\documentclass[12pt]{article}
\usepackage{amsmath,amsthm,amsfonts,amssymb}
\usepackage{euscript,graphicx,subfigure,times,version}
\usepackage{pstricks}
\include{amsmath_def}
\include{symbols}
\excludeversion{hide}
\parindent=0in

\title{MAT 280: Laplacian Eigenfunctions: Theory, Applications, and Computations\\
Lectures 14: Shape Recognition Using Laplacian Eigenvalues and
Computational Methods of Laplacian Eigenvalues/Eigenfunctions}
\author{Lecturer: Naoki Saito\\
        Scribe: Zhihua Zhang / Allen Xue}
\date{May 15 $\&$ 17 , 2007}

\begin{document}
\maketitle

\section{Shape recognition using Laplacian eigenvalues}
\par
In this section, we will introduce the work of Kbabou, Hermi, and
Rhonma (2007)\cite{KHABOU-HERMI-RHOUMA}. Their main idea is to use
the eigenvalues and their ratios of the Dirichlet-Laplacian for
various planar shapes as their features for classifying them.

\subsection{Recall some special geometric inequalities (2D)}

Let the sequence $ 0<\lambda_1<\lambda_2\leq \lambda_3\leq \cdots
\leq \lambda_k\leq \cdots \rightarrow \infty $ be the sequence of
eigenvalues of Dirichlet-Laplacian problem: $ -\varDelta u=\lambda
u$ in a given bounded planar domain $\Omega $ with Dirichlet
boundary condition $u=0$ on its boundary $\partial \Omega$.

\subsubsection{ Universal Inequalities }
\begin {itemize}
\item Payne-P$\acute{o}$lya-Weinberger
inequalities~\cite{PAYNE-POLYA-WEINBERGER}.
$$
\displaystyle\displaystyle \left\{
\begin {array}{lll}
\lambda_{m+1}-\lambda_m\leq 2\cdot
\fr{1}{m}\sum\limits_{j=1}^m\lambda_j,\\\\
\fr{\lambda_{m+1}}{\lambda_m}\leq 3.
\end {array}
\right.
$$
\end {itemize}
\begin {itemize}
\item Hile-Protter inequality~\cite{HILE-PROTTER}.
$$
\sum\limits_{j=1}^m\fr{\lambda_j}{\lambda_{m+1}-\lambda_j}\geq
\frac{m}{2}.\qquad\qquad
$$
\end {itemize}
\begin {itemize}
\item Yang inequalities~\cite{YANG}.
$$
\displaystyle\displaystyle\qquad\qquad\qquad\left\{
\begin {array}{lll}
\sum\limits_{j=1}^m(\lambda_{m+1}-\lambda_j)^2\leq 2\,
\sum\limits_{j=1}^m\lambda_j(\lambda_{m+1}-\lambda_j),\\\\
\lambda_{m+1}\leq 3\cdot \fr{1}{m}\sum\limits_{j=1}^m\lambda_j.
\end {array}
\right.
$$
\end {itemize}

\subsubsection{ Isoperimetric inequalities}

\begin {itemize}
\item Faber-Krahn inequality~\cite{CHAVEL}.
$$
\lambda_1\geq \frac{\pi
^2j_{0,1}^2}{|\Omega|^2},\qquad\qquad\qquad\qquad\quad
$$
\item Ashbaugh-Benguria inequality~\cite{ASHBAUGH}.
$$
\frac{\lambda_2}{\lambda_1}\leq \frac{j_{1,1}^2}{j_{0,1}^2}\approx
2.5387,\qquad\qquad\quad
$$
\end {itemize}
where $j_{0,1}\approx 2.4048$ and $j_{1,1}\approx 3.83171$ are the
first positive zeros of the Bessel functions $J_0(x)$ and $J_1(x)$,
respectively. And $|\Omega|$ is the area of $\Omega$. The equalities
are attained in the above two formulas if and only if the domain
$\Omega$ is a $2D$ disk.

