Bipartite density of cubic graphs

We first obtain the exact value for bipartite density of a cubic line graph on n vertices. Then we give an upper bound for the bipartite density of cubic graphs in terms of the smallest eigenvalue of the adjacency matrix. In addition, we characterize, except in the case n=20, those graphs for which the upper bound is obtained.     

An Inequality Between Dirichlet and Neumann Eigenvalues of the Heisenberg Laplacian

Let λ _k and μ _k be the eigenvalues of the Dirichlet and Neumann problems, respectively, in a domain of finite measure in ? ^d , d > 1. Filonov has proved in a simple way that the inequality μ _k+1 < λ _k holds for the Laplacian. We extend his result to the Heisenberg Laplacian in three-dimensional domains which fulfill certain geometric conditions.     

Cwikel and Quasi-Szegö Type Estimates for Random Operators

We consider Schrödinger operators with nonergodic random potentials. Specifically, we are interested in eigenvalue estimates and estimates of the entropy for the absolutely continuous part of the spectral measure. We prove that increasing oscillations in the potential at infinity have the same effect on the properties of the spectrum as the decay of the potential.     

一类边界条件带有谱参数且具有转移条件的四阶微分算子

本文研究一类在内部不连续点具有转移条件且特征参数不仅出现在微分方程中而且出现在四个边界条件中的四阶正则微分算子.通过构造与问题相关的新线性算子A,证明在适当的希尔伯特空间H中算子A是自共轭的当且仅当条件CTQ1C=ρQ1,θzATQ2A=θ1BTQ2B=θ1θ2Q1,θ2AQ1AT=θ1BQ1BT=θ1θ2Q2成立.利用微分方程的基本解证明λ是问题的特征值当且仅当det(Aλ+BλΦ(1,λ))=0.最后得到算子A仅有点谱.     

Stabilizing Semilinear Parabolic Equations

In this paper we prove the internal feedback stabilization of steady-state solutions of semilinear parabolic equations and introduce a controller synthesis methodology based on finite element approximations of the original PDEs. Numerical tests are given for some one- and two-dimensional nonlinear parabolic equations.     

An always convergent algorithm for the largest eigenvalue of an irreducible nonnegative tensor

In this paper we propose an iterative method to calculate the largest eigenvalue of a nonnegative tensor. We prove this method converges for any irreducible nonnegative tensor. We also apply this method to study the positive definiteness of a multivariate form.     

SUPER-GEOMETRIC CONVERGENCE OF A SPECTRAL ELEMENT METHOD FOR EIGENVALUE PROBLEMS WITH JUMP COEFFICIENTS

We propose and analyze a C^0 spectral element method for a model eigenvalue problem with discontinuous coefficients in the one dimensional setting. A super-geometric rate of convergence is proved for the piecewise constant coefficients case and verified by numerical tests. Furthermore, the asymptotical equivalence between a Gauss-Lobatto collocation method and a spectral Galerkin method is established for a simplified model.