays $\za_{\pm}=\left(\za_{\pm}^{0},\za_{\pm}^{1}\right)$, respectively, produces the 2D analytic\ signal s:
\begin{equation}\label{Th_za}
{\mathbf{T}}_{\pm}^{h}\mathbf{\cdot}(\za_{\pm}^{0},\za_{\pm}^{1})=2\bar{\mathbf{X}}_{\pm}=2(\mathbf{X}\pm i\,H(\mathbf{X})),
\end{equation}
where H(\textbf{X}) is the 2D signal\ consisting of the HTs of rows of the signal\ \textbf{X}.
Denote by $\tilde{\mathbf{T}}_{+}^{v}$ the direct 1D transform\ determined by the modulation matri x $\tilde{\mathbf{M}}_{+}^{q}$ applicable to columns of the corresponding signal s. The next step of the tensor product transform\ consists of the application of the 1D transform\ $\tilde{\mathbf{T}}_{+}^{v}$ to columns of the arrays ${\za}^{j},\;j=0,1.$ As a result, we get four transform\ coefficient s arrays:
\begin{eqnarray*}\label{tTv_za0}
\tilde{\mathbf{T}}_{+}^{v}\mathbf{\cdot}\za_{+}^{l} &=& \tilde{\mathbf{T}}_{+}^{v}\mathbf{\cdot}\left(\eta^{l}-i\,\xi^{l} \right)=\left\{
\begin{array}{l}
\left( \alpha^{0,l}-i\,\beta^{0,l} \right)-i\left( \g^{0,l}-i\,\delta^{0,l} \right) \\
\left( \alpha^{1,l}-i\,\beta^{1,l} \right)-i\left( \g^{1,l}-i\,\delta^{1,l} \right)
\end{array}
\right.
\\\nonumber&=&\left\{
\begin{array}{l}
\mathbf{Z}_{+[1]}^{0,l}= \mathbf{Y}_{+[1]}^{0,l}-i\, \mathbf{C}_{+[1]}^{0,l},\quad \mathbf{Y}_{+[1]}^{0,l}=\alpha^{0,l}-\delta^{0,l},\quad \mathbf{C}_{+[1]}^{0,l}=\beta^{0,l}+\g^{0,l} \\
\mathbf{Z}_{+[1]}^{1,l}= \mathbf{Y}_{+[1]}^{1,l}-i\, \mathbf{C}_{+[1]}^{1,l},\quad \mathbf{Y}_{+[1]}^{1,l}=\alpha^{1,l}-\delta^{1,l},\quad \mathbf{C}_{+[1]}^{1,l}=\beta^{1,l}+\g^{1,l}
\end{array}
\right.
,\\\nonumber
\alpha^{j,l}[k,n]&=&\sum_{\la,\mu=0}^{N-1}X[\la,\mu],{\psi}^{2r}_{[1],j}[\la -2k]{\psi}^{2r}_{[1],l}[\mu -2n],\\\nonumber
\delta^{j,l}[k,n]&=&\sum_{\la,\mu=0}^{N-1}X[\la,\mu],{\f}^{2r}_{[1],j}[\la -2k]{\f}^{2r}_{[1],l}[\mu -2n],\\\nonumber
\beta^{j,l}[k,n]&=&\sum_{\la,\mu=0}^{N-1}X[\la,\mu],{\psi}^{2r}_{[1],j}[\la -2k]{\f}^{2r}_{[1],l}[\mu -2n],\\\nonumber
\g^{j,l}[k,n]&=&\sum_{\la,\mu=0}^{N-1}X[\la,\mu],{\f}^{2r}_{[1],j}[\la -2k]{\psi}^{2r}_{[1],l}[\mu -2n],\quad j,l=0,1.
\end{eqnarray*}
Hence, it follows that
\begin{equation}
\label{Zjl+}
\begin{array}{lll}
Y_{+[1]}^{j,l}[k,n]&=&\sum_{\la,\mu=0}^{N-1}X[\la,\mu]\,{\vt}^{2r}_{+[1],j,l}[\la -2k,\mu -2n],\\ C^{j,l}_{+[1]}[k,n]&=&\sum_{\la,\mu=0}^{N-1}X[\la,\mu]\,{\th}^{2r}_{+[1],j,l}[\la -2k,\mu -2n],\\
Z_{+[1]}^{j,l}[k,n]&=&\sum_{\la,\mu=0}^{N-1}X[\la,\mu]\,{\Psi}^{2r}_{++[1],j,l}[\la -2k,\mu -2n],\quad j,l=0,1.
\end{array}
\end{equation}
\begin{rmk}\label{rem:+rec}Recall that the DFT spectr a of WPs ${\vt}^{2r}_{+[1],j,l}$ and ${\th}^{2r}_{+[1],j,l},\;j,l=0,1,$ which are the real and imaginary parts of the qWP ${\Psi}^{2r}_{++[1],j,l}$, are confined within the area $\mathbf{Q}_{+}$ of the frequency\ domain. It is seen from Eq. \rf{Zjl+} that if at least a part of the spectr um of a signal\ $\mathbf{X}\in\Pi[N,N]$ is located in the area $\mathbf{Q}_{-}$, then the signal\ $\mathbf{X}\in\Pi[N,N]$ cannot be fully restored from the transform\ coefficient s $Z_{+[1]}^{j,l}[k,n]$, although their number is the same as the number of samples in the signal\ $\mathbf{X}$. To achieve a perfect reconstruction, the coefficient s from the arrays $\mathbf{Z}_{-[1]}^{j,l}$ should be incorporated.\end{rmk}
The coefficient\ arrays $\mathbf{Z}_{-[1]}^{j,l}$ are derived in the same way as the arrays $\mathbf{Z}_{+[1]}^{j,l}$. The only differen ce is that, for the 1D transform\ $\tilde{\mathbf{T}}^{h}_{-}$ the modulation matri x
$\tilde{\mathbf{M}}_{-}^{q}$ is used instead of $\tilde{\mathbf{M}}_{+}^{q}$. For the transform\ $\tilde{\mathbf{T}}^{v}_{-}$, the modulation matri x
$\tilde{\mathbf{M}}_{+}^{q}$ is used. Consequently, to derive the coefficient\ arrays $\mathbf{Z}_{-[1]}^{j,l}$, the transform\ $\tilde{\mathbf{T}}_{+}^{v}$ should be applied to columns of the arrays $\za^{l}_{-}=(\za^{l}_{+})^{*}$. As a result, we get
\begin{eqnarray*}
\label{Zjl-}
\mathbf{Z}_{-[1]}^{j,l}= \mathbf{Y}_{-[1]}^{j,l}+i\,\mathbf{C}_{-[1]}^{j,l} =\sum_{\la,\mu=0}^{N-1}X[\la,\mu],{\Psi}^{2r}_{+-[1],j,l}[\la -2k,\mu -2n],\; j,l=0,1.
\end{eqnarray*}
\subsubsection{Inverse transforms with qWPs $\Psi^{2r}_{+\pm[1]}$}\label{sec:sss612}
Denote by $\mathbf{T}_{+}^{v}$ the 1D inverse transform\ with the synthesis\ modulation matri x ${\mathbf{M}}_{+}^{q}$ applicable to columns of the coefficient\ arrays. Denote by $H(\za_{\pm}^{l}),\;l=0,1,$
the HTs of the arrays consisting of columns of the coefficient\ arrays $\za_{\pm}^{j}$.
Proposition \ref{pro:Mq_z} implies that
\begin{eqnarray*}
\mathbf{T}_{+}^{v}\mathbf{\cdot} \left(
\begin{array}{c}
\mathbf{Z}_{+[1]}^{0,l} \\
\mathbf{Z}_{+[1]}^{1,l} \\
\end{array}
\right)=2\bar{\za}_{+}^{l},\quad \mathbf{T}_{+}^{v}\mathbf{\cdot} \left(
\begin{array}{c}
\mathbf{Z}_{-[1]}^{0,l} \\
\mathbf{Z}_{-[1]}^{1,l} \\
\end{array}
\right)=2\bar{\za}_{-}^{l}
\end{eqnarray*}
where $\;l=0,1$ and $\bar{\za}_{\pm}^{l}=\za_{\pm}^{l}+i\,H(\za_{\pm}^{l})$ are analytic\ coefficient\ arrays. Denote by $\mathbf{G}$ a signal\ from $\Pi[N,N]$ such that
\begin{equation}\label{sig_G}
\tilde{\mathbf{T}}^{h}_{\pm}\mathbf{\cdot}\mathbf{G}=\left(H(\za_{\pm}^{0}), H(\za_{\pm}^{1})\right)\Longrightarrow
{\mathbf{T}}_{\pm}^{h}\mathbf{\cdot}\left(H(\za_{\pm}^{0}), H(\za_{\pm}^{1})\right)=4(\mathbf{G}\pm i\,H(\mathbf{G}).
\end{equation}
Equations \rf{Th_za} and \rf{sig_G} imply that the applications of the transform s $ {\mathbf{T}}_{\pm}^{h}$ to rows of the respective coefficient\ arrays results in the following relations:
\begin{eqnarray}\label{Th_zahza}
\mathbf{X}_{+}&\stackrel{\Delta}{=}&{\mathbf{T}}_{+}^{h}\mathbf{\cdot}\left(\bar{\za}_{+}^{0}),\bar{\za}_{+}^{1}\right)=
4\left(\mathbf{X}+i\,H(\mathbf{X})+i\mathbf{G}-H(\mathbf{G}))\right),\\\nonumber
\mathbf{X}_{-}&\stackrel{\Delta}{=}&{\mathbf{T}}_{-}^{h}\mathbf{\cdot}\left(\bar{\za}_{-}^{0}),\bar{\za}_{-}^{1}\right)
=4\left(\mathbf{X}-i\,H(\mathbf{X})+i\mathbf{G}+H(\mathbf{G}))\right).
\end{eqnarray}
Finally, we have the signal\ \textbf{X} restored by $\mathbf{X}=\mathfrak{Re}(\mathbf{X}_{+}+\mathbf{X}_{-})/8$.
Figures \ref{xp_xm_x5} and \ref{xp_xm_x} illustrate the image ``Barbara" restoration by the 2D signal s $\mathfrak{Re}(\mathbf{X}_{\pm})$ and $\mathbf{X}=\mathfrak{Re}(\mathbf{X}_{+}+\mathbf{X}_{-})/8$. The signal\ $\mathfrak{Re}(\mathbf{X}_{+})$ captures edges oriented to \emph{north-east}, while $\mathfrak{Re}(\mathbf{X}_{-})$ captures edges oriented to \emph{north-west}. The signal\ $\mathbf{X}$ perfectly restores the image achieving PSNR=313.8596 dB.
\begin{figure}[H]
\centering
\includegraphics[width=5.2in]{png/xp_xm_x50.png}
\caption{Top left: Partially restored ``Barbara" image by $\mathfrak{Re}(\mathbf{X}_{+})$. Top right: Partially restored image by $\mathfrak{Re}(\mathbf{X}_{-})$.
Bottom left: Magnitude DFT spectr um of $\mathbf{X}_{+}$. Bottom right: Magnitude DFT spectr um of $\mathbf{X}_{-}$}
\label{xp_xm_x5}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=4.0in]{png/xp_xm_x2.png}
\caption{Left: Original ``Barbara" image. Right: Fully restored image by $\mathfrak{Re}(\mathbf{X}_{+}+\mathbf{X}_{-})/8$}
\label{xp_xm_x}
\end{figure}
\subsection{Multi-level 2D transform s}\label{sec:ss62}
It was established in Section \ref{sec:ss42} that the 1D qWP transform s of a signal\ $\mathbf{x}\in\Pi[N]$ to the second and further decomposition\ levels are implemented by the iterated application of
the filter bank s, that are determined by their analysis\ modulation matri ces $\tilde{\mathbf{M}}[2^{m}n],\;m=1,...,M-1,$ to the coefficient s arrays $\mathbf{z}_{\pm[m]}^{\la}$. The transform s applied to the arrays $\mathbf{z}_{\pm[m]}^{\la}$ produce the arrays $\mathbf{z}_{\pm[m+1]}^{\rr}$, respectively. The inverse transform\ consists of the iterated application of the
filter bank s that are determined by their synthesis\ modulation matri ces ${\mathbf{M}}[2^{m}n],\;m=1,...,M-1,$ to the coefficient s arrays $\mathbf{z}_{\pm[m+1]}^{\rr}$. In that way the first-level coefficient\ arrays $\mathbf{z}_{\pm[1]}^{\la},\;\la=0,1$ are restored.
The tensor-product of the 2D transform s of a signal\ $\mathbf{X}\in\Pi[N,N]$ consists of the subsequent application of the 1D transform s to columns and rows of the signal\ and coefficient s arrays. By application of filter bank s, which are determined by the analysis\ modulation matri x $\tilde{\mathbf{M}}[2n]$ to columns and rows of coefficient s array $\mathbf{Z}_{\pm[1]}^{j,l}$, we derive four second-level arrays
$\mathbf{Z}_{\pm[2]}^{\rr,\tau},\;\rr=2j,2j+1;\;\tau=2l,2l+1$. The arrays $\mathbf{Z}_{\pm[1]}^{j,l}$ are restored by the application of the
filter bank s that are determined by their synthesis\ modulation matri ces ${\mathbf{M}}[2n] $ to rows and columns of the coefficient s arrays $\mathbf{Z}_{\pm[2]}^{\rr,\tau},\;\rr=2j,2j+1;\;\tau=2l,2l+1$. The transition from the second to further levels and back are executed similarly using the modulation matri ces $\tilde{\mathbf{M}}[2^{m}n]$ and ${\mathbf{M}}[2^{m}n]$, respectively. The inverse transform s produce the coefficient s arrays $\mathbf{Z}_{\pm[1]}^{j,l},\;j,l=0,1,$ from which the signal\ $\mathbf{X}\in\Pi[N,N]$ is restored using the synthesis\ modulation matri ces ${\mathbf{M}}_{\pm}^{q}[n] $ as it is explained in Section \ref{sec:sss612}.
All the computations are implemented in the frequency\ domain using the FFT. For example, the Matlab execution of the 2D qWP transform\ of a $512\times 512$ image down to the sixth decomposition\ level takes 1.34 seconds. The four-level transform\ takes 0.28 second.
\paragraph{Summary}
The 2D qWP processing\ of a signal\ $\mathbf{X}\in\Pi[N,N]$ is implemented by a dual-tree scheme. The first step produces two sets of the coefficient s arrays: $\mathbf{Z}_{+[1]}=\left\{\mathbf{Z}_{+[1]}^{j,l}\right\}, \;j,l,=0,1,$ which are derived using the analysis\ modulation matri x $\tilde{\mathbf{M}}_{+}^{q}[n]$, and $\mathbf{Z}_{-[1]}=\left\{\mathbf{Z}_{-[1]}^{j,l}\right\}, \;j,l,=0,1,$ which are derived using the analysis\ modulation matri x $\tilde{\mathbf{M}}_{-}^{q}[n]$. Further decomposition\ steps are implemented in parallel on the sets $\mathbf{Z}_{+[1]}$ and $\mathbf{Z}_{-[1]}$ using the same analysis\ modulation matri x $\tilde{\mathbf{M}}[2^{m}n]$, thus producing two multi-level sets of the coefficient s arrays $\left\{\mathbf{Z}_{+[m]}^{j,l}\right\}$ and $\left\{\mathbf{Z}_{-[m]}^{j,l}\right\},\;m=2,...,M,\;j,l=0,2^{m}-1$.
By parallel implementation of the inverse transform s on the coefficient s from the sets $\left\{\mathbf{Z}_{+[m]}^{j,l}\right\}$ and $\left\{\mathbf{Z}_{-[m]}^{j,l}\right\}$ using the same synthesis\ modulation matri x ${\mathbf{M}}[2^{m}n]$, the sets $\mathbf{Z}_{+[1]}$ and $\mathbf{Z}_{-[1]}$ are restored, which, in turn, provide the signal s $\mathbf{X}_{+}$ and $\mathbf{X}_{-}$, using the synthesis\ modulation matri ces ${\mathbf{M}}_{+}^{q}[n]$ and ${\mathbf{M}}_{-}^{q}[n]$, respectively. Typical signal s $\mathbf{X}_{\pm}$ and their DFT spectr a are displayed in Fig. \ref{xp_xm_x5}.
Prior to the reconstruction, some structures, possibly differen t, are defined in the sets $\left\{\mathbf{Z}_{+[m]}^{j,l}\right\}$ and $\left\{\mathbf{Z}_{-[m]}^{j,l}\right\},\;m=1,...M,$ (2D wavelet\ or Best Basis structures, for example) and some manipulations on the coefficient s, (thresholding, $l_1$ minimization, for example) are executed.
\section{Numerical examples}\label{sec:s7}
In this section, we present two examples of application of the 2D qWPs to image restoration. These examples illustrate the ability of the qWPs to restore edges and texture details even from severely damaged images. Certainly, this ability stems from the fact that the designed 2D qWP transform s provide a variety of 2D waveform s oriented in multiple directions, from perfect frequency\ resolution of these waveform s and, last but not least, from oscillatory structure of many waveform s.
\subsection{Denoising examples}\label{sec:ss71}An image $\mathbf{I}$ is represent ed by the 2D signal\ $\mathbf{X}\in\Pi[N,N]$. The image, which is corrupted by additive Gaussian noise with STD=$\o$, is represent ed by the 2D signal\ $\mathbf{X}_{\o}$. We apply the following image denoising scheme:
\begin{itemize}
\item 2D transform\ of the signal\ $\mathbf{X}$ with directional qWPs $\Psi^{2r}_{++[m]}$ and $\Psi^{2r}_{+-[m]}$ down to level $M$ is implemented to generate two sets of the coefficient s arrays $\left\{\mathbf{Z}_{+[m]}^{j,l}\right\}$ and $\left\{\mathbf{Z}_{-[m]}^{j,l}\right\},\;m=1,...M,\;j,l=0,...,2^{m}-1$.
\item For comparison, the 2D WP transform\ of the signal\ $\mathbf{X}$ with non-directional WP $\psi^{2r}_{[m]}$ down to level $M$ is implemented thus generating the set $\left\{\mathbf{y}_{[m]}^{j,l}\right\},\;m=1,...M,\;j,l=0,...,2^{m}-1,$ of the coefficient s arrays.
\item In each of the sets $\left\{\mathbf{Z}_{\pm[m]}^{j,l}\right\}$ and $\left\{\mathbf{y}_{[m]}^{j,l}\right\}$ the ``Best Basis" is selected by a standard procedure of comparison the cost function (entropy or $l_1$ norm) of the ``parent" coefficient s block with the cost functions of its ``offsprings" (see \cite{coiw1}). The selected bases are designated by $\mathbf{B}_{\pm[M]}$ and $\mathbf{b}_{[M]}$, respectively.
\item The number of the transform\ coefficient s $\mathbf{Z}_{\pm[B]}$ and $\mathbf{y}_{[B]}$ associated with each basis is the same as the number $N^{2}$ of pixels in the image.
\item Denoising of the signal\ $\mathbf{X}_{\o}$ is implemented by hard thresholding of the coefficient s $\mathbf{Z}_{\pm[B]}$ and $\mathbf{y}_{[B]}$.
\item The thresholds for the coefficient s arrays are defined by the following naive scheme:
\begin{enumerate}
\item The absolute values of the coefficients from each set $\mathbf{Z}_{\pm[B]}$ and $\mathbf{y}_{[B]}$ are arranged in an ascending order thus forming the sequence s $\mathbf{A}_{\pm}$ and $\mathbf{A}$, respectively.
\item The values of the $L$-th term $T_{\pm}={A}_{\pm}[L]$ and $T={A}[L]$ are selected as the thresholds for the coefficient s arrays $\mathbf{Z}_{\pm[B]}$ and $\mathbf{y}_{[B]}$, respectively. The number $L$ is chosen depending on the noise intensity.
\end{enumerate}
\item The sets of the transform\ coefficient s $\mathbf{Z}_{\pm[B]}$ and $\mathbf{y}_{[B]}$ are subjected to thresholding with the selected values $T_{\pm}$ and $T$, respectively, and, after that the inverse transform s are applied to produce the 2D signal s $\mathbf{X}_{d}=\mathfrak{Re}(\mathbf{X}_{+}+\mathbf{X}_{-})/8$ from the directional qWPT and $\mathbf{X}_{nd}$ from the non-directional tensor product WPT.
\end{itemize}
\subsubsection{Example I: ``Pentagon" image}\label{sec:sss711}The ``Pentagon" image of size $1024\times 1024$ (1048576 pixels) was corrupted by additive Gaussian noise with STD=30 dB. As a result, the PSNR of the corrupted image was 18.59 dB. The corrupted image was decomposed by the directional qWPs $\Psi^{4}_{++[m]}$ and $\Psi^{4}_{+-[m]}$ originating from the fourth-order discrete spline s down to a fourth decomposition\ level. In this way, two sets $\left\{\mathbf{Z}_{+[m]}^{j,l}\right\}$ and $\left\{\mathbf{Z}_{-[m]}^{j,l}\right\},\;m=1,...4,\;j,l=0,...,2^{m}-1$, of the transform\ coefficient s were produced. For comparison, the corrupted image was decomposed by the non-directional WPs $\psi^{4}_{[m]}$, which resulted in the set $\left\{\mathbf{y}_{[m]}^{j,l}\right\},\;m=1,...4,\;j,l=0,...,2^{m}-1,$ of the coefficient s arrays.
The ``Best Bases" $\mathbf{B}_{\pm[4]}$ and $\mathbf{b}_{[4]}$ were designed for the coefficient\ arrays. The thresholds $T_{\pm}={A}_{\pm}[L]$ and $T={A}[L]$ were selected for each set of the coefficient\ arrays. $L=1010000$ was chosen. Thus, we had $T_{+}=130.42$, $T_{-}=126.8$ and $T=73.30$.
\begin{rmk}\label{rem:2thre}The threshold $T=73.30$ is approximat ely two times less than the threshold $T_{\pm}$. It happens because the transform\ coefficient s $\left\{\mathbf{Z}_{+[m]}^{j,l}\right\}$ are complex-valued and the threshold is operating with absolute values of the coefficient s.\end{rmk}
\begin{rmk}\label{rem_penta}qWPs oriented in 62 differen t directions were involved. The Matlab implementation of all the above procedures takes 4.2 seconds.\end{rmk}
Figure \ref{penta30} is the outputs of the image reconstruction\ by the directional qWPT and the non-directional tensor product WPT from the thresholded coefficient s arrays. The qWPT-restored image image $\mathbf{X}_{d}$ has PSNR=26.75 dB versus PSNR=23.75 dB for the WPT-restored image $\mathbf{X}_{nd}$. Visually, image $\mathbf{X}_{d}$ is cleaner in comparison to $\mathbf{X}_{nd}$ and more fine details are restored.
\begin{figure}[H]
\resizebox{17cm}{9cm}{
\includegraphics{png/penta30.jpg}
}
\caption{Top left: Original ``Pentagon" image. Top right: Image corrupted by noise with STD=30 dB. Bottom left: The qWPT-based restored image $\mathbf{X}_{d}$ . Bottom right: WPT-based restored image $\mathbf{X}_{nd}$}
\label{penta30}
\end{figure}
\subsubsection{Example II: ``Barbara" image}\label{sec:sss712}
We present two cases with the ``Barbara" image. In one case, the image was corrupted by an additive Gaussian noise with STD=30 dB and in the other, the noise was more intensive with STD=50 dB.
\begin{description}
\item[Noise with STD=30 dB:] In this case, the PSNR of the corrupted image was 18.59 dB.
In order to avoid boundary effects, the image of size $512\times 512$ was symmetr ally extended to the size $1024\times 1024$. After processing, the results were shrunk to the original size.
The corrupted image was decomposed by the directional qWPs $\Psi^{4}_{++[m]}$ and $\Psi^{4}_{+-[m]}$ originating from the fourth-order discrete spline s down to third decomposition\ level. In this way, the two sets $\left\{\mathbf{Z}_{+[m]}^{j,l}\right\}$ and $\left\{\mathbf{Z}_{-[m]}^{j,l}\right\},\;m=1,...3,\;j,l=0,...,2^{m}-1$, of the transform\ coefficient s were produced. For comparison, the corrupted image was decomposed by the non-directional WPs $\psi^{4}_{[m]}$, which resulted in the set $\left\{\mathbf{y}_{[m]}^{j,l}\right\},\;m=1,...3,\;j,l=0,...,2^{m}-1,$ of the coefficient s arrays.
The ``Best Bases" $\mathbf{B}_{\pm[3]}$ and $\mathbf{b}_{[3]}$ were designed for the coefficient\ arrays. The thresholds $T_{\pm}={A}_{\pm}[L]$ and $T={A}[L]$ were selected for each set of the coefficient\ arrays. $L=1011560$ was chosen. Thus, we had $T_{+}=144.51$, $T_{-}=144.66$ and $T=82.83$.
Figures \ref{barb30} and \ref{barb30F} are the outputs of the image reconstruction\ by using the directional qWPT and the non-directional tensor product WPT from the thresholded coefficient s arrays. The qWPT-restored image $\mathbf{X}_{d}$ has PSNR=28.52 dB versus PSNR=25.20 dB for the WPT-restored image $\mathbf{X}_{nd}$. Visually, the image $\mathbf{X}_{d}$ is much cleaner in comparison to $\mathbf{X}_{nd}$ and almost all edges and the texture structure are restored.
\begin{rmk}\label{rem_barb50}qWPs oriented in 31 differen t directions are involved. In this case, the image under processing\ has four times more pixels than the original image. The Matlab implementation of all the procedures including 3-level qWP and WP transform s, design of ``Best Bases", thresholding of the coefficient s arrays and the inverse transform s takes 3.5 seconds. Processing the image without the extension takes 0,88 seconds and produces the restores image with PSNR=28.32 dB.\end{rmk}
\begin{figure}[H]
\begin{center}
\resizebox{14cm}{9cm}{
\includegraphics{png/barb30.png}
}
\end{center}
\caption{Top left: Original ``Barbara" image. Top right: Image corrupted by noise with STD=30 dB. Bottom left: The qWPT-based restored image $\mathbf{X}_{d}$ . Bottom right: WPT-based restored image $\mathbf{X}_{nd}$}
\label{barb30}
\end{figure}
\begin{figure}[H]
\begin{center}
\resizebox{13cm}{9cm}{
\includegraphics{png/barb30F.png}
}
\end{center}
\caption{Fragments of the images shown in Fig. \ref{barb30}}
\label{barb30F}
\end{figure}
\item[Noise with STD=50 dB:] In this case, the PSNR of the corrupted image was 14.14 dB. The same operations as in the previous case were applied to the corrupted image. $L=1019600$ was chosen. Thus, we had $T_{+}=234.18$, $T_{-}=234.16$ and $T=137.29$.
Figures \ref{barb50} and \ref{barb50F} are the outputs of the image reconstruction\ by using the directional qWPT and the non-directional tensor product WPT from the thresholded coefficient s arrays. The qWPT-based restored image $\mathbf{X}_{d}$ has PSNR=25.66 dB versus PSNR=22.20 dB for the WPT-based restored image $\mathbf{X}_{nd}$. Visually, the image $\mathbf{X}_{d}$ is cleaner in comparison to $\mathbf{X}_{nd}$, which, in addition comprises many artifacts. Many edges and the texture structure are retained in the image $\mathbf{X}_{d}$.
\begin{figure}[H]
\begin{center}
\resizebox{14cm}{9cm}{
\includegraphics{png/barb50.png}
}
\end{center}
\caption{Top left: Original ``Barbara" image. Top right: Image corrupted by noise with STD=50 dB. Bottom left: The qWPT-based restored image $\mathbf{X}_{d}$ . Bottom right: WPT-based restored image $\mathbf{X}_{nd}$}
\label{barb50}
\end{figure}
\begin{figure}[H]
\begin{center}
\resizebox{13cm}{9cm}{
\includegraphics{png/barb50F.png}
}
\end{center}
\caption{Fragments of the images shown in Fig. \ref{barb50}}
\label{barb50F}
\end{figure}
\paragraph{Comment}In the above examples, we did not apply sophisticated adaptive denoising schemes such as Gaussian scale mixture model (\cite{porti_strela}) or bivariate shrinkage (\cite{sen_seles}). Our goal here was to compare performance of the directional qWP transform s with the performance of the standard tensor-product WP transform s.
\end{description}
\subsection{Image restoration examples}\label{sec:ss72}In this section we present a few cases of image restoration using directional qWPs. Images to be restored were degraded by blurring, aggravated by random noise and
random loss of significant number of pixels. In our previous work (\cite{Inv_frame} and Chapter 18 in \cite{ANZ_book1}) we developed the image restoration scheme utilizing 2D wavelet\ frames designed in Chapter 18 of \cite{ANZ_book1}. In the examples presented below we use, generally, the same scheme as in \cite{ANZ_book1} with the differen ce that the directional qWPs designed in Section \ref{sec:s6} are used instead of wavelet\ frames.
\subsubsection{Brief outline of the restoration scheme}\label{sec:sss721}
Images are restored by the application of the \emph{split
Bregman iteration} (SBI) scheme \cite{gold_os} that uses the
so-called \emph{analysis-based} approach (see for example \cite{ji_shen_xu}).
Denote by $\mathbf{u}=\left\{u[\k,\n]\right\}$ the original image array to be restored from the degraded array
\(
\mathbf{f}=\mathbf{K}\,\mathbf{u}+\e,
\) where $\mathbf{K}$ denotes the operator of 2D discrete\ convolution\ of the array
$\mathbf{u}$ with a kernel $\mathbf{k}=\left\{k[\k,\n]\right\}$, and
$\e=\left\{e_{k,n}\right\}$ is the random error array.
$\mathbf{K^*}$ denotes the conjugate operator of $\mathbf{K}$,
which implements the discrete\ convolution\ with the transposed kernel
$\mathbf{k}^{T}$. If some number of pixels are missing then the image
$\mathbf{u}$ should be restored from the available data
\begin{equation}\label{av_dat}
\mathbf{P}_{\Lambda}\,\mathbf{f}=\mathbf{P}_{\Lambda}\,(\mathbf{K}\,\mathbf{u}+\e),
\end{equation}
where $\mathbf{P}_{\La}$ denotes the projection on the
remaining set of pixels.
The solution scheme is based on the assumption that the original
image $\mathbf{u}$ can be
sparsely represented in the qWP domain. Denote by
$\mathbf{\tilde{F}}$ the operator of qWP expansion of the image
$\mathbf{u}$. To be specific, the 2D transform\ of the signal\ $\mathbf{X}$ with directional qWP $\Psi^{2r}_{++[m]}$ and $\Psi^{2r}_{+-[m]}$ down to level $M$ is implemented to generate two sets of the coefficient s arrays $\left\{\mathbf{Z}_{+[m]}^{j,l}\right\}$ and $\left\{\mathbf{Z}_{-[m]}^{j,l}\right\},\;m=1,...M,\;j,l=0,...,2^{m}-1$.
In each of the sets $\left\{\mathbf{Z}_{\pm[m]}^{j,l}\right\}$ either ``Best Basis" or ``basis", which consist of shifts of all the WPs from the decomposition\ level $M$, are selected. The bases are designated by $\mathbf{B}_{\pm[M]}$.
The number of the transform\ coefficient s $\mathbf{Z}_{\pm[B]}$ associated with each basis is the same as the number $N^{2}$ of pixels in the image. Thus, $\mathbf{C}\stackrel{\Delta}{=}\mathbf{\tilde{F}}\,\mathbf{u}=\mathbf{Z}_{+[B]}\bigcup\mathbf{Z}_{-[B]}$ is the set of the transform\ coefficient s.
Denote by $\mathbf{F}$ the reconstruction\ operator of the image $\mathbf{u}$
from the set of the transform\ coefficient s. We get
$\mathbf{F}\,\mathbf{C}=\mathbf{u}=\mathfrak{Re}(\mathbf{u}_{+}+\mathbf{u}_{-})/8$,
$\mathbf{F}\,\mathbf{\tilde{F}}=\mathbf{I}$, where $\mathbf{I}$ is
the identity operator.
An approximate solution to Eq. \rf{av_dat} is derived
via minimization of the function al
\begin{equation}\label{av_dat_min}
\min_{u}\frac{1}{2}\left\|\mathbf{P}_{\La}\,(\mathbf{K}\,\mathbf{u}-f)\right\|_{2}^{2}+\la\,\left\|\mathbf{\tilde{F}}\,\mathbf{u}\right\|_{1},
\end{equation}
where $\left\|\cdot\right\|_{1}$ and $\left\|\cdot\right\|_{2}$ are
the $l_{1}$ and the $l_{2}$ norms of the sequence s, respectively. If
$\mathbf{x}=\left\{x[\k,\n]\right\},\;{\k}=0,...,k,\;{\n}=0,...,n$,
then
\[\left\|\mathbf{x}\right\|_{1}\stackrel{\Delta}{=}\sum_{\k=0}^{k-1}\sum_{\n=0}^{n-1}|x[\k,\n]|,\quad \left\|\mathbf{x}\right\|_{2}\stackrel{\Delta}{=}\sqrt{\sum_{\k=0}^{k-1}\sum_{\n=0}^{n-1}|x[\k,\n]|^{2}}.\]
Denote by $\mathbf{T}_{\vt}$ the operator of soft thresholding:
\[\mathbf{T}_{\vt}\,\mathbf{x}=\left\{x_{\vt}[\k,\n]\right\},\quad x_{\vt}[\k,\n]\stackrel{\Delta}{=} \mathrm{sgn}(x[\k,\n])\,\max \left\{0, |x[\k,\n]|-\vt\right\}.\]
Following \cite{ji_shen_xu}, we solve the minimization problem in
Eq. \rf{av_dat_min} by an iterative SBI algorithm. We begin with the
initialization $\mathbf{u}^{0}=0,\;\mathbf{d}^{0}=\mathbf{b}^{0}=0$.
Then,
\begin{equation}\label{breg_iter}
\begin{array}{l}
\mathbf{u}^{k+1}:=(\mathbf{K^*}\,\mathbf{P}_{\La}
\,\mathbf{K} +\mu\,\mathbf{I})\,\mathbf{u}=\mathbf{K^*}\,\mathbf{P}_{\La}\,\mathbf{f}
+\mu\,\mathbf{F}\,(\mathbf{d}^{k}-\mathbf{b}^{k}),\\
\mathbf{d}^{k+1} =\mathbf{T}_{\la/\mu}(\mathbf{\tilde{F}}\,\mathbf{u}^{k+1}+\mathbf{b}^{k}),\\
\mathbf{b}^{k+1}=\mathbf{b}^{k}+(\mathbf{\tilde{F}}\,\mathbf{u}^{k+1}- \mathbf{d}^{k+1}).
\end{array}
\end{equation}
The linear system in the first line of Eq. \rf{breg_iter} is solved by
the application of the \emph{conjugate gradient} algorithm. The
operations in the second and third lines are straightforward. The
choice of the parameters $\la$ and $\mu$ depends on experimental
conditions.
\subsubsection{Examples}\label{sec:sss722}
\begin{description}
\item[Example I: ``Barbara" blurred, missing 50\% of pixels:] The ``Barbara" image was restored after it was blurred by a convolution\ with the Gaussian kernel (MATLAB function\ \\
\texttt{fspecial}('gaussian',[5 5])) and its PSNR became 23.32 dB.
Then, 50\% of its pixels were randomly removed. This reduced the
PSNR to 7.56 dB. Random noise was not added.
The image was restored by 50 SBI using the parameters $\la=0.0015,\;\mu=0.00014$ in Eq. \rf{breg_iter}. The conjugate gradient solver used 150 iterations.
qWPs originating from discrete spline s of sixth order were used. For ``bases", 8-samples shifts of all the WPs from the third decomposition\ level were selected. Matlab implementation of the restoration procedures took 59.6 seconds.
Figure \ref{barb50B} displays the restoration result. The image is deblurred and the fine texture is restored completely with PSNR=32.09 dB. Note that the best result in an identical experiment reported in \cite{ANZ_book1} achieved PSNR=30.32 dB.
\begin{figure}[H
%
\begin{center}
\resizebox{14cm}{9cm}{
\includegraphics{png/barb50_B.png}
}
\caption {Top left: Source input - ``Barbara" image. Top right: Blurred, PSNR=23.32 dB. Bottom right: After random removal of 50\% of its pixels. PSNR=7.56 dB.
Bottom left: The image restored by the directional qWPT. PSNR=32.09 dB}\label{barb50B}
%
\end{center}
\end{figure}
\item[Example II: ``Barbara" blurred, added noise, missing 50\% of pixels:] The ``Barbara" image was restored after it was blurred by a convolution\ with the Gaussian kernel (MATLAB function\
\texttt{fspecial}('gaussian',[5 5])). Random Gaussian noise with STD=10 dB was added and the image PSNR became 22.08 dB.
Then, 50\% of its pixels were randomly removed. This reduced the PSNR to 7.53 dB. The image was restored by 70 SBI using the parameters $\la=3,\;\mu=0.025$ in Eq. \rf{breg_iter}. The conjugate gradient solver used 15 iterations.
qWPs originating from discrete spline s of fourth order were used. For the ``bases", 16-samples shifts of all the WPs from the fourth decomposition\ level were selected. Matlab implementation of the restoration procedures took 51.9 seconds.
Figure \ref{barb50BN} displays the restoration result. The image is deblurred, noise is removed and the fine texture is partially restored producing PSNR=24.31 dB. Note that the best result in an identical experiment reported in \cite{ANZ_book1} achieved PSNR=24.19 dB.
\begin{figure}[H
%
\begin{center}
\resizebox{14cm}{9cm}{
\includegraphics{png/barbN50_B.png}
}
\caption {Top left: Source input - ``Barbara" image. Top right: Blurred and noised, PSNR=22.08 dB. Bottom right: After random removal of 50\% of its pixels. PSNR=7.53 dB.
Bottom left: The restored image by the directional qWPT. PSNR=24.31 dB}\label{barb50BN}
%
\end{center}
\end{figure}
\item[Example III: ``Barbara" blurred, missing 90\% of pixels:] The ``Barbara" image was restored after it was blurred by a convolution\ with the Gaussian kernel (MATLAB function\ \\
\texttt{fspecial}('gaussian',[5 5])) and 90\% of its pixels were randomly removed. This reduced the PSNR to 5.05 dB. The image was restored by 150 SBI using the parameters $\la=0.0025,\;\mu=0.000025$ in Eq. \rf{breg_iter}. The conjugate gradient solver used 150 iterations.
qWPs originating from discrete spline s of eighth order were used. For the ``bases", 16-samples shifts of all the WPs from the fourth decomposition\ level were selected. Matlab implementation of the restoration procedures took 226.8 seconds.
Figure \ref{barb90} displays the restoration result. The image is deblurred and the fine texture is partially restored. The output has PSNR=25.24 dB.
\begin{figure}[H
%
\begin{center}
\resizebox{14cm}{9cm}{
\includegraphics{png/barb902.png}
}
\caption {Top left: Source input - ``Barbara" image. Top right: Blurred, PSNR=23.32 dB. Bottom right: After random removal of 90\% of its pixels. PSNR=5.05 dB.
Bottom left: The image restored by the directional qWPT. PSNR=25.24 dB}\label{barb90}
%
\end{center}
\end{figure}
\end{description}
\section{Discussion}\label{sec:s8}We presented a library of complex discrete-time\ wavelet packet s operating in one- or two-dimensional spaces of periodic\ signal s. Seemingly, the requirement of periodic ity imposes some limitations on the scope of signal s available for processing, but actually these limitations are easily circumvented. Any limited signal\ can be regarded as one period of a periodic\ signal. In order to prevent boundary effects, the signal s can be symmetr ally extended beyond the boundaries before processing\ and shrunk to the original size after that. We used such a trick in the ``Barbara" denoising examples.
On the other hand, the periodic\ setting provides a lot of substantial opportunities for the design and implementation of WP transform s such as
\begin{itemize}
\item A unified computational scheme based on 1D and 2D FFT.
\item Opportunity to use filter s with infinite impulse response s, which enables us to design a variety of orthonormal\ WP systems where WPs can have any number of local vanishing moment s.
\item The number of local vanishing moment s does not affect the computational cost of the transform s implementation.
\item A simple explicit scheme of expansion of real WPs to analytic\ and quasi-analytic\ WPs with perfect frequency\ separation.
\end{itemize}
The library of qWP transform s described in the paper has a number of free parameters enabling to adapt the transform s to the problem under consideration:
\begin{itemize}
\item Order of the generating spline, which determines the number of local vanishing moment s.
\item Depth of decomposition, which in 2D case determines the direction ality of qWPs. For example, fourth-level qWPs are oriented in 62 differen t direction s.
\item Selection of an optimal structure, such as, for example, separate Best Bases in the real and imaginary parts of 1D qWP transform s, separate ``Best Bases" in positive and negative branches of 2D dual-tree qWP transform s, a wavelet-basis structure or the set of all wavelet packet s from a single level.
\item Controllable redundancy rate of the signal\ represent ation. The minimal rate is 2 when one of options listed in a previous item is utilized. However, several basis-type structures can be involved, for example, all wavelet packet s from several levels can be used for the signal\ reconstruction\ and results can be averaged.
\end{itemize}
The goal of the paper is to design qWPs with an efficient computational scheme for the corresponding transform s. A few experimental results highlight exceptional properties of these WPs. The directional qWPs are tested for image restoration examples. In the denoising examples, the goal was not to achieve the best output in RSNR values but rather to compare the performance of direction al versus standard WPs that are tensor-product-based. We did not use sophisticated adaptive denoising schemes such as for example Gaussian scale mixture model (\cite{porti_strela}) or bivariate shrinkage (\cite{sen_seles}). Instead, after decomposition\ of an image down to level $M$, we selected ``Best Bases" in the positive and negative branches of the qWP transform\ and in the WP transform. Then, we discarded all the coefficient s in these bases except for the $L$ largest coefficient s. Even with such naive scheme, the direction al qWPs significantly outperform the standard WPs in both PSNR values and in the visual perception. The edges, lines and oscillating texture structures in the ``Barbara" image, which was corrupted by Gaussian noise with STD=30 dB, were almost perfectly restored. When noise STD was 50 dB, an essential part of these structures was restored as well. We emphasise that the Matlab implementation of all the procedures including 3-level qWP and WP transform s, the design of ``Best Bases", thresholding of the coefficient s arrays and inverse transform s took 0.88 seconds. To eliminate boundary effects, the image was extended from $512\times512$ to $1024\times1024$ and the processing\ took 3.5 seconds.
The second group of experimental results dealt with the restoration of the ``Barbara" image which was blurred my the convolving the image with a Gaussian kernel and degraded by removing randomly either 50\% or 90\% of the pixels. The image was restored by using a constrained $l_{1}$ minimization of the qWP transform\ coefficient s from a certain decomposition\ level and implemented via the split Bregman Iterations procedure.
In Example I with missing 50\% of the pixels, the image was almost perfectly restored with PSNR-32.1 dB and practically all the fine structure reconstructed although it was blurred even before the removal of the pixels.
Addition of the Gaussian noise with STD 10 dB to the blurred image in Example II depleted the reconstruction\ result. Although the noise became suppressed and the image was deblurred, most of the fine structure was lost. Restoration results were better in Example III where, instead of adding noise, the number of pixels missing from the blurred image was raised to 90\%. The image was deblurred and an essential part of fine structure was restored.
Summarizing, we can state that, having such a versatile and flexible tool at hand, we are prepared to address multiple data processing\ problems such as signal\ and image deblurring and denoising, target detection, segmentation, inpainting, superresolution, to name a few. In one of the applications, whose results are to be published soon, direction al qWPs are used with Compressed Sensing methodology for the conversion of a regular digital photo camera to an hyperspectr al imager. Preliminary results appear in \cite{Hyper_camera}. Special efforts are on the way for the design of a denoising scheme which fully explores the direction ality properties of qWPs.
\paragraph{Acknowledgment}
This research was partially supported by the Israel Science Foundation (ISF, 1556/17),
Blavatnik Computer Science Research Fund
Israel Ministry of Science and Technology 3-13601 and by Academy of Finland (grant 311514).
\include{ANA_AP}
\bibliographystyle{plain}
|
1,116,691,497,589 | arxiv | \section{Introduction}
The motion of electrons in the periodic potential of ionic crystals is addressable by the celebrated Bloch solutions which give rise to band structure \cite{bloc28, ashc76}. Periodic potentials are also common in the arena of ultracold atoms where gases trapped in optical lattices can serve as a test-bed for questions in many-body physics \cite{bloc08,bloc12,kuhr16}. These scenarios are amenable to the same band structure approach.
Pulsed optical lattices are in common use as diffractive elements in atom optics and interferometry \cite{tino14} for diverse applications such as inertial sensing \cite{mcgu02,durf06,dutt16,geig11} and for tests of fundamental physics such as the equivalence principle \cite{fray04,schl14, tara14} and quantum electrodynamics \cite{bouc11,park18}. The atom-optics element of choice for beamsplitters and mirrors --- Bragg diffraction --- is traditionally analyzed using the two-state Rabi solution which predicts oscillatory behavior \cite{gilt95,gupt01}. To address the regime when the two-state approximation is invalidated for sufficiently short pulses, a host of numerical work has been performed \cite{durr99,horn99,mull08b} with the limiting case of Kapitza-Dirac diffraction allowing an analytic solution \cite{gupt01}.
In this work we apply the Bloch-bands approach to atom optics through the performance and analysis of a series of standing-wave diffraction and interferometry experiments with Bose-Einstein condensates. Our results impact three key directions. First, we experimentally demonstrate the equivalence between the band gap and the frequency for Bragg pendell{\"o}sung, and obtain accurate values distinct from the results of a commonly-used formula for Rabi frequency in Bragg diffraction. Second, we exploit the Bloch-bands approach for direct visualization and systematic analysis of diffraction phase effects, and provide useful methods for their suppression in precision atom interferometry. Finally, we invert the approach and determine atomic band structure in a periodic potential from measurements of phase shifts in an atom interferometer, thus introducing a new method for analyzing arbitrary periodic optical potentials.
\section{Atom optics in the Bloch-bands picture}
Our analysis is based on the Bloch solutions for a neutral atom interacting with the one-dimensional sinusoidal potential of an optical standing wave and is related to earlier theoretical work \cite{cham01,buch03}. In accord with parameters used in typical experiments, we work in a regime where the large one-photon detuning allows adiabatic elimination of the excited internal state. The atom-light interaction then reduces to a conservative (AC Stark shift) potential imposed on the atoms, which is proportional to the optical intensity \cite{gupt01}. We calculate the Bloch energy bands (Fig.\ref{fig:Gochnauer_BandPicture_Figure1}) by numerically diagonalizing the single-particle Hamiltonian for the potential $U = U_0 \sin^2(2kx)$. The energy and momentum are normalized to the recoil energy $E_{\rm rec}=\hbar^2 k^2/2m$ and recoil momentum $p_{\rm rec}=\hbar k$ respectively, where $\pi/k$ is the spatial periodicity of the lattice.
In a Brillouin-zone picture, the lattice opens up an avoided crossing at every intersection of free-particle energy levels, each of which can be identified with a Bragg diffraction process and is characterized by an energy gap which increases monotonically with $U_0$. This band gap is equivalent to $\hbar \Omega_R$, where $\Omega_R$ is the Rabi frequency for oscillations between the two Bragg-coupled states. In addition, there is also an energy shift $\hbar \Omega_D$ of the mean energy of the coupled states away from the original (unperturbed) crossing point. Both $\Omega_R$ and $\Omega_D$ can be seen as arising from ``level repulsion'' in second-order perturbation (see Fig.\ref{fig:Gochnauer_BandPicture_Figure1} inset).
\begin{figure}[h]
\center
\includegraphics[width=0.5\textwidth]{Gochnauer_BandPicture_Figure1}
\caption{(Color online) Atomic energy bands in quasimomentum space for a sinusoidal optical lattice. Solid (dashed) lines are calculated with a depth, $U_0$, of $6\hbar \omrec$ ($0\hbar \omrec$). The first through fourth order Bragg transitions are indicated by $N_B=1$ to 4. Inset shows a close-up for $N_B=2$. $\Omega_R$ and $\Omega_D$ correspond to the frequencies of Rabi flopping and diffraction phase evolution respectively.}
\label{fig:Gochnauer_BandPicture_Figure1}
\end{figure}
We now turn to determining the Bragg diffraction amplitude and phase in this picture. We explicitly consider an $N_B^{\rm th}$ order Bragg transition which can be seen as a $2N_B$ photon process connecting states $\ket{-N_B\hbar k}$ and $\ket{+N_B\hbar k}$. Here even (odd) $N_B$ corresponds to a crossing at the center (edge) of the Brillouin zone. In the band picture, the Bragg process is the behavior of an initially free particle state which is loaded into the lattice at the $N_B^{\rm th}$ avoided crossing as an equal superposition of the $N_B^{\rm th}$ and $(N_B-1)^{\rm th}$ excited bands (Bloch states). During the $2N_B$ photon pulse the population in the $\ket{\pm N_B\hbar k}$ states oscillate sinusoidally and out of phase with each other at angular frequency $\Omega_R(t)$. Since each of the $\ket{\pm N_B\hbar k}$ states spends equal time in each band, we may evaluate the corresponding phase by integrating the average energy of the two participating Bloch states, characterized by $\Omega_D$, over the duration of the pulse. Within an atom interferometer, diffraction pulses are frequently applied to a superposition of free particle states separated in momenta by multiples of $2\hbar k$, corresponding to different interferometer paths. The different $\Omega_D(t)$ for different paths during these processes can then lead to an observable interferometer phase shift called the {\it diffraction phase}.
In order to apply the Bloch-bands picture to pulsed standing waves, the time-dependence of $U_0(t)$ must preserve the two-state nature of the Bragg process. Practically, the desire for high diffraction efficiency means experiments work in exactly this regime, showing the suitability of the Bloch-bands approach.
For a time-varying standing-wave amplitude $U_0(t)$, adiabaticity mandates $\frac{1}{U_0}\frac{\partial U_0}{\partial t} \ll \Delta E/\hbar $ where $\Delta E$ is the energy separation from the eigenstate nearest to our two states of interest. Applying this criterion to the rise and fall times of a smooth (e.g., Gaussian) pulse shape with width $\tau$, resonant with an $N_B^{\rm th}$ order Bragg process, we arrive at $\tau \gg \frac{1}{4 N_B \omrec}$ where $\omrec=E_{\rm rec}/\hbar$ is the recoil frequency. We recognize this inequality to be equivalent to being in the Bragg regime of diffraction where states other than $\ket{\pm N_B\hbar k}$ are not populated.
We compare the Bloch band picture to both full numerical time evolution of the Schr{\"o}dinger equation and experiments. In Bloch picture simulations, we numerically solve for the band structure at many different lattice depths. We then use these saved band structures to numerically integrate $\Omega_R$ and $\Omega_D$ for any given diffraction pulse profile to obtain population fraction and diffraction phase predictions. With a single set of band structures, the effects of any pulse shape or duration may be quickly calculated, allowing for rapid prototyping of experimental sequences. These simulations enforce adiabaticity by assuming the atoms' wavefunctions remain confined to the two bands corresponding to their free-space momenta. By contrast, full numerical evolution solves the time-dependent Schr{\"o}dinger equation over a large basis of momentum states and uses the time evolution operator to extract population transfer and diffraction phase information (see Appendix B). A full time evolution must be calculated for every pulse shape or duration considered. Comparing to full time evolution allows us to quantify the non-adiabatic effects due to other bands missing from the adiabatic Bloch picture.
We test and verify the validity of the Bloch-bands picture of atoms optics in the $\tau \gtrsim \frac{1}{4 N_B \omrec}$ ``quasi-adiabatic'' regime (see Appendix A) and stay within the observed validity range for all the experimental work presented in this paper. Even when other states are negligibly populated, their presence has a significant effect for typical experimental parameters on both the splitting $\Omega_R$ and the shift $\Omega_D$ of the two coupled states. We now examine these effects individually.
\section{Accurate Rabi Frequencies for Bragg Diffraction}
We first report on our measurements of the Rabi frequency for various Bragg diffraction orders using a BEC atom source. Our results experimentally establish the Bloch-bands picture for atom optics and reveal the shortcomings of a commonly used result in the field (see Fig.\ref{fig:Gochnauer_BandPicture_Figure2}).
The experiments reported in this work were carried out with Bose-Einstein condensates (BECs) of $10^5$ ytterbium (\yb\!) atoms. We prepared the BEC in a $532\,$nm crossed-beam optical dipole trap and released them from the confinement after reducing the mean trap frequency to $\overline{\omega} = 2\pi \times 63\,$Hz \cite{plot18}. Upon release, the atoms are given $2\,$ms to expand before they encounter a diffraction pulse. The diffraction pulses are formed by a pair of horizontal, counterpropagating beams with a waist of $1.8\,$mm, blue detuned from the $556\,$nm intercombination line $(\g \rightarrow \tpl)$ by $+3500\Gamma$, where $\Gamma = 2\pi \times 182\,$kHz is the natural linewidth. The relative detuning between the two beams is controlled by direct digital synthesis electronics at the sub-Hz level. The size of the cloud during all the atom optics experiments is 34 $\mu$m, i.e., far less than the size of the diffraction beams. The depth of the optical lattice formed by these beams was calibrated with Kapitza-Dirac diffraction \cite{gupt01}. This method provided a depth calibration accurate at the $\pm2\%$ level.
\begin{figure}[h]
\center
\includegraphics[width=0.5\textwidth]{Gochnauer_BandPicture_Figure2}
\caption{(Color online) Measurement of band gaps (Rabi frequencies). (a) Sample pulse profile. (b) Rabi oscillations for $N_B=1$ in a $22.7\,\hbar \omrec$ lattice; upper panel shows corresponding sequence of time-of-flight absorption images. The pulse duration is defined as the extent of the intermediate flat region of the pulse profile. The diffracted fraction for zero pulse duration is due to the pulse rise and fall. (c) Measured Rabi frequency for various lattice depths and for $N_B=1,2,3,4$ (filled circles). Solid (dashed) lines are Bloch-bands (RHS of Eqn.\ref{eqn:RabiEqn}) predictions.}
\label{fig:Gochnauer_BandPicture_Figure2}
\end{figure}
We applied diffraction pulses with temporal intensity profiles consisting of Gaussian rise and fall $1/e$ times of $\tau_{\rm 1/e}=27\,\mu$s satisfying $4\omrec \tau_{\rm 1/e}=2.5$, with an intermediate flat profile of variable extent (Fig.\ref{fig:Gochnauer_BandPicture_Figure2}(a)). The relative detuning $\delta$ of the lattice beams was set to match the Bragg resonance condition $\delta = 4 N_B \omrec$. The population in each of the two coupled states was monitored by time-of-flight absorption imaging (Fig.\ref{fig:Gochnauer_BandPicture_Figure2}(b), upper panel). The fractional population in the final state oscillates as $P(t)=\sin^2[\frac{1}{2}\int_{t_0}^t\Omega_R^{(2N_B)}(t'){\rm d}t']$ where $\Omega_R^{(2N_B)}$ is the Rabi frequency for an $N_B^{\rm th}$ order Bragg process (Fig.\ref{fig:Gochnauer_BandPicture_Figure2}(b)). As shown in Fig.\ref{fig:Gochnauer_BandPicture_Figure2}(c), the measured $\Omega_R^{(2N_B)}$ is in good agreement with the Bloch-bands calculation.
Fig.\ref{fig:Gochnauer_BandPicture_Figure2}(c) also demonstrates the inadequacy of a commonly used \cite{gupt01,jans07,mazz15,huqi17} generalized Rabi frequency formula first derived in \cite{gilt95}, given by the RHS of the inequality below:
\begin{equation}
\Omega_R^{(2N_B)} < \frac {[\omega_R]^{2N_B}}{2^{4N_B-3}[(N_B-1)!]^2\Delta^{N_B}\omrec^{N_B-1}}
\label{eqn:RabiEqn}
\end{equation}
Here $\omega_R$ is the single photon Rabi frequency and $\Delta$ is the detuning from the excited state. The RHS is a perturbative result and therefore breaks down at large lattice depth, deviating significantly from the measured values. It is important to note that the standard pulse parameters used in Bragg diffraction experiments are comfortably outside this perturbative regime, stemming from the favoring
of short pulses in experiments in order to minimize state manipulation time in comparison to the longer interferometer interaction times.
\begin{figure}[h]
\center
\includegraphics[width=0.5\textwidth]{Gochnauer_BandPicture_Figure3}
\caption{(a) Space-time diagram for a CI with up to $16\hbar k$ momentum separation between the outer paths and free evolution time, $T=1\,$ms. (b) Representative readout signal for the CI (20 shot average) together with fitted sinusoid with a Gaussian envelope. (c) Varying the peak lattice depth of the mirror pulse changes both the diffracted fraction and the CI phase $\Phi$. The solid red lines show the corresponding Bloch band analysis predictions while the blue dashed lines show the full numerical integration theory, with both methods displaying good agreement with the data (open circles). The black dotted line is the prediction from the RHS of Eqn.\ref{eqn:RabiEqn}.}
\label{fig:Gochnauer_BandPicture_Figure3}
\end{figure}
\section{Application to Diffraction Phases in Atom Interferometry}
The Bloch-bands picture allows a straightforward understanding and assessment of diffraction phases in atom interferometry. In addition to the $\hbar \Omega_R$ band gap, the perturbation of the lattice produces a shift $\hbar \Omega_D$ of the mean energy of the two coupled states (Fig.\ref{fig:Gochnauer_BandPicture_Figure1}(b)). In the presence of a lattice of depth $U_0$, a particular path within an interferometer is characterized by a particular band number and quasimomentum $q$, accumulating an additional phase during a diffraction process:
\begin{equation}
\Phi_D = \frac{1}{\hbar}\int_{\rm pulse}({\bar{E}}(q,U_0)-E_f(q))dt
\label{eqn:dphase}
\end{equation}
where $E_f$ is the free particle energy. For an interferometer path at Bragg resonance during the pulse, ${\bar {E}}(t) = E_f+\hbar \Omega_D(t)$; however, away from a Bragg resonance it is the energy of the band the path is loaded into. Bragg diffraction within an atom interferometer thus results in differential phase shifts between interferometer paths and can lead to an overall diffraction phase, with important ramifications for precision measurements \cite{buch03,jami14,este15}.
We perform our experimental work on diffraction phases in a three-path contrast interferometer (CI) (see Fig.\ref{fig:Gochnauer_BandPicture_Figure3}(a)) with a \yb BEC source \cite{jami14,plot18}. After release of our BEC from the trap, atoms are first placed in an equal superposition of three momentum states ($\ket{+2\hbar k}$, $\ket{0\hbar k}$, $\ket{-2\hbar k}$) using a short standing-wave light pulse operating in the Kapitza-Dirac regime \cite{gupt01}. The three parts of the wavefunction separate for time $T$ after which the outer paths have their momenta reversed by a second-order Bragg $\pi$-pulse. The three paths are again spatially overlapped after an additional time $T$. The contrast of the resulting matterwave interference pattern is measured as the Bragg reflection signal of a traveling-wave light pulse. This readout signal (Fig.\ref{fig:Gochnauer_BandPicture_Figure3}(b)) has the oscillating form:
\begin{equation}
S(t)=C(t)\cos^2(\frac{\phi_1(t)+\phi_3(t)}{2} - \phi_2(t))
\label{eqn:CIsignal}
\end{equation}
where $\phi_i$ is the phase accrued by path $i$ and $C(t)$ is an envelope function determined by the coherence time of the condensate source \cite{plot18}. $S(t)$ oscillates at $8\omrec$ and is sensitive to the kinetic energy differences between the interfering paths and thus to the photon recoil frequency, and can therefore be used to precisely measure the fine structure constant \cite{plot18}. We fit such signals with the expression $C(t_r){\rm cos}^2(4\omrec t_r+\Phi)+S_0$ where $t_r$ is the time from the start of the readout pulse and $S_0$ is a vertical offset. The momentum separation between outer paths during free evolution can be increased to $n\hbar k$ by the insertion of acceleration pulses (Fig.\ref{fig:Gochnauer_BandPicture_Figure3}(a) shows $n=16$) with the resulting CI phase $\Phi=\frac{1}{2}n^2\omrec T+\Phi_{\rm offset}$. Diffraction phase effects are contained in $\Phi_{\rm offset}$, which we study by keeping $T$ fixed and monitoring $\Phi$ for varying diffraction pulse parameters. All acceleration pulses used in this work are either second or third order Bragg pulses with Gaussian-shaped temporal profiles and are incorporated into the theoretical model as integrals over a time-varying $U_0(t)$ in Eqn.\ref{eqn:dphase}.
\begin{figure}[h]
\center
\includegraphics[width=0.5\textwidth]{Gochnauer_BandPicture_Figure4}
\caption{(Color online) Suppression of diffraction phase effects with pulse intensity. Measurements of $U_0 \partial \Phi/\partial U_0$ at the $\pi-$point for different Bragg acceleration pulses are in agreement with the Bloch-bands calculation (curves). The pulses accelerated the outer paths from $\ket{\pm2\hbar k}$ to $\ket{\pm8\hbar k}$ (blue triangles), $\ket{\pm8\hbar k}$ to $\ket{\pm12\hbar k}$ (black squares), and $\ket{\pm8\hbar k}$ to $\ket{\pm14\hbar k}$ (red circles).}
\label{fig:Gochnauer_BandPicture_Figure4}
\end{figure}
As shown in Fig.\ref{fig:Gochnauer_BandPicture_Figure3}(c) for the Bragg (second order) mirror pulse, both the observed phase and the population oscillation are captured well by the Bloch-bands model. While paths 1 and 3 acquire diffraction phase according to $N_B=2$, path 2 remains in the lowest band. The observed sub-unity diffraction efficiency at the $\pi$-point (where the $\pi$-pulse condition is met) is due to the small but finite velocity width of the sample, which is not included in the model.
Since interferometry experiments are sensitive to diffraction phase through intensity noise (i.e., shot-to-shot variations in $U_0$), we characterize its effect using the measured quantity $U_0 \partial \Phi /\partial U_0$ at operating conditions, which for the CI geometry are $\pi$-pulses for both mirror and acceleration optics. This quantity serves as a good figure of merit for the typical situation where the uncertainty in depth scales with $U_0$, since it normalizes the phase fluctuations at each depth. From datasets similar to Fig.\ref{fig:Gochnauer_BandPicture_Figure3}(c), we determine the slopes at these $\pi-$points for several different pulse parameters of $2^{\rm nd}$ and $3^{\rm rd}$ order Bragg acceleration pulses (see Fig.\ref{fig:Gochnauer_BandPicture_Figure4}) and find good agreement with the Bloch-bands theory. When the momentum separation between the paths is large compared to recoil (i.e., multi-band separation), this quantity becomes negligible except for the path(s) undergoing the Bragg transition. Our analysis of diffraction phases in the Bloch-bands picture shows that $\Omega_D/\Omega_R$ monotically decreases with lattice depth in Bragg diffraction for $N_B>1$ (see Fig.\ref{fig:Gochnauer_BandPicture_Figure7}(b) in Appendix C). This is the reason for the observed decrease of $U_0 \partial \Phi / \partial U_0$ with increasing $U_0$ in Fig.\ref{fig:Gochnauer_BandPicture_Figure4}.
This result points to an important consequence for precision interferometry: for high-order Bragg diffraction as commonly used for large momentum separation interferometers \cite{chio11, plot18}, diffraction phase effects are minimized by operating at as high a lattice depth as possible, as long as additional states are not populated by the process. This can be understood as the slowed growth of $\Omega_D$ with $U_0$ from the level repulsion of higher energy states (see Appendix C). Another result of our analysis is that the diffraction phase can be significantly modified by the presence of other interferometer paths in a nearby band (see Eqn.\ref{eqn:CIsignal}) as can be seen in the difference between the red and blue data points in Fig.\ref{fig:Gochnauer_BandPicture_Figure4}. This method can be used to greatly suppress the diffraction phase effect (see blue data point at 27 recoils in Fig.\ref{fig:Gochnauer_BandPicture_Figure4}). We can also see that certain interferometer geometries are immune to diffraction phases from the symmetry of pulse application, e.g., the symmetric Mach-Zehnder. However interferometers that measure the recoil phase accrued between paths are generally sensitive to diffraction phases \cite{jami14,este15,park18,plot18}.
\section{Determining Band Structure from Interferometer Phase}
While the earlier discussion has been mainly focussed on the application of the Bloch-bands picture at avoided crossings, the picture applies equally well to all other points in quasimomentum space. This approach to atom diffraction and interferometry can thus naturally lend itself as a tool to determine band structure due to some unknown periodic potential. The transient presence of some unknown optical lattice manifests as an interferometric phase shift with different interferometer paths evolving phase according to the band number and quasimomentum into which they map. By varying these quantities with the atom optical elements of the interferometer, the complete band structure due to the unknown potential can be determined.
\begin{figure}[h]
\center
\includegraphics[width=0.5\textwidth]{Gochnauer_BandPicture_Figure5}
\caption{(a) Space-time diagram for band structure measurement with a modified $n=16$ CI. This is identical to Fig.\ref{fig:Gochnauer_BandPicture_Figure3}(a) except for the additional pulsed lattice (dark red stripe) which imparts the band structure to be determined. (b) CI phase at $q=0$ for various lattice strengths. (c) The ground band dispersion in a sinusoidal lattice from diffraction phase measurements. The solid line corresponds to the theoretical dispersion for a lattice depth of $6.5\hbar \omrec$.}
\label{fig:Gochnauer_BandPicture_Figure5}
\end{figure}
A clean implementation of this tool is furnished in the CI, modified as shown in Fig.\ref{fig:Gochnauer_BandPicture_Figure5}(a). For demonstration purposes we determine the ground band structure in a sinusoidal optical lattice with Gaussian temporal shape and $4\omrec \tau_{\rm 1/e}=2.5$. The value of quasimomentum $q$ in the standing-wave frame is chosen by precisely controlling the relative detuning of the counterpropagating lattice beams in the lab frame. During the pulse, the middle path is in the bottom band (band 0) and the outer paths are in bands 7 and 8. Diffraction phase accrued by the outer paths is negligible compared to that accrued by the middle path. Fig.\ref{fig:Gochnauer_BandPicture_Figure5}(b) shows the mesaured CI phase at $q=0$ for various lattice depths, in good agreement with the band theory prediction \cite{footrabi}. As we vary $q$ at a fixed depth, the measured CI phase converted to an energy shows the characteristic ground band dispersion (Fig.\ref{fig:Gochnauer_BandPicture_Figure5}(c)).
The data presented in Fig.\ref{fig:Gochnauer_BandPicture_Figure5}(c) was obtained through a series of experiments in which a non-diffracting lattice is applied within the interferometer at a time highlighted (red stripe) in Fig.\ref{fig:Gochnauer_BandPicture_Figure5}(a). Because the relative detuning of the two lattice beams was chosen such that the middle interferometer path loaded into the bottom band, the other paths --- having been accelerated --- were loaded into much higher bands and thus contributed negligibly to the total CI phase. The quasimomentum $q$ in the standing-wave frame could then be varied and any phase shift would be an observation of the lattice energy dispersion. The adiabaticity criterion prevents turning on a lattice at $q$ near the Brillouin zone edge without loading into a superposition of the bottom band and the first excited band. To measure the energy shift at a particular $q$ value in only the bottom band, we developed the following procedure: First we adiabatically turned on a lattice at $q=0$, reaching a depth of $6.5\hbar \omrec$ over 150 $\mu$s with a cubic spline temporal shape. Next we linearly ramped the relative detuning between diffraction beams for 56 $\mu$s until we reached $q=-0.95\hbar k$. Then we swept the relative detuning at the same rate in the opposite direction, stopping at the desired $q$ for 100 $\mu$s, and eventually reaching $q=+0.95\hbar k$. Finally, the relative detuning was brought back to zero and the lattice turned off with the same cubic spline shape.
This method thus allowed us to directly measure the diffraction phase $\Phi_D$ as a function of quasimomentum. In accord with Eqn.\ref{eqn:dphase}, we converted $\Phi_D$ to an energy shift and then added the calculated free-space energy at each $q$ to obtain the ground band dispersion. In the experimental sequence the diffraction phase acquired during each intensity ramp and frequency ramp was common for all experimental iterations, and resulted in a uniform offset phase in the experiment. This became an energy offset in the measured ground band dispersion, which we determined by fitting with one free parameter (the constant offset) to the calculated ground band energy dispersion for a $6.5\hbar \omrec$ depth lattice. The obtained ground band dispersion determined from the data in this way shows good agreement with the theoretical calculation (Fig.\ref{fig:Gochnauer_BandPicture_Figure5}(c)).
\section{Discussion and Conclusions}
We have investigated a Bloch-bands approach to analyzing atom diffraction and interferometry. Theoretical results for the amplitude and phase associated with standing wave diffraction show good agreement with measurements for arbitrary lattice strength. Significantly, our results span a range of atom optics parameters that extend beyond the weak lattice regime, and are thus of practical importance for current atom interferometry experiments. While analytic formulas for diffraction amplitude are known from earlier work \cite{gilt95,gupt01} and their inadequacy beyond the weak lattice limit recognized \cite{cham01,jans07}, our results constitute the first experimental study and its accurate analysis at arbitrary lattice depth. We have for the first time demonstrated the validity of a Bloch-bands approach to diffraction phases by direct comparison to interferometric phase shifts. Our work also points out general methods to control diffraction phases, an important systematic effect in precision measurements. All our results, presented as scaled to recoil frequency and momentum are generally valid for all atom diffraction and interferometry setups. Our interferometric method of band structure measurement is complementary to earlier methods \cite{klin10,geig18}, and while demonstrated here only for the ground band, can be extended to excited bands as well by loading path 2 into the desired band. Furthermore, the method can also be adapted to non-sinusoidal periodic potentials, as well as to higher-dimensional and time-dependent (e.g., Floquet) lattices \cite{fuji19}.
We thank Tahiyat Rahman for experimental assistance. This work was supported by NSF Grant No. PHY-1707575 and by the NASA Fundamental Physics Program.
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1,116,691,497,590 | arxiv | \section*{Informational Fourth Page}
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1,116,691,497,591 | arxiv | \section{Introduction}
Given the similarity between neutrons and protons, that gets ramified in the concepts of charge symmetry and charge invariance for nuclear interactions, the natural expectation is that there is little difference in how neutrons and protons distribute themselves across a~nucleus. In fact, it has been common to infer net nuclear density by scaling up the proton density deduced from electron scattering. Moreover, following expectations of incompressible nuclear matter, it has been common to find nuclear radii discussed as proportional to the cube root of mass number, particularly in textbooks.
Extrapolations from nuclei studied under laboratory conditions to neutron stars~\cite{horowitz_neutron_2001} spurred, though, an increased attention to potential differences in the distribution of neutrons and protons and, in particular, to the emergence of neutron skins and their interrelation with the symmetry energy, the energy penalty for neutron-proton imbalance in a nucleus. In theoretical considerations and calculations~\cite{dobaczewski_neutron_1996,Danielewicz:2003dd}, the size of neutron skin shrinks towards zero with vanishing imbalance. With limited relative imbalances in the larger nuclei suitable for exploring bulk nuclear properties, such as symmetry energy, the small expected sizes of the skins have made their precise determination, and the drawing of conclusions on symmetry energy, difficult. The radii of nucleonic distributions are primarily determined in elastic scattering off nuclei, of electrons, for proton distributions \cite{de_vries_nuclear_1987}, and of protons~\cite{PhysRevC.19.1855,PhysRevC.67.054605,Shlomo19795,PhysRevC.46.1825,PhysRevC.82.044611,PhysRevC.49.2118}, alphas~\cite{PhysRevC.29.1295} and pions~\cite{Friedman:2012pa}, for neutron and proton distributions. With a focus on parity breaking contribution to the differential cross section, the elastic electron scattering can also provide a~relatively direct information on neutron distributions~\cite{prex_collaboration_measurement_2012}, albeit at the cost of a~long running time for an experiment and elaborate efforts to suppress both statistical and systematic errors.
In our earlier work~\cite{Danielewicz200936}, hereafter referred to as I, we pointed out that the same considerations of the interaction symmetries, that produce the concept of symmetry energy, also give rise the concepts of universal isoscalar and isovector densities. Those densities may be expected to change weakly within an isobaric chain of nuclei and may be combined to yield neutron and proton densities. Within the Skyrme-Hartree-Fock (SHF) calculations of the density profiles for half-infinite nuclear matter in~I, we observed the isoscalar and isovector densities to vary little with the neutron-proton imbalance in the matter, particularly within the asymmetry range such as typical for heavier nuclei, consistent with the general expectations. However, the isovector densities on its own, and in relation to isoscalar, had in these calculations some unique characteristics tied to the symmetry-energy for the employed Skyrme interactions. In particular, when the symmetry energies exhibited a weak density-dependence at moderately subnormal densities, the isoscalar and isovector densities were similar to each other. On the other hand, when the energies had a strong density-dependence, the isovector density tended to have its surface pushed out relative to the isoscalar density by as much as $\sim$$1 \, \text{fm}$. Besides the slope of symmetry energy in density being correlated with the relative displacement of isovector and isoscalar surfaces, that slope was found to be correlated with a difference in steepness of the isoscalar and isovector densities, with larger slopes (or stiffer symmetry energies) yielding steeper isovector densities. The differences between isoscalar and isovector densities could be easily understood with the Thomas-Fermi (TF) approximation \cite{Danielewicz200936}. In fact, for a constant symmetry energy as a~function of density, the TF approximation would produce identical isoscalar and isovector densities.
In this work, we look for an evidence of the displacement of isovector surfaces relative to isoscalar, or of isovector skin, in direct-reaction cross-sections. We further assess the extent to which other characteristics of the densities observed in I for the half-infinite matter apply to finite nuclei. We confront our findings with results of SHF calculations for spherical nuclei and draw conclusions on the symmetry energy. The cross-section data stem from quasielastic (QE) charge-exchange (p,n) reactions, on one hand, and from elastic proton and neutron scattering, on the other, off the same set of target nuclei: $^{48}$Ca, $^{90}$Zr, $^{120}$Sn and~$^{208}$Pb. The~QE reactions are those to isobaric analog states (IAS) of the target ground states. The~above processes can be jointly~\cite{patterson_energy-dependent_1976,byrd_self-consistent_1980,wong_analysis_1984,jon_analog_1997,khoa_folding_2007} described in terms of a Lane-type potential~\cite{lane_new_1962,lane_isobaric_1962} with isoscalar and isovector parts. The components of such potential may be parametrized in a simple form \cite{patterson_energy-dependent_1976} or may be derived from folding a nucleon-nucleon (NN) potential with proton and neutron densities \cite{schery_folding_1980,khoa_folding_2007}. In arriving at conclusions on symmetry energy, we use a relation between the profiles of the isovector and isoscalar potentials, as a proxy for the relation between the profiles of isovector and isoscalar densities. That presumption would be generally satisfied in a folding model with the range of NN interaction idependent of the isospin channel, as commonly assumed in the literature~\cite{schery_folding_1980,khoa_folding_2007}. In the impulse approximation, in fact, the isovector and isoscalar potentials would just represent rescaled isoscalar and isovector densities.
Regarding the past work with a scope tied to the current effort, Schery \etal\ \cite{schery_nuclear_1976,schery_folding_1980} noticed a strong sensitivity of the QE (p,n) cross sections obtained from the folding model, to the difference of the rms proton and neutron radii, currently termed neutron skin. Using measured (p,n) cross sections they concluded on the presence~\cite{schery_p_1974,schery_nuclear_1976,schery_radius_1974} or absence~\cite{schery_evidence_1980} of the skins depending on target nuclei. Carlson \etal\ \cite{carlson_test_1973,carlson_optical_1975} demonstrated that the description of (p,n) data can be improved when assuming a different geometry for the isovector potential, than that following from the difference between proton and neutron potentials within the popular Becchetti-Greenlees (BG) potential parametrization \cite{becchetti_nucleon-nucleus_1969}. That was reaffirmed by Jolly \etal~\cite{jolly_$pn$_1973}. On the other hand, Patterson \etal\ \cite{patterson_energy-dependent_1976} described a comprehensive set of (p,n) data from their measurements, in parallel with reference elastic (p,p) and (n,n) cross sections, using single geometry of the BG proton potential for both the isovector and the isoscalar potentials. Jon \etal~\cite{jon_analog_1997,jon_isovector_2000} adjusted geometry of the imaginary part of isovector potential to reproduce their cross-section measurements for (p,n) reactions at $35 \, \text{MeV}$, while keeping the real part in the BG form, largely continuing the effort of Carlson \etal~\cite{carlson_test_1973,carlson_optical_1975}. However, they neither considered elastic cross sections in their analysis, nor allowed for a self-consistency between the adjusted isovector potential and the initial and final wavefunctions in the employed Distorted-Wave Born Approximation (DWBA). In our work, we find both to be of importance when deciding on the subtle details of the Lane potential, even though the contributions of the isovector potential to the nucleonic potentials are nominally small, $\sim |N-Z|/A$.
Both Jon \etal~\cite{jon_analog_1997,jon_isovector_2000} and Carlson \etal\ \cite{carlson_test_1973,carlson_optical_1975} were able to assign values of radii and surface diffuseness to isovector potentials on a nucleus-by-nucleus basis and these turned out to vary relatively smoothly with nuclear mass. Even though we work with larger data sets with smaller errors for individual nuclei, we find those data insufficient to constrain reliably such an abundance of parameters for isovector potentials, on a meaningful scale, as in the works above. The recent interest in the Lane potential, tied to the symmetry energy, focussed more on the strength of the isovector potential \cite{khoa_folding_2007,khoa_folding_2014,li_neutronproton_2015,liChen_neutronproton_2015} than on its geometry being central here. In one work~\cite{loc_charge-exchange_2014}, though, ($^3$He,t) on $^{90}$Zr and $^{208}$Pb reaction data were analyzed in terms of a folding model with the goal of learning on neutron skins, with the conclusion that these met expectations.
In the next section, we discuss various practicalities in theoretical description of elastic and quasielastic processes, such as nucleon optical potential parametrizations, impact of various components on the cross sections, DWBA etc. In Sec.~3, we fit the geometry of isovector and isoscalar potentials to data. The QE (p,n) data we rely on are those from the measurements by Doering, Patterson and Galonsky \cite{patterson_energy-dependent_1976,doering_microscopic_1975,doering_isobaric_1974} at the incident proton energies of 25, 35 and $45 \, \text{MeV}$. We supplement these with elastic data from the EXFOR database~\cite{otuka_towards_2014} for the same target nuclei. Over the target nuclei, incident energies and reaction types, we exploit 48 data sets in our fits. For exploring the potential geometry we modify the potential parametrization by Koning and Delaroche (KD) \cite{koning_local_2003}, following inspiration from~I. In~Sec.~4, we confront the results of our data analysis with results of spherical SHF calculations for different Skyrme interactions. Upon combining the results from that confrontation, with those from the confrontation of theoretical predictions for symmetry coefficients with data in~\cite{danielewicz_symmetry_2014}, hereafter refereed to as II, we arrive at constraints on symmetry energy and on energy of neutron matter. For details on the symmetry energy, we refer the reader to reviews such as~\cite{li_isospin_1998}. In arriving at the constraints we follow Bayesian inference. We conclude in Sec.~5.
\section{Description of Elastic and Quasielastic Cross Sections}
\subsection{Optical Potential}
The Lane optical potential~\cite{lane_new_1962,lane_isobaric_1962} is of the form
\begin{equation}
\label{eq:Lane}
{ U}({\pmb r}) = { U}_0({\pmb r}) + \frac{{\pmb \tau}\,{\pmb T}}{4A} \, { U}_1({\pmb r}) \equiv { U}_0({\pmb r}) + \frac{{\tau}_3 \,{T}_3}{4A} \, { U}_1({\pmb r}) + \frac{{\pmb \tau}_\perp \,{\pmb T}_\perp}{4A} \, { U}_1({\pmb r}) \, .
\end{equation}
Here, $U_0$ and $U_1$ are isoscalar and isovector optical potentials, respectively, and ${\pmb \tau}$ and ${\pmb T}$ are isospin operators for the projectile nucleon and target nucleus, respectively. The scalar product on the r.h.s.\ of \eqref{eq:Lane} can be represented as ${\pmb \tau}_\perp \,{\pmb T}_\perp = \tau_+ \, T_- + \tau_- \, T_+ $. The factors for $U_1$ in the Lane potential are organized so that the Lane potential reduces to a familiar form when employed in the elastic channels for proton and neutron scattering, specifically
\begin{equation}
\label{eq:Unp}
U_\text{n,p} ({\pmb r}) = { U}_0({\pmb r}) \pm \frac{N-Z}{A} \, { U}_1({\pmb r}) \, .
\end{equation}
Here, the upper sign pertains to neutrons and lower to protons and $N$, $Z$ and $A$ refer to the target nucleus. These potentials generally contain spin-orbit terms. The isoscalar and isovector potentials can be conversely deduced from the nucleonic potentials with
\begin{align}
\label{eq:U0np}
U_0 ({\pmb r}) & = \frac{1}{2} \big[ U_\text{n}({\pmb r}) + U_\text{p}({\pmb r}) \big] \, , \\[.5ex]
U_1 ({\pmb r}) & = \frac{A}{2(N-Z)} \, \big[ U_\text{n}({\pmb r}) - U_\text{p}({\pmb r}) \big] \, .
\label{eq:U1np}
\end{align}
The off-diagonal element of the Lane potential, driving the transition from the initial to final state in a (p,n) reaction, is
\begin{equation}
\langle n, Z+1| U({\pmb r}) |p, Z \rangle = 2 \frac{\sqrt{|N - Z|}}{A} \, { U}_1({\pmb r}) \, .
\label{eq:U1mtx}
\end{equation}
The original Lane~\cite{lane_new_1962,lane_isobaric_1962} optical potential \eqref{eq:Lane} is invariant under overall rotations in isospin space. We will argue that Coulomb interactions, breaking the isospin invariance, produce some $Z$-dependent difference between the $U_1$-factors multiplying the third and transverse isospin components in the Lane potential.
In the general considerations of average behavior of the potential, its isoscalar part~$U_0$ can depend only on scalar quantities in isospin space so its dependence on isovector quantities is quadratic or higher, i.e.~weak. The isovector part of the potential, that can couple to an external isospin ${\pmb \tau}$, transforms in isospin space in the same way as isospin density ${\pmb \rho}$, so can be written as that density multiplied by a scalar factor. The~simplest scalar factor is just a constant and a constant in particular allows to meet the requirement of the potential vanishing in the absence of matter and can provide, under any circumstances, a coarse approximation to the potential changing from zero outside of matter to a finite value prevailing across the matter interior. However, whether or not linear, the relation between the potential and density can also be weakly nonlocal. After the values of isospin components are factored out, as components of~${\pmb T}$, the form of~$U_1$ can be retained as a profile of isospin density being multiplied by a~scalar function, to provide insights into any variations on top of the variations of isospin content. In~II, we discussed that Coulomb interactions, while impacting the density of the third component of isospin, since displacing protons out relative to neutrons, yield no similar impact on the density of transverse isospin in an isobaric chain. With this, one expects a difference between the $U_1$-factor multiplying the third components of isospin in \eqref{eq:Lane} and the $U_1$-factor multiplying the transverse components there. That difference should develop with growing~$Z$, mirroring the difference in the densities for isospin components. With the isovector potential competing with isoscalar, in generating predictions for elastic differential cross sections, it can be a challenge to discern details in $U_1$ acting in elastic scattering to the level of telling them reliably from details in $U_1$ in the (p,n) reactions. Correspondingly, for now, in the context of the data analysis, as earlier in this section we will make no distinction between $U_1$ acting in elastic reactions and that acting in quasielastic (p,n) reactions and we will rather concentrate on revealing any difference in the geometry between $U_0$ and $U_1$, using both types of reactions. When confronting the results with structure calculations, we will get back to the consideration of different directions in isospin space, though.
In the literature, the nucleonic potentials $U_\text{p}$ and $U_\text{n}$ are commonly expressed in terms of Woods-Saxon formfactors and their derivative factors
\begin{align}
f(r,R,a) & = \frac{1}{\exp{\frac{r-R}{a}}+1} \, ,\\[.5ex]
f^d (r,R,a) & = -4 \, a \, \frac{\text{d}f}{\text{d}r} = 4\, f \, (1-f) \, .
\end{align}
For example, the popular and relatively recent parametrization of nucleonic optical potentials by Koning-Delaroche (KD) \cite{koning_local_2003} is of the form
\begin{equation}
\begin{split}
\label{eq:Usum}
U ({\pmb r}) = & -[V_V(E)+ i \, W_V(E)] \, f(r, R_V, a_V) - i \, W_D(E) \, f^d(r, R_D, a_D) \\
& - \frac{1}{2 \, a_\text{SO} \, m_\pi^2 } \,[V_\text{SO}(E) + i \, W_\text{SO}(E)] \, \frac{1}{r} \, f^d(r,R_\text{SO},a_\text{SO}) \, {\pmb s} \, {\pmb L}
\, .
\end{split}
\end{equation}
Here, ${\pmb s}$ is the spin operator for the incident nucleon and the potential strengths depend on incident energy~$E$.
For protons, the nuclear potential~$U$ is supplemented by a Coulomb potential that is conventionally taken in the form such as for a uniformly charged sphere:
\begin{equation}
V_C(r) =
\begin{cases}
\frac{Z \, e^2}{8 \pi \epsilon_0 \, R_C} \, \Big( 3 - \frac{r^2}{R_c^2} \Big) \, , & r < R_C \, ,\\
\frac{Z \, e^2}{4 \pi \epsilon_0 \, r} \, , & r > R_C \, .
\end{cases}
\end{equation}
When the geometric parameter sets $(R_X, a_X)$, $X=V$, $D$, $SO$, are the same for neutrons and protons~\cite{patterson_energy-dependent_1976,varner_global_1991}, then the isoscalar and isovector potentials from~\eqref{eq:U0np} and~\eqref{eq:U1np} have the same structure~\eqref{eq:Usum} as the nucleonic potentials, with the same set of geometric parameters for the different terms in the potentials. When the proton and neutron potentials have significant differences in their geometry, though, such as in the "best-fit" case of the BG parametrization~\cite{becchetti_nucleon-nucleus_1969}, the resulting isovector potential can have unusual structure~\cite{hoffmann_exact_1973} difficult to justify on physical grounds. Carlson \etal~\cite{carlson_optical_1975} and Jon \etal~\cite{jon_analog_1997,jon_isovector_2000}, in fact, postulated Woods-Saxon type formfactors for the isovector potentials and adjusted their parameters directly when describing QE (p,n) cross sections.
As the efforts here may be of interest to those focused on symmetry energy, but whose research areas lack overlap with the direct reactions, an introduction covering essential issues of relevance for the efforts here can be found in the review by Amado~\cite{amado_analytic_1985}. Briefly, within a direct reaction, leaving the final nucleus with little or no excitation, an important part of the nuclear processes occurs near the surface of the nucleus, with diffraction and refraction taking place there, cf.~Fig.~\ref{fig:DirectReaction}. Traversal of the nucleus through the volume generally populates more complicated final states and depletion of the incident flux into such channels is described in terms of an imaginary and, more generally nonhermitian, part of the optical potential utilized for the direct channels, such as \eqref{eq:Usum}.
The peripheral nature of direct processes limits the complexity that can be of relevance in the potentials.
\begin{figure}
\centerline{\includegraphics[width=.51\linewidth]{DirectReaction}}
\caption{Schematic view of a direct reaction.
}
\label{fig:DirectReaction}
\end{figure}
In our search for evidence of isovector skins, we use the KD potential parametrization~\cite{koning_local_2003} as the starting point. That parametrization is relatively recent and commonly used and it is fitted to a vast collection of elastic scattering data. Moreover it employs practically the same potential geometry for neutrons and protons. The only small difference is in the adopted values of the diffusivity in the imaginary potential. We actually reset that difference in the diffusivity to zero and, with the same geometry for neutrons and protons, the starting geometry becomes the same for the isocalar and isovector potentials.
\subsection{Cross Sections}
In connecting the potentials to elastic scattering data, the scattering wavefuction is decomposed in the standard manner into partial waves. The corresponding set of one-dimensional Schr\"{o}dinger equations is solved to yield the scattering matrix elements for individual partial waves, factorized as $S_{J\ell}= \text{e}^{2 i \sigma_\ell} \, S_{J\ell}^N$, where $\sigma_\ell$ are Coulomb phase shifts and $S_{J\ell}^N$ are nuclear factors. The cross section averaged over initial spin directions and summed up over final takes then the form
\begin{equation}
\frac{\text{d} \sigma}{\text{d} \Omega} = \frac{\text{d} \sigma_C}{\text{d} \Omega} + \frac{\text{d} \sigma_N}{\text{d} \Omega} + \frac{\text{d} \sigma_i}{\text{d} \Omega} \, ,
\end{equation}
where $\text{d} \sigma_C/\text{d} \Omega) = |f_C(\theta)|^2$ is the Rutherford cross section, and the two remaining terms are the nuclear cross section and the interference contribution. The interference contribution is
\begin{equation}
\frac{\text{d} \sigma_i}{\text{d} \Omega} = 2 \, \text{Re} \left[\overline{f}_N(\theta) \, f_C^*(\theta) \right] \, ,
\end{equation}
where $\overline{f}_N$ is nuclear scattering amplitude averaged over initial and summed over final spin directions
\begin{equation}
\overline{f}_N(\theta) = - \frac{i}{2k} \, \frac{1}{2s+1} \sum_{J\ell} (2J+1) \, \text{e}^{2i\sigma_\ell} \, (S_{J\ell}^N - 1) \, P_\ell(\cos{\theta}) \, ,
\end{equation}
where $s=\frac{1}{2}$. Finally, the nuclear cross section is, after \cite{frobrich_theory_1996,edmonds_angular_1996,biedenharn_properties_1952},
\begin{equation}
\label{eq:sigel}
\frac{\text{d} \sigma_N}{\text{d} \Omega} = \frac{1}{k^2} \, \frac{1}{2s+1} \sum_L (2L+1) \, {\mathcal A}_L^N \, P_L(\cos{\theta}) \, ,
\end{equation}
where the expansion coefficients for the cross section are
\begin{equation}
\label{eq:ALel}
\begin{split}
{\mathcal A}_L^N = & \frac{1}{4} \sum_{J' \, \ell'} (2J'+1)(2\ell'+1) \sum_{J \ell} (2J+1)(2\ell+1) \, \Bigg(\begin{matrix} \ell & \ell' & L \\ 0 & 0 & 0 \end{matrix}\Bigg)^2 \\
& \times \Bigg\lbrace\begin{matrix} \ell & \ell' & L \\ J' & J & s \end{matrix}\Bigg\rbrace^2 \, \text{Re} \left[\text{e}^{2i(\sigma_\ell - \sigma_{\ell'})} \, (S_{J'\,\ell'}^{N*} - 1) \,(S_{J\ell}^{N} - 1) \right] \, ,
\end{split}
\end{equation}
with 3j and 6j symbols being utilized. In elastic scattering of charged projectiles, the Coulomb cross section generally dominates in the forward direction with a~telltale sign being proximity of the net cross section to the Rutherford cross section. The nuclear processes generally take over past the grazing angle representing the classical Coulomb trajectory that just touches the nuclear surface. The interference contribution strongly eats into these cross sections in the vicinity of the grazing angle. With the exception of proton elastic scattering off $^{208}$Pb at the low end of considered incident energy range, the grazing angles for the data addressed in this paper are fairly low.
The (p,n) cross sections are typically described in the literature in DWBA~\cite{frobrich_theory_1996}. That approximation is justified when the final state is likely to be populated in one step, without other states serving as intermediaries, or multiple transitions occurring back-and-forth between the initial and final states. Construing towards DWBA validity are fast processes, yielding low transition probability, such as typical for peripheral reactions at high incident energy. The applicability of DWBA is expected to worsen with increasing inelasticity of a~reaction, with slowing down of the reaction, such as close to a~Coulomb barrier, and with inhibition of the probability of reaching the final state in one step, due to dynamics and/or conservation laws, as compared to a~multitude of steps, such as in emission into backward angles at high incident energies~\cite{tomita_analysis_2015}. Multiple back-and-forth transitions between the initial and final states are usually associated with a high transition probability. This will be addressed below. An alternative to DWBA is a solution to a coupled channel problem that can take on various levels of complexity and physics scope.
With \eqref{eq:Lane} and \eqref{eq:U1mtx}, the unpolarized (p,n) cross section in the DWBA approximation is~\cite{frobrich_theory_1996,satchler_optical-model_1964}
\begin{equation}
\label{eq:DWBA}
\frac{\text{d}\sigma_{(p,n)}}{\text{d} \Omega} = (2 \pi)^4 \, \mu_p \, \mu_n \, \frac{k_n}{k_p} \, \frac{1}{2s+1} \sum_{M_p \, M_n} \left| 2 \frac{\sqrt{|N-Z|} }{A} \int \text{d}{\pmb r} \, \chi^{(-)\dagger}_{n \, M_n}({\pmb r}) \, U_1({\pmb r}) \, \chi^{(+)}_{p \, M_p}({\pmb r}) \right|^2 \, ,
\end{equation}
where p and n are used as indices for the initial and final states, $\mu$ are reduced masses, $k$ are c.m.\ wavevectors and $\chi$ are the distorted waves that describe elastic scattering of a~proton and neutron in the initial and final channel, respectively, under the influence of the potentials \eqref{eq:Unp}. Upon partial wave decompositions, the DWBA cross section may be brought into the form similar to that for the elastic cross section, Eqs.~\eqref{eq:sigel} and \eqref{eq:ALel}, i.e.
\begin{equation}
\frac{\text{d} \sigma_\text{(p,n)}}{\text{d} \Omega} = \frac{1}{k_p^2} \, \frac{1}{2s+1} \sum_L (2L+1) \, {\mathcal A}_L^\text{(p,n)} \, P_L(\cos{\theta}) \, .
\end{equation}
Here the expansion coefficients for the differential cross section are
\begin{equation}
\begin{split}
{\mathcal A}_L^\text{(p,n)} = & 4 \, \mu_p \, \mu_n \, k_p \, k_n \sum_{J' \, \ell'} (2J'+1)(2\ell'+1) \sum_{J \ell} (2J+1)(2\ell+1) \\
& \times \Bigg(\begin{matrix} \ell & \ell' & L \\ 0 & 0 & 0 \end{matrix}\Bigg)^2 \Bigg\lbrace \, \begin{matrix} \ell & \ell' & L \\ J' & J & s \end{matrix}\Bigg\rbrace^2 \, \text{Re} \left[I_{J' \, \ell'}^* \, I_{J \ell} \right] \, ,
\end{split}
\end{equation}
where $I$ are the partial-wave integrals
\begin{equation}
\label{eq:IJl}
I_{J\ell} = 2 \frac{\sqrt{|N-Z|} }{A} \int_0^\infty \text{d}r \, r^2 \, u_{n \, J \ell}^{(+)}(r) \, U_1^{J \ell}(r) \,
u_{p \, J \ell}^{(+)}(r) \, ,
\end{equation}
and $u$ are radial wavefunctions for the initial and final channels.
The question whether a coupled-channel approach, combining the initial and final channels for a QE (p,n) reaction, within a Schr\"{o}dinger equation set~\cite{wesolowski_coupled-channel_1968,wong_analysis_1984,khoa_folding_2007}, could bring in some tangible benefits over DWBA, may be addressed by examining data. In Table~\ref{tab:csratio}, we show the ratios of total QE (p,n) cross sections from the measurements of Doehring \etal~\cite{doering_isobaric_1974}, to estimated geometric nuclear geometric cross sections, $\sigma_g = \pi \, R^2 \, [1-V_C(R)/E]$. The errors of the measured cross sections are about 10\%, but the primary uncertainty in the ratios, that may be interpreted as probabilities for p-n conversion, are tied to the ambiguity in the choice of $R$ for the geometric cross section. Here, we use the radii of the volume potential in KD parametrization, that tends to put the ratios on the high side. As alternative to total cross section ratio, one can examine normalized differential cross sections.
\begin{table}
\caption{Percent ratio of total quasielastic (p,n) cross section from measurements of Doehring \etal~\cite{doering_isobaric_1974}, to estimated nuclear geometric cross section, $\sigma_\text{(p,n)}/\sigma_g$, where $\sigma_g = \pi \, R^2 \, [1-V_C(R)/E]$.}
\label{tab:csratio}
\vspace*{.5ex}
\begin{tabular}{||c|| c| c| c | c ||}
\hline
\hline
& \multicolumn{4}{c ||}{Target Nucleus } \\
\cline{2-5}
$E_p$ & $^{48}$Ca & $^{90}$Zr & $^{120}$Sn & $^{208}$Pb \\
(MeV) & ($R=4.33 \, \text{fm}$) & ($R=5.44 \, \text{fm}$) & ($R=6.03 \, \text{fm}$) & ($R=7.32 \, \text{fm}$) \\
\hline
\hline
25 & 2.47 & 1.26 & 1.44 & 1.62 \\
35 & 2.14 & 0.74 & 0.75 & 0.75 \\
45 & 1.68 & 0.62 & 0.69 & 0.50 \\
\hline
\hline
\end{tabular}
\end{table}
For the elastic or inelastic scattering of charged projectiles, it is common to normalize the differential cross sections with the Rutherford cross sections. This brings in different benefits. First, the magnitude range needed for presenting the cross section values shrinks. Second, when diffraction effects are moderate, one can visually identify the range of angles where Coulomb scattering dominates. Finally, when both the diffraction and refraction effects are moderate, one can interpret the ratio in terms of a transmission probability along a classical trajectory for staying within the entrance channel or for moving to another. Normalizing the charge-exchange cross sections with the Rutherford cross-section, to reap any benefits, is pointless, though, as charges in the final state are different than in the entrance state and only half of the Coulomb deflection, behind a differential Coulomb cross section, is accumulated when the particles approach each other and the other half when they move away. However, one might normalize a charge-exchange cross section with a Coulomb cross section arrived at under the assumption that the charge exchange occurs around the point of closest approach, with the classical trajectory illustrated in Fig.~\ref{fig:Coulpn}. Under that assumption, the net classical deflection angle at impact parameter $b$ becomes
\begin{equation}
\theta(b) = \frac{1}{2} \, \theta_i(b) + \frac{1}{2} \, \theta_f(b) \, ,
\end{equation}
where the labels $i$ and $f$ pertain to the particles in the entrance and exit channels and where the r.h.s.\ angles are the Coulomb deflection angles at the impact parameter $b$ for the respective charge combinations in the entrance and exit channels.
\begin{figure}
\centerline{\includegraphics[width=.37\linewidth]{Coulpn}}
\caption{Charge-exchange Coulomb trajectory: at the point of closest approach charge-exchange takes place, making the incident particle continue along an altered Coulomb trajectory towards the detectors. The illustration here is for a (p,n) reaction on $^{208}$Pb at $25 \, \text{MeV}$ and $b= 3 \, \text{fm}$.
}
\label{fig:Coulpn}
\end{figure}
From the above, the classical Coulomb cross section, with charge exchange at the point of closest approach, becomes
\begin{equation}
\begin{split}
\frac{\text{d} \sigma^{(i,f)}_C}{\text{d} \theta \, \, } & = \frac{b}{\sin{\theta}} \, \frac{\text{d} b}{\text{d} \theta}
\\[.5ex]
& = \frac{(\kappa_i + \kappa_f)^2}{16 E^2} \, \frac{1}{\sin^4{\theta}} \, \frac{\left[\sqrt{1 - \left(\frac{\kappa_i-\kappa_f}{\kappa_i+\kappa_f}\right)^2 \, \sin^2{\theta}} + \cos{\theta} \right]^2}{\sqrt{1 - \left(\frac{\kappa_i-\kappa_f}{\kappa_i+\kappa_f}\right)^2 \, \sin^2{\theta}}} \, .
\end{split}
\end{equation}
Here $\kappa_{i,f}$ are Coulomb factors for the entrance and final channels, respectively, $\kappa_{i,f} = \frac{Z_{i,f} \, Z_{I,F} \, e^2}{4 \pi \epsilon_0}$. For $\kappa_i=\kappa_f$, the above cross section becomes the standard Rutherford cross section. For a (p,n) reaction, with $\kappa_n = 0$, the cross section becomes
\begin{equation}
\label{eq:dsigCpn}
\frac{\text{d} \sigma^{(p,n)}_C}{\text{d} \theta \, \, } = \frac{\kappa_p^2}{4E^2} \, \frac{\cos{\theta}}{\sin^4{\theta}} \, ,
\end{equation}
at $\theta < 90^\circ$ and at $\theta > 90^\circ$ the cross section is zero. The deflection angle is half of the standard Coulomb angle for a given $b$ and the largest possible classical scattering angle, under the assumptions, is then 90$^\circ$.
In normalizing the measured differential cross sections with \eqref{eq:dsigCpn}, we expect low values of the cross section ratio at angles lower than the grazing angle $\theta_g$ (half of the grazing angle for Rutherford scattering). Emission into those angles is actually most likely to origin from lower angular momenta than those expected for classical Coulomb trajectories populating the angles. Towards 90$^\circ$, the ratio is going to diverge, as refraction and diffraction due to nuclear processes have no problem populating the wide angles, towards 90$^\circ$ and beyond. Just past the grazing angle we expect cross-section ratios from the data that can genuinely reflect probabilities from moving from the entrance on local classical trajectories. In Fig.~\ref{fig:pn3R}, we show the (p,n) cross sections from the measurements of Doehring \etal~\cite{patterson_energy-dependent_1976,doering_microscopic_1975,doering_isobaric_1974}, normalized with Coulomb (p,n) cross section, for systems that combine large target mass with low incident energy. For those systems the spread of classical trajectories into wide angles is greater and, in the ratios, one can observe plateaus past the grazing angles.
\begin{figure}
\centerline{\includegraphics[width=.88\linewidth]{pn3R}}
\caption{Differential cross sections for quasielastic charge exchange (p,n) reactions, from measurements of Doehring \etal~\cite{patterson_energy-dependent_1976,doering_microscopic_1975,doering_isobaric_1974}, normalized with the Coulomb charge-exchange cross sections~\eqref{eq:dsigCpn}. Dashed lines indicate total cross section ratios from Table \ref{tab:csratio}. Arrows indicate grazing angles.
}
\label{fig:pn3R}
\end{figure}
The differential ratios in the plateau regions in Fig.~\ref{fig:pn3R} remain of the same order of magnitude as those in Table~\ref{tab:csratio}. We indicate the ratios from the table with dashed lines in Fig.~\ref{fig:pn3R}. Notably the integral of the Coulomb cross section, from the grazing angle on, is equal to the geometric cross section, $\int_{\theta_g} \text{d}\sigma_C^\text{(p,n)}=\sigma_g$. Small values of the transition probability between the channels indicate that any transition iterations between channels are not likely and the single step, represented by DWBA, should dominate. There is one caveat to the values in Table~\ref{tab:csratio} or in Fig.~\ref{fig:pn3R}, though, in that they actually reflect a product of (p,n) conversion probability and transition probabilities for getting in and out of the region where the conversion can take place. The latter can be in particular assessed from the mentioned ratios of elastic proton cross sections to Coulomb cross sections and are relatively large in the forward direction, of the order of one over few. With this, the corrected estimated local transition probabilities are still fairly low, of the order few percent.
\subsection{Influence of Potentials on Cross Sections}
For light projectiles at low incident energies, the differential cross sections for direct processes are significantly impacted by wave phenomena, evident, in particular in strong oscillations of these cross sections with emission angle, see the examples in Fig.~\ref{fig:ppn48Ca35}. The~major factor behind these oscillations is the interference of contributions to the scattering amplitude from different regions of the target surface, see Fig.~\ref{fig:DirectReaction}, in particular the near side and far side for any given scattering angle $\theta$. Coarsely, the positions of maximae and minimae in the interference pattern are tied to the mean radius of the bright ring on the surface shining towards the detectors. (We may notice in Fig.~\ref{fig:ppn48Ca35} a similar periodicity in the angle for the differential elastic and QE cross sections.) The volume of the nucleus contributes to the amplitude and the cross section as well, but it yields a more smudged out pattern. In the angular region where the wave phenomena dominate the cross section, i.e.\ wide angles, the general fall-off of the cross section with angle is tied to the sharpness of edges of the emitting regions~\cite{amado_analytic_1985}, with a sharper edge leading to a slower fall-off and softer to faster.
\begin{figure}
\centerline{\includegraphics[width=.65\linewidth]{ppn48Ca35}}
\caption{Differential cross section in elastic proton scattering (top panel) and quasielastic (p,n) charge-exchange reaction (bottom) on $^{48}$Ca at the incident energy of $35 \, \text{MeV}$. Filled circles represent measurements of McCamis \etal~\cite{mccamis_elastic_1986} and Doering \etal~\cite{doering_microscopic_1975}, respectively, for the two reactions. Solid lines represent calculations based on the KD potential \cite{koning_local_2003}.
}
\label{fig:ppn48Ca35}
\end{figure}
Figure~\ref{fig:wfiu35} further provides illustration for
the specific situation of the emission region in the context of the DWBA amplitude for a (p,n) reaction in Eq.~\eqref{eq:DWBA} with the KD parameterization of optical potential. Besides the magnitude of the transition potential $|U_1|$, to be further discussed, we show there the spin-averaged moduli of distorted wavefunctions, that can be obtained following expressions similar to Eqs.~(\ref{eq:DWBA})-(\ref{eq:IJl}):
\begin{equation}
\overline{|\chi({\pmb r})|^2} = \frac{1}{2s+1} \sum_{M_s} \chi_{M_s}^\dagger({\pmb r}) \, \chi_{M_s}({\pmb r}) = \frac{1}{2s+1} \sum_L (2L+1) \, {\mathcal A}_L^\rho(r) \, P_L(\cos{\theta}) \, .
\end{equation}
Here $\theta$ is the angle between wavevector ${\pmb k}$ and position vector ${\pmb r}$ for the wavefunction and
\begin{equation}
\begin{split}
{\mathcal A}_L^\rho = & \sum_{J' \, \ell'} (2J'+1)(2\ell'+1) \sum_{J \ell} (2J+1)(2\ell+1) \\
& \times \Bigg(\begin{matrix} \ell & \ell' & L \\ 0 & 0 & 0 \end{matrix}\Bigg)^2 \Bigg\lbrace \, \begin{matrix} \ell & \ell' & L \\ J' & J & s \end{matrix}\Bigg\rbrace^2 \, \text{Re} \left[i^{\ell - \ell'} \, u_{J' \, \ell'}^*(r) \, u_{J \ell}(r) \right] \, .
\end{split}
\end{equation}
For $r \rightarrow \infty$, the wavefunction moduli squared $|\chi|^2$ tend to~1. The final-state wavefunction for the DWBA approximation is computed using the $U_n$ potential from the KD-parametrization, at the equivalent incident energy for the final-state neutron.
\begin{figure}
\centerline{\includegraphics[width=.58\linewidth]{wfiu35}}
\caption{Spin-averaged square moduli of the distorted wavefunctions describing initial and final states in the quasielastic (p,n) reactions at the incident energy of $35 \, \text{MeV}$, on selected nuclei, as well as the modulus of isovector potential, as a function of the distance $r$ from target center. The results here are arrived at when utilizing the KD parametrization for the optical potential. Direction of the position vector here, for the respective wavefunctions, is taken as perpendicular to the initial and final wavevectors.
}
\label{fig:wfiu35}
\end{figure}
Regarding the isovector potential displayed in Fig.~\ref{fig:wfiu35}, the energy dependence in a potential, such as within the potential strengths the KD parametrization, complicates the situation, compared to that anticipated for Eqs.~\eqref{eq:U0np} and~\eqref{eq:U1np} for arriving at the isoscalar and isovector potentials. This is because a proton projectile slows down due to Coulomb repulsion when approaching the target nucleus~\cite{patterson_energy-dependent_1976,devito_neutron_2012}, making it inappropriate to combine the p and n potentials for the same asymptotic energy. This issue is reflected in the Q-values for QE (p,n) reactions being dominated by the Coulomb displacement energies and thus reduced energies of the emerging neutrons compared to the incident protons. In the literature, e.g.~\cite{patterson_energy-dependent_1976,devito_neutron_2012}, this has been ameliorated by reducing the neutron energy, compared to proton, for the nucleonic potential in the equations such as~\eqref{eq:U0np} and~\eqref{eq:U1np}. We account for the slowdown by determining the isoscalar and isovector potential components from the combination of entrance proton and exit neutron potentials in the (p,n) reaction, rather than the potentials on a single nucleus at one incident energy for neutron and proton.
With p and n referring then both to the particle and the channel in a (p,n) reaction, we determine then $U_0$ and $U_1$, from $U_p$ and $U_n$, with
\begin{align}
\label{eq:UE1np}
U_1 ({\pmb r}) & = \frac{A}{2(N-Z-1)} \, \big[ U_\text{n}({\pmb r}) - U_\text{p}({\pmb r}) \big] \, , \\[.5ex]
U_0 ({\pmb r}) & = U_\text{p}({\pmb r}) + \frac{N-Z}{A} \, U_1({\pmb r}) \, ,
\label{eq:UE0np}
\end{align}
rather than from \eqref{eq:U0np} and~\eqref{eq:U1np}. Here, $Z$ and $N$ refer to the target nucleus in the reaction and the incident energies in the nucleonic potential parametrizations represent the two channels.
The bulk of the nuclear potentials that shape, cf.~Eq.~\eqref{eq:Unp}, the wavefunctions illustrated in Fig.~\ref{fig:wfiu35}, is isoscalar.
In that figure one can see a depletion of the wavefunctions in the target region, for proton and neutron channels, stronger for $^{208}$Pb than $^{48}$Ca. The depletion is primarily due to the loss of probability flux to other channels and described by the imaginary part of the optical potential. Some of the depletion is due to acceleration, compared to the asymptotic region, as nucleons enter attractive nuclear potential. Finally, for protons some of the depletion is due to the deflection by the outside Coulomb potentials. Notably the reduction in the probability density, compared to the asymptotic regions is by a factor of few at most, consistent with the claim before, that the transmission probability into the inner nuclear region is of the order of one over few.
The enhancement in the probability densities at the center of nuclei is due to the waves coming from all sides from the surface and interfering constructively around $r = 0$. That enhancement is analogous to that in Arago-Fresnel-Poisson spot at the center of a circular shadow. The spatial extent of the enhancement shrinks as the projectile energy increases. Intuitively, at one incident energy, overall central enhancement is expected to be weaker for a target larger in size. The situation is reversed here at one incident energy for the two reactions due to the stronger slowing down of the projectile proton by the Coulomb potential for the larger target.
The surface ring contributions to the charge-exchange amplitude \eqref{eq:DWBA}, illustrated in Fig.~\ref{fig:DirectReaction}, stem from an interplay of the fall-off of nucleonic wavefunctions entering the nucleus and the isovector potential $U_1$ declining with the exit from a nucleus. In Fig.~\ref{fig:wfiu35}, it is apparent that the product of $U_1$ and the proton and neutron wavefunctions is not going to produce a significant surface enhancement in the production amplitude for the original KD parametrization of the optical potential. In the calculated (p,n) cross section in Fig.~\ref{fig:ppn48Ca35}, it is indeed observed that interference oscillations are very weak in disagreement with the data~\cite{doering_microscopic_1975}. Contrasting further with the situation for the charge-exchange reaction, the KD optical-potential parametrization yields a very good description of elastic data at the same incident energy of $35\, \text{MeV}$ as the discussed charge-exchange, cf.~in particular Fig.~\ref{fig:ppn48Ca35}.
Figure~\ref{fig:wfiu35} suggests, though, that the situation of describing charge-exchange data could be much improved, at least qualitatively, if the isovector potential were characterized by a~bit larger radii than the isoscalar. A resulting enhancement in the surface contribution to the amplitude~\eqref{eq:DWBA} could then plausibly produce interference oscillations comparable to the data. At the same time, given that the isovector potential enters the proton potential \eqref{eq:Unp} suppressed by the asymmetry factor, the quality of the description of elastic data might not change much.
The above idea is next tested in Fig.~\ref{fig:ppn48Ca35_fvr}. We show there elastic and QE cross sections generated when using a modified KD parametrization, with enhanced isovector radii. In the modification, the isoscalar potential is kept intact (Eq.~\eqref{eq:UE0np}), while the isovector potential is generated assuming that the radii in the volume, surface and spin-orbit terms of the nucleonic potentials are all enhanced by $\Delta R_1$ when used in~Eq.~\eqref{eq:UE1np}. The enhancement cycles through the values $\Delta R_1 = 0, 0.25, 0.5, 0.75$ and $1 \, \text{fm}$,. The potential depths and diffuseness are kept intact. The nucleonic potentials, for arriving at the elastic cross section and the initial and final states in DWBA, are then constructed from the isoscalar and the modified isovector potentials by combining those potentials in the standard manner in Eq.~\eqref{eq:Unp}. For $\Delta R_1 = 0$, the procedure just restores the original nucleonic potentials. For an identical starting Woods-Saxon geometry in the two nucleonic potentials, the procedure increases the isovector potential radii by $\Delta R_1$ and keeps the isoscalar radii identical to the original nucleonic radii.
\begin{figure}
\centerline{\includegraphics[width=.65\linewidth]{ppn48Ca35_fvr}}
\caption{Differential cross sections in elastic proton scattering (top panel) and quasielastic (p,n) charge-exchange reaction (bottom) on $^{48}$Ca at the incident energy of $35 \, \text{MeV}$, calculated while increasing radii in the isovector component $U_1$ of the KD optical potential parametrization in $0.25 \, \text{fm}$ increments, from 0 to $1 \, \text{fm}$. For both cross sections the pattern of oscillations generally shifts towards smaller angles and the magnitude of oscillations increases as the radii increase.
}
\label{fig:ppn48Ca35_fvr}
\end{figure}
It may be surprising that we test out values of $\Delta R_1$ in Fig.~\ref{fig:ppn48Ca35_fvr}, that are relatively large compared to the typical expectations regarding neutron skins in nuclei. For one, the neutron skins are typically quantified as a difference in the rms radii for neutrons and protons, and we address here a displacement of the surface radii, larger than the rms radii by a factor of $\sqrt{5/3}$ in the uniform-sphere model. Second, the difference in rms radii for neutrons and protons can be expressed in terms of the difference in rms radii for isovector and isocalar densities and, in that relation, the neutron skins are reduced by asymmetry factor:
\begin{equation}
\begin{split}
\langle r^2 \rangle_n^{1/2} - \langle r^2 \rangle_p^{1/2} & = \frac{N^2 - Z^2}{2 N Z} \, \Big( \langle r^2 \rangle_1^{1/2} - \langle r^2 \rangle_0^{1/2} \Big) \, \bigg[ 1 + {\mathcal O} \bigg( \frac{\langle r^2 \rangle_1^{1/2} - \langle r^2 \rangle_0^{1/2}}{\langle r^2 \rangle_0^{1/2}} \bigg) \bigg] \\
& \simeq 2 \, \frac{N-Z}{A} \, \Big( \langle r^2 \rangle_1^{1/2} - \langle r^2 \rangle_0^{1/2} \Big) \, .
\end{split}
\end{equation}
Here, the subscripts 0 and 1 refer to the rms radii computed with isoscalar and isovector densities, respectively, and the last result is obtained assuming that both the difference in radii and the asymmetry are small. For $^{48}$Ca, the combined amplification factor, of the surface isovector skin over the neutron rms skin, is of the order of 4 and, for $^{208}$Pb, it is of the order of 3. In the latter case, an additional dynamic source of difference between the skins are significant Coulomb forces that differently impact different directions in isospin space.
It is apparent in Fig.~\ref{fig:ppn48Ca35_fvr} that changes in $U_1$ radii have a strong impact on the charge-exchange cross section. Consistently with the expectation above, oscillations in the cross section become more pronounced with an increase in $\Delta R_1$. Separations in the angle between enhancements in the cross section decrease with the increase, consistently with a growth in the average radius of the surface ring emitting neutrons. If we simultaneously increase radii for $U_1$ and decrease for $U_0$, by the same amount, the separations between the enhancements stay about the same as the average radius of the ring stays about the same. On the other hand, the effect of changing $U_1$ radii on elastic proton cross section in Fig.~\ref{fig:ppn48Ca35_fvr} is rather minute. This is particularly shocking in that the proton elastic scattering cross sections have been used to tell the size of the neutron skin~\cite{Shlomo19795,PhysRevC.82.044611,karataglidis_discerning_2002}. Even with the effect being slight, as $\Delta R_1$ increases, the separation between maximae in proton cross section decreases. This can be attributed to the fact that for protons the $U_1$ contributions to the real and imaginary potential come in with the same sign as the dominating $U_0$ contributions and, thus, effectively an increase in $\Delta R_1$ pushes out the overall nuclear potential range.
We next show in Fig.~\ref{fig:ppn48Ca35_fsr} the impact on the cross sections of increasing the isoscalar potential radii. In analogy with the previous case, the isoscalar potential is constructed using the KD parametrization of nucleonic potentials modified by increasing the radii in the potentials on the r.h.s.\ of Eq.~\eqref{eq:UE0np} by the values of $\Delta R_0 = 0, 0.25, 0.5, 0.75$ and $1 \, \text{fm}$. The~isovector potential, on the other hand, is taken from the KD parametrization without any modification. The nucleonic potentials are then constructed in the standard manner from Eq.~\eqref{eq:Unp}.
\begin{figure}
\centerline{\includegraphics[width=.65\linewidth]{ppn48Ca35_fsr}}
\caption{Differential cross sections in elastic proton scattering (top panel) and quasielastic (p,n) charge-exchange reaction (bottom) on $^{48}$Ca at the incident energy of $35 \, \text{MeV}$, calculated while increasing radii in the isoscalar component $U_0$ of the KD optical potential parametrization in $0.25 \, \text{fm}$ increments, from 0 to $1 \, \text{fm}$. For the elastic cross section the pattern of oscillations generally shifts towards smaller angles and the magnitude of the oscillations increases as the radii increase. For the quasielastic cross section the opposite is true.
}
\label{fig:ppn48Ca35_fsr}
\end{figure}
We can see in Fig.~\ref{fig:ppn48Ca35_fsr} that the changes in the $U_0$ radii have a rather dramatic effect on the elastic scattering cross section. As the radii grow, the maximae in the scattering cross section move forward in the angle and the separations between the maximae decrease. The~changes in the $U_0$ radii also impact QE cross sections. Those changes, though, are opposite to the changes generated by the rise in the $U_1$ radii. Specifically, as the $U_0$ radii grow the oscillations in the charge-exchange cross section in the forward hemisphere dim out. This is consistent with the inner radius of the shining surface ring, cf.~Fig.~\ref{fig:DirectReaction}, growing and thus the ring thinning to extinction. It is obvious that the overall strength of the oscillations in a~charge-exchange cross section, particularly relative to the whole cross section, will depend on the difference of displacements in radii, $\Delta R = \Delta R_1 - \Delta R_0$, when the both sets of radii, isovector and isoscalar, get varied. Separations between the enhancements in the cross section at wide angles will coarsely depend, on the other hand, on the average of the two displacements.
\section{Data Interpretation}
\subsection{General Strategy}
\label{ssec:strategy}
Following the preliminary assessment, we turn to a systematic examination of elastic and quasielatic cross sections, with the goal of quantifying possible geometric differences between isovector and isoscalar potentials and, by proxy, isovector and isoscalar densities. The~data that we attempt to describe stem from elastic proton and neutron scattering and (p,n) QE reactions on the following nuclei: $^{48}$Ca, $^{90}$Zr, $^{120}$Sn and $^{208}$Pb. The (p,p) data stem from the similar incident energy range as the (p,n) reactions and the (n,n) data stem from the incident energy range similar to that for the neutrons in the final states of the analyzed (p,n) reactions.
In describing the data, our starting potential is that from the KD parametrization, minimally modified to make the proton and neutron potential geometry exactly the same and, thus, yielding exactly the same starting geometries for the isoscalar and isovector potentials. In fitting the data, we depart from those starting geometries to exhibit a preference of the data, or lack thereof, for the differences in the two geometries, inspired by the theory in~I. Rationale for specifics of our strategy is discussed below.
From the two types of cross sections, the QE cross sections and, in particular, the oscillations in that cross section best reflect the relative displacement of the isovector and isoscalar surfaces. Both the elastic and QE cross sections reflect geometric characteristics of the isoscalar potential. However, measurements of protons tend to be far more accurate than of neutrons, while spanning a larger range of energies and angles, so data on proton elastic scattering can be more effective in constraining the isoscalar potential than can the data on either QE charge-exchange reactions or on neutron elastic scattering. Along that line, one might insist on determining the isovector potential from proton elastic cross sections as well, but, when the impact of a potential on the cross sections is weak, one may worry about stability of the inversion procedure leading from the data to the deduced potential parameters. Even when the statistical measurement errors are small, there will be systematic errors present and these may be amplified by proximity to an instability in the inversion, there when a correlation between the parameters and data is weak. The systematic errors include those on the theory side, such as the use of the local form of isoscalar potential and the presumed details in the shapes of potential components.
The accuracy in neutron measurements interplays with the energy of the outgoing neutrons. When data is limited in terms of the energy span for the reactions, selection of outgoing angles and measurement accuracy, there is a limit on the number of parameters that may be reliably extracted from the data. In deciding on what information to extract, we combine the past experiences in the field, with a guidance from the theory. Thus, it has been possible in the past to describe elastic scattering data using a geometry of potential components that did not change with the incident energy. We assume this to hold as well for the isovector as for the isoscalar potential components. The theory suggests~(cf.~I) that isovector density is pushed out from a nucleus relative to isoscalar density, and this is of interest here, so we need to allow for the relative displacement of isovector and isoscalar radii. However, that relative displacement may depend on the specific potential component. E.g.~within a Thomas-Fermi consideration, the isovector and isoscalar densities and the dependence of symmetry energy on isoscalar density all interplay in generating the real part of the isovector potential. However, the outer parts of both the isoscalar and isovector potentials are primarily imaginary and impact oscillations in the cross sections the most. Even if their connection to the actual respective density is nonlocal, there is no reason to believe that the range of the nonlocality is significantly different for the two potentials. To refrain from making fits unstable, we assume the same displacement for all potential components and accept that the conclusions on the displacement will primarily pertain to the absorptive part of the potential. Still we allow the displacements to be different for individual isobars in the reactions we analyze. There might be diferent shell effects involved when moving from one isobar to another and, further, the universality of the displacements or lack thereof may tell whether we could be accessing a~genuine pronounced physical effect or whether we might be just adding a~fit parameter with no clear support from the data. Since the radii for nucleonic potentials in the original KD parametrization combine the effects of isovector and isoscalar potentials, we allow for a finite displacement of the isoscalar radii from the original KD parameterization, again the same for all potential components for a given isobar. Again, there might be shell effects playing a~role and, second, for getting a good description of proton elastic cross sections, any change in isovector radii may require an opposite, but smaller in magnitude, change in the isoscalar radii. Moreover, we allow for changes in the diffusivities for both isoscalar and isovector potentials, compared to starting values. However, we found that the neutron data have a~too weak constraining power to place reliable constraints on the adjustments in diffusivity of isovector potential on an isobar-by-isobar basis. In~consequence, we allow for global adjustments in the diffusivity across different components of isoscalar potentials for the individual isobars, but allow only for one universal shift in diffusivity across different components of isovector potential, relative to the isoscalar components, across all the analyzed isobars.
Besides geometry, the cross sections are obviously also sensitive to potential strengths. In~fact, coarsely the magnitudes of cross sections are often said to reflect the volume integrals of optical potentials. If we change radii, we need to allow for changes in potential magnitudes to permit a sensible description of the data. Since we allow only for a single universal adjustment of the radius per potential type across an isobar, isoscalar and isovector, we consistently allow only for an adjustment in the strength of that potential type by a single factor across that isobar.
In the end, the adjusted parameters per isobar for isoscalar potentials are the displacements of radii $\Delta R_0$ and of diffusivities $\Delta a_0$ and the strength renormalization ${\mathcal F}_0$, all relative to the KD parametrization where the diffusivity of the surface imaginary potential was set to a representative value of $a_D = 0.570 \, \text{fm}$, rather than varied by up to $\sim 0.1 \, \text{fm}$ with target mass depending on projectile~\cite{koning_local_2003}. The per-isobar-adjusted parameters for isovector potentials are the displacements of radii {\em relative to isoscalar potentials}, $\Delta R = \Delta R_1 - \Delta R_0$, and modifications of strengths {\em relative to modifications of isoscalar potentials} ${\mathcal F} = {\mathcal F}_1/{\mathcal F}_0$. I.e.~the displacements of isovector radii relative to the original KD parametrization become equal to $\Delta R_1 = \Delta R_0 + \Delta R$, and strength modifications become ${\mathcal F}_1 = {\mathcal F} \, {\mathcal F}_0$. Finally, we adjust a~global, across target masses, displacement of diffusivity for isovector potentials {\em relative to isoscalar potentials} $\Delta a = \Delta a_1 - \Delta a_0$. The changes in diffusivities relative to the starting parametrization are, similarly to the case of radii, $\Delta a_1 = \Delta a_0 + \Delta a$. In some fits, to be discussed, we required additional parameters, such as $\Delta R$, to be mass independent.
\subsection{Data Choice}
In drawing conclusions on relative geometry for isovector and isocalar potentials we exploit data on QE (p,n) reactions on $^{48}$Ca, $^{90}$Zr, $^{120}$Sn and $^{208}$Pb, at incident proton energies of 25, 35 and $45 \, \text{MeV}$, from the measurements by Doering \etal.~\cite{patterson_energy-dependent_1976,doering_microscopic_1975,doering_isobaric_1974}. Their advantage is the span over angles, incident energies and target nuclei, all arrived at within the same experimental setup, eliminating for us the issue of a relative normalization when different data are combined in analysis.
We tried to exploit results of other measurements of the (p,n) reactions on the same target nuclei, contained the EXFOR database \cite{otuka_towards_2014}, but they did not seem to seriously augment the information beyond that from the measurements by Doering \etal , due to large errors and/or sparse angular coverage.
We complement the experimental (p,n) results with those from elastic proton and neutron scattering off the same target nuclei as in the (p,n) reactions, requiring that the particle incident energy matches the general energy region for, respectively, either p or n in the quasieleastic (p,n) reaction. To find specific experiments and the final values of cross sections in those experiments we reach for the EXFOR database~\cite{otuka_towards_2014}.
From the p side, we include, in particular, the data on elastic scattering off $^{48}$Ca by McCamis \etal~\cite{mccamis_elastic_1986}, at the energies of 21, 25, 30, 35, 40, 45, and $48.4 \, \text{MeV}$ and the data on elastic scattering off $^{208}$Pb by Van Oehrs \etal~\cite{van_oers_optical-model_1974}, at the energies of 24.1, 30.3, 35, 45, and $47.3 \, \text{MeV}$. Moreover, we include the data by Mani \etal~\cite{mani_elastic_1971} on elastic scattering of $49.35 \, \text{MeV}$ protons off three of the targets: $^{90}$Zr, $^{120}$Sn and $^{208}$Pb. For the $^{90}$Zr target, we further include the elastic scattering data from the measurements by Van der Bijl \etal~\cite{van_der_bijl_high_1983}, at $21.05 \, \text{MeV}$, by De Swiniarski \etal~\cite{de_swiniarski_elastic_1977}, at $30 \, \text{MeV}$, and by Blumberg \etal~\cite{blumberg_polarizations_1966}, at $40 \, \text{MeV}$. For $^{120}$Sn, we include the scattering data from the measurements by Ridley and Turner at $30.3 \, \text{MeV}$ \cite{ridley_optical_1964}, by Boyd and Greenless at $39.6 \, \text{MeV}$ \cite{boyd_nuclear-matter_1968} and by Fricke \etal\ at $40 \, \text{MeV}$ \cite{fricke_polarization_1967}.
From the n side, we include the data on elastic scattering off $^{48}$Ca by Mueller \etal~\cite{mueller_asymmetry_2011} at $16.8 \, \text{MeV}$. Moreover, we include the data on elastic scattering off $^{90}$Zr by Bainum \etal~\cite{bainum_isospin_1978}, at $11 \, \text{MeV}$, and by Wang and Rapaport \cite{wang_neutron_1990}, at $24 \, \text{MeV}$. Further, we include data on elastic scattering off $^{120}$Sn by Rapaport \etal~\cite{rapaport_neutron_1980}, at $11 \, \text{MeV}$ and by Guss \etal~\cite{guss_optical_1989}, at 13.9 and $16.9 \, \text{MeV}$. For the $^{208}$Pb target, we include the scattering data by Roberts \etal~\cite{roberts_measurement_1991}, at $8 \, \text{MeV}$, by Delaroche \etal~\cite{delaroche_complex_1983}, at $10 \, \text{MeV}$, by Finlay \etal~\cite{finlay_energy_1984}, at 20, 22, and $24 \, \text{MeV}$, by Rapaport \etal~\cite{rapaport_neutron_1978} at $26 \, \text{MeV}$, and finally by DeVito \etal~\cite{devito_neutron_2012} at 30.4 and $40 \, \text{MeV}$.
\subsection{Fits to Data}
Figures \ref{fig:ca48}-\ref{fig:pb208} display the data discussed above, as filled circles, together with the predictions of the KD parametrization, as solid lines. It may be seen in these figures that the general description of both the elastic and QE cross sections in terms of the KD parametrization is quite good. Given that the KD potential parameters have been fitted to describe different elastic data, the quality in describing such data might not be surprising. However, the fact the quasielastic data, never considered in arriving at the KD parameter values, are also reasonably well described supports both the relevance of the isospin formalism in the Lane potential \eqref{eq:Lane} and the physics validity of the KD parametrization.
\begin{figure}
\centerline{\includegraphics[width=\linewidth]{ca48}}
\caption{
Differential cross section in elastic and quasielastic reactions on $^{48}$Ca as a function of c.m.\ angle. The left panel shows differential cross section, in mb/sr, in the reactions with an outgoing neutron, i.e.\ elastic n scattering and quasielastic (p,n) reactions. The right panel shows differential cross section in elastic p scattering, normalized with the Rutherford cross section. The~data are represented with filled circles and the theory is represented with lines, solid - for predictions of the original KD optical-potential parametrization \cite{koning_local_2003} and dashed - for the best-fit modified parametrization. For display purposes, the results for different indicated incident energies are multiplied by different indicated factors. The (n,n) data are from Ref.~\cite{mueller_asymmetry_2011}, the (p,n) data - from Refs.~\cite{patterson_energy-dependent_1976,doering_microscopic_1975,doering_isobaric_1974} and (p,p) - from Refs.~\cite{mccamis_elastic_1986}.
}
\label{fig:ca48}
\end{figure}
\begin{figure}
\centerline{\includegraphics[width=\linewidth]{zr90}}
\caption{
Differential cross section in elastic and quasielastic reactions on $^{90}$Zr as a function of c.m.\ angle. The left panel shows differential cross section, in mb/sr, in the reactions with an outgoing neutron, i.e.\ elastic n scattering and quasielastic (p,n) reactions. The right panel shows differential cross section in elastic p scattering, normalized with the Rutherford cross section. The~data are represented with filled circles and the theory is represented with lines, solid - for predictions of the original KD optical-potential parametrization \cite{koning_local_2003} and dashed - for the best-fit modified parametrization. For display purposes, the results for different indicated incident energies are multiplied by different indicated factors. The (n,n) data are from Refs.~\cite{bainum_isospin_1978,wang_neutron_1990}, the (p,n) data - from Refs.~\cite{patterson_energy-dependent_1976,doering_microscopic_1975,doering_isobaric_1974} and (p,p) - from Refs.~\cite{van_der_bijl_high_1983,de_swiniarski_elastic_1977,blumberg_polarizations_1966,mani_elastic_1971}.
}
\label{fig:zr90}
\end{figure}
\begin{figure}
\centerline{\includegraphics[width=\linewidth]{sn120}}
\caption{
Differential cross section in elastic and quasielastic reactions on $^{120}$Sn as a function of c.m.\ angle. The left panel shows differential cross section, in mb/sr, in the reactions with an outgoing neutron, i.e.\ elastic n scattering and quasielastic (p,n) reactions. The right panel shows differential cross section in elastic p scattering, normalized with the Rutherford cross section. The~data are represented with filled circles and the theory is represented with lines, solid - for predictions of the original KD optical-potential parametrization \cite{koning_local_2003} and dashed - for the best-fit modified parametrization. For display purposes, the results for different indicated incident energies are multiplied by different indicated factors. The (n,n) data are from Refs.~\cite{rapaport_neutron_1980,guss_optical_1989}, the (p,n) data - from Refs.~\cite{patterson_energy-dependent_1976,doering_microscopic_1975,doering_isobaric_1974} and (p,p) - from Refs.~\cite{ridley_optical_1964,boyd_nuclear-matter_1968,fricke_polarization_1967,mani_elastic_1971}.
}
\label{fig:sn120}
\end{figure}
\begin{figure}
\centerline{\includegraphics[width=\linewidth]{pb208}}
\caption{
Differential cross section in elastic and quasielastic reactions on $^{208}$Pb as a function of c.m.\ angle. The left panel shows differential cross section, in mb/sr, in the reactions with an outgoing neutron, i.e.\ elastic n scattering and quasielastic (p,n) reactions. The right panel shows differential cross section in elastic p scattering, normalized with the Rutherford cross section. The~data are represented with filled circles and the theory is represented with lines, solid - for predictions of the original KD optical-potential parametrization \cite{koning_local_2003} and dashed - for the best-fit modified parametrization. For display purposes, the results for different indicated incident energies are multiplied by different indicated factors. The (n,n) data are from Refs.~\cite{roberts_measurement_1991,delaroche_complex_1983,finlay_energy_1984,rapaport_neutron_1978,devito_neutron_2012}, the (p,n) data - from Refs.~\cite{patterson_energy-dependent_1976,doering_microscopic_1975,doering_isobaric_1974} and (p,p) - from Refs.~\cite{van_oers_optical-model_1974,mani_elastic_1971}.
}
\label{fig:pb208}
\end{figure}
When comparing details of the (p,n) predictions of the KD parametrization with data, though, one may notice that the predictions generally exhibit less oscillation with angle than do data. Given the discussion above, there are reasons to investigate whether this could be attributed to different geometric features of the isovector than isoscalar potentials - in the KD parametrization those features are practically identical. We progress fitting the parameters representing adjustments in the geometry and overall strength as discussed in Subsection \ref{ssec:strategy}.
In the context of estimating parameter values and their errors with a $\chi^2$ minimization, the following should be stated. The experimental errors on p elastic scattering cross sections can be quite small. The model relying on a simple potential form, with few parameters, has no chance of describing the elastic data with an arbitrary precision, i.e.\ yield $\chi^2$ values per degree of freedom (DOF) of the order of 1 no matter what the experimental errors are. It can be seen in Figs.~\ref{fig:ca48}-\ref{fig:pb208} that, in the rear direction, the model can struggle even at qualitative level, not just quantitative. When the cross sections from direct amplitudes are low, other processes can effectively compete. One of the standard strategies of statistical analysis in such a~situation is that of $\chi^2$ renormalization, effectively recognizing that an error residue represents limitations of a model that cannot be circumvented. If the renormalization is done, though, only after a $\chi^2$ value constructed with experimental errors gets minimized, the best-fit theoretical cross sections may exhibit awkward features. Namely the portions of cross sections where the experimental errors are very small may get reproduced very well within the fit, while the remainder may be left largely abandoned. E.g.\ if elastic proton and neutron elastic cross sections are combined in $\chi^2$ with experimental errors, the neutron cross sections may bear nearly no impact on the outcome of the fit. We find it more effective to assume some systematic theoretical errors from scratch in the fit, of magnitude that would yield a final $\chi^2/\text{DOF}$ of the order of few. Afterwards still the standard $\chi^2$ renormalization may be applied. In this way, the regions with larger experimental errors, that may be still fairly small on the scale of accuracy expected from the theory, may impact the fit as much as the regions with very small experimental errors, yielding a more democratic fit.
In our case we find that assumed theoretical errors of the order of (5--10)\% yield cross-section fits that are visually acceptable. Given the large experimental errors for the (p,n) reaction cross sections, the inflating of the errors ends up with quite limited impact on the parameters characterizing the relation between isoscalar and isovector potentials, i.e.\ $\Delta R$, $\Delta a$ and ${\mathcal F}$, and their errors. However, the impact is more significant on the parameters fine-tuning the characteristics of the isoscalar potentials, i.e.\ $\Delta R_0$, $\Delta a_0$ and ${\mathcal F}_0$, with the parameters values generally drifting either towards a better reproduction of (p,p) cross sections, for lower assumed theoretical errors, or towards a better reproduction of (n,n) cross sections, for higher errors. When quoting errors on the best-fit parameter values, we include there the specific drift under changing assumptions on the theoretical errors.
\begin{table}
\caption{Adjustment parameters for the isoscalar and isovector components of nucleonic potentials constructed from the KD parametrization \cite{koning_local_2003}, representing the best simultaneous description of elastic (p,p) and (n,n) and quasielastic (p,n) reactions on different targets, cf.\ Figs.~\ref{fig:ca48}--\ref{fig:pb208}. The~adjustment parameters for the isoscalar potential, ${\mathcal F}_0$, $\Delta R_0$ and $\Delta a_0$, are relative to the modified ($a_D = 0.570 \, \text{fm}$) KD parametrization. The adjustment parameters pertaining to the isovector potential, ${\mathcal F}$, $\Delta R$ and~$\Delta a$, are {\em relative to the adjustments in the isoscalar potential}, i.e.~e.g.~$\Delta R_1=\Delta R_0 + \Delta R$.\\[-2.5ex]}
\label{tab:BestFit}
\begin{tabular}{||c||c|c|c||c|c|c||}
\hline
\hline
Target & ${\mathcal F}_0$ & $\Delta R_0$ & $\Delta a_0$ & ${\mathcal F}$ & $\Delta R$ & $\Delta a$ \\
& & [fm] & [fm] & & [fm] & [fm] \\
\hline
\hline
$^{48}$Ca & 0.978$\, \pm \,$0.008 & -0.002$\, \pm \,$0.012 & -0.037$\, \pm \,$0.004 & 0.835$\, \pm \,$0.038 & 0.84$\, \pm \,$0.08 & \\
$^{90}$Zr & 1.045$\, \pm \,$0.011 & -0.130$\, \pm \,$0.034 & 0.019$\, \pm \,$0.016 & 0.814$\, \pm \,$0.030 & 0.87$\, \pm \,$0.09 & \\
$^{120}$Sn & 1.031$\, \pm \,$0.008 & -0.101$\, \pm \,$0.021 & 0.038$\, \pm \,$0.009 & 0.689$\, \pm \,$0.047 & 1.14$\, \pm \,$0.18 & \\
$^{208}$Pb & 1.010$\, \pm \,$0.008 & 0.032$\, \pm \,$0.034 & 0.034$\, \pm \,$0.016 & 0.799$\, \pm \,$0.028 & 1.08$\, \pm \,$0.17 & \\
\hline
All & & & & & & -0.104$\, \pm \,$0.033 \\
\hline
\hline
\end{tabular}
\end{table}
Optimal parameter values, when adjusting ${\mathcal F}_0$, $\Delta R_0$, $\Delta a_0$, ${\mathcal F}$ and $\Delta R$ on per-isobar basis, and $\Delta a$ globally, are listed in Table \ref{tab:BestFit}. The corresponding differential cross sections are represented with dashed lines in Figs.~\ref{fig:ca48}--\ref{fig:pb208}. While the description of the (p,n) data generally significantly improves across nuclei, compared to the original KD parametrization, this may come at the cost of some deterioration in the description of (p,p) and (n,n) data.
It may be seen in Table \ref{tab:BestFit} that the best-fit adjustments for the isoscalar potential are relatively minor, i.e.\ ${\mathcal F}_0$ tends to be close to 1 and $\Delta R_0$ and $\Delta a_0$ are small. Moderate evolution in $\Delta a_0$ from $^{48}$Ca to heavier nuclei mimicks the growth of $a_D$ for protons in the KD parametrization~\cite{koning_local_2003}. The prominent adjustments in the Table are those for the isovector potential, in favored weakening of the strength by $\sim 20 \%$ and pushing out of the surface, relative to isoscalar, by $\sim 1 \, \text{fm}$. The fit across all isobars favors also some steepening of the isovector surface relative to isoscalar, with relative drop in the slope parameters by $\sim 0.1 \, \text{fm}$. Besides the Table, we represent the deduced values of $\Delta R$ and $\Delta a$ in Figs.~\ref{fig:drx} and~\ref{fig:dax}.
\begin{figure}
\centerline{\includegraphics[width=.65\linewidth]{drs}}
\caption{
Displacement of isovector surface, relative to isoscalar, displayed vs mass number. Squares represent the displacement values arrived at when simultaneously adjusting isovector and isoscalar potential parameters, in the modified KD potential parametrization, to best describe the measured differential quasielastic (p,n) and elastic (p,p) and (n,n) cross sections on the same targets. The~remaining symbols represent the displacement of isovector surface relative to isoscalar arrived at for densities from SHF calculations with different indicated Skyrme parametrizations. In the parenthesis in the figure, values of the associated slope parameters $L$ of the symmetry energy are given in MeV, following the names of the Skyrme parametrizations. Arrows by the right axis indicate displacement values in the limit of half-infinite matter, after~I, for the specific parametrizations in the order of decreasing $L$-values, from top to bottom.
}
\label{fig:drx}
\end{figure}
\begin{figure}
\centerline{\includegraphics[width=.65\linewidth]{das}}
\caption{
Difference between diffusivities of isovector and isoscalar surfaces, displayed vs mass number. The shaded region represents a mass-independent difference between diffusivities of potential surfaces arrived at when simultaneously adjusting different isovector and isoscalar potential parameters, in the modified KD potential parametrization, to best describe the measured differential quasielastic (p,n) and elastic (p,p) and (n,n) cross sections on the same targets. The~remaining symbols represent the difference between diffusivities of the isovector and isoscalar surfaces arrived at for densities from SHF calculations with different indicated Skyrme parametrizations. In the parenthesis in the figure, values of the associated slope parameters $L$ of the symmetry energy are given in MeV, following the names of the Skyrme parametrizations. Arrows by the right axis indicate difference in diffusivity for the surfaces in the limit of half-infinite matter, after~I, for the specific parametrizations in the order of increasing $L$-values, from top to bottom.
}
\label{fig:dax}
\end{figure}
If we force the value of $\Delta R$ to be independent of $A$, we arrive at the optimal parameter values listed in Table~\ref{tab:BestFit1}. The assumption of universal surface displacement, yields $\Delta R = 0.88 \pm 0.05 \, \text{fm}$ and $\Delta a = -0.093 \pm 0.024 \, \text{fm}$. With parameters in Table \ref{tab:BestFit1} close those in Table \ref{tab:BestFit}, the description of differential cross sections is very similar to that in Figs.~\ref{fig:ca48}--\ref{fig:pb208}. With the by-eye differences relative to Figs.~\ref{fig:ca48}--\ref{fig:pb208} being hardly detectable, we refrain from producing another set of figures tied to Table \ref{tab:BestFit1}.
\begin{table}
\caption{Adjustment parameters for the isoscalar and isovector components of nucleonic potentials constructed from the KD parametrization \cite{koning_local_2003}, representing the best simultaneous description of elastic (p,p) and (n,n) and quasielastic (p,n) reactions on different targets, when forcing the displacement of isovector surface relative to isoscalar to be $A$-independent. The~listed adjustment parameters for the isoscalar potential, ${\mathcal F}_0$, $\Delta R_0$ and $\Delta a_0$, are relative to the modified ($a_D = 0.570 \, \text{fm}$) KD parametrization. The adjustment parameters pertaining to the isovector potential, ${\mathcal F}$, $\Delta R$ and~$\Delta a$, are {\em relative to the adjustments of the isoscalar potential}.\\[-2.5ex]}
\label{tab:BestFit1}
\begin{tabular}{||c||c|c|c||c|c|c||}
\hline
\hline
Target & ${\mathcal F}_0$ & $\Delta R_0$ & $\Delta a_0$ & ${\mathcal F}$ & $\Delta R$ & $\Delta a$ \\
& & [fm] & [fm] & & [fm] & [fm] \\
\hline
\hline
$^{48}$Ca & 0.976$\, \pm \,$0.006 & 0.003$\, \pm \,$0.007 & -0.041$\, \pm \,$0.005 & 0.808$\, \pm \,$0.035 & & \\
$^{90}$Zr & 1.048$\, \pm \,$0.010 & -0.150$\, \pm \,$0.028 & 0.023$\, \pm \,$0.010 & 0.795$\, \pm \,$0.032 & & \\
$^{120}$Sn & 1.036$\, \pm \,$0.007 & -0.115$\, \pm \,$0.014 & 0.027$\, \pm \,$0.005 & 0.738$\, \pm \,$0.031 & & \\
$^{208}$Pb & 1.012$\, \pm \,$0.006 & 0.016$\, \pm \,$0.027 & 0.024$\, \pm \,$0.010 & 0.826$\, \pm \,$0.035 & & \\
\hline
All & & & & & 0.88$\, \pm \,$0.05 & -0.093$\, \pm \,$0.024 \\
\hline
\hline
\end{tabular}
\end{table}
\section{Densities in Skyrme-Hartree-Fock Calculations}
\subsection{Isoscalar and Isovector Densities}
We discussed the isoscalar and isovector densities for half-infinite nuclear matter within SHF in I. In II, we used the densities from spherical SHF calculations, to arrive at symmetry-energy coefficients for individual nuclei. The latter coefficients can be expressed in terms of an integral of density of transverse isospin in a nucleus (termed also in I and II as asymmetric density). Coulomb interactions polarize the difference of neutron and proton densities making the density profile of the third component different from the profile of transverse components. While direct extraction of the difference of neutron and proton densities, $\rho_3 = \rho_n - \rho_p$, is obviously trivial in SHF, similar extraction of density for the transverse components is not, due to HF breaking isospin invariance. Fortunately, as indicated in II, owing to slow changes in Coulomb potential, compared to nuclear nonlocalities, in the bulk low-asymmetry limit, the density tied to transverse isospin can be obtained from the density of the third component through a renormalization with the local chemical potential for asymmetry:
\begin{equation}
\rho_\perp(r) \propto \frac{\rho_3(r)}{\mu_3 + \Phi(r)/2} \, .
\label{eq:rhoT}
\end{equation}
Here, $\mu_3$ is the global potential for asymmetry,
\begin{equation}
\mu_3 = \frac{\partial E}{ \partial (N-Z)} \bigg|_A \, ,
\end{equation}
and $\Phi$ is the local Coulomb potential for protons. The chemical potential itself can be arrived
at by using the discussed densities, cf.~II.
Besides the Coulomb field, the isovector densities are very sensitive to shell effects, far more than the net densities, because of their differential nature, as evident already in half-infinite matter in I, where the role of shell effects was taken over by the Friedel oscillations. Incidentally, the very use of the densities in assessing the chemical potential, and integrating then over isovector density, aims at reducing the impact of the shell effects on the potential as compared to direct energy differentials.
Commenting more, the renormalization \eqref{eq:rhoT} aims to produce the isospin distribution $\rho_\perp(r)$ without the effects of a~Coulomb polarization - at the microscopic level one for a pure state with isospin~$T$. The~distribution $\rho_3 (r)$ is for a state where, microscopically, the Coulomb interactions admix the states that differ in isospin from $T$, primarily the $T'=T+1$ state where an isospin-0 core is replaced by an isospin-1 giant monopole oscillation~\cite{lane_isobaric_purity_1962,auerbach_coulomb_1983}. Within e.g.\ $^{48}$Ca, that core may be perceived as $^{40}$Ca that gets polarized and for which the Coulomb-induced isovector modulation may be approximated in terms of the amplitude for a monopole oscillation. The excess neutrons, in the pure state~$T$ outside of the core in this microscopic transcription, distribute themselves according to $\rho_\perp(r)$, while the overall neutron-proton imbalance according to~$\rho_3(r)$.
Figure \ref{fig:rho_all}
exhibits densities arrived at from SHF calculations with the code of P.-G.~Reinhard~\cite{Reinhard:1991}. Different columns of panels there represent different nuclei and different rows represent different interaction parametrizations. In Fig.\ref{fig:rho_all}, the net density~$\rho$ is represented in its absolute normalization, with solid lines. The densities $\rho_3$ and $\rho_\perp$ are represented, respectively, with shorter- and longer-dashed lines. The density $\rho_\perp$ is normalized to the same average density in the interior as $\rho$. The density $\rho_3$ is normalized to the same average density as $\rho_\perp$, within the interior region immediately adjacent to the surface. For the average density in the normalizations, we first assess the position of the surface for the given density and then count, as the interior for that density, the volume with radius shorter by $0.8 \, \text{fm}$. As the interior immediately adjacent to the surface, we count the outer layer of the interior volume having $1 \, \text{fm}$ thickness.
The density normalizations are tied to the characteristics and use of those densities. The values of $\rho$ and $\rho_\perp$ are expected to stabilize within the interior of a large system, hence it makes sense to normalize $\rho_\perp$ to the same average value in the interior as $\rho$, when emphasizing any similarities and differences in the shape between the two. In the case of $\rho_3$, we want to emphasize that it is going to yield similar results as $\rho_\perp$ when used exclusively in the surface region, hence the normalization to the same average density as $\rho_\perp$ in the region immediately adjacent to the surface.
\begin{figure}
\centerline{\includegraphics[width=\linewidth]{rho_allM}}
\caption{
Densities from spherical SHF calculations with different Skyrme interaction parameterizations for different nuclei. For the panels from top to bottom the parametrizations are SkI5~\cite{1995NuPhA.584..467R}, s9s24~\cite{brown_constraints_2013}, SLy4~\cite{1998NuPhA.635..231C}, SKSC5~\cite{PhysRevC.50.460} and Z~\cite{PhysRevC.33.335}. Underneath the interaction names in the figure, values of the slope of the symmetry energy with respect to density are given. For the panels from left to right the nuclei are $^{48}$Ca, $^{90}$Zr, $^{120}$Sn and $^{208}$Pb. The net density $\rho$ (solid lines) is presented in absolute normalization. The isovector density $\rho_\perp$ (longer-dashed lines) is normalized to the same average density in the interior as $\rho$. The isovector density $\rho_3$ (shorter-dashed) is normalized to the same average density as $\rho_\perp$ in the region immediately adjacent to the surface.
}
\label{fig:rho_all}
\end{figure}
In examining the panels from left to right in Fig.~\ref{fig:rho_all}, one can see an evolution of the densities with system size. In examining the panels from top to bottom one can see the evolution of densities with changing slope $L$ of the symmetry energy with respect to density. From left to right in Fig.~\ref{fig:rho_all}, the strengthening of the Coulomb effects is evident in an~increasing departure of $\rho_3$ from~$\rho_\perp$, and in this $\rho_3$ growing on the average towards the nuclear center. For the $^{48}$Ca nucleus, with weak Coulomb effects, those two densities, $\rho_3$~and~$\rho_\perp$, tend to be nearly the same, no matter what the nuclear interaction. In Fig.~\ref{fig:rho_all} it is further apparent that the shell effects are much more pronounced in the isovector than in the isoscalar densities. The shell effects in density can be described as oscillations at about half of the Fermi wavelength. The wavelength is a bit longer for the lighter than heavier systems. Irrespectively of the shell effects giving rise to differences between the nuclei, due to the different number of wavelengths fitting into the size, the general expectation is that the surface in any nucleus synchronizes the rise in the particle wavefunctions towards the interior and the positions of the first nodes. There is a difference in the impact on neutrons and protons at finite asymmetry, due to a difference in the Fermi energy of the particles and due to any interaction differences tied to the symmetry energy. Because of the synchronization, the isovector densities may be expected to have a~universal appearance and a~universal relation to the isoscalar density in the surface region, no matter what the mass of a nucleus. However, the surface also has a different impact on the neutrons and protons due to the Coulomb interactions. To the latter impact can be attributed the gradual drift for $\rho_3$ in relation to $\rho$ in the surface region, that rises with charge number in Fig.~\ref{fig:rho_all}, from the advocated universality rule. After $\rho_3$ gets renormalized to yield $\rho_\perp$, though, we can observe that, for a~given interaction the same relation between $\rho_\perp$ and $\rho$ generally persists, no matter what the nuclear mass. This, in particular, underscores the significance of the half-infinite matter considerations such as in~I. It should be mentioned, though, that while the density $\rho_\perp$ should be a unique functional of $\rho$ in the bulk low-asymmetry limit, cf.~II, the Coulomb interactions have an impact on $\rho$. That impact is not removed in~\eqref{eq:rhoT}, while the Coulomb interactions are switched off with all their impact for the half-infinite matter.
In examining the evolution of the characteristics of the densities with changes in the symmetry energy in Fig.~\ref{fig:rho_all}, i.e.\ the panels top to down, we can see that the isovector and isoscalar surfaces approach each other as $L$ decreases. For high $L$, the $\rho_\perp$-surface is steeper than the $\rho$-surface, but for low $L$ the situation is generally reversed. Such findings where first arrived at within the examinations of half-infinite matter in~I.
It should be noted that these general characteristics of evolution with $L$ become evident for sufficiently large changes in~$L$. For smaller changes in $L$, the evolution in any particular characteristics of the densities, especially in a finite system, might not appear monotonic; obviously various characteristics of the densities depend also on aspects of the interactions that have no direct relation to the symmetry energy.