\subsubsection {Other Properties}
\par
\paragraph*{(a) Domain monotonicity property}
\par
As we have shown in the previous lecture, if $\Omega_1\subset
\Omega_2$, then
$$
\qquad\quad\lambda_k(\Omega_1)\geq \lambda_k(\Omega_2),\qquad
k\in\N.
$$
\par
\paragraph*{(b) Scaling Property: (see \cite{COURANT-HILBERT}, \cite{REUTER-WOLTER})}
$$
\qquad\quad\lambda_k(\alpha\,
\Omega)=\frac{\lambda_k(\Omega)}{\alpha^2}\qquad\forall\,\,\,
\alpha>0,\,\,\,k\in \mathbb{N}.
$$
where $\alpha\Omega$ is a scaling by factor $\alpha$ of $\Omega$.
This implies
$$
\qquad\qquad\frac{\lambda_k(\alpha \,\Omega)}{\lambda_m(\alpha\,
\Omega)}=\frac{\lambda_k(\Omega)}{\lambda_m(\Omega)}\qquad\forall\,\,\,
k,\,\,m\in \mathbb{N}.
$$
From this, we see that the ratios of Laplacian eigenvalues are {\em
scale invariant}.
\par
\paragraph*{(c) Laplacian eigenvalues are translation and rotation
invariant.}

\subsection{Features used by Khabou, Hermi, and Rhouma}

Let $\Omega$ be a domain represented by a binary image. Using the
Dirichlet-Laplacian eigenvalues for $\Omega$, define three sets of
features as follows.
$$
F_1(\Omega)\stackrel{\Delta}{=}
\left\{\,\,\left(\,\,\frac{\lambda_1}{\lambda_2},\quad
\frac{\lambda_1}{\lambda_3},\quad...,\quad\frac{\lambda_1}{\lambda_n}
\,\,\right)\,\,\right\},\qquad\qquad\qquad\quad
$$
$$
F_2(\Omega)\stackrel{\Delta}{=}
\left\{\,\,\left(\,\,\frac{\lambda_1}{\lambda_2},\quad
\frac{\lambda_2}{\lambda_3},\quad...,\quad
\frac{\lambda_{n-1}}{\lambda_n}
\,\,\right)\,\,\right\},\qquad\qquad\quad\quad
$$
$$
F_3(\Omega)\stackrel{\Delta}{=}
\left\{\,\,\left(\,\,\frac{\lambda_1}{\lambda_2}-\frac{d_1}{d_2},\quad
\frac{\lambda_1}{\lambda_3}-\frac{d_1}{d_3},\quad...,\quad
\frac{\lambda_1}{\lambda_n}-\frac{d_1}{d_n}\,\,\right)\,\,\right\},
$$

where $0<d_1<d_2=d_3<d_4=d_5< \cdots \leq d_n$ are the
Dirichlet-Laplacian (D-L) eigenvalues for the unit disk in $2D$.

First of all, we can get
$$
0<F_j(\Omega)_{\ell}\leq 1,\quad j=1,2,\qquad
|F_3(\Omega)_{\ell}|\leq 1,
$$
where $F_j(\Omega)_{\ell}$ is the $\ell$th component of each vector
in $F_j(\Omega)$.

It is interesting to consider that $F_3$ measures the deviation of
$\Omega$ from the 2D disk (not necessarily unit disk). The optimal
$n$ depends on the problem being addressed. Khabou, Hermi, and
Rhouma have shown that the three sets of features were tolerant of
boundary noise and deformation and have good inter-class
discrimination capabilities. The three sets of features were used
successfully to classify natural and man-made images with a high
degree of accuracy and using a relatively small number of features.
Reuter, Wolter, and Peinecke \cite{REUTER-WOLTER} introduces a
method to extract ``Shape-DNA'', a numerical fingerprint or
signature, of any 2D or 3D manifold (surface or solid) by taking the
eigenvalues (i.e., the spectrum) of its Laplace-Beltrami operator.
Since the spectrum is an isometry invariant, it is independent of
the object's representation including parametrization and spatial
position. Additionally, the eigenvalues can be normalized so that
uniform scaling factor for the geometric objects can be obtained
easily.

