text:
\begin{document}
\title{A generalised formulation of the Laplacian approach to resistor networks}

\author{N.Sh. Izmailian}
\email{izmail@yerphi.am; ab5223@coventry.ac.uk}
\affiliation{Applied Mathematics Research Centre, Coventry University, Coventry CV1 5FB, UK}
\affiliation{Yerevan Physics Institute, Alikhanian Brothers 2, 375036 Yerevan, Armenia}

\author{R. Kenna}
\email{r.kenna@coventry.ac.uk}
\affiliation{Applied Mathematics Research Centre, Coventry University, Coventry CV1 5FB, UK}

\date{\today}

\begin{abstract}
An analytic approach is presented to developing exact expressions for the two-point resistance between arbitrary nodes on certain non-regular resistor networks.
This generalises previous approaches, which only deliver results for networks of more regular geometry.
The new approach exploits the second minor of the Laplacian matrix associated with the given network to obtain the  resistance in terms its eigenvalues and eigenvectors.
The method is illustrated by application to the resistor network on the  globe lattice, for which the resistance between two arbitrary nodes is obtained in the form of single summation.
\end{abstract}

\pacs{01.55+b, 02.10.Yn}

\maketitle
\section{Introduction}
\label{Introduction}
The calculation of the resistance between two arbitrary nodes in resistor networks is the classic problem in electric circuit theory and was first studied by Kirchhoff in 1847 \cite{kirch}.
The problem has attracted the interest of numerous physicists over many years because it is intrinsically connected to a wide range of other physics problems, including   random walks \cite{random1,random2,random3,random4}, first-passage processes \cite{passage}, lattice Green's functions \cite{green1,green2,history} and classical transport in disordered media \cite{media1,media2,media3}.
The resistance $R_{i,j}$ between two  nodes $i$ and $j$ can be considered as a metric called ``resistance distance'' \cite{Klein}.
If there are many (few) paths between the two nodes, the resistance $R_{i,j}$  is small (large).
The total resistance distance of a graph, also called the ``Kirchhoff index'' \cite{Klein} (i.e., the sum of resistance distances between all pairs of nodes) is related to the network criticality \cite{network}, which characterises its robustness.

In the past, resistance-computation studies have focused mainly on infinite lattices \cite{history,green1,green2,asad1,asad2,asad3}.
Recently, attention has shifted  to the study of resistance on finite networks, as these are the configurations of relevance to real life.
For this reason, there has been a surge of research activity in recent years.
In 2004 Wu \cite{wu} derived a compact expression for the resistance between  two arbitrary nodes for finite, regular lattice networks in terms of
the associated Laplacian.
That approach, however, requires a complete knowledge of the eigenvalues and eigenvectors of the Laplacian.
This is straightforward to obtain for regular lattices in any dimensions, since the Laplacian for $d$-dimensional regular lattices can be represented as the sum of $d$ one-dimensional Laplacians, with known eigenvalues and eigenvectors.
However, the approach cannot readily deal with non-regular lattices.
For this reason, Izmailian, Kenna and Wu extended the approach to enable derivation of a closed-form expression for the resistance between two arbitrary nodes for finite networks in terms of the eigenvalues and eigenvectors of the {\emph{first minor}} of the Laplacian \cite{ikw}.
The new approach  has been applied to the cobweb and fan resistor networks \cite{ikw,ik}.
An alternative recent approach to the calculation of two-point resistances on distance-regular networks was based on the stratification of the network and the associated Stieltjes function \cite{jafar1}.
Still another approach  of computing resistances by using a method of direct summation has been developed in \cite{tan0,tan2globe}.

In this paper the recent approaches \cite{wu,ikw} are further generalized to compute two-point resistances based on the eigenvalues and eigenvectors of the {\emph{second}} minor of the Laplacian associated with the network.
The generalized approach is illustrated by application to the resistor network with configuration of a globe. In particular, the resistance between two arbitrary nodes on such a network is determined in the form of single summation, allowing determination to arbitrary precision.

\section{Resistor networks}
\label{Resistors}

Let us consider a resistor network consisting of $T$ nodes and let $r_{i,j}=r_{j,i}$ be the resistance  of the resistor connecting nodes $i$ and $j$. The resistance between arbitrary nodes $\alpha$ and $\beta$  can be written as \cite{wu}
\begin{equation}
R_{\alpha,\beta}=\sum_{i=2}^{T}\frac{\left|\psi_{i\alpha}-\psi_{i\beta}\right|^2}{\lambda_i},
\label{Rab}
\end{equation}
where $\lambda_i$ are nonzero eigenvalues with orthonormal eigenvectors  $\Psi_{i}=(\psi_{i1},\psi_{i2},...,\psi_{iT})$ of the Laplacian ${\bf L}_T$ of that network.

Determining the eigenvalues and eigenvectors of the Laplacian is usually a very difficult problem.
One way of approaching it is to reduce the original problem to that of 1D Laplacians with appropriate boundary conditions.
For regular rectangular lattices, this can be achieved in any dimension in a straightforward manner.
The eigenvalues and eigenvectors of the Laplacian for the 1D lattice with various boundary conditions are then easy to calculate and they given in Appendix 1.

Let us now consider, for example, the Laplacian for regular, two-dimensional,  rectangular lattices.
Denote by ${\bf L}_N^{\rm{free}}$, ${\bf L}_N^{\rm{per}}$, ${\bf L}_N^{\rm{DN}}$ and ${\bf L}_N^{\rm{DD}}$  the Laplacian of a 1D lattice with free, periodic, Dirichlet-Neumann and Dirichlet-Dirichlet boundary conditions, respectively and let $I_K$ be the $K \times K$ identity matrix.
Then, the 2D Laplacian of the resistor network ${\bf L}_{M \times N}$  consisting of a $M \times N$ rectangular lattice with free, cylindrical and toroidal boundary conditions and with resistors $r$ and $s$ in the two directions, can be expressed through the Laplacians of the 1D lattices  as \cite{wu}
\begin{eqnarray}
{\bf L}_{M \times N}^{\rm{free}}&=&r^{-1}{\bf L}_N^{\rm{free}}\otimes {\bf I}_M+s^{-1}{\bf I}_N \otimes {\bf L}_M^{\rm{free}},
\label{LapFree}\\
{\bf L}_{M \times N}^{\rm{cylinder}}&=&r^{-1}{\bf L}_N^{\rm{free}}\otimes {\bf I}_M+s^{-1}{\bf I}_N \otimes {\bf L}_M^{\rm{per}},
\label{LapCyl}\\
{\bf L}_{M \times N}^{\rm{torus}}&=&r^{-1}{\bf L}_N^{\rm{per}}\otimes {\bf I}_M+s^{-1}{\bf I}_N \otimes {\bf L}_M^{\rm{per}}.
\label{LapTor}
\end{eqnarray}
Thus, the 2D Laplacian ${\bf L}_{M \times N}$ can be diagonalize in the two subspaces separately, yielding eigenvalues and eigenvectors
\begin{eqnarray}
\lambda_{m,n}&=&r^{-1}\lambda_m+s^{-1}\lambda_n
\label{eigenvalue}\\
\psi_{(m,n);(x,y)}&=&\psi_{mx}^{(M)}\psi_{ny}^{(N)}
\end{eqnarray}
where $\lambda_m, \lambda_n$ and $\psi_{mx}^{(M)},\psi_{ny}^{(N)}$ are eigenvalues and eigenvectors of the appropriate 1D Laplacian.

But for non-regular lattices, such as the cobweb and fan networks consisting of $M \times N + 1$ sites, it is impossible to express the 2D Laplacian ${\bf L}_{M \times N+1}$ of the network through the Laplacians of such 1D lattices.
This means that it is difficult to apply Wu's method \cite{wu}.
Instead, on can apply the method of Izmailian, Kenna and Wu (``IKW method'') \cite{ikw} to compute resistance by using eigenvalues and eigenvectors of the {\emph{first minor}} of the 2D Laplacian.
Indeed, the first minor ${\bf \Delta}_{M \times N}$ of the 2D Laplacian ${\bf L}_{M \times N+1}$ can be reduced to the Laplacian of a 1D lattice and can be written as \cite{ikw,ik}
\begin{eqnarray}
{\bf \Delta}_{M \times N}^{\rm{cobweb}}&=&r^{-1}{\bf L}_N^{\rm{per}}\otimes {\bf I}_M+s^{-1}{\bf I}_N \otimes {\bf L}_M^{\rm{DN}},\label{D1cobweb}\\
{\bf \Delta}_{M \times N}^{\rm{fan}}&=&r^{-1}{\bf L}_N^{\rm{free}}\otimes {\bf I}_M+s^{-1}{\bf I}_N \otimes {\bf L}_M^{\rm{DN}}.
\label{D1fan}
\end{eqnarray}
Then the resistance between nodes $\alpha$ and $\beta$ can be written as \cite{ikw}
\begin{equation}
R_{\alpha,\beta}=\sum_{i=1}^{M\times N} \frac{\left|\psi_{i\alpha}-\psi_{i\beta}\right|^2}{\lambda_i},
\label{Rab}
\end{equation}
where $\lambda_i$ are eigenvalues with orthonormal eigenvectors  $\Psi_{i}=(\psi_{i1},\psi_{i2},...,\psi_{iT})$ of the minor ${\bf \Delta}_{M\times N}$. Note, that all eigenvalues of the minor ${\bf \Delta}_{M\times N}$ have nonzero value.

Therefore, to compute resistances  on regular $M \times N$ rectangular lattices with free, cylindrical and toroidal boundary conditions, one can use the Wu method \cite{wu}.
To compute resistances  on non-regular rectangular lattices, such as the cobweb and fan networks one can use IKW method \cite{ikw}.
The main difference between these two approaches is that in the Wu method one expresses the resistance through the eigenvalues and eigenvectors of the full Laplacian of the network, while in the IKW method the resistance is expressed through the eigenvalues and eigenvectors of the first minor of the Laplacian of the network.

There are, however, other non-regular rectangular lattices, such as the globe network comprising $M \times N + 2$ sites, for which it is impossible to express the Laplacian ${\bf L}_{M \times N+2}$ or the first minor of the Laplacian through Laplacians 1D lattices.
It is therefore difficult to apply either the Wu \cite{wu} or IKW \cite{ikw} methods to calculate resistances between nodes for such a network,
However, in this circumstance, the second  minor ${\bf {\cal L}}_{M \times N}$ of the Laplacian ${\bf L}_{M \times N+2}$ can be written as
\begin{eqnarray}
{\bf {\cal L}}_{M \times N}^{\rm{globe}}&=&r^{-1}{\bf L}_N^{\rm{per}}\otimes {\bf I}_M+s^{-1}{\bf I}_N \otimes {\bf L}_M^{\rm{DD}}.
\label{D2globe}
\end{eqnarray}
In what follows we shall show how to calculate resistances $R_{\alpha,\beta}$ on the globe  network using the modified Laplacian approach, i.e., by expressing the resistance through eigenvalues and eigenvectors of the second minor of the Laplacian of the network.
Extension to networks of similar geometries is possible in a straightforward manner.