\subsection{Comparison to Data}
In comparing the theory to data, we compare the geometric relation of $\rho_\perp$ to $\rho$ to the corresponding relation of $U_1$ to $U_0$, where $U_1$ is predominantly determined from the quasielastic charge exchange reactions. In a more sophisticated analysis one might ponder making~$U_1$ anisotropic in isospin space, with the $U_3$ component tied to $\rho_3$ and $U_\perp$ to $\rho_\perp$. While $\rho_3$ is not very different from $\rho_\perp$ in shape, when the two are considered exclusively in the surface region, cf.~Fig.~\ref{fig:rho_all}, we want to stress that it is $\rho_\perp$, rather than $\rho_3$, that is tied to the charge-exchange reactions leading to IAS. The similarity of the densities there is due to the fact that the Coulomb potential is not varying quickly enough across the surface region to make a difference even for heavy nuclei. Arguments for tying $\rho_\perp$ to the QE charge exchange include those from macroscopic and microscopic side. In classical considerations of isospin within an isobaric multiplet, the transverse components of isospin are distributed according to $\rho_\perp$ as already indicated earlier. When viewed from the microscopic side, that density represents transition density between adjacent members of the multiplet. From the microscopic side, when directly examining a matrix element of transverse isospin operator between a ground state and its isobaric analog, there will be a contribution from overlap of pure isotopic states, to be accounted for by $\rho_\perp$. Additional contribution might come from conversion of admixtures with different isospin to one state to those of the other. However, no good match can be achieved when changing the state of only one nucleon with the isospin operator and that contribution will be suppressed. Last but not least, the fit to the QE charge-exchange data gives no indication for a drop in isovector skin size with increase in target mass, suggested by Fig.~\ref{fig:rho_all} if $\rho_3$ were used. As one more remark concerning the densities, the renormalization~\eqref{eq:rhoT} can restore average properties of $\rho_\perp$ from $\rho_3$, but likely not the details of shell effects. Within the focus on the surface region, this should not be an issue, though.
For the sake of comparisons to data, in the combination of the net density $\rho$ for a nucleus and the isovector density $\rho_\perp$ normalized as in Fig.~\ref{fig:rho_all}, we determine the separation of the surfaces by finding the separation $\Delta R$ between the locations where these densities fall to the same value $\rho(0)/2$. We determine the diffusivity~$a_X$ for the density $\rho_X$, at the corresponding location above, from
\begin{equation}
a_X = \frac{\rho_X}{2 \, \text{d}\rho_X/\text{d} r} \, ,
\end{equation}
guided by a WS shape. Under the assumption that any significant differences in geometry for the isoscalar and isovector potentials explored in the direct reactions will primarily follow the differences in the geometry for the isoscalar and isovector densities, we plot in Figs.~\ref{fig:drx} and~\ref{fig:dax} the separations between the surfaces and differences in the diffusivity for the cases of the interactions in Fig.~\ref{fig:rho_all}. In addition to the results for finite nuclei, we provide in Figs.~\ref{fig:drx} and~\ref{fig:dax} the reference results for half-infinite matter, following I, represented in these figures with arrows by the right axes.
\section{Symmetry Energy and Neutron Matter}
\subsection{General Strategy}
As the values of $\Delta R$ and $\Delta a$ appear correlated with the slope of symmetry energy $L$ in the SHF calculations, cf.\ Figs.~\ref{fig:drx} and~\ref{fig:dax}, we attempt to use the values of $\Delta R$ and $\Delta a$ inferred from the reaction fits to constrain $L$ in the structure calculations. As discussed in~I, an even tighter correlation of $\Delta R$ is expected with $L/a_a^V$, than with $L$ alone. Here $a_a^V$ is the value of the symmetry energy at normal density, $a_a^V \equiv S(\rho_0)$. On its own, the geometric quantities poorly constrain the strength of the symmetry energy, though. To constrain both $L$ and $a_a^V$, the best strategy might be then to combine the conclusions following from $\Delta R$ or~$\Delta a$, with those following from the magnitude of symmetry coefficients for individual nuclei, such as deduced in II from the excitation energies to ground-state IAS. In II we combined the conclusions from the latter excitation energies with the conclusions following from the sizes of neutron skins~$\Delta r$. The accuracy of the experimental values for the latter has been a~source of concern - results from different sources in the literature disagreed with each other by more than expected on the basis of the claimed errors for those results. The vast majority of the results for $\Delta r$ stemmed from analysis of elastic scattering data for which the sensitivity of cross sections to skins is of the type such as illustrated in the top panel of Fig.~\ref{fig:ppn48Ca35_fvr}.
To simultaneously constrain $L$ and $a_a^V$, we progress then in parallel to II, using the symmetry coefficients $a_a(A)$ arrived at in II, proportional to the excitation energies to IAS, $E^*_\text{IAS}$, in combination with the geometric characteristics tied to the symmetry energy.
In II, as the geometric characteristics we utilized $\Delta r$ for different nuclei and here we utilize $\Delta R$ and $\Delta a$. Given the scarcity of Skyrme parametrizations that need to meet requirements, when many requirements are imposed, we assume that it is possible to find a path in the parameter space, between any two parametrizations that are close enough in their predictions, such that the corresponding predictions vary in a linear fashion with location along the path. With this we linearly interpolate and moderately (33\% in both directions) extrapolate physical predictions and other characteristics between and around the established interaction parametrizations that are close in their predictions. Similar strategies are employed in statistical analysis elsewhere, see e.g.~\cite{sangaline_toward_2016}, typically to reduce computational effort. The~interpolations allow, in particular, to explore better the boundaries of a constraint region in $(L,a_a^V)$ plane, when the number of parametrizations meeting the requirements is low.
In II, to arrive at specific conclusions on symmetry energy, we applied gates of consistency with data, within errors, to the Skyrme parameterisations. Here, instead, we employ Bayesian inference~\cite{dagostini_bayesian_2003}, constructing density of probability in the space of explored quantities, first uniform within the range of $(L,a_a^V)$ that is spanned by the Skyrme parametrizations in the literature. The Skyrme parametrizations and their mentioned combinations span for us possible connections between theoretical inputs and predictions. With inclusion of value~$\overline{E}$ for observable~$E$, determined with experimental error $\sigma_E$, the density of pobability in quantity~$x$ is updated according to
\begin{equation}
p(x|\overline{E}) \propto p_\text{prior}(x) \int \text{d}E \, \text{e}^{-\frac{(E-\overline{E})^2}{2 \sigma_E^2}} \, p(E|x) \, .
\end{equation}
Here, $p(a|b)$ stands for the conditional probability of $a$ subject to $b$ being true, with $\int \text{d}a \, p(a|b) = 1 $, for mutually exclusive and exhaustive occurances $a$. Specifically, $p(E|x)$ stands for the probability of arriving at a value $E$ of an observable for theoretical inputs~$x$, excluding yet impact of the uncertainty in a specific measurement. Typical theory activities give rise to such probabilities and, as mentioned before, we use the different Skyrme parametrizations and their combinations with symmetry energy characterized by similar $(L,a_a^V)$, to arrive at the span of observable values representing any specific $(L,a_a^V)$-region. After accounting for data on $E$, the resulting probability density $p(x|\overline{E})$ may be used as a prior density, when accounting for any subsequent data. In the limit of large number of data employed to constrain the probability, it is expected to be only weakly dependent on the assumed original prior. A~stronger dependence of the posterior on the prior is expected for limited data.
\subsection{Characteristics of Uniform Matter}
\begin{figure}
\centerline{\includegraphics[width=.7\linewidth]{aavl_bey}}
\caption{Contours of constant probability density, encircling 68\% of net probability, in the plane of symmetry-energy parameters at normal density, $L$ and $a_a^V$, when more and data is used to improve the inference. The outmost contour, with interior crossed diagonally, represents the probability obtained when incorporating results from II on the mass-dependence of symmetry-energy coefficients, following from systematics of excitation energies to ground-state IAS, $E^*_\text{IAS}$. The more inner contour, with interior cross-hatched, represents the probability arrived at after incorporating the result on the average difference of slopes for the isovector and isoscalar potential surfaces, $\Delta a$, cf.~Table~\ref{tab:BestFit}. The~most inner contour, with shaded interior, represents the probability arrived at after incorporating the results on the differences in the radii of isovector and isoscalar potentials for the four nuclei, $\Delta R$, cf.~Table~\ref{tab:BestFit} again.
}
\label{fig:aavl}
\end{figure}
In Fig.~\ref{fig:aavl}, we show the evolution of the density of probability in the $(L,a_a^V)$ plane~\cite{tsang_constraints_2012} as more and more information from data analyses is accounted for. Compared to II, we include more Skyrme parametrizations, but now we suppress, in the prior density, those with incompressibilities outside of the realistic region of $200 < K < 300 \, \text{MeV}$, with a~stronger suppression factor the farther they depart from that region. Even though we interpolate between the interactions and normalize the prior probability density to a constant value within that portion of the $(L,a_a^V)$ plane where pertinent Skyrme parametrizations are available, the~boundaries of the covered region are expectedly noisy and connections between observable quantities and symmetry-energy parameters exhibit graininess tied to the finite number of the starting Skyrme parametrizations. In spite of those limitations, it is very obvious in Fig.~\ref{fig:aavl} that the reproduction of the mass dependence of symmetry-energy coefficients in~II, $a_a^V(A)$, or equivalently the systematics of $E_\text{IAS}^*$, imposes a narrow positive correlation within the plane of $(L,a_a^V)$. Within that correlation region, the expectation of reproducing the difference in the slopes $\Delta a$ of isovector and isoscalar surfaces, deduced by fitting the elastic and QE (p,n) cross sections, shrinks the region compatible with data to higher values of symmetry-energy parameters, with $L$ above $\sim 50 \, \text{MeV}$. Inclusion of consistency
with the differences $\Delta R$ in the isovector and isoscalar radii for the four nuclei, from fitting the reaction data, shrinks the region of likely parameters of the symmetry energy farther out into high values, with $L$ above $\sim 70 \, \text{MeV}$.
In Figs.~\ref{fig:pl} and \ref{fig:pv}, we show the evolution of the probability density, with addition of data, when projected onto the $L$ and $a_a^V$ axes. We maintain the same normalization of the prior, to a constant value in the region of the $(L,a_a^V)$ plane that is covered and a~smooth, at the level of our discretization in the plane, transition to zero for the regions that are not covered. In Fig.~\ref{fig:pl}, in particular, oscillations are seen tied to noisy boundaries of the covered region. The oscillations persist after data are incorporated, due to graininess in the connection between symmetry-energy parameters and observables. As such they should be considered as artifacts of methodology, rather than of significance.
\begin{figure}
\centerline{\includegraphics[width=.7\linewidth]{PL}}
\caption{Evolution of the density of probability in the value $L$ of the slope of symmetry energy at $\rho_0$, as different data are accounted for. The prior density represents the projection of the prior density in the plane of $(L,a_a^V)$, that is normalized to a constant value in the regions of the plane for which pertinent Skyrme parametrizations or their linear combinations can be found. Oscillations in the prior represent a discontinuous behavior of the boundaries of the covered region in the $(L,a_a^V)$ plane. The shaded portion
of the final density, with all considered data included, represents the most narrow region in $L$ that contains 68\% of the probability.
}
\label{fig:pl}
\end{figure}
\begin{figure}
\centerline{\includegraphics[width=.7\linewidth]{PV}}
\caption{Evolution of the density of probability in the value of the symmetry energy at $\rho_0$, $a_a^V \equiv S(\rho_0)$, as different data are accounted for. The prior density represents the projection of the prior density in the plane of $(L,a_a^V)$, that is normalized to a constant value in the regions of the plane for which pertinent Skyrme parametrizations or their linear combinations can be found. The~shaded portion
of the final density, with all considered data included, represents the most narrow region in $a_a^V$ that contains 68\% of the probability.
}
\label{fig:pv}
\end{figure}
Within Bayesian inference, it may be seen in Figs.~\ref{fig:pl} and~\ref{fig:pv} that even accounting for the systematics of nuclear symmetry-energy coefficients pushes alone the probability towards higher values of $L$ and $a_a^V$. In~$L$, in fact even a strong push is seen against the high-end boundary of the prior. That boundary is a reflection of the limitation of the Skyrme parametrizations - relativistic mean field approaches tend to yield higher values of~$L$ and may eventually get invoked to improve inferences on the probability from the high-$L$ side, in the context of the specific set of observables. By the time the differences in the radii $\Delta R$ are accounted for in the inference, though, that push against the upper $L$-boundary subsides, though, see Fig.~\ref{fig:pl}. Looking for the most narrow interval along either the $L$- or the $a_a^V$-axis, that contains 68\% of the probability, we find that with this probability the slope and the value are within the limits of $70 < L < 101 \, \text{MeV}$ and $33.5 < a_a^V < 36.4 \, \text{MeV}$, respectively. The specific value of 68\% is obviously taken because this is the net probability within one error from the central value for a probability density in Gaussian form.
\begin{figure}
\centerline{\includegraphics[width=.7\linewidth]{aavl20}}
\caption{Constraints, from different sources, on the symmetry-energy parameters at $\rho_0$, $a_a^V \equiv S(\rho_0)$ and slope $L$, after \cite{tsang_constraints_2012,lattimer_constraining_2012}. Included are predictions from neutron-matter calculations by Gandolfi~\etal~\cite{gandolfi_maximum_2011}, within QMC, and within chiral effective field theory, in N$^\text{2}$LO order by Heberle \etal~\cite{hebeler_constraints_2010} and in N$^\text{3}$LO order by Tews \etal~\cite{tews_neutron_2013}. From Fig.~\ref{fig:aavl}, we further reproduce here our own $E^*_\text{IAS}$ + $\Delta R$ constraints, marked IAS, as combining conclusions from the excitation energies and cross sections to the ground-state IAS. Other illustrated constraints, deduced from observables, include those deduced from neutron skins by Chen \etal~\cite{chen_density_2010}, from nuclear masses by Korteleinen \etal~\cite{kortelainen_nuclear_2010}, pygmy dipole resonance (PDR) by Carbone~\etal~\cite{carbone_constraints_2010}, heavy-ion collisions~(HI) by Tsang~\etal\ and quoted in~\cite{horowitz_way_2014}, and from neutron-star (NS) observations by Steiner \etal~\cite{0004-637X-722-1-33}.
}
\label{fig:aavl18}
\end{figure}
Next, in Fig.~\ref{fig:aavl18} we compare our constraints on $(L, a_a^V)$ to some of those in the literature, with reasonable realism, specifically from nuclear-matter calculations within quantum Monte-Carlo (QMC) \cite{gandolfi_maximum_2011} and chiral effective field theory (CEFT) within N$^\text{2}$LO order~\cite{hebeler_constraints_2010} and N$^\text{3}$LO~\etal~\cite{tews_neutron_2013}, derived from neutron-star observations~\cite{0004-637X-722-1-33}, ground-state masses~\cite{kortelainen_nuclear_2010} and from various reaction \mbox{observables~\cite{chen_density_2010,carbone_constraints_2010,horowitz_way_2014}.} Relative to other inferences in the literature, ours occupy the high-value corner in both $L$ and $a_a^V$. In microscopic theory, the parameters of the symmetry energy are quite sensitive to the strength of three-nucleon (3N) interactions (primarily $c_3$ constant), implanted on top of the two-nucleon (2N) interactions, with both $a_a^V$ and $L$ increasing as the 3N strength is increased. In the case of QMC and CEFT N$^\text{2}$LO, the diagonally slanted regions in Fig.~\ref{fig:aavl18} explicitly illustrate the evolution of the parameters as the strength of the 3N interactions is increased. The~results of microscopic calculations, though, depend also on chosen strategies, with greater corresponding uncertainty the larger the density and the larger the order of the calculations. In Fig.~\ref{fig:aavl18} it appears that the best compromise between different inferences in the literature is around $a_a^V \sim 33 \, \text{MeV}$ and $L \sim 60 \, \text{MeV}$, with significant impact of the 3N forces on the parameters then. Our results alone, however, favor though even higher values of both parameters.
\begin{figure}
\centerline{\includegraphics[width=.7\linewidth]{symecvs1}}
\caption{Symmetry energy in uniform matter as a function of density. The shaded regions represent the most narrow ranges of the symmetry energy at a specific density $\rho$, $0.04 < \rho < 0.16 \, \text{fm}^\text{-3}$, that contain 68\% of probability in the Bayesian inference when SHF calculations are confronted against conclusions from data. The wider more lightly shaded region, generally encompassing the more narrow darker region, results from data on $E_\text{IAS}^*$ leading to the mass-dependent symmetry-coefficients~$a_a(A)$. The~more narrow and more darkly shaded region results when the $E_\text{IAS}^*$ systematics is combined with conclusions on difference in geometry for isovector and isoscalar potentials, quantified with $\Delta a$ and $\Delta R$. The~dashed lines represent extrapolations of the described regions to lower and higher densities. The~three symbols, diamond, circle, and square, represent the values of symmetry energy at moderately subnormal densities deduced, respectively, by Brown~\cite{brown_constraints_2013}, Roca-Maza \etal~\cite{RocaMaza:2012mh}, and by Zhang~\etal~\cite{zhangChen_constraining_2013}. The~vertical sizes of those symbols represent the claimed errors on the deduced values of the symmetry energy.
}
\label{fig:symecvs}
\end{figure}
Proceeding in a similar manner as in the case of $a_a^V$ and $L$, we next generate limits on the symmetry energy at individual densities, i.e.\ $S(\rho)$, seeking the most narrow intervals that contain 68\% of the probability at specific $\rho$ in Bayesian inference. We choose to normalize the prior in the same way as before, i.e.\ to a uniform density in the $(L,a_a^V)$ plane, even though the specific analysis does not progress through the density in that plane - in practice this amounts to a specific weighting of the individual Skyrme interactions so they end up giving uniform density of probability in $(L,a_a^V)$. The~constraints on $S(\rho)$, arrived at when confronting the Skyrme results against conclusions drawn from $E_\text{IAS}^*$, i.e.~on~$a_a(A)$, and when combining the latter conclusions with those from analysing differential cross sections, are shown in Fig.~\ref{fig:symecvs} as different shaded regions for the density interval $0.04 < \rho < 0.16 \, \text{fm}^\text{-3}$. Only within that density interval the symmetry energy gets in practice tested in a nucleus, cf.~I and~II. The~wider shaded region in the figure is arrived for fewer data incorporated and the narrower for more. Notably, this is by no means a rule in the Bayesian inference: a confidence region may widen when consecutive data appear to contradict each other. The dashed lines show extrapolations of the results to lower and higher densities. Our credibility regions in Fig.~\ref{fig:symecvs} are bracketed there by values of symmetry energy inferred for intermediate densities from different data, specifically by Brown~\cite{brown_constraints_2013}, Roca-Maza \etal~\cite{RocaMaza:2012mh}, and by Zhang~\etal~\cite{zhangChen_constraining_2013}. The astounding finding in employing the Bayesian inference, as compared to II, is in the arrival at a generally more narrow credibility region on $S(\rho)$, than in~II, when using just the excitation energies to ground state IAS to narrow the credibility. The sharp rise in the symmetry energy with density is favored within the $E_\text{IAS}^*$-analysis as explaining a strong variation of the symmetry coefficients with nuclear mass, cf.~II. Addition of information from analyzing the differential cross sections in the current paper only mildly improves the inference at intermediate to low densities. In the weakly subnormal density region, the last addition shifts the probability to the stiffest symmetry energies. It should be stressed, for the perspective, that the prior distribution is fairly broad at all individual densities within the range $0.04 < \rho < 0.16 \, \text{fm}^\text{-3}$, just as at $\rho_0$ in Fig.~\ref{fig:pv}.
\begin{figure}
\centerline{\includegraphics[width=.7\linewidth]{eaneu}}
\caption{Energy per neutron in uniform neutron matter as a function of density. The shaded regions represent the most narrow ranges of the energy at a specific density $\rho$, $0.04 < \rho < 0.16 \, \text{fm}^\text{-3}$, that contain 68\% of probability in the Bayesian inference when SHF calculations are confronted against conclusions from data. The wider and more lightly shaded region, mostly encompassing the more narrow darker region, stems from the data on $E_\text{IAS}^*$ leading to the mass-dependent symmetry-coefficients~$a_a(A)$. The~more narrow and more darkly shaded region results when the $E_\text{IAS}^*$ systematics is combined with the conclusions on differences in geometry for isovector and isoscalar potentials, quantified with $\Delta a$ and $\Delta R$. The~dashed lines represent extrapolations of the described regions to the lower and higher densities.
Further represented in the figure are predictions from the chiral effective field theory, specifically from the NLO lattice calculations by Epelbaum~\etal~\cite{epelbaum_ground-state_2009} - as crosses, and from the N$^\text{3}$LO calculations by Tews~\etal~\cite{tews_neutron_2013} with 2N interactions only - as~vertically hatched region, and with 3N interactions - as a region hatched diagonally with dots.
}
\label{fig:eaneu}
\end{figure}
Progressing as with $S(\rho)$, we proceed further to arrive at constraints on energy per neutron $\frac{E}{N}(\rho)$ in pure neutron matter, continuing to use the SHF calculations in connecting the theoretical inputs to observables. As the observables exploited here primarily constraint the symmetry energy portion of $E/N$, we must rely on the expectation that the Skyrme parameterizations on the average reasonably describe the energy per nucleon in symmetric matter and they provide a reasonable spread of values for the quartic term in the expansion in asymmetry for the energy, that leads to the symmetry energy. I.e.\ we need to rely now more heavily on the sensibility of the prior probability than in the case of $S(\rho)$. The~results for $\frac{E}{N}(\rho)$, when applying Bayesian inference, are shown in Fig.~\ref{fig:eaneu}. The lighter and darker shaded regions, in density interval of $0.04 < \rho < 0.16 \, \text{fm}^\text{-3}$, represent 68\% credibility intervals when, respectively, accounting for the $E^*_\text{IAS}$ systematics alone and that systematics in combination with the differences in geometry for the isoscalar and isovector potentials. The dashed lines in the figure show extrapolations to the lower and higher densities. Just as with $S(\rho)$, the large isovector skins favor a rapid rise of $\frac{E}{N}(\rho)$ with $\rho$. Besides our inferences, represented in Fig.~\ref{fig:eaneu} are the CEFT predictions, specifically from NLO lattice calculations Epelbaum~\etal~\cite{epelbaum_ground-state_2009} - as crosses, and from N$^\text{3}$LO calculations by Tews~\etal~\cite{tews_neutron_2013} with 2N interactions only - as~vertically hatched region, and with 3N interactions - as a region hatched diagonally with dots. (The four-nucleon interactions are also included in the latter calculations, but yield small contributions even at the highest considered densities.) At~lower densities in the figure, the different CEFT agree between each other and with our inferences. As density increases, however, the predictions that only rely on 2N interactions rise, though, too slowly compared to the inferences from data. Those that include 3N interactions overlap with the inferences, but when the isovector skins are accounted for, only the high-energy end of the latter predictions is consistent with the inferences. In the Tews~\etal\ calculations~\cite{tews_neutron_2013}, that end is built up by the calculations that rely on the Entem-Machleit 2N interactions~\cite{entem_accurate_2003} within the full interaction set.
\subsection{Neutron Skins}
The combination of $L$ and $a_a^V$, that we arrive at in the present work, using the isovector skins as observables, clearly represent stiffer symmetry energies than those arrived at using the neutron skins, see Fig.~\ref{fig:aavl18}, and Ref.~\cite{zhangChen_constraining_2013} or II. With the position of isovector surface, relative to isoscalar, testing sensitively the stiffness of the symmetry energy, as illustrated in Fig.~\ref{fig:rho_all}, the QE (p,n) charge exchange reactions principally test the stiffness better than the elastic reactions, cf.~Fig.~\ref{fig:ppn48Ca35_fvr}. The latter reactions were used in the past to determine the neutron skins. Even in other analyses of data, it has been common to transcribe any conclusions onto the skin values, most often for $^\text{208}$Pb. Another neutron skin of interest in the literature recently has been that of $^\text{48}$Ca. Earlier in this work, we estimated that the isovector skins differ from neutron skins by a factor of few for these specific nuclei. Clearly of interest can be what the current conclusions yield in detail. Evolution of the probability in the neutron skin values, $\Delta r = \langle r^2 \rangle_n^{1/2} - \langle r^2 \rangle_p^{1/2}$, as different data are incorporated, is~illustrated in Fig.~\ref{fig:PRNP}. In the top panel the evolution is shown for the lighter of the two nuclei and in the bottom -- for the heavier. Looking for the most narrow intervals that contain 68\% of probability, we arrive at the confidence limits of $0.191 < \Delta r < 0.213 \, \text{fm}$ and $0.205 < \Delta r < 0.241 \, \text{fm}$, for $^\text{48}$Ca and~$^\text{208}$Pb, respectively. These intervals lie above those arrived at in II for the two nuclei when summing up various analyses of data directly pertaining to the skins, primarily from elastic scattering. (The analysis of a later pion photoproduction experiment on $^\text{208}$Pb~\cite{crystal_ball_at_mami_and_a2_collaboration_neutron_2014} though, not included in II, gave as well a skin compatible with those in~II.) Regarding theoretical predictions, the CEFT one \cite{hagen_neutron_2016} of $0.13 < \Delta r < 0.15 \, \text{fm}$ for $^\text{48}$Ca is particularly low compared to the range inferred here and in the context of the density of probability in Fig.~\ref{fig:PRNP}, though not inconsistent with the analyses summed up in II. However then the symmetry energies for the interactions touted in \cite{hagen_neutron_2016} are particularly soft, with $37.8 < L < 47.7 \, \text{MeV}$ and $ 25.2 < a_a^V < 30.4 \, \text{MeV}$. The specific interaction promoted there, NNLO$_\text{sat}$, is in fact represented in the lower halves of these intervals and not realistic even in the context of the literature claims on $\Delta r$, summarized in II.
\begin{figure}
\centerline{\includegraphics[width=.7\linewidth]{PRNP}}
\caption{Evolution of the density of probability in the value of neutron skin, $\Delta r = \langle r^2 \rangle_n^{1/2} - \langle r^2 \rangle_p^{1/2}$, for $^\text{48}$Ca (top panel) and $^\text{208}$Pb (bottom), as different data are accounted for. Oscillations in the prior density are due to the fact that the prior is normalized to uniform in the plane of $(L,a_a^V)$ over a region with uneven boundaries. Oscillations in the posterior density are also tied to a finite number of original Skyrme parametrizations tying $\Delta r$ to specific observables. The shaded portion
of the final density, with all considered data included, represents the most narrow region in $\Delta r$ that contains 68\% of the probability.
}
\label{fig:PRNP}
\end{figure}
\subsection{Droplet-Model Context}
The size of isovector skins supported by data may seem startlingly large. Irrespectively of amplification factors relative to neutron skins and results from structure, such as in Fig.~\ref{fig:rho_all} where the skins evidence gets marred by shell effects, it can be important to gain an additional perspective as to whether the large skins make sense or not. Such a perspective may be provided by macroscopic models such as those used to describe average features of nuclear masses~\cite{myers_nuclear_1974,moller_nuclear_1995,Danielewicz:2003dd}.
Within a macroscopic model~\cite{Danielewicz:2003dd}, portion of neutron-proton imbalance is pushed out to the nuclear surface, reducing the energetic penalty for the imbalance in the interior. The ratio of the imbalance in the surface to the interior scales in proportion to surface and volume capacitances for asymmetry:
\begin{equation}
\frac{(N-Z)_S}{(N-Z)_V} = \frac{C_S}{C_V} = \frac{A^{2/3}/a_a^S}{A/a_a^V} = \frac{a_a^V}{a_a^S \, A^{1/3}} \, .
\end{equation}
On the other hand, we can write for the ratio of the surface-to-volume asymmetries:
\begin{equation}
\frac{(N-Z)_S}{(N-Z)_V} \simeq \frac{4 \pi \, r_0^2 \, A^{2/3} \, \Delta R}{(4\pi/3) \, r_0^3 \, A} = \frac{3 \Delta R}{r_0 \, A^{1/3}} \, .
\end{equation}
In the above, we assumed approximately the same isovector density in the interior and in the surface. Combining the two results we arrive at (see also I)
\begin{equation}
\frac{a_a^V}{a_a^S} \simeq \frac{3 \Delta R}{r_0} \simeq 2.3 \,.
\end{equation}
In the last step, we used the global fit value of $\Delta R = 0.88 \, \text{fm}$ from Table \ref{tab:BestFit1} and $r_0=1.14 \, \text{fm}$. The ratio of volume-to-surface symmetry coefficients of 2.3, equivalent to the surface skin $\Delta R \sim 0.88 \, \text{fm}$, is fairly mundane as far as the droplet-model considerations are concerned~\cite{myers_nuclear_1974,moller_nuclear_1995,Danielewicz:2003dd}.
A more accurate mapping of the current results onto the $(a_a^V,a_a^S)$ pair would require another Bayesian inference, beyond the scope of the present paper. We should mention that when the ratio $a_a^V/a_a^S$ is fitted to masses or excitation energies to IAS, an inflated value may emerge compared to $A \rightarrow \infty$ limit, cf.~II.
According to Fig.~\ref{fig:drx}, the geometric vector skins seem to approach the macroscopic limit much faster then the surface details in masses.
\section{Conclusions}
In this work, we simultaneously analyzed differential cross sections for elastic (p,p) and (n,n) reactions, and quasielastic (p,n) reactions to IAS, on four targets, $^{48}$Ca, $^{90}$Zr, $^{120}$Sn and $^{208}$Pb, within the energy range of (10--50)~MeV, following the concepts of isoscalar and isovector potentials combined into a Lane potential. The goal was to detect and quantify a~possible displacement of the isovector and isoscalar surfaces suggested by the macroscopic considerations and by structure calculations. For this purpose, we minimally modified the popular Koning-Delaroche potential to allow for controlled changes in the geometry of the isovector and isoscalar potentials. We demonstrated that the geometry of isovector potentials strongly impacts differential charge-exchange cross sections and only weakly elastic. Accordingly, we organized the fit procedure so that the elastic cross sections primarily governed the adjustments in the isoscalar potentials and the charge-exchange cross sections -- the adjustments in the isovector potentials relative to isoscalar. Fits to the data gave rise to large differences $\sim$$0.9 \, \text{fm}$ in the radii of isovector and isoscalar potentials, or isovector skins, changing little from a nucleus to a nucleus. In addition, the isovector surfaces were found to be slightly steeper, by $\sim$$0.1 \, \text{fm}$ in diffusivity, compared to isoscalar. The particular aspect of the charge-exchange cross sections that is sensitive to the isovector skins are the oscillations as a function of angle, strengthening and changing position as skin size increases.
In the impulse approximation and in the folding models, the isovector and isoscalar potentials are tied to the isovector and isoscalar densities, respectively. In the structure calculations and in Thomas-Fermi considerations, average differences between the latter densities are tightly tied to the density dependence of the symmetry energy. Expecting the differences in the geometry of the potentials to reflect the differences in the geometry of underlying densities, we attempted to use the differences from the fits to learn on the symmetry energy. Large isovector skins, such as found in analyzing cross sections, roughly independent of a~nucleus, with a steeper isovector surface than isoscalar are produced in Skyrme-Hartree-Fock calculations using relatively stiff symmetry energies. Relying on the current data analysis and on results of our prior work examining data on excitation energies to isobaric analog states of ground states, and employing Bayesian inference, we arrived at 68\% credibility limits for the parameters of symmetry energy at normal density, of
$70 < L < 101 \, \text{MeV}$ and $33.5 < a_a^V < 36.4 \, \text{MeV}$. We also arrived at credibility regions on symmetry energy and energy of neutron matter as functions of density. From the results on neutron matter in chiral effective field theory only those most stiff overlap with our constraint region. When transcribing our isovector skin results into neutron skins, within structure calculations, we arrived at large values, in particular $0.205 < \Delta r < 0.241 \, \text{fm}$ for $^{208}$Pb.
The greatest weakness of the analysis here is the attribution of the differences in the geometry of optical potentials, needed to explain differential cross sections, to the differences in the densities. Weak dependence of the inferred neutron skin on the nucleus seems to support this attribution. We should add that the Coulomb energy for a proton by the surface of the target strongly changes between $^{48}$Ca and $^{208}$Pb, so effectively we arrive at similar results for different energy brackets of a proton by the target nucleus.
Further work on this topic could be improved in different directions. Thus, the analysis critically depends on availability of quasielastic (p,n) data with a good resolution of oscillations in the differential cross sections. Data at high incident energies are usually favored investigations of densities. Fitting of potential parameters here was done following frequentist methodology and it is Bayesian methodology that is usually particularly effective when data of different type, such as from (p,p), (n,n) and (p,n) processes are combined. We, so far, did not exploit analyzing power in the analysis. In the paper we indicated that the isovector potentials multiplying the third and transverse isospin components should be different due to the Coulomb polarization, but we did not exploited that in the data analysis. On the structure side, we were short in the interactions with very stiff symmetry energies, such as common in relativistic mean field calculations.
\acknowledgments
Strong impetus for this work was provided by the talk by Dao Tien Khoa given in the context of the 2013 International Collaborations in Nuclear Theory Program on Symmetry Energy. In~the early stages of this work, the authors benefited from the expertise of Luke Titus. As work progressed, the authors benefited from discussions with many colleagues including Remco Zegers, Brent Barker, Ron Johnson, Achim Richter, Sam Austin, Naftali Auerbach and Bill Lynch. Alex Brown provided the authors with a family of Skyrme interaction parameters \cite{brown_constraints_2013}. Progress was further boosted through attendance of workshops co-sponsored by CUSTIPEN in 2015, on~reactions and on asymmetric nuclear matter.
This work was supported by the U.S.\ National Science Foundation under Grants PHY-1068571 and PHY-1403906 and by the University Grants Commission of India under Indo-US 21$^\text{st}$ Century Knowledge Initiative.
\newpage
|
1,116,691,497,592 | arxiv | \section{Introduction}
Magnetic random access memories are believed to be among the most promising candidates to deliver the future of scalable,
non-volatile, rapidly accessible data storage. At the heart of these devices are magnetic tunnel junctions (MTJs), which
store data on the relative orientation of the magnetisation vectors of two magnetic layers separated by an insulating barrier.
Reading and writing such junctions can be efficiently performed by applying an electric current through the device; exploiting
the tunnelling magneto-resistance (TMR)~\citesq{TMR} effect for reading and using spin-transfer torque (STT)~\citesq{STT}
to write. STT arises when a current passes across two ferromagnets having different magnetisation directions and it is caused
by the transfer of angular momentum between the two mediated by the current. The conduction electrons become spin polarised
by passing through the first magnetic layer and their angular momentum is then transferred to the second. The ideal insulating
barrier acts as a spin-filter maximising the spin-polarisation of the current and hence the torque.
Optimising the device structure to achieve low write currents is an important challenge in realising the potential of these devices.
Whilst early demonstrations of MTJs focused on devices with in-plane layers magnetisation, the write current can be reduced
significantly by adopting an out-of-plane geometry, where the magnetisation direction of both layers is oriented normally
to the barrier interface. In junctions with this configuration, known as perpendicular MTJs (pMTJs), a large perpendicular magnetic
anisotropy (PMA) is required to overcome the shape anisotropy of the thin film and enforce thermal
stability in scalable devices.
State-of-the-art devices are based upon CoFeB/MgO thin films, which can reach a TMR of up to 604\% at room temperature
and 1144\% at low temperature~\citesq{Ikeda2010}. Furthermore, a large PMA has been observed at the CoFeB/MgO interface which is sufficient to achieve a perpendicular geometry in ultra-thin layers~\citesq{Worledge2011a}. Alternatively, L$1_0$ FePt is
a popular material choice for high-density magnetic recording, since it has a large magneto-crystalline PMA, $K_u = \SI{7e6}{Jm^{-3}}$,
allowing stable grain sizes down to a few nanometres~\citesq{Weller2000}. Despite the large uniaxial anisotropy, switching has been
observed in FePt/Au giant magneto-resistance pillars with the aid of an applied magnetic field~\citesq{Seki2006}. Theoretical
calculations of a FePt/MgO MTJ predicts a TMR of 340\% for a Fe terminated interface~\citesq{Taniguchi2008}.
Unfortunately, growing FePt/MgO devices can be challenging since the lattice mismatch between L1$_0$ FePt and MgO
is large, $\sim 8.5\%$~\citesq{Cuadrado2014a}. This may cause issues during the growth process, such as the inability of
preserving the epitaxy across uneven layers. Strain can also cause a significant change in the magnetic properties of the
FePt layer. In particular calculations have shown that a strain of 4\% can reduce the PMA to about 10\% of its original
value~\citesq{Seki2006}. Practically, such strain can be reduced by inserting a seed-layer with a more amenable lattice
constant at the MgO/FePt interface.
In this work we investigate a series of FePt/MgO-based pMTJs in order to establish their potential for future device applications.
We utilise \emph{ab-initio} models to calculate the spin-transfer torque and the TMR for a range of FePt-based MTJ structures.
We begin by detailing our computational method, before presenting results on the atom-resolved STT in the zero-bias
limit for an Fe/MgO/Fe junction. This has an electronic structure analogous to that of CoFeB-based MTJs and hence provides
a useful starting point for the discussion. We then continue with the analysis of the torque acting on the MTJs with FePt/Fe
free layers and with a thin Fe seed layer intercalated at the MgO interface. In this case we vary the thickness of both the
FePt layer and the seed layer (including the case where there is no seed layer). We find that a MgO/FePt interface yields a
STT that decays more slowly in the free layer than in the MgO/Fe case, while the insertion of a Fe seed layer produces results similar
to the FePt-free case. We then present the outcome of our TMR calculations and the STT acting on the Fe reference layer for
some representative cases. Finally we replace the Fe atoms in the seed layer with Ni. This provides a comparison and
helps us to formulate an argument about the origin of the spatial dependence of the STT.
\section{Computational Method}
Our approach for calculating the spin-transfer torque follows the prescription provided by Haney et al. in reference~\citesq{Haney2007}
and is based on isolating the transport (non-equilibrium) contribution to the density matrix from the equilibrium part. The
influence that an electric current has on the system can be estimated from first principles by combining density functional theory
and the non-equilibrium Green's functions method for transport (DFT+NEGF). All calculations have been performed with the
{\sc Smeagol} code~\citesq{Rocha2005,Rocha2006,Rungger2008a,Rungger2009}, which implements the DFT+NEGF scheme
within the numerical atomic orbital framework of the {\sc Siesta} package~\citesq{Soler2002}.
\begin{figure}[t!]
\begin{center}
\includegraphics[width=\columnwidth]{./Fig1.pdf}
\caption{(Colour Online) Set up for a quantum transport calculation of a Fe/MgO/Fe/FePt/Fe junction. The dashed rectangle
delimits the scattering region from the leads. The green arrows indicate the different direction of the magnetisations of the
magnetic layers at the left- and right-hand side of the insulating barrier. The coloured spheres represent atoms of different
species: Fe atoms are in red, Pt in grey, O in light blue, Mg are small red spheres.}\label{setup}
\end{center}
\end{figure}
The system set up for the quantum transport calculation sandwiches the magnetic tunnel junction between two semi-infinite leads
(see Fig.~\ref{setup}). These are assumed to be made of bulk material and at equilibrium. Note that a certain portion of the
electrodes has to be included in the scattering region in order to ensure the continuity of the electrostatic potential. Here the
magnetisation of the reference or fixed layer, ${\bf M}_\text{ref}$, is considered to be magnetised along $z$ (the transport direction)
and the one of the free layer, ${\bf M}_\text{free}$, along $x$, so that the two form a $\pi/2$ angle. A voltage is applied in such a
way that the electron flux is flowing along the stacking direction, $z$, in our convention from the reference layer to the free one.
The component of the torque vector, ${\bf T}$, which is responsible for the switching between the parallel and the anti-parallel
magnetisation configurations is the one that lies in the plane defined by ${\bf M}_\text{free}$ and ${\bf M}_\text{ref}$,
namely the $x-z$ plane. In the free layer this component coincides with $T_z$, which is the main focus of our study.
In order to reduce the computational costs, we limit our analysis to the torque response to a small bias, the \emph{torkance},
meaning that all calculations are performed in the linear response approximation. At an atom $a$ in the free layer the torkance
is defined as
\begin{equation}\label{torque:def}
\tau_z^a \equiv \frac{dT^a_z}{dV}\Big|_{V=0} = \frac{1}{2}\text{Re}\sum_{i\in a}\sum_j \left(\boldsymbol\Delta_{ij}\times\frac{d\boldsymbol m_{ji}}{dV}\Big|_{V=0}\right)_z\:,
\end{equation}
and this can be estimated with a zero-bias calculation. Here $\boldsymbol\Delta$ denotes the exchange and correlation
field, namely the derivative of the exchange and correlation energy, $E_\mathrm{XC}$, with respect to the magnetisation
density, $\boldsymbol m$, $\boldsymbol\Delta={\delta E_\mathrm{XC}}/{\delta {\boldsymbol m}}$. Thus, the derivative
of $\boldsymbol m$ with respect to voltage embodies the spin contribution due to the rearrangement of the electronic
population under non-equilibrium conditions. Henceforth this will be referred to as the non-equilibrium spin density or
the spin accumulation. As such, the torque is the result of the interaction between the internal static field $\boldsymbol\Delta$
and the non-equilibrium spin density generated by the current flow. Further details on the calculation of the spin-transfer
torque and the torkance can be found in Refs.~\citesq{Stamenova2016, Ellis2017}.
A series of junctions are constructed, all having a barrier of 6 MgO layers sandwiched between two semi-infinite leads of bulk
{\it bcc} Fe oriented along the (001) direction. Periodic boundary conditions are applied in the plane perpendicular to the transport,
as a result of the perfect epitaxy of the junction. The in-plane lattice constant is taken to be $a_\text{Fe}=\SI{2.866}{\angstrom}$
throughout the system. The out-of-plane lattice constants of the remaining materials were chosen according to information provided
in reference~\citesq{Kohn2013,Cuadrado2014a}, in particular $c_\text{MgO}=4.05/\sqrt{2}$ $\SI{}{\angstrom}$,
$c_\text{FePt} = \SI{1.737}{\angstrom}$. The same studies assess that the most stable interfacial configuration is made of a
Fe-terminated FePt surface on top of O (Fe) for the FePt/ MgO (FePt/Fe) interface, with an inter-plane distance of $\SI{2.2}{\angstrom}$
($\SI{1.585}{\angstrom}$). The accuracy of such estimates was found satisfactory by relaxation of the different structures. The local
spin density approximation (LSDA) for the exchange correlation potential was adopted. A real-space mesh cut-off of \SI{900}~{Ry} along
with a 15$\times$15 $k$-point mesh in the plane orthogonal to transport were found to yield converged results.
We adopted double $\zeta$ polarised orbitals for each atomic species and the convergence of the radial cut-offs was verified
by comparing the band structure of bulk materials with the result of all-electron calculations. Since the introduction of spin-orbit
coupling effects did not yield a sizeable change to our calculated torques we have omitted relativistic corrections.
\section{Results}
\begin{figure}[!t]
\includegraphics[width=.5\textwidth]{./Fig2.pdf}
\caption{(Color Online) Real space profiles of the relevant components of (a) the exchange and correlation field,
$\boldsymbol\Delta$, (b) the non-equilibrium spin density, $\mathrm{d}\boldsymbol{m}/\mathrm{d}V$, and (c) the
torkance, $\boldsymbol\tau$, per unit $\mu_\mathrm{B}/e$ and area acting on the {\it bcc} Fe free layer. The coloured
background indicates the atomic species in the stack: red for Fe, blue for O, green for Mg.}\label{FeMgO}
\end{figure}
We begin by examining the properties of a Fe/MgO/Fe MTJ to later discuss their modification upon the introduction
of a FePt layer. As shown in Eq.~(\ref{torque:def}), the torkance is given
by the vector product of the exchange and correlation field and the non-equilibrium spin density. Since the free layer is
magnetised in the $x$-direction and within the LSDA the exchange and correlation field is proportional and locally parallel
to the magnetisation, the only relevant components to the torkance are $\Delta_x$ and $\mathrm{d}m_y/\mathrm{d}V$.
These two components and the resulting torkance, $\tau_z$, are shown in Fig.~\ref{FeMgO}.
In general, $\Delta_x$ peaks at the Fe/MgO interface and then presents small oscillations with the period of the interlayer
Fe separation, $a_\text{Fe}$. Such profile does correlate with the real space profile of the equilibrium magnetic moment
(not displayed), which is also enhanced at the Fe/MgO interface. In contrast, the non-equilibrium spin density [panel (b)] has an
appreciable magnitude only in the region around the Fe/MgO interface. This decays in the Fe layer and is almost fully attenuated
a few monolayers from the interface. Such behaviour will later be compared with that in FePt and in Ni. Finally note that
there is an appreciable non-equilibrium spin density also in the MgO, although it does not contribute to the torkance since the
exchange and correlation field vanishes in absence of a local magnetization [see panel (a)].
If we now consider the torkance we note that this is sharply peaked at the Fe/MgO interface and is attenuated in the Fe layer
at the same speed of the non-equilibrium spin density. In fact, for this Fe/MgO/Fe case the spatial dependance of the torkance
resembles closely that of the spin accumulation, given the fact that the exchange and correlation field has little spatial dependence
in Fe. Let us remark, however, that the point-by-point vector product of the quantities in panels (a) and (b) does not give the
torkance in panel (c), since the sum of the products of the matrix elements does not equal the product of the sums [namely,
$\sum_{ij} \left(\boldsymbol\Delta_{ij}\times\frac{d\boldsymbol m_{ji}}{dV}\right)\ne
\left(\sum_{i}\boldsymbol\Delta_{ii}\right)\times\left(\sum_{i}\frac{d\boldsymbol m_{ii}}{dV}\right)$ - see formula (\ref{torque:def})].
\begin{figure*}[!t]
\begin{center}
\includegraphics[width=17.78cm]{Fig3.pdf}
\end{center}
\caption{(Color Online) Study of the torkance in a FePt/Fe free layer. Left panel: the relevant components of (a) the exchange
and correlation field, $\boldsymbol\Delta$, (b) the non-equilibrium spin density, $\mathrm{d}\boldsymbol{m}/\mathrm{d}V$,
and (c) the torkance per unit of $\mu_\mathrm{B}/e$ and area, $\boldsymbol\tau$. Right Panel: comparison of the torkance per
unit $\mu_\mathrm{B}/e$ and area of MTJs with (d) 2, (e) 4 and (f) 6 FePt unit cells. In both subfigures the coloured background
indicates the atomic species: red for Fe, grey for Pt, blue for O, green for Mg.}\label{FePt:noSL}
\end{figure*}
We now explore the effects of inserting a layer of FePt at the MgO/free layer interface. Figures~\ref{FePt:noSL}(a)-(c) show,
as with the case of the Fe/MgO/Fe MTJ, the relevant components of $\boldsymbol\Delta$ and $\mathrm{d}\boldsymbol{m}/\mathrm{d}V$
contributing to the total torkance along $z$ for a FePt layer 4-unit-cell thick. From panel (a) it is clear that the exchange
and correlation field is enhanced at the Fe sites, and also finite at the Pt ones. This is because in L$1_0$ FePt there
is an induced magnetic moment on the Pt ions (this is about $0.4~\mu_\mathrm{B}$ as calculated from the M\"ulliken population
analysis), in agreement with previous \emph{ab-inito} calculations \citesq{Cuadrado2014a}. The oscillations in the $\Delta_x$ profile
remain constant in the FePt layer without any sign of decay, and then in the Fe layer the $\Delta_x$ profile returns to resemble
the one observed before in Fig.~\ref{FeMgO}. Note that $\boldsymbol\Delta$ is an equilibrium property, which essentially
depends on the presence of an exchange splitting in a given material. As such one does not expect a decay of $\boldsymbol\Delta$
unless there is a decay in the magnetisation.
\begin{figure*}[!t]
\begin{center}
\includegraphics[width=17.78cm]{Fig4.pdf}
\caption{(Color Online) Study of the torkance in a Fe/FePt/Fe free layer made of 4 FePt monolayers and a variable number of
Fe monolayers inserted between MgO and FePt. Left panel: the relevant components of (a) the exchange and correlation field,
$\boldsymbol\Delta$, (b) the non-equilibrium spin density, $\mathrm{d}\boldsymbol{m}/\mathrm{d}V$ and (c) the torkance per
unit of $\mu_\mathrm{B}/e$ and area, $\boldsymbol\tau$. Right Panel: comparison of the torkance per unit $\mu_\mathrm{B}/e$
and area of MTJs with with a seed layer comprising (d) 2, (e) 4 and (f) 8 Fe monolayers. In all panels the coloured background
indicates the atomic species: red for Fe, grey for Pt, blue for O, green for Mg.
}\label{FePt:FeSL}
\end{center}
\end{figure*}
In contrast to the pure Fe case, the non-equilibrium spin density has lower intensity in FePt than in Fe but a significantly less
attenuated decay [panel (b)]. The total non-equilibrium spin density shows regular oscillations within the FePt layer, whilst it is
enhanced at both the FePt/Fe and the MgO/FePt interfaces, and then vanishes within a few unit cells of the Fe lead.
Furthermore we observe that $\mathrm{d}{m_y}/\mathrm{d}V$ in Pt has opposite sign with respect to that of the first Fe layer
in contact to MgO. Finally, the torkance [panel (c)] is again peaked at the interface with MgO but its strength is reduced in
comparison to that computed for the Fe/MgO/Fe MTJ with the same MgO thickness. Within the FePt layer the torkance does
not attenuate as in Fe but persists to reach the Fe-only side of the free layer. Most interestingly the torkance has an oscillatory
behaviour in FePt, presenting small negative values at the Pt layers and positive at the Fe ones. Such oscillations are common
in antiferromagnets~[\cite{Stamenova2016}] and here are observed also in a ferromagnet with non-trivial magnetic texture. It
is also interesting to note that, despite the larger spin accumulation at Pt sites, the resulting torque is smaller than
that at the Fe ones. This is due to the fact that the exchange and correlation field in Pt is significant weaker than in Fe (because
the magnetisation is smaller).
The persistence of the torkance in the FePt layer remains as we change the FePt thickness, $n_\text{FePt}$ (number of unit cells).
This can be seen in the panels (d)-(f) of Fig.~\ref{FePt:noSL}. For a thin layer [panel (d)] the torque is enhanced at the FePt/Fe
interface, while it is attenuated for all the other cases [e.g. see $n_\text{FePt}=6$ in panel (f)]. Furthermore, for all the thicknesses
considered the torkance remains strikingly positive at all the Fe atomic planes of FePt, while it is small and negative at the Pt ones.
Moreover, the intensity of the peak at the MgO/FePt is not modified by the increase in thickness.
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{./Fig5.pdf}
\caption{(Colour Online) Panel (a): the total torkance per unit $\mu_\mathrm{B}/e$ and area acting on the free layer of
Fe/MgO/Fe/FePt/Fe junctions with 2, 4 and 6 FePt monolayers and a Fe seed layer of $n_\mathrm{SL}=0, 2, 4$ atomic
planes. Panel (b): the calculated TMR in Fe/MgO/FePt/Fe and Fe/MgO/Fe/FePt/Fe junctions with $n_\mathrm{SL}=2, 4, 8$
and 4 FePt unit cells. In both graphs the black dashed line represents the same quantity calculated for the Fe/MgO/Fe
junction.}\label{TotT+TMR}
\end{center}
\end{figure}
Although Fe/MgO/FePt/Fe junctions provide an interesting case of study, the significant lattice mismatch between MgO and
L1$_0$ FePt ($\sim$ 8.5\%) makes their experimental realisation troublesome.
This problem may be overcome by inserting a compatible seed layer at the MgO/FePt interface. Hence, we have
analysed the influence of incorporating a thin Fe seed layer (SL) between the MgO and the FePt, keeping the thickness of the FePt
layer constant at 4 unit cells. The Fe SL has different effects depending on its thickness (see Fig. \ref{FePt:FeSL}). We
notice from panel (a) that the exchange and correlation field profile in FePt is analogous to the previous case (since the equilibrium
magnetisation profile is also unchanged), while $\Delta_x$ is almost constant in the seed layer. The non-equilibrium spin density
still oscillates in FePt, although the amplitude of such oscillations is much smaller than that obtained in absence of the SL.
Consequently, the torkance [panel (c)] is peaked at the MgO/Fe interface with the SL and its intensity is comparable to that observed
for the Fe/MgO/Fe case (see figure~\ref{FePt:noSL}). The torkance, however, is not exactly zero away from the SL, in particular on
the Fe atoms of FePt and at the FePt/Fe interface. This does not happen for thicker Fe SLs [panels (e) and (f)], for which the total
torkance decays before reaching the interface with FePt. In general, however, the main effect of the seed layer is to suppress
the persistence of the torkance in FePt, so that all the angular momentum transfer takes place in the seed layer.
\begin{figure}[!t]
\begin{center}
\includegraphics[width=\columnwidth]{./Fig6.pdf}
\caption{(Colour Online) Torkance per unit $\mu_\mathrm{B}/e$ and area acting on the reference layer of Fe/MgO/Fe (a),
Fe/MgO/FePt(4)/Fe (b) and Fe/MgO/Fe(2)/FePt(4)/Fe (c) MTJs. The coloured background indicates the atomic species to
which each point corresponds to: red for Fe, blue for O, green for Mg. }\label{RefL}
\end{center}
\end{figure}
We now move to analyse the total torkance and the TMR of each junction. Figure \ref{TotT+TMR}(a) shows the total torkance
integrated over the free layer, $\tau_z^\mathrm{tot}=\sum_a^{\alpha\in\mathrm{FL}}\tau_z^a$, for different thicknesses of the
FePt layer. We present results for the situation where there is no SL (red squares), and for a Fe SL of respectively 2 (green
squares) and 4 atomic planes (blue squares). For each SL thickness, the torkance shows little dependence on the thickness of
the FePt layer. When there is no SL this is attributed to the oscillatory behaviour without attenuation of the torkance profile
as observed in Fig.~\ref{FePt:noSL}. In contrast, when a SL is present most of the torque resides at the first MgO/Fe interface
so that the thickness of the FePt becomes irrelevant (see Fig.~\ref{FePt:FeSL}). Interestingly, when a SL is present the total
torkance transferred into the Fe/MgO/Fe/FePt MTJ is larger than that of a simpler Fe/MgO/Fe MTJ with identical barrier (dashed
black line). This is no longer true when the SL is absent. Such finding means that the introduction of a Fe seed layer not only
helps in achieving a better epitaxy during the growth but also facilitates a larger spin transfer torque.
Figure~\ref{TotT+TMR}(b) shows the calculated TMR for each junction (in all cases the FePt layer comprises 4 layers) and a
comparison with that of a Fe/MgO/Fe MTJ with an identical barrier. We observe that the junction with no Fe SL presents
the largest TMR, despite having the lowest torkance. This is unexpected, since in FePt bands with $\Delta_1$ symmetry, namely
those with the largest transmission across MgO, are present for both spin channels~\citesq{Taniguchi2008}. Such feature returns a
predicted TMR for MTJs with FePt leads not exceeding 340\%~\citesq{Taniguchi2008}. However, here the situation is different since
in all our MTJs the leads are made of Fe, so that spin filtering is always in place. As such, in our case the addition of a FePt layer
(or a complex Fe/FePt layer) changes the details of the spin-dependent scattering potential, but does not alter the main spin-filtering
mechanism at play in Fe/MgO/Fe junctions. Interestingly, as the thickness of the Fe SL gets larger, the value of the TMR is
reduced.
\begin{figure*}[!t]
\begin{center}
\includegraphics[width=17.78cm]{Fig7.pdf}
\caption{(Color Online) Study of the torkance in a Fe/MgO/Ni/FePt/Fe junctions having a Ni seed layer of 2 (d), 4 (e) and 6 (f)
monolayers. Left panel: the relevant components of (a) the exchange and correlation field, $\boldsymbol\Delta$, (b) the
non-equilibrium spin density, $\mathrm{d}\boldsymbol{m}/\mathrm{d}V$ and (c) the torkance per unit of $\mu_\mathrm{B}/e$
and area, $\boldsymbol\tau$. Right Panel: comparison of the torkance per unit $\mu_\mathrm{B}/e$ and area of MTJs with
with a seed layer comprising (d) 2, (e) 4 and (f) 6 Ni monolayers. The coloured background represents the different atomic
species: red for Fe, grey for Pt, blue for O, green for Mg, purple for Ni.}\label{Ni}
\end{center}
\end{figure*}
So far the left electrode has been considered to be the fixed layer, namely the one producing the spin-polarised current. It is now
interesting to look at the opposite case, namely the one where the electrons flux flows from the right-hand side to the left-hand side
electrode. This is the situation where the FePt/Fe composite electrode acts as the fixed, current polarising, layer. Since in the right
electrode the magnetisation is along the $z$ direction, the relevant torque in this case is $\tau_x$. This is presented in Fig.~\ref{RefL}
for three representative junctions: (a) Fe/MgO/Fe, (b) Fe/MgO/FePt(4)/Fe,
and (c) Fe/MgO/Fe(2)/FePt(4)/Fe, where the numbers in parentheses indicate the number of unit cells. Since in this geometry the
current flows in the opposite direction than previously, we have plotted $-\tau_x$, namely the torque component that will lead to an
alignment of the magnetisations of the fixed and free layers. The trend of $-\tau_x$ is in all cases analogous to that of $\tau_z$ for
the Fe/MgO/Fe MTJ [see Fig.~\ref{FeMgO} (c)], namely the STT is peaked at the magnet/insulator interface and is negligible
elsewhere. The only significant difference between the three MTJs is the reduction of approximately a factor two of the peak intensity
for the Fe/MgO/FePt(4)/Fe stack [panel (b)].
\section{Discussion}
The results presented so far indicate that the STT (the torkance) varies strongly with the distance from the MgO interface, and
that the details depend subtly on the specific layer structure. In general, Fe seems capable of absorbing a significant amount of angular
momentum, so that only a few Fe monolayers are enough to make the STT decay sharply from the MgO interface. The main
cause of such effect has to be found in the intense Fe exchange field. In fact, the strong exchange interaction in Fe relaxes the
non-equilibrium spin density (the spin accumulation) toward the local direction of the magnetisation within a few atomic layers
from the interface, so that there is little $\mathrm{d}\boldsymbol{m}/\mathrm{d}V$ away from the interface itself. In addition the
exchange and correlation field remains almost constant within the Fe layer, resulting in a torque that persists little away from the
interface with MgO.
In L$1_0$ FePt the alternating planes of Fe and Pt lead to a magnetisation texture that is non-uniform at the atomic scale.
In particular ${\bf \Delta}$ is small at the Pt sites so that the average exchange and correlation field is reduced with respect to
that of the pure Fe case. As a consequence the spin accumulation can penetrate longer into the free layer so that the STT decays
less sharply. In order to further investigate the effects of the exchange field on the spatial decay of the torque we now consider
a Ni seed layer since it has a much smaller moment, and thus exchange field, than Fe. The calculation has been simplified by
maintaining the {\it bcc} structure and the lattice constant of Fe. As such our device stack does not correspond to a likely
experimental situation but just serves the purpose of comparing the different seed layers. The atomic resolved torkance for a
Fe/MgO/Ni/FePt/Fe stack with a Ni seed layer comprising 2, 4 and 6 atomic planes is shown in figure \ref{Ni}.
As in the case of a Fe seed layer, the torque [panel (c)] is strongly peaked at the Ni/MgO interface, but now it does not decay
entirely and thus a non-vanishing STT with an oscillatory behaviour persists into the FePt layer. A closer look at the profile
of ${\bf \Delta}$ across the junction [panel (a)] reveals that the exchange and correlation field in Ni is about half of that of
Fe [see Figure~\ref{FePt:FeSL}(a)]. As a consequence, in Ni the spin accumulation does not relax along the local direction of
the magnetization as efficiently as in Fe, a fact that can be appreciated by comparing Fig.~\ref{Ni}(b) with Fig.~\ref{FePt:FeSL}(b).
Interestingly, the attenuation of the spin accumulation and thus of the torque is not complete even for relatively thick Ni
seed layers, as can be seen in panels (d) through (f). A second interesting observation concerns the phase of the
oscillations of the STT in the FePt layer. In fact for a junction where FePt is in direct contact with the MgO barrier, the torque
is positive at the Fe planes and negative (although rather small) at the Pt ones. The same behaviour, although with a much
reduced torque is observed for Fe intercalation (in the presence of a Fe seed layer). In contrast when the seed layer is made
of Ni the sign of the STT on the FePt layer changes, becoming negative at the Fe planes and positive (although small) at the Pt
ones. As a result the total integrated torque over the entire free layer (seed layer plus FePt) for Ni intercalation is two thirds than
that obtained with Fe intercalation.
Finally, we wish to make a few general remarks on the spatial dependence of the STT. Macroscopic models combining the
Landau-Lifshitz-Gilbert equation for the magnetisation dynamics with a diffusion model for the spin accumulation
\citesq{Abert2015,Wang2006} suggest that the spin accumulation is maximised in regions where there is a large magnetisation
gradient, namely at interfaces. This is confirmed here at the microscopic level. In all cases investigated we find the maximum
spin accumulation, and hence torque, at the interface between the free layer and MgO regardless of the presence of a seed
layer. Furthermore, we also find an enhanced spin accumulation and torque at the second interface between the free layer
and the Fe lead, although this is small since the spin accumulation always decays in the free layer. The fine details of the
spin accumulation profile depend on how the entire stack responds to the application of an external bias. This in turn is
affected by the reorganisation in the occupation of the states around the Fermi surface, which is indeed a subtle effect.
In general a large exchange splitting causes the spin accumulation to relax faster along the local magnetisation direction.
As such we expect the spin accumulation to decay more severely in the free layer of stacks where there is a large torque
at the first few atomic layers in contact with the MgO barrier. This in turn depends on the strength of the exchange and
correlation field, which in the LSDA can be written as
\begin{equation}
{\bf\Delta}^\mathrm{LSDA}({\bf r})=\frac{\delta E^\mathrm{LSDA}_\mathrm{XC}}{\delta\mu_\mathrm{B}{\bf m}({\bf r})}=
-\frac{\partial \epsilon_\mathrm{XC}}{\partial m({\bf r})}\:\frac{n({\bf r})}{\mu_\mathrm{B}}\:\frac{{\bf m}({\bf r})}{m({\bf r})}\:,
\end{equation}
where ${\bf m}({\bf r})$ is the local magnetisation vector, $m({\bf r})=|{\bf m}({\bf r})|$, $E^\mathrm{LSDA}_\mathrm{XC}$
is the LSDA exchange and correlation energy, $\epsilon_\mathrm{XC}$ is the exchange and correlation energy density of
the homogeneous electron gas, $n({\bf r})$ is the charge density and $\mu_\mathrm{B}$ the Bohr magneton. Crucially the
LSDA ${\bf\Delta}$ is locally parallel to the magnetisation direction. As such one expects ${\bf\Delta}$ (and hence
the torque) to change sign as the local magnetisation changes sign (as in the case of antiferromagnets). Furthermore
one can show that $|{\bf\Delta}|\sim Im$, where $I$ is the Stoner parameter \citesq{Simoni2017}. This means that for similar
Stoner coupling the exchange and correlation field is more intense for materials presenting larger magnetization. This last
feature explains the difference in ${\bf\Delta}$ and torque between the Fe and the Ni seed layer. In fact Fe and Ni have rather
similar Stoner parameter but their magnetization differ by more than a factor three.
\section{Conclusions}
In conclusion, we have calculated the STT acting upon the free ferromagnetic layer in a series of FePt-based magnetic tunnel junctions.
For a simple Fe/MgO/Fe MTJ the torkance is peaked at the MgO interface and decays within 4 atomic planes. When the stack is modified
to include FePt [Fe/MgO/FePt/Fe] the torkance decays much slower and persists into the free layer up to at least 12 atomic planes.
Such retention is associated to torkance oscillations at the length scale of the Fe-Pt plane separation. Since the lattice mismatch
between MgO and FePt is large we have explored the option to intercalate a Fe seed layer at the interface between MgO and FePt.
Also in this case the torkance is significant only at the first MgO/Fe interface and it vanishes in FePt. This is the result of the strong
reduction of the spin accumulation beyond the Fe seed layer. Such strong attenuation appears to originate from the large exchange
and correlation field in Fe, which rapidly aligns the spin accumulation along the local direction of magnetization. Such hypothesis
is confirmed by calculations for the STT in some hypothetical MTJs incorporating a Ni seed layer. Since Ni has an exchange
and correlation field that is weaker than that of Fe, it is less effective at suppressing the spin accumulation (in absorbing angular
momentum) and thus the attenuation of the torkance is weaker. All together our results suggest that the atomic and materials details
of the MTJs stack play an important role in determining the total STT that a free layer can experience. This knowledge can help in
designing stacks with maximal torkance, so that a reduction in the critical current for switching can be achieved.
\section{Acknowledgements}
This work has been supported by the Science Foundation Ireland Principal Investigator award (grant no. 14/IA/2624 and
16/US-C2C/3287) and TCHPC (Research IT, Trinity College Dublin). The authors wish to acknowledge the DJEI/DES/SFI/HEA
Irish Centre for High-End Computing (ICHEC) for the provision of computational facilities and support.
|
1,116,691,497,593 | arxiv | \section{Introduction}
Let $C\subset\C$ be a compact infinite set in the complex plane and let $\P_n$ be the set of all polynomials with complex coefficients of degree $n$. Then $L_n(C)$, defined by
\begin{equation}
L_n(C)\equiv\max_{z\in{C}}|z^n+\sum_{k=0}^{n-1}c_{k}^{*}z^{k}|:=
\min_{c_{k}\in\C}\max_{z\in{C}}|z^n+\sum_{k=0}^{n-1}c_{k}z^{k}|,
\end{equation}
is usually called the \emph{minimum deviation} of degree $n$ on $C$. The polynomial
\[
M_{n}(z):=z^n+\sum_{k=0}^{n-1}c_{k}^{*}z^{k}\in\P_{n},
\]
for which the minimum is attained, is called the \emph{minimal polynomial} (or \emph{Chebyshev polynomial}) of degree $n$ on $C$. Fekete\,\cite{Fekete} proved that the limit
\begin{equation}
\CAP{C}:=\lim_{n\to\infty}\root{n}\of{L_n(C)}
\end{equation}
exists and the quantity $\CAP{C}$ is called the \emph{logarithmic capacity} (or \emph{Chebyshev constant} or \emph{transfinite diameter}). The logarithmic capacity is monotone, i.e.,
\begin{equation}\label{MonCap}
C_{1}\subseteq{C}_{2}\Longrightarrow\CAP{C_{1}}\leq\CAP{C_{2}}.
\end{equation}
Concerning many other properties of the logarithmic capacity and the connection to potential theory, we refer to \cite{Kirsch} and the references therein. For the logarithmic capacity $\CAP{C}$ and the minimum deviation $L_n(C)$, the inequality
\begin{equation}\label{IneqLnCap1}
L_n(C)\geq(\CAP{C})^n
\end{equation}
may be found in many textbooks and papers like \cite[Appendix\,B]{Simon} and goes back to Szeg{\"o}\,\cite{Szego} and Fekete\,\cite{Fekete}. Inequality \eqref{IneqLnCap1} is sharp: If $C$ is the unit circle then $\CAP{C}=1$, $M_n(z)=z^n$ and $L_n(C)=1$. In this paper, we prove that for all compact \emph{real} infinite sets $C\subset\R$
\begin{equation}\label{IneqLnCap2}
L_n(C)\geq2(\CAP{C})^n,
\end{equation}
where the constant $2$ is best possible. In addition, a large class of compact sets $C\subset\R$ is given such that equality is attained in \eqref{IneqLnCap2}. This is done in Section\,2. Some analogous results concerning compact infinite sets on the unit circle, lying symmetrically with respect to the real line, are given in Section\,3.
\section{Compact Subsets of the Real Line}
The first result gives an identity between the minimum deviation $L_{n}(A)$ and the logarithmic capacity $\CAP{A}$ in the case that the set $A$ is the inverse polynomial image of $[-1,1]$. As usual, for a polynomial $P_{n}\in\P_{n}$,
\begin{equation}
P_{n}^{-1}([-1,1]):=\bigl\{z\in\C:P_{n}(z)\in[-1,1]\bigr\}
\end{equation}
denotes the inverse polynomial image of $[-1,1]$ for the polynomial mapping $P_{n}$.
\begin{theorem}\label{Thm-LnCap}
Let $P_{n}\in\P_{n}$ and $A:=P_{n}^{-1}([-1,1])$. Then
\begin{equation}\label{LnCap}
L_{n}(A)=2\,(\CAP{A})^n.
\end{equation}
\end{theorem}
\begin{proof}
Let $P_{n}(z)=c_{n}z^{n}+\ldots\in\P_{n}$, $c_{n}\in\C\setminus\{0\}$, and let $T_{k}(z)=\cos(k\arccos(z))$ be the classical Chebyshev polynomial. In \cite{Peh-1996}, Peherstorfer proved that
\[
\frac{2}{(2c_{n})^{k}}\,T_{k}(P_{n}(z))=z^{kn}+\ldots,\quad~k=1,2,3,\ldots,
\]
is a sequence of minimal polynomials of degree $kn$ on $A$ with minimum deviation
\[
L_{kn}(A)=\frac{2}{(2|c_{n}|)^{k}}
\]
thus
\[
\CAP{A}=\lim_{k\to\infty}\sqrt[kn]{L_{kn}(A)}
=\lim_{k\to\infty}\sqrt[kn]{\frac{2}{(2|c_{n}|)^{k}}}
=\frac{1}{(2|c_{n}|)^{1/n}}=\frac{1}{\sqrt[n]{2}}\,\sqrt[n]{L_{n}(A)}.
\]
\end{proof}
\begin{example}\hfill{}
\begin{enumerate}
\item For $I:=[-1,1]$, we have $T_{n}^{-1}([-1,1])=I$, $L_{n}(I)=\frac{1}{2^{n-1}}$ and $\CAP{I}=\frac{1}{2}$, thus, for the set $I$, relation\,\eqref{LnCap} holds.
\item For $0<\alpha<1$ and $n$ even, the minimal polynomial on $E_{\alpha}:=[-1,-\alpha]\cup[\alpha,1]$ is
\[
M_{n}(z)=\frac{1}{2^{n-1}}(1-\alpha^{2})^{\frac{n}{2}}\,
T_{\frac{n}{2}}\bigl(\frac{2z^2-\alpha^2-1}{1-\alpha^2}\bigr)
\]
which gives
\[
L_{n}(E_{\alpha})=\frac{1}{2^{n-1}}(1-\alpha^{2})^{\frac{n}{2}}\quad\text{and}\quad
\CAP{E_{\alpha}}=\tfrac{1}{2}\sqrt{1-\alpha^{2}},
\]
thus, for the set $E_{\alpha}$, relation\,\eqref{LnCap} holds if $n$ is even.
\item Figure\,\ref{Fig_TPolynomial} shows a polynomial $P_{n}$ of degree $n=9$, for which the inverse polynomial image $A:=P_{n}^{-1}([-1,1])$ consists of four real intervals.
\end{enumerate}
\end{example}
\begin{figure}[ht]
\begin{center}
\includegraphics[scale=1.0]{Fig_TPolynomial}
\caption{\label{Fig_TPolynomial} Polynomial $P_{n}$ of degree $n=9$, for which the inverse polynomial image consists of four real intervals}
\end{center}
\end{figure}
\begin{lemma}\label{Lemma-MinPol}
Let $n\in\N$, let $C\subset\R$ be compact and infinite, let $M_{n}\in\P_{n}$ be the minimal polynomial of degree $n$ on $C$ with minimum deviation $L_{n}\equiv{L}_{n}(C)$, and let $\tilde{M}_{n}:=M_{n}/L_{n}$. Then $C':=\tilde{M}_{n}^{-1}([-1,1])$ is the union of $\ell'$ finite disjoint real intervals, where $1\leq\ell'\leq{n}$, and
\begin{equation}\label{Ca-B}
C\subseteq{C'}\subset\R.
\end{equation}
Moreover, $M_{n}$ is the minimal polynomial of degree $n$ on every compact set $C''$, for which $C\subseteq{C}''\subseteq{C'}$ holds.
\end{lemma}
\begin{proof}
Since $C$ is a real set, the minimal polynomial $M_n$ has only real coeffcients. By the alternation theorem, there are $n+1$ alternation points $x_0,x_1,\ldots,x_n\in{C}$, $x_0<x_1<\ldots<x_n$, with $M_n(x_j)=(-1)^{n-j}L_n$, $j=0,1,\ldots,n$. Thus $M_n$ has $n$ simple zeros $\xi_1<\xi_2<\ldots<\xi_n$ and
\[
x_0<\xi_1<x_1<\xi_2<x_2<\ldots<x_{n-1}<\xi_n<x_n.
\]
Thus, in each interval $(\xi_j,\xi_{j+1})$, $j=1,2,\ldots,n-1$, there is exactly one point $y_j\in(\xi_j,\xi_{j+1})$ with $M_n'(y_j)=0$ and $|M_n(y_j)|\geq{L}_n$, which gives that $\tilde{M}_{n}^{-1}([-1,1])$ is a subset of $\R$ and consists of a finite number (at most $n$) real intervals. The last assertion is an immediate consequence of the alternation theorem.
\end{proof}
With the help of Theorem\,\ref{Thm-LnCap} and Lemma\,\ref{Lemma-MinPol}, we are able to establish the main result.
\begin{theorem}\label{Thm-LB}
Let $C\subset\R$ be compact and infinite, then, for each $n\in\N$,
\begin{equation}\label{IneqLnCap3}
L_{n}(C)\geq2\,(\CAP{C})^{n},
\end{equation}
where equality is attained if there exists a polynomial $P_{n}\in\P_{n}$ such that\\ $P_{n}^{-1}([-1,1])=C$.
\end{theorem}
\begin{proof}
Let $C'$ be defined as in Lemma\,\ref{Lemma-MinPol}, then
\begin{align*}
L_{n}(C)&=L_{n}(C')\qquad\text{by~Lemma\,\ref{Lemma-MinPol}}\\
&=2\,(\CAP{C'})^{n}\qquad\text{by~Lemma\,\ref{Lemma-MinPol}~and~Theorem\,\ref{Thm-LnCap}}\\
&\geq2\,(\CAP{C})^{n}\qquad\text{by~\eqref{Ca-B}~and~\eqref{MonCap}}
\end{align*}
\end{proof}
\begin{remark}
Note that for each $n\in\N$, the constant $2$ in inequality \eqref{IneqLnCap3} is the \emph{best possible} constant which is independent of the set $C$.
\end{remark}
Of special interest is of course if $C$ is the union of several real intervals. For this reason, let us consider the case of $\ell$ intervals, $\ell\in\{2,3,4,\ldots\}$,
\begin{equation}\label{E}
E:=\bigcup_{j=1}^{\ell}[a_{2j},a_{2j-1}],
\end{equation}
where $a_{1},a_{2},\dots,a_{2\ell}$ are fixed with
\begin{equation}\label{aj}
-1=:a_{2\ell}<a_{2\ell-1}<\ldots<a_{2}<a_{1}:=1.
\end{equation}
A.Yu.\,Solynin\,\cite{Solynin} derived a very accurate lower bound for the logarithmic capacity of $E$:
\begin{theorem}[\cite{Solynin}]\label{Thm-Solynin}
Let $E$ be given in \eqref{E} and let
\begin{equation}\label{phipsi}
\varphi_{j}:=\arccos(a_{2j-1}),\quad\psi_{j}:=\arccos(a_{2j}),\quad j=1,2,\ldots,\ell,
\end{equation}
and let $\gamma_{j}$ and $\delta_{j}$ be characterised by
\begin{gather}
\gamma_{1}=0,\quad\gamma_{\ell}=\pi,\quad\varphi_{j}\leq\gamma_{j}\leq\psi_{j},\quad~j=2,3,\ldots,\ell-1,\label{gamma}\\
\psi_{j-1}\leq\delta_{j}\leq\varphi_{j},\quad~j=2,3,\ldots,\ell.\label{delta}
\end{gather}
Then
\begin{equation}\label{IneqCap1}
\CAP{E}\geq\frac{1}{2}\prod_{j=1}^{\ell-1}\Bigl(\sin\tfrac{(\psi_{j}-\gamma_{j})\pi}{2(\delta_{j+1}-\gamma_{j})}\Bigr)
^{\frac{2(\delta_{j+1}-\gamma_{j})^{2}}{\pi^{2}}}\cdot\Bigl(\sin\tfrac{(\gamma_{j+1}-\varphi_{j+1})\pi}{2(\gamma_{j+1}-\delta_{j+1})}\Bigr)
^{\frac{2(\gamma_{j+1}-\delta_{j+1})^{2}}{\pi^{2}}}.
\end{equation}
\end{theorem}
\begin{remark}\hfill{}
\begin{enumerate}
\item Note that the $\gamma_{j}$ and $\delta_{j}$ may be chosen arbitrarily subject to the inequalities \eqref{gamma} and \eqref{delta}. In order to get the best lower bound, one has to maximize the right hand side of \eqref{IneqCap1} over $\gamma_{j},\delta_{j}$, providing \eqref{gamma} and \eqref{delta}.
\item A possible choice for $\gamma_{j}$ and $\delta_{j}$ is
\begin{equation}
\begin{aligned}
\gamma_{j}&:=\tfrac{1}{2}(\varphi_{j}+\psi_{j}),\qquad j=2,3,\ldots,\ell-1,\\
\delta_{j}&:=\tfrac{1}{2}(\varphi_{j}+\psi_{j-1}),\qquad
j=2,3,\ldots,\ell,
\end{aligned}
\end{equation}
for which equality is attained in \eqref{IneqCap1} if $E$ has a special form (see \cite{Solynin}), and for which inequality \eqref{IneqCap1} reads as
\begin{equation}\label{IneqCap2}
\begin{aligned}
\CAP{E}\geq&\frac{1}{2}\prod_{j=1}^{\ell-1}
\Bigl(\sin\tfrac{(\psi_{j}-\varphi_{j})\pi}{2(\varphi_{j+1}-\varphi_{j})}\Bigr)
^{\frac{(\varphi_{j+1}-\varphi_{j})^{2}}{2\pi^{2}}}
\cdot\prod_{j=1}^{\ell-2}
\Bigl(\sin\tfrac{(\psi_{j+1}-\varphi_{j+1})\pi}{2(\psi_{j+1}-\psi_{j})}\Bigr)
^{\frac{(\psi_{j+1}-\psi_{j})^{2}}{2\pi^{2}}}\\
&\qquad\cdot\Bigl(\sin\tfrac{(\pi-\varphi_{\ell})\pi}{2\pi-\varphi_{\ell}+\psi_{\ell-1}}\Bigr)
^{\frac{(2\pi-\varphi_{\ell}+\psi_{\ell-1})^{2}}{2\pi^{2}}}
\end{aligned}
\end{equation}
\item In the simplest case of $\ell=2$ intervals, the capacity $\CAP{E}$ may be computed with the help of Jacobi's elliptic and theta functions, see \cite{Achieser-1930}. Numerical computations show that the lower bound \eqref{IneqCap1} (for optimal $\delta_{2}$) is very accurate and even the lower bound \eqref{IneqCap2} is very good. An upper bound in terms of elementary functions for the logarithmic capacity of two intervals is given by the author in \cite{Sch-2008}.
\item For the representation of the logarithmic capacity of three intervals with the help of theta functions, see \cite{FallieroSebbar-1999} and \cite{FallieroSebbar-2001}.
\item In \cite{PehSch-1999-1} and \cite{PehSch-1999-2}, polynomials $P_{n}\in\P_{n}$, for which $P_{n}^{-1}([-1,1])$ is the union of several intervals, are characterised by a polynomial system for the extremal points of $P_{n}$, see also \cite{Sch-2007}. With this polynomial system, we numerically computed inverse polynomial images $A$ consisting of $\ell=3,4$ and $5$ intervals, the minimum deviation $L_{n}(A)$ and, by \eqref{LnCap}, the capacity $\CAP{A}$, and compared $\CAP{A}$ with the lower bound \eqref{IneqCap1}. All numerical computations gave a relative error of less than $2\%$, which emphasizes the quality of the lower bound \eqref{IneqCap1}.
\item With inequality \eqref{IneqLnCap3}, together with \eqref{IneqCap1} or \eqref{IneqCap2}, one has an excellent lower bound for the minimum deviation of degree $n$ on $E$ in terms of elementary functions of the endpoints $a_{j}$.
\item A very good review of minimal polynomials on several intervals can be found in \cite{Peh-1997}.
\item Let us consider the set of all $E$ consisting of $\ell$ intervals, which are the inverse polynomial image of $[-1,1]$, i.e., for which there exists a polynomial $P_{n}\in\P_{n}$ such that $P_{n}^{-1}([-1,1])=E$ holds. Peherstorfer\,\cite{Peh-2001}, Totik\,\cite{Totik-2001} and Bogatyr{\"e}v\,\cite{Bogatyrev} proved that this set is \emph{dense} in the set of all $E$ consisting of $\ell$ intervals.
\item Let us take a brief look at how the inequality of Theorem\,\ref{Thm-LB} corresponds to the results of Achieser\,\cite{Achieser-1932},\,\cite{Achieser-1933} in the case of two intervals, i.e., $\ell=2$. For the sequence
\begin{equation}\label{Sequence}
\Bigl(\frac{L_{k}(E)}{(\CAP{E})^k}\Bigr)_{k=1}^{\infty}
\end{equation}
he proved the following:
\begin{enumerate}
\item If there exists a polynomial $P_{n}\in\P_{n}$, such that $P_{n}^{-1}([-1,1])=E$, then the sequence \eqref{Sequence} has a finite number of accumulation points from which the smallest one is equal to $2$.
\item If there is no polynomial $P_{n}\in\P_{n}$, such that $P_{n}^{-1}([-1,1])=E$, then the accumulation points of the sequence \eqref{Sequence} fill out an entire interval of which the left bound is equal to $2$.
\end{enumerate}
\end{enumerate}
\end{remark}
Deriving an upper bound for $L_n(E)$ is more complicated and has been achieved recently by Totik in \cite{Totik-2009}. Since we give an analogous result for several arcs of the unit circle (see Theorem\,\ref{Thm-UB}), we state here the result.
\begin{theorem}[Totik\,\cite{Totik-2009}]\label{Thm-Totik}
Let $E$ be given in \eqref{E}, then there exists a constant $K$ depending only on $E$ such that for each $n\in\N$
\begin{equation}\label{UpperBound-Totik}
L_{n}(E)\leq{K}(\CAP{E})^{n}.
\end{equation}
\end{theorem}
\section{Compact Subsets of the Unit Circle}
In this section, we give analogous bounds for the minimum deviation of the minimal polynomial on a compact subset of the unit circle, lying symmetrically with respect to the real axis.\\
Let $C\subseteq[-1,1]$ be a compact infinite set and let
\begin{equation}\label{Gamma}
\Gamma:=\bigl\{z\in\C:|z|=1,\re{z}\in{C}\bigr\},
\end{equation}
i.e., the set $C$ is the projection of $\Gamma$ onto the real axis. Consider the mapping $x=\tfrac{1}{2}(z+\tfrac{1}{z})$, $z\in\C$, $|z|=1$, $x\in[-1,1]$, for which
\begin{equation}\label{Tn}
T_{n}(x)=\frac{1}{2}\bigl(z^n+\frac{1}{z^n}\bigr)=\re(z^{n})
\end{equation}
holds for the classical Chebyshev polynomial $T_{n}$. Obviously
\begin{equation}\label{CiffGamma}
x\in{C} \iff z\in\Gamma.
\end{equation}
First, let us give a result of Robinson\,\cite{Robinson} concerning an identity between the capacities of $C$ and $\Gamma$.
\begin{theorem}[Robinson\,\cite{Robinson}]\label{Thm-Robinson}
Let $C$ and $\Gamma$ as above, then
\begin{equation}
\CAP\Gamma=\sqrt{2\,\CAP{C}}.
\end{equation}
\end{theorem}
\begin{theorem}
Let $C$ and $\Gamma$ as above and let $n\in\N$. Let $b_{n}:=1$, let
\[
P_{n}(z):=b_{n}z^n+b_{n-1}z^{n-1}+\ldots+b_1z+b_0
\]
be the minimal polynomial of degree $n$ on $\Gamma$ and let $k^*\in\{0,1,\ldots,n\}$ be defined by $k^*:=\min\{k:b_{k}\neq0\}$. If $k^*\neq{n}$, i.e., if $P_{n}(z)\neq{z}^n$, then
\begin{equation}
L_{n}(\Gamma)\geq\sqrt{2|b_{k^*}|}\,(\CAP\Gamma)^{n-k^*}.
\end{equation}
\end{theorem}
\begin{proof}
Since $\Gamma$ is symmetric with respect to the real axis, the coefficients $b_k$ have to be real. Then the square modulus of $P_n(z)$ may be written in the form ($z\in\Gamma$, $x\in{C}$)
\[
|P_n(z)|^2=A_0+2\sum_{\ell=1}^{n}A_{\ell}T_{\ell}(x)\qquad\text{where}\quad
A_{\ell}=\sum_{k=0}^{n-\ell}b_{k}b_{k+\ell}.
\]
By the definition of $k^*$,
\[
|P_n(z)|^2=A_0+2\sum_{\ell=1}^{n-k^*}A_{\ell}T_{\ell}(x)
\]
and $A_{n-k^*}=b_{n}b_{k^*}=b_{k^*}\neq0$. Thus
\begin{align*}
L_{n}(\Gamma)&=\max_{z\in\Gamma}|P_{n}(z)|\\
&=\max_{x\in{C}}\sqrt{\left|A_0+2\sum_{\ell=1}^{n-k^*}A_{\ell}T_{\ell}(x)\right|}\\
&=\sqrt{\max_{x\in{C}}\left|A_0+2\sum_{\ell=1}^{n-k^*}A_{\ell}T_{\ell}(x)\right|}\\
&\geq\sqrt{2^{n-k^*}|b_{k^*}|L_{n-k^*}(C)}\\
&\geq\sqrt{2^{n-k^*+1}|b_{k^*}|(\CAP{C})^{n-k^*}}\qquad\text{by~Theorem\,\ref{Thm-LB}}\\
&=\sqrt{2|b_{k^*}|}\,(\CAP\Gamma)^{n-k^*}\qquad\text{by~Theorem\,\ref{Thm-Robinson}}
\end{align*}
\end{proof}
\begin{theorem}\label{Thm-UB}
Let $E$ be given in \eqref{E} and define $\Lambda:=\{z\in\C:|z|=1,\re{z}\in{E}\}$, i.e., the set $E$ is the projection of $\Lambda$ onto the real axis. Then there exists a constant $B$ depending only on $\Lambda$ such that for each $n\in\N$
\begin{equation}
L_{n}(\Lambda)\leq{B}\,(\CAP\Lambda)^{n}.
\end{equation}
\end{theorem}
\begin{proof}
For the proof, we consider the cases $n$ even and $n$ odd. First let $n$ be even, say $n=2m$. Let
\[
M_{m}(x):=\sum_{k=0}^{m}b_{k}T_{k}(x)=x^{m}+\ldots,
\]
where $b_0,b_1,\ldots,b_{m-1}\in\R$, $b_m=1/2^{m-1}$, be the minimal polynomial on $E$ with corresponding minimum deviation $L_m(E)$. Further, define
\[
P_{2m}(z):=2^mz^m\sum_{k=0}^{m}\tfrac{b_k}{2}(z^k+\tfrac{1}{z^k})=z^{2m}+\ldots
\]
Then
\begin{align*}
L_{2m}(\Lambda)&\leq\max_{z\in\Lambda}|P_{2m}(z)|\\
&=2^m\max_{z\in\Lambda}|\sum_{k=0}^{m}\tfrac{b_k}{2}(z^k+\tfrac{1}{z^k})|
\qquad\text{by~definition}\\
&=2^m\max_{x\in{E}}|\sum_{k=0}^{m}b_{k}T_{k}(x)|
\qquad\text{by~\eqref{Tn}}\\
&=2^mL_{m}(E)\qquad\text{by~definition}\\
&\leq2^{m}K(\CAP{E})^m\qquad\text{by~Theorem\,\ref{Thm-Totik}}\\
&=K(\CAP\Lambda)^{2m}\qquad\text{by~Theorem\,\ref{Thm-Robinson}}
\end{align*}
Let $n$ be odd, say $n=2m+1$. Let $M_{m}$ be defined as above and define
\[
P_{2m+1}(z):=2^mz^{m+1}\sum_{k=0}^{m}\tfrac{b_k}{2}(z^k+\tfrac{1}{z^k})=z^{2m+1}+\ldots
\]
Then, by the same procedure as in the case $n$ even, we get
\[
L_{2m+1}(\Gamma)\leq{K}(\CAP\Gamma)^{2m}=\tfrac{K}{\CAP\Gamma}\,(\CAP\Gamma)^{2m+1}.
\]
Since $\CAP\Gamma\leq1$, for the constant $B$, we take $B=\frac{K}{\CAP\Gamma}$, and the theorem is proved.
\end{proof}
In conclusion, let us note that the connection of minimal polynomials on several intervals and on the corresponding arcs of the unit circle was investigated in \cite{ThiranDetaille} and \cite{PehSch-2002}.
{\bf Acknowledgment.} The author would like to thank Vilmos Totik and Barry Simon for valuable comments on earlier versions of this paper.
\bibliographystyle{amsplain}
|
1,116,691,497,594 | arxiv | \section{\large{\bf Introduction}}
SUSY hybrid inflation \cite{Dvali:1994ms,Copeland:1994vg,Linde:1993cn,Dvali:1997uq} provides fascinating framework to realize inflation within the grand unified theories (GUTs) of particle physics. Several GUT models such as, $SU(5)$ \cite{Georgi:1974sy}, Flipped $SU(5)$ \cite{Barr:1981qv,Antoniadis:1987dx,Rehman:2018nsn} and the Pati-Salam symmetry $SU(4)_C \times SU(2)_L \times SU(2)_R$ \cite{Pati:1974yy,Mohapatra:1974gc,Ahmed:2018jlv}, have been employed successfully to realize standard, shifted and smooth variants of hybrid inflation \cite{Ahmed:2022vlc,Rehman:2012gd,Rehman:2014rpa,Khalil:2010cp,urRehman:2006hu,Ahmed:2022wed}. The $SU(5) \times U(1)_{\chi}$ gauge symmetry is another suitable choice as a GUT model due to its various attractive features \cite{Kibble:1982ae,apal:2019}. The whole gauge group of the model is embedded in $SU(5) \times U(1)_{\chi} \subset SO(10)$, owing to special $U(1)_{\chi}$ charge assignment \cite{Kibble:1982ae}. In contrast to the $SU(5)$ \cite{Georgi:1974sy} model, a discrete $Z_2$ symmetry that avoids rapid proton decay, naturally arises after the breaking of $U(1)_{\chi}$ factor. This $Z_2$ symmetry not only serves as the Minimal Supersymmetric Standard Model (MSSM) matter parity, but also ensures the existence of a stable lightest supersymmetric particle (LSP) which can be a viable cold dark matter candidate. Furthermore, the right-handed neutrino mass is naturally generated by the breaking of $U(1)_{\chi}$ symmetry after one of the fields carrying $U(1)_{\chi}$ charge acquires a Vacuum Expectation Value (VEV) at some intermediate scale. The well-known advantages of $U(1)_{\chi}$ symmetry include seesaw physics to explain neutrino oscillations, and baryogenesis via leptogenesis \cite{Fukugita:1986hr,Lazarides:1990huy}.
The breaking of $SU(5)$ part of the gauge symmetry leads to copious production of magnetic monopoles \cite{Hill:1982iq} in conflict with the cosmological observations whereas, the breaking of $U(1)_{\chi}$ factor yields topologically stable cosmic strings \cite{Kibble:1982ae,Vilenkin:1984ib,Vilenkin:2000jqa}. The cosmic strings can be made to survive if $U(1)_{\chi}$ breaks after the end of inflation. In order to avoid the undesired monopoles, the shifted or smooth variant of hybrid inflation \cite{Khalil:2010cp} can be employed, where the gauge symmetry is broken during inflation and disastrous monopoles are inflated away. In the simplest SUSY hybrid inflationary scenario the potential along the inflationary track is completely flat at tree level. The inclusion of radiative corrections (RC) to the scalar potential provide necessary slope needed to drive inflaton towards the SUSY vacuum and in doing so the gauge symmetry $G$ breaks spontaneously to its subgroup $H$.
In this paper, we implement shifted hybrid inflation scenario in the $SU(5) \times U(1)_{\chi}$ GUT model \cite{apal:2019} where the $SU(5)$ symmetry is broken during inflation and the $U(1)_{\chi}$ symmetry radiatevely breaks to its $Z_2$ subgroup at some intermediate scale. The scalar spectral index $n_s$ lies in the observed range of Planck's results \cite{Planck:2018jri} provided the inflationary potential incorporates either the soft supersymmetry (SUSY) breaking terms~\cite{rehman,gravitywaves,Ahmed:2022rwy,Afzal:2022vjx,Ahmed:2021dvo,Ahmed:2022rwy}, or higher-order terms in the K\"ahlar potential \cite{urRehman:2006hu,bastero}. Without these terms, the scalar spectral index $n_s$ lies close to 0.98 which is acceptable only if the effective number of light neutrino species are slightly greater than 3 \cite{Ade:2015lrj}. We show that, by taking soft SUSY contribution into account along with the supergravity (SUGRA) corrections in a minimal K\"ahlar potential setup, the predictions of the model are consistent with the Planck’s latest bounds on scalar spectral index $n_s$ \cite{Ade:2015lrj}, although the values of tensor to scalar ratio remain small. By employing non-minimal K\"ahler potential, large tensor modes are easily obtained, approaching observable values potentially measurable by near-future experiments such as, PRISM \cite{Andre:2013afa}, LiteBird \cite{Matsumura:2013aja}, CORE \cite{Finelli:2016cyd}, PIXIE \cite{Kogut:2011xw}, CMB-S4 \cite{Abazajian:2019eic}, CMB-HD \cite{Sehgal:2019ewc} and PICO \cite{SimonsObservatory:2018koc}. Moreover, the $U(1)_{\chi}$ symmetry radiatively breaks after the end of inflation at an intermediate scale, yielding topologically stable cosmic strings. The Planck's bound \cite{Ade:2013xla,Ade:2015xua} on the strength of gravitational interaction of the strings, $G_N \mu_s$ are easily satisfied with the $U(1)_{\chi}$ symmetry breaking scale obtained in the model, which depends on the initial boundary conditions at the GUT scale. Furthermore, the Super-Kamiokande bounds \cite{Super-Kamiokande:2016exg} on dimension-5 proton decay lifetime are easily satisfied for SUSY breaking scale $M_{\text{SUSY}} \gtrsim 12.5$ TeV.
The rest of the paper is organised as follows. Sec. \ref{sec2} provides the description of the $SU(5) \times U(1)_{\chi}$ model. The implementation of shifted hybrid inflation including the mass spectrum of the model, gauge coupling unification and dimension-5 proton decay is discussed in Sec. \ref{sec3}. The results and inflationary predictions of the model with minimal K\"ahler potential are presented in Sec. \ref{sec5} and with non-minimal K\"ahler potential in Sec. \ref{sec6}. The radiative breaking of $U(1)_{\chi}$ symmetry and cosmic strings is discussed in Sec. \ref{sec7}. Finally we summarize our results in Sec. \ref{sec8}.
\section{\large{The $U(1)_R$ Symmetric $SU(5) \times U(1)_{\chi}$ Model}} \label{sec2}%
The $\mathbf{10}$, $\bar{\mathbf{5}}$ and $\mathbf{1}$ dimensional representations of the group $SU(5)$ constitute the $\mathbf{16}$ (spinorial) representation of $SO(10)$ and contains the MSSM matter superfields. Their decomposition with respect to the MSSM gauge symmetry is
\begin{equation}
\label{Mspectrum}
\begin{split}
F_i &\equiv (\mathbf{10}, -1) = Q(\mathbf{3}, \mathbf{2}, 1/6) + u^c(\bar{\mathbf{3}}, \mathbf{1}, -2/3) + e^c (\mathbf{1}, \mathbf{1}, 1)~, \\
\bar{f}_i &\equiv (\bar{\mathbf{5}}, +3) = d^c (\bar{\mathbf{3}}, \mathbf{1}, 1/3) + \ell(\mathbf{1}, \mathbf{2}, -1/2)~, \\
\nu_{i}^{c} &\equiv (\mathbf{1}, -5) = \nu^c(\mathbf{1}, \mathbf{1}, 0)~,
\end{split}
\end{equation}
where $i = 1, 2, 3$ denotes the generation index. The scalar sector of $SU(5) \times U(1)_{\chi}$ consists of the following superfields: a pair of Higgs fiveplets, $h\,\equiv (\mathbf{5}, 2)$, $\bar{h}\,\equiv (\bar{\mathbf{5}}, -2)$, containing the electroweak Higgs doublets ($h_d, h_u$) and color Higgs triplets ($D_h,\bar{D}_{\bar{h}}$); a Higgs superfield $\Phi$ that belongs to the adjoint representation ($\Phi\, \equiv 24_{0}$) and responsible for breaking $SU(5)$ gauge symmetry to MSSM gauge group; a pair of superfields ($\chi$, $\bar{\chi}$) which trigger the breaking of $U(1)_{\chi}$ into a $Z_2$ symmetry which is realized as the MSSM matter parity; and finally, a gauge singlet superfield $S$ whose scalar component acts as an inflaton.
The decomposition of the above $SU(5)$ representations under the MSSM gauge group is
\begin{align}
\label{Hspectrum}
\begin{split}
\Phi \equiv {}& (\mathbf{24}, 0) = \Phi_{24}(\mathbf{1}, \mathbf{1}, 0) + W_H (\mathbf{1}, \mathbf{3}, 0) + G_H (\mathbf{8}, \mathbf{1}, 0) \\
& \qquad \quad+ Q_H(\mathbf{3}, \mathbf{2}, -5/6) + \bar{Q}_H(\mathbf{3}, \mathbf{2}, 5/6), \\
h \equiv {}& (\mathbf{5}, 2) = D_h(\mathbf{3}, \mathbf{1}, -1/3) + h_u (\mathbf{1}, \mathbf{2}, 1/2)~, \\
\bar{h} \equiv {}& (\bar{\mathbf{5}}, -2) = \bar{D}_{\bar{h}}(\bar{\mathbf{3}}, \mathbf{1}, 1/3) + h_d(\mathbf{1}, \mathbf{2}, -1/2), \\
\chi \equiv {}& (\mathbf{1}, 10), \quad \bar{\chi} \equiv (\mathbf{1}, -10), \quad S \equiv (\mathbf{1}, 0),
\end{split}
\end{align}
where the singlets ($\chi$, $\bar{\chi}$) originate from the decomposition of $\mathbf{126}$ representation of $SO(10)$
\begin{equation}
\mathbf{126} = (\mathbf{1}, -10) + (\bar{\mathbf{5}}, -2) + (\mathbf{10}, -6) + (\bar{\mathbf{15}}, 6) + (\mathbf{45}, 2) + (\bar{\mathbf{50}}, -2).
\end{equation}
\begin{table}[t]
\setlength\extrarowheight{5pt}
\centering
\begin{tabular}{c c c}
\hline \hline \rowcolor{Gray}
\multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} \\ \rowcolor{Gray}
\multicolumn{1}{c}{\multirow{-2}{*}{Superfields}} & \multicolumn{1}{c}{\multirow{-2}{*}{\begin{tabular}[c]{@{}c@{}}Representations under\\ $SU(5) \times U(1)_{\chi}$\end{tabular}}} & \multicolumn{1}{c}{\multirow{-2}{*}{\begin{tabular}[c]{@{}c@{}}Global\\ $U(1)_R $\end{tabular}}} \\ \hline
\rowcolor{Gray2} \multicolumn{3}{c}{Matter sector} \\ \hline
\multicolumn{1}{c}{$F_i$} & $\left( \mathbf{10}, -1 \right)$ & $3/10$ \\
\multicolumn{1}{c}{$\bar{f}_i$} & $\left( \bar{\mathbf{5}}, 3 \right)$ & $1/10$\\
\multicolumn{1}{c}{$\nu_{i}^c$} & $\left( \mathbf{1}, -5 \right)$ & $1/2$\\ \hline
\rowcolor{Gray2} \multicolumn{3}{c}{Scalar sector} \\ \hline
\multicolumn{1}{c}{$\Phi$} & $\left( \mathbf{24}, 0 \right)$ & $0$\\
\multicolumn{1}{c}{$h$} & $\left( \mathbf{5}, 2 \right)$ & $2/5$\\
\multicolumn{1}{c}{$\bar{h}$} & $\left( \bar{\mathbf{5}}, -2 \right)$ & $3/5$\\
\multicolumn{1}{c}{$\chi$} & $\left( \mathbf{1}, 10 \right)$ & $0$\\
\multicolumn{1}{c}{$\bar{\chi}$} & $\left( \mathbf{1}, -10 \right)$ & $0$\\
\multicolumn{1}{c}{$S$} & $\left( \mathbf{1}, 0 \right)$ & $1$\\ \hline \hline
\end{tabular}
\caption{The representations of matter and scalar superfields under $SU(5) \times U(1)_{\chi}$ gauge symmetry and global $U(1)_R$ symmetry in shifted hybrid inflation model.}
\label{tab:field_charges}
\end{table}
Following \cite{Khalil:2010cp}, the $U(1)_R$ charge assignment of the superfields is given in Table \ref{tab:field_charges} along with their transformation properties.
The $SU(5) \times U(1)_{\chi}$ and $U(1)_R$, symmetric superpotential of the model with the leading-order non-renormalizable terms is given by
\begin{eqnarray}%
W &=& S\left[\kappa M^2-\kappa \Tr(\Phi^{2})-\frac{\beta}{m_P}
\Tr(\Phi^3) + \sigma_{\chi} \chi \bar{\chi}\right] + \gamma \bar{h} \Phi h + \delta \bar{h} h \nonumber \\
&+& y_{ij}^{(u)}\,F_i\,F_j\,h + y_{ij}^{(d,e)}\,F_i\,\bar{f}_j\,\bar{h}
+y_{ij}^{(\nu)}\,\nu_{i}^c\,\bar{f}_j\,h + \lambda_{ij} \chi \nu_{i}^{c} \nu_{j}^{c} , \label{superpotential}
\end{eqnarray} %
where $M$ is a superheavy mass and $m_P = 2.43 \times 10^{18}$ GeV is the reduced Planck mass. The terms in square bracket in the first line are relevant for shifted hybrid inflation while, the last two terms are involved in the solution of doublet-triplet splitting problem, as discussed in section \ref{sec4}.
The Yukawa couplings $y_{ij}^{(u)}$, $ y_{ij}^{(d,e)}$, $y_{ij}^{(\nu)}$ in the second line of \eqref{superpotential} generate Dirac masses for quarks and leptons after the electroweak symmetry breaking, whereas $m_{\nu_{ij}} = \lambda_{ij} \langle \chi \rangle$ is the right-handed neutrino mass matrix, generated after $\chi$ acquires a VEV through radiative breaking of $U(1)_{\chi}$ symmetry, as discussed in Sec. \ref{sec7}.
The superpotential $W$ exhibits a number of interesting features as a consequence of global $U(1)_{R}$ symmetry. First, it allows only linear terms in $S$ in the superpotential, omitting the higher order ones, such as $S^2$ which could generate an inflaton mass of Hubble size, invalidating the inflationary scenario. Second, the $U(1)_{R}$ symmetry naturally avoids the so called $\eta$ problem \cite{Linde:1997sj}, that appears when SUGRA corrections are included. Finally, several dangerous dimension-5 proton decay operators are highly suppressed.
\section{\large{\bf Shifted Hybrid $SU(5) \times U(1)_{\chi}$ Inflation}}\label{sec3}
In this section, the effective scalar potential is computed considering contributions from the $F$- and $D$-term sectors. The superpotential terms relevant for shifted hybrid inflation are
\begin{eqnarray}%
W &\supset& S\left[\kappa M^2-\kappa \Tr(\Phi^{2})-\frac{\beta}{m_P}
\Tr(\Phi^3)\right] + \gamma \bar{h} \Phi h + \delta \bar{h} h \nonumber \\
&+& \sigma_{\chi} S \chi \bar{\chi} +\lambda_{ij} {\chi} \nu_{i}^{c} \nu_{j}^{c}~. \label{superpotential_inflation}
\end{eqnarray} %
In component form, the above superpotential is expanded as follows,%
\begin{eqnarray}%
W \supset S\left[\kappa M^2-\kappa\frac{1}{2}
\sum_{i}\phi_{i}^{2}-\frac{\beta}{4 m_P}d_{ijk}\phi_{i}\phi_{j}
\phi_{k}\right] &+& \delta \bar{h}_{a}h_{a} + \gamma T_{a b}^{i}\phi^{i}\bar{h}_{a} h_{b} \nonumber \\
&+& \sigma_{\chi} S \chi \bar{\chi} +\lambda_{ij} {\chi} \nu_{i}^{c}\nu_{j}^{c},
\label{superpot-shift}%
\end{eqnarray} %
where $\Phi = \phi_i T^i$ with Tr$[T_i T_j] = \frac{1}{2}\delta_{ij}$ and $d_{ijk} = 2$Tr$[T_i\{T_j,T_k\}]$ in the $SU(5)$ adjoint basis. The $F$-term scalar potential obtained from the above superpotential is given by %
\begin{eqnarray}%
V_F &=& \left| \; \kappa M^2-\kappa\frac{1}{2}
\sum_{i}\phi_{i}^{2}-\frac{\beta}{4 m_P}d_{ijk}\phi_{i}\phi_{j}
\phi_{k}+ \sigma_{\chi} \chi \bar{\chi} \; \right|^{2} \nonumber \\
&+& \sum_{i}\left|\kappa S \phi_{i}+\frac{3
\beta}{4 m_P}d_{ijk} S \phi_{j} \phi_{k}
- \gamma T_{a
b}^{i}\bar{h_{a}}h_{b}\right|^{2} \nonumber \\
&+& \sum_{b}\left|\gamma
T_{a b}^{i}\phi^{i}\bar{h_{a}}+\delta
\bar{h_{b}}\right|^{2}+\sum_{b}\left|\gamma T_{a
b}^{i}\phi^{i}h_{a}+\delta
h_{b}\right|^{2} \nonumber\\
&+& \left| \, \sigma_{\chi} S \bar{\chi} + \lambda_{ij} \nu_{i}^{c}\nu_{j}^{c}\, \right|^2 + \left| \, \sigma_{\chi} S \chi \, \right|^2 + \left|2 \lambda_{ij} {\chi} \nu_{i}^{c}\right|^2 ,
\label{scalarpot-shift}
\end{eqnarray}%
where the scalar components of the superfields are denoted by the same symbols as the corresponding superfields. The VEV's of the fields at the global SUSY minimum of the above potential are given by,
\begin{gather}
S^0 = h_{a}^0 = \bar{h_{a}^0} = \nu_{i}^{c\,0}=0, \;\; \chi^0 = \bar{\chi}^0 = 0
\label{gmin}
\end{gather}
with $\phi_i^0$ satisfying the following equation: %
\begin{equation}%
\sum_{i=1}^{24}(\phi_{i}^{0})^{2} + \frac{\beta}{2 \kappa m_P}
d_{ijk}
\phi^0_i \phi^0_j \phi^0_k =2M^{2}.
\end{equation}%
The superscript `0' denotes the field value at its global minimum. The superfield pair ($\chi, \bar{\chi}$) break $U(1)_{\chi}$ to $Z_2$, the matter parity. This symmetry ensures the existence of a lightest supersymmetric particle (LSP) which could play the role of cold dark matter. Further, as discussed in~\cite{apal:2019}, this $Z_2$ symmetry yields topologically stable cosmic strings.
Using $SU(5)$ symmetry transformation the VEV matrix $\Phi^0 = \phi_i^0 T^i$ can be aligned in the $24$-direction,
\begin{equation}
\Phi_{24}^0 = \frac{\phi_{24}^0}{\sqrt{15}} \left( 1, 1, 1, - 3/2, - 3/2 \right).
\end{equation}
Thus the $SU(5)$ gauge symmetry is broken down to Standard Model gauge group $G_{\text{SM}}$ by the non-vanishing VEV of $\phi_{24}^0$ which is a singlet under $G_{\text{SM}}$.
The $D$-term scalar potential,
\begin{eqnarray} %
V_D &=& \frac{g_{5}^2}{2} \sum_{i} \left( f^{ijk} \phi_j \phi_{k}^{\dagger} + T^i \left( \left| h_a \right|^2 - \left| \bar{h}_a \right|^2 \right) \right)^2 \nonumber \\
&+& \frac{g_{\chi}^2}{2} \left( q_{\chi} \left| \chi \right|^2 + q_{\bar{\chi}} \left| \bar{\chi} \right|^2 + \left( q_{\bar{\chi}} + q_{\chi} \right) \varsigma \right)^2,
\label{dterm}
\end{eqnarray} %
also vanishes for this choice of the VEV (since $f^{i, 24, 24} = 0$) and for $\vert \bar{h}_a \vert = \vert h_a \vert$, $\vert \bar{\chi} \vert = \vert \chi \vert$.
The scalar potential in Eq.~(\ref{scalarpot-shift}) can be written in terms of the dimensionless variables%
\begin{equation} %
z = \frac{|S|}{M}~, ~~~~~~~~~~~~
y = \frac{\phi_{24}}{M\sqrt{2}},~~~~~~~~~~~~ %
\end{equation} %
as follows,
\begin{equation} %
\tilde{V} = \frac{V}{\kappa^2 M^4} = \left( 1-y^2+\alpha y^3\right)^2 + 2 z^2 y^2\left(1-3\alpha y/2\right)^2 ~,%
\label{VF0}
\end{equation} %
\begin{figure}[t]
\centering
\begin{subfigure}[b]{0.495\textwidth}
\centering
\caption{Standard, $\alpha = 0$}
\includegraphics[width=8cm]{plots/potential_a_0.pdf}
\label{fig:alpha_0}
\end{subfigure}
\begin{subfigure}[b]{0.495\textwidth}
\caption{Shifted, $\alpha = 0.25$}
\centering \includegraphics[width=8cm]{plots/potential_a_25.pdf}
\label{fig:alpha_025}
\end{subfigure}
\begin{subfigure}[b]{0.495\textwidth}
\caption{Shifted, $\alpha = 0.3$}
\centering \includegraphics[width=8cm]{plots/potential_a_30.pdf}
\label{fig:alpha_03}
\end{subfigure}
\caption{The tree-level, global dimensionless scalar potential $\tilde{V}=V/\kappa^2M^4$ versus $y$ and $z$ for; $\alpha = 0$ (a), $\alpha = 0.25$ (b) and $\alpha = 0.3$ (c). The standard track corresponds to $\alpha = 0$ whereas, $\alpha \neq 0$ corresponds to two shifted trajectories. The inflationary track feasible for realizing successful inflation is shown in panel (b) for $\alpha = 0.3$.}
\label{fig1}
\end{figure}
where $\alpha = \beta M/ \sqrt{30}\, \kappa m_P$. This dimensionless potential exhibits the following three extrema
\begin{equation}
y_1=0, \label{1st minima}
\end{equation}
\begin{equation}
y_2=\frac{2}{3 \alpha }, \label{2nd minima}
\end{equation}
\begin{eqnarray}
y_3 &=& \frac{1}{3\alpha} + \frac{1}{3 \sqrt[3]{2} \alpha }\Bigg( \sqrt[3]{2 -27 \alpha ^2+\sqrt{\left(2-27 \alpha ^2\right)^2+4 \left(9 \alpha ^2 z^2-1\right)^3}} \nonumber\\
&-& \sqrt[3]{-2 + 27 \alpha ^2+\sqrt{\left(2-27 \alpha ^2\right)^2+4 \left(9 \alpha ^2 z^2-1\right)^3}} \Bigg).
\end{eqnarray}
for a constant value of $z$ and is displayed in Fig. \ref{fig1} for different values of $\alpha$. The first extremum $y_1$ with $\alpha = 0$ corresponds to the standard hybrid inflation for which $y = 0, z > 1$ is the only inflationary trajectory that evolves at $z = 0$ into the global SUSY minimum at $y = \pm 1$ (Fig. \ref{fig:alpha_0}). For $\alpha \neq 0$, a shifted trajectory appears at $y = y_2$, in addition to the standard trajectory at $y = y_1 =0$, which is a local maximum (minimum) for $z < \sqrt{4/27\alpha^2-1}$ ($z > \sqrt{4/27\alpha^2-1}$). For $\alpha < \sqrt{2/27}\simeq 0.27$, this shifted trajectory lies higher than the standard trajectory (Fig. \ref{fig:alpha_025}). In order to have suitable initial conditions for realizing inflation along the shifted track, we assume $\alpha > \sqrt{2/27}$, for which the shifted trajectory lies lower than the standard trajectory (Fig. \ref{fig:alpha_03}). Moreover, to ensure that the shifted inflationary trajectory at $y_2$ can be realized before $z$ reaches zero, we require $\alpha < \sqrt{4/27} \simeq 0.38$. Thus, for $0.27 < \alpha < 0.38$, while the inflationary dynamics along the shifted track remain the same as for the standard track, the $SU(5)$ gauge symmetry is broken during inflation, hence alleviating the magnetic monopole problem. As the inflaton slowly rolls down the inflationary valley and enters waterfall regime at $z =\sqrt{4/27\alpha^2-1}$, its fast rolling ends inflation, and the system starts oscillating about the vacuum at $z= 0$ and $y=y_3$. In order to calculate one-loop radiative correction along $y_2$, we need to compute the mass spectrum of the model along this track where both $SU(5)$ gauge symmetry and SUSY are broken.
During inflation, the field $\Phi$ acquires a VEV in the $\phi_{24}$ direction which breaks the $SU(5)$ gauge symmetry down to SM gauge group $G_{SM}$ while, the $U(1)_{\chi}$ symmetry remains unbroken. The potential in Eq. \eqref{scalarpot-shift} generates the following masses for: 2 real scalars
\begin{equation}
m_{24_\pm}^2 = \pm \kappa^2 M_{\alpha}^2 + \kappa^2 \vert S \vert^2 ,
\end{equation}
22 real scalars
\begin{table}[t]
\setlength\extrarowheight{2pt}
\addtolength{\tabcolsep}{7pt}
\centering
\begin{tabular}{lr}
\hline\hline \rowcolor{Gray} {\bf Fields} & {\bf Squared Masses} \\
\hline 2 Majorana fermions & $\sigma_{\chi}^2 \vert S\vert^2$ \\
4 real and pseudo scalars & $\sigma_{\chi}^2 \vert S\vert^2 \pm \kappa \sigma_{\chi} M_{\alpha}^2$ \\
2 real scalars & $\kappa^2 \vert S\vert^2 \pm \kappa^2 M_{\alpha}^2$ \\
1 Majorana fermion & $\kappa^2 \vert S\vert^2 $ \\
22 real scalars & $25 \kappa^2 \vert S\vert^2 \pm 5\kappa^2 M_{\alpha}^2$ \\
11 Majorana fermions & $25 \kappa^2 \vert S\vert^2 $ \\
12 real scalars & $ \frac{25}{30} g_5^2 v_2^2 $ \\
12 Dirac fermions & $\frac{25}{30} g_5^2 v_2^2 $ \\
12 gauge bosons & $\frac{25}{30} g_5^2 v_2^2 $ \\
\hline
\end{tabular}%
\caption{The mass spectrum of the shifted hybrid $SU(5) \times U(1)_{\chi}$ model along the inflationary trajectory $y_2$.} \label{mass_spectrum_tab}
\end{table}
\begin{equation}
m_{i_\pm}^2 = \pm 5 \kappa^2 M_{\alpha}^2 + 25 \kappa^2 \vert S \vert^2 , \quad i = 1, \dots , 8, 21, 22, 23 ,
\end{equation}
and 4 real and pseudo scalars
\begin{equation}
m_{(\chi, \bar{\chi})_\pm}^2 = \pm \kappa \sigma_{\chi} M_{\alpha}^2 + \sigma_{\chi}^2 \vert S \vert^2,
\end{equation}
where $M_{\alpha}^2 = M^2 \left( \frac{4}{27 \alpha^2} - 1\right)$. The superpotential \eqref{superpot-shift} generates: a Majorana fermion with mass squared,
\begin{equation}
m_{24}^2 = \kappa^2 \vert S \vert^2 ,
\end{equation}
11 Majorana fermions with mass squared,
\begin{equation}
m_{i}^2 = 25 \kappa^2 \vert S \vert^2 , \quad i = 1, \dots , 8, 21, 22, 23,
\end{equation}
and 2 Majorana fermions with mass squared,
\begin{equation}
m_{\chi, \bar{\chi}}^2 = \sigma_{\chi}^2 \vert S \vert^2 .
\end{equation}
The scalar fields $\phi_i$ ($i = 9, ...., 20$) obtain a universal mass squared $\frac{25}{30} g_5^2 \upsilon_2^2$, from the $D$-term in Eq. \eqref{dterm}, while the mixing between chiral fermions $\psi_i$ ($i = 9, ...., 20$) and gauginos $\lambda_i$ ($i = 9, ...., 20$) yields 12 Dirac fermions with a mass squared $\frac{25}{30} g_5^2 \upsilon_2^2$. Finally the 12 guage bosons $A_{\mu}^i$ ($i = 9, ...., 20$) obtain a universal mass squared $\frac{25}{30} g_5^2 \upsilon_2^2$. The mass spectrum of the above model, along the shifted inflationary track, is summarized in Table \ref{mass_spectrum_tab}.
The 1-loop radiative correction to the inflationary effective potential is given by %
\begin{eqnarray} %
V_{\rm 1 loop} \!\!&\!=\!&\!\! \kappa^2 M_{\alpha}^4 \left(
\frac{\kappa^2}{16 \pi^2} \left[ F(M_{\alpha}^2,x^2)
+11\times 25\,F(5 M_{\alpha}^2,5\,x^2) \right] + \frac{\sigma_{\chi}^2}{8 \pi^2} F(M_{\alpha}^2,y^2) \right),%
\label{Vloop}
\end{eqnarray}%
with
\begin{equation} %
F(M_{\alpha}^2,x^2) =
\frac{1}{4}\left( \left( x^4+1\right) \ln \frac{\left( x^4-1\right)
}{x^4}+2x^2\ln \frac{x^2+1}{x^2-1}+2\ln \frac{\kappa ^{2}M_{\alpha}^{2}x^2}{%
Q^{2}}-3\right),
\end{equation}%
\begin{equation} %
F(M_{\alpha}^2,y^2) =
\frac{1}{4}\left( \left( y^4 + 1\right) \ln \frac{\left( y^4 - 1\right)
}{y^4}+2y^2\ln \frac{y^2 + 1}{y^2 - 1} + 2\ln \frac{\kappa \sigma_{\chi} M_{\alpha}^{2} y^2}{%
Q^{2}}-3\right),
\end{equation}%
where $x=|S|/ M_{\alpha}$, $y = \zeta x$, $\zeta = \kappa / \sigma_{\chi}$ and $Q$ is the renormalization scale.
Considering gravity-mediated SUSY breaking scenario, where SUSY is broken in the hidden sector and is communicated gravitationally to the observable sector, the soft potential is \cite{Nilles:1983ge}
\begin{eqnarray}
V_{\text{Soft}} = M_{z_{i}}^2\vert z_{i} \vert^2+m_{3/2}\left\{z_{i}W_{i}+\left(A-3\right)W +h.c\right\},
\end{eqnarray}
where $z_{i}$ is observable sector field, $W_{i}=\frac{\partial W}{\partial z_{i}}$, $m_{3/2}$ is the gravitino mass and $A$ is the complex coefficient of the trilinear soft-
SUSY-breaking terms. The effective contributions of soft SUSY breaking terms during inflation can be written as,
\begin{eqnarray}
V_{\text{Soft}} = a m_{3/2} \kappa M_{\alpha}^3 x + M_S^2 M_{\alpha}^2 x^2 + \frac{8 M_{\phi}^2 M_{\alpha}^2}{9 \left( 4/27 - \alpha^2 \right)},
\end{eqnarray}
with
\begin{eqnarray}
a=2\vert A-2 \vert \cos \left(\arg S + \arg \vert A-2 \vert\right) ,
\end{eqnarray}
where $a$ and $M_S$ are the coefficients of soft SUSY breaking linear and mass terms for $S$, respectively, $M_{\phi}$ is the soft mass term for $\phi_{24}$ and $m_{3/2}$ is the gravitino mass.
\subsection{Gauge Coupling Unification} \label{sec2_2}
After the breaking of the $SU(5)$ symmetry, the octet $G_H$ and triplet $W_H$ from the adjoint Higgs field remain massless, as shown in \cite{Barr:2005xya,Fallbacher:2011xg}. The presence of these flat directions is a generic feature of simple groups like the $SU(5)$ with a $U(1)_R$ symmetry, as discussed in \cite{Barr:2005xya,Fallbacher:2011xg}. These fields, however, acquire relatively light masses $\mathcal{O} (\sim \text{TeV})$ from the soft SUSY-breaking terms in our model, which spoils the unification of the gauge couplings. In order to preserve gauge-coupling unification we add the following combination of vectorlike particles
\begin{equation}
5 + \bar{5} + E + \bar{E} = \left( D + \bar{D}, L + \bar{L} \right) + E + \bar{E} \,,
\end{equation}
with the $R$-charge, $R \left(5\,\bar{5},E\,\bar{E}\right) = (1,1)$ and allow mass-splitting within a multiplet with some fine tuning. The superpotential of these vectorlike fermions is given by \cite{Masoud:2019gxx},
\begin{eqnarray}
W &\supset& \frac{\mathcal{A}_{ij}^{(E,\overline{E})}}{m_{P}} \Tr(\Phi^{2}) E_i\overline{E}_j +\frac{\mathcal{B}_{ij}^{(E,\overline{E})}}{m_{P}} \Tr(E_i \Phi^{2} \overline{E}_j) \nonumber \\
&+& \frac{\mathcal{A}_{ij}^{(5,\overline{5})}}{m_{P}} \Tr(\Phi^{2}) \Tr(5_i\overline{5}_j) +\frac{\mathcal{B}_{ij}^{(5,\overline{5})}}{m_{P}} \Tr(5_i \Phi^{2} \overline{5}_j), \nonumber \\
&\supset& M_{E} E \overline{E} + M_{D} D \overline{D} + M_{L} L \overline{L}\,.
\end{eqnarray}
Assuming $\mathcal{A}_{ij} = \delta_{ij}\mathcal{A}$ and $\mathcal{B}_{ij}= \delta_{ij}\mathcal{B}$ for convenience, we obtain the following masses of the MSSM field components of vectorlike particles,
\begin{align}
M_{E}& = \frac{40 \mathcal{A}^{(E,\overline{E})} + 12 \mathcal{B}^{(E,\overline{E})}}{45 \alpha^2}\left( \frac{M^{2}}{m_{P}}\right)\,,\\
M_{D}& = \frac{60\mathcal{A}^{(5,\overline{5})} + 8 \mathcal{B}^{(5,\overline{5})}}{135 \alpha^2}\left( \frac{M^{2}}{m_{P}}\right)\,,\\
M_{L}& = \frac{20 \mathcal{A}^{(5,\overline{5})} + 6 \mathcal{B}^{(5,\overline{5})}}{45 \alpha^2}\left( \frac{M^{2}}{m_{P}}\right)\,.
\end{align}
\begin{figure}[t]
\centering \includegraphics[width=8.00cm]{plots/xsu5_unif_ms_12_5.pdf}
\centering \includegraphics[width=8.00cm]{plots/xsu5_unif_ms_25.pdf}
\caption{The evolution of the inverse gauge couplings $\alpha_i^{-1}$ with the energy scale $\Lambda$ in $U(1)_R$ symmetric $SU(5) \times U(1)_{\chi}$ model, with SUSY breaking scale $M_{\text{SUSY}} = 12.5$ TeV (left) and $M_{\text{SUSY}} = 25$ TeV (right). Unification is achieved with three generations of vectorlike fermions and GUT scale at $M_{\text{GUT}} \sim 2 \times 10^{16}$ GeV in both cases.}
\label{fig:unification}
\end{figure}
The masses for $E + \overline{E}$ and $L + \overline{L}$ can be made light with fine tuning on the parameters such that
\begin{equation}
40 \mathcal{A}^{(E,\overline{E})} + 12 \mathcal{B}^{(E,\overline{E})} \sim 0, \qquad
20 \mathcal{A}^{(5,\overline{5})} + 6 \mathcal{B}^{(5,\overline{5})} \sim 0.
\end{equation}
The mass of $D + \overline{D}$ component is then given as
\begin{equation}
M_D \sim \frac{21 \mathcal{A}^{(5,\overline{5})}}{81 \alpha^2} \left( \frac{M^{2}}{m_{P}}\right) .
\end{equation}
\begin{table}[t]
\setlength\extrarowheight{5pt}
\centering
\begin{tabular}{ccccc}
\hline \hline \rowcolor{Gray}
& \multicolumn{3}{c}{ Vectorlike Fermion masses (GeV)} & \\
\rowcolor{Gray} \multirow{-2}{*}{ \begin{tabular}[c]{@{}c@{}}SUSY breaking \\scale $M_{\text{SUSY}}$\end{tabular} } & $M_D$ & $M_L$ & $M_E$ & \\ \cline{1-5}
12.5 TeV & $6.5 \times 10^{13}$ & $2.14 \times 10^{9}$ & $1.0 \times 10^{8}$ & \\
25 TeV & $6.5 \times 10^{13}$ & $3.16 \times 10^{9}$ & $2.5 \times 10^{8}$ & \multirow{-4}{*}{\begin{tabular}[c]{@{}c@{}} Unification scale \\ \\ $M_{\text{GUT}} \simeq 2 \times 10^{16}$ GeV \end{tabular}} \\ \hline
\end{tabular}
\caption{The effective SUSY breaking scales, $M_{\text{SUSY}}$ and corresponding mass splitting patterns of vectorlike families. Unification occurs at the same scale for both cases.}
\label{tab:vectorlike_mass_splitting}
\end{table}
Fig. \ref{fig:unification} shows successful gauge-coupling unification with three generations of the vectorlike families and different mass-splittings, as listed in Table \ref{tab:vectorlike_mass_splitting}, for two SUSY-breaking scales, $M_{\text{SUSY}} = (12.5,\,25)$ TeV. Here, we assume the masses of the octet and the triplet to be near the SUSY-breaking scale, $M_{\text{SUSY}} \simeq M_{G_H} \simeq M_{W_H}$. In both cases, the unification is achieved at $M_{\text{GUT}} = (5\sqrt{2}/9 \alpha)g_5 M \sim 2 \times 10^{16}$ GeV, where $g_5$, the gauge coupling of $SU(5)$ gauge group, is unified with $g_{\chi}$, the gauge coupling of $U(1)_{\chi}$ group.
\subsection{\large{\bf Dimension-5 Proton Decay}}\label{sec4}
In this section, the implementation of the douplet-triplet solution to the well known issue of the color triplets $D_h, \bar{D}_{\bar{h}}$ embedded in the same representations $\mathbf{5}$ and $\bar{\mathbf{5}}$ with the MSSM Higgs fields is briefly discussed.
The relevant superpotential terms are
\begin{equation}
W \supset \gamma \bar{h} \Phi h + \delta \bar{h} h~.
\end{equation}
After the $SU(5)$ symmetry breaking, these can be written in terms of the MSSM fields as follows
\begin{equation}
W \supset \left( \delta - \frac{3 \gamma \phi_{24}^0}{2\sqrt{15}} \right) h_u h_d + \left(\delta + \frac{\gamma \phi_{24}^0}{\sqrt{15}} \right) \bar{D}_{\bar{h}} D_h \supset \mu_H h_u h_d + M_{D_h} \bar{D}_{\bar{h}} D_h~.
\end{equation}
We observe that the doublet-triplet splitting problem is resolved by requiring fine-tuning of the involved parameters,
such that
$$\delta \sim \frac{3 \gamma \phi_{24}^0}{2\sqrt{15}} ~.$$
Here $\mu_H$ is identified with the MSSM $\mu$ parameter taken to be of the order of TeV scale while, $M_{D_h}$ is the color triplet Higgs mass parameter given by,
\begin{equation}
M_{D_h} \sim \frac{5 \, \gamma \phi_{24}^0}{2\sqrt{15}} = \frac{5 \gamma M_{\alpha} y_2}{\sqrt{30 \left( \frac{4}{27 \alpha^2} -1 \right)}}.
\end{equation}
The dominant contribution to dimension-5 proton decay amplitude comes from color-triplet Higgsinos and typically dominates the decay rate from gauge boson mediated dimension-6 operators. The proton lifetime for the decay $p \rightarrow K^+ \bar{\nu}$ mediated by color-triplet Higgsinos is approximated by \cite{Nagata:2013ive}:
\begin{equation}
\tau_p \simeq 4 \times 10^{35} \times \sin^4 2\beta \left( \frac{M_{\text{SUSY}}}{10^2 ~ \text{TeV}} \right)^2 \left( \frac{M_{D_h}}{10^{16} ~ \text{GeV}} \right)^2 \text{years}, \label{proton_lifetime}
\end{equation}
which depends on Higgino mass as well as the SUSY breaking scale $M_{\text{SUSY}}$. The Super-Kamiokande experiment places a lower bound on proton lifetime of $\tau_p = 5.9 \times 10^{33}$ years at $90\%$ confidence level for the channel $p \rightarrow K^+ \bar{\nu}$. With $M_{\alpha} \simeq 2 \times 10^{16}$ GeV, this translates into a lower bound on $M_{\text{SUSY}}$,
\begin{equation}
M_{\text{SUSY}} \gtrsim 12.5 ~ \text{TeV} .
\end{equation}
\begin{figure}[t]
\centering
\includegraphics[width=0.6\textwidth]{plots/p_decay.pdf}
\label{fig:superk_bound}
\caption{$SU(5)$ gauge symmerty breaking scale $M_{\alpha}$ as a function of SUSY breaking scale $M_{\text{SUSY}}$ for different values of $\tan \beta$. The curves are drawn for proton lifetime fixed at Super-Kamiokande bounds ($\tau_p = 5.9 \times 10^{33}$ years).}
\label{fig:proton_decay}
\end{figure}
This can also be seen in Fig. \ref{fig:proton_decay} where $SU(5)$ gauge symmerty breaking scale $M_{\alpha}$ is plotted against the SUSY breaking scale $M_{\text{SUSY}}$. The curves represent different values of $\tan \beta$ and drawn for proton lifetime fixed at Super-Kamiokande bounds \cite{Super-Kamiokande:2016exg}.
\section{\large{\bf Minimal K\"ahler Potential}}\label{sec5}
The minimal canonical K\"ahler potential is given as,
\begin{equation} \label{Ktree}
K = \vert S \vert^2 + Tr \vert \Phi \vert^2
+ \vert h \vert^2 + \vert \bar{h}\vert^2 + \vert \chi \vert^2 + \vert \bar{\chi} \vert^2+ \vert \nu_{i}^{c} \vert^2 .
\end{equation}
The F-term SUGRA scalar potential is given by
\begin{equation}
V_{\text{SUGRA}}=e^{K/m_P^{2}}\left(
K_{i\bar{j}}^{-1}D_{z_{i}}WD_{z^{*}_j}W^{*}-3 m_P^{-2}\left| W\right| ^{2}\right),
\label{VF}
\end{equation}
with $z_{i}$ being the bosonic components of the superfields $z%
_{i}\in \{S,\Phi,h,\bar{h}, \chi, \bar{\chi} ,\cdots\}$, and we have defined
\begin{equation}
D_{z_{i}}W \equiv \frac{\partial W}{\partial z_{i}}+m_P^{-2}\frac{%
\partial K}{\partial z_{i}}W , \,\,\,
K_{i\bar{j}} \equiv \frac{\partial ^{2}K}{\partial z_{i}\partial z_{j}^{*}},
\end{equation}
and $D_{z_{i}^{*}}W^{*}=\left( D_{z_{i}}W\right)^{*}.$
The SUGRA scalar potential during inflation becomes
\begin{eqnarray}
\begin{split}
V_{\text{SUGRA}}&=\kappa ^2 M_{\alpha}^4 \left[1+ \left(\frac{4}{9\left(4/27 - \alpha^2\right)}\right)\left(\frac{M_{\alpha }}{m_{p}}\right)^2 +\right.\\
&\left. \left(\frac{4 \left(2 + 9 x^2\left(4/27 - \alpha^2 \right) \right)}{81\left(4/27 - \alpha^2\right)^2} + \frac{1}{2} x^4\right)\left(\frac{M_{\alpha }}{m_{p}}\right)^4+.....\right]\, .
\end{split}
\end{eqnarray}
Putting all these corrections together, we obtain the following form of inflationary potential,
\begin{eqnarray}\nonumber
V &\simeq& V_{\text{SUGRA}} + V_{\text{1-loop}} + V_{\text{Soft}} \\ \nonumber
&\simeq& \kappa ^2 M_{\alpha}^4 \Bigg[1+ \left(\frac{4}{9\left(4/27 - \alpha^2\right)}\right)\left(\frac{M_{\alpha }}{m_{p}}\right)^2 \\ \nonumber
&+& \left(\frac{4 \left(2 + 9 x^2\left(4/27 - \alpha^2 \right) \right)}{81\left(4/27 - \alpha^2\right)^2} + \frac{1}{2} x^4\right)\left(\frac{M_{\alpha }}{m_{p}}\right)^4 \\ \nonumber
&+& \frac{\kappa^2}{16 \pi^2} \left[ F(M_{\alpha}^2,x^2) + 11\times 25\,F(5 M_{\alpha}^2,5\,x^2) \right] + \frac{\sigma_{\chi}^2}{8 \pi^2} F(M_{\alpha}^2,y^2) \\
&+& \frac{a m_{3/2} x}{\kappa M_{\alpha}} + \frac{M_S^2\, x^2}{\kappa^2 M_{\alpha}^2} + \frac{8 M_{\phi}^2}{9 \kappa^2 M_{\alpha}^2 \left( 4/27 - \alpha^2 \right)} \Bigg] .
\end{eqnarray}
The inflationary slow roll parameters are given by,
\begin{eqnarray}
\epsilon = \frac{1}{4}\left( \frac{m_P}{M_{\alpha}}\right)^2
\left( \frac{V'}{V}\right)^2, \,\,\,
\eta = \frac{1}{2}\left( \frac{m_P}{M_{\alpha}}\right)^2
\left( \frac{V''}{V} \right), \,\,\,
\alpha^2 = \frac{1}{4}\left( \frac{m_P}{M_{\alpha}}\right)^4
\left( \frac{V' V'''}{V^2}\right).
\end{eqnarray}
Here, the derivatives are with respect to $x=|S|/M_{\alpha}$, whereas the canonically normalized field $\sigma \equiv \sqrt{2}|S|$. In the slow-roll (leading order) approximation, the tensor-to-scalar ratio $r$, the scalar spectral index $n_s$, and the running of the scalar spectral index $dn_s / d \ln k$ are given by
\begin{eqnarray}
r &\simeq& 16\,\epsilon, \\
n_s &\simeq& 1+2\,\eta-6\,\epsilon, \\
\frac{d n_s}{d\ln k} &\simeq& 16\,\epsilon\,\eta
-24\,\epsilon^2 - 2\,\xi^2 .
\end{eqnarray}
\begin{figure}[t]
\centering \includegraphics[width=8.035cm]{plots/Ma_ns_nosoft.pdf}
\centering \includegraphics[width=8.035cm]{plots/r_ns_nosoft.pdf}
\centering \includegraphics[width=8.035cm]{plots/k_ns_nosoft.pdf}
\centering \includegraphics[width=8.035cm]{plots/s0mP_ns_nosoft.pdf}
\caption{The scalar spectral index $n_s$ vs the $SU(5)$ symmetry breaking scale $M_{\alpha}$, the tensor-to-scalar ratio $r$, $\kappa$ and $S_0/m_P$ for minimal K\"ahler potential without the soft mass terms.}
\label{fig:minimal_nosoft}
\end{figure}
\begin{figure}[t]
\centering \includegraphics[width=8.035cm]{plots/Ma_MS.pdf}
\centering \includegraphics[width=8.035cm]{plots/Ma_r.pdf}
\centering \includegraphics[width=8.035cm]{plots/k_MS.pdf}
\centering \includegraphics[width=8.035cm]{plots/s0mp_MS.pdf}
\caption{The scalar spectral index $n_s$ vs the $SU(5)$ symmetry breaking scale $M_{\alpha}$, the tensor-to-scalar ratio $r$, $\kappa$ and $S_0/m_P$ for minimal K\"ahler potential without the soft mass terms.}
\label{fig:minimal_soft}
\end{figure}
The last $N_0$ number of e-folds before the end of inflation is,
\begin{eqnarray}
N_0 = 2\left( \frac{M_{\alpha}}{m_P}\right) ^{2}\int_{x_e}^{x_{0}}\left( \frac{V}{%
V'}\right) dx,
\label{efolds}
\end{eqnarray}
where $x_0$ is the field value at the pivot scale $k_0$, and
$x_e$ is the field value at the end of inflation, defined by $|\eta(x_e)| = 1$. Assuming a standard thermal history, $N_0$ is related to $T_r$ as \cite{Garcia-Bellido:1996egv}
\begin{equation}\label{n0}
N_0=54+\frac{1}{3}\ln\Big(\frac{T_r}{10^9\text{ GeV}}\Big)+\frac{2}{3}\ln\Big(\frac{V(x)^{1/4} }{10^{15}\text{ GeV}}\Big),
\end{equation}
where $T_r$ is the reheat temperature and in numerical calculation we set $T_r=10^9$ GeV. This could easily be reduced to lower values if the gravitino problem is regarded to be an issue.\footnote{For a recent discussion on the gravitino overproduction problem in hybrid inflation see Ref.\cite{Nakayama:2010xf}} The amplitude of the curvature perturbation is given by \cite{Liddle:1993fq}
\begin{eqnarray}
A_{s}(k_0) = \frac{1}{24\,\pi^2}
\left. \left( \frac{V/m_P^4}{\epsilon}\right)\right|_{x = x_0},
\label{perturb}
\end{eqnarray}
where $A_{s}= 2.137 \times 10^{-9}$ is the Planck normalization at $k_0 = 0.05\, \rm{Mpc}^{-1}$. In our numerical calculations, we have taken $M_S = M_{\phi}$ and have set the dimensionless couplings equal, $\kappa = \sigma_{\chi}$, such that $\zeta = 1$. Fig. \ref{fig:minimal_nosoft}, shows our results without soft SUSY mass terms where various parameters are plotted against the scalar spectral index $n_s$. It can be seen that without the soft mass terms, the scalar spectral index $n_s$ cannot be achieved within Plank 2-$\sigma$ bounds. With the inclusion of soft mass terms, the scalar spectral index $n_s$ can easily be obtained within Planck's 2-$\sigma$ bounds.
The soft mass terms, seem to play an important role in inflationary predictions. Fig. \ref{fig:minimal_soft} shows our numerical results with soft mass terms where the behavior of $SU(5)$ guage symmetry breaking scale $M_{\alpha}$ (upper left panel), $\kappa$ (lower left panel) and $S_0/m_P$ (lower right panel) is depicted as a function of soft mass parameter $\vert M_S \vert $ for different values of the gravitino mass $m_{3/2}$. The behavior of $SU(5)$ guage symmetry breaking scale $M_{\alpha}$ with respect to the tensor to scalar ratio $r$ is shown in the upper right panel. In obtaining these results, we have fixed the scalar spectral index $n_s$ at the central value (0.9655) of Planck's latest bounds. The soft mass squared parameter $M_S^2$ and the combination $a m_{3/2}$ can be either positive or negative. We consider the following possible cases in our numerical calculations,
\begin{gather}\nonumber
a m_{3/2} > 0 \quad M_S^2 > 0 ,\\ \nonumber
a m_{3/2} < 0 \quad M_S^2 < 0 ,\\ \nonumber
a m_{3/2} < 0 \quad M_S^2 > 0 ,\\
a m_{3/2} > 0 \quad M_S^2 < 0 .
\end{gather}
The first case with $M_S^2 > 0$ and $a m_{3/2} > 0$ ($a = +1$) is inconsistent with Planck's results. A red tilted scalar spectral index ($n_s < 1$) compatible with Planck's latest bounds is obtained for the rest of the cases. The yellow curves are drawn for $a m_{3/2} < 0$ ($a = -1$) where the solid lines correspond to the case when $M_S^2 < 0$ and dashed lines correspond to $M_S^2 > 0$. For lower values of $\vert M_S \vert \simeq (1 - 10^4)$ TeV, the radiative corrections provide dominant contribution, whereas both SUGRA corrections and soft mass squared terms are suppressed. The suppression of supergravity corrections in this region is supported by small values of $S_0/m_P \simeq 10^{-3}$, as shown in lower right panel of Fig. \ref{fig:minimal_soft}. For $\vert M_S \vert \gtrsim 10^4$ TeV, the soft mass squared term begins to take over, which drives the curve upward for $M_S^2 < 0$, and downward for $M_S^2 > 0$. For $M_S^2 < 0$ and $\vert M_S \vert \gtrsim 10^4$ TeV, $\kappa$ takes on large values, $M_{\alpha}$ approaches $\sim 1.5 \times 10^{16}$ GeV and supergravity corrections become important.
It is useful to analytically examine some approximate equations to understand the behavior depicted in Fig. \ref{fig:minimal_soft}. In the slow-roll approximation, the amplitude of the power spectrum of scalar curvature perturbation $A_{s}$ and the scalar spectral index $n_s$ is given by,
\begin{eqnarray}\label{cur_analytic}
A_{s}(k_{0}) &\simeq& \frac{\kappa^2}{6\,\pi^2}\left(\frac{M_{\alpha}}{m_{p}}\right)^{6} \left( 2 x_0^3\left(\frac{M_{\alpha}}{m_P}\right)^4 - \frac{2 M_S^2 x_0}{\kappa^2 M_{\alpha}^2} + \frac{a m_{3/2}}{\kappa M_{\alpha}} + \frac{278 \kappa^2}{16\,\pi^2} F^{'}(5 x_0) \right)^{-2}, \\ \label{ns_analytic}
n_s &\simeq& 1 + \left(\frac{m_{p}}{M_{\alpha}}\right)^{2} \left( 6 x_0^2\left(\frac{M_{\alpha}}{m_P}\right)^4 - \frac{2 M_S^2}{\kappa^2 M_{\alpha}^2} + \frac{278 \kappa^2}{16\,\pi^2} F^{''}(5 x_0) \right) .
\end{eqnarray}
Taking the contributions of the soft linear mass term comparable to mass squared term, we obtain the following analytical expressions for $M_{\alpha}$, $\kappa$ and $\vert M_S \vert$;
\begin{gather}
\kappa \simeq \left( \left( \frac{8 \pi^3}{139} \right)^3 \frac{\left(1 - n_s\right)}{\vert F^{'} (5 x_0) \vert^2 \vert F^{''} (5 x_0) \vert} \right)^{1/8} \left( \frac{m_{3/2}}{m_P} \right)^{1/4}, \\
M_{\alpha} \simeq \left( \frac{139 \vert F^{''} (5 x_0) \vert^3}{8 \pi^2 \left( 1 - n_s \right)^3 \vert F^{'} (5 x_0) \vert^2} \right)^{1/8} \left( m_{3/2} m_P^3 \right)^{1/4} , \\
\vert M_S \vert \simeq \kappa^2 M_{\alpha} \sqrt{\frac{139}{32 \pi^2} F^{'} (5 x_0)} \,.
\end{gather}
It can be checked that, for $m_{3/2} \simeq 10$ TeV, $n_s = 0.9655$, $x_0 \sim 1$, we obtain $\kappa \simeq 1.3 \times 10^{-4}$, $M_{\alpha} \simeq 2 \times 10^{15}$ GeV and $\vert M_S \vert \simeq 2 \times 10^4$ TeV. Also, for $m_{3/2} \simeq 1000$ TeV, $n_s = 0.9655$, $x_0 \sim 1$, we obtain $\kappa \simeq 4.5 \times 10^{-4}$, $M_{\alpha} \simeq 5.5 \times 10^{15}$ GeV and $\vert M_S \vert \simeq 7 \times 10^5$ TeV. These estimates are in excellent agreement with the numerical results shown in Figs. \ref{fig:minimal_soft}. Therefore, for $m_{3/2} \lesssim 1000$ TeV and $M_S \lesssim 10^4$ TeV, only radiative corrections and linear soft mass term dominate, whereas the SUGRA corrections and soft mass-squared term are suppressed. It should be noted that larger values of $m_{3/2}$ shift the contribution of the soft mass-squared term towards larger values of $M_S$. For example, with $m_{3/2} \simeq 10$ TeV, the soft mass-squared term begins to take over for $M_S \gtrsim 10^4$ TeV, whereas with $m_{3/2} \simeq 1000$ TeV, the soft mass-squared term becomes important for $M_S \gtrsim 5 \times 10^5$ TeV. Furthermore, for larger values of $m_{3/2}$ ($\gtrsim 1000$ TeV), the SUGRA corrections become important and large values of $M_{\alpha}$ can be obtained, independent of $M_S$. The curves exhibit similar behavior for $\vert M_S \vert \gtrsim 10^6$ TeV for all three cases but decouple for value of $\vert M_S \vert$ below $10^6$ TeV.
The fourth case with $a m_{3/2} > 0$ ($a = +1$) and $M_S^2 <0$, generates large $M_{\alpha}$ that easily approaches $M_{\text{GUT}}$. For $\vert M_S \vert \lesssim 10^6$ TeV, $M_{\alpha}$ takes on large values, whereas the radiative corrections become suppressed owing to small values of $\kappa$. This is in contrast to the other two cases where the contribution of soft mass-squared term becomes negligible below $\vert M_S \vert \lesssim 10^6$ TeV. With radiative corrections suppressed, Eqs. \eqref{cur_analytic} and \eqref{ns_analytic} simplify to the following form,
\begin{eqnarray}\label{cur_analytic_2}
A_{s}(k_{0}) &\simeq& \frac{\kappa^2}{6\,\pi^2}\left(\frac{M_{\alpha}}{m_{p}}\right)^{6} \left( 2 x_0^3\left(\frac{M_{\alpha}}{m_P}\right)^4 - \frac{2 M_S^2 x_0}{\kappa^2 M_{\alpha}^2} + \frac{a m_{3/2}}{\kappa M_{\alpha}} \right)^{-2}, \\ \label{ns_analytic_2}
n_s &\simeq& 1 + \left(\frac{m_{p}}{M_{\alpha}}\right)^{2} \left( 6 x_0^2\left(\frac{M_{\alpha}}{m_P}\right)^4 - \frac{2 M_S^2}{\kappa^2 M_{\alpha}^2} \right) .
\end{eqnarray}
Taking the soft mass-squared term to be comparable to linear soft SUSY-breaking term, we obtain the following analytical expressions for $\kappa$ and $M$ in terms of $m_{3/2}$ and $M_S$;
\begin{eqnarray}
\kappa \simeq 2 \left(2 \left(1 - n_s \right)\right)^{1/2} \frac{\vert M_S \vert^3}{m_{3/2} m_P} \, ,\\
M \simeq \left( \frac{1}{2 \left(1 - n_s\right)} \right)^{1/2} \left( \frac{m_{3/2} m_P}{\vert M_S \vert} \right) .
\end{eqnarray}
Using $n_s \simeq 0.9655$, $m_{3/2} \simeq 10$ TeV and $\vert M_S \vert \simeq 4.6 \times 10^3$ TeV, we obtain $\kappa \sim 2 \times 10^{-7}$ and $M_{\alpha} \sim 2 \times 10^{16}$ GeV. Similarly for $n_s \simeq 0.9655$, $m_{3/2} \simeq 1000$ TeV and $\vert M_S \vert \simeq 4.5 \times 10^5$ TeV, we obtain $\kappa \sim 2 \times 10^{-5}$ and $M_{\alpha} \sim 2 \times 10^{16}$ GeV. These estimates are in good agreement with our numerical results displayed in Fig. \ref{fig:minimal_soft}.
The behavior of $M_{\alpha}$ with respect to the tensor to scalar ratio $r$ is shown in the upper right panel of Fig. \ref{fig:minimal_soft} and can be understood from the following approximate relation between $r$, $M_{\alpha}$ and $\kappa$, obtained by using the Planck's normalization constraint on $A_s$,
\begin{equation} \label{eq:rkappaexplicit}
r \simeq \left( \frac{2 \kappa^2}{3 \pi^2 A_s (k_0)} \right) \left(\frac{M_{\alpha}}{m_P}\right)^4 .
\end{equation}
This shows that $r$ is proportional to both $M_{\alpha}$ and $\kappa$ and large values of $r$ are obtained for large $M_{\alpha}$ and $\kappa$. It can readily be checked that for $M_{\alpha} \simeq 2.4 \times 10^{15}$ GeV and $\kappa \simeq 1.3 \times 10^{-4}$, the above equation gives $r \simeq 4.7 \times 10^{-7}$. Similarly, for $M_{\alpha} \simeq 1.3 \times 10^{16}$ GeV and $\kappa \simeq 0.01$, we obtain $r \simeq 2.5 \times 10^{-6}$. These approximate values are very close to the actual values obtained in our numerical calculations. The above equation therefore gives a valid approximation of our numerical results. The tensor to scalar ratio $r$ varies in the range $4.7 \times 10^{-13} \lesssim r \lesssim 2.5 \times 10^{-6}$ and is beyond the current measuring limits of various upcoming experiments.
\section{\large{\bf Non-Minimal K\"ahler Potential}}\label{sec6}
In this section we employ a non-minimal K\"ahler potential including non-renormalizable terms up to sixth order;
\begin{equation}
\label{eq:nonminK}
\begin{split}
K &= \vert S \vert^2 + \Tr \vert \Phi \vert^2 + \vert h \vert^2 + \vert \bar{h}\vert^2 + \vert \chi \vert^2 + \vert \bar{\chi} \vert^2
\\
& +\kappa_{S\Phi} \frac{\vert S\vert^2 \, \Tr \vert \Phi \vert^2}{m_P^2}
+ \kappa_{S h} \frac{\vert S \vert^2 \vert h \vert^2}{m_P^2}
+ \kappa_{S \bar{H}} \frac{\vert S \vert^2 \vert \bar{H} \vert^2}{m_P^2}
+ \kappa_{S {\chi}} \frac{\vert S \vert^2 \vert \chi \vert^2}{m_P^2} + \kappa_{S \bar{\chi}} \frac{\vert S \vert^2 \vert \bar{\chi} \vert^2}{m_P^2}
\\
& + \kappa_{H \Phi} \frac{\vert h \vert^2 \, \Tr \vert \Phi \vert^2}{m_P^2}
+ \kappa_{h \chi} \frac{\vert h \vert^2 \vert \chi \vert^2}{m_P^2} + \kappa_{h \bar{\chi}} \frac{\vert h \vert^2 \vert \bar{\chi} \vert^2}{m_P^2} + \kappa_{\bar{h} \Phi} \frac{\vert \bar{h} \vert^2 \, \Tr \vert \Phi \vert^2}{m_P^2}
\\
& + \kappa_{\bar{h} \chi} \frac{\vert \bar{h} \vert^2 \vert \chi \vert^2}{m_P^2} + \kappa_{\bar{h} \bar{\chi}} \frac{\vert \bar{h} \vert^2 \vert \bar{\chi} \vert^2}{m_P^2} + \kappa_{h \bar{h}} \frac{\vert h \vert^2 \vert \bar{h} \vert^2}{m_P^2}
+ \kappa_{\chi \bar{\chi}} \frac{\vert \chi \vert^2 \vert \bar{\chi} \vert^2}{m_P^2} + \kappa_S \frac{\vert S\vert^4}{4 m_P^2}
\\
& + \kappa_{\Phi} \frac{ (\Tr \vert \Phi \vert^2)^2}{4 m_P^2}
+ \kappa_{H} \frac{ \vert h \vert^4}{4 m_P^2} + \kappa_{\bar{h}} \frac{ \vert \bar{h} \vert^4}{4 m_P^2}
+ \kappa_{\chi} \frac{ \vert \chi \vert^4}{4 m_P^2}
+ \kappa_{\bar{\chi}} \frac{ \vert \bar{\chi} \vert^4}{4 m_P^2}
\\
& + \kappa_{SS} \frac{\vert S\vert^6}{6 m_P^4} + \kappa_{\Phi \Phi} \frac{ (\Tr \vert \Phi \vert^2)^3}{6 m_P^4}
+ \kappa_{h h} \frac{ \vert h \vert^6}{6 m_P^4}
+ \kappa_{\bar{h} \bar{h}} \frac{ \vert \bar{h} \vert^6}{6 m_P^4}
+ \kappa_{\chi \chi} \frac{ \vert \chi \vert^6}{6 m_P^4} + \kappa_{\bar{\chi} \bar{\chi}} \frac{ \vert \bar{\chi} \vert^6}{6 m_P^4}
+ \cdots.