\section{Computational Method of Laplacian Eigenvalues }

In this section, we will discuss three methods to compute Laplacian
eigenvalues and eigenvectors..

\subsection{Finite Difference Scheme
\cite{KHABOU-HERMI-RHOUMA}.}

Its main idea is to discretize the equation $-\Delta u=\lambda u$ by
the finite difference approximation
$$
-\frac{1}{h^2}\left[\,\,u_{i+1,j}+u_{i,j+1}+u_{i-1,j}+u_{i,j-1}-4u_{ij}\,\,\right]=\lambda
u_{ij}.
$$
Here the domain $\Omega$ is cut into squares of side $h$, $u_{ij}$
is the value of the eigenfunction corresponding to $\lambda$ at the
lattice point $(ih,jh)$. The convergence rate of this method is
$O(h^2)$. Moreover, it is slow and inaccurate for $\Omega$ of a
complicated shape with a large number of grid points.

\subsection{Finite Element Method \cite{REUTER-WOLTER}.}

Many people use this in practice. It is better for general $\Omega$.
But it is still very cumbersome. Here we just briefly describe the
idea of finite element method.

\ \\
From Equation $-\varDelta u=\lambda u$, it follows that for any
trial function $\varphi$ with $\varphi=0$ on the boundary of
$\Omega$,
$$
-\iint_\Omega \varphi\, \varDelta u\dd{\bx}=\lambda \iint_\Omega
\varphi\, u\dd{\bx}.
$$
By Green's first identity, $ -\iint_\Omega \varphi \varDelta
u\dd{\bx}=\iint_\Omega \nabla \varphi\cdot \nabla u\dd{\bx}$. This
implies that
$$
\iint_\Omega \nabla \varphi\cdot \nabla u\dd{\bx}=\lambda
\iint_\Omega \varphi \,u
\dd{\bx}.\\
\eqno{(1)}
$$
We choose $n$ linearly independent functions $F_1,..., F_n$ which
are the linear, quadratic or cubic polynomials on every tetrahedral
element of $\Omega$ and satisfy $F_k=0$ on the boundary of $\Omega$.
Take their linear combination as approximation of the solution $u$,
i.e.,
$$
u\approx\sum\limits_{i=1}^n U_i \,F_i,\quad U_i\in \mathbb{R}.\\
\eqno{(2)}
$$
Again, let $\varphi=F_k$. So
$$
\iint_\Omega \nabla \varphi\cdot \nabla
u\dd{\bx}=\sum\limits_{i=1}^n a_{ki}\,U_i,\qquad k=1,...,n,
$$
where
$$\,\,a_{ki}=\iint_\Omega\,\sum_{j=1}^n \( \pdif{F_k}{x_j} \) \( \pdif{F_i}{x_j} \)
\dd{\bx}.$$

On the other hand,
$$
\iint_\Omega \varphi\, u\dd{\bx}=\sum\limits_{i=1}^n
b_{ki}\,U_i,\quad k=1,...,n,\quad$$ where
$$
b_{ki}=\iint_\Omega F_k\,F_i\dd{\bx}.
$$
From this and (1), one obtains that
$$
\sum\limits_{i=1}^na_{ki}\,U_i=\lambda
\sum\limits_{i=1}^nb_{ki}\,U_i,\quad k=1,...,n.
$$
\par
Define the matrix $A=(a_{ki})$ and $B=(b_{ki})$. Then (1) can be
written as
$$
A \bU=\lambda B \bU,\quad\text{where}\quad \bU=(U_1,...,U_n)^T.
$$
So $\lambda$ and $\bU$ can be calculated. Furthermore, the
eigenfunction $u$ is obtained by (2).