\section{Modified Laplacian approach}
\label{Resistors2}

Let us consider a network, in which the total number of nodes is $M  N +2$.
Let us denote the nodes by the index $i$, wherein $i$ takes values $0, 1, 2, ..., M  N +1$. Denote the electric potential at the $i$th node by $V_i$ and the current flowing into the network at the $i$th node by $I_i$.
We write Kirchhoff's law as
\begin{equation}
\sum_{j=0}^{M N+1}c_{i,j}(V_i-V_j)=I_i, \qquad i=0,1,2,...,M N, M N+1,
\label{Kirchhoff}
\end{equation}
with the constraint
\begin{equation}
\sum_{i=0}^{MN+1}I_i=0.
\label{constraint}
\end{equation}
Here $c_{i,j}=c_{j,i}$ is the conductance, which can be expressed through the resistance of the resistor connecting nodes $i$ and $j$ $r_{i,j}=r_{j,i}$ as
\begin{equation}
c_{i,j}=r_{i,j}^{-1}.
\label{condactance}
\end{equation}
Eq.~(\ref{Kirchhoff}) can be written in matrix form as
\begin{equation}
{\bf L} {\vec V} = {\vec I},
\label{matrix0}
\end{equation}
where ${\bf L}$ is the Laplacian of the system
$$
{\bf L}=\left( \begin{array}{ccccccc}
c_0 & -c_{0,1} & -c_{0,2} &\ldots &-c_{0,M N}&-c_{0,M N+1} \\
-c_{1,0} & c_1 & -c_{1,2}&\ldots  &-c_{1,M N}&-c_{1,M N+1}\\
\vdots & \vdots & \vdots &\ddots &\vdots & \vdots & \vdots \\
-c_{M N,0}&-c_{M N,1}&-c_{M N,2}&\ldots&c_{M N}&-c_{M N, M N+1}\\
-c_{M N+1,0}&-c_{M N+1,1}&-c_{M N+1,2}&\ldots&-c_{M N+1,M N}&c_{M N+1}
\end{array} \right),
$$
and $c_i$ is given by
\begin{equation}
c_i=\sum_{j=0 \above0pt  j\neq i}^{M N+1}c_{i,j} \qquad i=0,1,2,...,M N+1.
\label{ci}
\end{equation}
Here, ${\vec V}$ and ${\vec I}$ are vectors
$$
{\vec V}=\left( \begin{array}{ccccccc}
V_0\\
V_1\\
V_2\\
\vdots \\
V_{MN}\\
V_{MN+1}
\end{array} \right),
\qquad
{\vec I}=\left( \begin{array}{ccccccc}
I_0\\
I_1\\
I_2\\
\vdots \\
I_{MN}\\
I_{MN+1}
\end{array} \right).
$$
Under the constraint (\ref{constraint}) we actually have only $M N+1$ independent equations in Eq. (\ref{Kirchhoff}).
Without loss of generality, therefore, we choose to delete the equation numbered $i=M N +1$ and choose the potential at node $M N+1$ to be zero: $V_{MN+1}=0$.
Then the $M N +2$ equations in (\ref{Kirchhoff}) are reduced to the set of independent $M N +1$ equations
\begin{equation}
\sum_{j=0}^{M N+1}c_{ij}(V_i-V_j)=I_i, \qquad i=0,1,2,...,MN.
\label{Kirchhoff1}
\end{equation}
To this point, we have followed the ref.~\cite{ikw}.
Next we partition the set of equations (\ref{Kirchhoff1}) into two parts.
The first is a single equation and the second is a set of $M N$ equations, {\emph{viz.}}
\begin{eqnarray}
\sum_{j=1}^{M N+1}c_{0j}(V_0-V_j)&=&I_0, \label{single}\\
\sum_{j=0}^{M N+1}c_{ij}(V_i-V_j)&=&I_i, \qquad i=1,2,...,MN.
\label{Kirchhoff2}
\end{eqnarray}
The set of equations (\ref{Kirchhoff2}) can be written in the matrix form as
\begin{equation}
{\bf {\cal L}} {\vec {\cal V}} = {\vec {\cal I}},
\label{matrix}
\end{equation}
where ${\bf {\cal L}}$ is the second minor of the Laplacian ${\bf L}$ and is given by
$$
{\bf {\cal L}}=\left( \begin{array}{ccccccc}
c_1 & -c_{1,2} & -c_{1,3} &\ldots &-c_{1,M N} \\
-c_{2,1} & c_2 & -c_{2,3}&\ldots  &-c_{2,M N}\\
\vdots & \vdots & \vdots &\ddots &\vdots & \vdots & \vdots \\
-c_{M N,1}&-c_{M N,2}&-c_{M N,3}&\ldots&c_{M N}
\end{array} \right),
$$
and ${\vec {\cal V}}$ and ${\vec {\cal I}}$ are vectors, which are now given by
$$
{\vec {\cal V}}=\left( \begin{array}{ccccccc}
V_1\\
V_2\\
V_3\\
\vdots \\
V_{MN}
\end{array} \right),
\qquad
{\vec {\cal I}}=\left( \begin{array}{ccccccc}
{\cal I}_1\\
{\cal I}_2\\
{\cal I}_3\\
\vdots \\
{\cal I}_{MN}
\end{array} \right).
$$
Here ${\cal I}_i=I_i+c_{i,0}V_0$, for $i=1,2,3,...,MN$.

Eq.~(\ref{matrix}) can now be straightforwardly solved for ${\vec V}$ since ${\bf {\cal L}}^{-1}$ is not singular. Multiplying from the left by ${\bf {\cal L}}^{-1}$, we obtain the solution ${\vec V}={\bf {\cal L}}^{-1} {\vec {\cal I}}$. Explicitly, this reads
\begin{eqnarray}
V_i&=&\sum_{j=1}^{M N} {\cal L}^{-1}_{i,j} {\cal I}_j =V_0\sum_{j=1}^{MN}{\cal L}^{-1}_{i,j}c_{j,0}+\sum_{j=1}^{M N} {\cal L}^{-1}_{i,j} I_j, \qquad i=1,2,...,M N,
\label{Vi1}
\end{eqnarray}
where ${\cal L}^{-1}_{i,j}$  is the $(i,j)$th elements of the inverse matrix ${\bf {\cal L}^{-1}}$.

Since we choose  the potential at node $M N+1$ to be zero, $V_{MN+1}=0$, Eq.~(\ref{single}) can be transformed as
\begin{eqnarray}
c_0 V_0&=&I_0+\sum_{i=1}^{MN}c_{0,i}V_i \label{single1}\\
&=&I_0+\sum_{i=1}^{MN}\sum_{j=1}^{MN}{\cal L}_{ij}^{-1}c_{0,i}I_j+V_0\sum_{i=1}^{MN}\sum_{j=1}^{MN}{\cal L}_{ij}^{-1}c_{j,0}c_{0,i}.
\label{single2}
\end{eqnarray}
From Eq. (\ref{single2}) we can find $V_0$,
\begin{equation}
V_0=\frac{I_0+\sum_{i=1}^{MN}\sum_{j=1}^{MN}{\cal L}_{ij}^{-1}c_{0,i}I_j}{c_0-\sum_{i=1}^{MN}\sum_{j=1}^{MN}{\cal L}_{ij}^{-1}c_{j,0}c_{0,i}}.
\label{V00}
\end{equation}
Plugging this expression for $V_0$ back to  Eq. (\ref{Vi1}) we obtain for $V_k$ the expression
\begin{eqnarray}
V_k&=&\frac{I_0+\sum_{i=1}^{MN}\sum_{j=1}^{MN}{\cal L}_{ij}^{-1}c_{0,i}I_j}{c_0-\sum_{i=1}^{MN}\sum_{j=1}^{MN}{\cal L}_{ij}^{-1}c_{j,0}c_{0,i}}\sum_{j=1}^{MN}{\cal L}^{-1}_{k,j}c_{j,0}+\sum_{j=1}^{M N} {\cal L}^{-1}_{k,j} I_j, \qquad k=1,2,...,M N.
\label{Vi2}
\end{eqnarray}
Thus Eqs. (\ref{V00}) and (\ref{Vi2}) give us expressions for $V_i$ ($i=0,1,2,...,MN$) in terms of the elements of the inverse matrix ${\bf {\cal L}^{-1}}$.

To compute the resistance $R_{\alpha \beta}$ between arbitrary  two nodes $\alpha$ and $\beta$, we connect $\alpha$ and $\beta$ to an external battery and measure the current $I$ going through the battery with no other nodes are connected to external sources. Let the potentials at the two nodes be, respectively, $V_{\alpha}$ and $V_{\beta}$. Then, by Ohm's law, the desired resistance is
\begin{equation}
R_{\alpha \beta} = \frac{V_{\alpha}-V_{\beta}}{I}.
\label{resistance}
\end{equation}
The computation of $R_{\alpha \beta}$ is now reduced to solving Eqs. (\ref{single}) and (\ref{Kirchhoff2}) for $V_{\alpha}$ and $V_{\beta}$ with the current given by
\begin{equation}
I_j = I (\delta_{j\alpha}-\delta_{j\beta}).
\label{current}
\end{equation}
Combining Eqs. (\ref{resistance}) and (\ref{current}) with Eq. (\ref{Vi2}) we obtain the resistance between any two nodes $\alpha$ and $\beta$ other than the zeroth node ($\alpha = 0$ or $\beta = 0$) as
\begin{equation}
R_{\alpha,\beta}={\cal L}^{-1}_{\alpha,\alpha} + {\cal L}^{-1}_{\beta,\beta}  - {\cal L}^{-1}_{\alpha,\beta}- {\cal L}^{-1}_{\beta \alpha}+\frac{\sum_{i=1}^{MN}\left({\cal L}_{i,\alpha}^{-1}-{\cal L}_{i,\beta}^{-1}\right)c_{0,i}\sum_{j=1}^{MN}\left({\cal L}_{\alpha,j}^{-1}-{\cal L}_{\beta,j}^{-1}\right)c_{j,0}}{c_0-\sum_{i=1}^{MN}\sum_{j=1}^{MN}{\cal L}_{ij}^{-1}c_{j,0}c_{0,i}}.
\label{resistorab}
\end{equation}
Here we have used the fact that, in the case of any two nodes $\alpha$ and $\beta$ other than $0$, the current $I_0=0$, as  follow from Eq. (\ref{current}).
Up to now all our considerations have been quite general.
We next illustrate the method by application to the globe resistor network.

\section{Application of new Approach to the Globe  Resistor Network}
\label{Resistors1}

\begin{figure}[tbp]
\includegraphics[width=0.3\textwidth]{sphere.eps}
\caption{Illustration of a spherical $M \times N$ + 2 globe network.
Here there are $M=9$ latitudinal rows and $N= 7$ longitudinal ones.
Periodic boundary conditions are imposed in the latitudinal direction.
Each site on the bottom row is  connected to the site labelled $O$ and each site on the top row is connected to $O^\prime$. } \label{fig1}
\end{figure}

Here we consider an example of a network for which the Wu and IKW approaches fail.
One such network is that with geometry of a  globe, an example of which is illustrated in Figure~1.
The  network ${\cal L}_{\rm globe}$ is an $M \times N$ rectangular lattice with periodic boundary condition in one direction and with nodes on each of the two boundaries in the other direction connected to two single nodes.
Topologically ${\cal L}_{\rm globe}$ is in the form of a globe consisting of N longitudinal lines and M lines of latitude, with two poles $O$ and $O'$.
The total number of node in globe  network is $M  N +2$.
Bonds in longitudinal and latitudinal directions have resistances $s$ and $r$, respectively. The elements $c_{0,i}=c_{i,0}$ of the globe  network  have the following values
\begin{eqnarray}
c_{0,i}=c_{i,0}&=&s^{-1} \qquad \mbox{for} \quad i=1,2,3,...,N , \nonumber\\
c_{0,i}=c_{i,0}&=&0 \qquad \mbox{for} \quad i=N+1,N+2,N+3,...,MN , \nonumber
\end{eqnarray}
and $c_0$ is given by
\begin{equation}
c_0=\sum_{j=1}^{MN+1}c_{0,j}=N s^{-1} .
\end{equation}
Then Eq.~(\ref{resistorab}) can be transformed as
\begin{equation}
R_{\alpha,\beta}={\cal L}^{-1}_{\alpha,\alpha} + {\cal L}^{-1}_{\beta,\beta}  - {\cal L}^{-1}_{\alpha,\beta}- {\cal L}^{-1}_{\beta \alpha}+\frac{\sum_{i=1}^{N}\sum_{j=1}^{N}\left({\cal L}_{i,\alpha}^{-1}-{\cal L}_{i,\beta}^{-1}\right)\left({\cal L}_{\alpha,j}^{-1}-{\cal L}_{\beta,j}^{-1}\right)}{N s-\sum_{i=1}^{N}\sum_{j=1}^{N}{\cal L}_{ij}^{-1}}.
\label{resistorab1}
\end{equation}
Here, ${\cal L}^{-1}_{i,j}$  is the $(i,j)$th element of the inverse matrix ${\bf {\cal L}^{-1}}$ which  is given by
\begin{equation}
{\cal L}^{-1}_{i,j}=\sum_{k=1}^{MN}\frac{\psi_{k,i}\psi_{k,j}^{*}}{\Lambda_k},
\label{Delta1}
\end{equation}
in which $\Lambda_k$ and $\psi_{k,i}$ are the eigenvalues and eigenvectors of the second minor ${\bf {\cal L}}$ of the Laplacian.

The second minor of the Laplacian for the globe network is given by
\begin{eqnarray}
{\bf {\cal L}}_{M \times N}^{globe}&=&r^{-1}{\bf L}_N^{\rm{per}}\otimes {\bf I}_M+s^{-1}{\bf I}_N \otimes {\bf L}_M^{\rm{DD}},
\label{MtimesNGlobe}
\end{eqnarray}
where ${\bf L}_N^{\rm{per}}$ and ${\bf L}_N^{\rm{DD}}$ are the Laplacians of 1D lattices with periodic and Dirichlet-Dirichlet  boundary conditions, respectively, and ${\bf I}_M$ and ${\bf I}_N$ are identity matrices.
The eigenvalues and eigenvectors of ${\bf L}_N^{\rm per}$ and ${\bf L}_M^{\rm{DD}}$ are known (see Appendix A). This leads to the following eigenvalues and eigenvectors for  the second minor of the Laplacian for the globe network:
\begin{eqnarray}
\Lambda_k \equiv
\Lambda_{m,n}&=&2r^{-1}(1-\cos{2\theta_n})+2s^{-1}(1-\cos{2\varphi_m}),
\label{Lambdamn}\\
\psi_{k,i} \equiv \psi_{(m,n);(x,y)}&=&\sqrt{\frac{2}{N(M+1)}}\exp\left(2 i x \theta_n\right)\sin(2 y \varphi_m),\label{varphiki}
\end{eqnarray}
where $\theta_n$ and $\varphi_m$ are given by Eqs. (\ref{thetan}) and (\ref{varphin}) respectively.

