Keller's Theorem Research Articles

Anisotropic diffusion in media with absorbing traps (application to the Keller–Dykhne theorem)

Anisotropic diffusion in multidimensional media with absorbing traps is investigated. It is shown that the diffusion asymptotic forms of the survival probability over long time intervals are determined by a new effective diffusion coefficient. Exact expressions derived for the effective diffusion coefficient generalize the well-known Dykhne–Keller 2D results to the 3D case. In other words, the dynamic diffusion approach is used for obtaining a generalization of the Dykhne–Keller theorem to the 3D case.

On the convergence of general stationary iterative methods for range‐Hermitian singular linear systems

AbstractGeneral stationary iterative methods with a singular matrix M for solving range‐Hermitian singular linear systems are presented, some convergence conditions and the representation of the solution are also given. It can be verified that the general Ortega–Plemmons theorem and Keller theorem for the singular matrix M still hold. Furthermore, the singular matrix M can act as a good preconditioner for solving range‐Hermitian linear systems. Numerical results have demonstrated the effectiveness of the general stationary iterations and the singular preconditioner M. Copyright © 2009 John Wiley & Sons, Ltd.

On Keller theorem for anisotropic media

The Keller theorem in the problem of effective conductivity in anisotropic two-dimensional (2D) many-component composites makes it possible to establish a simple inequality σise(σi−1)⋅σise(σk)>1 for the isotropic part σise(σk) of the second rank symmetric tensor σ̂i,je of effective conductivity.

Projected iterative algorithms with application to multicomponent transport

We investigate projected iterative algorithms for solving constrained symmetric singular linear systems. We discuss the symmetry of generalized inverses and investigate projected standard iterative methods as well as projected conjugate-gradient algorithms. Using a generalization of Stein's theorem for singular matrices, we obtain a new proof of Keller's theorem. We also strengthen a result from Neumann and Plemmons about the spectrum of iteration matrices. As an application, we consider the linear systems arising from the kinetic theory of gases and providing transport coefficients in multicomponent gas mixtures. We obtain low-cost accurate approximate expressions for the transport coefficients that can be used in multicomponent flow models. Typical examples for the species diffusion coefficients and the volume viscosity are presented.

TOPOLOGICAL PROOFS OF KELLER'S THEOREM AND AN EQUIVARIANT VERSION OF IT

The known proofs of Keller's theorem that any infinite-dimensional compact convex set in Hilbert space is homeomorphic to the Hilbert cube are analytic. Here a topological proof of this theorem is given. A new approach to the old theorem leads to a proof of an equivariant version of Keller's theorem.

A THEORETICAL APPROACH TO THERMAL PROPERTY OF ARRAY OF CYLINDERS EMBEDDED IN HOMOGENEOUS MATRIX

We investigate in this article the thermal conductivity of array of cylinders embedded in a homogeneous matrix. Using Green's function, we confirm that the method invented by Rayleigh can be generalized to deal with thermal property of these systems. A technique for calculating effective thermal conductivities of these systems is proposed. As an example, we consider a system with square symmetry, and a neat formula for effective thermal conductivity is derived. We show that the method also includes the proof of Keller theorem.