\subsection{Method of Particular Solutions (MPS)
\cite{BETCKE-TREFETHEN}, \cite{FOX-HENRICI-MOLER}.}

\subsubsection{History of MPS}

In 1967, Fox, Henrici, and Moler (FHM) \cite{FOX-HENRICI-MOLER}
described the MPS. The famous example was the Dirichlet-Laplacian
eigenfunctions of a $L-$shaped region, which turns be the logo of
MATLAB. Note that the actual MATLAB logo fails to satisfy the
Dirichlet boundary condition, which was intentionally done for
visual purpose.

\ \\
Since the MPS runs into difficulties when dealing with more
complicated regions, after the early 1970s, the MPS got less
attention. Later on, Descloux and Tolley presented other method
based on local expansions near each vertex \cite{DESCLOUX-TOLLEY}.
Betcke and Trefethen proposed a modification of MPS
\cite{BETCKE-TREFETHEN}. The crucial changes are to work not only
with boundary points, but also interior points of the domain. The
modified MPS can deal with quite complicated domains.

\begin{figure}[!h]
 \begin{center}
 \psset{unit=1cm}
 \begin{pspicture}(0,0)(4,4)
\psline{<->}(0.5,4)(0.5,0.5)(4,0.5) \uput[180](0.5,3.8){$y$}
\uput[270](3.8,0.5){$x$} \psline(0.5,0.5)(3,3)
\uput[270](2,0.5){$\partial\Omega$}
\uput[90](1.8,1.9){$\partial\Omega$} \uput[0](0.8,0.75){\small
$\pi/\alpha$} \psarc[showpoints=false](0.5,0.5){0.3}{0}{45}
%\pscurve[showpoints=false](0.75,0.5)(0.72,0.6187)(0.6768,0.6768)
\uput[0](2,1.4){$\Omega$}
\end{pspicture}
 %\includegraphics[width=3in, trim=0.5in 0 0 0]{lecture14-1.eps} %[trim=l b r t]
 \caption{An unbounded domain $\Omega$ with interior angle of the wedge.}
 \label{fig:angle}
\end{center}
\end{figure}

\subsubsection{The Dirichlet-Laplacian problem in an unbounded wedge}

As shown in Figure~\ref{fig:angle}, if the interior angle of the
wedge is $\frac{\pi }{\alpha}$, then we have the D-L eigenfunctions
$$
\varphi(r, \theta)=J_{\alpha k}(\sqrt{\lambda}\,\,r)\sin \alpha k
\theta,\quad k\in N\setminus \{0\}
$$
for any $\lambda>0$, where $J_{\alpha k}$ is a Bessel function of
the first kind with order $\alpha k$. The spectrum is continuous.
\\\\
In general, the Dirichlet-Laplacian eigenfunctions are in
$C^\infty(\Omega)\bigcup C(\overline{\Omega})$. If there exists a
corner in $\partial \Omega$ with angle $\frac{\pi }{k},\,\,\,k\in
\mathbb{N}$, then we can do odd (even for the Neumann-Laplacian
problem) reflection $k-1$ times followed by one reflection to get a
continuous extension of the eigenfunction $\varphi$ in the whole
neighborhood of the corner. Such a corner is called {\em regular}
corner; otherwise, i.e., $\frac{\pi}{\alpha}$,
$\alpha\in\Rf_{+}\setminus\N$, is called {\em singular}. For a
singular corner, it is not possible to extend the eigenfunction
analytically to a whole neighborhood of that corner.

\subsubsection{The Original Method of Particular Solutions (MPS) }

Its main idea is as follows.

\ \\
(1) Consider various solutions of $-\varDelta u=\lambda u$ in
$\Omega$ for a given value of $\lambda$. They are called {\em
particular solutions}.

\ \\
(2) Try to vary $\lambda$ until we can find a linear combination of
such solutions that satisfies the Dirichlet boundary condition $u=0$
on $\partial\Omega$, at a number of sample points of $\partial
\Omega$.