We next calculate the two double sums in Eq. (\ref{resistorab1}), namely $S_1=\sum_{i=1}^{N}\sum_{j=1}^{N}{\cal L}_{ij}^{-1}$ and $S_2=\sum_{i=1}^{N}\sum_{j=1}^{N}\left({\cal L}_{i,\alpha}^{-1}-{\cal L}_{i,\beta}^{-1}\right)\left({\cal L}_{\alpha,j}^{-1}-{\cal L}_{\beta,j}^{-1}\right)$.
Our objective is not to actually determine the two double summations exactly.
Rather it is to reduce each of them a single summation, because a single summation allows one to determine the resistance to arbitrary precision numerically.

Let us start first with double sum $S_1$.
Since the coordinates of the nodes from $i=1,2,...,N$ are given by $(x,y)$, where the $x$ coordinate takes values from 1 to N ($x=1,2,3,...,N$) and the $y$ coordinate takes value 1 ($y=1$), the first sum can be written as
\begin{equation}
S_1=\sum_{i=1}^{N}\sum_{j=1}^{N}{\cal L}_{ij}^{-1}=\sum_{i=1}^{N}\sum_{j=1}^{N}\sum_{k=1}^{MN}\frac{\psi_{k,i}\psi_{k,j}^*}{\Lambda_k}=
\sum_{x_1=1}^{N}\sum_{x_2=1}^{N}\sum_{m=0}^{M-1}\sum_{n=0}^{N-1}\frac{\psi_{(m,n);(x_1,1)}
\psi_{(m,n);(x_2,1)}^*}{\Lambda_k}.
\end{equation}

In the case of the globe network we should use $\Lambda_k$ and $\psi_{k,i}$ given by Eqs. (\ref{Lambdamn}) and (\ref{varphiki}) and for $S_1=S_1^{globe}$ we obtain
\begin{equation}
S_1^{\rm{globe}}=\frac{1}{2N(M+1)}\sum_{x_1=1}^{N}\sum_{x_2=1}^{N}\sum_{n=0}^{N-1}\sum_{m=0}^{M-1}\frac{\exp{(2 i (x_1-x_2)\theta_n)} \sin^2(2\varphi_m)}{r^{-1}\sin^2\theta_n+s^{-1}\sin^2\varphi_m}
\label{S1G}.
\end{equation}
Using the  identities
\begin{eqnarray}
\sum_{x=1}^N \exp{(2 i x \theta_n)}&=&\sum_{x=1}^N \exp{(2 \pi i x n/N)}=N\delta_{n,0}\label{identity1},\\
\sum_{m=0}^{M-1}\cos^2\varphi_m&=&\sum_{m=0}^{M-1}\cos^2\left(\frac{\pi(m+1)}{2(M+1)}\right)=\frac{M}{2}\label{identity2},
\end{eqnarray}
This reduces to the simple expression
\begin{equation}
S_1^{\rm{globe}}=\frac{M N s}{M+1}\label{S1GH}.
\end{equation}

The second double sum $S_2$  can be written as a product of two single sum, namely
$$
S_2=\sum_{i=1}^{N}\left({\cal L}_{i,\alpha}^{-1}-{\cal L}_{i,\beta}^{-1}\right) \times \sum_{j=1}^{N}\left({\cal L}_{\alpha,j}^{-1}-{\cal L}_{\beta,j}^{-1}\right)
$$
Let us choose the coordinates of nodes $\alpha$ and $\beta$ as $\alpha=(x_1,y_1)$ and $\beta=(x_2,y_2)$.
Then the first single sum in the case of the globe network can be written as
\begin{eqnarray}
\lefteqn{\sum_{i=1}^{N}\left({\cal L}_{i,\alpha}^{-1}-{\cal L}_{i,\beta}^{-1}\right)}
\\
&=&\sum_{x=1}^{N}\sum_{m=0}^{M-1}\sum_{n=0}^{N-1}\frac{\psi_{(m,n);(x,1)}
\left(\psi_{(m,n);(x_1,y_1)}^*-\psi_{(m,n);(x_2,y_2)}^*\right)}{\Lambda_{m,n}}\nonumber\\
&=&\frac{1}{2N(M+1)}\sum_{x=1}^{N}\sum_{m=0}^{M-1}\sum_{n=0}^{N-1}
\frac{e^{2 i x \theta_n}\sin(2\varphi_m)\left(e^{-2 i x_1 \theta_n}\sin(2y_1\varphi_m)-e^{-2 i x_2 \theta_n}\sin(2y_2\varphi_m)\right)}{r^{-1}\sin^2\theta_n+s^{-1}\sin^2\varphi_m}\nonumber\\
&=&\frac{1}{2(M+1)}\sum_{m=0}^{M-1}\sum_{n=0}^{N-1}
\frac{\delta_{n,0}\sin(2\varphi_m)\left(e^{-2 i x_1 \theta_n}\sin(2y_1\varphi_m)-e^{-2 i x_2 \theta_n}\sin(2y_2\varphi_m)\right)}{r^{-1}\sin^2\theta_n+s^{-1}\sin^2\varphi_m}\nonumber\\
&=&\frac{s}{M+1}\sum_{m=0}^{M-1}\cot\varphi_m\left(\sin(2y_1\varphi_m)-\sin(2y_2\varphi_m)\right)\nonumber\\
&=&\frac{s}{M+1}(y_2-y_1).\label{sum1G3}
\end{eqnarray}
Here we have use the  identity
\begin{equation}
\sum_{m=0}^{M-1}\cot\varphi_m \sin(2y\varphi_m)=\sum_{m=0}^{M-1}\cot\frac{\pi (m+1)}{2(M+1)}\sin\frac{\pi (m+1) y}{M+1}=M+1-y ,\label{identity4}
\end{equation}
which holds for integer values of $y$.
The second single sum can be obtain in the similar manner:
\begin{eqnarray}
\sum_{j=1}^{N}\left({\cal L}_{\alpha,j}^{-1}-{\cal L}_{\beta,j}^{-1}\right)&=&
\frac{s}{M+1}(y_2-y_1)\label{sum1G4}
\end{eqnarray}
Thus for the second double sum $S_2$ in the case of globe network we obtain
\begin{equation}
S_2=S_2^{globe}=\frac{s^2(y_2-y_1)^2}{(M+1)^2}.
\end{equation}

Substituting Eq. (\ref{Delta1}) into Eq. (\ref{resistorab1}) and plugging the double sums $S_1$ and $S_2$ back to Eq. (\ref{resistorab1}) we finally obtain for the resistance between two nodes $\alpha=(x_1,y_1)$ and $\beta=(x_2,y_2)$ of the globe network the  expression
\begin{equation}
R_{\alpha,\beta}=\frac{s(y_2-y_1)^2}{N(M+1)}+\sum_{m=0}^{M-1}\sum_{n=0}^{N-1}
\frac{\left|\psi_{(m,n);(x_1,y_1)}-\psi_{(m,n);(x_2,y_2)}\right|^2}{\Lambda_{m,n}}.
\label{resistorabfinal}
\end{equation}

From Eq. (\ref{resistorabfinal}) and using expressions  (\ref{Lambdamn}) and (\ref{varphiki}) for the eigenvalues and eigenvectors of the second minor of the Laplacian of the globe network, we obtain for the resistance $R^{\rm{globe}}_{\alpha,\beta}$ between
two nodes at $\alpha = \{x_1, y_1\}$ and $\beta = \{x_2, y_2\}$,
\begin{eqnarray}
R ^{\rm{globe}}_{\alpha,\beta}&=&\frac{s(y_2-y_1)^2}{N(M+1)}+\sum_{m=0}^{M-1}\sum_{n=0}^{N-1}
\frac{\left|\phi_{(m,n);(x_1,y_1)}-\phi_{(m,n);(x_2,y_2)}\right|^2}
{\Lambda_{m,n}}\nonumber\\
&=&\frac{s(y_2-y_1)^2}{N(M+1)}+\frac{r}{N(M+1)}\sum_{m=0}^{M-1}\sum_{n=0}^{N-1}
\frac{S_1^2+S_2^2-2S_1S_2\cos[2(x_1-x_2)\theta_n]}
{(1-\cos{2\theta_n})+ h (1-\cos{2\varphi_m})},
\label{R12}
\end{eqnarray}
where
\begin{equation}
h=r/s, \qquad S_1=\sin\left(2y_1\varphi_{m}\right), \qquad S_2=\sin\left(2y_2\varphi_{m}\right). \nonumber
\end{equation}

It is convenient to introduce the quantity $\varLambda(\varphi_m)$ by writing
\begin{equation}
1 + h (1-\cos \varphi_m ) = {\rm ch} 2 \varLambda (\varphi_m) \nonumber
\end{equation}
or,
\begin{equation}
{\rm sh} \varLambda(\varphi_m) = \sqrt h \sin \varphi_m\, .
\end{equation}
We can then carry out the summation over $n$ in Eq.~(\ref{R12}) by using the summation identity
\begin{equation}
\frac 1 N \sum_{n=0}^{N-1} \frac {\cos (2 \ell \theta_n)}
 {{\rm ch} 2 \varLambda - \cos 2\theta_n   } = \, \frac{{\rm ch}[(N -2 \ell)\varLambda)]}
{{\rm sh} (2\varLambda ){\rm sh}(N \varLambda)},
\label{sumidentity}
\end{equation}
with  $\ell = 0, \, |x_1-x_2|$, to obtain
\begin{eqnarray}
R ^{\rm{globe}}_{\alpha,\beta}&=& \frac{r} {M+1}
\sum_{m=0}^{M-1}\frac{S_1^2+S_2^2-2S_1 S_2
{\rm ch}\big[2|x_1-x_2|\,\varLambda_m\big]}
{{\rm sh}(2\,\varLambda_m)}\,\coth ( N\,\varLambda_m ) \nonumber\\
&+&\frac{r} {M+1}  \sum_{m=0}^{M-1}
\frac{ 2S_1S_2{\rm sh}\big[2|x_1-x_2|\,\varLambda_m \big]}
{{\rm sh} (2\,\varLambda_m)}+\frac{s(y_2-y_1)^2}{N(M+1)}, \label{R12y}
\end{eqnarray}
where $\varLambda_m = \varLambda (\varphi_m)$.
This is our desired expression -- the resistance as a single summation.
Note, that in Ref.~\cite{wu} the exact expression for the two-point resistance on regular lattices was obtained in the form of a double summation only.
One requires the summation identities given by Eq. (\ref{sumidentity}) to reduce the  expression  to the form of single summation.

In the special case of $x_1 = x_2=x$, i.e., two nodes in the same $y$ column at $y_1$ and $y_2$, Eq.~(\ref{R12y}) reduces to
\begin{equation}
R ^{\rm{globe}}(x_1=x_2=x) = \frac{s(y_2-y_1)^2}{N(M+1)}+\frac{r} {M+1}\sum_{m=0}^{M-1}\frac {\coth (N \varLambda_m)}
  {{\rm sh} (2\varLambda_m)} \big[\sin (2y_1 \varphi_m) -\sin (2y_2 \varphi_m)\big]^2.
\label{x1x2}
\end{equation}
similarly, in the special case of $ y_1 = y_2=y$, wherein two nodes are in the same $x$ row at $x_1$ and $x_2$, Re.~(\ref{R12y}) reduces to
\begin{equation}
R ^{\rm{globe}}(y_1=y_2=y) = \frac{4r} {M+1}\sum_{m=0}^{M-1}
\frac { {\rm sh} \big[|x_1-x_2| \varLambda_m\big]
        {\rm sh} \big[ \big(N-|x_1-x_2| \big) \varLambda_m \big]  }
      {{\rm sh} (2 \varLambda_m ) {\rm sh} (N \varLambda_m) }  \sin^2 (2 y \varphi_m). \label{y1y2}
\end{equation}

Note that our results (\ref{R12y})-(\ref{y1y2}) coincides with corresponding results of \cite{tan2globe}, obtained by alternative approach of computing resistances by using a method of direct summation.

