Polarization tensors of planar domains as functions of the admittivity contrast

(Electric) polarization tensors describe part of the leading order term of asymptotic voltage perturbations caused by low volume fraction inhomogeneities of the electrical properties of a medium. They depend on the geometry of the support of the inhomogeneities and on their admittivity contrast. Corresponding asymptotic formulas are of particular interest in the design of reconstruction algorithms for determining the locations and the material properties of inhomogeneities inside a body from measurements of current flows and associated voltage potentials on the body’s surface. In this work, we consider the two-dimensional case only and provide an analytic representation of the polarization tensor in terms of spectral properties of the double layer integral operator associated with the support of simply connected conductivity inhomogeneities. Furthermore, we establish that an (infinitesimal) simply connected inhomogeneity has the shape of an ellipse, if and only if the polarization tensor is a rational function of the admittivity contrast with at most two poles whose residues satisfy a certain algebraic constraint. We also use the analytic representation to provide a proof of the so-called Hashin–Shtrikman bounds for polarization tensors; a similar approach has been taken previously by Golden and Papanicolaou and Kohn and Milton in the context of anisotropic composite materials.     

One‐dimensional q‐Dirac equation

In this paper, we introduce a q‐analog of 1‐dimensional Dirac equation. We investigate the existence and uniqueness of the solution of this equation. Later, we discuss some spectral properties of the problem, such as formally self‐adjointness, the case that the eigenvalues are real, orthogonality of eigenfunctions, Green function, existence of a countable sequence of eigenvalues, and eigenfunctions forming an orthonormal basis of . Finally, we give some examples.     

双调和算子特征值问题的混合三角谱元方法

本文针对双调和算子特征值问题设计了基于混合变分形式的三角谱元逼近格式,其基函数采用指标为(-1,-1,-1)的广义Koornwinder多项式.在H~1-及H_0~1-正交谱元投影的逼近理论基础上,我们建立了双调和算子特征值与特征函数的收敛性估计;它关于网格尺寸h是最优的,关于多项式次数M是次优的.然而,在H_0~2-正交谱元投影的最优估计假设前提下,关于M的次优收敛阶估计则提升为最优.此外,Koornwinder分片多项式逼近的结果还表明,在带权Besov空间范数的度量下,对于存在着区域角点奇性的双调和算子特征值问题,谱元方法的收敛阶能达到h-型有限元方法的2倍.最后,本文的数值实验结果展示了谱元逼近格式的高效性,同时也验证了相关理论的正确性.     

Normal Forms for High Co-Dimension Bifurcations of Nonlinear Time-Periodic Systems with Nonsemisimple Eigenvalues

The normal forms for time-periodic nonlinear variational equations witharbitrary linear Jordan form undergoing bifurcations of highco-dimension are found. First, the equations are transformed via theLyapunov–Floquet (L–F) transformation into an equivalent form in whichthe linear matrix is constant with degenerate nonsemisimple lineareigenvalues while the nonlinear monomials have periodic coefficients. Byconsidering the resulting coupling of the bases of the near identitytransformation, the solvability condition for an arbitrary Jordan matrixis then derived. It is shown that time-independent and/or time-dependentnonlinear resonance terms remain in the normal form for various Jordanmatrices. Specifically, the normal forms for quadratic and cubicnonlinearities with the following linear Jordan forms are explicitlyderived: double zero eigenvalues (co-dimension two bifurcation), triplezero eigenvalues (co-dimension three bifurcation), and two repeatedpairs of purely imaginary eigenvalues (co-dimension two bifurcation). Acommutative system with cubic nonlinearities and a double inverted pendulum with a periodicfollower force are used as illustrative examples.     

Perturbations of embedded eigenvalues for the planar bilaplacian

Operators on unbounded domains may acquire eigenvalues that are embedded in the essential spectrum. Determining the fate of these embedded eigenvalues under small perturbations of the underlying operator is a challenging task, and the persistence properties of such eigenvalues are linked intimately to the multiplicity of the essential spectrum. In this paper, we consider the planar bilaplacian with potential and show that the set of potentials for which an embedded eigenvalue persists is locally an infinite-dimensional manifold with infinite codimension in an appropriate space of potentials.     

The bialternate matrix product revisited

The well known bialternate product of two square matrices is re-examined together with another matrix product defined by means of the permanent function and having similar properties. Old and new results concerning both products are presented in a unified manner. A simple and elegant relation with the Kronecker product of matrices is also given.     

Estimates for eigenvalues of stochastic matrices

It is well-known that the eigenvalues of stochastic matrices lie in the unit circle and at least one of them has the value one. Let {1, r 2 , ··· , r N } be the eigenvalues of stochastic matrix X of size N × N . We will present in this paper a simple necessary and sufficient condition for X such that |r j | < 1, j = 2, ··· , N . Moreover, such condition can be very quickly examined by using some search algorithms from graph theory.     

Asymptotic behavior of a differential operator with discontinuities at two points

In this paper, we study a Sturm–Liouville operator with eigenparameter‐dependent boundary conditions and transmission conditions at two interior points. By establishing a new operator A associated with the problem, we prove that the operator A is self‐adjoint in an appropriate space H, discuss completeness of its eigenfunctions in H, and obtain its Green function. Copyright © 2010 John Wiley & Sons, Ltd.     

The Rayleigh-Ritz method, refinement and Arnoldi process for periodic matrix pairs

We extend the Rayleigh-Ritz method to the eigen-problem of periodic matrix pairs. Assuming that the deviations of the desired periodic eigenvectors from the corresponding periodic subspaces tend to zero, we show that there exist periodic Ritz values that converge to the desired periodic eigenvalues unconditionally, yet the periodic Ritz vectors may fail to converge. To overcome this potential problem, we minimize residuals formed with periodic Ritz values to produce the refined periodic Ritz vectors, which converge under the same assumption. These results generalize the corresponding well-known ones for Rayleigh-Ritz approximations and their refinement for non-periodic eigen-problems. In addition, we consider a periodic Arnoldi process which is particularly efficient when coupled with the Rayleigh-Ritz method with refinement. The numerical results illustrate that the refinement procedure produces excellent approximations to the original periodic eigenvectors.     

一个Krasnoselski定理的推广

主要讨论了非线性方程F(λ,u)=λu-G(u)=θ的分歧问题,其中G:X→X为非线性可微映射,X为Banach空间.在G′(θ)为紧算子,N(λ~*I-G′(θ))\R(λ~*I-G′(θ))≠{θ}的条件下,利用Lyapunov-Schmidt约化过程和隐函数定理证得了方程F(λ,u)=θ在多重特征值处的分歧定理,推广了Krasnoselski的经典分歧定理.