\end{split}
\end{equation}
Including the one loop radiative corrections and soft SUSY mass terms, the full scalar potential during inflation then reads as,
\begin{eqnarray}
V &\simeq& V_{\text{SUGRA}} + V_{\text{1-loop}} + V_{\text{Soft}} \nonumber \\
&\simeq& \kappa ^2 M_{\alpha}^4 \Bigg[1+ \left(\frac{4(1-\kappa_{S\Phi})}{9\,(4/27 - \alpha^2)} -\kappa_S\,x^2 \right) \left( \frac{M_{\alpha}}{m_P}\right)^{2} \nonumber \\
&+& \left( \frac{4((1-2\kappa_{S\Phi})^2+1+\kappa_{\Phi})}{81\,(4/27 - \alpha^2)^2} \right. \nonumber \\
&+& \left. \frac{4((1-\kappa_{S\Phi})^2-\kappa_{S}(1-2\kappa_{S\Phi}))x^2}{9\,(4/27 - \alpha^2)} +\frac{\gamma _{S}\,x^4}{2}\right) \left( \frac{M_{\alpha}}{m_P}\right)^{4} \nonumber \\
&+& \frac{\kappa^2}{16 \pi^2} \left[ F(M_{\alpha}^2,x^2) + 11\times 25\,F(5 M_{\alpha}^2,5\,x^2) \right] + \frac{\sigma_{\chi}^2}{8 \pi^2} F(M_{\alpha}^2,y^2) \nonumber \\
&+& \frac{a m_{3/2} x}{\kappa M_{\alpha}} + \frac{M_S^2\, x^2}{\kappa^2 M_{\alpha}^2} + \frac{8 M_{\phi}^2}{9 \kappa^2 M_{\alpha}^2 \left( 4/27 - \alpha^2 \right)} \Bigg] ,
\end{eqnarray}
\noindent where $\gamma_S = 1 - \frac{7 \kappa_S}{2} + 2 \kappa_S^2 - 3 \kappa_{SS}$. The results of our numerical calculations with a non-minimal K\"ahler potential are displayed in Figs. \ref{fig:nonminimal_1} - \ref{fig:nonminimal_3}. In obtaining these results, we have used up to second order approximation on the slow-roll parameters and the $SU(5)$ gauge symmetry breaking scale $M_{\alpha}$ is fixed at $M_{\text{GUT}} \simeq 2 \times 10^{16}$ GeV. We have also fixed the soft SUSY masses at $m_{3/2} \simeq M_S \simeq 10$ TeV, with $a = 1$ and $M_S^2 > 0$.
\begin{figure}[!htb]
\centering \includegraphics[width=8.035cm]{plots/ns_r.pdf}
\centering \includegraphics[width=8.035cm]{plots/ns_k.pdf}
\caption{Behavior of $\kappa$ (right) and tensor-to-scalar ratio $r$ (left) with respect to scalar spectral index $n_s$ for $SU(5)$ breaking scale $M_{\alpha} \simeq M_{\text{GUT}} = 2 \times 10^{16}$ GeV. The lighter (darker) shaded region represents the Planck 2-$\sigma$ (1-$\sigma$) bounds, whereas the red and blue curves correspond to the $S_0 = m_P$ and $\kappa_{SS} = 1$ constraints, respectively.}
\label{fig:nonminimal_1}
\end{figure}
\begin{figure}[!htb]
\centering \includegraphics[width=8.035cm]{plots/ks_r.pdf}
\centering \includegraphics[width=8.035cm]{plots/gs_r.pdf}
\caption{Behavior of tensor-to-scalar ratio $r$ with respect to the non-minimal coupling $\kappa_S$ (left) and quartic coupling $\gamma_S$ (right) for $SU(5)$ breaking scale $M_{\alpha} \simeq M_{\text{GUT}} = 2 \times 10^{16}$ GeV. The lighter (darker) shaded region represents the Planck 2-$\sigma$ (1-$\sigma$) bounds, whereas the red and blue curves correspond to the $S_0 = m_P$ and $\kappa_{SS} = 1$ constraints, respectively.}
\label{fig:nonminimal_2}
\end{figure}
\begin{figure}[t]
\centering \includegraphics[width=8.035cm]{plots/ks_kss.pdf}
\centering \includegraphics[width=8.035cm]{plots/ks_gs.pdf}
\caption{Behavior of non-minimal coupling $\kappa_{SS}$ (left) and quartic coupling $\gamma_S$ (right) with respect to the non-minimal coupling $\kappa_S$ for $SU(5)$ breaking scale $M_{\alpha} \simeq M_{\text{GUT}} = 2 \times 10^{16}$ GeV. The lighter (darker) shaded region represents the Planck 2-$\sigma$ (1-$\sigma$) bounds, whereas the red and blue curves correspond to the $S_0 = m_P$ and $\kappa_{SS} = 1$ constraints, respectively.}
\label{fig:nonminimal_3}
\end{figure}
As compared to the minimal case, the non-minimal K\"ahler potential increases the parametric space and with the addition of new parameters, we now expect to obtain $n_s$ within the latest Planck bounds with large values of tensor-to-scalar ratio $r$. The radiative corrections and SUGRA corrections parameterized by $\kappa_{S}$ and $\kappa_{SS}$, dominate the global SUSY potential while the soft mass terms with $m_{3/2} \simeq M_S \simeq 10$ TeV are adequately suppressed. To keep the SUGRA expansion under control we impose $S_0 \leq m_P$. We also restrict the non-minimal couplings $\vert \kappa_{S} \vert \leq 1$ and $\vert \kappa_{SS} \vert \leq 1$. These two constraints are shown in Figs. \ref{fig:nonminimal_1} - \ref{fig:nonminimal_3} by the red ($S_0 = m_P$) and blue ($\kappa_{SS} = 1$) curves. The lighter (darker) yellow region represents the Planck 2-$\sigma$ (1-$\sigma$) bounds on scalar spectral index $n_s$. By employing non-minimal K\"ahler potential, there is a significant increase in the tensor-to-scalar ratio $r$ and both $\kappa_S$ and $\gamma_{S}$ play vital role to bring the scalar spectral index $n_s$ within Planck 2-$\sigma$ data bounds, with a large value of tensor to scalar ratio $r \simeq 10^{-3}$.
The behavior of tensor-to-scalar ratio $r$ and $\kappa$, as displayed in Fig. \ref{fig:nonminimal_1}, can be understood from the explicit relation \eqref{eq:rkappaexplicit} between $r$, $\kappa$ and $M_{\alpha}$ which shows that larger values of $r$ are expected when $\kappa$ or $M_{\alpha}$ is large. Since $M_{\alpha}$ is fixed, larger $r$ values should be obtained for large $\kappa$. For fixed $M_{\alpha} \simeq 2 \times 10^{16}$ GeV, the largest value of $r$ ($\sim 1.5 \times 10^{-3}$) obtained in our numerical results occurs for $\kappa \simeq 0.1$. The behavior of tensor to scalar ratio $r$ with respect to $\kappa_{S}$ and $\gamma_{S}$ is presented in Fig. \ref{fig:nonminimal_2}, while Fig. \ref{fig:nonminimal_3} depicts the behavior of $\kappa_{SS}$ and $\gamma_{S}$ with respect to $\kappa_{S}$. It can be seen that the large $r$ values are obtained with non-minimal couplings $\kappa_S < 0$, $\kappa_{SS} > 0$ and the quartic coupling $\gamma_S <0 $. Moreover, in the large $r$ limit, both $\kappa_{S}$ and $\kappa_{SS}$ are tuned to make $\gamma_{S}$ very small ($\sim -0.003$). Note that large tensor modes can be obtained for any value of scalar spectral index $n_s$ within Planck 2-$\sigma$ bounds. Finally, smaller $r$ values ($\sim 10^{-6}$) are obtained for $S_0 \lesssim 0.05 \, m_P$ and $\kappa_{SS} \simeq 1$ for which $\gamma_{S}$ is negative and fairly large ($\sim -2$).
The spectral index $n_s$ and tensor to scalar ratio $r$ in the leading order slow-roll approximation are given by
\begin{equation}
n_s \simeq 1 - 2 \kappa_S + \left( 6 \gamma_S x_0^2 + \frac{8 \left(1 - \kappa_S \right)}{9 \left( 4/27 - \alpha^2 \right)} \right) \left(\frac{M_{\alpha}}{m_P}\right)^2 + \frac{278 \kappa^2 F^{''}(5 x_0)}{16 \pi^2} \left(\frac{m_P}{M_{\alpha}}\right)^2 ,
\end{equation}
\begin{equation}
r \simeq 4 \left( \frac{m_P}{M_{\alpha}} \right)^2 \left( - 2 \kappa_S x_0 \left(\frac{M_{\alpha}}{m_P}\right)^2 + \left( 2 \gamma_S x_0^3 + \frac{8 \left(1 - \kappa_S \right)}{9 \left( 4/27 - \alpha^2 \right)} \right) \left(\frac{M_{\alpha}}{m_P}\right)^4 + \frac{278 \kappa^2 F^{'}(5 x_0)}{16 \pi^2} \right)^2.
\end{equation}
Solving these two equations simultaneously for $S_0 \simeq m_P$, $r \simeq 10^{-3}$, and $n_s \simeq 0.9655$ we obtain $\kappa_S \simeq -0.006$ and $\gamma_{S} \simeq -0.005$. Similarly in the small $r$ region for $S_0 \simeq (0.05) m_P$, $r \simeq 3 \times 10^{-6}$, and $n_s \simeq 0.9655$ we obtain $\kappa_S \simeq -0.005$ and $\gamma_{S} \simeq -2$. These approximate
values are very close to the actual values obtained in the numerical calculations. The above analytical equations therefore gives a valid approximation of our numerical results displayed in Figs. \ref{fig:nonminimal_1} - \ref{fig:nonminimal_3}. For non-minimal couplings ($-0.011 \lesssim \kappa_S \lesssim - 0.00063$) and ($0.34 \lesssim \kappa_{SS} \lesssim 1$), we obtain the scalar spectral index $n_s$ within the Planck 2-$\sigma$ bounds and tensor to scalar ratio $r$ in the range ($9.7 \times 10^{-7} \lesssim r \lesssim 1.5 \times 10^{-3}$).
\section{\large{\bf Radiative Breaking of $U(1)_{\chi}$} Symmetry}\label{sec7}
After the end of inflation, the effective unbroken gauge symmetry is $SU(3)_C \times SU(2)_L \times U(1)_Y \times U(1)_{\chi}$. The charge assignments of the fields under this symmetry are displayed in Table \ref{tab:table_radiative}.
\begin{table}[!htb]
\setlength\extrarowheight{3pt}
\centering
\begin{tabular}{c c}
\hline \hline \rowcolor{Gray}
\multicolumn{1}{c}{} & \multicolumn{1}{c}{} \\ \rowcolor{Gray}
\multicolumn{1}{c}{\multirow{-2}{*}{Superfields}} & \multirow{-2}{*}{\begin{tabular}[c]{@{}c@{}}Representations under\\ $SU(3)_C \times SU(2)_L \times U(1)_Y \times U(1)_{\chi}$\end{tabular}} \\
\hline
\rowcolor{Gray2} \multicolumn{2}{c}{Matter sector} \\ \hline
\multicolumn{1}{c}{$Q$} & $\left( \mathbf{3}, \mathbf{2}, 1/6, -1 \right)$ \\
\multicolumn{1}{c}{$u^c$} & $\left( \bar{\mathbf{3}}, \mathbf{1}, -2/3, -1 \right)$\\
\multicolumn{1}{c}{$d^c$} & $\left( \bar{\mathbf{3}}, \mathbf{1}, 1/3, 3 \right)$ \\
\multicolumn{1}{c}{$\ell$} & $\left( \mathbf{1}, \mathbf{2}, -1/2, 3 \right)$ \\
\multicolumn{1}{c}{$e^c$} & $\left( \mathbf{1}, \mathbf{1}, 1, -1 \right)$ \\
\multicolumn{1}{c}{$\nu^c$} & $\left( \mathbf{1}, \mathbf{1}, 0, -5 \right)$ \\ \hline
\rowcolor{Gray2} \multicolumn{2}{c}{Scalar sector} \\ \hline
\multicolumn{1}{c}{$h_u$} & $\left( \mathbf{1}, \mathbf{2}, 1/2, 2 \right)$ \\
\multicolumn{1}{c}{$h_d$} & $\left( \mathbf{1}, \mathbf{2}, -1/2, -2 \right)$ \\
\multicolumn{1}{c}{$\chi$} & $\left( \mathbf{1}, \mathbf{1}, 0, 10 \right)$ \\
\multicolumn{1}{c}{$\bar{\chi}$} & $\left( \mathbf{1}, \mathbf{1}, 0, -10 \right)$ \\
\multicolumn{1}{c}{$S$} & $\left( \mathbf{1}, \mathbf{1}, 0, 0 \right)$ \\ \hline \hline
\end{tabular}
\caption{Superfields and their representations under the effective unbroken guage symmetry $SU(3)_C \times SU(2)_L \times U(1)_Y \times U(1)_{\chi}$ after the end of inflation.}
\label{tab:table_radiative}
\end{table}
The superpotential terms relevant for $U(1)_{\chi}$ symmetry breaking are given by
\begin{eqnarray}
\label{W_u1_breaking} \nonumber
W &=& W_{\text{MSSM}} + W_{\chi} \\
W_{\chi} &=& \sigma_{\chi} S \chi \bar{\chi}
+ \lambda_{ij} \chi \nu_{i}^c \nu_{j}^c \,.
\end{eqnarray}
From the above equation, we obtain
\begin{eqnarray} \nonumber
F_S^{\dagger} &=& \frac{\partial W}{\partial S} = \sigma_{\chi} \chi \bar{\chi} = 0 ,\\ \nonumber
F_{\bar{\chi}}^{\dagger} &=& \frac{\partial W}{\partial \bar{\chi}} = \sigma_{\chi} S \chi = 0 ,\\ \nonumber
F_{\chi}^{\dagger} &=& \frac{\partial W}{\partial \chi} = \sigma_{\chi} S \bar{\chi} + \lambda_{ij} \nu_{i}^c \nu_{j}^c = 0 ,\\
F_{\nu^c}^{\dagger} &=& \frac{\partial W}{\partial \nu^c} = 2 \lambda_{ij} \nu_{i}^c \nu_{j}^c = 0 .
\end{eqnarray}
This leads to the following vacua;
\begin{equation}
\langle \chi \rangle = \langle \bar{\chi} \rangle= 0, \qquad \langle \nu_i^c \rangle = 0 , \qquad \langle S \rangle = \text{Arbitrary}.
\end{equation}
In order to break the $U(1)_{\chi}$ symmetry, a non-zero VEV of the field $\chi$ is desired; $\langle \chi \rangle = \langle \bar{\chi} \rangle \neq 0$. Including the soft SUSY breaking mass terms,
\begin{eqnarray} \nonumber
V_{\text{Soft}} &=& m_S^2 \vert S \vert^2 + m_{\chi}^2 \vert \chi \vert^2 + m_{\bar{\chi}}^2 \vert \bar{\chi} \vert^2 + m_{\nu_i}^2 \vert \nu_i^c \vert^2 \\
&+& A_{\nu} \lambda_{ij} \chi \nu_{i}^c \nu_{j}^c + A_{l} y_{ij}^{\nu} \nu_i^c l_j h_u + A_{\chi} \sigma_{\chi} S \chi \bar{\chi} + \frac{1}{2} M_{\chi} Z_{\chi} Z_{\chi} ,
\end{eqnarray}
where $A_{\nu}$ and $A_{\chi}$ are coefficients of linear soft mass terms, $Z_{\chi}$ is the $U(1)_{\chi}$ gaugino and $M_{\chi}$ is the gaugino mass. The full scalar potential is then given by,
\begin{eqnarray}
\nonumber
V &=& V_{F} + V_{D} + V_{\text{Soft}} \\ \nonumber
&=& \sigma_{\chi}^2 \vert \chi \bar{\chi} \vert^2 + \left| \, \sigma_{\chi} S \bar{\chi} + \lambda_{ij} \nu_{i}^{c}\nu_{j}^{c}\, \right|^2 + \left| \, \sigma_{\chi} S \chi \, \right|^2 + \left|2 \lambda_{ij} {\chi} \nu_{i}^{c}\right|^2 \\ \nonumber
&+& 50 g_{\chi}^2 \left( \vert \chi \vert^2 - \vert \bar{\chi} \vert^2 \right)^2 \\ \nonumber
&+& m_S^2 \vert S \vert^2 + m_{\chi}^2 \vert \chi \vert^2 + m_{\bar{\chi}}^2 \vert \bar{\chi} \vert^2 + m_{\nu_i}^2 \vert \nu_i^c \vert^2 \\
&+& A_{\nu} \lambda_{ij} \chi \nu_{i}^c \nu_{j}^c + A_{\chi} \sigma_{\chi} S \chi \bar{\chi} + \frac{1}{2} M_{\chi} Z_{\chi} Z_{\chi}.
\end{eqnarray}
The potential minima can be obtained as follows;
\begin{eqnarray} \nonumber
\frac{\partial V}{\partial S^{\dagger}} &=& \sigma_{\chi}^2 S\left( \vert \chi \vert^2+\vert \bar{\chi} \vert^2\right)+\sigma_{\chi} \lambda_{ij} \nu_{i}^{c}\nu_{j}^{c}\bar{\chi}^{\dagger} + m_{S}^2 S = 0 ,\\ \nonumber
\frac{\partial V}{\partial \bar{\chi}^{\dagger}}&=&\sigma_{\chi}^2\left( \chi\vert \chi \vert^2 +\bar{\chi} \vert S \vert^2\right)-100 g_{\chi}^2 \bar{\chi}\left( \vert \chi \vert^2 - \vert \bar{\chi} \vert^2 \right)+\sigma_{\chi} \lambda_{ij} \nu_{i}^{c}\nu_{j}^{c}S^{\dagger} + m_{\bar{\chi}}^2 \bar{\chi}= 0 ,\\ \nonumber
\frac{\partial V}{\partial \chi^{\dagger}}&=& \sigma_{\chi}^2\left( \bar{\chi}\vert \chi \vert^2 +\chi \vert S \vert^2\right)+100 g_{\chi}^2 \chi \left( \vert \chi \vert^2 - \vert \bar{\chi} \vert^2 \right)+4 \lambda_{ij}^2 \vert \nu_i^c \vert^2\chi + m_{\chi}^2 \chi= 0,\\
\frac{\partial V}{\partial {\nu_i^c}^{\dagger}} &=& 2{\nu_i^c}^{\dagger}\left(\sigma_{\chi} S \bar{\chi} +\lambda_{ij} \nu_{i}^c \nu_{j}^c\right) +4 \lambda_{ij}^2 \nu_{i}^{c} \vert \chi \vert^2 + m_{\nu_i}^2 \nu_i^c = 0 .
\end{eqnarray}
Conservation of $R$-parity requires, $\langle \nu_i ^c \rangle = 0$. The VEV of the fields $\chi$, $\bar{\chi}$ is found to be,
\begin{equation}
\langle \vert \bar{\chi} \vert\rangle=\langle \vert \chi \vert\rangle = \sqrt{-\frac{m_S^2}{2 \, \sigma_{\chi}^2}}\;.
\end{equation}
The negative mass squared, $m_{S}^2 < 0$ should be satisfied at an intermediate scale $M_{*}$ below the GUT scale to realize the correct $U(1)_{\chi}$ symmetry breaking. A negative mass squared can be achieved through the RG running from the GUT scale to an intermediate scale with a large enough Yukawa coupling even if the mass squared is positive at the GUT scale.
We consider the $U(1)_{\chi}$ renormalization group equations and analyze the running of the scalar masses $m_{\chi}^2$, $m_{\nu_i^c}^2$, $m_{\bar{\chi}}^2$ and $m_{S}^2$. A negative mass-squared $m_{S}^2$ will trigger the radiative breaking of $U(1)_{\chi}$ symmetry. We show that the mass-squared of the fields $\chi$, $\bar{\chi}$, $\nu^c$ and $S$ evolve in such a way that $m_S^2$ becomes negative whereas $m_{\nu_i^c}^2$, $m_{\chi}^2$ and $m_{\bar{\chi}}^2$ remain positive.
The renormalization group equations are given by
\begin{eqnarray}
16 \pi^2 \frac{d g_{\chi}}{d t} &=& \frac{57}{5} g_{\chi}^3 , \\
16 \pi^2 \frac{d M_{\chi}}{d t} &=& \frac{114}{5} g_{\chi}^2 M_{\chi} , \\
16 \pi^2 \frac{d \lambda_{i}}{d t} &=& \lambda_{i} \left( 8 \lambda_{i}^2 + 2 \Tr \lambda^2 + \sigma_{\chi}^2 - \frac{15}{2} g_{\chi}^2 \right) , \\
16 \pi^2 \frac{d \sigma_{\chi}}{d t} &=& \sigma_{\chi} \left( 3 \sigma_{\chi}^2 + 2 \Tr \lambda^2 - 10 g_{\chi}^2 \right) , \\
16 \pi^2 \frac{d m_{\chi}^2}{d t} &=& 2 \sigma_{\chi}^2 \left( m_{\chi}^2 + m_{\bar{\chi}}^2 + m_S^2 \right) + 4 m_{\chi}^2 \Tr \lambda^2 + 8 \Tr \left(m_{\nu^c}^2\lambda^2\right) \\ &+& 4 T_{\sigma_{\chi}}^2 - 20 g_{\chi}^2 M_{\chi}^2 , \\
16 \pi^2 \frac{d m_{\bar{\chi}}^2}{d t} &=& 2 \sigma_{\chi}^2 \left( m_{\chi}^2 + m_{\bar{\chi}}^2 + m_S^2 \right) + 2 T_{\sigma_{\chi}}^2 - 20 g_{\chi}^2 M_{\chi}^2 , \\
16 \pi^2 \frac{d m_{\nu^c_{i}}^2}{d t} &=& 8 \lambda_i^2 \left( m_{\chi}^2 + 2 m_{\nu^c_{i}}^2 \right) + 8 T_{\nu i}^2 - 5 g_{\chi}^2 M_{\chi}^2 , \\
16 \pi^2 \frac{d m_{S}^2}{d t} &=& 2 \sigma_{\chi}^2 \left( m_{\chi}^2 + m_{\bar{\chi}}^2 + m_S^2 \right) + 2 T_{\sigma_{\chi}}^2 , \\
16 \pi^2 \frac{d T_{\sigma_{\chi}}}{d t} &=& T_{\sigma_{\chi}} \left( 9 \sigma_{\chi}^2 + 2 \Tr \lambda^2 - 10 g_{\chi}^2 \right) + 4 \sigma_{\chi} \left( \Tr \lambda^2 + 5 g_{\chi}^2 M_{\chi}\right) ,\\
16 \pi^2 \frac{d T_{\nu_i}}{d t} &=& T_{\nu_i} \left( \sigma_{\chi}^2 + 2 \Tr \lambda^2 + 24 \lambda_i^2 - \frac{15}{2} g_{\chi}^2 \right) \\ &+& 2 \lambda_i \left( 2 \Tr \lambda^2 + 15 g_{\chi}^2 M_{\chi} + T_{\sigma_{\chi}} \sigma_{\chi} + 2 \Tr (\lambda_i T_{\nu_i}) \right),
\end{eqnarray}
\begin{figure}[!htb]
\centering \includegraphics[width=10.0cm]{plots/radiative_breaking.pdf}
\caption{The evolution of scalar squared masses $m_{\chi}^2$, $m_{\nu^c}^2$, $m_{\bar{\chi}}^2$ and $m_{S}^2$ from the GUT to TeV scale.}
\label{RGEs}
\end{figure}
where $\lambda_{ij} = \diag \left( \lambda_1, \lambda_2, \lambda_3\right)$. The evolution of these parameters depends on the boundary conditions at GUT scale, $M_{\text{GUT}}=2\times 10^{16}$ GeV. We assume universal soft SUSY breaking at this scale,
\begin{gather} \label{mzb}
\quad m_{\chi}^2 = m_{\bar{\chi}}^2 = m_{\nu_i^c}^2 = m_{S}^2 = m_0^2, \quad M_{\chi} =16 ~ \text{TeV}, \nonumber \\
g_{\chi}^2=1.04,\quad \lambda_{3} = 0.4, \quad \sigma_{\chi} = 0.3, \quad m_0 = 3.2 ~ \text{TeV}, \\ \nonumber
T_{\nu_i} = \lambda_{i} A_{\nu}, \quad T_{\sigma_{\chi}} = \sigma_{\chi} A_{\chi} .
\end{gather}
Here we let the tri-linear couplings ($A_{\nu} = A_{\chi} = 0$) vanish as they have negligible effect on the overall running of the parameters. Also, for simplicity, we neglect the couplings of the first two right handed neutrino (RHNs) generations ($\lambda_1 = \lambda_2 = 0$). Fig. \ref{RGEs} shows the running of scalar masses from GUT scale. It can be seen that the mass-squared $m_{S}^2$ turns negative at scale $\sim 8 \times 10^{12}$ GeV, whereas $m_{\bar{\chi}}^2$ rapidly increase and approaches $\sim 10^9 ~ \text{GeV}^2$ around TeV scale. Note that, although the running of mass squared $m_{\nu_i^c}^2$ and $m_{\chi}^2$ decreases from $M_{\text{GUT}}$, they always remain positive.
The breaking of $U(1)_{\chi}$ at the end of inflation yields topologically stable cosmic strings on which, the observational bounds are given in terms of the dimensionless quantity $G_N \mu_s$, which characterizes the strength of the gravitational interaction of the strings. Where $G_N$ is the Newton’s constant, and $\mu_s \simeq 2 \pi \langle \chi \rangle^2$ denotes the mass per unit length of the string. The Planck bound on $G_N \mu_s$ derived from constraints on the string contribution to the CMB power spectrum is given by \cite{Ade:2013xla,Ade:2015xua}
\begin{equation}
G_N \mu_s \lesssim 2.4 \times 10^{-7}.
\end{equation}
This then translates to the following upper bound on $U(1)_{\chi}$ breaking scale
\begin{equation}
\langle \chi \rangle \lesssim 2.35 \times 10^{15} ~ \text{GeV},
\end{equation}
which is easily satisfied as can be seen from Fig. \ref{RGEs} and depends on the initial boundary conditions at GUT scale. After the $U(1)_{\chi}$ symmetry breaking, the RHNs and the $U(1)_{\chi}$ gauge boson $(Z^{\prime})$ acquire the following masses;
\begin{equation}
M_{Z^{\prime}}^2\simeq g_{\chi}^2\langle \chi \rangle^2, \quad m_{\nu_3}^2\simeq \lambda_{3}^2 \langle \chi \rangle^2 ,
\end{equation}
which for the particular boundary condition in \eqref{mzb} yields, $M_{Z^{\prime}}\simeq 4.89$ TeV and $m_{\nu_3} \simeq 2.44$ TeV. The bound on $M_{Z^{\prime}}$ is constantly being updated by comprehensive analyses. The severest bound on $M_{Z'}$ comes from negative results of LEP data, $M_{Z'}/g_{\chi} \geq 6$ TeV \cite{Cacciapaglia:2006pk}. Even though considering its decay modes can lower the mass bound on $Z^{\prime}$ \cite{Accomando:2013sfa}, setting $M_{Z^{\prime}} \geq 4$ TeV \cite{ATLAS:2017wce} guarantees avoiding possible exclusions limit due to the light $Z^{\prime}$ mass.
The mass of RHNs predicted by the above model is of the order of TeV scale. Since they are singlet under the SM gauge group, a mixing between the RHNs and the SM neutrinos is generated through the Dirac Yukawa coupling in the seesaw mechanism. As a result, the RHN mass eigenstates couple to the weak gauge bosons $W,Z$ through this mixing.
Although in general, this mixing can be made sizable even for TeV-scale RHNs, contrary to the naive seesaw expectations, under special textures of the Dirac and RHN Majorana mass matrices~\cite{Pilaftsis:1991ug, Tommasini:1995ii, Gluza:2002vs, Xing:2009in, Gavela:2009cd, He:2009ua, Adhikari:2010yt, Deppisch:2010fr, Mitra:2011qr, Dev:2013oxa, Chattopadhyay:2017zvs}, it has been shown \cite{Das:2017nvm} that
this mixing has an upper bound of ${\cal O}(0.01)$ to satisfy various experimental constraints, such as the neutrino oscillation data, the electroweak precision measurements, neutrinoless double beta decay
and the charged lepton flavor violating (LFV) processes.
Hence, the canonical production cross section of TeV-scale RHNs through either the weak gauge bosons~\cite{Datta:1993nm, Panella:2001wq, Han:2006ip, delAguila:2007qnc, Dev:2013wba, Alva:2014gxa, Das:2015toa, Das:2016hof, Pascoli:2018heg} or the Higgs boson~\cite{Dev:2012zg, Cely:2012bz, Hessler:2014ssa, Das:2017zjc, Das:2017rsu} at the LHC is expected to be very small within the minimal seesaw.
In the above model, all SM fermions as well as the RHNs have non-zero $U(1)_\chi$ charges, and therefore, the RHNs can be efficiently produced at colliders, in particular, through the resonant production of $Z^\prime$ boson, if kinematically allowed, and its subsequent decay into a pair of RHNs.
\section{\large{\bf Summary}}\label{sec8}
We have explored shifted hybrid inflation in the framework of supersymmetric $SU(5) \times U(1)_{\chi}$ model where $SU(5)$ gauge symmetry is spontaneously broken during inflation, inflating the disastrous magnetic monopoles away. The $U(1)_{\chi}$ symmetry is radiatevely broken after the end of inflation at an intermediate scale, yielding topologically stable cosmic strings. The symmetry breaking scale of $U(1)_{\chi}$ depends on the initial boundary conditions at the GUT scale and easily satisfies Planck's bound on $G_N \mu_s$. The $d = 5$ proton lifetime for the decay $p \rightarrow K^+ \bar{\nu}$, mediated by color-triplet Higgsinos is found to satisfy Super-Kamiokandae experimental bounds for SUSY breaking scale $M_{\text{SUSY}} \gtrsim 10$ TeV. We have shown that with minimal K\"ahler potential, the soft supersymmetry breaking terms play a vital role in bringing the scalar spectral index $n_s$ within the Planck's latest bounds and the $SU(5)$ guage symmetry breaking scale is obtained in the range ($2 \times 10^{15} \lesssim M_{\alpha} \lesssim 2 \times 10^{16}$) GeV with small values of tensor-to-scalar ratio $r \lesssim 2.5 \times 10^{-6}$. In a non-minimal K\"ahler potential setup, large values of tensor to scalar ratio are obtained ($r \lesssim 10^{-3}$) with non-minimal couplings ($-0.011 \lesssim \kappa_S \lesssim - 0.00063$) and ($0.34 \lesssim \kappa_{SS} \lesssim 1$) and symmetry breaking scale $M_{\alpha} \simeq M_{\text{GUT}} = 2 \times 10^{16}$ GeV.
\section*{Acknowledgements}
The authors would like to thank Mansoor Ur Rehman, Lorenzo Callibi, Shabbar Raza and Qaisar Shafi for helpful discussions.
|
1,116,691,497,595 | arxiv | \section{Acknowledgements}
The authors acknowledge discussions with
A. Macdonald, D. Ralph, Kelly Luo, Vishakha Gupta, Rakshit Jain, Nai Chao Hu, Bowen Shen, and Zui Tao.
Work at UCSB was primarily supported by the Army Research Office under award W911NF-20-2-0166 and by the Gordon and Betty Moore Foundation EPIQS program under award GBMF9471.
Work at Cornell was funded by the Air Force Office of Scientific Research under award no. FA9550-19-1-0390.
K.W. and T.T. acknowledge support from JSPS KAKENHI (Grant Numbers 19H05790, 20H00354 and 21H05233).
ER and TA were supported by the National Science Foundation through Enabling Quantum Leap: Convergent Accelerated Discovery Foundries for Quantum Materials Science, Engineering and Information (Q-AMASE-i) award number DMR-1906325.
CLT acknowledges support from the Hertz Foundation and from the National Science Foundation Graduate Research Fellowship Program under grant 1650114.
\section{Methods}
\subsection{Device fabrication}
AB-stacked 2L-MoTe$_2$/WSe$_2$ devices were fabricated using the layer-by-layer dry transfer method discussed in detail in \cite{li_quantum_2021}. An optical image of the device is presented in Fig. \ref{fig:sfigschematic}a. A dashed line identifies the extent of the few-layer graphene bottom gate. A black rectangle identifies a region illustrated in schematic form in Fig. \ref{fig:sfigschematic}b. Contact is made to the moir\'e superlattice formed by the MoTe$_2$ and WSe$_2$ crystals with $\approx5$~nm platinum contacts prepatterned onto a hBN flake; these are themselves contacted with gold wires outside the encapsulated region of the heterostructure. The contacts used for the measurements presented here are labelled in Fig. \ref{fig:sfigschematic}b and are referred to throughout the main text using these labels. Contacts f, g, h, and i are used to probe $R_{xy}$ and $R_{xx}$ in the Chern magnet. The relative locations of the WSe$_2$ monolayer, MoTe$_2$ bilayer, and few-layer graphene top gate are marked in red, blue, and light gray, respectively. The region of overlap between these three flakes defines the device- the bottom gate is omitted for simplicity; it covers this region of overlap entirely, and thus does not define any edges of the dual-gated moir\'e superlattice. A dashed rectangle identifies the region imaged using nanoSQUID magnetometry in Fig. \ref{fig:fig2}. The precise locations of the contacts relative to the device were determined using atomic force microscopy (AFM); this data is presented with an overlaid outline of the top gate (black line) and contacts (dashed lines) in \ref{fig:sfigschematic}c.
Optical images of the WSe$_2$ monolayer, the few-layer graphene top gate, and the MoTe$_2$ bilayer are presented in Fig. \ref{fig:sfigschematic}d-f. The crystal axes of the MoTe$_2$ and WSe$_2$ flakes were identified optically using angle-resolved second harmonic generation (SHG) and aligned with a 60$^{\circ}$ offset. In the case of the MoTe$_2$ bilayer (for which SHG cannot provide useful information) the crystal axes were identified for an attached monolayer (Fig. \ref{fig:sfigschematic}f). The relative positions of the top gate and WSe$_2$ flake were determined using optical microscopy during the stacking process (Fig. \ref{fig:sfigschematic}g).
\subsection{Electrical transport measurements}
The measurements presented here were conducted in a pumped liquid helium cryostat at a base temperature of 1.6K.
AC transport data was acquired using a finite frequency excitation $\delta I\approx 300$~pA at $f\approx17$~Hz for Fig. \ref{fig:fig1}b and \ref{fig:fig2}b and $f\approx3$~kHz for Fig. \ref{fig:fig3}a-b.
Data in Figs. \ref{fig:fig1}b and \ref{fig:fig2} b are field symmetrized, so the plotted resistivity R$_{xx}(B)$ = \((R_{meas}(B) + R_{meas}(-B))/2\) and
R$_{xy}(B)$ = \((R_{meas}(B) - R_{meas}(-B))/2\).
\subsection{Magnetic imaging}
Magnetic imaging is performed using a nanoscale superconducting quantum interference device (nanoSQUID).
The typical static magnetic field in the region of space accessible by the nanoSQUID is below the DC noise floor of our sensor. Therefore, it is necessary to generate signals at finite frequency. We employ several different methods to generate the data presented in the main text and extended data figures including bottom gate modulation, spatial modulation of the nanoSQUID position, and modulation of the current in the device. Additional descriptions of these several techniques used to generate the data in the main text are available in the literature\cite{vasyukov_imaging_2017,anahory_squid--tip_2020,uri_nanoscale_2020,tschirhart_imaging_2021}.
\subsubsection{Bottom gate modulation magnetometry}
As discussed in the main text, the magnetic signal may be modulated via carrier density variation, producing an AC response in the local magnetic field detected by the nanoSQUID. In practice, this is implemented using the circuit shown in Fig. \ref{fig:fig2}a. For the data shown in Figs. \ref{fig:fig2}a, c, d-e, Fig. \ref{fig:sfigureacbgmagnetization}, the bottom gate modulation has peak-peak amplitude of $\delta V_{BG}=35$~mV applied at frequency $f\approx 3$~kHz. We assume the resulting spatial map of $\delta B$ obeys $\delta B = \frac{dB_z}{dV_{BG}} \delta V_{BG}$.
We reconstruct the magnetization under the assumption that it is entirely out-of-plane, so that $\vec m=m_z \hat z$. Reconstructed $m_z$ is shown in Figs. \ref{fig:fig2}d-e, \ref{fig:sfigureacbgmagnetization}, and \ref{fig:sfiguremagnetizationinhole}. To do this, we first determine $B_z$, which is accomplished by acquiring $\delta B$ over a continuous range of $V_{BG}$ that spans the entire range of the magnetism. As described in Fig. \ref{fig:sfigset}, the nanoSQUID is also sensitive to electric fields due to parasitic conduction through quantum dots near the tip. However, the $V_{BG}$ dependence of this signal is screened by the top gate, and varies slowly with $V_{BG}$ in the device regions outside the extent of the top gate. This spurious signal is eliminated by assuming that $B_z=0$ for values of $V_{BG}$ both lower and higher than the narrow domain of $V_{BG}$ where we observe magnetic structure in the bulk and in transport. Reconstruction of the magnetization from $B_z$ may then be done by Fourier transform techniques identical to those described in \cite{tschirhart_imaging_2021}. A schematic of this analysis is shown in Fig. \ref{fig:sfigureacbgmagnetization}, and video format data of the bottom gate evolution are available as supplementary data.
\subsubsection{Spatial gradient magnetometry}
Figs. \ref{fig:fig3}c-d show a reconstruction of the steady state magnetization under an applied DC current. To avoid convolving modulations of the resistivity by the bottom gate with our detected SQUID signal, the magnetization is measured using gradient magnetometry. In this technique, we contact the nanoSQUID tip with a piezoelectric tuning fork (TF) which is modulated at $f\approx 32$~kHz.
The resulting modulation of the in-plane nanoSQUID tip displacement produces a signal $\delta B_{TF} \approx \delta \vec r \cdot \vec \nabla_r B_z$, where $\delta \vec r$ is the vector describing the spatial modulation of the tip. As described in both \cite{tschirhart_imaging_2021} and Fig. \ref{fig:sfigureTFdomains}, tuning fork measurements produce a number of additional spurious signals arising from electric fields and mechanical interactions with the surface. However, as in the bottom gate modulation magnetometry, these do not vary with the independent variable of interest here, the applied DC current $I_{SD}$. We thus analyze the difference images between zero and finite DC current, which contain only $\delta B_{TF}$. To convert $\delta B_{TF}$ to $B_z$, we integrate the signal along the direction of the oscillation, producing a map of $B_z$. This may then be converted to $m_z$ using the same standard Fourier domain analysis techniques described above and in \cite{tschirhart_imaging_2021}. We did not precisely calibrate $\delta \vec r$ during this experimental run, and so provide the extracted $m_z$ in arbitrary units. However, prior work with the identical setup\cite{tschirhart_imaging_2021} allows us to estimate both the magnitude $|\delta \vec r|\approx 100$~nm and direction of $\delta \vec r$ (see Fig. \ref{fig:sfigureTFdomains}i).
\subsubsection{Current modulation magnetometry}
As described in the main text, AC currents may also modulate magnetic structure and thus the local magnetic field signal. Data in Figs. \ref{fig:fig3}e,f,h,i, \ref{fig:fig4}c-g, \ref{fig:sfigbdependence}a,c,e-h, \ref{fig:sfigure3explained}e-f, \ref{fig:sfigureAllDomains}b-g, are all acquired in this way, with a current modulation $\delta I_{SD}$ applied at $f\approx3$~kHz. The contact configuration and amplitude of the applied current vary between data sets. Applying AC and DC bias to the source contact and grounding the drain, the parameters are: \begin{itemize}
\item Fig. \ref{fig:fig3}e, Fig. \ref{fig:sfigure3explained}f
\subitem Source=j ; Drain=a,b,c,d
\subitem $\delta I_{SD} = 30$~nA
\item Fig. \ref{fig:fig3}f, Fig. \ref{fig:sfigure3explained}c
\subitem Source=a,b,c,d ; Drain=j
\subitem $\delta I_{SD} = 30$~nA
\item Fig. \ref{fig:fig3}h-i, Fig. \ref{fig:sfigbdependence}e-h
\subitem Source=a,b,c,d ; Drain=j
\subitem $\delta I_{SD} = 5$~nA
\item Fig. \ref{fig:sfigbdependence}a
\subitem Source=a,b,c,d ; Drain=j
\subitem $\delta I_{SD} = 10$~nA
\item Fig. \ref{fig:sfigbdependence}c
\subitem Source=a,b,c,d ; Drain=j
\subitem $\delta I_{SD} = 89$~nA
\item Fig. \ref{fig:fig4}c-f
\subitem Source= j; Drain=a,b,c,d
\subitem $\delta I_{SD} = 125$~nA
\item Fig. \ref{fig:fig4}g
\subitem Source= j; Drain=a,b,c,d
\subitem $\delta I_{SD} \in [0$ $270$~nA$]$
\item Fig. \ref{fig:sfigureAllDomains}b,d,f
\subitem Source= a,b,c,d; Drain=j
\subitem $\delta I_{SD}$ indicated on figure.
\item Fig. \ref{fig:sfigureAllDomains}b,d,f
\subitem Source= j; Drain=a,b,c,d
\subitem $\delta I_{SD}$ indicated on figure.
\end{itemize}
Throughout the main text figures, we standardize the phase of the AC current such that positive corresponds to increasing magnitude of current.
\clearpage
\normalem
\let\oldaddcontentsline\addcontentslin
\renewcommand{\addcontentsline}[3]{
|
1,116,691,497,596 | arxiv | \section{Introduction}
\label{sec:intro}
The $\Lambda$ cold dark matter ($\Lambda$CDM) model has become
well established as the standard model of cosmology,
due to its very impressive fit
to a variety of cosmological observations, including
CMB anisotropy \citep{wmap9h,planck-xvi}, large-scale
galaxy clustering including the baryon acoustic oscillation
(BAO) feature \citep{boss13}, and the Hubble diagram for distant supernovae
(SNe; \citealt{betoule14}). In $\Lambda$CDM and close relatives, the
mass-energy content of the Universe
underwent a transition from matter domination to dark
energy domination in the recent past at a redshift $z_{me} \sim 0.33$;
the transition from decelerating to accelerating expansion,
hereafter $\zacc$, was somewhat
earlier, at a redshift $\zacc \approx 0.67$. In $\Lambda$CDM,
these are given by $1 + \zacc = \sqrt[3]{ 2 \Omega_\Lambda / \Omm }$
and $1 + z_{me} = \sqrt[3]{\Omega_\Lambda / \Omm }$, so
$1 + \zacc = \sqrt[3]{2} (1 + z_{me})$.
We see later that the value of $\zacc$ is relatively insensitive
to dark energy properties, assuming standard GR and simple
parametrizations of the dark energy equation of state.
The most direct evidence for recent accelerated expansion comes
from the many observations of distant SNe at $0.02 < z \simlt 1.5$;
the early SN results
in 1998 \citep{hiz98, scp99} began a rapid acceptance
of dark energy, due also to previous indirect evidence from large-scale
structure \citep{esm90},
the cluster baryon fraction
\citep{white93} and the Hubble constant \citep{ferr96}.
Strong independent support came from observation of the
first CMB acoustic peak defining a near-flat
universe \citep{boom00, maxima00},
combined with decisive evidence for a low value of $\Omm$ from
the 2dF Galaxy Redshift Survey \citep{peacock01, psp02}.
In the past decade there has been a rapid improvement
in the precision of observations in all these areas (see references above),
most recently from the {\em Planck}, Baryon Oscillation Spectroscopic
Survey (BOSS) and Supernova Legacy Survey (SNLS) projects.
Current joint constraints are impressively consistent
with $\Lambda$CDM with $\Omm \simeq 0.30$
and $H_0 \simeq 68.3 \hunit$ \citep{boss13, betoule14}.
Many deductions in cosmology are based on
six, seven or eight-parameter fits of extended $\Lambda$CDM to
observational data, which generally show good consistency with
the six-parameter model and place upper limits on the
additional parameters.
However, given our substantial ignorance of the
nature of dark energy, it is clearly interesting to ask what
we can deduce with fewer assumptions, e.g. keeping
the cosmological principle while dropping the assumption of
standard gravity.
In particular, fitting models of GR with dark energy to the data
produces a reasonably sharp prediction for the value of $\zacc$;
however, if the apparent cosmic acceleration is due to another cause
such as modified gravity \citep{cfps}, a giant local void \citep{celerier}
or other, this may not necessarily hold; therefore, it is of considerable
interest to see what constraints we can place on $\zacc$ {\em without}
assuming specific models.
It has been shown by e.g. \cite{shap-turn} that the SN
brightness/redshift relation does provide
evidence for accelerated expansion independent of GR; but
direct evidence for past deceleration is less secure.
A number of other authors have explored GR-independent constraints
on the cosmic expansion history, {\newtwo dark energy evolution}
and/or $\zacc$;
{ \newtwo e.g.
\cite{ss06} provide a broad review mainly focused on
dark energy reconstruction; \citet{catt-viss} explore various
distance definitions related to $z$ or $y = z/(1+z)$;
\cite{cl08} derived constraints on $\zacc$ from SNe
assuming simple parametrizations of deceleration parameter $q(z)$;
\cite{clark-zunck} provide a method for non-parametric
reconstruction of $w(z)$ (mainly from future high-quality data);
\cite{mort-clark} provide non-parametric
estimates of $H(z)$;
and \cite{ngb13} give a comparison of several methods
for estimating $w(z)$ from SNe data. }
Our work is partly related to these,
but focusing more on the possibility of non-parametric
constraints specifically on $\zacc$;
where we overlap we are generally in agreement.
The plan of the paper is as follows: in Section~\ref{sec:dl}
we discuss the value of $\zacc$ and the SN Hubble diagram,
and the cause of the downturn in the latter.
In Section~\ref{sec:flatna} we point out several advantages of
comparing SN residuals relative to
a flat non-accelerating model.
We discuss some future prospects in Section~\ref{sec:disc},
and we summarize our conclusions in Section~\ref{sec:conc}.
Our default model is $\Lambda$CDM with $\Omm = 0.300$; $H_0$ generally
cancels except where stated.
\section{Relation between luminosity distances and $\zacc$}
\label{sec:dl}
\subsection{The expected value of $\zacc$}
\label{sec:zacc}
Here we note that the value \footnote{ In highly non-standard models,
it is not guaranteed that $\zacc$ (defined by $\ddot{a} = 0$)
is single-valued;
e.g. if there were short-period low-amplitude
oscillations in $\dot{a}$, or a past accelerating
phase transitioned back to deceleration at a very low redshift,
then in principle $\zacc$ may be multi-valued.
These possibilities appear improbable and hard to test
observationally,
so we assume $\zacc$ is single-valued (after the CMB era)
for the remainder of this paper; see also \cite{linder10}.}
of $\zacc$ is now constrained rather
well in flat $w$CDM models with constant dark energy
equation of state $w$; for this model family, $\zacc$ depends on
only $\Omm$ and $w$, and is given by
\begin{equation}
\label{eq:zacc}
1 + \zacc = \left[ (-1 - 3w) (1-\Omm) / \Omm \right]^{-1/3w} \
\end{equation}
(e.g. \citealt{tr02}).
This is shown in a contour plot in Fig.~\ref{fig:zacc}.
It is interesting that in the neighbourhood of $\Omm \sim 0.3, w \sim -1$,
the contours of constant $\zacc$ are nearly vertical,
thus $\zacc$ is nearly independent of $w$ and is well
approximated by
\begin{equation}
\zacc \simeq 0.671 - 2.65 (\Omega_m - 0.3) \ .
\end{equation}
Qualitatively, this occurs because as $w$ increases above $-1$,
there is less negative pressure hence less acceleration
per unit $\rho_{DE}$,
but larger $w$ gives higher $\rho_{DE}$ in the past; these effects
happen to cancel (largely coincidentally) near the concordance model,
so $\zacc$ is rather insensitive to $w$.
This has positive and negative consequences: on the one hand, measuring
$\zacc$ is not useful for constraining $w$; on the other hand,
the range $0.60 \le \zacc \le 0.75$ appears to be a robust prediction of
$w$CDM, so if future data
{\newtwo (e.g. direct measurements
of $H(z)$ from BAOs or cosmic chronometers,
or new more precise SN data)}
were to empirically measure $\zacc$ {\em outside} this
range, it could essentially falsify the whole class of $w$CDM models.
(Models with time-varying $w$ such as the common model $w(a) = w_0 + w_a(1-a)$
allow a wider range of $\zacc$, but these generally require $\zacc < 1$
unless $w_a$ is dramatically negative, $w_a \simlt -1$, which is
disfavoured in most quintessence-type models).
In Fig.~\ref{fig:zacc} we also show contours of $(1+\zacc)/E(\zacc)$,
which is equivalent to the ``net speedup'' or integrated acceleration
between $\zacc$ and today; this is discussed later in \S~\ref{sec:flatna}.
\begin{figure*}
\includegraphics[angle=-90,width=12cm]{fig_zacc_cont2.eps}
\caption{
A contour plot of the acceleration redshift $\zacc$,
and $(1+\zacc)/E(\zacc)$, as functions of
$\Omm, w$ for flat wCDM models. The dotted horizontal line
shows $w = -1$. \hfill\break
The solid black contours show $\zacc$, in linear steps of 0.1 from
0.35 (right) to 0.95 (left). The dashed green contours show
$(1+\zacc)/E(\zacc)$ (i.e. total net speed-up) in linear steps
of 0.05 from 1.05 (upper right) to 1.35 (lower left). Selected contours
are labelled.
\label{fig:zacc}
}
\end{figure*}
\subsection{SN data}
\label{sec:sn}
For comparison with models, we use the ``Union 2.1'' compilation of
type-Ia SN distance moduli \citep{union21}, which contains
580 SNe of good quality spanning the range
$0.01 < z < 1.6$. For plotting purposes
we divide the sample into bins of approximately equal width in
$\ln(1+z)$, while adjusting bin widths so that each bin contains
$\ge 20$ SNe except at the highest redshifts; then, the mean
distance modulus residual and weighted average redshift are computed
for each bin. The resulting binned data points are
shown as `Union 2.1' in subsequent figures.
We show a fit of this data set to flat $w$CDM models (with
$\Omega_m$ and constant $w$ as the fit parameters; results of this
fit are shown in Fig.~\ref{fig:sn-omw},
with a best-fitting point near $\Omm = 0.28, w = -1.01$.
This shows the well-known
degeneracy track between $\Omega_m$ and $w$; here we note that
the long axis of the track is quite similar to the
contour $(1+\zacc)/E(\zacc) \approx 1.15$ in Fig.~\ref{fig:zacc};
this is discussed in later sections.
We note that a more recent SN Ia compilation has
been produced by \citet{betoule14} which includes
more intermediate redshift SNe,
more detailed photometric calibration and expanded treatment of
systematic errors; however, the best-fitting
parameters from the latter paper are within $1\sigma$
of those above, so the slight difference is not important for
the remainder of this paper.
\subsection{Fiducial models and $\Delta\mu$}
\label{sec:dmu}
The observations of Type Ia SNe are sensitive to
the standard luminosity distance $D_L(z)$ for each SN,
plus some scatter due to the intrinsic dispersion in absolute
magnitude per SN. In practice, the distant $z \simgt 0.1$
SNe are compared to a local sample ``in the Hubble flow''
typically at $z \sim 0.02$ to $0.05$; for the local sample,
peculiar velocities are assumed to be relatively small compared
to the cosmological redshift, so the value of $H_0$ cancels
with the (unknown) characteristic
luminosity $L_c$ of a standardized SN. Thus, quasi-local
SNe really constrain the degenerate combination $h^2 L_{c}$
or equivalently $M_c + 5 \log_{10} h$; and comparison of distant
and local SN samples actually constrains the distance ratio
$D_L(z) / D_L(z \sim 0.03)$, rather than the absolute distance.
The value of $D_L(z)$ spans a very wide range over the redshift
interval covered by SNe: from
$z \sim 0.03$ to $z \sim 2$ is a factor of $\approx 118$
in distance or $10.3$ magnitudes,
while the differences between models are relatively modest:
e.g. 15\,percent differences between
$\Lambda$CDM and a zero$-\Lambda$ open model,
down to differences $\sim 2\,$percent between
$\Lambda$CDM and a $w = -0.9$ model.
This implies that plotting $D_L(z)$ versus $z$ directly is not very informative
since model differences are very small compared to the plot range;
therefore it is common to present SN results
as residuals relative to some fiducial model; residuals are often
presented in distance modulus or magnitude units, i.e.
\begin{equation}
\Delta\mu(z) \equiv 5 \log_{10} { D_L(z) \over D_{L,{\rm fid}}(z) }
\end{equation}
where $D_{L,{\rm fid}}$ is the value for some fiducial model.
The choice of fiducial model is essentially arbitrary (up to small
binning effects second-order in bin size); however, this choice can have
a strong effect on the shape of the results and
intuitive deductions, as shown below.
\begin{figure}
\hspace*{-5mm}\includegraphics[angle=-90,width=10cm]{fig_cont_wcdm2.eps}
\caption{
The allowed region in the $(\Omm, w)$ plane
from fitting flat constant-$w$ models to the Union 2.1 SN
sample. Contours show the values of $\Delta \chi^2 = 2.3, 6.0, 10.6$,
corresponding to 68, 95 and 99.8 percent confidence regions.
\label{fig:sn-omw}
}
\end{figure}
\begin{figure*}
\includegraphics[angle=-90,width=15cm]{fig_dmu_milne3.eps}
\caption{
Distance modulus residuals relative to the Milne model for various
cosmological models. The solid black lines show $\Lambda$CDM with
$\Omm = 0.27$ (upper) and $0.30$ (lower).
Long-dashed red lines show the corresponding D0 models (Equation~\ref{eq:d0})
with deceleration artificially turned off above $\zacc$.
Dashed green lines show Friedmann models
of historical interest: from top to bottom,
a pure-vacuum model ($\Omlam = 1$); an open model with $\Omm = 0.27$,
$\Omlam = 0$; and an Einstein-de Sitter model $(\Omm = 1)$.
Dotted blue lines show constant-$q$ models with
$q_c = -0.6, -0.4, -0.33$ respectively (top to bottom).
\label{fig:dmue}
}
\end{figure*}
One obvious choice of fiducial is $\Lambda$CDM itself; however, this
makes observed residuals (almost) flat--line, which does not translate readily
into inferences on deceleration or acceleration.
Another common choice of fiducial model is the empty or Milne model,
with $\Omm = 0$, $\Omlam = 0$, $\Omk = 1$,
as used by many notable papers e.g. \cite{hiz98, leib01, hiz04, gl11}.
The zero matter density means
this is clearly not a viable model for the real Universe, but
it is a convenient fiducial model for two reasons:
\begin{enumerate}
\item It has a very simple analytic form for $D_L(z)$, given by
\begin{equation}
\label{eq:dle}
D_{L,E}(z) = \frac{c}{H_0} z \left(1 + \frac{z}{2}\right) \ ;
\end{equation}
hereafter we define $\Delta \mu_E$ to be distance modulus residuals
relative to this.
\item For a given $H_0$, the Milne model has the maximum luminosity
distance among all Friedmann models with zero
dark energy (assuming non-negative matter density). Therefore,
observational evidence for distance ratios larger
than the Milne model (positive $\Delta \mu_E$) at any redshift
is direct evidence that we do not live in a Friedmann
model with zero dark energy.
\end{enumerate}
{\newtwo However, using the Milne model
as fiducial has some drawbacks which we discuss
in the next subsection;
we suggest an improved fiducial model in Section~\ref{sec:flatna}
(see also \citealt{mort-clark}).
}
\vfill
\newpage
\subsection{Downturn in distance residuals}
It is very well known that observed SN distance residuals
are all significantly positive at $0.2 \simlt z \simlt 0.6$, in agreement
with the $\Lambda$CDM accelerating expansion.
It is also fairly well known that $\Lambda$CDM models exhibit
a turning point (a maximum) in the $\dme(z)$ relation.
Fig.~\ref{fig:dmue} shows that this turning point, hereafter $z_{tp}$,
occurs at $z \simeq 0.50$ for the $\Omm = 0.300$ concordance model,
and the predicted residuals then decline to a zero-crossing
at $z \simeq 1.26$. It is seen in Fig.~\ref{fig:dmue} that
the actual supernova data do hint at the existence of a turnover,
with the three data points at $z > 0.9$ all slightly low compared
to their predecessors.
The actual evidence for this turnover is not decisive, but it is
clearly somewhat preferred by the data.
The turnover occurs quite close to the theoretical transition
epoch $\zacc \approx 0.67$, and it is therefore widely believed
(at least anecdotally) that supernovae have directly detected the
predicted cosmic deceleration at $z \simgt 1$.
{\newtwo We discuss some prior claims to this effect in
Appendix~\ref{app:decel}. }
We demonstrate in the next subsection that the latter conclusion does not
follow; specifically, while a downturn in $\dme$ {\em is} favoured by
the data, the downturn predicted by $\Lambda$CDM is {\em mostly}
caused by the negative space curvature in the fiducial Milne model,
and cosmic deceleration makes only a
minority contribution to the downturn.
The fairly close match between $z_{tp}$ and $\zacc$ is found to be
largely coincidental.
\subsection{Cause of the turnover in $\dme$}
\label{sec:turnover}
Assuming homogeneity, the luminosity distance $D_L(z)$ is given
by
\begin{equation}
\label{eq:dl}
D_L(z) = \frac{c}{H_0} (1+z) \, \frac{1}{\sqrt{\vert \Omk \vert} }
S_k\left( \sqrt{\vert \Omk \vert} \int_0^{z} \frac{ dz'}{E(z')} \right)
\end{equation}
with $E(z) \equiv H(z)/H_0$, and
the function $S_k(x) = \sin x, x, \sinh x$
for $k = +1, 0, -1$ respectively, where $k$ is the sign of the
curvature (opposite to the sign of $\Omk$, in the usual convention
where $\Omk = 1 - \Omega_{tot}$).
It is convenient to factorize this so that
\begin{eqnarray}
\label{eq:dldr}
D_L(z) & = & (1+z) \, D_R(z) \, \left( \frac{S_k(x)}{x} \right) \\
D_R(z) & = & \frac{c}{H_0} \int_0^z \frac{dz'}{E(z')} \quad =
c \int_0^z \frac{dz'}{H(z')} \\
x & \equiv & \frac{H_0 \, D_R(z) }{c} \, \sqrt{\vert \Omk \vert }
\end{eqnarray}
where $D_R(z)$ is the comoving radial distance to redshift $z$;
and $x$ is the dimensionless ratio between $D_R(z)$ and the
cosmic curvature radius, which in a Friedmann model is
$ R_c = c / H_0 \sqrt{\vert \Omk \vert } $.
We note that these distance results are still valid
in a homogeneous and isotropic non-GR model,
as long as the Robertson-Walker metric applies and
we define $\Omk$ from the curvature radius
via $\Omk \equiv \pm (H_0 R_c / c)^{-2}$,
which is then not necessarily equal to $1 - \Omega_{tot}$.
Looking at equation~(\ref{eq:dldr}), the first $(1+z)$ factor
is parameter-independent
and due to time-dilation and loss of photon energy;
these each give one power of $(1+z)^{-1}$ in flux, hence combine
to $(1+z)$ in equivalent distance.
The parameter dependence of $D_L(z)$ then factorizes into
two parts, the $D_R(z)$ term dependent only on expansion history,
and the factor $S_k(x)/x$ which depends mainly on
curvature and also (more weakly) on expansion history;
this is asymptotically $1 - k x^2/6$ for $x \ll 1$, or
$1 + \Omega_k z^2 / 6$ for $z \ll 1$.
The factorization above is helpful to understand the relative importance
of curvature versus acceleration/deceleration on the distances and
distance ratios. In the non-flat $\Lambda$CDM model, the
combination of {\em Planck}+BAO
data requires $\vert \Omega_k \vert < 0.008$ at 95 percent
confidence\footnote{
We note that in non-GR models the standard
limits on $\Omega_k$ does not apply;
however, if the true cosmology were a curved non-GR model, if
$\vert \Omega_k \vert \simgt 0.05$ we would then require a rather
close cancellation between curvature and non-GR effects
in order to make the non-flat $\Lambda$CDM fits turn out so
close to $\Omega_k = 0$.
If we discard this possibility as an unnatural conspiracy,
it is reasonable to assume
$\vert \Omega_k \vert < 0.05$, and in that case the curvature factor
$S_k(x)/x \approx 1 \pm 0.01$ for $z < 1.5$ for
reasonable expansion histories.},
{\newtwo (see equations~68a and b of \cite{planck-xvi})},
which implies that
the curvature factor is within 0.2~percent of 1 at
the redshift range $z \simlt 1.5$ of current SNe.
It is now interesting to compare terms in equation~(\ref{eq:dldr})
for the $\Lambda$CDM and empty models.
In the case of the empty model, $D_R(z) = (c/H_0) \ln(1+z)$,
$\Omega_k = +1$, so equation~(\ref{eq:dldr}) becomes
\begin{equation}
\label{eq:dlefac}
D_{L,E}(z) = (1+z) {c \over H_0} \ln(1+z)
\, { \sinh(\ln(1+z)) \over \ln(1+z)}
\end{equation}
which easily simplifies to equation~(\ref{eq:dle}).
However, it is more informative to keep the longer form
of equation~(\ref{eq:dlefac}) since
the rightmost fraction is a pure curvature effect; it is
well approximated by $1 + (\ln(1+z))^2/6$ at $z \simlt 1$.
We next show that this term, {\em not} the transition
to deceleration, is the dominant cause of
the downturn in $\Delta \mu_E$ for models similar to $\Lambda$CDM.
Considering the distance modulus residual $\Delta\mu$ for any flat
model relative to the empty model,
we then have
\begin{eqnarray}
\label{eq:dmue}
\Delta \mu_E(z) &=& 5 \log_{10} \left[ {\int_0^z {1 \over E(z')} dz'
\over \ln(1+z) } \, {\ln(1+z) \over \sinh(\ln(1+z)) } \right]
\\
\label{eq:dmusplit}
& \equiv & \Delta \mu_{H}(z) - \Delta\mu_{k}(z) \\
\nonumber
\end{eqnarray}
where we have broken the $\Delta\mu_E$ into two additive terms,
$\Delta\mu_H (z) \equiv 5 \log_{10} [ \int_0^z (1/E(z')) dz' /
\ln(1+z) ]$ due to expansion histories, and
$\Delta\mu_{k} (z) \equiv 5 \log_{10} [\sinh(\ln(1+z))/\ln(1+z) ] $
is the term due to curvature in the empty model (here defined so
$\Delta\mu_k$ is positive,
thus it is subtracted in equation~(\ref{eq:dmusplit}) above).
For illustration, we evaluate each of these terms for $\Lambda$CDM
(with $\Omm = 0.30$) at two specific redshifts: we
choose $z_a = 0.50$ close to the turning point,
and $z_b = 1.26$ to be the downward zero-crossing where $\Delta\mu_E(z) = 0$.
We then find $\Delta\mu_E(0.50) = 0.1231 = 0.1822 - 0.0592$ where the
latter two are $\Delta\mu_H$ and $\Delta\mu_k$ respectively.
At $z_b = 1.26$ we find $-0.0005 = 0.2350 - 0.2355$. Note that
$\Delta\mu_H$ grows from $z = 0.50$ to $z = 1.26$, since although
the expansion is decelerating over most of this interval,
the expansion {\em rate} $\dot{a}$ remains smaller than the present-day
value; see below.
For comparison purposes, it is useful to evaluate
how much the predicted deceleration contributes to $\Delta\mu_H$: for
this we define another model set, hereafter D0,
which exactly matches $\Lambda$CDM back to $\zacc$ but with deceleration
artificially switched off $(q = 0)$ at $z > \zacc$: specifically,
we define model D0 by
\begin{eqnarray}
\label{eq:d0}
H(z) & =& H_{\Lambda CDM}(z) \ \ {\rm if \ } z \le \zacc \\
& = & H_{\Lambda CDM}(\zacc) (1+z) / (1+\zacc)
\ \ {\rm if \ } z > \zacc
\nonumber
\end{eqnarray}
The D0 models are somewhat artificial, but have a continuous
$q(z)$ and are useful to isolate
the relative contribution of $\Lambda$CDM deceleration
on the observables. Also, they represent in a sense the closest
possible match to $\Lambda$CDM among all possible non-decelerating
models, {\newtwo so they are an interesting target to attempt to
exclude observationally}.
The D0 model (for $\Omm = 0.30$) is identical to the
corresponding $\Lambda$CDM at $z_a$,
and at $z_b$ we find
$\Delta\mu_H = 0.2464$ (and $\Delta\mu_k = 0.2355$ again).
Therefore, the actual brightening effect attributable to deceleration in
$\Lambda$CDM is just the difference in $\Delta\mu_H$ between
$\Lambda$CDM and D0, which is only $-0.011$ mag. This is smaller
by a factor of 20 than the curvature effect; so, the bottom line
of this subsection is that at $z = 1.26$, 95\% of this downturn
is due to curvature in the empty fiducial model (or 90\% if
we divide by the value $\Delta\mu_E = 0.1231$~mag at its maximum). Either
way, it is clear that the open curvature in the Milne model
greatly dominates over deceleration as the source of the downturn
in $\Delta \mu_E$.
\section{An improved fiducial model}
\label{sec:flatna}
\subsection{The flat non-accelerating model}
We have argued above that the presentation of distance residuals
from the Milne or empty model is potentially confusing, since
it leads to a generic curvature-induced downturn in the residuals
at $z \simgt 0.5$ which occurs {\em independent} of whether the
expansion really decelerated prior to that epoch.
In this section we look at an improved fiducial model and
demonstrate several advantages.
In particular, the above discussion suggests
a natural fiducial model is one with
a constant expansion rate (deceleration
parameter $q(z) = 0$, and $H(z) = H_0(1+z)$
at all redshifts, as for the Milne model), but
simply setting curvature to zero (equivalent to
striking out the $\sinh$ in the equations above).
This is equivalent to
a Friedmann model with $\Omm = 0$, $\Omega_{DE} = 1$ and
$w = -1/3$; hereafter model N for short. (This reference model has been
employed previously by \citet{seik-schw} and \citet{mort-clark},
but appears to be rather uncommon in the literature.)
Again, this model is not realistic due to the zero
matter density, but it is useful since it has both zero deceleration
and zero curvature. This model straightforwardly gives
\begin{equation}
\label{eq:dln}
D_{L,N}(z) = \frac{c}{H_0} (1+z)\, \ln(1+z) \ .
\end{equation}
We now define the distance ratio for any other model, $\yd(z)$, as the ratio
$D_L(z)/D_{L,N}(z)$, therefore
\begin{equation}
\label{eq:yd}
\yd(z) \equiv {H_0 D_R(z) \over c \ln(1+z)} \, {S_k(x) \over x}
\end{equation}
For an almost-flat model at $z \simlt 1.7$ we can again
neglect the curvature term as very close to 1 (as per footnote
in Sect.~\ref{sec:turnover}
Thus, for flat models the distance ratio becomes
\begin{eqnarray}
\label{eq:ydz}
\yd(z) &=& { 1 \over \ln(1+z) } \, \int_0^{z} \frac{dz'}{E(z')}
\end{eqnarray}
For many purposes below, it is more convenient to change the
redshift variable to $u = \ln(1+z)$, which gives
\begin{eqnarray}
\label{eq:ydu}
\yd(u) & = & \frac{1}{u} \, {\int_0^u \frac{1+z'}{E(z')} \; du' } \ ;
\end{eqnarray}
as usual $u', z'$ are
dummy integration variables, not derivatives, and
$\yd(u)$ means $\yd(z = e^u - 1)$.
\begin{figure*}
\includegraphics[angle=-90,width=15cm]{fig_ydu2.eps}
\caption{ As Fig.~\ref{fig:ydz}, but with the horizontal axis
now linear in $u = \ln(1+z)$.
\label{fig:ydu} }
\includegraphics[angle=-90,width=15cm]{fig_ydz2.eps}
\caption{ The distance ratio $\yd(z)$ defined in equation~(\ref{eq:yd})
for various cosmological models. As in Fig.~\ref{fig:dmue},
solid black lines are $\Lambda$CDM models with $\Omm = 0.27$ (upper)
and $0.30$ (lower). Long-dashed red lines are corresponding D0 models,
with deceleration artificially switched off.
The short-dashed green lines are four Friedmann models of historical interest:
from top to bottom,
vacuum-dominated ($\Omm = 0, \Omlam = 1$); empty (Milne);
open ($\Omm = 0.27, \Omde = 0$); and Einstein-de Sitter ($\Omm = 1$).
Dotted blue lines are three
constant-$q$ models with $q = -0.6, -0.4, -0.33$ (top to bottom).
Points with errorbars show the binned Union 2.1 SNe data.
\label{fig:ydz} }
\end{figure*}
This $\yd$ is directly related to $\Delta\mu_H$ above via
$\Delta\mu_H(z) = 5 \log_{10} \yd(z)$,
but several results below are simplified if we choose
{\em not} to apply this log.
Since $\yd(z)$ is fairly close to $1$ in reasonable models,
this is anyway rather close to a linear stretch
$\Delta \mu_H \approx 2.17 (\yd -1)$.
Since $E(z)/(1+z)$ is just the
expansion rate at $z$ relative to the present day,
i.e. $\dot{a}(z) / \dot{a}(z=0)$,
the integrand of equation~(\ref{eq:ydu}) is just the inverse of this; i.e.
$\yd(z)$ measures the average value of $(\dot{a})_0 / \dot{a}$
with respect to $\ln(1+z)$, over the interval from the source
to the present. It is more convenient to work with averages of $(1+z)/E(z)$
rather than $1/E(z)$, since the former varies much more slowly
with redshift:
for our default $\Lambda$CDM model,
$(1+z)/E(z)$ reaches a maximum value of $1.153$ at $\zacc \simeq 0.67$,
crosses 1 again at $ z \simeq 2.08$,
and declines to 0.895 at $z = 3$.
Note also that since $(1+z)/E(z)$ contains the inverse of $\dot{a}$,
while $z$ increases backwards in time, derivatives
of $(1+z)/E(z)$ have the same sign as $\ddot{a}$, i.e. positive for
acceleration.
In fact the standard deceleration parameter
$q \equiv -\ddot{a}/(aH^2(a))$ is given by
\begin{equation}
\label{eq:qu}
q(u) = - \frac{d}{du} \ln \left( {1+z \over E(z)} \right)
\end{equation}
which is useful below.
\subsection{Useful properties of $\yd$}
\label{sec:ydprops}
The above definition of $\yd$ is simple and intuitive,
and we show below that it enables a number of useful
non-parametric deductions, as follows:
\begin{enumerate}
\item
It is clear above that a value of $\yd(z) > 1$ at any $z$ implies the
past-average of $\dot{a}$ was less than the present value, i.e.
acceleration has dominated over deceleration over
this interval (note, this is not strictly the same as requiring $\ddot{a} > 0$
at the present day); this feature is similar to the Milne fiducial
model above.
\item
It is easy to see that if $q(z)$ is always negative
over some interval $0 \le z \le z_1$, then $(1+z)/E(z)$ is a
strictly increasing function of $z$,
and therefore so is $\yd(z)$; i.e. a flat model which is
non-decelerating at $0 < z < z_1$
cannot have a turnover in $\yd$ at $z \le z_1$, regardless of the
specific expansion history.
The converse of this is that {\em if} a turnover in $\yd(z)$
is observed, this implies a transition to deceleration must have occurred
within the interval, i.e. we can definitely conclude
$ \zacc < z_{tu} $ independent of the functional form of $E(z)$.
Also, if a turnover exists at $z_{tu}$, differentiating
equation~(\ref{eq:ydu}) implies that the value
of $1/\dot{a}$ at $z_{tu}$ was equal to its
average value (w.r.t. $u$) across the interval
from $z_{tu}$ to today.
\item
We can improve on the results above using the Mean Value Theorem:
specifically, if had a known value $\yd(z_1) = y_1$, this theorem
implies that there exists some $z < z_1$ with $(1+z)/E(z) \ge y_1$;
i.e. the cosmic expansion rate has speeded up by at least a factor
of $y_1$ since some $z < z_1$, independent of the functional
form of $E(z)$. For a more realistic case where
we measure an average value of $\yd$ in a finite bin, e.g.
$\langle \yd \rangle = \hat{y}$ averaged between $z_1 < z < z_2$, we can
use the Mean Value Theorem {\em twice}: first, there exists some $z_m$
within this bin with $\yd(z_m) = \hat{y}$, and secondly there exists
some $z_3 \le z_m \le z_2$ satisfying $(1+z_3)/E(z_3) \ge \hat{y}$.
The above argument applies for exact knowledge of $\hat{y}$, neglecting
error bars; however, it is clear that the same argument also applies
if we insert an observational lower bound for $\hat{y}$.
\item
Also, it is interesting to ask a reverse question:
if the expansion was decelerating at all $z > \zacc$, does
this imply that a turnover in $\yd(z)$ must exist ?
The answer appears to be `almost always': it is possible to
build a contrived expansion history
where $q(z)$ crosses from negative to a small positive value,
then asymptotes back to zero from above at high $z$, so
$(1+z)/E(z)$ tends to a constant from above; in this
contrived case we can have deceleration at all $z > \zacc$ while
$\yd(z)$ monotonically increases to the same constant.
However, if we assume non-infinitesimal deceleration,
$q(z) \ge +\epsilon$ for all $z > z_1$ and some positive
value $\epsilon$, it is readily proved that
$\yd(z)$ must have a turnover at some $z$
(though not necessarily in a readily observable range).
\item
Differentiating equation~(\ref{eq:ydu}) and rearranging gives
\begin{equation}
\label{eq:dyddu}
{ 1 + z \over E(z) } = \yd(u) + u \frac{d\yd}{du} \ .
\end{equation}
This gives us a direct graphical implication: taking the tangent to the
curve of $\yd(u)$ at any point $u_1$
and extrapolating the tangent line to $u = 2u_1$ gives us directly
the value of $(1 + z) / E(z)$ at $z_1 = \exp(u_1)-1$.
Differentiating again shows that the transition to
acceleration occurs when $d^2\yd / du^2 = -(2/u) d\yd / du$;
however, as is well known the need to take a second derivative
of noisy data implies that this is not a very useful method
for directly estimating $\uacc$.
\item
Substituting from equation~(\ref{eq:qu}) above leads to the compact results
\begin{eqnarray}
\label{eq:ydq}
\yd(u) & = & \frac{1}{u} \int_0^u
\exp\left[ -\int_0^{u'} q(u'') \, du'' \right] \; du' \ , \\
q(u) & = & { - 2 \frac{d\yd}{du} - u \frac{d^2 \yd}{du^2} \over
\yd(u) + u \frac{d\yd}{du} } \ ;
\end{eqnarray}
this shows that $q_0 = -2 (d\yd/du)(0)$, but also that
as $u$ increases we get increasing weight from the second-derivative
term, so it becomes increasingly more challenging to constrain
$q(u)$ directly from numerical derivatives of data with realistic noise.
Even for optimistic $1\%$ error bars on $\yd$ in bins $\Delta u = 0.1$,
we get order-unity errors on $d^2 \yd/du^2$, so
free-form reconstruction of $q(u)$ is essentially impossible
given realistic errors;
the best we can do is assume some smooth few-parameter model
for $q(u)$ and fit.
\item
From equation~(\ref{eq:ydu}) it clearly follows that for two
measurements at redshifts corresponding to $u_1, u_2$ we have
\begin{equation}
\label{eq:ydrange}
{ u_2 \yd(u_2) - u_1 \yd(u_1) \over u_2 - u_1 } =
\frac{1}{u_2 - u_1} \int_{u_1}^{u_2} { 1 + z' \over E(z') } \, du'
\end{equation}
where the right-hand side (RHS)
is the average of $(1+z)/E(z)$ between the endpoints;
therefore we can estimate this average
as a linear combination of the two values at the ends;
this is simple with respect to combination of error bars,
and does not assume $u_2 - u_1$ is small.
\item
We now show another useful property of $\yd$:
for any flat model with $q(z) =\ $constant (of either sign),
the second derivative $d^2 \yd / du^2$ with respect to $u$ is everywhere
non-negative.
For such a model, denoting $q_c$ as the constant value of $q$,
we have $H(z) = H_0 (1+z)^{1+q_c}$. This easily leads to
\begin{eqnarray}
D_L(z) &=& \frac{c}{H_0} (1+z) \frac{-1}{q_c} \left[(1+z)^{-q_c} -1 \right] \\
\yd(z) &=& \frac{-1}{q_c} \frac{ (1+z)^{-q_c} -1 } {\ln(1+z)} \\
\yd(u) &=& \frac{1 - e^{-q_c u}}{q_c u}
\end{eqnarray}
Now differentiating twice with respect to $u$ gives
\begin{eqnarray}
\frac{d^2 \yd}{du^2} & = & \frac{-1}{q_c} \left[ \frac{ e^{-q_c u}
(u^2 q_c^2 + 2 u q_c +2) -2 }{u^3} \right] \\
& = & q_c^2 \left[ \frac{2 - e^{-p}(p^2 + 2p +2)}{p^3} \right]
\end{eqnarray}
where we define $p \equiv q_c u$.
The function in square brackets above is positive for all $p$,
thus the above second derivative is everywhere non-negative
for any value of $q_c$ with either sign, and is zero only if
$q_c = 0$ and $\yd \equiv 1$.
For the cases of interest here, we are mainly
interested in $-0.6 < q_c < 0$ at $0 < u < 1$, hence
$-0.6 < p < 0$; the square-bracket term
evaluates to $1/3$ for $p = 0$ and $0.53$ for $p = -0.6$,
so for any reasonable $q_c$ model the second derivative
is then between $0.33 q_c^2$ and $0.53 q_c^2$, i.e. small, positive
and slowly varying with $u$.
This has a useful consequence: if $q(u)$ were in fact any constant,
then the graph of $\yd(u)$ versus $u$ must always
show positive curvature (concave from above).
Conversely, if the observed data points
for $\yd(u)$ exhibit significant negative curvature over
some interval, we can conclude
that $q(u)$ increased with $u$ at some point within the observed
interval, again regardless of the specific functional form.
(Note this does not necessarily imply that $q(u)$ became positive,
merely that it increased with $u$ i.e. was less negative in the past.)
\end{enumerate}
We note that in the above points, items (i)-(iv) apply
whether we choose $z$ or $u$ as the redshift variable, but items
(v)-(viii) only apply with $u$ as the variable; this suggests
the latter is preferred.
For an illustration of the current data, we plot
$\yd(u)$ against $u = \ln(1+z)$ in Fig.~\ref{fig:ydu}.
Although this is a simple transformation of the $x-$axis from
Fig.~\ref{fig:ydz}, the qualitative appearance is somewhat
different due to the non-linear transformation, i.e.
higher redshifts become squashed. The apparent ``knee''
in the $\Lambda$CDM models around $z \sim 0.5$ in Fig.~\ref{fig:ydz}
is significantly smoothed out with the $u-$axis,
and both $\Lambda$CDM models now look very close to simple parabolas
(see below).
Also, the constant-$q$ models change curvature from
negative in Fig.~\ref{fig:ydz} to small and positive in
Fig.~\ref{fig:ydu}, as derived above.
Comparing to the data, it is clear that
the SNe data points do marginally prefer a negative curvature
in $\yd(u)$, but not overwhelmingly so.
To quantify this, we fit three models to the $\yd(u)$ data points:
a linear model, a quadratic, and the family of constant-$q$ models above;
we find that the quadratic model is preferred over the linear
model by $\Delta\chi^2 = 3.5$ for 1 extra degree of freedom (d.o.f.), while
the quadratic is preferred over the best constant-$q$ model by
$\Delta \chi^2 = 5.7$ for 1 extra d.o.f. This indicates
that negative curvature in $\yd$ (increasing $q$) is preferred, but
only at around the $2\sigma$ significance level. We expand on the
quadratic model below.
\subsection{A quadratic fitting function for $(1+z)/E(z)$}
\label{sec:quad}
Here we note that it is interesting to consider a fitting function
where $1/\dot{a}$ is a quadratic function of $u$,
specifically
\begin{equation}
\label{eq:quadu}
{ 1 + z \over E(z) } = 1 + b_1 u - b_2 u^2
\end{equation}
with arbitrary constants $b_1, b_2$, and $u \equiv \ln(1+z)$ as before.
The minus sign above is chosen so that positive
$b_1, b_2$ leads to recent acceleration and
past deceleration as anticipated, with $\uacc = b_1 / 2b_2$ from
equation~(\ref{eq:qu}).
This fitting function is not physically motivated,
but is useful since it provides a
very good approximation to models similar to $\Lambda$CDM
at $u < 1, (z < 1.72)$ (see Appendix~\ref{app:qjerk} for an approximate
explanation of this property),
and it gives several simple analytic results below.
Fitting this function to the default
$\Lambda$CDM $(1+z)/E(z)$ over $0 < u < 1$ $(z < 1.72)$
gives best-fitting values $b_1 = 0.569$, $b_2 = 0.530$ with
an rms error of 0.28~percent,
and a worst-case error of $-0.8$~percent.
(This fit becomes significantly worse above $z \simgt 2$,
and has a catastrophic zero-crossing at $u \sim 2$ ($z \sim 6.4$),
but it is good over the range accessible to medium-term SN data.)
The functional form (\ref{eq:quadu}) gives simple relations between
$\uacc$ and the turnover in $\yd$; it easily gives
\begin{eqnarray}
\label{eq:ydquad}
\yd(u) & = & 1 + \frac{1}{2} b_1 u - \frac{1}{3} b_2 u^2 \ ; \\
\label{eq:qquad}
q(u) & = & { -b_1 + 2 b_2 u \over 1 + b_1 u - b_2 u^2 } \ ; \\
\label{eq:dlquad}
D_L(u) & = & {c \over H_0} (1+z)
(u + \frac{1}{2} b_1 u^2 - \frac{1}{3} b_2 u^3) \ ;
\end{eqnarray}
so $\yd(u)$ is also an exact quadratic in this case.
The $q(u)$ behaviour is approximately linear at moderate $u$,
so this model is fairly similar to the model $q(a) = q_0 + q_a(1-a)$
used elsewhere.
Equation~(\ref{eq:dlquad}) with values $b_1, b_2$ as above
matches the exact numerical $D_L(z)$ for $\Lambda$CDM with
very high accuracy, a maximum error only 0.13~percent back to $u = 1$;
this error is substantially smaller than for $E(z)$, due to the integral for
$D_L$.
We find that the results above also work well for $w$CDM models in the region
$0.2 < \Omm < 0.4, -1.2 < w < -0.8$; thus, it is interesting (and
partly a coincidence) that any $w$CDM model within the
presently-favoured range leads to a $\yd(u)$ curve virtually
indistinguishable from a quadratic, to around the 0.2 percent
level i.e. comparable to the line thickness in Fig.~\ref{fig:ydu}.
This gives another helpful feature: any proof of `percent-level'
deviation of $\yd(u)$ from a simple quadratic would signify a failure of
$w$CDM.
We now look at the relation between $\zacc$ and the turning point in $\yd$.
In the above model equation~(\ref{eq:quadu}) with $b_1,b_2 > 0$, recall
the acceleration epoch is $u_{acc} = b_1 / 2 b_2$, hence
$(1+\zacc)/E(\zacc) = 1 + b_1^2 / 4b_2$ ; while the maximum
in $\yd$ occurs at $u_{tp} = 3 b_1/ 4 b_2$, at height
$\yd(\utp) = 1 + 3 b_1^2 / 16 b_2$.
So, in this model $\zacc$ is
directly related to the location $\utp$ of the maximum,
and $(1+\zacc)/E(\zacc)$ is directly related to its height, via
\begin{eqnarray}
\label{eq:zaztp}
\uacc & = & \frac{2}{3} \utp \ , \qquad
\zacc = (1+\ztp)^{2/3} - 1 \ ; \\
{ 1+\zacc \over E(\zacc)} & = & 1 + \frac{4}{3} (\yd(\utp ) - 1 )
\label{eq:ezacc}
\end{eqnarray}
without requiring to solve for $b_1, b_2$.
This suggests that for other reasonably smooth parametrizations of $E(z)$
such as $w$CDM models,
we may expect equations~(\ref{eq:zaztp}) and (\ref{eq:ezacc})
to hold approximately, rather
than exactly as above.
{\newtwo In our default $\Lambda$CDM model,
the exact values are $\zacc = 0.671$, $ (1+\zacc)/E(\zacc) = 1.1530$,
while from numerical evaluation
of $\utp$ and $\yd(\utp)$ the RHS of the above equations
evaluate to 0.693 and 1.1525 respectively;
thus equation~(\ref{eq:zaztp}) is quite good, while
equation~(\ref{eq:ezacc}) is an excellent approximation.
More generally, we have tested these for
wCDM models (constant $w$) with the results shown in
Fig.~\ref{fig:ezacc} ; this shows that equation~(\ref{eq:ezacc})
remains very accurate for a substantial
range around the concordance model. }
\begin{figure}
\hspace{-1.5cm}\includegraphics[angle=-90,width=11cm]{fig_zturn_fin2.eps}
\caption{
This figure shows
the peak value of $\yd$ against the integrated acceleration
$(1+\zacc)/E(\zacc)$, for a grid of $w$CDM models. The differing
point styles show $w = -1.2, -1.1, -1.0, -0.9, -0.8$ as indicated
in the key. For each value of $w$ we show seven points with
$\Omm = 0.24, 0.26,
\ldots, 0.36$ in linear steps of $0.02$;
in each case these run from $\Omm = 0.24$
at upper-right to $0.36$ at lower-left,
so the central point is $\Omm = 0.30$.
The dotted line (not a fit) is equation~(\ref{eq:ezacc}).
\label{fig:ezacc}
}
\end{figure}
We have also tested linear-$q$ models
$q(a) = q_0 + q_a(1-a)$,
and find that equation~(\ref{eq:ezacc}) is accurate to
better than 0.01 for reasonable values of $q_0, q_a$,
while equation~(\ref{eq:zaztp}) is somewhat worse but generally good
to a few percent.
For varying-$w$ models of the form $w(a) = w_0 + w_a(1-a)$,
these approximations remain good for $w_a \ge 0$ but
become somewhat less accurate for negative $w_a$, especially for
$w_a < -0.5$.
The summary here is that equation~(\ref{eq:ezacc}) is generally an
excellent approximation for constant-$w$ models, and a good
approximation for varying-$w$ if $w_a$ is not too negative; while
equation~(\ref{eq:zaztp}) is fairly good at the few-percent level.
{\newtwo These approximations are useful since the right-hand-side
of equations~(\ref{eq:zaztp}) and (\ref{eq:ezacc}) are in
principle directly observable: }
it is clear from Fig.~\ref{fig:ydu}
that the location of the possible maximum in $\yd$ is relatively
poorly constrained, but {\em if} the suggestion of negative curvature
in $\yd$ is real and persists as expected to higher redshifts,
then the SNe datapoints imply that
$\yd(u)$ is probably approaching a maximum value $\sim 1.10 - 1.14$
at $\utp \simlt 1$; if so,
this would give a direct and reasonably model-independent
inference of the integrated acceleration
$(1+\zacc)/E(\zacc) \approx 1.13 - 1.18$.
{\newtwo This provides a useful intuitive explanation of the
ridge-line of $\Omm$ versus $w$ observed in Fig.~\ref{fig:sn-omw}.
}
To summarize this subsection, we find that $w$CDM models with
constant $w$ near the concordance model
are very well approximated by the above fitting functions,
i.e. very close to simple quadratics in $\yd(u)$, and thus
equations~(\ref{eq:zaztp}) and (\ref{eq:ezacc}) provide
quite accurate approximations relating the observable turning point
in $\yd$ to $\zacc$ and the net acceleration.
Finally, in Appendix~\ref{app:dlapp} we use the fitting function
of equation~(\ref{eq:quadu}) to provide
a simple and accurate `computer-free' approximation to the luminosity
distance in $w$CDM models.
\subsection{Linear q(a) models}
\label{sec:linqa}
Here we briefly consider the two-parameter model family with
deceleration parameter $q$ given by a linear function of scale factor
$a$, i.e.
\begin{equation}
\label{eq:qa}
q(a) = q_0 + q_a(1-a)
\end{equation}
for constants $q_0, q_a$. This model has been used before
by various authors (e.g. \citealt{cl08}, \citealt{sca11}),
since it is simple, fairly flexible and can produce
a fairly good approximation to the behaviour of
many dark energy models at $z \simlt 2$.
We have fitted this parameter pair to the Union~2.1 SN data,
with best-fit values at $(-0.62, +1.40)$ and
the resulting likelihood contours shown in Fig.~\ref{fig:q0qa};
as expected, negative $q_0$ is required at very high significance.
(This agrees well with a similar figure in \citet{sca11}).
The figure also shows lines bounding the regions of no past
deceleration $q_0 + q_a < 0$,
and the region $\zacc < 2$ equivalent to $q_0 + 2q_a / 3 > 0$;
the wedge between these lines corresponds to
a transition redshift $\zacc > 2$.
This plot shows that the no-deceleration
region is disfavoured at around the $1.3\sigma$ confidence level,
but there is a region inside the wedge $\zacc > 2$ which is allowed
at around $0.8\sigma$. In this wedge, no deceleration occurs
within the redshift range of observed SNe,
so the inference of deceleration relies
on a linear extrapolation of the $q(a)$ model beyond the range of
SNe.
This generally agrees with our previous conclusions, that
a trend of less negative $q$ at higher redshift is clearly preferred,
but there is negligible evidence from SN data alone
for an actual transition to deceleration within the observed range.
\begin{figure}
\hspace{-1cm}
\includegraphics[angle=-90,width=11cm]{fig_cont_qlin2.eps}
\caption{
The allowed region in the $(q_0, q_a)$ plane
from fitting models with $q(a) = q_0 + q_a(1-a)$
to the Union 2.1 supernova data.
Elliptical contours show the values of $\Delta \chi^2 = 2.3, 6.0, 10.6$
corresponding to 68, 95 and 99.8 percent confidence regions.
The sloping lines bound the region of no deceleration and the region
$\zacc < 2$, with the wedge between these giving $\zacc > 2$.
The line along the major axis of the ellipse is illustrative and
gives a pivot value
$q(a = 0.815) = -0.36$ at $a = 0.815$ ($z = 0.227$).
\label{fig:q0qa}
}
\end{figure}
\section{Discussion}
\label{sec:disc}
It is instructive to blink back and forth between Figs~\ref{fig:dmue},
\ref{fig:ydz}, \ref{fig:ydu} above: although from a parameter-fitting
perspective there is no difference since the residuals
(data--model) are all the same, from the perspective of visual
intuition about expansion rate there are rather striking differences
between these three Figures.
Clearly, Fig.~\ref{fig:dmue}
shows a fairly convincing turnover in the data points; while
in Fig.~\ref{fig:ydz} the data shows negligible evidence for a turnover,
but a reasonably convincing change in slope to a broad near-flat ``plateau''
above $z \simgt 0.6$. Finally, in
Fig.~\ref{fig:ydu} the $\Lambda$CDM models are extremely close to
parabolic (i.e. near-constant negative second derivative),
while the data points show near-linear behaviour with
a reasonable but non-decisive indication of negative curvature;
the constant-$q$ models show weak positive
curvature as derived earlier in equation~(\ref{eq:dyddu}).
As we argued earlier, the turnover in Fig.~\ref{fig:dmue} is largely
attributable to the negative space curvature in the Milne model,
not due to actual deceleration. Figs~\ref{fig:ydz} and \ref{fig:ydu}
show a much more gradual turnover in the $\Lambda$CDM models,
while the D0 models show the expected gradual rise;
clearly the current data are completely unable to discriminate
between $\Lambda$CDM and D0 models.
We suggest that Fig.~\ref{fig:ydu} is the most
informative due to the various useful intuitive
properties outlined in \S~\ref{sec:ydprops} above.
The above conclusions seem somewhat unexpected: there
is a widespread view {\newtwo (see Appendix~\ref{app:decel}) }
that the SN data has convincingly
verified the expected deceleration of the universe
at $z \simgt 1$. However from the discussion above,
the SNe data are almost entirely inconclusive
on the sign of $q$ at $z > 0.7$, and even
a constant-$q$ model with $q(z) \approx -0.4$ back to
$z \simgt 1$ is only excluded at the $\sim 2.5\sigma$ level which is
significant but not overwhelming.
Thus, there is moderately good evidence for $q$ increasing in the past,
but concluding that $q$ actually crossed zero
to a positive value relies strongly
on a smooth extrapolation of this trend, and
is therefore model-dependent.
Conversely, {\em if} we assume GR, almost all the acceptable
models imply significant deceleration at $z > 1$.
Essentially, {\em if} we assume GR with the weak energy
condition and a value of $\Omega_m > 0.2$, then the eightfold increase
in $\rho_m$ back to $z=1$ combined with the much slower increase in
dark energy guarantees matter domination and
deceleration at $z > 1$; in this case deceleration at $z > 1$
is mainly a prediction of GR, rather than a feature
directly required by data.
For the value of $\zacc$ it is important
to keep clear the distinction between an extrapolation based
on GR parameter-fitting, or an actual detection
purely based on data.
It is clear that the CMB does provide much stronger constraints
due to the long distance lever-arm: if we assume the standard
sound horizon length inferred from {\em Planck}, then we deduce
$\yd(z \simeq 1090) \simeq 0.44$, which clearly requires a turnover
and hence deceleration.
However, since
the CMB only gives us one integrated distance to $z \sim 1090$
spanning seven $e-$folds of expansion,
while the supernova data constrains only the last one $e-$fold of expansion,
it would be straightforward to construct `designer' expansion histories
with some extra deceleration hidden in the un-observed six $e-$folds to
offset an absence of deceleration back to $z \sim 1.7$.
This is clearly contrived,
but would not directly conflict with any available $D_L(z)$ data.
Therefore, even adopting the standard distance constraint from
the CMB, we do not yet have a GR-independent proof
that the expansion was actually decelerating at $1 \simlt z \simlt 2$;
this is clearly the most probable and least contrived interpretation,
but loopholes remain.
We note that recent BAO results do
provide significant evidence for deceleration; from the first detection
of BAOs in the Ly-$\alpha$ forest by \citet{busca13} and comparison
with lower--redshift measurements, \citet{busca13} quote
\begin{equation}
{ E(z=2.3)/3.3 \over E(z=0.5)/1.5 } = 1.17 \pm 0.05
\end{equation}
which is a $3.4\,\sigma$ detection of deceleration between
the above two redshifts (though this
does assume an external WMAP7 curvature constraint,
which introduces some slight level of GR-dependence).
However, the desirable goal of verifying that $\zacc < 1$ as expected
is considerably more challenging, since the
expected change in $\dot{a}$ between $z = 0.67$ and $1$
is only 1.7 percent in our default model.
The {\em Euclid} spacecraft \citep{euclid-red} is predicted
to get sub--percent measurements of $r_s H(z)$ at a range
of redshifts $0.9 < z < 1.8$, which looks very promising for a direct
model-independent result, {\newtwo
while improved ground-based measurements spanning
$0.3 < z < 0.9$ would also be highly desirable.}
\section{Conclusions}
\label{sec:conc}
We summarize our conclusions as follows:
\begin{enumerate}
\item The {\em predicted} value of $\zacc$ is rather well constrained
by current data within $w$CDM models, and is mainly sensitive
to $\Omm$ rather than $w$;
this implies that a direct measurement of $\zacc$ is not helpful
for measuring $w$, but is potentially an interesting
test of $w$CDM versus alternate models such as modified gravity.
\item Contrary to intuition, the (probable) downturn in SN
residuals relative to the empty Milne model does {\em not} provide
convincing evidence for deceleration. The predicted downturn is
strongly dominated by the negative space curvature in the Milne model, and
the actual deceleration in $\Lambda$CDM makes only a small
minority contribution to the downturn.
\item There are many advantages to presenting SNe distance
residuals relative
to a flat coasting model ($\Omm = 0$, $\Omde = 1$, $w = -1/3$),
and also in changing the horizontal axis from $z$ to $u = \ln(1+z)$
as in Fig.~\ref{fig:ydu}.
This presentation enables a number of robust non-parametric deductions
about expansion history based on the {\em global shape} of the
observed residuals $\yd(u)$, without needing specific
numerical derivatives of data or fitting functions. Notably,
a turnover in this plot is decisive evidence for deceleration,
while any negative curvature in the data points is evidence for
higher $q$ in the past.
\item If a turning point in $\yd(u)$ is observed, then we can
infer $\zacc$ from its location and $(1+\zacc)/E(\zacc)$ from its
height from Eqs.~(\ref{eq:zaztp},\ref{eq:ezacc});
the latter relation holds to very good accuracy in the case of $w$CDM models,
slightly degrading in the case of large negative $w_a$.
\item For the case of $w$CDM models near the concordance range,
the model curves of $\yd(u)$ are remarkably close to simple quadratics
to an rms accuracy $\simlt 0.3$ percent, significantly better than
present data. This provides a simple intuitive visual test for
potential deviations from $w$CDM.
\item For constraining expansion history, there are
significant complementarities between SNe and BAO (or cosmic chronometers):
the SNe have a precise local anchor at $z \le 0.05$ and therefore
place strong constraints on the {\em integrated}
acceleration, e.g. giving robust lower bounds on
the value of $1.7/E(0.7) \ge 1.1$.
However, the combination of the integral in
SNe distances and the broad maximum in $(1+z)/E(z)$ around
the acceleration transition implies that SNe are weak at giving
model-independent constraints on $\zacc$.
In contrast, BAOs offer direct access to $H(z)$ without differentiation
and are therefore potentially stronger at constraining $\zacc$;
but they have limited precision due to cosmic variance at
$z \simlt 0.25$, and they are therefore weaker at constraining
the total integrated acceleration, most of which occurs
at $0 < z < 0.3$.
It is clearly important to get a good cross-anchor between SN
measurements and BAO measurements for constraining the absolute
distance scale; as argued by e.g. \cite{suth12}, precision measurements
of {\em both} SNe and BAO at matched redshifts would be very useful for
this; {\newtwo see also \cite{blake11} for a slightly different but related
approach}.
\end{enumerate}
\section*{Acknowledgements}
We thank the anonymous referee for helpful comments which have
improved the clarity of this paper.
This is a pre-copyedited, author-produced version of
an article accepted for publication in MNRAS. The version of
record is available online, at Digital Object Identifier
DOI:10.1093/mnras/stu2369 \ .
\vfill
|
1,116,691,497,597 | arxiv | \section{Introduction}
\label{sec:intro}
Convolutional neural networks have led to significant progress in object detection by learning with a large number of training images with annotations~\cite{ren2015faster,redmon2017yolo9000,cai2018cascade,carion2020end}. However, humans can easily localize and recognize new objects with only a few examples. Few-shot object detection (FSOD) is a task to address this setting~\cite{kang2019few,wang2020frustratingly,fan2020fsod,sun2021fsce,qiao2021defrcn}. FSOD is desirable for many real-world applications in diverse domains due to lack of training data, difficulties in annotating them, or both, \emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, identifying new logos, detecting anomalies in the manufacturing process or rare animals in the wild, \emph{etc}\onedot} \def\vs{\emph{vs}\onedot. These diverse tasks naturally have vast differences in class distribution and style of images. Moreover, large-scale pre-training datasets in the same domain are not available for many of these tasks. In such cases, we can only rely on existing natural image datasets, such as COCO~\cite{COCO} and OpenImages~\cite{OpenImages} for pre-training.
Despite the diverse nature of FSOD tasks, FSOD benchmarks used in prior works are limited to a homogeneous setting~\cite{kang2019few,wang2020frustratingly,fan2020fsod,sun2021fsce,qiao2021defrcn}, such that the pre-training and few-shot test sets in these benchmarks are from the same domain, or even the same dataset, \emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, VOC~\cite{VOC} $15+5$ and COCO~\cite{COCO} $60+20$ splits. The class distributions of such few-shot test sets are also fixed to be balanced.
While they provide an artificially balanced environment for evaluating different algorithms,
it might lead to skewed conclusions for applying them in more realistic scenarios. Note that few-shot classification suffered from the same problem in the past few years~\cite{vinyals16mini,ren2018tiered};
Meta-dataset~\cite{triantafillou2020meta} addressed the problem with 10 different domains and a sophisticated scheme to sample imbalanced few-shot episodes.
\begin{figure}[t]
\centering
\includegraphics[width=0.80\linewidth]{datasets_0.pdf}
\caption{Sample images in the proposed FSOD benchmark.}
\label{fig:datasets_0}
\end{figure}
\begin{figure*}[t]
\centering
\begin{subfigure}[t]{0.45\linewidth}
\includegraphics[width=\linewidth]{cross_domain.pdf}
\caption{Multi-domain datasets.}
\label{fig:cross_domain}
\end{subfigure}
\hspace{\fill}
\begin{subfigure}[t]{0.45\linewidth}
\includegraphics[width=\linewidth]{cocod_ap.pdf}
\caption{Domain distance vs 5-shot performance.}
\label{fig:domain_dis}
\end{subfigure}
\caption{
(a) Real-world applications of FSOD are not limited to the natural image domain; we propose to pre-train models on large-scale natural image datasets and transfer to target domains for few-shot learning. (b) We measure the domain distance between datasets (see \Cref{sec:mofsod} for details) in the benchmark and COCO, and plot against 5-shot AP50 of these datasets fine-tuned from a model pre-trained on COCO. VOC is added for reference. The figure shows the benchmark covers a wide range of domains.
}
\label{fig:motivation}
\end{figure*}
Inspired by Meta-dataset~\cite{triantafillou2020meta}, we propose a Multi-dOmain FSOD (MoFSOD) benchmark consisting of 10 datasets from 10 different domains, as shown in \Cref{fig:datasets_0}. The diversity of MoFSOD datasets can be seen in \Cref{fig:motivation} where the domain distance of each dataset to COCO~\cite{COCO} is depicted. Our benchmark enables us to estimate the performance of FSOD algorithms across domains and settings and helps in better understanding of various factors, such as pre-training, architectures, \emph{etc}\onedot} \def\vs{\emph{vs}\onedot, that influence the algorithm performance. In addition, we propose a simple natural $K$-shot sampling algorithm that encourages more diversity in class distributions than balanced sampling.
Building on our benchmark, we extensively study the impact of freezing parameters, detection architectures, pre-training datasets, and the effectiveness of several state-of-the-art (SOTA) FSOD algorithms~\cite{wang2020frustratingly,sun2021fsce,qiao2021defrcn}. Our empirical study leads to rethinking conventions in the field and interesting findings that can guide future research on FSOD.
Conventionally, in FSOD or general few-shot learning, it is believed that freezing parameters to avoid overfitting is helpful or even crucial for good performance~\cite{wang2020frustratingly,sun2021fsce,snell2017proto,wang2019meta}. If we choose to tune more parameters, specific components or designs must be added, such as weight imprinting~\cite{wu2020meta} or decoupled gradient~\cite{qiao2021defrcn}, to prevent overfitting. Our experiments in the MoFSOD show that these design choices might be helpful when pre-training and few-shot learning are in similar domains, as in previous benchmarks. However, if we consider a broader spectrum of domains, unfreezing the whole network results in better overall performance, as the network has more freedom to adapt. We further demonstrate a correlation between the performance gain of tuning more parameters and domain distance (see \Cref{fig:domain_ft}). Overall, \emph{fine-tuning (FT) is a strong baseline} for FSOD on MoFSOD without any bells and whistles.
Using FT as a baseline allows us to explore the impact of different architectures on FSOD tasks. Previous FSOD methods~\cite{wang2020frustratingly,fan2020fsod,sun2021fsce,qiao2021defrcn} need to make architecture-specific design choices; hence focus on a single architecture -- mostly Faster R-CNN~\cite{ren2015faster}, while we conduct extensive study on the impact of different architectures, \emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, recent development of anchor-free~\cite{zhou2021centernet2,zhou2019centernet} and transformer-based architectures~\cite{carion2020end,zhu2020deformable}, on few-shot performance. Surprisingly, we find that even with similar performance on COCO, different architectures have very different downstream few-shot performances. This finding suggests the potential benefits of specifically designed few-shot architectures for improved performance.
Moreover, unlike previous benchmarks, which split the pre-training and few-shot test sets from the same datasets (VOC or COCO), MoFSOD allows us to freely choose different pre-training datasets and explore the potential benefits of large-scale pre-training. To this end, we systematically study the effect of pre-training datasets with ImageNet~\cite{deng2009imagenet}, COCO~\cite{COCO}, LVIS~\cite{lvis}, FSOD-Dataset~\cite{fan2020fsod}, Unified~\cite{zhou2021uni}, and the integration of large-scale language-vision models. Similar to observations in recent works in image classification~\cite{bit} and NLP~\cite{brown2020bert,clip}, we find that large-scale pre-training can play a crucial role for downstream few-shot tasks.
Finally, motivated by the effectiveness of the unfreezing parameters and language-vision pre-training, we propose two extensions: FSCE+ and LVIS+. FSCE+ extends FSCE~\cite{sun2021fsce} to fine-tune more parameters with a simplified fully-connected (FC) detection head. LVIS+ follows the idea of using CLIP embedding of class names as the classifier, but instead of using it in zero-shot/open-vocabulary setting as in~\cite{zhou2021detic,gu2021zero}, we extend it to few-shot fine-tuning. Both methods achieve SOTA results with/without extra pre-training data.
We summarize our contributions as follows:
\begin{itemize}
\item We propose a Multi-dOmain Few-Shot Object Detection (MoFSOD) benchmark to simulate real-world applications in diverse domains.
\item We conduct extensive studies on the effect of architectures, pre-training datasets, and hyperparameters with fine-tuning and several SOTA methods on the proposed benchmark.
We summarize the observations below:
\begin{itemize}
\item \textbf{Unfreezing more layers} do not lead to detrimental overfitting and improve the FSOD performance across different domains.
\item \textbf{Object detection model architectures} have a significant impact on the FSOD performance even when the architectures have a similar performance on the pre-training dataset.
\item \textbf{Pre-training datasets} play an important role in the downstream FSOD performance. Effective utilization of the pre-training dataset can significantly boost performance.
\end{itemize}
\item Based on these findings, we propose two extensions that outperform SOTA methods by a significant margin on our benchmark.
\end{itemize}
\section{Related Work} \label{sec:related}
\textbf{Meta-learning-based methods} for FSOD are inspired by few-shot classification.
Kang et al.~\cite{kang2019few} proposed a meta feature extractor with feature reweight module, which maps support images to mean features and reweight query features with the mean features, inspired by protoypical networks~\cite{snell2017proto}. Meta R-CNN~\cite{yan2019meta} extended the idea with an extra predictor-head remodeling network to extract class-attentive vectors. MetaDet~\cite{wang2019meta} proposed meta-knowledge transfer with weight prediction module.
\textbf{Two-stream methods} take one query image and support images as inputs, and use the correlations between query and support features as the final features to the detection head and the Region Proposal Network (RPN). Several works in this direction~\cite{fan2020fsod,zhang2021,han2021iccv} have shown competitive results. These methods require all classes to have at least one support image to be fed to the model, which makes the overall process slow.
\textbf{Fine-tuning-based methods} update only the linear classification and regression layers~\cite{wang2020frustratingly}, the whole detection head and RPN with an additional contrastive loss~\cite{sun2021fsce}, or decoupling the gradient of RPN and the detection head while updating the whole network~\cite{qiao2021defrcn}. These methods are simple yet have shown competitive results. We focus on benchmarking them due to their simplicity, efficiency, and higher performance than other types.
\textbf{Multi-domain few-shot classification benchmarks.} In few-shot classification, miniImageNet~\cite{vinyals16mini} and tieredImageNet~\cite{ren2018tiered} have been used as standard benchmarks. Similar to benchmarks in FSOD, they are divided into two splits and used for pre-training and few-shot learning, respectively, such that they are in the same natural image domain. Recent works have proposed new benchmarks to address this issue: Tseng~\emph{et al}\onedot~\cite{tseng2020cross} proposed a cross-domain few-shot classification benchmark with five datasets from different domains. Triantafillou~\emph{et al}\onedot~\cite{triantafillou2020meta} proposed Meta-Dataset, which is a large-scale few-shot classification benchmark with ten datasets and a sophisticatedly designed sampling algorithm to sample realistically imbalanced few-shot training datasets. Although not specifically catering to few-shot applications, Wang~\emph{et al}\onedot~\cite{wang2019towards} proposed universal object detection, which aims to cover multi-domain datasets with a single model for high-shot object detection.
\section{MoFSOD: A Multi-Domain FSOD Benchmark}
In this section, we first describe existing FSOD benchmarks and their limitations. Then, we propose a Multi-dOmain FSOD (MoFSOD) benchmark.
\subsection{Existing Benchmarks and Limitations}
Recent FSOD works have evaluated their methods in PASCAL VOC $15+5$ and MS COCO $60+20$ benchmarks proposed by Kang~\emph{et al}\onedot~\cite{kang2019few}. From the original VOC~\cite{VOC} with 20 classes and COCO~\cite{COCO} with 80 classes, they took 25\% of classes as novel classes for few-shot learning, and the rest of them as base classes for pre-training. For VOC $15+5$, three splits were made, where each of them consists of $15$ base classes for pre-training, and the other 5 novel classes for few-shot learning. For each novel class, $K = \{1, 3, 5, 10\}$ object instances are sampled, which are referred to as shot numbers. For COCO, 20 classes overlapped with VOC are considered novel classes, and $K=\{10, 30\}$-shot settings are used. Different from classification, as an image usually contains multiple annotations in object detection, sampling exactly $K$ annotations per class is difficult. Kang~\emph{et al}\onedot~\cite{kang2019few} proposed pre-defined support sets for few-shot training, which would cause overfitting~\cite{huang2021survey}. Wang~\emph{et al}\onedot~\cite{wang2020frustratingly} proposed to sample few-shot training datasets with different random seeds to mitigate this issue, but the resulting sampled datasets often contain more than $K$ instances. While these benchmarks contributed to the research progress in FSOD, they have several limitations.
First, these benchmarks do not capture the breadth of few-shot tasks and domains as they sample few-shot task instances from a single dataset, as we discussed in \Cref{sec:intro}. Second, these benchmarks contain only a fixed number of classes, 5 or 20.
However, real-world applications might have a varying number of classes, ranging from one class, \emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, face/pedestrian detection~\cite{widerface,widerperson}, to thousands of classes, \emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, logo detection~\cite{logodet3k}. Last but not least, these benchmarks are constructed with the balanced $K$-shot sampling. For example, in the 5-shot setting, a set of images containing exactly 5 objects~\cite{kang2019few} is pre-defined. Such a setting is unlikely in real-world few-shot tasks.
We also demonstrate that such a sampling strategy can lead to high variances in the performance of multiple episodes (see \Cref{tab:sota_balanced}). Moreover, different from classification, object detection datasets tend to be imbalanced due to the multi-label nature of the datasets. For example, COCO~\cite{COCO} and OpenImages~\cite{OpenImages} have a more dominant number of person instances than any other objects. The benchmark datasets should also explore these imbalanced scenarios.
\subsection{Multi-Domain Benchmark Datasets}
\label{sec:mofsod}
While FSOD applications span a wide range of domains, gathering enough pre-training data from these domains might be difficult. Hence, it becomes important to test the few-shot algorithm performance in settings where the pre-training and few-shot domains are different. Similar to Meta-dataset~\cite{triantafillou2020meta} in few-shot classification, we propose to extend the benchmark with datasets from a wide range of domains rather than a subset of natural image datasets. Our proposed benchmark consists of 10 datasets from 10 domains:
VisDrone~\cite{visdrone} in aerial images,
DeepFruits~\cite{deepfruits} in agriculture,
iWildCam~\cite{iwildcam} in animals in the wild,
Clipart in cartoon,
iMaterialist~\cite{imaterialist} in fashion,
Oktoberfest~\cite{oktoberfest} in food,
LogoDet-3K~\cite{logodet3k} in logo,
CrowdHuman~\cite{crowdhuman} in person,
SIXray~\cite{sixray} in security, and
KITTI~\cite{kitti} in traffic/autonomous driving.
We provide statistics of these datasets in \Cref{tab:stats}. The number of classes varies from 1 to 352, and that of boxes per image varies from 1.2 to 54.4, covering a wide range of scenarios.
\input{table_pretraining}
In \Cref{fig:domain_dis}, we illustrate the diversity of domains in our benchmark by computing the domain distances between these datasets and COCO~\cite{COCO} and plotting against the 5-shot performance of fine-tuning (FT) on each dataset from a model pre-trained on COCO. Specifically, we measure the domain similarity by calculating the recall of a pre-trained COCO model on each dataset in a class-agnostic fashion, similar to the measurement of unsupervised object proposals~\cite{hosang2014proposal}.
Intuitively, if a dataset is in a domain similar to COCO, then objects in the dataset are likely to be localized well by the model pre-trained on COCO. As a reference, VOC has a recall of 97\%. For presentation purpose, we define $(1 - \text{recall})$ as the domain distance. We can see diverse domain distances in the benchmark, ranging from 0.1 to 0.8.
Interestingly, the domain distance also correlates with the FSOD performance. Although this is not the only deciding factor, as the intrinsic properties (such as the similarity between training and test datasets) of a dataset also play an important role, we can still see the linear correlation between the domain distance and 5-shot performance with the Pearson correlation coefficient -0.43. Oktoberfest and CrowdHuman are outliers in our analysis possibly as they are relatively easy.
\textbf{Natural $K$-Shot Sampling.}
We use a natural $K$-shot sampling algorithm to maintain the original class distribution for this benchmark. Specifically,
we sample $C \times K$ images from the original dataset without worrying about class labels, where $C$ is the number of classes of the original dataset. Then, we check missing images to ensure we have at least one image for each class of all classes. We provide the details in
\section{Experiments}
\label{sec:exp}
In this section, we conduct extensive experiments on MoFSOD and discuss the results.
For better presentations, we highlight compared methods and architectures in \textit{italics} and pre-training/few-shot datasets in \textbf{bold}.
\subsection{Experimental Setup}
\textbf{Model architecture.}
We conduct experiments on six different architectures. For simplicity, we use ResNet-50~\cite{he2016deep} as the backbone of all architectures. We also employ deformable convolution v2~\cite{zhu19deconvv2} in the last three stages of the backbone. Specifically, we benchmark two-stage object detection architectures: 1) \textit{Faster R-CNN}~\cite{ren2015faster} 2) \textit{Cascade R-CNN}~\cite{cai2018cascade}, and the newly proposed 3) \textit{CenterNet2}~\cite{zhou2021centernet2},
and one-stage architectures: 4) \textit{RetinaNet}~\cite{lin2017retina}, as well as transformer-based 5) \textit{Deformable-DETR}~\cite{zhu2020deformable}. Note that all architectures utilize Feature Pyramid Networks (FPN)~\cite{lin2017feature} or similar multi-scale feature aggregation techniques. In addition, we also experiment the combination of the FPN-P67 design from \textit{RetinaNet} and \textit{Cascade R-CNN}, dubbed 6) \textit{Cascade R-CNN-P67}~\cite{zhou2021uni}. We conduct our architecture analysis pre-trained on \textbf{COCO}~\cite{COCO} and \textbf{LVIS}~\cite{lvis}. \Cref{tab:pre-perf} summarizes the architectures and their pre-training performance.
\textbf{Freezing parameters.}
Based on the design of detectors, we can think of three different levels of fine-tuning the network:
1)~only the last classification and regression fully-connected (FC) layer~\cite{wang2020frustratingly},
2)~the detection head consisting of several FC and/or convolutional layers~\cite{fan2020fsod}, and
3)~the whole network, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, standard fine-tuning.\footnote{%
When training an object detection model, the batch normalization layers~\cite{ioffe2015batch} and the first two macro blocks of the backbone (\texttt{stem} and \texttt{res2}) are usually frozen, even for large-scale datasets. We follow this convention in our paper.
}
We study the effects of these three ways of tuning on different domains in MoFSOD with \textit{Faster R-CNN}~\cite{ren2015faster}.
\textbf{Pre-training datasets.}
To explore the effect of pre-training dataset, we conduct experiments on five pre-training datasets: \textbf{ImageNet}\footnote{%
This is ImageNet-1K for classification, which is commonly used for pre-training standard object detection methods, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, we omit pre-training on an object detection task.%
}~\cite{deng2009imagenet}, \textbf{COCO}~\cite{COCO}, \textbf{FSODD}\footnote{%
The name of the dataset is also FSOD, so we introduce an additional D to distinguish the dataset from the task.%
}~\cite{fan2020fsod}, \textbf{LVIS}~\cite{lvis}, and \textbf{Unified}, which is a union of OpenImages v5~\cite{OpenImages}, Object365 v1~\cite{O365}, Mapillary~\cite{mapillary}, and \textbf{COCO}, combined as in~\cite{zhou2021uni}. To reduce the combinations of different architectures and pre-training datasets, we conduct most of the studies on the best performing architecture \textit{Cascade R-CNN-P67}. Also, inspired by \cite{zhou2021detic,gu2021zero}, we experiment with the effect of CLIP~\cite{clip} embeddings to initialize the final classification layer of detector when pre-training on \textbf{LVIS} dubbed \textbf{LVIS+}. In addition, \textbf{LVIS++} uses the backbone pre-trained on the ImageNet-21K classification task instead of ImageNet-1K before pre-training on \textbf{LVIS}. All experiments are done with Detectron2~\cite{wu2019detectron2}.
\textbf{Hyperparameters.}
For pre-training, we mostly follow standard hyperparameters of the corresponding method, with the addition of deformable convolution v2~\cite{zhu19deconvv2}.
On \textbf{COCO} and \textbf{LVIS},
for \textit{Faster R-CNN}, \textit{Cascade R-CNN}, and \textit{Cascade R-CNN-P67}, we use the 3$\times$ scheduler with 270k iterations, the batch size of 16, the SGD optimizer with initial learning rate of 0.02 decaying by the factor of 0.1 at 210k and 250k.
For \textit{RetinaNet}, the initial learning rate is 0.01~\cite{lin2017retina}.
For \textit{CenterNet2}, following~\cite{zhou2021centernet2}, we use the \textit{CenterNet}~\cite{zhou2019centernet} as the first stage and the Cascade R-CNN head as the second stage, where the other hyperparameters are the same as above.
For \textit{Deformable-DETR}~\cite{zhu2020deformable}, we follow the two-stage training of 50 epochs, the AdamW optimizer, and the initial learning rate of 0.0002 decaying by the factor of 0.1 at 40 epochs. On \textbf{FSODD}, we train for 60 epochs with the learning rate of 0.02 decaying by the factor of 0.1 at 40 and 54 epochs and the batch size of 32. On \textbf{Unified}, following~\cite{zhou2021uni}, the label space of four datasets are unified, the dataset-aware sampling and equalization loss~\cite{tan2020eql} are applied to handle long-tailed distributions, and training is done for 600k iterations with the learning rate of 0.02 decaying by the factor of 0.1 at 400k and 540k iterations, and the batch size of 32.
For few-shot training, we train models for 2k iterations with the batch size of 4 on a single V100 GPU.\footnote{The batch size could be less than 4 if the sampled dataset size is less than 4, \emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, when the number of classes and shot number $K$ is 1, and the batch size has to be 1.}
For \textit{Faster R-CNN}, \textit{Cascade R-CNN}, \textit{Cascade R-CNN-P67}, and \textit{CenterNet2}, we train models with the SGD optimizer and different initial learning rates in \{0.0025, 0.005, 0.01\} and choose the best, where the learning rate is decayed by the factor of 0.1 after 80\% of training.
For \textit{RetinaNet}, we halve the learning rates to \{0.001, 0.0025, 0.005\}, as we often observe training diverges with the learning rate of 0.01. For \textit{Deformable-DETR} and \textit{CenterNet2 with CLIP}, we use the AdamW optimizer and initial learning rates in \{0.0001, 0.0002, 0.0004\}.\footnote{%
The learning rates are chosen from our initial experiments on three datasets. Note that training \textit{CenterNet2} with AdamW results in worse performance than SGD.%
}
\textbf{Compared methods.}
\textit{TFA~\cite{wang2020frustratingly}} or \textit{Two-stage Fine-tuning Approach} has shown to be a simple yet effective method for FSOD.
TFA fine-tunes the box regressor and classifier on the few-shot dataset while freezing the other parameters of the model.
For this method, we use the FC head, such that TFA is essentially the same as tuning the final FC layer.\footnote{%
Replacing the FC head with the cosine-similarity results in a similar performance.}
\textit{FSCE~\cite{sun2021fsce}} or \textit{Few-Shot object detection via Contrastive proposals Encoding} improves TFA by 1)~additionally unfreezing detection head in the setting of TFA, 2)~doubling the number of proposals kept after NMS and halving the number of sampled proposals in the RoI head, and 3)~optimizing the contrastive proposal encoding loss. While the original work did not apply the contrastive loss for extremely few-shot settings (less than 3), we explicitly compare two versions in all shots: without (\textit{FSCE-base}, the same as tuning the detection head) and with the contrastive loss (\textit{FSCE-con}).
\textit{DeFRCN~\cite{qiao2021defrcn}} or \textit{Decoupled Faster R-CNN} can be distinguished with other methods by 1)~freezing only the R-CNN head, 2)~decoupling gradients to suppress gradients from RPN while scaling those from the R-CNN head, and 3)~calibrating the classification score from an offline prototypical calibration block (PCB), which is a CNN-based prototype classifier pre-trained on ImageNet~\cite{deng2009imagenet}. We note that PCB does not re-scale input images in their original implementation, unlike the object detector, so we manually scaled images to avoid GPU memory overflow if they are too large.
\textit{FT} or \textit{Fine-tuning}
does not freeze model parameters as done for other methods. Though it is undervalued in prior works, we found that this simple baseline outperforms state-of-the-art methods in our proposed benchmark. All experiments are done with this method unless otherwise specified.
\input{table_overall}
\subsection{Experimental Results and Discussions}
\label{exp:analysis}
\textbf{Effect of tuning more parameters.}
We first analyze the effect of tuning more or fewer parameters on MoFSOD. In \Cref{tab:sota_natural} and \Cref{tab:tune}, we examine three methods freezing different number of parameters when fine-tuning: \textit{TFA} as tuning the last FC layers only, \textit{FSCE-base} as tuning the detection head, and \textit{FT} as tuning the whole network. We observe that freezing fewer parameters improves the average performance: tuning the whole network (\textit{FT}) shows better performance than others, while tuning the last FC layers only (\textit{TFA}) shows lower performance than others. Also, the performance gap becomes larger as the size of few-shot training datasets increases. For example, \textit{FT} outperforms \textit{FSCE-base} and \textit{TFA} by 1.0\% and 8.1\% in 1-shot, and 2.4\%, 17.4\% in 10-shot, respectively.
This contrasts with the conventional belief that freezing most of the parameters generally improves the performance of few-shot learning, as it prevents overfitting~\cite{snell2017proto,finn2017maml,wang2020frustratingly,sun2021fsce}. However, this is not necessarily true for FSOD. For example, in the standard two-stage object detector training, RPN is class-agnostic, such that its initialization for training downstream few-shot tasks can be the one pre-trained on large-scale datasets, preserving the pre-trained knowledge on objectness. Also, the detection head utilizes thousands of examples even in few-shot scenarios, because RPN could generate 1--2k proposals per image. Hence, the risk of overfitting is relatively low.
\begin{figure*}[t]
\centering
\begin{subfigure}[t]{0.47\linewidth}
\includegraphics[width=\linewidth]{domain_tuning2.pdf}
\caption{Performance gain by tuning \textit{the whole network} \textit{FT}) rather than \textit{the last FC layer only} (\textit{TFA}) vs. domain distance.}
\label{fig:domain_ft}
\end{subfigure}
\hspace{\fill}
\begin{subfigure}[t]{0.45\linewidth}
\includegraphics[width=\linewidth]{cocod_pre2.pdf}
\caption{Performance gap between \textbf{Unified} and \textbf{COCO} vs. domain distance.}
\label{fig:domain_pre}
\end{subfigure}
\caption{We demonstrate the correlation between tuning more parameters and domain distance in \Cref{fig:domain_ft} and the correlation between pre-training datasets and domain distance in \Cref{fig:domain_pre}.}
\label{fig:domain_dist}
\end{figure*}
However, fine-tuning more parameters does not always improve performance. \Cref{fig:domain_ft} illustrates the performance gain by tuning more parameters with respect to the domain distance. There is a linear correlation, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, the performance gain by fine-tuning more parameters increases when the domain distance increases. This implies that fine-tuning fewer parameters to preserve the pre-trained knowledge is better when the the few-shot dataset is close to the pre-training dataset. Hence, for datasets close to \textbf{COCO}, such as \textbf{KITTI} and \textbf{iWildCam}, \textit{tuning Last FC layers} (\textit{TFA}) is the best performing method. From these observations, an interesting research direction might be exploring a sophisticated tuning of layers based on the few-shot problem definition and the domain gap between pre-training and few-shot tasks.
One way to design such sophisticated tuning is to develop a better measure for domain distance. In fact, the proposed measure with class-agnostic object recall has limitations. If we decouple the object detection task into localization (background vs. foreground) and classification (among foreground classes), then the proposed domain distance is biased towards measuring localization gaps. Therefore, it ignores the potential classification gaps that would also require tuning more layers. For example, although we can get a good coverage with the proposals of \textbf{COCO} pre-training for \textbf{Clipart} (classic domain adaption dataset) and \textbf{DeepFruits} (infrared images), resulting in relatively small domain distances, there exist significant gaps of feature discrimination for fine-grained classification. In this case, we need to tune more parameters for better performance.
\input{table_arch_5shot}
\textbf{Effect of model architectures.}
A benefit of using \textit{FT} as the baseline is that we can systematically study the effects of model architectures without the constraints of specifically designed components. Many different architectures have been proposed to solve object detection problems; each has its own merits and drawbacks. One-stage methods, such as YOLO~\cite{redmon2017yolo9000} and RetinaNet~\cite{lin2017retina}, are known for their fast inference speed. However, different from two-stage methods, they do not have the benefit of inheriting the pre-trained class-agnostic RPN. Specifically, the classification layers for discriminating background/foreground and foreground classes need to be reinitialized, as they are often tied together in one-stage methods. We validate this hypothesis by comparing with two-stage methods in \Cref{tab:arch_overall}. Compared to \textit{Faster R-CNN}, \textit{RetinaNet} has 4--6\% low performance on MoFSOD. Per-dataset performance in \Cref{tab:arch} shows that \textit{RetinaNet} is worse in all cases.
The same principle of preserving as much information as possible from pre-training also applies for two-stage methods,
\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, reducing the number of randomly reinitialized parameters is better.
Specifically, we look into the performance of \textit{Cascade R-CNN} vs \textit{Faster R-CNN}. For \textit{Cascade R-CNN}, we need to reinitialize and learn three FC layers as there are three stages in the cascade detection head, while we only need to reinitialize the last FC layer for \textit{Faster R-CNN}. However, \textit{Cascade R-CNN} is known to have better performance,
as demonstrated in \Cref{tab:pre-perf}. In FSOD, these two factors cancelled out, such that their performance is on a par with each other as shown in \Cref{tab:arch_overall}.
Based on these insights, we extend \textit{Cascade R-CNN} by applying the FPN-P67 architecture~\cite{lin2017retina}, similar to \cite{zhou2021uni,zhou2021centernet2}. Specifically, assuming ResNet-like architecture~\cite{he2016deep}, we use the last three stages of the backbone, namely $\left[\texttt{res3}, \texttt{res4}, \texttt{res5}\right]$, instead of the last four in standard FPN. Then, we add P6 and P7 of the FPN features from P5 with two different FC layers, such that RPN takes in features from $\left[\texttt{P3}, \texttt{P4}, \texttt{P5}, \texttt{P6}, \texttt{P7}\right]$ to improve the class-agnostic coverage, which can be inherited for down-stream tasks. While the detection head still uses $\left[\texttt{P3}, \texttt{P4}, \texttt{P5}\right]$ only. As shown in \Cref{tab:arch_overall}, the resulting architecture, \textit{Cascade R-CNN-P67} improves \textit{Cascade R-CNN} by 3--4\% on the downstream few-shot tasks.
Moreover, recent works proposed new directions of improvement, such as utilizing point-based predictions~\cite{zhou2019centernet,zhou2021centernet2} or transformer-based set predictions~\cite{carion2020end,zhu2020deformable}. These methods are unknown quantities in FSOD, as no previous FSOD work has studied them. In our experiments, while \textit{CenterNet2}~\cite{zhou2021centernet2} outperforms \textit{Faster R-CNN} on \textbf{COCO} by 2.6\% as shown in \Cref{tab:pre-perf}, its FSOD performance on MoFSOD is lower, \emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, 2.4\% in 1-shot. In the case of \textit{Deformable-DETR}~\cite{zhu2020deformable}, it outperforms \textit{Faster R-CNN} in both pre-training and few-shot learning, by 3.6\% on \textbf{COCO} and 2.1\% on MoFSOD in 10-shot. These results show that the upstream performance might not necessarily translate to the downstream FSOD performance. We note that we could not observe a significant correlation between the performance gap of different architectures and domain distances.
\textbf{Effect of pre-training datasets.}
MoFSOD consists of datasets from a wide range of domains, allowing us to freely explore different pre-training datasets while ensuring domain shifts between pre-training and few-shot learning. We examine the impact of the pre-training datasets with the best performing \textit{Cascade R-CNN-P67} architecture.
In \Cref{tab:pre_training_overall}, we first observe that pre-training on \textbf{ImageNet} for classification results in low performance, as it does not provide a good initialization for downstream FSOD, especially for RPN. On the other hand, compared to \textbf{COCO}, \textbf{FSODD}, \textbf{LVIS}, and \textbf{Unified} have more classes and/or more annotations, while they have a fewer, similar, and more number of images, respectively. Pre-training on these larger object detection datasets does not improve the FSOD performance significantly, as shown in \Cref{tab:pre_training_overall}. For example, pre-training on \textbf{Unified} improves the performance over \textbf{COCO} when the domain distance from \textbf{COCO} is large, such as \textbf{Deepfruits} and \textbf{LogoDet-3K} as shown in \Cref{fig:domain_pre}.
However, pre-training on \textbf{Unified} results in lower performance for few-shot datasets close to \textbf{COCO}, such that the overall performance is similar. We hypothesize that this could be due to the non-optimal pre-training of \textbf{LVIS} and \textbf{Unified}, as these two datasets are highly imbalanced and difficult to train. It could also be the case that even \textbf{LVIS} and \textbf{Unified} do not have better coverage for these datasets.
On the other end of the spectrum, we can combine the idea of preserving knowledge and large-scale pre-training by utilizing a large-scale language-vision model. Following~\cite{gu2021zero,zhou2021detic}, we use CLIP to extract text features from each class name and build a classifier initialized with the text features.
In this way, we initialize the classifier with the CLIP text embeddings for downstream few-shot tasks, such that it has strong built-in knowledge of text-image alignment, better than random initialization.
As demonstrated in \Cref{tab:pre_training_overall}, For \textbf{LVIS+}, we can see this improves performance significantly by 7.5\% for \textit{CenterNet2}, and 3.3\% for \textit{Cascade R-CNN-P67} in 5-shot. \textbf{LVIS++} pre-trains the backbone on ImageNet-21K instead of ImageNet-1K (before pre-training on \textbf{LVIS}), and it further improves over \textbf{LVIS+} by 0.8\% in 5-shot.
However, the benefit of CLIP initialization is valid only when class names are matched with texts presented in CLIP; an exceptional case is \textbf{Oktoberfest}, which has German class names, such that \textbf{LVIS+} does not help.
\textbf{Comparison with SOTA methods.}
\Cref{tab:sota_natural} and \Cref{tab:sota_balanced} compare SOTA methods and our proposed methods.
For balanced $K$-shot sampling, we follow Wang~\emph{et al}\onedot~\cite{wang2020frustratingly} to sample $K$ instances for each class whenever possible greedily. Here \textit{FT} and \textit{FSCE+} employ a similar backbone/architecture and pre-trained data as all SOTA methods for a fair comparison. We confirm that \textit{FT} is indeed a strong baseline, such that it performs better than other SOTA methods in both natural $K$-shot and balanced $K$-shot settings.
In addition to \textit{FT}, based on the insights above, we propose several extensions:
1)~\textit{FSCE+} is an extension of \textit{FSCE} by tuning the whole network parameters, similar to \textit{FT}. We keep the contrastive proposal encoding loss, but we simplify the classification head from the cosine similarity head to the FC head. We can see the improvement by 2--3\% compared to \textit{FSCE} for both natural $K$-shot and balanced $K$-shot scenarios and performs slightly better than \textit{FT}.
2)~\textit{FT+} replaces \textit{Faster R-CNN} with \textit{Cascade R-CNN-P67}, and it improves over \textit{FT} by 3--4\% without sacrificing inference speed or memory consumption much.
3)~\textit{FT++} replaces \textit{Faster R-CNN} with \textit{CenterNet2} and uses \textbf{LVIS++} for initialization, and it further improves the performance by around 3\% in 3-, 5-, and 10-shot.
We also observe that while the overall trend of performance is similar for both natural and balanced $K$-shot sampling, the standard deviation of the natural $K$-shot performance is less than that of the balanced $K$-shot.
\section{Conclusion}
\label{sec:conclusion}
We present the Multi-dOmain Few-Shot Object Detection (MoFSOD) benchmark consisting of 10 datasets from different domains to evaluate FSOD methods. Under the proposed benchmark, we conducted extensive experiments on the impact of freezing parameters, different architectures, and different pre-training datasets. Based on our findings, we proposed simple extensions of the existing methods and achieved state-of-the-art results on the benchmark. In the future, we would like to go beyond empirical studies and modifications, to designing architectures and smart-tuning methods for a wide range of FSOD tasks.
\section{Natural $K$-Shot Sampling}
\label{sec:natural_k_shot}
In this section, we describe how we perform natural $K$-shot sampling in detail:
\textbf{Step 1. Sample $C \times K$ images.} $C$ is the number of classes of the original dataset $\mathcal{S}$. In this step, without worrying about class labels, we sample $\mathcal{S}$ from the entire dataset $\mathcal{D}$. Unlike the standard $K$-shot sampling algorithm in recent FSOD works~\cite{kang2019few,wang2020frustratingly,sun2021fsce}, we do not apply stratified sampling. This is because an image usually contains multiple annotations, such that stratified sampling might result in an artificial class distribution~\cite{kang2019few}.
\textbf{Step 2. Check missing classes.} The initial sampled dataset $\mathcal {} $ might not contain some classes, particularly those present only in a few images in the original dataset. To compensate for this, we check if there are any missing classes and update the sampled dataset. Specifically, we manage two datasets: $\mathcal{P}$ is a set of images to be added, and $\mathcal{Q}$ is a set of images to be kept. Then, for each class, if no image in $\mathcal{S}$ contains the class, we sample an image from the $\mathcal{D}$ containing the class and put it in $\mathcal{P}$; otherwise, we sample an image from $\mathcal{S}$ containing the class and put it in $\mathcal{Q}$.
\textbf{Step 3. Update the sampled dataset.} As the final step, we adjust the initial dataset $\mathcal{S}$ to guarantee that all classes are present.
To match the number of added and removed images, we sample a set of images to be removed $\mathcal{R}$ from $\mathcal{S} - \mathcal{Q}$ where the size of $\mathcal{R}$ is the same as $\mathcal{P}$. Here, $\mathcal{Q}$ guarantees that any class in $\mathcal{S}$ does not become empty.
\input{alg_natural}
Finally, we add $\mathcal{P}$ and remove $\mathcal{R}$ from $\mathcal{S}$.
The complete algorithm is in \Cref{alg:natural}.
\section{Dataset Size Reduction}
We initially collected more than 100 public detection datasets, and then selected 32 datasets based on availability, diversity of domains, annotation quality, and number of citations. After initial experiments on them, to reduce the computational burden for future research, we picked 10 datasets out of the 32, which show similar performance trends with the 32 datasets, while covering a variety of domains based on the domain distance.
In the proposed MoFSOD benchmark, several datasets contain a large number of classes and testing images, such as LogoDet-3K. With the proposed natural $K$-shot sampling, the training time is proportional to the number of classes. To address concerns on computational cost and speed up overall experiment time, we limited the number of classes to 50 and the number of test samples to 1k.
Specifically, we randomly sample 50 classes and remove images containing all the rest classes in each episode, such that the intention of the original datasets is kept, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, all remaining logos or traffic signs should be detected. We note that all classes in these datasets are mostly isolated to certain images, such that removing images containing a class does not hurt the distribution of other classes. We confirmed that the performance differences between sampled and full
test sets are less than 1.5\% for all datasets.
\section{Additional Experimental Results}
\subsection{Dataset Statistics}
More detailed statistics of the ten datasets of MoFSOD can be found in~\Cref{tb:coreset}.
\input{table_coreset}
\input{table_arch_1shot}
\input{table_arch_3shot}
\input{table_arch_10shot}
\subsection{Detailed 1-, 3- and 10-shot Results}
In addition to per-dataset 5-shot results in Table~\textcolor{red}{3}~ of the main paper, we present per-dataset 1-, 3- and 10-shot results in \Cref{tab:1_shot}, \ref{tab:3_shot}, and \ref{tab:10_shot}, respectively.
\subsection{Extended 32 Datasets Results}
We evaluate \textit{FT} against SOTA methods in an extended 32 dataset benchmark on 17 Domains. These datasets are:
CARPK~\cite{carpk}, DOTA~\cite{dota}, and VisDrone~\cite{visdrone} in aerial images,
DeepFruits~\cite{deepfruits} and MinneApple~\cite{minneapple} in agriculture,
ENA24~\cite{ena24} and iWildCam~\cite{iwildcam} in animal in the wild,
Clipart, Comic, and Watercolor~\cite{cartoon} in cartoon,
SKU110K~\cite{sku110k} in dense product,
DeepFashion2~\cite{deepfashion2} and iMaterialist~\cite{imaterialist} in fashion,
WIDER FACE~\cite{widerface} in face,
Kitchen~\cite{kitchen} and Oktoberfest~\cite{oktoberfest} in food,
HollywoodHeads~\cite{hollywoodheads} in head,
LogoDet-3K~\cite{logodet3k} and OpenLogo~\cite{openlogo} in logo,
ChestX-Det10~\cite{chest10} and DeepLesion~\cite{deeplesion} in medical imaging,
CrowdHuman~\cite{crowdhuman} and WiderPerson~\cite{widerperson} in person,
PIDray~\cite{pidray} and SIXray~\cite{sixray} in security,
table-detection~\cite{table} in table,
COCO-Text~\cite{cocotext} in text in the wild, and
Cityscapes~\cite{cityscapes}, KITTI~\cite{kitti}, LISA~\cite{lisa}, and TT100K~\cite{tt100k} in traffic,
DUO~\cite{duo} in underwater.
\input{table_dataset}
Their statistics can be found in~\Cref{tb:dataset}. Sample images from these datasets can be found in~\Cref{fig:datasets_12}.
\begin{figure*}[t]
\centering
\includegraphics[width=.495\linewidth]{datasets_1.pdf}
\includegraphics[width=.495\linewidth]{datasets_2.pdf}
\caption{Image samples of the additional datasets.}
\label{fig:datasets_12}
\end{figure*}
\Cref{tab:result_all} compares \textit{FT} and SOTA methods. \textit{TFA-cos} is a variation of \textit{TFA} where the classification head is replaced with the cosine similarity. Note that, while the comparison within tables is fair, the results are NOT directly comparable to the results in the main paper, as they are experimented in different settings. Specifically, the architecture uses deformable convolution v1 while the one in the main paper uses v2, and is trained with a non-standard scheduler. We can observe that \textit{FT} is a strong baseline, outperforming all other methods.
\input{table_all}
\input{table_arch}
We then present the results between different architectures and pre-training datasets in~\Cref{tab:result_arch}.
Although the ablation study here is not as comprehensive as the main paper, we can still see that \textit{Cascade R-CNN-P67} outperforms \textit{Faster R-CNN}. The margin here is smaller mainly due to the less optimal learning rate scheduler we used for \textit{Cascade R-CNN-P67} and possibly the lack of deformable convolution in this model. Once we use the same learning rate scheduler and backbone architecture with deformable convolution v2~\cite{zhu19deconvv2} for both \textit{Cascade R-CNN-P67} and \textit{Faster R-CNN} as in the main paper, the performance gap for different shots actually increases. On the other hand, the comparison between all \textit{Cascade R-CNN-P67} experiments is fair. We can see that over a larger range of domains, \textbf{Unified} provides better results than \textbf{COCO} by a significant margin. However, this performance gap could be due to the non-optimal training of \textbf{COCO}. These suggest that besides the size/quality of the pre-training datasets, how to train for downstream tasks optimally is also an important factor.
|
1,116,691,497,598 | arxiv | \section{Introduction}
Reionization was the last major phase transition of the intergalactic medium (IGM), and scrutinizing the detailed evolution of the IGM is a key frontier in observational cosmology. High-redshift star-forming galaxies are thought to be the primary sources of ionizing photons \citep[e.g.,][]{Robertson2013a, Robertson2015a, Finkelstein2012a, Finkelstein2015a}. Although bright quasi-stellar objects (QSOs) and active galactic nuclei (AGNs) are able to ionize their proximate areas as well and keep the IGM highly ionized during the last phase of cosmic reionization \citep[e.g.,][]{Giallongo2015a, Worseck2014a, DAloisio2017a, DAloisio2018a, Chardin2017a, Yoshiura2017a, Mitra2018a}, the number density of these objects rapidly decreases in the early universe.
Ionizing photon production and the escape fraction ($f_{\text{esc}}$) of these photons are key to modeling cosmic reionization. The ionizing photon budgets are estimated from the cosmic star-formation history, which is tied to the shape of the galaxy luminosity function \citep[e.g.,][]{Bouwens2015a, Finkelstein2012a, Finkelstein2015a, Livermore2017a}, while the interstellar medium (ISM) and circumgalactic medium (CGM) determine $f_{\text{esc}}$ \citep[e.g.,][]{Paardekooper2015a, Kakiichi2017a, Katz2018a, Kimm2013a, Kimm2014a, Kimm2017a, Laursen2011a, Mason2017a, Rosdahl2018a}. Thus, the interactions between high-redshift galaxies and the IGM have a significant impact on the evolution of the galaxies, and revealing detailed timelines of cosmic reionization and investigating IGM properties is key to gaining knowledge of galaxy evolution in the early Universe.
\textit{Wilkinson Microwave Anisotropy Probe (WMAP)} and \textit{Planck} observations constrain the midpoint of reionization to be $z\sim8-9$ \citep{Larson2011a, Planck-Collaboration2016a} from the measure of the large-scale polarization of the cosmic microwave background (CMB), while quasar observations studying the Ly$\alpha$ forest and Gunn-Peterson effects at high redshift suggest a highly-ionized IGM at $z\sim6$ \citep[e.g.,][]{Becker2001a, Fan2006a, Bolton2011a, Mortlock2011a, McGreer2015a}. Complementary measurements of the end of reionization based on the Ly$\alpha$ emitter (LAE) luminosity function at $z\gtrsim6$ agree with those from the CMB and quasar observations \citep[e.g.,][]{Malhotra2004a, Ota2008a, Ota2017a, Ouchi2010a, Ouchi2018a, Zheng2017a}. However, robust studies of the IGM during reionization are still limited, as it is, for example, difficult to map the neutral fraction of the IGM during reionization with quasar spectroscopy due to the lack of a large population of quasars at $z>7$.
An immediately accessible method for studying the IGM in the reionization era is searching for Ly$\alpha$ emission from continuum-selected galaxies with follow-up spectroscopy. Due to the resonant nature of Ly$\alpha$ scattering by neutral hydrogen, the presence of neutral hydrogen in the IGM easily attenuates Ly$\alpha$-emission-line strengths. The fraction of continuum-selected Lyman break galaxies (LBGs) with strong spectroscopically-detected Ly$\alpha$ emission (known as the ``Ly$\alpha$ fraction'') was found to increase from $z =$ 3 to $z =$ 6 \citep{Stark2010a}. It was thus expected that the Ly$\alpha$ fraction at $z \sim$ 7 would be at least as high as at $z =$ 6 \citep{Stark2011a}. However, initial studies have found an apparent deficit of strong Ly$\alpha$ emission at $z >$ 6.5 \citep[e.g.,][]{Fontana2010a, Pentericci2011a, Pentericci2014a, Curtis-Lake2012a, Mallery2012a, Caruana2012a, Caruana2014a, Finkelstein2013a, Ono2012a, Schenker2012a, Schenker2014a, Treu2012a, Treu2013a, Tilvi2014a, Schmidt2016a, Vanzella2014a}. The dust content of UV-selected galaxies has been found to decrease with increasing redshift \citep{Finkelstein2012b, Bouwens2014a, Marrone2018a}, thus the increased fraction of strong Ly$\alpha$ emission from $z=3\rightarrow6$ is likely due to decreasing dust attenuation in galaxies. A joint effect from metal poor stellar populations in the galaxies at higher redshift is likely as well, as it fosters the escape of Ly$\alpha$ photons by enlarging HII regions from the generation of hard ionizing photons \citep[e.g.,][]{Finkelstein2011a, Nakajima2013a, Song2014a, Trainor2016a}. Therefore, the perceived drop in Ly$\alpha$ emission at $z>6$ is unlikely due to dust and implies that the neutral hydrogen fraction in the IGM increases significantly from $z\sim6$ $\rightarrow$ 7, although other galaxy evolutionary features and the uncertainties of the Lyman continuum escape fraction and the gas covering fraction need to be taken into account \citep[see][]{Papovich2011a, Finkelstein2012b, Dijkstra2014b}.
Despite this tantalizing evidence, measuring the Ly$\alpha$ fraction depends on the sensitivity of the observed spectra and the completeness of the detected LAEs. \cite{De-Barros2017a} report a Ly$\alpha$ fraction at $z\sim6$ lower than the values previously reported in the literature from a large sample of LAEs, and \cite{Caruana2018a} find no dependence of the Ly$\alpha$ fraction on redshift at $3<z<6$ based on the analysis of 100 LAEs from the Multi-Unit Spectroscopic Explorer (MUSE)-Wide survey \citep{Herenz2017a}.
As discussed above, while many previous studies have used the Ly$\alpha$ fraction as a measure of the evolution of Ly$\alpha$ emission, it is a somewhat less constraining measure since it often does not account for the continuum luminosity of the host galaxy. For this reason, we implement a more detailed analysis of our dataset, where we place constraints on the evolution of the Ly$\alpha$ equivalent width (EW) distribution, using detailed simulations to include the effects of incompleteness. This distribution function has been well studied at $0.3 < z < 6$, and has been found to have the form of an exponential distribution, $\text{dN/dEW}\propto \text{exp(-EW/}W_0)$, with a characteristic EW $e$-folding scale ($W_0$) of $\sim60\text{\AA}$ over the epoch $0.3 < z < 3$ \citep[e.g.,][]{Gronwall2007a, Guaita2010a, Ciardullo2012a, Wold2014a, Wold2017a}; and possibly higher at higher redshift \citep[e.g.,][]{Ouchi2008a, Zheng2014a}. Particularly, in the epoch of reionization, the neutral hydrogen atoms in the IGM are expected to diminish these EWs, lowering the $e$-folding scale ($W_0$) of the observed Ly$\alpha$ EW distribution \citep[e.g.,][]{Bolton2013a,Mason2017a}. This gives us our research question: at what confidence can new observations rule out the $e$-folding scale ($W_0$) of $\sim60\text{\AA}$? More importantly, understanding the evolution of the $e$-folding scale is key to predicting the number of Ly$\alpha$ emitting galaxies with a given Ly$\alpha$ EW distribution.
\begin{figure*}[t]
\centering
\includegraphics[width=0.57\paperwidth]{mask-eps-converted-to.pdf}
\caption{Mask designs of our DEIMOS and MOSFIRE configurations overlaid in the GOODS-S (left) and GOODS-N (right) WFC3/F160W CANDELS images. Observed areas are marked by green rectangles: larger solid rectangles (5$\arcmin$$\times$16.7$\arcmin$) show DEIMOS observations, and smaller dashed squares (6$\arcmin$$\times$4$\arcmin$) represent MOSFIRE observations. Cyan and yellow circles are observed galaxies with DEIMOS and MOSFIRE, respectively. While this figure shows our entire spectroscopic survey program with DEIMOS and MOSFIRE, we discuss our analysis with DEIMOS in this paper, and our follow-up paper will include the MOSFIRE data. }
\label{fig:mask_design}
\end{figure*}
To place observational constraints on the Ly$\alpha$ EW $e$-folding scale, we performed Keck/DEIMOS (optical) spectroscopic observations of Ly$\alpha$ emitting galaxies using a robust sample of candidate galaxies with photometric redshifts $z=5.5-8.3$. We comprehensively accounted for incompleteness due to the noise level in the data (from a combination of telescope$+$instrument throughput, and also integration time) and the night sky lines which are ubiquitous at these wavelengths, and also due to galaxy photometric redshift probability distribution functions.
In this paper, we present our measure of the $e$-folding scale of the Ly$\alpha$ EW distribution at $6.0<z<7.0$ measured from Keck/DEIMOS spectra. We describe our spectroscopic data in Section 2, and the detected emission lines are summarized in Section 3. Our analysis of the Ly$\alpha$ EW distribution at $z\sim6.5$ is explained in Section 4. Section 5 discusses the redshift dependence of the Ly$\alpha$ EW $e$-folding scale, while our findings are summarized in Section 6. We assume the \textit{Planck} cosmology \citep{Planck-Collaboration2016a} in this study, with $H_0$ = 67.8\,km\,s$^{-1}$\,Mpc$^{-1}$, $\Omega_{\text{M}}$ = 0.308 and $\Omega_{\Lambda}$ = 0.692, and a \cite{Salpeter1955a} initial mass function with lower and upper stellar-mass limits of 0.1 to 100 $M_{\odot}$ is assumed. The \textit{Hubble Space Telescope (HST)} F435W, F606W, F775W, F814W, F850LP, F098M, F105W, F125W, F140W and F160W bands are referred as $B_{435}$, $V_{606}$, $i_{775}$, $I_{814}$, $z_{850}$, $Y_{098}$, $Y_{105}$, $J_{125}$, $JH_{140}$ and $H_{160}$, respectively. All magnitudes are given in the AB system \citep{Oke1983a}. All errors presented in this paper represent 1$\sigma$ uncertainties (or central 68\% confidence ranges), unless stated otherwise.
\section{Data}
\subsection{Spectroscopic Survey and Sample Selection}
The target galaxies were selected from \cite{Finkelstein2015a} which found a sample of 7446 high-redshift candidate galaxies at $3.5<z<8.5$, using a photometric redshift ($z_{\text{phot}}$) measurement technique, in the CANDELS GOODS-South and -North fields. We note that our entire observing program, the \textit{Texas Spectroscopic Search for Ly$\alpha$ Emission at the End of Reionization}, utilizes 10 nights of MOSFIRE (near-infrared) observations as well as 4 nights of DEIMOS (optical) observations on the Keck telescopes. Observations were conducted from Apr 2013 to Feb 2015, and two Ly$\alpha$ detections at $z>7.5$ from the MOSFIRE observations are already published in \cite{Finkelstein2013a} and \cite{Song2016b}. In our survey program, observations with both instruments (optical + near-infrared) enable us to put strong constraints on the observability of Ly$\alpha$ emission, covering the broad range of galaxy photometric redshift probability distributions of $z\sim7$ candidate galaxies, as the DEIMOS and MOSFIRE combined wavelength coverage corresponds to redshifted Ly$\alpha$ emission at $5<z<8$. We first present the analysis of our DEIMOS observations in this paper, focusing on a Ly$\alpha$ emission search at $5<z<7$, and a follow-up paper will include the MOSFIRE data to provide a comprehensive analysis, covering the entire wavelength of Ly$\alpha$ emission at $z\sim5-8$.
\begin{figure}[t]
\centering
\includegraphics[width=\columnwidth]{muv_redshift_plot-eps-converted-to.pdf}
\caption{$M_{\text{UV}}$ distribution of target galaxies in our DEIMOS observations as a function of redshift. Open circles are target galaxies in GOODS-S (red) and GOODS-N (blue), and larger filled circles are Ly$\alpha$ detections. The line detections are discussed further in Section 3.}
\label{fig:muv_z}
\end{figure}
Figure \ref{fig:mask_design} shows the entire spectroscopic survey program. The DEIMOS mask designs are solid rectangles, and the MOSFIRE masks are shown as dashed squares. Cyan and yellow circles represent our target galaxies, observed by DEIMOS and MOSFIRE, respectively. We also display the rest-frame ultraviolet (UV) magnitude ($M_{\text{UV}}$) distribution of our objects on slits as a function of redshift in Figure \ref{fig:muv_z}. The rest-frame $M_{\text{UV}}$ is obtained from \cite{Finkelstein2015a}, who derive $M_{\text{UV}}$ through galaxy spectral energy distribution (SED) fitting. $M_{\text{UV}}$ is measured based on an continuum flux of the best-fit model SED, which is averaged over a 100\AA-wide top-hat window centered at rest-frame 1500\AA. As shown in the figure, our targeted galaxies cover range of $-22.0 \lesssim M_{UV} \lesssim -18.5$, except for a couple of faint objects with $M_{UV} \gtrsim -18$ found at $z\lesssim6$.
\subsection{DEIMOS Spectroscopy and Data Reduction}
We carried out four nights of observations for 118 galaxies with DEIMOS on the Keck 2 telescope: two nights in GOODS-S and two nights in GOODS-N (PI: Rachael Livermore). However, we would note that the second night of observation for GOODS-N had relatively bad seeing and throughput ($\lesssim 20\%$ of that from the first night), thus we do not include this data in our analysis. Also, unfortunately, for our GOODS-S data, we could not achieve properly flux-calibrated spectra due to the bad observing conditions. We discuss the issue later in this section and in Appendix.
We used the same slitmask for the two nights on each field. The slitmasks were designed using DSIMULATOR, a DEIMOS mask design tool. We used the OG550 filter/830G grating centered at 9000\AA, which effectively covers a wavelength range, $\lambda\sim7000-10000$\AA\ (corresponding to Ly$\alpha$ emission at $5\lesssim z\lesssim7$), and the spectral resolution ($\lambda/\Delta\lambda$) is $\sim3600$ with a 1.0$\arcsec$ slit width. We use an A-B-B'-A (0.0$\arcsec$, +1.0$\arcsec$, -1.0$\arcsec$, 0.0$\arcsec$) dither pattern to reduce systematics (e.g., improving sky subtraction) in the final combined spectra, and clean cosmic-rays and detector defects. We targeted 52 objects ($z_{\text{phot}}\gtrsim5.5$) in GOODS-S and 66 in GOODS-N: 58 at $5.5<z_{\text{phot}}<6.5$, 54 at $6.5<z_{\text{phot}}<7.5$, and 6 at $7.5<z_{\text{phot}}<8.3$. The total exposure times are 22860 seconds (6.35hrs) for GOODS-S and 14400 seconds (4hrs) for GOODS-N.