\section{Summary}
\label{Summary}

We have revisited the problem of the evaluation of two-point resistances
in a resistor network using the Laplacian approach considered in \cite{wu,ikw}.
We reformulated the problem in terms of the eigenvalues and eigenfunctions of the second minor of the Laplacian $\cal L$.
We showed that this strategy can deliver solutions in circumstances outside the reach of previous Laplacian based approaches.
As an example, the new formulation is applied to the globe resistor network: a cylindrical lattice with sites on end boundaries connected to two external common nodes $O$ and $O'$.  Our analysis leads to an exact expression (\ref{R12y}) for the resistance between arbitrary two nodes on the globe network, from which numerical results may be determined to arbitrary precision.

\section{Acknowledgments}
\label{Acknowledgments}
The work was supported by a Marie Curie IIF (Project no. 300206-RAVEN)
and IRSES (Projects no. 295302-SPIDER and 612707-DIONICOS) within 7th European Community Framework
Programme and by the grant of the Science Committee of the Ministry of Science and
Education of the Republic of Armenia under contract 13-1C080. In addition, we would like to thank John Essam for discussions.

\appendix

\section{1D Laplacians: eigenvalues and eigenvectors}
\label{1DLaplac}

On a domain of rectangular shape the $d$-dimensional Laplacian ${\bf L^{(d)}}$ is a sum of $d$ one-dimensional Laplacians ${\bf L^{(1)}}$ if the boundary conditions in  one direction do not depend on the coordinates in other directions. If we call $\alpha_1,\alpha_2,...,\alpha_d$ the boundary conditions in the $1,2,...,d$ directions, the d-dimension Laplacian ${\bf L^{(d)}}_{\alpha_1,\alpha_2,...,\alpha_d}$ can be written as
\begin{equation}
{\bf L^{(d)}}_{\alpha_1,\alpha_2,...,\alpha_d}= L^{(1)}_{\alpha_1} \otimes {\bf I}\otimes...\otimes {\bf I}+{\bf I} \otimes L^{(1)}_{\alpha_2} \otimes ...\otimes {\bf I}+...+{\bf I} \otimes .... \otimes {\bf I} \otimes L^{(1)}_{\alpha_d}.
\nonumber
\end{equation}
If ${\mathbf f_{\alpha_i}(x)}$ is an eigenfunction of ${\bf
L_{\alpha_i}^{(1)}}$, with eigenvalue ${\bf \lambda_{\alpha_i}}$, then the
product $\prod_{i=1}^d {\bf f_{\alpha_i}(x)}$ is an eigenvalue of
${\bf L_{\alpha_1,\alpha_2,...,\alpha_d}^{(d)}}$ with eigenvalue $\sum_{i=1}^d{\bf
\lambda_{\alpha_i}}$.
Thus the spectra of one-dimensional Laplacians is all one needs to
diagonalize higher dimensional versions.

Let us consider a 1D lattice with $N$ sites, with coordinate
${\bf x}$ between $1$ and $N$.
The lattice version of the Laplacian is given by
\begin{eqnarray}
{\bf L} f(x)= 2\; f(x) - f(x + 1) - f (x - 1),\nonumber
\end{eqnarray}
where $f(x)$ is the vector
$$
{\bf f(x)}=\left( \begin{array}{ccccccc}
f(1)\\
f(2)\\
f(3)\\
\vdots \\
f(N-1)\\
f(N)
\end{array} \right).
$$
Now we are going to solve the following equation:
\begin{equation}
{\bf L} \; f(x) = \lambda \; f(x), \hspace{1cm} x = 1,2,..., N .\nonumber
\end{equation}
In the lattice version,
\begin{equation}
2\; f(x) - f(x + 1) - f (x - 1) = \lambda \; f(x), \hspace{1cm} x = 1,2,..., N \nonumber
\end{equation}
where $f(x)$ is an eigenfunction and $\lambda$ is an eigenvalue. The solution is given by
\begin{equation}
f(x) = a \; e^{i \theta x} + b \; e^{- i \theta x}, \hspace{1cm}
\lambda = 2 - 2 \cos{\theta},  \nonumber
\end{equation}
in which the coefficients $a$, $b$ and $\theta$ are fixed by specific boundary conditions and by normalization conditions for $f(x)$: $\sum_{x=1}^{N}f(x)^*f(x)=1$

Let us now consider the following boundary conditions of the 1D lattice

\subsection{Periodic: $f(1)=f(N+1)$}

The Laplacian  for periodic boundary conditions is given by
$$
{\bf L}_N^{\rm{per}}=\left( \begin{array}{ccccccc}
2 & -1 & 0 &\ldots &0&0&-1 \\
-1 & 2 & -1&\ldots &0 &0&0\\
\vdots & \vdots & \vdots &\ddots &\vdots & \vdots & \vdots \\
0&0&0&\ldots&-1&2&-1\\
-1&0&0&\ldots&0&-1&2
\end{array} \right).
$$
The eigenvectors $f_n(x)$ and eigenvalues $\lambda_n$ are given by
\begin{eqnarray}
f_n(x) &=& \sqrt{\frac{1}{N}}\; {\rm exp}(2\,i  \theta_n \, x ), \qquad x=1,2,...,N \nonumber\\
\lambda_n &=& 2 - 2\cos (2 \, \theta_n) , \qquad n=0,1,...,N-1,
\nonumber
\end{eqnarray}
where $\theta_n$ is given by
\begin{equation}
\theta_n=\frac{\pi n}{N}.
\label{thetan}
\end{equation}

\subsection{Free (Neumann-Neumann): $f(0)=f(1)$ and $f(N)=f(N+1)$}

The Laplacian  for free (Neumann-Neumann) boundary conditions is given by
$$
{\bf L}_N^{\rm{free}}=\left( \begin{array}{ccccccc}
1 & -1 & 0 &\ldots &0&0&-1 \\
-1 & 2 & -1&\ldots &0 &0&0\\
\vdots & \vdots & \vdots &\ddots &\vdots & \vdots & \vdots \\
0&0&0&\ldots&-1&2&-1\\
-1&0&0&\ldots&0&-1&1
\end{array} \right).
$$
The eigenvectors $f_n(x)$ and eigenvalues $\lambda_n$ are given by
\begin{eqnarray}
f_n(x)  &=& \frac{1}{\sqrt{N}} \qquad \mbox{for} \quad n=0 \nonumber\\
    &=& \sqrt{\frac{2}{N}}\;\cos((x-1/2)\,\theta_n), \qquad \mbox{for} \quad n=1,2,...,N-1 \nonumber\\
\lambda_n &=& 2 - 2\cos (\theta_n) , \qquad n=0,1,...,N-1,
\nonumber
\end{eqnarray}
where $\theta_n$ is given by Eq. (\ref{thetan}).

\subsection{Dirichlet-Dirichlet: $f(0)=f(N+1)=0$}

The Laplacian  for Dirichlet-Dirichlet boundary conditions is given by
$$
{\bf L}_N^{\rm{(DD)}}=\left( \begin{array}{ccccccc}
2 & -1 & 0 &\ldots &0&0&0 \\
-1 & 2 & -1&\ldots &0 &0&0\\
\vdots & \vdots & \vdots &\ddots &\vdots & \vdots & \vdots \\
0&0&0&\ldots&-1&2&-1\\
0&0&0&\ldots&0&-1&2
\end{array} \right),
$$
The eigenvectors $f_n(x)$ and eigenvalues $\lambda_n$ are given by
and
\begin{eqnarray}
f_n(x) &=& \sqrt{\frac{2}{N+1}}\sin (2 \, x \varphi_n )\nonumber\\
\Lambda_n &=& 2 - 2\cos (2 \varphi_n) , \qquad n=0,...,N-1,
\label{lambda}
\end{eqnarray}
where $\varphi_n$ is given by
\begin{equation}
\varphi_n=\frac{\pi (n+1)}{2(N+1)}
\label{varphin}
\end{equation}

\subsection{Dirichlet-Neumann: $f(0)=0$ and $f(N)=f(N+1)$}

If one chooses, for instance, a left Dirichlet and a right
Neumann boundary, then the Laplacian is given by
$$
{\bf L}_N^{\rm{(DN)}}=\left( \begin{array}{ccccccc}
2 & -1 & 0 &\ldots &0&0&0 \\
-1 & 2 & -1&\ldots &0 &0&0\\
\vdots & \vdots & \vdots &\ddots &\vdots & \vdots & \vdots \\
0&0&0&\ldots&-1&2&-1\\
0&0&0&\ldots&0&-1&1
\end{array} \right).
$$
The eigenvectors $f_n(x)$ and eigenvalues $\lambda_n$ are given by
and
\begin{eqnarray}
f_n(x) &=& \frac{2}{\sqrt{2N+1}}\sin (2 \, x \phi_n )\, \nonumber\\
\lambda_n &=& 2 - 2\cos (2 \phi_n) , \qquad n=0,...,N-1,
\label{lambda}
\end{eqnarray}
where $\phi_n$ is given by
\begin{equation}
\phi_n=\frac{\pi \left(n+\frac{1}{2}\right)}{2N+1}.
\label{varphin}
\end{equation}


TITLE: math notation question regarding location within ordered set
QUESTION [0 upvotes]: if $\{x_i\}$ for $i=1...N$ are a set of ordered values such that $x_1\ge x_2 \ge ... \ge x_N$ how do I notate most efficiently for a given number $y$ where $x_1 \ge y \ge x_N$ the maximum subscript $j$ such that $x_j \ge y \ge x_{j+1}$
thanks in advance

REPLY [1 votes]: I would write $$j^* = \max \{j : x_j \geq y \geq x_{j+1}\}.$$  Of course, there's also nothing wrong with the way you defined it in words.


text:
\chapter{Introduction}
Data compression and its fundamental limits are one of the main questions in communication theory \cite{cover2012elements}. Classical communication theory treats data acquisition and data compression as two separate problems and has studied each of these {modules} extensively, for example  in information theory it is well known that the fundamental limit of data compression is given by the entropy of source. With the emergence of big data applications and the prohibitive  cost of sensing mechanisms for high resolution data, smarter sensing mechanisms seem to be inevitable.

The theory of compressive sensing  \cite{foucart2013mathematical} takes its name from the premise that in many applications data acquisition and compression can be performed simultaneously. This is done by utilizing the sparse structure of many natural signals such as images, sound etc. For example, natural images are known to be sparse in Fourier bases. By taking advantage of the sparse structure one could go beyond the fundamental limits imposed by physical constraints and uncertainty principles.

Research in compressive sensing can be categorized in three major branches: mathematical theory, algorithm design and applications.  In this thesis we focus on all three aspects. From a mathematical perspective, we use tools from empirical process theory and statistical signal processing in order to study sampling complexity tradeoffs of compressive sensing for dynamic and non i.i.d compressive sensing. This is motivated by the fact that interdependence and dynamic structures are ubiquitous features of natural signals. From the algorithmic perspective, we provide fast algorithms for signal recovery under these conditions, and finally we will apply our theory in several applications of interest.

In order to facilitate reading of this of thesis we will provide a very brief introduction to compressive sensing and a few key results which will be used recurrently throughout the thesis. A more detailed treatment can be found in \cite{foucart2013mathematical}.

\section{Sparse Solutions of Underdetermined Systems}
A vector $\boldsymbol{\theta} \in \mathbb{R}^p$ is $s$-sparse if it has at most $s$ nonzero entries, i.e. if
\[
\|\boldsymbol{\theta}\|_0 := \text{card}\left(\support(\boldsymbol{\theta}) \right) \leq s.
\]
We define
\begin{equation}
\sigma_s(\boldsymbol{\theta}) := \|\boldsymbol{\theta}-\boldsymbol{\theta}_s\|_1
\end{equation}
and \begin{equation}
\varsigma_s(\boldsymbol{\theta}) := \|\boldsymbol{\theta}-\boldsymbol{\theta}_s\|_2
\end{equation}
which are scalar functions of $\boldsymbol{\theta}$ and $s$, and capture the compressibility of the parameter vector $\boldsymbol{\theta}$ in the $\ell_1$ and $\ell_2$ sense, respectively. Note that by definition
$\varsigma_s(\boldsymbol{\theta}) \leq \sigma_s(\boldsymbol{\theta})$. For a fixed $\xi \in (0,1)$, we say that $\boldsymbol{\theta}$ is \emph{$(s,\xi)$-compressible} if $\sigma_s(\boldsymbol{\theta}) = \mathcal{O}(s^{1-\frac{1}{\xi}})$ \cite{needell2009cosamp}. Note that when $\xi = 0$, the parameter vector $\boldsymbol{\theta}$ is exactly $s$-sparse.

