Estimates for the Norm of Green's Matrix Based on the Integral Performance Criterion for Dichotomy and Hausdorff Set Bounds

For a system of ordinary differential equations of the form , , we obtain new upper bounds for the second norm of Green's matrix using the integral performance criterion for dichotomy and bounds of the Hausdorff set of the matrix A. These estimates are considerably better for many applications than some well-known ones.     

On the Additive D-Stability of Matrices on the Basis of the Kharitonov Criterion

On the basis of the Kharitonov theorem, sufficient conditions on an matrix A are presented for the matrix to be stable for arbitrary , .     

Solutions of symmetry‐constrained least‐squares problems

In this paper, two new matrix‐form iterative methods are presented to solve the least‐squares problem: and matrix nearness problem: where matrices and are given; ??₁ and ??₂ are the set of constraint matrices, such as symmetric, skew symmetric, bisymmetric and centrosymmetric matrices sets and S_XY is the solution pair set of the minimum residual problem. These new matrix‐form iterative methods have also faster convergence rate and higher accuracy than the matrix‐form iterative methods proposed by Peng and Peng (Numer. Linear Algebra Appl. 2006; 13 : 473–485) for solving the linear matrix equation AXB+CYD=E. Paige's algorithms, which are based on the bidiagonalization procedure of Golub and Kahan, are used as the framework for deriving these new matrix‐form iterative methods. Some numerical examples illustrate the efficiency of the new matrix‐form iterative methods. Copyright © 2008 John Wiley & Sons, Ltd.     

Symmetries of the complex Dirac-Kähler equation

We consider a complex version of a Dirac-Kähler-type equation, the eight-component complex Dirac-Kähler equation with a nonvanishing mass, which can be decomposed into two Dirac equations by only a nonunitary transformation. We also write an analogue of the complex Dirac-Kähler equation in five dimensions. We show that the complex Dirac-Kähler equation is a special case of a Bhabha-type equation and prove that this equation is invariant under the algebra of purely matrix transformations of the Pauli-Gürsey type and under two different representations of the Poincaré group, the fermionic (for a two-fermion system) and bosonic -representations. The complex Dirac-Kähler equation is also written in a manifestly covariant bosonic form as an equation for the system ( , , ) of irreducible self-dual tensor, scalar, and vector fields. We illustrate the relation between the complex Dirac-Kähler equation and the known 16-component Dirac-Kähler equation.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 143, No. 1, pp. 64–82, April, 2005.     

A Canonical Directly Infinite Ring

Let be the set of nonnegative integers and the ring of integers. Let be the ring of N × N matrices over generated by the following two matrices: one obtained from the identity matrix by shifting the ones one position to the right and the other one position down. This ring plays an important role in the study of directly finite rings. Calculation of invertible and idempotent elements of yields that the subrings generated by them coincide. This subring is the sum of the ideal consisting of all matrices in with only a finite number of nonzero entries and the subring of generated by the identity matrix. Regular elements are also described. We characterize all ideals of , show that all ideals are finitely generated and that not all ideals of are principal. Some general ring theoretic properties of are also established.     

The Phase Space of an Initial-Boundary Value Problem for the Hoff Equation

The Hoff equation describes the H-beam buckling dynamics. We show that the phase space of the Hoff equation is a simple Banach manifold modeled on a subspace complementary to the kernel .     

Oscillatory Properties of Second Order Half-Linear Difference Equations

We study oscillatory properties of the second order half-linear difference equation

Augmentation and Modification Problems for Hermitian Matrices

We obtain necessary and sufficient conditions for the solvability of the augmentation and modification problems of order for Hermitian matrices. The augmentation problem consists in the construction of a Hermitian -matrix with a given -block in block -representation and with the prescribed eigenvalues. The modification problem consists in the construction of a Hermitian -matrix of rank not greater than so that the obtained matrix, being added to a given Hermitian -matrix , will have the required spectrum. We give an estimate for the minimal number of different eigenvalues of the solutions to these problems.     

On the Solution of the Dirichlet Problem for an Elliptic Equation Degenerating on the Boundary

The Dirichlet problem for the equation is studied in the semi-circle . The restrictions on F are established under which the problem is uniquely solvable in the definite generalized sense.     

Oscillation and Nonoscillation Criteria for a Second Order Linear Equation

New oscillation and nonoscillation criteria are established for the equation

Quantum inverse scattering method and (super)conformal field theory

We consider the possibility of using the quantum inverse scattering method to study the superconformal field theory and its integrable perturbations. The classical limit of the considered constructions is based on the (1|2) super-KdV hierarchy. We introduce the quantum counterpart of the monodromy matrix corresponding to the linear problem associated with the L-operator and use the explicit form of the irreducible representations of _q(1|2) to obtain the fusion relations for the transfer matrices (i.e., the traces of the monodromy matrices in different representations).Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 142, No. 2, pp. 252–264, February, 2005.     

Existence of Positive Solutions for a Class of Higher Order Neutral Functional Differential Equations

The higher order neutral functional differential equation

One Case of Appearance of Positive Solutions of Delayed Discrete Equations

When mathematical models describing various processes are analysed, the fact of existence of a positive solution is often among the basic features. In this paper, a general delayed discrete equation

Kolmogorov equation and large-time behaviour for fractional brownian motion driven linear sde’s

We consider a stochastic process X _t ^x which solves an equation

Bounded Oscillation of Nonlinear Neutral Differential Equations of Arbitrary Order

The paper is concerned with oscillation properties of n-th order neutral differential equations of the form

Some Results on Stability of Retarded Functional Differential Equations Using Dichotomic Map Techniques

This paper deals with the study of the stability of nonautonomous retarded functional differential equations using the theory of dichotomic maps. After some preliminaries, we prove the theorems on simple and asymptotic stability. Some examples are given to illustrate the application of the method. Main results about asymptotic stability of the equation and of itsnonlinear generalization are established.     

A Technique for Constructing Symmetric Designs

Let M be a set of incidence matrices of symmetric (v,k,)-designs and G a group of mappings M M. We give a sufficient condition for the matrix W M, where M M and W is a balanced generalized weighing matrix over G, to be the incidence matrix of a larger symmetric design. This condition is then applied to the designs corresponding to McFarland and Spence difference sets, and it results in four families of symmetric (v,k, )-designs with the following parameters k and (m and d are positive integers, p and q are prime powers): (i) ; (ii) ; (iii) ; (iv) .     

Algebras Generated by Semicentral Idempotents

Let be a unital K-algebra, where K is a commutative ring with unity. An idempotent is {\it left semicentral\/} if , and is {\it SCI-generated\/} if it is generated as a K-module by left semicentral idempotents. This paper develops the basic properties of SCI-generated algebras and characterizes those that are also prime, semiprime, primitive, or subdirectly irreducible. Minimal ideals and the socle of SCI-generated algebras are investigated. Conditions are found to describe a large class of SCI-generated algebras via generalized triangular matrix representations. SCI-generated piecewise domains are characterized. Examples are given that illustrate the breadth and diversity of the class of SCI-generated algebras.