The \texttt{spec2d} IDL pipeline developed for the DEEP2 Redshift Shift Survey \citep{Cooper2012a, Newman2013a} is publicly available for DEIMOS data reduction. However, the public pipeline has technical issues with our dithered observational data. Thus, we first obtained the sets of reduced individual science frames for individual slit objects from the pipeline, and then performed post-processing manually. The extracted frames from the pipeline are already flat-fielded, rectified, and response corrected. Every object spectrum spans two of the CCDs (blue and red sides) in the DEIMOS 4$\times$2 CCD configuration, and the pipeline reduces the spectrum independently for the blue and red sides. Taking the individual rectified science frames, we cleaned cosmic-rays (CRs) using the IDL procedure \texttt{L.A.Cosmic} \citep{van-Dokkum2001a}. Sky backgrounds were calculated by averaging two surrounding science frames, and we subtracted the sky backgrounds from all science frames before combining the individual frames. Similar to \cite{Kriek2015a}, when combining CR-cleaned and sky-subtracted science images, we measured relative weights through different science frames, based on the maximum fluxes estimated from Gaussian fitting for continuum sources (e.g., alignment stars). One-dimensional (1D) spectra of objects were extracted from the fully-reduced and combined two-dimensional (2D) spectra with $\sim$1.2 $\times$ the mean Gaussian FWHM along the spatial direction ($\sim 1.0 ''$). This follows an optimal extraction scheme \citep{Horne1986a}, which includes a spatial weight using a Gaussian profile in addition to an inverse-variance weight. Since DEIMOS is not equipped with an atmospheric dispersion compensator (ADC), differential refraction is problematic in cases where observations span a large airmass, causing up to a few pixels of offset in y-axis (spatial direction) on tracing object positions in the blue and red sides of 2D spectra \citep{Szokoly2005a, Newman2013a}. To correct this in our observations, we calculate the spatial offset of our four guide stars between the expected and the actual locations, independently in the blue and red sides of 2D spectra. Centering guide stars is done by Gaussian fitting along the spatial (y-axis) direction at every pixel in the wavelength direction (x-axis), and we measured the median offsets in the blue and red sides of the 2D spectra for all four guide stars. Lastly, by averaging the pre-obtained median offsets from the guide stars, we obtained the mean blue and red side spatial offsets, -0.66 and +0.64 pixels, respectively. Thus, we applied these offsets on locating object spatial positions when we extract 1D spectra.
Flux calibration and telluric absorption correction was done by using the model stellar spectrum \citep{Kurucz1993a} that has the same spectral type (B2IV) to the standard star (BD+33d2642 for GOODS-N). The standard star was observed in three science frames with a long slit and a 45-second exposure in each frame. Data reduction was done in the same manner as was used for reducing the spectra of our science objects as described earlier. The response profiles of the stellar 1D spectra were derived for both the blue and red sides separately, dividing the reduced 1D spectra of the standard star by the Kurucz model spectra. Absolute flux calibration was done simultaneously by measuring a scaling of the spectra to match their known $z$-band magnitudes. To test our flux calibration, we calculate $HST$/ACS $z_{850}$-band fluxes of mask alignment stars, and compare the measured fluxes to the $HST$ CANDELS imaging data. Slit losses were taken into account in this procedure, assuming our target galaxies are point sources, as the high-redshift galaxies are unresolved under the seeing of our observations. The estimated accuracy level of the flux calibration was $\sim20\%$ in flux (see Appendix), thus we include 20\% systematic errors in our emission line flux measurement.
As mentioned earlier, our flux calibration of GOODS-S data was less successful. This is due to the bad observing conditions (a large variation of the seeing between science objects and standard star observations, and a large drift on centering 2D spectra due to atmospheric dispersion, which is problematic to center 2D spectra precisely in the spatial direction). As the $z_{850}$ magnitudes of the GOODS-S continuum sources, which are calculated with the $HST$/ACS 850LP filter curve, are significantly different from those from \cite{Finkelstein2015a}, the calibrated fluxes for the GOODS-S dataset are unreliable. Thus, we were unable to use the calibrated Ly$\alpha$ line flux of the GOODS-S objects in our analysis in later sections. The details on flux calibration are discussed in Appendix.
\begin{table*}[t]
\centering
\begin{center}
\caption{Summary of Ly$\alpha$ Emitters Observed with Keck/DEIMOS}
\label{tab:candidates}
\begin{tabular}{ccccccccc}
\tableline
\tableline
\quad {ID\tablenotemark{a}} & {R.A.} & {Decl.} & {$z_{\text{spec}}$} & {$z_{\text{phot}}$} & {$J_{125}$} & {$M_{\text{UV}}$} & {$f_{\text{Ly$\alpha$ }}$} & {EW$_{\text{Ly}\alpha}$} \\
\quad {} & {(J2000.0)} & {(J2000.0)} & {} & {} & {} & {} & {($10^{-18}$erg s$^{-1}$cm$^{-2}$)}& {(\AA)} \\
\tableline
\quad{z6\_GND\_28438} & {189.177979} & {62.223713} & {6.551 $\pm$ 0.002} & {6.12 $^{+0.21}_{-4.48}$} & {26.51 $^{+0.09}_{-0.08}$} & {-19.93} & {3.08 $\pm$ 0.56} & {20.50 $\pm$ 3.90}\\
\quad{z6\_GND\_5752} & {189.199585} & {62.320965} & {6.583 $\pm$ 0.004} & {5.70 $^{+0.32}_{-4.72}$} & {27.32 $^{+0.20}_{-0.17}$} & {-19.18} & {3.34 $\pm$ 0.67} & {36.72 $\pm$ 7.35}\\
\quad{z7\_GND\_10402} & {189.179276} & {62.275894} & {6.697 $\pm$ 0.001} & {7.00 $^{+0.17}_{-0.25}$} & {25.75 $^{+0.06}_{-0.06}$} & {-21.16} & {3.08 $\pm$ 0.59} & {16.86 $\pm$ 2.51}\\
\quad{z5\_MAIN\_4396\tablenotemark{b}} & {53.138580} & {-27.790218} & {5.479 $\pm$ 0.001} & {5.18 $^{+0.06}_{-0.08}$} & {25.93 $^{+0.02}_{-0.02}$} & {-20.56} & {-} & {-}\\
\quad{z6\_GSD\_10956} & {53.124886} & {-27.784111} & {5.780 $\pm$ 0.002} & {5.51 $^{+0.26}_{-0.21}$} & {26.78 $^{+0.12}_{-0.10}$} & {-19.90} & {-} & {-}\\
\tableline
\end{tabular}
\end{center}
\begin{flushleft}
\tablenotetext{a}{The object IDs are from \cite{Finkelstein2015a}, encoded with their photometric redshifts and the fields in the CANDELS imaging data.}
\tablenotetext{b}{Ly$\alpha$ detection with $z=5.42\pm0.07$ for this object is reported in \cite{Rhoads2005a} from their $HST$/ACS grism survey.}
\end{flushleft}
\label{tab:LAEs}
\end{table*}
\begin{figure*}[t]
\centering
\includegraphics[width=0.84\paperwidth]{lae_goodsn.pdf}
\caption{(Top panels) One- and two-dimensional spectra of line-detected objects in GOODS-N. Red curves show the best-fit asymmetric Gaussian curves. (Bottom) Galaxy SED fitting results. Each panel shows two SEDs for high-$z$ (Ly$\alpha$) and low-$z$ (OII) solutions (black and blue solid curves, respectively). Red diamonds are observed fluxes with their associated errors. Photometric redshift probability distributions, $P(z)$ taken from \cite{Finkelstein2015a}, are displayed as inset figures, and the best-fit photometric redshifts are shown with vertical dashed lines.}
\label{fig:emissions1}
\end{figure*}
\begin{figure}[t]
\centering
\includegraphics[width=\columnwidth]{lae_goodss.pdf}
\caption{Same as Figure \ref{fig:emissions1} but for the GOODS-S objects.}
\label{fig:emissions2}
\end{figure}
\section{Results}
\subsection{Emission line detection}
We searched for emission lines utilizing both the 1D and 2D spectra. After a primary visual inspection of the 2D spectra to search for significant lines, we generated a list of potential emission lines from an automated emission line search routine. In this search we fit an asymmetric Gaussian function to 1D spectra at intervals of $\sim3$\AA, which is comparable to the instrumental spectral resolution, using \texttt{MPFIT} \citep{Markwardt2009a}. We derive the signal-to-noise (S/N) levels of the machine-detected lines via 1000 Monte-Carlo (MC) simulations, modulating the 1D spectrum within the 1D noise level. If any emission feature is detected with S/N $\gtrsim 4$, it is recorded in the list of potential emission lines. Candidates among the machine-detected lines are confirmed via secondary visual inspection of the 2D spectra.
With the systematic emission line search we find five (probable) Ly$\alpha$ emission lines at $z\gtrsim5.5$. The five LAEs are summarized in Table \ref{tab:LAEs}: three detections in GOODS-N and two in GOODS-S. The object IDs in the table are encoded with their photometric redshifts and the fields where they are detected in imaging data. The 1D and 2D spectra of the LAEs are shown in Figure \ref{fig:emissions1} (GOODS-N) and Figure \ref{fig:emissions2} (GOODS-S). We check if any of our detected LAEs are previously reported in the literature, and find that z5\_MAIN\_4396 in the Hubble Ultra Deep Field (HUDF) was previously detected as a Ly$\alpha$ emitting galaxy with a measured Ly$\alpha$ redshift \citep[$z=5.42\pm0.07$;][]{Rhoads2005a} from the $HST$/ACS grism survey program, the Grism ACS Program for Extragalactic Science (GRAPES) \citep[PI: S. Malhotra;][]{Pirzkal2004a}. Our spectroscopic redshift ($z=5.479$) is within their 1$\sigma$ error. However, as the GOODS-S data cannot be properly flux-calibrated, the two emission galaxies (z5\_MAIN\_4396 and z6\_GSD\_10956) in GOODS-S are excluded from further analysis in this paper, which requires the calibrated Ly$\alpha$ line flux. The line fluxes are measured from asymmetric Gaussian fitting (red curves in the figures). For calculating the EW of the detected Ly$\alpha$ line we use a continuum flux of the best-fit model galaxy SED, which is averaged over a 100\AA\ window redward of the Ly$\alpha$ line. The model SEDs are constructed from the \cite{Bruzual2003a} stellar population synthesis model, and our SED fitting is further described in Section 3.2.
We check the possibility of the detected emission lines being low-$z$ contaminants. First, to rule out the possibility of being [OII] $\lambda\lambda$3726, 3729, we compare the high-$z$ solutions of the Ly$\alpha$ lines to galaxy SED fitting results at the redshift of [OII] (bottom panels in Figure \ref{fig:emissions1} and Figure \ref{fig:emissions2}). In the case of z7\_GND\_10402 and z5\_MAIN\_4396, due to their strong Lyman-break our SED fitting strongly rejects the low-$z$ solutions, while for z6\_GND\_28438, z6\_GND\_5752, and z6\_GSD\_10956, we cannot rule out the low-$z$ solutions in the SED fitting results. However, if the emission lines we find are one of the OII doublets, the DEIMOS spectral resolution, $\sim3$\AA, can distinguish the two peaks with a gap of $\sim$7--8\AA\ at $z\sim1.5$. Inspecting the 2D spectral images of the two objects, we cannot find any significant second peak of the doublet nearby the detected emission line. Although there are small bumps found in their 1D spectra at 9172\AA\ for z6\_GND\_28438 and at 9230\AA\ for z6\_GND\_5752 (Figure \ref{fig:emissions1}), those are not considerably favored as the line intensity ratios of the bumps to the detected emission lines (5.36 for z6\_GND\_28438 and 0.22 for z6\_GND\_5752) do not satisfy the physically motivated low- and upper-limits of the line intensity ratio of the OII doublet $I(3729)/I(3726)$, which is from 0.35 to 1.5 \citep{Pradhan2006a}.
We also visually inspect the 1D and 2D spectra of the emission galaxies to find any features of H$\beta$ and [OIII] $\lambda\lambda$4959, 5007. In the case that any of our detected lines are one of H$\beta$ and the OIII doublet, the expected gaps between the lines range from 230 -- 282\AA\ at their low-$z$ solutions of $z=0.57$ -- 0.93. We check the expected locations of the three emission lines (H$\beta$ and [OIII] $\lambda\lambda$4959, 5007) individually, assuming that the detected emission is one of the three lines. We also check the possibility of being H$\alpha$ emission, searching for nearby emission features, [NII] $\lambda\lambda$6548, 6584. From visual inspection, we find no significant emission features, although there were some cases that the expected line locations fall close to sky emission lines so that we are unable to completely ignore the possibility. H$\beta$ and [OIII] $\lambda$4959 may be weaker than [OIII] $\lambda$5007, and [NII] $\lambda\lambda$6548, 6584 are weaker than H$\alpha$ as well, thus the expected S/N levels of the lines could be low, making it difficult to detect them. Thus, we additionally compare galaxy SED fitting results with the low-$z$ solutions of [OIII] $\lambda$5007 and H$\alpha$ to those with the high-$z$ Ly$\alpha$ solutions in order to check the low-$z$ possibility, but SED analysis alone cannot completely rule out the possibility. Although we still cannot completely rule out all the scenarios of low-$z$ interlopers for our sample (except for z5\_MAIN\_4396) in the given S/N levels, the low-$z$ solutions are highly unlikely as discussed above (e.g., no significant spectral features found). Therefore, we consider all the five detections as Ly$\alpha$ in our further analysis.
\begin{table}[t]
\centering
\begin{center}
\caption{Summary of the Physical Properties of the Ly$\alpha$ Emitters}
\label{tab:SED_fitting}
\begin{tabular}{cccc}
\tableline
\tableline
\quad {ID} & {log $M_{*}/M_{\odot}$} & {SFR ($M_{*}/M_{\odot}$)} & {$E(B-V)$} \\
\tableline
\quad {z6\_GND\_28438} & {9.43$^{+0.12}_{-0.17}$} & {22.1 $^{+9.5}_{-7.3}$} & {0.19 $^{+0.03}_{-0.04}$} \\
\quad {z6\_GND\_5752} & {9.30$^{+0.28}_{-0.29}$} & {11.6$^{+11.2}_{-5.6}$} & {0.16$^{+0.07}_{-0.06}$}\\
\quad {z7\_GND\_10402} & {9.42$^{+0.47}_{-0.43}$} & {16.7$^{+24.5}_{-10.2}$} & {0.31$^{+0.10}_{-0.10}$}\\
\quad {z5\_MAIN\_4396} & {8.78$^{+0.02}_{-0.01}$} & {12.4$^{+0.2}_{-0.2}$} & {0.004$^{+0.002}_{-0.002}$}\\
\quad {z6\_GSD\_10956} & {9.23$^{+0.09}_{-0.09}$} & {8.7$^{+1.9}_{-1.4}$} & {0.06$^{+0.02}_{-0.02}$}\\
\tableline
\end{tabular}
\end{center}
\tablecomments{Physical quantities of each object are derived from galaxy spectral energy distribution (SED) fitting with photometric data.}
\end{table}
\subsection{Galaxy physical properties}
To derive physical galaxy properties, we perform galaxy SED fitting with the $HST$/ACS (F435W, F606W, F775W, F814W and F850LP) + WFC3 (F105W, F125W, F140W and F160W) and $Spitzer$/IRAC 3.6$\mu$m and 4.5$\mu$m band fluxes of the line-detected galaxies. We use the Ly$\alpha$ emission line-subtracted fluxes in SED fitting, subtracting the measured Ly$\alpha$ emission fluxes from the $z_{850}$ and $Y_{105}$ band continuum fluxes. The details of our SED fitting are described in \cite{Jung2017a}. Briefly, it uses a Markov Chain Monte Carlo (MCMC) algorithm to fit the observed multi-wavelength photometric data with the model galaxy SEDs based on the \cite{Bruzual2003a} stellar population synthesis models. The physical properties of the five LAEs derived from our SED fitting are summarized in Table \ref{tab:SED_fitting}, and the derived stellar masses and UV-corrected SFRs show that our LAEs are typical star-forming galaxies, distributed within the $\sim1\sigma$ scatter of the typical $M_{*}-$SFR relation at $z\sim6$ \citep{Salmon2015a, Jung2017a}.
\begin{figure}[t]
\centering
\includegraphics[width=\columnwidth]{lya_sensitivity.pdf}
\caption{The $5\sigma$ detection limit of an emission line flux across the instrument wavelength coverage, measured with 3\AA\ spacing using a Monte-Carlo simulation, inserting mock emission lines. The colored dots show the measured detection limit from the different galaxies, and the median detection limit is drawn as the red curve. On the bottom, grey shaded regions represent sky emission lines. Between the sky emission lines, the typical $5\sigma$ detection limit is $\sim$ 3--5 $\times 10^{-18}$ erg s$^{-1}$ cm $^{-2}$. We derive a linear relation between the line strength and its S/N level across the instrument wavelength coverage, and the detection limit of each simulated Ly$\alpha$ is interpolated from the pre-calculated linear relation.}
\label{fig:sensitivity}
\end{figure}
\section{Measuring the L\lowercase{y}$\alpha$ EW distribution}
The EW distribution of Ly$\alpha$ emission is often described by an exponential form, $\text{dN/dEW}\propto \text{exp(-EW/}W_0)$, characterized by the $e$-folding scale, $W_0$ \cite[e.g.,][]{Cowie2010a}. The measured $e$-folding scale at $0.3 < z < 3.0$ is $\sim 60$\AA\ \citep[e.g.,][]{Gronwall2007a, Nilsson2009a, Guaita2010a, Blanc2011a, Ciardullo2012a, Wold2014a, Wold2017a}, and increases to higher redshift \citep[e.g.,][]{Zheng2014a, Hashimoto2017a}. One would expect to see a relatively reduced EW of observed Ly$\alpha$ emission from galaxies before cosmic reionization is completed, compared to the quantity observed from galaxies at lower redshift in the absence of galaxy evolution \citep{Bolton2013a, Mason2017a}, because Ly$\alpha$ photons emitted from high-redshift galaxies are resonantly scattered by neutral hydrogen atoms in the IGM. Thus, a measure of the $e$-folding scale of the Ly$\alpha$ EW distribution at the end of reionization is a key observable, which reflects the ionization status of the IGM. In this section, we provide our measure of this $e$-folding scale at $6.0<z<7.0$.
\begin{figure*}[t]
\includegraphics[width=1.05\columnwidth]{ndetection_w100_z65-eps-converted-to.pdf}
\includegraphics[width=1.05\columnwidth]{ndetection_z65-eps-converted-to.pdf}
\caption{(Left) 100 Monte-Carlo simulations of the expected number of detections as a function of S/N level ($\mathcal{S}$) with $W_0=100$\AA\ at $z\sim6.5$. We show only 100 of the 1000 simulation results for clarity. Each simulation is denoted by a different color, and the dashed curve shows the mean value, averaged over the 1000 simulations. (Right) the expected number of detections as a function of S/N level ($\mathcal{S}$) with various EW distributions at $z\sim6.5$. A larger choice of the $e$-folding scale ($W_0$) of the Ly$\alpha$ EW distribution (redder color) predicts a larger number of Ly$\alpha$ detections.}
\label{fig:n_detection}
\end{figure*}
\subsection{Simulating the expected number of detections}
A simple but key observable from our spectroscopic survey is the number of detected Ly$\alpha$ emission lines. This depends not only on the observed Ly$\alpha$ EW distribution of the observed galaxies, but also the completeness of the observations.
In this study, we wish to test the hypothesis that a uniform quenching of Ly$\alpha$ emission from a partially-neutral IGM is evolving the $e$-folding scale of the EW distribution towards lower values at $z\gtrsim6$. To facilitate this, we develop simulations which assess the likelihood of detecting a Ly$\alpha$ emission line of a given strength accounting for all sources of incompleteness (e.g., spectroscopic depth, sky lines, wavelength coverages, $P(z)$ distribution). \cite{Song2016b} described this as a Ly$\alpha$ visibility test, comparing the number of Ly$\alpha$ detections above a specific S/N level to that expected, with the latter calculated by assigning mock Ly$\alpha$ emission profiles in 1D spectra of target galaxies in a Monte-Carlo fashion. The Ly$\alpha$ wavelengths were drawn from the photometric redshift probability distribution function, $P(z)$, and \cite{Song2016b} adopted the intrinsic EW distribution from \cite{Schenker2014a}, which is based on published data at $3<z<6$, when the IGM was ionized.
We advance the Ly$\alpha$ visibility test of \cite{Song2016b} by setting the EW $e$-folding scale $W_0$ as a free parameter which we constrain with our observations.
To estimate the expected S/N levels for the simulated Ly$\alpha$ lines, we derive the detection limit for every individual galaxy at each wavelength by adding a mock emission line to the galaxy 1D spectra. We assume this mock line has an intrinsic line profile equal to the best-fit asymmetric Gaussian profile of our highest S/N Ly$\alpha$ emission detected in z5\_MAIN\_4396 (FWHM$_{\text{blue}}=0.88$\AA\ and FWHM$_{\text{red}}=9.69$\AA), obtained via \texttt{MPFIT}. We add in this emission line at each wavelength step at a range of emission line fluxes, and then measure the resultant S/N level of each line in the same Monte Carlo fashion as done on the real lines. We derive a linear relation between the line strength and its S/N level at all wavelengths with 3\AA$\ $spacing, a comparable size to the spectral resolution in our observational setting. This measurement allows us to determine the expected S/N levels for a given Ly$\alpha$ emission strength across the entire wavelength range. A typical 5$\sigma$ detection limit of Ly$\alpha$ flux is $\sim$3--5$\times$10$^{-18}$ erg s$^{-1}$ cm$^{-2}$ between sky emission lines as shown in Figure \ref{fig:sensitivity}. To check any dependence of the derived Ly$\alpha$ detection limit on the shape of the mock emission profile (specifically, FWHM), we re-did the simulations assuming a narrower mock line profile with $\text{FWHM}=5$\AA. In \cite{Mallery2012a}, the size of the Ly$\alpha$ FWHM at $3.8<z<6.5$ ranges from 5.71\AA\ to 10.88\AA\ (68\% confidence), so our tested range of FWHM from 5\AA\ to 10\AA\ reasonably considers the typical Ly$\alpha$ line profile. The smaller choice of $\text{FWHM}=5$\AA\ makes the line profile sharper, and slightly lowers the derived Ly$\alpha$ line detection limit. However, the overall difference of the estimated detection limit of an emission line flux between $\text{FWHM}=$5\AA\ and 10\AA\ is below the $\sim10\%$ level.
Using these S/N values, we then calculate the expected number of detections for a range of potential $W_0$ values. In this simulation a Monte-Carlo aspect is needed, as the broad photometric redshift distributions require us to sample a broad wavelength range, and the line strength of the simulated Ly$\alpha$ emission lines are sampled through the assumed EW distribution. For each mock emission line, we i) assign a wavelength for the Ly$\alpha$ line by drawing randomly from the photometric redshift distribution: $\lambda_{\text{Ly$\alpha$ }}=(1+z)\times1215.67$\AA, ii) assign the line strength by drawing from the assumed Ly$\alpha$ EW distribution: $P(\text{EW}) \propto \text{exp}^{-\text{EW}/W_0}$, which is based on the inferred continuum magnitudes near the wavelength of Ly$\alpha$ (averaged over a 100\AA\ window redward of Ly$\alpha$ emission), and iii) determine the S/N level of the simulated Ly$\alpha$ line at that wavelength using the values from the simulations described above. By doing so, we account for incompleteness due to the noise level in the data (from a combination of telescope+instrument throughput, and also integration time), and also due to the night sky lines which are ubiquitous at these wavelengths.
We perform this emission line simulation above for our GOODS-N observations, measuring the posterior distribution of the expected number of detections as a function of S/N for $e$-folding scales of $W_0=5-200$\AA. For each choice of $W_0$, we carry out 1000 Monte Carlo simulations. We illustrate the results for one value of $W_0$ (100 \AA) in the left panel of Figure \ref{fig:n_detection}, which shows the measured number of detections from 100 of the 1000 simulations, highlighting the dispersion in expected number, necessitating the need for a large number of simulations to robustly measure the posterior distribution. The right panel of Figure \ref{fig:n_detection} displays the mean expected number of detections, averaged over each set of 1000 simulations, as a function of S/N for a range of EW distributions for $6.0 < z < 7.0$. As seen in this figure, a larger choice of $W_0$ understandably predicts a larger number of detections, as we expect the galaxies to show stronger Ly$\alpha$ emission on average. One strong advantage of this method is that, in addition to our detected Ly$\alpha$ emission lines, the large number of non-detections are highly constraining as well.
\begin{figure*}[t]
\includegraphics[width=1.05\columnwidth]{w0_fit_z65-eps-converted-to.pdf}
\includegraphics[width=1.05\columnwidth]{w0_pdf_z65-eps-converted-to.pdf}
\caption{(Left) the probability distribution of the expected number of Ly$\alpha$ detections as a function of a S/N level ($\mathcal{S}$) at $z\sim6.5$, which is obtained from the $10^5$ MCMC chain steps. Higher probability regions are denoted by the brighter colors. The black solid curve shows the mean of the expected number of detections from our simulations as a function of S/N ($\mathcal{S}$), and the dashed curves are 1$\sigma$ uncertainties. Our three detections are drawn as the red solid line. (Right) the cumulative probability of the EW $e$-folding scale ($W_{0}$) from our MCMC-based fitting algorithm. The 1$\sigma$ and 2$\sigma$ upper limits are denoted with dotted and dashed red vertical lines, respectively.}
\label{fig:w0_fit}
\end{figure*}
\subsection{An $e$-folding scale of Ly$\alpha$ EW Distribution at $z\sim6.5$}
As counting the number of Ly$\alpha$ line detections can be described as a general Poisson problem, the likelihood of obtaining the particular results (counting the number of Ly$\alpha$ detections) is the Poisson likelihood. A well-known statistic related to the Poisson likelihood is the ``Cash statistic" \citep{Cash1979a}, which is described as follows.
\begin{eqnarray}
\begin{split}
C &= -2\ ln\mathcal{L}\\
&= -2\sum_{i=1} ({N_{o,i} ln(N_{m,i}) - N_{m,i} - ln N_{m,i}!}),
\end{split}
\label{eqn:likelihood}
\end{eqnarray}
where $\mathcal{L}$ is the Poisson likelihood, $N_{\text{o,i}}$ and $N_{\text{m,i}}$ are the observed and expected number of detections in a corresponding S/N bin, $i$, and $C$ is the goodnees-of-fit statistic so that the expected number of detections matches the observed number of detections in all S/N bins. $N_{\text{m,i}}$ is calculated based on the choice of $W_0$ as described in the previous section (refer the left panel of Figure \ref{fig:n_detection}). To construct the probability distribution of $W_0$ with the goodness-of-fit, we carry out MCMC sampling, which uses a Metropolis-Hastings algorithm \citep{Metropolis1953a,Hastings1970a}. In each chain step, a new candidate value for $W_0$ is randomly drawn from a Gaussian distribution, and we calculate the Poisson distribution log-likelihood of the candidate to go through the acceptance-rejection step. If the log-likelihood of the candidate $W_0$ exceeds that of the previous one by more than a uniform random variate drawn between 0 and 1, the candidate is accepted and recorded. Otherwise, the candidate is thrown away and retaken by the previous step. The random Gaussian width for choosing a new candidate $W_0$ in each step is tuned to have an optimal acceptance rate of 23.4\% \citep{Roberts1997a} to achieve the maximum efficiency of our MCMC sampling. Before recording the MCMC chains, we also run a burn-in stage to check the convergence of the MCMC sampling. We employ the Geweke diagnositc \citep{Geweke1992a}, comparing the mean and the variance of the first 10\% of chain steps to those of the last half of samples. Once the convergence criteria are satisfied, we record the MCMC chains. After the burn-in stage, we generate $10^5$ MCMC chains, which fully sample the $W_0$ probability distribution.
Performing this MCMC sampling with the three detections (S/N = 5.1, 5.5, and 6.7, respectively) from our observations, we calculate the posterior distribution of the Ly$\alpha$ EW $e$-folding scale at $6<z<7$. Our low number of detections make us unable to robustly constrain the median of $W_0$, thus we find a 1$\sigma$ (84\%) upper limit of $W_{0}<36.4$\AA\ (125.28\AA\ for a 2$\sigma$ limit; see the right panel of Figure \ref{fig:w0_fit}). In the left panel of Figure \ref{fig:w0_fit}, the background colors represent the probability of the expected number of detections obtained from the $10^5$ MCMC chain steps; higher probability region is denoted by brighter color. Black solid and dashed curves show the mean and 1$\sigma$ errors of the expected number of detections, and our observational data are shown as a red solid line.
\section{Discussion}
\begin{figure*}[t]
\centering
\includegraphics[width=0.65\paperwidth]{ew_evolution-eps-converted-to.pdf}
\caption{The redshift dependence of the Ly$\alpha$ EW $e$-folding scale ($W_{0}$). All data are shown without an IGM absorption correction. The black dashed line describes the best-fit redshift evolution from \cite{Zheng2014a}, compiling $0<z<7$ LAEs from literature: \cite{Guaita2010a} at $z=2.1$, \cite{Nilsson2009a} at $z=2.25$, \cite{Gronwall2007a} at $z=3.1$, \cite{Ciardullo2012a} at $z=3.1$, \cite{Ouchi2008a} at $z=3.1, 3.7$, \cite{Zheng2014a} at $z=4.5$, \cite{Kashikawa2011a} at $z=5.7, 6.5$, and \cite{Hu2010a} at $z=5.7, 6.5$ shown as filled circles. Blue diamonds are the measurements of \cite{Hashimoto2017a} using the LAEs ($M_{UV}<-18.5$) from the MUSE HUDF Survey \citep{Bacon2017a}, which are consistent with \cite{Zheng2014a} at that redshift range. At lower redshift, the $W_0$ measurements of \cite{Wold2017a} at $z\sim0.3$ and \cite{Wold2014a} at $z\sim0.9$ (orange triangles) suggest a relatively unevolving EW $e$-folding scale of Ly$\alpha$ across $z\sim0.3 - 3.0$, considering the other measurements described above, including \citet[black triangle]{Blanc2011a} at $z\sim2.85$.}
\label{fig:ew_evolution}
\end{figure*}
\subsection{Redshift dependence of the Ly$\alpha$ EW $e$-folding scale}
As discussed in the literature, the Ly$\alpha$ EW $e$-folding scale, $W_0$, is expected to decrease with increasing neutral hydrogen in the IGM. \citet[Z14 hereafter]{Zheng2014a} evaluate the redshift dependence of $W_0$ from compiled data at $0<z<7$, which show that larger EW LAEs are found at higher redshift. More recently, \cite{Hashimoto2017a} report their measurements of $W_0$ at $3<z<6$ using the MUSE HUDF Survey \citep{Bacon2017a}, which are consistent with Z14.
In Figure \ref{fig:ew_evolution}, we compare the redshift dependence of the Ly$\alpha$ EW $e$-folding scale from previous studies to our measure. Compared to the derived evolution of Z14 (black dashed curve), our measurement (red arrows) shows that, at 1$\sigma$ confidence, this quantity must begin to drop at $z >$ 6. As we expect that a higher fraction of neutral hydrogen in the IGM would reduce the strength of Ly$\alpha$ emission and lower the EW $e$-folding scale, this drop can be interpreted as a signal of an increasing amount of neutral hydrogen in the IGM, although the literature measurements of $W_0$ at $4 < z < 6$ are consistent with our measure at the 2$\sigma$ level.
However, the recent study of \cite{De-Barros2017a} presents a lower Ly$\alpha$ fraction at $z\sim6$ than the values previously reported in the literature. Although they measure the Ly$\alpha$ fraciton and not the EW distribution, this could mitigate the tension between our 1$\sigma$ upper limit and previous results, implying perhaps no significant evolution from $z =$ 6 to 6.5, though increasing the confidence of significant evolution at $z =$ 4 to 6. This is confirmed by \cite{Mason2017a}, who parameterize the $z\sim6$ Ly$\alpha$ EW distribution of \cite{De-Barros2017a} as a function of $M_{\text{UV}}$, Eq. (4) in their paper, and find an $e$-folding scale of the $z\sim6$ Ly$\alpha$ EW distribution from their parameterization ranges from $W_0 =$19 -- 43\AA\ (with $M_{\text{UV}}=$-17.5 to -22.5), also lower than those from other studies in the literature at $4 < z < 6$. With the drop of the EW $e$-folding scale at $z\sim6.5$ from our measurement, the recent measurements of the EW $e$-folding scale imply the smoother evolution of the neutral hydrogen fraction in the IGM between $z < 6$ and $z > 6$.
Also, it is worth mentioning the known effect that UV-selected LAEs have larger EWs with fainter UV magnitudes \citep[e.g.,][]{Ando2006a, Stark2010a, Stark2011a, Schaerer2011a, Cassata2015a, Furusawa2016a, Wold2017a}. \cite{Hashimoto2017a} systematically test the effect of sample selection on measuring the Ly$\alpha$ EW $e$-folding scale and find that including UV-fainter LAEs increases the measured $e$-folding scale of the Ly$\alpha$ EW distribution (refer to Figure 8 in their paper). Shown in Figure \ref{fig:muv_z}, our photo-$z$ selected galaxies have UV magnitudes $\lesssim-18.5$ (in GOODS-N), missing very UV-faint galaxies. Therefore, our measure of the EW $e$-folding scale can be biased toward a small value. However, the drop at $z\sim6.5$ of the measured EW $e$-folding scale found in this study is not fully explained by the sample selection effect and is still significant, compared to those at lower redshifts which use the similar UV magnitude cut ($M_{\text{UV}}<-18.5$) \citep[e.g.,][]{Hashimoto2017a}. Incorporating additional data from our MOSFIRE observations in our follow-up paper will update this result and its statistical confidence, and for further constraints, a more comprehensive analysis accounting for the UV magnitude dependence of the Ly$\alpha$ strength is needed in future study.
\subsection{Testing our measure of the Ly$\alpha$ EW $e$-folding scale}
\begin{figure}[t]
\includegraphics[width=\columnwidth]{w0_test-eps-converted-to.pdf}
\caption{Fractional error of the recovered $e$-folding scale compared to the input as a function of the number of mock detections used in our fitting procedure. We fit the $e$-folding scale with mock detections which follow the assumed EW distributions with the $e$-folding scale ranging from 5 to 200\AA. For each assumed $e$-folding scale, we create 1000 sets of mock detections in a Monte-Carlo fashion and recover the $e$-folding scale. The medians of the fractional errors of the $e$-folding scale are shown as diamonds with the error bars denoting the standard deviation. With $\lesssim10$ detections, our simulation recovers the true $e$-folding scale within $\lesssim30\%$, though there is a bias towards recovering more galaxies than those input due to up-scattering by noise near to the detection limit. However, an increased number of detections mitigate this bias, ensuring a more accurate recovery of the true $e$-folding scale. }
\label{fig:w0_test}
\end{figure}
We provide a measure of the Ly$\alpha$ EW $e$-folding scale at the end of reionization using our comprehensive simulations for predicting the expected number of Ly$\alpha$ detections. A novel way of accounting for the data incompleteness allows us to constrain the observed EW distribution of Ly$\alpha$ lines with a handful of detections as all non-detections are highly constraining in our simulations. This is very promising for upcoming additional spectroscopic studies of high-redshift Ly$\alpha$ emitting galaxies. Here we test the ability of our simulation to recover the Ly$\alpha$ EW $e$-folding scale using sets of virtual observations, and show how much future spectroscopic searches can improve the constraining power of measuring the EW $e$-folding scale.
To do the recovery test, we create sets of virtual observations. In each set of virtual observations, we first generate mock emission lines with $N_{\text{detection}} = 1$ -- 50, following the likelihood of the expected detections as a function of S/N level, which is derived from our simulation in Section 4.1. With the virtual set of detections, we fit the EW $e$-folding scale as described in Section 4.2. For each set of \textit{virtually} detected emission lines ($N_{\text{detection}}=1-50$), we create 1000 sets of virtual observations in a Monte-Carlo fashion, and recover the EW $e$-folding scale from each virtual dataset in the same was as done on our real data. Figure \ref{fig:w0_test} shows median and standard deviation of the fractional error of the recovered $e$-folding scale to the input as a function of the number of virtual detections used in our fitting procedure. With $\lesssim10$ detections, our simulation recovers the true $e$-folding scale with $\lesssim30\%$ of fractional errors. This does show a systematic bias on the derived EW $e$-folding scale, in that the recovered values are often lower than the true values with small numbers ($\lesssim10$) of detections. This bias is lessened with increasing numbers of detections, and the spread in this fractional error also decreases. One cause of this bias could be that in the limit where few lines are seen, these lines are likely close to the detection limit, where noise fluctuations can up-scatter lines below the limit to the detectable level, in a form of Eddington bias. However, an increasing number of mock detections mitigate this bias, ensuring the recovery of the assumed $e$-folding scale with smaller biases ($\lesssim10\%$ with $\gtrsim20$ detections) and smaller errors. Due to our small number of detected lines ($N_{\text{detection}}=3$), our measure of the Ly$\alpha$ EW $e$-folding scale at $z\sim6.5$ may be subject to this bias. However, the direction of this bias would cause us to overestimate the number of detected lines, resulting in our quoted upper limits on $W_0$ being conservatively high.
\subsection{Ly$\alpha$ Detection Probability as a Function of Redshift}
\begin{figure}[t]
\centering
\includegraphics[width=\columnwidth]{ndetection_z55-eps-converted-to.pdf}
\caption{The expected number of detections as a function of S/N level ($\mathcal{S}$) with various EW distributions at $z=5.5-6.0$. Unlike the predicted number of detections, we do not detect any Ly$\alpha$ emission in this range. A large spectroscopic survey with Ly$\alpha$ detections is needed to constrain the Ly$\alpha$ EW distribution.}
\label{fig:n_detection_z55}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=\columnwidth]{pz_dist-eps-converted-to.pdf}
\caption{Ly$\alpha$ detection probability (blue dashed curve) as a function of redshift in our spectroscopic dataset. The probability is measured, accounting for the probability distribution functions, $P(z)$, of the photometric redshifts of our target galaxies (grey curves) and the DEIMOS instrument throughput (red dashed curve). The estimated Ly$\alpha$ detection probability is high at $z\sim5.5-6.5$, and declines at $z\gtrsim7$.}
\label{fig:pz_dist}
\end{figure}
We have no Ly$\alpha$ detections at $5.5 < z < 6.0$ in our GOODS-N data, although the DEIMOS observations are sensitive to Ly$\alpha$ in that redshift range. As shown in Figure \ref{fig:n_detection_z55}, our simulations estimate that at $5.5 < z < 6.0$ we should have detected at least a couple of Ly$\alpha$ lines, as many as eight detections with a large $e$-folding scale of $W_0=200$\AA. At the published values at $5.5 < z < 6.0$ of $W_0\sim100$\AA\ (see Figure \ref{fig:ew_evolution}) our simulations predict $N_{\text{detection}}=6.62\pm1.99$. In Figure \ref{fig:pz_dist} we show the galaxy photometric redshift probability distribution functions for our sample (grey curves) and the DEIMOS instrument throughput (red dashed curve). Combining the two we calculate the Ly$\alpha$ detection probability (blue dashed curve), which is the normalized-expected number of Ly$\alpha$ detections among our target galaxies as a function of redshift accounting for this throughput, which is high at $z\sim5.5-6.5$, and declines at $z\gtrsim7$. This shows that non-detections are understandable at $z\sim7$, but the lack of detections at $z<6$ is somewhat unexpected. This could be due to the inhomogeneous nature of cosmic reionization at the very end, but it also reiterates the need for a more comprehensive spectroscopic survey over larger area to marginalize over these small number statistics.
\subsection{Systematic Effects of Photometric Redshifts}
\begin{figure}[t]
\centering
\includegraphics[width=\columnwidth]{ndetection_z65_pz_test-eps-converted-to.pdf}
\centering
\includegraphics[width=\columnwidth]{pz_test-eps-converted-to.pdf}
\caption{Top: the expected number of detections measured as described in Section 4.1, but when increasing the photometric redshift uncertainties by 10\% (left) and 50\% (right). Increasing the $P(z)$ uncertainty reduces $N_{\text{detected}}$ as the Ly$\alpha$ emission lines have more chance to be found outside of the instrumental wavelength coverage; the predicted $N_{\text{detected}}$ is almost half the number of our fiducial result (Figure \ref{fig:n_detection}) in the case that $P(z)$ has an additional 50\% systematic error. Bottom: The Ly$\alpha$ EW $e$-folding scale as a function of the augmented systematic errors in $P(z)$. As the augmented systematic error on $P(z)$ increases, the fitted Ly$\alpha$ EW $e$-folding scale in our simulation increases as well. If the true uncertainties were $\gtrsim$50\% of those assumed, our observations could be consistent with no evolution in the Ly$\alpha$ EW distribution.}
\label{fig:pz_test}
\end{figure}
As spectroscopic confirmation of galaxies at $z\gtrsim6$ is observationally expensive, photometric redshift selection provides a powerful means to construct extensive high-redshift galaxy catalogs based on multi-wavelength imaging survey data \citep[e.g.,][]{Stark2009a, Papovich2011a, Dunlop2013a, McLure2013a, Schenker2013a, Bouwens2015a, Finkelstein2015a, Salmon2015a, Song2016a, Livermore2017a}. However, photometric redshift measurements alone cannot completely remove the possibility that high-redshift candidate galaxies can be low-redshift interlopers. More pressingly, a comprehensive analysis on the accuracy of the photometric redshift measurements, specifically a calibration of these PDFs, at $z\gtrsim6$ is lacking. Thus statistical studies using photometric redshifts could be biased if there are uncovered systematic effects in photometric redshifts. Interestingly, a recent spectroscopic confirmation of Ly$\alpha$ emission from the $HST$ Faint Infrared Grism Survey (FIGS) \citep[PI: S. Malhotra;][]{Pirzkal2017a} finds a Ly$\alpha$ emission line at $z=$7.452 which is different from the photometric redshift at the 2$\sigma$ level \citep{Larson2017a}. While these differences should happen a small fraction of the time, here we consider the effects of underestimating the photometric redshift PDF.
To test how accurate our measure of the Ly$\alpha$ EW $e$-folding scale is to an increased photometric redshift uncertainty, we smooth the probability distribution functions of photometric redshifts to simulate a potential systematic underestimation of the errors of the current photometric redshift measurement, and perform our simulations on this altered dataset. The top panels of Figure \ref{fig:pz_test} show the likelihood of detecting Ly$\alpha$ emission lines with errors augmented by 10\% (left) and 50\% (right), which predict fewer detections than that with the current errors on the photometric redshifts. If the errors on the photometric redshifts are underestimated, our simulations predict too many Ly$\alpha$ detections, and our measured EW $e$-folding scale could be biased toward smaller values. A systematic test of the increased errors of photometric redshifts is shown in the bottom panel of Figure \ref{fig:pz_test}. If the true uncertainty of photometric redshift is 50\% larger than the current estimate, our constrained EW $e$-folding scale would be increased by a factor of a few: $W_0<154.68$\AA\ at 1$\sigma$ confidence. If this is the case, then our lack of detections is consistent with \emph{no} evolution in the Ly$\alpha$ EW distribution at $z >$ 6, highlighting the importance of calibrating the photometric redshift uncertainties with a dedicated spectroscopic survey, likely to come with the advent of the {\it James Webb Space Telescope}.
\section{Summary}
We have collected four nights of spectroscopic observations over 118 galaxies at $z\sim5-7$ in the GOODS fields with DEIMOS on the Keck telescope to search for Ly$\alpha$ emission in the early universe. We use these data to provide a new constraint for the $e$-folding scale of the Ly$\alpha$ EW distribution at the end of reionization. We simulate the predicted number of Ly$\alpha$ detections at a given expected S/N value in our observational data with a range of Ly$\alpha$ EW distributions (parameterized by the $e$-folding scale, $W_0$). We comprehensively account for incompleteness due to observational conditions (e.g., integration time, sky emission, and instrument throughput) as well as galaxy photometric redshift probability distribution functions. With our three detected Ly$\alpha$ lines in the GOODS-N field, we constrain the characteristic $e$-folding scale of the Ly$\alpha$ EW distribution at $z\sim6.5$. Our main results are summarized as follows.
\begin{enumerate}
\item Performing an automated search for emission lines in 1D spectra as well as visual inspection of 2D spectra, we detect five emission lines above a 5$\sigma$ significance level from four nights of Keck/DEIMOS observation among a sample of 118 high-$z$ candidate galaxies in the GOODS-S and GOODS-N fields. Our tests of the possibility of low-$z$ interlopers indicates that the detected lines are likely Ly$\alpha$ emission at $z\gtrsim5.5$.
\item We simulate the expected number of Ly$\alpha$ detections from our observational dataset, comprehensively taking into account noise in the dataset and galaxy photometric redshift probability distributions. In the simulations, we construct the probability distribution of the expected number of detections as a function of S/N level with various $e$-folding scales ($W_0$) of the Ly$\alpha$ EW distribution, where a larger value of $W_0$ predicts a larger number of Ly$\alpha$ detections.
\item Our dataset constrains the Ly$\alpha$ EW $e$-folding scale at $z\sim6.5$ to be $<$ 36.40\AA\ at 1$\sigma$ confidence (125.28\AA\ at 2$\sigma$). This is lower than previous measurements at lower redshifts, providing weak evidence for an increasing fraction of neutral hydrogen in the IGM at this epoch. Additional data from our MOSFIRE observations at $z>7$ will update this result in a future paper with a higher statistical confidence.
\item We test the ability of our simulation to recover the Ly$\alpha$ EW $e$-folding scale as a function of the number of detections, and find $\gtrsim20$ detections allow us to recover the true value of the Ly$\alpha$ EW $e$-folding scale to $\lesssim10\%$ accuracy; these simulations imply that our current results provide conservative upper limits. We also find that systematic errors in the photometric redshift uncertainties would have a significant impact on constraining the EW $e$-folding scale, suggesting that a comprehensive analysis of photometric redshift uncertainties in the early universe is necessary.
\end{enumerate}
As mentioned in Section 2.1 and shown in Figure \ref{fig:mask_design}, in addition to DEIMOS, our entire spectroscopic dataset utilizes MOSFIRE as well to search Ly$\alpha$ emission at $z>7$, and we will analyze these MOSFIRE observations in a follow-up paper, using the DEIMOS observations in this paper as the $z =$ 6.5 anchor. Furthermore, analyzing both the DEIMOS and MOSFIRE data allows us to search for Ly$\alpha$ emission more comprehensively, which guarantees to cover the entire wavelength range of the probable locations of Ly$\alpha$ emission from galaxies at $5<z<8.2$. Particularly for $z\sim7$ candidate galaxies, Ly$\alpha$ emission from these galaxies cannot be entirely searched by a single instrument, either DEIMOS or MOSFIRE (which observes Ly$\alpha$ emission at $5<z<7$ and $7<z<8.2$, respectively). Thus, a complete search for Ly$\alpha$ emission at $z\sim7$ with a systematic spectroscopic survey with both DEIMOS and MOSFIRE will make significant progress.
Our measurement of the Ly$\alpha$ EW distribution alone is not enough to calculate directly the IGM neutral hydrogen fraction during reionization. We require detailed models, which account for a variety of effects, to calculate a likely range for the neutral fraction based on our observations \cite[see discussion in][]{Bolton2013a, Dijkstra2014a}. Combined with these models our Ly$\alpha$ visibility can constrain the neutral fraction of hydrogen in the IGM, but this constraining power depends on several factors, which need to be considered in future studies: the amount of residual neutral hydrogen in the circumgalactic medium, the number of self-shielding Lyman-limit systems in the IGM, where Ly$\alpha$ is self-shielded by overdense gas, and the Lyman continuum escape fraction. By providing our estimate of the Ly$\alpha$ EW distribution at $z>6$, we provide observational constraints to reionization models, leading to improved predictions during the end of reionization over $6 < z < 8$.
\acknowledgments
The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Mauna Kea has always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain. We also thank M. Dijkstra for constructive discussions. I.J. acknowledges support from the NASA Headquarters under the NASA Earth and Space Science Fellowship Program - Grant 80NSSC17K0532. I.J. and S.F. acknowledge support from NSF AAG award AST-1518183. RCL acknowledges support from an Australian Research Council Discovery Early Career Researcher Award (DE180101240). M.S.'s research was supported by an appointment to the NASA Postdoctoral Program at the NASA Goddard Space Flight Center, administered by Universities Space Research Association under contact with NASA.
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1,116,691,497,599 | arxiv | \section{Introduction and Conclusions}
Supersymmetry breaking string vacua (including 10d non-supersymmetric strings) are generically affected by tadpole sources for dynamical fields, unstabilizing the vacuum \cite{Fischler:1986ci,Fischler:1986tb}. We refer to them as {\em dynamical tadpoles} to distinguish them from {\em topological tadpoles}, such as RR tadpoles, which lead to topological consistency conditions on the configuration (note however that dynamical tadpoles were recently argued in \cite{Mininno:2020sdb} to relate to violation of swampland constraints of quantum gravity theories). Simple realizations of dynamical tadpoles arose in early models of supersymmetry breaking using antibranes in type II (orientifold) compactifications \cite{Sugimoto:1999tx,Antoniadis:1999xk,Aldazabal:1999jr,Uranga:1999ib}, or in 10d non-supersymmetric string theories \cite{Polchinski:1998rr}.
Dynamical tadpoles indicate the fact that equations of motion are not obeyed in the proposed configuration, which should be modified to a spacetime-dependent solution (more precisely, solution in which some fields do not preserve the maximal symmetry in the corresponding spacetime dimension, but we stick to the former nomenclature), e.g. rolling down the slope of the potential. This approach has been pursued in the literature (see e.g. \cite{Dudas:2000ff,Blumenhagen:2000dc,Dudas:2002dg,Dudas:2004nd,Mourad:2016xbk}), although the resulting configurations often contain metric singularities or strong coupling regimes, which make their physical interpretation difficult.
In this work we present large classes of spacetime\footnote{Actually, we restrict to configurations of fields varying over spatial dimensions (rather than time); yet we abuse language and often refer to them as spacetime-dependent.} dependent field configurations sourced by dynamical tadpoles, which admit a simple and tractable smoothing out of such singularities. Remarkably, these examples reveal a set of notable physical principles and universal scaling behaviours. We argue that the presence of a dynamical tadpole implies the appearance of ends of spacetime (or walls of nothing) at a finite spacetime distance, which is (inversely) related to the strength of the tadpole. These ends of spacetime moreover correspond to cobordism defects (or end of the world branes) of the theory implied by the swampland cobordism conjecture \cite{McNamara:2019rup,Montero:2020icj}. In most setups the cobordism defects end up closing off the space into a compact geometry (possibly decorated with branes, fluxes or other ingredients), thus triggering spontaneous compactification.
We can sum up the main features described above, and illustrated by our examples, in two lessons:
\smallskip
$\bullet$ {\em {\bf{\em Finite Distance:}} In the presence of a dynamical tadpole controlled by an order parameter ${\cal{T}}$, the spacetime-dependent solution of the equations of motion cannot be extended to spacetime distances beyond a critical value $\Delta$ scaling inversely proportional to ${\cal T}$, with a scaling relation}
\begin{eqnarray}
\Delta^{-n}\sim {\cal T}\ .
\label{bound-lesson1}
\end{eqnarray}
In our examples, $n=1$ or $n=2$ for setups with an underlying AdS-like or Minkowski vacuum, respectively.
\smallskip
$\bullet$ {\bf {\em Dynamical Cobordism:}} {\em The physical mechanism cutting off spacetime dimensions at scales bounded by the $\Delta$ above, is a cobordism defect of the initial theory (including the dynamical tadpole source).}
\smallskip
To be precise, when there are multiple spacetime directions to be closed off, the actual defect is the cobordism defect corresponding to circle or toroidal compactifications of the initial theory, with suitable monodromies on non-trivial cycles. This is analogous to the mechanism by which F-theory on half a ${\bf {P}}_1$ provides the cobordism defect for type IIB on ${\bf {S}}^1$ with $SL(2,{\bf {Z}})$ monodromy \cite{McNamara:2019rup} (see also \cite{Dierigl:2020lai}).
As explained, we present large classes of models illustrating these ideas, including (susy and non-susy) 10d string theories and type II compactifications with D-branes, orientifold planes, fluxes, etc. For simplicity, we present models based on toroidal examples (and orbifolds and orientifolds thereof), although many of the key ideas easily extend to more general setups. This strongly suggests that they can apply to general string theory vacua. Very remarkably, the tractability of the models allows to devise spontaneous compactification whose endpoint corresponds to some of the (supersymmetric extensions of the) SM-like D-brane constructions in the literature. As will be clear, our examples can often be regarded as novel reinterpretations of models in the literature.
Although our examples are often related to supersymmetric models, supersymmetry is not a crucial ingredient in our discussion. Dynamical tadpoles correspond to sitting on the slope of potentials, which, even in theories admitting supersymmetric vacua, correspond to non-supersymmetric points in field space. On the other hand, supersymmetry of the final spacetime-dependent configuration is a useful trick to guarantee that dynamical tadpoles have been solved, but it is possible to build solutions with no supersymmetry but equally solving tadpoles.
\medskip
Our results shed new light on several features observed in specific examples of classical solutions to dynamical tadpoles, and provide a deeper understanding of the appearance of singularities, and the stringy mechanism smoothing them out and capping off dimensions to yield dynamical compactification. In particular, we emphasize that our discussion unifies several known phenomena and sheds new light on the strong coupling singularities of type I' in \cite{Polchinski:1995df} and in heterotic M-theory \cite{Witten:1996mz} (and its lower bound on the 4d Newton's constant). There are several directions which we leave for future work, for instance:
$\bullet$ As is clear from our explicit examples, many constructions of this kind can be obtained via a reinterpretation of known compactifications. This strongly suggests that our lessons have a general validity in string theory. It would be interesting to explore the discussion of tadpoles, cobordism and spontaneous compactifications in general setups beyond tori.
$\bullet$ A general consequence of (\ref{bound-lesson1}) is a non-decoupling of scales between the geometric scales controlling the order parameter of the dynamical tadpole and the geometric size of the spontaneously compactified dimensions. This is reminiscent of the swampland AdS distance conjecture \cite{Lust:2019zwm}. It would be interesting to explore the generation of hierarchies between the two scales, possibly based on discrete ${\bf {Z}}_k$ gauge symmetries as in \cite{Buratti:2020kda}.
$\bullet$ Our picture can be regarded as belonging to the rich field of swampland constraints on quantum gravity \cite{Vafa:2005ui} (see \cite{Brennan:2017rbf,Palti:2019pca,vanBeest:2021lhn} for reviews). It would be interesting to study the interplay with other swampland constraints. In particular, the relation between the strength of the dynamical tadpole and the size of the spacetime dimensions is tantalizingly reminiscent of the first condition on $|\nabla V|/V$ of the de Sitter conjecture \cite{Obied:2018sgi,Garg:2018reu,Ooguri:2018wrx}, with ${\cal T}=|\nabla V|$ and if we interpret $V$ as the inverse Hubble volume and hence a measure of size or length scale in the spacetime dimensions. It would be interesting to explore cosmological setups and a possible role of horizons as alternative mechanisms to cut off spacetime. Also, the inequality admittedly works in different directions in the two setups, thus suggesting they are not equivalent, but complementary relations.
$\bullet$ It would be interesting to apply our ideas to the study of other setups in which spacetime is effectively cut off, such as the capping off of the throat in near horizon NS5-branes due to strong coupling effects, or the truncation in \cite{Montero:2015ofa} of throats of the euclidean wormholes in pure Einstein+axion theories \cite{Giddings:1987cg}.
$\bullet$ Finally, we have not discussed time-dependent backgrounds\footnote{For classical solutions of tadpoles involving time dependence, see e.g. \cite{Dudas:2002dg}.}. These are obviously highly interesting, but their proper understanding is likely to require new ingredients, such as {\em end (or beginning) of time} defects (possibly as generalization of the spacelike S-branes \cite{Kruczenski:2002ap,Jones:2004rg}).
\medskip
Until we come back to these questions in future work, the present paper is organized as follows. In Section \ref{sec:conifold} we reinterpret the Klebanov-Strassler (KS) warped throat supported by 3-form fluxes as a template illustrating our two tadpole lessons. Section \ref{sec:conifold-tadpole} explains that the introduction of RR 3-form flux in type IIB theory on AdS$_5\times T^{1,1}$ produces a tadpole. The varying field configuration is the Klebanov-Tseytlin solution, which leads to a metric singularity at a finite distance scaling as (\ref{bound-lesson1}), as we show in section \ref{sec:conifold-singu}. In section \ref{sec:conifold-cobordism} we relate the KS smoothing of this singularity with cobordism defects. In section \ref{sec:conifold-generalized} we extend the discussion to other warped throats. In Section \ref{sec:3form-flux} we present a similar discussion in toroidal compactifications with fluxes. Section \ref{sec:3form-flux-tadpole} introduces a ${\bf {T}}_5$ compactification with RR 3-form flux, whose tadpole backreacts producing singularites at finite distance as we show in section \ref{sec:3form-flux-singu}. In section \ref{sec:3form-flux-cobordism} we argue they are smoothed out by capping off dimensions and triggering spontaneous compactification. In Section \ref{sec:magnetization} we build examples in the context of magnetized D-branes. In section \ref{sec:magnetization-local} we describe the tadpole backreaction and its singularities, which are removed by spontaneous compactification in section \ref{sec:magnetization-global}. In Section \ref{sec:10d} we turn to the dilaton tadpole of several 10d strings. In section \ref{sec:massive-iia} we consider massive type IIA theory, where the running dilaton solutions produce dynamical cobordisms by introduction of O8-planes as cobordism defects of the IIA theory, eventually closely related to type I' compactifications. In section \ref{sec:hw} we discuss a similar picture for M-theory on K3 with $G_4$ flux, and a Horava-Witten wall as its cobordism defect. In section \ref{sec:sugimoto} we consider the 10d non-supersymmetric $USp(32)$ theory, in two different approaches. In section \ref{sec:sugimoto-dm} we build on the classical solution in \cite{Dudas:2000ff} and discuss its singularities in the light of the cobordism conjecture. In section \ref{sec:sugimoto-magnetization} we describe an explicit (and remarkably, supersymmetry preserving) configuration solving its tadpole via magnetization and spontaneous compactification on ${\bf {T}}^6$. In Section \ref{sec:the-sm} we discuss an interesting application, describing a 6d model with tadpoles, which upon spontaneous compactification reproduces a semi-realistic MSSM-like brane model. Finally, Appendix \ref{app:tadpole-wgc} discusses the violation of swampland constraints of type IIB on AdS$_5\times T^{1,1}$ when its tadpole is not duly backreacted, in a new example of the mechanism in \cite{Mininno:2020sdb}.
\section{The fluxed conifold: KS solution as spontaneous cobordism}
\label{sec:conifold}
In this section we consider the question of dynamical tadpoles and their consequences in a particular setup, based on the gravity dual of the field theory of D3-branes at a conifold singularity. The discussion is a reinterpretation, in terms useful for our purposes, of the construction of the Klebanov-Tseytlin (KT) solution \cite{Klebanov:2000nc} and its deformed avatar, the Klebanov-Strassler (KS) solution \cite{Klebanov:2000hb}. This reinterpretation however provides an illuminating template to discuss dynamical tadpoles in other setups in later sections.
We consider type IIB on AdS$_5\times T^{1,1}$, where $T^{1,1}$ is topologically ${\bf {S}}^2\times {\bf {S}}^3$ \cite{Klebanov:1998hh}. This is the near horizon geometry of D3-branes at the conifold singularity \cite{Klebanov:1998hh} (see also \cite{Morrison:1998cs,Uranga:1998vf,Dasgupta:1998su}), which has been widely exploited in the context of holographic dualities. The vacuum is characterized by the IIB string coupling $e^\phi=g_s$ and the RR 5-form flux $N$. The model has no scale separation, since the $T^{1,1}$ and AdS$_5$ have a common scale $R$, given by
\begin{eqnarray}
R^4\,=\, 4\pi\, g_sN\alpha'{}^2\ .
\label{ads-radius}
\end{eqnarray}
In any event, we will find useful to discuss the model, and its modifications, in terms of the (KT) 5d effective theory introduced in \cite{Klebanov:2000nc}. This is an effective theory not in the Wilsonian sense but in the sense of encoding the degrees of freedom surviving a consistent truncation. In particular, it includes the dilaton $\phi$ (we take vanishing RR axion for simplicity), the NSNS axion $\Phi=\int_{{\bf {S}}^2}B_2$ and the $T^{1,1}$ breathing mode $q$ (actually, stabilized by a potential arising from the curvature and the 5-form flux), which in the Einstein frame enters the metric as
\begin{eqnarray}
&&ds^2_{10} = R^2 \big( \, e^{- 5q}\, ds^2_5 + \, e^{3q}ds^2_{T^{1,1}} \big)\ .
\label{metric-ansatz-eft}
\end{eqnarray}
%
This approach proved useful in \cite{Buratti:2018xjt} in the discussion of the swampland distance conjecture \cite{Ooguri:2006in} in configurations with spacetime-dependent field configurations (see \cite{Lust:2019zwm} for a related subsequent development, and \cite{Baume:2020dqd,Perlmutter:2020buo}).
\subsection{The 5d tadpole and its solution}
\label{sec:conifold-tadpole}
Let us introduce $M$ units of RR 3-form flux in the ${\bf {S}}^3$, namely
\begin{eqnarray}
F_3\,=\, M\, \omega_3\ ,
\end{eqnarray}
where $\omega_3$ is defined in eq. (27) in \cite{Klebanov:2000hb}. We do not need its explicit expression, it suffices to say that it describes a constant field strength density over the ${\bf {S}}^3$. The introduction of this flux sources a backreaction on the dilaton and the metric, namely a dynamical tadpole for $\phi$ and $q$. In addition, as noticed in \cite{Buratti:2018xjt}, it leads to an axion monodromy potential for $\Phi$ \cite{Silverstein:2008sg,Kaloper:2008fb,Marchesano:2014mla,Hebecker:2014eua}. The situation is captured by the KT effective action (with small notation changes) for the 5d scalars $\phi$, $\Phi$ and $q$, collectively denoted by $\varphi^a$
\begin{eqnarray}
S_{ 5}
= - \frac{2}{\kappa_{5}^2} \int d^{5} x
\ \sqrt{-g_{5}} \,\bigg[
{ \frac 1 4} R_5
- { \frac 1 2} G_{ab}(\varphi) \partial \varphi^a \partial \varphi^b
- V(\varphi)\bigg],
\label{ktaction}
\end{eqnarray}
with the kinetic terms and potential given by
\begin{eqnarray}
&& G_{ab}(\varphi) \partial \varphi^a \partial \varphi^b \, =\, 15 (\partial q)^2 + { \frac{1}{4}}(\partial \phi )^2 + { \frac{1}{4}} e^{ -\phi- 6 q}(\partial \Phi )^2 \ , \label{kinetic}\\
&& V(\varphi) \, = \,-5 e^{ - 8 q} + {\frac{1}{8}} M^2\, e^{\phi- 14 q}
+ {\frac{1}{8}} (N + M \phi)^2 e^{ - 20 q}\ . \label{potential}
\end{eqnarray}
Clearly $g_s M^2$ is an order parameter of the corresponding dynamical tadpole. In the following we focus on the case\footnote{This implies that the configuration is uncharged under a discrete ${\bf {Z}}_M$ symmetry, measured by $N$ mod $M$, and associated to the redundancy generated by transformation $\phi\to \phi+1$, $N\to N-M$, see footnote \ref{foot:non-multiple}.\label{foot:multiple}} of $N$ being a multiple of $M$.
Ignoring the backreaction of the dynamical tadpole (i.e. considering constant profiles for the scalars over the 5d spacetime) is clearly incompatible with the equations of motion. Furthermore, as argued in \cite{Mininno:2020sdb}, it can lead to violations of swampland constraints. In particular, since the introduction of $F_3$ breaks supersymmetry, if the resulting configuration was assumed to define a stable vacuum, it would violate the non-susy AdS conjecture \cite{Ooguri:2016pdq}; also, as we discuss in Appendix \ref{app:tadpole-wgc}, it potentially violates the Weak Gravity Conjecture \cite{ArkaniHamed:2006dz}.
Hence, we are forced to consider spacetime-dependent scalar profiles to solve the equations of motion. Actually, this problem was tackled in \cite{Klebanov:2000hb}, with the scalars running with $r$, as we now review in the interpretation in \cite{Buratti:2018xjt}. There is a non-trivial profile for the axion $\Phi$, given by
\begin{eqnarray}
\Phi\, =\, 3g_sM\, \log (r/r_0)\ .
\end{eqnarray}
This implies the cancellation of the dilaton tadpole, which can be kept constant $e^{\phi}=g_s$, as follows from its equation of motion from (\ref{kinetic}), (\ref{potential})
\begin{eqnarray}
\nabla \phi\sim -e^{-6q-\phi}(\partial\Phi)^2+e^{-14q+\phi}M^2\ .
\end{eqnarray}
\subsection{Singularity at finite distance}
\label{sec:conifold-singu}
The varying $\Phi$ corresponds to the introduction of an NSNS 3-form flux in the configuration
\begin{eqnarray}
H_3\,=\, -g_s \,*_{6d} F_3\ ,
\end{eqnarray}
where the 6d refers to $T^{1,1}$ and the AdS$_5$ radial coordinate $r$, and the Hodge duality is with the AdS$_5\times T^{1,1}$ metric. This is precisely such that the complexified flux combination $G_3=F_3-\tau H_3$ satisfies the imaginary self duality (ISD) constraint making it compatible with 4d Poincar\'e invariance in the remaining 4d coordinates (and in fact, it also preserves supersymmetry). The backreaction on the metric thus has the structure in \cite{Dasgupta:1999ss,Giddings:2001yu}. The metric (\ref{metric-ansatz-eft}) takes the form
\begin{eqnarray}
ds_{10}^{\, 2}\,=\, Z^{-\frac 12} \eta_{\mu\nu} dx^\mu dx^\nu\, +\,Z^{\frac 12}\, \, (\, dr^2 \, +\, r^2 ds_{T^{1,1}}^2 \, )\ ,
\end{eqnarray}
where $Z$ obeys a Laplace equation in AdS$_5$, sourced by the fluxes, and reads
\begin{eqnarray}
Z(r)\,=\,\frac{1} {4r^{4}}\,(g_s M)^2\, \log (r/r_0)\ .
\end{eqnarray}
The warp factor also enters in the RR 5-form flux, which decreases with $r$ as
\begin{eqnarray}
N(r)\,=\, \int_{{\bf {S}}^5} F_5\,=\, g_sM^2 \log (r/r_0)\ .
\label{running-flux}
\end{eqnarray}
This matches nicely with the monodromy for the axion $\Phi$ as it runs with $r$ \cite{Buratti:2018xjt}.
These features (as well as some other upcoming ones) were nicely explained as the gravity dual of a Seiberg duality cascade in \cite{Klebanov:2000hb}.
This 5d running solution in \cite{Klebanov:2000nc} solves the dynamical tadpole, but is not complete, as it develops a metric singularity at $r=r_0$. This is a physical singularity at finite distance in spacetime, whose parametric dependence on the parameters of the initial model is as follows
\begin{eqnarray}
& \Delta(r) = &\int_{r_0}^r Z(r)^{\frac 14}\, dr\sim \int_{r_0}^r (g_sM)^{\frac 12} \big[ \log (r/r_0)\big]^{\frac 14}\,\frac{dr}r \nonumber \\
&& \sim (g_sM)^{\frac 12} \big[ \log (r/r_0)\big]^{\frac 54}= (g_s N)^{\frac 14}\,\frac {N}{g_sM^2} \sim R\, \frac {N}{g_sM^2}\ .
\end{eqnarray}
In the last equalities we used (\ref{running-flux}), (\ref{ads-radius}). Hence, starting with an AdS$_5\times T^{1,1}$ theory with $N$ units of RR 5-form flux, the introduction of $M$ units of RR 3-form flux leads to a breakdown of the corresponding spacetime-dependent solution at a distance scaling as $\Delta\sim M^{-2}$. Recalling that the dynamical tadpole is controlled by an order parameter ${\cal T}=g_sM^2$, this precisely matches the scaling relation (\ref{bound-lesson1}) of the Finite Distance Lesson.
\subsection{Dynamical cobordism and the KS solution}
\label{sec:conifold-cobordism}
As is well known, the singularity in the KT solution is smoothed out in the KS solution \cite{Klebanov:2000hb}. This is given by a warped version of the deformed conifold metric, instead of the conical conifold singularity, with warp factor again sourced by an ISD combination of RR 3-form flux on ${\bf {S}}^3$ and NSNS 3-form flux on ${\bf {S}}^2$ times the radial coordinate. At large $r$ the KS solution asymptotes to the KT solution, but near $r\sim r_0$, the solutions differ and the KT singularity is replaced by the finite size ${\bf {S}}^3$ of the deformed conifold.
Hence, the Finite Distance Lesson still applies even when the singularity is removed, and the impossibility to extend the coordinate $r$ to arbitrary distances is implemented by a smooth physical end of spacetime. The purpose of this section is to highlight a novel insight on the KS solution, as a non-trivial realization of the swampland cobordism conjecture \cite{McNamara:2019rup,Montero:2020icj}. The latter establishes that any consistent quantum gravity theory must be trivial in (a suitably defined version of) cobordism. Namely in an initial theory given by an $n$-dimensional internal compactification space (possibly decorated with additional ingredients, like branes or fluxes), there must exist configurations describing an $(n+1)$-dimensional (possibly decorated) geometry whose boundary is the initial one. The latter describes an end of the world defect (which we will refer to as the `cobordism defect') for the spacetime of the initial theory. Since the arguments about the swampland cobordism conjecture are topological, there is no claim about the unprotected properties of the cobordism defect, although in concrete examples it can preserve supersymmetry; for instance, in maximal dimensions, the Horava-Witten boundary is the cobordism defect for 11d M-theory, and similarly the O8-plane is the cobordism defect of type IIA theory\footnote{Other 10d theories are conjectured to admit cobordism branes, but they cannot be supersymmetric and their nature is expected to be fairly exotic, and remains largely unknown. We will come back to this point in section \ref{sec:sugimoto-dm}.}.
In our setup, the initial theory is AdS$_5\times T^{1,1}$ with $N$ units of RR 5-form flux and $M$ units of RR 3-form flux on ${\bf {S}}^3$. From the above discussion, it is clear that the KS solution is just the cobordism defect of this theory\footnote{Recalling footnote \ref{foot:multiple}, the case of $N$ multiple of $M$ implies the vanishing of a ${\bf {Z}}_M$ charge, and allows the cobordism defect to be purely geometrical; otherwise the cobordism defect ending spacetime must include explicit D3-branes, which are the defect killing the corresponding cobordism class \cite{McNamara:2019rup}.\label{foot:non-multiple}}. The remarkable feature is that the end of spacetime is triggered dynamically by the requirement of solving the equations of motion after the introduction of the RR 3-form flux, hence it is fair to dub it dynamical cobordism. Hence, this is a very explicit illustration of the Dynamical Cobordism Lesson.
This powerful statement will be realized in many subsequent examples in later sections, and will underlie the phenomenon of spontaneous compactification, when the cobordisms close off the spacetime directions bounding them into a compact variety.
\subsection{More general throats}
\label{sec:conifold-generalized}
A natural question is the extension of the above discussion to other AdS$_5\times {\bf {X}}_5$ vacua with 3-form fluxes. This question is closely related to the search for general classes of gravity duals to Seiberg duality cascades and their infrared deformations, for which there is a concrete answer if ${\bf {X}}_5$ is the real base of a non-compact toric CY threefold singularity ${\bf {Y}}_6$, which are very tractable using dimer diagrams \cite{Hanany:2005ve,Franco:2005rj} (see \cite{Kennaway:2007tq} for a review).
From our perspective, the result in \cite{Franco:2005fd} is that the ${\bf {X}}_5$ compactification with 3-form flux $F_3$ admits a KS-like end of the world (cobordism defect\footnote{We note in passing that the regions between different throats in the multi-throat configurations \cite{Franco:2005fd} can be regarded as domain walls interpolating between two different, but bordant, type IIB vacua.}) if ${\bf {Y}}_6$ admits a complex deformation which replaces its conical singularity by a finite-size 3-cycle corresponding to the homology dual of the class $[F_3]$. In cobordism conjecture terms, in these configurations the corresponding global symmetry is gauged, and spacetime may close off without further ado (as the axion monodromy due to the 3-form fluxes allows to eat up the RR 5-form flux before reaching the end of the world). Such complex deformations are easily discussed in terms of the web diagram for the toric threefold, as the splitting of the web diagram into consistent sub-diagrams \cite{Franco:2005fd}. Simple examples include the deformation of the complex cone over dP$_2$ to a smooth geometry, or the deformation of the complex cone over dP$_3$ to a conifold, or to a smooth geometry.
There are however singularities (or 3-form flux assignments), for which the complex deformations are simply not available. One may then wonder about how our Dynamical Cobordism lesson applies. The answer was provided in particular examples in \cite{Berenstein:2005xa,Franco:2005zu,Bertolini:2005di}: the infrared end of the throat contains an explicit system of fractional D-branes, which in the language of the cobordism conjecture kill the corresponding cobordism classes, and allow the spacetime to end. As noticed in these references, the system breaks supersymmetry, and in \cite{Franco:2005zu} it was moreover noticed (as later revisited in \cite{Intriligator:2005aw}) to be unstable and lead to a runaway behaviour for the field blowing up the singularity. Hence, this corresponds to an additional dynamical tadpole, requiring additional spacetime dependence, to be solved. Simple examples include the complex cone over dP$_1$, and the generic $Y^{p,q}$ theories. We will not enter the discussion of possible mechanisms to stabilize these models, since following \cite{Buratti:2018onj} they are likely to require asymptotic modifications of the warped throat ansatz (i.e. at all positions in the radial direction, including the initial one).
\section{Type IIB fluxes and spontaneous compactification}
\label{sec:3form-flux}
In this Section we construct an explicit 5d type IIB model with a tunable dynamical tadpole, and describe the spacetime-dependent solution solving its equations of motion, which is in fact supersymmetry preserving. The configuration displays dynamical cobordism resulting in spontaneous compactification to 4d. The resulting model is a simple toroidal compactification with ISD NSNS and RR 3-form fluxes \cite{Dasgupta:1999ss,Giddings:2001yu}, in particular it appeared in \cite{Marchesano:2004xz,Marchesano:2004yq}. With this perspective in hindsight, one can regard this section as a reinterpretation of the latter flux compactification. Our emphasis is however in showing the interplay of the dynamical tadpoles in the 5d theory and the consequences in the spacetime configuration solving them.
\subsection{The 5d tadpole and its solution}
\label{sec:3form-flux-tadpole}
Consider type IIB on ${\bf {T}}^5$, which for simplicity we consider split as ${\bf {T}}^2\times{\bf {T}}^2\times {\bf {S}}^1$. We label the coordinates of the ${\bf {T}}^2$'s as $(x^1,y^1)$ and $(x^2,y^2)$, with periodicity 1, and introduce complex coordinates as $z^1=x^1+\tau_1 y^1$, $z^2=x^2+ \tau_2 y^2$. We also use a periodic coordinate $x^3\simeq x^3+1$ to parametrize the ${\bf {S}}^1$. For simplicity, we do not consider moduli deviating from this rectangular structure\footnote{As usual, they can be removed in orbifold models, although we will not focus on this possibility.}, and also take the ${\bf {T}}^5$ to have an overall radius $R$,
\begin{eqnarray}
ds^2\,=\, R^2\, [\, (dz^1)^2\,+\, (dz^2)^2\, +\, (dx^3)^2\,]\, .
\end{eqnarray}
The result so far is a standard 5d supersymmetric ${\bf {T}}^5$ compactification.
We introduce a non-trivial dynamical tadpole source by turning on an RR 3-form flux (using conventions in \cite{Giddings:2001yu})
\begin{eqnarray}
F_3\,=\, (2\pi)^2\alpha'\, N\, dx^1\,dx^2\,dx^3\, .
\end{eqnarray}
The introduction of this flux does not lead to RR topological tadpoles, but induces dynamical tadpoles for diverse fields. In the following we focus on the dynamics of the 5d light fields $R$, $\tau_1$, $\tau_2$, the dilaton $\phi$ and the NSNS axion $\Phi$ defined by
\begin{eqnarray}
B_2\,=\, \Phi \,dy^1\,dy^2\,.
\end{eqnarray}
The discussion of the dynamical tadpole is similar to the $T^{1,1}$ example in Section \ref{sec:conifold}, so we sketch the result. There is a dilaton tadpole, arising from the dimensional reduction of the 10d kinetic term for the 3-form flux,
\begin{eqnarray}
\nabla^2\, \phi\, =\, \frac{1}{12}\, e^\phi\, (F_3)^2\ .
\end{eqnarray}
Since $(F_3)^2$ is a constant source density, which does not integrate to zero over ${\bf {T}}^5$, there is no solution for this Laplace equation if we assume the solution to be independent of the 5d spacetime coordinates. One possibility would be to allow for 5d spacetime dependence of $\phi$ (at least in one extra coordinate, as in \cite{Dudas:2000ff}). Here we consider a different possibility, which is to let the NSNS axion $\Phi$ acquire a dependence on one of the 5d coordinates, which we denote by $y$, as follows
\begin{eqnarray}
\Phi \,=\, -\,(2\pi)^2\alpha'\, \frac{N}{t_3}\, y\quad \Rightarrow\, \quad H_3\, = -\,(2\pi)^2\alpha'\, \frac{N}{t_3}\, dy^1\,dy^2\,dy\,.
\label{toroidal-axion-solution}
\end{eqnarray}
We have thus turned on NSNS 3-form field strength in the directions $y^1$, $y^2$ in ${\bf {T}}^5$ and the 5d spacetime coordinate $y$. Here the sign has been introduced for later convenience, and $t_3$ is a positive real parameter allowing to tune the field strength density, whose meaning will become clear later on.
Including this new source, the dilaton equation of motion becomes
\begin{eqnarray}
\nabla^2\, \phi\, =\, \frac{1}{12}\, \left[\,e^\phi\, (F_3)^2\,-\,e^{-\phi}\, (H_3)^2\,\right]\, .
\end{eqnarray}
Hence, the spacetime-dependent profile (\ref{toroidal-axion-solution}) can cancel the right hand side and solve the dilaton tadpole when
\begin{eqnarray}
e^{2\phi}\, (F_3)^2\,=\,(H_3)^2\, .
\label{isd-square}
\end{eqnarray}
We can thus keep the dilaton constant $e^{\phi}=g_s$. Taking for simplicity purely imaginary $\tau_1=it_1$ and $\tau_2=it_2$, the condition (\ref{isd-square}) is simply
\begin{eqnarray}
g_s\, t_1\, t_2\, t_3\, =\, 1\ .
\label{flux-moduli-condition}
\end{eqnarray}
In addition to the dilaton, the 3-form fluxes backreact on the metric and other fields, which we discuss next.
\subsection{The singularities}
\label{sec:3form-flux-singu}
We now discuss the backreaction on the metric and other fields. For convenience, we use the complex coordinates $z^1$, $z^2$ and $z^3=x^3+iy$. In terms of these, we can write the combination
\begin{eqnarray}
G_3\,=\, F_3 -\tau H_3\,=\, \frac{(2\pi)^2}{4}\alpha'\, N\, (\,
d{\overline {z}_1}\,dz_2\,dz_3+ dz_1d{\overline{z}_2}dz_3 + dz_1dz_2d{\overline{z}_3} + d{\overline{z}_1}d{\overline{z}_2}d{\overline{z}_3}\,) \quad .
\label{complex-flux}
\end{eqnarray}
Regarding ${\bf {T}}^5\times {\bf {R}}^1_y$ as a (non-compact) CY, this is a combination of $(2,1)$ and $(0,3)$ components, which is thus ISD. There is a backreaction on the metric and RR 4-form of the familiar black 3-brane kind. In particular, the metric includes a warp factor $Z$
\begin{eqnarray}
ds_{10}^{\, 2}\,=\, Z^{-\frac 12} \eta_{\mu\nu} dx^\mu dx^\nu\, +\,Z^{\frac 12}\, R^2\, [ \, dz^1d{\overline z}^1\, +\,dz^2d{\overline z}^2\,+\,dz^3d{\overline z}^3
\,]\ ,
\end{eqnarray}
where $x^\mu$ runs through the four Poincar\'e invariant spacetime coordinates. The warp factor is determined by the Laplace equation
\begin{eqnarray}
-{\tilde \nabla}^2 Z = \frac{g_s}{12}G_3\cdot {\overline G}_3\, =\, \frac{g_s}6\, (F_3)^2\,,
\label{laplace-toroidal}
\end{eqnarray}
with the tilde indicating the Laplacian is computed with respect to the unwarped, flat metric, and in the last equation we used (\ref{isd-square}).
Note that, since $y$ parametrizes a non-compact dimension, there is no tadpole problem in solving (\ref{laplace-toroidal}) i.e. we need not add background charge. One may then be tempted to conclude that this provides a 5d spacetime-dependent configuration solving the 5d tadpole. However, the solution is valid locally in $y$, but cannot be extended to arbitrary distances in this direction. Since the local flux density in ${\bf {T}}^5$ is constant, we can take $Z$ to depend only on\footnote{In fact, this is the leading behaviour at long distances, compared with the ${\bf {T}}^5$ size scale $R$.} $y$, hence leading to a solution
\begin{eqnarray}
-\frac{d^2 Z}{dy^{\,2}}\,=\, \frac{g_s}6\, (F_3)^2\quad \Rightarrow \quad
Z\,=\, 1 \, -\, \frac{g_s}{12} (F_3)^2\, y^2\, ,
\end{eqnarray}
where we have set an integration constant to 1. The solution hits metric singularities at
\begin{eqnarray}
y^{-2}\,=\, \frac{1}{12} g_s(F_3)^2\, ,
\end{eqnarray}
showing there is a maximal extent in the direction $y$. Let us introduce the quantity ${\cal T}=\frac{1}{12} g_s(F_3)^2$, which controls the parametric dependence of the tadpole. Then, the distance between the singularities is
\begin{eqnarray}
\Delta\,=\, \int_{-{\cal T}^{-1/2}}^{{\cal T}^{-1/2}} \, Z^{\frac14}\, dy\, =\, \frac 2{\sqrt{{\cal T}}}\, \int_0^1 (1-t^2)^{\frac 14}\, dt\ ,
\end{eqnarray}
with $t=\sqrt{{\cal T}}y$. We thus recover the scaling (\ref{bound-lesson1}) with $n=2$,
\begin{eqnarray}
\Delta^{-2}\sim {\cal T}\ .
\label{scaling}
\end{eqnarray}
Hence the appearance of the singularities as a consequence of the dynamical tadpole is as explained in the introduction.
\subsection{Cobordism and spontaneous compactification}
\label{sec:3form-flux-cobordism}
The appearance of singularities is a familiar phenomenon. In this section we argue that they must be smoothed out, somewhat analogously to the KS solution in section \ref{sec:conifold}. The fact that it is possible follows from the swampland cobordism conjecture \cite{McNamara:2019rup,Montero:2020icj}, namely there must exist an appropriate cobordism defect closing off the extra dimension into nothing.
Since there are two singularities, the formerly non-compact dimension becomes compact, in an explicit realization of spontaneous compactification\footnote{Spontaneous compactification has been discussed in the context of dynamical tadpoles in \cite{Dudas:2000ff}.}.
In the following, we directly describe the resulting geometry, which turns out to be a familiar ${\bf {T}}^6$ (orientifold) compactification with ISD 3-form fluxes. Consider type IIB theory on ${\bf {T}}^2\times {\bf {T}}^2\times {\bf {T}}^2$, with
\begin{eqnarray}
F_3\,=\, (2\pi)^2\alpha'\, N\, dx^1\,dx^2\,dx^3\ , \quad H_3\,=\, (2\pi)^2\alpha'\, N\, dy^1\,dy^2\,dy^3\ .
\end{eqnarray}
We use $z^i=x^i+it_i y^i$, hence the above defined $t_3$ is the complex structure modulus for the ${\bf {T}}^2$ involving the newly compact dimension. For moduli satisfying (\ref{flux-moduli-condition}) the ${\bf {T}}^6$ flux combination $G_3$ is given by (\ref{complex-flux}), which is ISD and indeed compatible with 4d Poincar\'e invariance as usual. Notice that in this case, it is possible to achieve a large size for the new compact dimension $t_3\gg 1$ by simply e.g. taking small $g_s$. This corresponds to the regime of small 5d tadpole, with the relation
\begin{eqnarray}
t_3^{-2}\sim g_s^{\,2}\sim \cal T\ ,
\end{eqnarray}
in agreement with the maximal distance relation in the previous section.
Consistency, in the form of $C_4$ RR tadpole cancellation, requires the introduction of O3-planes at fixed points of the involution ${\cal R}:z^i\to -z^i$ (together with mobile D3-branes). From the perspective of the 5d theory, the additional dimension is compactified on an interval, with two end of the world defects given by the O3-planes, which constitute the cobordism defects of the configuration (possibly decorated with explicit D3-branes if needed).
\section{Solving dynamical tadpoles via magnetization}
\label{sec:magnetization}
In this Section we consider a further setup displaying dynamical tadpoles, based on compactifications with magnetized D-branes \cite{Bachas:1995ik,Angelantonj:2000hi,Blumenhagen:2000wh,Aldazabal:2000dg,Aldazabal:2000cn}. In toroidal setups, these have been (either directly or via their T-dual intersecting brane world picture) widely used to realize semi-realistic particle physics models in string theory. In more general setups, magnetized 7-branes are a key ingredient in the F-theory realization of particle physics models \cite{Donagi:2008ca,Beasley:2008dc,Beasley:2008kw}.
\subsection{Solving dynamical tadpoles of magnetized branes}
\label{sec:magnetization-local}
We consider a simple illustrative example. Consider type IIB theory compactified on ${\bf {T}}^2 \times {\bf {T}}^2$ (labelled 1 and 2, respectively) and mod out by $\Omega {\cal R}_1(-1)^{F_L}$, where ${\cal R}_1:z_1\to -z_1$. This introduces 4 O7$_1$-planes spanning $({\bf {T}}^2)_2$ and localized at the fixed points on $({\bf {T}}^2)_1$. We also have 32 D7-branes (as counted in the covering space), split as 16 D7-branes (taken at generic points) and their 16 orientifold images. This model is related by T-duality on $({\bf {T}}^2)_1$ to a type I toroidal compactification, but we proceed with the D7-brane picture.
We introduce $M$ units of worldvolume magnetic flux along $({\bf {T}}^2)_2$ for the $U(1)$ of a D7-brane\footnote{If $N$ the D7-branes are coincident, it is also possible to use the overall $U(1)\subset U(N)$. We will stick to the single D7-brane for the moment, but such generalization will arise in later examples.\label{foot-u1}}
\begin{eqnarray}
\frac{1}{2\pi \alpha'}\,\int_{{\bf {T}}^2} F_2\,=\, M\, .
\label{magnetic-flux-quant}
\end{eqnarray}
The orientifold requires we introduce $-M$ units of flux on the image D7-brane\footnote{For simplicity we consider vanishing discrete NSNS 2-form flux \cite{Blumenhagen:2000ea}, although such generalization will arise in later examples.}. This also ensures that there is no net induced ${\bf {Z}}$-valued D5-brane charge in the model, and hence no associated RR tadpole, in agreement with the fact that the RR 6-form is projected out. In addition, there is a ${\bf {Z}}_2$ K-theory charge \cite{Uranga:2000xp} which is cancelled as long as $M\in 2{\bf {Z}}$.
The introduction of the worldvolume flux leads to breaking of supersymmetry. As is familiar in the discussion of supersymmetries preserved by different branes \cite{Berkooz:1996km}, we introduce the angle
\begin{eqnarray}
\theta_2\,=\, \arctan ( 2\pi \alpha'\, F)\,=\, \arctan \left( M\chi \right)\ ,
\label{angle-6d}
\end{eqnarray}
where $F$ is the field strength and $\chi$ is the inverse of the ${\bf {T}}^2$ area, in string units.
This non-supersymmetric configuration introduces dynamical tadpoles. For small $\theta_2$, the extra tension can be described in effective field theory as an FI term controlled by $\theta$ \cite{Kachru:1999vj,Cvetic:2001nr,Cvetic:2001tj}. In fact, in \cite{Cremades:2002te} a similar parametrization was proposed for arbitrary angles. By using the DBI action, the extra tension has the structure
\begin{eqnarray}
V\, \sim\, \frac {1}{g_s}\, \left(\, \sqrt{1+(\tan\theta_2)^2} -1\,\right)
\,.
\end{eqnarray}
This leads to a tadpole for the dilaton and the $({\bf {T}}^2)_2$ K\"ahler modulus.
We now consider solving the tadpole by allowing for some spacetime-dependent background. Concretely, we allow for a non-trivial magnetic field $-F$ on two of the non-compact space coordinates, parametrized by the (for the moment, non-compact) coordinate $z_3$. In fact this leads to a configuration preserving supersymmetry since, defining the angle $\theta_3$ in analogy with (\ref{angle-6d}), we satisfy the $SU(2)$ rotation relation $\theta_3+\theta_2=0$ \cite{Berkooz:1996km}. In other words, the field strength flux has the structure
\begin{eqnarray}
F_2\,=\, F (\,dz_2d{\overline z}_2\, - \, dz_3d{\overline z}_3\,)\ ,
\end{eqnarray}
which is $(1,1)$ and primitive (i.e. $J\wedge F_2=0$), which are the supersymmetry conditions for a D-brane worldvolume flux.
Hence, it is straightforward to find spacetime-dependent solutions to the tadpole of the higher-dimensional theory, at the price of breaking part of the symmetry of the lower-dimensional spacetime. In the following we show that, as in earlier examples, this eventually also leads to spontaneous compactification.
\subsection{Backreaction and spontaneous compactification}
\label{sec:magnetization-global}
The spacetime field strength we have just introduced couples to gravity and other fields, so we need to discuss its backreaction.
In fact, this is a particular instance of earlier discussions, by considering the F-theory lift of the D7-brane construction. This can be done very explicitly by taking the configuration near the $SO(8)^4$ weak coupling regime \cite{Sen:1996vd}. The configuration without magnetic flux $M=0$ simply lifts to F-theory on K3$\times {\bf {T}}^2\times {\bf {R}}^2$, where the $({\bf {T}}^2)_1$ (modulo the ${\bf {Z}}_2$ orientifold action) is the ${\bf {P}}_1$ base of K3, and the ${\bf {T}}^2$ and ${\bf {R}}^2$ explicit factors correspond to the directions $z_2$ and $z_3$, respectively. As is familiar, the 24 degenerate fibers of the K3 elliptic fibration form 4 pairs, reproducing the 4 orientifold planes, and 16 D7-branes in the orientifold quotient. Actually, the discussion below may be carried out for F-theory on K3 at generic points in moduli space, even not close to the weak coupling point.
The introduction of magnetization for one 7-brane corresponds to the introduction of a $G_4$ flux along the local harmonic $(1,1)$-form supported at an $I_1$ degeneration (or enhanced versions thereof, for coincident objects), of the form
\begin{eqnarray}
G_4\,=\, \omega_2\wedge F (\,dz_2d{\overline z}_2\, - \, dz_3d{\overline z}_3\,)\ .
\end{eqnarray}
This flux is self-dual, and in fact $(2,2)$ and primitive, which is the supersymmetry preserving condition for 4-form fluxes in M/F-theory \cite{Dasgupta:1999ss,Becker:2001pm}. The backreacted metric is described by a warp factor satisfying a Laplace equation sourced by the fluxes, similar to (\ref{laplace-toroidal}). Considering the regime in which the warp factor is taken independent of the internal space and depends only on the coordinates in the ${\bf {R}}^2$ parametrized by $z_3$, the constant flux density leads to singularities at a maximal length scale $\Delta$
\begin{eqnarray}
\Delta^{-2}\,\sim\, F^2\, .
\end{eqnarray}
This is another instance of the universal relation (\ref{bound-lesson1}) with ${\cal T}\sim F^2$, hence $n=2$.
This is in complete analogy with earlier examples. Hence, we are led to propose that the smoothing out of these singularities is provided by the compactification of the corresponding coordinates, e.g. on a ${\bf {T}}^2$, with the addition of the necessary cobordism defects, namely orientifold planes and D-branes\footnote{To be precise, the cobordism defects of an ${\bf {S}}^1$ compactification of the model. This is analogous to the mechanism by which F-theory on half a ${\bf {P}}_1$ provides the cobordism defect for type IIB on ${\bf {S}}^1$ with $SL(2,{\bf {Z}})$ monodromy \cite{McNamara:2019rup} (see also \cite{Dierigl:2020lai}). In fact, since magnetized branes often lead to chiral theories in the bulk, this extra circle compactification allows them to become non-chiral and admit an end of the world describable at weak coupling, see the discussion below (\ref{scaling-distance-dm}) in section \ref{sec:sugimoto}. We will nevertheless abuse language and refer as cobordism defect to the structures involved in the final spontaneous compactification under discussion.}.
To provide an explicit solution, we introduce the standard notation (see e.g. \cite{Blumenhagen:2000wh,Aldazabal:2000dg}) of $(n,m)$ for the wrapping numbers and the magnetic flux quanta on the $({\bf {T}}^2)_i$'s for the directions $i=1,2,3$. In this notation, the O7$_1$-planes and unmagnetized D7$_1$-branes are associated to $(0,1)\times (1,0)\times (1,0)$, while the magnetized D7$_1$-branes\footnote{If the magnetization is in the $U(1)\subset U(N)$ of a stack of $N$ coincident branes, see footnote \ref{foot-u1}, the corresponding wrapping goes as $(N,M)$.} correspond to $(0,1)\times (1,M)\times (1,-M)$, and $(0,1)\times (1,-M)\times (1,M)$ for the orientifold images. In other words, we require a flux quantization condition on $({\bf {T}}^2)_3$ as in (\ref{magnetic-flux-quant}), up to a sign flip.
Since now the last complex dimension is compact, there is an extra RR tadpole cancellation condition, which requires the introduction of 16 O7$_3$-planes, wrapped on $({\bf {T}}^2)_1\times ({\bf {T}}^2)_2$ and localized at fixed points in $({\bf {T}}^2)_1$, namely with wrapping numbers $(1,0)\times (1,0)\times (0,1)$. This introduces an extra orbifold action generated by $(z_1,z_2,z_3)\to (z_1,-z_2,-z_3)$, so the model can be regarded as a (T-dual of a) magnetized version of the D9/D5-brane ${\bf {T}}^4/{\bf {Z}}_2$ orientifolds in \cite{Pradisi:1988xd,Gimon:1996rq}. Allowing for $n$ additional mobile D7$_3$-branes (as counted in the covering space, and arranged in orbifold and orientifold invariant sets), the RR tadpole cancellation conditions is
\begin{eqnarray}
2M^2+n = 32\ .
\end{eqnarray}
The supersymmetry condition is simply that the ${\bf {T}}^2$ parameters satisfy $\chi_2=\chi_3$.
From the perspective of the original 6d configuration, the tadpole in the initial ${\bf {T}}^2\times {\bf {T}}^2$ configuration has triggered a spontaneous compactification. Since the additional O-planes and D-branes required to cancel the new RR tadpoles are localized in $z_3$, they can be interpreted as the addition of I-branes to cancel the cobordism charge of the original model.
\medskip
It should be possible to generalize the above kind of construction to global K3-fibered CY threefolds with O7-planes. The local fibration in a small neighbourhood of a generic point of the the base provides a local 6d model essentially identical to our previous one. On the other hand, the global geometry defining how the two extra dimensions compactify would correspond to another possible spontaneous compactification, with the ingredients required for the cancellation of the new RR tadpoles.
However, a general drawback of this class of models is that the scales of the compact spaces in the directions 2 and 3 are of the same order\footnote{In the toroidal example, if the magnetization along $({\bf {T}}^2)_2$ is on the overall $U(1)\subset U(16)$ of 16 coincident D7-branes, the magnetic field along $z_2$ is $F\sim M/16$; this weakened tadpole implies an increase of the critical size of the spontaneously compactified dimensions by a factor of 4.}. Thus, there is no separation of scales, and no reliable regime in which the dynamics becomes that of a 6d model. This is easily avoided in more involved models, as we will see in the examples in coming sections.
\section{Solving tadpoles in 10d strings}
\label{sec:10d}
In this section we consider dynamical tadpoles arising in several 10d string theories, and confirm the general picture. We illustrate this with various examples, with superymmetry (massive type IIA and M-theory on K3), and without it (non-supersymmetric 10d $USp(32)$ theory).
\subsection{Massive IIA theory}
\label{sec:massive-iia}
We consider 10d massive type IIA theory \cite{Romans:1985tz}. This can be regarded as the usual type IIA string theory in the presence of an additional RR 0-form field strength $F_0\equiv m$. The string frame effective action for the relevant fields is
\begin{eqnarray}
S_{10}\,=\, \frac{1}{2\kappa_{10}^{\,2}}\int d^{10}x \,\sqrt{-G}\, \{\, e^{-2\phi}\, [\, R+4(\partial \phi)^2\,]-\frac 12 (F_0)^2 -\frac 12 (F_4)^2\,\}\, +\, S_{\rm top}\, ,
\end{eqnarray}
where $S_{\rm top}$ includes the Chern-Simons terms. In the Einstein frame $G_E=e^{-\frac{\phi}{2}} G$, we have
\begin{eqnarray}
S_{10,E}\,=\, \frac{1}{2\kappa^{\,2}}\int d^{10}x \,\sqrt{-G_E}\, \{\, [\, R-\frac 12(\partial \phi)^2\,]-\frac 12 e^{\frac 52\phi} m^2 -\frac 12 e^{\frac 12 \phi}(F_4)^2\,\}\, .
\end{eqnarray}
Here we have used $m$ to emphasize this quantity is constant. This theory is supersymmetric, but at a given value of $\phi$, it has a tadpole controlled by
\begin{eqnarray}
{\cal T}\,\sim \, e^{\frac 52\phi} m^2\ .
\label{tadpole-massiveiia}
\end{eqnarray}
This is in particular why the massive IIA theory does not admit 10d maximally symmetric solutions. In the following we discuss two different ways of solving it, leading to Minkowski or AdS-like configurations.
\subsubsection{Solution in 9d and type I' as cobordism}
\label{sec:type-iprime}
To solve the tadpole (\ref{tadpole-massiveiia}) we can consider a well-known 1/2 BPS solution with the dilaton depending on one coordinate $x^9$. Since the flux $m$ can be regarded as generated by a set of $m$ distant D8-branes, this is closely related to the solution in \cite{Bergshoeff:1995vh}. We describe it in conventions closer to \cite{Polchinski:1995df}, for later use. In the Einstein frame, the metric and dilaton background have the structure
\begin{eqnarray}
(G_{E})_{MN}\, =\, Z(x^9)^{\frac 1{12}}\,\eta_{MN}\ ,\quad
e^{\phi}\,=\, Z(x^9)^{-\frac 56}\ , \quad{\rm with}\; \;Z(x^9)\sim B-mx^9\ ,
\end{eqnarray}
where $B$ is some constant (in the picture of flux generated by distant D8-branes, it relates to the D8-brane tensions).
The solution hits a singularity at $x^9=B/m$. Starting at a general position $x^9$, the distance to the singularity is
\begin{eqnarray}
\Delta\,=\, \int_{x^9}^{\frac Bm} Z(x^9)^{\frac{1}{24} }\,dx^9\,\sim Z(x^9)^{\frac{25}{24}}\,m^{-1}\, \sim \, m^{-1} e^{-\frac 54}\ ,
\end{eqnarray}
where in the last equality we have traded the position for the value the dilaton takes there.
Recalling (\ref{tadpole-massiveiia}), this reproduces the Finite Distance scaling relation (\ref{bound-lesson1}) with $n=2$,
\begin{eqnarray}
\Delta^{-2}\sim {\cal T}\ .
\end{eqnarray}
It is easy to propose the stringy mechanism capping off spacetime before or upon reaching this singularity, according to the Dynamical Cobordism lesson. This should be the cobordism defect of type IIA theory, which following \cite{McNamara:2019rup} is an O8-plane, possibly with D8-branes.
In fact, this picture is implicitly already present in \cite{Polchinski:1995df}, which studies type I' theory, namely type IIA on an interval, namely IIA on ${\bf {S}}^1$ modded out by $\Omega{\cal R}$ with ${\cal R}:x^9\to -x^9$, which introduces two O8$^-$-planes which constitute the interval boundaries. There are 32 D8-branes (in the covering space), distributed on the interval, which act as domain walls for the flux $F_0=m$, which is piecewise constant in the interval. The metric and the dilaton profile are controlled by a piecewise linear function $Z(x^9)$. The location of the boundaries at points of strong coupling was crucial to prevent contradiction with the appearance of certain enhanced symmetries in the dual heterotic string (the role of strong coupling at the boundaries for the enhancements was also emphasized from a different perspective in \cite{Seiberg:1996bd,Kachru:1996nd}). In our setup, we interpret the presence of (at least, one) O8-plane as the cobordism defect triggered by the presence of a dynamical tadpole in the bulk theory.
\subsubsection{A non-supersymmetric Freund-Rubin solution}
\label{sec:freund-rubin}
We now consider for illustration a different mechanism to cancel the dynamical tadpole, which in fact underlies the spontaneous compactification to (non-supersymmetric\footnote{Thus, it should be unstable according to \cite{Ooguri:2016pdq}. However, being at a maximum of a potential is sufficient to avoid dynamical tadpoles, so the solution suffices for our present purposes.}) AdS$_4\times {\bf {S}}^6$ in \cite{Romans:1985tz}). The idea is that, rather than solving for the dilaton directly, one can introduce an additional flux $F_4$ along three space dimensions and time (or its dual $F_6$ on six space dimensions) to balance off the dilaton sourced by $F_0$. This can be used to fix $\phi$ to a constant, and following \cite{Romans:1985tz} leads to a scaling
\begin{eqnarray}
F_4\sim \,m^2\, d(vol)_4\, ,
\end{eqnarray}
where $d(vol)_4$ is the volume form in the corresponding 4d. Using arguments familiar by now, the constant $F_4$ backreaction on the metric is encoded in a solution of the 4d Laplace equation with a constant source, leading to a solution quadratic in the coordinates (to avoid subtleties, we take solutions depending only on the space coordinates). This develops a singularity at a distance scaling as
\begin{eqnarray}
\Delta^2\sim |F_4|^{-2}\sim m^{-4}\sim {\cal T}\ ,
\end{eqnarray}
where in comparison with (\ref{tadpole-massiveiia}) we have taken constant dilaton.
The singularities are avoided by an AdS$_4\times {\bf {S}}^6$ compactification, whose curvature radius is $R\sim m^2$, in agreement with the above scaling. From our perspective, the compactification should be regarded as a dynamical cobordism (where the cobordism is actually that of the 10d theory on an ${\bf {S}}^5$ (i.e. equator of ${\bf {S}}^5$)).
\subsection{An aside on M-theory on K3}
\label{sec:hw}
In this section we relate the above system to certain compactifications of M-theory and to the Horava-Witten end of the world branes as its cobordism defect. Although the results can be obtained by direct use of M-theory effective actions, we illustrate how they can be recovered by applying simple dualities to the above system.
Consider the above massive IIA theory with mass parameter $m$, and compactify on ${\bf {T}}^4/{\bf {Z}}_2$. This introduces O4-planes, and requires including 32 D4-branes in the configuration, either as localized sources, or dissolved as instantons on the D8-branes. Actually this can be considered as a simple model of K3 compactifications, where in the general K3 the O4-plane charge is replaced by the contribution to the RR $C_5$ tadpole arising from the CS couplings of D8-branes and O8-planes to ${\rm tr \,} R^2$.
We now perform a T-duality in all the directions of the ${\bf {T}}^4/{\bf {Z}}_2$ (Fourier-Mukai transform in the case of general K3). We obtain a similar model of type I' on ${\bf {T}}^4/{\bf {Z}}_2$, but now with the tadpole being associated to the presence of $m$ units of non-trivial flux of the RR 4-form field-strength over ${\bf {T}}^4$ (namely, K3). Also, the dilaton of the original picture becomes related to the overall K\"ahler modulus of K3. Finally, we lift the configuration to M-theory by growing an extra ${\bf {S}}^1$ and decompactifying it. We thus end up with a 7d compactification of M-theory on K3, with $m$ units of $G_4$ flux,
\begin{eqnarray}
\int_{\rm K3} G_4\,=\,m\ .
\end{eqnarray}
This leads to a dynamical tadpole, cancelled by the variation of the overall K\"ahler modulus (i.e. the K3 volume) along one the 7d space dimensions, which we denote by $x^{11}$. As in previous sections, this will trigger a singularity at a finite distance in $x^{11}$, related to the tadpole by $\Delta^{-2}\sim {\cal T}$. The singularity is avoided by the physical appearance of a cobordism defect, which for M-theory is a Horava-Witten (HW) boundary \cite{Horava:1995qa}. This indeed can support the degrees of freedom to kill the $G_4$ flux, as follows. From \cite{Horava:1996ma}, the 11d $G_4$ is sourced by the boundary as
\begin{eqnarray}
dG_4\,=\, \delta(x^{11})\, (\, {\rm tr \,} F^2\,-\,\frac 12 {\rm tr \,} R^2\,)\ ,
\end{eqnarray}
where $\delta(x^{11})$ is a bump 1-form for the HW brane, and $F$ is the field-strength for the $E_8$ gauge fields in the boundary. Hence, the $m$ units of $G_4$ in the K3 compactification can be absorbed by a HW boundary with an $E_8$ bundle with instanton number $12+m$ (the 12 coming from half the Euler characteristic of K3 $\int_{\rm K3} {\rm tr \,} R^2=24$).
The above discussion is closely related to the picture in \cite{Witten:1996mz}, which discusses compactification of HW theory (namely, M-theory on an interval with two HW boundaries) on K3 and on a CY threefold. It includes a K\"ahler modulus varying over the interval according to a linear function\footnote{In the presence of explicit M5-branes, it is a piecewise linear function. It is straightforward to include them in our cobordism description if wished, with explicit branes considered as part of the cobordism defect.} and the appearance of a singularity at finite distance. In that case, the HW brane was located at the strong coupling point, based on heuristic arguments, and this led, in the CY$_3$ case, to a lower bound on the value of the 4d Newton's constant.
Our perspective remarkably explains that the location of the HW wall is not an arbitrary choice, but follows our physical principle of Dynamical Cobordism, and the bound on the Newton's constant is a consequence of that of Finite Distance!
\subsection{Solving tadpoles in the non-supersymmetric 10d \texorpdfstring{$USp(32)$}{USP(32)} theory}
\label{sec:sugimoto}
The previous examples were based on an underlying supersymmetric vacuum, on top of which the dynamical tadpole is generated via the introduction of fluxes or other ingredients. In this section we consider the opposite situation, in which the initial theory is strongly non-supersymmetric and displays a dynamical tadpole from the start. In particular we consider the non-supersymmetric 10d $USp(32)$ theory constructed in \cite{Sugimoto:1999tx}, in two different ways: first, we use our new insights to revisit the spacetime-dependent solution proposed in \cite{Dudas:2000ff} (see also \cite{Blumenhagen:2000dc} for other proposals); then we present a far more tractable solution involving magnetization, which in fact provides a supersymmetric compactification of this non-supersymmetric 10d string theory.
\subsubsection{The Dudas-Mourad solution and cobordism}
\label{sec:sugimoto-dm}
The non-supersymmetric 10d $USp(32)$ theory in \cite{Sugimoto:1999tx} is obtained as an $\Omega$ orientifold of type IIB theory. The closed string sector is as in type I theory, except that the O9$^-$-plane is replace by an O9$^+$-plane. Cancellation of RR tadpoles requires the introduction of open strings, which must be associated to 32 ${\overline{\rm D9}}$-branes. The closed string sector is a 10d ${\cal N}=1$ supergravity multiplet; the orientifold action on the D9-branes breaks supersymmetry, resulting in an open string sector with $USp(32)$ gauge bosons and gauginos in the two-index antisymmetric representation. All anomalies cancel, a remarkable feat from the field theory viewpoint, which is just a consequence of RR tadpole cancellation from the string viewpoint.
Although the RR tadpoles cancel, the NSNS tadpoles do not, implying that there is no maximally symmetric 10d solution to the equations of motion. In particular there is a dynamical dilaton tadpole of order the string scale, as follows from the terms in the 10d (Einstein frame) action
\begin{eqnarray}
S_E\,=\, \frac{1}{2\kappa^2}\int d^{10}x \sqrt{-G}\,[\, R-\frac{1}{2}(\partial\phi)^2\,] \,-\, T_9^E\int d^{10}x \sqrt{-G}\, 64\, e^{\frac{3\phi}2}\, ,
\label{action-sugimoto}
\end{eqnarray}
where $T_9^E$ is the (anti)D9-brane tension. The tadpole scales as ${\cal T}\sim T_9^E g_s^{3/2}$, with the dilaton dependence arising from the fact that the supersymmetry breaking arises from the Moebius strip worldsheet topology, with $\chi=3/2$.
Ref. \cite{Dudas:2000ff} proposed solutions of this dynamical tadpole with 9d Poincar\'e invariance, and the dilaton varying over one spacetime dimension. In the following we revisit the solution with dependence on one spatial coordinate $y$, from the vantage point of our Lessons.
The 10d solution is, in the Einstein frame,
\begin{eqnarray}
\phi &=& \frac 34 \alpha_E y^2\,+\,\frac 23\log|\sqrt{\alpha_E}y|\,+\,\phi_0 \ ,\nonumber\\
ds_E^{\,2}&=& |\sqrt{\alpha_E}y|^{\frac 19}\, e^{-\frac{\alpha_E y^2}8} \eta_{\mu\nu} dx^\mu dx^\nu + |\sqrt{\alpha_E}y|^{-1} e^{-\frac{3\phi_0}2} e^{-\frac{9\alpha_E y^2}{8}}\, dy^2\, ,
\label{solution-dm}
\end{eqnarray}
where $\alpha_E= 64 k^2T_9$. There are two singularities, at $y=0$ and $y\to\infty$, which despite appearances are separated by a finite distance
\begin{eqnarray}
\Delta\,\sim\, \int_0^\infty |\sqrt{\alpha_E}y|^{-\frac 12} e^{-\frac{3\phi_0}4} e^{-\frac{9\alpha_E y^2}{16}}\, dy\,\sim \,e^{-\frac{3\phi_0}4} \alpha_E^{-\frac 12}\ .
\end{eqnarray}
The fact that the solution has finite extent in the spatial dimension on which the fields vary is in agreement with the Finite Distance Lesson, and in fact satisfying its quantitative bound (\ref{bound-lesson1})
\begin{eqnarray}
\Delta^{-2}\sim {\cal T}\ .
\label{scaling-distance-dm}
\end{eqnarray}
We can now consider how the Dynamical Cobordism Lesson applies in the present context. Following it, we expect the finite extent in the spatial dimensions to be physically implemented via the cobordism defect corresponding to the 10d $USp(32)$ theory. In general the cobordism defect of bulk chiral 10d theories are expected to be non-supersymmetric, and in fact rather exotic, as their worldvolume dynamics must gap a (non-anomalous) set of chiral degrees of freedom. In fact, on general grounds they can be expected to involve strong coupling\footnote{We are indebted to Miguel Montero for this argument, and for general discussions on this section.}. An end of the world defect imposes boundary conditions on bulk supergravity fields, which at weak coupling should be at most linear in the fields, to be compatible with the superposition principle. A typical example are boundary conditions that pair up bulk fermions of opposite chiralities. However, the anomaly cancellation in the 10d $USp(32)$ theory involves fields of different spins, which cannot be gapped by this simple mechanism, and should require strong coupling dynamics (a similar phenomenon in a different context occurs in \cite{Razamat:2020kyf}).
This strong coupling fits nicely with the singularity at $y\to \infty$, but the singularity at $y=0$ lies at weak coupling. The simplest way out of this is to propose that the singularity at $y=0$ is actually smoothed out by perturbative string theory (namely, $\alpha'$ corrections, just like orbifold singularities are not singular in string theory), and does not turn into an end of the world defect. Hence the solution (\ref{solution-dm}) extends to $y<0$, and, since the background is even in $y$, develops a singularity at $y\to -\infty$. This is still at finite distance $\Delta$ scaling as (\ref{scaling-distance-dm}), and lies at strong coupling, thus allowing for the possibility that the singularity is turned into the cobordism defect of the 10d $USp(32)$ theory.
It would be interesting to explore this improved understanding of this solution to the dilaton tadpole. Leaving this for future work, we turn to a more tractable solution in the next section.
\subsubsection{Solving the tadpole via magnetization}
\label{sec:sugimoto-magnetization}
We now discuss a more tractable alternative to solve the dynamical tadpole via magnetization, following section \ref{sec:magnetization}.
Stabilizing the tadpole via magnetization is, ultimately, equivalent to finding a compactification (on a product of ${\bf {T}}^2$'s) which is free of tadpoles, for instance by demanding it to be supersymmetric. Hence we need to construct a supersymmetric compactification of the non-supersymmetric 10d $USp(32)$ theory \cite{Sugimoto:1999tx}.
As explained above, the 10d model is constructed with an O9$^+$-plane and 32 ${\overline{\rm D9}}$-branes. Hence, we need to introduce worldvolume magnetic fields in different 2-planes, in such a way that the corresponding angles add up to 0 mod $2\pi$. It is easy to convince oneself that this requires magnetization in at least three complex planes, ultimately triggering a ${\bf {T}}^2\times {\bf {T}}^2\times {\bf {T}}^2$ compactification. In order to preserve supersymmetry, we need the magnetization to induce D5-brane charges, rather than ${\overline{\rm D5}}$-brane charge, hence we need the presence of three independent kinds of negatively charged O5$^-_i$-planes, where $i=1,2,3$ denotes the ${\bf {T}}^2$ wrapped by the corresponding O5-plane. We are thus considering an orientifold of ${\bf {T}}^6/({\bf {Z}}_2\times{\bf {Z}}_2)$ with an O9$^+$-plane, and 8 O5$^-_i$-planes\footnote{For such combinations of orientifold plane signs, see the analysis in \cite{Klein:2000qw}, in particular its table 6. We will not need its detailed construction for our purposes.}.
The wrapping numbers for the O-planes, and for one simple solution of all constraints for the D9-branes (and their explicitly included orientifold image D9-branes), are
\begin{center}
\begin{tabular}{|c|c||c|c|c|}
\hline
Object & $N_\alpha$ & $(n_\alpha^{1},m_\alpha^{1})$ & $(n_\alpha^{2},m_\alpha^{2})$ & $(n_\alpha^{3},m_\alpha^{3})$\\
\hline\hline
O9$^+$ & $32$ & $(1,0)$ & $(1,0)$ & $(1,0)$\\
\hline
O5$^-_1$ & $-32$ & $(1,0)$ & $(0,1)$ & $(0,-1)$\\
\hline
O5$^-_2$ & $-32$ & $(0,1)$ & $(1,0)$ & $(0,-1)$\\
\hline
O5$^-_3$ & $-32$ & $(0,1)$ & $(0,-1)$ & $(1,0)$\\
\hline
D9 & $16$ & $(-1,1)$ & $(-1,1)$ & $(-1,1)$\\
\hline
D9$'$ & $16$ & $(-1,-1)$ & $(-1,-1)$ & $(-1,-1)$\\
\hline \end{tabular}
\end{center}
It obeys the RR tadpole conditions for the ${\bf {Z}}$-valued D9- and D5-brane charges, and the discrete ${\bf {Z}}_2$ RR tadpole conditions for D3- and D7$_i$-brane charges \cite{Uranga:2000xp}.
The supersymmetry condition determined by the O-plane wrappings is
\begin{eqnarray}
\sum_i\arctan (-\chi_i)\equiv \theta_1+\theta_2+\theta_3=0\; {\rm mod}\; 2\pi \,
\end{eqnarray}
The model is in fact T-dual (in all ${\bf {T}}^6$ directions) to that in section 5 of \cite{Marchesano:2004xz}.
It is easy to see that the above condition forces at least one of the ${\bf {T}}^2$ to have ${\cal O}(1)$ area in $\alpha'$ units. From our perspective, this a mere reflection of the fact that the 10d dynamical tadpole to be canceled is of order the string scale, hence it agrees with the scaling $\Delta^{-2}\sim {\cal T}$. Happily, the use of an $\alpha'$ exact configuration, which is moreover supersymmetric, makes our solution reliable. This is an improvement over other approaches e.g. as in section \ref{sec:sugimoto-dm}.
Although we have discussed the compactification on (an orientifold of) ${\bf {T}}^6$ directly, we would like to point out that it is easy to describe it as a sequence of ${\bf {T}}^2$ spontaneous compactifications, each eating up a fraction of the initial 10d tadpole until it is ultimately cancelled upon reaching ${\bf {T}}^6$. However, this picture does not really correspond to a physical situation, given the absence of decoupling of scales. This is true even in setups which seemingly allow for one ${\bf {T}}^2$ of parametrically large area. Indeed, consider for instance the regime $\chi_3\sim 2\lambda$ and $\chi_1,\chi_2\sim \lambda^{-1}$, for $0<\lambda\ll 1$, which corresponds to $\theta_1,\theta_2\sim \frac \pi{2}+\lambda$, $\theta_3\sim \pi- 2\lambda$. This corresponds to a compactification on substringy size $({\bf {T}}^2)_1\times ({\bf {T}}^2)_2$ and a parametrically large $({\bf {T}}^2)_3$. However, the fact that the $({\bf {T}}^2)_1$, $ ({\bf {T}}^2)_2$ can be T-dualized into large area geometries shows that there is not true decoupling of scales: in the original picture, the small sizes imply that there are towers of light winding modes, whose scale is comparable with the KK modes of $({\bf {T}}^2)_3$. Hence, the lack of decoupling is still present, as expected from our general considerations in the introduction.
\section{The SM from spontaneous compactification}
\label{sec:the-sm}
In this section we explore an interesting application of the above mechanism, and provide an explicit example of a 6d theory with brane-antibrane pairs, and a dynamical tadpole triggering spontaneous compactification to a 4d (MS)SM-like particle physics model. Interestingly, the complete chiral matter and electroweak sector, including the Higgs multiplets, are generated as degrees of freedom on cobordism branes. Only the gluons are present in some form in the original 6d models.
Consider the type IIB orientifold of ${\bf {T}}^4/{\bf {Z}}_2$ with orientifold action $\Omega$ constructed in \cite{Pradisi:1988xd,Gimon:1996rq}, possibly with magnetization. To describe it, we introduce the notation in \cite{Blumenhagen:2000ea,Ibanez:2001nd} of wrapping numbers $(n_\alpha^i,m_\alpha^i)$, where $n_\alpha^i$ and $m_\alpha^i$ provide the wrapping number and magnetic flux quantum of the D-brane $\alpha$ on the $i^{\rm th}$ ${\bf {T}}^2$, respectively. We consider the following stacks of D-branes (and their orientifold images, not displayed explicitly)
\smallskip
\begin{center}
\begin{tabular}{|c||c|c|}
\hline
$N_\alpha$ & $(n_\alpha^{1},m_\alpha^{1})$ & $(n_\alpha^{2},m_\alpha^{2})$ \\
\hline\hline $N_{a+d} = 6+2$ & $(1,3)$ & $(1,-3)$ \\
\hline
\hline $N_{h_1}= 4$ & $(1,-3)$ & $(1,-4)$ \\
\hline $N_{h_2}= 4$ & $(1,-4)$ & $(1,-3)$ \\
\hline $40 $ & $(0,1)$ & $(0,-1)$ \\
\hline \end{tabular}
\end{center}
\smallskip
The O9- and O5-planes correspond to the wrapping numbers $(1,0) \times (1,0)$ and $(0,1)\times (0,-1)$ respectively.
The stacks $a$ and $d$ are taken different and separated by Wilson lines, but they can be discussed jointly for the time being. They correspond to 8 D9-branes with worldvolume magnetic fluxes 72 units of D5-brane charge. The stacks $h_1$ and $h_2$ correspond to 8 additional D9-branes, with 96 with units of induced ${\overline{\rm D5}}$-branes charge. The addition of 40 explicit D5-branes leads to RR tadpole cancellation (once orientifold images are included). In terms of the wrapping numbers, we have
\begin{eqnarray}
&\sum_\alpha N_\alpha n_\alpha^2n_\alpha^3=1
\quad , \quad
&\sum_\alpha N_\alpha m_\alpha^2m_\alpha^3=-16\ .
\end{eqnarray}
The model is far from supersymmetric due to the presence of D5-${\overline{\rm D5}}$ pairs, and in fact has a decay channel to supersymmetric model by their annihilation. On the other hand, even at the top of the tachyon potential, the theory is not at a critical point of its potential due to dynamical tadpole for the closed string moduli, namely the area moduli of the ${\bf {T}}^2$'s. In other words, the excess tension depends on these, as they enter the angles determining the deviation from the supersymmetry condition
\begin{eqnarray}
\arctan \Big( \frac {m_\alpha^1}{n_\alpha^1}\chi_1\Big)\, +\, \arctan \Big( \frac {m_\alpha^2}{n_\alpha^2}\chi_2\Big)\,=\,0\, .
\end{eqnarray}
For instance, we can make the stacks $a$, $d$ supersymmetric, by choosing
\begin{eqnarray}
\chi_1=\chi_2\ ,
\end{eqnarray}
but the D-branes $h_1$ and $h_2$ break supersymmetry. Hence, there is a dynamical tadpole associated to the excess tension of these latter objects.
The dynamical tadpole can be solved by introducing magnetization along two of the 6d spacetime dimensions. The backreaction of this extra flux forces these two dimensions to be compactified on a ${\bf {T}}^2$, with the addition of cobordism I-branes \cite{Montero:2020icj}, which in general includes orientifold planes and D-branes, as in the examples above. We take these extra branes to be arranged in two new stacks $b$ and $c$. Overall, we end up with an orientifold of ${\bf {T}}^6/({\bf {Z}}_2\times{\bf {Z}}_2)$, with D-brane stacks and topological numbers given by
\smallskip
\begin{center}
\begin{tabular}{|c||c|c|c|}
\hline
$N_\alpha$ & $(n_\alpha^{1},m_\alpha^{1})$ & $(n_\alpha^{2},m_\alpha^{2})$ & $(n_\alpha^{3},m_\alpha^{3})$\\
\hline\hline $N_{a+d} = 6+2$ & $(1,3)$ & $(1,-3)$ & $(1,0)$\\
\hline $N_b=2$ & $ (0,1)$ & $(1,0)$ & $(0,1)$ \\
\hline $N_c=2$ & $(-1,0)$ & $(0,-1)$ & $(0,1)$\\
\hline
\hline $N_{h_1}= 2$ & $(1,-3)$ & $(1,-4)$ & $(2,-1)$\\
\hline $N_{h_2}= 2$ & $(1,-4)$ & $(1,-3)$ & $(2,-1)$\\
\hline $40 $ & $(0,1)$ & $(0,-1)$ & $(0,1)$\\
\hline \end{tabular}
\end{center}
The model satisfies the RR tadpole conditions
\begin{eqnarray}
& \sum_\alpha\,N_\alpha n_\alpha^1n_\alpha^2n_\alpha^3\,=\, 16\ ,\quad & \sum_\alpha\,N_\alpha n_\alpha^1m_\alpha^2m_\alpha^3\,=\, 16\ , \nonumber \\
& \sum_\alpha\,N_\alpha m_\alpha^1n_\alpha^2m_\alpha^3\,=\, 16\ ,\quad & \sum_\alpha\,N_\alpha m_\alpha^1m_\alpha^2n_\alpha^3\,=\, -16\ .
\end{eqnarray}
This corresponds to O9-planes along $(1,0)\times (1,0)\times (1,0)$, and O5-planes along $(0,1)\times (0,-1)\times (1,0)$, as already present in the 6d theory, and cobordism O5-planes along $(0,1)\times (1,0)\times(0,1)$ and $(1,0)\times (0,1)\times (0,1)$.
The model still contains only 3 stacks of D-branes with non-trivial angles, so that they are just enough to fix the 2 parameters $\chi_i$ of the ${\bf {T}}^2$'s. The O-planes fix the supersymmetry condition signs to
\begin{eqnarray}
\arctan \Big( \frac {m_\alpha^1}{n_\alpha^1}\chi_1\Big)\, +\, \arctan \Big( \frac {m_\alpha^2}{n_\alpha^2}\chi_2\Big)\,-\, \arctan \Big( \frac {m_\alpha^3}{n_\alpha^3}\chi_3\Big)\,=\,0\, .
\end{eqnarray}
Using the branes above, we get
\begin{eqnarray}
\chi_1=\chi_2\quad ,\quad \chi_3\,=\, \frac{14\chi_1}{1-12\chi_1^{\,2}}\,.
\end{eqnarray}
The regime of large $({\bf {T}}^2)_3$ corresponds to small $\chi_3$, which is also attained for small $\chi_1$. Note that in this context the last condition $\chi_1\sim \chi_3$ encodes the relation between the 6d tadpole and the inverse area of the spontaneously compactified ${\bf {T}}^2$.
The model is, up to exchange of directions in the ${\bf {T}}^6$ and overall sign flips, precisely one of the examples of 4d MSSM-like constructions in \cite{Marchesano:2004xz,Marchesano:2004yq}. The gauge group is $U(3)_a\times USp(2)_b\times U(1)_c \times U(1)_d$, where we break the naive $USp(2)_c$ by Wilson lines or shifting off the O-plane for the corresponding D5-branes. Taking into account the massive $U(1)$'s due to $BF$ couplings, this reproduces the SM gauge group. In addition, open strings between the different brane stacks reproduce a 3-family (MS)SM chiral matter content, and the MSSM Higgs doublet pair. Hence, we have described the spontaneous compactification of a 6d model to a semi-realistic MSSM-like 4d theory.
A fun fact worth emphasizing is that most of the SM spectrum is absent in the original 6d model, and arises only after the spontaneous compactification. In particular, all the MSSM matter and Higgs chiral multiplets, as well as the electroweak gauge sector, arise from open string sectors involving the $b$ and $c$ branes, which arises as cobordism branes. It is remarkable that cobordism entails that spontaneous compactification implies not just the removal of spacetime dimensions, but also the dynamical appearance of novel degrees of freedom. It is tantalizing to speculate on the potential implications of these realizations in cosmological or other dynamical setups.
\section*{Acknowledgments}
We are pleased to thank Inaki Garc\'ia-Etxebarria, Luis Ib\'anez, Fernando Marchesano, Miguel Montero and Irene Valenzuela for useful discussions. This work is is supported by the Spanish Research Agency (Agencia Espa\~nola de Investigaci\'on) through the grants IFT Centro de Excelencia Severo Ochoa SEV-2016-0597, the grant GC2018-095976-B-C21 from MCIU/AEI/FEDER, UE.
\newpage
|
1,116,691,497,600 | arxiv | \section{Introduction}
\IEEEPARstart{T}{he} interplay between social networks and complex human behavioral dynamics has been the subject of extensive research. Social ties, a.k.a. friendships and/or acquaintanceships, have been connected to various complex human phenomena. Some studies, for example, demonstrated the social capital brought about by strong and weak ties~\cite{granovetter1977strength}. Others illuminated on the benefits that social ties can bring to mental and physical health, such as increased longevity, reduced loneliness, and lower levels of stress~\cite{kawachi2001social, cohen2004social, umberson2010social, thoits2011mechanisms}. Recent studies compared online and offline social ecosystems, studying the formation of online social ties and showing how these ties evolve into social networks~\cite{backstrom2006group, ellison2007benefits, szell2010multirelational, wilson2012review}, including in video games~\cite{zhong2011effects, trepte2012social}.
Researchers have also been concerned with understanding how physical and mental factors impact human behavior and performance~\cite{leonard2008richard}. However, little attention has been devoted to study the interplay between social ties and human performance dynamics, which is the subject of this study.
In the age of big data, humans leave behind traces of their online activity in the form of digital behavioral data, which facilitates our research and bestow us with new data-centric perspectives to study social ties, in addition to established methods like interviews, surveys, or ethnographic observations. In this paper, our main interest is the influence that social ties have on human performance in collaborative team-based settings, more specifically in Multiplayer Online Battle Arena (MOBA) video games. Understanding the relationship between social ties, in particular preexisting connections within team members, and (individual and/or team) performance is a question of broad relevance across education, psychology, and management sciences~\cite{wuchty2007increasing, guimera2005team, borner2010multi, contractor2013some, Mukherjee2018}, and could lead us to better understand what underlies human behaviors in such systems.
Our research will focus on a popular MOBA collaborative team-based game, \textit{Dota 2}, a rich dataset that will allow us to study millions of players and matches.
Dota 2 is one of the most successful MOBA games: according to the official Dota 2 website, more than ten million unique players participate in the games each month.\footnote{Statistics on Dota 2 Official Website: \url{http://blog.dota2.com/}}
Dota 2 not only hosts a huge user base but also innately incorporates mechanisms that stress the impact of social ties. Since two opposing teams, each consisting of 5 players, compete against each other, preexisting friendships are put to test, and strangers are brought together, to collaborate as a team in order to prevail over the rivals. Each player has the autonomy to befriend other players, and these constructed social ties are stored in a \textit{friendship list} on \textit{Steam}, the online game distribution platform that hosts Dota 2 and hundreds of other games and associated communities. In each list, both the time of formation and the actors involved in each dyad are recorded. In this paper, we jointly leverage the behavioral data provided by the log of Dota 2 matches and the social network data (friendship lists) provided by the Steam community.\footnote{Steam Community Website: \url{https://steamcommunity.com/}}
Motivated by the need for a thorough investigation of the influence of social ties on performance, and in light of the recent advancement in network science and team science, we analyze our data considering four different perspectives: \textit{(RQ1)} We first look into the actions of individuals that may be affected by the presence of friendship ties in the team. \textit{(RQ2)} Next, we proceed to divide teams into four categories based on both collective experience (experts vs newbies) and performance (high vs low) and analyze them separately. After that, we focus on explaining the dynamics governing performance in short-term sessions of consecutive matches, and propose two additional research questions: \textit{(RQ3)} We plan to understand how individuals perform in consecutive games when they only play with friends. \textit{(RQ4)} We aim to investigate how social ties affect team performance in consecutive games.
\bigskip
In summary, in this paper we will address the following four research questions (RQs):
\begin{itemize}
\item[\textbf{RQ1:}] \textit{What is the influence of social ties on individual players' activity?}. We will test whether the presence of social ties affects the activity of individuals within a team. Our hypothesis is that the presence of preexisting friendship ties within a team will increase teammates activity. We will set to test whether there exists a spillover effect by which even individuals who do not have friendship connections with other teammates, but who play in a team where some players are friends among each other, experience such effect. If social ties have an effect, we will also characterize which dimensions of activity it affects.
\item[\textbf{RQ2:}] \textit{What is the influence of social ties on team dynamics?}. We will investigate whether preexisting social ties will affect the performance of the team as a whole. We will further investigate the subsets of teams composed by high/low experience players, and high/low performing players. Our hypothesis is that preexisting social ties improve team performance. We will test whether this is the case, and if so, we will characterize how performance is affected.
\end{itemize}
While the former two questions focus on measuring effects within single matches, the next two questions focus on effects that span over the course of a gaming session (i.e., a nearly-uninterrupted sequence of consecutive matches):
\begin{itemize}
\item[\textbf{RQ3:}] \textit{What is the influence of social ties on individuals over gaming sessions?}. We will study whether playing game sessions within teams with preexisting social ties affects individuals' short-term activity. We hypothesize that the presence of such ties can mitigate known effects of deterioration in individual performance over the course of the sessions.
\item[\textbf{RQ4:}] \textit{What is the influence of social ties on teams over gaming sessions?}. We will determine whether short-term performance of teams as a whole is affected by the presence of social ties. Our hypothesis is again that social ties can influence team performance and mitigate known session-level deterioration effects.
\end{itemize}
This paper is organized as follows: We will first explain data gathering and preprocessing steps (see Section~\S\ref{sec:data}). In Section~\S\ref{sec:methods}, we will elucidate the methods we employ when answering our four research questions. The results will be presented and discussed in Section~\S\ref{sec:results}. We will also provide an overview of literature concerning social ties, online games, and performance dynamics in Section~\S\ref{sec:related}. In Section~\S\ref{sec:conclusions}, we will conclude our study and shed light on its potential applications and future extensions.
\section{Data \& Statistics}\label{sec:data}
\subsection{Match log data from Dota 2}
\textit{Defense of the Ancients 2} (Dota 2), is a multi-player online battle arena (MOBA) video game. In this paper, we only study matches consisting of all five human players (as opposed to matches that mix humans and bots, or 1-vs-1 matches). In such matches, two teams each composed by five players compete to destroy the opponent team's fortified home-base known as the ``Ancient''. Each player has the choice to draft a virtual avatar known as a \textit{hero}, to participate in each match. The game is designed with an internal nudging mechanism to foster cooperation, since heroes have complementary abilities (e.g., \textit{Pudge} is popular for its strength, \textit{Sniper} is recognized for agility, \textit{Invoker} is known for intelligence, etc.). Thus, to increase the probability of winning,
teammates have to coordinate to form balanced teams during the draft phase that precedes each match, and fill various desirable roles (e.g. Carry, Disabler, Support, etc.).
We acquired match log data of Dota 2 from the OpenDota API.\footnote{OpenDota API: \url{https://docs.opendota.com/}} This service provides information such as match duration, team members, action statistics of each players, matchmaking type, etc., for millions of Dota 2 matches. The Dota 2 gaming system provides four mechanisms to construct the two opposing teams (matchmaking), i.e., normal match, ranked match, practice 1-vs-1 match, and bot match. Since ranked matches are governed by the Matchmaking Rating (MMR) system, friends cannot freely group together---the goal is to create artificially-balanced opposing teams, thus teams are often composed by random strangers. Practice 1-vs-1 matches and bot matches do not meet our need to evaluate how social ties affect human performance in team-based human environments. Therefore, in our analysis, we only focus on the normal matches, where ten human players participate in one 5-vs-5 match. To win the match, teammates need to collaborate, coordinate, and support each other to harvest resources (e.g., collect gold by killing AI-controlled mobs called creeps), defend their base and towers, attack and defeat the enemies, and destroy their towers and base.
Due to data privacy, some users' match records we collected are incomplete. After discarding these unusable match records, our dataset contains 3,566,804 matches, comprising 1,940,047 unique players, and spans from July 17, 2013 to December 14, 2015. Figure \ref{fig:match_distribution} shows the number of matches per player in our dataset. The average match duration of the matches in our data is about 41.8 minutes, and its distribution is displayed in Figure \ref{fig:duration_distribution}.
\begin{figure*}[t]
\centering
\subfloat[Match Distribution]
{\label{fig:match_distribution}
\includegraphics[width = .6\columnwidth]{./match_distribution.png}
\qquad
\subfloat[Duration Distribution]{\label{fig:duration_distribution}
\includegraphics[width = .6\columnwidth]{./duration_distribution.png}}%
\qquad
\subfloat[Individual's Matches Time Gap Distribution]{\label{fig:Matchgap}
\includegraphics[width = .6\columnwidth]{./Matchgap.png}}%
\caption{\textbf{Distribution plot of our dataset.} (a) Number of matches per player and (b) Duration (in seconds) per match and (c) Duration (in seconds) per match.}%
\end{figure*}
\subsection{Friendship data from Steam}
Steam is currently the world's largest digital game distribution platform where registered users can not only purchase and manage a variety of games, but also join gaming communities. It is worth noting that the Steam platform and Dota 2 are synchronized via the \textit{convertible Steam account ID} linked to the \textit{Dota 2 player account ID}. Therefore, we are able to connect each Dota 2 player's in-game behavioral data acquired from the Dota 2 API with their friend list (and other account metadata) on Steam.
According to Steam's official Website,\footnote{Steam Official Statistics: \url {https://store.steampowered.com/stats/}} over 10 million players are active on the platform on a daily basis.
The Steam platform, with its open API, has provided researchers with access to a massive amount of data, that has been leveraged to analyze various aspects of players' behaviors. For instance, \cite{sifa2015large} analyzed play-time related, cross-games behavior of Steam users. \cite{hamari2011framework} proposed analytical abstractions between the different components of game achievements. Other than cross-game behaviors, the gamers' social network provided by the Steam Community has also caught the attention of the research community: \cite{becker2012analysis} studied the evolution patterns of the Steam community network; \cite{blackburn2012branded} utilized the network structure of Steam to identify cheaters in gaming social networks. Despite of these macro-level analyses, the influence of social ties (i.e., online friendships) on individual and team performance remains largely unexplored: Therefore, we will utilize the friendship lists of Dota 2 players provided by Steam to reconstruct the player social network and closely examine the impact of social ties on players' in-game performance and behavior.
The collection of players' friendship lists from the Steam API follows the process of identifying Dota 2 players from match log data. After making sure that each friendship pair was formed before the starting time of each match, we can construct the exact team-wise friendship network structure within each match. We describe teams in each match as a network with 5 nodes, i.e., 5 players are represented by 5 nodes and each pairwise friendship formed before the starting time of the match is recorded as an edge.
Similar to the Dota 2 API, the Steam API also respects each player's data privacy preferences. We requested friendship information for all 1,940,047 distinct Dota 2 players, and acknowledged that data for 227,045 players was unavailable due to privacy restrictions.
\subsection{Final dataset}
We now finally combine information provided by both the Dota 2 API and the Steam API. To this aim, we make sure that for each team in our final dataset all information about all players' friendships and match actions is openly accessible (i.e., not restricted by privacy settings). Our final dataset therefore contains 954,731 players, 673,864 teams, and 621,629 matches. This is the final dataset used in all our experiments, discussed next. Here, we have records of 365,412 teams consisting of 652,215 unique players who participated in 337,043 normal matchmaking matches. This dataset starts on July 14, 2014 and ends on December 14, 2015. It includes match features, player's individual actions, and social ties related to each team.
\begin{figure*}[t]\centering
\subfloat[Individuals playing with friend(s) \textit{(in-friendship players)} vs \textit{null players} (those players in all-stranger teams).]{\label{fig:overall_in_friendship}\includegraphics[width = .9\columnwidth, clip, trim=4 1 225 1]{./lby0_aggregated_with_friends.png}}
\qquad
\subfloat[Individuals playing without friend(s) (\textit{out-friendship players}) vs \textit{null players} (those players in all-stranger teams).]{\label{fig:overall_out_friendship}\includegraphics[width = .9\columnwidth, clip, trim=4 1 225 1]{./lby0_aggregated_without_friends.png}}
\caption{\textbf{Social ties' impact on individuals playing with friends and individuals playing without friends}. Violin plots convey the statistical distribution distinctions between (a) Individuals playing \textit{with friends}, and (b) Individuals playing \textit{without friends}, , in \textit{teams of friends} and \textit{mixed teams}, as well as comparing against \textit{null players} (those who play in all-stranger teams). Shuffled null players are displayed as orange violins (right violin of each pair), observed individuals with and without friends are shown in green violins (left violin of each pair). Stars representing t-test statistical significance are shown in all subplots ($^{***}$ means p-value $<=$ 0.001).}
\label{fig:Pannoramic_Individual}
\end{figure*}
\section{Methods}\label{sec:methods}
\subsection{RQ1: Overview of the influence of social ties on individuals}
We first introduce the types of social structures we will study. There exist three types of teams in our setting:
\begin{itemize}
\item \textit{Teams of strangers}, where no preexisting friendship ties exist among any players prior to the match;
\item \textit{Teams of friends}, where each team member has at least another friend in the team; and,
\item \textit{Mixed teams}, where some members have at least on friend among their teammates, while some others do not.
\end{itemize}
It is worth noting that in our categorization, we do not have a distinct definition for teams that are cliques, i.e., where each player is friend with everyone else, because these instances are exceptionally rare in the data at hand.
Furthermore, we consider individuals playing in teams of friends, as well as in mixed teams, as in conditions potentially affected by the influence of social ties. Conversely, teams of strangers are used as control groups where players cannot be affected by social ties influence, due to their absence.
Since our first research question focuses on analyzing the influence of social ties on individual players, for each instance of a match, we divide players into three types:
\begin{itemize}
\item \textit{Null players}, i.e., those playing in a \textit{team of strangers}; due to the absence of social ties, these players are used as null models (thus the name ``null players''), or baselines, to compare and contrast with other player types.
\item \textit{In-friendship players}, i.e., those playing in \textit{teams of friends}, as well as those playing in \textit{mixed teams}, who have preexisting social ties with some teammates (both sets of players may directly experience social ties' influence).
\item \textit{Out-friendship players}, i.e., those playing in \textit{mixed teams} who do not have preexisting social ties with any of their teammates (yet may indirectly benefit from their teammates' preexisting friendships).
\end{itemize}
In Section~\S\ref{sub:rq1}, we analyze the influence of social ties on players with friends, and players without friends, separately. Take the study on \textit{teams of friends}, for instance: We compare the statistic distributions of performance on in-game actions (kills, assists, and deaths) for \textit{in-friendship players} with that of the \textit{null players}, contrasting the distributions by both statistical analysis and visual analysis (using so-called \textit{violin plots}). We carry out t-test(s) to prove or reject our hypothesis. If the t-test results are statistically significant across all observed distribution pairs, our hypothesis is confirmed and thus we observe an effect of social ties on in-game activity.
It is worth noting that the data of null players is randomly sampled with reshuffling to yield samples of the exact same size as the samples of in-friendship players. Likewise, we use the same null model strategy to analyze the impact of social ties on out-friendship players.
\subsection{RQ2: Overview of the influence of social ties on teams}
After answering our first research question, we proceed to include not only in-game actions but also performance and experience into our analysis. We use the \textit{kill-death-assist ratio} (KDA) to measure both the performance of individual players and the performance of teams. KDA can be formalized as $(k+a)/max\{1,d\}$, where $k$ is the number of kills, $a$ is the number of assists, and $d$ is the number of deaths of a player (or a team of players) in a given match.
The teams composed of players with the very high/low experience and very high/low skills are of particular interest for our analysis, provided that they may exhibit noteworthy behavioral patterns: for example, they may exacerbate the effect of social ties' influence in one direction or another.
To this purpose, we consider the top and bottom 25th percentiles of teams composed by players ranked by average team experience (i.e., number of played matches) and by average team performance (as measured by the average team's KDA score).
To achieve that, we calculate the average KDA and the experience of each team as match-based features. For each match, the average KDA is calculated by averaging all five players’ KDA in the current match (for each of the opposing teams), while the experience is calculated by summing over the number of past matches of all five players until the current match. We select the top 25\% players in each ranking as high-level category and the bottom 25\% as low-level category. Having divided the data in the four categories of teams, in each category we further compare the actions and performance of the whole team, the players with friends (\textit{in-friendship players}) as well as the players without friends in mixed teams (\textit{out-friendship players}) with that of the null players. We compute the difference in each case as $(Y-X)/X$, where $Y$ is the mean of actions or performance of players (or teams) who may be subject to the influence of social ties, and $X$ is the mean of actions or performance of null players (teams of strangers).
Thus, to summarize, we select four categories of teams as follows:
\begin{itemize}
\item \textit{High Experience \& High KDA}: these are teams composed by players that are in the top 25th percentile by experience (no. played matches) as well as by performance (KDA).
\item \textit{High Experience \& Low KDA}: these are teams composed by players that are in the top 25th percentile by experience (no. played matches) and in the bottom 25th percentile by performance (KDA).
\item \textit{Low Experience \& High KDA}: these are teams composed by players that are in the bottom 25th percentile by experience (no. played matches) and in the top 25th percentile by performance (KDA).
\item \textit{Low Experience \& Low KDA}: these are teams composed by players that are in the bottom 25th percentile by experience (no. played matches) as well as by performance (KDA).
\end{itemize}
\subsection{RQ3: Influence of social ties on individuals over sessions}
More often than not, individual players tend to complete a sequence of matches rather than a single match before they decide to stop their gaming session. Playing consecutive matches may bring tiredness and boredom to players, which could in turn affect their performance. On the other hand, playing consecutive matches could also help train proficiency. Due to this dichotomy, this aspect warrants further investigation.
Therefore, we formalize consecutive playing patterns as individual gaming sessions. Provided that we don't know exactly when players start or interrupt a gaming session, we need to infer such sessions from the start/end times recorded in each match metadata.
We set 1 hour as the threshold to split gaming sessions: if the time gap between the end of a match and the beginning of the next match, for each player, is shorter than one hour, we assume that these two matches belong to the same gaming session; otherwise, two separate gaming sessions are extracted. We calculate all the time gaps---the time intervals between the end of a match and the beginning of the subsequent match---for each player and concatenate all players' time gaps together. Fig. \ref{fig:Matchgap} shows the distribution of time gaps that is less than 24 hours in our dataset: amongst these 84K time gaps, the median is 1.265 hours, supporting our choice of 1 hour threshold to split sessions. Each gaming session consists of a list of consecutive matches ordered by their starting time. Such sequence index in a session is named as \textit{match position}. For example, in a session of four matches, the first match is called \textit{match in position one}, and the last match is referred to as \textit{match in position four}.
To isolate the effect of social ties on gaming sessions, in RQ3, we focus on individuals who only play with friends throughout the entire session. This will allow us to reduce the variability that may arise in case of inclusion of mixed sessions where users played games both with and without friends. Of course, this filter also reduces the number of sessions suitable for analysis.
We use two strategies to analyze social ties’ impact on these individuals:
\begin{itemize}
\item
First, we study the individuals' KDA trajectories throughout the gaming sessions. We only utilize gaming sessions with length 1 to 4, as data about sessions of length larger than 4 is very sparse (for reference, a gaming session of length 4 usually spans between 3 and 5 consecutive hours of uninterrupted playing; anecdotally, in our data, we observe isolated instances of sessions that last up to 20 consecutive hours).
For gaming sessions with different length, we separately aggregate the KDA on each match position, and use separate line plots to visualize the trajectories over the course of the sessions. Then, we randomly reshuffle the sequence of matches in all gaming sessions, and reconstruct the trajectories based on the shuffled data---this is used as a randomized null model. By comparing the trajectories in sessions with original match positions against trajectories in sessions with randomized match positions, we exclude the possibility that any emerging trend is a produced just by chance.
\item
Second, we compute the KDA difference between the last and the first match of a session, expressed as $(Y-X)/X$, where $Y$ is the KDA performance of the last match in a gaming session and $X$ is the KDA performance of the first match in a gaming session. This measure is adopted to capture the variation of overall performance throughout the whole session, i.e., the overall size of such an effect.
\end{itemize}
\subsection{RQ4: Influence of social ties on teams over sessions}
\label{sub:rq4def}
In the previous section, we introduced the notion of gaming sessions of individual players. Here, we define a team’s gaming session as the average gaming session of its 5 individual players. For example, for a given team in a given match, three players may be playing the first match of their session, two players may be playing the second match of their session, and one player may be playing the fourth match of their session: in this case, the average session length for this team would be $(1+1+1+2+4)/5=1.8$.
Therefore, due to the employed averaging strategy, the length of a team’s gaming session will be expressed in a ranges, i.e., gaming sessions of length [1-2), [2-3) and [3-4). Sessions of average length greater than 4 are exceptionally rare and therefore excluded from our analysis.
To answer RQ4, we analyzed the kills, assists, and deaths as well as KDA performance of players subject to the influence of social ties, considering team’s gaming sessions. We use one pair of violin plots to visualize each type of action or KDA’s distributions of \textit{in-friendship players} versus \textit{out-friendship players} in each team. We then use t-test(s) to verify the statistical difference of each pair. To investigate the trend of teams’ actions and performance when the length of gaming session increases, we organize the plots by comparing each type of action (or KDA) across teams of different session lengths. Consider, for example, the plot tracking team kills (take a glance at Fig.~\ref{fig:team_session_kills}): from left to right, the first pair of violin plots belongs to teams with avg. session length 1-2, the second pair of violin plots belongs to teams with avg. session length 2-3, and the third pair of violin plots belongs to teams with avg. session length 3-4.
\begin{figure}[t]
\centering
\includegraphics[width = .95\columnwidth]{./lby0_individual_session.png}
\caption{\textbf{KDA trajectories of individuals who only play with friends throughout the entire gaming session.} The left plot shows the actual data suggesting the presence of individual performance deterioration over the course of gaming sessions. The right plot shows the reshuffled null model where the effect of match position is disrupted (therefore, the lines are expected to become flat).}
\label{fig:individual_session}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.5\textwidth]{./lby0_individual_session_percentage.png}
\caption{\textbf{Changing rate of KDA performance in sessions of different lengths by individuals who only play with friends throughout the entire gaming session.} This plot shows the KDA change percentage of last game in the session from the first game in the session from in-friendship players (those who played the entire session with some friend(s) in their team).}
\label{fig:individual_session_percentage}
\end{figure}
\section{Results}\label{sec:results}
In this section, we present the results of our analysis aimed to address the research questions defined above. We present our summary in four parts, corresponding to the four proposed research questions. For RQ1 (\S\ref{sub:rq1}), we provide an overview of the interplay between individual players' activity and the effect of social ties. To answer RQ2 (\S\ref{sub:rq2}), we focus on friendships' influence on team's actions and performance by categorizing teams according to average experience and KDA performance. We explore RQ3 (\S\ref{sub:rq3}) by identifying performance dynamics of individual players who only play with (and without) friends throughout their entire gaming sessions. Lastly, for RQ4 (\S\ref{sub:rq4}) we extend our analysis to actions and performance trajectories of teams within gaming sessions. For each research question, we will aim at validating or rejecting the hypotheses formalized above.
\begin{table*}[t]
\centering \small
\begin{tabular}{@{}c|l|lll|l@{}}
\toprule
\multicolumn{1}{l|}{Team Category} & Condition & Kills & Deaths & Assists & KDA \\ \midrule\midrule
\multirow{3}{*}{\begin{tabular}[c]{@{}c@{}}Low Experience\\ Low KDA\end{tabular}} & Whole Team & 87\% & 129\% & 19\% & 79\% \\
& In-Friendship & 96\% & 15\% & 27\% & 474\% \\
& Out-Friendship & 79\% & 97\% & 9\% & 990\% \\
\midrule
\multirow{3}{*}{\begin{tabular}[c]{@{}c@{}}High Experience\\ Low KDA\end{tabular}} & Whole Team & 109\% & 125\% & 18\% & 89\% \\
& In-Friendship & 185\% & 209\% & 54\% & 437\% \\
& Out-Friendship & 48\% & 56\% & -1\% & 392\% \\
\midrule
\multirow{3}{*}{\begin{tabular}[c]{@{}c@{}}Low Experience\\ High KDA\end{tabular}} & Whole Team & 17\% & 23\% & 94\% & -36\% \\
& In-Friendship & 25\% & 35\% & 109\% & -32\% \\
& Out-Friendship & 1\% & 8\% & 79\% & -28\% \\
\midrule
\multirow{3}{*}{\begin{tabular}[c]{@{}c@{}}High Experience \\ High KDA\end{tabular}} & Whole Team & 39\% & 52\% & 151\% & -32\% \\
& In-Friendship & 67\% & 85\% & 208\% & -26\% \\
& Out-Friendship & 9\% & 13\% & 89\% & -23\% \\ \bottomrule
\end{tabular}
\caption{Percentage difference of 4 categories of teams' action/performance compared with null model of all-stranger teams.}
\label{tab:team_percentage}
\end{table*}
\subsection{RQ1: Influence of social ties on individual players' activity}\label{sub:rq1}
To study how social ties influence players' activity in the game, we compare \textit{teams of friends} and \textit{mixed teams} against \textit{teams of strangers}. We call \textit{teams of friends} and \textit{mixed teams} as teams with preexisting social ties. \textit{Teams of strangers} are consider as our null model since no social ties exist in this condition. In other words, we consider players in teams of strangers as our control group, while players in teams with preexisting ties are assembled as observed conditions in the following analysis aimed to tackle the first research question.
We evaluate observed players' actions (kills, assists, and deaths) against the null model, i.e., comparing individual's action patterns under the presence of social ties in contrast with all-strangers team scenario. Data of players in all-stranger teams are randomly shuffled and then under-sampled (or over-sampled) to match the number of players in our observed conditions data.
As long as there exists some social tie in the team, individual players could be divided into two types, namely individuals playing with some friend(s) and individuals playing without any friend(s). We named the first group \textit{in-friendship players}, and the second group as \textit{out-friendship players}. Our hypothesis is that if social ties have some form of influence on players' activity, \textit{in-friendship players} will experience this effect directly (since these are the players who are playing with some friends), while \textit{out-friendship players} may experience it indirectly, even without playing with friends, yet by playing with teammates who are friends among each other.
The plot for players with friends (a.k.a., in-friendship) is shown in Figure \ref{fig:overall_in_friendship}. The plot for players without friends (a.k.a., out-friendship) can be found in Figure \ref{fig:overall_out_friendship}. Note that in Figure \ref{fig:overall_in_friendship}, we use ``With Friends'' to label the distributions associated with in-friendship players, while in \ref{fig:overall_out_friendship}, we use ``Without Friends'' to mark the distribution associated with out-friendship players. Stars in the all plots represent t-test statistical significance obtained by comparing observed conditions versus the null model (\textit{null players}, i.e., those in all-stranger teams where no social ties exist).
By inspecting Figure \ref{fig:overall_in_friendship}, we observe that, in comparison to null players (players in all-strangers teams), individuals playing with friends have higher number of kills and assists. However, deaths also arise along with kills and assists. In other words, they are more engaged and active in the game, which leads to an increased number of in-game actions, both positive (kills and deaths) and negative (deaths). This may suggest that players with friends in their team may tend to adopt more aggressive or impulsive strategies.
Figure \ref{fig:overall_out_friendship} shows a reverse patterns: contrary to players with friends (in-friendship players), players without friends (out-friendship ones) have relatively fewer actions in comparison with null players. Such decrease suggests that players in the out-friendship condition may tend to act in the best interest of themselves, adopting a more conservative play style. Alternatively, they may also experience being left out of the coordination and therefore being exposed to less game action, thus having fewer opportunities throughout a match to accomplish both positive and negative actions.
\subsection{RQ2: Influence of social ties on team dynamics}\label{sub:rq2}
To better understand how social ties impact teams as a whole, we divide teams into four categories: \textit{(i) High Experience \& High KDA}, \textit{(ii) High Experience \& Low KDA}, \textit{(iii) Low Experience \& High KDA}, and \textit{(iv) Low Experience \& Low KDA} teams. Moreover, by comparing them with our null model, all-stranger teams, we analyze the actions (kills, deaths, assists) and performance (KDA) of each category. To compare the two groups we calculate the percentage-difference of actions as: $(Y-X)/X$, where $Y$ is the mean of actions/performance of observed players in teams with social ties, and $X$ is the mean of actions/performance of the null model. In Table \ref{tab:team_percentage}, we report the results we obtained.
