Deficiency numbers of operators generated by infinite Jacobi matrices

New conditions for minimality, maximality, and nonmaximality of deficiency numbers of the minimal operator generated by the infinite Jacobi matrix with m × m matrix entries in the Hilbert space of mdimensional vectors are presented. Special attention is given to the case m = 1, i.e., to conditions on the elements of a tridiagonal numerical Jacobi matrix under which the determinate case of the classical power moment problem is realized.     

A noncommutative version of the bispectral problem

We consider a matrix valued version of the bispectral problem involving a block tridiagonal doubly infinite matrix and a first-order differential operator with matrix coefficients. We give a set of necessary conditions that the coefficients need to satisfy and solve these equations under a variety of conditions. The situations discussed here should make it plain that while the corresponding problem in the scalar case is relatively trivial and devoid of any interest, the noncommutative version of the problem is much richer and subtle. The results here should be useful, for instance, in the study of a noncommutative version of the nonlinear Toda lattice.     

Characterization,parity, and power sequences of locally finite doubly stochastic matrices

We show that infinite locally finite doubly stochastic matrices are particular limits of sequences of finite doubly stochastic matrices and reciprocally. Thereby, we define the parity in the set of infinite locally finite doubly stochastic matrices. In particular, convexity and stability properties of the even matrix of this set are investigated, as well as the differences between the finite case and the infinite case. Moreover, the limits of the powers of locally finite infinite doubly stochastic matrices in this context are determined.     

一类特殊无限方阵的逆

研究了除环上一类特殊无限方阵的逆方阵,用紧致性论证给出了除环上无限上三角阵具有左（右）逆和双侧逆方阵的充分条件。     

The Sobolev-type moment problem

We propose necessary and sufficient conditions for a bisequence of complex numbers to be a moment one of Sobolev type over the real line, the unit circle and the complex plane. We achieve this through converting the moment problem in question into a matrix one and utilizing some techniques coming from operator theory. This allows us to consider the Sobolev type moment problem in its full generality, not necessarily in the diagonal case and even of infinite order.     

除环上无限方阵的对角化

讨论了除环上无限方阵的对角化问题,证明了除环上行列有限方阵等价于一类特殊对角矩阵Dn的两个等价条件。     

Monic divisors with one invariant factor of polynomial matrices

For a nonsingular polynomial matrix over an infinite field, necessary and sufficient conditions for separation of a monic factor from a matrix polynomial with canonical diagonal form diag (1,…,1, φ(x)) are established.     

Infinite horizon multiobjective optimal control problems in the discrete time case

This article studies multiobjective optimal control problems in the discrete time framework and in the infinite horizon case. The functions appearing in the problems satisfy smoothness conditions. This article generalizes to the multiobjective case results obtained for single-objective optimal control problems in that framework. The dynamics are governed by difference equations or difference inequations. Necessary conditions of Pareto optimality are presented, namely Pontryagin maximum principles in the weak form and in the strong form. Sufficient conditions are also provided. Other notions of Pareto optimality are defined when the infinite series do not necessarily converge.     

Full spark frames and totally positive matrices

In this paper we establish a connection between full spark frames and totally nonsingular matrices. Then we provide a method for constructing infinite totally positive matrices which make up a subclass of the class of totally nonsingular matrices. Using this method we then construct a family of infinite totally positive matrices parameterized by non-negative numbers which contains, as the simplest case, the infinite Pascal matrix. The paper ends with some examples and comments on full spark frames.     

A General Framework of Multivariate Wavelets with Duals

A comprehensive development of multivariate wavelets along with their duals is presented in this paper. The basic ingredients, such as duality relations, reconstruction and decomposition formulas, and the notion of infinite direct sums, are formulated and established in the general nonorthogonal and multivariate setting. Special emphases include duality criteria and stability conditions. As an application, new results are contributed, particularly in dual wavelets, to the existing literature on low-dimensional wavelets. In addition, the special case when the dilation matrix has determinant 2 is studied in some detail.     