Let
\begin{align}
\label{eq:comp1}
\mathbf{y} = \mathbf{A}\boldsymbol{\theta} + \mathbf{v},
\end{align}
where $\mathbf{A}\in \mathbf{R}^{n \times p}$ is a known measurement matrix and $\mathbf{y} \in \mathbb{R}^n $ consists of $n$ linear measurements of $\boldsymbol{\theta}$ and $\mathbf{v}$ is the bounded measurement noise satisfying
\[
\| \mathbf{v}\|_2 \leq \epsilon.
\]

Compressive sensing problem is concerned with solving (\ref{eq:comp1}) in the underdetermined setting, i.e. when $n < p$.  In this setup, the key assumption of $\boldsymbol{\theta}$ being (close to) an $s$-sparse vector can be used to recover $\boldsymbol{\theta}$. In the absence of observation noise $\boldsymbol{\theta}$ can be \textit{exactly} recovered from $\mathbf{y}$ if rank$(A) \geq 2s$, which requires at least $n \geq 2s$ measurements. Special designs of measurement matrices such as Vandermonde matrices or Fourier matrices have resulted in elegant reconstruction algorithms such as Prony's method \cite{parks1987digital}.

In the presence of noise one would need higher number of measurements . Ideally one would like to solve the optimization of finding the \textit{sparsest} $\boldsymbol{\theta}$ which satisfies the bounded noise constraints, i.e.
\begin{align}
\label{eq:comp2}
\begin{tabular}{l}$\minimize_{\boldsymbol{\theta} \in \mathbb{R}^p} \quad \|\boldsymbol{\theta}\|_0$,\\
$\subjectto \quad \|\mathbf{Ax}-\mathbf{y}\|_2 \leq \epsilon.$
\end{tabular}
\end{align}

Unfortunately  ($\ref{eq:comp2}$) is an NP-hard problem and due to the combinatorial nature of $\ell_0$norm. The convex relaxation of (\ref{eq:comp2}) is therefore used most often, which is achieved by replacing the $\ell_0$ norm with an $\ell_1$ norm, i.e.

\begin{align}
\label{eq:comp3}
\begin{tabular}{l}$\minimize_{\boldsymbol{\theta} \in \mathbb{R}^p} \quad \|\boldsymbol{\theta}\|_1$,\\
$\subjectto \quad \|\mathbf{A} \boldsymbol{\theta}-\mathbf{y}\|_2 \leq \epsilon.$
\end{tabular}
\end{align}

\section{Restricted Isometry Property}
For the convex optimization problem (\ref{eq:comp3}) the following property is \textit{sufficient} for stable recovery of $\boldsymbol{\theta}$:
\begin{definition}[Restricted Isometry Property \cite{candes2008introduction}]
The matrix $\mathbf{A} \in \mathbb{R}^{n \times p}$ satisfies the restricted isometry property (RIP) \cite{candes2006compressive} of order $s$, if for all $s$-sparse $\boldsymbol{\theta}\in \mathbb{R}^p$, we have
\begin{equation}
\label{eq:tv_rip}
(1-\delta_s) \|\boldsymbol{\theta}\|_2^2 \leq \|\mathbf{A}\boldsymbol{\theta}\|_2^2 \leq (1+\delta_s)\|\boldsymbol{\theta}\|_2^2,
\end{equation}
where $\delta_s \in (0,1)$ is the smallest constant for which Eq. (\ref{eq:tv_rip}) holds.
\end{definition}
Intuitively speaking, RIP requires that the linear measurements acts as an almost isometry on $s$-sparse vectors, resulting in \textit{invertibility} of the underdetermined system of equations (\ref{eq:comp1}). The following formalizes this idea:

\begin{thm}[Implications of the RIP \cite{RIP_orig} ]
\label{thm:thm1}
Suppose that $\mathbf{A}$ satisfies the RIP of order $2S$ with $\delta_{2s} < {\sqrt{2}} -1$. Then any solution $\widehat{\boldsymbol{\theta}}$ to (\ref{eq:comp3}) satisfies
\begin{align}
\|\boldsymbol{\theta}-\widehat{\boldsymbol{\theta}}\|_1 \leq c_1 \sigma_s(\boldsymbol{\theta}) + c_2 \sqrt{s} \epsilon,
\end{align}
and
\begin{align}
\|\boldsymbol{\theta}-\widehat{\boldsymbol{\theta}}\|_2 \leq c_1' \frac{\sigma_s(\boldsymbol{\theta})}{\sqrt{s}} + c_2'  \epsilon.
\end{align}
\end{thm}

For an arbitrary value $\delta_s  \in (0,1)$ one can prove existence of a measurement matrix satisfying the RIP as long as $n > c_{\delta_s} s \log (p/s)$ namely:

\begin{thm}[RIP for Random Matrices \cite{foucart2013mathematical} ]
Let $\mathbf{A}$ be an $n \times p$ subgaussian random matrix. Then there exits a constant $c>0$ depending only on subgaussian parameters, such that the restricted isometry constant of $\frac{1}{\sqrt{n} \mathbf{A}}$ satisfies $\delta_s \leq \delta$ with probability at least $1-\epsilon$ provided
\[
n \geq \frac{c}{\delta^2} \left( s \log\left( e p/s \right) + \log({2}/{\epsilon}) \right).
\]
\end{thm}
Setting $\epsilon = 2 \exp(-\delta^2 n/2c)$ yields the condition
\[
n \geq 2c/\delta^2 s \log \left( ep/s \right),
\]
which guarantees that $\delta_s \leq \delta$ with probability at least $1-2 \exp(-\delta^2n/2c)$.

We will next discuss the commonly used reconstruction algorithms for sparse recovery.

\section{Sparse Recovery Algorithms}
Popular sparse recovery algorithms in compressive sensing can be divided into three main categories: optimization methods, greedy methods, and thresholding-based methods \cite{foucart2013mathematical}. In this thesis we focus on developing algorithms for the optimization methods and greedy methods.

\subsection{Optimization Problems}
Three of the most popular optimization problems for sparse recovery are the quadratically constrained basis pursuit, lasso \cite{tibshirani1996regression} and basis pursuit denoising. The quadratically constrained basis pursuit algorithm solves the optimization problem
\begin{align}
\label{eq:qcbp}
\begin{tabular}{ll}$\minimize_{\boldsymbol{\theta} \in \mathbb{R}^p}$ & $\|\boldsymbol{\theta}\|_1$,\\
$\subjectto$ & $\|\mathbf{A}\boldsymbol{\theta}-\mathbf{y}\|_2 \leq \epsilon.$
\end{tabular}
\end{align}
Alternatively one can find the solution which is closest to the measurements while maintaining a controlled sparsity level. Such formulation is known as the lasso given by

\begin{align}
\label{eq:lasso_def}
\begin{tabular}{ll}$\minimize_{\boldsymbol{\theta} \in \mathbb{R}^p}$ & $\|\mathbf{A}\boldsymbol{\theta}-\mathbf{y}\|_2  $,\\
$ \subjectto$ & $ \|\boldsymbol{\theta}\|_1 \leq \tau.$
\end{tabular}
\end{align}

A closely related optimization problem via the lagrangian formulation of both problems is known as basis pursuit denoising, given by

\begin{align}
\label{eq:lasso_def}
\begin{tabular}{ll}$\minimize_{\boldsymbol{\theta} \in \mathbb{R}^p}$ & $\|\mathbf{A} \boldsymbol{\theta}-\mathbf{y}\|_2 + \lambda \|\boldsymbol{\theta}|\|_1.$
\end{tabular}
\end{align}

\subsection{Greedy Algorithms}

Although there exist fast solvers to convex problems of the type given by Eq. (\ref{eq:qcbp}), these algorithms are polynomial time in $n$ and $p$, and may not scale well with high-dimensional data.  This motivates us to consider greedy solutions for the estimation of sparse parameters. In particular, in this thesis we will consider generalizations of the Orthogonal Matching Pursuit (OMP) \cite{OMP} for general convex cost functions.  The main idea behind the OMP is in the greedy selection stage, where the absolute value of the gradient of the cost function at the current solution is considered as the selection metric. The OMP algorithm adds one index to the estimated support of $\boldsymbol{\theta}$ at each step. A flowchart of the algorithm is given in Table (\ref{tab:OMP_main}).

\begin{table}
\centering
\framebox{$\begin{array}{l}
\text{Input: } \mathbf{A}, \mathbf{y} \\
\text{Output: } \widehat{\boldsymbol{\theta}}_{\sf OMP}\\
\text{Initialization:}\Big\{\begin{array}{l}
\text{Start with the index set } S^{(0)}=\emptyset\\
\text{and the initial estimate }\widehat{\boldsymbol{\theta}}^{(0)}_{{\sf OMP}} = \mathbf{0}
\end{array}\\
\textbf{for } k=1,2,\cdots,s^\star\\
\text{  }\begin{array}{l}
j = \argmax \limits_i \left| \left(  \mathbf{A}' \left(\mathbf{y}-\mathbf{A}\widehat{\boldsymbol{\theta}}_{{\sf OMP}}^{(k-1)}\right) \right)_i\right|\\
S^{(k)}=S^{(k-1)}\cup \{j\}\\
\widehat{\boldsymbol{\theta}}_{{\sf OMP}}^{(k)} = \argmin \limits_{\support ({\boldsymbol{\theta}}) \subset S^{(k)}} \|\mathbf{y}-\mathbf{A}\boldsymbol{\theta}\|_2
\end{array}\\
\textbf{end }\\
\end{array}$
}
\caption{\small{Orthogonal Matching Pursuit (OMP)}}
\label{tab:OMP_main}
\end{table}

The choice of the maximum number of iterations $s^\star$ will be discussed in detail later. One can repeat the process until a stopping criterion is met. The advantage of the OMP algorithm is breaking the high ($p$)-dimensional optimization problem into several low ($\leq s^\star$)-dimensional problem which can usually be solved much faster.

\section{Theoretical Guarantees for  Convex Cost Functions}

In many applications of interest the measurements are not linear, or the noise is not additive or Gaussian. In these scenarios the objective function is not in the form of squared errors. We will now introduce the theoretical requirements which generalize RIP to general convex cost functions, and the corresponding theoretical guarantees.

\subsection{Restricted Strong Convexity}
General convex cost functions are usually in the form of a loss function or a negative log-likelihood. Estimation problems for such cost functions are known as M-estimation problems. The general form of an M-estimation problem is given by
\begin{equation}
\label{eq:M_Est}
\widehat{\boldsymbol{\theta}}:=\argmin\limits_{\boldsymbol{\theta}\in \boldsymbol{\Theta}} \quad \mathfrak{L}(\boldsymbol{\theta}),
\end{equation}
and its $\ell_1$-regularized counterpart is given by
\begin{equation}
\label{eq:sp_est_pp_L}
\widehat{\boldsymbol{\theta}}_{{\sf sp}}:=\argmin\limits_{\boldsymbol{\theta}\in \boldsymbol{\Theta}} \quad \mathfrak{L}(\boldsymbol{\theta})+ \gamma_n\|\boldsymbol{\theta}\|_1.
\end{equation}
where, $\mathfrak{L}(\boldsymbol{\theta})$ is a cost function which is convex in $\boldsymbol{\theta}$, $\gamma_n > 0$ is a regularization parameter and $\boldsymbol{\Theta}$ is the convex set of admissible solutions.

The main sufficient condition for theoretical guarantees of general convex cost functions is the notion of Restricted Strong Convexity (RSC) \cite{Negahban}. By the convexity of the cost function, it is clear that a small change in $\boldsymbol{\theta}$ results in a small change in the cost function. However, the converse is not necessarily true. Intuitively speaking, the RSC condition guarantees that the converse holds: a small change in the cost implies a small change in the parameter vector, i.e., the cost function is not too \textit{flat} around the true parameter vector. A depiction of the RSC condition for $p=2$, adopted from \cite{Negahban}, is given in Figure \ref{fig:rsc}. In Figure \ref{fig:rsc}(a), the RSC does not hold since a change along $\theta_2$ does not change the log-likelihood, whereas the log-likelihood in Figure \ref{fig:rsc}(b) satisfies the RSC.

\begin{figure}[h]
\centering
\includegraphics[width=.8\columnwidth]{rscfig}
\caption{\small{Illustration of RSC (a) RSC does not hold (b) RSC does hold.}}\label{fig:rsc}
\end{figure}
More formally, if the log-likelihood is twice differentiable at $\boldsymbol{\theta}$, the RSC is equivalent to existence of a lower quadratic bound on the negative log-likelihood:

\begin{equation}
\label{RSC}
\mathfrak{D_L}({\boldsymbol{\psi}},\boldsymbol{\theta}):= \mathfrak{L}(\boldsymbol{\theta}+{\boldsymbol{\psi}})-  \mathfrak{L}(\boldsymbol{\theta})-{\boldsymbol{\psi}}'\nabla\mathfrak{L}(\boldsymbol{\theta})\geq \kappa\|{\boldsymbol{\psi}}\|_2^2,
\end{equation}

for a positive constant $\kappa > 0$ and all { ${\boldsymbol{\psi}}\in\mathbb{R}^p$} in a carefully-chosen neighborhood of $\boldsymbol{\theta}$ depending on $s$ and $\xi$. Based on the results of \cite{Negahban}, when the RSC is satisfied, sufficient conditions akin to that in Theorem \ref{thm:thm1} can be obtained by estimating the {Euclidean} extent of the solution set around the true parameter vector.
Here we  restate the main result of \cite{Negahban} concerning RSC and its implications in controlling the estimation error for general convex cost functions:

\begin{prop}[Implications of RSC (Theorem 1 of \cite{Negahban})]
\label{negahban_thm}
For a negative log-likelihood $\mathfrak{L}(\boldsymbol{\theta})$ which satisfies the RSC with parameter $\kappa$, every solution to the convex optimization problem (\ref{eq:sp_est_pp_L}) satisfies
\begin{equation}
\label{error_bound}
\left\|\widehat{\boldsymbol{\theta}}_{\sf {sp}}-\boldsymbol{\theta}\right\|_2 \leq \frac{2\gamma_n \sqrt{s}}{\kappa}+\sqrt{\frac{2\gamma_n \sigma_s(\boldsymbol{\theta})}{\kappa}}
\end{equation}
with a choice of the regularization parameter

\begin{equation}
\label{reg}
\gamma_n \geq   2\left\|\nabla\mathfrak{L}(\boldsymbol{\theta})\right\|_\infty.
\end{equation}
\end{prop}

The first term in the bound (\ref{reg}) is increasing in $s$ and corresponds to the estimation error of the $s$ largest components of $\boldsymbol{\theta}$ in magnitude, whereas the second term is decreasing in $s$ and represents the cost of replacing $\boldsymbol{\theta}$ with its best $s$-sparse approximation.