We can observe that low-experience \& low-KDA teams have kills, assists, and deaths actions all higher than those of all-strangers teams. In terms of having positive percentage gain of actions, in-friendship players are the biggest winners since they almost double the amount of kills and also have a 27 percent raise on assists. However, out-friendship players turn out to be the largest beneficiary of KDA performance. Although in-friendship players gained a 4.74 times performance boost by collaborating with friends, the out-friendship players received twice as much benefits by an indirect effect. Such effect is not consistent with regard to high-experience \& low-KDA teams, whose in-friendship players are the biggest gainer in all actions as well as KDA performance. Moreover, their out-friendship players show increased unwillingness to help other teammates since their assists are even less than null players. For the remaining two categories with high KDA, in-friendship players have the largest actions percentage gain, but the whole team is the biggest loser in terms of KDA performance.
When comparing across the four categories, players with friends, and the whole team, have consistently positive gains on kills, deaths, and assists. We observe that in-friendship players in high-experience categories double the actions in comparison to their low-experience counterparts. However, it is not the case for out-friendship players, namely their actions are not obviously affected by experience difference. This reveals that, for players with friends, experience boosts activity but it is not effective on performance. We also observe that for high-KDA teams, their KDA drops drastically when exposed to a team with preexisting social ties. The sharpest rate of decline goes to the whole teams' performance, followed by the decline rate of 20 percent concerning the in-friendship players and out-friendship players. While for low performing teams, the KDA improves drastically when compared with teams of all strangers, such that on a whole team level their performance rose by almost 80 percent, while in-friendship players exhibit over 4 times higher KDA.
As a summary, low-performance players, regardless of their experience, exhibit the highest gains in KDA when preexisting social ties are present in the team. Conversely, high-skill players exhibit significant decreases in KDA, regardless of their experience, when social ties are present in the team. In other words, playing with friends benefit almost exclusively low-performance players, who drag down the performance of their better-skilled friends.
\subsection{RQ3: Influence of social ties on individuals over sessions}\label{sub:rq3}
To answer RQ3, we focus on analyzing the impact of social ties on individuals who only play with friends throughout the entire gaming session.
Figure \ref{fig:individual_session} displays the KDA performance trajectory of individuals who only play with friends throughout the entire gaming session: The left plot shows the actual data suggesting the presence of individual performance deterioration over the course of gaming sessions. For example, for sessions of length 3, the average KDA in the first match of such sessions is above 3.3, while the average drops to below 3.2 in the third and last match of such sessions. This effect is visible across the three conditions of session length greater than one.
Then we verified our findings via randomization to exclude the possibility that the performance deterioration phenomenon was created by chance (random effect). The right plot of Figure \ref{fig:individual_session} shows the reshuffled data, where the effect of match position is disrupted: as we would expect, the lines flatten out suggesting that indeed the position of a match in a session has an effect on performance, corroborating the performance deterioration hypothesis in line with recent research results~\cite{sapienza2017performance, sapienza2018non, sapienza2018individual}.
To quantify the effect size of such performance deterioration, we computed the percentage-change in KDA between the last and the first game in each session. Figure \ref{fig:individual_session_percentage} shows the percentage change in KDA performance for sessions of length 1, 2, 3 and 4. The greater effect size is experienced for sessions of length 3 and 4, where the drop in KDA is approximately 5\% (the randomized model as expected shows a flat line suggesting the absence of such effect in the null model).
Summarizing: Results in Figure \ref{fig:individual_session} and \ref{fig:individual_session_percentage} reveal that players who only play with friends in a gaming session display an apparent trend of performance deterioration. However, such strong trend was not apparent for players who only play without friends and players who only play with strangers in a gaming session.
\begin{figure*}[t]\centering
\subfloat[Kills of In-friendship players vs Out-friendship players ]{\label{fig:team_session_kills}
\includegraphics[width = .95\columnwidth]{./lby0_team_session_kills.png}}
\qquad
\subfloat[Assists of In-friendship players vs Out-friendship players ]{\label{fig:team_session_assists}
\includegraphics[width = .95\columnwidth]{./lby0_team_session_assists.png}}
\qquad
\subfloat[Deaths of In-friendship players vs Out-friendship players ]{\label{fig:team_session_deaths}
\includegraphics[width = .95\columnwidth]{./lby0_team_session_deaths.png}}
\qquad
\subfloat[Average KDA of In-friendship players vs Out-friendship players ]{\label{fig:team_session_KDA}
\includegraphics[width = .95\columnwidth]{./lby0_team_session_KDA.png}}
\caption{\textbf{Social ties' impact on teams over gaming sessions.} Violin plots convey the statistic distributions between players with friends and players without friends in teams with preexisting social ties. Four aspects were examined: (a) kills, (b) assists, (c) deaths, and (d) KDA performance. In-friendship players' data are displayed as green violins (left violins), while out-friendship players' data are shown in orange violins (right violins). Stars represent t-test statistical significance ($^*$ means p-value$<$0.05, $^{**}$ means p-value$<$0.01, $^{***}$ means p-value$<$0.001, $ns$ means not-significant).}
\label{fig:Team_Session}
\end{figure*}
\subsection{RQ4: Influence of social ties on teams over sessions}\label{sub:rq4}
We now analyze how actions and performance change in relation to different session positions in presence of social ties. Here we focus on the analysis of actions and performance over the entire team
As mentioned in Section~\S\ref{sub:rq4def}, the average gaming session's length of a team falls into three ranges, namely [1-2), [2-3), and [3-4). Data of sessions beyond that average length are excluded due to high sparsity and low significance.
Furthermore, we concentrate exclusively on teams with social ties. In Figure \ref{fig:Team_Session}, we compare individuals playing with friends (in-friendship players) with individuals playing without friends (out-friendship players). Figure \ref{fig:team_session_kills}, Figure \ref{fig:team_session_assists}, Figure \ref{fig:team_session_deaths}, and Figure \ref{fig:team_session_KDA} each convey the kills, assists, deaths, and average KDA performance distributions of in-friendship players vs out-friendship players, on different session positions. All the distributions of out-friendship players are labeled as ``out'', whereas in-friendship players are labeled as ``in''.
Our results overall reveal that, as time goes by throughout a gaming session, individuals playing with friends have gradually increasing kills, assists, deaths and KDA scores, while individuals playing without friends have decreasing actions and performance. Such results suggest that, except for experience, the presence of social ties in a team can also help players mitigate performance deterioration over the short term throughout a gaming session.
\section{Related Work}\label{sec:related}
\subsection{Social ties and performance}
Our research is solely based on objective measurements of behavioral data from Dota 2 players where connections between players are studied as social ties. However, most existing social tie studies heavily rely on interviews, surveys, or ethnographic observations to collect self-reported relational data from study participants where each individuals are asked about their proximity to and friendship with others.
\cite{eagle2009inferring} collected both self-reported data and mobile phones usage behavioral data from 94 subjects. The authors revealed that self-reports are biased toward recent and more vivid events.
Some empirical studies examined the effect of friends (vs. strangers) on individual and team performance.
In the context of business psychology and organizational development management, for example, workplace ties have for long been the subject of debate.
~\cite{chung2018friends} proposed that relative to acquaintance groups, when friends work together, coordination will be improved through increased collaboration, communication, and conflict management; motivation will be increased via increased commitment, goal-setting, and goal pursuit. Their results revealed that friendship has a significant positive effect on group task performance. Although many scholars and practitioners have assumed that friendships lead to desirable organizational outcomes,~\cite{pillemer2018friends} explained the downsides associated with workplace friendships.
Researchers have also investigated the influence of social ties, especially peer-effects in academic environments, where youngsters in classrooms become ideal study targets. ~\cite{wentzel1997friendships} reached a conclusion that aspects of peer relationships are related to GPA indirectly, by way of significant relations with pro-social behavior i.e., the behavior of helping other children learn. Since their study,~\cite{zimmerman2003peer} found that students in the middle of the SAT distribution may have somewhat worse grades if they share a room with a student who is in the bottom 15 percent of the verbal SAT distribution. \cite{ding2007peers} investigated that high-ability students benefit more from having higher-achieving schoolmates and from having less variation in peer quality than students of lower ability. \cite{burke2013classroom} found that low ability students benefit about twice as much from an increase in the share of top-quality peers as they do from an increase in the share of low ability peers. Middle students will benefit from mixing with top-quality peers as well.
Our paper focuses on analyzing the effect of social ties on the performance of online games. ~\cite{leenders2016once} pointed out that while much progress has been made in detailing different types of team processes, empirical evidence of their predictive validity is generally underwhelming and they pointed to the need for a more specific temporally rich theoretical formulation of process. While identifying these critical rarely analyzed aspects, in this paper we used gaming sessions to capture behavior patterns considering temporal limitations.
\subsection{Social ties in games and virtual teams}
Teams science is essential to organizations, informal groups and individuals \cite{wuchty2007increasing, guimera2005team, borner2010multi, contractor2013some, Mukherjee2018}. Considerable attention has been paid to teams across a range of interdisciplinary challenges. However, the factors affecting team performance in complex, realistic task environments remain yet scarcely understood, both in theory and in practice. In this paper, we focus on individual and team engagement in online games, specifically in a MOBA game called Dota 2.
Research has been conducted on a Massively Multiplayer Online Role Playing Games (MMORPG), Dragon Nest~\cite{wax2017self}, which revealed that self-assembled teams form via three mechanisms: homophily, familiarity, and proximity; the authors show that successful and unsuccessful teams were homogeneous in terms of different characteristics, but successful teams are more often formed based on friendship than those unsuccessful teams.
A recent work by Mukherjee \textit{et al.}~\cite{Mukherjee2018} further corroborated this hypothesis by analyzing both sports (football, cricket, baseball) and esports (Dota 2) suggesting that success shared in prior team experiences is an excellent predictor of future team success. This would imply that social ties and prior experience between team members play a more important role in successful team dynamics than the so-called ``superstar effect''---the idea that well-constructed teams can be outperformed by poorly-constructed teams that have a superstar player whose skills are far better than everyone else.
Other research investigated \textit{social-network games}. A social-network game is a type of online game that is played through social networks. They typically feature multiplayer gameplay mechanics. Social-network games were originally implemented as browser games. As mobile gaming took off, the games moved to mobile as well.
However, we found a significant impact of social online gaming frequency on the probability of meeting exclusively online friends. Different social motives played an important role for modality switching processes. Players with a pronounced motive to gain social capital and to play in a team had the highest probability to transform their social relations from online to offline context. We found that social online gamers are well integrated that they use the game to spend time with old friends as well as to recruit new ones.
\subsection{Performance deterioration dynamics}
A recent research thread is concerned with quantifying the temporal dynamics of performance in techno-social systems.
Short-term deterioration of individual performance was previously observed in real world (offline) tasks. Recent studies investigate this phenomenon by drawing a parallel with online platforms: research shows that the quality of comments posted by users on Reddit~\cite{singer2016evidence}, the answers provided on StackExchange question-answering forums~\cite{ferrara2017dynamics}, and the messages written on Twitter~\cite{kooti2016} and Facebook~\cite{kooti2017understanding} decline over the course of an activity session. Other than individual online behaviors, short-term deterioration effects has also been found in virtual teams in MOBA games. ~\cite{sapienza2017performance, sapienza2018non, sapienza2018individual}.
These results pose the basis for RQ3 and RQ4: our analysis revealed that social ties can play a role in mitigating performance deterioration over the course of an activity session, however such a mitigation is not homogeneous across all individuals, but tend to benefit more the low-skilled or low-experienced ones, while high-skill or high-experience individuals may not be affected by social ties when it comes to mitigating performance deterioration.
\section{Conclusions}\label{sec:conclusions}
In this paper, we investigated four research questions. The first two questions
focused on measuring effects within single matches: What is the influence of social ties on \textit{(RQ1)} individual players', and \textit{(RQ2)} on team's activity?.
The other two questions focused on effects that span over the course of a gaming session (i.e., a nearly-uninterrupted sequence of consecutive matches):
What is the influence of social ties on \textit{(RQ3)} individuals and \textit{(RQ4)} teams, over the course of entire gaming sessions?.
We set ourselves to test whether the presence of social ties affected the activity of individuals within a team (RQ1), and of the team as a whole (RQ2). We also investigated whether there exists a spillover effect by which individuals who do not play with friends, yet play in a team where some players are friends among each other, may indirectly benefit of the presence of social ties in their team. We further investigated teams composed by high/low experienced players, and high/low performing players.
Our research revealed that individuals playing with friends have higher kills and assists frequency than players gaming without friends. Moreover, death differences suggest that players with teammates friends may have more aggressive or impulsive behaviors than those without preexisting friendships.
Moreover, kills, deaths, and assists actions are found to increase for all four categories of teams. However, for teams with preexisting ties and consisting of highly-skilled players, KDA performance dropped compared to teams with no friends. Conversely, for teams with preexisting ties but consisting of lowly-skilled players, all actions and KDA performance improve drastically compared to teams with no friends. As a summary, low-performance players (regardless of their experience) benefit the most, while high-skill players exhibit significant performance decreases, by the presence of team social ties. In other words, playing with friends benefits almost exclusively low-performance players and negatively affects high-skill players.
We also studied whether playing gaming sessions within teams with preexisting social ties affects individuals' (RQ3) and teams' (RQ4) performance over the sessions course.
For players who only play with friends throughout a gaming session, there exists evident performance deterioration, in line with recent results~\cite{sapienza2017performance, sapienza2018non, sapienza2018individual}. However, for players that only play with strangers in a game session, results are not fully conclusive that such performance deterioration occurs.
As time goes by, teams under with social ties have gradually increasing actions and performance while teams of strangers have decreasing actions and performance. The results suggest that, except for experience, social relationships within teams can help players mitigate performance deterioration.
Analogous analyses along the lines of what we proposed in this paper applied to other techno-social systems only requires data that capture who interacts with whom at what point in time (or in what order). Preferably (yet not necessarily), researchers would also collect some performance/outcome data, to test whether certain interaction patterns are associated with differential levels of performance (of groups, of the individuals in the groups, or of systems of groups). Since there is often no sound theoretical argument as to exactly when or for how long an outcome is expected to occur, ideal data would include sufficiently long longitudinal observations.
We plan to carry out some of such studies ourselves in the future, targeting other types of games, virtual teams in online and offline task-specific settings, virtual reality individual- and team-based settings, and more in general both competitive and collaborative endeavors, to study how social networks dynamics may affect human behavior and performance.
\section*{Acknowledgements}
The authors are grateful to DARPA for support (grant \#D16AP00115). This project does not necessarily reflect the position/policy of the Government; no official endorsement should be inferred. Approved for public release; unlimited distribution.
\section*{Author contributions}
All authors designed the research project. YZ collected the data. YZ and AS performed the experiments. All authors analyzed the results, wrote and reviewed the manuscript.
\bibliographystyle{IEEEtran}
|
1,116,691,497,601 | arxiv | \section{Introduction}
Machine learning models are increasingly deployed for critical decision-making tasks (e.g., Bail decision\cite{angwin2016machine} ). It is important to verify that they do not contain gender or racial biases picked up from training data.
Typical approaches to achieve fairness revolve around efforts to curate training data, with post-hoc fairness evaluation on testing data (which is often proprietary).
In contrast, we propose techniques to \emph{prove} fairness using formal methods that verify properties of neural network models.
Beyond the strength of guarantee implied by a formal proof, our methods have the advantage that we do not need explicit training or testing data to analyze a given trained model.
Formal approaches to neural network verification have been developed over the past few years~\cite{liu2019algorithms,tran2020verification,katz2017reluplex,gehr2018ai2} that can in certain cases prove properties over all possible executions of a specific network.
Existing work has focused on networks used in safety-critical control algorithms and robotics~\cite{ivanov2019verisig,xiang2020reachable}, or guaranteed robustness for visual perception networks~\cite{tran2019safety}.
In this paper, we apply such methods to the task of validating fairness properties of classification and regression models.
We systematically explore classes of fairness specifications that can be evaluated statistically as well as using our developed formal approaches.
We consider using formal methods to analyze fairness for \emph{classification tasks with explicit input labels}. Here one of the input fields for each record explicitly encodes a sensitive property, such as race, gender, or age.
Informally, fairness dictates that a given model $M$ should perform ``the same'' over all possible values of these sensitive fields.
Note that excluding such inputs when training a model does not solve the fairness problem, as sensitive properties are often inferable from other input variables (e.g. the zip code of an African-American neighborhood).
\begin{figure*}[t]
\centering
\includegraphics[width=1.0\columnwidth]{figs/overview.png}
\caption{In our verification approach, each partition $\Theta$ of the network input space defines a domain for integration over a given input probability density function $P$, calculating a probability of the input space with a fixed output classification.
%
By repeating and summing over all partitions, we can evaluate fairness metrics for the network.}
\vspace{-3mm}
\label{fig:input_split}
\end{figure*}
Formal analysis approaches are most often applied to fully connected networks with ReLU activation functions~\cite{vnncomp2021}.
For such networks, a binary decision model partitions the space of possible inputs into a collection of geometric polytopes, such that all points within a particular polytope are labeled homogeneously as all positive or all negative.
The fairness properties which we are concerned with revolve around showing that these geometric regions are comparable for two distinct labels or groups, as illustrated in Figure~\ref{fig:input_split}.
There are several possible provable properties depending upon how we define that $M$ performs “the same” for different groups, including:
\begin{itemize}
\item {\em Volumetric symmetric difference} ---
Here, we say that $M$ is provably fair if the volume of the non-overlapping acceptance regions for the protected classes is below a certain threshold.
\item {\em Probability-weighted symmetric difference} ---
Here we say that $M$ is provably fair with respect to input distributions $A$ and $B$ if the percentage of each population which would change classified label by changing label of the protected class is below a certain threshold.
\item{\em Net Preference} --- Here we say that $M$ is provably fair with respect to input distributions $A$ and $B$ if there is not a large difference in acceptance probability if the criteria used to accept the population from distribution $A$ is used evaluated on the population from distribution $B$.
\end{itemize}
Our major contributions in this paper are:
\begin{itemize}
\item {\em Provable Fairness Guarantees \textbf{without} Training or Evaluation Data} -- We define a class of fairness guarantees which can be formally computed on neural classification models for real tasks.
%
A particularly exciting aspect of our approach is that verification requires only black box access to the trained model. Previous approaches to fairness evaluation require access to
training or evaluation data, usually deemed confidential and proprietary for consumer product models. One of the metrics we introduce require no training and evaluation data, nor knowledge of the distribution of the class features. The other two metrics we introduce only require knowledge of the probability distribution of the class features in the applied setting, which may be estimated either from the training and evaluation data or using external empirical studies demographic in nature.
\item {\em Addressing Scaling Issues} -- Verifying the properties of neural networks requires sophisticated geometric algorithms in high-dimensional spaces.
The number and complexity of polytopes defining our analysis grow exponentially with the size of the model, and its parameter space.
%
Still, the state-of-the-art in formal verification has advanced steadily despite theoretical intractability~\cite{katz2017reluplex}.
We identify the bottleneck in our approach as the volume computation step and propose alternate algorithms for volume integration that can further push the scalability limits of our approach.
\end{itemize}
\xingzhiswallow{
This paper is organized as follows.
Section \ref{sec:prior-work} reviews previous work in formal verification techniques for neural networks, and also in the field of algorithmic fairness.
We discuss our formal verification methods in Section \ref{sec:methods}.
Our experimental results demonstrating the trade off between model quality and quantified fairness are presented in Section \ref{sec:experimental-results}.
Finally, we do a critical performance analysis on the scalability of our methods in Section \ref{sec:scalability}, pointing to future research directions in the verification of model fairness.
Our source code and datasets will be released upon publication, and included in this submission at \textcolor{blue}{\url{https://bit.ly/3MC64Ke}}
}
\section{Related Work}
\label{sec:prior-work}
\paragraph{Formal verification methods for neural networks}: The formal verification methods~\cite{xiang2018verification,liu2019algorithms,albarghouthi-book} perform set-based analysis of the network, rather than executing a network for individual input samples.
Given a neural network $f_{NN}: \mathbb{R}^n \rightarrow \mathbb{R}^m$, an input set $X \subseteq \mathbb{R}^{n}$, and an unsafe set of outputs $U \subseteq \mathbb{R}^{m}$, the \emph{open-loop NN verification problem} is to prove that for all allowed inputs $x \in X$, the network output is never in the unsafe set, $f_{NN}(x) \notin U$.
One way to solve the verification problem is to compute the range of the neural network function for a specific input domain $X$, and then check if the range intersects the unsafe states $U$~\cite{bak2020cav, vincent2021reachable}.
In this work, we reuse this range computation approach in order to evaluate how probability distributions propagate through the networks.
Technically, two core operations are performed in our work: (1) computing the range of the neural network as a union of polytopes, and (2) computing the volume of the output polytopes. Differing from the formal verification methods used for safety and motion planning, we explore the possibility to apply it for neural network fairness measure.
\paragraph{Fairness in neural networks}: Machine learning fairness has also received increasing attention recently~\cite{mehrabi2021survey,caton2020fairness,bellamy2018ai}.
Existing work on fairness typically uses the input data and samples the network to provide estimates of fairness~\cite{bastani2019probabilistic}. However, such testing samples may not cover whole population (e.g., biased testing samples), and thus cannot guarantee the model fairness.
In contrast, our techniques use external distributions over input data and perform set-based range computation of neural networks, without the need for any testing data for fairness evaluation.
A limited amount of existing research exists on provable fairness.
One framework focuses on proving \emph{dependency
fairness}~\cite{urban2020perfectly,galhotra2017fairness}, which strives to prove the outputs are not affected by certain input features.
This method is based on forward and a backward static analysis as well as input feature partitioning.
Another recent approach~\cite{ruoss2020learning} focuses on \emph{individual fairness}, which essentially means that similar individuals get similar treatments.
In contrast, our provable fairness metrics are defined over geometric properties of the network outputs, such as the symmetric difference between the ranges with different values of sensitive inputs.
\section{Methods: Provable Fairness using Formal Verification}
\label{sec:methods}
\subsection{Acceptance Region}
We evaluate the fairness of the models by looking carefully at the exact acceptance regions of the models.
Let $f(\vec{x})$ be the label assigned by the neural network for the value $\vec{x}$ in the input space. The acceptance region for class C, which we denote as $\rho(C)$ is the region in the input space for which $\vec{x}\in C\land f(\vec{x})=\ell$. The label $\ell$ that defines the acceptance region is user-definable based on the properties of the neural network they are interested in exploring. We denote the acceptance region of a class $C$ as $\rho(C)$.
\subsection{Proposed Metric}
\label{sec:metric}
We next introduce the metrics that we use to evaluate the fairness of our models. These metrics serve as quantifications of the legal notions of disparate treatment and disparate impact \cite{legaldef}.
The weighted symmetric difference (WSD) quantifies the total difference in the model’s behaviour towards different classes. The WSD quantifies disparate treatment.
It is weighted in order to discount differences in behaviour for unrealistic sets of features that are improbable to ever appear in the input. Formally, the metric is defined as:
\begin{align*}
\WSD(C_1, C_2) &= \Advantage(C_1, C_2) + \Advantage(C_2, C_1) \\
\Advantage(C_1, C_2) &= \int_{\rho(C_1)} P(\vec{X}|C_1)d\vec{X} - \int_{\rho(C_1)\cap\rho(C_2)} P(\vec{X}|C_2)d\vec{X} \\
\end{align*}
The $\Advantage(C_1, C_2)$ metric quantifies how many examples from class $C_1$ would have not been in the acceptance region of the model if they were of class $C_2$, with all other features held equal. It is a special case of the discrimination score~\cite{discscore}.
The volumetric symmetric difference (VSD) also quantifies the total difference in the model’s behaviour towards different classes. It may be used when information about the input feature distribution for the two classes is unavailable. The VSD also quantifies disparate treatment.
\begin{align*}
\VSD(C_1, C_2) &= |\rho(C_1) - \rho(C_2)| + |\rho(C_2) - \rho(C_1)| \\
|\rho(C_1) - \rho(C_2)| &= \int_{\rho(C_1)} 1\cdot d\vec{X} - \int_{\rho(C_1)\cap\rho(C_2)} 1\cdot d\vec{X} \\
\end{align*}
Finally, the net preference (NP) quantifies how much the model prefers the features of one class over the other when assigning a label. The NP quantifies disparate impact. It is useful for investigating whether the model makes its decisions on variables strongly correlated with the protected class. When evaluating the preference, the acceptance region of $C_1$ is used in order to discount possible differences in the acceptance regions, which can be investigated using the $\WSD$ and the $\VSD$. Net preference is similar to that disparate impact metric given by Feldman et. al\cite{feldman}, with one key advantage: by taking into account the possible difference in acceptance regions between the two classes, NP focuses on capturing disparity that arises from criteria that a priori might seem agnostic to the sensitive class. Feldman et al's metric captures disparities that arise both from disparate treatment and from disparate impact, making it difficult to further investigate the source of the problem.
\begin{align*}
\NP(C_1, C_2) &= \max\{|\Preference(C_1, C_2)|, |\Preference(C_2, C_1)|\} \\
\Preference(C_1, C_2) &= \int_{\rho(C_1)} P(\vec{X}|C_1)d\vec{X} - \int_{\rho(C_2)} P(\vec{X}|C_2)d\vec{X} \\
\end{align*}
The preference function here quantifies the difference in the acceptance probability of the two classes when evaluated by the acceptance criteria of the first class.
\begin{comment}
\subsection{Definitions of Model Fairness}
The definition of fairness we strive to prove is based on calculated probability distributions for different groups.
We will consider the following types of fairness challenges that can be verified using formal methods:
\begin{itemize}
\item \textbf{Classification with explicit input labels.} Here one of the input fields for each record explicitly encodes a sensitive property, such as race, gender, or age. Fairness dictates that a given model M performs “the same’’ for all possible values of these sensitive fields.
\item \textbf{Classification with differential input distributions.} Here sensitive information is not explicitly passed to the model but may be inferable from other input variables (e.g. the zip code of an African-American neighborhood). In addition to a given model M we are given input probability distributions for two or more distinct groups. Fairness dictates that M performs “the same” over records drawn from these distinct input distributions.
\end{itemize}
Every binary decision model partitions the space of possible inputs into a collection of geometric polytopes, such that all points within any particular polytope $\mathcal{D}$ are labeled homogeneously as all positive or all negative.
The fairness properties which we are concerned with revolve around showing that these geometric regions are comparable for two distinct labels or groups. There are several possible provable properties depending upon how we define that M performs “the same” for different groups, including:
\begin{itemize}
\item \textbf{Volumetric identity.} Here we say that M is provably fair when the positive regions are identical for different groups A and B, ignoring the input variable explicitly encoding the identity of group A and B.
Volumetric magnitude – Here we say M is provably fair when the volume of the positive regions are comparable for different groups A and B. The symmetric difference of the positive regions of A and B may be greater than zero, but the difference of the size of the input volumes is provably bounded.
\item \textbf{Probability-weighted magnitude.} Here we say that M is provably fair with respect to input distributions A and B when the weighted volumes of the positive regions are the same for A and B, or that the difference of these weighted volumes is provably bounded. The weighted volume of polytope $\mathcal{D}$ is defined by the integral of the probability of each point in $\mathcal{D}$.
\end{itemize}
\end{comment}
\subsection{Verification Approach}
Our technical approach to verification is based on set-based execution of neural networks, extended to include probability distributions.
Given a set of possible inputs, we propagate the set through the network to see the range of the network, the possible outputs.
In our approach, we use the linear star set representation~\cite{duggirala2016parsimonious,tran2020cav} to propagate sets of states through the network.
After analysis, the entire input set is partitioned into a collection of polytopes that map to outputs with an identical label.
\begin{comment}
We plan to extend existing set propagation methods with star sets to work with layers in LSTM networks, and then using the result to reason over how probability distributions can be propagated through the network.
\end{comment}
This representation of sets of inputs allows for defining a simple procedure for verifying the fairness of a neural network. There is an assumption that it is possible to encode $C_1$ and $C_2$ directly into the input features by fixing one or more of the features in the input to the appropriate values. The procedure is as follows:
\begin{enumerate}
\small
\item Use neural-network reachability analysis in order to calculate the acceptance regions for each of the classes: $\rho(C1)$ and $\rho(C2)$, then calculate $\rho(C_1)\cap \rho(C_2)$.
\item Calculate $\WSD(C_1, C_2)$ using the probability distributions $P(\vec{X}|C1)$ and $P(\vec{X}|C_2)$ over the data.
\end{enumerate}
The calculation of $\WSD(C_1, C_2)$ requires a probability distribution over the data.
For our experiments, we estimate the probability distribution from the input data, although external demographic data can be used to create these distributions if input data is not available.
The details of this method are in the appendix.
The neural network reachability analysis returns two lists of polytopes, encoding the acceptance region of each class: $\rho(C_1)=\{A_1, A_2, A_3, \ldots A_N\}$ and $\rho(C_2) = \{B_1, B_2, B_3,\ldots B_M\}$.
Calculating the intersection between the two regions is thus a matter of calculating the non-empty intersection between all pairs of polytopes in set induced by the
Cartesian product $\rho(C_1) \times \rho(C_2)$.
As the polytopes are
$\mathcal{H}$-polytopes, they are
specified as a list of constraints $C\alpha\leq d$.
The constraints of two polytopes are simply concatenated to create the intersection of the two polytopes.
Linear programming can be used to determine if the intersection is empty.
\subsection{Integration over the Probability Distribution}
The final step for calculating the $\WSD(C_1, C_2)$ for a model requires the integration the probability distributions $P(\vec{X}|C_1)$ and $P(\vec{X}|C_2)$ over a union of polytopes. In our experiments, direct integration of the probability distribution over the polytope proved too slow once more then two dimensions needed to be integrated over. However, determining the volume of the polytope has a better time complexity then integrating over the volume of the polytope.
Although we use \textsc{QHull} to find the volume due to its accessibility, algorithms that are linear in the number of vertices exist. We take advantage of this fact, computing the probability by discretizing the input space into a evenly-spaced grid. We calculate the intersection of each polytope with each region of the grid. We find the volume of this intersection and multiply it by the probability density at that point in the grid. This gives us a large speed-up in runtime for only a small cost in precision.
\begin{comment}
Given a subset of the input space represented as a polytope in the domain $\mathcal{D}$ with a corresponding single output label, we can compute the probability density corresponding to that label integrating over the input domain, $\int_\mathcal{D} P(x) ~ \diff x$, where the probability density function $P(x)$ is defined over the entire input space.
Repeating this over the entire input domain (all the star set partitions), we can compute the probability of each output over the entire domain of possible inputs.
The process is illustrated in Figure~\ref{fig:input_split} on a hypothetical 2-d input distribution, where a single partition over the input space given as a polytope $\mathcal{D}$ is used to compute a probability of a specific output.
\end{comment}
\section{Experimental Results}
\label{sec:experimental-results}
\xingzhiswallow{
To investigate the fairness of the neural networks, we train multiple models on three datasets that are commonly used in model fairness studies, and measure their fairness with our method according to the metrics in Section~\ref{sec:metric}. Note that our measurement does not require expensive data sampling or availability of testing data, making it different
compared to other works as described in Section~\ref{sec:prior-work}.
Moreover, we also study two fairness-oriented data-augmentation strategies, effectively producing more \textit{fair} models at little expense of predictive power.
In the following subsections, we present the statistics of the datasets and models, report both the performance and the fairness of the models trained by various data-augmentation strategies, including non-/fairness-enhanced.
}
We train various models on three datasets that are commonly used in model fairness studies, and measure their fairness with our proposed metrics described in Section~\ref{sec:metric}. Note that our measurement does not require expensive data sampling or availability of testing data, making it different compared to other works as described in Section~\ref{sec:prior-work}.
\subsection{Dataset and Model Training }
We selecte three common datasets (COMPAS \footnote{Correctional Offender Management Profiling for Alternative Sanctions: \url{https://www.propublica.org/article/how-we-analyzed-the-compas-recidivism-algorithm/}},
ADULTS \cite{Dua:2019},
HEALTH \footnote{\url{https://www.kaggle.com/c/hhp}})
that contains both categorical and continuous variables as the input features.
All the datasets contains at least one protected feature (e.g., Race or Sex), which we consider as a potential factor that incurs unfair inductive bias during model training.
In the following, we describe four training strategies
\footnote{The details of strategies 3 and 4 can be found in Algorithm \ref{algo:fair-data-aug} in Appendix.}
that increasingly alleviate such unfairness, demonstrating that our proposed metrics well capture the improvement in fairness.
We present the details in Table \ref{tab:res-adults} and Table \ref{tab:dataset-stats} in Appendix due to space limitations
\xingzhiswallow{
\begin{table*}[ht]
\centering
\label{tab:dataset-stats}
\caption{The statistics of dataset. For each dataset we split data into train/val/test set with the ratio 70\%/15\%/15\%, respectively. For each continuous feature, we applied min-max normalization. For simplification, we use binary label for the protected feature: Race (White v.s. African American) and Sex (Male v.s. Female).}
\begin{tabular}{p{0.18\textwidth} p{0.10\textwidth} p{0.1\textwidth} p{0.15\textwidth}}
\toprule
\textbf{Dataset} & \textbf{COMPAS} & \textbf{ADULTS} & \textbf{HEALTH} \\
\midrule
\#Samples & 2,363 & 43,031 & 124,086 \\
\#Categorical Feat. & 2 & 5 & 11 \\
\#Continuous Feat. & 2 & 3 & 11 \\
Protected Feat. & Race & Race & Sex \\
\#Classes & Binary & Binary & Binary \\
\bottomrule
\end{tabular}
\end{table*}
}
\begin{itemize}
\item[1.] {\em Baseline}: Use the original data to train the model, without any augmentation or filtering.
\item[2.] {\em Protected Feature Permuted}: Randomly shuffle the protected feature (Race or Sex) in training data, then train the model once. The approach breaks the correlation between the feature and the label, for example, making prediction class-agnostic.
\item[3.] {\em Data-Removed}: Re-train the model with the \textbf{removed} data, iteratively deleting the training data which may cross the decision boundary (threshold 0.5) when the protected feature is flipped. This is a simplified version of the technique presented in \cite{dataremoval}, without employing the ranking algorithm.
\item[4.] {\em Data-Augmented}: Re-train the model with the \textbf{augmented} data, iteratively creating training data which may cross the decision boundary (threshold 0.5) when the protected feature is flipped. The max number of training data is limited to three times of the original data size. This is a combination of techniques applied in~\cite{dataaug1} and~\cite{dataaug2}.
\end{itemize}
\subsection{Results}
\begin{comment}
Present tables showing the fairness and model quality for each challenge and training choice \todo{XZ: 1. statistical/empirical fairness (on testing set); 2.add curve of acc/fairness}.
Also discuss the run time of the analytical method -- and even better a runtime performance comparsion of multiple methods of volume computation.
\end{comment}
\xingzhiswallow{
There are several key results. The first is that without any measures to ensure fairness during the training process, the trained models demonstrate a significant degree of unfairness. Inspecting the WSD of various models trained on the COMPAS dataset in Table~\ref{tab:res-compas}, there is a significant number of individuals for whom the model decision would have switched had only their race been different. This demonstrates the importance of taking measures to ensure model fairness, especially in domains that have historically been race-sensitive.
}
\paragraph{The baseline models are unfair without fairness training}: Inspecting the WSD of various models trained on the COMPAS dataset in Figure~\ref{fig:auc-vs-fairness-compas}, there are a significant number of individuals for whom the model decision would have switched had only their race been different. Figure \ref{fig:auc-vs-fairness-adults-long} and Table \ref{tab:res-adults} in Appendix show the consistent pattern in all trials on two datasets.
\begin{figure}[ht]
\centering
\includegraphics[width=0.9\textwidth]{figs/fig-compas-fairness-vs-auc-scatter-diff.pdf}
\vspace{-3mm}
\caption{ COMPAS Dataset: Model performance/AUC vs Model fairness/NP under three fairness-sensitive training methods. Arrows down and to the left reflect models that are fairer but less accurate than the original mode. We obtain large improvements in fairness at little cost in accuracy. }
\vspace{-3mm}
\label{fig:auc-vs-fairness-compas}
\end{figure}
\xingzhiswallow{
The WSD and NP of various models on the ADULTS dataset in Table~\ref{tab:res-adults} demonstrate another important result. In the presence of many features on which to base its decision, a model might have small WSD, indicating that there are few individuals for which had their race been different, the model decision would have been different. However, a well-known problem is the ability to use a \textit{proxy variable}, such as ZIP code and other correlated variables, in order to determine the race of an individual. This can be seen in the fact that the WSD is small but the NP is large for the original models in Table~\ref{tab:res-adults}, indicating that the model has a strong preference for features correlated with a certain race.
}
\xingzhiswallow{
Comparisons of the $\WSD$ and the $\VSD$ show that when the probability distribution on the input features is not available, the $\VSD$ is a suitable proxy for the $\WSD$. Reductions in the $\WSD$ are accompanied by similar reductions in the $\VSD$. In 11 out of the 16 trials, the model with the smallest $\WSD$ was also the model with the smallest $\VSD$.
}
\paragraph{$\VSD$ is a suitable proxy for the $\WSD$ when the probability distribution on the input features is not available}: Reductions in the $\WSD$ are accompanied by similar reductions in the $\VSD$. In 11 out of the 16 trials, the model with the smallest $\WSD$ was also the model with the smallest $\VSD$.
\xingzhiswallow{
Finally, our results demonstrate the effectiveness of our data-augmentation method in order to improve fairness. The data-augmentation method leads to a significant improvement in the fairness of the models, with only a small drop in model accuracy (and sometimes an improvement). The data-augmentation method is often able to significantly outperform or achieve parity with the baseline of permuting the race in many cases for the WSD metric. However, unlike the race-permutation method, the method is also effective in greatly reducing the NP. Thus, the models rely much less on proxy variables for race.
}
\paragraph{The trade-off between fairness and accuracy}: The data-augmentation method leads to a significant improvement in the fairness of the models, with only a small drop in model accuracy (and sometimes an improvement). The data-augmentation method is often able to significantly outperform or achieve parity with the baseline of permuting the race in many cases for the WSD metric. However, unlike the race-permutation method, the method is also effective in greatly reducing the NP. The data-augmented models rely much less on proxy variables for race.
\swallow{
\begin{table*}[ht]
\centering
\caption{For COMPAS dataset: Model performance/AUC vs Model fairness/NP under three fairness-sensitive training methods. Arrows down and to the left reflect models that are fairer but less accurate than the original mode. We obtain large improvements in fairness at little cost in accuracy.Model performance(\textit{AUC}) \textit{v.s} Model Fairness (\textit{Weighted Symmetric Difference}) over 5 runs for the small/medium models on COMPAS dataset.}
\small
\begin{tabular}{lllrrll}
\toprule
\multirow{2}{4em}{Model Size} & \multirow{2}{3em}{Trial ID} & \multirow{2}{4em}{Measure} & \multirow{2}{4em}{Original} & \multirow{2}{4em}{Race Permuted} & \multirow{2}{4em}{Data Removed} & \multirow{2}{4em}{Data Augmented} \\
\\
\midrule
Small & 0 & AUC & \textbf{0.8169} & 0.8127 & 0.8143 & 0.8130 \\
& & WSD & 0.1663 & 0.0457 & 0.0687 & \textbf{0.0303}\\
& & VSD & 0.1248 & \textbf{0.0193} & 0.0420 & 0.0212 \\
& & NP & 0.3919 & 0.1943 & 0.2480 & \textbf{0.1118}\\
& 1 & AUC & \textbf{0.7928} & 0.7830 & 0.7892 & 0.7925 \\
& & WSD & 0.1340 & \textbf{0.0058} & 0.0956 & 0.0059 \\
& & VSD & 0.0648 & \textbf{0.0028} & 0.0342 & 0.0047 \\
& & NP & 0.3598 & 0.2232 & 0.3197 & \textbf{0.1629}\\
& 2 & AUC & \textbf{0.8019} & 0.7935 & 0.8018 & 0.7937 \\
& & WSD & 0.1330 & \textbf{0.0188} & 0.1052 & 0.0310 \\
& & VSD & 0.0524 & \textbf{0.0093} & 0.0517 & 0.0284 \\
& & NP & 0.3617 & 0.2082 & 0.3232 & \textbf{0.0689}\\
& 3 & AUC & \textbf{0.8307} & 0.8298 & 0.8280 & 0.8273 \\
& & WSD & 0.1845 & 0.0450 & 0.1263 & \textbf{0.0246}\\
& & VSD & 0.1129 & 0.0264 & 0.0432 & \textbf{0.0219}\\
& & NP & 0.4093 & 0.2374 & 0.2891 & \textbf{0.0134}\\
& Average & AUC & \textbf{0.8106} & 0.8047 & 0.8083 & 0.8066 \\
& & WSD & 0.1545 & 0.0288 & 0.0990 & \textbf{0.0230}\\
& & VSD & 0.0887 & \textbf{0.0144} & 0.0428 & 0.0191 \\
& & NP & 0.3807 & 0.2158 & 0.0990 & \textbf{0.0230}\\
Medium & 0 & AUC & \textbf{0.8125} & 0.8072 & 0.8125 & 0.8005 \\
& & WSD & 0.2021 & \textbf{0.0228} & 0.1659 & 0.1093 \\
& & VSD & 0.1399 & \textbf{0.0138} & 0.0852 & 0.0282 \\
& & NP & 0.4361 & 0.2050 & 0.3823 & \textbf{0.0582}\\
& 1 & AUC & 0.7888 & 0.7824 & \textbf{0.7938} & 0.7893 \\
& & WSD & 0.1196 & 0.0203 & 0.0991 & \textbf{0.0064}\\
& & VSD & 0.0552 & 0.0083 & 0.0524 & \textbf{0.0044}\\
& & NP & 0.3467 & 0.2055 & 0.3247 & \textbf{0.1249}\\
& 2 & AUC & \textbf{0.8002} & 0.7923 & 0.7987 & 0.7970 \\
& & WSD & 0.1147 & \textbf{0.0172} & 0.0878 & 0.0651 \\
& & VSD & 0.0452 & \textbf{0.0046} & 0.0317 & 0.0257 \\
& & NP & 0.3443 & 0.2140 & 0.2988 & \textbf{0.1111}\\
& 3 & AUC & 0.8205 & 0.8257 & 0.8251 & \textbf{0.8344}\\
& & WSD & 0.2641 & \textbf{0.0095} & 0.0935 & 0.0161 \\
& & VSD & 0.1139 & \textbf{0.0050} & 0.0486 & 0.0073 \\
& & NP & 0.4549 & 0.2168 & 0.3064 & \textbf{0.0966}\\
& Average & AUC & 0.8055 & 0.8019 & \textbf{0.8075} & 0.8053 \\
& & WSD & 0.1751 & \textbf{0.0175} & 0.1116 & 0.0492 \\
& & VSD & 0.0885 & \textbf{0.0079} & 0.0545 & 0.0164 \\
& & NP & 0.3955 & 0.2103 & 0.1116 & \textbf{0.0492}\\
\bottomrule
\end{tabular}
\label{tab:res-compas}
\end{table*}
}
\swallow{
\begin{table*}[ht]
\centering
\caption{For ADULTS dataset: Model performance(\textit{AUC}) \textit{v.s} Model Fairness (\textit{Weighted Symmetric Difference}) over 4 runs for the small/medium models on Adult Income dataset. }
\small
\begin{tabular}{lllrrll}
\toprule
\multirow{2}{4em}{Model Size} & \multirow{2}{3em}{Trial ID} & \multirow{2}{4em}{Measure} & \multirow{2}{4em}{Original} & \multirow{2}{4em}{Race Permuted} & \multirow{2}{4em}{Data Removed} & \multirow{2}{4em}{Data Augmented} \\
\\
\midrule
Small & 0 & AUC & \textbf{0.8373} & 0.8341 & 0.8371 & 0.8354 \\
& & WSD & 0.0225 & 0.0118 & 0.0222 & \textbf{0.0034}\\
& & VSD & 0.2042 & \textbf{0.0846} & 0.2042 & 0.1041 \\
& & NP & 0.1155 & 0.0945 & 0.1155 & \textbf{0.0168}\\
& 1 & AUC & 0.8319 & 0.8341 & \textbf{0.8349} & 0.8248 \\
& & WSD & 0.0194 & 0.0285 & 0.0118 & \textbf{0.0100}\\
& & VSD & 0.2661 & 0.2338 & \textbf{0.1007} & 0.1533 \\
& & NP & 0.1030 & 0.1321 & 0.0728 & \textbf{0.0412}\\
& 2 & AUC & \textbf{0.8409} & 0.8366 & 0.8354 & 0.8382 \\
& & WSD & 0.0480 & 0.0238 & 0.0327 & \textbf{0.0151}\\
& & VSD & 0.2564 & \textbf{0.1308} & 0.1958 & 0.1434 \\
& & NP & 0.1553 & 0.0923 & 0.1020 & \textbf{0.0530}\\
& 3 & AUC & 0.8325 & 0.8302 & \textbf{0.8336} & 0.8303 \\
& & WSD & 0.0650 & 0.0283 & 0.0372 & \textbf{0.0058}\\
& & VSD & 0.4439 & 0.1769 & 0.2569 & \textbf{0.0857}\\
& & NP & 0.2052 & 0.1677 & 0.1057 & \textbf{0.0104}\\
& Average & AUC & \textbf{0.8357} & 0.8337 & 0.8353 & 0.8322 \\
& & WSD & 0.0387 & 0.0231 & 0.0260 & \textbf{0.0086}\\
& & VSD & 0.2927 & 0.1565 & 0.1894 & \textbf{0.1216}\\
& & NP & 0.1448 & 0.1217 & 0.0260 & \textbf{0.0086}\\
Medium & 0 & AUC & \textbf{0.8384} & 0.8372 & 0.8210 & 0.8384 \\
& & WSD & 0.0457 & 0.0138 & \textbf{0.0027} & 0.0457 \\
& & VSD & 0.3317 & 0.1420 & \textbf{0.0769} & 0.2913 \\
& & NP & 0.1336 & 0.0927 & 0.0194 & \textbf{0.0168}\\
& 1 & AUC & \textbf{0.8352} & 0.8320 & 0.8336 & 0.8289 \\
& & WSD & 0.0295 & 0.0230 & 0.0162 & \textbf{0.0055}\\
& & VSD & 0.1818 & 0.3919 & 0.1134 & \textbf{0.0615}\\
& & NP & 0.0624 & 0.1153 & 0.0547 & \textbf{0.0320}\\
& 2 & AUC & 0.8351 & 0.8351 & \textbf{0.8374} & 0.8180 \\
& & WSD & 0.0459 & 0.0322 & 0.0235 & \textbf{0.0104}\\
& & VSD & 0.4267 & 0.3616 & 0.2495 & \textbf{0.0954}\\
& & NP & 0.1478 & 0.0905 & 0.1076 & \textbf{0.0117}\\
& 3 & AUC & 0.8306 & \textbf{0.8324} & 0.8297 & 0.8298 \\
& & WSD & 0.1292 & 0.0076 & 0.0447 & \textbf{0.0020}\\
& & VSD & 0.5266 & \textbf{0.0560} & 0.2816 & 0.1179 \\
& & NP & 0.2723 & 0.1373 & 0.1365 & \textbf{0.0136}\\
& Average & AUC & \textbf{0.8348} & 0.8342 & 0.8304 & 0.8288 \\
& & WSD & 0.0626 & 0.0192 & 0.0218 & \textbf{0.0159}\\
& & VSD & 0.3667 & 0.2379 & 0.1804 & \textbf{0.1415}\\
& & NP & 0.1540 & 0.1089 & 0.0218 & \textbf{0.0159}\\
\bottomrule
\end{tabular}
\end{table*}
}
\xingzhiswallow{
\begin{figure*}
\centering
\includegraphics[page=1,width=0.95\textwidth]{figs/fig-compas-fairness-vs-auc-scatter-diff.pdf}
\includegraphics[page=2,width=0.95\textwidth]{figs/fig-compas-fairness-vs-auc-scatter-diff.pdf}
\includegraphics[page=3,width=0.95\textwidth]{figs/fig-compas-fairness-vs-auc-scatter-diff.pdf}
\caption{COMPAS Dataset: Model performance/AUC vs Model fairness/NP under three fairness-sensitive training methods. Arrows down and to the left reflect models that are fairer but less accurate than the original mode. We obtain large improvements in fairness at little cost in accuracy. }
\label{fig:auc-vs-fairness-compas}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[page=1,width=0.95\textwidth]{figs/fig-adults-fairness-vs-auc-scatter-diff.pdf}
\includegraphics[page=2,width=0.95\textwidth]{figs/fig-adults-fairness-vs-auc-scatter-diff.pdf}
\includegraphics[page=3,width=0.95\textwidth]{figs/fig-adults-fairness-vs-auc-scatter-diff.pdf}
\caption{ADULTS Dataset: Model performance/AUC vs Model fairness/NP under three fairness-sensitive training methods. Arrows down and to the left reflect models that are fairer but less accurate than the original mode. We obtain large improvements in fairness at little cost in accuracy. }
\label{fig:auc-vs-fairness-adults}
\end{figure*}
}
\begin{comment}
\begin{figure*}
\centering
\includegraphics[width=0.95\textwidth]{figs/fig-adults-fairness-vs-auc-scatter.pdf}
\caption{ADULTS Dataset: Model performance/AUC vs Model fairness/NP under three fairness-sensitive training methods. Arrows down and to the left reflect models that are fairer but less accurate than the original mode. We obtain large improvements in fairness at little cost in accuracy.}
\label{fig:auc-vs-fariness-adults}
\end{figure*}
\end{comment}
\begin{comment}
\begin{figure*}
\centering
\includepdf[pages={1-},width=0.95\textwidth]{figs/fig-compas-fairness-vs-auc-scatter-preference.pdf}
\caption{COMPAS Dataset: Model performance/AUC vs Model fairness/NP under three fairness-sensitive training methods. Arrows down and to the left reflect models that are fairer but less accurate than the original mode. We obtain large improvements in fairness at little cost in accuracy. }
\label{fig:auc-vs-fariness-compas-preference}
\end{figure*}
\end{comment}
\subsection{Limitations of the Current Method}
\label{sec:scalability}
\xingzhiswallow{
\paragraph{Scalability}: The health dataset was omitted due to scalability issues of the current method. Performing the reachability analysis and identifiying the union of polytopes corresponding to the acceptance region of each of the classes took a total of less than two minutes ($\sim 100$s) for the 197 models tested for the health dataset. However, attempts to integrate the probability distribution over the probability distribution in a reasonable time frame failed. The integration method requires discretizing the input space, dividing each continuous dimension into $x$ divisions. If there are $d$ continuous dimensions, this creates $x^d$ total divisions, which are iterated over. As there are $11$ continuous dimensions in the health dataset, this generates $x^{11}$ total divisions. The precision of the integration is determined by the number of divisions. For a reasonably precise answer, at least $10$ divisions are necessary. With at least $10^{11}$ divisions per polytope, this naive integration method was impractical.
Another approximation for the integral of the probability distribution over the polytope was to sample the probability distribution at the center of the polytope and scaling it by the volume of the polytope. Our method for finding the volume requires enumerating the polytope vertices. This becomes infeasible for $11$-dimensional polytopes as well.
There are alternative methods to finding the volume of the polytope that do not require
enumeration of the vertices. Emeris and Fisikopoulos~\cite{volapprox} give a polytope
volume approximation algorithm that takes polynomial time in the desired precision. As the number of vertices of a polytope can grow exponentially with the dimension, this improves the practicality of our method.
}
\paragraph{Integration}:
The procedure for discretizing the input space into a grid makes analysis infeasible for more than a small number of continuous inputs.
If there are $11$ continuous variables, and each axis is broken into $10$ bins, this yields $10^{11}$ total regions.
Thus, the number of continuous variables must be limited.
There are workarounds to this approach. A direct discretization of the polytope based on the gradient of the $P(\vec{X}|C)$ would greatly reduce the number of regions while even allowing for improvements in precision.
Additionally, efficient methods for integration of \textit{polynomials} over polytopes exist, which can be exploited if the $P(\vec{X}|C)$ takes the form of a polynomial.
\paragraph{Handling of One-Hot Features}:
The manner in which we handle one-hot features in the reachability analysis propagates every combination of the one-hot features through the neural network. This makes reachability analysis infeasible for more than a moderate number of one-hot features: for example, if there are $11$ one-hot features, each with $10$ different options, there are $10^{11}$ different combinations that must be handled.
\paragraph{Vertex Enumeration}:
Our method performs vertex enumeration by computing the convex-hull of the dual of the constraints. This is fast enough for the limited number of dimensions on which we try our method, as currently the limiting factor is the exponential growth of the size of the discretized grid. However, it still bears a significant time penalty and this method can be improved by using the Avis-Fukuda method cite{avis1991pivoting}, which runs in linear time on the number of vertices.
\paragraph{Polytope Volume Computation}:
\textsc{QHull} is used to compute the volume of the polytope due to its accessibility and robustness. However, this becomes suboptimal as the input dimensionality and the number of vertices grows. Lawrence's algorithm runs in linear time in the number of vertices, providing a large potential speed-up.
\begin{comment}
An experiment demonstrating the potential improvement in scalability based on the polytopes computed during our analysis is shown in figure (\ref{fig:volume_run_time}).
\begin{figure}[ht]
\centering
\includegraphics[width=0.95\columnwidth]{figs/runtime.png}
\caption{Comparison of \textsc{QHull} and Lawrence's Algorithm for Volume Computation (10 Constraints in 10D Space)}
\label{fig:volume_run_time}
\end{figure}
\end{comment}
\paragraph{Explicit Encoding of Classes}:
Our analysis methods assumes that the classes analyzed can be represented as polytopes in the input-space. This limits the applicability of the method to analyzing fairness of latent variables. However, if one has an estimate for the form of the latent variable, this can be approximated with a union of polytope in the input space and analyzed.
\vspace{-3mm}
\section{Conclusion}
We introduce a method to formally analyze the fairness of neural network models. Our fairness analysis methods do not require access to the model's training data, thus enabling model fairness evaluation in a broad range of contexts. We use our method to evaluate the fairness of several models, and demonstrate that models trained without intervention often demonstrate a significant degree of unfairness. We demonstrate that for circumstances in which the probability distribution on the inputs is not accessible, a probability-free approach provides a suitable substitute.
Our results show that our data-augmentation method is a simple yet effective method for improving the model's fairness. Critically, it is also effective at reducing reliance on proxy variables of the protected class.
We have explored the limits of scalability of our method: showing that there is still much room to improve runtime performance, by replacing the algorithms that enumerate polytope vertices and find the volume of the polytope from these vertices with more optimal ones.
\label{sec:conclusion}
\bibliographystyle{abbrv}
|
1,116,691,497,602 | arxiv | \section{Introduction}
\label{intro}
The goal of this study is three-fold:
\begin{enumerate}
\item Construct an algorithm to approximate a solution to a system of congruence equations pertaining to the zeta function.
\item Use the approximations in item (1) to locate regions along the line $\sigma=1$ where the real part of $\zeta(\sigma+it)$ may become negative.
\item Attempt to locate and verify instances where the real part of this function becomes negative along this line by a method other than trial and error.
\end{enumerate}
The value of $\zeta(s)$ for $s=\sigma+it$ in the half-plane $\sigma>1$ is commonly given by it's Dirichlet series
\begin{equation}
\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^s},\quad \text{Re}(s)>1
\end{equation}
or it's Euler Product form
\begin{equation}
\zeta(s)=\prod_{p}\frac{1}{1-1/p^s},\quad \text{Re}(s)>1
\label{eqn001}
\end{equation}
where $p$ is over the set of primes.
From these two expressions, little can be deduced about the range of $\zeta(s)$. However, Titchmarsh in \cite{Titchmarsh2}, provides an analysis which gives an upper bound. We first take the logarithm of both sides of \eqref{eqn001} to obtain
\begin{equation}
\log\zeta(\sigma+it)=-\sum_{n=1}^{\infty} \log\left(1-p_n^{-s}\right).
\label{eqn002}
\end{equation}
For $s=\sigma_0+it$ with $\sigma_0>1$, let:
$$U=\{u: u=\log\zeta(\sigma_0+it)=-\sum_{n=1}^{\infty} \log\left(1-p_n^{-\sigma_0} e^{-it\log(p_n)}\right)$$
and
$$V=\{v: v=\Phi(\sigma_0,\theta_1,\theta_2,\cdots)=-\sum_{n=1}^{\infty} \log\left(1-p_n^{-\sigma_0} e^{-i\theta_n\log(p_n)}\right)$$
for independent variables $(\theta_1,\theta_2,\cdots)$ with $0\leq \theta_n \leq 2\pi$.
Note first that $U\subset V$ since $t\log p_n \equiv \theta_n \text{mod}\; 2\pi$.
Using the set $V$, Titchmarsh placed approximate bounds on the range of the set $U$ as a function of $\sigma$ for $\log \zeta(s)$ in the half-plane $\sigma > 1$. These ranges are in the form of circles with centers at $1/2\log \zeta(2 \sigma)$ given by \eqref{eqnveqn}.
\begin{equation}
U = \left\{u: u\in \text{D}\left(1/2\log \zeta(2\sigma), \frac {1}{2}\log\frac {\zeta^2(\sigma)}{\zeta(2\sigma)}\right)\right\}.
\label{eqnveqn}
\end{equation}
Expression \eqref{eqnveqn} is monotonic in $\sigma$, and the larger $\sigma$ becomes, the smaller this circular region becomes. For example, when $\sigma = 2$, the radius of $U$ is approximately $0.5$. This means the values of $\log\zeta(\sigma + it)$ for $\sigma\geq 2$ are all (approximately) contained in a circular region of radius $.5$ and center at $(0.04,0)$. For all values of $\sigma > 2$, the absolute value of $\log\zeta(s)$ is smaller than $0.504 < \pi/2$. Consider $\zeta(s) = - a + i\gamma$ for positive $a$ somewhere in the half-plane $\sigma > 1$. Then $\log( - a + i\gamma) = \ln| - a + i\gamma| + i(\theta + 2n\pi)$ with $|\theta + 2n\pi| > \pi/2$. But $\pi/2>1/2$ and so $\text{Re}\,\zeta(\sigma+it)$ is never negative for $\sigma>2$. In order for the real part of $\zeta(\sigma+it)$ to be negative we must have
\[
\frac {1}{2}\log\frac {\zeta^2(\sigma)}{\zeta(2\sigma)}\geq \pi/2,
\]
or $\sigma< 1.197$. Once $\sigma$ becomes smaller than this value, the Titchmarsh circles extend beyond the $\pi/2$ line and so $\text{Re}\,\zeta(s)$ can and does become negative. For the purpose of this study, we only investigate $\zeta(1+it)$ since by continuity, if the function has real part negative on the line $\sigma=1$, then it is also negative for some neighborhood $1<\sigma<1+\epsilon$.
\section{Initial Considerations}
Consider the expression:
\begin{equation}
-\sum_{n=1}^{N} \log\big(1-p_n^{-\sigma}e^{-i t}\big)
\label{eqn00a}
\end{equation}
for a constant $\sigma$. What value of $t$ causes the fewest terms of this sum to reach an imaginary part outside the interval $(-\pi/2,\pi/2)$? It's easy to see that one such value is $t=\pi/2$ since in that case, the terms in \eqref{eqn00a} become $\log\left(1+i \frac{1}{p^{\sigma}}\right)$ and thus give rise to the largest argument. Table \ref{table1} summarizes how this build-up of arguments proceeds. By the time $N=14$, the imaginary component has exceeded $|\pi/2|$. Now consider the partial sum:
$$-\sum_{n=1}^{N}\log\left(1-p_n^{-1} e^{-it\log(p_n)}\right).$$
When $p=2$ and $t=\frac{\pi}{2\log(2)}$ for example, the imaginary part of this term is at its maximum value or $\arctan(1/2)\approx 0.463$. And in the case of a value of $t$ which simultaneously reduces $t\log(2)$ and $t\log(3)$ $\mod 2$ to a value close to $\pi/2$, the net imaginary part of these two terms would be approximately $0.79$. This is the case for example, when $t=\frac{3186 \pi }{\text{Log}\left[\frac{3}{2}\right]}$. And when $t$ is larger still, additional "confluences" of this sort can push the first $n$ terms of the sum even closer to a negative imaginary part.
\begin{table}[h!b!p!]
\caption{Table (default)}
\begin{tabular}{|l|l|}
\hline
N & $\text{Im}\log(1+i p_n)$ \\
\hline
1 & 0.46365 \\
2 & 0.78540 \\
3 & 0.98279 \\
4 & 1.12469 \\
5 & 1.21535 \\
6 & 1.29212 \\
7 & 1.35088 \\
8 & 1.40346 \\
9 & 1.44691 \\
10 & 1.48138 \\
11 & 1.51363 \\
12 & 1.54065 \\
13 & 1.56503 \\
14 & 1.58829 \\
15 & 1.60956 \\
\hline
\end{tabular}
\label{table1}
\end{table}
These considerations do not guarantee the infinite sum will have an imaginary component outside the range $(-\pi/2,\pi/2)$ but does give us some indication how the real part of zeta might grow negative in this part of the half-plane: a value of t which when reduced modulo $2\pi$ for the first $n$ terms, builds up to an imaginary part greater than $\pi/2$ that is simultaneously not offset by the remaining terms in the sum.
The algorithm described in this paper searches for values of $t$ which maximizes this build-up of initial argument for the first $n$ set of primes, then searches this set of points for values that give rise to a negative real part of $\zeta(1+it)$.
\section{Description of the Algorithm}
\label{sectionk}
All algorithms for this study were written in Mathematica 7.0. The code locates values of t which approximately satisfy \eqref{eqn005} for a set of primes within a tolerance of $\delta$. One might initially guess that this would reduce to nothing more than a brute-force search for values meeting the tolerances. However, the modular space of \eqref{eqn005} contains a marvelous symmetry that allows us to locate approximate solutions to $\eqref{eqn005}$ in much less time than a brute-force search. The solutions to $\eqref{eqn005}$ in this paper are called "confluence points" and denoted by $C_n(\theta,\delta)$ representing a solution to the first $n$ equations of \eqref{eqn005} for particular values of $\theta$ within $\pm\delta$. In this paper, sometimes when referring to these points, only $C_n$ or "confluence order" is used or if $\theta$ is implied, only $C_n(\delta)$ is stated.
\begin{equation}
M_n=\begin{cases}
t\log(2)&\equiv D\left(\frac{\pi}{2},\delta\right)\mod(2\pi)\\
&\vdots\\
t\log(p_{n})&\equiv D\left(\frac{\pi}{2},\delta\right)\mod(2\pi)
\end{cases}
\label{eqn005}
\end{equation}
A value of $t$ which satisfies the first $n$ equations in \eqref{eqn005} will push the imaginary term of the log sum into a region outside the interval $(-\pi/2,\pi/2)$. If the net effect of the remaining terms of the sum do not appreciably build-up in the opposite direction, then this value of $t$ will cause $\text{Re}\,\zeta$ to become negative. The plan then is to locate a large number of $C_n$ points and check the value of $\zeta(1+it)$ in a neighborhood of each to determine if its real part is negative. We search along the line $\sigma=1$ since the greater $\sigma$ is, the smaller the argument becomes for each term in the series and the less likely the initial terms will have any affect on the final sum.
\section{Mapping the modular space of $M_n$}
The algorithm described below is designed around a "base confluence set". We start by determining the confluence points for two primes. The following description explains a base set in terms of the first two primes. The algorithm is conceptually the same for at least the pair $(5,7)$ and likely so for other pairs.
Consider the modular equations
\begin{figure}
\centering
\includegraphics{modplot1.pdf}
\caption{Modulus space $M_n$}
\label{fig:modplot1}
\end{figure}
\begin{equation}
\begin{aligned}
m_2(t)&=\mod(t\log(2),2\pi)-\pi/2\\
m_3(t)&=\mod(t\log(3),2\pi)-\pi/2,
\label{eqn0010}
\end{aligned}
\end{equation}
the plots of which are show in Figure \ref{fig:modplot1} with $m_2(t)$ in blue and $m_3(t)$ in red. Where the two lines cross, we have a solution to the system:
\begin{equation}
\begin{aligned}
t\log(2)&\equiv x\mod(2\pi)-\pi/2\\
t\log(3)&\equiv x\mod(2\pi)-\pi/2
\label{eqn0011}
\end{aligned}
\end{equation}
for some $-\pi/2\leq x<3\pi/2$. One immediately notices a symmetry in Figure \ref{fig:modplot1}: the confluence points, shown as black dots, line up. If we connect the dots with diagonal black lines running from the upper left corner to the lower right corner and expand the range of $t$, we obtain the plot in Figure \ref{fig:modplot2} (to avoid clutter, the modular $m_n$ lines have been removed). Equations for the black lines are given by
$$L_n(t)=(3\log(2)-2\log(3))t+2n\pi-\pi/2,\quad n=1,2,3,\cdots.$$
\begin{figure}
\centering
\includegraphics{modplot2.pdf}
\caption{Diagram of $L_n$ and $G_n$}
\label{fig:modplot2}
\end{figure}
The lines $L_n(t)$ serve as a first approximation for locating the next order of confluence points: we simply calculate where the black lines cross the $t$-axis, check if a base confluence point is near-by, then check for a $C_3$ match within a specified tolerance. However, the $L_n$ lines are too-closely spaced to offer an efficient algorithm for locating a large number of $C_3$ points quickly. We can improve the algorithm by considering a second set of symmetries in Figure \ref{fig:modplot2} which is shown as the green lines represented by the equations:
$$G_n(t)=1/3(8\log(2)-5\log(3))t-2n\pi/3+3\pi/2,\quad n=1,2,3,\cdots$$
\begin{figure}
\centering
\includegraphics{modplot3.pdf}
\caption{Diagram of $G_n$ and $R(n,m,t)$}
\label{fig:modplot3}
\end{figure}
In order to further reduce the number of points that need to be checked, we can again form a system of diagonal lines connecting the base confluence points. These are shown as the red lines in Figure \ref{fig:modplot3} where this time, both the mod lines $m_n$ and black lines $L_n$ were removed for clarity. The red lines are given by the equations:
$$R(n,m,t)=\log\left(4/3(2/3)^{22/31}\right) t+\pi/62(16 n+4m-47)$$
for $m=1,2,3,4$ and $n=0,1,2,\cdots$.
And although we could continue this process to obtain even further improvements, we stop at the $R(n,m,t)$ lines for this study. Note in Figure \ref{fig:modplot3}, the $R$ lines cross the $t$-axis with a very small slope with the two nearest $C_2$ confluence points along each red line being very close to the $t$-axis. Where each $R$ lines crosses the $t$-axis, the nearest-neighbor set denoted by $B_{(2,3)}$, satisfy
\begin{equation}
B_{(2,3)}(\pi/2,\delta)=\begin{cases}
t\log(2)\equiv D\left(\frac{\pi}{2},\delta\right)\mod(2\pi)\\
t\log(3)\equiv D\left(\frac{\pi}{2},\delta\right)\mod(2\pi)
\end{cases}
\end{equation}
for small $\delta$. An example nearest-neighbor set for one $R$ line is show in the figure as the two red dots. It is not difficult to determine the zeros of the lines $R(m,n,t)$ and then subsequently,the two nearest neighbors. This allows for an efficient search of the modular space for the points $C_2(\pi/2,\delta)$ requiring only a few computations for every $2000$ increment in $t$.
Each confluence point $C_2$, along the $R$ lines is at a distance of $\Delta_z(2,3)=\frac{62\pi}{\log(3)-\log(2)}\approx 480$. The distance between each successive zero of the $R$ lines is $\frac{2\pi}{53 \log(3)-84 \log(2)}\approx 3009$ and the slope of each $R$ line is $m_R=\log(4/3(2/3)^{22/31})\approx -6.7\times 10^{-5}.$ Thus, knowing the zero point of each $R(m,n)$ and $\Delta_z(2,3)$, we can determine with relatively few calculations, the nearest $C_2$ confluence points closest to the t-axis and then check only these for higher-ordered confluence points. But it turns out that we can do much better than this because the pattern of confluence points in one range of values of $t$ goes a long way in determining the pattern in a following range of values. Some drift is of course expected but in actual runs of the software, this turned out to be surprising little over a very large interval. We can exploit this property by using a set of lower-ordered confluence points as a map for the next higher set of confluence points thereby greatly reducing the time to locate these points.
It's easy to see that the maximum height reached by either the previous or next $C_2$ point along each $R_n$ line is $m_R \frac{62\pi}{\log(3)-\log(2)}\approx 0.032$. Therefore, if the tolerance $\delta$ is set greater than $0.032$, than both the previous and next point will be below this value and can be chosen without checking the residue mod $2\pi$.
The following Mathematica code implements this approach to locate the base-confluence points $B_{(2,3)}$.
$$
\begin{aligned}
&\hspace{-15pt}\texttt{For[n=1,n$\leq$nMax,n++,} \\
&\hspace{-10pt}\texttt{For[m=1,m$\leq$4,m++,} \\
&\texttt{zeropt}=\frac{(-31+4m+16n)\pi}{2(53\log(3)-84\log(2))}\texttt{;}\\
&\texttt{startpt}=\frac{2(31-7m+3n)\pi}{\log(3)-\log(2)}; \\
&\texttt{intpt}=\text{IntegerPart}[\frac{62}{k}\left(\texttt{zeropt}-\texttt{startpt}\right)];\\
&\texttt{prevpt}=\left(\texttt{startpt}+\frac{62\pi\texttt{inpt}}{k}\right);\\
&\texttt{nextpt}=\left(\texttt{prevpt}+\frac{62\pi}{k}\right);\\
&\texttt{If[theDegree$<$maxDifference,} \\
&\hspace{15pt}\texttt{thePointList23=Append[thePointList23,prevpt];} \\
&\hspace{15pt}\texttt{thePointList23=Append[thePointList23,nextpt];} \\
&\texttt{,}\\
&\hspace{15pt}\texttt{testprev=Mod[prevpt Log[2],2$\pi$]-$\pi/2$;}\\
&\hspace{15pt}\texttt{testnext=Mod[nextpt Log[2],2$\pi$]-$\pi/2$;}\\
&\hspace{20pt}\texttt{If[0 $<$ testprev $<$ theDegree,}\\
&\hspace{25pt}\texttt{thePointList2=Append[thePointList2,prevpt];}\\
&\hspace{20pt}\texttt{];}\\
&\hspace{20pt}\texttt{If[0 $<$ testnext $<$ theDegree,}\\
&\hspace{25pt}\texttt{thePointList2=Append[thePointList2,nextpt];}\\
&\hspace{20pt}\texttt{];}\\
&\texttt{];}\\
&\hspace{-10pt}\texttt{];}\\
&\hspace{-15pt}\texttt{];}
\end{aligned}
$$
From this starting set of points, we go on to find the set of $C_3$ points by calculating the gaps between successive $C_2$ points and then use those gaps to predict with good success, the location of the next $C_3$. In particular, if we use the above code to locate the first $10,000$ $C_2(\pi/2,0.01)$ points, we find only three possible gap sizes:
$$\left\{g_1,g_2,g_3\right\}=\left\{\frac{778\pi}{\log(3/2)},\frac{7360\pi}{\log(3/2)},\frac{8138\pi}{\log(3/2)}\right\}$$We can then use this table to search for $C_3$ point by simply checking three possible locations. For example, if a $C_2$ point was located at $t_2$, then we need only check the three points $t_2+g_1$, $t_2+g_2$, $t_2+g3$.
The following code implements the next stage of the search by locating the $C_3$ points using the previously-calculated $C_2$ gap table. In the code, we have a While loop which checks for the $C_2$ points against the corresponding gap table. As the confluences grow in order, we add conditionals for additional primes. The following code is only the main routine. In actual practice, pre-processing of the previous confluence data is necessary. The variable \texttt{mintval} is then initially set to the last $C_{n}$ point (if any) found in the pre-processed data. After each acquisition, the data must then be post-processed into a sorted gap table for the next confluence search.
$$
\begin{aligned}
&\texttt{thePointList3}=\{\};\\
&\texttt{mintval=lastin23};\\
&\hspace{10pt}\texttt{n=1}; \\
&\texttt{For mypoints=1},\texttt{mypoints}\leq \texttt{maxN,mypoints++} \\
&\hspace{10pt}\texttt{n=1}\\
&\hspace{10pt}\texttt{testpt}=\texttt{mintval}+\texttt{gaptable23}[[n]];\\
&\hspace{10pt}\texttt{While[} \\
&\hspace{15pt}\texttt{Not[0$\leq$ Mod[test Log[2],$2\pi$]-$\pi/2<$theDegree]},\\
&\hspace{20pt}\texttt{n++};\\
&\hspace{20pt}\texttt{If[n>}\texttt{gaptable23len},\\
&\hspace{25pt}\texttt{Print["Gap table 23 exhausted"]};\\
&\hspace{25pt}\texttt{Abort[];}\\
&\hspace{25pt}\texttt{,}\\
&\hspace{25pt}\texttt{testpt}=\texttt{mintval}+\texttt{gaptable23[[n]]};\\
&\hspace{25pt}\texttt{];}\\
&\hspace{20pt}\texttt{];}\\
&\hspace{15pt}\texttt{mintval}=\texttt{testpt};\\
&\hspace{15pt}\texttt{If[Mod[mintval Log[5],2$\pi$]-$\pi/2$<theDegree,}\\
&\hspace{15pt}\texttt{thePointList3}=\texttt{Append[thePointList3,mintval]};\\
&\hspace{10pt}\texttt{];}\\
&\hspace{5pt}\texttt{];} \\
\end{aligned}
$$
With this code and some additional pre-processing and post-processing code, we are able to find a large number of $C_3(\pi/2,\delta)$ points quickly. For example, we can check about 50,000 $C_2$ points in about $40$ seconds with the machine this code was tested on. We next calculate the gap table for a sufficient number of $C_3$ points. In actual practice, this was usually around $1500$ points. In one run of the $C_3$ code, we obtained the following gap table:
$$\texttt{gaptable3}=\left\{\frac{91260\pi}{\log(3/2)},\frac{360014\pi}{\log(3/2)},\frac{451274\pi}{\log(3/2)},\frac{921936\pi}{\log(3/2)},\frac{1013196\pi}{\log(3/2)},\frac{1373210\pi}{\log(3/2)},\frac{1464470\pi}{\log(3/2)}\right\}$$
Notice the gap table has grown in size from the first table. This is quite expected considering we are working with residues $\mod 2\pi$. The next step of course is to use $\texttt{gaptable3}$ to locate the $C_4(\pi/2,\delta)$ points. We then add another conditional to the While loop to check for a $C_3$ confluence point. We then create a gap table of those points and continue the search and if necessary, adjust the tolerance. The routine is coded to terminate a run if after checking all values suggested by the gap table for a particular point, the routine fails to find the desired confluence point at the specified tolerance. In this case, a small change to the tolerance was found to correct for the drift at least in the range studied.
\section{Acquisition of Data}
All code was run on a $2.2$ GHz machine with one Gb RAM. In order to study this algorithm, three runs were performed. The first two were run with a base-confluence set $B_{(2,3)}$. The details of constructing this set were described in the previous sections. The third set was done with a base-confluence set $B_{(5,7)}$, and it's construction is virtually identical to the $B_{(2,3)}$ set except the equations for $L_n,G_n,$ and $R_n$ are different reflecting the use of different primes in their construction. In all cases, $\theta=\frac{\pi}{2}$.
\subsection{Run 1}
Using base-confluence $B_{(2,3)}$, and $\delta$ set to 0.1, the algorithm located approximately $1000$ points for each confluence up to $C_{10}$ in approximately $90$ minutes. The variable \texttt{nMax} was set to $50,000$. This means that $50,000$ $C_{n-1}$ terms were checked in order to obtain the $C_n$ set within the currently set tolerance. In all cases, the tolerance was "globally" set meaning that each modular calculation was checked against the same value of $\delta$. Table \ref{table1} summarizes the result of this run. "First" and "Last" are the first and last points of the set, "Gap" is the average distance between $C_n$ points as determined by this algorithm, and $k=\frac{\pi}{\log(3)-\log(2)}$, that is, the first $C_3$ point is located at $\frac{8274\pi}{\log(3)-\log(2)}$.
\begin{table}[ht]
\caption{Run 1}
\centering
\begin{tabular}{|c|c|r|r|c|}
\hline
Type & Points & First & Last & Gap \\
\hline
$B_{(2,3)}(.1)$ & 7992 & $432k$ & $1551976k$ & $10^3$ \\
$C_{3}(.1)$ & 923 & $8274k$ & $9342654k$ & $10^5$ \\
$C_{4}(.1)$ & 817 & $171406k$ & $450025676k$ & $10^6$ \\
$C_{5}(.2)$ & 1601 & $6538724k$ & $11407973800k$ & $ 10^8$ \\
$C_{6}(.2)$ & 1625 & $180883380k$ & $101681208846k$ & $ 10^9$ \\
$C_{7}(.2)$ & 1659 & $1524427526k$ & $556034603992k$ & $10^9$ \\
$C_{8}(.2)$ & 1689 & $34995876276k$ & $3927590605616k$ & $10^{10}$ \\
$C_{9}(.3)$ & 2438 & $232692763660k$ & $20585726758472k$ & $10^{11}$ \\
$C_{10}(.4)$ & 3197 & $5206151247198k$ & $53443637744022k$& $ 10^{11}$ \\
\hline
\end{tabular}
\label{table1b}
\end{table}
Note the tolerance was relaxed at $C_5$ and $C_9$ due to gap table exhaustion. Figure \ref{fig:run1c6} represents a "confluence portrait" for one $C_6$ value obtained in this run and serves to graphically illustrate the solutions for $M_6$ of \eqref{eqn005}. Each line in the figure represents one equation of \eqref{eqn005} plotted in the range $(t-\alpha,t+\alpha)$ for some $\alpha$. Notice how all the equations confluence at this $C_6$ point hence the name given to these points.
Consider the six terms of the partial sum of \eqref{eqn00a} which for simplicity we take $\sigma=1$ since we are only interested in an approximation:
\begin{figure}
\centering
\includegraphics{run1c6.pdf}
\caption{$C_6(\pi/2,0.2)$ at $t_6=20\,430\,730\,768 k$ }
\label{fig:run1c6}
\end{figure}
$$-\sum_{n=1}^{6}\log\left(1-\frac{1}{p} e^{-i t_6 \log(p)}\right)\approx -0.126968 - 1.32215 i, $$
and note the imaginary part is already close to $|\pi/2|$ and even although this gives no indication of how the remaining terms will contribute to the sum, it illustrate how the confluences are at least affecting the first few terms of the log sum.
\subsection{Run 2}
The second set of data was initially acquired at $\delta=0.05$ (except the base-set) and with a sampling size of $200,000$. The gap tables successfully located on average $1600$ confluence points until $C_9$ was reached. Then $\delta$ was relaxed to $0.07$ and the $C_9$ set successfully acquired. Alternatively, we could have re-acquired a larger gap table for $C_8$. The acquisition of $C_{10}$ exhausted the $C_9$ gap table. This set of confluence points was re-acquired at $\delta=0.08$ until after the acquisition of $1273$ terms, the gap table was again exhausted. Total execution time was approximately five hours. Figure \ref{fig:c10run2} shows a typical $C_{10}$ confluence portrait at this value of $\delta$. Note the equations at the origin confluence at a smaller tolerance compared to those in Figure \ref{fig:run1c6}.
\begin{figure}
\centering
\includegraphics{c10run2.pdf}
\caption{$C_{10}(\pi/2,0.08)$ at $t=34\,494\,360\,555\,864\,694k$}
\label{fig:c10run2}
\end{figure}
\begin{table}[ht]
\caption{Run 2}
\centering
\begin{tabular}{|c|c|r|r|c|}
\hline
Type & Points & First & Last & Gap \\
\hline
$B_{(2,3)}(.01)$ & 1239 & $3186k$ & $1551556k$ & $10^4$ \\
$C_{3}(.05)$ & 1599 & $118922k$ & $242220222k$ & $10^6$ \\
$C_{4}(.05)$ & 1602 & $66656806k$ & $19751098706k$ & $10^8$ \\
$C_{5}(.05)$ & 1642 & $1095953254k$ & $1684031570650k$ & $10^{10}$ \\
$C_{6}(.05)$ & 1624 & $10971337166k$ & $46958108569642k$ & $10^{11}$ \\
$C_{7}(.05)$ & 1563 & $5837564637802k$ & $963484780912880k$ & $10^{12}$ \\
$C_{8}(.05)$ & 1608 & $235403791542126k$ & $41141345684941152k$ & $10^{14}$ \\
$C_{9}(.07)$ & 2199 & $1409137033297936k$ & $838179310625884816k$ & $10^{15}$ \\
$C_{10}(.08)$ & 1273 & $3449430555864694k$ & $5206649605555820266k$& $10^{16}$ \\
\hline
\end{tabular}
\label{table2}
\end{table}
\subsection{Run 3}
Since one objective of this study was to devise an efficient algorithm for numerically solving the system of congruences defined in \eqref{eqn005}, one would like to know how well it worked for extremely large values of $t$. This third run attempted to determine this by running the algorithm at an initial tolerance of $\delta=0.01$ and sampling size of $500,000$. As Table \ref{tabler3} suggests, the algorithm was stable enough to acquire $1765$ $C_{14}$ points with an average gap interval of $10^{26}$ at a tolerance of $0.05$. This allowed the acquisition of $121$ $C_{15}$ before exhausting the $C_{14}$ gap table which by then had grown to about $2000$ entries. This data took approximately 24 hours to acquire. The final $121$ $C_{15}$ points were checked for a minimum $C_{16}$ confluence which was found at a tolerance of $0.063$ and is shown in Figure \ref{fig:firstc16}. In Table \ref{tabler3}, $k=\frac{\pi}{\log(7)-\log(5)}$.
\begin{table}[ht]
\caption{Run 3}
\centering
\begin{tabular}{|c|c|r|r|c|}
\hline
Type & Points & First & Last & Gap \\
\hline
$B_{(2,3)}(.01)$ & 835 & $1188k$ & $105918k $ & $10^3$ \\
$C_{3}(.01)$ & 239 & $415346k$ & $188449608k $ & $10^5$ \\
$C_{4}(.01)$ & 1760 & $951398k$ & $341342656k $ & $10^6$ \\
$C_{5}(.01)$ & 785 & $945400159026k$ & $248625146191138k $ & $10^{12}$ \\
$C_{6}(.01)$ & 782 & $358872295235106k$ & $53873791685357248k $ & $10^{15}$ \\
$C_{7}(.01)$ & 2449 & $25465934548192130k$ & $24901118319251218470k $ & $10^{17}$\\
$C_{8}(.01)$ & 1607 & $4228669878993511528k$ & $3026698278169709217444k $ & $10^{19}$ \\
$C_{9}(.01)$ & 1581 & $443049036634051704488k$ & $106247940250479965840452k $ & $10^{20}$\\
$C_{10}(.02)$ & 1608 & $15784886571659169335556k$ & $2201816585968897375148202k $ & $10^{22}$ \\
$C_{11}(.03)$ & 2418 & $325647381902527914552536k$ & $37724324922892546494833526k$ & $10^{23}$\\
$C_{12}(.03)$ & 1800 & $4833028477320576769793928k$ & $502072051726121146180272476k$ & $10^{24}$ \\
$C_{13}(.04)$ & 3924 & $60624317376187697337278868k$ & $5796475948831968271200664166k$ & $10^{25}$ \\
$C_{14}(.04)$ & 1765 & $510082174019371223753727784k$ & $19898784954288834658068768846k$ & $10^{26}$ \\
$C_{15}(.05)$ & 121 & $6450267346766950732885822366k$ & $26894796143731914838108040998k$ & $10^{27}$ \\
\hline
\end{tabular}
\label{tabler3}
\end{table}
\begin{figure}
\centering
\includegraphics{firstc16.pdf}
\caption{$C_{16}(\pi/2,0.063)$ at $t=12253527959225463513391519458k$}
\label{fig:firstc16}
\end{figure}
\section{Processing the Data Sets}
For the first run, we tested each entry in the confluence tables up to $C_6$ by calculating $\zeta(1+i(t\pm1))$ at $20$ equally-spaced points in the interval and obtained a total of ten regions along the line $\text{Re}(s)=1$ for which the real part of zeta became negative. Sample points in the indicated $C_n$ region are given in Table \ref{table15}. The value of zeta in the regions was determined using the Mathematica "N" command using arbitrary precision arithmetic with the final answer reported to thirty digits of accuracy to the right of the decimal place using the following command for each of the ten points in Table \ref{table15}:
$$\texttt{N[zeta[1+it],$\{\infty,30\}$]}$$
The smallest value of $t$ detected in the data sets described above in which $\text{Re}\,\zeta(1+it)<0$ was found in the vicinity of $3.4\times 10^7$ as shown in Table \ref{table15} (recall the factor $k$). All the values in Table \ref{table15} had negative imaginary part due to the search criteria selecting $\pi/2$ residues. No points in the first four confluence tables of the remaining runs were found to give rise to a negative real part of $\zeta$ at least using the method described above. The remaining confluence tables were not inspected at this time because of the impracticality of actually computing the zeta function (on the machine used to analyze the data) at the relatively large values of $t$ in these tables.
\begin{table}[h!b!p!]
\caption{Points where $\text{Re}\,\zeta(1+it)<0$}
\begin{tabular}{|c|c|l|l|}
\hline
ID & Type &t & $\zeta(1+it)$ \\
\hline
1 &$C_3$ & $4\,378\,640k\phantom{111k}-2/5$ & $-0.009-1.22i$\\
2 & $C_4$ & $415\,782\,314k\phantom{1k}-2/5$ & $-0.024-1.23i$\\
3 & $C_6$ & $20\,430\,730\,768k-1/10$ &$-0.015-1.08i$ \\
4 &$C_6$ & $25\,705\,015\,862k-1/5$ &$-0.002-1.20i$ \\
5 & $C_6$ & $47\,668\,373\,108k-1/10$ & $-0.027-0.96i$ \\
6 & $C_6$ & $53\,761\,507\,682k$ & $-0.009-1.00i$ \\
7 & $C_6$ & $62\,484\,882\,686k-1/5$ & $-0.013-1.13i$ \\
8 & $C_6$ & $65\,421\,460\,042k-1/10$ & $-0.003-0.91i$ \\
9 & $C_6$ & $97\,190\,286\,104k-1/10$ & $-0.018-1.00i$ \\
10 & $C_6$ & $99\,154\,858\,182k-3/10$ & $-0.026-1.25i$\\
\hline
\end{tabular}
\label{table15}
\end{table}
\section{Conclusions}
One can easily locate a value of $t$ where $\text{Re}\,\zeta(1+it)<0$ by trial an error by simply sampling the $t$-axis at intervals of $1/10$ starting at $t=0$. Doing so, one first finds a negative real part at $t=682112.9$. However, this study attempted to predict where the real part of the function would turn negative without a trial and error approach. A surprising result to come from this study is the relative stability of the method. One would have guessed that such a method would quickly degrade due to chaotic drift in the modular space rendering it virtually impossible to predict any future values let alone values up to $10^{29}$. It would be interesting in a further study to try and explain what is the source of this stability.
Additionally:
\begin{enumerate}
\item
What are the precise shapes of these negative contours where $\text{Re}\,\zeta(\sigma+it)<0$ and how do they change as we go up the $t$-axis?
\item
This study focused on two base-confluence algorithms: $(2,3)$ for the first two trials, and $(5,7)$ for the third trial. It is not known how the algorithms are affected by different base prime pairs.
\item
This study demonstrated some structure in the modular space of \eqref{eqn005}. Is there a similar periodicity in the domains where $\text{Re}\,\zeta(\sigma+it)$ dips below the $\sigma-t$ plane? Can one predict where such domains might be located analogous to how the various confluence points were "predicted" using the gap tables?
\item
This study did not attempt to investigate in detail the drift of values in the modular space. When a gap table became exhausted, the tolerance was simply increased or a greater number of $C_{n-1}$ values were located and a new larger gap table created. It would be interesting to better understand this drift.
\end{enumerate}
|
1,116,691,497,603 | arxiv | \section{Introduction}
\label{sec1} |
1,116,691,497,604 | arxiv | \section{Introduction}
Consider the following problem faced by a group of $n$ manufacturers of some good. Each manufacturer produces the same undifferentiated product, which sells for some exogenously given price. The manufacturers are constrained by the production process into producing goods of the same given average quality; however, they can choose the distribution of the good's quality--by being, say, more by-the-book and rigid a producer can ensure a more constant quality; or by being more flexible and hands-off he can achieve a wider spread of quality realizations. There is some buyer who wants to buy the good, and she naturally would like the good with the highest quality. Moreover, before she makes her choice of product, she may inspect the goods in order to accurately pick out the best one. What distribution over qualities should a producer choose in order to maximize the change that his product is best?
One might suspect, \textit{ex ante}, that optimal choice of distribution would be the distribution with the highest variance. However, as we show in this paper, that is not the case, and instead it is the uniform distribution that is king.
This model can be formalized as the following $n$-player game. We fix a mean, $\mu$ and have each player simultaneously choose the distribution of a random variable with realizations constrained to a common interval--without loss, $[0,1]$--such that the expectation of the random variable is the given $\mu$. The winner of this zero-sum game is the player with the highest realization of his random variable. Each player's objective; therefore, is to maximize the probability that the realized value of his random variable is higher than that of his opponents.
Our results are as follows. There is a unique symmetric equilibrium: if $\mu = 1/n$ then both players play a distribution with cdf $F(x) = x^{1/(n-1)}$ supported on the whole interval $[0,1]$, and if $\mu < 1/n$ they each play a distribution with the same curvature on a smaller interval of support, $[0,n\mu]$. The key is that for $j \neq i$, the distribution of $\max_{j \neq i} X_{j}$ is the uniform distribution. On the other hand, if the given mean is greater than $1/n$ then each player places a point mass on $1$ and the remainder of the distribution is continuous, supported on a subset of $[0,1]$. Holding $\mu$ fixed, as the number of players increases, the weight placed on $1$ increases. As $n$ goes to infinity, the unique symmetric equilibrium converges to one where each player chooses a distribution consisting of two point masses on $0$ and $1$. Finally, we show that these results can be extended to a modified case where the maximal support of the distribution is any interval that is a subset of $[0,+ \infty)$.
\subsection{Applications and Discussion}
We believe that this model has numerous applications. Perhaps foremost among these uses is that of competitive information design, or persuasion. This setup models the situation where a group of principals compete over information provision to a risk neutral agent where the principals and the agent share a common prior with binary support. Indeed, our problem is equivalent to one where agents each choose an experiment in the Kamenica and Gentzkow \cite{kam} sense, given the common binary prior. Real world examples of this problem include sellers choosing how much information to convey about their products to a buyer with unknown tastes, schools competitively choosing grading policies for their respective candidates (as in \cite{bol}), and political parties attempting to persuade a voter to choose their candidate (as in \cite{Albrecht}).
More generally, when the probability space is one dimensional this problem of choosing a signal structure becomes one of choosing a Blackwell experiment (see \cite{blackwell}).\footnote{This approach was introduced as a method for examining persuasion problems in Gentzkow and Kamenica \cite{gent}, and a number of other papers have followed that interpret the information design problem as one of choosing an optimal Blackwell experiment. Some other recent examples include \cite{kol} and \cite{skreta}.} As a result, this paper can be interpreted as solving the same problem but with fewer constraints. Put a different way, while the general competitive persuasion question looks for the equilibrium where each player chooses a distribution of posteriors subject to the constraint that the distribution of posteriors be Bayes-plausible, this paper looks for the equilibrium where the constraints are now the two conditions that the maximal support and mean of the posterior and the prior must be the same.
For a prior distribution with binary support, our formulation is precisely the competitive persuasion problem, without any loss of generality. However, it is easy to see that this is not generally true for other prior distributions. For instance, suppose that the prior consisted merely of a weight of $1$ on $1/2$. Obviously, there is no signal structure that could convince a decision maker that the expected value of the object were anything other than $1/2$, and certainly we could not achieve a distribution on posteriors that yielded a uniform distribution on expected values. This intuition is apposite for any distribution with support on $n \neq 2$ points, where $n$ is allowed to be uncountably infinite (i.e. where the prior is non-atomic)--if there is too much weight towards the barycenter of the prior, a uniform distribution of expected rewards is simply not Bayes-plausible.
In part, this paper bears resemblance to Condorelli and Szentes \cite{Condor}. In \cite{Condor}, the authors characterize the equilibrium of a simple game consisting of a monopolistic seller and a buyer. The buyer may choose the cdf of her valuation; and then the seller, after observing the distribution but not the realization, makes an offer to the buyer. This of course is related to Roesler and Szentes \cite{Roes}, where instead of observing her valuation, the buyer merely receives a signal about it. The authors then characterize the properties of this buyer-optimal signal structure. Concordantly, here, we look at the (relatively) unconstrained problem, which provides insight as to the solution to the general competitive persuasion problem.
The approach taken in this paper differs greatly from that taken by other papers in the information design literature. Because the distribution chosen by our players is free from the Bayes' plausibility constraints, we need not utilize the usual approaches--where one concavifies the value function or uses the experiments-as-convex-functions idea \cite{gent, kol, skreta}. Instead, we can solve for the equilibrium of the game directly, in part using insights gleaned from Hulko and Whitmeyer \cite{Hulko}. The unique equilibrium here has the same intuitive property as the unique equilibrium of the two-player dice game in \cite{Hulko}. As shown by Hulko and Whitmeyer, the famous non-transitivity of generalized dice\footnote{Or, ``Efron's Dice"; see \cite{Con, Savage, Tenney}.}, results in cycles of best responses a la ``Rock-Paper-Scissors"; e.g. die $A$ beats die $B$ beats die $C$ beats die $A$. The only die impervious to this is the standard die (the analog of the uniform distribution in this paper), which guarantees a payoff of $1/2$ to both players. Thus, while the standard die does not beat any other die, it does not lose to any other die either.
In this paper, we arrive at a similar result in the two-player case, which is a corollary of our $n$-player result. Indeed, this result for two-players features as a lemma (Lemma 4.1) in Boleslavsky and Cotton (2015) \cite{cotton}, who look at a two-player problem in which schools must choose students to admit to their respective schools and then design a competitive grading policy. For a given mean less than or equal to $1/2$, the unique equilibrium is the uniform distribution, which guarantees a payoff of at least $1/2$ to the player who chooses it. Then, when $\mu > 1/2$, we have to modify this somewhat because the uniform distribution is no longer an equilibrium. Surprisingly, the unique equilibrium must have a point mass at $1$ and then a portion with a linear distribution.
For a general number of players $n$, the crucial cutoff is $1/n$; and moreover, the equilibrium distribution is the $(n-1)$-th root of $x$. Thus, as $n$ grows the continuous portion of the distribution becomes increasingly concave. As in the two-player case, if the mean is too high (greater than $1/n$) then at equilibrium each player puts a point mass on $1$. Then, for a fixed mean, as the number of players increases, each player places increasingly more weight as a point mass on $1$. As $n$ goes to infinity, the equilibrium distribution converges to one consisting merely of point masses on $1$ and $0$, of weight $\mu$ and $1-\mu$, respectively.
Finally, we would be remiss should we not mention the other papers in the realm of competitive information design. The area of Bayesian Persuasion and information design is growing rapidly, but there are still relatively few papers that look at competitive information provision. As mentioned above, this paper generalizes a result of Boleslavsky and Cotton \cite{cotton}, and the methods used in this paper--the use of calculus of variations techniques to directly solve for the equilibrium distribution--are novel in the literature. Moreover, because we derive a full characterization of the equilibrium for a general $n$, we are able to fully characterize the effect of an increase in population size on competitive information provision.\footnote{It has recently come to our attention that Spiegler \cite{spieg}, in looking at firms selling to a boundedly rational consumer, formulates a model that is extremely similar to ours in which the given mean is endogenous. The equilibrium in that study as well as the resulting insights are virtually the same as the results here, although our approach is novel.}
An additional paper similar to this one is that by Koessler, Laclau, and Tomala \cite{Koessler}. Each player in their game designs a signal structure that contains information about their respective (independent) pieces of information, with the goal of persuading the decision-maker to take the player's preferred action. Similarly, in a pair of papers, Au and Kawai \cite{Au1, Au2} also look at the situation where a number of persuaders compete through information provision. Board and Lu \cite{Board}, in turn, look at information in a search setting. In concert with the insight derived in \cite{Au2} and \cite{Board} we establish that competition elicits greater information provision. In another paper, Boleslavsky and Cotton \cite{bol} look at another two-player game of competitive persuasion in which agents provide information in order to secure funding for proposals. Similarly, Albrecht \cite{Albrecht}, looks a two-player game consisting of two parties vying for the support of a voter. His paper is another real world manifestation of the two-player case of our problem
\subsection{Formulating the Problem}
Formally, let the sample space, $\Omega$, be the closed interval $[0,1]$ and $\mathcal{B}$ be the $\sigma$-Algebra of Borel sets on $[0,1]$. Define random variable $X_{i}$, $i \in \big\{1,2,\dots, n\big\}$ as the identity random variable. We define a strategy for a player $i$ as follows:
\begin{definition}
Fix mean $\mu \in (0,1)$. A \textcolor{Maroon}{Strategy} for player $i$ consists of a choice of probability distribution $F_{i}$ such that $\mathbb{E}_{F_{i}}\big[X_{i}\big] = \mu$. Write a player $i$'s strategy as the duple $S_{i} \coloneqq (F_{i},X_{i})$.
\end{definition}
The game we shall analyze is constant sum and symmetric; and the payoff for Player $1$, $u_{1}(S_{1},S_{-1})$, is given by
\[\begin{split}
u_{1}(S_{1},S_{-1}) = \Pr(X_{1} > \max_{j \neq 1}(X_{j}) &+ \frac{1}{n}\Pr(X_{1} = X_{2} = \dots = X_{n})\\
&+ \binom{n-1}{n-2}\frac{1}{n-1}\Pr(X_{1} = X_{2} = \dots = X_{n-1} > X_{n})\\
&\vdots\\
&+ \binom{n-1}{1}\frac{1}{2}\Pr(X_{1} = X_{2} > \max_{j \neq 1,2}X_{j})
\end{split}\]
In other words, players want their random variable to have the highest realization, and ties are broken fairly. Before continuing we wish to make the following remark,
\begin{remark}
The set of mixed strategies is equal to the set of pure strategies.
\end{remark}
To see this, note that every mixed strategy consists of some randomization over a set of pure strategies, which is a probability distribution over probability distributions of the random variable. However, this itself is clearly a probability distribution of the random variable and so it is a pure strategy.
\section{The \texorpdfstring{$n$}{TEXT}-Player Game}
First, we write the following lemmata. The intuition behind them is straightforward; simply, there are no symmetric equilibria in which players choose discrete distributions. A player may always deviate profitably from such distributions by shaving a small amount of weight from the highest point in the support and moving this weight to the other points in the support.
\begin{lemma}\label{discrete}
There are no symmetric Nash Equilibria where players choose discrete distributions supported on $N(<\infty)$ points.
\end{lemma}
\begin{proof}
It is easy to see that there is no symmetric equilibrium in which each player chooses a distribution consisting of a single point mass. Such a distribution could only consist of distribution with weight $1$ placed on $\mu$ and would yield to each player a payoff of $1/n$. However, there is a profitable deviation for a player to instead place weight $1 - \epsilon$ on $\mu + \eta$ and weight $\epsilon$ on $0$ ($\epsilon, \eta > 0$). In doing so, this player could achieve a payoff arbitrarily close to $1$.
Now, assume $\infty > N \geq 2$. Observe that a strategy consists of a choice of probabilities $\big\{p_{1}, p_{2}, \dots, p_{N}\big\}$, $p_{i} \in [0,1]$ $\forall i$,\quad $\sum_{i=1}^{N}p_{i} = 1$ \quad and support \quad $a_{1} < a_{2} < \cdots < a_{N} \in [0,1]$ \quad such that $\sum_{i=1}^{N} a_{i}p_{i} = \mu$.
The expected payoff to each player from playing an arbitrary strategy, $S_1 = S_2=\dots=S_n = E$, is
\[
u_i(S_i,S_{-i}) = \sum_{j=0}^{N-1}\left(\sum_{i=0}^{n-1} \binom{n-1}{i} \frac{1}{m-i} p_{N-j}^{n-i}\left(\sum_{k=1}^{N-j-1}p_k\right)^i\right)
\]
We claim deviating to the following strategy is profitable: $S_1' $ where $a_{N}^{'} = a_{N}$ is played with probability $p_{N} - \epsilon$ and $a_{j}^{'} = a_j+\eta$ is played with probability $p_{j} + \epsilon_{j}$, for $j \neq N$, where $\sum_{j}^{N-1}\epsilon_{j} = \epsilon$ (Again, $\epsilon, \eta, \epsilon_{j} > 0$ $\forall j$).\footnote{Note that we can always find such an $\eta > 0$.} The expected payoff to player 1 playing strategy $S_1'$ is
\[\begin{split}
u_{1}(S_1',S_{-1}) &= \sum_{j=0}^{N-1}\left(\sum_{i=0}^{n-1} \binom{n-1}{i} \frac{1}{n-i} p_{N-j}^{n-i}\left(\sum_{k=1}^{N-j}p_k\right)^i\right)\\
&- \epsilon \sum_{i=0}^{n-1} \binom{n-1}{i} \frac{1}{n-i} p_{N}^{n-i-1} \left( 1-p_N \right)^i\\
&+\sum_{j=1}^{N-1}\left(\sum_{i=0}^{n-1} \binom{n-1}{i} \frac{n-i-1}{n-i} p_{N-j}^{n-i}\left(\sum_{k=1}^{N-j}p_k\right)^i\,\right)
\end{split}\]
Note that the deviation is profitable for the player 1 if
\[
\epsilon \sum_{i=0}^{n-1} \binom{n-1}{i} \frac{1}{n-i} p_{N}^{n-i-1} \left( 1-p_N \right)^i < \sum_{j=1}^{N-1}\left(\sum_{i=0}^{n-1} \binom{n-1}{i} \frac{n-i-1}{n-i} p_{N-j}^{n-i}\left(\sum_{k=1}^{N-j}p_k\right)^i\,\right)\,,
\]
Which holds for a sufficiently small vector $(\epsilon_{1},...,\epsilon_{N-1})$.
\end{proof}
Moreover, we can also show that there can be no distributions with point masses on any point in $[0,1)$; to wit,
\begin{lemma}
There are no symmetric Nash equilibria with point masses on any point in the interval $[0,1)$. Moreover if $\mu < 1/n$, then all symmetric equilibria must be atomless.
\end{lemma}
\begin{proof}
Using an analogous argument to that used in Lemma \ref{discrete}, it is easy to see that there cannot be multiple point masses. Accordingly, it remains to show that there cannot be a single point mass. First, we will show that there cannot be an atom at any point $b \in (0,1)$. Suppose for the sake of contradiction that that there is a symmetric equilibrium where each player plays a point mass of size $p$ on point $b$. That is, each player plays strategy $S$ that consists of a distribution $F$ and a point mass of size $p$ on point $b$. Let $H(x) = F^{n-1}$. Then, player $1$'s payoff is
\[u_{1}(S_{1}, S_{-1})= \int_{0}^{1}\int_{0}^{y}h(x)f(y)dxdy + p\sum_{i=0}^{n-1}\binom{n-1}{i}\bigg(\frac{1}{n-i}\bigg)F(b)^{i}p^{n-1-i}\]
Then, let player $1$ deviate by introducing a tiny point mass of size $\epsilon$ at $0$ and moving the other point mass to $b + \eta$ and reducing its size slightly to $p - \epsilon$ ($\epsilon, \eta > 0$); call this strategy $S_{1}^{'}$. The payoff to player $1$ is
\[u_{1}(S_{1}^{'}, S_{-1})= \int_{0}^{1}\int_{0}^{y}h(x)f(y)dxdy + (p-\epsilon)\sum_{i=0}^{n-1}\binom{n-1}{i}F(b + \eta)^{i}p^{n-1-i}\]
Suppose that this is not a profitable deviation. This holds if and only if
\[\begin{split}
p\sum_{i=0}^{n-1}\binom{n-1}{i}\bigg(\frac{1}{n-i}\bigg)F(b)^{i}p^{n-1-i} &\geq (p-\epsilon)\sum_{i=0}^{n-1}\binom{n-1}{i}F(b + \eta)^{i}p^{n-1-i}\\
\end{split}\]
Or,
\[\begin{split}
\epsilon p^{n-1} &+ p\sum_{i=1}^{n-1}\binom{n-1}{i}\bigg(\frac{1}{n-i}\bigg)F(b)^{i}p^{n-1-i}\\
&\geq \frac{n-1}{n}p^{n} + (p-\epsilon)\sum_{i=1}^{n-1}\binom{n-1}{i}F(b + \eta)^{i}p^{n-1-i}
\end{split}\]
Clearly, as $\epsilon$ and $\eta$ go to zero we achieve a contradiction. Hence, there is a profitable deviation and so this is not an equilibrium. It is clear that there cannot be an equilibrium with a point mass on $0$ and so we omit a proof. Finally, we may conclude from the analysis in the sections \textit{infra} that there may not be a point mass on $1$ if $\mu \leq 1/n$.
\end{proof}
As will become clear, the value of $\mu$ is important in determining the equilibrium of this game. We divide our analysis into the following two cases:
\begin{enumerate}[label=\textbf{\arabic*})]
\item $\mu \geq \frac{1}{n}$ (Section \ref{=}); and
\item $\mu < \frac{1}{n}$ (Section \ref{<}).
\end{enumerate}
\subsection{\texorpdfstring{$\mu \geq \frac{1}{n}$}{text}}\label{=}
The main result of this section is the following theorem:
\begin{theorem}\label{n>}
In the game with $n$ players, if $\mu \geq 1/n$ then the unique symmetric Nash equilibrium is for each player to play $\mathcal{F}_{i}$, defined as
\[F_{i}(x) = (1-a)\bigg(\frac{x}{s}\bigg)^{1/(n-1)} \hspace{1cm} \text{for} \hspace{.2cm} x \in [0, s]\]
where $a = \mu - \mu(1-a)^{n}$ and $s = n\mu(1-a)^{n-1}$; and $\Pr(X=1) = a$.
\end{theorem}
\begin{proof}
First, we show that this is an equilibrium. Accordingly, we need to show that there can be no unilateral profitable deviation. Define $Z$ as $\max_{i \neq 1}X_{i}$, which, recall has a point mass on $1$. Moreover, define $H$ as the corresponding continuous portion of the distribution of $Z$; $H \coloneqq F_{i}^{n-1}$:
\[H(z) = (1-a)^{n-1}\frac{x}{s}\]
with associated density
\[h(z) = \frac{(1-a)^{n-1}}{s}\]
Evidently, it suffices to show that our candidate strategy achieves a payoff of at least $1/n$ to the player who uses it, irrespective of the strategy choice by the other players. Suppose for the sake of contradiction that there is a profitable deviation, that is, player $1$ deviates profitably by playing strategy $\mathcal{G}$. Clearly, we can represent $\mathcal{G}$ as having a point mass of size $c$, $0 \leq c \leq \mu$ on $1$ (naturally, if $c = 0$, then there is no point mass there). Written out, $\mathcal{G}$ consists of
\[\label{gee2}\tag{$\mathcal{G}$}
G(y) \hspace{1cm} \text{for} \hspace{.2cm} x \in [0, 1)
\]
and $\Pr(Y=1)= c$. Define $K \coloneqq \int_{0}^{1} dG = 1-c$. Naturally, $K \leq 1$. Then, player $1$'s utility from this deviation, $u_{1}(\mathcal{G},S_{-1})$, is\footnote{Note that the first term, $c(1-(1-a)^n)/na$, is derived below, in the proof of Lemma \ref{blah}.}
\[\begin{split}
u_{1} &= c\frac{1 - (1-a)^{n}}{na} + (1-a)^{n-1}\big(K - G(s)\big) + \int_{0}^{s}\int_{0}^{y}h(x)g(y)dxdy\\
&= \frac{c}{n\mu} + (1-a)^{n-1}\big(K - G(s)\big) + \int_{0}^{s}\int_{0}^{y}h(x)g(y)dxdy
\end{split}\]
Evidently, this is a profitable deviation if and only if $u_{1} > 1/n$; that is,
\[\begin{split}
\frac{c}{n\mu} + (1-a)^{n-1}\big[K - G(s)\big] + \int_{0}^{s}\frac{(1-a)^{n-1}}{s}yg(y)dy &> \frac{1}{n}\\
\end{split}\]
After some clever manipulation\footnote{See Appendix \ref{a01} for the detailed derivation.}, this reduces to
\[\tag{$1$}\label{101}\begin{split}
K &> \int_{s}^{1}\frac{1}{s}yg(y)dy + G(s)\\
\end{split}\]
It is clear that $\int_{s}^{1}\frac{1}{s}yg(y)dy \geq \int_{s}^{1}g(y)dy$ and thus we have
\[\begin{split}
K > \int_{s}^{1}\frac{1}{s}yg(y)dy &+ G(s) \geq \int_{0}^{s}g(y)dy + \int_{s}^{1}g(y)dy = K
\end{split}\]
We have established a contradiction and thus the result is shown.
\end{proof}
It remains to show uniqueness. To do this, we derive the candidate strategy presented in the theorem above. First, through analogous arguments to those presented \textit{supra}, it is clear that there cannot be multiple mass points, nor can there be any mass point in the interval $[0,1)$. Hence, we allow for there to possibly be a point mass on $1$, and show it must satisfy the following inequality.
\begin{lemma}\label{blah}
Suppose that in a symmetric equilibrium each player puts a point mass of size $a \geq 0$ on $1$. Then, $a$ must satisfy $a \geq \mu\big[1 - (1-a)^{n}\big]$.
\end{lemma}
\begin{proof}
Let each player play strategy $S_{i} = S$ where they each put weight $a$ on $1$. Suppose that player $1$ deviates and plays strategy $\hat{S}_{1}$ consisting of random variable $Y$ distributed with value $1$ with probability $\mu$ and $0$ with probability $1-\mu$.
Then, player $1$'s payoff is
\[\tag{$2$}\label{6}u_{1}(\hat{S}_{1}, S_{-1}) = \mu \sum_{i=0}^{n-1}\binom{n-1}{i}\bigg(\frac{1}{n-i}\bigg)(1-a)^{i}a^{n-1-i}\]
Through judicious use of the binomial theorem\footnote{See Appendix \ref{a0}.}, we can write \ref{6} as
\[\tag{$3$}\label{l}u_{1}(\hat{S}_{1}, S_{-1}) = \mu\frac{1 - (1-a)^{n}}{na}\]
This must be less than or equal to $1/n$:
\[\label{b3}\tag{$4$}\begin{split}
\frac{1}{n} &\geq \mu\frac{1 - (1-a)^{n}}{na}\\
a &\geq \mu\big[1 - (1-a)^{n}\big]\\
\end{split}\]
\end{proof}
There must also be a continuous portion of the distribution on some interval $[t,s]$ with $t \geq 0$, $s \leq 1$. Accordingly, our candidate equilibrium strategy, $\mathcal{F}_{i}$, is of the following form
\[\mathcal{F}_{i} = \begin{cases}\label{f}\tag{$\mathcal{F}_{i}$}
0 & x \in [0,t)\\
F_{i} & x \in [t,s) \\
1-a & x = s
\end{cases}
\]
with $0 \leq t < s \leq 1$ and $\Pr(X=1)=a$.\footnote{It is clear that the weight on $1$, $a$, cannot be $\mu$ in the symmetric equilibrium. To see this, note that such a value for $a$ would beget a distribution with binary support, which we already ruled out in Lemma \ref{discrete}.}
We look for a symmetric equilibrium. Observe that distributions $F_{i}$ must be such that
\[\int_{t}^{s}xf_{i}(x)dx = \mu - a\]
Fix $F_{j}$ for $j \neq i$ and define $H$ as $F_{j}^{n-1}$. Given this distribution, we have the necessary condition that $F_{i}$ maximizes
\[\frac{1 - (1-a)^{n}}{n} + \int_{t}^{s}f_{i}(x)H(x)dx \]
We use calculus of variations techniques and so we define the functional $J[f]$ as the Euler-Lagrange equation
\[J[f_{i}] = \int_{t}^{s}f_{i}(x)H(x)dx - \lambda_{0}\bigg[\int_{t}^{s}f_{i}(x)dx - (1-a)\bigg] - \lambda\bigg[\int_{t}^{s}xf_{i}(x)dx - \mu + a\bigg]\]
The first constraint ensures the distribution satisfies Kolmogorov's second axiom, and the second constraint guarantees that the expectation is $\mu$.
The functional derivative is then
\[\begin{split}
\left.\frac{\delta J(f(x))}{\delta f(x)}\right. &= H(x) - \lambda_{0} - \lambda x
\end{split}\]
This must equal $0$ at a maximum, so we have,
\[H(x) = \lambda_{0} + \lambda x\]
Then, by symmetry, $H(\cdot) = F_{i}^{n-1}(\cdot)$. Moreover, we have two initial conditions that allow us to obtain $t$ and $s$. Using the conditions $F_{i}(t) = 0$ and $F_{i}(s) = (1-a)$, the equilibrium distribution, $F_{i}$, must be
\[F_{i}(x) = (1-a)\bigg(\frac{x-t}{s-t}\bigg)^{1/(n-1)}\]
with the corresponding pdf,
\[f_{i}(x) = \frac{1}{x-t}\bigg(\frac{1-a}{n-1}\bigg)\bigg(\frac{x-t}{s-t}\bigg)^{1/(n-1)}\]
Note that we also need $\int_{t}^{s}xf_{i}(x)dx = \mu - a$, which reduces to
\[\tag{$5$}\label{2}\begin{split}
a = \frac{n\mu - \big[s + (n-1)t\big]}{n - \big[s + (n-1)t\big]}
\end{split}\]
We now show that $t$ must be $0$:
\begin{lemma}\label{lem7}
The lower bound of the continuous portion of the distribution, $t$, must be $0$.
\end{lemma}
\begin{proof}
We leave the detailed proof for Appendix \ref{a1}. Our proof is through contradiction; we suppose that there is a symmetric equilibrium in which $t > 0$, and show that there exists a profitable deviation.
\end{proof}
Finally, we pin down the size of the weight on $1$:
\begin{lemma}\label{lem8}
The weight on $1$, $a$, is given by $a = \mu - \mu(1-a)^{n}$.
\end{lemma}
\begin{proof}
The detailed proof may be found in Appendix \ref{a2}.
\end{proof}
\begin{corollary}\label{core}
If $\mu = 1/n$, then the unique symmetric Nash equilibrium is for each player to play strategy $F_{i} \coloneqq \big(\frac{x}{n\mu}\big)^{1/(n-1)}$ supported on $[0,n\mu]$.
\end{corollary}
\begin{proof}
We need to show that $a = 0$ and $s = 1$. Recall that we have $a = \mu - \mu(1-a)^n$, which becomes $0 = (1-a)^{n} + na - 1$ when $\mu = 1/n$. It is easy to see that this polynomial has a root at $a = 0$. For convenience, define $b = 1-a$, and after rearranging we obtain
\[\tag{$6$}\label{4}\begin{split}
\mu &= \frac{1-b}{1-b^{n}}\\
\end{split}\]
or, for $\mu = 1/n$,
\[\begin{split}
\frac{1}{n} &= \frac{1-b}{1-b^{n}}\\
\end{split}\]
Define $\varphi \coloneqq (1-b)/(1-b^{n}) - \mu$. Clearly, $\varphi$ is decreasing in $b$ for $b \in [0,1]$, and is therefore increasing in $a$ over the same domain. As a result, $a$ must be $0$. We can substitute this into $s = (1-a)^{n-1}$ and obtain that $s = 1$.
\end{proof}
We write the following result, which describes the effect of an increase in the number of players.
\begin{theorem}\label{lemlem}
Fix $\mu>1/n$. Then, if the number of players is increased, the weight placed on $1$ in the symmetric equilibrium must increase. That is, $a$ is strictly increasing in $n$. Moreover, $s$ is strictly decreasing in $n$.
In the limit, as the number of players, $n$, becomes infinitely large, the weight on $1$ converges to $\mu$. That is, the equilibrium distribution converges to a distribution with support on two points, $1$ and $0$.
\end{theorem}
\begin{proof}
Define $b = 1-a$ and $\varphi$ as above (the right hand side of \ref{4}). Recall that for $b \in [0,1]$, $\varphi$ is decreasing in $b$ and therefore increasing in $a$ over the same interval. Moreover, we make take the partial derivative with respect to $n$:
\[\frac{\partial{\varphi}}{\partial{n}} = \frac{(1-b)b^{n}\ln(b)}{(b^{n}-1)^{2}} < 0\]
Thus, as $n$ increases, the $a$ needed to satisfy the above expression must increase. That is, more and more weight is put on $1$. Concurrently, $s$, or the upper bound of the continuous portion of the distribution is shrinking, since, recall \[s = \frac{n(\mu-a)}{1-a}\]
and thus \[\frac{\partial{s}}{\partial{a}} = \frac{-n(1-\mu)}{(1-a)^{2}}\]Furthermore, as $n$ goes to infinity, we see that $a$ goes to $\mu$.
\end{proof}
An illustration of the relationship between $n$ and the symmetric equilibrium values of $a$ and $s$ is given in Figure \ref{figgypudding}.
\begin{figure}
\begin{center}
\includegraphics{randomvariables2.jpg}
\end{center}
\caption{\label{figgypudding} Effect of $n$ on $s$ and $a$ at Equilibrium for $\mu = 1/2$}
\end{figure}
\subsection{\texorpdfstring{$\mu < \frac{1}{n}$}{text}}\label{<}
We write, simply
\begin{theorem}\label{n<}
If $\mu < 1/n$ then the unique symmetric Nash equilibrium is for each player to play strategy $F_{i} \coloneqq \big(\frac{x}{n\mu}\big)^{1/(n-1)}$ supported on $[0,n\mu]$.
\end{theorem}
\begin{proof}
From Corollary \ref{core}, it is clear that this distribution is the unique symmetric Nash equilibrium for distributions restricted to $[0,n\mu]$. However, here we wish to show that this distribution is the unique symmetric Nash equilibrium even for deviations on the whole of $[0,1]$.
Consider the case where all $n$ players play strategy $S_{i}$, in which $X_{i}$ has distribution $F_{i}(x) = \big(\frac{x}{n\mu}\big)^{1/(n-1)}$. Suppose a player, say player $1$, deviates, and plays any other strategy $\hat{S}_{1} = G(y)$ supported on $[0,1]$ with mean $\mu$. We wish to show that the probability that $\max_{i \neq 1}X_{i} < Y$ is less than or equal to $1/n$.
For convenience define the new random variable $Z \coloneqq \max_{i \neq 1}X_{i}$. Evidently, $Z$ has distribution $H(z) = \frac{z}{n\mu}$, with associated density $h(z) = \frac{1}{n\mu}$.
Suppose for the sake of contradiction that this deviation is profitable for player $1$: $u_{1}(\hat{S}_{1},S_{-1}) > 1/n$. Then,
\[\begin{split}
u_{1}(\hat{S}_{1},S_{-1}) &= \big(1 - G(n\mu)\big) + \int_{0}^{n\mu}\int_{0}^{y}g(y)h(z)dzdy\\
&= \big(1 - G(n\mu)\big) + \int_{0}^{n\mu}\frac{y}{n\mu}g(y)dy\\
\end{split}\]
Thus, our supposition above holds if and only if
\[\begin{split}
\big(1 - G(n\mu)\big) + \int_{0}^{n\mu}\frac{y}{n\mu}g(y)dy &> \frac{1}{n}\\
\frac{1}{n} - \int_{n\mu}^{1}\frac{y}{n\mu}g(y)dy &> G(n\mu) - \frac{n-1}{n}\\
\end{split}\]
Or,
\[\begin{split}
1 &> G(n\mu) + \int_{n\mu}^{1}\frac{y}{n\mu}g(y)dy \geq G(n\mu) + \int_{n\mu}^{1}g(y)dy = 1\\
\end{split}\]
We have established a contradiction and thus we conclude that there is no profitable deviation. Uniqueness is immediate, following similar arguments to those used in proving Theorem \ref{n>}.
\end{proof}
\subsection{Any Positive Support}
We finish by solving for the unique equilibrium when players may choose any positive support for the random variable $X_{i}$. We write the following theorem.
\begin{theorem}Fix $\mu > 0$. The unique symmetric Nash equilibrium is for each player to play strategy $F_{i} \coloneqq \big(\frac{x}{n\mu}\big)^{1/(n-1)}$ supported on $[0,n\mu]$.
\end{theorem}
The proof is analogous to that for Theorem \ref{n<} and so is omitted. This problem, of course, is sensible should our interpretation be one of sellers choosing a distribution of qualities for a product. If on the other hand, our problem is one of competitive persuasion (with the Bayes-plausibility constraint relaxed), this environment (allowing for the support chosen to be any subset of $\Re_{+}$) is not appropriate.
\subsection{Arbitrary Ranking}
We can further generalize the model by characterizing the unique symmetric equilibrium for the situation where each player's objective is to have at least the $k$th highest realization (with ties settled randomly). This would correspond to the scenario with $k \geq 1$ buyers.
\section{Brief Discussion}
To put succinctly, in the game examined in this paper, the uniform distribution (or more generally, the distribution $F(x) = x^{1/(n-1)}$) is supreme. What is important is that the distribution of the random variable $\max_{j \neq i}X_{j}$; the distribution each player $i$ faces, is uniform. The intuition behind this result is simple; the uniform does not allow for one's opponent to achieve a payoff higher than $1/n$. If the exogenously given mean is too high; however, any continuous distribution on the interval falls vulnerable to deviation by putting a point mass at $1$ and so to counter this at equilibrium players must put a point mass on $1$.
As the number of players grows, the cutoff beyond which a point mass is necessary shrinks and the size of the point mass on $1$ grows. Moreover, as the number of players becomes infinitely large, each player will have to put all of the weight $\mu$ on the point $1$. If however, the players can choose any positive support for their distribution then they can continue to enlarge the interval of support as $n$ grows and they never have to include a point mass
Thus, we can see that there is interesting intuition that can be gleaned from this problem. If the mean is relatively small ($\mu < 1/n$), then the players do not use the top portion of their intervals at all, since that portion of the distribution is too ``valuable", so-to-speak, and is better spread out over the lower portion of the interval. If we think of the problem as one of a posterior distribution over prizes induced by signals, this means that there is no fully informative signal for the highest realization of the prize. This runs counter to the seminal result from \cite{kam}, where the highest state always induces the ``high" signal.
Then, as the number of players grows, players will use more and more of the top portion of the intervals. Furthermore, if the number of players is sufficiently high, or equivalently, if the mean becomes sufficiently large, then the players must put a point mass at $1$. As the number of players grows beyond the cutoff, each player will increase the weight on $1$ and the support of the continuous portion of the distribution will shrink. Finally, in the limit, equilibrium will consist of two point masses. Looking again at the problem as the choice of posteriors, we see that the players become increasingly ``honest" about the high realization. Consonantly, as $n$ grows (in the persuasion setting with a binary prior) we get full revelation in the limit--competition forces the players to reveal everything.
From the point of view of the consumer; then, an increase in the number of players is a good thing. They would ideally draw from distributions with the maximum variance, which is the distribution supported on the end points of the interval. Thus, as the number of players becomes infinitely large, the equilibrium converges to one that is consumer-optimal.
Overall, this paper can seen as an analysis of a competitive persuasion problem for the situation in which the agents have a common binary prior, or for the general competitive persuasion problem where the Bayes-plausibility constraint is relaxed. Moreover, our results apply to a variety of problems beyond information design. Our setup models the unconstrained version of competition between agents who each must mix or choose a mixture of some type or quality, which mixtures are each then randomly sampled from and the highest chosen.
|
1,116,691,497,605 | arxiv | \section{Introduction}
Topological phases have been the subject of extensive studies over the past few decades~\cite{Thouless1982,Thouless1983,Kane_and_Mele2005, ExperimentalRealization2007,chen2009experimental,chang2013experimental,Khanikaev2013,Gao2013,lv2015experimental,xu2015discovery,yang2015weyl,Chiu2016,he2017chiral,gao2018experimental,imhof2018topolectrical,schindler2018higher,xue2019acoustic,hofmann2020reciprocal}. Among them, Weyl semimetals are of particular interest~\cite{WSMPyrochloreIridates,TopologicalNodalSemimetals,WSMHeterostructure1,TimeReversalInvariantWSM,WSMTransitionMetalMonophosphides,WSMTaAs,WSMTaAs2,EMResponseWSM}. They are three-dimensional (3D) gapless topological phases of matter analogous to graphene, in which electrons behave at low energies as relativistic massless fermions. In a Weyl semimetal, the conduction and valence energy bands touch at a finite number of nodes. These Weyl nodes always come by pair, each carrying opposite chirality and having linear energy dispersion. These Weyl nodes are also robust against general perturbations which only shift their position in quasimomentum space~\cite{Yan2017WSM}, a different behaviour from Dirac cones in graphene where a general perturbation can open a gap. Additionally, Weyl semimetals are an intensely researched topic due to their exotic transport properties~\cite{Wang20173DQuantumHallSemiMetal,Igarashi2017MagnetotransportWSM} such as the chiral anomality effect~\cite{Nielsen1983WeylFermions,Aji2012ABJAnomaly,Son2013ChiralAnomaly,Burkov2014ChiralAnomaly,Gorbar2014ChiralAnomaly,Burkov2015ChiralAnomaly}, a large intrinsic anomalous Hall effect~\cite{Kuroda2017HallEffectWSM}, and perhaps the even more intriguing presence of open Fermi-arc surface states~\cite{WSMPyrochloreIridates,FermiArcSurfaceStates,WSMTaAs2,lv2015experimental,SuYang2015FermiArcs}. These properties rely on the topological nature of the Weyl nodes, which act as monopoles of the Berry curvature. Weyl semimetals were experimentally realized and the Fermi arcs were observed first in TaAs~\cite{WSMTaAs,WSMTaAs2} but since then have been realized in many other compounds~\cite{WSMTaP1,WSMTaP2,WSMNbAs}, heterostructures using topological insulator multilayers~\cite{WSMHeterostructure1,WSMHeterostructure2} or even more diverse metamaterials like in photonics~\cite{Lu2013PhotonicCrystalWSM,Feng2019PhotonicsWSM}.
Interaction effect is ubiquitous in nature, and therefore it is of utmost importance to explore its impact and interplay with Weyl points~\cite{InteractingWSMLattice,WSMStrongCoulombLimit,3DTopoPhaseBulkToBoundary,InteractingWeylFermions,EMResponseInteractingWSM,DynamicsElecTransportInteractingWSM}. However, treating interaction effect is challenging due to the exponential increase in the Hilbert space dimension with the system size. Therefore, a mean field approximation is employed to turn a strongly interacting Weyl semimetal into an effective single particle but nonlinear Weyl semimetal. This kind of mean-field approach is especially relevant to describe the behavior of cold-atom systems such as Bose-Einstein condensates~\cite{Gross1961BEC_GP,Pitaevskii1961BEC_GP,Burger1999DarkSolitonsBEC,Denschlag2000GenerateSolitonsPhaseEngiBEC,Biao2003Review_BEC,Bleu2016ExampleMeanFieldBEC,Watanabe2016ReviewUltracoldAtoms}. Apart from being a mean-field approximation to a strongly interacting system, such nonlinear treatment also naturally appears in photonics setting with the optical Kerr effect, and has received considerable attention~\cite{Lu2013PhotonicCrystalWSM,Lumer2013SelfLocStatesPhotnicTI,Plotnik2013PhotonicGraphene,Morimoto2016TopologicalNatureNLOpticalEffectsSolids,Xin2017KerrEffectNonlinearTopologicalPhotonics,Hadad2018ExampleNonlinearProtectionPossiblePhaseTransition,Feng2019PhotonicsWSM,Smirnova2019NonlinearTopologicalPhotonics}. However, a comprehensive study of nonlinear Weyl semimetals has remained elusive~\cite{WU2011AnomalousMonopoles,NLandTopology}.
\textcolor{black}{In view of the above,} we consider here a few variations of a minimal Weyl semimetal lattice model with two Weyl nodes, then investigate the effect of adding an on-site nonlinearity. \textcolor{black}{While such systems are inherently 3D, the advent of synthetic dimensions~\cite{Jukic20134DPhotonics,Luo2015Synthetic2DPhotonics,Yuan2016SyntheticDimensions,Ozawa2016SyntheticDimensions,Yuan2018SyntheticDimensions,Lustig2019SyhtheticDimensions,Dutt2020SyntheticDimensions} allows the simulation of some quasimomenta with artificial periodic parameters. As a result, the systems under study can in principle be realized in lower dimensional photonic waveguides with Kerr nonlinearity.}
By investigating the systems in momentum space under periodic boundary conditions (PBC), \textcolor{black}{we find that nonlinearity breaks down a Weyl point into nodal lines and nodal surfaces. By evaluating the Chern number of a two-dimensional (2D) surface enclosing these nodal structures, we further confirm that the topological charge of the original Weyl point is preserved. Depending on the system under study,} additional nodal lines may emerge at high nonlinear strength. \textcolor{black}{Our further inspection reveals that these nodal lines are in fact a higher dimensional generalization of the nonlinear Dirac cones discovered in Ref.~\cite{NLDC} and will thus be referred to as \emph{nonlinear Dirac lines}}.
\textcolor{black}{To capture the topological properties of the obtained nodal structures, we propose two complementary methods. First, by adapting the theory of adiabatic pumping~\cite{Thouless1982,Thouless1983,Ke2016ThoulessPumping,Lohse2016ThoulessPumping,Nakajima2016ThoulessPumping,Tangpanitanon2016NLTopoPumping,Hayward2018NLPumping,TopoPumpingBlochOscillations} to the nonlinear setting, we show that the pumped charge is a sum of a term proportional to the system's Chern number and another term that depends on the nonlinear strength. In this case, adiabatic pumping can thus be utilized not only to probe the topological charge of a given nodal structure, but also to determine the strength of nonlinearity in the system, which in turn informs the shape of the nodal structure based on our energy band analysis. However, such a method is not suitable to capture the presence of nonlinear Dirac lines, which carry zero Chern number. Motivated by Ref.~\cite{NLDC}, we thus propose to employ an Aharonov-Bohm (AB) interference experiment as an alternative method to probe such nonlinear Dirac lines.}
This paper is organized as follows. In Sec.~\ref{section:Nonlinear Weyl semimetals}, {\textcolor{black} we introduce our nonlinear Weyl semimetals, and present our results detailing the effect of nonlinearity on the systems' band structure. Major results include the breaking down of Weyl points into nodal lines and nodal surfaces, understanding how their shapes develop as nonlinearity increases, as well as the robustness of these nodal structures against general perturbation. In some of the systems under study, we further reveal the emergence of nonlinear Dirac lines, which are topologically different from the nodal lines arising from the breaking down of Weyl points.} In Sec.~\ref{section:Experimental characterization of nodal structures}, we propose two complementary experimental setups to probe the nodal structures in nonlinear Weyl semimetals.
\textcolor{black}{Specifically, by evaluating the particle's displacement in an adiabatic pumping experiment and comparing with the analytical expression derived below, the topological charge and shape of a particular nodal structure that originates from a Weyl point can in principle be obtained. By performing an AB interference experiment, nonlinear Dirac lines can further be probed}. In Sec.~\ref{section:Concluding remarks}, we summarize the main findings of this paper and discuss prospects for possible future work.
\section{Nonlinear Weyl semimetals}
\label{section:Nonlinear Weyl semimetals}
\subsection{Nonlinearity as a mean-field approximation to interaction}
\label{section:Nonlinearity as a mean-field approximation to interaction}
We consider a nonlinear version of a two-band Weyl semimetal phase, where two sublattices serve as the pseudospin. It can be described by the stationary Gross-Pitaevskii (GP) equation, which writes in momentum space as
\begin{equation}
\label{eqn:GPequation}
H(\mathbf{k},\ket{\psi(\mathbf{k})}) = E(\mathbf{k}) \ket{\psi(\mathbf{k})}
\end{equation}
where $\mathbf{k} = (k_x,k_y,k_z)$ is the 3D quasimomentum and
\begin{multline}
H(\mathbf{k},\ket{\psi(\mathbf{k})}) = h_x(\mathbf{k}) \, \sigma_x + h_y(\mathbf{k}) \, \sigma_y + h_z(\mathbf{k}) \, \sigma_z \\ + g\begin{pmatrix} \left|\psi_{1}(\mathbf{k})\right|^2 & 0 \\ 0 & \left|\psi_{2}(\mathbf{k})\right|^2 \end{pmatrix}
\label{eqn:NWSM}
\end{multline}
where $\sigma$'s are the Pauli matrices in the standard representation, $\ket{\psi(\mathbf{k})} = (\psi_{1}(\mathbf{k}),\psi_{2}(\mathbf{k}))^{T}$ is a Bloch state with two pseudospinor components, and $g$ is the nonlinear strength. In this work, all physical variables are assumed to be scaled, and therefore are in dimensionless units. The nonlinear term in the above Hamiltonian can be used to represent the effect of bosonic mean-field interactions, or the Kerr effect naturally arising in photonic systems. By defining $\Sigma(\mathbf{k}) = \left|\psi_{2}(\mathbf{k})\right|^2 - \left|\psi_{1}(\mathbf{k})\right|^2$, we can rewrite the nonlinear Hamiltonian in the more convenient form
\begin{equation}
H(\mathbf{k},\ket{\psi(\mathbf{k})}) = \frac{g}{2} I_2 + h_x(\mathbf{k}) \, \sigma_x + h_y(\mathbf{k}) \, \sigma_y + (h_z(\mathbf{k}) - \frac{g}{2}\Sigma(\mathbf{k})) \sigma_z.
\label{eqn:NWSMSigma}
\end{equation}
\textcolor{black}{In the following, three distinct nonlinear Weyl semimetal systems will be considered. In the first system, which will be referred to as \emph{the perpendicular case}, its Weyl points lie along a line of $h_y=h_z=0$. In the second system, which will be referred to as \emph{the parallel case}, its Weyl points lie along a line of $h_x=h_y=0$. Finally, the third system describes the general case in which the Weyl points lie along a line of $h_x,h_y,h_z\neq 0$. It is worth noting that, in the linear limit, the three systems above are physically equivalent and are unitarily related to one another. Remarkably, in the presence of onsite nonlinearity, the three systems exhibit fundamentally different band structure properties as further elaborated below.}
\subsection{Perpendicular case}
\label{section:Perpendicular case}
This part focuses on a nonlinear Weyl semimetal whose Hamiltonian in quasimomentum space is given by Eq.~(\ref{eqn:NWSMSigma}) with $h_x(\mathbf{k}) = (M + \cos{k_x} + \cos{k_y} + \cos{k_z})$, $h_y(\mathbf{k}) = \sin{k_y}$, $h_z(\mathbf{k}) = \sin{k_z}$ and $M = 2$.
The corresponding linear model is a Weyl semimetal phase, exhibiting two Weyl points at $A\equiv (\frac{\pi}{2},\pi,\pi)$ and $B\equiv (-\frac{\pi}{2},\pi,\pi)$. Points A and B respectively have negative and positive chirality, the Chern number around them being -1 and +1.
Based on the known eigenstate solutions for two-level systems, a stationary state of this Hamiltonian is found to satisfy the self consistency equation (see Appendix~\ref{app:A})
\begin{equation}
(\frac{g}{2})^2 \Sigma^4 - g h_z \Sigma^3 + [h_x^2 + h_y^2 + h_z^2 - (\frac{g}{2})^2] \Sigma^2 + g h_z \Sigma - h_z^2 = 0,
\label{eqn:SelfConsistency}
\end{equation}
showing that the nonlinear system can have up to 4 energy bands.
\begin{figure}
\includegraphics[width=0.9\linewidth]{BandsPerpendicularLegends.png}
\caption{\textcolor{black}{The system's energy bands at different nonlinear strengths.} \textcolor{black}{The third band $E_3$, in yellow, is not visible as it is a flat band, degenerate with $E_4$ at every point in the Brillouin zone where it exists.} For all sub-figures we fixed $k_z = \pi$. a) is $g=1$. b) is $g=3$. c) is $g=6$.}
\label{fig:PerpendicularBandsZ=pi}
\end{figure}
In Fig.~\ref{fig:PerpendicularBandsZ=pi}, we show the system’s band structure at three representative nonlinear strengths. \textcolor{black}{As soon as we add nonlinearity ($g>0$), we notice the following two phenomena. First, while the system can now support up to four energy bands $E_1 \leq E_2 \leq E_3 \leq E_4$, the bands $E_2$ and $E_3$ \textcolor{black}{do not span across the whole the Brillouin zone: they exist} only for a small region near the original Weyl points (see green colored bands in Fig.~\ref{fig:PerpendicularBandsZ=pi}-a)). As the nonlinear strength increases, these intermediate bands progressively grow until they merge together as shown in Fig~\ref{fig:PerpendicularBandsZ=pi}-b). Upon further increase in the nonlinear strength, the newly merged band structure grows and eventually spans the entire Brillouin zone as shown in Fig.~\ref{fig:PerpendicularBandsZ=pi} -c).}
\textcolor{black}{Second, the proliferation of the number of bands in the system from $2$ to $4$ also yields the splitting of each Weyl point into two different nodal structures separated in energy. The shape of these nodal structures is presented in Fig.~\ref{fig:PerpendicularNodal} for two representative nonlinear strength. Specifically, we find that the lower band touching structure, i.e., $E_1=E_2$, occurs along a 1D line, whereas the upper band touching structure, i.e., $E_3=E_4$, forms a 2D surface. As the nonlinear strength increases, these nodal lines and nodal surfaces eventually merge, as shown in Fig.~\ref{fig:PerpendicularNodal}-b). The analytical derivation of these nodal structures is presented in appendix~\ref{app:A}.}
\begin{figure}
\includegraphics[width=\linewidth]{NodalPerpendicular.png}
\caption{The shape of the nodal lines (blue) and nodal surfaces (red) in the 3D Brillouin zone at (a) $g=1$ and (b) $g=3$. The nodal lines (nodal surfaces) correspond to the band touching points at $E_1 = E_2 = \frac{g}{2}$ ($E_3 = E_4 = g$).}
\label{fig:PerpendicularNodal}
\end{figure}
\textcolor{black}{We will now discuss the properties of the above nodal structures in more detail}. In a conventional Weyl semimetal such as the corresponding linear Hamiltonian to the one under study, the energy band dispersion around a Weyl point is given by
\begin{equation}
E_{\pm} = E_0 \pm \sqrt{\kappa_x^2 + \kappa_y^2 + \kappa_z^2}
\end{equation}
where $E_0$ is the energy at the Weyl point and $(\kappa_x,\kappa_y,\kappa_z)$ are small displacements away from the Weyl points along the 3 axis of the 3D Brillouin zone. Accordingly, the dynamics around such a Weyl points are described by the effective Hamiltonian
\begin{equation}
h_{\text{eff}} = \kappa_x \, \sigma_x + \kappa_y \, \sigma_y + \kappa_z \, \sigma_z,
\end{equation}
which is closely related to Weyl's equation from particle physics, effectively describing a massless fermion, dubbed Weyl fermion.
Using perturbative expansion around the original Weyl points, we are able to study the effect of nonlinearity on the energy bands' dispersion. Let us consider the band touching point $(\frac{\pi}{2},\pi,\pi)$, where we can find degenerate energy solutions $E_1 = E_2 = \frac{g}{2}$ \textcolor{black}{and $E_3=E_4=g$}. \textcolor{black}{As detailed in Appendix~\ref{app:B},} at a point $(\frac{\pi}{2}+\kappa_x,\pi+\kappa_y,\pi+\kappa_z)$ near this band touching point, the two lowest and two highest energy bands are respectively modified to become
\begin{eqnarray}
\label{eqn:DispersionPerpendicular}
E_{\pm}^{(l)} &=& \frac{g}{2} \pm \sqrt{\kappa_x^2 + \kappa_y^2} \;, \nonumber \\
E_{\pm}^{(h)}&=& g \pm \kappa_z \;
\end{eqnarray}
\textcolor{black}{up to first order in $\kappa_x,\kappa_y$, and $\kappa_z$}, where + and - respectively designate the upper energy bands ($E_2$ and $E_4$) and the lowest energy bands ($E_1$ and $E_3$). These energy solutions can be regarded as eigenstates of the effective Hamiltonians
\begin{eqnarray}
\label{eqn:heffPerpendicular}
h_{\text{eff},\pm}^{(l)} &=& \frac{g}{2} I_2 -\kappa_x \, \sigma_x - \kappa_y \, \sigma_y \pm \frac{2 \kappa_z}{g} \sqrt{\kappa_x^2 + \kappa_y^2} \sigma_z\;, \nonumber \\
h_{\text{eff},\pm}^{(h)} &=& \frac{g}{2} I_2 -\kappa_x \, \sigma_x - \kappa_y \, \sigma_y + (-\kappa_z \pm \frac{g}{2}) \sigma_z \;.
\end{eqnarray}
That each energy solution has its own effective Hamiltonian is simply a consequence of the state-dependence of the nonlinear Hamiltonian. While not explicitly shown in this paper, we get similar energy bands dispersion and effective Hamiltonian around the other Weyl point at $(-\frac{\pi}{2},\pi,\pi)$.
\textcolor{black}{It is particularly interesting to note that the nodal lines only exhibit linear dispersion in the $x$- and $y$-directions, whereas the nodal surfaces only exhibit linear dispersion in the $z$-direction. When viewed together, both nodal lines and nodal surfaces carry the linear dispersion along all three quasimomenta directions as expected from their parent Weyl points. It is also worth noting that while these nodal structures individually only exhibit partial linear dispersion, they inherit the robustness of the original Weyl points. Indeed, a perturbation proportional to any of the Pauli matrices will simply displace the locations of the original Weyl points away from $(\pm \frac{\pi}{2},\pi,\pi)$. The same analysis presented in Appendix~\ref{app:B} can then be repeated in the vicinity of the displaced Weyl points to yield exactly the same energy dispersion and effective Hamiltonians as Eqs.~(\ref{eqn:DispersionPerpendicular}) and (\ref{eqn:heffPerpendicular}) respectively. This robustness can also be attributed from the fact that each of these nodal structures carries a topological charge, which can be evaluated through the Chern number of a 2D surface enclosing it.}
\subsection{Parallel case}
\label{section:Parallel case}
\textcolor{black}{We now turn to the second system, which is again described by the Hamiltonian of the form Eq.~(\ref{eqn:NWSM}), but with $h_x(\mathbf{k}) = -\sin{k_z}$, $h_y(\mathbf{k}) = \sin{k_y}$, and $h_z(\mathbf{k}) = (M + \cos{k_x} + \cos{k_y} + \cos{k_z})$ and $M=2$. Similarly to the perpendicular case studied in Sec.~\ref{section:Perpendicular case}, at $g=0$, the system exhibits two Weyl points along the $k_x$-axis at $(\frac{\pi}{2},\pi,\pi)$ and $(-\frac{\pi}{2},\pi,\pi)$, which respectively have -1 and +1 chirality.}
Using the self-consistency equation given in Eq.~(\ref{eqn:SelfConsistency}), \textcolor{black}{we find that as we increase nonlinear strength, the energy bands structure develops in a similar manner as the perpendicular case, where the two full energy bands proliferate into four bands, two of which initially only exist in the vicinity of the Weyl points, splitting the latter into pairs of nodal lines and nodal surfaces (see Fig.~\ref{fig:ParallelNodal}-a)) at $E_1 = E_2 = \frac{g}{2}$ and $E_3 = E_4 = g$ respectively. As the nonlinearity increases, these nodal lines and nodal surfaces grow and eventually merge together as depicted in Fig.~\ref{fig:ParallelNodal}-b).}
\begin{figure}
\includegraphics[width=\linewidth]{NodalParallel.png}
\caption{The shape of the nodal lines (blue) and nodal surfaces (red) in the 3D Brillouin zone at (a) $g=1$ and (b) $g=3$. The nodal lines (nodal surfaces) correspond to the band touching points at $E_1 = E_2 = \frac{g}{2}$ ($E_3 = E_4 = g$).}
\label{fig:ParallelNodal}
\end{figure}
\textcolor{black}{Despite the aforementioned similarities, two striking differences between the present system and that of Sec.~\ref{section:Perpendicular case} exist. First, the nodal lines are aligned along a direction parallel (perpendicular) to the separation between Weyl points in the present system (the system of Sec.~\ref{section:Perpendicular case}), thus explaining the name ``parallel case" (``perpendicular case") when referring to such a system. Mathematically,} taking once again the example of a Weyl point at $(\frac{\pi}{2},\pi,\pi)$, for a point $(\frac{\pi}{2}+\kappa_x,\pi+\kappa_y,\pi+\kappa_z)$ near this original Weyl point, the energies can be pertubatively expanded as
\textcolor{black}{
\begin{eqnarray}
\label{eqn:DispersionParallel}
E_{\pm}^{(l)} &=& \frac{g}{2} \pm \sqrt{\kappa_y^2 + \kappa_z^2} \;, \nonumber \\
E_{\pm}^{(h)}&=& g \pm \kappa_x \;
\end{eqnarray}
}\textcolor{black}{up to first order in $\kappa_x,\kappa_y$, and $\kappa_z$.} \textcolor{black}{This corresponds to the effective Hamiltonians
\begin{eqnarray}
\label{eqn:heffParallel}
h_{\text{eff},\pm}^{(l)} &=& \frac{g}{2} I_2 + \kappa_z \, \sigma_x - \kappa_y \, \sigma_y \pm \frac{2 \kappa_x}{g} \sqrt{\kappa_y^2 + \kappa_z^2} \sigma_z\;, \nonumber \\
h_{\text{eff},\pm}^{(h)} &=& \frac{g}{2} I_2 + \kappa_z \, \sigma_x - \kappa_y \, \sigma_y + (-\kappa_x \pm \frac{g}{2}) \sigma_z \;.
\end{eqnarray}
}
Second, as shown in Fig.~\ref{fig:ParallelNodal}-b), the parallel case sports additional nodal lines (nonlinear Dirac lines) at large enough nonlinear strength, not connected to one of the original Weyl points. A snapshot of the system's band structure at a fixed $k_x$, as presented in Fig.~\ref{fig:ParallelBands}, highlights the resemblance of such nodal lines with the nonlinear Dirac cones discovered in Ref.~\cite{NLDC}. In Fig.~\ref{fig:ParallelNodal}-b), we show that these nonlinear Dirac lines are located at $(k_y,k_z) = (0,\pi)$ and $(k_y,k_z) = (\pi,0)$. At an even higher nonlinear strength, an additional nonlinear Dirac line emerges at $(k_y,k_z) = (0,0)$. \textcolor{black}{The energy dispersion around these nonlinear Dirac lines is the same as for the two original nodal lines. For example, for a point $(\pi + \kappa_x , \kappa_y , \pi + \kappa_z)$ near one of these nonlinear Dirac lines, the energy dispersion is
\begin{equation}
\label{eqn:EnergyDispersionNDL}
E^{(\text{NDL})}_{\pm} = \frac{g}{2} \pm \sqrt{\kappa_y^2 + \kappa_z^2},
\end{equation}
while the effective Hamiltonian is
\begin{equation}
\label{eqn:EffectiveHNDL}
h_{\text{eff},\pm}^{(\text{NDL})} = \frac{g}{2} I_2 + \kappa_z \, \sigma_x + \kappa_y \, \sigma_y \mp \frac{2}{g} \sqrt{\kappa_y^2 + \kappa_z^2} \sigma_z.
\end{equation}}
\begin{figure}
\includegraphics[width=\linewidth]{Bands_x=pi_g=3,0_with_legends.png}
\caption{Multiple loop energy bands due to the apparition of nonlinear Dirac lines at high nonlinear strength in the parallel case. Parameters are $k_x = \pi$ and $g=3$. }
\label{fig:ParallelBands}
\end{figure}
While the nonlinear Dirac lines have the same orientation as the nodal lines that originate from Weyl points, they are different in nature. This is confirmed by evaluating their topological charge, via Chern number calculation through an enclosing surface. In the case of the nodal lines corresponding to a linear Weyl point, appearing as soon as $g > 0$ at $(k_y,k_z)=(\pi,\pi)$, the topological charge is the same as the original Weyl point, -1 for the line going through point $(\frac{\pi}{2},\pi,\pi)$ and +1 for the one going through point $(-\frac{\pi}{2},\pi,\pi)$. On the other hand, repeating the same calculation with respect to a nonlinear Dirac line yields a result of 0, showing that it holds no topological charge. However, \textcolor{black}{as these additional nodal lines are the higher-dimensional generalization of the nonlinear Dirac cones in Ref.~\cite{NLDC}, they are expected to be equally robust against perturbations.}
\subsection{General case}
\label{section:General case}
It is possible to extend the two models presented above to a more general one, by using Pauli transformation in the linear model, before applying the onsite nonlinearity. Taking the perpendicular case Hamiltonian as our starting point, and applying a rotation of angle $\theta$ on $\sigma_x$ and $\sigma_z$ to get $\sigma_x' = \cos{\theta} \, \sigma_x + \sin{\theta} \, \sigma_z$ and $\sigma_z' = \cos{\theta} \, \sigma_z - \sin{\theta} \, \sigma_x$. This again yields the Hamiltonian of Eq.~(\ref{eqn:NWSMSigma}), but with
\begin{equation}
\begin{aligned}
h_x(\mathbf{k},\theta) &= (M + \cos{k_x} + \cos{k_y} + \cos{k_z}) \cos{\theta} - \sin{k_z} \sin{\theta} ,\\
h_y(\mathbf{k}) &= \sin{k_y} ,\\
h_z(\mathbf{k},\theta) &= \sin{k_z} \cos{\theta} + (M + \cos{k_x} + \cos{k_y} + \cos{k_z}) \sin{\theta},
\end{aligned}
\end{equation}
and $M = 2$. It is immediate to verify that for $\theta = 0$ and $\theta = \frac{\pi}{2}$, we respectively recover the perpendicular case and the parallel case presented earlier.
\begin{figure}
\includegraphics[width=\linewidth]{NodalAngles_g=1_and_2,5.png}
\caption{\textcolor{black}{Nodal structures of the general nonlinear Weyl semimetal in the 3D Brillouin zones. In a-d), the 1D nodal curves in blue correspond to the band touching points between the lowest full band and the lower loop band at $E_1 = E_2 = \frac{g}{2}$, and the 2D nodal surfaces, in red, represent the band touching surfaces between the higher loop band and the highest full band at $E_3 = E_4 = g$. In subfigures a) and b), we take $g=1$, and in subfigures c) and d) we take $g=2.5$, a nonlinear strength high enough for the nodal structures to merge. a) and c) are $\theta = \frac{\pi}{6}$. b) and d) are $\theta = \frac{\pi}{3}$. e) is a schematic representation of the nodal lines' direction around both original Weyl points A and B.} }
\label{fig:NodalAngles}
\end{figure}
\textcolor{black}{While such a Pauli rotation does not change the physics of the linear system, it leads to the change in orientation} of both the nodal lines and nodal surfaces in the presence of nonlinearity, as shown in Fig.~\ref{fig:NodalAngles}-a) and b). Figure~\ref{fig:NodalAngles}-c) more explicitly highlights the orientation of the nodal lines in the $x-z$ plane in terms of the angle $\theta \notin \{0,\frac{\pi}{2}\}$. \textcolor{black}{Up to first order in $\kappa_x,\kappa_y,$ and $\kappa_z$,} the energy dispersions around the two points $A\equiv (\frac{\pi}{2},\pi,\pi)$ and $B\equiv (-\frac{\pi}{2},\pi,\pi)$ are
\begin{equation}
\begin{aligned}
E^{(l)}_{A,\pm} &= \frac{g}{2} \pm \sqrt{(-\kappa_x \cos{\theta} + \kappa_z \sin{\theta})^2 + \kappa_y^2} , \\
E^{(l)}_{B,\pm} &= \frac{g}{2} \pm \sqrt{(\kappa_x \cos{\theta} + \kappa_z \sin{\theta})^2 + \kappa_y^2} ,
\end{aligned}
\end{equation}
which means that \textcolor{black}{the nodal lines are aligned along}:
\begin{equation}
\text{For A: } \mathbf{k_A} = \begin{pmatrix} \sin{\theta} \\ 0 \\ \cos{\theta} \end{pmatrix} \quad
\text{For B: } \mathbf{k_B} = \begin{pmatrix} -\sin{\theta} \\ 0 \\ \cos{\theta} \end{pmatrix}.
\end{equation}
\textcolor{black}{
The corresponding effective Hamiltonians are
\begin{equation}
\label{eqn:heffLowNodal}
\begin{aligned}
&h_{\text{eff},A,\pm}^{(l)} = \frac{g}{2} I_2 + (-\kappa_x \cos{\theta} + \kappa_z \sin{\theta}) \sigma_x - \kappa_y \, \sigma_y \\ & \pm \frac{2 (\kappa_z \cos{\theta} + \kappa_x \sin{\theta})}{g} \sqrt{(-\kappa_x \cos{\theta} + \kappa_z \sin{\theta})^2 + \kappa_y^2} \, \sigma_z \;, \\
&h_{\text{eff},B,\pm}^{(l)} = \frac{g}{2} I_2 + (\kappa_x \cos{\theta} + \kappa_z \sin{\theta}) \sigma_x - \kappa_y \, \sigma_y \\ & \pm \frac{2 (\kappa_z \cos{\theta} - \kappa_x \sin{\theta})}{g} \sqrt{(\kappa_x \cos{\theta} + \kappa_z \sin{\theta})^2 + \kappa_y^2} \, \sigma_z \;.