\ \\
\begin{ex}
A convenient set of particular solutions near a corner of angle
$\frac{\pi }{\alpha}$ are
$$
\left\{\,\,\varphi^{(k)}(r,\,\theta)=J_{\alpha
k}(\sqrt{\lambda}\,\,r)\sin (\alpha k\theta)\,\,\right\}_{k\in\N}.
$$
We call these the {\em Fourier-Bessel} functions. These functions
happen to satisfy the Dirichlet boundary condition along the line
segments.
\end{ex}

\begin{figure}[!h]
 \begin{center}
 \psset{unit=1cm}
 \begin{pspicture}(0,0)(4,4)
\psline{<->}(0.5,4)(0.5,0.5)(4,0.5) \uput[180](0.5,3.8){$y$}
\uput[270](3.8,0.5){$x$} \psline(0.5,0.5)(3,3)
\uput[270](2,0.5){$\partial\Omega$}
\uput[90](1.8,1.9){$\partial\Omega$} \uput[0](0.8,0.75){\small
$\pi/\alpha$} \psarc[showpoints=false](0.5,0.5){0.3}{0}{45}
\psarc[showpoints=false, linestyle=dotted](0.5,0.5){3.5}{0}{45}
%\pscurve[showpoints=false](0.75,0.5)(0.72,0.6187)(0.6768,0.6768)
\uput[0](2,1.4){$\Omega$}
\end{pspicture}
%\includegraphics[width=3in, trim=0.5in 0 0 0]{lecture14-2.eps} %[trim=l b r t]
\caption{A bounded domain $\Omega$ with interior angle of the
wedge.} \label{fig:dot-boundary}
\end{center}
\end{figure}

Consider an approximate eigenfunction
$$
\varphi^*(r,\,\theta)=\sum\limits_{k=1}^Nc_k^{(N)}\varphi^{(k)}(r,\,\theta),
$$
with $c^{(N)}_k$ to be determined so as to satisfy the Dirichlet
boundary condition on the remainder of the boundary (e.g., the
dotted line part in Figure~\ref{fig:dot-boundary}).

\ \\
One approach is to set the collocation points, i.e., sampling
points, on the boundary $(r_i,\,\theta_i),\,\,i=1,...,N$ such that

\[\varphi^*(r_i,\,\,\theta_i)=0,\quad \forall i=1,\ldots,N.\]

This leads to a nonlinear system of equations
$$
A(\lambda){\bc}=\bzero,
$$
where
$$
{\bc}=(c_1^{(N)},\,\,...,\,\,c_N^{(N)})^T\in\Rf^N,
$$
$$
A(\lambda)=(\,a_{jk}(\lambda)\,)\in\Rf^{N\times N},
$$
$$
a_{jk}(\lambda)\stackrel{\Delta}{=} J_{\alpha
k}(\sqrt{\lambda}\,\,r_j)\sin (\alpha k\, \theta_j),\quad
j,k=1,\ldots,N.$$

\ \\
The FHM approach \cite{FOX-HENRICI-MOLER} is to look for a zero of
$\mbox{det}\, A(\lambda)$, and then solve for ${\bc}$ (modulo
constant multiplications).

\ \\
An alternative approach proposed by Moler (1969) is to choose
$M\,\,(M>N)$ samples on the boundary. Then $ A(\lambda)\in
\mathbb{R}^{M\times N}$. Again, look for a zero or near zero of {\em
the smallest singular value} of $A(\lambda)$. This was superior to
the original FHM method.
\\\\
Unfortunately, the FHM method fails to obtain more than four digits
of accuracy, the method breaks down after $N=14$ per remaining edge
of the boundary of the L-shaped region.

