1. CMB 2016 (vol 59 pp. 606)
 Mihăilescu, Mihai; Moroşanu, Gheorghe

Eigenvalues of $ \Delta_p \Delta_q $ Under Neumann Boundary Condition
The
eigenvalue problem $\Delta_p u\Delta_q u=\lambdau^{q2}u$
with $p\in(1,\infty)$, $q\in(2,\infty)$, $p\neq q$ subject to
the
corresponding homogeneous Neumann boundary condition is
investigated on a bounded open set with smooth boundary from
$\mathbb{R}^N$ with $N\geq 2$. A careful analysis of this problem leads
us to a complete description of the set of eigenvalues as being
a
precise interval $(\lambda_1, +\infty )$ plus an isolated point
$\lambda =0$. This comprehensive result is strongly related to
our
framework which is complementary to the wellknown case $p=q\neq
2$ for which a full description of the set of eigenvalues is
still
unavailable.
Keywords:eigenvalue problem, Sobolev space, Nehari manifold, variational methods Categories:35J60, 35J92, 46E30, 49R05 

2. CMB 2011 (vol 54 pp. 506)
 Neamaty, A.; Mosazadeh, S.

On the Canonical Solution of the SturmLiouville Problem with Singularity and Turning Point of Even Order
In this paper, we are going to investigate the canonical property of solutions of
systems of differential equations having a singularity and turning
point of even order. First, by a replacement, we transform the system
to the SturmLiouville equation with turning point. Using of the
asymptotic estimates provided by Eberhard, Freiling, and Schneider
for a special fundamental system of solutions of the SturmLiouville
equation, we study the infinite product representation of solutions of the systems. Then we
transform the SturmLiouville equation with
turning point to the
equation with singularity, then we study the asymptotic behavior of its solutions. Such
representations are relevant to the inverse spectral problem.
Keywords:turning point, singularity, SturmLiouville, infinite products, Hadamard's theorem, eigenvalues Categories:34B05, 34Lxx, 47E05 

3. CMB 2011 (vol 55 pp. 88)
 Ghanbari, K.; Shekarbeigi, B.

Inequalities for Eigenvalues of a General Clamped Plate Problem
Let $D$ be a
connected bounded domain in $\mathbb{R}^n$. Let
$0<\mu_1\leq\mu_2\leq\dots\leq\mu_k\leq\cdots$ be the eigenvalues
of the following Dirichlet
problem:
$$
\begin{cases}\Delta^2u(x)+V(x)u(x)=\mu\rho(x)u(x),\quad x\in
D
u_{\partial D}=\frac{\partial u}{\partial n}_{\partial
D}=0,
\end{cases}
$$
where $V(x)$ is a nonnegative potential,
and $\rho(x)\in C(\bar{D})$ is positive.
We prove the following inequalities:
$$\mu_{k+1}\leq\frac{1}{k}\sum_{i=1}^k\mu_i+\Bigl[\frac{8(n+2)}{n^2}\Bigl(\frac{\rho_{\max}}
{\rho_{\min}}\Bigr)^2\Bigr]^{1/2}\times
\frac{1}{k}\sum_{i=1}^k[\mu_i(\mu_{k+1}\mu_i)]^{1/2},
$$
$$\frac{n^2k^2}{8(n+2)}\leq
\Bigl(\frac{\rho_{\max}}{\rho_{\min}}\Bigr)^2\Bigl[\sum_{i=1}^k\frac{\mu_i^{1/2}}{\mu_{k+1}\mu_i}\Bigr]
\times\sum_{i=1}^k\mu_i^{1/2}.
$$
Keywords:biharmonic operator, eigenvalue, eigenvector, inequality Category:35P15 

4. CMB 2009 (vol 52 pp. 9)
 Chassé, Dominique; SaintAubin, Yvan

On the Spectrum of an $n!\times n!$ Matrix Originating from Statistical Mechanics
Let $R_n(\alpha)$ be the $n!\times n!$ matrix whose matrix elements
$[R_n(\alpha)]_{\sigma\rho}$, with $\sigma$ and $\rho$ in the
symmetric group $\sn$, are $\alpha^{\ell(\sigma\rho^{1})}$ with
$0<\alpha<1$, where $\ell(\pi)$ denotes the number of cycles in $\pi\in
\sn$. We give the spectrum of $R_n$ and show that the ratio of the
largest eigenvalue $\lambda_0$ to the second largest one (in absolute
value) increases as a positive power of $n$ as $n\rightarrow \infty$.
Keywords:symmetric group, representation theory, eigenvalue, statistical physics Categories:20B30, 20C30, 15A18, 82B20, 82B28 

5. CMB 2007 (vol 50 pp. 356)
 Filippakis, Michael E.; Papageorgiou, Nikolaos S.

Existence of Positive Solutions for Nonlinear Noncoercive Hemivariational Inequalities
In this paper we investigate the existence of positive solutions
for nonlinear elliptic problems driven by the $p$Laplacian with a
nonsmooth potential (hemivariational inequality). Under asymptotic
conditions that make the Euler functional indefinite and
incorporate in our framework the asymptotically linear problems,
using a variational approach based on nonsmooth critical point
theory, we obtain positive smooth solutions. Our analysis also
leads naturally to multiplicity results.
Keywords:$p$Laplacian, locally Lipschitz potential, nonsmooth critical point theory, principal eigenvalue, positive solutions, nonsmooth Mountain Pass Theorem Categories:35J20, 35J60, 35J85 

6. CMB 2006 (vol 49 pp. 560)
 Luijk, Ronald van

A K3 Surface Associated With Certain Integral Matrices Having Integral Eigenvalues
In this article we will show that there are infinitely many
symmetric, integral $3 \times 3$ matrices, with zeros on the
diagonal, whose eigenvalues are all integral. We will do this by
proving that the rational points on a certain nonKummer, singular
K3 surface
are dense. We will also compute the entire NÃ©ronSeveri group of
this surface and find all low degree curves on it.
Keywords:symmetric matrices, eigenvalues, elliptic surfaces, K3 surfaces, NÃ©ronSeveri group, rational curves, Diophantine equations, arithmetic geometry, algebraic geometry, number theory Categories:14G05, 14J28, 11D41 

7. CMB 2006 (vol 49 pp. 358)
 Khalil, Abdelouahed El; Manouni, Said El; Ouanan, Mohammed

On the Principal Eigencurve of the $p$Laplacian: Stability Phenomena
We show that each point of the principal eigencurve of the
nonlinear problem
$$
\Delta_{p}u\lambda m(x)u^{p2}u=\muu^{p2}u \quad
\text{in } \Omega,
$$
is stable (continuous) with respect to the exponent $p$ varying in
$(1,\infty)$; we also prove some convergence results
of the principal eigenfunctions corresponding.
Keywords:$p$Laplacian with indefinite weight, principal eigencurve, principal eigenvalue, principal eigenfunction, stability Categories:35P30, 35P60, 35J70 

8. CMB 1999 (vol 42 pp. 169)
 Ding, Hongming

Heat Kernels of Lorentz Cones
We obtain an explicit formula for heat kernels of Lorentz cones, a
family of classical symmetric cones. By this formula, the heat
kernel of a Lorentz cone is expressed by a function of time $t$ and
two eigenvalues of an element in the cone. We obtain also upper and
lower bounds for the heat kernels of Lorentz cones.
Keywords:Lorentz cone, symmetric cone, Jordan algebra, heat kernel, heat equation, LaplaceBeltrami operator, eigenvalues Categories:35K05, 43A85, 35K15, 80A20 