\end{aligned}
\end{equation}
}\textcolor{black}{Nonlinearity thus provides a means to generate exotic nodal lines along any arbitrary direction in the Brillouin zone.}
\textcolor{black}{In the vicinity of the original Weyl points, the nodal surfaces also undergo the same rotation, remaining orthogonal to their corresponding nodal line, with the energy dispersions
\begin{equation}
\begin{aligned}
E^{(h)}_{A,\pm} &= g \pm (-\kappa_z \cos{\theta} - \kappa_x \sin{\theta}), \\
E^{(h)}_{B,\pm} &= g \pm (-\kappa_z \cos{\theta} + \kappa_x \sin{\theta}),
\end{aligned}
\end{equation}
corresponding to the effective Hamiltonians
\begin{equation}
\label{eqn:heffHighNodal}
\begin{aligned}
h_{\text{eff},A,\pm}^{(h)} &= \frac{g}{2} I_2 + (-\kappa_x \cos{\theta} + \kappa_z \sin{\theta}) \sigma_x - \kappa_y \, \sigma_y \\ & + (-\kappa_z \cos{\theta} - \kappa_x \sin{\theta} \pm \frac{g}{2}) \, \sigma_z\;, \\
h_{\text{eff},B,\pm}^{(h)} &= \frac{g}{2} I_2 + (\kappa_x \cos{\theta} + \kappa_z \sin{\theta}) \sigma_x - \kappa_y \, \sigma_y \\ & + (-\kappa_z \cos{\theta} + \kappa_x \sin{\theta} \pm \frac{g}{2}) \, \sigma_z \;.
\end{aligned}
\end{equation}
As shown in Fig.~\ref{fig:NodalAngles}-c) and d), the nodal structure eventually \textcolor{black}{ends up} merging as nonlinear strength increases, however, the process can wildly differ depending on the choice of angle $\theta$. Although in some cases like $\theta = \frac{\pi}{3}$ it leads to the apparition of nonlinear Dirac lines just like in the parallel case, as shown in Fig.~\ref{fig:NodalAngles}-d), in others such as $\theta = \frac{\pi}{6}$ we witness the creation of additional nodal surfaces not connected to one of the original Weyl points as shown in Fig.~\ref{fig:NodalAngles}-c). Just like the nonlinear Dirac lines, these additional nodal surfaces do not hold any topological charge.
}
\subsection{Conservation of Fermi-arcs}
\textcolor{black}{One of the most peculiar properties of Weyl semimetals is the presence of Fermi-arc surface states connecting the Weyl points. It is then natural to investigate \textcolor{black}{the profiles of such Fermi-arcs in our nonlinear Weyl semimetal model,especially how they change with nonlinearity strength.} For this purpose we consider the system in the general case, i.e., $\theta\in [0,\pi/2]$, but this time we take an infinite slab, infinite along the $x$ and $y$ directions, and finite along the $z$ direction. Using Bloch theorem with the good quantum numbers $k_x,k_y$, we can then study the model as a 1D lattice in real space, along the $z$ direction, whose equations of motion are given in appendix~\ref{app:D}. In the linear case, regardless of $\theta$, the Fermi-arcs consist of degenerate zero energy states along the $x$-axis connecting the two Weyl points. Using an iterative method also described in appendix~\ref{app:D}, we are able to obtain the energy spectrum of the 1D chain in the nonlinear case. We did so under two types of boundary conditions along the $z$ direction, open boundary condition (OBC) and periodic boundary condition (PBC).}
\begin{figure}
\includegraphics[width=\linewidth]{FermiArcs.png}
\caption{Energy spectrum of the 1D chain along the $z$-axis in real space, assuming PBC and Bloch wave solutions along the $y$ and $z$ axis. Each row corresponds to a different value of the nonlinear strength $g$, while the left and right columns correspond respectively to OBC and PBC along the $x$-axis. Parameters are $\theta = 0$ and $k_y = \pi$, taking $N=20$ unit cells along the $z$-axis. \textcolor{black}{Similar results are observed for other values of $\theta$.}}
\label{fig:FermiArcs}
\end{figure}
\textcolor{black}{The energy spectra for increasing nonlinear strength and both types of boundary conditions are shown in Fig.~\ref{fig:FermiArcs}. The one striking result we observe is the persistence of degenerate edge states, which are however no longer pinned at zero energy, for $\frac{\pi}{2} \leq k_x \leq \frac{3 \pi}{2}$. The fact that these edge states can be observed under OBC but not under PBC shows that they are topological in nature, and is further proof that the Weyl points indeed retain their topological features in the presence of nonlinearity.}
\section{Experimental characterization of nodal structures}
\label{section:Experimental characterization of nodal structures}
\subsection{Thouless pumping for detection of nonlinear Weyl points}
\label{section:Thouless pumping for detection of nonlinear Weyl points}
Weyl nodes in Weyl semimetals can already be interpreted as magnetic monopoles for the Berry curvature, and their topological charge can be evaluated by integrating the flux of said Berry curvature through a closed surface enclosing a Weyl node. There is however another way to understand this 2D closed surface of integration; it can be interpreted as describing a band insulator in two dimensions. Indeed, the two parameters needed to describe the enclosing surface can be considered as quasimomenta for the Hamiltonian of a 2D insulator. The corresponding system will be gapped everywhere since the surface is taken to avoid Weyl points. The topology of such 2D insulator can then be characterized by the Chern number, which will be exactly the flux of the Berry curvature through the surface.
Here, we extend this idea to the 2D surfaces enclosing the nodal structures in Weyl semimetals. For small enough nonlinear stength, it is still possible to define a closed surface around one of the original Weyl points that encloses the full nodal structures, whether they are 1D (nodal lines) or 2D (nodal surfaces).
As an example, let us consider the perpendicular case of Sec.~\ref{section:Perpendicular case}, around point $B(\frac{3 \pi}{2},\pi,\pi)$. As shown on Fig.~\ref{fig:PerpendicularNodal}-a), a straightforward choice for the enclosing surface would be a cylinder centered on point $B\equiv(-\frac{\pi}{2},\pi,\pi)$, with axis of revolution along the $k_z$ direction, and a radius $\rho$ taken large enough to enclose the entire nodal structures. Points on such a cylinder can be described with two parameters $(\phi,z)$ as $k_x = \frac{3 \pi}{2} + \rho \cos{\phi} \, , \, k_y = \pi + \rho \sin{\phi} \, , \, k_z = z$. Due to the periodicity of quasimomenta, this cylinder is actually a torus, and the two parameters $(\phi,z)$ can be immediately taken as the two quasimomenta in a 2D Brillouin zone. This gives us the following Hamiltonian for a 2D Chern insulator
\begin{multline}
H_{\text{2D}}(k_x,k_y,\ket{\psi(k_x,k_y)}) = \frac{g}{2} I_2 + h_x(k_x,k_y) \, \sigma_x + h_y(k_x) \, \sigma_y \\ + (h_z(k_y) - \frac{g}{2}\Sigma(k_x,k_y)) \sigma_z
\label{eqn:H2DChern}
\end{multline}
where
\begin{equation}
\begin{aligned}
h_x(k_x,k_y) &= M + \sin{(\rho \cos{k_x})} - \cos{(\rho \sin{k_x})} + \cos{k_y} \\
h_y(k_x) &= -\sin{(\rho \sin{k_x})} \\
h_z(k_y) &= \sin{k_y} \\
\end{aligned}
\end{equation}
and $M=2$.
One convenient way to realize such a system is to actually substitute one of the spatial dimensions for a slow time modulation, realizing a Chern insulator in 1+1 dimensions. \textcolor{black}{In the following, we take $k_y$ as the physical quasimomentum and $k_x = \omega t$ as the time periodic parameter. The resulting system is known as an adiabatic charge pump, and was originally proposed by Thouless~\cite{Thouless1982,Thouless1983}.}
\textcolor{black}{In the linear setting, it is known that the average displacement of a particle during one adiabatic cycle in time is equal to the Chern number of the associated 2D Chern insulator. However, nonlinearity is known to nontrivially modify the adiabatic evolution of a system~\cite{NLZP}. In another recent work, we have indeed shown that such modification leads to a nonlinear correction term that generally no longer quantizes the adiabatic pumping result}\textcolor{black}{~\cite{NonlinearAdiabaticPumping}. Considering the average displacement of a particle as
\begin{equation}
\Delta \left< x \right> = \int \, dt \left< \Bar{v} \right>
\label{eqn:Delta}
\end{equation}
with
\begin{equation}
\label{eqn:vbar}
\begin{aligned}
\left< \Bar{v} \right> &= \frac{1}{2 \pi} \int_{-\pi}^{\pi} \, dk \left< v \right> \\
&= \frac{1}{2 \pi} \int_{-\pi}^{\pi} \, dk \left< \frac{\partial H}{\partial k} \right>
\end{aligned}
\end{equation}
where $\left< ... \right>$ represents the average over the state at a given $(k,t)$, we show in Ref.~\cite{NonlinearAdiabaticPumping} that due to nonlinear dynamics, we have
\begin{equation}
\begin{aligned}
\Delta \left< x \right> &= \frac{1}{2 \pi} \int dt \int_{-\pi}^{\pi} dk \left[ \mathcal{B}(k,t) + \mathcal{D}(k,t) \right] \;,
\end{aligned}
\label{eqn:Displacement}
\end{equation}
where $\mathcal{B}(k,t)$ is the Berry curvature and
\begin{equation}
\mathcal{D}(k,t) = \frac{g \sin^3{\theta}}{2E - g \sin^2{\theta}} \frac{\partial \phi}{\partial t} \frac{\partial}{\partial k} \left( \frac{\theta}{2} \right)
\label{eqn:Drift}
\end{equation}
is a correction to the Berry curvature for the general 2D state $\Psi = (\cos{\frac{\theta(k,t)}{2}},\sin{\frac{\theta(k,t))}{2}} e^{i \phi(k,t)})^{T}$. This correction causes the average displacement to drift away from the integral of the conventional Berry curvature.}
\begin{figure}
\includegraphics[width=\linewidth]{DisplacementThouless.png}
\caption{Average displacement of a particle over one adiabatic cycle in the nonlinear Thouless pumping setup. The average displacement without accounting for the additional nonlinear drift is represented by the full blue line, while the red dotted line corresponds to the displacement while accounting for it. Parameters are $\rho = \frac{\pi}{2}$ and $\omega = 10^{-2}$.}
\label{fig:DisplacementThouless}
\end{figure}
In Fig.~\ref{fig:DisplacementThouless}, we show the average displacement of a particle through numerical simulation of a Thouless pumping in the nonlinear adiabatic pump presented in Eq.~(\ref{eqn:H2DChern}), both with and without taking the additional nonlinear drift into account. When simply integrating the Berry curvature without the nonlinear drift, we expectedly retain a quantized result, equal to the Chern number of the corresponding 2D Chern insulator and the topological charge of the original linear Weyl point. However, this quantization is immediately broken by the addition of the additional drift due to nonlinear dynamics, making it a dead giveaway of the nonlinear nature of the system. \textcolor{black}{Moreover, by experimentally noting down this nonlinear correction and comparing it against Eq.~(\ref{eqn:Displacement}), the nonlinear strength and, consequently, the shape of the enclosed nodal lines and nodal surfaces can in principle be determined.}
There are still obstacles to the realization of such a nonlinear Thouless pump, the main one being that the average over quasimomenta $k$ in Eq.~(\ref{eqn:vbar}) is usually obtained in linear systems by taking the initial state to be a Wannier state involving a superposition of Bloch states with all different possible $k$. The superposition principle being lost in the case of nonlinear systems, we are forced to look into other ways to realize this average. One possibility worthy of further investigation is the use of Bloch oscillations to realize this average overall all quasimomenta \cite{TopoPumpingBlochOscillations}. \textcolor{black}{On the other hand, more recent studies highlight that the motion of solitons in a nonlinear Thouless pump is tied to the motion of Wannier states~\cite{Jurgensen2022,mostaan2021quantized}, completely bypassing the need \textcolor{black}{for preparing} the latter.} \textcolor{black}{However, we warn the reader that the soliton pumping and $k$-averaged pumping generally yield different results. Nonetheless, we show in Ref.~\cite{NonlinearAdiabaticPumping} that both exhibit similar behavior with increase in nonlinearity.}
\subsection{AB-effect experiment for detection of nonlinear Dirac lines}
\label{section:Experimental proposal}
We showed in the previous section that nonlinear structures connected to original Weyl points of the corresponding linear Weyl semimetal can be detected through an adiabating pumping setup, the displacement being equal to the sum of the topological charge of the original Weyl point and an additional drift giving away the presence of nonlinear effects. \textcolor{black}{However, given that the nonlinear Dirac lines identified in Sec.~\ref{section:Parallel case} do not carry a topological charge, an alternative detection method is deemed necessary. Motivated by the similarity between nonlinear Dirac lines and nonlinear Dirac cones of Ref.}~\cite{NLDC}, we propose a means for their detection via an interference setup akin to an AB-effect experiment~\cite{ABInterferometer}.
\begin{figure}
\includegraphics[width=\linewidth]{AB.png}
\caption{Interfering paths traced adiabatically around a nodal line during the interference experiment. The blue line represents a nodal line, that can be connected to an original Weyl point or not. The green and red arrows respectively represent paths A and B, which describe to semi-circles with opposite senses of rotation, in a plan of fixed $k_x$. The radius of the semi-circles $\rho$ is taken to be very small.}
\label{fig:ABsetup}
\end{figure}
For this purpose, we study the phase difference between two states adiabatically evolved along two different paths in quasimomentum space. The two interfering paths are shown on Fig.~\ref{fig:ABsetup}, taken to be two semi-circles sharing the same starting point, and going around a nonlinear nodal line in a symmetric manner, with one clockwise and the other one counterclockwise. These paths are designed in the quasimomentum space, in the $k_x = \pi$ plane. During this adiabatic evolution, each state will pick up a geometric phase, made of the conventional, linear geometric phase corresponding to the integral over the path of the Berry connection, and an additional geometric contribution due to the nonlinear dynamics. By taking the phase difference between two states evolving along paths with opposite senses of rotation, we make sure that the nonlinear contributions cancel out so that only the original geometric phases remain.
\begin{figure}
\includegraphics[width=\linewidth]{ABdisconnected.png}
\caption{Adiabatic AB phases associated with two interfering paths going around a nonlinear Dirac line with zero topological charge, located at different planes of fixed $k_x$. The system is made to move adiabatically along each path at a frequency $\omega = 10^{-5}$, and each interfering path is a semi-circle of radius $\rho = 10^{-3}$. The nodal line considered here is the one at $(k_y,k_z) = (0,\pi)$, but the nodal line at $(k_y,k_z) = (\pi,0)$ yields the same results}
\label{fig:ABDisconnected}
\end{figure}
The results of this interference setup around a nodal line with zero topological charge are shown on Fig.~\ref{fig:ABDisconnected}. Due to the mutual cancellation of the additional nonlinear contributions to the geometric phase, the results obtained are quantized to a multiple of $\pi$. \textcolor{black}{By fixing the interfering path at a specific $k_x$ value, the AB phase is zero at sufficiently small nonlinearity due to the absence of nonlinear Dirac lines.} As the nonlinear strength increases, the nonlinear Dirac lines eventually appear and grow in size until it first goes through the ring at $k_x=\pi$, then at $k_x = \frac{2\pi}{3}$ and later at $k_x = \frac{\pi}{2}$, thus explaining the jump in the AB phase from $0$ to $\pi$ in Fig.~\ref{fig:ABDisconnected}.
\begin{figure}
\includegraphics[width=\linewidth]{ABconnected.png}
\caption{Adiabatic AB phases associated with two interfering paths going around a nonlinear Dirac line with non-zero topological charge, located at different planes of fixed $k_x$. The system is made to move adiabatically along each path at a frequency $\omega = 10^{-5}$, and each interfering path is a semi-circle of radius $\rho = 10^{-3}$.}
\label{fig:ABConnected}
\end{figure}
One might then wonder what this AB phase would be if the two interfering paths would be instead located around one of the two nodal lines that arise from the breaking down of a linear Weyl point as shown in Fig.~\ref{fig:ParallelNodal}-a), whose topological charge is non-zero. As shown in Fig.~\ref{fig:ABConnected}, a similar conclusion is obtained; it shows zero AB phase when the ring does not go around a nodal line, and an AB phase of $\pi$ as soon as a nodal line goes through it. \textcolor{black}{That the AB phase measurement does not make a distinction between nonlinear Dirac lines and nodal lines that originate from a linear Weyl point implies the necessity of using both AB phase measurement and adiabatic pumping as complementary detection schemes.}
\section{Concluding remarks}
\label{section:Concluding remarks}
\textcolor{black}{We have earlier elucidated how nonlinearity breaks down a Weyl point into a pair of nodal line and nodal surface, both of which carry the topological charge (Chern number) of the original Weyl point. This then raises a natural question regarding the distribution of this topological charge over the nodal structures. Since nodal lines and nodal surfaces typically hold zero Chern number in linear systems, one may na\"{i}vely think that this topological charge is still localized at the point where the original Weyl point is located. However, our careful analysis below shows otherwise}
\begin{figure}
\includegraphics[width=\linewidth]{PiercedSphere.png}
\caption{\textcolor{black}{Pierced sphere \textcolor{black}{enclosing} a nonlinear Dirac line used for integration of the flux of the Berry curvature. For readability, the holes are larger in the figure than in numerical calculations where their solid angle is $\Omega = \SI{1.57e-4}{\steradian}$.}}
\label{fig:PiercedSphere}
\end{figure}
\textcolor{black}{Specifically,} to verify if the Berry curvature was indeed still emitted by a point-like monopole at the original Weyl point's location, we numerically integrated its flux through a ``pierced sphere", centered on the original Weyl point B, and with diminishing radius. This sphere is pierced at two points along the $k_x$ axis, where the nodal lines go through, to avoid the band touching point where the Berry curvature is ill-defined, as shown in Fig.~\ref{fig:PiercedSphere}.
\begin{figure}
\includegraphics[width=\linewidth]{FluxPiercedSphere.png}
\caption{Flux of the Berry curvature through a pierced sphere of radius $\rho$ centered on the original Weyl point $B(\frac{3 \pi}{2},\pi,\pi)$. The sphere is pierced in two points along the line $k_y = k_z = \pi$ to avoid the ill-defined gap closing points. Nonlinear strength is $g=1$, and the solid angle of each hole is $\Omega = \SI{1.57e-4}{\steradian}$.}
\label{fig:FluxPiercedSphere}
\end{figure}
The numerically integrated flux is shown in Fig.~\ref{fig:FluxPiercedSphere}. As expected, for $\rho \geq \frac{\pi}{6}$, where the entire nodal line is contained in the pierced sphere, the integrated flux is equal to the Chern number with very good precision (up to $10^{-5}$). However, as soon as the pierced sphere does not contain the entire nodal line, i.e. for $\rho < \frac{\pi}{6}$, the integrated flux starts to linearly decrease, all the way to $0$ as the radius goes to $0$. \textcolor{black}{This demonstrates that, contrary to the na\"{i}ve expectation above, the topological charge is in fact uniformly distributed throughout the nodal line. Such a feature is unique to nonlinear systems and has no counterpart in linear nodal-line semimetals.}
To conclude, in this work, we have studied in detail the effects of an on-site nonlinearity on the band structure and gap closing points in some Weyl semimetal lattice models exhibiting a single pair of Weyl nodes. \textcolor{black}{One main finding of our work is the breaking down of Weyl nodes into pairs of nodal lines and nodal surfaces, the orientation of which depends on the underlying linear Hamiltonian. At large nonlinearity, additional nodal lines (termed nonlinear Dirac lines) which hold zero topological charge may emerge.} We completed this investigation by \textcolor{black}{identifying} a proposal for a nonlinear adiabatic pump that allows to access the topological charge of a corresponding nonlinear Weyl point, \textcolor{black}{showing in the process that the additional contribution to the pumping number over an adiabatic cycle due to nonlinear dynamics can be used as an effective probe for the strength of nonlinearity and the shape of the accompanying nodal structure.} Finally, we successfully adapted an AB interference setup~\cite{ABInterferometer,NLDC} to detect the presence of nonlinear Dirac lines.
\textcolor{black}{As a possible future direction, the additional two-dimensional nodal surfaces appearing at high nonlinear strength in the general case are yet to be studied and remain elusive, as neither the integration of the flux of the Berry curvature, nor AB interference experiment is able to detect them.}
As its interpretation from a mean-field theoric point of view is different from the one-dimensional nodal structures~\cite{WU2011AnomalousMonopoles}, we expect to witness new exciting features. \textcolor{black}{In addition, our observation that nonlinearity deforms the shape of the system's Fermi arcs is also expected to further motivate} the study of the interplay between nonlinearity and Fermi arcs in Weyl semimetals.
\textcolor{black}{Finally, the study of nonlinear effects on other systems is still in its infancy and surely many remains to be discovered. Such systems include higher-order topological systems~\cite{Benalcazar2017,BenalcazarHingeState2017,Langbern2017,LiuHOTP2017,Song2017,Khalaf2018,Schindler2018,Leeli2019HighOrderTopological}, Floquet systems~\cite{Cayssol2013,Gomez2013FloquetBloch,Asboth2014,Grushin2014Floquet,Zhou2014,Bomantra2016Floquet,Linhu2018Floquet}, and non-Hermitian systems~\cite{Longhi2017,Gong2018NonHermitian,MartinezAlvarez2018,Shen2018,Yao2018NonHermitian,Yao2018NonHermitianChern,Ghatak:2019zke,Kawabata2019,Kawabata2019NHSym,Lee2019,Li2019NH,Liu2019NH,Zhou2019NHTopoBands,Ozdemir2019,Okuma2020NonHermitian,Zhao2020NonlinearNonHermitian,Shiqi2021NonlinearNonHermitian}. Even among topological semimetals, many different phases could be investigated, such as nodal line semimetals~\cite{Fang2015NodalLineSemimetals,Fang2015NodalLineSemimetals}, type-II Weyl semimetals~\cite{Soluyanov2015TypeIIWSM,Deng2016TypeIIWSM,Jiang2017TypeIIWSM}, multi-Weyl semimetals~\cite{Fang2012MultiWSM,Umer2021MultiWSM}, and triple-fermion semimetals~\cite{Lv2017TripleFermionSM}.}
\begin{acknowledgements}
{\bf Acknowledgement}: R.W.B is supported by the Australian Research Council Centre of Excellence for Engineered Quantum Systems (EQUS, CE170100009). J.G. is funded by the Singapore National Research Foundation Grant No. NRF-NRFI2017-04 (WBS No.
R-144-000-378- 281).
\end{acknowledgements}
|
1,116,691,497,606 | arxiv | \section*{Acknowledgments}
I would like to warmly thank my collaborator and friend Fran\c cois Englert with whom almost all the original results
presented here have been derived. I am also grateful to Sophie de Buyl, Marc Henneaux and Nassiba Tabti for enjoyable collaborations.
|
1,116,691,497,607 | arxiv | \section{Introduction}
The theory of the (eigenvalue) matrix models (MAMO) occupies more and more space in the
context of modern theoretical physics,
what could seem unexpected and even unjustified.
However, this impression is basically wrong:
MAMO are indeed crucially significant, because they appear to capture
the most interesting theoretical properties of generic field/string theory.
Most important, they allow to study the "non-perturbative" properties
of integrals and their interplay with the group/symmetry structures --
avoiding additional (and rarely significant) complexities of {\it functional}
integration.
The main property of exact non-perturbative partition function is {\it integrability},
which reflects invariance of the (functional) integral under arbitrary change
of integration variables (fields).
It appears related to the set of equations for partition function
as a function of (an infinite set of) the coupling constants,
in particular, to the definition of an integral with parameters (couplings)
as a $D$-module.
$D$-module is associated with linear equations (often called Virasoro or $W$-constraints)
and leads to a $W$-representation of partition function as an action
of an exponentiated differential/difference {\it cut-and-join} operator on unity,
while integrability implies also existence of less trivial non-linear (Hirota-like)
equations, typically quadratic.
Not only are matrix integrals perfectly suited to the study of this phenomenon,
they allow to further develop it to the concept of {\it superintegrability},
which is a possibility to find a special basis in the space of couplings,
where correlators are simple, deeply factorized quantities.
Despite certain partial results, exact inter-relations inside the quadruple
superintegrability - Hirota - Virasoro -W-representation
remain an open problem and a subject for future research.
We restrict consideration to Hermitian matrix model in Gaussian background,
where all the four phenomena are well studied,
linear constraints are exactly Virasoro and integrability is of the most familiar
KP/Toda type.
Superintegrability, Virasoro-like constraints and $W$-representation
survive various deformations,
and integrability can be used to describe non-Gaussian backgrounds.
Analogous results can be obtained for other matrices -- unitary, orthogonal, simplectic
and, probably, belonging to arbitrary representations of arbitrary Lie algebras
(including exceptional).
Of special interest are further generalizations to quantum algebras,
currently available up to DIM, which involve Macdonald polynomials and
their further Kerov and $3$-Schur analogues, which are still under investigation.
\section{Eigenvalue Matrix Models}
The usual starting point in the theory of MAMO is
an integral goes over $N\times N$ Hermitian matrices $H$
\begin{eqnarray}
Z_G\{t\}:=\frac{1}{V_N}\int dH \exp\left(-\frac{1}{2}{\rm tr}\, H^2 + \sum_{k=0}^\infty t_k{\rm tr}\, H^k\right)
\label{GMint}
\end{eqnarray}
where $V_N$ is the volume of unitary group $U(N)$.
This quantity is considered as a formal series in powers of the couplings $t_k$,
often called "time-variables" -- this terminology comes from the theory of KP hierarchy, see below.
In modern theory (\ref{GMint}) is considered as a particular integral representation of the
{\it non-perturbative partition function} $Z_G\{t\}$, which
depends on the class of fields (one Hermitian matrix), background action (Gaussian),
boundary conditions for fields (zero at infinity) and the set of moduli (coupling constants $t_k$).
In fact all these dependencies are inter-related, what allows to dream about a common non-perturbative
partition function as a kind of a universal object in a universal moduli space,
unifying all field/string theories.
\bigskip
Four approaches are currently available to the study of such quantities.
\bigskip
{\bf 1. Perturbation theory, genus expansion and superintegrability.}
It studies various correlators
\begin{eqnarray}
\frac{\partial^m}{\partial t_{i_1}\ldots \partial t_{i_m}} Z = \left<{\rm Tr}\, H^{i_1}\ldots {\rm Tr}\, H^{i_m}\right>
\end{eqnarray}
In the case of Gaussian measure $<\ldots>:= \int dH e^{-\frac{1}{2}{\rm tr}\, H^2} \ldots$
they are easily calculated, satisfy Wick theorem and can be handled by Feynman diagram technique.
Also the topological $1/N$ expansion and t'Hooft calculus in terms of {\it fat} diagrams
is applicable.
Less obvious, there is a special basis in the space of these correlators,
where they acquire an additional peculiar property of {\it superintegrability},
see s.\ref{si} below.
\bigskip
{\bf 2. Ward identities, D-module and Virasoro constraints.}
One can perform an arbitrary change of variables $H\longrightarrow f(H)$ in the integral,
what leaves it intact
(provided $f(H)$ is polynomial, and the boundary terms at infinity are damped by the
Gaussian factor $e^{-{\rm tr}\, H^2}$).
In the simples basis for the infinitesimal variations $\delta H \sim H^{n+1}$
we obtain
\begin{eqnarray}
\hat L_n Z_G = \left( - \frac{\partial}{\partial t_{n+2}} +
\sum_{k=1}^\infty kt_k \frac{\partial}{\partial t_{k+n}} + \sum_{a+b=n}\frac{\partial^2}{\partial t_a\partial t_b}
\right)Z_G = 0,
\ \ \ \ n\geq -1
\ \ \ \ \ \ \frac{\partial}{\partial t_0} Z_G = NZ_G
\label{virconG}
\end{eqnarray}
what is called the system of {\it Virasoro constraints}, because $[\hat L_m,\hat L_n] = (n-m)\hat L_{n+m}$
and $\hat L_n$ with $n\geq -1$ form a Borel subalgebra of Virasoro algebra.
This describes $Z_G\{t\}$ as a solution to the compatible system of linear differential equations
with no reference to any integral, i.e. provides a $D$-module description.
The way to restore partition function from Ward identities is also known as
the AMM-CEO topological recursion and is applicable far beyond matrix models.
\bigskip
{\bf 3. $W$-representation.}
The system of Virasoro constraints imply that their special linear combination
\begin{eqnarray}
\sum_{n=1}^\infty nt_nL_{n-2} Z_G = 0
\end{eqnarray}
is split into a sum of two operators: the dilatation
\begin{eqnarray}
\hat l := \sum nt_n\frac{\partial}{\partial t_n}
\end{eqnarray}
and the $2$-descendant of the cut-and-join
\begin{eqnarray}
\hat W_{2} :=
\sum_{k,l}\left( kl t_kt_l\frac{\partial}{\partial t_{k+l-2}} + (k+l+2)t_{k+l+2}\frac{\partial^2}{\partial t_k\partial t_l}\right)
\end{eqnarray}
which have a very special commutation relation
\begin{eqnarray}
\phantom. [\hat l_0, \hat W_{2}] = 2\hat W_{2}
\end{eqnarray}
This implies that
\begin{eqnarray}
Z_G = e^{\frac{1}{2}\hat W_{2}} \cdot 1
\end{eqnarray}
i.e. represents partition function as a result of action of an operator
on a trivial state.
In other words, non-perturbative partition functions can be substituted
by cut-and-join operators, acting in the moduli space.
Of certain interest is the interplay between this and the $D$-module descriptions.
\bigskip
{\bf 4. Vandermonde, determinant representation and KP-integrability.}
The angular part of the matrix $H$ decouples in (\ref{GMint}) and the integral reduces to that over $N$
eigenvalues $h_a$ of $H$:
\begin{eqnarray}
Z_G\{t\} = \int \prod_{a<b}^N (h_a-h_b)^2
\prod_{a=1}^N dh_a \exp\left(-\frac{1}{2}h_a^2 + \sum_{k=0}^\infty t_k h_a^k\right)
\end{eqnarray}
Since Vandermonde is actually a determinant,
this provides a determinantal representation for the partition function
\begin{eqnarray}
Z_G\{t\} = {\rm det}_{a,b} \phi_{a,b}
\label{ZGdet}
\end{eqnarray}
with
\begin{eqnarray}
\phi_{a,b}:= \int dh \,h^{a+b-2}
\exp\left(-\frac{1}{2}h^2 + \sum_{k=0}^\infty t_k h^k\right)
= \frac{\partial^2}{\partial t_{a-1}\partial t_{b-1}}{\cal H}_G\{t\},
\end{eqnarray}
\begin{eqnarray}
{\cal H}_G\{t\} = dh
\exp\left(-\frac{1}{2}h^2 + \sum_{k=0}^\infty t_k h^k\right)
\end{eqnarray}
Such determinants with the property
\begin{eqnarray}
\frac{\partial}{\partial \phi_c} \phi_{a,b} = \phi_{a+c,b} = \phi_{a,b+c}
\label{phider}
\end{eqnarray}
satisfy bilinear Hirota equations
and are known as KP/Toda $\tau$-functions.
In the particular case of Gaussian Hermitian model there is an additional
symmetry between indices $a$ and $b$, and the relevant one is
the {\it Toda-chain} integrable hierarchy.
The appearance of determinant reflects the relation to the special class
of antisymmetric representations of $\widehat{Sl(N)}$ algebras with unit central charge
(the free-fermion formalism).
Of most interest is generalization of KP/Toda determinantal technique
to arbitrary representations and to other algebras --
bilinear Hirota-like equations are in fact a consequence of
Tanaka-Krein formalism (multiplication of representations)
and are not restricted to fermionic sector.
However, the lack of consensus on notations in general situation
slows down the development of general theory.
\section{Integrability}
The notion of integrability in quantum field theory does not yet have a
commonly accepted definition.
The usual idea in mechanics refers to sufficient number of commuting integrals of motion,
what in general setting leads to the idea of some hidden algebraic structure.
Another. in fact, closely related approach is "a projection method",
which implies that description is adequate
in terms of free fields with possible screening operators (playing the role of projectors).
Still, the universality of integrability for non-perturbative partition functions
is best understood in terms of diffeomorphism invariance of (functional) integrals --
what also refers to a hidden symmetry and group structure, but it is, perhaps,
less straightforward than the naive one, associated with commuting integrals.
Roughly speaking, the naive integrability is formulated in terms of $Sl(N)$ algebras,
while diffeomorphisms are rather related to Virasoro algebra and its further
$W$-generalizations -- and this is what happens at the simplest level of matrix models.
The true structures in true field/string theory are still a mystery --
still there is mounting evidence that they exist in a very broad context
and thus deserve investigation and understanding.
In the present-day theory of matrix models integrability usually appears in a very
restricted traditional context -- of KP/Toda type, associated with free fermions.
Namely, the tau-function is defined as generic Gaussian correlator
\begin{eqnarray}
\tau\{t|A\} = \left<0\ \Big|\
\overbrace{ e^{\oint t(z)\tilde \psi(z) \psi(z)dz} }^{U\{t\}}\cdot
\underbrace{e^{A_{mn}\tilde\psi_m\psi_n}}_{{\cal O}_A}\ \Big|\ 0\right>
\label{Hireq}
\end{eqnarray}
and the main secret is the "bosonisation property"
\begin{eqnarray}
\exp\left( \int_{z_1}^{z_2} \tilde\psi(z) \psi(z)dz \right) \sim \tilde\psi(z_1)\psi(z_2)
\label{bosprop}
\end{eqnarray}
It looks especially simple for free fermions and implies a distinguished way to separate
the small set of time-variables $t(z) = \sum t_k z^k$ from the generic "Grassmannian point"
labeled by semi-infinite matrix $A$.
Though elementary, explicit calculations with free fermions are somewhat lengthy and
borrowing, and in the study of matrix models one usually uses some standard corollaries.
Most useful so far were the following five:
\begin{itemize}
\item {\bf Hirota equations.} These are bilinear equations, solved by a $\tau$-function
at arbitrary point $A$:
\begin{eqnarray}
\int \frac{dz}{z} e^{\sum_k (t_k-t'_k)z^k} \sum_n D^+_n(t)\tau\{t|A\} \cdot D^-_n(t') \tau\{t'|A\}
= 0
\end{eqnarray}
with the shit operators
\begin{eqnarray}
\sum_n D^\pm_n(t)z^{-n} = \exp\left(\pm\sum_k \frac{1}{kz^k}\frac{\partial}{\partial t_k}\right)
\end{eqnarray}
Their role is to use (\ref{bosprop}) and generate an operator $\sum_n \tilde \otimes \psi_n$
which commutes with ${\cal O}_A\otimes {\cal O}_A$ and can be moved to the right in (\ref{Hireq})
to annihilate the vacuum $\sum_n \tilde \psi_n| 0\big> \otimes \psi_n |0\big> = 0$.
It can not be moved to the left, because it does not commute with the bi-evolution operator
$U\{t\}\otimes U\{t'\}$, involving two different sets of time-variables $t$ and $t'$.
As already mentioned, all these arguments can be straightforwardly generalized to arbitrary
representations of arbitrary algebras -- but many technical details were never worked out.
\item {\bf Schur expansion, Plucker relations and Casimir parametrization.}
As a series in powers of $t$, the $\tau$-function can be expanded over full basis od Schur functions,
labeled by Young diagrams $R$:
\begin{eqnarray}
\tau\{t|A\} = \sum_R G_R\{A\}\cdot{\rm Schur}_R\{t\}
\end{eqnarray}
then Hirota equations are equivalent to bilinear Plucker relations on the coefficients $K_R$,
Generic solution to plucker relations is parameterized by arbitrary matrix
\begin{eqnarray}
G_R = \det_{ij} B_{i,R_j-j}
\end{eqnarray}
where $R_j$ is the length if the $j$-th row in the diagram.
This leads to an interesting $\tau$-functions, parameterized by a special basis of Casimir operators
$C_n(R) = \sum_j \Big(\left(R_j-j+\frac{1}{2}\right)^n - \left(-j+\frac{1}{2}\right)^n\Big)$:
\begin{eqnarray}
\tau\{t,\bar t\big|g\} = \sum_{R} e^{\sum g_n C_n(R)} \cdot {\rm Schur}_R\{t\} \cdot {\rm Schur}_R\{\bar t\}
\label{Castau}
\end{eqnarray}
or, alternatively, by arbitrary function $f$
\begin{eqnarray}
\tau\{t,\bar t\big|f\} = \sum_R f_R \cdot {\rm Schur}_R\{t\} \cdot {\rm Schur}_R\{\bar t\}
\label{hyptau}
\end{eqnarray}
with $f_R = \prod_{(i,j)\in R} f(i-j)$.
The latter class is sometime called hypergeometric $\tau$-functions.
In the las examples we also added Schur functions of auxiliary times $\bar t$, which complement
KP $\tau$-function to the Toda-lattice one.
They can be eliminated by putting $\bar t_k=\delta_{k,1}$, which, however, leaves behind an essential
$R$-dependent combinatorial factor $d_R:=S_R\{\delta_{k,1}\}$.
For $g_k=0$ in (\ref{Castau}) or $f=1$ in (\ref{hyptau}) Cauchy identity for the Schur functions
provides a trivial $\tau$-function
\begin{eqnarray}
\tau\{t,\bar t|0\} = \sum_R {\rm Schur}_R\{t\}\cdot {\rm Schur}_R\{\bar t\} =
\exp\left(\sum_k \frac{t_k\bar t_k}{k}\right)
\end{eqnarray}
It is not fully understood what are the implications for integrability
of the substitution of other interesting
systems of polynomials (Jack, Macdonald, 3-Schur, Shiraishi, \ldots)
in place of Schur functions, and what are exact modifications
of free fermion formalism and Hirota equations -- this is one of the interesting questions
in the theory of matrix models, where this kind of deformations is quite straightforward.
\item {\bf Hirota equations in Miwa variables.}
Miwa transform restricts time-variable to an $N$-dimensional subspace
\begin{eqnarray}
t_k = \frac{1}{k}\,{\rm tr}\, X^k = \frac{1}{k}\sum_{\alpha = 1}^N x_i^k
\end{eqnarray}
This converts Schur functions and their generalizations into orthogonal polynomials
with Vandermonde measures,
\begin{eqnarray}
\prod_{\alpha=1}^N \oint_0 \frac{dx_\alpha}{x_\alpha^2}
\prod_{\alpha<\beta}^N (x_\alpha- x_\beta) \cdot {\rm Schur}_R[X] \cdot {\rm Schur}_{R'}[X^{-1}]
\end{eqnarray}
In this sense matrix a more natural choice for $X$ is unitary or orthogonal rather than symmetric matrix.
Hirota equations can be written in terms of Miwa variables, then they describe a pair of
$\tau$ with one added and one subtracted $x$-variables.
Solutions have peculiar determinant form
\begin{eqnarray}
\tau[X] = \frac{\det_{\alpha,\beta} \psi_{\alpha}(x_\beta)}{\Delta(X)}
\end{eqnarray}
where Vandermonde in denominator is $\Delta[X] = \prod_{\alpha<\beta}^N (x_\alpha-x_\beta)
= \det_{\alpha,\beta} x_\beta^{\alpha-1}$, and
the functions
\begin{eqnarray}
\psi_\alpha(x) = x^{\alpha-1}\Big(1+ O(x^{-1})\Big)
\end{eqnarray}
can be arbitrary series with the given asymptotic.
Such representations of KP $\tau$-functions naturally arise in matrix models with
background fields, like Generalized Kontsevich Model.
\item {\bf Forced hierarchies.}
Our original example, the Gaussian model (\ref{ZGdet}) belongs to a special class
of $\tau$ functions, which possess determinant representation with the property
(\ref{phider}).
They are expanded in Schur {\it polynomials} -- a special sub-class of Schur functions,
associated with symmetric representations (single-line Schur diagrams)
and generated by the expansion of $\exp\left(\sum_k t_kz^k\right) = \sum_r S_{[r]}\{t\}z^r$.
In integrability theory they are associated with so called {\it forced} hierarchies
and possess an additional index $N$, which is sometime called the "zero-time" or its conjugate --
a name, natural from the point of view of the equations (\ref{virconG}).
\item {\bf Riemann surfaces.}
An important class of $\tau$-functions appears as $\det \bar\partial$ on Riemann surfaces
with singularities or boundaries, where $t_k$ are associated with boundary conditions,
while Grassmannian point $A$ -- with the moduli space.
This is a very important class, which opens a way to study universal moduli space
and non-perturbative string theory, but it is still not used at a large scale in
the theory of matrix models.
\end{itemize}
\section{Superintegrability
\label{si}}
The term {\it superintegrability} usually implies that the problem is fully solvable
in terms of the elementary single-valued functions, without any ramifications
and other transcendentalities.
The typical example are the motions in harmonic and Coulomb potentials.
where the orbits are closed and described by curves of the second order (quadrics).
In these examples there are additional conservation laws, associated with Runge-Lenz vectors,
but most important is the very fact that the motion is exhaustively described
by some well-behaved functions.
In quantum theory superintegrability means that there is a full set of correlators,
which can be explicitly calculated in generic and elementary terms.
In matrix model case the main example are Gaussian models,
where explicitly calculable are averages of Schur functions
${\rm Schur}_R[H] := {\rm Schur}_R\left\{t_k=\frac{1}{k}{\rm tr}\, H^k\right\}$
\begin{eqnarray}
\left<{\rm Schur}_R\right> := \int {\rm Schur}_R[H]\cdot e^{-\frac{1}{2}{\rm tr}\, H^2} dH
= \frac{{\rm Schur}_R[N] \cdot {\rm Schur}_R\left\{\frac{1}{2}\delta_{k,2}\right\}}
{{\rm Schur}_R\{\delta_{k,1}\}}
\end{eqnarray}
Moreover, there is additional set of functions $K_R[H]=S_R[H]+\ldots$
which form an orthogonal set,
\begin{eqnarray}
\left <K_{R} K_{R'}\right> = \frac{{\rm Schur}_R[N] }{ {\rm Schur}_R\{\delta_{k,1}\}}\cdot\delta_{R,R'}
\end{eqnarray}
and do not change the averages $S_R$ in the following sense:
\begin{eqnarray}
\left< K_{Q}\ {\rm Schur}_R \right> =
\frac{{\rm Schur}[N]\cdot {\rm Schur}_{R/Q}\left\{\frac{1}{2}\delta_{k,2}\right\}}
{{\rm Schur}_R\{\delta_{k,1}\}}
= \frac{ {\rm Schur}_{R/Q}\left\{\frac{1}{2}\delta_{k,2}\right\}}
{ {\rm Schur}_R\left\{\frac{1}{2}\delta_{k,2}\right\}}
\left<{\rm Schur}_R\right>
\end{eqnarray}
\section{Perspectives}
There are plenty of MAMO examples beyond Gaussian Hermitian model,
where different pieces of above pattern are already revealed and investigated --
what leaves little doubts about their universal nature and applicability.
The main research direction now is the further extension of integrability approach
to new frontiers beyond matrix models, especially to the full fledged field-theory examples.
This already seems to be within reach for non-eigenvalue matrix and tensor models,
for 2d conformal, 3d Chern-Simons, 4d SYM, supersymmetric low-energy gauge
and DIM-controlled brane theories.
The task is to find either quadratic equations for non-pertrurbative partiion functions
or the special basis in the space of correlators, where they are well factorized
and calculable -- what would suit into integrability or superintegrability paradigm respectively
An important technical problem is to extend integrability concept beyond KP/Toda setting
and determinantal (fermionic) examples, what requires a progress in the basics of
representation theory and non-linear algebra.
In the language of quantum field theory these problems are closely related to the subjects like
exact renormalization group, equations for Feynman diagrams and resurgency.
\section*{Further reading}
The literature list below contains a few selected papers on different aspects of integrability story
for matrix models (a solid piece of knowledge) and quantum field theory (projects and dreams).
It contains both old and fresh presentations, which can be compared to understand the
degree and directions of progress in the field.
The list is far from being complete and even representative.
Much more references can be found in the cited papers.
\section*{Acknowledgements}
I appreciate years of common research on these subjects with many colleagues and friends.
This work is partly supported by the grant of the Foundation for the Advancement of Theoretical Physics
“BASIS” and by the joint grants 21-52-52004-MNT-a and 21-51-46010-ST-a.
|
1,116,691,497,608 | arxiv | \section{Introduction}
It is well known that for an absolutely irreducible polynomial
$f\in\Z[x,y]$ the reduction $f\bmod p$ is also absolutely irreducible
if the prime $p$ is large enough. For small $p$ the polynomial $f\bmod
p$ may be reducible.
E.g. $f=x^9y-9x^9-2x+9y+2$ is absolutely irreducible over
$\Q$ but reducible modulo $p=186940255267545011$ where
$x-93470127633772547$ divides $f\bmod p$.
It is natural to ask how large $p$
has to be to be sure that $f\bmod p$ is absolutely irreducible.
In \cite{R} we showed that
$$p>d^{3d^2-3}\cdot H(f)^{d^2-1}$$
is sufficient for absolute irreducibility mod $p$ where $d$ is the total
degree of $f$ and $H(f)$ the height\footnote{the height of a
polynomial $f=\sum_{i,j}a_{ij}x^iy^j\in \Z[x,y]$ as we use it is
defined by $H(f)=\max_{i,j}|a_{ij}|$.} of $f$.
Sometimes it is more natural to consider the polynomial having
degree $m$ in $x$ and
$n$ in $y$. For this case Zannier \cite{Z} has shown that
$$p>e^{12n^2m^2}(4n^2m)^{8n^2m}\cdot H(f)^{2(2n-1)^2m}$$
is sufficient for absolute irreducibility $\bmod p$.
Our aim is to improve Zannier's estimate by showing the following
theorem:
\begin{Theorem} Let $f\in\Z[x,y]$ be an absolutely irreducible
polynomial with degree $m\ge 1$ in $x$, $n\ge 1$ in $y$ and
height $H(f)$. If $p$ is a prime with
$$p>[m(n+1)n^2+(m+1)(n-1)m^2]^{mn+\frac{n-1}{2}}\cdot H(f)^{2mn+n-1}$$
then the reduced polynomial $f\bmod p$ is also absolutely irreducible.
\end{Theorem}
The basic ingredient of the proof is the structure
theorem for closed $1$-forms as it was already used in \cite{R}.
In section 2 the connection between closed $1$-forms and
reducibility is given in two lemmas and applied to prove the theorem.
The lemmas are proved in section 3.
To test the quality of the estimate in the theorem we
construct examples of polynomials $f\in\Z[x,y]$ in section 4 with
a certain reducibility behavior. Assuming the
Bouniakowsky conjecture (which will also be explained in section
4) one gets the following result:
\begin{Proposition} Let $m,n\ge 1$ be integers.
If the Bouniakowsy conjecture is true there are infinitely many
polynomials $f\in\Z[x,y]$ with $\deg_xf=m$ and $\deg_yf=n$ which are
absolutely irreducible over $\Q$ but reducible for a prime $p$ with
$$p\ge H(f)^{2m}.$$
\end{Proposition}
In case $n=1$ the inequality in the theorem is $p>(2m)^m\cdot
H(f)^{2m}$. The proposition shows then that the exponent $2m$ is
best possible.
In case $n=2$ the exponent in the theorem is $4m+1$. In
\cite{R2} it is shown that the exponent can be improved to $6$ (for
$m=2$), $6\frac{2}{3}$ (for $m=3$) and $2m$ (for $m\ge 4$). This
supports my belief that the best exponent in the theorem will
be smaller than $2mn+n-1$ if $n\ge 2$.
\section{A criterion for reducibility}
If $f(x,y)$ is a polynomial with $\deg_xf=m$ and
$\deg_yf=n$ we write $\deg f=(m,n)$. The notation $\deg f\le
(m,n)$ will mean that $\deg_x f\le m$, $\deg_yf\le n$. If it happens
that we write $\deg f\le (m,n)$ with $m<0$ or $n<0$ then $f=0$.
The following lemmas contain our criterion for reducibility.
\begin{Lemma} Let $k$ be an arbitrary algebraically closed field and
$f(x,y)\in k[x,y]$ a reducible polynomial with $\deg f=(m,n)$. Then
there are polynomials $r,s\in k[x,y]$ with $\deg r\le(m-1,n)$ and
$\deg s\le(m,n-2)$ such that
$$\frac{\partial}{\partial y}\left(\frac{r}{f}\right)=
\frac{\partial}{\partial x}\left(\frac{s}{f}\right)\quad\mbox{ and
}\quad (r,s)\ne (0,0).$$
\end{Lemma}
\begin{Lemma} Let $k$ be an arbitrary algebraically closed field
of characteristic $0$ and
$f(x,y)\in k[x,y]$ with $\deg f=(m,n)$ and $n\ge 1$. If
there are polynomials $r,s\in k[x,y]$ with $\deg r\le(m-1,n)$ and
$\deg s\le(m,n-2)$ such that
$$\frac{\partial}{\partial y}\left(\frac{r}{f}\right)=
\frac{\partial}{\partial x}\left(\frac{s}{f}\right)\quad\mbox{ and
}\quad (r,s)\ne (0,0)$$
then $f$ is reducible.
\end{Lemma}
The proof of the lemmas will be postponed to the next section.
We remark that the example $f=x$, $r=1$, $s=0$ shows that $n\ge 1$
is a necessary condition in lemma 2.
\bigskip
We reformulate the lemmas: Let $f\in k[x,y]$ have degree $(m,n)$
and assume that $m,n\ge 1$. When do we find $r,s\in k[x,y]$ with
$\deg r\le (m-1,n)$ and $\deg s\le (m,n-2)$ such that the equation
\begin{equation}\frac{\partial}{\partial y}\left(\frac{r}{f}
\right)=\frac{\partial}{\partial x}\left(\frac{s}{f}\right)
\end{equation}
holds? We write
$$f=\sum_{\substack{0\le i\le m\\0\le j\le n}}a_{ij}x^iy^j,\quad
r=\sum_{\substack{0\le i\le m-1\\0\le j\le n}}u_{ij}x^iy^j,\quad
s=\sum_{\substack{0\le i\le m\\0\le j\le n-2}}v_{ij}x^iy^j$$
with unknowns $u_{ij}$ $(0\le i\le m-1,0\le j\le n)$ and
$v_{ij}$ $(0\le i\le m,0\le j\le n-2)$.
(There are $m(n+1)+(m+1)(n-1)=
2mn+n-1$ unknowns $u_{ij}$ and $v_{ij}$ if $m,n\ge 1$.)
Equation (1) can be written as
$$\frac{\partial r}{\partial y}f-r\frac{\partial f}{\partial y}-
\frac{\partial s}{\partial x}f+s\frac{\partial f}{\partial x}=0.$$
We have
$$\frac{\partial r}{\partial y}f-r\frac{\partial f}{\partial y}-
\frac{\partial s}{\partial x}f+s\frac{\partial f}{\partial x}=
\sum_{k,l}g_{kl}x^ky^l$$
with
$$g_{kl}=\sum_{(i,j)\in A_{kl}}(-l+2j-1)a_{k-i,l-j+1}u_{ij} +
\sum_{(i,j)\in B_{kl}}(k-2i+1)a_{k-i+1,l-j}v_{ij}$$
where
\begin{eqnarray*}
A_{kl}&=&\{(i,j):0\le k-i\le m,\quad 0\le l-j+1\le n,\quad
0\le i\le m-1, \quad 0\le j\le n\},\\
B_{kl}&=&\{(i,j):0\le k-i+1\le m,\quad 0\le l-j\le n,\quad
0\le i\le m, \quad 0\le j\le n-2\}.
\end{eqnarray*}
One sees that $\deg \sum g_{kl}x^ky^l\le (2m-1,2n-2)$.
Equation (1) is satisfied iff we find $u_{ij},v_{ij}\in k$ with
$$g_{00}=\dots=g_{2m-1,2n-2}=0.$$
We can write this as a matrix equation
$$\left(\begin{array}{c}g_{00}\\ \vdots \\ g_{2m-1,2n-2}\end{array}
\right)=
M(f)\cdot \left(\begin{array}{c}u_{00}\\ \vdots \\ u_{m-1,n} \\
v_{00} \\ \vdots \\ v_{m,n-2} \end{array}\right)=0$$
where the entries of the matrix $M(f)$ are coefficients of
certain $g_{kl}$ with respect to $u_{ij}$ and $v_{ij}$.
With these notations it is clear that equation (1) has a
nontrivial solution iff $M(f)$ has rank $<(2mn+n-1)$, i.e. all
$(2mn+n-1)\times (2mn+n-1)$-submatrices of $M(f)$ vanish. Now
we can reformulate the two lemmas for $f\in k[x,y]$ in terms of the
matrix $M(f)$:
\begin{itemize}
\item If $f$ is reducible then ${\rm rank } M(f)<2mn+n-1$.
\item If $k$ has characteristic $0$ and ${\rm rank }M(f)<2mn+n-1$
then $f$ is reducible.
\end{itemize}
\bigskip
We apply this to prove the theorem: Let $f\in \Z[x,y]$ be absolutely
irreducible of degree $(m,n)$. Then the matrix $M(f)$ has rank $2mn+n-1$, i.e. there
is a $(2mn+n-1)\times (2mn+n-1)$-submatrix $M_0$ of $M(f)$ with $\det
M_0\ne 0$.
We will estimate $|\det M_0|$ using Hadamard's estimate for
determinants. To do this we have to know the $L_2$-norm of the
rows of $M_0$. A row of $M_0$ is given by the coefficients of a
linear form $g_{kl}$ with respect to the variables $u_{ij}$ and
$v_{ij}$. We have
\begin{eqnarray*}
||g_{kl}||_2^2&=&\sum_{(i,j)\in A_{kl}}(-l+2j-1)^2a_{k-i,l-j+1}^2
+\sum_{(i,j)\in B_{kl}}(k-2i+1)^2a_{k-i+1,l-j}^2\le \\
&\le&(\sum_{(i,j)\in A_{kl}}(-l+2j-1)^2 +\sum_{(i,j)\in B_{kl}}
(k-2i+1)^2)\cdot H(f)^2.
\end{eqnarray*}
If $(i,j)\in A_{kl}$ then $0\le l-j+1\le n$ and $0\le j\le n$ so that
$-n\le -(l-j+1)+j\le n$ and $(-l+2j-1)^2\le n^2$. Furthermore $\#
A_{kl}\le m(n+1)$.
If $(i,j)\in B_{kl}$ then $0\le k-i+1\le m$ and $0\le i\le m$ so that
$-m\le (k-i+1)-i\le m$ and $(k-2i+1)^2\le m^2$. Furthermore $\#
B_{kl}\le (m+1)(n-1)$.
This implies
$$||g_{kl}||_2^2 \le (n^2\cdot \#A_{kl}+m^2\cdot \#B_{kl})\cdot H(f)^2\le [m(n+1)n^2+(m+1)(n-1)m^2]\cdot H(f)^2$$
so that the $L_2$-norm of a row of $M_0$ is $\le
\sqrt{[m(n+1)n^2+(m+1)(n-1)m^2]\cdot H(f)^2}$ and therefore
using Hadamard
\begin{eqnarray*}
|\det M_0|&\le&\sqrt{[m(n+1)n^2+(m+1)(n-1)m^2]\cdot H(f)^2}^{2mn+n-1}
\\ &=&[m(n+1)n^2+(m+1)(n-1)m^2]^{mn+\frac{n-1}{2}}\cdot H(f)^{2mn+n-1}.
\end{eqnarray*}
Now if $p$ is any prime with
\begin{equation*}
p>[m(n+1)n^2+(m+1)(n-1)m^2]^{mn+\frac{n-1}{2}}\cdot H(f)^{2mn+n-1}
\end{equation*}
then $0<|\det M_0|<p$ which implies that $\det M_0\not\equiv 0\bmod p$
so that $M(f)$ considered as a matrix over $\F_p$ has rank $2mn+n-1$ and
$f\bmod p$ is absolutely irreducible by the above criterion.
This proves our theorem.
\section{Proof of Lemma 1 and 2}
We start with a remark: If $k$ is an algebraically closed field and
$g\in k[x,y]$ satisfies $\dd{g}{x}=\dd{g}{y}=0$ then $g$ is constant
in characteristic $0$ or a $p$-power in characteristic $p$. In
each case, $g$ is not irreducible.
\bigskip
{\it Proof of Lemma 1:} Let $f\in k[x,y]$ be reducible of degree
$(m,n)$. We have to construct a nontrivial solution for the equation
$\dd{}{y}(\frac{r}{f})=\dd{}{x}(\frac{s}{f})$ with $\deg r\le (m-1,n)$
and $\deg s\le (m,n-2)$. We distinguish different cases:\\
{\sl Case I:} $f$ is squarefree. We write $f=gh$ with
$\deg_yg=\ell$ and we can assume that $h$ is irreducible. Writing
$$g=b_0(x)+b_1(x)y+\dots+b_{\ell}(x)y^{\ell},\quad
h=c_0(x)+c_1(x)y+\dots +c_{n-\ell}(x)y^{n-\ell}$$
gives
\begin{eqnarray*}
\frac{\partial g}{\partial y}h&=&
b_1(x)c_0(x)+\dots+\ell b_{\ell}(x)c_{n-\ell}(x)y^{n-1},\\
g\frac{\partial h}{\partial y}&=&
b_0(x)c_1(x)+\dots+(n-\ell)b_{\ell}(x)c_{n-\ell}(x)y^{n-1}.
\end{eqnarray*}
{\sl Case I.1:} $\ell\ne 0$ in $k$. Take
$$r=(n-\ell)\dd{g}{x}h-\ell g\dd{h}{x}\quad \mbox{ and }\quad
s=(n-\ell)\dd{g}{y}h-\ell g\dd{h}{y}.$$
One sees at once that
$\dd{}{y}(\frac{r}{f})=\dd{}{x}(\frac{s}{f})$ holds and that by
construction $\deg r\le (m-1,n)$, $\deg s\le (m,n-2)$.
If we had $r=s=0$ then $h$ would divide $\dd{h}{x}$ and
$\dd{h}{y}$ which would imply $\dd{h}{x}=\dd{h}{y}=0$,
contradicting the irreducibility of $h$. Therefore $(r,s)\ne (0,0)$
and we are done. \\
{\sl Case I.2:} $\ell=0$ in $k$. Then $\deg_y \dd{g}{y}h\le n-2$.
Take $$r=\dd{g}{x}h,\quad s=\dd{g}{y}h.$$
Then the equation $\dd{}{y}(\frac{r}{f})=\dd{}{x}(\frac{s}{f})$
is satisfied with $\deg r\le (m-1,n)$ and $\deg s\le (m,n-2)$. Also
$(r,s)\ne (0,0)$ else $g$ would be a $p$-power contradicting the
fact that $f$ is supposed to be squarefree. \\
{\sl Case II:} $f$ is not squarefree. We write $f=g^2h$ and we
can assume that $g$ is irreducible. Take
$$r=h\dd{g}{x}\quad\mbox{ and }\quad s=h\dd{g}{y}.$$
Then $(r,s)\ne (0,0)$ because
$g$ is irreducible and
$$\frac{r}{f}=\frac{1}{g^2}\dd{g}{x}=\dd{}{x}(-\frac{1}{g}),\quad
\frac{s}{f}=\frac{1}{g^2}\dd{g}{y}=\dd{}{y}(-\frac{1}{g})$$
shows that $\dd{}{y}(\frac{r}{f})=\dd{}{x}(\frac{s}{f})$
holds. It is clear that $\deg r\le (m-1,n)$ and $\deg s\le
(m,n-2)$.
\rule{2mm}{2mm}\bigskip
{\it Proof of Lemma 2:} Suppose that $k$ is algebraically closed of
characteristic $0$, $f\in k[x,y]$ is irreducible with $\deg
f=(m,n)$ and
$$\frac{\partial }{\partial y}\left(\frac{r}{f}\right)=
\frac{\partial }{\partial x}\left(\frac{s}{f}\right)$$
with $\deg r\le(m-1,n)$, $\deg s\le(m,n-2)$ and $(r,s)\ne (0,0)$.
The equation implies that
$$\omega=\frac{r}{f}dx+\frac{s}{f}dy$$
is a nontrivial closed differential form. Now the structure theorem
for closed $1$-forms (cf. \cite[Satz 2, p.172]{R}) says that
$\omega$ has the form
$$\omega=\sum_{i=1}^u\lambda_i\frac{dp_i}{p_i}+d(\frac{g}{q_1^{e_1}\dots
q_v^{e_v}})$$
where $p_i,q_j\in k[x,y]$ are irreducible, $g\in k[x,y]$,
$\lambda_i\in k$, $e_j\ge 0$, $p_1,\dots,p_u$ are pairwise prime,
$q_1,\dots,q_v,g$ are pairwise prime. Comparing the coefficients of
$dx$ and $dy$ gives
\begin{eqnarray*}
\frac{r}{f}&=&\frac{\lambda_1\dd{p_1}{x}}{p_1}+\dots+\frac{\lambda_r
\dd{p_u}{x}}{p_u}+\frac{\dd{g}{x}}{q_1^{e_1}\dots q_v^{e_v}}
-\frac{e_1g\dd{q_1}{x}}{q_1^{e_1+1}q_2^{e_2}\dots q_v^{e_v}}-\dots
-\frac{e_vg\dd{q_v}{x}}{q_1^{e_1}\dots q_{s-1}^{e_{s-1}}q_v^{e_v+1}}\\
\frac{s}{f}&=&\frac{\lambda_1\dd{p_1}{y}}{p_1}+\dots+\frac{\lambda_u
\dd{p_u}{y}}{p_u}+\frac{\dd{g}{y}}{q_1^{e_1}\dots q_v^{e_v}}
-\frac{e_1g\dd{q_1}{y}}{q_1^{e_1+1}q_2^{e_2}\dots q_v^{e_v}}-\dots
-\frac{e_vg\dd{q_v}{y}}{q_1^{e_1}\dots q_{s-1}^{e_{s-1}}q_v^{e_v+1}}
\end{eqnarray*}
$k[x,y]$ is factorial and therefore we have for each $p_i$ and $q_j$ a
valuation $v_{p_i}$ and $v_{q_j}$. \\
If $g\ne 0$ and $e_j\ge 1$ for some $j$ we would get
$v_{q_j}(\frac{r}{f})=-e_j-1\le -2$ or $v_{q_j}(\frac{s}{f})=-e_j
-1\le -2$ as $(\dd{q_j}{x},\dd{q_j}{y})\ne (0,0)$, a contradiction
to the irreducibility of $f$. Therefore we can assume
$e_1=\dots=e_v=0$. \\
If $\lambda_i\ne 0$ and $p_i$ is prime to $f$ then
$(\dd{p_i}{x},\dd{p_i}{y})\ne (0,0)$ would imply
$v_{p_i}(\frac{r}{f})=-1$ or $v_{p_i}(\frac{s}{f})=-1$, a
contradiction. We can write now
$$\omega=\lambda\frac{df}{f}+dg$$
with $\lambda\in k$ which gives
$$r=\lambda\dd{f}{x}+f\dd{g}{x}\quad\mbox{ and }\quad
s=\lambda\dd{f}{y}+f\dd{g}{y}.$$
If $\dd{g}{x}\ne 0$ then $r$ would have degree $\ge m$ in $x$, a
contradiction, if $\dd{g}{y}\ne 0$ then $s$ would have degree $\ge n$
in $y$, a contradiction. Therefore we get
$$r=\lambda\dd{f}{x}\quad\mbox{ and }\quad s=\lambda\dd{f}{y}$$
with $\lambda\ne 0$. As $n\ge 1$ we can write
$f=a_0(x)+\dots+a_n(x)y^n$ with $a_n(x)\ne 0$ and get
$\dd{f}{y}=a_1(x)+\dots+na_n(x)y^{n-1}$
which shows that $s$ has degree $n-1$ in $y$, a
contradiction. Therefore $f$ can not be irreducible.
This proves the lemma. \rule{2mm}{2mm}\bigskip
\section{Examples}
In the following lemma families of polynomials are constructed
with an explicit reducibility condition.
\begin{Lemma}
\begin{enumerate}
\item Let $k$ be an algebraically closed field of characteristic
$\ne 2$, $m,n\ge 1$ integers and $t\in k$. The polynomial
$f_t(x,y)=(tx^m-2x+2)+(x^m-t)y^n\in k[x,y]$
is reducible if and only if $(t^2+2)^m-2^mt=0$.
In this case the factor $x-\frac{t^2+2}{2}$ splits off.
\item The polynomial $g_m(t)=(t^2+2)^m-2^mt\in \Z[t]$
is irreducible over $\Q$ and $\gcd\{g_m(\ell):\ell\in\N\}=1$.
\end{enumerate}
\end{Lemma}
{\it Proof:}
\begin{enumerate}
\item Suppose first that $tx^m-2x+2$ and $x^m-t$ are relatively prime
and $f_t$ is reducible. Then $f_t$ is reducible as a
polynomial in $y$ with coefficients in $k(x)$ and therefore
$\frac{-tx^m+2x-2}{x^m-t}$ is a nontrivial power in $k(x)$.
Then $-tx^m+2x-2$ and $x^m-t$ have to be nontrivial powers in $k[x]$
and therefore inseparable. But $x^m-t$ is inseparable only if
$m=0$ or $t=0$ in $k$ and for both cases $-tx^m+2x-2$ is
separable. So this case can not happen. \\
If $tx^m-2x+2$ and $x^m-t$ have a common factor
$x-u$ for some $u\in k$ then $f_t$ is clearly reducible.
This happens iff
$tu^m-2u+2=u^m-t=0$ which is equivalent to
$u=\frac{t^2+2}{2}$ and $(t^2+2)^m-2^mt=0$ which proves part 1
of the lemma.
\item Let $\alpha\in\overline{\Q}$ be any root of $g_m$ over $\Q$,
i.e. $\alpha=(\frac{\alpha^2+2}{2})^m$. Define
$\beta=\frac{\alpha^2+2}{2}\in\Q(\alpha)$. Then
$\alpha=\beta^m\in\Q(\beta)$ and therefore $\Q(\alpha)=\Q(\beta)$.
Finally
$0=\alpha^2+2-2\beta=\beta^{2m}-2\beta+2$
shows that $\beta$ is a root of the irreducible Eisenstein polynomial
$t^{2m}-2t+2$, which implies that $\Q(\alpha)=\Q(\beta)$ has degree
$2m$ over $\Q$. Therefore $g_m=(t^2+2)^m-2^mt$ is irreducible over
$\Q$. From $g_m(0)=2^m$ and $g_m(1)\equiv 1\bmod 2$ one sees
that $\gcd\{g_m(\ell):\ell\in\N\}=1$. \rule{2mm}{2mm}\bigskip
\end{enumerate}
To construct infinitely many examples with the right reduction
behavior we use the very plausible Bouniakowsky conjecture which was
generalized by Schinzel as hypothesis H (cf. \cite{B},\cite{S}):
\begin{Conjecture}[Bouniakowsky] If $g(t)\in\Z[t]$ is
irreducible and $N=\gcd\{g(\ell):\ell\in\N\}$ then there are
infinitely many $\ell\in\N$ such that $\frac{1}{N}|g(\ell)|$
is a prime.
\end{Conjecture}
Now we prove our proposition of section 1. We use the
notations and results of the previous lemma.
Let $m,n\ge 1$ be integers and take
$$f_{\ell}(x,y)=(\ell x^m-2x+2)+(x^m-\ell)y^n\in \Z[x,y]$$
with $\ell\in\Z,\ell\ge 2$. Then $H(f_{\ell})=\ell$.
As $g_m(\ell)\ne 0$ in $\Q$ the polynomial $f_{\ell}$ is absolutely
irreducible over $\Q$. If $p_{\ell}=g_m(\ell)$ is a prime,
then $g_m(\ell)\equiv 0\bmod p_{\ell}$ and
$f_{\ell}\bmod p_{\ell}$ is reducible and
$$p_{\ell}=g_m(\ell)\ge \ell^{2m}=H(f_{\ell})^{2m}.$$
Now the Bouniakowsky conjecture says that there are infinitely
many $\ell$ such that $g_m(\ell)$ is prime. This proves the
proposition.
|
1,116,691,497,609 | arxiv | \section{Introduction}
Plasma diagnostics using $K_{\alpha}$ fluorescence spectra allows to investigate properties of warm dense matter.
This work focuses on the bulk temperature distribution of plasmas created from Ar droplets irradiated by high energy laser pulses
with a power of 10$^{19}$ W/cm$^2$ \cite{Ar}. Due to the high intensity, matter at temperatures between some 10 eV to 250 eV (bulk) and up to 1 MeV (blow-off) is created. Most of the atoms are ionized and a wide range of different ion species is observed.
The atomic density of the Ar droplet is $n_{\textrm{\scriptsize tot}}= 2.2 \times 10^{22} $ cm$^{-3}$.
Accordingly, free electron densities vary between $10^{22}-10^{24}$ cm$^{-3}$ depending on the degree of ionization of the Ar plasma.
Due to laser plasma interaction the argon plasma radiates $K_{\alpha}$ lines with emission energies at about 3 keV, which can be used straightforwardly to infer the plasma parameters.
To propagate through the plasma, radiation frequencies have to be larger than the plasma frequency $\omega_{\textrm{\scriptsize plasma}}=\sqrt{\frac{n_{e}e^{2}}{\epsilon_{0} m}}$,
where $n_{e}$ is the density and $m$ is the mass of free electrons \cite{saha}. Considering the expected high free electron densities, x-rays (like the $K_{\alpha}$-lines) are needed to investigate the created plasmas.
To describe the plasma microscopically, including its emission spectra, one has to consider in detail the influence of the environment on the ionic potential, the electron-electron and the electron-ion interaction \cite{ion}. To introduce plasma effects, the electron-ion plasma screening is described within an ion sphere model approximation \cite{ion} and the electron-electron plasma screening is described within a quantum statistical approach by the Montroll-Ward self-energy contribution. To understand the $K_{\alpha}$ emissions, which correspond to an electron going from a 2p to an 1s level, also the fine structure of the emitting states has to be taken into account. Moreover, the creation of inner (K-) shell vacancies is accompanied by ionization and excitation of outer shell electrons (L- and M-shell) resulting in additional satellite lines. The emission energies of the satellites significantly increase with the number of vacancies and excited states. The intensity ratio of different lines is determined according to their statistical weight (LS coupling) and the abundance of the corresponding emitter species in dependence on the plasma parameters.
Using coupled Saha equations, we consider the different charge states in thermodynamic equilibrium \cite{saha}. Finally, we add Lorentz profiles and convolute with a Gaussian instrument function to calculate synthetic spectra from the plasma shifted emission energies.
We compare our results with the experimental data presented in \cite{Ar} as well as with theoretical results obtained with the FLYCHK code, also given in \cite{Ar}.
FLYCHK\cite{FLYCHKpublication,FLYCHK,FLYCHKManual} provides emission spectra and ionization distributions within plasmas by solving rate equations for level populations considering collisional and radiative atomic processes. The code is well benchmarked for long-pulse laser experiments. However, there can appear jumps in the density of nearly neutral ion species if bound states are pressure ionized. When constructing synthetic X-ray spectra, line components that involve vanishing states have to be removed by hand if there is no steady fade out of its contribution
with respect to the plasma parameters\cite{Stambulchik09}. We apply an approach where pressure ionized states shift steadily into the continuum and no extra treatment is needed \cite{cpp_chen,Sengebusch09}.
\section{Unperturbed emitter}
To describe the influence of a plasma environment on an emitter, a perturbative ansatz, $H = H_0 + H'$, is chosen. The system’s Hamiltonian $H$ is split into a part $H_0$ describing the isolated emitting particle and a perturbing plasma potential $H'$. The isolated emitter can be described by means of atomic physics.
In contrast to previous publications, we changed and expanded the emitter configurations that are taken into account. Previously, to determine emission and ionization energies of various ionic configurations, we solved the corresponding self-consistent Roothaan-Hartree-Fock equations applying the chemical ab initio code Gaussian 03, see e.g. \cite{cpp_chen,Sengebusch09}. However the calculations prevent straightforward observation of fine-structure splitting and only a very restricted number of excitation levels could be taken into account. As there are several hundred transitions located closely together, it is common to limit the number of levels in the calculations especially for lower charge states. FLYCHK for example applies super-configuration transition arrays, which result in effective bound states and respective transition rates \cite{Ar}. However, our observations show that the details of the energy level configurations are crucial to the determination of the plasma composition and emission spectra.
Despite the calculational efforts we decided to take all tabulated values of energy levels and emission energies into account to describe the unperturbed particles as precisely as possible. In particular, we consider ionization for all shells but 1s, i.e. for argon ($Z=18$) we apply 16 ionization stages from Ar$^+$ to Ar$^{16+}$. For those ions we considered altogether 111 different electronic configurations and 713 different ionization energies. The latter can be found in the NIST Atomic Spectra Database \cite{NIST}.
To get the difference between the two numbers across, let's consider an example: The ground state electronic configuration of Ar$^{+}$ is (1s$^2$2s$^2$2p$^6$3s$^2$3p$^5$). According to LS coupling we have two different states $J=1/2$ and $J=3/2$ with two different ionization energies for the outermost electron. Another possibility is the excited configuration (1s$^2$2s$^2$2p$^6$3s$^2$3p$^4$3d$^1$). Here we already have 28 different ionization energies corresponding to the different terms from $^2$S to $^4$F and J-states from 1/2 to 9/2.
To calculate the plasma composition also the total binding energies of the considered ions were necessary and taken from NIST as well. Moreover, we took into account a total of 1211 different K$_\alpha$ emission lines, as tabulated by Palmeri et al. \cite{palmeri}, to construct synthetic emission spectra. All those level and emission energies build up the unperturbed basis to which a perturbation according to the plasma is added as described below.
According to first order perturbation theory, the unperturbed wave functions are required to calculate changes of energy levels. If we look at single levels, e.g. in ionization processes, the energy shift is simply given by the averaging over the perturbation $H'$
\begin{equation}
\Delta{E^{(1)}}=\langle\varphi_{val}|H'|\varphi_{val}\rangle~.
\end{equation}
However, when we consider shifts of emission energies two levels are involved. Then the shift is determined by
\begin{equation}
\Delta{E^{(1)}}=\langle\varphi_i|H'|\varphi_i\rangle - \langle\varphi_f|H'|\varphi_f\rangle~.
\end{equation}
Both equations apply the unperturbed wave functions. Whereas $\varphi_{val}$ denotes the wave function of the valence orbital of the electron that is ionized, $\varphi_i$ and $\varphi_f$ denote the initial and final orbitals of the emission transition, respectively.
The orbital wave functions are separated into radial orbitals $R_{nl}(r)$ and spherical harmonics $Y_{lm}(\theta, \phi)$,
\begin{equation}
\varphi_{nlm}(\vec{r})=R_{nl}(r)\cdot Y_{lm}(\theta, \phi)~.
\end{equation}
Further, the radial atomic orbitals are expanded as a finite superposition of Slater orbitals. The suitable superposition can be determined self-consistently within an iterative process to minimize the energy of Hamiltonian $H_0$. In the following we use the radial wave functions determined by Bunge \textit{et al}. All parameters and results can be found in detail in \cite{Bunge}.
\section{Plasma polarization}
In the following we describe the determination of the perturbing Hamiltonian $H'$ and calculate shifts of the energy levels caused by the surrounding charges in the plasma environment
within the approach described above. To obtain the distribution of the free plasma electrons around the quasi-static ionic emitters,
an ion sphere or confined atom model is used \cite{cpp_chen}. The ion sphere contains a nucleus of charge $Z=18$ and the respective number of electrons, so that the system in total is neutral. The electron density inside the sphere is divided into the density of bound and free electrons, $n_b(r)$ and $n_f(r)$ respectively.
Accordingly, the radial Poisson equation for the potential $\phi(r)$ reads
\begin{equation}
\Delta{\phi(r)}=4\pi e~n_b(r)+4\pi e~n_f(r)-4\pi~Ze~\delta(r).\label{poisson}
\end{equation}
The perturbing potential is given by the difference of the sphere's potential with and without free plasma electrons,
\begin{equation}
H'=-e\left[\phi(r)-\phi(r, n_f=0)\right]~.
\end{equation}
The free electron density is determined self-consistently,
\begin{eqnarray}
n_f(r)&=&\frac{4 n_f(R)}{\sqrt{\pi}}\int_{p_0}^{\infty} \frac{dp~p^2}{\left(2mk_BT\right)^{3/2}}~\nonumber\\
&&\times
\textrm{exp}\left[\left( \frac{-p^2}{2mk_BT}+\frac{e~{\phi(r)}}{k_BT} \right)\right],\label{nfree}
\end{eqnarray}
with the minimum momentum $p_0=\sqrt{2me~\phi(r)}$ and the Wigner-Seitz radius $R =\sqrt[3]{\frac{3~Z_\textrm{\scriptsize ion}}{4\pi n_e}}$ which is the boundary of the ion sphere.
Equation (\ref{nfree}) is obtained from the equation of state of non-degenerate, free electrons in momentum representation. The fraction before the integral represents a normalization factor to assure that the number of free electrons within the sphere equals the ionic charge.
The bound electron density is given by the radial wave functions of the unperturbed emitter,
\begin{equation}
n_b(r)=\frac{1}{4\pi}\sum_{nl}r^2\left|R_{nl}(r)\right|^2.\label{nbound}
\end{equation}
The sum $\sum_{nl}$ depends on the specific electronic configuration of the considered ion (every electron contributes one bound state orbital). Since the configurations of initial and final state of a transition differ, we also obtain different potentials $\phi(r)$ and $H'$ of the ion sphere before and after the transition. For the $R_{nl}(r)$ functions we take the results of Bunge \textit{et al.} \cite{Bunge}, as described above.
The iterative calculation of $\phi(r)$ and $n_f(r)$ starts with the assumption of a constant free electron density $n_{\textrm{\scriptsize f}}(r)=n_e$ throughout the sphere.
We solve (\ref{poisson}) and (\ref{nfree}) iteratively and finally obtain a self-consistent free electron density which forms a polarization cloud close to the nucleus and dilutes for larger radii.
Due to screening caused by the free electrons close to the nucleus, the energy levels and hence the emission and ionization energies are shifted.
Our calculational results are exemplarily shown in figures (\ref{shift_n}) and (\ref{shift_T}). We obtain emission line shifts to lower energies (red shift) in the order of some eV depending on both the plasma temperature $k_BT$ and the average free electron density $n_e$. This effect is referred to as plasma polarization shift and can be explained as follows \cite{Sengebusch09}:
The free electrons screen the nucleus resulting in lower absolute values of the negative binding energies. As the 1s level is localized closer to the nucleus than 2p, it is more affected by the screening of the nucleus and thus experiences the larger shift, i.e. the gap between the two levels narrows. Since the emission energy is given by the difference of the two involved levels the spectral line is red shifted to lower emission energies. The red shift increases with rising free electron density as the screening of the nucleus rises as well. However, the red shift decreases with rising plasma temperature. This is due to the fact that the self-consistently determined free electron density within the ion sphere is not constant but depends on the distance to the emitter's core. The higher the temperature the more the screening cloud is spatially extended ('smeared out') and the actual free electron density close to the nucleus decreases, resulting in less screening and a lower red shift.
Plasma polarization is not only important to calculate the accurate emission line positions but also to determine the plasma composition. In contrast to emission lines, a single level surrounded by plasma experiences a blue shift to higher energies when its absolute value of the binding energy is lowered by screening, i.e. the level shifts closer to the continuum ($E=0$). Depending on the plasma parameters this shift can be that large that the continuum is reached and the bound electron is set free (pressure ionization, Mott effect \cite{quantum}). Hence, we obtain another free electron and a new ion species. In the following we will outline, how the plasma composition is calculated taking into account the polarization shifts we determined within the ion sphere model.
\begin{figure}
\includegraphics[width=0.4\textwidth]{ArKA_shift1.pdf}
\caption{Shift of $K_{\alpha}$ emission energy with rising temperature for two different electron densities and ground state ionization stages. }\label{shift_n}
\end{figure}
\begin{figure}
\includegraphics[width=0.4\textwidth]{ArKA_shift2.pdf}
\caption{Shift of $K_{\alpha}$ emission energy with rising free electron density for two different plasma temperatures and ground state ionization stages.}\label{shift_T}
\end{figure}
\section{Plasma composition}
Assuming a local thermal equilibrium, we apply Saha equations to determine the abundance of the different ion species \cite{CPP_PL},
\begin{equation}
\frac{n_{(m+1)}n_e}{n_{(m)}}=\frac{\sigma_e^{in}}{\lambda_e^3}~\frac{\sigma_{(m+1)}^{in}}{\sigma_{(m)}^{in}}~.
\end{equation}
Here we use a representation of the chemical potential of species $c$ which is of the same form as the one for the classical ideal gas,
$\mu_c=k_BT~\textrm{ln}\left[\frac{n_c\lambda_c^3}{\sigma_{c}^{in}}\right]$. However, the denominator in the logarithm is given by the internal partition function $\sigma_{c}^{in}$
instead of the statistical weight of the corresponding particle. The internal partition functions contain the particle interactions and are discussed in detail in the next paragraph. Coupling the Saha equations of successive ionization stages we obtain
\begin{equation}
n_{(m)}=\left[\frac{1}{n_e\lambda_e^3}\right]^m~\sigma_e^{in}(m)~\frac{\sigma_{(m)}^{in}}{\sigma_{(0)}^{in}}~n_{(0)}~.
\end{equation}
Here, $m$ denotes the $m$-th ionization stage with respect to the uncharged atom (reference state).
Accordingly, $n_{(0)}$ and $n_{(m)}$ are the densities of particles in these states.
Keeping the total particle density fixed at the bulk value $n_{\textrm{\scriptsize tot}}$ the following conservation laws have been used to solve the system of Saha equations
\begin{equation}
n_{\textrm{\scriptsize tot}}=\sum_{m=0}^Z n_{(m)} \textrm{~~and~~} n_e=\sum_{m=0}^Z m\cdot n_{(m)}\label{ne-equation}~.
\end{equation}
Here, we limit ourselves to a maximum number of ionization stages sufficient for temperatures we are interested in ($T \le 250$ eV) ,
i.e. the sum doesn't run until 18 but is considered up to the Lithium like ion ($m=15$).
Let us now consider the electronic partition function $\sigma_e^{in}$.
Since the electrons do not have any internal degrees of freedom, we split the partition function into the ideal part represented by the statistical weight $g_e=2$ and a Boltzmann factor containing the interaction energy $\Delta_e$,
\begin{equation}
\sigma_e^{in}=g_e~e^{-\Delta_e/k_BT}~.
\end{equation}
The electron interaction energy $\Delta_e$ is treated as a rigid, momentum independent quasi-particle shift of the free particle energies and is divided into an electron-ion interaction term and an electron-electron interaction term. According to \cite{CPP_PL} the total internal partition function of the non-degenerate free electrons reads
\begin{eqnarray}
\sigma_{e}^{in}(m) &=&2^{m} \exp\left[-\frac{\Delta_{e}^{\textrm{MW}}}{k_{B}T} \right]^{m}\\
&&\times\exp \left[\frac{1}{k_{B}T} \frac{e^{2}}{(4\pi)^{2/3}\varepsilon_{0}} \left(\frac{n_{e}}{3}\right)^{1/3} \sum_{x=1}^m x^{2/3} \right].\nonumber
\end{eqnarray}
The first factor is given by the statistical weight, the second factor is given by the electron-electron interaction and the third factor originates from the electron-ion-interaction.
For the contribution of the electron-electron interaction we use the Montroll-Ward approximation \cite{saha}
\begin{eqnarray}
\Delta_{e}^{\textrm{MW}}&=&-\frac{e^2}{2}~\sqrt{\frac{n_ee^2}{\varepsilon_0k_BT}}\nonumber\\
&&+\frac{\sqrt{2\pi^{2}}n_{e}\lambda_{e}e^{4}}{8(k_BT)^{2}}-\frac{n_{e}\lambda_{e}^{3}}{8\sqrt{2}}
+\frac{n_{e}\lambda_{e}^{2}e^{2}}{4k_BT}.
\end{eqnarray}
Taking a closer look at the the internal partition functions of the ions $\sigma_{(m)}^{in}$ we have to consider internal degrees of freedom, i.e. differently excited bound states.
The screening due to the plasma environment leads to a shift of the ionization energies till the bound states vanish at the Mott densities (pressure ionization)\cite{Lin,quantum}. To avoid discontinuities due to pressure ionization, we apply a Planck-Larkin renormalization to the internal partition function \cite{CPP_PL}. It than reads for the $m$-th ionization stage as follows
\begin{eqnarray}
\sigma_{(m)}^{in}&=&\textrm{exp}\left[-\frac{E_0^{(m+1)}}{k_BT}\right]\\
&&\times~\sum_i\left(\textrm{exp}\left[-\frac{\Delta E_i^{(m)}}{k_BT}\right]-1+\frac{\Delta E_i^{(m)}}{k_BT}\right).\label{PL}\nonumber
\end{eqnarray}
To obtain this equation we separate the energies $E_i^{(m)}$ of possible bound states of the $m$-th ion
into the ground state energy of the next ionization stage $E_0^{(m+1)}$ and possible excitations of the outermost valence electron
$\Delta E_i^{(m)}$,
\begin{equation}
E_i^{(m)} =E_0^{(m+1)} + \Delta E_i^{(m)}~.
\end{equation}
Within this approximation a Mott transition would refer to a state where $\Delta E_i^{(m)}\rightarrow 0$ due to plasma polarization.
Keeping the total density fixed at the bulk value of $n_{\textrm{\scriptsize tot}} = 2.2 \times 10^{22} $ cm$^{-3}$ we calculate
the abundance of different ion species (figure (\ref{compo})) and the corresponding average charge state (solid line in figure (\ref{charge})) depending on the plasma
temperature. We compare our results for the charge state distribution with results of Neumayer \textit{et al.}
obtained from the FLYCHK code \cite{Ar} (dashed line in figure (\ref{charge})).
We see a sharp rise of the charge state and a rapid sequence of ionization stages are passed through for temperatures below 50 eV.
This is the regime of M-shell ionization, binding energies are rather low and pressure ionization plays an important role. Comparing with FLYCHK, at the same temperature our calculations give not only a higher charge state but also larger energy shifts of the emission energies in this regime. For temperatures above 50 eV, where the L-shell ionization regime sets in, our results swing back to the FLYCHK results and oscillate around the graph.
The differences in figure (\ref{charge}) arise from the different treatment of the variety of bound states and pressure ionization within the approaches.
As there are several hundred transitions closely distributed for lower charge states usually a simplification is applied for better handling.
FLYCHK uses super-configuration transition arrays (effective bound states) weighted by respective transition rates rather than the complete level structure\cite{Ar}.
We do not average over several levels, but take all low lying excited states into account.
Further, in the FLYCHK code continuum lowering is treated within a Stewart-Pyatt like model to implement medium corrections to the ionization energies.
The model interpolates between the two limiting cases: Debye screening for dilute and hot plasma on the one hand and screening within an ion sphere model with constant free electron density for dense and cool plasmas an the other hand\cite{FLYCHKManual}. If bound states merge into the continuum, the calculations get discontinuous since these states suddenly disappear
from the implemented rate equations. Using effective bound states reduces these events. Within our quantum statistical approach, there is no such discontinuity thanks to the use of the Planck-Larkin renormalization of bound states \cite{CPP_PL}. As shown above, we also use an ion sphere model to calculate the continuum lowering. However, the free electron density within the sphere is not constant but radially dependent.
All in all, we observe several Mott transitions in solid density argon at temperatures below 50 eV which lead to a higher degree of ionization than proposed by FLYCHK. Looking at figure (\ref{compo}) magnesium and neon like states seem to be more stable, which is due to their closed shell structure. Hence, the slope of the charge state in figure (\ref{charge}) flattens for rising plasma temperatures. As the ion charge rises pressure ionization becomes less important and our results converge to the FLYCHK values. The oscillation around the values is in accordance with the variing shell structure of the ions. In agreement with the expectations, less than half filled shells seem to be the least stable states and closed shells show a high stability against temperature increase. In contrast to our detailed slope, the FLYCHK code creates a rather generic output.
\begin{figure}
{\includegraphics[width=0.4\textwidth]{Ar_compo.pdf}}
\caption{Plasma composition depending on bulk temperature: percental abundance of different Ar ions.}\label{compo}
\end{figure}
\begin{figure}
{\includegraphics[width=0.4\textwidth]{Ar_Charge.pdf}}
\caption{Average charge state of the plasma depending on bulk temperature. (solid) this work, (dashed) from Neumayer \textit{et al.} obtained by FLYCHK\cite{Ar}.}\label{charge}
\end{figure}
\section{Synthetic spectra and temperature distribution}
To construct synthetic spectra from the shifted emission energies every line as well as its fine structure components
are assigned a Lorentz profile with natural line width $\gamma$ and maximum intensity $I_\textrm{max}$,
\begin{equation}
I(E)=\frac{I_\textrm{max}~(\gamma/2)^2}{(E-E_0)^2+(\gamma/2)^2}~.
\end{equation}
The central line position $E_0$ is given by the plasma shifted line positions. The maximum intensity is mainly given by the emitter abundance in the plasma.
We determine all intensities relative to a reference state,
\begin{equation}
\frac{I_\textrm{max}}{I_\textrm{ref}}\simeq \frac{n}{n_\textrm{ref}} \left( \frac{E_{0}}{E_{0}^\textrm{ref}}\right)^{4} .\label{intensity}
\end{equation}
For the electron transitions considered here, we assume nearly constant dipole matrix elements $\left|\frac{1}{\Omega_0}\int d^3r ~\varphi_\textrm{\scriptsize in}^* ~\vec{r}~\varphi_\textrm{\scriptsize fin}^*\right|^2$ with respect to excitation and ionization of outer shells since the one particle wave functions $\varphi_\textrm{\scriptsize in}$ (initial state, 2p) and $\varphi_\textrm{\scriptsize fin}$ (final state, 1s) change only very slightly \cite{Sengebusch09}. Hence, the Einstein coefficients for spontanious emisssion are simply $\propto E_{0}^3$ and equation (\ref{intensity}) can easily be derived.
The Lorentzians are summed up and convoluted with a Gaussian profile of width $\Gamma$ to take into account instrumental broadening
of the measurements. According to Neumayer \textit{et al.} \cite{Ar} we varied the broadening parameters within the range of some eV
and found the most suitable combination in a universal natural linewidth of $\gamma=1$ eV and a Gaussian width of $\Gamma= 2.5$ eV.
A selection of spectra obtained at different plasma temperatures is shown in figure (\ref{Spektren}).
Keeping the total density and plasma temperature fixed, the free electron density follows from the composition.
Shown are some typical spectra: At low temperatures, up to some 10 eV, we are in the regime of M-shell ionization.
Since the differences of the various emission energies are small, the lines of Ar$^+$ to Ar$^{9+}$ group indistinguishably within one large main peak which
shifts to higher energies with rising temperatures. With further increase of the temperature, we enter the regime of L-shell ionization.
The main peak becomes less prominent and several distinguishable peaks appear, which correspond to the emission energies
of the respective ions. We calculated spectra up 250 eV where the Li-like satellites are dominant.
Experimentally observed higher emission energies mainly arise from argon ions with only one or two electrons (Ly-$\alpha$, He-$\alpha$,
intercombination line). These emissions stem from the blow-off plasma created by the laser impact.
\begin{figure}
{\includegraphics[width=0.4\textwidth]{ArKA_Spec.pdf}}
\caption{Selected normalized synthetic spectra for different plasma temperatures.
The total density is fixed at the bulk value, the free electron density corresponds to the plasma composition.}\label{Spektren}
\end{figure}
We apply a superposition of our calculated spectra for different temperatures to model the experimental results obtained by Neumayer \textit{et al.} \cite{Ar}.
To obtain the best fit the weight of the different spectra is determined within a variational approach using the method of least squares.
Results for the best fit and the corresponding temperature distribution are shown in figure (\ref{bestfit}) and figure (\ref{temperaturedistribution}), respectively.
The superimposed spectrum shows a good agreement especially for the M-shell satellites. The position and width of the main peak is rather sensitive to the plasma bulk temperature.
At a first glimpse, the agreement with the L-shell satellites is less satisfying. The peaks are at correct positions but the widths seem to small. We attribute this to our choice of the natural line widths. We use a universal value of $\gamma$ which implies the same lifetime for all considered states. But especially highly ionized and excited states experience shorter lifetimes and hence show a larger broadening. Accordingly, the L-shell emission lines would blur and give a better fit to the energy spectrum above 2980 eV.
\begin{figure}
{\includegraphics[width=0.4\textwidth]{ArKA_Int1.pdf}}
\caption{Best fit of superposition of theoretical spectra to space-time-integrated measurements
of Neumayer \textit{et al.} \cite{Ar}.}\label{bestfit}
\end{figure}
\begin{figure}
{\includegraphics[width=0.4\textwidth]{ArKA_Temp1.pdf}}
\caption{Temperature distribution which gives the best fit of theoretical to experimental spectra.
The relative weights are determined via the method of least squares.}\label{temperaturedistribution}
\end{figure}
\section{Discussion and conclusion}
The temperature distribution, which best fits to the experimentally obtained argon K$_{\alpha}$ spectra, is a two-peaked curve with local maxima at our lowest temperature value of 10 eV and at our highest temperature value of 250 eV and a steady decrease of relative weights for intermediate temperatures. We chose temperature steps of 10 eV to keep the variation manageable. Smaller steps are possible, however the two peaked slope of the temperature distribution would not change significantly. The two peaks can be interpreted as a result of the two differently heated parts of the argon droplet, namely the cold bulk and hot blow off plasma. The blow off plasma is directly created by the laser-plasma interaction. It is rather hot and dilute and is indicated by the H- and Ne-like emission lines above 3100 eV. However, as the experimental data is spatially and temporally integrated, the measurements contain also emissions of not fully heated or already cooled down particles. Due to such emitters we obtain the high temperature tail above 200 eV.
In fact, this work is more related to the bulk temperature distribution due to electronic heating. The electrons are accelerated in the laser field and flow directedly trough the droplet, mainly heating the area close to the laser focus and the volume behind. As the electrons scatter and the ions tend to equilibrate, the heat is transferred radially to cooler regions of the droplet. The maximum of the temperature distribution at lowest temperatures originates on the one hand from emitters of the outer spacial regions and on the other hand from particles that already cooled down before emission.
There are significant differences between the temperature distribution proposed by Neumayer \textit{et al.} \cite{Ar} and the results shown here: according to their FLYCHK results, they also obtained a two peaked distribution, only our first maximum lies at much lower plasma temperature and our temperature range is about 100 eV broader. These differences are due to the different results of the composition calculations as show in the previous chapter and figure (\ref{charge}). As we obtain higher degrees of ionization at lower temperatures, we see the general shift of emission lines to higher energies (blue shift) already at lower temperatures and hence we obtain a temperature of about 10 to 20 eV instead of 50 eV to fit the main peak of the measured spectrum. Neumayer \textit{et al.} discussed several mechanisms like resistive heating or radiative heating that might help to explain the absence of temperatures below 50 eV. However, our results suggest, that these influences might not be as substantial as the authors assumed. In order to gain more insight into the heating and cooling processes within the plasma, kinetic codes and hydrodynamic simulations are favourable.
The details of the synthetic spectra need further discussion. It turns out, that using the same broadening for all lines slightly overestimates the width for the main peak (Ar$^{+}$ to Ar$^{9+}$) and underrates the width for the higher lying peaks as pointed out above. A more detailed discrimination of the widths would improve the agreement between theory and experimental data. In particular, the theoretically obtained low minima between the peaks will possibly be smeared out, if the applied broadening is dependent on the emitter configuration as well as on the plasma parameters. A quantum statistical approach to the line width would be desirable, which is well elaborated for H-like systems \cite{broadening} but needs further development for mid-$Z$ materials.
The inclusion of plasma effects, especially shifts and the merging of bound states with the continuum, is important to discuss the composition of the plasma and the density of free electrons.
We have shown that such effects have significant influence on the inferred temperature distribution. In general we have shown that K$_{\alpha}$ spectroscopy is an interesting means for plasma diagnostics and allows an estimate of the temperature profile in warm dense matter.
\ack
We would like to thank P. Neumayer for fruitful discussions and the experimental data.
This work has been supported by the German Research Foundation through the Collaborative Research Center 652 - Strong
correlations and collective effects in radiation fields: Coulomb systems, clusters and particles.
\section*{References}
|
1,116,691,497,610 | arxiv | \section{Introduction}
Our aim was to obtain for categorical groups an analogous description in terms of certain crossed module type objects as we have it for $\mathcal{G}$-groupoids obtained by Brown and Spencer \cite{BS1}, which are strict categorical groups, or equivalently, group-groupoids or internal categories in the category of groups. Under categorical groups we mean a coherent 2-group in the sense of Baez and Lauda \cite{baez-lauda-2-groups}. It is important to note that it is well known that a categorical group is equivalent to a strict categorical group \cite{Sinh 2,JoyalStreet,baez-lauda-2-groups}, but we do not have an equivalence between the corresponding categories. This idea brought us to a new notion of group up to congruence relation. In this way we came to the definition of c-group and the corresponding category. Then we defined action in this category and introduced a notion of c-crossed module. Among of this kind of objects we distinguished connected, strict and special c-crossed modules denoted as cssc-crossed module. We proved that every categorical group gives rise to a cssc-crossed module. In the following paper we will prove that there is an equivalence between the category of categorical groups and the category of cssc-crossed modules. We hope that this result will give a chance to consider for categorical groups the problems analogous to those considered and solved in the case of strict categorical groups in terms of group-groupoids and internal categories in \cite{BM1,Dat1,Dat2,Dat3,Dat4}.
In Section 2 we give a definition of a categorical group, group groupoid and crossed module in the category of groups. In Section 3 we give a definition of group up to congruence relation, shortly c-group, give examples and consider the corresponding category of c-groups denoted as $\cGr$. We define $\cKer f$, $\cIm f,$ for any morphism $f$ in $\cGr,$ and normal c-subgroup in any c-group. In Section 4 we define split extension and action in $\cGr$. After this we give a definition of c-crossed module and give examples. We introduce the notions of special, strict and connected c-crossed modules and give examples. We prove that every categorical group defines a cssc-crossed module.
\section{Preliminaries}
Recall the definition of a monoidal category given by Mac Lane \cite{Mac71}.
\begin{Def}\label{moncat}
A \textbf{monoidal category} is a category $\mathsf{C}=(C_0, C_1, d_0, d_1, i, m)$ equipped with a bifunctor $+\colon\mathsf{C}\times\mathsf{C}\rightarrow\mathsf{C}$ called the monoidal sum, an object $0$ called the zero object, and three natural isomorphisms $\alpha, \lambda$ and $\rho$. Explicitly, \[\alpha=\alpha_{x,y,z}\colon (x+y)+z\stackrel{\approx}{\rightarrow} x+(y+z)\] is natural for all $x,y,z\in C_0$, and the pentagonal diagram
\[\xymatrix{
((x+y)+z)+t \ar[d]_{\alpha+1} \ar[r]^-{\alpha} & (x+y)+(z+t) \ar[r]^-{\alpha} & x+(y+(z+t)) \\
(x+(y+z))+t \ar[rr]_-\alpha & & x+((y+z)+t) \ar[u]^{1+\alpha} }\]
commutes for all $x,y,z,t\in C_0$. Again, $\lambda$ and $\rho$ are natural $\lambda_x\colon 0+x\stackrel{\approx}{\rightarrow} x,$ $\rho_x\colon x+0\stackrel{\approx}{\rightarrow} x,$ for all $x\in C_0$, the diagram
\[\xymatrix{
(x+0)+y \ar[r]^-{\alpha} \ar[d]_-{\rho+1} & x+(0+y) \ar[d]^-{1+\lambda} \\
x+y \ar@{=}[r] & x+y }\]
commutes for all $x,y\in C_0$ and also $\lambda_0=\rho_0\colon 0+0 \stackrel{\approx}{\rightarrow} 0$. Moreover "all" diagrams involving $\alpha,\lambda$, and $\rho$ must commute.
\end{Def}
In this definition we use the term monoidal sum and denote it as $+,$ instead of monoidal product, used in the original definition, and write the operation additively. From the definition it follows that $1_0+f\approx f+1_0 \approx f,$ for any morphism $f.$
In what follows the isomorphisms $\alpha, \lambda$ and $\rho$ involved in group like identities, their inverses, compositions and their monoidal sums will be called as \emph{special isomorphisms}.
Since $+$ is a bifunctor in a monoidal category we have $d_j(f+g)=d_j(f)+d_j(g), j=0,1, i(x+y)=i(x)+i(y)$ and interchange law $(f'+g')(f+g)=f'f+g'g,$ whenever the composites $f'f$ and $g'g$ are defined, for any $x, y\in C_0, f,f',g,g'\in C_1$.
Any category $\mathsf{C}$ with finite products can be considered as a monoidal category where $+$ assigns any two objects to their product and $0$ is the terminal object. The category of abelian groups $\mathsf{Ab}$ is a monoidal category where tensor product $\otimes$ is monoidal sum and $\mathbb{Z}$ is the unit object. There are other examples as well \cite{Mac71}.
In a monoidal category, if the special isomorphisms $\alpha, \lambda$, and $\rho$ are identities, then $\mathsf{C}$ is called a \textbf{strict monoidal category}.
Let $\mathsf{C}$ and $\mathsf{C'}$ be two monoidal categories. A \emph{(strict) morphism} of monoidal categories $T:(C,+,0,\alpha,\lambda,\rho)\rightarrow (C',+',0',\alpha',\lambda',\rho')$ is a functor $T:C\rightarrow C'$, such that for all objects $x, y, z$ and morphisms $f$ and $g$ there are equalities $T(x+y)=Tx+'Ty, T(f+g)=Tf+'Tg, T0=0', T\alpha_{x,y,z}=\alpha'_{Tx,Ty,Tz}, T\lambda_x=\lambda'_{Tx}, T\varrho_x=\rho'_{Tx}.$
\begin{Def}\label{invobj}\cite{baez-lauda-2-groups}
If $x$ is an object in a monoidal category, an inverse for $x$ is an object $y$ such that $x+y\approx 0$ and $y+x\approx 0.$ If $x$ has an inverse, it is called invertible.
\end{Def}
As it is noted in \cite{baez-lauda-2-groups,JoyalStreet} and it is easy to show, that if any object has a one side inverse in a monoidal category, then any object is invertible.
\begin{Def}\label{catgrp}
A \textbf{categorical group} $\mathsf{C}=(C_0, C_1, d_0, d_1, i, m)$ is a monoidal groupoid, where all objects are invertible and moreover, for every object $x\in C_0$ there is an object $-x\in C_0$ with a family of natural isomorphisms
\[\epsilon_x\colon -x+x\approx 0,\]
\[\delta_x\colon x+(-x)\approx 0\] such that the following diagrams are commutative:
\[\xymatrix@C=2.3pc{
0 + x \ar[r]^<<<<<<{\delta^{-1}_x + 1} \ar[d]_{\lambda _x}
& (x + (-x)) + x \ar[r]^<<<<<{a_{x, -x ,x}}
& x+( -x + x) \ar[d]^{1_x + \epsilon_x} \\
x \ar[rr]_{\rho ^{-1}_x}
& & x + 0 }
\]
\[
\[email protected]{
-x + 0
\ar[r]^<<<<<<{1 + \delta^{-1}_x}
\ar[d]_{\rho _{-x}}
& -x + (x + (-x))
\ar[r]^{a^{-1}_{-x, x, -x}}
& (-x + x)+ (-x)
\ar[d]^{\epsilon_x + 1_{-x}} \\
-x
\ar[rr]_{\lambda^{-1}_{-x}}
&& 0 + (-x) }
\]
\end{Def}
It is important, and a well-known fact, that from the definition of categorical group it follows that for any morphism $f:x\rightarrow x' \in C_1$ there is a morphism $-f\colon -x\rightarrow -x'$ with natural isomorphisms $-f+f\approx 0$ and $f+ (-f)\approx 0,$ where $0$ morphism is $1_0$ (see e.g. \cite{Sinh 1}).
Like the case of monoidal category the natural transformations $\alpha,\lambda$, $\rho, \epsilon, \delta,$ identity transformation $1_ \mathsf{C}\rightarrow 1_
\mathsf{C},$ their compositions and sums will be called \emph{special isomorphisms}. Categorical group defined above is coherent \cite{Laplaza,baez-lauda-2-groups}, which means that all diagrams commute involving special isomorphisms. For a monoidal category one can see in \cite{Mac71}, Coherence Chapter VII Section 2.
A categorical group is called strict if the special isomorphisms $\alpha$, $\lambda$, $\rho$, $\epsilon,$ and $\delta$ are identities. Strict categorical groups are known as group-groupoids (see below the definition), internal categories in the category of groups or 2-groups in the literature.
The definition of categorical group we gave is Definition 7 by Baez and Lauda in \cite{baez-lauda-2-groups}, where the operation is multiplication and which is called there coherent 2-group. Sinh \cite{Sinh 1} calls them ``gr-categories" and this name is also used by other authors as well, e.g. Breen \cite{Breen}. It is called ``categories with group structure" by Ulbrich \cite{Ulbrich} and Laplaza \cite{Laplaza} in which all morphisms are invertible. The term categorical group for strict categorical groups is used by Joyal and Street \cite{JoyalStreet}, and it is used by Vitale \cite{Vitale1,Vitale2} and others for non strict ones.
From the functorial properties of addition $+$ it follows that in a categorical group we have $-1_x=1_{-x},$ for any $x\in C_0.$ Since an isomorphism between morphisms $\theta :f\thickapprox g$ means that there exist isomorphisms $\theta_i:d_i(f)\rightarrow d_i(g), i=0,1$ with $\theta_1 f=g\theta_0$, from natural property of special isomorphisms there exist special isomorphisms between the morphisms in $C_1$. But if $\theta_i, i=0,1$ are special isomorphisms, it doesn't imply that $\theta$ is a special isomorphism; in this case we will call $\theta$ \emph{weak special isomorphism}. It is obvious that a special isomorphism between the morphisms in $C_1$ implies the weak special isomorphism. Note that if $f\approx f'$ is a weak special isomorphism, then from the coherence property it follows that $f'$ is a unique morphism weak special isomorphic to $f$ with the same domain and codomain objects as $f'$.
\begin{Exam} Let $X$ be a topological space and $x\in X$ be a point in $X$. Consider the category $\Pi_2(X,x),$ whose objects are paths $x\rightarrow x$, and whose morphisms are homotopy classes of paths between paths, where $f,g:x\rightarrow x$. This category is a categorical group, for the proof see
\cite{baez-lauda-2-groups} and the paper of Hardie, Kamps and Kieboom \cite{HKK}.
\end{Exam}
One can see more examples in \cite{baez-lauda-2-groups}, and also we will give them in the following paper, where we will construct a categorical group for any cssc-crossed module defined below in Section 5.
We define (strict) morphisms between categorical groups, which satisfy conditions of (strict) morphism of monoidal categories. Note that from this definition follow: $T(-x)=-{T(x)}$ and $T(-f)=-{T(f)}$, for any object $x$ and arrow $f$ in a categorical group. Categorical groups form a category with (strict) morphisms between them.
For any categorical group $\mathsf{C}=(C_0, C_1, d_0, d_1, i, m)$ denote $\Ker d_0=\{f\in \mathsf{C}|d_0(f)\approx 0\}$ and $\Ker d_1=\{f\in \mathsf{C}|d_1(f)\approx 0\}$
\begin{Lem}\label{comm} Let $\mathsf{C}=(C_0, C_1, d_0, d_1, i, m)$ be a categorical group. For any $f\in \Ker d_1$ and $g\in \Ker d_0$ we have a weak special isomorphism $f+g\approx g+f.$
\end{Lem}
\begin{proof}Suppose $d_0(g)=0'$ and $d_1(f)=0'',$ where $0'\approx 0 \approx 0''.$ By interchange law we have $(1_{0''}+g)(f+1_{0'})=f+g$ and $(g+1_{0''})(1_{0'}+f)=g+f$. Let $\gamma \colon 0''\approx 0'$ be a special isomorphism. Applying the coherence property of a categorical group, we easily obtain that the left sides of the noted both equalities are isomorphic to $g\gamma f$, and both are weak special isomorphisms. From this it follows that there is a weak special isomorphism $f+g\approx g+f.$ \qed
\end{proof}
The analogous statement is well known for group-groupoids, where instead of the isomorphisms we have equalities in the definitions of $\Ker d_0$ and $\Ker d_1$ and in the final result \cite{BS1}.
Below we recall the definition of crossed module introduced by Whitehead in \cite{Wth2} defining homotopy system. A {\em crossed module} $(A,B,\mu)$ consists of a group homomorphism
$\mu\colon A\rightarrow B$ together with an action $(b,a)\mapsto
{b\cdot a}$ of $B$ on $A$ such that for $a,a_1\in A$ and $b\in B$
\begin{enumerate}[leftmargin=1.5cm]
\item [CM1.] $\mu(b\cdot a)=b+\mu(a)-b$, and
\item [CM2.] $\mu(a)\cdot a_1=a+a_1-a$.
\end{enumerate}
For an extensive treatment of crossed modules, see \cite[Part I]{Br-Hi-Si}.
Here are some examples of crossed modules.
\begin{itemize}
\item[$\bullet$] The inclusion of a normal subgroup $N\rightarrow G$
is a crossed module with the action by conjugation of $G$ on $N$. In particular any group $G$ can be regarded as a crossed module $1_G\colon G\rightarrow G$.
\item[$\bullet$] For any group $G$, modules over the group ring of $G$ are crossed
modules with $\mu= 0.$
\item[$\bullet$] For any group $G$ the object $\mu\colon G\rightarrow \Aut G$ is a crossed module, where $\mu(g)\cdot g'=\mu(g)(g')$ for any $g, g' \in G.$
\end{itemize}
A {\em morphism} $(f,g)\colon (A,B,\mu)\rightarrow (A^{\prime },B^{\prime },\mu^{\prime })$ of crossed module
is a pair $f\colon A\rightarrow A^{\prime }$,
$g\colon B\rightarrow B^{\prime }$ of morphisms of groups such that
$g\mu=\mu^{\prime }f$ and $f$ is an operator morphism over $g$,
i.e., $f(b\cdot a)=$ $g(b)\cdot f(a)$ for $a\in A$, $b\in B$. So crossed modules and morphisms of them, with the obvious composition of morphisms $(f',g')(f,g)=(f'f,g'g)$ form a category.
\begin{Def}\label{Defgroup-groupoid}
A {\em group-groupoid} $G$ is a \emph{group object} in the category of groupoids, which means that it is a groupoid $G$ equipped with functors
\begin{enumerate}[label=(\roman{*}), leftmargin=1cm]
\item $+\colon G\times G\rightarrow G$, $(a,b)\mapsto a+b$;
\item $u\colon G\rightarrow G$, $a\mapsto -a$;
\item $0\colon \{\star\}\rightarrow G$, where $\{\star\}$ is a singleton, \qed
\end{enumerate}
which are called respectively sum, inverse and zero, satisfying the usual axioms for a group.
\end{Def}
The definition we gave was introduced by Brown and Spencer in \cite{BS1} under the name {\em $\mathcal{G}$-groupoid}, where the group operation is multiplication. The term group-groupoid was used later in \cite{BM1}. It is interesting that the group object in the category of small categories called {\em $\mathcal{G}$-category} is a group-groupoid. As it is noted by the authors this fact was known to Duskin.
\begin{Exam}
If $X$ is a topological group, then the fundamental groupoid $\pi_1X$ of the space $X$ is a group-groupoid \cite{BS1}.
\end{Exam}
\begin{Exam}
For a group $X$, the direct product $G=X\times X$ is a group-groupoid. Here the domain and codomain homomorphisms are the projections; the object inclusion homomorphism is defined by the diagonal homomorphism $i(x)=(x,x),$ for any $x\in X$ and the composition of arrows is defined by $(x,y)\circ(z,t)=(z,y)$ whenever
$x=t$, for any $x, y, z, t\in X.$
\end{Exam}
\begin{Theo} \label{Theocatequivalence}\cite{BS1}
The categories of crossed modules and of group-groupoids are equivalent.
\end{Theo}
According to the authors this result is due to Verdier, which was used by Duskin and which was discovered independently by them. It was proved by Porter that the analogous statement is true in more general setting of a category of groups with operations \cite{Por}.
\section{Groups up to congruence relation}
Let $X$ be a non-empty set with an equivalence relation $R$ on $X$. Denote such a pair by $X_R$. Define a category whose objects are the pairs $X_R$ and morphisms are functions $f\colon X_R\rightarrow Y_S,$ such that $f(x)\sim_S f(y),$ whenever $x\sim_R y$. Denote this category by $\tilde{\Sets}$.
Note that for $X_R,Y_S\in Ob(\tilde{\Sets})$, the product $X_R\times Y_S$ is a product object in $\Sets$ with the equivalence relation $R\times S$
defined by \[(x,y)\sim_{R\times S}(x_1,y_1) \Leftrightarrow x\sim_R x_1 \q \t{and} \q y\sim_Sy_1 \]
We now define \emph{group up to congruence relation} or briefly {\em c-group} concept as follows.
\begin{Def}
Let $G_R$ be an object in $\tilde{\Sets}$ and
\[ \begin{array}{cccl}
m \colon &G\times G& \longrightarrow & G\\
&(a,b) & \longmapsto & a+b
\end{array}\]
a morphism in $\tilde{\Sets}$, i.e, $m\in \tilde{\Sets}((G\times G)_{R\times R},G_R)$. $G_R$ is called a {\em c-group} if the following axioms are satisfied.
\begin{enumerate}[label=(\roman{*}), leftmargin=1cm]
\item $a+(b+c)\sim_R(a+b)+c$ for all $a,b,c\in G$;
\item there exists an element $0\in G$ such that $a+0\sim_R a\sim_R 0+a,$ for all $a\in G;$
\item for each $a\in G$ there exists an element $-a$ such that $a-a\sim_R 0$ and $-a+a\sim_R0$.
\end{enumerate}
\end{Def}
In a c-group $G$, $0\in G$ is called {\em zero element} and for any $a\in G$ the element $-a\in G$ is called {\em inverse} of $a$. The congruence relations involved in group like identities and their compositions and sums will be called \emph{special congruence relations}.
\begin{Rem}Let $G_R$ be a c-group. Then we have the following:
\begin{enumerate}[label=(\roman{*}), leftmargin=1cm]
\item if $a\sim_R b$ and $c\sim_R d$ for $a, b, c, d\in G$ then $a+c\sim_R b+d;$
\item if $0$ and $0'$ are different zero elements in $G_R$, then $0\sim_R 0'$;
\item if $-a$ and $a'$ are different inverses of $a\in G$, then $a'\sim_R -a$;
\item if $a\sim_Rb$ then $-a\sim_R -b$.
\end{enumerate}
\end{Rem}
\begin{Exam}
Every group $G$ is a c-group where the equivalence relation is equality.
\end{Exam}
So the concept of c-group generalizes group notion.
\begin{Exam}
This example comes from the Mac Lane's paper \cite{Mac Lane}, where the author regards the quotient group as a group with congruence relation. Let $G$ be a group and $H$ a normal subgroup in $G.$ The quotient group $G/H$ can be regarded as a group with the same elements as the group $G$ and with the congruence relation - $g\sim g'$ if and only if $g-g'\in H.$ The operations are defined in the same way as in $G$ and they preserve the congruence relation. Such a group is a c-group, where group identities are satisfied up to equality.
\end{Exam}
\begin{Exam}
Let $X$ be a topological space and $x\in X$. The set $P(X,x)$ of all closed paths at $x$ is a c-group with the composition of paths. Here the congruence relation is the homotopy of the paths.
\end{Exam}
\begin{Exam}
Let $\mathbb{Z}^{\ast}=\mathbb{Z}\backslash\{0\}$. Define an equivalence relation on $\mathbb{Z}^{\ast}$ by $x\sim_Ry \Leftrightarrow xy>0$. Then $\mathbb{Z}^{\ast}$ becomes a c-group with respect to the multiplication. The unit is the number $1$ and the inverse for any number is itself this number.
\end{Exam}
\begin{Exam}
In a categorical group $\mathsf{C}$ the set $C_1$ of morphisms and the set $C_0$ of objects are both c-groups. The congruence relations are isomorphisms between arrows and between objects respectively.
\end{Exam}
\begin{Exam}\label{gr} Any group can be endowed with a c-group structure. To show this recall that every group $G$ can be regarded as a part of a certain crossed module in the category of groups, for example $G \rightarrow \Aut G.$ According to Theorem \ref{Theocatequivalence} there exists a group-groupoid $C=(C_0, C_1, d_o, d_1, i, m),$ for which $d_1|_{\Ker d_0}\colon \Ker d_0 \rightarrow C_0,$ is a crossed module and is isomorphic to $G \rightarrow \Aut G.$ $\Ker d_0$ is a c-group, the congruence relation on it is induced from the congruence relation in $C_1,$ which is the relation being isomorphic between the morphisms. From this follows that $G$ has also a c-group structure, group identities are satisfied up to equality, and naturally all special isomorphisms are equalities.
\end{Exam}
\begin{Def}
Let $G_R$ be a c-group. If $a+b\sim_R b+a$ for all $a,b\in G$, then $G_R$ is called {\em c-abelian (or c-commutative)} c-group.
\end{Def}
\begin{Def}
Let $G_R$ and $H_S$ be c-groups. A morphism $f\in \tilde{\Sets}(G_R,H_S)$ such that $f(a+b)=f(a)+f(b)$ for any $a,b\in G$ is called a {\em c-group morphism} from $G_R$ to $H_S$.
\end{Def}
From the definition it follows that a morphism between c-groups preserves the congruence relation; moreover we obtain that $f(0)\sim 0$ and $f(-a)=-f(a)$, for any $a\in G,$ where the second equality means that $f(-a)$ is one of the inverse element of $f(a).$ As a result we obtain that a morphism between c-groups carries special congruence relations to special congruence relations
\begin{Rem}
If $f\colon G_R\rightarrow H_S$ and $g\colon H_S\rightarrow N_T$ are two c-group morphisms, then $gf\colon G_R\rightarrow N_T$ is also a c-group morphism. Further for each c-group $G_R$ there is a unit morphism $1_G\colon G_R\rightarrow G_R$ such that $1_G$ is the identity function on $G_R$. Therefore we have a category of c-groups with c-group morphisms; denote this category by $\cGr$.
\end{Rem}
Let $G_R$ and $H_S$ be c-groups, and $f\colon G_R\rightarrow H_S$ a morphism of c-groups.
\begin{Def}
The subset $\cKer f=\{a\in G_R~|~ f(a)\sim_S 0_H \}$ is said to be {\em c-kernel} of the c-group morphism $f$.
\end{Def}
Note that $\cKer d_0$ is a c-group with the congruence relation induced from the isomorphisms in $C_1$.
\begin{Def}
The subset $\cIm f=\{b\in H_S~|~ \exists a\in G_R, f(a)\sim_S b\}$ is said to be the {\em c-image} of the morphism $f$.
\end{Def}
\begin{Lem}\label{grp}
Let $G $ be a c-group with congruence relation $R$. Then the quotient set $G/R$ becomes a group with the operation defined by the induced map
\[ \begin{array}{cccl}
m^{\ast} \colon & G/R\times G/R & \longrightarrow & G/R \\
& ([a],[b]) & \longmapsto & [a]+[b]=[a+b]
\end{array}\]
\end{Lem}
\begin{Def}
Let $G_R$ be a c-group and $H$ be a subset of the underlying set of $ G$. $H$ is called a {c-subgroup} in $G_R$ if $H_S$ is a c-group with the addition and congruence relation $S$ induced from $G_R$.
\end{Def}
Let $G_R$ be a c-group and $H$ be a subset of $G$. If for an element $a\in G$ there exists an element $b\in H$ such that $a\sim_R b$ then we write $a\inc H$.
If $H$ and $H'$ are two subsets of $G_R$, then we write $H\tilde{\subset}H'$ if for any $h\in H$ we have $h \inc H'$. If $H\tilde{\subset}H'$ and $H'\tilde{\subset}H,$ then we write $H\sim H'$.
\begin{Def}
Let $G_R$ be a c-group and $H_S \subseteq G_R $ a c-subgroup in $G_R$. Then $H_S$ is called \emph{normal c-subgroup} if $g+h-g\inc H_S$ for any $h\in H_S$ and $g\in G.$
\end{Def}
The condition given in the definition is equivalent to the condition $g+H_S-g\tilde{\subset} H_S,$ and it is equivalent itself to the condition $g+H_S\sim H_S+g$ for any $g\in G.$
\begin{Def}
Let $G_R$ be a c-group and $H_S \subseteq G_R $ be a c-subgroup in $G_R$. Then $H_S$ is called \emph {perfect c-subgroup} if from $g\inc H$ it follows that $g\in H,$ for any $g\in G.$
\end{Def}
\begin{Def}
A c-group $G_R$ is called \emph {connected} if $g\sim g'$ for any $ g, g'\in G.$
\end{Def}
\begin{Lem}\label{connected}
Let $G_R$ and $H_S$ be c-groups and let $f\colon G_R\rightarrow H_S$ be a morphism of c-groups. Then
\begin{enumerate}[label=(\roman{*}), leftmargin=1cm]
\item $\cKer f$ is perfect and normal c-subgroup in $G_R$;
\item $\cIm f$ is perfect c-subgroup in $H_S$.
\end{enumerate}
\end{Lem}
\begin{proof}
Follow from the definitions.\qed
\end{proof}
Now we shall construct the quotient object $G/H$, where $H$ is a normal c-subgroup of a c-group $G$. Consider the classes $\{g+H|g\in G\}.$ If $g+H \cap g'+H\neq \emptyset,$ then we obtain $-g+g'\inc H,$ which implies that $g+H\sim g'+H.$ Now consider classes of these classes $\{cl(g+H)|g\in G\},$ where $cl(g+H)=\cup\{x\in G|x\inc g+H\}.$ We define $G/H=\{cl(g+H)|g\in G\}.$ An addition operation in this set is defined by $cl(g+H)+cl(g'+H)=cl((g+g')+H),$ for any $g,g'\in G.$ It is easy to see that this operation is defined correctly, it is associative and we have the unit element $cl(0+H)$. Actually constructed object is a group, the congruence relation in $G/H$ is the equality $``=".$ We have a usual surjective morphism $p: G \rightarrow G/H.$
\begin{Lem} (i) If $G$ is a c-group and $H$ is a normal c-subgroup in $G,$ then for any group $G'$ and c-group morphism $f:G\rightarrow G'$, if $f(h)=0$ for any $h\in H,$ there exists a unique morphism $\theta :G/H \rightarrow G',$ in $\cGr$ such that $\theta p=f$.
(ii) If $H$ is a perfect normal c-subgroup in $G,$ then $H=\cKer p.$
\end{Lem}
\begin{proof}
Easy checking.\qed
\end{proof}
\section{Actions and crossed modules in $\cGr$}
An extension in the category $\cGr$ is defined in a similar way as in the category of groups.
\begin{Def}
Let $A$, $B\in\cGr$. An \emph{extension} of $B$ by $A$ is a sequence
\begin{equation} \label{extension}
\xymatrix{0\ar[r]&A\ar[r]^-{i}&E\ar[r]^-{p}&B\ar[r]&0}
\end{equation} in which $p$ is surjective and $i$ is the c-kernel of $p$ in $\cGr$. We say that an extension is \emph{split} if there is a morphism $s\colon B\rightarrow E,$ such that $ps=1_B$.
\end{Def}
We shall identify $a\in A$ with its image $i(a)$. We shall use the notation $b\cdot a=s(b)+(a-s(b))$. Then a split extension induces an action (from the left side) of $B$ on $A$. We have the following conditions for this action:
\begin{enumerate}[label=(\roman{*}), leftmargin=1cm]
\item $b\cdot(a+a_1)\sim (b\cdot a)+(b\cdot a_1)$,
\item $(b+b_1)\cdot a\sim b\cdot(b_1\cdot a)$,
\item $0\cdot a\sim a$,
\item If $a\sim a_1$ and $b\sim b_1$ then $b\cdot a \sim b_1\cdot a_1$,\end{enumerate}
for $a,a_1\in A$ and $b,b_1\in B$.
Here and in what follows we omit congruence relations symbols for $A$ and $B$.
Let $A, B \in \cGr$ and suppose $B$ acts on $A$. Consider the product $B \times A$ in $\cGr$. We have the operation in $B \times A$, defined in analogous way as in the case of groups:
$(b',a')+(b,a)=(b'+b, a'+b'\cdot a)$
for any $b, b'\in B, a, a'\in A$.
This operation is associative up to the relation defined by $(b,a)\sim (b',a')$ if and only if $b\sim b'$ and $a\sim a'$, which is a congruence relation.
Obviously we have a zero element $(0,0)$ in $B \times A$ and the opposite element for any pair $(b,a)\in B \times A$ is $(-b, -b\cdot (- a)).$ Therefore we have a semidirect product $B \ltimes A$ in $\cGr$.
\begin{Def}\label{c-iso}
Let $f:D\rightarrow D'$ be a morphism in $\cGr$. $f$ is called an isomorphism up to congruence relation or c-isomorphism if there is a morphism $f':D'\rightarrow D$, such that $ff'\sim 1_{D'}$ and $f'f\sim 1_D$.
\end{Def}
We will denote such isomorphism by $\tilde{\approx}.$
We have a natural projection $p': B \ltimes A \rightarrow B$. The c-kernel of $p'$ is not isomorphic to $A$ as it is in the case of groups, but we have an isomorphism up to congruence relation $\cKer p'\tilde{\approx}A$.
Let $0\rightarrow A\rightarrow E \rightarrow B\rightarrow 0$ be a split extension of $B$ by $A$ in $\cGr$. Then we have an action of $B$ on $A$ and the corresponding semidirect product $B \ltimes A$. In this case we obtain a c-isomorphism $E\tilde{\approx}B\ltimes A$ given by the correspondences analogous to the group case.
\begin{Def} Let $G$ and $H$ be two c-groups, $\partial\colon G\rightarrow H$ morphism of c-groups and $H$ acts on $G$. We call $(G,H,\partial)$ a \emph{c-crossed module} if the following conditions are satisfied:
\begin{enumerate}[label=(\roman{*}), leftmargin=1cm]
\item $\partial(b\cdot a)= b+(\partial(a)-b)$,
\item $\partial(a)\cdot a_1\sim a+(a_1-a).$
\end{enumerate}
for $a,a_1\in G$ and $b\in H$.
\end{Def}
Let $(G,H,\partial)$ and $(G',H',\partial')$ be two c-crossed modules. A \emph{c-crossed module morphism} is a pair of morphisms $<f,g>\colon(G,H,\partial)\rightarrow(G',H',\partial')$ such that the diagram
\[\xymatrix{
G \ar[r]^-{\partial} \ar[d]_-{f} & H \ar[d]^-{g} \\
G'\ar[r]^-{\partial'} & H' }\]
is commutative, and for all $b\in H$ and $a\in G$, we have $f(b\cdot a)= g(b)\cdot f(a)$, where $f$ and $g$ are morphisms of c-groups.
c-crossed modules and morphisms of c-crossed modules form a category.
\begin{Exam} Any crossed module in the category of groups can be endowed with the structure of c-crossed module, the proof is analogous to the one in Example \ref{gr}.
\end{Exam}
For other examples see Section 5.
Let $G\in \cGr$ and $H$ be a normal c-subgroup in $G$. It is easy to see that in general we do not have a usual action by conjugation of $G$ on $H.$
\begin{Lem}
If $H$ is a perfect normal c-subgroup of a c-group $G$, then we have an action of $G$ on $H$ in the category $\cGr$ and the inclusion morphism defines a c-crossed module.
\end{Lem}
\begin{proof}
Easy checking.\qed
\end{proof}
For a categorical group $\mathsf{C}=(C_0, C_1, d_0, d_1, i, m)$, we have a split extension
\begin{equation} \label{extension}
\xymatrix{0\ar[r]&\cKer d_0\ar[r]^-{j}&C_1\ar[r]^-{d_0}&C_0\ar[r]&0}
\end{equation}
where $\cKer d_0$ is a c-group, a congruence relation is defined naturally as an isomorphism between the arrows in $\cKer d_0$. Now we define an action of $C_0$ on $\cKer d_0$ by
\[ \begin{array}{ccl}
C_0\times \cKer d_0& \rightarrow & \cKer d_0\\
(r,c) & \mapsto & r\cdot c=i(r)+(j(c)-i(r)).
\end{array}\]
\begin {Prop}
The action of $C_0$ on $\cKer d_0$ satisfies action conditions in $\cGr$.
\end {Prop}
\begin{proof}First we shall show the congruence relation $r\cdot(c+c')\sim r\cdot c+r\cdot c'.$
\begin{alignat*}{2}
r\cdot(c+c') &= i(r)+((c+c')-i(r)) \\
& \sim (i(r)+((c-i(r))+i(r))+(c'-i(r))) \\
& \sim (i(r)+(c-i(r)))+(i(r)+(c'-i(r))) \\
& = r\cdot c+r\cdot c'.
\end{alignat*}
Next we shall show that $(r+r')\cdot c\sim r\cdot (r'\cdot c)$. We shall omit some brackets since we deal with congruence relation.
We have
\begin{alignat*}{2}
(r+r')\cdot c &= i(r+r')+(c-i(r+r')) \\
& \sim i(r)+(i(r')+c-i(r'))-i(r) \\
& =r\cdot (r'\cdot c).
\end{alignat*}
It is trivial that $0\cdot c\sim c$ and $r\cdot 0\sim 0$. Now we shall show that if $r\sim r'$ and $c\sim c'$ then $r\cdot c\sim r'\cdot c'$. We have
\begin{alignat*}{2}
r\cdot c &= i(r)+(c-i(r))\\
&\sim i(r')+(c'-i(r'))\\
&=r'\cdot c'.
\end{alignat*} \qed
\end{proof}
\section{cssc-crossed modules and the main theorem}
\begin{Def}\label{connected cr}
A c-crossed module $(G,H,\partial)$ will be called connected if $G$ is a connected c-group.
\end{Def}
Denote $d=d_1{|\cKer d_0}.$
\begin{Prop}
For a categorical group $\mathsf{C}=(C_0, C_1, d_0, d_1, i, m)$, $(\cKer d_0,C_0,d)$ is a connected c-crossed module.
\end{Prop}
\begin{proof} $\cKer d_0$ is a connected c-group, which follows from the fact that any two arrows in $\cKer d_0$ have the domains isomorphic to $0$ and that $\mathsf{C}$ is a groupoid. Note that the congruence relation in $C_0$ is generated by the isomorphisms between the objects in $C_0.$ Therefore $d$ preserves the congruence relation in $\cKer d_0$ since $f\approx f'$ implies that $d_1 f\approx d_1 f'.$ For the first condition of crossed module we have $d(r\cdot c)=d(i(r)+(c-i(r)))=r+(dc-r),$ for any $c\in\cKer d_0$ and $r\in C_0.$
For the second condition of crossed module we have to prove that $(dc)\cdot c'\sim c+(c'-c)$ for $c, c'\in \cKer d_0,$ which follows from the fact that $\cKer d_0$ is a connected c-group. \qed
\end{proof}
Now we shall introduce another sort of object denoted as $c\textbf{K}er d_0$ for any categorical group $\mathsf{C}=(C_0, C_1, d_0, d_1, i, m)$. By definition $c\textbf{K}er d_0 =\{f\in C_1| d_0(f)=0\}$. An addition operation is defined by $f+f'=(f+f')\gamma,$ where $f+f':0+0\rightarrow d_1(f)+d_1(f')$ is a sum in $C_1$, i.e. the same as the sum in $\cKer d_0$ and $\gamma : 0 \rightarrow 0+0$ is a unique special isomorphism in $C_1$). $\sim$ -relation in $c\textbf{K}er d_0$ is induced from the relation in $C_1$, which is a relation of being isomorphic in $C_1$ and it is a congruence relation in $c\textbf{K}er d_0$. It is obvious that $d\mid_{c\textbf{K}er d_0}$ preserves the congruence relation. The operation in $c\textbf{K}er d_0$ is associative up to congruence relation. Zero element in $c\textbf{K}er d_0$ is a zero arrow $0$; we have $f+0\sim f,$ $0+f\sim f,$ for any $f\in c\textbf{K}er d_0.$ The opposite morphism of $f$ in $C_1$ is $-f: -0\rightarrow -d_1f.$ There is a unique special isomorphism $\kappa: 0\approx -0$. Define the opposite morphism $-f$ in $c\textbf{K}er d_0$ as $-f\kappa.$ One can easily see that $f+(-f)\approx 0$ in $c\textbf{K}er d_0$ and the $\sim$ -relation is a congruence relation. Therefore $c\textbf{K}er d_0$ is a c-group. $C_0$ is also a c-group, where a congruence relation is generated by isomorphisms between the objects. Now we will define an action of $C_0$ on $c\textbf{K}er d_0.$ By definition $r\cdot c=(i(r)+(c-i(r)))\gamma$ for any $r\in C_0, c\in c\textbf{K}er d_0,$ where $\gamma$ is a special isomorphism $0\approx r+(0-r),$ which is unique as we know already. Here we check action identities. We have
\begin{alignat*}{2}
r\cdot (c+c')&=(i(r)+((c+c')-i(r))\gamma \\
& \sim(i(r)+(c-i(r)))\gamma_1+ i(r)+(c'-i(r)))\gamma_2 \\
& =r\cdot c+ r\cdot c'
\end{alignat*}
for any $r\in C_0, c,c' \in c\textbf{K}er d_0$. Other three conditions of action for c-groups are checked in analogous way.
\begin{Def}\label{strict cr}
A c-crossed module $(G,H,\partial)$ will be called strict if it satisfies c-crossed module conditions, where instead of $\sim$-relation in the second condition is equality, i.e.
\begin{enumerate}[label=(\roman{*}), leftmargin=1cm]
\item $\partial(b\cdot a)= b+(\partial(a)-b)$,
\item $\partial(a)\cdot a_1= a+(a_1-a)$,
\end{enumerate}
for $a,a_1\in G$ and $b\in H$.
\end{Def}
\begin{Def}\label{weak special} In a c-crossed module $(G,H,\partial)$ a congruence relation $g\sim g'$ in $G$ will be called weak special relation if $\partial (g)\sim\partial(g')$ is a special congruence relation in $H$.
\end{Def}
Since in a c-crossed module $(G,H,\partial)$ the morphism $\partial$ carries any special congruence relation to the special congruence relation, in a crossed module every special congruence relation in $G$ is a weak special congruence relation.
Let $(G,H,\partial)$ be a c-crossed module.
\begin{Cond}\label{special cr}
For any congruence relation $\gamma :\partial c\sim r$, there exists $c'\sim c$, such that $\partial c' =r$, where $c, c'\in G$ and $r\in H$. If $\gamma $ is a special congruence relation, then $c'$ is a unique element in $G$ which is weak equivalent to $c$.
\end{Cond}
\begin{Def}\label{special}
A c-crossed module will be called \emph{special} if it satisfies Condition \ref {special cr}.
\end{Def}
If a c-crossed module is connected, strict and special we will write shortly that it is a cssc-crossed module. This kind of crossed modules are exactly those we were looking for the description of categorical groups up to equivalence of the corresponding categories, which will be proved in the next paper.
\begin{Theo}
For a categorical group $\mathsf{C}=(C_0, C_1, d_0, d_1, i, m)$ the triple $(c\textbf{K}er d_0, C_0, d)$ is a cssc-crossed module.
\end{Theo}
\begin{proof}
First we shall show that we have equality in the first condition of the crossed module $(c\textbf{K}er d_0, C_0, d)$. We have $d(r\cdot c)=d((i(r)+(c-i(r))\varepsilon )=d(i(r)+(c-i(r)))=r+(dc-r),$ where $\varepsilon: 0\rightarrow r+(0-r)$ is a special isomorphism. Now we shall show that we have equality in the second condition of a crossed module. First we compute the left side of the condition. We have $dc\cdot c'=(i(dc)+(c'-i(dc)))\gamma$ where $\gamma: 0\rightarrow dc+(0-dc)$ is a special isomorphism and $i(dc)+(c'-i(dc))$ is a morphism $dc+(0-dc)\rightarrow dc+(dc'-dc)$. We have $-c+idc\in \cKer d_1$ and $c'\in \cKer d_0;$ by Lemma \ref{comm} we obtain that there is a weak special isomorphism $(-c+idc)+c'\approx c'+(-c+idc);$ from this it follows that $idc+c'\approx c+c'-c+i(dc),$ which implies $id(c)+(c'-i(dc))\approx c+(c'-c);$ from which we obtain the weak special isomorphism $id(c)\cdot c'\approx c+(c'-c).$ By the definition of a sum in $c\textbf{K}er d_0$ for the right side we have $c+(c'-c)=(c+(c'-c)\varphi)\psi,$ where $\varphi:0\rightarrow 0-0$ and $\psi:0\rightarrow 0+0$ are special isomorphisms. Here we have in mind that $d(-c)=-dc$ and $i(-dc)=-i(dc).$ Obviously we have a weak special isomorphism $(c+(c'-c)\varphi\approx i(dc)+(c'-i(dc))$, from which it follows that there is a special isomorphism between the domains of these morphisms $\theta: 0+0\rightarrow dc+(0-dc),$ such that $(i(dc) +(c'-i(dc)))\theta=c+(c'-c)\varphi.$ Here we applied that the codomains of these morphisms are equal. Since $\psi, \theta $ and $\gamma$ are special isomorphisms, we obtain that $\theta \psi=\gamma,$ from this it follows that $(c+(c'-c)\varphi)\psi=(i(dc)+(c'-i(dc)))\gamma,$ which means that for the c-crossed module $(c\textbf{K}er d_0, C_0, d)$ we have an equality in the second condition for c-crossed modules.
The crossed module is connected by the definition of the object $c\textbf{K}er d_0$. Now we shall prove that this crossed module is a special c-crossed module. Let $c\in c\textbf{K}er d_0,$ and there is a congruence $\gamma: dc\sim r$, which means that $\gamma$ is an isomorphism in $C_1$. Take $c'=\gamma c$, then we will have $c'\approx c$ in $C_1,$ which means that $c'\sim c$ in $c\textbf{K}er d_0.$ Suppose $\gamma$ is a special congruence relation, then it is a special isomorphism in $C_1.$ From the coherence property of $\mathsf{C}$ we have that $\gamma$ is a unique special isomorphism from $dc$ to $r$ and therefore there is a unique morphism $d_0c\rightarrow r$, which is weak special isomorphic to $c$ and it is a composition $\gamma c$. Therefore $c'$ is unique with this property and $(c\textbf{K}er d_0, C_0, d)$ is a special c-crossed module. \qed
\end{proof}
\section*{Acknowledgements}
The first author is grateful to Ercyies University (Kayseri, Turkey) and Prof. Mucuk for invitations and to the Rustaveli National Science Foundation for financial support, grant GNSF/ST09 730 3 -105.
|
1,116,691,497,611 | arxiv | \section*{Acknowledgments}
E.~L.~Berger is supported
by the United States Department of Energy, Division of High Energy
Physics, under Contract W-31-109-ENG-38. J.~W.~Qiu is supported in part by
the United States Department of Energy under Grant No. DE-FG02-87ER40371.
Y.~Wang is supported in part by the United States Department of Energy
under Grant No. DE-FG02-01ER41155.
|
1,116,691,497,612 | arxiv | \section{Introduction}
Pluto, Titan and Triton all have a nitrogen-based atmosphere containing a significant fraction of methane, an efficient recipe known to lead to the formation of organic haze in the atmosphere, as confirmed by observations \citep{Toma:05,RagePoll:92,HerbSand:91,Ster:15} and laboratory experiments \citep{Trai:06,Rann:10,Lavv:08}. Here, we use the Global Climate Model of Pluto (herein referred to as GCM), developed at the Laboratoire de M\'et\'eorologie Dynamique (LMD) and designed to simulate the atmospheric circulation and the methane cycle on Pluto and to investigate several aspects of the presence of haze at a global scale on Pluto \citep{Forg:16,Bert:16}.
What controls haze formation on Pluto? At which altitudes and latitudes does it form and where does sedimentation occur? What amount of particles forms the haze, and what is its opacity? To address those key questions we have developed a simple parametrization of haze in the GCM. The parametrization is based on a function of aerosols production, which directly depends on the amount of the Lyman-$\alpha$ UV flux. The photolysis reaction of CH$_{4}$ is photon-limited. That is, all incident photons are absorbed by the CH$_{4}$ molecules present in Pluto's atmosphere.
During the flyby of the Pluto system on July 14, 2015, the New Horizons spacecraft recorded data about the structure, composition and variability of Pluto's atmosphere. In particular, Alice, the UV spectrometer on-board, observed solar occultations of Pluto's atmosphere which help to determine the vertical profiles of the densities of the present atmospheric constituents and provide key information about the haze. Within this context, our work aims to help the analysis of the New Horizons observations with model predictions of the possible evolution, spatial distribution and opacity of haze in Pluto's atmosphere and on its surface.
We begin in Section \ref{background} with a background on haze formation processes as understood on Titan, Triton and Pluto. In Section \ref{model} we describe the GCM.
The parametrization of organic haze, as well as its implementation in the model are described step by step in Section \ref{param}. Finally, results are shown in Section \ref{results} for two climate scenarios: with and without South Pole N$_{2}$ condensation.
\section{Background on planetary haze formation}
\label{background}
One of Titan's most fascinating features is the dense and widespread organic haze shrouding its surface and containing a large variety of molecules which strongly impact the global climate. This makes Titan a perfect place to study organic chemistry and the mechanisms involved in a planetary haze formation.
Since 2004, the exploration of Titan's haze by the Cassini/Huygens mission has provided a large amount of observational data, revealing complex chemistry, particularly at high altitudes. This has stimulated more interest in understanding this phenomenon.
The haze on Titan is vertically divided into two regions: a main haze up to 300 km altitude, and a thinner, overlying detached haze typically between 400-520 km \citep{Lavv:09}, whose origin is thought to be dynamic \citep{Rann:02}, although other scenarios were suggested \citep{Lars:15}. Both layers contain solid organic material resulting from photochemistry and microphysical mechanisms, some of which remain unknown \citep{Lebo:02, WilsAtre:03, Lavv:08}.
First, methane and nitrogen molecules are dissociated and ionized in the upper atmosphere (up to 1000 km above the surface) by solar UV radiation, cosmic rays and energetic electrons from Saturn's magnetosphere \citep{Sitt:10}. It is commonly thought that the molecules resulting from photolysis chemically react with each other, which leads to the formation of larger and heavier molecules and ions such as hydrocarbons, nitriles and oxygen-containing species \citep[e.g.]{Niem:10, Crav:06, Coat:07, Wait:07, Crar:09}. While CH$_{4}$ is easily destroyed by photolysis and provides most of the organic materials, N$_{2}$ is dissociated as well by extreme UV radiation which explains the rich composition of Titan's upper atmosphere.
In particular, observations from Cassini and Huygens spacecrafts show the presence of hydrocarbons and nitriles, such as C$_{2}$H$_{2}$, C$_{2}$H$_{4}$, C$_{2}$H$_{6}$, C$_{4}$H$_{2}$, C$_{6}$H$_{6}$, and HCN, as well as other more complex organics \citep{Shem:05}. These species, formed after photolysis in the upper atmosphere, are the precursors of the haze.
Then, through multiple processes of sedimentation, accumulation and aggregation, the precursors are thought to turn into solid organic aerosols which become heavy enough to form the orange haze surrounding the moon as seen in visible wavelengths \citep{West:91, Rann:95, Yell:06, Lavv:09}. These aerosols are thought to be aggregates (modeled as fractal-like particles) composed of many spherical particles (monomers) that bond to each other. On Titan, the aerosols start to become large enough to be visible in the detached haze layer around 500 km altitude. Typically, they grow spherical up to radius 40-50 nm and then form fractal particles with monomer sizes of around 50~nm \citep{Lavv:09}.
What are the haze's dominant pathways? What are the chemical natures of complex haze particles?
Several microphysical models \citep{Toon:92,Rann:97,Lavv:09} and photochemical models \citep{WilsAtre:04, Lavv:08, Hebr:13} have been developed, combining both transport and chemistry effects. The formation mechanisms of aerosol particles in Titan's atmosphere have also been investigated using laboratory experiments. By performing UV irradiation of CH$_{4}$ in a simulated Titan atmosphere, several experiments have been successful in producing solid particles and have found that they contain mostly high-molecular-weight organic species \citep[e.g.,][]{Khar:84,Khar:02,Coll:99, Iman:04, Szop:06,Gaut:12}. Experimental results from \citet{Trai:06} also show a linear relationship between the rate of aerosol production and the rate of CH$_{4}$ photolysis. In addition, they found that an increased CH$_{4}$ concentration could lead to a decrease in aerosol production in photon-limited reactions (this could be due to reactions between CH$_{4}$ and precursors forming non-aerosol products).
Titan's atmosphere is not the unique place where organic haze can form. First, similar processes of haze formation are also thought to occur on Triton but yield less haze. During the Voyager 2 flyby in 1989, evidence of a thin haze was detected in Triton's atmosphere from limb images taken near closest approach \citep{Smit:89, Poll:90, RagePoll:92} and from Voyager 2 UVS solar occultation measurements \citep{HerbSand:91, Kras:92, Kras:93}. These data enabled the mapping of the horizontal and vertical distribution of CH$_{4}$ and haze as well as estimation of radiative and microphysical properties of the haze material.
Analyses showed that the haze is present nearly everywhere on Triton, from the surface up to 30 km at least \citep{Poll:90}, where it reached the limit of detectability. Vertical optical depth derived from observations were found to be in the range 0.01-0.03 at UV wavelength 0.15~$\mu$m, and 0.001-0.01 at visible wavelength 0.47~$\mu$m. Haze particle sizes were estimated to be spherical and small, around 0.1-0.2~$\mu$m \citep{Kras:92, RagePoll:92, Poll:90}.
As on Titan, complex series of photochemical reactions may be involved in the formation of this haze, starting with CH$_{4}$ photolysis by the solar and the interstellar background Lyman-$\alpha$ radiation in the atmosphere of Triton at altitudes between 50-100 km, producing hydrocarbons such as C$_{2}$H$_{2}$, C$_{2}$H$_{4}$, C$_{2}$H$_{6}$ \citep{Stro:90, KrasCrui:95}. Dissociation of N$_{2}$ molecules is also suggested in the upper atmosphere around 200-500 km. Transitions between haze precursors to solid organic particles are still incompletely known, but it is commonly thought that it involves similar mechanisms to those on Titan.
Secondly, organic chemistry has also been studied in the Early Earth climate context, where a scenario of a N$_{2}$/CH$_{4}$ atmosphere is plausible to form a hydrocarbon haze \citep{Trai:06}.
\begin{table}[h]
\begin{center}
\begin{tiny}
\begin{tabular}{>{\bfseries}llll}
& \textbf{Titan} & \textbf{Triton} & \textbf{Pluto (2015)} \\
\hline
Distance from Sun (UA) & 9.5 & 30 & 32.91 \\
Solar Flux (ph\,m$^{-2}$\,s$^{-1}$) & $4.43\times10^{13}$ & $4.44\times10^{12}$ & $3.69\times10^{12}$\\
CH$_{4}$ mixing ratio & $1.5\%^{a}$ & $0.02\%^{b}$ & $0.6\%^{c}$\\
CO mixing ratio & $0.0045\%$ & $0.07\%^{b}$ & $0.05\%^{c}$\\
P$_{est}$ (kg\,m$^{-2}$\,s$^{-1}$) & $2.94\times10^{-13}$ & $7.47\times10^{-14}$ & $5.98\times10^{-14}$\\
P$_{lit}$ (kg\,m$^{-2}$\,s$^{-1}$) & $0.5-3\times10^{-13}$ $^{d}$ & $6.0\times10^{-14}$ $^{e}$ & $9.8\times10^{-14}$ $^{f}$ \\
\hline
\multicolumn{4}{l}{$^{a}$\textit{above the tropopause, \citet{Niem:10}}} \\
\multicolumn{4}{l}{$^{b}$\textit{\citet{Lell:10}}} \\
\multicolumn{4}{l}{$^{c}$\textit{\citet{Lell:11}}} \\
\multicolumn{4}{l}{$^{d}$\textit{\citet{WilsAtre:03,McKa:01}}} \\
\multicolumn{4}{l}{$^{e}$\textit{\citet{StroSumm:95}}} \\
\multicolumn{4}{l}{$^{f}$\textit{\citet{Glad:16}}} \\
\hline
\end{tabular}
\end{tiny}
\caption{Comparison of the incident UV flux and fraction of methane for a first order estimation of aerosol production rates on Titan, Triton and Pluto. The estimated rate P$_{est}$ is compared to the observed rate P$_{lit}$, as detailed in the literature.\newline}
\label{Table1}
\end{center}
\end{table}
\begin{table}[h]
\begin{center}
\begin{tiny}
\begin{tabular}{>{\bfseries}llll}
& \textbf{Titan (at 400km)} & \textbf{Triton} & \textbf{Pluto}\\
\hline
Gravity (m$^{2}$\,s$^{-2}$) & 1.01 & 0.779 & 0.62\\
Pressure (Pa) & 1.5 & 1.4-1.9 & 1-1.1$^{a}$\\
Visible normal opacity & 0.07$^{b}$ & 0.003-0.008$^{c}$ & 0.004$^{a}$\\
\hline
\multicolumn{4}{l}{$^{a}$\textit{\citet{Ster:15}}} \\
\multicolumn{4}{l}{$^{b}$\textit{\citet{Cour:11}}} \\
\multicolumn{4}{l}{$^{c}$\textit{\citet{RagePoll:92,Kras:92}}} \\
\hline
\end{tabular}
\caption{Gravity,surface pressure and visible aerosol opacity on Pluto and Triton, compared to the the values encountered in the detached haze layer on Titan \newline}
\label{Table2}
\end{tiny}
\end{center}
\end{table}
Finally, the presence of a haze on Pluto was suspected \citep{Elli:89,Stan:89,Forg:14} and confirmed in 2015 by New Horizons.
At high phase angles, Pluto's atmosphere revealed an extensive haze reaching up to 200 km above the surface, composed of several layers \citep{Ster:15}. Observations show that the haze is not brightest to the sub solar latitude, where the incoming solar flux is stronger, but to Pluto North Pole.
The haze is strongly forward scattering in the visible with a blue color, while at the same time there is haze extinction optical depth exceeding unity in the UV. The blue color and UV extinction are consistent with a small size of about 10 nm for monomers, whereas the high forward scatter to back scatter ratio in the visible suggests a much larger overall size of at least 200 nm. Although the haze may contain particles of diverse sizes and shapes depending on the altitude, these properties may also be consistent with fractal aggregate particles composed of 10 nm monomers \citep{Glad:16,Chen:16}.
Although the specific mechanisms of haze formation are not fully understood, it seems that the main parameters controlling the formation of haze in a N$_{2}$/CH$_{4}$ atmosphere are the fractional amount of CH$_{4}$ (enough CH$_{4}$ is required to avoid CH$_{4}$-limited reactions, that is when the CH$_{4}$ concentration in the atmosphere is not sufficient to absorb all incoming photons) and the UV flux available to photolyze it.
One can compare the UV flux and the fraction of methane for Titan, Triton and Pluto to estimate the haze formation rate to first order.
Here we assume that the impact of cosmic rays and energetic electrons from Saturn's magnetosphere is negligible for this first order comparison.
As shown on \autoref{Table1} and \autoref{Table2}, Pluto's atmosphere contains 10 times less CH$_{4}$ and receives 10 times less solar UV flux than Titan (relative to the atmospheric mass). Consequently, it is likely that CH$_{4}$ photolysis on Pluto leads to the formation of haze aerosols (and precursors) in lower quantities than on Titan. Compared to Triton, Pluto has similar surface pressure and gravity and its atmosphere contains 10 times more CH$_{4}$, for a comparable UV flux. Thus, similar amounts of haze are expected on Pluto and Triton, depending on the accelerating or decelerating role of larger CH$_{4}$ amount. \citet{Ster:15} reported a visible normal opacity of 0.004 on Pluto, which is in the range of what has been observed on Triton, although it also depends on the scattering properties of haze particles.
On Titan, the pressure corresponding to the location of the detached haze layer at about 400 km altitude is about 1 Pa, which is similar to the surface pressure on Pluto in 2015.
While \citet{Rann:03} predicted the peak of production of haze in Titan's GCMs at a pressure around 1.5 Pa, Cassini observations \citep{Wait:05, Tean:12} pointed to active chemistry and haze formation at lower pressures. In addition, the amounts of methane at these altitudes on Titan and in Pluto's atmosphere are of the same order of magnitude. Thus, Pluto has sufficient pressure and material in its atmosphere so that complex and opaque organic aerosols form, in a manner similar to the detached haze layer on Titan.
Consequently, in this paper, we use the microphysical and single scattering optical properties of Titan detached haze around 400 km altitude as a reference to define the haze properties on Pluto while the mass of aerosols is calculated by the model without any empirical assumption.
\section{Model description}
\label{model}
The LMD Pluto General Circulation Model (GCM) contains a 3D Hydrodynamical core inherited and adapted from the LMD Mars GCM \citep{Forg:99}. It is described in more details in \citet{Forg:16}. The large-scale atmospheric transport is computed through a "grid point model"
composed of 32 longitude and 24 latitude points. A key difference with the \citet{Forg:16} version of the model is that we use 28 layers instead of 25 to extend the model top up to about 600 km, with most of the layers in the first 15 km in order to obtain a finer near-surface resolution, in the boundary layer. The horizontal resolution at the equator is typically around 170 km.
The physical part of the model, which forces the dynamics, takes into account the N$_{2}$ and the CH$_{4}$ cycles (condensation and sublimation in both the atmosphere and the ground), the vertical turbulent mixing and the convection in the planetary boundary layer, the radiative effect of CH$_{4}$ and CO, using the correlated-k method to perform a radiative transfer run and taking into account NLTE effects, a surface and subsurface thermal conduction model with 22 layers and the molecular conduction and viscosity in the atmosphere.
\section{Modeling haze on Pluto}
\label{param}
Here we describe our representation of the organic haze formation and transport in the GCM.