\subsubsection{Failure of the MPS}

The aim of the MPS is to find a value $\lambda$ such that there
exists a nontrivial linear combination of the Fourier-Bessel
functions that is 0 at the collocation points on $\partial \Omega$.
If $\lambda$ is not close to an actual eigenvalue, then we expect
$$
A(\lambda){\bc}=\bzero
$$
to have no nontrivial solution. The problem is that Fourier-Bessel
functions behave similarly to a power basis $z^k$ and the condition
number of $A(\lambda)$ grows exponentially as $N$ increases. This
also happens for any value of $\lambda$, whether or not it is an
eigenvalue. Consequently, when $N$ is large, it is always possible
to find a linear combination of columns of $A(\lambda)$ that are
close to zero. If $\lambda$ is not close to an eigenvalue, this
results in approximating the zero function which is not a D-L
eigenfunction. Since the MPS method examines only boundary points,
it cannot distinguish zero functions and the D-L eigenfunction.

\subsubsection{A modified method}

\begin{figure}[!h]
 \begin{center}
 \includegraphics[width=3in, trim=0.5in 0 0 0]{lecture14-3.eps} %[trim=l b r t]
 \caption{In the modified method, the Fourier-Bessel functions are sampled at interior as well as boundary points. The interior points are chosen randomly.}
 \label{fig:interior-boundary}
\end{center}
\end{figure}

The idea of Betcke and Trefethen is to impose constraints using some
interior points from $\Omega$ (see
Figure~\ref{fig:interior-boundary}). Let

\begin{equation}\nonumber
\begin{array}{lcl}
m_B & \define & \#\ \text{of boundary collocation
points},\\
m_I & \define & \#\ \text{of interior collocation
points},\\
m & \define & m_B+m_I.
\end{array}
\end{equation}

Let $\bvphi\in \mathbb{R}^m$ be a vector containing the values of
eigenfunction $\varphi(r,\,\,\theta)$ at these $m$ points. We order
the entries of $\bvphi$ such that the first $m_B$ entries correspond
to the boundary collocation points and the remaining $m_I$ entries
correspond to the interior collocation points. Now the matrix
$A(\lambda)$ has more rows:
$$
\displaystyle\displaystyle A(\lambda)=\left[
\begin {array}{ccc}
A_B(\lambda)\\\\
A_I(\lambda)
\end {array}
\right]\in \mathbb{R}^{m\times N}
$$
with $B$ and $I$ corresponding to boundary and interior.
Let $\mathcal{A}(\lambda)$ be the space of trial functions sampled
at boundary and interior points, i.e.,
$$
\mathcal{A}(\lambda)\stackrel{\Delta}{=}\mbox{range}\,\,(A(\lambda))\subset
\mathbb{R}^m.
$$
Construct an orthonormal basis of $\mathcal{A}(\lambda)$ via QR
factorization of $A(\lambda)$. Set
$$
\displaystyle\displaystyle Q(\lambda)=\left[
\begin {array}{ccc}
Q_B(\lambda)\\\\
Q_I(\lambda)
\end {array}
\right]\in \mathbb{R}^{m\times N}.
$$
Now, for $\bvphi\in \mathcal{A}(\lambda),\,\,\parallel
\bvphi\parallel=1$, there exists a vector $\bpsi\in \mathbb{R}^N$
such that
$$
\bvphi=Q(\lambda)\bpsi,
$$
$$
\parallel \bpsi\parallel=1.
$$
Because $Q(\lambda)$ is orthonormal,
$$
Q^T(\lambda)\,Q(\lambda)=I_N,\quad \text{if }m>N.
$$
Consider the minimization problem

\begin{equation}\label{minf}
\min\limits_{\bpsi\in \mathbb{R}^N,\ \parallel
\bpsi\parallel=1}\parallel Q_B(\lambda )\bpsi\parallel,
\end{equation}