Similarly the counterpart of OMP for general cost functions was introduced in \cite{zhang_omp} and is summarized in Table \ref{tab:OMP_gen}.

\begin{table}
\centering
\framebox{$\begin{array}{l}
\text{Input: } \mathfrak{L}(\boldsymbol{\theta}) , s^\star\\
\text{Output: } \widehat{\boldsymbol{\theta}}_{\sf OMP}^{(s^\star)}\\
\text{Initialization:}\Big\{\begin{array}{l}
\text{Start with the index set } S^{(0)}=\emptyset\\
\text{and the initial estimate }\widehat{\boldsymbol{\theta}}^{(0)}_{{\sf OMP}} = 0
\end{array}\\
\textbf{for } k=1,2,\cdots,s^\star\\
\text{  }\begin{array}{l}
j = \argmax \limits_i \left| \left( \nabla \mathfrak{L} \; \left(\widehat{\boldsymbol{\theta}}_{{\sf OMP}}^{(k-1)}\right) \right)_i\right|\\
S^{(k)}=S^{(k-1)}\cup \{j\}\\
\widehat{\boldsymbol{\theta}}_{{\sf OMP}}^{(k)} = \argmin \limits_{\support ({\boldsymbol{\theta}}) \subset S^{(k)}} \mathfrak{L}(\boldsymbol{\theta})
\end{array}\\
\textbf{end }\\
\end{array}$
}
\caption{\small{Generalized Orthogonal Matching Pursuit }}
\label{tab:OMP_gen}
\end{table}

The main theoretical result regarding the generalized OMP is given by the following Proposition stating that the greedy procedure is successful in obtaining a reasonable $s^\star$-sparse approximation, if the cost function satisfies the RSC:

\begin{prop} [Guarantees of OMP (Theorem 2.1 of \cite{zhang_omp})]
\label{prop:omp_gen}
Suppose that $\mathfrak{L}(\boldsymbol{\theta})$ satisfies RSC with a constant $\kappa > 0$. Let $s^\star$ be a constant such that
\begin{equation}
\label{sstar}
s^\star = \mathcal{O}(s\log s),
\end{equation}
Then, we have
\begin{equation*}
\left \|\widehat{\boldsymbol{\theta}}^{(s^\star)}_{{\sf OMP}}-\boldsymbol{\theta}_s \right \|_2 \leq \frac{\sqrt{6} \epsilon_{s^\star}}{\kappa},
\end{equation*}
where $\epsilon_{s^\star}$ satisfies
\begin{equation}
\label{eps_bound}
\epsilon_{s^\star} \leq \sqrt{s^\star+s} \|\nabla\mathfrak{L}(\boldsymbol{\theta}_s)\|_\infty .
\end{equation}
\end{prop}

\section{Roadmap of the Thesis}

As noted earlier most of the theoretical guarantees of compressive sensing provide sharp trade-offs between the number of measurement, sparsity, and estimation accuracy when random i.i.d measurements are at disposal. However, in most practical applications of interest, the measured signals and the corresponding covariates are highly interdependent and follow specific dynamics. In Chapters \ref{chap:ar} and \ref{chap:hawkes} we will generalize these theoretical guarantees to two large classes of stationary processes, namely the autoregressive (AR) processes and generalized linear models (GLM), where the covariates are highly nonlinear and non-i.i.d. Our theoretical results provide insight into the tradeoffs in sampling requirements and a measure of dependence in these models.

In Chapters \ref{chap:css} and \ref{chap:slapmi} we provide fast iterative algorithms for compressible state-space models as well as general convex optimization problems with positivity constraints. We will use ideas from projection microscopy to develop a two-photon imaging technique that scans lines of excitation across the sample at multiple angles, recovering high-resolution images from relatively few incoherently multiplexed measurements. By using a static image of the sample as a prior, we recover neural activity from megapixel fields of view at rates exceeding 1kHz.

config: arxiv


TITLE: Evaluate $\lim_{x\to 1}\frac{x^{x^a}-x^{x^b}}{\ln^2 x}$
QUESTION [1 upvotes]: Evaluate $$\lim_{x\to 1}\frac{x^{x^a}-x^{x^b}}{\ln^2 x}$$

I've tried to use fundamental limits to solve it and conclude that the limit is $a-b$ as follows:
\begin{align*}
\lim_{x\to 1}\frac{e^{x^a\ln x}-e^{x^b\ln x}}{\ln^2 x}&=\lim_{x\to 1}\frac{e^{x^a\ln x}-1-(e^{x^b\ln x}-1)}{\ln^2 x}\\
&=\lim_{x\to 1}\left(\frac{e^{x^a\ln x}-1}{\ln^2 x}-\frac{e^{x^b\ln x}-1}{\ln^2 x}\right)\\
&=\lim_{x\to 1}\frac{x^a-x^b}{\ln x}=\lim_{x\to 1}\left(\frac{a(e^{a\ln x}-1)}{a\ln x}-\frac{b(e^{b\ln x}-1)}{b\ln x}\right)\\
&= a-b
\end{align*}
I uses the fact that the $\lim_{x\to a}\dfrac{e^{u(x)}-1}{u(x)}=1$ where  $\lim_{x\to a} u(x)=0$

REPLY [0 votes]: It's not clear to me what happens in the step from blue to red:

$$\begin{align*}
\lim_{x\to 1}\frac{e^{x^a\ln x}-e^{x^b\ln x}}{\ln^2 x}&=\lim_{x\to 1}\frac{e^{x^a\ln x}-1-(e^{x^b\ln x}-1)}{\ln^2 x}\\
&=\color{blue}{\lim_{x\to 1}\left(\frac{e^{x^a\ln x}-1}{\ln^2 x}-\frac{e^{x^b\ln x}-1}{\ln^2 x}\right)}\\
& = \color{red}{\lim_{x\to 1}\frac{x^a-x^b}{\ln x}}=\lim_{x\to 1}\left(\frac{a(e^{a\ln x}-1)}{a\ln x}-\frac{b(e^{b\ln x}-1)}{b\ln x}\right)\\
&= a-b
\end{align*}$$

If you take $x = e^u$, then:
$$\lim_{x\to 1}\frac{e^{x^a\ln x}-e^{x^b\ln x}}{\ln^2 x}
=\lim_{u\to 0}\frac{e^{ue^{au}}-e^{ue^{bu}}}{u^2}$$
Now if $u \to 0$, then $ue^{au} \to 0$ so you can use a Taylor expansion on $e^{ue^{au}}$ and $e^{ue^{bu}}$.


text:
\begin{document}

\maketitle

\begin{abstract}
Suppose that $F$ is a non-Archimedean local field of characteristic not $2$ and $D$ is a central division algebra over $F$.
Let $n$ be a positive integer.
We show a classification modulo essentially square-integrable representations of standard modules of $\GL_n(D)$ which have non-zero linear periods.
By this classification, the conjecture of Prasad and Takloo-Bighash is reduced to the case of essentially square-integrable representations.
\end{abstract}

\section{Introduction}\label{sec:Introduction}

Suppose that $F$ is a non-Archimedean local field of characteristic not $2$ and $E$ is a quadratic extension of $F$.
Take a positive integer $n$ and a central division algebra $D$ over $F$ of dimension $d^2$.
Set $G=G_n=\GL_{n}(D)$, which is the multiplicative group of $\mat_n(D)$.
Let $\nu=\nu_n$ denote the composition of the reduced norm of $\mat_n(D)$ and the normalized absolute value of $F$.
Assume that $E$ embeds in $\mat_n(D)$ and let $H=C_{G}(E^\times)$ be the centralizer of $E^\times$ in $G$.
Note that the centralizer $C_{\mat_n(D)}(E)$ of $E$ in $\mat_n(D)$ is a central simple algebra over $E$.
Let $\chi$ be a character of $E^\times$ and regard it as a character of $H$ by composing with the reduced norm of $C_{\mat_n(D)}(E)$.
An admissible representation $\pi$ of $G$ is said to be \textit{$(H, \chi)$-distinguished} if $\Hom_H(\pi, \chi)\neq0$.
Prasad and Takloo-Bighash proposed a conjecture \cite[Conjecture 1]{PTB11} about $(H, \chi)$-distinguished representations.
The following is $\chi=1$ case of that conjecture.
\begin{conj}\label{conj:PTB}
Let $\pi$ be an irreducible, admissible representation of $G$ with trivial central character.
\begin{enumerate}
\item If $\pi$ is $H$-distinguished,
    \begin{enumerate}
    \item[(i)] The Langlands parameter of $\pi$ takes values in the symplectic group
    $\mathrm{Sp}_{nd}(\C)$.

    \item[(ii)] The root number satisfies
        $\varepsilon(\pi)\varepsilon(\pi\otimes\eta_{E/F})
        =(-1)^n\eta_{E/F}(-1)^{nd/2}$.
    \end{enumerate}

\item If $\pi$ is an essentially square integrable representation satisfying (i) and (ii), then $\pi$ is $H$-distinguished.
\end{enumerate}
\end{conj}

\begin{remark}
The original conjecture of Prasad and Takloo-Bighash is stated under the assumption that $\pi$ corresponds to a generic representation of $\GL_{nd}(F)$.
We will clarify the meaning of this assumption in Section \ref{subsec:Jacquet-Langlands transfer}.
One of the new observations of this paper is that this assumption is not necessary.
\end{remark}

This conjecture is based on the result of Tunnell \cite{Tun83} for the case of $D=F$ with odd residual characteristic and $n=2$.
There are a lot of research to generalize this result.
Saito gave a simpler proof of Tunnell's result  in \cite{Sai93} including the case of residual characteristic $2$ when the characteristic of $F$ is $0$.
Prasad \cite{Pra07} obtained a different proof of Saito's result based on a global theory.

The next three results of \cite{PTB11}, \cite{FMW18} and \cite{Xue} are for the case of characteristic $0$.
Prasad and Takloo-Bighash proved Conjecture \ref{conj:PTB} in \cite{PTB11} for representations of $\GL_2(D)$ where $D$ is a quaternion algebra.
They used the theta correspondence between $\mathrm{GSO}_6$ and $\mathrm{GSp}_4$.
In \cite{FMW18}, Feigon, Martin and Whitehouse obtained (1) when $D=F$ and $\pi$ is supercuspidal, by using a relative trace formula.
Using the same relative trace formula in a different way, Xue proved (1) for all essentially square-integrable representations and (2) for supercuspidal representations with supercuspidal Jacquet-Langlands transfer to $\GL_{nd}(F)$ (\cite{Xue}).
When $D$ is a quaternion algebra or $D=F$, Xue also proved (2) for all supercuspidal representations.

The following results of \cite{Cho19} and \cite{CM20} cover the case of all non-Archimedean local fields of odd residual characteristic.
In \cite{Cho19}, Chommaux proved the original conjecture (for a general character $\chi$) for Steinberg representations by Mackey machinery.
This result also covers the case where $F$ has characteristic $0$ and the residual characteristic $2$.
Chommaux and Matringe showed the original conjecture when $D=F$ and $\pi$ is a supercuspidal representation of depth zero (\cite{CM20}).

