The Equality Constrained Indefinite Least Squares Problem: Theory and Algorithms

We present theory and algorithms for the equality constrained indefinite least squares problem, which requires minimization of an indefinite quadratic form subject to a linear equality constraint. A generalized hyperbolic QR factorization is introduced and used in the derivation of perturbation bounds and to construct a numerical method. An alternative method is obtained by employing a generalized QR factorization in combination with a Cholesky factorization. Rounding error analysis is given to show that both methods have satisfactory numerical stability properties and numerical experiments are given for illustration. This work builds on recent work on the unconstrained indefinite least squares problem by Chandrasekaran, Gu, and Sayed and by the present authors.     

The Cauchy Problem for the System of Maxwell Equations

We consider the problem of analytic continuation of a solution to the system of Maxwell equations in a bounded spatial domain from data on part of the boundary of the domain. We construct an approximate solution to the problem using the Carleman matrix method.     

Stability Conditions of the MMAP [ K ]/ G [ K ]/1/ LCFS Preemptive Repeat Queue

In this paper, we study the stability conditions of the MMAP[K]/G[K]/1/LCFS preemptive repeat queue. We introduce an embedded Markov chain of matrix M/G/1 type with a tree structure and identify conditions for the Markov chain to be ergodic. First, we present three conventional methods for the stability problem of the queueing system of interest. These methods are either computationally demanding or do not provide accurate information for system stability. Then we introduce a novel approach that develops two linear programs whose solutions provide sufficient conditions for stability or instability of the queueing system. The new approach is numerically efficient. The advantages and disadvantages of the methods introduced in this paper are analyzed both theoretically and numerically.     

THE INVERSE PROBLEM FOR PART SYMMETRIC MATRICES ON A SUBSPACE

In this paper, the following two problems are considered:Problem Ⅰ. Given S∈E Rn×p,X,B 6 Rn×m, find A ∈ SRs,n such that AX = B, where SR8,n = {A∈ Rn×n|xT(A - AT) = 0, for all x ∈ R(S)}.Problem Ⅱ. Given A* ∈ Rn×n, find A ∈ SE such that ||A-A*|| = minA∈sE||A-A*||, where SE is the solution set of Problem Ⅰ.The necessary and sufficient conditions for the solvability of and the general form of the solutions of problem Ⅰ are given. For problem Ⅱ, the expression for the solution, a numerical algorithm and a numerical example are provided.     

Fracture Model for a Thin Layer of a Fibrous Composite

A model for a flat isolated layer of a unidirectional fibrous composite with a regular structure is constructed to investigate the possible variants of its failure development. An integrodifferential equation for determining the forces in fibers is obtained. Primary attention is focused on examining the failure process after the rupture of one fiber. This causes a drastic redistribution of stresses, which can lead to a failure of adjacent fibers owing to the increased load on them, to an interfacial shear fracture, and to the matrix cracking. It is shown that the development of layer failure is determined by the strength of fibers, the crack resistance of the matrix in axial tension and transverse shear, and also by the adhesion strength of the matrix-fiber interface. The sufficient conditions of applicability of the brittle fracture model are formulated.     

Asymptotic Spectra of Dense Toeplitz Matrices Are Unstable

The paper is concerned with the limiting set of the eigenvalues of the truncations of an infinite Toeplitz matrix whose symbol is continuous but not rational. This limiting set is shown to be unstable with respect to small perturbations of the symbol in the uniform norm, which reveals that the numerical computation of the asymptotic spectra of dense Toeplitz matrices is a genuine mathematical challenge.     

The Effect of an Active Diluent on the Properties of Epoxy Resin and Unidirectional Carbon-Fiber-Reinforced Plastics

The influence of an active diluent on the properties of an epoxy matrix and carbon-fiber-reinforced plastics (CFRP) is investigated. The physicomechanical properties of an ED-20 epoxy resin modified with diglycidyl ether of diethylene glycol (DEG-1), the adhesion strength at the epoxy matrix–steel wire interface, and the mechanical properties of unidirectional CFRP are determined. The concentration of DEG-1 was varied from 0 to 50 wt.%. The properties of the matrix, the interface, and the composites are compared. It is stated that the matrix strength affects the strength of unidirectional CFRP in bending and not their strength in tension, compression, and shear. The latter fact seems somewhat unexpected. The interlaminar fracture toughness of the composites investigated correlates with the ultimate elongation of the binder. A comparison between the concentration dependences of adhesion strength and the strength of CFRP shows that the matrices utilized provide such a high interfacial strength that the strength of CFRP no longer depends on the adhesion of its constituents.     

Calculation of the Probabilities of Molecular Transitions in Modeling Interisomeric Structural Transformations

The problems of a computational procedure in modeling structural isomer–isomer transitions in multiatomic molecules are considered. A number of approximations that allow one to considerably simplify calculation of the probabilities of these transformations are suggested. In comparison with traditional calculations of potential surfaces and determination of their saddle points, in the proposed approach the relationship between the electronic and vibrational components of a general problem is virtually changed. A particular emphasis is put on the vibrational part, which seems to be more physical. It is shown that the problem of structural isomer–isomer conversions can be solved without resorting at all to explicit calculation of the contribution of an electronic matrix element to the probability of transition. There is no need to study in detail the shapes of potential surfaces; it suffices only to know the positions of their minima. The calculation of multidimensional vibrational integrals is replaced by calculation of one-dimensional integrals. The results obtained form a basis for constructing methods for modeling the dynamic spectra of complex molecules considering their structural transformations.     

Hermitian quark matrices

Assuming a relation between the quark mass matrices of the two sectors a unique solution can be obtained for the CKM flavor mixing matrix. A numerical example is worked out which is in excellent agreement with experimental data.