which is exactly to minimize the boundary part of $\bvphi$. The
minimizer of \eqref{minf} is the right singular vector of
$Q(\lambda)$ corresponding to the smallest singular value of
$Q_B(\lambda)$. Let this singular value be denoted by
$\sigma_B(\lambda)$ and the minimizer by $\widetilde{\bpsi}$. Then
$$
\sigma_B(\lambda)=\min\limits_{\bpsi\in \mathbb{R}^N,\ \parallel
\bpsi\parallel=1}
\parallel Q_B(\lambda)\bpsi\parallel=\parallel Q_B(\lambda)\widetilde{\bpsi}\parallel.
$$
Let $\widetilde{\bvphi}=Q(\lambda)\widetilde{\bpsi}$. Then
$$
1=\parallel\widetilde{\bvphi}\parallel^2=\bigg\Vert
\left[\,\frac{Q_B(\lambda
)}{Q_I(\lambda)}\,\right]\widetilde{\bpsi}\bigg\Vert^2=\sigma_B^2(\lambda)+\parallel
Q_I(\lambda)\widetilde{\bpsi}\parallel^2.
$$
Here $\sigma_B(\lambda)\approx 0$ and $\parallel
Q_I(\lambda)\widetilde{\bpsi}\parallel\approx 1.$

\ \\
So, at the interior points, $\widetilde{\bvphi}$ are not zeros
automatically. Using this method, Betcke and Trefether could compute
the D-L eigenvalues with 14 digits accuracy.

\begin{thebibliography}{10}

\bibitem{COURANT-HILBERT}
{\sc R. Courant, D. Hilbert}, {\em Methods of Mathematical Physics},
Vol. I, Wiley-Interscience, 1953.

\bibitem{KHABOU-HERMI-RHOUMA}
{\sc M. A. Khabou, L. Hermi, M. B. H. Rhouma}:
Shape Recognition Using Eigenvalues of the Dirichlet Laplacian,
{\em Pattern Recognition}, vol.~40, pp.~141--153, 2007.

\bibitem{REUTER-WOLTER}
{\sc M. Reuter, F. E. Wolter, N. Peinecke}: Laplace-Beltrami Spectra
as Shape-DNA of Surfaces and Solids, {\em Computer-Aided Design},
vol.~38, pp.~342--366, 2006.

\bibitem{BETCKE-TREFETHEN}
{\sc T. Betcke, L. N. Trefethen}: Reviving the Method of Particular
Solutions, {\em SIAM Review}, vol.~47, no.~3, pp.~469--491, 2005.

\bibitem{FOX-HENRICI-MOLER}
{\sc L. Fox, P. Henrici, C. Moler}: Approximations and
Bounds for Eigenvalues of Elliptic Operators, {\em SIAM J. Numer. Anal.}
vol.~4, no.~1, pp.~89--102,1967.

\bibitem{DESCLOUX-TOLLEY}
{\sc J. Descloux, M. Tolley}: An accurate algorithm for computing
the eigenvalues of a polygonal membrane, {\em Comput. Methods Appl.
Mech. Engrg.}, vol.~39, pp.~37--53, 1983.

\bibitem{PAYNE-POLYA-WEINBERGER}
{\sc L. E. Payne, G. P\'olya, H. F. Weinberger}: On the ratio
of consecutive eigenvalues, {\em J. Math. Phys.}, vol.~35,
pp.~289--298, 1956.

\bibitem{HILE-PROTTER}
{\sc G. N. Hile, M. H. Protter}: Inequalities for eigenvalues of the
Laplacian, {\em Indiana Univ. Math. J.}, vol.~29, pp.~523--538, 1980.

\bibitem{YANG}
{\sc H. C. Yang}: Estimate of the difference between consecutive
eigenvalues, 1995 preprint (revision of International Centre for
Theoretical Physics preprint IC/91/60, Trieste, Italy, April 1991).

\bibitem{CHAVEL}
{\sc I. Chavel}: Eigenvalues in Riemannian Geometry, {\em Academic
Press}, New York, 1984.

\bibitem{ASHBAUGH}
{\sc M. S. Ashbaugh}: The universal eigenvalue bounds of
Payne-P\'olya-Weinberger, Hile-Protter, and H.C. Yang, in:
Spectral and inverse spectral theory, {\em Proc. Indian Acad. Sci.
Math. Sci.}, vol.~112, pp.~3--30, 2002.

\end{thebibliography}


\end{document}
