Some matrix functional equations

We investigate the pair of matrix functional equations G(x)F(y) = G(xy) and G(x)G(y) = F(y/x), featuring the two independent scalar variables x and y and the two N×N matrices F(z) andG(z) (with N an arbitrary positive integer and the elements of these two matrices functions of the scalar variable z). We focus on the simplest class of solutions, i.e., on matrices all of whose elements are analytic functions of the independent variable. While in the scalar (N = 1) case this pair of functional equations only possess altogether trivial constant solutions, in the matrix (N > 1) case there are nontrivial solutions. These solutions satisfy the additional pair of functional equations F(x)G(y) = G(y/x) andF(x)F(y) = F(xy), and an endless hierarchy of other functional equations featuring more than two independent variables.     

On nonoscillation of canonical or noncanonical disconjugate functional equations

Qualitative comparison of the nonoscillatory behavior of the equations

Variationally trivial Lagrangians and locally variational differential equations of arbitrary order

We continue our investigation of the Lagrangian formalism on jet bundle extensions using Fock space methods. We are able to provide the most general form of a variationally trivial Lagrangian of arbitrary order and we also give a generic expression for the most general locally variational differential equation. As anticipated in the literature, these expressions involve some special combinations of the highest order derivatives, called hyper-Jacobians.     

On matrix equivalence and matrix equations

Roth's theorem on the solvability of matrix equations of the form AX?YB=C is proved for matrices with coefficients from a division ring or a ring which is module-finite over its center.     

More on oscillation ofnth-order equations

In this paper we prove that a higher-order differential equation with one middle term has every bounded solution oscillatory. Moreover, the behavior of unbounded solutions is given. Two other results dealing with positive solutions are also given.     

Generating equations approach for quadratic matrix equations

We show how Van Loan's method for annulling the (2,1) block of skew‐Hamiltonian matrices by symplectic‐orthogonal similarity transformation generalizes to general matrices and provides a numerical algorithm for solving the general quadratic matrix equation: For skew‐Hamiltonian matrices we find their canonical form under a similarity transformation and find the class of all symplectic‐orthogonal similarity transformations for annulling the (2,1) block and simultaneously bringing the (1,1) block to Hessenberg form. We present a structure‐preserving algorithm for the solution of continuous‐time algebraic Riccati equation. Unlike other methods in the literature, the final transformed Hamiltonian matrix is not in Hamiltonian–Schur form. Three applications are presented: (a) for a special system of partial differential equations of second order for a single unknown function, we obtain the matrix of partial derivatives of second order of the unknown function by only algebraic operations and differentiation of functions; (b) for a similar transformation of a complex matrix into a symmetric (and three‐diagonal) one by applying only finite algebraic transformations; and (c) for finite‐step reduction of the eigenvalues–eigenvectors problem of a Hermitian matrix to the eigenvalues– eigenvectors problem of a real symmetric matrix of the same dimension. Copyright © 1999 John Wiley & Sons, Ltd.     

Convolution equations and nonlinear functional equations

The survey is devoted to applications of nonlinear integral equations to linear convolution equations, their discrete analogues, and also the connection of these equations with problems of radiative transfer, in particular, with the Ambartsumyan equations.Translated from Itogi Nauki i Tekhniki, Seriya Matematicheskii Analiz, Vol. 22, pp. 175–244, 1984.     

Quartic functional equations

In this paper, we solve a new functional equation

f(2x+y)+f(2x−y)=4f(x+y)+4f(x−y)+24f(x)−6f(y)     

Obstructions to embeddings of bundles of matrix algebras in a trivial bundle

We evaluate the cohomology obstructions to the existence of fiber-preserving unital embedding of a locally trivial bundle A _k → X whose fiber is a complex matrix algebra M _k (?) in a trivial bundle with fiber M _kl (?) under the assumption that (k, l) = 1. It is proved that the first obstruction coincides with the obstruction to the reduction of the structure group PGL _k (?) of the bundle A _k to SL _k (?), which coincides with the first Chern class c ₁(ξ _k ) reduced modulo k under the assumption that A _k ≌ End(ξ _k ) for some vector ? _k -bundle ξ _k → X. If the first obstruction vanishes, then A _k ≌ End( $\tilde \xi _k $ ) for some vector ? ^k bundle ξ _k → X with structure group SL _k (?), and the second obstruction is c ₂( $\tilde \xi _k $ )modk ∈ H ⁴(X, ?/k?). Further, the higher obstructions are defined using a Postnikov tower, and each of the obstructions is defined on the kernel of the previous obstruction.     

Working with differential,functional and difference equations using functional networks

In this paper we first analyze the problem of equivalence of differential, functional and difference equations and give methods to move between them. We also introduce functional networks, a powerful alternative to neural networks, which allow neural functions to be different, multidimensional, multiargument and constrained by link connections, and use them for predicting values of magnitudes satisfying differential, functional and/or difference equations, and for obtaining the difference and differential equation associated with a set of data. The estimation of the differential or difference equation coefficients is done by simply solving systems of linear equations, in the cases of equally or unequally spaced or missing data points. Some examples of applications are given to illustrate the method.     

Measure functional differential equations and functional dynamic equations on time scales

We study measure functional differential equations and clarify their relation to generalized ordinary differential equations. We show that functional dynamic equations on time scales represent a special case of measure functional differential equations. For both types of equations, we obtain results on the existence and uniqueness of solutions, continuous dependence, and periodic averaging.     

Stability in the first approximation of the trivial solution of stochastic functional-differential equations

A theorem is proved on the asymptotic stability (in the sense of convergence with probability one) in the first approximation of the trivial solution of stochastic functional-differential equations.Translated from Matematicheskie Zametki, Vol. 23, No. 5, pp. 733–738, May, 1978.     

More on generalizations of matrix monotonicity

Following Berman and Plemmons [5], Werner [17], Poole and Barker [13], and others, we investigate some generalizations of matrix monotonicity and consider their relation to MP matrices. We prove some observations on almost monotone, MP, and group monotone matrices. However, of much interest to us is the problem whether a symmetric positive semidefinite matrix is MP if and only if it is monotone on its range. To consider this question we obtain several results on the structure of range monotone matrices.