Positive Semidefiniteness and Positive Definiteness of a Linear Parametric Interval Matrix

We consider a symmetric matrix, the entries of which depend linearly on some parameters. The domains of the parameters are compact real intervals. We investigate the problem of checking whether for each (or some) setting of the parameters, the matrix is positive definite (or positive semidefinite). We state a characterization in the form of equivalent conditions, and also propose some computationally cheap sufficient\,/\,necessary conditions. Our results extend the classical results on positive (semi-)definiteness of interval matrices. They may be useful for checking convexity or non-convexity in global optimization methods based on branch and bound framework and using interval techniques

Characterizing and approximating eigenvalue sets of symmetric interval matrices

We consider the eigenvalue problem for the case where the input matrix is symmetric and its entries perturb in some given intervals. We present a characterization of some of the exact boundary points, which allows us to introduce an inner approximation algorithm, that in many case estimates exact bounds. To our knowledge, this is the first algorithm that is able to guaran- tee exactness. We illustrate our approach by several examples and numerical experiments

A filtering method for the interval eigenvalue problem

We consider the general problem of computing intervals that contain the real eigenvalues of interval matrices. Given an outer estimation of the real eigenvalue set of an interval matrix, we propose a filtering method that improves the estimation. Even though our method is based on an sufficient regularity condition, it is very efficient in practice, and our experimental results suggest that, in general, improves significantly the input estimation. The proposed method works for general, as well as for symmetric matrices

Efficient Approaches for Enclosing the United Solution Set of the Interval Generalized Sylvester Matrix Equation

In this work, we investigate the interval generalized Sylvester matrix equation ${\bf{A}}X{\bf{B}}+{\bf{C}}X{\bf{D}}={\bf{F}}$ and develop some techniques for obtaining outer estimations for the so-called united solution set of this interval system. First, we propose a modified variant of the Krawczyk operator which causes reducing computational complexity to cubic, compared to Kronecker product form. We then propose an iterative technique for enclosing the solution set. These approaches are based on spectral decompositions of the midpoints of ${\bf{A}}$, ${\bf{B}}$, ${\bf{C}}$ and ${\bf{D}}$ and in both of them we suppose that the midpoints of ${\bf{A}}$ and ${\bf{C}}$ are simultaneously diagonalizable as well as for the midpoints of the matrices ${\bf{B}}$ and ${\bf{D}}$. Some numerical experiments are given to illustrate the performance of the proposed methods

Localization Analysis of an Energy-Based Fourth-Order Gradient Plasticity Model

The purpose of this paper is to provide analytical and numerical solutions of the formation and evolution of the localized plastic zone in a uniaxially loaded bar with variable cross-sectional area. An energy-based variational approach is employed and the governing equations with appropriate physical boundary conditions, jump conditions, and regularity conditions at evolving elasto-plastic interface are derived for a fourth-order explicit gradient plasticity model with linear isotropic softening. Four examples that differ by regularity of the yield stress and stress distributions are presented. Results for the load level, size of the plastic zone, distribution of plastic strain and its spatial derivatives, plastic elongation, and energy balance are constructed and compared to another, previously discussed non-variational gradient formulation.Comment: 41 pages, 24 figures; moderate revision after the first round of review, Appendix A re-written completel

Subgap tunneling through channels of polarons and bipolarons in chain conductors

We suggest a theory of internal coherent tunneling in the pseudogap region where the applied voltage is below the free electron gap. We consider quasi 1D systems where the gap is originated by a lattice dimerization like in polyacethylene, as well as low symmetry 1D semiconductors. Results may be applied to several types of conjugated polymers, to semiconducting nanotubes and to quantum wires of semiconductors. The approach may be generalized to tunneling in strongly correlated systems showing the pseudogap effect, like the family of High Tc materials in the undoped limit. We demonstrate the evolution of tunneling current-voltage characteristics from smearing the free electron gap down to threshold for tunneling of polarons and further down to the region of bi-electronic tunneling via bipolarons or kink pairs.Comment: 14 pages, 8 postscript figure