In \cite{Sec20}, S\'echerre proved the conjecture for all supercuspidal representations when $F$ has characteristic $0$  and odd residual characteristic, by using Bushnell-Kutzko type.
Recently, Xue and the author proved that Conjecture \ref{conj:PTB} for essentially square-integrable representations reduces to the case of supercuspidal representations when the characteristic of $F$ is $0$ (\cite{SX20}).
See also \cite[Section 1.7]{Sec20} for a historical review on this conjecture and other related research.

The goal of this paper is to prove the next theorem.

\begin{theorem}\label{thm:classification}
Suppose $P=MU$ is a standard parabolic subgroup of $G$ corresponding to the partition $n=(n_1, \ldots, n_t)$ of $n$.
Let $\widetilde{\pi}=\Delta_1\times\cdots\times\Delta_t$ be a standard module of $G$ induced from segments on $M$.
Then, $\widetilde{\pi}$ is $H$-distinguished if and only if there is an involution $\sigma\in\fS_t$ such that $\Delta_{\sigma(i)}\simeq\Delta_i^\vee$ for all $i$ and if $\sigma(i)=i$, $E$ embeds in $\mat_{n_i}(D)$ and $\Delta_i$ is $C_{G_{n_i}}(E^\times)$-distinguished.
Here, $C_{G_{n_i}}(E^\times)$ is the centralizer of $E^\times$ in $G_{n_i}$.
\end{theorem}

From this, we can reduce Conjecture \ref{conj:PTB} (1) to the case of essentially square-integrable representations.

\begin{theorem}\label{thm:PTB}
Conjecture \ref{conj:PTB} (1) for essentially square-integrable representations implies the general case.
\end{theorem}

Combining this theorem with \cite{Sec20} and \cite{SX20}, we see that Conjecture \ref{conj:PTB} holds when the characteristic of $F$ is zero and  the residual characteristic is different from $2$.
When $D$ is a quaternion algebra or $D=F$,  we can remove the condition on the residual characteristic by applying \cite{Xue} instead of \cite{Sec20}.

\section{Notation}\label{sec:Notation}

Suppose that $F$ is a non-Archimedean local field of characteristic not $2$ and $E$ is a quadratic extension of $F$.
Take a positive integer $n$ and a central division algebra $D$ over $F$ of dimension $d^2$.
Set $G=G_n=\GL_{n}(D)$, which is the multiplicative group of $\mat_n(D)$.
Let $\nu=\nu_n$ denote the composition of the reduced norm of $\mat_n(D)$ and the normalized absolute value of $F$.
Assume that $E$ embeds in $\mat_n(D)$ and let $H=C_{G}(E^\times)$ be the centralizer of $E^\times$ in $G$.
Note that the centralizer $C_{\mat_n(D)}(E)$ of $E$ in $\mat_n(D)$ is a central simple algebra over $E$.

\subsection{Parabolic subgroups}\label{subsec:Parabolic subgroups}

For an ordered partition $\lambda=(n_1, \ldots, n_t)$ of $n$, let $P=P_\lambda$ be the parabolic subgroup of $G$ consisting of matrices of the form
    \[
    \left(\begin{array}{ccc}
    g_1&\ast     &\ast \\
          &\ddots&\ast \\
          &           &g_t \\
    \end{array}\right)\in G,\quad g_i\in G_{n_i}.
    \]
Write by $M=M_{\lambda}$ the Levi subgroup of $P$ consisting of matrices of the form $\diag(g_1, \ldots, g_t)$ with $g_i\in G_{n_i}$.
Let $U=U_\lambda$ be the unipotent radical of $P$.
Let $P_0$ be the minimal parabolic subgroup corresponding to the partition $\lambda_0=(1, \ldots, 1)$ and set $M_0=M_{\lambda_0}$.
A parabolic subgroup of $G$ is called \textit{standard parabolic subgroup} if it contains $P_0$.
A standard parabolic subgroup of $G$ has a unique Levi component which contains $M_0$ and is called a \textit{standard Levi component}.
By a parabolic subgroup (resp. Levi component) we shall always mean a standard parabolic subgroup (resp. Levi component).
A Levi subgroup of $G$ will be a Levi component of a parabolic subgroup of $G$.
Throughout the paper, the letters $P$, $M$ and $U$ are reserved for parablic subgroups, their Levi components and their unipotent radicals respectively.
For a Levi subgroup $M$ of $G$, we say that a parabolic subgroup of $M$ is standard if it contains $P_0\cap M$.

Let $P=MU$ be a parabolic subgroup of $G$ and $\rd p$ a left Haar measure on $P$. There is a continuous character $\delta_P\,\colon P\rightarrow\R_{>0}$, called the \textit{modulus character} of $P$, which satisfies
    \[
    \rd(pp')=\delta_P(p')^{-1}\rd p
    \]
for all $p'\in P$.

\subsection{Representations}\label{subsec:Representations}

Let $P=MU$ be a parabolic subgroup of $G$ and $\rho$ an admissible representation of $M$ on a complex vector space $V_\rho$.
Denote by $\Ind_{P}^G(V_\rho)$ the space of locally constant functions $f\,\colon G\rightarrow V_\rho$ satisfying
    \[
    f(pg)=\delta_P^{1/2}(p)\rho(p)f(g)
    \]
for all $p\in P$ and $g\in G$.
The representation of $G$ on this space given by right translation is called the \textit{parabolically induced representation} from $(\rho, V_\rho)$ and denoted by $(\Ind_P^G(\rho), \Ind_P^G(V_\rho))$.

Let $\lambda=(n_1, \ldots, n_t)$ be the partition corresponding to $P$.
Take a representation $(\rho_i, V_i)$ of $G_{n_i}$ for each $i$ and set $(\rho, V_\rho)=(\rho_1\boxtimes\cdots\boxtimes\rho_t, V_1\otimes\cdots\otimes V_t)$.
This is a representation of $M$.
We extend $\rho$ to a representation of $P$ so that $U$ acts trivially on $V_\rho$ and write the parabolically induced representation $\Ind_{P}^G(\rho)$ by $\rho_1\times\cdots\times\rho_t$.

Let $P'=M'U'$ be a standard parabolic subgroup of $M$.
We denote by $V_\rho(P')$ the subspace of $V_\rho$ spanned by elements of the form $\rho(u')v-v$ with $v\in V_\rho$ and $u'\in U'$.
Set $r_{M', M}(V_\rho)=V_\rho/V_\rho(P')$ and we call this the \textit{Jacquet module} of $V_\rho$ with respect to $P'$.
We write the natural projection $V_\rho\rightarrow r_{M', M}(V_\rho)$ by $j_{P'}$ and define the representation $r_{M', M}(\rho)$ of $M'$ on $r_{M', M}(V_\rho)$ by
    \[
    r_{M', M}(\sigma)(m)j_{P'}(v)=\delta_{P'}^{-1/2}(m')j_{P'}(\sigma(m')v),
    \quad v\in V,\ m'\in M'.
    \]

We briefly recall basic facts about representations of $G$.
We refer to \cite[Section 3]{LM16} for details.

Let $k$ and $t$ be positive integers such that $n=tk$ and $\rho$ an irreducible supercuspidal representation of $G_k$.
There is a positive integer $l_\rho$ such that the induced representation $\rho\times\nu^l\rho$ for $l\in\R$ is reducible if and only if $l\in\{\pm l_\rho\}$.
For a pair of integers $(a, b)$ such that $b-a=t-1$, the set
\[
[a, b]_\rho=\{\rho\nu^{al_\rho}, \ldots, \rho\nu^{bl_\rho}\}
\]
is called a \textit{segment} of $G$.
Let $\fL([a, b]_\rho)$ denote the unique irreducible quotient of $\rho\nu^{al_\rho}\times\cdots\times\rho\nu^{bl_\rho}$.
For an irreducible essentially square-integrable representation $\pi$ of $G$, there is a unique segment $\Delta$ such that $\pi=\fL(\Delta)$.

Let $\Delta=[a, b]_\rho$ and $\Delta'=[a', b']_{\rho'}$ be two segments.
We say that $\Delta$ and $\Delta'$ are \textit{linked} if $\Delta\cup\Delta'$ is a segment but neither $\Delta\subset\Delta'$ nor $\Delta'\subset\Delta$.
If $\Delta$ and $\Delta'$ are linked and $\rho'\nu^{a'l_{\rho'}}=\rho\nu^{(a+j)l_\rho}$ with some $j>0$, we say that $\Delta$ \textit{precedes} $\Delta'$.
Let $\lambda=(n_1, \ldots, n_t)$ be a partition of $n$ and $\Delta_i$ a segment of $G_{n_i}$.
Let $\fm$ denote the multi-set $\{\Delta_1, \ldots, \Delta_t\}$.
Such a multi-set is called a \textit{multisegment}.
The induced representation $\Delta_1\times\cdots\times\Delta_t$ is called a \textit{standard module} if for every $1\leq i< j\leq t$, $\Delta_i$ does not precede $\Delta_j$.
A standard module has a unique irreducible quotient $\fL(\fm)$.
Note that the isomorphsim class of $\fL(\fm)$ depends only on $\fm$.
For an irreducible admissible representation $\pi$ of $G$, there is a unique multisegment $\fm$ such that $\pi=\fL(\fm)$.

A multisegment $\fm=\{\Delta_1, \ldots, \Delta_t\}$ is said to be \textit{totally unlinked} if $\Delta_i$ and $\Delta_j$ are not linked for all $i, j=1, \ldots, t$.
For a multisegment $\fm=\{\Delta_1, \ldots, \Delta_t\}$ of $G$, the induced representation $\Delta_1\times\cdots\times\Delta_t$ is irreducible if and only if $\fm$ is unlinked.
If this is the case, the induced representation is independent of the ordering of the segments.
When $D=F$, this is equivalent to say that $\fL(\fm)$ is generic.

Let $\widetilde{\pi}=\Delta_1\times\cdots\times\Delta_t$ be a standard module of $G$.
For a supercuspidal representation $\rho$, set $\Z_\rho=\{\nu^{ml_\rho} \mid m\in\Z\}$ and $\pi_\rho=\Delta_{i_1}\times\cdots\times\Delta_{i_l}$, where $\Delta_{i_j}\subset\Z_\rho$ and $\Delta_j\not\subset\Z_\rho$ for all $j\not\in\{i_1, \ldots, i_l\}$.
Reordering the segments, we may write $\widetilde{\pi}\simeq\pi_{\rho_1}\times\cdots\pi_{\rho_p}$.
Further we assume that $\pi_{\rho_i}=[a_1^i, b_1^i]_{\rho_i}\times\cdots\times[a_{q_i}^i, b_{q_i}^i]_{\rho_i}$ with $b_1^i\geq\cdots\geq b_{q_i}^i$ for each $i$.
We call the expression $\pi_{\rho_1}\times\cdots\pi_{\rho_p}$ a \textit{right-ordered form} of the standard module $\widetilde{\pi}$.

\subsection{Jacquet-Langlands transfer}\label{subsec:Jacquet-Langlands transfer}

In this subsection, we summarize basic facts about Jacquet-Langlands transfer.
For details, see \cite[Section 2]{Bad08}.

Jacquet-Langlands transfer is a bijection from the set of irreducible essentially square-integrable representations of $G$ to that of $\GL_{nd}(F)$.
We write this map by $\JL$.
For an irreducible supercuspidal representation $\rho$ of $G_k$, we have $\JL(\rho)=\fL\left(\left[\frac{1-l_\rho}{2}, \frac{l_\rho-1}{2}\right]_{\rho'}\right)$ fore some irreducible supercuspidal representation $\rho'$ of $\GL_{kd/l_\rho}(F)$.

For a segment $\Delta=[a, b]_\rho$ of $G$, let $\JL(\Delta)$ be the segment of $\GL_{nd}(F)$ such that $\JL(\fL(\Delta))=\fL(\JL(\Delta))$.
We have
    \[
    \JL(\Delta)=\left[\frac{(2a-1)l_\rho+1}{2}, \frac{(2b+1)l_\rho-1}{2}\right]_{\rho'}
    \]
and it is easy to check that for two segments $\Delta$ and $\Delta'$, $\Delta$ precedes $\Delta'$ if and only if $\JL(\Delta)$ precedes $\JL(\Delta')$.

Let $\fm=\{\Delta_1, \ldots, \Delta_t\}$ be a multisegment of $G$ and $\pi=\fL(\fm)$ the associated irreducible admissible representation of $G$.
Define a multisegment $\JL(\fm)$ of $\GL_{nd}(F)$ by $\JL(\fm)=\{\JL(\Delta_1), \ldots, \JL(\Delta_t)\}$.
We set $\JL(\fL(\fm))=\fL(\JL(\fm))$  and call it the \textit{representation of $\GL_{nd}(F)$ corresponding to $\fL(\fm)$}.
In \cite[Section 2.7]{Bad08}, $\JL(\fL(\fm))$ is denoted by $Q_n(\fL(\fm))$.
From the above remark, we see that $\JL(\Delta_1)\times\cdots\times\JL(\Delta_t)$ is a standard module attached to $\JL(\fm)$.
Moreover, the corresponding representation is generic if and only if $\fm$ is totally unlinked.