The driving force of the photochemical reactions occurring in a N$_{2}$-CH$_{4}$ atmospheric layer is the UV flux received by this layer. First we consider the photolysis of CH$_{4}$ by Lyman-$\alpha$ only (Section \ref{photolysis}), using the results from \citet{Glad:15} to calculate the incident Lyman-$\alpha$ flux at Pluto (Section \ref{lymanalpha}). We assume that each incident photon ultimately interacts with one molecule of methane, to form by photolysis haze precursors which can be transported by the circulation (Section \ref{mechanism}). Finally we convert haze precursors into organic haze using a constant characteristic decay time (Section \ref{conversion}). Haze particles properties used in this study are detailed in Section \ref{hazeproperties}. In order to validate this approach, we estimate the total aerosol production thus obtained on Pluto, Titan and Triton and compare with literature values in Section \ref{hazeproduction}.
\subsection{Photolysis of CH$_{4}$ by Lyman-$\alpha$}
\label{photolysis}
We consider only the photolysis of CH$_{4}$ by the Lyman-$\alpha$ component of the UV spectrum. This is because the Hydrogen Lyman-$\alpha$ line at 121.6 nm is the strongest ultraviolet emission line in the UV solar spectrum where absorption by CH$_{4}$ happens. In fact, the solar irradiance between 0 and 160 nm (far ultraviolet) is dominated by the Lyman-$\alpha$ emission by a factor of 100. The UV solar irradiance grows significantly at wavelengths values higher than 200 nm (middle and near-ultraviolet) but N$_{2}$, CH$_{4}$ and CO do not absorb at these wavelengths.
Both N$_{2}$ and CH$_{4}$ absorb with similar efficiency in the UV but not at the same wavelengths. N$_{2}$ is the primary absorber at wavelength between 10 and 100 nm, while CH$_{4}$ absorbs mainly between 100 and 145 nm. Thus the interaction between CH$_{4}$ and Lyman-$\alpha$ emission dominates the other interactions between the UV flux and the N$_{2}$-CH$_{4}$ atmosphere by a factor of 100.
On Pluto, CO may also contribute to the formation of haze. It absorbs in the far UV spectrum at similar rates that N$_{2}$. However, at 121.6 nm, it absorbs 10 times less than CH$_{4}$. Here we chose to neglect the effect of N$_{2}$ and CO absorption.
This first assumption enables us to write Beer's law as the following:
\begin{equation}
I(\lambda,P) = I_{0} \, e^{-\int_{0}^{P}\frac{\sigma_{CH4}\,N_{a}\,q_{CH4}}{M_{CH4}\,g}\,\frac{dP}{cos(\theta)}}
\label{Beerlaw}
\end{equation}
where $I_{0}$ is the incident intensity (in ph\,m$^{-2}$\,s$^{-1}$) and $I(\lambda,P)$ the intensity after absorption for a given wavelength $\lambda$ and pressure $P$, $\sigma_{CH4}$ is the absorption cross section of CH$_{4}$ at wavelength $\lambda$ (here in m$^{2}$\,molec$^{-1}$ but usually given in cm$^{2}$\,molec$^{-1}$), $q_{CH4}$ is the mass mixing ratio of CH$_{4}$ at pressure $P$ (kg\,kg$^{-1}_{air}$), $M_{CH4}$ is the methane molecular mass (kg\,mol$^{-1}$), $N_{a}$ is the Avogadro constant, $\theta$ is the flux incident angle and g the surface gravity. We use $\sigma_{CH4}=1.85\times10^{-17}$ cm$^{2}$ at Lyman-$\alpha$ wavelength \citep{Kras:04} and $q_{CH4}$ as calculated by the GCM for each vertical layer. The calculation of the Lyman-$\alpha$ flux radiative transfer is performed independently for the solar and the interplanetary medium fluxes in order to take into account different values for the incident flux $I_{0}$ and the incident angle $\theta$ (see Section \ref{lymanalpha}).
\subsection{Sources of Lyman-$\alpha$}
\label{lymanalpha}
The sources of Lyman-$\alpha$ flux at Pluto are adopted from \citet{Glad:15}, which takes into account the solar as well as the interplanetary medium (IPM) Lyman-$\alpha$ fluxes. The IPM emission corresponds to interplanetary hydrogen atoms passing through the solar system which resonantly scatter solar Lyman-$\alpha$ photons and thus diffuse Lyman-$\alpha$ emission. Therefore the total Lyman-$\alpha$ flux at any pressure level $P$ in Pluto's atmosphere is:
\begin{equation}
I_{tot}(P) = I_{sol}(P) + I_{IPM}(P)
\label{Itot}
\end{equation}
The solar Lyman-$\alpha$ flux at Pluto is inversely proportional to the square of the Sun-Pluto distance. It is obtained by considering a constant solar Lyman-$\alpha$ flux at Earth of $4\times10^{15}$ ph\,m$^{-2}$\,s$^{-1}$ and a constant extinction factor of 0.875 due to the interaction with interplanetary hydrogen between Pluto and the Sun, which are values estimated by \citep{Glad:15} for 2015. The solar Lyman-$\alpha$ flux $I^{sol}_{0}$ thus estimated at Pluto is $3.23\times10^{12}$ ph\,m$^{-2}$\,s$^{-1}$. The incident angle $\theta^{sol}$ corresponds to the solar zenith angle.
The IPM Lyman-$\alpha$ source at Pluto is not isotropic, as shown on figure 4 in \citet{Glad:15}, which presents the all-sky brightness of IPM emissions at Pluto in Rayleigh units in 2015. The brightness is stronger near the subsolar point and is minimal in the anti-sunward hemisphere. In order to take into account this property in the parametrization and compute the number of photons entering Pluto's atmosphere at a given location, we integrated the all-sky IPM brightness estimated in 2015 from \citet{Glad:15} over the half celestial sphere as seen at the considered location. The flux $I^{IPM}_{0}$ obtained varies with the local time but does not strongly depend on the Sun-Pluto distance (we use the flux estimated in 2015 for all other years). \autoref{fit} shows the final result: we find a maximum flux at subsolar point of 1.15$\times10^{12}$ ph\,m$^{-2}$\,s$^{-1}$, a minimum flux at anti-subsolar point of 4.90$\times10^{11}$ ph\,m$^{-2}$\,s$^{-1}$ and an average flux over the planet of 7.25$\times10^{11}$ ph\,m$^{-2}$\,s$^{-1}$.
We consider that the incident angle for the IPM flux $\theta^{IPM}$ is equal to the solar zenith angle during daytime, when the IPM flux is dominated by the forward scattered halo of the solar flux. When the solar zenith angle is greater than $\pi$/3 (nighttime), we consider that the IPM flux is more isotropic and we set the incident angle to $\pi$/3.
At the Sun-Pluto distance during New Horizon flyby (32.91 UA), this IPM source of Lyman-$\alpha$ is significant compared to the solar source. Considering the solar Lyman-$\alpha$ flux, the energy of a photon at Lyman-$\alpha$ wavelength (121.6 nm) and its dissipation over the whole surface of Pluto (the initial flux is divided by a factor of 4), the power of solar Lyman-$\alpha$ source at Pluto obtained is 22.93 MW. The same calculation can be performed for the IPM flux. \citet{Glad:15} gives an averaged IPM brightness at Pluto of 145 R (1 R = 1/ $4\pi$ $\times10^{10}$ ph\,m$^{-2}$\,s$^{-1}$\,sr$^{-1}$), which corresponds to a flux of $1.45\times10^{12}$ ph\,m$^{-2}$\,s$^{-1}$ once integrated on the celestial sphere. This leads to a contribution of IPM Lyman-$\alpha$ source at Pluto of 10.30 MW. Consequently, solar and IPM sources at Pluto account for respectively $70\%$ and $30\%$ of the total power source.
\begin{figure}[h]
\begin{center}
\includegraphics[width=6in]{figures/IPMflux.pdf}
\caption{An instantaneous map of interplanetary Lyman-$\alpha$ emission (10$^{10}$ ph\,m$^{-2}$\,s$^{-1}$) on Pluto in July 2015, estimated by integrating the all-sky IPM brightness given by figure 4 in \citet{Glad:15} over the half celestial sphere at each point of the map. In this example, the subsolar longitude is the sub Charon longitude (0\char23) \newline}
\label{fit}
\end{center}
\end{figure}
\newpage
\subsection{Production of haze precursors}
\label{mechanism}
In the parametrization, we consider that each absorbed Lyman-$\alpha$ photon destroys one molecule of methane by photolysis, thus forming haze precursors (CH$_{3}$, CH$_{2}$ , CH + N, etc.) converted later into aerosols. Using equation \ref{Beerlaw} and \ref{Itot}, the precursors production rate (in kg\,kg$^{-1}_{air}$\,s$^{-1}$) is calculated as:
\begin{equation}
P_{prec}(P) = \frac{M_{CH4}\,g}{N_{a}}\,\frac{dI_{tot}}{dP}
\label{procprod}
\end{equation}
In the model, all possible precursors which can form during this reaction are represented by a unique gas. The equation of the reactions is:
\begin{equation}
CH_{4} + h\nu \rightarrow precursors \rightarrow haze\; aerosols
\label{reaction}
\end{equation}
This mechanisms correlates linearly the rate of haze precursors production with the rate of CH$_{4}$ photolysis. It has also been used by \citet{Trai:06} to estimate aerosols production on Titan and Early Earth.
In reality, the reactions are more complex and could lead to the irreversible production of HCN, or to the production of molecules such as C$_{2}$H$_{2}$ or C$_{2}$H$_{6}$ which can later be photolyzed themselves as well. In addition, CH$_{4}$ molecules may be chemically dissociated by reacting directly with the precursors. Consequently, these reactions could lead either to an increase in the amount of carbon atoms available as haze material, increasing the haze production, or to non-aerosol products, slowing down the haze production \citep{Trai:06}.
In the parametrization, the haze production is regulated by a factor $K_{CH4}$, that corresponds to the ratio between the total number of carbon atoms in the tholins and the number of carbon atoms coming from CH$_{4}$ photolysis. $K_{CH4}$ would range from 1 to 2 (respectively all or half of the carbon in the tholins are formed by direct CH$_{4}$ photolysis) if direct reactions between precursors and CH$_{4}$ occur and contribute to provide tholins with carbon atoms. However, the ratio could be lower than 1 considering the formation of other non-aerosol products (see Section \ref{sec:sensib_mass}).
Additionally, nitrogen may contribute to the chemical reactions and provide material for aerosol formation. In order to take into account this process, the haze production is also boosted by a factor $K_{N}$=1+$N/C$, $N/C$ representing the mass ratio between nitrogen and carbon atoms contribution observed in the tholins (since molar masses of nitrogen and carbon are quite similar, the mass ratio is close to the number ratio). Different values of this ratio have been observed in laboratory experiments, ranging from 0.25 to 1 depending on the pressure (the higher the pressure, the lower the ratio), the temperature and the amount of methane in the simulated atmosphere \citep[e.g.][]{Coll:99,Tran:08,Nnam:13}. In the model, we adopt $N/C$ = 0.5, in line with the values obtained in \citet{Nnam:13} at low pressure, and $K_{CH4}$ = 1, so that the total production of tholins remains in the range of estimated values on Titan and Pluto (see Section \ref{hazeproduction}).
\subsection{Conversion of haze precursors to aerosols}
\label{conversion}
As the mechanisms at the origins of formation of organic haze are not well known, another assumption is made in the parametrization: we consider that the precursors become solid organic particles (by a set of processes of aggregation and polymerization that are not represented) after a given time. In practice, the amount of precursors is subject to exponential decay and is converted into aerosols with characteristic decay time $\tau$ (or characteristic time for aerosol growth).
In other words, $\tau$ is the mean lifetime of the precursors before they become solid aerosols. This time is difficult to estimate as it depends on atmospheric conditions (concentration, pressure...). However, Titan's atmospheric models show that the time needed for precursors to evolve from the photolysis area to the detached layer is typically around 10$^{6}$-10$^{8}$ s \citep{Lavv:11,Rann:93}.
Consequently, we used in our reference GCM simulations a value of 10$^{7}$ s for Pluto aerosols and we examine the sensitivity of the results to this parameter in Section \ref{sec:sensib_tau}.
Once produced, the aerosols are transported by the atmospheric circulation, mixed by turbulence, and subject to gravitational sedimentation (see Section \ref{hazeproperties}).
\subsection{Discussion on total aerosol production}
\label{hazeproduction}
Equation \ref{reaction} enables us to estimate the total haze production rate $P$ (kg\,m$^{-2}$\,s$^{-1}$) in a N$_{2}$/CH$_{4}$ atmosphere:
\begin{equation}
P=(F_{SOL}+F_{IPM})\,\frac{M_{CH4}}{N_{a}}\,K_{CH4}\,K_{N} \quad with \quad
F_{SOL}=\frac{I_{Earth}}{4\,{d_{P}}^{2}}\,E_{H}
\label{production}
\end{equation}
where F$_{SOL}$ and F$_{IPM}$ are the solar and IPM Lyman-$\alpha$ flux respectively (in ph\,m$^{-2}$\,s$^{-1}$), $M_{CH4}$ is the molar mass of methane ($M_{CH4}=16\times10^{-3}$ kg\,mol$^{-1}$), $N_{a}$ is the Avogadro constant, I$_{Earth}$ is the initial Lyman-$\alpha$ flux at Earth (we set $I_{Earth}$=$4\times10^{15}$ ph\,m$^{-2}$\,s$^{-1}$), d$_{P}$ is the distance in astronomical units of the considered planet $P$ to the Sun and E$_{H}$ is a constant extinction factor due to interaction with interplanetary hydrogen between the planet $P$ and the Sun. Here E$_{H}$ is set to 0.875 for the case of Pluto \citep{Glad:15} and to 1 for the other cases. The solar flux F$_{SOL}$ is equal to the incident solar flux $I^{sol}_{0}$ divided by a factor of 4 to take into account the distribution on the planetary sphere.
It is important to note that the haze production rate is independent of the CH$_{4}$ concentration, even for CH$_{4}$ concentrations several orders of magnitude lower than on Pluto (see Section \ref{sec:sensib}). The reactions are photon-limited, i.e. that enough CH$_{4}$ is present in Pluto's atmosphere for all photons to be absorbed by CH$_{4}$.
In order to validate the approach described by equation \ref{reaction}, we apply equation \ref{production} to Titan, Triton and Pluto and compare the haze production rates obtained with the literature. The values, obtained with $K_{CH4}$=1 and $K_{N}$=1.5, are summarized in \autoref{Table1}.
For Titan's case, we consider that the IPM flux is negligible compared to the solar flux. Using an average Sun-Titan distance $d_{Titan}$=9.5 UA, we find for Titan's atmosphere a Lyman-$\alpha$ flux of $1.11\times10^{13}$ ph\,m$^{-2}$\,s$^{-1}$ (dissipated on the planetary sphere) and a production rate of $2.94\times10^{-13}$ kg\,m$^{-2}$\,s$^{-1}$. This is comparable to values found by \citet{WilsAtre:03} and \citet{McKa:01}, as shown on \autoref{Table1}.
For Triton's case, we consider an averaged IPM flux of 340 R \citep{Broa:89,Kras:95}, which correspond to an IPM flux of $170\times10^{10}$ ph\,m$^{-2}$\,s$^{-1}$ distributed on the planetary sphere. Using an average Sun-Triton distance $d_{Titan}$=30 UA, we find for Triton's atmosphere a total Lyman-$\alpha$ flux (solar and IPM) of $2.81\times10^{12}$ ph\,m$^{-2}$\,s$^{-1}$ and a photolysis rate of $7.47\times10^{-14}$ kg\,m$^{-2}$\,s$^{-1}$, which is also in line with the literature references.
Since this approach provides good estimation of Titan's and Triton's total aerosol production, we used it to estimate the aerosol production rate for Pluto's atmosphere. \autoref{production} gives a production rate of $5.98\times10^{-14}$ kg\,m$^{-2}$\,s$^{-1}$ using the solar and IPM flux as calculated in Section \ref{lymanalpha}.
This value is one order of magnitude lower than the one on Titan (due to the UV flux one order of magnitude lower) and comparable to the value found on Triton. It is of the same order of magnitude as the value estimated on Pluto from photochemical models \citep{Glad:16} shown in \autoref{Table1}.
\subsection{Properties of haze particles for sedimentation and opacity estimations}
\label{hazeproperties}
Haze precursors and particles are transported in the model by atmospheric circulation and are not radiatively active. In addition, the haze is considered too thin to affect the surface energy balance and does not change its ground albedo (in line with haze and surface observations on Triton as discussed in \citet{HillVeve:94}).
The density of the aerosol material in the model is set to 800 kg\,m$^{-3}$, which is in the range of values typically used on Titan \citep{Soti:12,Lavv:13,Trai:06}.
The size of the haze particles affects their sedimentation velocity and thus the haze distribution in Pluto's atmosphere. In the GCM, we prescribe a uniform size distribution of particles. For the reference simulations (with and without South Pole N$_{2}$ condensation), we assumed spherical particles with a radius of 50 nm, consistent with the properties of the detached haze layer on Titan (see Section \ref{background}). We also examine the sensitivity of the results to different sizes of particles in Section \ref{sec:sensib_r}, in order to bracket the different possible scenarios for Pluto's haze.
We consider two lower radii of 30 nm and 10 nm, which is in the range of recent estimations \citep{Glad:16}, and one larger radius of 100 nm.
The particles fall with their Stokes velocity $\omega$, corrected for low pressures \citep{Ross:78}:
\begin{equation}
\omega=\frac{2}{9}\,\frac{r^{2}\,\rho\,g}{v}\,(1+\alpha\,Knud) \quad with \quad
Knud=\frac{k_{B}\,T}{\sqrt{2}\,\pi\,d^{2}\,p\,r}
\label{stokes}
\end{equation}
with $r$ the particle radius, $\rho$ the particle density, $g$ the Pluto's gravitational constant, $v$ the viscosity of the atmosphere, $Knud$ the Knudsen number, $p$ the considered pressure, $T$ the atmospheric temperature, $d$ the molecular diameter, $k_{B}$ the Boltzmann's constant and $\alpha$ a correction factor.
On Pluto, the Knudsen number is significant and thus the sedimentation velocity is proportional to the particle radius. Consequently, in an ideal atmosphere without atmospheric circulation, a 100 nm particle will fall twice faster than a 50 nm particle, leading to a twice lower column mass of haze. Assuming an atmospheric temperature of 100 K and a surface pressure of 1 Pa, the sedimentation velocities above Pluto's surface are about 4.6$\times10^{-4}$, 1.4$\times10^{-3}$, 2.3$\times10^{-3}$ and 4.6$\times10^{-3}$ m\,s$^{-1}$ for an aerosol radius of 10, 30 50 and 100 nm respectively.
One can note that the Stokes velocity is proportional to the inverse of the pressure. Theoretically, the lower the pressure, the higher the sedimentation velocity of the aerosol and thus the lower the mass of haze in the atmosphere.
The choice of the size and the shape of aerosol particles is also critical to estimate their optical properties and thus their detectability. In Section \ref{sec:sensib_r}, we compare the opacities obtained with different particle radii. In Section \ref{opacity}, we examine the case of fractal particles by considering that they fall at the velocity of their monomers, due to their aggregate structure, which is only true for a fractal dimension equal to 2 \citep{Lavv:11,Lars:14}.
\subsection{Description of the reference simulations}
In this paper, we compare two reference simulations which correpond to the two climate scenarios detailed in \citet{Forg:16}:
One is the case of Sputnik Planum as the only reservoir of N$_{2}$ ice without N$_{2}$ condensation elsewhere (referred as No South Pole N$_{2}$ condensation), and the other is the case with a latitudinal band of N$_{2}$ ice at northern mid latitudes, as an additional reservoir of N$_{2}$ ice with Sputnik Planum, and an initially colder South Pole, allowing the N$_{2}$ ice to condense (with South Pole N$_{2}$ condensation).
The reference simulations study are defined as follows. A seasonal volatile model of Pluto is used to simulate the ice cycles over thousands of years and obtain consistent ices distribution, surface and subsurface temperatures as initial conditions for the GCM (see \citet{Bert:16} for more details). Then, GCM runs are performed from 1988 to 2015 included so that the atmosphere has time to reach equilibrium before 2015 (the spin up time of the model is typically 10-20 Earth years). The initial conditions, the settings of the model, as well as discussions about the sensitivity of the predictions to those settings can be found in \citet{Forg:16}.
The model is run with the haze parametrization using a precursor characteristic time for aerosol growth of 10$^{7}$ s (about 18 sols on Pluto), a fraction K$_{CH4}$=1 and K$_{N}$=1.5. The density and sedimentation effective radius of haze particles are set uniformly to 800 kg\,m$^{-3}$ and 50 nm respectively (see Section \ref{conversion}). \autoref{tab:inicond} summarizes the surface conditions and haze parameters used in the reference simulations \citep{Forg:16}. \newline
\begin{table}[h]
\begin{center}
\begin{tiny}
\begin{tabular}{llll}
\hline
\textbf{Global Thermal Inertia (J s$^{-0.5}$ m$^{-2}$ K$^{-1}$)} & 50 (diurnal) & 800 (seasonal) & \\
\textbf{Albedo} & 0.68 (N$_{2}$ ice) & 0.50 (CH$_{4}$ ice) & 0.15 (Tholins)\\
\textbf{Emissivity} & 0.85 (N$_{2}$ ice) & 0.85 (CH$_{4}$ ice) & 1 (Tholins)\\
\hline
\end{tabular}
\begin{tabular}{>{\bfseries}ll}
\hline
Characteristic time for aerosol growth $\tau$ (s) & 10$^{7}$\\
K$_{CH4}$ & 1 \\
K$_{N}$ & 1.5 \\
Effective radius of haze particles (nm) & 50\\
Density of haze particles (kg.m$^{-3}$) & 800\\
\hline
\end{tabular}
\caption{Surface conditions and settings for haze parametrization set for the GCM reference simulations \newline}
\label{tab:inicond}
\end{tiny}
\end{center}
\end{table}
\section{Results}
\label{results}
This section presents the results obtained with the GCM coupled with the haze parametrization. All figures and maps are shown using the new IAU convention, spin north system for definition of the North Pole \citep{Buie:97,Zang:15}, that is with spring-summer in the northern hemisphere during the 21th Century. Here we focus on model predictions in July 2015. We first compare the two reference simulations, then we show the corresponding ranges of UV and VIS opacities and we perform sensibility studies.
\subsection{Reference simulation 1: No South Pole N$_{2}$ condensation}
The predictions of the state of the atmosphere in July 2015 remain unchanged compared to what is shown in \citet{Forg:16}, since haze particles are not radiatively active and since their sedimentation on Pluto's surface does not impact the surface albedo. These processes could be taken into account in future GCM versions.
In July 2015, the modeled surface pressure is found to be around 1 Pa. The nitrogen reservoir in Sputnik Planum at mid northern latitudes is under significant insolation during the New Horizon flyby (the subsolar latitude in July 2015 is 51.55\char23N), as well as the mid and high northern CH$_{4}$ frosts which sublime and become an important source of atmospheric CH$_{4}$, as described by \citet{Forg:16}.
According to equation \ref{reaction}, methane photolysis occurs at all latitudes but is more intense at locations where strong incoming flux of Lyman-$\alpha$ photons occurs, that is at high northern latitudes in July 2015. This is confirmed by \autoref{photprecsection}, showing the CH$_{4}$ photolysis rate as simulated in the GCM. All Lyman-$\alpha$ photons are absorbed above 150 km altitude. The maximum photolysis rate is is typically around 1.3$\times10^{-21}$ g\,cm$^{-3}$\,s$^{-1}$ and is obtained at 250 km altitude above the North Pole.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.3]{figures/photprecsection_gcm3_ref50.pdf}
\caption{Photolysis rate of CH$_{4}$ (g\,cm$^{-3}$\,s$^{-1}$) obtained with the reference simulation without South Pole N$_{2}$ condensation for July 2015 (color bar in log scale) \newline}
\label{photprecsection}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.2]{figures/prechazesection_allref.pdf}
\caption{Zonal mean latitudinal section of haze precursor density (g\,cm$^{-3}$) obtained with the reference simulation without (left) and with (right) South Pole N$_{2}$ condensation (color bar in log scale) \newline}
\label{prechazesection}
\end{center}
\end{figure}
Haze precursors formed by CH$_{4}$ photolysis are then transported by general circulation in the GCM. As shown by \citet{Forg:16}, the fact that N$_{2}$ ice is entirely sequestered in the Sputnik Planum basin and does not condense elsewhere leads to very low meridional wind velocities in the atmosphere and a weak meridional circulation.
Consequently, haze precursors are not transported fast towards the surface by circulation. In 2015, with a lifetime of 18 sols, the haze precursors are still confined to high altitudes above 140 km, and are in larger amount in northern latitudes where most of the photolysis of CH$_{4}$ occurs (\autoref{prechazesection}).
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.2]{figures/hazesection_allref.pdf}
\caption{Zonal mean latitudinal section of haze aerosol density (g\,cm$^{-3}$) obtained with the reference simulation for July 2015 without (top) and with (bottom) South Pole N$_{2}$ condensation (color bar in log scale). The right panels correspond to a zoom in the lowest 15 km above the surface.\newline}
\label{hazesection}
\end{center}
\end{figure}
\autoref{hazesection} shows the zonal mean latitudinal section of haze density predicted in July 2015.
The aerosols formed above 150 km slowly fall towards the surface, and accumulate in the first kilometers above the surface, due to the decrease of sedimentation velocity with atmospheric pressure.
The haze obtained extends at high altitudes. The density decreases with the altitude but remains non-negligible with values up to 4$\times10^{-19}$ g\,cm$^{-3}$ at 500 km altitude.
In this case, the meridional circulation is quite weak: the diurnal condensation and sublimation of N$_{2}$ ice in Sputnik region only impacts the circulation in the first km above the surface, and at higher altitudes, the circulation is forced by the radiative heating (the northern CH$_{4}$ warms the atmosphere, leading to a transport of this warm air from the summer to the winter hemisphere) inducing low meridional winds. Consequently, the general circulation does not impact the haze distribution, which is dominated by the incoming flux and the sedimentation velocity. In other words, the vertical and meridional atmospheric motions are not strong enough to signicantly push and impact the latitudinal distribution of the haze composed of 50 nm particles: the haze density in the atmosphere is always higher at the summer pole, where a stronger CH$_{4}$ photolysis occurs.
In the summer hemisphere, the haze density is typically 2-4$\times10^{-15}$ g\,cm$^{-3}$ at 100 km altitude while it reaches 1-2$\times10^{-13}$ g\,cm$^{-3}$ above the surface.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.35]{figures/allevolhaze.pdf}
\caption{Evolution of the mean column atmospheric mass of haze aerosols (g\,cm$^{-2}$) from 1988 to 2016 obtained with different particle radius in the reference simulation without South Pole N$_{2}$ condensation: 10 nm (blue), 30 nm (green), 50 nm (red) and 100 nm (black). The dashed lines correspond to similar simulations started with a higher initial amount of haze. With 50 nm particles (red curve), the mass of haze reaches an equilibrium within less than one year. The dash-dotted line corresponds to the 10 nm case with the real variable initial Lyman-$\alpha$ flux (at Earth).\newline}
\label{evolhaze}
\end{center}
\end{figure}
\autoref{evolhaze} shows the evolution of the mean column atmospheric mass of haze aerosols since 1988. Assuming a constant initial flux of Lyman-$\alpha$ (at Earth) and a particle radius of 50 nm, the column mass of haze reaches a peak of 1.8$\times10^{-7}$ g\,cm$^{-2}$ in 2015.
Because the transport of haze is dominated by its sedimentation, the column mass of haze directly depends on the sedimentation velocity of the haze particles. As shown by equation \ref{stokes}, the sedimentation velocity decreases when pressure increases, hence the increase of column mass of haze, in line with the threefold increase of surface pressure since 1988. Note that this trend still applies when considering the real and variable initial Lyman-$\alpha$ flux at Earth between 1988 and 2015, as shown by \autoref{evolhaze}.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.2]{figures/massmaphaze_ref.pdf}
\caption{Column atmospheric mass map of haze aerosols (g\,cm$^{-2}$) obtained with the reference simulation without (left) and with (right) South Pole N$_{2}$ condensation \newline}
\label{massmaphaze}
\end{center}
\end{figure}
\autoref{massmaphaze} shows the column atmospheric mass of haze aerosols. In line with the previous results, the column mass obtained is higher at the North Pole than at the South Pole by one order of magnitude, due to the maximum haze production in the summer hemisphere. The column mass of haze reaches 3.9$\times10^{-7}$ g\,cm$^{-2}$ at the North Pole.
\subsubsection{Reference simulation 2: with South Pole N$_{2}$ condensation}
\label{sec:polarcap}
The sublimation of N$_{2}$ in mid northern latitudes (Sputnik region and the latitudinal band) and its condensation in the winter hemisphere induce an atmospheric flow from the northern to the southern hemisphere, and thus a stronger meridional circulation than in the reference simulation without South Pole N$_{2}$ condensation, although the latitudinal winds remain relatively weak \citep{Forg:16}.
Although the atmospheric methane is more mixed in the atmosphere in this case, the state of the atmosphere remains similar to the reference simulation without South Pole N$_{2}$ condensation. The surface pressure is increasing before 2015 and reaches 1 Pa in 2015.
Because of the condensation flow from the northern to the southern hemisphere, the air in the upper atmosphere is transported along with the haze precursors from the summer atmosphere to the winter atmosphere.
As shown on \autoref{prechazesection}, the characteristic decay time of haze precursors (18 sols) is sufficient for some of the precursors to be transported from the summer to the winter hemisphere where the descending branch bring them at lower altitudes down to the surface.
As a consequence of that, more haze is formed in the winter hemisphere than in the reference simulation without N$_{2}$ condensation flow, which compensates the haze production in the summer hemisphere due to the higher CH$_{4}$ photolysis rate. It leads to a similar haze density at all latitudes, as shown by \autoref{hazesection}.
The haze density is typically 4$\times10^{-15}$ g\,cm$^{-3}$ at an altitude of 100 km, which is similar to the reference simulation without the condensation flow. The haze remains latitudinally well dispersed down to 3 km, where the meridional circulation driven by the N$_{2}$ condensation flow affects the haze distribution: the haze is pushed towards southern latitudes by the N$_{2}$ ice sublimation above the N$_{2}$ frost latitudinal band and Sputnik Planum, avoiding an accumulation of haze at the mid and high northern latitudes. Between -70\char23S and -90\char23S, haze particles in the first layers are suctioned towards the surface of the N$_{2}$ polar cap.
The haze reaches a density of about 5-20$\times10^{-12}$ g\,cm$^{-3}$ below 1 km in the winter hemisphere, and 3-6$\times10^{-14}$ g\,cm$^{-3}$ in the summer hemisphere, which is twice less compared to the reference simulation without the condensation flow.
In line with the previous results, the column mass of haze in the simulation with condensation flow shown on \autoref{massmaphaze} (right figure) is well dispersed on Pluto, with small variations: in the summer atmosphere, the mass is about 2$\times10^{-7}$ g\,cm$^{-2}$, but it is slightly less at low and mid latitudes because the haze above the surface is transported towards the south polar cap, and slightly more at the North Pole because the haze is not impacted by the N$_{2}$ ice sublimation and transport which occur at lower latitudes.
As in the previous simulation without South Pole N$_{2}$ condensation, the mean column mass of haze increases with surface pressure. In 2015, a similar averaged column mass of haze is obtained. Slight discrepancies are found due to slightly different surface pressures to first order \citep{Forg:16}, and to the different circulation to second order.
\subsection{Haze opacity}
\label{opacity}
In order to better quantify the amount of haze formed on Pluto and compare with the observations as well as with the Titan and Triton cases, one can compute the total column opacity and the line of sight opacity of the haze (as a diagnostic of the results). Here we focus on the opacity at UV ($\lambda$ = 150 nm) and visible ($\lambda$ = 550 nm) wavelengths for sake of comparison with the data recorded by the UV spectrometer Alice and the Ralph and LORRI instruments on board New Horizons. Assuming a homogeneous size and extinction efficiency for the aerosols in Pluto's atmosphere, the opacity $\tau_{\lambda}$ for a given wavelength $\lambda$ is directly proportional to the atmospheric column mass of aerosols:
\begin{equation}
\tau_{\lambda}=\alpha.M \quad \quad with \quad \quad
\alpha=\frac{3}{4}\frac{Q_{ext,\lambda}}{\rho_{aer}r_{eff}}
\label{tau}
\end{equation}
where $Q_{ext}$ is the aerosol extinction efficiency, $r_{\mbox{eff}}$ the aerosol particle effective radius, $\rho_{aer}$ the aerosol density and $M$ is the atmospheric column mass of aerosol in kg\,m$^{-2}$.
\subsubsection{Spherical particles}
Assuming that the haze on Pluto is composed of spherical particles and behaves like the detached haze layer on Titan, we used a Mie code to generate single scattering extinction properties for different spherical particle sizes. The code takes into account a modified gamma size distribution of particles with the considered effective radius and an effective variance $\nu_{eff}$ = 0.3, as well as the optical indices of \citet{Rann:10}. These indices have been updated from \citet{Khar:84} thanks to new sets of Cassini observations. For 50 nm particles, we obtain an extinction efficiency Q$_{ext}$ of 2.29 in UV and 0.19 in visible wavelengths. Using equation \ref{tau} with a density of aerosol material of 800 kg\,m$^{-3}$, we find that the haze column opacity in July 2015 reaches 0.077-0.17 (UV) and 0.0064-0.014 (VIS) in the summer hemisphere, in the reference simulation without South Pole N$_{2}$ condensation. In the simulation with South Pole N$_{2}$ condensation, the opacities are 0.064-0.086 (UV) and 0.0053-0.0071 (VIS) in the summer hemisphere.
\subsubsection{Fractal particles}
The case of fractal particles can also be discussed. On Titan, an upper limit of the maximum equivalent mass sphere radius (or bulk radius) of fractal particles in the detached haze layer has been estimated to 300 nm, containing up to 300 monomers \citep{Lars:14}, while larger particles containing a higher number of monomers are mostly found in the main haze atmosphere of Titan, at lower altitudes.
In fact, some aerosols of the detached haze layer on Titan are large aggregates that grow within the main haze layer at lower altitudes and that are lift up back to the detached layer by ascending currents occurring in the summer hemisphere \citep{Rann:02,Lebo:09}. On Pluto, such mechanisms are not likely to occur because of the thin atmosphere, and the size of fractal particles, if formed, should be limited.
Consequently, we consider only a small fractal particle with a limited amount of monomers.
Fractal particles have a different optical behavior compared to spherical particles. As shown by the figure 10 in \citet{Lars:14}, the optical depth of a 1 $\mu$m fractal particle is strongly dependent on the considered wavelength and decreases from the UV to the near infrared, while the optical depth of a similar sized spherical particle remains quite constant with the wavelength.
One can use equation \ref{tau} to calculate the opacity of fractal particles with $Q_{ext}$ the aerosol extinction efficiency (referred to the equivalent mass sphere), $r_{\mbox{eff}}$ the equivalent mass sphere radius of the particle and $\rho_{aer}$ the density of the material (or density of the monomers).
Here we used a mean field model of scattering by fractal aggregates of identical spheres \citep{Bote:97,Rann:97} to estimate the extinction efficiency of fractal particles.
From the number of monomers N and the monomers radius $r_{m}$, on can calculate the equivalent mass sphere radius of the corresponding fractal particle, given by $R_{s}$ = N$^{\frac{1}{3}} \times r_{m}$. Using these parameters and the fractal dimension of the particle, the model computes $Q_{ext}$ by dividing the extinction cross section of the particle by the geometrical cross section of the equivalent mass sphere ($\pi$\,$R_{s}^{2}$).
Here we compare the opacities obtained in the reference simulations when considering spherical or fractal particles. We consider fractal particles composed of 50 nm monomers, with a fractal dimension equal to 2 and with a bulk radius of 100 nm and 232 nm (N=8 and N=100 monomers respectively).
The model gives an extinction efficiency Q$_{ext}$ of 4.1 in the UV and 0.49 in the visible wavelengths for the 100 nm fractal particle and 7.2 in the UV and 1.93 in the visible wavelengths for the 232 nm fractal particle. The resulting nadir opacities are summarized in \autoref{tabop_r} and limb opacities are shown on \autoref{opacity_limb_ref}.
The opacities obtained for fractal particles are higher than for spherical particles in the visible, with a factor of 1.3 for the 100 nm and 2.2 for the 232 nm particle but lower in the UV with a factor of 0.9 and 0.7 respectively for the 100 nm and the 232 nm particle. This is shown by \autoref{opacity_limb_ref}.
As shown in \autoref{tabop_r}, the visible nadir opacity obtained in the summer hemisphere are in the range of what is estimated from New Horizons observations (0.004-0.012, \citet{Ster:15,Glad:16}) in both the spherical and the 100 nm fractal cases, and in both reference simulations. Values of the 232 nm fractal case are outside the observational range. The case of fractal particles composed of 10 nm particles is discussed in Section \ref{sec:sensib_r}.
\subsubsection{Line of sight opacity profiles}
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.2]{figures/lineofsight_ref.pdf}
\caption{Line of sight opacity profiles obtained with the GCM for the spherical and fractal cases, at the ingress (-163\char23E, 17\char23S, solid lines) and egress point (16\char23E, 15\char23N, dashed lines) of Pluto's solar occultation, for the reference simulation without (top) and with (bottom) South Pole N$_{2}$ condensation. Left and right are the results in UV and VIS wavelength respectively. The red curve is the reference simulation with 50 nm spherical particles. The blue and green curves correspond to the fractal cases with R$_{s}$=100 nm / N=8 and R$_{s}$=232 nm / N=100 respectively. \newline}
\label{opacity_limb_ref}
\end{center}
\end{figure}
\autoref{opacity_limb_ref} shows the line of sight opacity profiles in the UV and in the visible wavelengths obtained for both reference simulations at the ingress and the egress points of Pluto's solar occulation by New Horizons. The profiles are computed using an onion peeling method and considering that the line of sight only crosses one GCM atmospheric column.
Generally speaking, few differences are obtained between both reference simulations. The difference of opacity between the egress point (which is above the equator at the latitude 15\char23N) and the ingress point (which is below the equator at the latitude 17\char23S) are larger for the simulation without South Pole N$_{2}$ condensation, because of the higher haze density in the summer hemisphere shown in \autoref{hazesection}.
\newpage
\subsection{Sensitivity studies}
\label{sec:sensib}
The poor constraint on haze properties on Pluto gives us a flexibility to explore further other scenarios for Pluto's haze. In this section, the haze parametrization is tested with different precursor lifetimes and sedimentation radius. We also discuss the possible values for K$_{CH4}$ in the parametrization.
One objective is to investigate if another set of haze parameters can cause a more realistic aerosol distribution and concentration in the sunlit equatorial and summer atmosphere, compared to the observations.
In addition, the sensitivity study aims to “bracket” the reality of Pluto's haze by analyzing extreme cases and compare them to both reference simulations.
First, it has been checked that the haze production is insensitive to the amount of CH4 present in the upper atmosphere. Although the amount of CH4 molecules decreases in the upper atmosphere due to the absorption of incident photons and photolysis reactions, this loss remains negligible compared to the total amount of CH4 in Pluto's atmosphere. In addition, the production of haze precursors still occurs at high altitudes above 100 km even for low values of CH$_{4}$ mixing ratio. The ratio between the production rate of precursors at 100 km and the rate at 220 km (top of the model) becomes higher than 1\% for a mean CH$_{4}$ mixing ratio of 0.04\%, which is one order of magnitude less than the typical values found on Pluto. This confirms that the reaction is photon-limited and that different (and realistic) CH4 mixing ratio will not impact haze production and distribution.
\subsubsection{Sensitivity to characteristic time for aerosol growth}
\label{sec:sensib_tau}
The characteristic time for aerosol growth, defined in Section \ref{conversion}, is challenging to estimate. Here we consider two possible extreme values in the model. If this time is set to 1 second, this means that precursors are instantaneously converted into haze aerosols in the upper atmosphere where CH$_{4}$ photolysis occurs. This remains acceptable since photolysis and photochemistry can actually occur at much higher altitudes above the model top. An upper value up to several terrestrial years seems reasonable considering the number of years simulated and will allow precursors to be more mixed in the entire atmosphere.
Here we compare simulation results obtained with different characteristic times for aerosol growth (\autoref{masshaze_tau} and \autoref{opacity_limb_tau}): 1 s (haze directly formed from photolysis reactions), 10$^{6}$ s (1.81 Pluto sols), 10$^{7}$ s (18.12 sols, reference simulations), 10$^{8}$ s (181.20 sols, that is about 3 terrestrials years). The rest of the settings remain similar to both reference simulations.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.3]{figures/sensib_tau/masshaze_tau_all.pdf}
\caption{Zonal mean of column atmospheric mass of haze aerosols (kg\,m$^{-2}$) obtained for July 2015 with different times for aerosol growth $\tau$ (s), for the simulations without (solid lines) and with (dashed lines) South Pole N$_{2}$ condensation. \newline}
\label{masshaze_tau}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.2]{figures/opacity_limb_tau_vis.pdf}
\caption{Line of sight opacity profiles in VIS wavelength obtained with the GCM with different times for aerosol growth, at the ingress (-163\char23E, 17\char23S, solid lines) and egress point (16\char23E, 15\char23N, dashed lines) of Pluto's solar occultation, for the simulations without (left) and with (right) South Pole N$_{2}$ condensation \newline}
\label{opacity_limb_tau}
\end{center}
\end{figure}
In the simulations without South Pole N$_{2}$ condensation, using 1-10$^{7}$ s leads to similar column mass of haze, as shown by \autoref{masshaze_tau}. With a lifetime of 10$^{8}$ s, the precursors have enough time to be transported by the circulation induced by radiative heating from the summer to the winter hemisphere, and at lower altitudes. It results in a better dispersed haze density at all latitudes, a lower mass in the summer hemisphere, and thus similar egress and ingress line of sight opacities, as shown on \autoref{opacity_limb_tau}.
In the simulations with South Pole N$_{2}$ condensation, the longer the precursor lifetime, the more they are transported by radiative heating towards the
winter hemisphere and by the descending circulation branch towards the surface of the winter polar cap. Thus, the haze tends to accumulate in the winter hemisphere and in lower amounts if long lifetimes are considered, and in the summer hemisphere in larger amounts otherwise.
The difference of opacity obtained between the egress and the ingress points is larger for low lifetimes and conversely, as shown on \autoref{opacity_limb_tau}.
\subsubsection{Sensitivity to particle radius}
\label{sec:sensib_r}
The uniform and constant radius of aerosol particles is a parameter that strongly controls the aerosol sedimentation and opacity in the GCM. As shown by equation \ref{stokes} in Section \ref{hazeproperties}, a smaller particle radius induce a lower haze sedimentation velocities and thus a higher mass of haze in the atmosphere.
Here we compare eight simulations: the reference simulations (50 nm particles, with and without condensation flow) and simulations performed with particle sizes of 10, 30 and 100 nm (with and without condensation flow). We compare the column atmospheric mass obtained (\autoref{masshaze_r}), the limb opacities (\autoref{opacity_limb_r}) and the nadir opacities (\autoref{tabop_r}). These simulations correspond to the four first lines of \autoref{tabop_r}. The six last lines of \autoref{tabop_r} show the nadir opacities obtained from the simulations with 10 nm and 50 nm particles, but considering fractal particles (four cases with 10 nm monomers and two cases with 50 nm monomers). Haze aerosol density is also shown for the simulation with condensation flow and with a particle radius of 10 nm (\autoref{hazesection_r10_cap}).
Aerosol particles with radii of 10, 30, 50 and 100 nm typically fall from 200~km down to the surface in 1110, 370, 220 and 111 Earth days respectively. Basically, this corresponds to the time needed to reach an equilibrated mass of haze in the atmosphere.
As shown by \autoref{masshaze_r}, the latitudinal mass distribution is not impacted by the considered size of the particle. The column mass of haze is driven by the sedimentation velocity and the mass ratios correspond to the particle size ratios. This is also shown by \autoref{evolhaze}.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.3]{figures/sensib_r/masshaze_r_all.pdf}
\caption{Zonal mean of column atmospheric mass of haze aerosols (kg\,m$^{-2}$, log scale) obtained with different particle radii, for the simulations without (solid lines) and with (dashed lines) South Pole N$_{2}$ condensation.}
\label{masshaze_r}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.2]{figures/opacity_limb_r_vis.pdf}
\caption{Line of sight opacity profiles in VIS wavelength obtained with the GCM for different spherical particle radii, at the ingress (-163\char23E, 17\char23S, solid lines) and egress point (16\char23E, 15\char23N, dashed lines) of Pluto's solar occultation, for the simulations without (left) and with (right) South Pole N$_{2}$ condensation \newline}
\label{opacity_limb_r}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.2]{figures/hazesection_cap10all.pdf}
\caption{Zonal mean latitudinal section of haze aerosol density (g\,cm$^{-3}$) obtained with the simulation for July 2015 with condensation flow and a particle radius of 10 nm (color bar in log scale). The right panel correspond to a zoom in the lowest 15 km above the surface.\newline}
\label{hazesection_r10_cap}
\label{lastfig}
\end{center}
\end{figure}
As shown by \autoref{tabop_r} and \autoref{opacity_limb_r}, the nadir and limb opacities remain in the same order of magnitude for the simulations performed with different particle radii. Lower opacities are obtained with a particle radius of 30 nm. We also investigated nadir opacities for fractal particles with a bulk radius of 22, 46, 100 and 200 nm, respectively composed of 10, 100, 1000 and 8000 monomers of 10 nm radius. As discussed in Section \ref{background}, the 200 nm fractal particle is the best hypothesis for the particle shape and size in order to fit the observations. Here we find that the nadir visible opacities obtained in this case are higher than the upper observational limit (see \autoref{tabop_r}). Realistic values are obtained for the other smaller particles.
\autoref{opacity_limb_r} show the line of sight visible opacities obtained for different spherical particle radii. Generally speaking, the profiles have similar shapes because changing the particle radius does not affect the haze distribution but only the mass of haze in the atmosphere, due to the change of sedimentation velocity.
However, for 10 nm particles, the opacities at ingress are significantly higher than at egress below 50 km, which is not the case for higher radii.
This is because the particles are lighter and have more time to be transported by the circulation towards the winter hemisphere before sedimentation to the surface. Thus, the change of haze distribution due to the condensation flow below 50 km altitude is more pronounced for this 10 nm case. This is highlighted by \autoref{hazesection_r10_cap} which shows the 10 nm haze particles density in the simulation with condensation flow. In the first kilometers above the surface, a peak of density is obtained at the South Pole. In addition, above 2 km altitude, the haze also accumulates at the North Pole, pushed away by the condensation flow.
\begin{table}[h]
\begin{center}
\begin{tiny}
\begin{tabular}{m{1.5cm}m{0.5cm}m{0.8cm}m{0.8cm}m{1.5cm}m{1cm}m{1cm}m{1.5cm}m{1cm}m{1cm}}
\hline
& & & & \multicolumn{3}{c}{Without winter polar cap} & \multicolumn{3}{c}{With winter polar cap} \\
\hline
Radius & N$_{m}$ & Q$_{ext}$ UV & Q$_{ext}$ VIS & Aerosol mass (g\,cm$^{-2}$) & UV opacity & VIS opacity & Aerosol mass (g\,cm$^{-2}$) & UV opacity & VIS opacity \tabularnewline
r = 10 nm & 1 & 0.35 & 0.007 & $9.5-18\times10^{-7}$ & 0.31-0.59 & 0.0062-0.012 & $4.9-7.8\times10^{-7}$ & 0.16-0.26 & 0.0032-0.0051 \\
r = 30 nm & 1 & 1.54 & 0.05 & $3.0-6.5\times10^{-7}$ & 0.14-0.31 & 0.0047-0.010 & $2.5-3.4\times10^{-7}$ & 0.12-0.17 & 0.0039-0.0053 \\
r = 50 nm (reference) & 1 & 2.29 & 0.19 & $1.8-3.9\times10^{-7}$ & 0.077-0.17 & 0.0064-0.014 & $1.5-2.0\times10^{-7}$ & 0.064-0.086 & 0.0053-0.0071 \\
r = 100 nm & 1 & 2.67 & 1.01 & $0.9-1.9\times10^{-7}$ & 0.023-0.048 & 0.0085-0.018 & $0.75-1.1\times10^{-7}$ & 0.019-0.028 & 0.0071-0.010 \\
R$_{s}$ = 22 nm r~=~10~nm& 10 & 0.84 & 0.018 & $9.5-18\times10^{-7}$ & 0.34-0.64 & 0.0073-0.014 & $4.9-7.8\times10^{-7}$ & 0.18-0.28 & 0.0038-0.0060 \\
R$_{s}$ = 46 nm r~=~10~nm& 100 & 2.06 & 0.052 & $9.5-18\times10^{-7}$ & 0.40-0.76 & 0.010-0.019 & $4.9-7.8\times10^{-7}$ & 0.21-0.33 & 0.0052-0.0083 \\
R$_{s}$ = 100 nm r~=~10~nm& 1000 & 4.65 & 0.15 & $9.5-18\times10^{-7}$ & 0.41-0.78 & 0.013-0.025 & $4.9-7.8\times10^{-7}$ & 0.21-0.34 & 0.0069-0.0110 \\
R$_{s}$ = 200 nm r~=~10~nm& 8000 & 9.44 & 0.38 & $9.5-18\times10^{-7}$ & 0.42-0.80 & 0.017-0.032 & $4.9-7.8\times10^{-7}$ & 0.22-0.35 & 0.0087-0.0139 \\
R$_{s}$ = 100 nm r~=~50~nm& 8 & 4.10 & 0.49 & $1.8-3.9\times10^{-7}$ & 0.069-0.15 & 0.0083-0.018 & $1.5-2.0\times10^{-7}$ & 0.058-0.077 & 0.0069-0.0092 \\
R$_{s}$ = 232 nm r~=~50~nm& 100 & 7.20 & 1.93 & $1.8-3.9\times10^{-7}$ & 0.052-0.11 & 0.014-0.030 & $1.5-2.0\times10^{-7}$ & 0.044-0.058 & 0.0117-0.0156 \\
\hline
\end{tabular}
\caption{Haze aerosol opacities obtained at nadir in the summer hemisphere in the GCM, for four particle radii and for both climate scenarios with and without South Pole N$_{2}$ condensation. The time for aerosol growth used is $10^{7}$ s. The particles with a number of monomers N$_{m}$ equal to 1 are spherical particles, otherwise they are fractal particles (R$_{s}$ is the bulk radius, $r$ is the monomer radius). The first four fractal particles are composed of 10 nm monomers, and the last two are composed of 50 nm monomers. \newline}
\label{tabop_r}
\label{lasttable}
\end{tiny}
\end{center}
\end{table}
\newpage
\subsubsection{Sensitivity to the mass of aerosols}
\label{sec:sensib_mass}
The haze production rate used in the reference simulations corresponds to an optimal scenario where the photolysis of one molecule of CH$_{4}$ gives one carbon atom available for the production of haze ($K_{CH4}$=1). However, the carbon atoms collected from CH$_{4}$ photolysis may form different gaseous species and slow down tholins production. As an example, \citet{McKa:01} suggest that the tholins production is about 25 less than the photolysis rate of methane. Therefore, lower values of $K_{CH4}$ remain possible and would lead to a decrease of aerosol mass and thus of opacity.
\section{Summary}
The parametrization of haze aerosols in the Pluto GCM consists of several steps: the photolysis of methane by the solar and IPM flux, the creation of haze precursors and their transport in the atmosphere, the conversion of precursors to haze aerosols and the sedimentation of the aerosols.
The haze parametrization has been tested with 50 nm particles, a time for aerosol growth of 10$^{7}$ s, and for the two climate scenarios described in \citet{Forg:16}: with and without South Pole N$_{2}$ condensation (reference simulations). The sensitivity of the model to other particle sizes and times for aerosol growth has been explored.
Results show that the CH$_{4}$ photolysis occurs at all latitudes, with a maximum rate at high northern latitudes and around 250 km in altitude. In all simulations, the haze extends to high altitudes, comparable to what has been observed by New Horizons. From 200 km altitude upwards, the density decreases with the altitude by one order of magnitude every 100 km, leading to a density scale height of typically 40 km above 60 km altitude. This is comparable to the typical haze brightness scale height of 50 km observed by New Horizons \citep{Glad:16}.
Without South Pole N$_{2}$ condensation, the meridional atmospheric circulation is dominated by the radiative heating but remains weak, even in the first kilometers above the surface. The haze precursors remains at high altitudes and in larger amount at high northern latitudes. This leads to a higher density of haze in the summer hemisphere, decreasing with the latitudes.
With South Pole N$_{2}$ condensation, the circulation is also weak in the upper atmosphere, except above the South Pole where a descending branch of air driven by the condensation of N$_{2}$ transports the precursors to lower altitudes. This leads to a distribution of haze latitudinally more homogeneous with a slight peak of haze density above the South Pole. This peak is reiforced by the circulation in the first kilometers above the surface, which is more intense and able to move light aerosols from the northern hemisphere towards the South Pole.
In both climate scenarios, because of the generally weak meridional circulation, the computed mean atmospheric column mass of haze remains similar, and primarily depends on the sedimentation velocity and thus on the pressure and the considered monomer radius. In our simulations, the initial flux of Lyman-$\alpha$ at Earth remains constant between 1990 and 2015, but even if we consider the variable initial flux of Lyman-$\alpha$, the flux of Lyman-$\alpha$ at Pluto remains relatively constant. Consequently, the mean column mass of haze follows the trend in surface pressure, that is an increase by a factor of 3 between 1990 and 2015. Haze particles with a small radius remain longer in the atmosphere before reaching the surface. In our simulations, the sedimentation fall of 10 nm particles lasts about 3 terrestrial years, which could be enough time to form fractal aggregates.
The mean column atmospheric mass of haze on Pluto is difficult to assess because it depends on many parameters. First, it is depending on the photolysis rate and the complex recombinations of carbon and nitrogen atoms. The parametrization uses K$_{CH4}$ and K$_{N}$ equal to 1 and 1.5 to take these mechanisms into account. However, the production could be overestimated. In fact, New Horizons detected the presence of C$_{2}$H$_{2}$, C$_{2}$H$_{4}$ and maybe other carbon-based gas in Pluto atmosphere, which suggests another pathway for carbon atoms formed by CH$_{4}$ photolysis. In addition, HCN has been detected, and the irreversible nature of its formation may lead to less nitrogen atoms available for the haze formation.
The column mass of haze also strongly depends on the sedimentation radius of the haze particle, and to a lesser extent on the lifetime of the haze precursors.
However, we computed the UV and VIS opacities of the haze as a diagnostic of our simulation results and in all simulation cases, the column visible opacities have similar values (same order of magnitude) around 0.001-0.01, and slightly higher values when considering large fractal particles. This is because the extinction factor of smaller particles is lower but is compensated by a larger mass of haze. These opacities are in the range of what has been estimated on Pluto, that is 0.003-0.012 \citep{Glad:16,Ster:15}, and thus suggest an acceptable order of magnitude for the mass of haze obtained.
Comparing the haze distribution (obtained with and without South Pole N$_{2}$ condensation) with the observations (made by imaging with the instruments Ralph/MVIC and LORRI and by UV occultation with the Alice spectrometer) can help to reveal the presence or the absence of N$_{2}$ ice at the South Pole. A latitudinally homogeneous haze density with a slight peak above the North and particularly above the South Pole is typical of our simulation with South Pole N$_{2}$ condensation. Conversely, simulations without South Pole N$_{2}$ condensation show a more extensive haze in the summer hemisphere.
Comparing the line of sight opacity profiles at the egress and the ingress points can also help to distinguish both cases. The opacity at the egress point is at least twice the opacity at the ingress point in the case without South Pole N$_{2}$ condensation, and no significant difference is obtained in the case without.
However, a latitudinally homogeneous haze density can also be the results of a long characteristic time for precursors growth (several terrestrial years), that allows precursors to be transported towards southern latitudes by radiative heating and meridional circulation.
Finally, another way to distinguish both cases is to compare the haze distribution in the first kilometers above the surface. \autoref{hazesection_r10_cap} shows that the condensation flow induced by the presence of N$_{2}$ ice in the winter hemisphere leads to a lack of haze above the surface in the summer hemisphere, and an accumulation of haze between 3 and 20 kilometers in the winter hemisphere, which is more pronounced for small particle radii.
Although the simulations were done with uniform particle sizes, in reality the haze particle size may be locally distributed and vary in space and time, especially in the vertical. Thus it may be more realistic to consider a distribution of haze particle sizes, in order to take into account the gravitational segregation.
Compared to the uniform size case, if 10 nm spherical particles in the upper atmosphere become fractal particles in the lower atmosphere, with same monomer radius, then there will be a change in opacity but not in haze vertical distribution (because the sedimentation velocity remains the same). If 10 nm spherical particles grow up to 100 nm during their fall down towards the surface, then the sedimentation velocity of the particle would change. The increase of the particle size during the fall would compensate the increase of atmospheric pressure and lead to a more homogeneous haze density with altitude. In addition, at the altitudes where transitions of particle size occur, layers of haze could form.
\ack
We wish to thank P. Rannou and E. Lellouch for helpful conversations and suggestions. We are grateful to P. Rannou for supplying a mean model of absorption and scattering by fractal particles of identical spheres. The authors thank the NASA \textit{New Horizons} team for their excellent work on a fantastic mission and their interest in this research.
\label{lastpage}
\bibliographystyle{plainnat}
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1,116,691,497,613 | arxiv | \section*{Notes}}
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\section{Introduction}
Reasoning about knowledge and common knowledge has been shown to be
widely applicable in distributed computing, AI, and game theory. (See
\cite{FHMV} for numerous examples.)
Complete axioms for reasoning about knowledge and common knowledge are
well known in the case of a fixed finite set of agents. However, in
many applications, the set of agents is not known in advance and
has no {\em a priori\/} upper bound (think of software agents on the web
or nodes on the Internet, for example); it is often easiest to model the
set of agents
as an infinite set.
Infinite sets of agents also arise in game theory and economics
(where
reasoning about knowledge and common knowledge is quite standard; see,
for example, \cite{Au,Gea94}).
{F}or example, when analyzing a game played with two teams,
we may well want to say that everyone on team 1 knows that
everyone on team 2 knows some fact $p$, or that it is common knowledge
among the agents
on
team 1 that $p$ is common knowledge among the agents
on
team 2. We would want to say this even if the teams consist of
infinitely many agents.
Since economies are often modeled
as consisting of infinitely many (even uncountably many) agents, this
type of situation arises when economies are viewed as teams in a
game.
The logics for reasoning about the knowledge of groups of agents contain
modal operators $K_i$ (where $K_i \phi$ is read ``agent $i$ knows
$\phi$''), $E_G$ (where $E_G \phi$ is read ``everyone in group $G$ knows
$\phi$''), and $C_G$ (where $C_G \phi$ is read ``$\phi$ is common
knowledge among group $G$'').
The operators $E_G$ and $C_G$ make perfect sense even if we allow the
sets $G$ to be infinite---their semantic definitions remain unchanged.
If the set of agents is finite, so that,
in particular, $G$ is finite, there is a simple axiom connecting $E_G
\phi$ to $K_i \phi$, namely,
$E_G \phi \Leftrightarrow \land_{i \in G} K_i \phi$.
Once we allow infinite groups $G$ of agents, there is no
obvious analogue for this axiom. Nevertheless, in this paper,
we show that there exist natural sound and complete axiomatizations for
reasoning about knowledge and
common knowledge even if there are infinitely many agents.
It is also well known that if there are finitely many agents, then there
is a decision procedure that decides if a formula $\phi$ is satisfiable
(or valid)
that runs in time exponential in $|\phi|$, where $\phi$ is the length of
the formula viewed as a string of symbols.
We prove a similar result for a language with
infinitely
many
agents. However,
two issues arise (that, in fact, are also relevant
even if there are only finitely many agents, although they have not been
considered before):
\begin{itemize}
\item
In the statement
of the complexity result in \cite{FHMV}, $E_G$ and $C_G$ are both viewed
as having length $2 + 2|G|$
(where $|G|$ is the cardinality of $G$).
Clearly we cannot use this definition here if we want to get interesting
complexity results,
since $|G|$ may be infinite. Even if we restrict
our attention
to finite sets $G$,
we would like a decision procedure that treats these sets in a uniform
way,
independently
of their cardinality.
Here we view $E_G$ as having
length 1 and $C_G$ as having length 3, independent of the cardinality of
$G$.
(See, for example, the proof of Proposition~\ref{Shore}
for the role of
independence and the definition of $Sub(\phi)$ in the proof
of Theorem~\ref{dec}
for an indication as to why $C_G$ has length
3 rather than $1$.)
Even with this definition of length, we prove that the complexity
of the satisfiability problem is still essentially exponential time.
(We discuss below what ``essentially'' means.)
Thus our results improve previously-known results even if there are
only finitely many agents.
\item
In the earlier proofs, it is implicitly
assumed
that the sets
$G$ are presented in such a way that there is no difficulty in testing
membership in $G$. As we show here, in order to decide if certain
formulas are
satisfiable, we need to be able to test if certain subsets of agents of
the form $G_0 - (G_1 \cup \ldots \cup G_k)$ are empty, where $G_0,
\ldots, G_k$ are sets
of agents.
In fact, if we are interested in a notion of knowledge that
satisfies {\em positive introspection\/}---that is, if agent $i$ knows
$\phi$, then she knows that she knows it---then we also
must be able to check whether such subsets are singletons.
And if we are interested in a notion of knowledge that
satisfies {\em negative introspection\/}---that is, if agent $i$
does not know
$\phi$, then she knows that she does not know it---then
we
must be able to
check whether such subsets have cardinality
$m$, for certain finite $m$. The difficulty of deciding these
questions depends in part on how $G_0,
\ldots, G_k$ are presented and which sets of agents we can talk about in
the language. For example, if $G_0, \ldots, G_k$ are
recursive sets, deciding if $G_0 - (G_1 \cup \ldots \cup G_k)$ is
nonempty may not even be recursive. Here, we
provide a
decision procedure for satisfiability that runs in time exponential in
$|\phi|$ provided
that we have oracles for testing appropriate properties of sets of the
form $G_0 - (G_1 \cup \ldots \cup G_k)$. Moreover, we show that
any decision procedure
must
be able to answer the questions
we ask.
In fact, we actually prove a
stronger result, providing a tight bound
on the complexity of deciding satisfiability that takes into account the
complexity of answering questions about the cardinality of $G_0 - (G_1
\cup \ldots \cup G_k)$.
Again, this issue is of significance even if there are only finitely
many agents. For example, in the SDSI approach to security \cite{RL96},
there are names, which can be viewed as representing sets of agents.
SDSI provides a (nondeterministic) algorithm for computing
the set of agents represented by a name. If we want to make statements
such as ``every agent represented by name ${\tt n}$ knows $\phi$''
(statements that we believe will be useful in reasoning about
security \cite{HM99,HMS}) then the results of this paper show that to
decide validity in the resulting logic, we need more than just an
algorithm for
resolving the agents represented by a
given name. We also need
algorithms for resolving
which agents are
represented by one name and not
another. More generally, if we assume that we have a separate language
for representing sets of agents, our results characterize the properties
of sets that we need to be able to decide in order to reason about the
group knowledge of these agents.
\end{itemize}
In the next section, we briefly review the syntax and semantics of
the logic of common knowledge.
In Section~\ref{simplified} we state the main results and prove them
under some simplifying assumptions that allow us to bring out the main
ideas of the proof.
We drop these assumptions in Section~\ref{proofs}, where we provide
the proofs of the full results.
\section{Syntax and Semantics: A Brief Review}\label{review}
\paragraph{Syntax:} We start with a (possibly infinite) set ${\cal A}$ of
agents. Let ${\cal G}$ be a set of nonempty subsets of ${\cal A}$. (Note that we
do not require ${\cal G}$ to be closed under union, intersection, or
complementation; it can be an arbitrary collection of subsets.)
We get the language ${\cal L}_{\G}^C(\Phi)$ by starting with a set $\Phi$ of
primitive propositions, and closing under $\land$, $\neg$, and the modal
operators $K_i$,
for
$i \in{\cal A}$, and $E_G, C_G$, for $G \in {\cal G}$. Thus, if
$p, q
\in\Phi$, $i \in {\cal A}$, and $G, G' \in{\cal G}$, then $K_i C_G (p \land E_{G'}
q) \in {\cal L}_{\G}^C(\Phi)$.
Let ${\cal L}_{\G}^E$ be the sublanguage of
${\cal L}_{\G}^C$ that does not include the $C_G$ operators.
Let $|\phi|$ be the length of the formula viewed as a string of
symbols, where the modal operators $K_i$ and $E_G$ are counted as having
length 1 and $C_G$ is
counted as having length 3 (even if $G$ is an infinite set of agents)
and all primitive propositions are counted as having length 1.
In \cite{FHMV,HM2}, ${\cal A}$ is taken to be the set $\{1,\ldots,n\}$;
in \cite{HM2}, ${\cal G}$ is taken to be the singleton
$\{\{1,\ldots,n\}\}$
(so that we can only talk about every agent in ${\cal A}$ knowing $\phi$
and common knowledge among
the agents in ${\cal A}$), while in
\cite{FHMV}, ${\cal G}$ is taken to consist of all nonempty subsets of ${\cal A}$.
\paragraph{Semantics:} As usual, formulas in ${\cal L}_{\G}^C$
are either true or false at a world in a Kripke structure. Formally, a
Kripke structure $M$ over ${\cal A}$ and $\Phi$
is a tuple $(S, \pi, \{{\cal K}_i: i \in {\cal A}\})$, where $S$ is a set of states
or possible worlds, $\pi$ associates with each state in $S$ a truth
assignment to the primitive propositions in $\Phi$ (so that $\pi(s):
\Phi \rightarrow \{{\bf true},{\bf false}\}$), and ${\cal K}_i$ is a binary
relation on $S$ for each agent $i \in {\cal A}$. We occasionally write
${\cal K}_i(s)$ for $\{t: (s,t) \in {\cal K}_i\}$.
We define the truth relation $\models$ as follows:
\begin{description}
\item[]
$(M,s)\models p$ (for $p\in\Phi$) iff $\pi(s)(p)={\bf true}$
\item[]
$(M,s)\models\varphi\land \psi$ iff both $(M,s)\models \varphi$ and
$(M,s)\models \psi$
\item[]
$(M,s)\models \neg \varphi$ iff $(M,s)\not\models \varphi$
\item[]
$(M,s)\models K_i\varphi$ iff $(M,t)\models \varphi$ for all $t \in
{\cal K}_i(s)$
\item[]
$(M,s) \models E_G \phi$ iff $(M,s) \models K_i \phi$ for all $i \in G$
\item[]
$(M,s) \models C_G \phi$ iff $(M,s) \models E_G^k \phi$ for $k = 1, 2, 3,
\ldots$, where $E_G^k$ is defined inductively by taking
$E_G^1\phi \eqdef E_G \phi$ and $E_G^{k+1} \phi \eqdef
E_G E_G^k \phi$.
\end{description}
We say that
$t$ is {\em $G$-reachable from $s$ in $M$\/} if
there exist
$s_0, \ldots, s_k$ with $s = s_0$, $t= s_k$, and $(s_i, s_{i+1}) \in
\cup_{i \in G}{\cal K}_i$. For later use, we extend this
definition so that if $S' \subseteq S$, we say that $t$ is
{\em $G$-reachable from $s$ in $S'$\/} if $s_0, \ldots, s_k \in S'$.
The following characterization of common knowledge is well known
\cite{FHMV}.
\begin{lemma}\label{G-reach} $(M,s) \models C_G \phi$ iff $(M,t) \models \phi$ for all
$t$
that
are $G$-reachable from $s$ in $M$.
\end{lemma}
Let ${\cal M}_{\cal A}(\Phi)$\glossary{\glosmznap}
be the class of all Kripke
structures over ${\cal A}$ and $\Phi$ (with no restrictions on the
${\cal K}_i$ relations).
We are also interested in
various subclasses of ${\cal M}_{\cal A}(\Phi)$, obtained by restricting the ${\cal K}_i$
relations.
In particular, we consider
${\cal M}_n^r(\Phi)$, $\M_n^{\mbox{\scriptsize{{\it rt}}}}(\Phi)$, $\M_n^{\mbox{\scriptsize{{\it rst}}}}(\Phi)$, and
$\M_n^{\mbox{\scriptsize{{\it elt}}}}(\Phi)$,
the class of all structures over ${\cal A}$ and $\Phi$ where the ${\cal K}_i$
relations are reflexive (resp.,\ reflexive
and transitive; reflexive, symmetric, and transitive; Euclidean,%
\footnote{Recall that a relation $R$ is Euclidean if $(s,t), (s,u) \in
R$ implies that $(t,u) \in R$.}
serial, and transitive).
For the remainder of this paper, we
take $\Phi$ to be fixed, and do not mention it, writing, for example
${\cal L}_{\G}^C$ and ${\cal M}_n$ rather than ${\cal L}_{\G}^C(\Phi)$ and ${\cal M}_n(\Phi)$.
As usual, we define a formula to be {\em valid in a class ${\cal M}$\/} of
structures
if $(M,s) \models \phi$ for all $M \in{\cal M}$ and all states $s$ in $M$;
similarly, $\phi$ is {\em satisfiable in ${\cal M}$\/} if $(M,s) \models \phi$
for some $M \in {\cal M}$ and some $s$ in $M$.
\paragraph{Axioms:}
The following are the standard axioms
and rules
that have been considered for
knowledge;
They hold for all $i \in {\cal A}$.
\begin{description}
\item[Prop.]
All substitution instances of tautologies of propositional calculus.
\item[K1.]
$(K_i\varphi\land K_i(\varphi\Rightarrow \psi)) \Rightarrow
K_i\psi$
\item[K2.] $K_i \phi \Rightarrow \phi$
\item[K3.] $\neg K_i \mbox{{\it false}}$
\item[K4.] $K_i \phi \Rightarrow K_i K_i \phi$
\item[K5.] $\neg K_i \phi \Rightarrow K_i \neg K_i \phi$
\item[MP.]
{F}rom $\varphi$ and $\varphi\Rightarrow\psi$
infer $\psi$
\item[KGen.]
{F}rom $\varphi$ infer $K_i\varphi$
\end{description}
Technically, Prop and K1--K5 are axiom schemes, rather than single
axioms.
K1, for example, holds for all formulas $\phi$ and $\psi$. A formula
such as $K_1 q \lor \neg K_1 q$ is an instance of axiom Prop
(since it is a substitution instance of the propositional tautology
$p \lor \neg p$, obtained by substituting $K_1 q$ for $p$).
We will be interested in the following axioms and rule
for reasoning about everyone knows,
which hold for all $G \in {\cal G}$.
\begin{description}
\item[E1.] $E_G \phi \Rightarrow K_i \phi$ if $i \in G$
\item[E2.] $(\land_{i \in {\cal A}'} K_i \phi \land \land_{G' \in {\cal G}'}
E_{G'} \phi)
\Rightarrow E_{G} \phi$ if ${\cal A}'$ is a finite subset of ${\cal A}$, ${\cal G}'$ is a
finite subset of ${\cal G}$,
and $G \subseteq
({\cal A}' \cup (\cup {\cal G}'))$.
\item[E3.]
$(E_G\varphi\land E_G(\varphi\Rightarrow \psi)) \Rightarrow
E_G\psi$.
\item[E4.] $E_G (E_G \phi \Rightarrow \phi)$.
\item[E5.] $E_G \phi \Rightarrow \phi$.
\item[E6.] $\neg \phi \Rightarrow E_G \neg E_G \phi$.
\item[E7.] From $\neg(\phi_1 \land \ldots \land \phi_k)$ infer
$\neg(E_{G_1} \phi_1 \land \ldots \land E_{G_k} \phi_k)$ if $G_1 \cap
\ldots \cap G_k \ne \emptyset$.
\item[EGen.] From $\varphi$ infer $E_G\varphi$.
\end{description}
E2 can be viewed as a generalization of the axiom
$E_G \phi \Rightarrow E_{G'} \phi$ if $G' \subseteq G$ (of which E1 is a
special case if we identify $K_i \phi$ with $E_{\{i\}} \phi$, as we
often do in the paper). Essentially it says that if $K_i \phi$ hold for
all agents $i \in G$ (and perhaps some other agents $i \notin G$) then
$E_G \phi$ holds. Since, if $G$ is infinite, we cannot write the
infinite conjunction of $K_i \phi$ for all $i \in G$, we approximate
as well as we can within the constraints of the language. As long as
$E_{G'} \phi$ and $K_i \phi$ holds for sets $G'$ and agents $i$ whose
union contains $G$, then certainly $E_G \phi$ holds.
If ${\cal A}$ is finite (so that all the sets in ${\cal G}$ are finite)
we can simplify E1 and E2 to
\begin{description}
\item[E.] $E_G \phi \Leftrightarrow \land_{i \in G} K_i \phi$.
\end{description}
It is easy to see that E follows from E1 and E2 (in the presence of
Prop and MP) and every instance of E1 and E2 follows from E if ${\cal A}$ is
finite. E is used instead of E1 and E2 in \cite{FHMV,HM2}. Note that
E2 is recursive iff deciding if $G - ({\cal A}' \cup (\cup {\cal G}')
) = \emptyset$ is recursive.
(We
determine precisely which such questions we must be able to
answer in Proposition~\ref{reduction}.)
E3 and EGen are the obvious analogues of K1 and KGen for $E_G$. We do
not need them in the case that ${\cal A}$ is finite; it is easy to see
that they follow from K1, KGen, and E. In the case that ${\cal A}$ is
infinite, however, they are necessary.
Axiom E4 is sound in ${\cal M}_n^r$, $\M_n^{\mbox{\scriptsize{{\it rt}}}}$, $\M_n^{\mbox{\scriptsize{{\it rst}}}}$, and $\M_n^{\mbox{\scriptsize{{\it elt}}}}$.
It is easy to see that E4 follows from K2, E1, and EGen, so will not be
needed in systems that contain these axioms. Moreover,
it is not hard to show that E4 follows from E1, E2, and K5 if the set of
agents is finite. However, it does not follow from these axioms if the
set of agents is infinite.
Axiom E5 follows from K2 and E1. Moreover, we use
it only in systems
that already include K2 and E1. Nevertheless, for
technical reasons, it is useful to
list it
separately.
Similarly, it is not hard to see that E7 is a derivable rule in any system
that includes Prop, MP, K1, K3, E1, E4, and EGen
(we prove this in Section~\ref{sec:elt}). While we use E7 only in such
systems, like E5, it is useful to list it separately.
Axiom E6 (with $E_G$ replaced by $K_i$) is the standard axiom
used to characterize symmetric ${\cal K}_i$ relations \cite{FHMV}. It follows
easily from K2, K5, E1, and E2 if ${\cal A}$ is finite. However, like E4,
it must be
specifically included
if ${\cal A}$ is infinite.
{F}inally, we have the following well-known axiom and inference rule for
common knowledge:
\begin{description}
\item[C1.] $C_G \phi\Rightarrow E_G(\phi \land C_G\phi$).
\item[RC1.]
{F}rom $\phi \Rightarrow E_G(\psi \land \phi)$ infer $\phi
\Rightarrow C_G\psi$
\end{description}
Historically,
in the case of one agent,
the system with axioms and rules Prop, K1, MP, and KGen has been
called~K; adding K2 to K gives us T; adding K4 to T gives
us S4; adding K5 to S4 gives us S5; replacing K2 by K3 in S5 gives us
KD45. We use the subscript ${\cal G}$ to emphasize the fact that we are
considering systems with sets of agents coming from ${\cal G}$ rather than
only one
agent and the superscript $C$ to emphasize that we add E1--E3, EGen, C1,
and RC1 to the system. In this way, we get the systems
${\rm K}_n^C$\glossary{\gloskc},
${\rm T}_n^C$\glossary{\glostnc}, and
${\rm S4}_n^C$\glossary{\glosszec}.
Thus,
${\rm K}_n^C$ consists of Prop, K1, MP, KGen, E1, E2, E3, EGen,
C1, and RC1; we get
${\rm S4}_n^C$
by adding
K2 and K4
to ${\rm K}_n^C$.
We get ${\rm KD45}_n^C$\glossary{\gloskdefc} by adding K3--K5 and E4
to ${\rm K}_n^C$ and we get
${\rm S5}_n^C$\glossary{\glosszfc} by adding
K2, K4, K5
and E6 to ${\rm K}_n^C$.
One of the two main results of this paper shows that each
of these axiom systems is sound and complete with respect to an
appropriate class of structures. For example, ${\rm K}_n^C$ is a sound and
complete axiomatization
with respect to ${\cal M}_n$
and ${\rm S5}_n^C$ is a sound and complete axiomatization with respect to
$\M_n^{\mbox{\scriptsize{{\it rst}}}}$. In the case that ${\cal A}$ is finite, this result is well known
(see \cite{FHMV,HM2}---as mentioned earlier, E is used in the
axiomatization instead of E1--E3 and EGen). What is perhaps surprising
is that
E1--E3 and EGen suffice even if ${\cal A}$ is infinite. For example, suppose
that ${\cal G}$ just
consists of the singleton ${\cal A}$. In that case, E2 becomes vacuous.
Thus, while the axioms force $E_{\cal A} \phi$ to imply
that each agent in ${\cal A}$ knows $\phi$, we have no way of expressing the
converse.
Indeed,
it is easy to construct a structure for the axioms with the
standard interpretations of all the $K_i$ relations but a
nonstandard one of $E_{\cal A}$, where all the agents in ${\cal A}$ know
$\phi$ and yet $E_{\cal A} \phi$ does not hold.
Consider, for example, a structure with a single state $s$ for the
language with an infinite set ${\cal A}$ of agents.
Suppose that every primitive proposition $p$ is true at $s$,
${\cal K}_i$ is empty for all $i \in {\cal A}$,
and
$K_i$ is interpreted in the usual way for all $i \in {\cal A}$ (so that $K_i
\phi$ is true at $s$ for all formulas $\phi$).
{F}or $E_{\cal A}$, however, we say
that $E_{\cal A} \phi$ holds at $s$ if and only if it is provable in, say,
${\rm K}_n^C$. Of course, there are obviously standard models in which $E_{\cal A}
p$ does not hold and so (by the soundness of the
axioms for standard interpretations) $E_{\cal A} p$ is not provable. Thus,
in this interpretation, $E_{\cal A} p$ does not hold at $s$ while
$K_i p$ does for every $i \in {\cal A}$. Finally, it is clear that all the
axioms of ${\rm K}_n^C$ are true in this structure.
Similar examples can be given to
show
that E4 and
E6 do not follow from the specified other axioms when the set of agents
is infinite.
\section{The Main Results and a Proof in a Simplified Setting}
\label{simplified}
In this section, we state
the two main results of this paper---complete
axiomatizations and decision procedures.
We then provide a proof of a simpler version of these results that
illustrates some of the main ideas. We first state the
completeness
results.
\begin{theorem}\label{complete} For formulas in the language ${\cal L}_{\G}^C$:
\begin{enumerate}
\item[(a)] ${\rm K}_n^C$ is a sound and complete axiomatization
with respect to ${\cal M}_n$,
\item[(b)] ${\rm T}_n^C$ is a sound and complete axiomatization
with respect to ${\cal M}_n^r$,
\item[(c)] ${\rm S4}_n^C$ is a sound and complete axiomatization with
respect to $\M_n^{\mbox{\scriptsize{{\it rt}}}}$,
\item[(d)] ${\rm S5}_n^C$ is a sound and complete axiomatization with
respect to $\M_n^{\mbox{\scriptsize{{\it rst}}}}$,
\item[(e)] ${\rm KD45}_n^C$ is a sound and complete axiomatization with respect
to $\M_n^{\mbox{\scriptsize{{\it elt}}}}$.
\end{enumerate}
\end{theorem}
Before stating the
results
regarding complexity,
we first show that questions about certain facts regarding
sets of the form $G_0 - (G_1 \cup \ldots \cup G_k)$ are reducible to
satisfiability.
We are not just interested in sets of the form $G_0 - (G_1 \cup \ldots
\cup G_k)$ for $G_1, \ldots, G_k \in {\cal G}$. For example, when dealing
with ${\cal M}^{rt}$, it turns out that we are interested in sets $H$ of this
form if $|H|=1$. But if $H_1$ is such a set, then we are also
interested in sets of the form $H_2 = G_0 - (G_1 \cup \ldots G_k
\cup H_1)$. And if $|H_2| = 1$, then we can also include $H_2$ in the
union, and so on. The following definition makes this precise.
\begin{definition}\label{Gm} Given a set ${\cal J}$ of subsets of ${\cal A}$ and
an integer $m \ge 1$, define a sequence ${\cal J}^m_0, {\cal J}^m_1,
\ldots$ of sets of subsets of
${\cal A}$ inductively as follows. Let ${\cal J}^m_0 = {\cal J}$. Suppose that we have
defined ${\cal J}^m_0, \ldots, {\cal J}^m_{k}$. Then
${\cal J}^m_{k+1} = {\cal J} \cup
\{G - \cup {\cal H}:
G \in {\cal J},\,
{\cal H} \subseteq
{\cal J}^m_{k},\,
{\cal H} \mbox{ finite}, \, |G - \cup
{\cal H}| \le m\}$.
Let ${\cal J}^m = \cup_i {\cal J}_i^m$; let $\widehat{{\cal J}}^m = \{G - \cup {\cal H}: G \in {\cal J}, {\cal H}
\subseteq {\cal J}^m, {\cal H}$ finite$\}$. For uniformity, we take $\widehat{{\cal J}}^0 =
\{G - \cup {\cal H}: G \in {\cal J}, {\cal H} \subseteq {\cal J}, {\cal H}$ finite$\}$.
\vrule height7pt width4pt depth1pt\end{definition}
Let ${\cal J}^*$ be the algebra generated by ${\cal J}$ (that is, the Boolean
combinations
of sets in ${\cal J}$).
It is useful to talk about the length of a description of various
sets in ${\cal J}^*$ (particularly those in $\widehat{{\cal J}}^m$ for some $m$). Formally,
we assume we have a language
whose primitive objects
consist
of the elements of ${\cal J}$ and the symbols
$\cup$ and $-$ (for set difference). The length of a
description is then the number
of symbols of ${\cal J}$ that appear in it. Notice that, in general, an
element of ${\cal J}^*$ may have several different descriptions. We
are not always careful to distinguish a set from its description. (We
hope that the reader will be able to tell which is intended from
context.) We use $l(G)$ to denote the length of
the description of $G \in {\cal J}^*$.
Let ${\cal G}_{\cal A} = {\cal G} \cup \{\{i\}: i \in {\cal A}\}$. Throughout the paper
(and, in particular, in the proof of the next proposition),
for ease of exposition, we identify
$E_{\{i\}}$ with $K_i$, for $i \in {\cal A}$
(which allows us to write $E_G$ for each $G \in {\cal G}_{\cal A}$).
\begin{proposition}\label{reduction}
\begin{itemize}
\item[(a)] The question of whether $|G| > 0$ for $G \in \widehat{{\cal G}}_{\cal A}^0$
is
reducible (in time linear in
$l(G)$) to the
satisfiability problem for the language ${\cal L}_{\G}^E$
with respect to all of ${\cal M}_n$, ${\cal M}_n^r$, $\M_n^{\mbox{\scriptsize{{\it rt}}}}$,
$\M_n^{\mbox{\scriptsize{{\it rst}}}}$, and $\M_n^{\mbox{\scriptsize{{\it elt}}}}$.
\item[(b)]
The
questions of whether $|G| >0$ and $|G| > 1$ for $G \in
\widehat{{\cal G}}_{\cal A}^1$
are
each
reducible (in time linear in
$l(G)$) to the satisfiability
problem for the language ${\cal L}_{\G}^E$
with respect to all of $\M_n^{\mbox{\scriptsize{{\it rt}}}}$, $\M_n^{\mbox{\scriptsize{{\it rst}}}}$, and $\M_n^{\mbox{\scriptsize{{\it elt}}}}$.
\item[(c)]
{F}or
all $m \ge 1$, the question of whether
$|G| > m$ for $G \in \widehat{{\cal G}}_{\cal A}^m$ is reducible
(in time
linear in $l(G) + m$) to
the satisfiability problem for ${\cal L}_{\G}^E$ with respect to $\M_n^{\mbox{\scriptsize{{\it rst}}}}$ and
$\M_n^{\mbox{\scriptsize{{\it elt}}}}$.
\item[(d)]
The
question of whether $|G_1 \cap \ldots \cap G_k| >
0$, for $G_1, \ldots, G_k \in {\cal G}_{\cal A}$
is reducible (in time linear in $k$) to the satisfiability problem
for ${\cal L}_{\G}^E$ with respect to $\M_n^{\mbox{\scriptsize{{\it elt}}}}$.
\end{itemize}
\end{proposition}
\noindent{\bf Proof:} For part (a), suppose that $G \in \widehat{{\cal G}}_{\cal A}^0$. Thus, $G
= G_0 - (G_1 \cup \ldots \cup G_k)$
for some $G_0, \ldots, G_k \in {\cal G}_{\cal A}$. Consider the formula
$\phi_a \eqdef
\neg E_{G_0}p \land E_{G_1} p \land \ldots \land E_{G_k} p $, where $p$
is a
primitive proposition. Clearly
$\phi_a$ is satisfiable in ${\cal M}_n$, ${\cal M}_n^r$, $\M_n^{\mbox{\scriptsize{{\it rt}}}}$, $\M_n^{\mbox{\scriptsize{{\it rst}}}}$, or $\M_n^{\mbox{\scriptsize{{\it elt}}}}$
iff $|G_0 - (G_1 \cup \ldots \cup G_k)| > 0$.
{F}or part (b), given $G$,
we construct two formulas $\phi_{G,p}$
and $\psi_{G}$ with the following properties.
\begin{itemize}
\item $\phi_{G,p}$ is satisfiable iff $|G| > 0$.
\item If $(M,s) \models \phi_{G,p}$, then
$(M,s) \models \neg K_j p$ for some
$j \in G$.
\item $\psi_{G}$ is satisfiable iff $|G| > 1$.
\item $|\phi_{G,p}|$ and $|\psi_G|$ are both linear in $l(G)$.
\end{itemize}
This, of course, suffices to prove the result.
We construct the formulas $\phi_{G,p}$ by induction on the least $h$
such that $G = G' - \cup {\cal H}$ and ${\cal H} \subseteq
({\cal G}_{\cal A})^1_h$.
(We are here thinking of $G$ as specified by its description.)
If ${\cal H} \subseteq
({\cal G}_{\cal A})^1_0 = {\cal G}_A$, suppose that ${\cal H} = \{G_1, \ldots, G_k\}$. Then we
take
$\phi_{G,p}$ to be $\neg E_{G'} p \land
E_{G_1} p \land \ldots \land E_{G_k}p$. This clearly has the desired
properties.
Now suppose that
${\cal H} \subseteq ({\cal G}_{\cal A})^1_h$ for $h \ge 1$.
Without loss of generality, we can assume
that
${\cal H} = \{G_1,
\ldots, G_{k'}, G_{k'+1} , \ldots , G_k\}$, where
$G_1, \ldots, G_{k'} \in
{\cal G}_{\cal A}$ and, for $j = k'+1, \ldots, k$,
$G_j \in ({\cal G}_{\cal A})^1_h - {\cal G}_{\cal A}$ is of the
form $G_j' - \cup {\cal H}_j$ with $G'_j \in {\cal G}_{\cal A}$, ${\cal H}_j \subseteq
({\cal G}_{\cal A})^1_{h-1}$, and $|G_j| = 1$.
Define $\phi_{G,p}$ as
$$\neg E_{G'} \neg (\neg p \land
\!\!\!\ \bigwedge_{j=k'+1}^k \!\!\!\!
\phi_{G_j,p_j}) \land
E_{G_1} p \land \ldots \land E_{G_{k'}} p \land
\!\!\! \bigwedge_{j=k'+1}^k\!\!\!\!
E_{G_j'} p_j,$$
where we assume that the sets of primitive propositions that appear in
$\phi_{G_j,p_j}$, $j= k'+1, \ldots, k$, are mutually exclusive and do
not include $p$.%
\footnote{Here we are implicitly assuming that the set of primitive
propositions is infinite, so that this can be done. With more effort,
we can prove a similar result even if the set is finite, using the
techniques of \cite{Hal12}.}
Now suppose that $\phi_{G,p}$ is
true
at some state $s$ in
$M \in \M_n^{\mbox{\scriptsize{{\it rt}}}}$. Then for some $i \in G'$, we must have $(M,s)
\models \neg K_i \neg (\neg p \land \bigwedge_{j=k'+1}^k \phi_{G_j,p_j})$.
We cannot have $i \in G_1 \cup \ldots \cup G_{k'}$, since $(M,s)
\models E_{G_j} p$ for $j = 1, \ldots, k'$. Nor can we have $i \in G_j$
for $j = k'+1, \ldots, k$. For suppose that $G_j = \{i_j\}$, $j
\in \{k'+1,\ldots,k\}$. Then $(M,s) \models
\neg K_i \neg \phi_{G_j,p_j} \land E_{G_j'} p_j$,
which implies that
$(M,s) \models
\neg K_i K_{i_j} p_j \land K_{i_j} p_j$. Thus, we cannot have $i =
i_j$. It follows that $G \ne \emptyset$.
Conversely, if $G \ne
\emptyset$, we show that $\phi_{G,p}$ is satisfiable in $\M_n^{\mbox{\scriptsize{{\it rst}}}}$
(and hence also in $\M_n^{\mbox{\scriptsize{{\it rt}}}}$ and $\M_n^{\mbox{\scriptsize{{\it elt}}}}$). We actually prove a stronger
result. We show that if $G_1, \ldots, G_k$ are nonempty and the formulas
$\phi_{G_1,p_1}, \ldots, \phi_{G_k,p_k}$
involve disjoint sets of primitive propositions, then
$\phi_{G_1,p_1} \land \ldots \land \phi_{G_k,p_k}$ is satisfiable in
a structure in $\M_n^{\mbox{\scriptsize{{\it rst}}}}$ of a certain form. To make this precise, suppose
that $M = (S,\pi, \{{\cal K}_i, i \in {\cal A}\})$, $s \in S$, $S'$ is
a set of states disjoint from $S$, and $s' \in S'$. We say that $M$ is
{\em embedded
in
the structure $M' = (S \cup S', \pi', \{{\cal K}_i', \in {\cal A}\})$ at
$(s,s')$\/} if
\begin{enumerate}
\item $\pi'|_S = \pi$ and ${\cal K}_i'|_{S \times S} = {\cal K}_i$ for $i \in {\cal A}$,
\item if $(t,t') \in {\cal K}_i'$ for $t \in S$ and
$t' \in S'$, then $t=s$ and $t' = s'$.
\end{enumerate}
We show by induction on $h$ that if $G_1, \ldots, G_k \in
({\cal G}_{\cal A})^1_h$, $|G_i| > 0$ for $i = 1, \ldots, k$, and the formulas
$\phi_{G_1,p_1}, \ldots, \phi_{G_k,p_k}$
involve disjoint sets of primitive propositions, then for all $i_1,
\ldots, i_k$ such that $i_j \in G_j$, there exists a
structure $M \in \M_n^{\mbox{\scriptsize{{\it rst}}}}$ and a state $s$ in $M$ such that:
\begin{enumerate}
\item $(M,s) \models \phi_{G_1,p_1} \land \ldots \land \phi_{G_k,p_k}$,
\item $\exists t_1, \dots, t_k$ such that $(s,t_j) \in {\cal K}_{i_j}$ and
$(M,t_j)\models \neg p_j$,
\item ${\cal K}_i(s) = \{s\}$ for $i \notin \{i_1, \ldots, i_k\}$,
\item for all structures $M'$ and states $s'$ in $M'$ such that $M$ is
embedded in $M'$ at $(s,s')$ and
$(M',s') \models
p_1 \land \ldots \land p_k$, we have that $(M',s) \models \phi_{G_1,p_1}
\land \ldots \land \phi_{G_k,p_k}$.
\end{enumerate}
If $h=1$, then it is easy to construct
such a structure. Given $i_1, \ldots, i_k$ such that $i_j
\in G_j$ (where the $i_j$ are not necessarily distinct)
we construct a structure $M$ with states $s, t_1,
\ldots, t_k$ (where $t_j = t_{j'}$ if $i_j = i_{j'}$)
such that $(M,t_j) \models \bigwedge_{\{j': i_{j'} = i_j\}} \neg p_{j'} \land
\bigwedge_{\{j': i_{j'} \ne i_j\}} p_{j'}$, $(M,s)
\models p_1 \land \ldots \land p_k$, and ${\cal K}_{i}$ is the least equivalence
relation that includes $(s,t_{j})$ if $i = i_j$. It
is easy to check that $M$ has the required properties. For the
inductive step, suppose that we are given $i_1, \ldots, i_k$ such that
$i_j
\in G_j$. Note that the first conjunct of $\phi_{G_j,p_j}$ has the
form $\neg E_{G_j'} \neg (\neg p_j \land \bigwedge_{k=1}^{m_j}
\phi_{G_{jk},p_{jk}})$. By the induction hypothesis, we can find a
structure $M_j$ with state space $S_j$ and a state $t_j$ in $S_j$ with
the properties above
such that $(M,t_j) \models p_j \land
\bigwedge_{k=1}^{m_j}\phi_{G_{jk},p_{jk}}$. If
$i_j
= i_{j'}$, we can also assume without loss of generality that $M_j =
M_{j'}$ and $t_j
= t_{j'}$. Let $S$ consist
of $S_1 \cup \ldots \cup S_k$ together with a new state $s$. We
define $M \in \M_n^{\mbox{\scriptsize{{\it rst}}}}$ so that each of the structures $M_j$ is embedded
in $M$ at
$(s,t_j)$ and the relation in ${\cal K}_{i_j}$ in $M$ is the least
equivalence relation that makes this true such that $(s,t_j) \in
{\cal K}_{i,j}$. We leave it to the reader to check that we can define an
interpretation $\pi'$ with all the required properties. Of course, the
fact that $\phi_{G,p}$ is satisfiable if $|G| > 0$ is now immediate.
{F}inally, define $\psi_G$ to be $\phi_{G,p} \land E_{G'}(q \land
(\neg p \Rightarrow \phi_{G,q}))$, where we assume that the primitive
propositions that
appear in $\phi_{G,p}$ and $\phi_{G,q}$ are disjoint.
We claim that
$\psi_G$ is not satisfiable if $|G| \le 1$. Clearly it is not
satisfiable if $|G| = 0$, since $\phi_{G,p}$ is not. So suppose, by way
of contradiction, that $G = \{i\}$ and $(M,s) \models \psi_G$ for some $M
\in \M_n^{\mbox{\scriptsize{{\it rt}}}}$. Then,
thanks to the properties of $\phi_{G,p}$ and $\phi_{G,q}$, we must have
$(M,s) \models \neg K_i p \land K_i(q \land (\neg p \Rightarrow \neg K_i q))$.
It is easy to see that this gives us a contradiction. On the other
hand, if $|G| > 1$, we can construct a structure satisfying $\psi_G$ as
follows. Suppose that $i, j \in G$ and $\phi_{G,p}$ is of the form
$$\neg E_{G'} \neg (\neg p \land \bigwedge_{j=k'+1}^k \phi_{G_j,p_j}) \land
E_{G_1} p \land \ldots \land E_{G_{k'}} p \land \bigwedge_{j=k'+1}^k
E_{G_j'} p_j.$$
We know that $|G_{k'}| = \cdots = |G_k| = 1$, so by
our previous argument, we can find a
structure $M' =(S', \ldots) \in \M_n^{\mbox{\scriptsize{{\it rst}}}}$ and states $s',t' \in S'$ such
that $(M,s') \models \phi_{G,q} \land \bigwedge_{j=k'+1}^k \phi_{G_j,p_j}$,
$(s',t') \in {\cal K}_j$, $(M',t') \models \neg q$, and ${\cal K}_i(s') = \{s'\}$.
Since $p$ does not appear in $\phi_{G,q}$, we can assume
without loss of generality that $(M,s) \models \neg p$. Now let $M \in
\M_n^{\mbox{\scriptsize{{\it rst}}}}$ be a structure whose state space
is $S' \cup \{s\}$, where $s$ is a fresh
state not in $S'$, such that $M'$ is embedded in $M$ at $(s,s')$,
$(s,s') \in {\cal K}_i$, $(M,s) \models p \land q$, and ${\cal K}_{i'}(s) = \{s\}$ for
$i' \ne i$. It is easy to see that $(M,s) \models \psi_G$.
{F}or part (c),
we construct formulas
$\phi_{m,G,p}$ such that
\begin{itemize}
\item if $(M,s) \models \phi_{m,G,p}$ for $M \in \M_n^{\mbox{\scriptsize{{\it elt}}}}$ (and hence also
for
$M \in \M_n^{\mbox{\scriptsize{{\it rst}}}}$), then there exist $m+1$ distinct agents $i_1, \ldots,
i_{m+1} \in G$ such that $(M,s) \models \neg K_{i_j} \neg p$, $j = 1,
\ldots, m+1$;
\item $|\phi_{m,G,p}|= O(l(G) + m)$;
\item if $|G| > m$, then $\phi_{m,G,p}$ is satisfiable in
$\M_n^{\mbox{\scriptsize{{\it rst}}}}$ (and hence in
$\M_n^{\mbox{\scriptsize{{\it elt}}}}$).
\end{itemize}
We first define an auxiliary family of formulas.
If $G', G_1, \ldots, G_k
\subseteq {\cal G}_{\cal A}$,
let $\psi_{m,G',G_1, \ldots,
G_k,p}$ be the formula
$$\begin{array}{ll}
E_{G_1} q_0 \land \ldots \land E_{G_k} q_0 \land\\
\neg E_{G'}\neg (p_0 \land p_1 \land q_1 \land E_{G'}(p_0
\Rightarrow p_1 \land q_1)) \land\\
\ \ \ \ldots \land \neg E_{G'} \neg (p_0 \land p_{m+1} \land q_{m+1}
\land E_{G'}(p_0 \Rightarrow p_{m+1} \land q_{m+1})) \land\\
E_{G'} ((p_0 \Rightarrow (p \land \neg q_0)) \land
(q_1 \Leftrightarrow \neg p_2 \land q_2) \land (q_2 \Leftrightarrow \neg
p_3
\land q_3) \land \ldots \land (q_{m+1} \Leftrightarrow {\em true})),
\end{array}$$
where $p_0, \ldots, p_{m+1}, q_0, \ldots, q_{m+1}$ are fresh primitive
propositions distinct from $p$.
Observe that $|\psi_{m,G',G_1,\ldots,G_k,p}|$ is $O(k + m)$.
It is easy to check that the last clause forces $q_i$,
for $1 \leq i \leq m$,
to be equivalent
to $\neg p_{i+1} \land \ldots \land \neg p_{m+1}$. at least in the
worlds $G'$-reachable in one step.
Thus, in these worlds, the formulas
$p_i \land q_i$, $i=1, \ldots, m+1$, are mutually exclusive. Clearly if
$(M,s) \models
\psi_{m,G',G_1,\ldots,G_k,p}$
for $M \in \M_n^{\mbox{\scriptsize{{\it elt}}}}$, then
there must be agents $i_1,
\ldots, i_{m+1}$ in $G' - (G_1 \cup \ldots \cup G_k)$ such that $(M,s)
\models \neg K_{i_j} \neg (p_0 \land p_j \land q_j \land
E_{G'}(p_0 \Rightarrow p_j \land q_j))$.
(Note that we cannot have $i_j \in
G' \cap (G_1 \cup \ldots \cup G_k)$ since $(M,s) \models E_{G_j}q_0
\land E_{G'}(p_0 \Rightarrow \neg q_0)$). Thus, there must
exist states $t_j$, $j= 1,\ldots, m+1$ such that $(s,t_j) \in {\cal K}_{i_j}$
and $(M,t_j) \models p_0 \land p_j \land q_j
\land E_{G'} (p_0 \Rightarrow p_j
\land q_j)$. To see that these
agents $i_j$ must be distinct, suppose that $i_j = i_{j'}$ for $j < j'$.
By the Euclidean property, we have $(t_j,t_{j'}) \in {\cal K}_{i_j}$. Since
$(M,t_j) \models E_{G'} (p_0 \Rightarrow p_j \land q_j)$,
we must have $(M,t_{j'}) \models p_j \land q_j$. But since $(M,t_{j'})
\models q_j \Leftrightarrow (\neg p_j \land \ldots \land \neg p_{m+1})$, this is
inconsistent with the fact that $(M,t_{j'}) \models p_{j'}$.
Since $(M,s) \models E_{G'} (p_0 \Rightarrow p)$, it follows that $(M,s) \models
\neg K_{i_j} \neg p$ for $j = 1, \ldots, m+1$.
Conversely, it is easy to see that if $|G' - (G_1 \cup \ldots \cup
G_k)| > m$ then
$\psi_{m,G',G_1,\ldots,G_k,p}$
is
satisfiable in $\M_n^{\mbox{\scriptsize{{\it rst}}}}$. We leave
the
details to the reader.
We now construct the formulas $\phi_{m,G,p}$
by induction on the least $h$
such that $G = G' - \cup {\cal H}$ and ${\cal H} \subseteq
({\cal G}_{\cal A})^m_h$. If ${\cal H} = \{G_1, \ldots, G_k\} \subseteq
({\cal G}_{\cal A})^m_0 = {\cal G}_A$, then we take $\phi_{m,G,p} =
\psi_{m,G',G_1,\ldots,G_k,p}$.
Now suppose that ${\cal H} \subseteq ({\cal G}_{\cal A})^m_h$ for $h >1$.
Without loss of generality, we can assume that ${\cal H} = \{G_1,
\ldots, G_{k'}, G_{k'+1}, \ldots, G_k\}$, where
$G_1, \ldots, G_{k'} \in
{\cal G}_{\cal A}$ and, for $j = k'+1, \ldots, k$,
$G_j \in ({\cal G}_{\cal A})^m_{h-1}$ is of the
form $G_j' - \cup {\cal H}_j$ with $G'_j \in {\cal G}_{\cal A}$, ${\cal H}_j \subseteq
({\cal G}_{\cal A})^m_{h-1}$, and $|G_j| \le m$. Suppose that $|G_j| = m_j$.
By induction, for $j = k'+1, \ldots,k$,
we can construct formulas $\phi_{m_j-1,G_j,p}$
such that if $(M,s)\models \phi_{m_j-1,G_j,p}$, then for each
agent $i \in G_j$,
we have $(M,s) \models \neg K_i \neg p$ and
the
formula $\psi_{m,G',G_1,
\ldots, G_{k'},p}$. Without loss of generality, we
can
assume that, other than $p$, the sets of primitive propositions
mentioned in the formulas $\phi_{m_j-1,G_j, p}$ are disjoint, and these
sets are all disjoint from the set of primitive propositions in
$\psi_{m,G',G_1,\ldots, G_{k'},p}$.
Let $\phi_{m,G,p}$ be the formula
$$
\psi_{m,G',G_1,\ldots,G_{k'},p'} \land
\bigwedge_{j=k'+1}^m \phi_{m_j-1,G_j, p} \land E_{G'}(p' \Rightarrow E_{G'} \neg
p).
$$
The argument that this formula has the required properties is almost
identical to that for $\psi_{m,G',G_1, \ldots, G_k,p}$; we leave details
to the reader.
{F}inally, for part (d), consider the formula $\phi_d$ defined as
$$E_{G_1}p_1 \land
\ldots \land E_{G_{k-1}} p_{k-1} \land E_{G_k} (\neg p_1 \lor \ldots
\lor
\neg p_{k-1}).$$
We leave it to the reader to check that $\phi_d$ is satisfiable
in $\M_n^{\mbox{\scriptsize{{\it elt}}}}$ iff $G_1 \cap \ldots \cap G_k = \emptyset$.
\vrule height7pt width4pt depth1pt\vspace{0.1in}
We already saw that for axiom E2 to be recursive, we need to be able to
decide whether
$|G_0 - (G_1 \cup \ldots \cup G_k)| \ge 1$ (or, equivalently,
whether $G_0 \subseteq G_1 \cup \ldots \cup G_k$) for $G_0, \ldots,
G_k \in
{\cal G}_{\cal A}$.
Proposition~\ref{reduction} shows that if there is no recursive
algorithm for answering such questions, the satisfiability problem for
the logic (even without $C_G$ operators) is also not decidable.
{F}or simplicity here, we assume we have oracles that can answer the
questions that we need to answer (according to
Proposition~\ref{reduction}) in unit time; we consider the
complexity of querying the oracle in more detail in
Section~\ref{oracle}.
More precisely, let $O_m$ be an oracle that, for a set $G \in \widehat{{\cal G}}_{\cal A}^m$,
tells us whether $|G| > k$, for any $k < m$. (Thus, queries to oracle
$O_m$ have the form $(G,k)$.)
Let $O'$ be an oracle that tells us whether
$G_1 \cap \ldots \cap G_k = \emptyset$, for $G_1, \ldots, G_k \in
{\cal G}_{\cal A}$.
\begin{theorem}\label{complexity} There is a constant $c > 0$
(independent of ${\cal A}$)
and an algorithm that,
given as input a formula $\phi \in {\cal L}_{\G}^C$,
decides if $\phi$
is satisfiable in
${\cal M}_n$ (resp.,\
${\cal M}_n^r$, $\M_n^{\mbox{\scriptsize{{\it rt}}}}$, $\M_n^{\mbox{\scriptsize{{\it rst}}}}$, $\M_n^{\mbox{\scriptsize{{\it elt}}}}$) and runs in time
$2^{c|\phi|}$
given oracle $O_0$ (resp.,\ $O_0$, $O_1$, $O_{|\phi|}$,
both $O_{|\phi|}$
and
$O'$), where queries to the oracle take unit time. Moreover, if ${\cal G}$
contains
a subset with at least two elements, then there exists a constant $d > 0$
(independent of ${\cal A}$)
such that every algorithm for deciding the satisfiability of formulas
in ${\cal M}_n$ (resp.,\ ${\cal M}_n^r$, $\M_n^{\mbox{\scriptsize{{\it rt}}}}$, $\M_n^{\mbox{\scriptsize{{\it rst}}}}$, $\M_n^{\mbox{\scriptsize{{\it elt}}}}$) runs in time at
least $2^{d|\phi|}$, even given access to oracle $O_0$
(resp.,\ $O_0$, $O_1$, $O_{|\phi|}$, both $O_{|\phi|}$ and $O'$),
for infinitely many formulas $\phi$.
\end{theorem}
Before proving
Theorems~\ref{complete} and~\ref{complexity}, we prove a somewhat
simpler theorem that
allows us to both
explain intuitively why the
results are
true and
point out some of the
difficulties in proving them.
\begin{proposition}\label{Shore} If there is an oracle that
decides if $G = \emptyset$
for each Boolean combination $G$ of elements in ${\cal G}_{\cal A}$,
then,
for every
formula $\phi \in {\cal L}_{\G}^C$, we can effectively find a formula $\phi^\sigma$
in a language ${\cal L}_{\G'}^C$, where ${\cal G}'$ consists of all subsets
of
a set
${\cal A}'$ of
at most $2^{|\phi|}$ agents, such that $|\phi^\sigma| = |\phi|$ and
$\phi$
is satisfiable in ${\cal M}_n$ iff $\phi^\sigma$ is satisfiable in ${\cal M}_{{\cal A}'}$.
\end{proposition}
\noindent{\bf Proof:}
Given $\phi$, let ${\cal G}_\phi$ be the set of subsets
$G$ of agents such that $E_G$ or $C_G$ appears in $\phi$.
(Recall that we are identifying $K_i$ with $E_{\{i\}}$,
so that $\{i\} \in {\cal G}_\phi$ if $K_i$ appears in $\phi$.)
Note that $|{\cal G}_\phi| \le |\phi|$.
Suppose that ${\cal G} = \{G_1, \ldots, G_N\}$.
An
{\em atom over
${\cal G}$\/} is a nonempty set of the form $G_1' \cap \ldots \cap
G_N'$, where $G_i' = G_i$ or $G_i' = \overline{G_i}$.
Clearly there are at most $2^N$ atoms over ${\cal G}$.
Let ${\cal A}'$ consist of the
nonempty atoms over ${\cal G}_\phi$. Note that $|{\cal A}'| \le 2^{|\phi|}$.
Define $\sigma: {\cal A}
\rightarrow {\cal A}'$ by taking $\sigma(i)$ to be the unique atom over
${\cal G}_\phi$ containing $i$. We extend $\sigma$ to a map from $2^{{\cal A}}
\rightarrow 2^{{\cal A}'}$ by taking $\sigma(G) = \{\sigma(i): i \in G\}$
($= \{H\in {\cal G}_\phi: H \subseteq G\}$).
Translate $\phi$ to
$\phi^\sigma$ by replacing all occurrences of
$E_G$ and $C_G$ in
$\phi$ by
$E_{\sigma(G)}$, and
$C_{\sigma(G)}$, respectively.
Clearly $|\phi| = |\phi^\sigma|$. (Note that it is important here that
we take the length of $E_G$ and $C_G$ to be
independent of $G$.)
If $\phi $ is satisfiable, let $(M,s)$ witness that fact. Convert $M$
into a structure $M^\sigma$ over ${\cal A}'$ with the same state space by
setting $(s,t)\in {\cal K}_A$ iff
$(s,t)\in \cup_{j \in A}{\cal K}_{j}$
for each $A \in {\cal A}'$.
An easy induction shows that for every formula $\psi $
with
sets
(of agents)
chosen from
${\cal G}_\phi$,
we have $(M,s)\models \psi $ if and only if $%
(M^\sigma,s)\models \psi^\sigma$. The only point that needs any comment
is that
$E_{G}$ (and so also $C_{G})$ has the same meaning in $M$ (in terms of
reachability)
as $E_{\sigma(G)}$ ($C_{\sigma(G)}$) in $M^{\sigma}$,
by the definition
of $\sigma(G)$ and the ${\cal K}_{A}$ relations. Thus $(M^{\sigma},s)\models
\phi^{\sigma}$ as required.
{F}or the other direction, suppose that $(M',s)\models \phi^\sigma$ for
some
structure $M'$ over ${\cal A}'$. We define a
structure $M$ over ${\cal A}$ by defining ${\cal K}_i = {\cal K}_{\sigma(i)}$.
Again an easy induction shows that for every formula $\psi $ with
sets chosen from
${\cal G}_\phi$,
$(M^{\prime },s)\models \psi $ if and only if $(M,s)\models
\psi^{\sigma}$. Once again, the only point to notice is that
$E_{G}$
(and so also $C_{G}$) has the same meaning in $M^{\prime }$ (in
terms
of reachability) as $E_{\sigma(G)}$ ($C_{\sigma(G)}$) in $M$ by the
definition of $\sigma(G)$ and the relations ${\cal K}_{j}$. Thus $(M,s)\models
\phi $ as required. \vrule height7pt width4pt depth1pt\vspace{0.1in}
\begin{corollary}\label{Shore1} Given an oracle that
decides, for each Boolean combination $G$ of elements in
${\cal G}_{\cal A}$,
whether
$G = \emptyset$, there is
a constant $c > 0$ (independent of ${\cal A}$) and
an algorithm that,
given as input a formula $\phi \in {\cal L}_{\G}^C$,
decides if $\phi \in {\cal L}_{\G}^C$ is satisfiable in ${\cal M}_n$ and runs in time
$2^{c2^{|\phi|}}$. \end{corollary}
\noindent{\bf Proof:} Clearly, to check if $\phi$ is satisfiable, it suffices to check if
$\phi^\sigma$ is satisfiable. In \cite{HM2}, there is an exponential
time
algorithm for checking satisfiability. However, this algorithm presumes
that the set of agents is fixed. A close look at the algorithm actually
shows that it runs in time $2^{cm|\phi|}$, where $m$ is the number of
agents. In our translation, the set of agents is exponential in
$|\phi|$,
giving us a double-exponential
time
algorithm.
\vrule height7pt width4pt depth1pt\vspace{0.1in}
\begin{corollary}\label{Shore2} If ${\cal G}$ is closed under
intersection
and complementation,
then ${\rm K}_n^C$ is a sound and complete axiomatization for the language
${\cal L}_{\G}^C$ with respect to ${\cal M}_n$. \end{corollary}
\noindent{\bf Proof:} Soundness is straightforward, so we focus on completeness.
Suppose that $\phi$ is valid. By Proposition~\ref{Shore}, so is
$\phi^\sigma$. Since ${\cal A}'$ is finite,
${\rm K}_{\G'}^C$ is a complete axiomatization for ${\cal L}_{\G'}^C$ with respect to
${\cal M}_{{\cal A}'}$. Thus, ${\rm K}_{\G'}^C \vdash \phi^\sigma$. We
can translate this proof step by step to a proof of $\phi$ in ${\rm K}_n^C$.
We simply replace every formula $\psi$ that appears in the proof of
$\phi^{\sigma}$ by $\psi^{\tau}$, where $\psi^{\tau}$ is
obtained by replacing each occurrence
of
$K_A$ in $\psi$
by $E_{A}$ unless $A = \{i\}$ is a singleton, in which case
we replace $K_A$ by $K_i$, and replacing each occurrence of
$E_G$, and $C_G$ in $\psi$ by
$E_{\cup G}$, and $C_{\cup G}$, respectively.
Since we have assumed ${\cal G}$ is closed under complementation and
intersection, it is also closed under union, and hence $\psi^\tau$ is
a formula in ${\cal L}_{\G}^C$.
It is easy to check that the translated proof is still a proof
over the language ${\cal L}_{\G}^C$:
Tautologies become tautologies as $(\phi \vee \psi )^{\tau}=\phi
^{\tau}\vee
\psi^{\tau}$ and similarly for negations.
Instances of MP in the proof of $\phi^\sigma$ become instances of
MP in the proof of $\phi$
because $(\phi \rightarrow \psi )^{\tau}=\phi^{\tau}\rightarrow
\psi^{\tau}$.
Instances of KGen in the proof of $\phi^\sigma$ become instances of EGen
or KGen in the proof of $\phi$; similarly, instances of K1 are
converted to
instances of K1 or E1.
It is easy to see
that instances of E1, E2,
E3, EGen, C1, and RC1 are converted to legitimate instances of the same
axiom.
\vrule height7pt width4pt depth1pt\vspace{0.1in}
While Corollaries~\ref{Shore1} and~\ref{Shore2} are close to our desired
theorems, they also make clear the difficulties we need to overcome in
order to prove Theorems~\ref{complete} and~\ref{complexity}.
Specifically,
\begin{itemize}
\item we need to cut the complexity down from double-exponential
to single exponential;
\item we need to prove completeness without assuming that ${\cal G}$ is
closed under complementation and intersection;
\item we want to use an oracle that tests only whether a set of
the form $G_0 - (G_1 \cup \ldots \cup G_k)$ is nonempty, rather
than one that applies to arbitrary Boolean combinations;
\item we want to extend these results to the case that the ${\cal K}_i$
relations satisfy properties like transitivity.
\end{itemize}
With regard to the last point, while in general it is relatively
straightforward to extend completeness and complexity results to deal
with relations that have properties like transitivity, it is not so
straightforward in this case. For example, even if $M \in \M_n^{\mbox{\scriptsize{{\it rt}}}}$, the
relations in the structure $M^\sigma$ constructed in
Proposition~\ref{Shore} are not necessarily transitive. As shown in
Proposition~\ref{reduction},
we need a different oracle to deal with transitivity.
\section{Proving the Main Results}\label{proofs}
In this section, we prove
Theorems~\ref{complete} and~\ref{complexity}.
The structure of the proof is similar to that of
Corollaries~\ref{Shore1} and~\ref{Shore2}; we describe step by step the
modifications required to deal with the problems raised in the previous
section.
It is convenient to split the proof into four cases,
depending on the class of structures considered.
\subsection{The Proof for ${\cal M}_n$ and ${\cal M}_{\cal A}^r$}
In Proposition~\ref{Shore} we showed that we
could translate a formula $\phi$ to a formula $\phi^\sigma$ such that
$\phi$
was satisfiable in ${\cal M}_n$ iff $\phi^\sigma$ was satisfiable in
${\cal M}_{{\cal A}'}$,
where ${\cal A}'$
consisted
of the atoms over ${\cal G}_\phi$. Our goal is to
maintain the translation idea, but use as our target set of agents
a set whose elements we can determine with the oracles at our disposal
(for testing the
nonemptiness of certain set differences). As a first step, we
try
to abstract the key ingredients of Proposition~\ref{Shore}. Suppose
that we have a set ${\cal A}'$ of agents and a
partial map
$\sigma: {\cal A} \rightarrow {\cal A}'
$.
Again, we can extend $\sigma$ to a map from $2^{\cal A}$ to
$2^{{\cal A}'}$: $\sigma(G) = \{\sigma(i): i \in {\cal G}
\}$. Given a formula $\phi$, let $\phi^\sigma$ be the
formula
that results by replacing all the occurrences of $G$ in $\phi$ by
$\sigma(G)$.
In Proposition~\ref{Shore}, ${\cal A}'$ is the set of
atoms over ${\cal G}_\phi$ and $\sigma(i)$ is the unique atom containing
$i$. We were able to show
that,
for that choice of ${\cal A}'$ and $\sigma$,
the formulas $\phi$ and $\phi^\sigma$ were
equisatisfiable.
What does
it take to obtain such a result in general?
The following result shows that we need to be able to
find a mapping $\tau: {\cal A}' \rightarrow 2^{\cal A} - \{\emptyset\}$ with one key
property.
\begin{proposition}\label{trans} Given a formula $\phi$ and a
partial map
$\sigma: {\cal A}
\rightarrow {\cal A}'$ such that $\sigma(G) \ne \emptyset$ for all $G \in
{\cal G}_\phi$, suppose that there is a mapping $\tau: {\cal A}' \rightarrow
2^{\cal A} -
\{\emptyset\}$
such that for all $G \in {\cal G}_\phi$, we have
$\cup \{\tau(A): A \in \sigma(G)\} = G$.
Then $\phi$ is satisfiable in ${\cal M}_n$ (resp.,\ ${\cal M}_n^r$) iff $\phi^\sigma$
is satisfiable in ${\cal M}_{{\cal A}'}$ (resp.,\ ${\cal M}_{{\cal A}'}^r$).
\end{proposition}
\noindent{\bf Proof:}
Given $\phi$ and $\sigma$, suppose there exists a mapping $\tau$
with the property above. We show that $\phi$ and $\phi^{\sigma}$ are
equisatisfiable.
{F}irst suppose that $(M,s)\models\phi$, where $M \in {\cal M}_n$. We convert
$M = (S,\pi, \{{\cal K}_i: i \in {\cal A}\})$
into a structure
$M' = (S,\pi, \{{\cal K}_A: A \in {\cal A}'\})$
by defining ${\cal K}_A =
\cup \{{\cal K}_i:i\in\tau(A)\}$. Notice that the
assumed
property of
$\tau$ implies that for all $G \in {\cal G}_\phi$, we have
$$\cup_{A \in \sigma(G)} {\cal K}_A = \cup_{A \in \sigma(G)} \cup_{i \in
\tau(A)} {\cal K}_i = \cup_{i \in G} {\cal K}_i.$$
An easy induction on
the structure of $\psi$ now shows that $(M,t)\models\psi$ if and
only if $(M',t)\models\psi^{\sigma}$ for all $t \in S$ and all formulas
$\psi
\in {\cal L}_{{\cal G}_\phi}^C$.
Also note that if $M \in {\cal M}_n^r$, then $M' \in {\cal M}_{{\cal A}'}^r$ (since the
union of reflexive relations is reflexive).
{F}or the opposite direction, suppose $(M',s)\models\phi^{\sigma}$ for
some
$M' = (S,\pi, \{{\cal K}_A: A \in {\cal A}'\}) \in {\cal M}_{{\cal A}'}$. Define $M =
(S,\pi,\{{\cal K}_i: i \in {\cal A}\}) \in {\cal M}_n$ by setting
${\cal K}_i={\cal K}_{\sigma(i)}$
if $\sigma(i)$ is defined and the empty relation otherwise.
Note that for all $G \in {\cal G}_\phi$ we have
$$\cup_{i \in G}{\cal K}_i = \cup_{i \in G} {\cal K}_{\sigma(i)} = \cup_{A
\in \sigma(G)} {\cal K}_A.$$
Again, an easy induction on
the structure of $\psi$
shows that
$(M,t)\models\psi$ if and
only if $(M',t)\models\psi^{\sigma}$ for all $t \in S$ and all formulas
$\psi \in {\cal L}_{{\cal G}_\phi}^C$.
If $M' \in {\cal M}_n^r$, we modify the construction slightly by taking
${\cal K}_i = \{(t,t): t\in S\}$ if $\sigma(i)$
is undefined.
Since
$\sigma(G)
\ne \emptyset$ for $G \in {\cal G}_\phi$, it is easy to check that we still
have $\cup_{i\in G} {\cal K}_i = \cup_{i \in G} {\cal K}_{\sigma(i)}$, so the
modified construction works for the reflexive case.
\vrule height7pt width4pt depth1pt\vspace{0.1in}
{F}or the mapping $\sigma$ of Proposition~\ref{Shore} we
can take $\tau$ to be the identity, but this requires an oracle for
nonemptiness of atoms.
We now show how to choose ${\cal A}'$ and define maps $\sigma$ and $\tau$ in a
way that requires
only information about whether sets of the form $G_0 - (G_1 \cup
\ldots \cup G_k)$
are empty.
\begin{definition}\label{Gmax}
Given a set ${\cal G}$ of sets of agents and
$G \in {\cal G}$,
a set ${\cal H} \subseteq {\cal G}$ is a {\em $G$-maximal\/} subset of ${\cal G}$
if $G - \cup {\cal H} \ne \emptyset$ and
$G - ((\cup {\cal H})
\cup G')
=\emptyset$ for all
$G'
\in {\cal G} -{\cal H}$.
Let ${\cal R}({\cal G})=\{(G,{\cal H}):G\in{\cal G}, {\cal H} \mbox{ is a $G$-maximal subset of
${\cal G}$}\}$.~\vrule height7pt width4pt depth1pt\end{definition}
Note that we can check whether ${\cal H}$ is a $G$-maximal subset of ${\cal G}$ by
doing at most
$|{\cal G}|$ tests of the form $(G - \cup {\cal H}') = \emptyset$, and we can find
all pairs $(G,{\cal H})$ in $\R(\G_\phi)$ by doing at most $|{\cal G}|2^{|{\cal G}|-1}$ such
tests.
The following lemma gives some technical properties of ${\cal R}({\cal G})$ that
will be used frequently.
\begin{lemma}\label{unique}
Suppose that $(G,{\cal H}) \in{\cal R}({\cal G})$ for some set ${\cal G}$ of
subsets of ${\cal A}$.
\begin{itemize}
\item[(a)] $G - \cup {\cal H}$ is an atom over ${\cal G}$ and, in fact,
$G - \cup {\cal H} = \cap({\cal G} - {\cal H}) \cap
(\cap_{H \in {\cal H}} \overline{H})$.
\item[(b)] If $(G',{\cal H}) \in {\cal R}({\cal G})$, then
$(G-\bigcup{\cal H})=(G'-\bigcup{\cal H})$.
\item[(c)] If $(G',{\cal H}') \in {\cal R}({\cal G})$
and
${\cal H} \ne {\cal H}'$, then
$(G-\bigcup {\cal H})\cap(G'-\bigcup{\cal H}')=\emptyset$.
\end{itemize}
\end{lemma}
\noindent{\bf Proof:}
{F}or part (a), first observe that
since ${\cal H}$ is a $G$-maximal
subset of ${\cal G}$, for $H \notin {\cal H}$, we have $G - \cup
({\cal H} \cup \{H\}) = \emptyset$; i.e.,~ $G - \cup {\cal H} \subseteq H$.
Thus, if $H \notin {\cal H}$, we have $G - \cup {\cal H} = (G \cap H) - \cup
{\cal H}$. Thus, $G -\cup {\cal H} = G \cap
(\cap_{H \in {\cal H}} \overline{H}) = \cap({\cal G} - {\cal H}) \cap
(\cap_{H \in {\cal H}} \overline{H})$, as desired. By definition, $G - \cup
{\cal H}$ is an atom over ${\cal G}$.
Part (b) is immediate from part (a), since it is clear
that
$G - \cup {\cal H}$
is independent of $G$ and depends only on ${\cal H}$.
{F}or part (c), suppose that ${\cal H} \ne {\cal H}'$. Without loss
generality, there is some $H \in {\cal H} - {\cal H}'$. It follows
immediately from part
(a) that $G - \cup {\cal H}$ and $G' - \cup {\cal H}'$ are distinct atoms (hence
disjoint), since $G - \cup {\cal H} \subseteq \overline{H}$ and $G' \cup {\cal H}'
\subseteq H$.
\vrule height7pt width4pt depth1pt\vspace{0.1in}
If $(G,{\cal H}) \in {\cal R}({\cal G})$, let $A_{{\cal H}}^{\cal G}$ denote the atom associated with
${\cal H}$ defined in Lemma~\ref{unique}(a). It is independent of $G$ by
Lemma~\ref{unique}(b). We omit ${\cal G}$,
writing simply $A_{\cal H}$,
when it is clear from the context which set ${\cal G}$
we have in mind.
We now show how to define a translation satisfying the hypotheses of
Proposition \ref{trans} using the elements of $\R(\G_\phi)$ identified
according to the second coordinate alone.
Given a formula $\phi$, let ${\cal A}^\phi=\{{\cal H}: \exists
G[(G,{\cal H})\in\R(\G_\phi)]\}$. Define $\sigma_1: {\cal A} \rightarrow {\cal A}^\phi$
by setting
$\sigma_1(i) = {\cal H}$ if
$i \in A_{\cal H}$ (as defined after Lemma~\ref{unique})
and undefined otherwise.
As before, we extend $\sigma_1$ to $2^{\cal A}$
by
defining $\sigma_1(G) =
\{\sigma_1(i): i \in G
\}$.
\begin{lemma}\label{K}
Define $\tau: {\cal A}^\phi \rightarrow 2^{\cal A}$ by setting $\tau({\cal H}) = \cap
({\cal G}_\phi - {\cal H})$. Then
\begin{itemize}
\item[(a)] $\sigma_1(G) =
\{{\cal H} \in {\cal A}^\phi: \exists G' \in {\cal G}_\phi((G',{\cal H}) \in \R(\G_\phi)), \, G
\notin {\cal H}\}$,
\item[(b)] $\sigma_1(G) \ne \emptyset$ for $G \in {\cal G}_\phi$,
\item[(c)] $\tau({\cal H}) \ne \emptyset$ for ${\cal H}
\in {\cal A}^\phi$,
\item[(d)] $\cup \{\tau({\cal H}): {\cal H} \in \sigma_1(G)\} = G$.
\end{itemize}
\end{lemma}
\noindent{\bf Proof:} For part (a),
first suppose that $G \notin {\cal H}$
and $(G',{\cal H}) \in {\cal R}({\cal G}_\phi)$ for some $G' \in {\cal G}_\phi$.
Then by Lemma~\ref{unique}(a), it
follows that $A_{\cal H} \subseteq G$. Since $A_{\cal H} \ne \emptyset$, there is
some $i \in A_{\cal H}$. Since $i \in G$ and $\sigma_1(i) = {\cal H}$, it follows
that
${\cal H}
\in \sigma_1(G)$. For the opposite inclusion, suppose that
${\cal H}
\in \sigma_1(G)$. Then
${\cal H}
= \sigma_1(i)$ for some $i \in G
\cap A_{\cal H}$. Since $G \cap A_{\cal H} \ne\emptyset$, it follows from the
definition of $A_{\cal H}$ that $G \notin {\cal H}$.
{F}or part (b), given $G$, note that there must be some $G$-maximal
subset ${\cal H}$.
Thus, $(G,{\cal H}) \in {\cal R}({\cal G}_\phi)$.
Since $G - \cup {\cal H} \ne \emptyset$, we must have $G
\notin {\cal H}$. By part (a), ${\cal H} \in \sigma_1(G)$, so $\sigma_1(G) \ne
\emptyset$.
{F}or part (c), suppose that ${\cal H} \in
{\cal A}^\phi$.
Then there exists some $G$ such
that $(G,{\cal H}) \in \R(\G_\phi)$, and hence $G - \cup {\cal H} \ne \emptyset$. It
suffices to show that $\cap ({\cal G}_\phi - {\cal H}) \supseteq G - \cup {\cal H}$.
Since $G - ((\cup {\cal H}) \cup G') = \emptyset$ for all $G'
\in {\cal G}_\phi - {\cal H}$, it follows that
$G - \cup {\cal H} \subseteq G'$ for each $G' \in {\cal G}_\phi - {\cal H}$.
Thus, $\cap ({\cal G}_\phi - {\cal H}) \supseteq G - \cup {\cal H}$.
{F}or part (d), we first show that
$\cup \{\tau({\cal H}): {\cal H} \in \sigma_1(G)\} \subseteq G$. Note that if ${\cal H}
\in \sigma_1(G)$, then by part (a),
$G \in {\cal G}_\phi - {\cal H}$. Thus, $\tau({\cal H}) =
\cap ({\cal G}_\phi -{\cal H}) \subseteq G$.
{F}or the oppposite containment, suppose that $i \in G$. Let ${\cal H}^i =
\{G' \in {\cal G}_\phi: i \notin G'\}$. Since $i \in G - \cup {\cal H}^i$,
there must be a $G$-maximal subset ${\cal H}$ of ${\cal G}_\phi$ containing ${\cal H}^i$.
By part (a), we have ${\cal H} \in \sigma_1(G)$.
Moreover,
since ${\cal H}^i \subseteq {\cal H}$, for all $H' \in {\cal G}_\phi - {\cal H}$, we have $i \in
H'$. Thus, $i \in \cap ({\cal G}_\phi - {\cal H})$. It follows that $ i \in
\cup_{{\cal H} \in \sigma_1(G)} \cap ({\cal G}_\phi - {\cal H})$, as desired. \vrule height7pt width4pt depth1pt\vspace{0.1in}
Since $|{\cal A}| \le 2^{|\phi|}$, we have now reduced satisfiability with
infinitely many agents to satisfiability with finitely many agents, at
least for ${\cal M}_n$ and ${\cal M}_n^r$, using only tests that we know we need
to be able to perform in any case. We next must deal with the problem
we observed in the proof of Corollary~\ref{Shore1}, that
is,
there may be
exponentially
many
agents in the subgroups mentioned in $\phi^{\sigma_1}$. This is
done in the following result. In this result, we assume that the
complexity of checking whether $i \in G$ is no worse than linear in
$|{\cal A}|$. While we do not assume this in general, it is true for the
${\cal A}'$ and sets $G$ that arise in the translation of
Proposition~\ref{trans}, which suffices for our application of
the result to the proof of Theorem~\ref{complexity}.
\begin{theorem}\label{dec}
If ${\cal A}$ is finite and there is an algorithm for deciding if $i \in G$
for $G \in {\cal G}$ that runs in time linear in $|{\cal A}|$, then
there is
a constant $c > 0$ (independent of ${\cal A}$)
and an algorithm that,
given as input a formula $\phi \in {\cal L}_{\G}^C$,
decides if $\phi$ is satisfiable in
${\cal M}_n$ (resp.,\ ${\cal M}_n^r$)
and runs in time $O(|{\cal A}|2^{c|\phi|})$.
\end{theorem}
\noindent{\bf Proof:} We first present an algorithm that
decides if $\phi$ is satisfiable in
${\cal M}_n$; we then show how to modify it to deal with ${\cal M}_n^r$. The
algorithm is just
a slight modification of standard decision procedures \cite{FHMV,HM2}.
(Far more serious modifications are needed to prove
the analogous result for the $\M_n^{\mbox{\scriptsize{{\it rt}}}}$, $\M_n^{\mbox{\scriptsize{{\it rst}}}}$, and $\M_n^{\mbox{\scriptsize{{\it elt}}}}$; see
Theorems~\ref{decrt}, \ref{decrst}, and~\ref{decelt}.)
Let $\mbox{\it Sub}(\phi)$ be the set of subformulas of $\phi$ together with
$E_{G}(\psi \land C_G \psi)$ and $\psi \land C_{G}\psi$ for each
subformula $C_{G}\psi$ of
$\phi$. $\mbox{\it Sub}^{+}(\phi)$ consists of the formulas in $\mbox{\it Sub}(\phi)$ and
their negations. An easy induction
on $|\phi|$ shows that $|\mbox{\it Sub}(\phi)| \le |\phi|$, so $|Sub^+(\phi)| \le
2|\phi|$. (Here we need to use the fact that we take the length of
$C_G$ to be 3.)
Let $S^1$ consist of all
subsets $s$ of $\mbox{\it Sub}^+(\phi)$ that are
{\em maximally consistent\/} in that
(a) for each formula $\psi
\in \mbox{\it Sub}(\phi)$, either $\psi \in s$ or $\neg \psi
\in s$,
(b)
they are
propositionally consistent
(for example, we cannot have
all of $\psi \land \psi'$, $\neg \psi$, and $\neg
\psi'$ in $s$), and
(c) they contain $E_G (\psi \land C_G \psi)$
iff
they contain $C_G \psi$.
Note that there are at most $2^{|\phi|}$ sets in $S^1(\phi)$.
{F}or $s \in S^{1}$ and $G \in
{\cal G}_{\cal A}$, we define $s/E_G=\{\psi:
E_G\psi\in s\}$ (again, we identify $K_i$ with $E_{\{i\}}$).
Define $s/\overline{K_i} = \cup_{i \in G} (s/E_G)$.
Define
a
binary relation ${\cal K}_i$ on $S^1$ for each $i \in {\cal A}$ by taking
$(s,t) \in{\cal K}_i$ iff $s/\overline{K_i} \subseteq t$.
We now define a
sequence $S^{j}$ of subsets of $S^{1}$.
Suppose that we have defined $S^1, \ldots, S^j$. $S^{j+1}$ consists of
all states in $S^j$ that {\em seem consistent}, in that
the following two
conditions hold:
\begin{enumerate}
\item If $\lnot E_{G}\psi\in s$, then there is some $t \in S^j$
such that $(s,t) \in \cup_{i \in G}{\cal K}_i$ and $\neg \psi \in t$.
\item If $\lnot C_{G}\psi\in s$,
then
there is some $t \in S^j$ such that
$t$ is $G$-reachable from $s$ in $S^j$ and $\neg \psi \in t$.
\end{enumerate}
If $S^j \ne S^{j+1}$
then we continue the construction. Otherwise the construction
terminates; in this case, the algorithm returns
``$\phi$ is satisfiable'' if $\phi \in s$ for some state
$s \in S^{j+1}$ and returns ``$\phi$ is unsatisfiable'' otherwise.
Since $S^{j} \supseteq S^{j+1}$, $S^{1}$ has at most
$2^{|\phi|}$ elements, and there are $|{\cal A}|$
relations,
it is easy to see that the whole procedure can be carried out
in time $O(|{\cal A}|2^{c|\phi|})$ for some $c > 0$.
It remains to show that the algorithm is correct.
{F}irst suppose that
$\phi$ is satisfiable. In that case, $(M,s_0)\models \phi$ for some
structure $M = (S,\pi, \{{\cal K}_i': i \in {\cal A}\})
\in {\cal M}_n$. We can associate with each state $s \in S$ the state $s^*$ in
$S^1$ consisting of all the formulas $\psi \in \mbox{\it Sub}(\phi)$ such that
$(M,s) \models \psi$. It is easy to see that if $(s,t) \in {\cal K}_i'$ then
$(s^*,t^*) \in {\cal K}_i$. A straightforward
induction shows that the states $s^*$ for $s \in S$ always seem
consistent, and thus are in $S^j$ for all $j$. Moreover, $\phi \in
s_0^*$. Thus, the algorithm declares that $\phi$ is satisfiable, as
desired.
Conversely, suppose that the algorithm declares that $\phi$ is
satisfiable.
We construct a structure $M = (S,\pi, \{{\cal K}_i': i \in {\cal A}\})$ over ${\cal A}$
and $\Phi$ in which
$\phi$ is satisfied as follows. Let $j$ be the stage at which the
algorithm terminates. Let $S = S^j$.
Define $\pi$ so that $\pi(s)(p) = {\bf true}$ iff $p
\in s$, for $s \in S$ and $p \in \Phi$. For each $i \in {\cal A}$,
we take ${\cal K}_i'$ to be the restriction of ${\cal K}_i$ to $S^j$.
A straightforward induction on the structure of formulas shows that
for all formulas $\psi \in \mbox{\it Sub}(\phi)$ and states $s \in S$, we have
$(M,s) \models\psi$ iff $\psi \in s$.
(The cases for $E_G \psi$ and $C_G \psi$ use the appropriate clauses of
the definition of seeming inconsistent and the choice of $j$.)
Since $\phi \in s$ for some $s^* \in
S$, it follows that $(M,s^*) \models \phi$, so $\phi$ is satisfiable.
To deal with ${\cal M}_n^r$, the only change necessary is that in going
from $S^1$ to $S^2$ in the construction, we also eliminate $s \in S^1$ if
$(s,s) \notin {\cal K}_i$ for some $i \in {\cal A}$. This guarantees that the
${\cal K}_i$ relations are reflexive. The remainder of the
proof goes through unchanged.
\vrule height7pt width4pt depth1pt\vspace{0.1in}
\bigskip
\noindent
{\bf Proof of Theorem~\ref{complexity} for ${\cal M}_n$ and ${\cal M}_n^r$:}
The deterministic exponential time lower bound in
Theorem~\ref{complexity} follows from the lower bound in the case
where ${\cal A}$ is finite, which is proved in \cite[Theorem~6.19]{HM2} using
techniques developed by Fischer and Ladner \cite{FL} for PDL.
The sets $G$ that arise in the lower bound proof have cardinality 2, so
oracles are of no help here.
{F}or the upper bound, suppose that we are given a formula $\phi$.
We first compute the set $\R(\G_\phi)$. This can be done with
at most $|\phi|2^{|\phi|}$
calls to oracle $O_0$,
since $|{\cal G}_\phi| \le |\phi|$ and we
need only check,
for each $G \in {\cal G}_\phi$ and ${\cal H} \subseteq {\cal G}_\phi$, whether $G - {\cal H} =
\emptyset$.
Consider the mapping $\sigma_1$ of Lemma~\ref{K}. By part (a)
of Lemma~\ref{K}, we can compute the formula $\phi^{\sigma_1}$
using $\le |\phi|2^{|\phi|}$ calls to oracle $O_0$. By
Proposition~\ref{trans} and
Lemma~\ref{K}, the formulas $\phi$ and $\phi^{\sigma_1}$ are
equisatisfiable. By Theorem~\ref{dec}, we can decide if $\phi^{\sigma_1}$
is satisfiable in time $O(2^{c|\phi|})$ for some $c > 0$ (since
$|\phi^{\sigma_1}| = |\phi|$ and the set ${\cal A}$ of agents that appear in
$\phi^{\sigma_1}$ has
size
at most $2^{|\phi|}$). \vrule height7pt width4pt depth1pt\vspace{0.1in}
We now want to prove Theorem~\ref{complete} in the case of ${\cal M}_n$ and
${\cal M}_n^r$. The idea is the same as that of Corollary~\ref{Shore2}. If
$\phi$ is valid, then so is $\phi^{\sigma_1}$. We can then appeal to
completeness in the case of finitely many agents to get a proof of
$\phi^{\sigma_1}$ that we can then ``pull back'' to a proof of $\phi$.
There is only one difficulty that we encounter
when
trying to put this idea
into practice. Exactly how do we pull back the proof? For example,
suppose that the proof of $\phi^{\sigma_1}$ involves a formula $\psi$
with an
operator $K_{\cal H}$. In general, there will be many agents $i \in {\cal A}$
such that $\sigma_1(i) = {\cal H}$. One option is to replace $K_{\cal H}$ by
$E_{\sigma_1^{-1}({\cal H})}$, that is, replace ${\cal H}$ by all $i$ such that
$\sigma_1(i) = {\cal H}$. (This is what was done in the proof of
Corollary~\ref{Shore2}.) The problem with this is that there is no
guarantee that the resulting set is in ${\cal G}$. Alternatively, we could
replace $K_{\cal H}$ by $K_i$ for some $i$ such that $\sigma_1(i) ={\cal H}$. But if
so, which one?
We actually take the latter course here. We solve the problem of which
$i$ to choose by showing that there is a proof of $\phi^{\sigma_1}$ in
which the only modal operators that arise in any formula used
in the proof are modal operators that appear in $\phi^{\sigma_1}$
(Lemma~\ref{proof}). For these operators,
there is a canonical way to do the replacement
(Lemma~\ref{replace}). While it may seem almost trivial that the only
operators that should be needed in the proof of $\phi^{\sigma_1}$ are
ones that already appear in the formula, this is not the case for the
standard completeness proof \cite{FHMV,HM2}, since in the proof of the
validity of a formula of the form $E_G \psi$, the modal operators $K_i$
are used for $i \in G$, although these operators may not appear in
$\psi$. It is important that we use the axioms E1 and E2 in doing the
proof, rather than the axiom E; otherwise the result would not hold.
Indeed, the result does not quite hold in the case of ${\rm T}_n^C$; we need
to augment it with E5.
\begin{lemma}\label{replace} The mapping $\sigma_1$ (when viewed as a map with
domain $2^{\cal A}$) is injective on ${\cal G}_\phi$.
\end{lemma}
\noindent{\bf Proof:} Suppose that $G \ne G'$. Without loss of generality, suppose that
$i
\in G - G'$. Then there is a $G$-maximal set ${\cal H}$ that includes $G'$.
By Lemma~\ref{K}(a), we have ${\cal H} \in \sigma_1(G)$. Since $G' \in {\cal H}$, it
follows from Lemma~\ref{K}(a) that ${\cal H} \notin \sigma_1(G')$. Thus,
$\sigma_1(G) \ne\sigma_1(G')$. \vrule height7pt width4pt depth1pt\vspace{0.1in}
{F}or the next lemma, we write $AX \vdash_\phi \psi$ if there is a proof
of $\phi$ in AX that involves only modal operators that appear in
$\phi$. Let $({\rm T}_\G^C)^+$ consist of ${\rm T}_n^C$
augmented with the axiom E5.
Although E5 follows from E1 and K2, using E5
allows us to be able to write proofs of $\phi$ that use only the modal
operators in $\phi$.
\begin{lemma}\label{proof} If ${\cal A}$ is finite and $\phi \in
{\cal L}_{\G}^C$ is
valid with respect to ${\cal M}_n$ (resp.,\ ${\cal M}_n^r$), then ${\rm K}_n^C
\vdash_\phi \phi$ (resp.,\ $({\rm T}_\G^C)^+ \vdash_\phi \phi$). \end{lemma}
\noindent{\bf Proof:} We first consider the case of ${\cal M}_n$.
Since $\phi$ is
valid, $\neg \phi$ is not satisfiable. That means, when we apply the
construction in the proof of Theorem~\ref{dec} to $\neg \phi$, all the
sets containing $\neg \phi$ are eliminated.
{F}or each state $s \in S^1$, let $\phi_s$ be
the conjunction of all the formulas in $s$.
We prove the result by showing, by induction on $j$, that
\begin{equation}\label{eq0}
\mbox{if a state $s \in S^j$ does not seem
consistent, then $\phi_s$ is ${\rm K}_n^C$-inconsistent, i.e.,~ ${\rm K}_n^C
\vdash_\phi \neg \phi_s$.}
\end{equation}
To see that (\ref{eq0}) suffices to
prove the lemma,
note that
standard propositional reasoning
(i.e., using Prop and MP) shows that,
for any formula $\psi \in \mbox{\it Sub}(\neg \phi)$,
$$
{\rm K}_n^C \vdash_\phi \psi
\Leftrightarrow \lor_{\{s \in S^1: \psi \in s\}} \phi_s.
$$
(Here we need the observation that by C1 and RC1, nothing is lost by
our assumption that $C_G\psi \in s$ iff $E_G(\psi \land C_G\psi) \in
s$.) Negating both sides of $\Leftrightarrow$, we get
\begin{equation}\label{eq-1}
{\rm K}_n^C \vdash_\phi \neg \psi
\Leftrightarrow \land_{\{s \in S^1: \psi \in s\}} \neg \phi_s.
\end{equation}
Thus, if ${\rm K}_n^C \vdash_\phi \neg \phi_s$ for each set $s$ containing
$\neg
\phi$, it follows by standard propositional reasoning that ${\rm K}_n^C
\vdash_\phi \phi$, as desired.
While this general approach to proving completeness is quite standard,
we must take extra care because of our insistence on restricting to
symbols that appear in $\phi$, particularly when dealing
with the case when a state seems inconsistent due to a formula of the
form $\neg E_G \psi$
or $\neg C_G \psi$
not being satisfied. This is where the axioms E1
and E2 come into play.
To prove (\ref{eq0}),
we first need a number of basic facts of epistemic logic and some
preliminary observations.
The basic facts (which are easily proved using Prop, E3 (or K1 when $G=\{i\}$),
MP, and EGen (or KGen);
see \cite[p.~51, 94]{FHMV}) are that if $\psi$ and $\psi'$ involve only
modal operators in $\phi$,
then
\begin{equation}\label{basic3}
{\rm K}_n^C \vdash_\phi E_G(\psi \land \psi') \Leftrightarrow E_G\psi \land E_G\psi'
\end{equation}
and
\begin{equation}\label{basic4}
\mbox{if ${\rm K}_n^C \vdash_\phi \psi \Rightarrow \psi'$ then
${\rm K}_n^C \vdash_\phi E_G \psi \Rightarrow E_G \psi'$.}
\end{equation}
Assume by induction that for all $s \in S^1 - S^j$, we have ${\rm K}_n^C
\vdash_\phi \neg \phi_s$.
We now show that if $s \in S^j$ does not seem consistent
then ${\rm K}_n^C \vdash_\phi \neg \phi_s$, by considering in turn each of the
two ways $s$ may seem inconsistent.
{F}irst suppose that $s$ does not seem consistent because $\neg E_G \psi \in s$
and there is no state $t \in S^j$ such that $(s,t) \in \cup_{i \in G}{\cal K}_i$
and $\neg \psi \in t$. We show
that
\begin{equation}\label{eq5}
{\rm K}_n^C \vdash_\phi \phi_s \Rightarrow E_G\psi.
\end{equation}
Since $\neg E_G \psi$ is a conjunct of $\phi_s$ (since $\neg E_G
\psi \in s$, by assumption),
(\ref{eq5})
shows that $\phi_s$ is
${\rm K}_n^C$-inconsistent, as desired.
To prove (\ref{eq5}),
we first show that if $G \in {\cal G}_\phi$, then
\begin{equation}\label{eq2}
\mbox{if $(s,t) \notin \cup_{i \in G}{\cal K}_i$,
then ${\rm K}_n^C \vdash_\phi \phi_s \Rightarrow E_G \neg \phi_t$.}
\end{equation}
To prove~(\ref{eq2}), suppose that $(s,t) \notin \cup_{i \in G} {\cal K}_i$.
{F}or each $i \in G$, there must be some $G^{i,t}
\in {\cal G}_\phi$ and formula $E_{G^{i,t}}
\theta$ such that $i \in G$, $E_{G^{i,t}} \theta \in s$ and $\neg \theta \in t$.
Since $E_{G^{i,t}} \theta \in s$ and $\neg \theta \in t$
it is immediate that
${\rm K}_n^C \vdash_\phi \phi_s \Rightarrow E_{G^{i,t}} \theta$ and
${\rm K}_n^C \vdash_\phi \theta \Rightarrow \neg \phi_t$. Now applying
(\ref{basic4}) and propositional reasoning, we
get that ${\rm K}_n^C \vdash_\phi
\phi_s \Rightarrow E_{G^{i,t}} \neg \phi_t$. Since we can find such a $G^{i,t}$
for each $i \in G$, we have that $G \subseteq \cup_{i \in G} G^{i,t}$.
Since $G$ is finite, by E2,
we have ${\rm K}_n^C \vdash_\phi \phi_s \Rightarrow E_G \neg \phi_t$, as desired.
Returning to the proof of (\ref{eq5}), note that
(since $E_G \psi \in s$) if $\neg \psi \in t$ then $(s,t) \notin
\cup_{i \in G} {\cal K}_i$. Thus, from (\ref{eq2}) and
(\ref{basic3}),
we have
\begin{equation}\label{eq4.5}
{\rm K}_n^C \vdash_\phi \phi_s \Rightarrow E_G(\land_{\{t \in S^j: \neg \psi \in t\}}
\neg \phi_t).
\end{equation}
By the induction hypothesis,
for all states in $t \in S^1 - S^j$, we have that ${\rm K}_n^C \vdash_\phi
\neg
\phi_t$. Thus, using (\ref{eq-1}), we have
\begin{equation}\label{eq4}
{\rm K}_n^C \vdash_\phi \psi \Leftrightarrow \land_{\{t \in S^j: \neg \psi \in t\}}
\neg\phi_t.
\end{equation}
(\ref{eq5}) now follows from (\ref{basic4}), (\ref{eq4.5}),
and (\ref{eq4}).
{F}inally, we must show that if $\neg C_G \psi \in s$ and
there is no state $t \in S^j$ $G$-reachable from $s$ in $S^j$ such that
$\neg \psi \in t$, then ${\rm K}_n^C \vdash_\phi\phi_s \Rightarrow C_G \psi$, again
showing that $\phi_s$ is ${\rm K}_n^C$-inconsistent. This follows by a
relatively straightforward modification of the completeness proof
given in \cite{FHMV,HM2}, so we just sketch
the details here. Let $T_1 = \{ t \in S^j : \neg C_G \psi \in t$ and
there is no
state $t' \in S^j$ $G$-reachable from $t$ in $S^j$ such that $\neg \psi
\in t'\}$ and $T_2 = \{t \in S^j: C_G\psi \in t\}$. Let $T_i'$
consist of those states in $T_i$ that also contain $\psi$, $i = 1,2$.
Let $T = T_1 \cup T_2$ and let $T' = T_1' \cup T_2'$.
We claim that
there is no pair $(t,t') \in \cup_{i \in G} {\cal K}_i$ such
that $t \in T$ and $t' \in S^j - T'$. It is immediate that if $t
\in T_2$ then (since $\psi \land C_G \psi \in t/E_G \subseteq t'$)
$t' \in T_2'$. If $t \in T_1$ and $t' \in S^j - T'$, then
either $\neg \psi \in t'$ or $\neg C_G \psi \in t'$ and there is a state
$t''$ $G$-reachable from $t'$ in $S^j$ such that $\neg \psi \in t''$.
This means that either $t'$ or $t''$ is a state $G$-reachable from
$t$
in $S^j$ containing $\neg \psi$. This contradicts the fact that
$t \in
T_1$.
It now follows from
(\ref{eq2})
that for all $t \in T$ and
$t' \in S^j - T'$, we have
\begin{equation}\label{eq6.5}
{\rm K}_n^C \vdash_\phi\phi_{t} \Rightarrow E_G \neg \phi_{t'}.
\end{equation}
Let $\phi_T = \lor_{t \in T} \phi_{t}$ and let $\phi_{T'} = \lor_{t'
\in T'} \phi_{t'}$. By propositional reasoning, we have
${\rm K}_n^C \vdash_\phi
\phi_{T'} \Leftrightarrow (\phi_T \land \psi)$.
It easily follows from
(\ref{basic3}), (\ref{basic4}), and (\ref{eq6.5}) that ${\rm K}_n^C \vdash_\phi
\phi_{t} \Rightarrow E_G
\phi_{T'}$. Since this is true for all $t \in T$, we have
\begin{equation}\label{eq7}
{\rm K}_n^C \vdash_\phi\phi_T \Rightarrow E_G(\phi_T \land \psi).
\end{equation}
By applying RC1 and the fact that $s \in T$, we have
${\rm K}_n^C \vdash_\phi\phi_s \Rightarrow C_G\psi$. Since $\neg C_G \psi \in s$, it
follows that $\phi_s$ is ${\rm K}_n^C$-inconsistent.
This completes the completeness proof in the case of ${\cal M}_n$. To deal
with ${\cal M}_n^r$, we must just show that if $s$ is eliminated because $(s,s)
\notin {\cal K}_i$ for some $i \in{\cal A}$, then ${\rm T}_n^C\vdash_\phi \neg \phi_s$;
all other cases are identical. But if $(s,s) \notin {\cal K}_i$, then there
must be some $G$ and $\psi$ such that $i \in G$, $E_G \psi \in s$, and
$\neg \psi \in s$. Since $({\rm T}_\G^C)^+$ includes the axiom $E_G \psi \Rightarrow
\psi$, we have that
$({\rm T}_\G^C)^+ \vdash_\phi \neg \phi_s$, as desired.
\vrule height7pt width4pt depth1pt\vspace{0.1in}
\bigskip
\noindent
{\bf Proof of Theorem~\ref{complete} for ${\cal M}_n$ and ${\cal M}_n^r$:} We have
already observed that the axioms are sound. For completeness,
suppose that $\phi$ is valid with respect to ${\cal M}_n$. By
Proposition~\ref{trans}, so is $\phi^{\sigma_1}$. By Lemma~\ref{proof},
there is a proof of $\phi^{\sigma_1}$ in ${\rm K}_{\G_\phi}^C$ that mentions only the
modal operators in $\phi^{\sigma_1}$.
Given a formula $\psi$ in which the only modal operators that appear are
modal operators that appear in $\phi^{\sigma_1}$ (and thus have the form
$E_{\sigma_1(G)}$, $C_{\sigma_1(G)}$, and $K_{\sigma_1(i)}$, for sets $G$
and $\{i\}$ in ${\cal G}_\phi$) let $\psi^{\tau_1}$ be the unique formula all
of whose modal operators appear in $\phi$ such that
$(\psi^{\tau_1})^{\sigma_1} = \psi$. Lemma~\ref{replace} assures us that
$\psi^{\tau_1}$ is well defined.
We can
pull the proof of $\phi^{\sigma_1}$ back to a proof of $\phi$, by
replacing
each occurrence of a formula $\psi$ in the proof by $\psi^{\tau_1}$.