On infinite doubly substochastic matrices

Summary An infinite matrix is said to be doubly substochastic if it has nonnegative components and each row and each column sum is at most 1. Let x and y be two real sequences which converge to 0 or which are absolutely summable. This paper introduces necessary and sufficient conditions for existence of an infinite doubly substochastic matrix A such that x=Ay concerning partial order and convex hull for sequences.     

Integral equations of the first kind in the theory of oscillating plates

A modified matrix of fundamental solutions is used to derive and solve first-kind integral equations for the problem of high-frequency harmonic oscillations of an infinite elastic plate with a hole when Dirichlet or Neumann conditions are prescribed on the boundary curve.     

关于高等数学中几个易错命题的举例说明

本文通过构造具体的典型例子对高等数学中的几个易错命题进行了阐述和说明.对无限多个无穷小量的和与积的性质进行了探讨,举例说明了无限多个无穷大量的乘积不一定是无穷大量.给出了无限乘积运算时仍然是无穷大量或无穷小量的充分条件.这有助于更好地理解无穷大量和无穷小量两个概念的本质内涵,也有助于认识无限运算和有限运算的根本差异.     

Nonlinear Hyperbolic Equations in Infinite Homogeneous Waveguides

ABSTRACT

In this paper we prove global and almost global existence theorems for nonlinear wave equations with quadratic nonlinearities in infinite homogeneous waveguides. We can handle both the case of Dirichlet boundary conditions and Neumann boundary conditions. In the case of Neumann boundary conditions we need to assume a natural nonlinear Neumann condition on the quasilinear terms. The results that we obtain are sharp in terms of the assumptions on the dimensions for the global existence results and in terms of the lifespan for the almost global results. For nonlinear wave equations, in the case where the infinite part of the waveguide has spatial dimension three, the hypotheses in the theorem concern whether or not the Laplacian for the compact base of the waveguide has a zero mode or not.     

Stability of Periodic Points of Diffeomorphisms of Multidimensional Space

We study the diffeomorphism of a multidimensional space into itself with a hyperbolic fixed point at the origin and a nontransversal homoclinic point. From the works of Sh. Newhouse, B.F. Ivanov, L.P. Shilnikov, and other authors, it follows that there is a method of tangency for the stable and unstable manifold such that the neighborhood of a nontransversal homoclinic point can contain an infinite set of stable periodic points, but at least one of the characteristic exponents of those points tends to zero as the period increases. In this paper, we study diffeomorphisms such that the method of tangency for the stable and unstable manifold differs from the case studied in the works of the abovementioned authors. This paper continues previous works of the author, where diffeomorphisms are studied such that their Jacobi matrices at the origin have only real eigenvalues. In those previous works, we find conditions such that the neighborhood of a nontransversal homoclinic point of the studied diffeomorphism contains an infinite set of stable periodic points with characteristic exponents separated from zero. In the present paper, it is assumed that the Jacobi matrix of the original diffeomorphism at the origin has real eigenvalues and several pairs of complex conjugate eigenvalues. Under this assumption, we find conditions guaranteeing that a neighborhood of a nontransversal homoclinic point contains an infinite set of stable periodic points with characteristic exponents separated from zero.     

Letter to the Editor: A New Criterion for Gabor Frames

We give a new criterion for the invertibility of the Gabor frame operator in the irrational case. It is sufficient to verify the invertibility of a single (infinite) matrix.     

Some remarks on matrix norms,condition numbers,and error estimates for linear equations

Inequalities between some norms of rectangular matrices and the corresponding relationships between condition numbers are established and clearly arranged in two simple diagrams. Furthermore, some well-known error estimates for linear equations in the nonsingular case are shown to be valid for all submultiplicative matrix norms. The new proofs work without using infinite series.     

Morita equivalence for infinite matrix rings

In this paper, weak distinguished subcategory and distinguished subcategory of modules are introduced. Left(right) local unital rings are particularly considered. Also, representable equivalent functors between categories. By using the replacement techniques of modules, a general theory of Morita equivalence for infinite matrix rings is established. This theory not only extends the classical Morita theory of equivalence from finite matrix rings to infinite matrix rings and also contains some new results which are useful in studying the algebraic structures for infinite matrix rings. Some results of classical Morita theory are included as its special cases.