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1. CMB 2011 (vol 54 pp. 506)

 On the Canonical Solution of the Sturm-Liouville 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 Sturm-Liouville equation with turning point. Using of the asymptotic estimates provided by Eberhard, Freiling, and Schneider for a special fundamental system of solutions of the Sturm-Liouville equation, we study the infinite product representation of solutions of the systems. Then we transform the Sturm-Liouville 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, Sturm-Liouville, infinite products, Hadamard's theorem, eigenvaluesCategories:34B05, 34Lxx, 47E05

2. 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, inequalityCategory:35P15

3. CMB 2009 (vol 52 pp. 9)

Chassé, Dominique; Saint-Aubin, 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 physicsCategories:20B30, 20C30, 15A18, 82B20, 82B28

4. 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 TheoremCategories:35J20, 35J60, 35J85

5. 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 non-Kummer, singular K3 surface are dense. We will also compute the entire NÃ©ron-Severi group of this surface and find all low degree curves on it. Keywords:symmetric matrices, eigenvalues, elliptic surfaces, K3 surfaces, NÃ©ron--Severi group, rational curves, Diophantine equations, arithmetic geometry, algebraic geometry, number theoryCategories:14G05, 14J28, 11D41

6. 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 $$-\Delta_{p}u-\lambda m(x)|u|^{p-2}u=\mu|u|^{p-2}u \quad \text{in } \Omega,$$ 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, stabilityCategories:35P30, 35P60, 35J70

7. 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, Laplace-Beltrami operator, eigenvaluesCategories:35K05, 43A85, 35K15, 80A20