The argument for ${\cal M}_n^r$ is identical, except that the proof
uses
instances of the axiom E5. These can be eliminated by using E1 and K2,
as we observed earlier (although now the proof of $\phi$ may use modal
operators $K_i$ that do not appear in $\phi$). \vrule height7pt width4pt depth1pt\vspace{0.1in}
\subsection{Dealing with ${\cal M}_{\cal A}^{\it rt}$}
Proposition~\ref{trans} as it stands does not hold for $\M_n^{\mbox{\scriptsize{{\it rt}}}}$.
There is no
guarantee that the translated formula is satisfiable in
$\M_n^{\mbox{\scriptsize{{\it rt}}}}$, even if $\phi$ is. Indeed, suppose that ${\cal G}$ is closed under
intersection and complementation, so that we can use the function
$\sigma$ of Proposition~\ref{Shore}. Suppose that $\phi$ is the
formula $E_G p \land \neg E_G E_G p$, where $|G| \ge 2$. The formula
$\phi^{\sigma}$
looks syntactically identical, except that
$\sigma(G)$
is
a single agent in ${\cal A}'$. We cannot make the ${\cal K}_G$ relation transitive
and still satisfy
$\phi^{\sigma}$.
More generally, to deal with
$\M_n^{\mbox{\scriptsize{{\it rt}}}}$,
we must be careful in how we deal with singleton sets.
As a first step, we define {\em mixed\/} structures.
Since we also
need these to deal with $\M_n^{\mbox{\scriptsize{{\it rst}}}}$ and $\M_n^{\mbox{\scriptsize{{\it elt}}}}$, we define three types of
mixed structures at once. We say that a binary relation ${\cal K}$ is {\em
secondarily reflexive\/} \cite{Chellas} if $(s,t) \in {\cal K}$ implies $(t,t)
\in {\cal K}$.
Let $\M_{\A_1+\A_2}^{\mbox{\scriptsize{{\it rt}}}}$
(resp.,\ $\M_{\A_1+\A_2}^{\mbox{\scriptsize{{\it rst}}}}$; $\M_{\A_1+\A_2}^{\mbox{\scriptsize{{\it elt}}}}$)
consist of structures $M = (S,\pi, \{{\cal K}_i: i \in
{\cal A}_1 \cup {\cal A}_2\})$ where the relations ${\cal K}_i$ for $i \in {\cal A}_1$
are reflexive and transitive
(resp.,\ reflexive, symmetric and transitive; Euclidean, serial and transitive)
and the relation ${\cal K}_i$ for $i \in {\cal A}_2$ are reflexive
(resp.,\ reflexive and symmetric; serial and secondarily reflexive).
We can now define our translation in the case of
$\M_n^{\mbox{\scriptsize{{\it rt}}}}$.
Although we can in fact get an analogue to Proposition~\ref{trans} for
$\M_n^{\mbox{\scriptsize{{\it rt}}}}$,
it turns out to be easier to provide a translation that combines
Proposition~\ref{trans} and Lemma~\ref{K}, rather than separating them.
As suggested by Proposition~\ref{reduction}, the translation involves
${\cal R}({\cal G}_\phi^1)$, rather than ${\cal R}({\cal G}_\phi)$.
Given a formula $\phi$,
let ${\cal A}^{\phi,rt} = \{{\cal H} : \exists G [(G,{\cal H}) \in {\cal R}({\cal G}_\phi^1)]\}$. Let
${\cal A}_1 = \{{\cal H}: \exists G [(G,{\cal H}) \in {\cal R}(G_\phi^1), \, |G- \cup {\cal H}| =
1]\}$; let ${\cal A}_2 = {\cal A}^{\phi,rt} - {\cal A}_1$.
Define $\sigma_2: {\cal A} \rightarrow {\cal A}^{\phi,rt}$
as before: $\sigma_2(i) = {\cal H}$ if $i \in A_{\cal H}$
and $\sigma_2(i)$
is undefined
otherwise.
Given ${\cal H} \in {\cal A}^{\phi,rt}$, we define
$\tau_2({\cal H}) = \cap ({\cal G}_\phi^1 - {\cal H})$. Since it is easy to see that
${\cal R}({\cal G}_\phi^1) = {\cal R}({\cal G}_\psi)$ for some
appropriate
$\psi$, it
is immediate that Lemma~\ref{K} applies
to
$\sigma_2$ and $\tau_2$.
\begin{proposition}\label{transrt}
$\phi$ is satisfiable in $\M_n^{\mbox{\scriptsize{{\it rt}}}}$ iff $\phi^{\sigma_2}$
is satisfiable in $\M_{\A_1+\A_2}^{\mbox{\scriptsize{{\it rt}}}}$.
\end{proposition}
\noindent{\bf Proof:}
{F}irst suppose that $(M,s)\models\phi$, where $M \in \M_n^{\mbox{\scriptsize{{\it rt}}}}$. We convert
$M = (S,\pi, \{{\cal K}_i: i \in {\cal A}\})$
into a structure $M' = (S,\pi, \{{\cal K}_{\cal H}: {\cal H} \in {\cal A}^{\phi,rt}\})$ as
before, by defining ${\cal K}_{{\cal H}} = \cup \{{\cal K}_i: i \in
\tau_2({\cal H})\}$. Since
Lemma~\ref{K} applies, the proof that
$(M',s) \models \phi$ is identical to that in Proposition~\ref{trans}. We
must only show that $M' \in \M_{\A_1+\A_2}^{\mbox{\scriptsize{{\it rt}}}}$. Since the union of reflexive
relations is reflexive, it is immediate that ${\cal K}_{{\cal H}}$ is reflexive for
${\cal H} \in {\cal A}_2$. If ${\cal H} \in {\cal A}_1$, then
$|A_{\cal H}| = 1$.
Suppose that $A_{\cal H} = \{i\}$. We claim that $\tau_2({\cal H}) = \{i\}$.
By construction, $\{i\} \in {\cal G}_\phi^1$. We cannot
have $\{i\} \in {\cal H}$, since $i \notin \cup {\cal H}$. Thus
$\{i \} \in {\cal G}_\phi^1 - {\cal H}$, so
$\tau_2({\cal H}) =
\cap({\cal G}_\phi^1 - {\cal H}) \subseteq \{i\}$. Since $\tau_2({\cal H}) \ne \emptyset$
by Lemma~\ref{K}(c), we must have $\tau_2({\cal H}) = \{i\}$. Thus, ${\cal K}_{\cal H} =
{\cal K}_i$, so ${\cal K}_{\cal H}$ is reflexive and transitive.
{F}or the opposite direction we need to work a little harder than before,
because we must ensure that all the ${\cal K}_i$ relations are reflexive and
transitive for all $i \in {\cal A}$. Supppose
$(M,s)\models\phi^{\sigma_2}$
for some $M = (S,\pi, \{{\cal K}_{\cal H}: {\cal H} \in {\cal A}^{\phi,rt}\}) \in \M_{\A_1+\A_2}^{\mbox{\scriptsize{{\it rt}}}}$.
Let $S_0$ and $S_1$ be two disjoint copies of $S$. For a state $s \in
S$, let $s_i$ be the copy of $s$ in $S_i$, $i = 0,1$.
Let $M' = (S',\pi', \{{\cal K}_i: i \in {\cal A}\})$ be defined as follows:
\begin{itemize}
\item $S' = S_0 \cup S_1$.
\item $\pi'(s_i) = \pi(s)$ for $i = 0,1$.
\item
If $\sigma_2(i) \in {\cal A}_1$, define ${\cal K}_i = \{(s_i, t_j): (s,t) \in
{\cal K}_{\sigma_2(i)}, i, j \in \{0,1\}\}$. ${\cal K}_i$ is clearly reflexive and
transitive in this case, since ${\cal K}_{\sigma_2(i)}$ is.
\item If $\sigma_2(i) = {\cal H} \in {\cal A}_2$,
note that $|A_{\cal H}| \ge 2$.
It is immediate from the definition that $\sigma_2(i) = {\cal H}$ for all
$i \in A_{\cal H}$. Pick some $i_{\cal H} \in A_{\cal H}$. If $i = i_{\cal H}$,
then define
${\cal K}_{i}=\{(s_0,t_1): (s,t) \in {\cal K}_{\cal H}\} \cup \{(s_j,s_j): j \in
\{0,1\}\}$; if $i \ne i_{\cal H}$, define ${\cal K}_i
= \{(s_1,t_0): (s,t) \in {\cal K}_{\cal H}\} \cup \{(s_j,s_j): j \in
\{0,1\}\}$. Clearly ${\cal K}_i$ is reflexive and transitive.
\end{itemize}
This construction guarantees that
\begin{equation}\label{eq8}
\mbox{$(s,t)
\in {\cal K}_{\cal H}$ iff
$(s_0,t_1), (s_1,t_0) \in \cup_{\{i:
\sigma_2(i) = {\cal H}\}}{\cal K}_i$}
\end{equation}
and
\begin{equation}\label{eq9}
\mbox{$(s_1,t_0) \in \cup_{\{i: \sigma_2(i) = {\cal H}\}}{\cal K}_i$ iff
$(s_0,t_1) \in \cup_{\{i: \sigma_2(i) = {\cal H}\}}{\cal K}_i$.}
\end{equation}
A straightforward argument by induction on structure now
shows that if $\psi \in {\cal L}_{{\cal G}_\phi^1}^C$, then the following are
equivalent for all $t \in S$:
\begin{itemize}
\item $(M,t) \models \psi^{\sigma_2}$,
\item both $(M',t_0) \models \psi$ and $(M',t_1) \models \psi$,
\item $(M',t_0) \models \psi$ or $(M',t_1) \models \psi$.
\end{itemize}
Of course, the interesting cases are if $\psi$ is of the form $K_i
\psi'$, $E_G \psi'$, or $C_G\psi'$. These follow immediately from
observations (\ref{eq8}) and (\ref{eq9}).
\vrule height7pt width4pt depth1pt\vspace{0.1in}
The next step is to get an analogue of Theorem~\ref{dec} for
$\M_{\A_1+\A_2}^{\mbox{\scriptsize{{\it rt}}}}$.
The basic idea of the proof is the same as that of
Theorem~\ref{dec}. However, in our construction, we need to make
the ${\cal K}_i$ relations transitive. To see the difficulty, suppose that
$\phi$
is $K_1 p \land E_G q$, where $G$ is a set of agents containing 1.
Recall that in Theorem~\ref{dec}, states are consistent subsets of
$\mbox{\it Sub}^+(\phi)$.
Let $s$, $t$, and $u$ be states such that $s= \{K_1 p, E_G q, p, q\}$,
$t = \{K_1 p, \neg E_G q, p, q\}$, and $u = \{K_1 p, \neg E_G q, p, \neg
q\}$. With our previous construction, we would have both $(s,t) \in
{\cal K}_1$
and $(t,u) \in {\cal K}_1$. By transitivity, we should also have $(s,u)
\in {\cal K}_1$. But since $E_G q \in s$ and $\neg q \in u$, we have
$(s,u) \notin {\cal K}_1$. Nevertheless,
each of $s$, $t$, and $u$
individually seems consistent.
Which
state should we eliminate in order to preserve transitivity?
To deal with this problem, we need to put more information
(i.e., more formulas) into each state. Intuitively, if $(s,t) \in
{\cal K}_i$, then we should have $K_i q \in t$, because if $E_G q \in s$, then
$K_i q$ should also be in $s$, as should $K_i K_i q$ by K4. It would
then follow that $K_i q$ should be in $t$. This, in turn, would
guarantee that $(t,u) \notin {\cal K}_i$, since $q \notin u$.
What we would like to do now is to augment $\mbox{\it Sub}(\phi)$ by including all
formulas $K_i \psi$ such that $E_G \psi \in \mbox{\it Sub}(\phi)$ and $i \in G
\cap {\cal A}_1$. (We restrict to ${\cal A}_1$ since these are the only relations
that are required to be transitive.)
While this approach can be used to force the ${\cal K}_i$ relations
to be transitive, the resulting set of formulas can
have size
$O(|{\cal A}_1||\phi|)$, which means the resulting state space (the analogue
of $S^1$)
could then
have size $2^{|{\cal A}_1||\phi|}$. This would not give us the desired
complexity bounds. Thus, we must proceed a little more cautiously.
\begin{theorem}\label{decrt}
If ${\cal A} = {\cal A}_1 \cup {\cal A}_2$ is finite and there is an algorithm for
deciding if $i
\in G$ for $G \in {\cal G}$ that runs in time linear in $|{\cal A}|$, then
there is a constant $c > 0$ (independent of ${\cal A}$) and
an algorithm
that, given a formula
$\phi$ of ${\cal L}_{\G}^C$, decides if $\phi$ is satisfiable
in
$\M_{\A_1+\A_2}^{\mbox{\scriptsize{{\it rt}}}}$
and runs in time $O(|{\cal A}|2^{c|\phi|})$.
\end{theorem}
\noindent{\bf Proof:}
We assume for ease of exposition that ${\cal A}_1 \ne \emptyset$; we leave the
straightforward modification in case ${\cal A}_1 = \emptyset$ to the reader.
{F}or each $i \in {\cal A}_1$, let $\mbox{\it ESub}_i(\phi)$
be the least set containing $\mbox{\it Sub}(\phi)$ such that if $E_G \phi \in
\mbox{\it Sub}(\phi)$ and $i \in G$, then $K_i \phi \in
\mbox{\it ESub}_i(\phi)$.
It is easy to see that
$|\mbox{\it ESub}_i(\phi)| \le 2|\mbox{\it Sub}(\phi)|$, since we add at most
one formula for each formula in $|\mbox{\it Sub}(\phi)|$.
Let $S^1_i$ consist of all the subsets of $\mbox{\it ESub}^+_i(\phi)$ that are
maximally consistent, and now let $S^1 = \cup_{i \in {\cal A}_1} S^1_i$.
Note that, as modified, $|S^1| \le 2^{2|\phi|}$. Thus,
this modification
keeps us safely within the
desired exponential time bounds.
We keep the definition of ${\cal K}_i$ unchanged for $i \in {\cal A}_2$
(i.e.,~ $(s,t) \in {\cal K}_i$ iff $s/\overline{K}_i \subseteq t$),
but we
need to modify it for $i
\in {\cal A}_1$. We redefine ${\cal K}_i$ for $i \in {\cal A}_1$ by defining
$(s,t) \in {\cal K}_i$ iff
$s/\overline{K_i} \cup \{K_i \psi: K_i \psi \in s\} \subseteq
t \cap (t/\overline{K_i} \cup \{K_i \psi: K_i \psi \in t\})$.
It is easy to check that this modification forces the ${\cal K}_i$
relations to be transitive. We force all the ${\cal K}_i$ relations to be
reflexive just as with ${\cal M}_n^r$, by eliminating $s \in S^1$ if
$(s,s) \notin {\cal K}_i$ for some $i \in {\cal A}_1 \cup {\cal A}_2$.
The remainder of the construction---eliminating the states that do not
seem consistent---is unchanged.
We now need to show that the algorithm is correct. First suppose that
$\phi$ is satisfiable in $\M_{\A_1+\A_2}^{\mbox{\scriptsize{{\it rt}}}}$.
In that case, $(M,s_0)\models \phi$ for
some structure $M = (S,\pi, \{{\cal K}_i': i \in {\cal A}_1 \cup {\cal A}_2\})
\in \M_{\A_1+\A_2}^{\mbox{\scriptsize{{\it rt}}}}$. We can associate with each state $s \in S$ and $i \in
{\cal A}_1$ the state $s^*_i$ in
$S^1_i$ consisting of all the formulas $\psi \in
\mbox{\it ESub}_i(\phi)$
such that
$(M,s) \models \psi$. It is easy to see that if $(s,t) \in {\cal K}_i'$ then
$(s^*_j,t^*_i) \in {\cal K}_i$ for all $j$.%
\footnote{Note that it is not necessarily the case
that $(s^*_j,t^*_{j'}) \in {\cal K}_i$ for $j' \ne i$. For example, suppose
$\phi$ is the formula $E_G p$, $i \in G \cap {\cal A}_1$, and $M$ is such
that $(M,s)
\models E_G p \land p$, $(M,t) \models \neg E_G p \land p$, and $(s,t) \in
{\cal K}_i$. Then for $i \ne j, j'$ and
$j\notin G$, we have $s^*_j = \{E_G
p, p\}$ and $t^*_{j'} = \{p,\neg E_G p\}$. Since $p \in s^*_j/\overline{K_i} -
t^*_{j'}/\overline{K_i}$, we have that $(s^*_j,t^*_{j'}) \notin {\cal K}_i$.}
Using this observation, a straightforward
induction shows that the states $s^*_i$ for $s \in S$ always seem
consistent, and thus are in $S^j$ for all $j$ and all $i \in
{\cal A}_1$. Moreover, $\phi \in
(s_0)^*_i$ for all $i \in {\cal A}_1$. Thus, the algorithm will declare
that $\phi$ is satisfiable, as desired.
Conversely, suppose that the algorithm declares that $\phi$ is
satisfiable.
We construct a structure $M = (S,\pi, \{{\cal K}_i': i \in {\cal A}_1 \cup {\cal A}_2\})
\in \M_{\A_1+\A_2}^{\mbox{\scriptsize{{\it rt}}}}$ in which $\phi$ is satisfied just as
Theorem~\ref{dec}. Our modified construction
guarantees that the ${\cal K}_i'$ relations are all reflexive and the ones in
${\cal A}_1$ are transitive. \vrule height7pt width4pt depth1pt\vspace{0.1in}
We are almost ready to prove Theorem~\ref{complexity} for $\M_n^{\mbox{\scriptsize{{\it rt}}}}$.
However, we first we need to characterize the complexity of
translating from $\phi$ to $\phi^{\sigma_2}$. In particular, we need a
bound on the number of elements in ${\cal R}({\cal G}_\phi^1)$ and the
number of oracle calls required to compute them.
To do this, we first define two
auxiliary sequences of sets ${\cal D}_i^m({\cal J})$
and ${\cal E}_i^m({\cal J})$, $i = 1, 2, 3, \ldots$. (We omit the parenthetical ${\cal J}$
when it is clear
from context.) Fix $m$. Let ${\cal D}_0^m = {\cal J}$ and ${\cal D}_{i+1}^m =
{\cal J} \cup
\{G - \cup{\cal H}:
(G,{\cal H}) \in {\cal R}({\cal D}_i^m) \mbox{ and } |G - \cup{\cal H}|\leq m\}$;
let ${\cal E}_i^m = {\cal D}_i^m - {\cal D}_0^m$.
Set ${\cal D}^m = \cup_i {\cal D}_i^m$ and ${\cal E}^m = \cup_i {\cal E}_i^m$.
{F}inally, denote ${\cal R}({\cal D}^m)$ by ${\cal R}^m({\cal J})$.
It is easy to check that ${\cal D}_0^m \subseteq {\cal D}_1^m \subseteq \ldots$
and that ${\cal R}^m({\cal J}) = \cup_i {\cal R}({\cal D}^m_i)$.
The next lemma provides partial motivation for these definitions.
\begin{lemma}\label{equiv}
${\cal R}^m({\cal J}) = {\cal R}({\cal J}^m)$.
\end{lemma}
\noindent{\bf Proof:} An easy induction on $i$ shows that ${\cal D}^m_i({\cal J}) \subseteq {\cal J}^m_i$
(as defined in Definition~\ref{Gm}) for all $i$, so ${\cal D}^m\subseteq
{\cal J}^m$. We next show that
every set in ${\cal J}^m_i$ is the union of sets in ${\cal D}^m$, by induction on
$i$. This is immediate if $i=0$, since ${\cal J}^m_0 = {\cal D}^m_0 = {\cal J}$. Suppose
that the result holds for ${\cal J}^m_i$; we show it for ${\cal J}^m_{i+1}$.
Suppose that
$H \in {\cal J}^m_{i+1}$. If $H \in {\cal J}$, then clearly $H \in {\cal D}^m$. Thus,
without loss of generality, $H \in {\cal J}^m_{i+1} - {\cal J}$, which means that
$|H| \le m$. Let $H'$ be the union of all sets in ${\cal D}^m$ contained in
$H$. If $H' = H$, then we are done. Suppose by way of contradiction
that $H
- H' \ne \emptyset$. We obtain a contradiction to the choice of
$H'$ by showing that $H - H'$ contains a set in ${\cal D}^m$.
Since $H'$ is finite, it can be written as a finite union
of
sets in
${\cal D}^m$, say of
${\cal H}_1 = H_1, \ldots, H_k$. Since $H \in {\cal J}^m_{i+1} - {\cal J}$,
$H = G - \cup {\cal H}_2$ for
some $G \in {\cal J}$ and ${\cal H}_2 \subseteq {\cal J}^m_i$. By the induction
hypothesis,
there exists some ${\cal H}_3 \subseteq {\cal D}^m$ such that $\cup {\cal H}_2 = \cup
{\cal H}_3$. There must exist some set ${\cal H}_4 \supseteq {\cal H}_1
\cup {\cal H}_3$ such that $(G, {\cal H}_4) \in {\cal R}({\cal D}^m)$. But then $H - H'
\supseteq G - \cup {\cal H}_4 \in {\cal D}^m$, and we obtain the desired
contradiction.
It now easily follows that ${\cal R}({\cal J}^m) = {\cal R}({\cal D}^m) = {\cal R}^m({\cal J})$. \vrule height7pt width4pt depth1pt\vspace{0.1in}
The following result will be used to help compute the elements of
${\cal R}^m({\cal J})$.
\begin{lemma}\label{Gmcount}
Let ${\cal J}$ be a set of subsets of ${\cal A}$ with $|{\cal J}| = n$.
\begin{itemize}
\item[(a)] If $(G,{\cal H}) \in {\cal R}({\cal D})$, where ${\cal J} \subseteq {\cal D} \subseteq
{\cal J}^*$, then $G - \cup {\cal H}$ is an atom over ${\cal J}$.
\item[(b)] ${\cal J} \subseteq {\cal D}_i^m \subseteq {\cal J}^*$ for all $i, m$.
\item[(c)] $|\{{\cal H}: \exists G \in {\cal D}^m((G,{\cal H}) \in {\cal R}^m({\cal J})\}| \le 2^n$.
\item[(d)] If $(G,{\cal H}) \in {\cal R}({\cal D}_i^m)$, then
either $G \in {\cal J}$ and ${\cal E}_i^m\subseteq {\cal H}$ or
$A_{\cal H} \in {\cal E}_i^m$ and ${\cal E}_i^m - \{A_{\cal H}\} \subseteq {\cal H}$.
Moreover,
if $(G,{\cal H}) \in {\cal R}^m({\cal J})$, then
either $G \in {\cal J}$ and ${\cal E}^m\subseteq {\cal H}$ or $(G - \cup{\cal H}) \in
{\cal E}^m$ and ${\cal E}^m - \{G- \cup {\cal H}\} \subseteq {\cal H}$.
\item[(e)] ${\cal D}^m = {\cal D}_n^m$ and ${\cal E}^m = {\cal E}_n^m$.
\end{itemize}
\end{lemma}
\noindent{\bf Proof:}
{F}or part (a), we know from Lemma~\ref{unique}(a) that if $(G,{\cal H}) \in
{\cal R}({\cal D})$, then $G - \cup {\cal H}$ is an atom over ${\cal D}$. Since
${\cal J} \subseteq {\cal D} \subseteq
{\cal J}^*$, it is immediate that it must in fact be an atom over ${\cal J}$ as
well.
(Recall that ${\cal J}^*$ is the algebra generated by ${\cal J}$.)
Part (b) follows immediately from (a),
since an easy induction on $i$ shows that ${\cal E}_i^m \subseteq {\cal J}^*$.
For part (c), by Lemma~\ref{unique}(a), it follows that
$A_{\cal H}$ is an atom over
${\cal D}^m$. But since ${\cal J} \subseteq {\cal D}^m = \cup_i {\cal D}_i^m \subseteq {\cal J}^*$
by part
(b), it follows that $A_{\cal H}$ is actually at atom over ${\cal J}$. Moreover if
$(G',{\cal H}') \in {\cal R}^m({\cal J})$ and ${\cal H} \ne {\cal H}'$, then it follows from
Lemma~\ref{unique}(c) that $A_{\cal H} \ne A_{{\cal H}'}$. Since there are at most
$2^n$ atoms over ${\cal J}$, part (c) follows.
{F}or part (d), if $(G, {\cal H})
\in {\cal R} (D_i^{m})$ then,
by Lemma~\ref{unique}(a), $A_{\cal H} = G - \cup {\cal H}$ is an atom over
${\cal D}_i^m$ and has the form
$\cap ({\cal D}_i^m - {\cal H})\cap\cap \{\overline H :H \in {\cal H}\} $.
By the arguments of part (c), $A_{\cal H}$ is also an atom over ${\cal J}$.
We say that the
sets in ${\cal D}_i^m - {\cal H}$ {\em appear positively\/} in $A_{\cal H}$ and the sets
in
${\cal H}$ {\em appear negatively\/} in $A_{\cal H}$.
If one of the sets $G' \in {\cal E}_i^m$ appears positively in $A_{\cal H}$
then clearly $A_{\cal H} \subseteq G'$.
But since the elements of ${\cal E}_i^m$ are also atoms over ${\cal J}$, it follows
that in this case $A_{\cal H} = G' \in {\cal E}_i^m$ and,
since ${\cal H}$ is $G$-maximal,
${\cal E}_i^m - \{A_{\cal H}\} \subseteq {\cal H}$.
Otherwise, ${\cal E}_i^m\subseteq {\cal H}$ as required;
moreover, since ${\cal D}_i^m = {\cal E}_i^m \cup {\cal J}$ and $G \notin {\cal H}$, we must
have $G \in {\cal J}$. The argument for the second half of (d) is identical.
Clearly the two claims in part (e) are equivalent. We prove the second.
As observed in the proof of (c), every set in
${\cal E}^m$
is an atom $A$ over ${\cal J}$.
It is easy to see that there are no
atoms in
${\cal E}^m$
where all $n$ sets in ${\cal J}$ appear negatively, since
every set in
${\cal E}^m$
is a nonempty subset of some $G \in {\cal J}$.
(This can be proved by induction on $i$ for each ${\cal E}^m_i$.)
We prove by induction on $i$ that if
$A \in {\cal E}^m$ and $n-i$ sets appear negatively in $A$ for $i
\ge 1$, then $A \in {\cal E}^m_{i}$.
Clearly if $i=1$, then $A = G - (H_1 \cup \ldots \cup H_{n-1})$,
and ${\cal H} = \{H_1, \ldots, H_{n-1}\}$ is a $G$-maximal subset of ${\cal J}$.
Thus, $(G,{\cal H}) \in {\cal D}_1^m$ and $A \in {\cal E}_1^m$. Suppose that the
result is true if $i=k$ and suppose that $n-(k+1)$ sets appear
negatively in $A$. As $A \in {\cal E}^m$,
there must be some minimal $j$ such that
$A \in {\cal E}_{j+1}^m$.
By definition, $A = {\cal A}_H$ for some $(G,{\cal H}) \in {\cal R}({\cal D}_j^m)$.
By (d), either
$A = G - (\cup{\cal H}' \cup {\cal E}_j^m)$ and ${\cal H}' \subseteq {\cal J}$ or $A \in
{\cal E}_j^m$.
The latter case contradicts our choice of $m$,
so we may assume that
$A = G - (\cup{\cal H}' \cup
{\cal E}_j^m)$ and ${\cal H}' \subseteq {\cal J}$.
It is easy to see that ${\cal H}'$ must
consist of precisely the sets in ${\cal J}$ that appear negatively in $A$.
(If it did not include all the sets that appear negatively in $A$
then ${\cal H}'
\cup {\cal E}_j^m$ would not be a $G$-maximal subset of ${\cal J} \cup {\cal E}_j^m$;
if it includes any sets that appear postively then $A$ would be
empty.) Let ${\cal E}'$ consist of all the atoms $A'$ in ${\cal E}_j^m$ in
which the set of sets in ${\cal J}$ that appear negatively in $A'$ is a
strict superset of
${\cal H}'$. It is easy to see that $G - (\cup{\cal H}' \cup {\cal E}_j^m) =
G - (\cup{\cal H}'
\cup {\cal E}')$, since all the sets in ${\cal E}_j^m - {\cal E}'$ must be disjoint
from $G - \cup{\cal H}'$.
(This is clear for the $B \in {\cal E}_j^m - {\cal E}'$ for which some set appearing
negatively in
$A$ does not appear negatively in $B$. On the other hand, if the same
sets appear negatively in $B$ as in $A$ then $B=A$ and we contradict
the minimality of $j$.)
By the induction hypothesis, ${\cal E}' \subseteq
{\cal E}^m_{n-k}$. Thus,
$A = G - ({\cal H}' \cup {\cal E}_{n-k}^m) \in {\cal E}_{n-k+1}^m$,
as desired. \vrule height7pt width4pt depth1pt\vspace{0.1in}
We remark that a simpler proof, just using the fact that there are at
most $2^n$ atoms over ${\cal J}$, can be used to show that ${\cal E}^m_{n'} =
{\cal E}^m_{2^n}$ for $n' > 2^n$. This simpler proof would suffice for the
purposes of this subsection. However, we shall use the added
information in (e) in Section~\ref{oracle}.
\bigskip
\noindent
{\bf Proof of Theorem~\ref{complexity} for $\M_n^{\mbox{\scriptsize{{\it rt}}}}$:}
Again, the lower bound follows from standard results in \cite{HM2}.
{F}or the upper bound, suppose that
we are given a formula $\phi$ such that $n
= |\phi|$
and ${\cal H} \in {\cal A}^{\phi,rt}$. By definition, there exists a
$G$ such that $(G,{\cal H}) \in {\cal R}({\cal G}_\phi^1)$.
By Lemma~\ref{equiv}, ${\cal R}^1({\cal G}_\phi) = {\cal R}({\cal G}_\phi^1)$.
Thus, ${\cal H} \subseteq {\cal D}^1({\cal G}_\phi) = {\cal G}_\phi \cup E^1_n({\cal G}_\phi)$.
By Lemma~\ref{Gmcount}(d), either ${\cal E}^1_n({\cal G}_\phi) \subseteq {\cal H}$ or
${\cal H}$ contains all but one element of ${\cal E}^1_n({\cal G}_\phi)$. Thus,
we can uniquely characterize ${\cal H}$ by a pair
$({\cal H}',X)$, where
${\cal H}' = {\cal H} \cap {\cal G}_\phi$
and $X = {\cal E}^1_n({\cal G}_\phi) - {\cal H}$ (so that $X$ is either the empty set or a
singleton).
It should be clear that we can compute
compute the set ${\cal E}^1_{n}({\cal G}_\phi)$ in time $O(n^2 2^{cn})$ and
which of these (at most $2^{2n} + 2^n$)
pairs
is in ${\cal A}_1$ and
${\cal A}_2$ using at most $2n(2^{2n} + 2^n)$ calls to the oracle $O_1$.
By
Lemmas~\ref{K}(a) and~\ref{Gmcount}, we can
similarly
compute the
formula $\phi^{\sigma_2}$ in time
$O(2^{cn})$ using $O(2^{cn})$ oracle calls. We now
apply Proposition~\ref{transrt} and Theorem~\ref{decrt}, just as we
applied Proposition~\ref{trans} and Theorem~\ref{dec} in the case of
${\cal M}_n$. \vrule height7pt width4pt depth1pt\vspace{0.1in}
We next want to prove Theorem~\ref{complete} for $\M_n^{\mbox{\scriptsize{{\it rt}}}}$. Just as with
${\cal M}_n$ and ${\cal M}_n^r$, we want to pull a proof of $\phi^{\sigma_2}$ back to
a proof of $\sigma$. However, it is no longer true that we can
necessarily prove $\phi^{\sigma_2}$ using only the modal operators that
appear in $\phi^{\sigma_2}$. We may also need to use $K_{\cal H}$ for ${\cal H} \in
{\cal A}_1$. Fortunately, this does not cause us problems. The following
extension of Lemma~\ref{replace} is immediate.
\begin{lemma}\label{replacert} The mapping $\sigma_2$ (when viewed as a map with
domain $2^{\cal A}$) is injective on
${\cal G}_\phi^1$.
\end{lemma}
Let $({\rm S4}_\G^C)^{\A_1+\A_2}$ consist of the axioms in $({\rm T}_\G^C)^+$ (so that, in
particular, E5 is included), together with every instance of K4 ($K_i
\phi \Rightarrow K_i K_i \phi$) for $i \in {\cal A}_1$. We write $({\rm S4}_\G^C)^{\A_1+\A_2}
\vdash_\phi \psi$ if there is a proof of $\psi$ in $({\rm S4}_\G^C)^{\A_1+\A_2}$ using
only the modal operators that appear in $\phi$ and $K_i$ for $i \in
{\cal A}_1$.
\begin{lemma}\label{proofrt} If ${\cal A}$ is finite and $\phi \in {\cal L}_{\G}^C$ is
valid with respect to $\M_{\A_1+\A_2}^{\mbox{\scriptsize{{\it rt}}}}$, then $({\rm S4}_\G^C)^{\A_1+\A_2}
\vdash_\phi \phi$. \end{lemma}
\noindent{\bf Proof:} The proof is similar to that of Lemma~\ref{proof} for
${\cal M}_n^r$, except
that since the definition of the ${\cal K}_i$ relation is different, we must
still check that the results still hold with the modified definition.
Suppose that $s \in S^j$ does not seem consistent because $\neg E_G \psi
\in s$ and there is no state $t \in S^j$ such that
$(s,t) \in \cup_{i \in G}{\cal K}_i$ and $\neg \psi \in t$.
We want to show that $({\rm S4}_\G^C)^{\A_1+\A_2} \vdash_\phi \phi_s \Rightarrow E_G \psi$.
As before this suffices.
{F}or each $i \in G$ and, by induction, each $j$, we have a provable
equivalence for $\psi$ similar to the one
before: $({\rm S4}_\G^C)^{\A_1+\A_2} \vdash_\phi \psi
\Leftrightarrow \land_{\{t \in {S_i}^j): \neg\psi \in t\}} \neg\phi_t$. So it suffices
to find, for each
such $i$ and each $t \in S^j_i$ with $\neg\psi \in t$,
a $G^{i,t}$ containing $i$ such that $({\rm S4}_\G^C)^{\A_1+\A_2} \vdash_\phi
\phi_s \Rightarrow E_{G^{i,t}} \neg\phi_t$.
{F}or $i \in {\cal A}_2$, this follows just as before.
{F}or $i \in {\cal A}_1$, we show that $({\rm S4}_\G^C)^{\A_1+\A_2} \vdash_\phi
\phi_s \Rightarrow K_i \neg \phi_t$.
By our assumption $(s,t) \notin {\cal K}_i$.
Thus, there exists some formula $\theta \in s/\overline{K_i} \cup \{K_i
\theta: K_i \theta \in s\} - (t \cap (t/\overline{K_i} \cup \{K_i \theta: K_i \theta
\in t\}))$. If $\theta \in s/\overline{K_i}$, then
$({\rm S4}_\G^C)^{\A_1+\A_2} \vdash_\phi \phi_s \Rightarrow K_i \theta$.
If $\theta = K_i \theta'$ is in $s$, then $({\rm S4}_\G^C)^{\A_1+\A_2} \vdash_\phi \phi_s \Rightarrow
K_i \theta'$. By K4, we have that $({\rm S4}_\G^C)^{\A_1+\A_2} \vdash_\phi \phi_s \Rightarrow K_i
K_i \theta'$. Thus, in either case, we have $({\rm S4}_\G^C)^{\A_1+\A_2} \vdash_\phi \phi_s
\Rightarrow K_i \theta$.
Since $\theta \in s/\overline{K_i} \cup \{K_i \theta: K_i \theta \in s\}$,
it follows that $K_i \theta \in \mbox{\it ESub}_i(\neg \phi)$. We cannot have $K_i
\theta \in t$, for then (since $(t,t) \in {\cal K}_i$, so $t/\overline{K_i} \subseteq t$)
we would have $\theta \in t \cap t/\overline{K_i}$, contradicting our choice of
$\theta$. Thus
we must have that $\neg K_i \theta \in t$. It follows that $({\rm S4}_\G^C)^{\A_1+\A_2}
\vdash_\phi K_i \theta \Rightarrow \neg \phi_t$. Using (\ref{basic4}), we get
that $({\rm S4}_\G^C)^{\A_1+\A_2} \vdash_\phi K_i K_i \theta \Rightarrow K_i \neg \phi_t$.
Since $({\rm S4}_\G^C)^{\A_1+\A_2} \vdash_\phi K_i \theta \Rightarrow K_i K_i \theta$ and, as
shown earlier,
$({\rm S4}_\G^C)^{\A_1+\A_2} \vdash_\phi \phi_s \Rightarrow K_i \theta$, it follows that
$({\rm S4}_\G^C)^{\A_1+\A_2} \vdash_\phi \phi_s \Rightarrow K_i \neg \phi_t$, as desired.
{F}inally, we must show that if $\neg C_G \psi \in s$ and
there is no state $t \in S^j$ $G$-reachable from $s$ in $S^j$ such that
$\neg \psi \in t$, then ${\rm S4}_n^C \vdash_\phi\phi_s \Rightarrow C_G \psi$.
This argument is identical to that given in the proof of
Lemma~\ref{proof}, so we do not repeat it here.
\vrule height7pt width4pt depth1pt\vspace{0.1in}
\bigskip
\noindent
{\bf Proof of Theorem~\ref{complete} for $\M_n^{\mbox{\scriptsize{{\it rt}}}}$:} Again, we have
already observed that the axioms are sound. For completeness, suppose
that $\phi$ is valid with respect to ${\cal M}_n$. By
Proposition~\ref{transrt}, $\phi^{\sigma_2}$ is valid with respect to
$\M_{\A_1+\A_2}^{\mbox{\scriptsize{{\it rt}}}}$.
By Lemma~\ref{proofrt},
there is a proof of $\phi^{\sigma_2}$ in ${\rm K}_{\G_\phi}^C$ that mentions only the
modal operators in $\phi^{\sigma_2}$ and the operators $K_{\cal H}$ for ${\cal H}
\in{\cal A}_1$. Using Lemma~\ref{replacert}, it follows that we can
pull this back to a proof of $\phi$ in ${\rm S4}_n^C$. \vrule height7pt width4pt depth1pt\vspace{0.1in}
\subsection{Dealing with ${\cal M}_{\cal A}^{\it rst}$}
$\M_n^{\mbox{\scriptsize{{\it rst}}}}$ and $\M_n^{\mbox{\scriptsize{{\it elt}}}}$ introduce additional complications. The
translation used in Proposition~\ref{transrt} no longer suffices.
We need to deal with the fact that in $\M_n^{\mbox{\scriptsize{{\it rst}}}}$, we can test not only
that whether a set is a singleton, but whether it has size $k$ for any
$k$.
Given a formula $\phi$, suppose that $|\phi| = n$.
We want to map ${\cal A}$ to a finite set of agents and prove an analogue
of
Propositions~\ref{transrt}.
The obvious analogue of ${\cal A}^{\phi,rt}$ would be to consider the sets
${\cal H}$ such that $(G,{\cal H}) \in
{\cal R}({\cal G}_\phi^n)$.
We essentially do this,
except that we replace all sets of cardinality $\le n$ by the singletons
in them.
Given a set ${\cal J}$ of subsets of ${\cal A}$, let $\widetilde{{\cal J}}^m = {\cal D}^m({\cal J})
\cup \{\{i\}: \exists G \in {\cal D}^m({\cal J})
(|G| \le m, \, i \in G\}$.
Let ${\cal A}^{\phi,rst} = \{{\cal H} : \exists G ((G,{\cal H}) \in
{\cal R}(\widetilde{{\cal G}}_\phi^{n})\}$. Let
${\cal A}_1 = \{{\cal H}: \exists G [(G,{\cal H}) \in {\cal R}(\widetilde{{\cal G}}_\phi^{n}), \, |G- \cup
{\cal H}| = 1\}$; let ${\cal A}_2 = {\cal A}^{\phi,rst} - {\cal A}_1$.
Define $\sigma_3: {\cal A} \rightarrow {\cal A}_1 \cup {\cal A}_2
$ as before: $\sigma_3(i) = {\cal H}$ if
$i \in A_{\cal H}$ and $\sigma_3(i)$
is undefined
otherwise.
Much
as before, we define
$\tau_3({\cal H}) = \cap (\widetilde{{\cal G}}_\phi^{n} - {\cal H})$. Since it is easy to
see that
${\cal R}(\widetilde{{\cal G}}_\phi^{n}) = {\cal R}({\cal G}_\psi)$ for some appropriately chosen
$\psi$, it
is immediate that Lemma~\ref{K} applies without change to
$\sigma_3$ and $\tau_3$.
\begin{lemma}\label{separation}
If ${\cal H} \in {\cal A}_2$, then $|A_{\cal H}| \ge n+1$. \end{lemma}
\noindent{\bf Proof:} Suppose, by way of contradiction, that ${\cal H} \in {\cal A}_2$ and $1 \le
|A_{\cal H}| \le n$. We must have $|A_{\cal H}| > 1$, for otherwise ${\cal H} \in {\cal A}_1$.
Since ${\cal A}_2 \subseteq {\cal A}^{\phi,rst}$,
there must exist $G \in {\cal D}^n({\cal G}_\phi)$ such that ${\cal H}$ is $G$-maximal.
But if $|A_{\cal H}| \le n$, then every singleton subset of $A_{\cal H}$ is in
$\widetilde{{\cal G}}_\phi^n$. This contradicts the fact that ${\cal H}$ is $G$-maximal,
because if ${\cal H}'$ is ${\cal H}$ together with one of
these singleton subsets,
we must have $G - \cup {\cal H}' \ne \emptyset$. \vrule height7pt width4pt depth1pt\vspace{0.1in}
\begin{proposition}\label{transrst}
$\phi$ is satisfiable in
$\M_n^{\mbox{\scriptsize{{\it rst}}}}$
iff
$\phi^{\sigma_3}$
is satisfiable in $\M_{\A_1+\A_2}^{\mbox{\scriptsize{{\it rst}}}}$.
\end{proposition}
\noindent{\bf Proof:}
{F}irst suppose that $(M,s)\models\phi$, where $M \in \M_n^{\mbox{\scriptsize{{\it rst}}}}$. We
convert
$M = (S,\pi, \{{\cal K}_i: i \in {\cal A}\})$
into a structure $M' = (S,\pi, \{{\cal K}_{\cal H}: {\cal H} \in {\cal A}^{\phi,rst}\})$ as
before, by defining ${\cal K}_{{\cal H}} = \cup \{{\cal K}_i: i \in \tau_3(A)\}$.
As the union of symmetric relations is symmetric,
the proof that this works is
essentially
identical to that in Lemma~\ref{transrt}
for the case of $\M_n^{\mbox{\scriptsize{{\it rt}}}}$.
{F}or the opposite direction,
suppose that
$(M,s)\models\phi^{\sigma_3}$
for some $M = (S,\pi, \{{\cal K}_{\cal H}: {\cal H} \in {\cal A}^{\phi,rst}\}) \in \M_{\A_1+\A_2}^{\mbox{\scriptsize{{\it rst}}}}$.
We must
construct
a structure $M' \in \M_n^{\mbox{\scriptsize{{\it rst}}}}$ that satisfies $\phi$.
The state space for
the
structure $M'$ will again consist of copies of $S$,
but two copies no longer suffice to guarantee that the ${\cal K}_i$ relations
are equivalence relations. In fact, we
use
countably many copies.
By Lemma~\ref{separation}, for each ${\cal H} \in {\cal A}_2$, there exist at least
$n+1$ agents in
$A_{\cal H}$. Choose $n+1$ such agents, and call them $i_{\cal H}^0, \ldots,
i_{\cal H}^n$.
Partition $A_{\cal H}$ into $n+1$ disjoint sets $G_{{\cal H},j}$
with $i_{\cal H}^j \in G_{{\cal H},j}$.
We build copies of $M$ in a tree-like manner.
We index the copies of $M$ with strings of the form $((s_1,t_1), i_1,
\ldots, (s_k,t_k), i_k)$, such that $s_j,t_j \in S$, $i_j$ is
$i_{\cal H}^{j'}$ for some
${\cal H} \in {\cal A}_2$ and $0 \le j' \le n$, $(s_j,t_j) \in {\cal K}_{\cal H}$,
and $i_j \ne i_{j+1}$. Roughly speaking,
between $M_{\sigma}$ and $M_{\sigma\cdot((s_k,t_k),i_k)}$ we have
edges for the ${\cal K}_{i}$ relations for $\{i\} \in {\cal A}_1$ and also edges
between $s_{k}$ and $t_k$ in ${\cal K}_{i_k}$; however, there are no
edges in ${\cal K}_j$ if $\{j\} \notin {\cal A}_1$ and $j \ne i_k$; moreover, there
are no other edges in ${\cal K}_{i_k}$
except
those required
to assure reflexivity.
Before we can construct $M'$, we need some preliminary observations. We
can suppose
that the states in $S$ are numbered. Thus, for each state $s \in S$, if
$(M,s) \models \neg C_G \psi$, there is a lexicographically minimal
shortest path $(s_0,
\ldots, s_k)$ such
that $(s_i, s_{i+1}) \in {\cal K}_{\cal H}$ for some ${\cal H} \in G$
and
$(M,s_k) \models \neg
\psi$. Note that,
for each $i \leq k$, $(M,s_i) \models \neg C_G \psi$ and
$(s_i, \ldots, s_k)$
is also the lexicographically
minimal shortest $G$-path from
$s_i$
leading to a state that satisfies
$\neg \psi$. For each $s \in S$
and $B = E$ or $C$,
let
$\neg B_{G_1} \psi_1,\ldots, \neg B_{G_k} \psi_k$
be the formulas in $Sub^+(\phi)$ such that
$(M,s) \models (\neg B_{G_j} \psi_j)^{\sigma_3}$.
{F}or each state $s \in
S$, we can associate a set $F(s)$ of at most $n$ pairs $({\cal H},t)$ such
that $(s,t) \in {\cal K}_{\cal H}$ and for every formula
$B_{G}
\psi \in \mbox{\it Sub}(\phi)$, if $(M,s) \models
(\neg B_G
\psi)^{\sigma_3}$, then there exists a pair $({\cal H},t) \in F(s)$ such that
$t$ is the first state
after $s$ on the lexicographically minimal $\sigma_3(G_j)$-path from
$s$
to a state satisfying $\neg\psi$.
We can now define a set
$\Sigma$ of strings inductively. Let $\Sigma_0$ be the empty string.
Suppose that we have constructed $\Sigma_k$ consisting of strings
$((s_1,t_1), i_1, \ldots, (s_k,t_k), i_k)$ with the properties given
above.
{F}or each $\sigma = ((s_1,t_1), i_1, \ldots, (s_k,t_k), i_k)
\in \Sigma_k$, $s \in S$,
$({\cal H},t)
\in F(s)$, such that ${\cal H} \in
{\cal A}_2$, there is
exactly one string $\sigma \cdot ((s,t),i) \in \Sigma_{k+1}$. We choose
$i \in A_{\cal H}$ in such a way that $i \ne i_k$, $i$ is one of $i_{\cal H}^0,
\ldots, i_{\cal H}^n$, and a different $i$ is chosen for each
$({\cal H},t)
\in
{F}(s)$. Since $|F(s)| \le n$ and we can choose among $n+1$ agents
$i_0^{\cal H}, \ldots, i_n^{\cal H}$, this can clearly be done. Let $\Sigma =
\cup_k \Sigma_k$.
Let $M' = (S',\pi', \{{\cal K}_i: i \in {\cal A}\})$ be defined as follows:
\begin{itemize}
\item $S' = \cup_{\sigma \in \Sigma} S_\sigma$, where
each $S_\sigma$ is
a disjoint copy of $S$.
We denote by $s_\sigma$ the copy of state
$s \in S$ in
$S_\sigma$.
\item $\pi'(s_\sigma) = \pi(s)$ for $s \in S$, $\sigma \in
\Sigma$.
\item
If $\sigma_3(i) \in {\cal A}_1$, define ${\cal K}_i = \{(s_\sigma, t_{\sigma'}):
(s,t) \in{\cal K}_{\sigma_3(i)}, \sigma, \sigma' \in \Sigma\}$. ${\cal K}_i$ is clearly
reflexive, symmetric, and transitive in this case, since
${\cal K}_{\sigma_3(i)}$ is.
\item If
$\sigma_3(i) =
{\cal H} \in {\cal A}_2$
and $i \in G_{{\cal H},j}$,
then ${\cal K}_i =
\{(s_\sigma,s_\sigma): s \in S, \sigma \in \Sigma)\} \cup
\{(s_\sigma,t_{\sigma'}),(t_{\sigma'},s_\sigma): \sigma' = \sigma\cdot
((s,t),i^j_{\cal H}) \mbox{ and } (s,t) \in{\cal K}_{\sigma_3(i)}\}$.
Again, it is clear from the construction that ${\cal K}_i$
is reflexive, symmetric, and transitive.
\item If $\sigma_3(i)$ is undefined, then ${\cal K}_i =
\{(s_\sigma,s_\sigma): s \in S, \sigma \in \Sigma)\}.$
Of course, in this case ${\cal K}_i$
is also reflexive, symmetric, and transitive.
\end{itemize}
We claim that for each formula $\psi \in Sub^+(\phi)$, the following are
equivalent:
\begin{itemize}
\item[(a)] $(M,s) \models
\psi^{\sigma_3}$,
\item[(b)] $(M',s_\sigma) \models \psi$ for all $\sigma \in \Sigma$,
\item[(c)] $(M',s_\sigma) \models \psi$ for some $\sigma \in \Sigma$.
\end{itemize}
The argument proceeds by a straightforward induction on the structure of
$\psi$. The argument that (a) implies (b) is easy using the induction
hypothesis, and the implication from (b) to (c) is trivial. For the
argument that (c) implies (a), the only interesting cases are when
$\psi$ is of
the form $K_i \psi'$, $E_G \psi'$ or $C_G \psi'$. For $K_i \psi'$, the
argument is easy because it is easy to see that $\{i\} \in {\cal A}_1$. For
$E_G \psi'$,
suppose that $(M',s_\sigma)
\models E_G \psi'$. Then we must have $(M,s) \models (E_G
\psi')^{\sigma_3}$. For suppose not. Then
there is some
$({\cal H},t)
\in F(s)$ such that ${\cal H} \in \sigma_3(G)$ and
$(s,t) \in {\cal K}_{\cal H}$. Our construction guarantees that
$\sigma' = \sigma\cdot ((s,t),i)
\in \Sigma$ for some $i \in A_{\cal H}$. From Lemmas~\ref{unique}(a)
and~\ref{K}(a), it follows that $i \in G$. Moreover, by our
construction, $(s_\sigma,t_{\sigma'}) \in {\cal K}_i$. The induction
hypothesis
now
guarantees that $(M',t_{\sigma'}) \models \neg \psi'$. But this
contradicts the assumption that $(M',s_\sigma) \models E_G \psi'$.
{F}inally, suppose that $(M',s_\sigma) \models C_G \psi'$. Again, for a
contradiction, suppose that $(M,s) \models \neg (C_G \psi')^{\sigma_3}$.
Now we proceed by a subinduction on the length of the shortest
$\sigma_3(G)$-path
in $M$ leading to a state satisfying $(\neg \psi')^{\sigma_3}$ to show
that $(M',s_\sigma) \models \neg C_G \psi'$. We leave
the straightforward details to the reader.
\vrule height7pt width4pt depth1pt\vspace{0.1in}
Next, we want an analogue of Theorem~\ref{decrt} for $\M_n^{\mbox{\scriptsize{{\it rst}}}}$. The
reader will not be surprised to learn that there are new complications
here as well, although the basic result still holds.
\begin{theorem}\label{decrst}
If ${\cal A} = {\cal A}_1 \cup {\cal A}_2$ is finite and there is an algorithm for
deciding if $i
\in G$ for $G \in {\cal G}$ that runs in time linear in $|{\cal A}|$, then
there is a constant $c > 0$ and
an algorithm that, given a formula
$\phi$ of ${\cal L}_{\G}^C$, decides if $\phi$ is satisfiable
in
$\M_{\A_1+\A_2}^{\mbox{\scriptsize{{\it rst}}}}$
and runs in time $O(|{\cal A}|2^{c|\phi|})$
\end{theorem}
\noindent{\bf Proof:} We start as in the proof of Theorem~\ref{decrt}. Again, we assume
for ease of exposition that ${\cal A}_1 \ne \emptyset$. For $i \in {\cal A}_1$,
let $S^1_i$ consist of all the subsets of $\mbox{\it ESub}^+_i(\phi)$
that are maximally consistent and let $S^1 = \cup_{i \in {\cal A}_1} S^1_i$.
The definition of the ${\cal K}_i$ relations depends on whether $i \in {\cal A}_1$
or $i \in {\cal A}_2$. For $i \in {\cal A}_1$, we define the ${\cal K}_i$ relations on
$S^1$ so that $(s,t) \in {\cal K}_i$ iff
$s/\overline{K_i} \cup \{K_i \psi: K_i \psi \in s\} \subseteq t$ and
$s/\overline{K_i} \cup \{K_i \psi: K_i \psi \in s\} =
t/\overline{K_i} \cup \{K_i \psi: K_i \psi \in t\}$.
It is easy to check that this modification forces these ${\cal K}_i$
relations to be Euclidean and transitive.
{F}or $i \in {\cal A}_2$ we define ${\cal K}_i$ so that $(s,t) \in {\cal K}_i$ iff
$s/\overline{K_i} \subseteq t$ and $t/\overline{K_i} \subseteq s$.
Clearly this modification forces these ${\cal K}_i$ relations to be symmetric.
We force all the ${\cal K}_i$ relations to be
reflexive just as with ${\cal M}_n^r$, by eliminating $s \in S^1$ if
$(s,s) \notin {\cal K}_i$ for some $i \in {\cal A}_1 \cup {\cal A}_2$.
We now must also change the definition of $s$ seeming consistent.
Define the
relations
$\preceq_i$ on $S^1 \times S^1_i$ by taking
$s\preceq_{i}s^{\prime}$ if $s^{\prime}\in S_{i}^{1}$ and $s\cap
ESub_{i}(\phi)\subseteq s^{\prime}$. Suppose that we have defined $%
S^{1},\ldots,S^{m}$. $S^{m+1}$ consists of all states $s\in S^{m}$ that {\em %
seem consistent}, in that
the following three conditions hold (where we assume
that all states considered are in $S^{m}$):
\begin{enumerate}
\item For all $i\in {\cal A}_{1}$, there exists an $s^{\prime }\in S^{j}$
such that $s\preceq _{i}s^{\prime }$.
\item There exist distinct agents $i_{1},\ldots ,i_{k}\in {\cal A}_{1}$ and
states $s_{1},\ldots ,s_{k}$ such that $s\preceq _{i_{h}}s_{h}$ for $h\in
\{1,\ldots ,k\}$ and for every formula of the form $\lnot E_{G}\psi \in s$,
there is a $t$ such that either
\begin{enumerate}
\item $(\exists i\in G\cap {\cal A}_{2})((s,t)\in {\cal K}_{i}\,\wedge
\,\lnot \psi \in t)$ or
\item $(\exists h\leq k)(i_{h}\in G\,\wedge \,(s_{h},t)\in {\cal K}%
_{i_{h}}\,\wedge \,\lnot \psi \in t)$.
\end{enumerate}
\item If $\lnot C_{G}\psi \in s$ then there exist states $%
s_{0},s_{0}^{\prime },s_{1},s_{1}^{\prime },\ldots ,s_{k}$ such that $s=s_{0}
$, $\lnot \psi \in s_{k}$, and there exist $j_{0},\ldots ,j_{k-1}$ in $G$
such that, for each $i\leq k$, $(s_{i}^{\prime },s_{i+1})\in {\cal K}_{j_{i}}
$ and either $j_{i}\in {\cal A}_{2}$ and $s_{i}=s_{i}^{\prime }$ or $%
j_{i}\in {\cal A}_{1}$, $s_{i}\preceq _{j_{i}}s_{i}^{\prime }$ and $%
s_{i}^{\prime }$ is acceptable for $s_{i}$, where we say that $s^{\prime }$
is {\em acceptable} for $s$ if there are
states $s_{h}$ and agents $i_{h}$, $h = 1, \ldots, k$, as described in
condition 2 for $s$, and $s^{\prime }=s_{i}$ for some $i\leq k$.
\end{enumerate}
We need to show that we can check whether $s$ seems consistent in time $O(|%
{\cal A}|2^{|\phi|})$.
The only difficulty is to determine, for given $s$ and
$s^{\prime}$, if $s^{\prime}$ is acceptable for $s$. It is clear that $%
k\leq|\phi|$, since we need at most one state and agent for each formula of
the form $\lnot E_{G}\psi\in s$. However, if we simply check each subgroup
of states containing $s^{\prime}$ and of agents containing $j$ where $%
s^{\prime }\in S_{j}^{1}$ that are of size $\leq|\phi|$ in the naive way,
this check will take time
at least $C(2^{|\phi|},|\phi|)C(|{\cal A}|,|\phi|)$
(where $C(n,k)$ is $n$ choose $k$),
which is
unacceptable for our desired time bounds. Instead, we proceed as follows.
Suppose that
$s' \in S_{i_1}^1$ and
$s\preceq_{i_{1}}s^{\prime}$. (If it is not the case that
$s\preceq_{i_{1}}s^{\prime}$,
then clearly $s^{\prime}$ is not acceptable for $s^{\prime}$.) Let $%
{F}(s,s^{\prime })$ consist of all formulas $E_{G}\psi$ such that
\begin{enumerate}
\item $\lnot E_{G}\psi \in s$,
\item $\lnot \exists t,i(i\in G\cap {\cal A}_{2}\,\wedge (s,t)\in {\cal K}%
_{i}\,\wedge \,\lnot \psi \in t)$, and
\item $|A(s,s^{\prime },E_{G}\psi )|<|\phi |$, where $A(s,s^{\prime
},E_{G}\psi )=\{i\in G\cap {\cal A}_{1}:i=i_{1}\vee \,\exists t(s\preceq
_{i}t\,\wedge \,\lnot K_{i}\psi \in t)\}$.
\end{enumerate}
Intuitively, $F(s,s')$ consists of the potentially ``problematic''
formulas that may prevent $s'$ from being acceptable for $s$.
Let $T=\cup{_{E_G \psi \in F(s)}A(s,s}^{\prime},{E_{G}\psi)}$. Note that
$|T|<|\phi|^{2}$.
Suppose that $T=\{i_{1},\ldots,i_{N}\}$. We construct sets
$B_{1},\ldots,B_{N}$
of subsets of $F(s,s^{\prime})$ with the property that a set $X\in B_{k}$
iff there exist states $t_{1},\ldots,t_{k}$ such that $s\preceq_{i_{j}}t_{j}$
for $j=1,\ldots,k$ , $t_{1}=s^{\prime}$ and, for each formula $E_{G}\psi\in
X $, there exists a $j$ such that $\lnot K_{i_{j}}\psi\in t_{j}$.
Given a state $t\in S_{i}^{1}$, let $F_{t}(s,s^{\prime})=\{E_{G}\psi\in
{F}(s,s^{\prime}):\lnot K_{i}\psi\in t, \, i \in G\}$. Intuitively,
$F_t(s,s')$ consists of the formulas in $F(s,s')$ that can be ``taken
care of'' by state $t$.
Let $B_{1}=\{F_{s^{%
\prime}}(s,s^{\prime})\}$. Suppose that we have defined
$B_{1},\ldots,B_{k}$. Let
$B_{k+1}=\{X\cup{F_{t}(s,s}^{\prime}{):X\in B_{k}\,\wedge s\preceq}_{i_{k+1}}%
{\,t\}}$. It is easy to check that $B_{k+1}$ has the required property.
Moreover, we can compute the sets $B_{1},\ldots,B_{N}$ in time $O(2^{cn})$.
To see this, note that since $|F(s,s^{\prime})|\leq|\phi|$, clearly $%
|B_{j}|\leq2^{|\phi|}$. Thus, given $B_{k}$, we can clearly compute $B_{k+1}$
in time $O(2^{cn})$ for some $c>0$. Since $N<|\phi|^{2}$, the result
follows. Finally, we claim that $s^{\prime}$ is acceptable for $s$ iff $%
{F}(s,s^{\prime})\in B_{N}$.
Clearly if $F(s,s^{\prime})\notin B_{N}$, then
it is almost immediate from the definition that
$s^{\prime}$ is not
acceptable for $s$. Conversely, if $F(s)\in B_{N}$, then there exist states $%
t_{1},\ldots,t_{N}$ such that $s^{\prime}=t_{1}$, $s\preceq_{i_{j}}t_{j}$
and, for each formula in $E_{G}\psi\in F(s)$, there exists $j$ such that $%
{\cal K}_{i_{j}}\psi\in t_{j}$. We clearly do not need all of these states
and agents; we just need at most one for each formula in $F(s,s^{\prime})$.
That is, there exists a set ${\cal A}^{\prime}$ of agents (contained in $%
\{i_{1},\ldots,i_{N}\}$) with $|{\cal A}^{\prime}|\leq|F(s,s^{\prime})|$ and
a state $u_{i}$ corresponding to each agent $i\in{\cal A}^{\prime}$
(contained in $\{t_{1},\ldots,t_{N}\}$) such that for each formula $%
E_{G}\psi\in F(s,s^{\prime})$, there exists an agent $i\in{\cal A}^{\prime}$
such that $s\preceq_{i}u_{i}$ and $\lnot K_{i}\psi\in u_{i}$. We now wish to
extend ${\cal A}^{\prime}$ to a set showing that $s^{\prime}$ is acceptable
for $s$. If we consider any $\lnot E_{G}\psi\in s$, either condition
2(a) is
satisfied or there is already an $i\in A^{\prime}$ satisfying 2(b) or $%
|A(s,s^{\prime},E_{G}\psi)|\geq|\phi|$. In the last case, it is immediate
that we can extend ${\cal A}^{\prime}$ to include
an
agent satisfying
2(b) for $E_{G}\psi$.
To show that this algorithm is correct, first suppose that $\phi$ is
satisfiable. In that case, $(M,s_{0})\models\phi$ for some structure $%
M=(S,\pi,\{{\cal K}_{i}^{\prime}:i\in{\cal A}\})\in{\cal M}^{rst}$. As for $%
{\cal M}^{rt}$, we can associate with each state $s\in S$ and $i\in{\cal A}%
_{1}$ the state $s_{i}^{\ast}$ in $S_{i}^{1}$ consisting of all the formulas
$\psi\in ESub_{i}(\phi)$ such that $(M,s)\models\psi$. It is easy to see
that if $(s,t)\in{\cal K}_{i}^{\prime}$ then $(s_{i}^{\ast },t_{i}^{\ast})\in%
{\cal K}_{i}$. Using this observation, a straightforward induction shows
that the states $s_{i}^{\ast}$ for $s\in S$ always seem consistent, and thus
are in $S^{j}$ for all $j$ and all $i\in{\cal A}_{1}$. Moreover, $%
\phi\in(s_{0})_{i}^{\ast}$ for all $i\in{\cal A}_{1}$. Thus, the algorithm
will declare that $\phi$ is satisfiable, as desired.
{F}or the converse, we need to show that if the algorithm declares that $\phi$
is satisfiable, then it is indeed satisfiable
in ${\cal M}_{%
{\cal A}_{1}+{\cal A}_{2}}^{rst}$. We need to work a little harder than in
the previous proofs. Now we can no longer just view the object constructed
by our algorithm as the required structure. Rather, it serves as a
``blueprint'' for building the required structure.
Suppose that the algorithm terminates at stage $N$ with a state
$u\in S_{i_{u}}=S_{i_{u}}^{N}$ containing $\phi$. Before we go on,
we make one observation that will prove useful in the sequel.
Notice that if $s \preceq_i s'$, then $E_G \psi \in s$ iff $E_G \psi \in
s'$ for $G \ne \{i\}$, and if
$j \in {\cal A}_2$, then $(s,t) \in {\cal K}_j$
iff $(s',t) \in {\cal K}_j$.
A {\em complete
state} is a vector $\vec{s} =
(s^i: i \in {\cal A}_i \wedge s^i \in S^N_i)$
such
that
\begin{itemize}
\item $s^i \preceq_j s^j$ for all $i, j \in {\cal A}_1$ and
\item for every formula of the form $\neg E_G \psi \in \cup_{i \in
{\cal A}_1} s^i$, there exists an agent $j \in G$ and a state $t \in S^N$
such that $\neg \psi \in t$ and either
$j \in {\cal A}_1 \cap G$, $\neg K_j \psi \in s^j$, and $(s^j,t) \in
{\cal K}_j$ or $j \in {\cal A}_2$ and
$(s^i,t) \in {\cal K}_j$ for some $i \in {\cal A}_1$ (and hence $(s^i,t) \in {\cal K}_j$
for all $i \in {\cal A}_1$).
\end{itemize}
By consistency condition 2, every state $s \in S^N$ must be a
component of some (perhaps many) complete states.
Define
a structure $M^* = (S^*, \pi^*, \{{\cal K}_i^*: i \in {\cal A}_1 \cup {\cal A}_2\}$ as
follows:
\begin{itemize}
\item $S^*$ consists of all complete states;
\item $\pi^*(\vec{s})(p) = {\bf true}$ iff $p \in \cup_{i \in {\cal A}_1}
s^i$;
\item $(\vec{s},\vec{t}) \in {\cal K}_i^*$ for $i \in {\cal A}_1$ iff $s^i = t^i$
or $(s^i,t^i) \in {\cal K}_i$;
\item $(\vec{s},\vec{t}) \in {\cal K}_i^*$ for $i \in {\cal A}_2$ iff $(s^j,t^j)
\in {\cal K}_i$ for some $j \in {\cal A}_1$ (it is easy to check that if $(s^j,t^j)
\in {\cal K}_i$ for some $j \in {\cal A}_1$ then $(s^j,t^j) \in {\cal K}_j$ for all $j \in
{\cal A}_1$).
\end{itemize}
It is easy to check that $M^* \in \M_{\A_1+\A_2}^{\mbox{\scriptsize{{\it rst}}}}$. We now show that for
all $\psi \in \cup_{i \in {\cal A}_i} \mbox{\it ESub}^+_i(\phi)$, we have
$$
\mbox{$(M^*,\vec{s}) \models \psi$ iff $\psi \in \cup_{i \in {\cal A}_1} s^i$.}
$$
We proceed, as usual, by induction on the structure of $\psi$.
If $\psi$ is a primitive proposition, a conjunction, or a negation,
the argument is easy. Suppose that $\psi$ is of the form $E_G\psi'$.
If $E_G \psi'
\in \cup_{i \in {\cal A}_1} s^i$, then the construction of the ${\cal K}_j$
relations guarantees that $\psi' \in \cup_{i \in
{\cal A}_1}
t^i$ for all
$\vec{t} \in S^*$ such that $(\vec{s},\vec{t}) \in {\cal K}_j^*$ for some $j
\in G$. Thus, by the induction hypothesis, we have
that $(M^*,\vec{s}) \models E_G \psi'$. For the converse, suppose that
$\neg
E_G \psi' \in \cup_{i \in {\cal A}_1} s^i$. Then from the definition of
complete
state and consistency condition 2, there must be some complete state
$\vec{t}$ and $j \in G$ such that $(\vec{s},\vec{t}) \in {\cal K}_j$ and $\neg
\psi' \in \cup_{i \in {\cal A}_1} t^i$.
{F}inally, suppose that $\psi$ is of the form $C_G\psi'$. If $C_G \psi'
\in \cup_{i \in {\cal A}_1} s^i$ then, since $E_G (\psi' \land C_G \psi')$
must also be in $\cup_{i \in {\cal A}_1} s^i$, an easy induction on the
length of the path shows that for every complete state $\vec{t}$
$G$-reachable from $\vec{s}$, we must have $\psi' \in \cup_{i \in
{\cal A}_1} t^i$ so, by the induction hypothesis, we have
$(M^*,\vec{s}) \models C_G \psi'$. For the converse, suppose that $\neg
C_G \psi \in \cup_{i \in {\cal A}_1} s^i$. Then $\neg C_G \psi \in s^j$ for
some
(in fact, all) $j \in {\cal A}_1$. If $G \cap {\cal A}_1 \ne \emptyset$, choose
$j \in G \cap {\cal A}_1$; otherwise, choose $j$
$\in G$
arbitrarily.
{F}rom consistency condition 3, it
easily follows that there exist complete states $\vec{s}_0, \ldots,
\vec{s}_k$ and $j_0, \ldots, j_{k-1} \in G$
such that $s_0^j = s^j$, $(\vec{s}_h,\vec{s}_{h+1}) \in {\cal K}_{j_h}^*$
$h = 0, \ldots, k-1$, and $\neg \psi' \in \cup_{i \in {\cal A}_1} s_k^i$.
If $j \in {\cal A}_1$, then $(\vec{s}, \vec{s}_0) \in {\cal K}_j^*$; if $j \notin
{\cal A}_1$, then $j_0 \in {\cal A}_2$, and it follows from our initial observation
that $(\vec{s},\vec{s}_1) \in {\cal K}_{j_0}^*$.
In either case, $\vec{s}_k$ is $G$-reachable from $\vec{s}$, so
$(M^*,\vec{s})
\models \neg C_G \psi'$, as desired. \vrule height7pt width4pt depth1pt\vspace{0.1in}
We can now prove Theorem~\ref{complexity} for $\M_n^{\mbox{\scriptsize{{\it rst}}}}$.
\bigskip
\noindent
{\bf Proof of Theorem~\ref{complexity} for $\M_n^{\mbox{\scriptsize{{\it rst}}}}$:}
Again, the lower bound follows from standard results in \cite{HM2}.
{F}or the upper bound, suppose that we are given a formula $\phi$ such
that
$n
= |\phi|$.
We can compute the set ${\cal E}^m_{n}({\cal G}_\phi)$
defined just before Lemma~\ref{Gmcount} in time $O(n^2 2^{cn})$,
using at most $n^2 2^n$ calls to the oracle $O_m$, just as we computed
${\cal E}^1({\cal G}_\phi)$. Similarly, we can characterize the sets
${\cal H}$ such that $(G,{\cal H})$ is in
${\cal R}({\cal G}_\phi^m) = {\cal R}^m({\cal G}_\phi)$
by a pair
$({\cal H}',X)$, where ${\cal H}' \subseteq {\cal G}_\phi$ and $X$ is either $\emptyset$ or an
element of ${\cal E}^1_m({\cal G}_\phi)$ and compute which of the pairs actually
represent sets in ${\cal H}$ such that $(G,{\cal H}) \in {\cal R}({\cal G}_\phi^m)$ using
at most $2n(2^{2n} + 2^n)$ calls to the oracle $O_m$.
We cannot compute the individual elements of the sets $A_{\cal H}$
such that $|A_{\cal H}| \le m$, but it
does not matter. It suffices that we know the cardinality of these
atoms (which our oracle will tell us). We let ${\cal A}_1$ consist of the
agents $i_1^{\cal H}, \ldots, i_{|A_{\cal H}|}^{\cal H}$ for each ${\cal H}$ such that $|A_{\cal H}|
\le |\phi|$ ($i_1^{\cal H}, \ldots, i_{|A_{\cal H}|}^{\cal H}$ are just fresh names for
agents); let ${\cal A}_2$ consist of all ${\cal H}$ such that $|A_{\cal H}| > |\phi|$.
It is now straightforward to compute the formula $\phi^{\sigma_3}$ in
time $O(2^{cn})$ using $O(2^{cn})$ oracle calls.
We now
apply Proposition~\ref{transrst} and Theorem~\ref{decrst}, just as we
applied Proposition~\ref{trans} and Theorem~\ref{dec} in the case of
${\cal M}_n$. \vrule height7pt width4pt depth1pt\vspace{0.1in}
We now turn our attention to proving Theorem~\ref{complete} for
$\M_n^{\mbox{\scriptsize{{\it rst}}}}$. Again, the basic structure is the same as for ${\cal M}_n$ and
$\M_n^{\mbox{\scriptsize{{\it rt}}}}$.
\begin{lemma}\label{replacerst} The mapping $\sigma_3$ (when viewed as a map
with domain $2^{\cal A}$) is injective on $\widetilde{{\cal G}}_\phi^n$.
\end{lemma}
Let $({\rm S5}_\G^C)^{\A_1+\A_2}$ consist of the axioms in $({\rm T}_\G^C)^+$ (including E5)
together with E6 and every instance of K4 and K5 for $i \in {\cal A}_1$.
We write $({\rm S5}_\G^C)^{\A_1+\A_2}
\vdash_\phi \psi$ if there is a proof of $\psi$ in $({\rm S5}_\G^C)^{\A_1+\A_2}$ using
only the modal operators that appear in $\phi$ and $K_i$ for $i \in
{\cal A}_1$.
\begin{lemma}\label{proofrst} If ${\cal A}$ is finite and $\phi \in {\cal L}_{\G}^C$ is
valid with respect to $\M_{\A_1+\A_2}^{\mbox{\scriptsize{{\it rst}}}}$, then $({\rm S5}_\G^C)^{\A_1+\A_2}
\vdash_\phi \phi$. \end{lemma}
\noindent{\bf Proof:} The proof is similar in spirit to that of Lemma~\ref{proofrt} for
$\M_n^{\mbox{\scriptsize{{\it rt}}}}$, except that since we have a different definition of the ${\cal K}_i$
relations and of seeming
consistent, we must check that states eliminated under this definition
are inconsistent. Again we must consider each of the three ways that a
state $s$ can be eliminated.
{F}irst, suppose that $s \in S^j$ and, for some $i \in {\cal A}_1$, there is no
$s'$ such that $s \preceq_i s'$. As before, propositional reasoning
shows that
$({\rm S5}_\G^C)^{\A_1+\A_2} \vdash_\phi \phi_s \Leftrightarrow \lor_{\{s' \in S^1_i: s \preceq_i
s'\}} \phi_{s'}$. Thus, it
easily follows that $({\rm S5}_\G^C)^{\A_1+\A_2} \vdash_\phi \neg \phi_s$.
Next, suppose that $s\in S^{j}$ does not satisfy the second condition of
seeming consistent.
There must be a formula $\lnot E_{G}\psi \in s$ such that
\begin{enumerate}
\item for all $i\in {\cal A}_{2}$ and all $t\in S^{j}$ such that $(s,t)\in
{\cal K}_{i}$, we have $\psi \in t$ and
\item for all $i\in G\cap {\cal A}_{1}$, and all $t\in S^{j}$ such that $%
(s^{i},t)\in {\cal K}_{i}$, we have $\psi \in t$.
\end{enumerate}
Define an {\em extension of $s$\/} to be a vector $\vec{s%
}=(s^{i}:i\in {\cal A}_{1})$ of states, where $s\preceq _{i}s^{i}$. Let $EX(s)$ be
the set of all extensions of $s$. If $\vec{s}$ is an extension of $s$, let $%
\phi _{\vec{s}}$ be the conjunction over all $i\in {\cal A}_{1}$ of the formulas
in $\phi _{s^{i}}$. By straightforward propositional reasoning, we have $(%
{\rm S5}_{\cal G}^C)^{{\cal A}_1+{\cal A}_2}\vdash _{\phi }\phi _{s}\Leftrightarrow\vee _{\vec{s}\in
EX(s)}\phi _{\vec{s}} $.
Thus, to show that
$({\rm S5}_\G^C)^{\A_1+\A_2} \vdash_\phi \neg \phi_s$ if
$s$ is eliminated
by the second condition of seeming consistent, it suffices to show that
$({\rm S5}_{\cal G}^C)^{{\cal A}_1+{\cal A}%
_2}\vdash _{\phi }\lnot \phi _{\vec{s}}$ for each $\vec{s}\in EX(s)$.
This we do by showing that
$({\rm S5}_{G}^{C})^{{\cal A}_{1}+{\cal A}_{2}}\vdash_{\phi }\phi _{\vec{s}}\Rightarrow
E_{G}\psi $
for each $\vec{s} \in EX(s)$.
So suppose that $\vec{s} \in EX(s)$.
The proof follows the lines of the analogous argument in the proof of Lemma~%
\ref{proofrt}. As before, it suffices to find, for each $i\in G$ and each
$t\in S_{i}^{j}$ with $\lnot \psi \in t$, a
set $G^{i,t}$ of agents containing $i$ such
that $({\rm S5}_{G}^{C})^{{\cal A}_{1}+{\cal A}_{2}}\vdash _{\phi }
\phi _{\vec{s}}\Rightarrow %
E_{G^{i,t}}\lnot \phi _{t}$. For $i\in {\cal A}_{2}$, this follows as before if
the reason that $(s,t)\notin {\cal K}_{i}$ is that $s/\overline{K_i} \not\subseteq
t
$. If instead $t/\overline{K_i} \not\subseteq s$, then there is some $E_{G^{\prime
}}\theta \in t$ with $i\in G^{\prime }$ such that $\lnot \theta \in s$ and
so $\lnot \theta \in s^{i}$ for each $i$. Thus $({\rm S5}_{G}^{C})^{{\cal A}_{1}+{\cal A}%
_{2}}\vdash _{\phi }\phi _{\vec{s}}\Rightarrow\lnot \theta $ and, by E6,
$({\rm S5}_{G}^{C})^{{\cal A}_{1}+{\cal A}_{2}}\vdash _{\phi }\lnot \theta \Rightarrow
E_{G'}\lnot E_{G'}\theta $. Since $E_{G'}\theta \in t$
we have that $({\rm S5}_{G}^{C})^{{\cal A}%
_{1}+{\cal A}_{2}}\vdash _{\phi }\phi _{\vec{s}}\Rightarrow E_{G'}\lnot \phi _{t}$.
That is, we can take $G^{i,t} = G'$ in this case.
{F}or $i\in {\cal A}_{1}$, we show that $({\rm S5}_{G}^{C})^{{\cal A}_{1}+{\cal A}_{2}}\vdash
_{\phi }\phi _{\vec{s}}\Rightarrow K_{i}\lnot \phi _{t}$
(so that we can take $G^{i,t} = \{i\}$).
By our assumption, $(s^{i},t)\notin {\cal K}_{i}$ for all $t \in S^i_j$.
Thus, if $t \in S^i_j$, there is some formula $\theta$ such that
either $K_i \theta \in s^i$
and $\neg K_i \theta \in t$ or $K_i \theta \in t$ and $\neg K_i \theta
\in s^i$. Here we are
implicitly using the following facts: (1) if $E_{G'} \theta \in s$
for some
$G'$ such that $i \in G'$ then $K_i \theta \in s^i$, since $s^i \in
S_i^1$, and similarly for $t$, (2) if $K_i \theta \notin s$, then
$\neg K_i \theta \in s$, since $s^i \in S_i^j$, and similarly for $t$,
and (3) if $K_i \theta \in s^i$ then $\theta \in S$ since $(s,s) \in
{\cal K}_i$, and similarly for $t$. If $K_i \theta \in s$ and $\neg K_i
\theta \in t$, it follows that $({\rm S5}_\G^C)^{\A_1+\A_2} \vdash_\phi \phi_s \Rightarrow
K_i \neg \phi_t$ just as in the case of $({\rm S4}_\G^C)^{\A_1+\A_2}$. If $K_i \theta \in
t$ and $\neg K_i \theta \in s$, then by K5 we have
$({\rm S5}_\G^C)^{\A_1+\A_2} \vdash_\phi \phi_s \Rightarrow K_i \neg K_i \theta$
$({\rm S5}_\G^C)^{\A_1+\A_2} \vdash_\phi \neg K_i \theta \Rightarrow
\neg\phi_t$.
The desired
result now follows by standard arguments.
We have now shown that for all $i \in G$ and $t \in S_j^i$ such that
$\psi
\in t$, there exists some set $G^{i,t}$ with $i \in G^{i,t}$ such that
$({\rm S5}_\G^C)^{\A_1+\A_2} \vdash_\phi \phi_s \Rightarrow E_{G^{i,t}} \neg \phi_t$.
We can now conclude that $({\rm S5}_\G^C)^{\A_1+\A_2} \vdash_\phi \phi_s \Rightarrow \neg E_G
\psi$ just as in the case of $({\rm S4}_\G^C)^{\A_1+\A_2}$, showing that $\phi_s$ is
inconsistent, as desired.
{F}inally, if $s \in S^j$ does not satisfy the third condition of
seeming consistent, the argument that $({\rm S5}_\G^C)^{\A_1+\A_2} \vdash_\phi \neg
\phi_s$ is similar to that of Lemma~\ref{proof}. We replace $G$-reachability
by the existence of sequences as in
condition 3 in the definition of seeming consistent
in Theorem~\ref{decrst} and note that we have essentially already proved the
analogue of (\ref{eq2}) from Lemma~\ref{proof}. We leave the remaining
details to the reader. \vrule height7pt width4pt depth1pt\vspace{0.1in}
\bigskip
\noindent
{\bf Proof of Theorem~\ref{complete} for $\M_n^{\mbox{\scriptsize{{\it rst}}}}$:} The proof follows as
for $\M_n^{\mbox{\scriptsize{{\it rt}}}}$ using the analogous lemmas proved above for $\M_n^{\mbox{\scriptsize{{\it rst}}}}$.
\vrule height7pt width4pt depth1pt\vspace{0.1in}
\subsection{Dealing with ${\cal M}_{\cal A}^{\it elt}$}\label{sec:elt}
For $\M_n^{\mbox{\scriptsize{{\it elt}}}}$, we proceed much as for $\M_n^{\mbox{\scriptsize{{\it rst}}}}$. There is one new
subtlety. Consider the construction
in the proof of Proposition~\ref{transrst}, which uses $\sigma_3$.
Recall that
$\sigma_3(i)$ may be undefined for some $i$. For such $i$, we defined
${\cal K}_i$ to consist of all pairs $(s_\sigma,s_\sigma)$, making it
reflexive. This approach will not work for $\M_n^{\mbox{\scriptsize{{\it elt}}}}$. More precisely,
the analogue of Proposition~\ref{transrst} for $\M_n^{\mbox{\scriptsize{{\it elt}}}}$ will not hold
using this construction (even if we drop the reflexivity requirement).
For example, if $\phi = \neg p\land E_{G_1} p \land
E_{G_2} p$ and $G_1 \cap G_2 \ne \emptyset$, then $\phi^{\sigma_3}$
is satisfiable in $\M_{\A_1+\A_2}^{\mbox{\scriptsize{{\it elt}}}}$ but $\phi$ is not satisfied in the
structure $M'$ constructed in Proposition\ref{transrst}, since for all
$i \in G_1 \cap G_2$, the construction will make ${\cal K}_i$ reflexive.
We solve this problem by defining a mapping $\sigma_4$ much like
$\sigma_3$, except that we ensure that $\sigma_4$ is never undefined.
Let ${\cal B}$ be the set of
maximal subsets ${\cal T}$ of ${\cal G}_{\phi }$ such that $\cap {\cal T}%
\neq \emptyset $ and such that the corresponding atom over ${\cal
G}^{\phi }$, ${\cal A}_{{\cal T}}=(\cap {\cal T})\cap
(\cap_{G \in {\cal G}_\phi - {\cal T}\,} G)$
($=\cap {\cal T}$ by the maximality of ${\cal T}$), is not one of the
ones $%
{\cal A}_{{\cal H}}$ for ${\cal H}\in {\cal A}^{\phi ,rst}$. Let ${\cal
A}^{\phi ,elt}={\cal A}^{\phi ,rst}\cup {\cal B}$, ${\cal
A}_{1}={\cal B%
}\cup \{{\cal H}\in {\cal A}^{\phi ,rst}:|{\cal A}_{{\cal H}}|=1\}$, $%
{\cal A}_{2}={\cal A}^{\phi ,elt}-{\cal A}_{1}$. The definitions of
$\sigma
_{4}:{\cal A}\rightarrow {\cal A}^{\phi ,elt}$ and $\tau _{4}:{\cal A}^{\phi
,elt}\rightarrow 2^{{\cal A}}$ need some care. If $i\in {\cal A}_{{\cal H}}$
for some ${\cal H}\in {\cal A}^{\phi ,rst}$, let $\sigma _{4}(i)={\cal H}$
as before. Otherwise, choose ${\cal T}\in {\cal B}$ such that
${\cal T} \supseteq \{G\in {\cal G}_{\phi }: i\in G\}$ and let
$\sigma (i)={\cal T}$.
Note that, by construction, $\sigma_4$ is defined for all $i$.
For ${\cal H}\in {\cal A}^{\phi ,rst}$, $\tau_{4}({\cal H})
=\cap \{\widetilde{{\cal G}}_{\phi}^{n}-{\cal H}\}$ as before. For ${\cal T}%
\in {\cal B}$, choose some $i_{{\cal T}}\in {\cal A}_{{\cal T}}$ (it
does not matter which) and set
$%
\tau _{4}({\cal T})=\{i_{{\cal T}}\}$.
\begin{proposition}\label{transelt}
$\phi $ is satisfiable in ${\cal M}_{{\cal A}}^{elt}$ iff $\phi ^{\sigma
_{4}}$ is satisfiable in ${\cal M}_{{\cal A}_{1}+{\cal A}_{2}}^{elt}$.
\end{proposition}
\noindent{\bf Proof:}
{F}irst suppose that $(M,s)\models \phi $, where $M\in {\cal M}^{elt}$.
We convert $M$
to $M' \in \M_{\A_1+\A_2}^{\mbox{\scriptsize{{\it elt}}}}$
as before by defining ${\cal K}_{{\cal I}}=\cup \{{\cal K}_{i}:i\in \tau
_{4}({\cal I})\}$ for ${\cal I}\in {\cal A}^{\phi ,elt}$.
To apply Proposition~\ref{trans}, we need to show that
$\cup\{\tau({\cal I}): {\cal I} \in \sigma_4(G)\} = G$ for all $G \in {\cal G}_\phi$.
We know from the
analysis of the ${\cal M}^{rst}$ case that
$\cup \{\tau_3({\cal H}): {\cal H} \in \sigma_3(G)\} = G$ for all $G \in {\cal G}_\phi$.
Since $\sigma _{4}(G)\supseteq \sigma _{3}(G)$ and
$\tau_4({\cal H}) = \tau_3({\cal H})$ for ${\cal H} \in {\cal A}^{\phi,rst}$, we have
that $\cup \{\tau_4({\cal I}): {\cal I} \in \sigma_4(G)\} = \cup \{\tau_3({\cal H}):
{\cal H} \in \sigma_3(G) \cup \cup\{\tau_3({\cal I}): {\cal I} \in \sigma_4(G) -
\sigma_3(G)\}$.
It is clear from the definitions, however, that
if ${\cal I} \in \sigma_4(G) - \sigma_3(G)$, then there exists some $i \in
G$ such that ${\cal I} = \sigma_4(i)$ and $\sigma_3(i)$ is undefined.
Moreover, ${\cal I} = {\cal A}_{{\cal T}}\subseteq G$, so $\tau_4({\cal I}) \in G$.
Thus, $\cup\{\tau_3({\cal I}): {\cal I} \in \sigma_4(G) - \sigma_3(G)\}
\subseteq G$, so
$\cup \{\tau_4({\cal I}): {\cal I} \in \sigma_4(G)\} = \cup \{\tau_3({\cal H}):
{\cal H} \in \sigma_3(G)\} = G$, as desired.
Applying Proposition~\ref{trans}, we get that
to see that $(M^{\prime },s)\models \phi ^{\sigma _{4}}$.
It remains to verify that $M' \in \M_{\A_1+\A_2}^{\mbox{\scriptsize{{\it elt}}}}$. For this, we need to
show that the ${\cal K}_{{\cal I}}$ relations for ${\cal I}\in {\cal A}_{1}$
are Euclidean, serial and transitive and that those in ${\cal A}_{2}$
are serial and secondarily reflexive. For the ones in ${\cal A}_{1}$,
note
that $\tau _{4}({\cal I})$ is a singleton and so the desired properties hold
since they hold for all agents in $M$. For the ones in ${\cal A}_{2}$, we
just note that the union of serial relations is serial and the union of
Euclidean relations is secondarily reflexive.
{F}or the other direction, we proceed much as in the proof of
Proposition~\ref{transrst}.
In addition to the concerns dealt with there for ${\cal M}^{rst}$,
our primary new one is to make sure that
the ${\cal K}_i$ relations for all agents
are serial. The problem
arises for those $i$ for which $\sigma _{3}(i)$ was undefined. The new
agents in ${\cal B}$ are used to deal with this problem.
We proceed much as in Proposition~\ref{transrst}, with two changes.
{F}irst, we replace the automatic forcing of reflexivity by forcing secondary
reflexivity for $\sigma _{3}(i)\in {\cal A}_{2}$. Second, we
modify the definition of the ${\cal K}_i$ relation in $M'$ as follows.
\begin{itemize}
\item If $\sigma_4(i) \in {\cal A}_1 \cap {\cal A}^{\phi,rst}$ then, as before,
${\cal K}_i = \{(s_\sigma, t_{\sigma'}):
(s,t) \in{\cal K}_{\sigma_3(i)}, \sigma, \sigma' \in \Sigma\}$.
\item If $\sigma _{4}(i)\in {\cal A}_{2}$ and $i\in G_{{\cal H},j}$,
then ${\cal K}_{i}=\{(s_{\sigma },t_{\sigma ^{\prime }}),(t_{\sigma'},
t_{\sigma ^{\prime }}):\sigma ^{\prime }=\sigma \cdot
((s,t),
i^j_{\cal H})\}$.
\item If $\sigma _{4}(i)={\cal T}\in {\cal B}$, then ${\cal K}%
_{i}=\{(s_{\sigma },t_{\sigma ^{\prime }}):(s,t)\in {\cal K}_{\sigma
_{4}(i)},\sigma ,\sigma ^{\prime }\in \Sigma \}$.
\end{itemize}
Now note that every relation ${\cal K}_{i}$ is Euclidean, serial and
transitive. For the ones corresponding to agents in ${\cal A}_{1}$ this is
immediate from the fact that the agents in ${\cal A}_{1}$ have these
properties. For those with $\sigma_{4}(i)\in{\cal A}_{2}$, seriality follows
from the fact that the agents in ${\cal A}_{2}$ are serial and the
construction. Transitivity and the Euclidean property follow from the
construction. In particular, if there is a ${\cal K}_{i}$ edge coming into
some $t_{\sigma}$ then there is none going out by construction except for
the one from $t_{\sigma}$ to itself.
The verification that $M^{\prime}$ satisfies $\phi$ now proceeds as in
Proposition~\ref{transrst}.
\vrule height7pt width4pt depth1pt\vspace{0.1in}
\begin{theorem}
\label{decelt} If ${\cal A}={\cal A}_{1}+{\cal A}$ is finite and there is an
an algorithm for deciding if $i\in G$ for $G\in {\cal G}$ that runs in
time
linear in $|{\cal A}|$, then there is
a constant $c > 0$ (independent of $|{\cal A}|$) and an algorithm
that, given a formula $%
\phi $ of ${\cal L}_{{\cal G}}^{C}$, decides if $\phi $ is satisfiable
in
${\cal M}_{{\cal A}_{1}+{\cal A}_{2}}^{elt}$ and runs
in time $O(|{\cal A}|2^{c|\phi |})$.
\end{theorem}
\noindent{\bf Proof:}
The argument here is like that for the ${\cal M}_{{\cal A}_{1}+{\cal A}%
_{2}}^{rst}$ case in Theorem~\ref{decrst}. We keep the definition of
${\cal K}_{i}$
for $i\in {\cal A}_{1}$ and, as we noted there, this makes these
relations
Euclidean and transitive. We change the definition of ${\cal K}_{i}$ for $%
i\in {\cal A}_{2}$ by putting $(s,t)$ in ${\cal K}_{i}$ iff
$s/\overline{K_i} \subseteq t$ and $t/\overline{K_i} \subseteq t$. This latter definition
clearly makes the ${\cal K}_{i}$ secondarily reflexive for $i\in {\cal A}_{2}
$. We ensure
seriality
by adding a clause to the definition of a state $s$
seeming consistent:
\begin{enumerate}
\item[4] For every agent $i\in {\cal A}_{2}$ there is a state $t$ such that
$(s,t)\in {\cal K}_{i}$ and for every agent $i\in {\cal A}_{1}$ there are
states $s^{\prime }$ and $t$ such $s\preceq _{i}s^{\prime }$ and $(s^{\prime
},t)\in {\cal K}_{i}$.
\end{enumerate}
The proof now proceeds as before. \vrule height7pt width4pt depth1pt\vspace{0.1in}
\bigskip
\noindent
{\bf Proof of Theorem~\ref{complexity} for $\M_n^{\mbox{\scriptsize{{\it elt}}}}$:} The argument here
is essentially the same as for $\M_n^{\mbox{\scriptsize{{\it rst}}}}$. Just note that using the oracle
$O'$ we can determine the members of $\cal{B}$ within the appropriate
time bound and so compute $\phi^{\sigma_4}$ as required.
\vrule height7pt width4pt depth1pt\vspace{0.1in}
We now turn our attention to proving Theorem~\ref{complete} for
$\M_n^{\mbox{\scriptsize{{\it elt}}}}$. The basic structure is the same as for $\M_n^{\mbox{\scriptsize{{\it rst}}}}$.
\begin{lemma}\label{replaceelt} The mapping $\sigma_4$ (when viewed as a map
with domain $2^{\cal A}$) is injective on $\widetilde{{\cal G}}_\phi^n$.
\end{lemma}
Let $({\rm KD45}_\G^C)^{\A_1+\A_2}$ consist of the axioms in ${\rm K}_n^C$
together with K3, E4, E7, and every instance of K4 and K5 for $i \in
{\cal A}_1$. We write $({\rm KD45}_\G^C)^{\A_1+\A_2}
\vdash_\phi \psi$ if there is a proof of $\psi$ in $({\rm KD45}_\G^C)^{\A_1+\A_2}$ using
only the modal operators that appear in $\phi$ and $K_i$ for $i \in
{\cal A}_1$.
\begin{lemma}\label{proofelt} If ${\cal A}$ is finite and $\phi \in {\cal L}_{\G}^C$ is
valid with respect to $\M_{\A_1+\A_2}^{\mbox{\scriptsize{{\it elt}}}}$, then $({\rm KD45}_\G^C)^{\A_1+\A_2}
\vdash_\phi \phi$. \end{lemma}
\noindent{\bf Proof:} The proof is similar to that of Lemma~\ref{proofrst} for $\M_n^{\mbox{\scriptsize{{\it rst}}}}$.
Again we must check that all states eliminated in the construction are
provably inconsistent, but now using the axioms of $({\rm KD45}_\G^C)^{\A_1+\A_2}$ and the
modified
definition of the ${\cal K}_i$ relations, and dealing with the additional
clause in the definition of seeming consistent.
The argument for the first condition for seeming consistent
is the same as that for $\M_n^{\mbox{\scriptsize{{\it rst}}}}$.
Before dealing with the second condition, we prove a fact that will
also be useful in dealing with the fourth condition. Let $T_i = \{t
\in S_i^j: (t,t) \in {\cal K}_i\}$. It is easy to see that
\begin{equation}
\label{eqelt}
\mbox{if $t \notin T_i$, then $({\rm KD45}_\G^C)^{\A_1+\A_2} \vdash_\phi \phi_s \Rightarrow E_G
\neg \phi_t$ for some $G$ such that $i \in G$.}
\end{equation}
For if $t \notin T_i$, then there exists $E_G \theta \in t$ such
that $i \in G$ and $\neg \theta \in t$. But then $(E_G \theta \Rightarrow
\theta) \Rightarrow \neg \phi_t$ is propositionally valid (and so provable by
Prop). Since $({\rm KD45}_\G^C)^{\A_1+\A_2} \vdash_\phi \phi_s \Rightarrow E_G(E_G\theta \Rightarrow
\theta)$, we can easily obtain (\ref{eqelt}) using (\ref{basic4}).
Now suppose that $s$ is
eliminated because it does not satisfy the second condition for seeming
consistent due to $E_{G}\psi$.
It again suffices to show that for each $i \in G$ and $t \in S_i^j$ such
that $\psi \in t$, there is a set $G^{i,t}$ of agents containing $i$ such
that $({\rm KD45}_{G}^{C})^{{\cal A}_{1}+{\cal A}_{2}}\vdash _{\phi }
\phi _{\vec{s}}\Rightarrow E_{G^{i,t}}\lnot \phi _{t}$.
First suppose $i \in {\cal A}_2$.
If $(s,t)\notin {\cal K}_{i}$ because $s/\overline{K_i} \not\subseteq t$ then
the argument given in Lemma~\ref{proof} works to get a $G^{i,t}$
as desired. If $s/\overline{K_i}$ $\subseteq t$ but $t/\overline{K_i}\not\subseteq t$ then
the existence of the required $G^{i,t}$ is immediate from (\ref{eqelt}).
Now suppose $i \in {\cal A}_1$
and $t$ is such that $(s^{i},t)\notin {\cal K}_{i}$. If
$s/K_{i}\not\subseteq t$, then there is some formula $\theta$ such that
$K_{i}\theta \in s^{i}$ and $\lnot \theta \in t$; it easily follows that
$({\rm KD45}_\G^C)^{\A_1+\A_2} \vdash_\phi \phi_{\vec{s}} \Rightarrow K_i \neg \phi_t$, as required.
If $\{K_i \theta
:K_{i}\theta \in s\}\not\subseteq t$, then there is some $\theta $ such
that $%
K_{i}\theta \in s$ but $\lnot K_{i}\theta \in t$; the result now easily
follows using K4, just as in the argument for $({\rm S4}_\G^C)^{\A_1+\A_2}$.
If
both of these conditions hold (but still $(s^{i},t)\notin {\cal K}_{i}$),
then it must be that there is a $\theta $ with $K_{i}\theta \in t$ and $%
K_{i}\theta \notin s$. In this case $\lnot K_{i}\theta \in s$,
and the result follows using K5, just as in the argument for
$({\rm S5}_\G^C)^{\A_1+\A_2}$.
The argument in the case that $s$ is
eliminated because it does not satisfy the third condition for seeming
consistent is the same as in the proof of Lemma~\ref{proofrst}.
Finally, suppose that $s$ does not satisfy
the new (fourth) condition of seeming consistent. Then either
\begin{itemize}
\item there is an $i\in {\cal A}_{2}$ for which there is no $t$ with $%
(s,t)\in {\cal K}_{i}$ or
\item there is an $i\in {\cal A}_{1}$ for which there is no pair $s^{\prime
},t$ such that $s\preceq _{i}s^{\prime }$ and $(s^{\prime },t)\in {\cal K}%
_{i}$.
\end{itemize}
{F}or the first case,
for each $t \in T_i$, it must be the case that $s/\overline{K_i} \not\subseteq
t$, so that there must be some $G^{i,t}$ with $i \in G^{i,t}$ such that
$({\rm KD45}_\G^C)^{\A_1+\A_2} \vdash \phi_s \Rightarrow E_{G^{i,t}} \neg \phi_t$, as usual. By
(\ref{eqelt}), for each $t \notin T_i$, there is some $G^{i,t}$ with $i
\in G^{i,t}$ such that
$({\rm KD45}_\G^C)^{\A_1+\A_2} \vdash_\phi \phi_s \Rightarrow E_{G^{i,t}} \neg \phi_t$. Thus,
$({\rm KD45}_\G^C)^{\A_1+\A_2} \vdash_\phi \phi_s \Rightarrow \land_{t \in S^j} E_{G^{i,t}} \neg
\phi_t$. But since
$({\rm KD45}_\G^C)^{\A_1+\A_2} \vdash_\phi \neg (\land_{t \in S^j} \neg \phi_t)$ by
induction and
propositional reasoning, it follows from E7 that
$({\rm KD45}_\G^C)^{\A_1+\A_2} \vdash_\phi \neg (\land_{t \in S^j} E_{G^{i,t}} \neg
\phi_t)$. Thus we get
$({\rm KD45}_\G^C)^{\A_1+\A_2} \vdash_\phi \neg \phi_s$, as desired.
{F}or the second case, we know as in the proof of Lemma \ref{proofrst}
that $\phi _{s}$ is provably equivalent to the disjunction of $\phi
_{s^{\prime }}$ for those $s^{\prime }$ such that $s\preceq _{i}s^{\prime }$
and similarly for any $t$. Thus to prove
$({\rm KD45}_\G^C)^{\A_1+\A_2} \vdash_\phi \lnot \phi _{s}$ it suffices
to prove
$({\rm KD45}_\G^C)^{\A_1+\A_2} \vdash_\phi \lnot \phi _{s^{\prime }}$ for every $s^{\prime }\in
S_{i}^{j}$ such
that $s\preceq _{i}s^{\prime }$. For each such $s^{\prime }$ we know that
there is no $t^{\prime }\in S_{i}^{j}$ such that $(s^{\prime },t^{\prime
})\in {\cal K}_{i}$.
Given $s'$, if $t' \in S_i^j$ and
$(s',t')
\notin {\cal K}_i$, then the same
argument as in the proof of Lemma~\ref{proofrst} shows that
$({\rm KD45}_\G^C)^{\A_1+\A_2} \vdash_\phi \phi_{s'} \Rightarrow K_i \neg \phi_{t'}$, since the
${\cal K}_i$
relations are defined the same way for agents in ${\cal A}_1$ in both
the
$\M_n^{\mbox{\scriptsize{{\it elt}}}}$
and $\M_n^{\mbox{\scriptsize{{\it rst}}}}$
cases,
and the proof in Lemma~\ref{proofrst} used only
axioms K4 and K5 (as well as Prop, K1, and MP), and these axioms are in
both $({\rm S5}_\G^C)^{\A_1+\A_2}$ and $({\rm KD45}_\G^C)^{\A_1+\A_2}$.
By (\ref{basic3}), we have that $({\rm KD45}_\G^C)^{\A_1+\A_2} \vdash \phi_{s'} \Rightarrow
K_i(\land_{t' \in S_i^j} \neg \phi_{t'})$. Since $({\rm KD45}_\G^C)^{\A_1+\A_2} \vdash_\phi
(\land_{t' \in S_i^j} \neg \phi_{t'}) \Rightarrow \mbox{{\it false}}$ by
induction and
propositional
reasoning, we conclude that $({\rm KD45}_\G^C)^{\A_1+\A_2}
\linebreak[2]
\vdash_\phi \phi_{s'} \Rightarrow K_i
\mbox{{\it false}}$. Now using K3, we get $({\rm KD45}_\G^C)^{\A_1+\A_2} \vdash_\phi \neg \phi_{s'}$, as
desired. \vrule height7pt width4pt depth1pt\vspace{0.1in}
\bigskip
\noindent
{\bf Proof of Theorem~\ref{complete} for $\M_n^{\mbox{\scriptsize{{\it elt}}}}$:} The proof follows as
for $\M_n^{\mbox{\scriptsize{{\it rt}}}}$ using the analogous lemmas proved above for $\M_n^{\mbox{\scriptsize{{\it elt}}}}$.
We must just show that E7 is derivable from the other axioms in ${\rm KD45}_n^C$.
Suppose that $i \in G_1 \cap \ldots G_k$. Then, using E1, ${\rm KD45}_n^C
\vdash E_{G_1}
\phi_1 \land \ldots \land E_{G_k} \phi_k \Rightarrow K_i \phi_1 \land \ldots
\land K_i \phi_k$. By (\ref{basic3}), we have
${\rm KD45}_n^C \vdash K_i \phi_1 \land \ldots \land K_i \phi_k \Rightarrow K_i(\phi_1
\land \ldots \land \phi_k)$. Thus, ${\rm KD45}_n^C \vdash \neg K_i(\phi_1 \land
\ldots \land \phi_k) \Rightarrow
\neg (E_{G_1} \phi_1 \land \ldots \land E_{G_k} \phi_k)$.
It thus suffices to show that in ${\rm KD45}_n^C$, from $\neg(\phi_1
\land \ldots \land \phi_k)$ we can infer $\neg K_i(\phi_ \land \ldots
\land \phi_k)$. But since $\neg (\phi_1 \land \ldots \land \phi_k)$ is
equivalent to $(\phi_1 \land \ldots \phi_k) \Rightarrow \mbox{{\it false}}$, this follows
easily using (\ref{basic4}) and K3.
\vrule height7pt width4pt depth1pt\vspace{0.1in}
\subsection{The complexity of querying the oracles}\label{oracle}
Up to now we have assumed that we are charged one for each query to an
oracle. In this section, we reconsider our results, trying to take into
account more explicitly the cost of the oracle queries.
Let $f(m,k)$ be the worst-case time complexity
of deciding whether a set with description $G \in \widehat{{\cal G}}_{\cal A}^m$
such that $l(G) \le k$ has cardinality greater $m'\le m$ (where we take
the worst case over all $G \in \widehat{{\cal G}}_{\cal A}^m$ such that $l(G) \le
k$ and over all $m' \le m$).
Let $g(k)$ to be the worst-case complexity of deciding if
$G_1 \cap \ldots \cap G_k = \emptyset$ for $G_1, \ldots, G_k \in
{\cal G}_{\cal A}$.
We take $f(m,k)$ (resp.,\ $g(k)$) to be $\infty$ if these questions are
undecidable.
We can think of $f(m,k)$ (resp.,\ $g(k)$) as the worst-case cost of
querying the oracle $O_m$ (resp.,\ $O'$)
on a set
with a
description of length $\le k$.
Using these definitions,
we can sharpen Theorem~\ref{complexity} as follows.
\begin{theorem}\label{complexity1} There is a constant $c > 0$ and an algorithm
that decides if
a formula $\phi \in {\cal L}_{\G}^C$ is satisfiable in
${\cal M}_n$ (resp.,\
${\cal M}_n^r$, $\M_n^{\mbox{\scriptsize{{\it rt}}}}$, $\M_n^{\mbox{\scriptsize{{\it rst}}}}$, $\M_n^{\mbox{\scriptsize{{\it elt}}}}$) and runs in time
$2^{c|\phi|}f(0,|\phi|)$
(resp.,\
$2^{c|\phi|}f(0,|\phi|)$,
$2^{c|\phi|}f(1,2^{c|\phi|^2})$,
$2^{c|\phi|}f(|\phi|,2^{c|\phi|^2})$,
$2^{c|\phi|}(f(|\phi|,2^{c|\phi|^2}) + g(|\phi|))$)
Moreover, if ${\cal G}$
contains
a subset with at least two elements, then there exists a constant $d >
0$ such that every algorithm for deciding the satisfiability of formulas
in ${\cal M}_n$ (resp.,\ ${\cal M}_n^r$, $\M_n^{\mbox{\scriptsize{{\it rt}}}}$, $\M_n^{\mbox{\scriptsize{{\it rst}}}}$, $\M_n^{\mbox{\scriptsize{{\it elt}}}}$) runs in time at
least
$\max(2^{d|\phi|},f(0,d|\phi|))$ (resp.,\
($\max(2^{d|\phi|},f(0,d|\phi|))$,
$\max(2^{d|\phi|},f(1,d|\phi|))$,
$\max(2^{d|\phi|},f(d|\phi|,d|\phi|))$,
$\max(2^{d|\phi|},f(d|\phi|,d|\phi|),g(d|\phi|))$)
for infinitely many formulas $\phi$.
\end{theorem}
\noindent{\bf Proof:} The upper bound is almost immediate from the proof of
Theorem~\ref{complexity}. The only point that needs discussion is the
second argument---$2^{c|\phi|^2}$---of $f$ in the cases $\M_n^{\mbox{\scriptsize{{\it rt}}}}$,
$\M_n^{\mbox{\scriptsize{{\it rst}}}}$, and $\M_n^{\mbox{\scriptsize{{\it elt}}}}$. This follows from Lemma~\ref{Gmcount}.
An easy induction on $i$ shows that the sets in
the set
${\cal E}_i^{|\phi|}$
constructed just before Lemma~\ref{Gmcount} have
description length at most
$\le 2^{2i|\phi|}$
(using the
fact that
$|{\cal E}_i^{|\phi|}|
\le 2^{|\phi|}$ for all $i$). Thus, all the sets that
we need to deal with have description length $\le 2^{2|\phi|^2}$, since
they are all in
${\cal E}_{|\phi|}^{|\phi|}$,
by
Lemma~\ref{Gmcount}(e).
The lower bound is immediate from the results of \cite{HM2} and
Proposition~\ref{reduction}. \vrule height7pt width4pt depth1pt\vspace{0.1in}
Note that if $f_0(k) = f(0,k)$ is
{\em well behaved}, in that
there exist $c'$, $k_0$ such that $f_0(k) \le
2^{c'k}$ for all $k \ge k_0$ or $f_0(k) \ge 2^{c'k}$ for all $k \ge
k_0$, then it is easy to see that there is some
$c'' > 0$ such that
$2^{c|\phi|}f(0,|\phi|) \le
\max(2^{c''|\phi|},c''f(0,|\phi|))$.
Thus, if $f_0$ is well behaved, then the lower and upper bounds
of Theorem~\ref{complexity} match, and we have tight bounds in the case
of ${\cal M}_n$ and ${\cal M}_n^r$. This is not the case for $\M_n^{\mbox{\scriptsize{{\it rt}}}}$, $\M_n^{\mbox{\scriptsize{{\it rst}}}}$, and
$\M_n^{\mbox{\scriptsize{{\it elt}}}}$, because the sets that arise have exponential-length
descriptions.
Do we really have to answer queries of about such complicated formulas
if we are to deal with $\M_n^{\mbox{\scriptsize{{\it rt}}}}$, $\M_n^{\mbox{\scriptsize{{\it rst}}}}$, and $\M_n^{\mbox{\scriptsize{{\it elt}}}}$? To some extent,
this is an artifact of our insistence that the sets be described using
union and set difference. In fact, all the sets that we need to
consult the oracle about in our
algorithm are atoms, and so have very simple descriptions
($O(|\phi|)$) if we are allowed to used intersections and complementation.
Indeed, suppose that we define an ordering $\prec$ on
atoms such that $A_{\cal H} \prec A_{{\cal H}'}$ if ${\cal H} \supset {\cal H}'$. It follows
easily
from Lemma~\ref{unique} and Lemma~\ref{K} that in order to compute
$\sigma_2(G)$ (resp.,\ $\sigma_3(G)$, $\sigma_4(G)$), we start by
considering all atoms $A_{\cal H}$ such that $G$ appears positively in $A_{\cal H}$
and all other sets in ${\cal G}_\phi$ appear negatively; we then need to check
whether $|A_{\cal H}| > 0$ and $|A_{\cal H}| > 1$ (resp.,\ $|A_{\cal H}| > 0$, \ldots, and
$|A_{\cal H}| > |\phi|$) only for those atoms $A_{\cal H}$ such that for all ${\cal H}'
\prec {\cal H}$, we have $|A_{{\cal H}'}| \le 1$ (resp.,\ $|A_{{\cal H}'}| \le |\phi|$).
(In addition, in the case of $\sigma_4$, we have also have to check
whether $G_1 \cap \ldots
\cap
G_k = \emptyset$, but again, these are sets
with simple descriptions if we allow intersection.)
Thus, as long as we can check the required properties of sets
described in terms of intersection and complementation relatively
efficiently, then the queries to the oracle pose no problem.
Unfortunately, the bounds in Proposition~\ref{reduction} depends on the
descriptions involving only set difference and union, so we cannot get
tight bounds for Theorem~\ref{complexity} (at least, with our current
techniques) using descriptions that involve intersection and
complementation.
It remains an open question whether we can get tight bounds in all cases
taking into account the cost of querying the oracle.
\section{Conclusions}\label{discussion}
We have characterized the complexity of satisfiability for epistemic
logics when the set of agents is infinite. Our results emphasize the
importance of how the sets of agents are described and
provide new information
even in the case where the sets involved are finite.
In this paper we have focused on a language that has operators $E_G$ and
$C_G$. There are two interesting directions to consider extending our
results.
\begin{itemize}
\item We could restrict the language so that it has only $E_G$
operators. If the set of agents is finite (and all sets $G$ are
presented in such a way that it is easy to check if $i \in G$), then
there are well-known results that show the complexity of the decision
problem in this case is PSPACE complete \cite{HM2}. However, again,
this result counts $E_G$ as having length $|G|$. Although we have not
checked details, it seems relatively straightforward to combine the
techniques of \cite{HM2} with those presented here to get PSPACE
completeness for ${\cal L}_{\G}^E$, taking $E_G$ to have length 1, using the same
types of oracle calls as in Theorem~\ref{complexity}.
(Note that
Proposition~\ref{reduction} applies to the language ${\cal L}_{\G}^E$; we did not
use the $C_G$ operators in this proof.)
\item We could add the distributed knowledge operator $D_G$ to the
language \cite{FHMV,FHV2,HM2}. Roughly speaking, $\phi$ is distributed
knowledge if the agents could figure out that $\phi$ is true by pooling
their knowledge together. Formally, we have
\begin{quote}
$(M,s) \models D_G \phi$ if
$(M,t) \models \phi$ for all $t \in \cap_{i \in G} {\cal K}_i(s)$.
\end{quote}
It is known that if ${\cal A}$ is finite (and there is no difficulty in
telling if $i \in G$), then adding $D_G$ to the language poses no
essential new difficulties \cite{FHMV,HM2}. We can get a complete
axiomatization, the satisfiability problem for the language with $D_G$
and $E_G$ operators is PSPACE complete, and once we add common
knowledge, the satisfiability problem becomes exponential-time complete.
Once we allow infinitely many agents, adding $D_G$ introduces new
subtleties.
For example,
even if we place no assumptions on the ${\cal K}_i$ relations,
once we have both $E_G$ and $D_G$ in the language, we need to be able
to distinguish between sets of
cardinality one and those with larger cardinality since
$E_Gp \Leftrightarrow D_Gp$ is valid if and only if $G$ is a singleton. New
issues also arise
once we make further assumptions about the ${\cal K}_i$ relations
because different
properties are preserved for the new agents, say
$K_{{\cal A}^D}$ and $K_{{\cal A}^E}$, which are to be added on as in
Proposition~\ref{Shore} to represent $D_{\cal A}$ and $E_{\cal A}$, respectively.
Intuitively,
${\cal K}_{{\cal A}^E}$ corresponds to the union of the
relations $K_i$ for $i \in G$
while ${\cal K}_{{\cal A}^D}$ corresponds to their intersection. Thus, while both
$K_{{\cal A}^D}$ and $K_{{\cal A}^E}$ inherit reflexivity and symmetry from the
$K_i$ relations,
$K_{{\cal A}^D}$ inherits transitivity and the Euclidean property
while $K_{{\cal A}^E}$ does not.
There are also additional
relations between these agents that must be taken into account.
Examples in S4 and S5 include $K_{{\cal A}^E}\phi \Rightarrow K_{{\cal A}^D} \phi$,
$K_{{\cal A}^E}K_{{\cal A}^D}\phi \Rightarrow K_{{\cal A}^E}\phi$ and
$K_{{\cal A}^D}K_{{\cal A}^E}\phi \Rightarrow K_{{\cal A}^E}\phi$.
These are issues for future work.
\end{itemize}
\bibliographystyle{alpha}
|
1,116,691,497,614 | arxiv | \section*{Main Text}
\subsection*{Introduction}
Breath analysis examines the hundreds or thousands of low-concentration molecular species present in exhaled human breath and correlates them with particular health conditions or diseases. Some examples include formaldehyde being correlated with breast cancer, ammonia with asthma, and methane with intestinal problems~\cite{Wang_laserBreathReview}. This research field is gaining considerable interest due to its potential for non-invasive, low-cost, and real-time diagnosis. Unleashing the full potential of breath analysis requires the identification of a panel of biomarkers that are associated with a certain medical condition, and how their concentrations co-vary in the presence or absence of a particular condition. Two major road blocks hindering the adoption of breath analysis to medical diagnosis have been the difficulties in i) reaching the necessary capability to identify breath biomarkers with superior detection specificity and sensitivity, and ii) data analysis to correlate the measurement results with certain medical conditions for unambiguous and reliable predictions.
Cavity-enhanced direct frequency comb spectroscopy (CE-DFCS) offers great advantages for detecting molecules~\cite{Adler_CEDFCS}. It combines ultra-sensitive and high-resolution detection, broadband spectral coverage, and high data acquisition efficiency all in one platform, while also providing both isomer- and isotopologue-specific detection capabilities. The simplicity of in-situ optical absorption together with massively-parallel signal collection provide direct and objective measurements of biomarkers, ideal for building data bases to facilitate cross-field studies.
The application of CE-DFCS to breath analysis was initially demonstrated in the near-infrared spectral region~\cite{Thorpe_breathalyzer}. In 2021, a more than 2-orders-of-magnitude improvement in sensitivity was achieved by extending this technology to the mid-infrared molecular fingerprint region, where strong fundamental vibrational transitions are accessed and detection sensitivities as high as parts-per-trillion ($10^{−12}$) were achieved~\cite{Liang-PNAS}. We have thus finally reached the full sensitivity requirement for breath analysis where most trace species are present at the $10^{−9}$ to $10^{−12}$ levels~\cite{Wang_laserBreathReview}. However, the massive density of spectral features carrying richer-than-ever, molecular-specific information poses a serious complexity in data processing to establish robust predictive modeling for medical diagnosis. We overcome this complexity using supervised machine learning dedicated to extremely high-dimensional data analysis and verification. The high-quality comb data has allowed the use of a pattern-based analysis method to establish recognizable digital spectral fingerprints for breath-based disease diagnosis. From these digital fingerprints we can also identify a panel of responsible molecules. This study constitutes the first real-world examination of CE-DFCS’s capability for medical diagnostics.
We target the detection of Coronavirus Disease 2019 (COVID-19), a highly infectious disease with airborne transmission via aerosol and droplets~\cite{Liu_airborneCOVID} that has tragically caused many deaths globally~\cite{Ritchie_COVIDDeaths}. In some exploratory COVID-19 breath testing efforts, studies have been conducted with small sample sizes of research subjects of different age groups (mainly above 40) and infection severity~\cite{Subali_ReviewBreathCovid} using nanomaterial-based sensors~\cite{Shan_COVID_NanoSensor,ZAMORAMENDOZA_COVID_NanoSensor}, ion-mobility spectrometry~\cite{RUSZKIEWICZ_COVID_GCIMS,Chen_COVID_GCIMS}, and mass-spectrometry~\cite{Ibrahim_COVID_Massspec,GRASSINDELYLE_COVID_PTRMassSpec}. We have performed breath analysis of COVID infected individuals in a unique manner. We target a group of young people with a high vaccination rate. We focus on collecting highest quality molecular data from breath and performing rigorous cross-validation and baseline confirmation.
From May 2021 to January 2022, we collected breath samples from a total of 170 research subjects with a class distribution for COVID-19 infection of 83 positives (\SI{49}{\%}) and 87 negatives (\SI{51}{\%}), as identified by RT-PCR. Ninety-six percent of the research subjects were born after 1980, all above 18 years old and affiliated with the University of Colorado Boulder as a student and/or a university employee. Fifty-nine percent reported 5 days or more of school/work commuting needs per week. The general population on campus is $>$\SI{90}{\%} vaccinated. The recruited subjects thereby constitute a young and high transmission potential study group.
Inspired by the DOME recommendations calling for a standard to enable improved cross-study diagnostic comparisons~\cite{Walsh_DOME}, we analyze this large sample group and use 10,000 cross-validation runs based on stratified random sampling to obtain converging indicators required for medical diagnostic studies. Excellent cross-validated prediction performance for COVID-19 is achieved with an area under the Receiver-Operating-Characteristic (ROC) curve of 0.849(4). Interestingly, this comb-machine learning breath analysis also identifies significant differences based on a number of variables collected in tandem for this study including male vs. female and smoker vs. nonsmoker. The baseline probability is checked by comparing the COVID-19 and gender results against a binary response classification for parity (even/odd) of birth months.
\subsection*{Data analysis}
The working principle of the mid-infrared comb breathalyzer for COVID-19 classification is illustrated in Fig.~\ref{fig:Principle}a. A research subject provides a breath sample in a standard Tedlar bag. The sample is loaded and analyzed by the breathalyzer to generate high-resolution broadband spectroscopic data that resolves rotational and vibrational molecular absorption features (see sample spectrum in Fig.~\ref{fig:Principle}b). The massive amount of data collected, consisting of absorption signals measured by a total of 14,836 frequency comb lines uniformly sampled in a spectral window ranging from \SI{2810}{cm^{-1}} to \SI{2945}{cm^{-1}}, with a frequency uncertainty of $\sim \SI{0.0000017}{cm^{-1}}$, is then processed by a machine learning algorithm to predict the COVID-19 infection status (either positive or negative) of the research subject.
The learning algorithm is constructed from a set of observations, each representing the spectroscopic data collected from a single research subject combined with its true class label, also named response. The spectroscopic data used for prediction purposes are called predictor variables. We utilize a Molecular Science (MS) approach, which uses the spectra of 16 known compounds as predictor variables, and a Data Science (DS) approach, which uses all 14,836 frequency comb lines directly as the predictor variables (see Methods). We use the Partial Least-Squares Discriminant Analysis (PLS-DA) algorithm~\cite{DEJONG_SIMPLS,Chuen_PLSDA-review} to discriminate between opposing response classes. Linear combinations of the predictor variables are optimized based on maximizing their covariance with the response. For model assessment, a subset of observations is randomly selected and isolated from the algorithm training process and used only for testing the prediction performance (see Fig.~\ref{fig:CrossVad}). To generate the prediction results, the predictor variables for an observation in the testing set are regressed to yield a probability value for grouping into the opposing classes.
We evaluate the prediction performance by using ROC curves~\cite{MANDREKAR_AUCinterpret,FAWCETT_ROC,Chen_ROC_tilt} (see Methods). The entire model assessment, called cross-validation, is repeated 10,000 times to generate an averaged ROC curve obtained from all individual runs. In each run, stratified random sampling is used for the selection of a new training set and testing set so as to preserve the class distribution in the complete data set. The predictive power is quantified by the area under the curve (AUC) of the averaged ROC curve, along with the prediction sensitivity (true positive rate) and specificity (true negative rate) quoted for the data point on the averaged ROC curve giving the highest pair average of the two (see Methods for more details). We investigate binary classification for COVID-19 infection (positive vs. negative) and compare it with a few other response type classifications including birth month (odd vs. even), gender (female vs. male), smoking status, and abdominal pain.
\subsection*{Results and discussion}
We present a number of important findings. First, the DS approach outperforms the MS approach for classification of COVID-19 infection. We show distributions of the subjects’ data projected on the first three PLS components in Fig.~\ref{fig:PLS}. Here, the complete data set (N = 170) is used for constructing the PLS components and all observations are plotted to provide a complete picture. Viewed from the first three PLS components, the DS approach (Fig.~\ref{fig:PLS}b) clearly demonstrates larger separability between the positive and the negative classes over the MS approach (Fig.~\ref{fig:PLS}a). The total percentage variance in the response explained by the first three PLS components is \SI{30}{\%} for the MS approach and \SI{76}{\%} for the DS approach (see Methods for details).
Second, we find the use of 15 PLS components sufficient to saturate the total percentage variance explained for both the MS and the DS approaches, with the values obtained at \SI{31.222}{\%} and \SI{99.996}{\%}, respectively. We thus use the 15 PLS components to evaluate the relative importance of the predictor variables through the Variable Importance in the Projection (VIP) score~\cite{CHONG_VIP}.
Third, based on the VIP score, MS and DS approaches identify their respective important predictor variables. Figures~\ref{fig:PLS}c and \ref{fig:PLS}d give their VIP scores. For MS, formaldehyde (H$_2$CO) is identified as the most important predictor variable for detecting COVID-19 infection owing to its greatest explained variance for the response. In contrast, important predictor variables for the DS approach are distributed nearly uniformly across the entire spectrum, showing no obvious weighting in this spectral region. This broad distribution of discriminating signals across the spectrum illustrates the power of CE-DFCS for distinguishing biological conditions, with future improvements expected from expansion of the spectral range.
Fourth, our method does not generate over-optimistic prediction results. The classification result on birth month (Fig.~\ref{fig:ROC}a) examines and tries to predict whether the research subjects were born in the odd (January etc.) or even (February etc.) months. Of course, one would not expect such response type classification to yield a correlated result based on the exhaled human breath. Indeed, we find this a rigorous check to ensure the algorithm does not report unwarranted results. We find the AUC, sensitivity, and specificity to be 0.516(4), \SI{55}{\%} and \SI{48}{\%} respectively based on the DS approach. For MS, the AUC is 0.487(9). These results confirm that both classifiers are indeed random guessers when relying on the CE-DFCS breath data to distinguish the even/odd parity of birth month.
Strikingly, in a completely unexpected result, our approach is able to distinguish between breath of men and women. Specifically, we find the DS-based AUC of 0.667(12), sensitivity of \SI{52}{\%}, and specificity of \SI{72}{\%} (Fig.~\ref{fig:ROC}b), revealing a significant gender-based difference in exhaled breath. For MS, the AUC is a comparable value of 0.638(9). The gender-based difference may increase when we probe a wider spectral range. Similar analyses on other personal attributes such as smoking and abdominal pain has also revealed significant differences in breath.
Most importantly, our spectra are able to distinguish COVID-19 infected and uninfected individuals. The classification results for COVID-19 (Fig.~\ref{fig:ROC}c) give an AUC of 0.849(4), sensitivity of \SI{76}{\%}, and specificity of \SI{82}{\%}. A classifier with an AUC $>$0.8 is considered to have excellent discriminating capability~\cite{MANDREKAR_AUCinterpret}. This demonstrates the power of CE-DFCS for breath analysis for COVID-19 diagnosis. Since RT-PCR tests for COVID-19 are not \SI{100}{\%} accurate, it is possible that our diagnostic results are even better than reported. It will be very valuable to conduct future large-scale tests that pair CE-DFCS with other standard COVID-19 testing techniques for daily screening.
For COVID-19 diagnosis, the DS approach clearly outperforms the MS approach, the latter of which produces an AUC of 0.769(7). To understand this result, we note that the CE-DFCS technology is sensitive to the presence of both i) small molecules with their rotational and vibrational structure resolved by absorption signals, and ii) large molecules with unresolved spectroscopic features. The presence of the large molecules will introduce frequency-dependent optical losses inside the cavity, which are encoded in the measured spectroscopic data through alteration in absorption patterns of the resolved molecules. Simple fitting of the spectroscopy features to a set of known molecules can neither guarantee the accuracy of the fitted concentrations nor confirm the exhaustion of all relevant molecules. For breath analysis, this additional sensitivity is actually very valuable as it reveals the presence of more molecular species. The DS approach senses all the variations in the spectroscopic signals related to the presence of unresolved molecules, providing detection sensitivity to both large and small molecules. Thus, in contrast to the MS approach, the variability in the CE-DFCS data measured for different research subjects is entirely preserved by the DS approach. This suggests the superior prediction performance achieved by the DS approach in the COVID-19 diagnosis can be attributed to the presence of unresolved molecules and potentially even the coronavirus itself~\cite{Sawano_BreathViralLoad}. This discussion also highlights a key reason for the powerful capability of CE-DFCS for breath analysis: its high detection sensitivity to both resolved and unresolved molecules. Both are important for the goal of providing highly accurate medical diagnosis based on exhaled breath contents.
It is also interesting to note that the classification performance of COVID-19 infection is substantially better than that for gender. This result is likely related to the fact that the former is distinguishing healthy from unhealthy subjects whereas the latter is trying to distinguish an innate biological difference and can thus be more difficult to classify. Comparing the prediction results for the three response types investigated, a natural progression can be seen. First, the random guess birth month check ensures that the model does not generate unwarranted prediction results. A statistically significant prediction of differences in gender reveals the unique capability of CE-DFCS plus learning model. Finally, a much more robust prediction for COVID-19 highlights the remarkable potential of CE-DFCS-based breath analysis paired with supervised learning for reliable medical diagnostics. It is exciting to anticipate future powerful applications of CE-DFCS for other disease state response classifications, by using exhaled breath contents to differentiate between the presence or absence of medical conditions in a non-invasive manner.
For our study group (N = 170), the between-type response covariances evaluated over the complete data set are negligible (see SI for details). The lack of correlations confirms that the classification of one of the three response types is not affected by the natural forces driving the separability for the other two response types. We cannot definitely rule out the possibility of other response types being potentially correlated with a specific response investigated in this work, for example non-COVID-19-related respiratory infections such as the common cold, influenza, or strep throat, which could affect the binary classification results. However, correlations with the response types not monitored in this study can in principle be averaged away statistically with more data collected, as is the case in all machine learning analysis, where a larger data set is always preferred.
It is important to stress the crucial capability of breath analysis for simultaneous multi-response classifications if sufficiently large group sizes are available, together with multiple non-correlated response types. A massive database can be constructed. To demonstrate this, we present multi-response analysis for a total of eleven personal attributes in the SI, which includes smoking, abdominal pain, and constipation.
We emphasize the importance of baseline checks to prevent over-interpretation and using the rigorous cross-validation protocol to ensure a converging AUC value with minimal uncertainty. These permit accurate and unbiased comparisons of the diagnostic power between non-interfering response types. There is an exponential growth of machine learning studies in the bio-medical field and the community has called for standards to be developed to enable meaningful cross-study assessment and comparisons. The community-proposed recommendations~\cite{Walsh_DOME} so far focus on only supervised learning, although research in breath analysis has broadly employed both supervised and unsupervised learning, along with many other published studies that provide only correlation analysis with no diagnostic power assessment~\cite{Subali_ReviewBreathCovid}. Vast differences in the cross-validation protocols and in the characteristics of the research subjects of various studies make it difficult to compare the diagnostic power of different technologies. With that in mind, our experimental findings together with other published studies strongly encourage the practical use of breath analysis for COVID-19 testing.
\subsection*{Future outlook}
The diagnostic power of our comb breathalyzer can be substantially enhanced by expanding the spectral coverage to detect more molecular species, based on the recent extension of frequency combs towards longer wavelengths~\cite{Iwakuni_10um,Picque2019,Kippenberg2020,Lesko_octave}. Second, by miniaturization and simplification of the technology, CE-DFCS can become transportable~\cite{Xiang_ChipComb,Fathy_ChipFTIR} and deployed in point-of-care settings with minimal operator training. Finally, alternative AI approaches that use neural networks~\cite{Amato_NeuralMedical} or weighted analyses with non-classifications might improve the predictive power of spectral analyses.
While the study here is limited to binary response classifications, we have also gathered information from the research subjects such as the number of COVID-19 symptoms experienced. This information can be used for designing a continuously varying test score for machine learning regression analysis as opposed to classification analysis where the response is discrete. In this way, we could assess whether the disease progression could be monitored by the exhaled breath contents. Another future endeavor is to correlate breath analysis data with viral loads quantified by RT-PCR to give more direct interpretation of the COVID-19 infection status and establish the sensitivity limit of CE-DFCS ~\cite{Sawano_BreathViralLoad}.
\subsection*{Conclusion}
Supervised machine learning analysis is used to tackle the critical complexity of the extreme high-dimensional frequency comb breath data carrying record-breaking richness in molecular-specific information, allowing the first realization of robust medical diagnosis. With a sample size of 170, the use of 14,836 absorption features as the predictor variables produces an excellent cross-validated prediction performance, yielding AUC of 0.849(4) for COVID-19 classification. At the same time, the same analysis reveals significant breath differences on a number of other personal attributes. The availability of tens of thousands of molecular spectral features for rigorous machine learning and classification provides a powerful new medical capability that promises to surpass traditional techniques of breath analysis including the use of animals, and highlights methodologies for diagnosing a diverse set of disease states, including breast cancer, asthma, and intestinal problems\cite{Liang-PNAS}. With an optical frequency comb serving as a massively parallel data generator, analyzer, and processor, we anticipate other exciting and important applications in the modern scientific era of big data with increased size and complexity~\cite{Xu_CombNeural,Feldmann_CombCompute}.
\newpage
\begin{figure} [p]
\includegraphics[width=\textwidth]{figures/Figure_Principle.png}
\caption{\textbf{Comb-based breathalyzer for COVID-19 diagnosis.} {\bf a}, Schematic representation of the working principle of the device. An exhaled human breath sample is collected by a Tedlar bag and then loaded into a high-finesse optical cavity resonantly coupled with a mid-IR frequency comb laser. Each comb line performs an ultra-sensitive measurement of the molecular absorption signals at an isolated optical frequency. The broadband molecular absorption spectrum, containing more than ten thousand molecular spectral features, is then used for supervised machine learning to identify the binary response class for the research subject (either positive or negative). {\bf b}, Sample absorption spectrum collected from a research subject’s exhaled breath (black). The spectrum is fitted with a total of sixteen molecular species. Inverted in sign and plotted with different colors are the four major species (CH$_3$OH, H$_2$O, HDO, and CH$_4$) that give the most dominant absorption features.}
\label{fig:Principle}
\end{figure}
\newpage
\begin{figure} [p]
\includegraphics[width=\textwidth]{figures/Figure_CrossVad.png}
\caption{\textbf{Cross-validation workflow.} The complete data set (N = 170) is first divided into a training set and a testing set using stratified random sampling. The predictor variables and the response for each observation in the training set are used for building a classification model using the PLS-DA algorithm. After the model is built, the predictor variables for each subject in the testing set is fed to the machine to yield a predicted class (positive or negative), which is then compared against the true class for assessment of the prediction performance. Note that the machine is blind to the true class of the testing set before all predictions having been made.}
\label{fig:CrossVad}
\end{figure}
\newpage
\begin{figure} [p]
\includegraphics[width=\textwidth]{figures/Figure_PLS.png}
\caption{\textbf{Variables construction for the supervised machine learning.} Two different predictor variables construction approaches are evaluated: the MS approach (predictor variables: concentrations of the 16 molecular species), and the DS approach (predictor variables: absorption signals measured by 14,836 comb lines). For both approaches, Partial Least-Squares (PLS) regression analysis of the response on the predictor variables leads to a set of PLS components generated from linear combinations of the predictor variables. The magnitudes and signs are determined based on the covariance of each predictor variable with the response. PLS regression allows dimensionality reduction of the prediction variables, maximized separability between the two opposing response classes, and evaluation of the relative variables importance in predictions. Results shown for the ({\bf a}, {\bf c}) MS and ({\bf b}, {\bf d}) DS approaches are calculated using the complete data set (N = 170) for the response type of COVID-19 (positive or negative). {\bf a}, {\bf b}, distribution of the subjects’ data for the first three PLS components, with red and blue triangles representing positive and negative research subjects, respectively. {\bf c}, {\bf d}, the VIP scores evaluated over a set of fifteen PLS components. The number of PLS components is chosen to ensure saturation of the total percentage variance explained in the response, which are \SI{31.222}{\%} and \SI{99.996}{\%} respectively for the MS and the DS approaches. Predictor variables with VIP scores above (or below) unity are plotted in purple (or black) and considered as important (or unimportant) for predictions.}
\label{fig:PLS}
\end{figure}
\newpage
\begin{figure} [p]
\includegraphics[width=\textwidth]{figures/Figure_ROC.png}
\caption{\textbf{Prediction performance.} Cross-validation results analyzed by the DS approach using a set of 15 PLS components are presented in the form of the ROC curves. Shown are the results averaged over 10,000 cross-validation runs from repeated stratified random sampling at the fixed partition ratio of the training and testing set (140 vs. 30). AUC values are reported for the averaged ROC curves. See Methods for details on the averaging. Prediction results using different numbers of PLS components and different partition ratios of the training and testing set are presented in the SI, from which the uncertainties of the AUC values are determined. Three different binary response types are examined: ({\bf a}) birth month (positive: odd; negative: even), ({\bf b}) gender (positive: female; negative: male), and ({\bf c}) COVID-19 (positive: infected; negative: not infected). For birth month and gender, the respective assignment of the response classes to positive and negative is done at random and does not carry any particular meaning. The class distributions for the three response types are 83(\SI{49}{\%}):87(\SI{51}{\%}), 87(\SI{51}{\%}):83(\SI{49}{\%}), and 83(\SI{49}{\%}):87(\SI{51}{\%}), respectively.}
\label{fig:ROC}
\end{figure}
\clearpage
\newpage
\section*{Methods}
\begin{flushleft}
\textbf{Subject recruitment.} The study was approved by the Institutional Review Board (protocol no. 21-0088) of the University of Colorado. Research subjects are all affiliates of the University of Colorado Boulder, at least 18 years old, recruited after taking a saliva-based or nasal swab COVID-19 RT-PCR test provided by the University. After receiving their COVID-19 test results, potential subjects received a recruitment email for this study and were asked to contact the research team within 24 hours if interested in participating. They then reviewed and signed the informed consent form, completed a questionnaire, and scheduled an appointment with the research team members to collect their breath samples. For the COVID-19-positive subjects, the average time delay between the completion of the COVID-19 test and the collection of the breath samples was $2.05\pm0.95$ days (error denotes one standard deviation), where the disease is likely to progress over this period of time. The information gathered through the questionnaire includes date of birth, gender, and commuting needs. Other gathered information includes COVID-19 symptoms (for COVID-19 positive subjects only), abdominal symptoms, commuting behaviors, and typical alcohol consumption and smoking habits. Analysis with respect to these additional attributes is not discussed here and will be presented in a future publication. A total of 170 research subjects were recruited from May 2021 to January 2022. Forty-eight percent of the subjects were born after 1990, and \SI{96}{\%} born after 1980. We note that all information gathered through the questionnaire is self-identified. The question about gender in the questionnaire gives three options, ``male'', ``female'' and ``other''. Two research subjects selected two options simultaneously, with ``other'' alongside either ``male'' or ``female''. Since there are no research subjects who selected both ``male'' and ``female'' simultaneously, the classification analysis on gender performed in this work uses the subject's selection for ``male'' or ``female''as the binary true class labels. This results in a gender distribution of \SI{51}{\%} ``female'' and \SI{49}{\%} ``male''. All data (i.e. informed consent form, questionnaire, and Tedlar bag ID) are collected and managed using the REDCap electronic data capture tool \cite{Harris_REDCap1,Harris_REDCap2} hosted by the University of Colorado Denver.
\end{flushleft}
\begin{flushleft}
\textbf{Sample collection and analysis.} Standard Tedlar bags (1-liter, part no. 249-01-PP, SKC Inc.) were used to collect exhaled breath. During the appointment for sample collection, research subjects were asked to hold their noses and breathe in/out through the mouth. They were instructed to inhale to full lung capacity for 1-3 s, followed by exhaling the first half of their breath to the surroundings and the second half into the bag until the latter was above $\sim$\SI{80}{\%} full. The location for the sample collection was selected to be an outdoor parking lot within the university campus. The participants were not instructed or required to limit or control their smoking, food or alcohol intake prior to the sample collection. Immediately after the sample collection, the breath sample was transported via an air-tight container to the indoor lab where the comb breathalyzer was set up. Here, each sample was heated at a temperature of $\sim$\SI{37}{\degree C} for 20 minutes to mimic body temperature and avoid molecules condensing onto the bag. It was then loaded into a cleaned vacuum chamber held at room temperature (\SI{20}{\degree C}) by steadily flowing the sample gas through the chamber at $\sim$ 1 liter per minute. When the sample was about to exhaust, the input gas valve to the chamber was closed and the intracavity chamber pressure finely controlled by timely closure of the output valve to reach a static pressure of \SI{50}{Torr} (\SI{67}{mbar}). The entire gas flowing process took about 1 minute. After collection, the breath sample was pumped out while the Tedlar bag was autoclaved and disposed. The same collection and analysis protocol was used for all samples in an effort to minimize inter-person systematic variations.
\end{flushleft}
\begin{flushleft}
\textbf{Cavity-enhanced direct frequency comb spectroscopy.} The principle of CE-DFCS is detailed in Ref.~\cite{Adler_CEDFCS} and the characterization of the breathalyzer used in this study is reported in Ref.~\cite{Liang-PNAS}. In brief, the mid-infrared frequency comb source used in this study is generated by a singly-resonant optical parametric oscillator, with a repetition rate of \SI{136}{MHz} and power of $\sim$\SI{100}{mW} used. The Pound-Drever-Hall technique is used to resonantly lock the mid-IR comb light to an optical cavity, which is formed by a pair of high reflectivity mirrors with finesse ranging from 6,000 to 8,000 depending on the wavelength. The cavity transmission light is analyzed with a home-built Fourier-transform infrared spectrometer sampled at a frequency interval that is matched to half the transmitted comb spacing. By utilizing the known instrument function (a sinc function), the absorption of each comb light is extracted \cite{Maslowski_SincFunc}. Cavity mirror dispersion restricts the spectral bandwidth of the incident comb light coupled into the cavity. Thus, spectra collected at different center wavelengths are concatenated to form a complete spectrum.
\end{flushleft}
\begin{flushleft}
\textbf{Supervised Machine Learning.} A classification machine is built from learning the correlation of the predictor variables with the response and then constructing a decision metric for prediction of the class labels of new data based on the predictor variables alone. Feeding knowledge of the true class labels to the machine makes the learning ``supervised'', as opposed to ``unsupervised'' where only the predictor variables are fed into it. The Molecular Science (MS) approach and the Data Science (DS) approach are two different predictor variables construction methods employed and compared in this work. The MS approach aims at identifying the panel of molecular species that can be used to discriminate between opposing classes. Absorption spectra are fitted to the High-Resolution Transmission Molecular Absorption (HITRAN) database \cite{GORDON_HITRAN2016}, out of which we choose a total of 16 molecular species with cross sectional data available in the probed spectral region. A total of 16 predictor variables for each observation are then formed by concentrations of the 16 molecular species. On the other hand, the DS approach focuses on carrying out only the necessary analysis steps based on the raw spectroscopy data to enable the construction of a prediction model. Absorption signals measured by a total of 14,836 frequency comb lines are directly used as the predictor variables, without the knowledge of the exact molecular species present. In this work, the Partial Least-Squares Discriminant Analysis (PLS-DA) algorithm \cite{DEJONG_SIMPLS,Chuen_PLSDA-review} is adopted based on its good capability for high-dimensional data modeling. We provide the PLS-DA basics in a separate section in the Methods. A model assessment run begins by dividing the complete data set into a training set used for building the model, and a non-overlapping complement set called the testing set and used for the model evaluation. Each predictor variable in the training set is normalized to give a standard deviation of unity. All observations from the training set are used for construction of a set of PLS components. The linear combination coefficients used for building the set of PLS components determines a set of regression coefficients relating the predictor variables and the response for observations in the training set. These regression coefficients are used for predicting the class labels for new data. To generate the prediction results, the predictor variables for an observation in the testing set are each normalized by the standard deviation calculated for the same predictor variable in the training set, multiplied by a regression coefficient, then summed together to yield a numerical predicted value. We then translate such numerical value into posterior probabilities for the observation to be grouped into the opposing classes, based on how it proportionally compares to the values used for representing the two true classes in the training process. If the predicted value is outside of the range specified by the true classes values, the observation is assigned with \SI{100}{\%} posterior probability to be associated with one of the two classes. The posterior probabilities are then compared with a decision threshold value for predicted class assignments, which can later be checked against the true class labels for model evaluation.
\end{flushleft}
\begin{flushleft}
\textbf{Principle of PLS-DA.} The principle of PLS regression and its usage for discriminant analysis, namely the PLS-DA algorithm, is briefly introduced here. The PLS regression toolbox used in our work is developed by Matlab and implemented using the SIMPLS formulation. We discuss only the univariate response classification, corresponding to what is used in this work, but interested readers may consult Ref.~\cite{DEJONG_SIMPLS} for more details beyond this classification type and how the actual algorithm is implemented. We use bold upper case, bold lower case, and un-bold letters to denote matrices, vectors, and scalars, respectively. Matrix transpose are denoted by primes ($'$). Collected data used for the training process are represented by the $n \times p$ predictor variables matrix ${\bf X}_0$ and the $n \times 1$ univariate response variable vector ${\bf y}_0$. Here, $n$ is the total number of research subjects, $p$ is the total number of predictor variables. Both ${\bf X}_0$ and ${\bf y}_0$ have been column-centered so that the covariance of different predictor variables with the response can be expressed by a $p \times 1$ column vector ${\bf s}_0 = {\bf X}_0'{\bf y}_0$. PLS regression relates ${\bf X}_0$ and ${\bf y}_0$ based on ${\bf y}_0 = {\bf X}_0 {\bf b} + {\bf e}$, where ${\bf b}$ is the $p \times 1$ coefficients estimate, ${\bf X}_0 {\bf b}$ is the explained component, ${\bf e}$ is the fit residual. In contrast to least squares regression, where the coefficients estimate ${\bf b}$ is constructed by minimizing the residual sum of squares ${\bf e}'{\bf e}$, PLS regression constructs it based on the covariance ${\bf s}_0 = {\bf X}_0'{\bf y}_0$ to get more stabilized values of ${\bf b}$ and achieve more reliable predictive power. The formulation begins by projecting the predictor variables matrix ${\bf X}_0$ onto a new coordinate system ${\bf T}= {\bf X}_0 {\bf R}$ of reduced dimensionality spanned by a total of $A$ $(\leq p-1)$ PLS components, where ${\bf R}$ denotes the $p \times A$ weight transfer matrix and ${\bf T}$ denotes the $n \times A$ projected scores matrix. The construction of ${\bf R}$ is subject to two constraints: 1) the covariance vector ${\bf T}'{\bf y}_0$ is maximized for each entry, meaning each PLS component exhibits the largest possible covariance with the response; 2) the PLS components are orthonormal, i.e., columns of ${\bf T}$ satisfy ${\bf t}_i'{\bf t}_j = \delta_{ij}$ for any $i,j = 1,2,...,A$, where $\delta_{ij}$ is the Kronecker delta. The coefficients estimate ${\bf b}$ can be determined once ${\bf R}$ is known, since ${\bf y}_0={\bf T}\T'{\bf y}_0={\bf X}_0{\bf R}\R'{\bf X}_0'{\bf y}_0={\bf X}_0{\bf b}$, and thus ${\bf b}={\bf R}\R'{\bf X}_0'{\bf y}_0={\bf R}\R'{\bf s}_0$. The process of determining ${\bf R}$ proceeds column by column. For the first iteration step $k = 1$, the maximization of the covariance of the first PLS component (${\bf t}_k = {\bf X}_0 {\bf r}_k $) with the response, ${\bf t}_k'{\bf y}_0 = {\bf r}_k'{\bf X}_0'{\bf y}_0 = {\bf r}_k'{\bf s}_0 = \max$, constrains the first weight vector ${\bf r}_k$ ($k = 1$) to be along the direction of ${\bf s}_0$. For steps $k > 1$, the orthogonality condition, ${\bf t}_k'{\bf t}_i = {\bf r}_k'({\bf X}_0'{\bf t}_i) = 0$ for $i=1,2,...,k-1$, requires the newly constructed ${\bf r}_k$ to be orthogonal to each of the $p \times 1$ vectors ${\bf X}_0'{\bf t}_i$ for $i = 1,2,...,k-1$. We define ${\bf p}_i \equiv {\bf X}_0'{\bf t}_i$ called the loading vectors. One may use the Gram-Schmidt process to find the orthonormal basis of the subspace $V_{k-1}$ spanned by the loading vectors ${\bf p}_i$ ($i = 1,2,...,k-1$) and then determine the $p \times p$ projection operator $\bf{P}^{\perp}$ for the orthogonal complement space $V^{\perp}_{k-1}$. This loosely constrains the direction of ${\bf r}_k$ to be within $V^{\perp}_{k-1}$, requiring ${\bf r}_k = \bf{P}^{\perp} {\bf r}_k$. Now, with the covariance maximization criteria, ${\bf t}_k'{\bf y}_0 = {\bf r}_k'(\bf{P}^{\perp'} {\bf s}_0) = \max$, the direction of ${\bf r}_i$ is ultimately determined to be along the direction of the vector $\bf{P}^{\perp'} {\bf s}_0$, which is the projection of the covariance vector ${\bf s}_0$ onto the subspace $V^{\perp}_{k-1}$. The iteration process proceeds until the directions of all ${\bf r}_k$ is determined, where the normalization condition ${\bf T}'{\bf T} = 1$ governs the magnitudes of ${\bf r}_k$. Finally, the coefficients estimate is determined and can be used for prediction of the response class for new observation based on ${\bf y}^{pred}_0 = {\bf X}^{new}_0 {\bf b}$, where the $m \times p$ matrix ${\bf X}^{new}_0$ is the testing data for a total of $m$ research subjects. The $m \times 1$ predicted values ${\bf y}^{pred}_0$ are translated proportionally into posterior probabilities and compared with a threshold value for response class assignment.
\end{flushleft}
\begin{flushleft}
\textbf{Variable importance in the projection scores.} In PLS-DA, assessment of the importance of the predictor variables needs to take into account 1) the weighting of a given predictor variable to form different PLS components and 2) the importance of different PLS components in explaining the response. Regarding 1), the formation of the $a$-th PLS component ($a = 1,2, ..., A$) takes the contribution from the $j$-th predictor variable with the normalized weight given by $w_{ja}/\|w_a\|$, where $w_{ja}$ is the $j$-th row $a$-th column element from the $p \times A$ weight matrix ${\bf R}$, and $\|w_a\| = (\sum_{j=1}^{p} w_{ja}^2)^{1/2}$ is the normalization. Regarding 2), we first note that the variance of the response among all observations ${\bf y}_0'{\bf y}_0$ is explained by the total of $A$ PLS components to the extent of ${\bf \hat{y}}_0'{\bf \hat{y}}_0$, where ${\bf \hat{y}}_0 = {\bf X}_0 {\bf b} = {\bf y}_0 - {\bf e}$. The total percentage variance explained in the response, $({\bf \hat{y}}_0'{\bf \hat{y}}_0/{\bf y}_0'{\bf y}_0) \times \SI{100}{\%}$, can be used for estimating the minimum number of PLS components needed for reliable predictions (see SI for details). The explained variance ${\bf \hat{y}}_0'{\bf \hat{y}}_0 = {\bf \hat{y}}_0'{\bf T}\T'{\bf \hat{y}}_0 = \sum_{a=1}^{A} ({\bf \hat{y}}_0'{\bf t}_a)^2$ is further broken down into a summation of the square of the covariance of all PLS components with ${\bf \hat{y}}_0$. We can thus evaluate the importance of the $a$-th PLS component by its variance explained ${\bf q}_a^2 \equiv ({\bf \hat{y}}_0'{\bf t}_a)^2$, a quantity assigning larger importance to the PLS components that have larger covariance with the explained component, with the total variance explained by the $A$ PLS components given by $\sum_{a=1}^{A} {\bf q}_a^2$. Taking both 1) and 2) into account, the variable importance for the predictor variable $j$ summing over all the $A$ PLS components is proportional to $[\sum_{a=1}^{A} q_a^2 \cdot (w_{ja}/\|w_{a}\|)^2]^{1/2}$. From this, we define its $\text{VIP}$ score \cite{CHONG_VIP}, a metric for characterizing its importance, by:
\begin{equation}
\text{VIP}_j = \sqrt{\frac{p \cdot \sum_{a=1}^{A} [q_a^2 \cdot (w_{ja}/\|w_{a}\|)^2]}{\sum_{a=1}^{A} q_a^2}}
\end{equation}
The normalization is taken to ensure the mean square sums of the VIP scores among all predictor variables equals to unity, $p^{-1} \sum_{j=1}^{p} \text{VIP}_j^2 = 1$. Because of this normalization, predictor variables with $\text{VIP}$ scores above (or below) unity can be regarded as important (or unimportant) variables.
\end{flushleft}
\begin{flushleft}
\textbf{Receiver-Operating-Characteristics curve.} The ROC curve is a model assessment tool and is generated by comparing the posterior probabilities with a varying decision threshold value scanned from zero to unity, where an observation is assigned positive if its positive class posterior probability is above the threshold. This evaluation method allows an overall assessment of the classifier's performance for all levels of decision making aggressiveness, from totally liberal by assigning all observations to be positive (threshold = 0) to totally conservative by assigning all to be negative (threshold = 1). Prediction performance is quantified by the area under the curve, where a value of 0.5 means random guessing, 0.7-0.8 means the discrimination capability is acceptable, 0.8-0.9 means excellent, and 0.9-1.0 means outstanding~\cite{MANDREKAR_AUCinterpret}. We perform the averaging of the ROC curves using the non-parametric method adapted from Ref.~\cite{Chen_ROC_tilt}. This method ensures that: 1) the AUC of the averaged curve equals the average AUC of individual cross-validation runs, and 2) the averaged AUC for a perfect (or random) classifier is equal to 1 (or 0.5). Proof for statement 1) can be found in the appendix of \cite{Chen_ROC_tilt}, while statement 2) can be straightforwardly deduced from 1). In our work, we average the individual ROC curves vertically in the tilted space formed by rotating the ($\text{FP}$,$\text{TP}$) axes counter-clockwise by an angle $\theta < \pi/2$, where $\text{FP}$ and $\text{TP}$ denotes false positive rates and true positive rates, respectively. This enables the averaging to be taken over singular functions. Any data point from an individual ROC curve can take its FP values from $\{(0, 1, 2, ... , \text{N})/\text{N}\}$, and TP values from $\{(0, 1, 2, ... , \text{P})/\text{P}\}$. Since we are using stratified sampling at the fixed testing set size $\mathrm{L_{test}} = \text{P}+\text{N}$, different cross-validation runs preserve the total number of positives $\text{P}$ and negatives $\text{N}$. Hence, we choose $\theta = \arctan{(\text{P}/\text{N})}$ such that the curve averaging in the tilted space will be performed to yield a total of $(\mathrm{L_{test}}+1)$ sample points for plotting the ROC curve. The $j$-th $(j = 0, 1, 2, ..., \mathrm{L_{test}})$ sample point represents the $j$-th observation in the testing set scanned over by the threshold line, and is obtained from the statistical mean over a total of the number of cross-validation runs of the $j$-th observation from each run.
\end{flushleft}
\section*{Acknowledgements}
We thank Holly Gates-Mayer, Mark T. Hernandez, Aaron Gilad Kusne, and Lee R. Liu for helpful discussions. This work is supported by AFOSR 9FA9550-19-1-0148; DOE DE‐SC0002123, NSF CHE-2053117, NSF QLCI OMA–2016244, NSF PHY-1734006, and NIST. J.T. was supported by the Lindemann Trust in the form of a Postdoctoral Fellowship.
\section*{Authors contributions}
All authors contributed to the experimental design, results interpretation, and manuscript writing. Q.L. and Y.C. collected and analyzed the data.
\section*{Competing interests}
The authors declare no competing interests.
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