\section{Proof}\label{sec:Proof}

Let $P=MU$ be a standard parabolic subgroup of $G$ corresponding to the partition $(n_1, \ldots, n_t)$ of $n$.
We recall some facts on coset representatives for $P\bs G/H$ from \cite{Cho19}.
Let $I(P)$ be the set of symmetric matrices $s=(n_{i,j})\in\mat_t(\Z)$ with non-negative entries such that the sum of $i$-th row equals $n_i$ for each $i=1, \ldots, t$ and when $d$ is odd, all diagonal entries are even.
The set $P\bs G/H$ is in bijection with $I(P)$ and an explicit choice of the coset representative corresponding to $s\in I(P)$, which we write by $u_s$, is given in \cite{Cho19}.
For a subgroup $C$ of $G$, we set $C^{u_s}=C\cap u_sHu_s^{-1}$.
We say that $u_s$ or the coset represented by it is \textit{$P$-admissible} if $s$ is a monomial.
Take $s=(n_{i, j})\in I(P)$ and set $u=u_s$.
Let $P_s=M_sU_s$ be the standard parabolic subgroup of $G$ corresponding to the partition $(n_{1,1}, n_{1,2}, \ldots, n_{t, t-1}, n_{t,t})$ of $n$.
Note that $n_{i,j}$ could be $0$ and such entries are ignored when we say $(n_{1,1}, n_{1,2}, \ldots, n_{t, t-1}, n_{t,t})$ is a partition.
Let $\widetilde{\pi}=\Delta_1\times\cdots\times\Delta_t$ be a standard module of $G$ induced from a segment $\Delta=\Delta_1\boxtimes\cdots\boxtimes\Delta_t$ on $M$.

For $s=(n_{i, j})\in I(P)$, the Jacquet module $r_{M, M_s}(\Delta)$ of $\Delta$ with respect to the standarad parabolic subgroup $P_s\cap M=M_s(U_s\cap M)$ of $M$ is written as
    \[
    \Delta_{1,1}\boxtimes\Delta_{1,2}\boxtimes\cdots
    \boxtimes\Delta_{t, t-1}\boxtimes\Delta_{t, t},
    \]
where $\Delta_{i, j}$ is a segment on $G_{n_{i, j}}$ for $i, j=1, \ldots, t$ such that $\Delta_i=\Delta_{i, 1}\sqcup\cdots\sqcup\Delta_{i, t}$ with the convention that $G_{n_i,j}=\{1\}$ and $\Delta_{i, j}$ is the trivial representation if $n_{i,j}=0$.

The following is \cite[Proposition 5.5]{BM19}, which is a consequence of the Mackey theory.
\begin{lemma}\label{lem:Mackey}
Let $\widetilde{\pi}=\Delta_1\times\cdots\times\Delta_t$ be a standard module of $G$ induced from segments on $M$.
Suppose that $\widetilde{\pi}$ is $H$-distinguished.
Then there is $s\in I(P)$ such that the Jacquet module $r_{M, M_s}(\Delta)$ on $M_s$ is $M_s^{u_s}$-distinguished.
If we write $r_{M, M_s}(\Delta)$ as above, this is equivalent to $\Delta_{i,j}\simeq\Delta_{j, i}^\vee$ for all $i, j$ and $\Delta_{i,i}$ is $C_{G_{n_{i,i}}}(E^\times)$-distinguished for all $i$.
\end{lemma}

By a standard argument, we can show that only $P$-admissible cosets contribute to $H$-invariant linear forms.

\begin{lemma}\label{lem:admissible}
Let $\widetilde{\pi}=\Delta_1\times\cdots\times\Delta_t$ be a right-ordered form of a standard module of $G$ induced from segments on $M$.
If $u=u_s$ is not $P$-admissible, then the Jacquet module $r_{M, M_s}(\Delta)$ is not $M_s^u$-distinguished.
\end{lemma}

\begin{proof}
This result is stated in \cite[Lemma 3.3]{Gur15} for Galois involution, which is a generalization of \cite{Mat11}.
The proof holds almost verbatim for our case.
We include the proof for completeness.

Assume that $u=u_s$ is not $P$-admissible and $r_{M, M_s}(\widetilde{\pi})$ is $M_s^u$-distinguished.
Since $u$ is not $P$-admissible, $M_s$ is strictly contained in $M$.
There exists $1\leq i\leq t$ such that $\{n_{i, j} \mid j=1, \ldots, t\}-\{0\}$ is not a singleton.
Let $i_0$ be the minimal such index and we write $\{j\mid n_{i_0, j}\neq0\}=\{ j_1<j_2<\cdots< j_r\}$.
We can write $\Delta_{i_0}=[a, b]_\rho$ with some supercuspidal representation $\rho$ and integers $a<b$.
There are some integers $c$ and $d$ such that $a\leq c\leq d<b$ and $\Delta_{i_0, j_1}=[d+1, b]_\rho$ and $\Delta_{i_0, j_2}=[c, d]_\rho$.
Note that we have $\Delta_{j_1, i_0}\simeq\Delta_{i_0, j_1}^\vee\simeq[-b, -d-1]_{\rho^\vee}$ and $\Delta_{j_2, i_0}\simeq\Delta_{i_0, j_2}^\vee\simeq[-d, -c]_{\rho^\vee}$.
Since $\Delta=\Delta_1\times\cdots\times\Delta_t$ is a right-ordered form, $\Delta_{j_1}\simeq[a', b']_{\rho^\vee}$ for some integers $a'\leq -b$ and $b'\geq -c\geq-d>-d-1$.
In particular, $i_0>1$.
Write $\Delta_{j_1,1}\simeq[e, b']_{\rho^\vee}$ with some $-d\leq e\leq b'$.
Then we get $\Delta_{1, j_1}\simeq[-b', -e]_{\rho}$  and since $i_0>1$, we have $\Delta_1=\Delta_{1, j_1}$.
This contradicts the fact that $\Delta=\Delta_1\times\cdots\times\Delta_t$ is a right-ordered form since $-e<b$.
\end{proof}

The main theorem is a straightforward consequence of Lemma \ref{lem:admissible}.

\begin{proof}[Proof of Theorem \ref{thm:classification}]
Suppose that $\Delta$ is $H$-distinguished.
From Lemma \ref{lem:Mackey}, there is $s\in I(P)$ such that $r_{M, M_s}(\Delta)$ is $M_s^{u_s}$-distinguished.
By Lemma \ref{lem:admissible}, $u=u_s$ is $P$-admissible.
In other words, $s$ is a monomial matrix in $\mat_t(\Z_{\geq0})$.
Let $\sigma\in\fS_t$ be the corresponding permutation.
Then $\sigma$ is an involution and satisfies the desired conditions.

By the same argument as the proof of \cite[Proposition 2.9]{MO18}, we obtain the converse implication.
\end{proof}

Now we deduce Theorem \ref{thm:PTB} from Theorem \ref{thm:classification}.

\begin{proof}[Proof of Theorem \ref{thm:PTB}]
Let $\widetilde{\pi}=\Delta_1\times\cdots\times\Delta_t$ be a standard module of $G$ which has a surjection onto $\pi$.
Since $\pi$ is $H$-distinguished, so is $\widetilde{\pi}$.
Let $\sigma\in\fS_t$ be the permutation of Theorem \ref{thm:classification}.
It is easy to check that the $L$-parameter of $\pi$ takes values in the symplectic group.
What we have to show is the equation of Conjecture \ref{conj:PTB} (ii).
We have
    \[
    \varepsilon(\pi)\varepsilon(\pi\otimes\eta_{E/F})
    =\prod_{i=1}^t\varepsilon(\Delta_i)\varepsilon(\Delta_i\otimes\eta_{E/F}).
    \]
Suppose that $\Delta$ is $H$-distinguished.
Let $\sigma\in\fS_t$ be the involution in Theorem \ref{thm:classification}.
For each $i$, let $\omega_i$ be the central character of $\Delta_i$.
Since $\Delta_{\sigma(i)}\simeq\Delta_i^\vee$, we have
    \[
    \varepsilon(\Delta_i)\varepsilon(\Delta_{\sigma(i)})=\omega_i(-1), \quad
    \varepsilon(\Delta_i\otimes\eta_{E/F})
    \varepsilon(\Delta_{\sigma(i)}\otimes\eta_{E/F})=\omega_i(-1)\eta_{E/F}(-1)^{n_id}.
    \]
Hence we get
    \[
    \varepsilon(\pi)\varepsilon(\pi\otimes\eta_{E/F})
    =\prod_{i;\, \sigma(i)<i}\eta_{E/F}(-1)^{n_id}\prod_{i;\, \sigma(i)=i}\varepsilon(\Delta_i)
    \varepsilon(\Delta_i\otimes\eta_{E/F}).
    \]
Let $i$ be a fixed point of $\sigma$.
Since $\Delta_i$ is $C_{G_{n_i}}(E^\times)$-distinguished, we have
    \[
    \varepsilon(\Delta_i)\varepsilon(\Delta_i\otimes\eta_{E/F})
    =(-1)^{n_i}\eta_{E/F}(-1)^{n_id/2}
    \]
from the essentially square-integrable case.
Therefore the above product equals $(-1)^n\eta_{E/F}(-1)^{nd/2}$.
\end{proof}

\begin{coro}
\begin{enumerate}
\item Conjecture \ref{conj:PTB} holds without the condition that $\pi$ corresponds to
    a generic representation of $\GL_{nd}(F)$.

\item For representations corresponding to generic representations,
    \textit{i.e.} representations induced from totally unlinked multisegments,
    Theorem \ref{thm:classification} provides a complete classification of
    $H$-distinguished representations modulo essentially square-integrable
    representations.
\end{enumerate}
\end{coro}

\begin{proof}
First we consider the first assertion.
In the proof of Theorem \ref{thm:PTB}, we did not use the assumption that $\pi$ corresponds to a generic representation.
Hence the first part of Conjecture \ref{conj:PTB} holds for any irreducible admissible representations.
Note that Jacquet-Langlands transfer of essentially square integrable representations are essentially square integrable, hence generic.
Therefore the assumption in the original conjecture of Prasad and Takloo-Bighash is unnecessary.

The second assertion is obvious from the fact that standard module is irreducible if and only if it is induced from totally unlinked multisegments.
\end{proof}

\if0
\begin{example}
Let $1_n$ be the trivial representation of $G=G_n$.
We check that $\pi=1_n$ satisfies the equation of Conjecture \ref{conj:PTB} (ii).
The standard module which has a surjection onto $1_n$ is
    \[
    \nu_1^{(n-1)d/2}\times\nu_1^{(n-3)d/2}\times\cdots\times\nu_1^{(1-n)d/2}.
    \]
From \cite[Theorem 6.1 (a)]{Bad08}, we have
    \[
    \varepsilon(1_n)=\prod_{j=0}^{n-1}\varepsilon\left(
    \frac12+\left(\frac{n-1}{2}-j\right)d, 1_1, \psi\right)
    =\begin{cases}
    1 & \text{ if $n$ is even}\\
    \varepsilon(1_1) & \text{ if $n$ is odd}.
    \end{cases}
    \]
Similarly, we get
    \[
    \varepsilon(1_n\otimes\eta_{E/F})=\begin{cases}
    \eta_{E/F}(-1)^{nd/2} & \text{ if $n$ is even}\\
    \eta_{E/F}(-1)^{nd/2}\varepsilon(1_1\otimes\eta_{E/F}) & \text{ if $n$ is odd}.
    \end{cases}
    \]
When $n$ is even, we obtain the desired equation.
Suppose that $n$ is odd.
From Conjecture \ref{conj:PTB} for essentially square-integrable representation $1_1$ of $D^\times$, we have $\varepsilon(1_1)\varepsilon(1_1\otimes\eta_{E/F})=-\eta_{E/F}(-1)^{d/2}$.
Hence the equation of Conjecture \ref{conj:PTB} (ii) holds.
\end{example}
\fi

\begin{Ac}
I would like to thank Hang Xue  for constant support and helpful discussion.
In the previous version of this paper, I only treated representations induced from totally unlinked multisegments.
I am very grateful to Nadir Matringe who suggested considering the general standard modules by the argument of \cite{Gur15}.
I am also grateful to Dipendra Prasad for answering many questions and  Alberto M\'ingeuz for useful comments.
I appreciate the anonymous referee for careful reading and pointing out several mistakes.
This work is partially supported by JSPS Research Fellowship for Young Scientists No. 20J00434.
\end{Ac}

\end{document}