Mixed Löwner and Nevanlinna-Pick Interpolation

A mixed type, L?wner and Nevanlinna-Pick directional two-sided interpolation problem is considered. A necessary and sufficient condition for the problem to have a solution is established, in terms of properties of the Pick kernel to the problem. As well, a parametrization of the set of all real rational solutions of minimal degree is given. The corresponding Nevanlinna-Pick boundary-interior interpolation problem is also considered and a solvability condition for it is obtained. The approach to the problem is via functional Hilbert spaces.     

On an operator approach to interpolation problems for Stieltjes functions

A general interpolation problem for operator-valued Stieltjes functions is studied using V. P. Potapov's method of fundamental matrix inequalities and the method of operator identities. The solvability criterion is established and under certain restrictions the set of all solutions is parametrized in terms of a linear fractional transformation. As applications of a general theory, a number of classical and new interpolation problems are considered.     

On the nonuniqueness of the solution of a mixed boundary value problem for the Laplace equation

We study the solvability of the mixed boundary value problem for the Laplace equation with three distinct boundary conditions, two of which include two directional derivatives with distinct tilt angles and the remaining one is the first boundary condition. An example of a nontrivial solution of the homogeneous problem is given, and conditions under which the problem has a unique solution are established. The solvability of the problem with a nonhomogeneous first boundary condition is studied.     

The mass-transfer vector problem

The mass-transference problem is studied for Banach valued cost functions and operator-valued measures. The solvability of the primal problem is stated under certain natural conditions, for general measurable functions and measures of bounded variation. The continuous case is also studied, and the solvability and the absence of a duality gap are established for continuous vector functions and regular operator valued measures.     

Solvability of Newton equations in smoothing-type algorithms for the SOCCP

In this paper, we first investigate the invertibility of a class of matrices. Based on the obtained results, we then discuss the solvability of Newton equations appearing in the smoothing-type algorithm for solving the second-order cone complementarity problem (SOCCP). A condition ensuring the solvability of such a system of Newton equations is given. In addition, our results also show that the assumption that the Jacobian matrix of the function involved in the SOCCP is a P₀-matrix is not enough for ensuring the solvability of such a system of Newton equations, which is different from the one of smoothing-type algorithms for solving many traditional optimization problems in ℜⁿ.     

ON HASEMAN BOUNDARY VALUE PROBLEM FOR A CLASS OF META-ANALYTIC FUNCTIONS

In this article, Haseman boundary value problem for a class of meta-analytic functions is studied. The expression of solution and the condition of solvability for Haseman boundary value problem are obtained by changing the problem discussed into the equivalent Haseman boundary value problem of bi-analytic function. And the expression of solution and the condition of solvability depend on the canonical matrix.     

On some issues related to the moment problem for the band matrices with operator elements

The connection between the classical moment problem and the spectral theory of second order difference operators (or Jacobi matrices) is a thoroughly studied topic. Here we examine a similar connection in the case of the second order operator replaced by an operator generated by an infinite band matrix with operator elements. For such operators, we obtain an analog of the Stone theorem and consider the inverse spectral problem which amounts to restoring the operator from the moment sequence of its Weyl matrix. We establish the solvability criterion for such problems, find the conditions ensuring that the elements of the moment sequence admit an integral representation with respect to an operator valued measure and discuss an algorithm for the recovery of the operator. We also indicate a connection between the inverse problem method and the Hermite-Padé approximations.     

J-inner matrix functions, interpolation and inverse problems for canonical systems, IV: Direct and inverse bitangential input scattering problems

Bitangential input scattering problems are formulated and analyzed for canonical integral systems. Special attention is paid to the case when the input scattering matrix is ap×q matrix valued function of Wiener class. Formulas for the solution of the inverse input scattering problem are obtained by reproducing kernel Hilbert space methods. A number of illustrative examples are presented. Additional examples for the case when the input scattering matrix is of Wiener class/rational will be presented in a future publication.     

The Bitangential Inverse Input Impedance Problem for Canonical Systems,I: Weyl-Titchmarsh Classification,Existence and Uniqueness

The inverse input impedance problem is investigated in the class of canonical integral systems with matrizants that are strongly regular J-inner matrix valued functions in the sense introduced in [ArD1]. The set of solutions for a problem with a given input impedance matrix (i.e., Weyl- Titchmarsh function) is parameterized by chains of associated pairs of entire inner p × p matrix valued functions. In our considerations the given data for the inverse bitangential input impedance problem is such a chain and an input impedance matrix, i.e., a p × p matrix valued function in the Carathéodory class. Existence and uniqueness theorems for the solution of this problem are obtained by consideration of a corresponding family of generalized bitangential Carathéodory interpolation problems. The connection between the inverse bitangential input scattering problem that was studied in [ArD4] and the bitangential input impedance problem is also exploited. The successive sections deal with: 1. The introduction, 2. Domains of linear fractional transformations, 3. Associated pairs of the first and second kind, 4. Matrix balls, 5. The classification of canonical systems via the limit ball, 6. The Weyl-Titchmarsh characterization of the input impedance, 7. Applications of interpolation to the bitangential inverse input impedance problem. Formulas for recovering the underlying canonical integral systems, examples and related results on the inverse bitangential spectral problem will be presented in subsequent publications.D. Z. Arov thanks the Weizmann Institute of Science for hospitality and support, partially as a Varon Visiting Professor and partially through the Minerva Foundation. H. Dym thanks Renee and Jay Weiss for endowing the chair which supports his research and the Minerva Foundation.     

A solvability theorem and minimax fractional programming

This paper presents a stable solvability theorem for general inequality systems under a local closedness condition. It is shown how this mild regularity condition can be characterized by the validity of the solvability theorem for all local perturbations. Based on this solvability theorem zero duality gap and stability are established for general minimax fractional programming problems.The research was initiated while the first named author was a visitor at the University of New South Wales and was completed while the second named author was a visitor at the Technische Hochschule Darmstadt.     

Theoretical Analyses on Discrete Formulae of Directional Differentials in the Finite Point Method

For the five-point discrete formulae of directional derivatives in the finite point method,overcoming the challenge resulted from scattered point sets and making full use of the explicit expressions and accuracy of the formulae,this paper obtains a number of theoretical results:(1)a concise expression with definite meaning of the complicated directional difference coefficient matrix is presented,which characterizes the correlation between coefficients and the connection between coefficients and scattered geometric characteristics;(2)various expressions of the discriminant function for the solvability of numerical differentials along with the estimation of its lower bound are given,which are the bases for selecting neighboring points and making analysis;(3)the estimations of combinatorial elements and of each element in the directional difference coefficient matrix are put out,which exclude the existence of singularity.Finally,the theoretical analysis results are verified by numerical calculations.The results of this paper have strong regularity,which lay the foundation for further research on the finite point method for solving partial differential equations.     

解浅水方程离散化线性方程组的PEk方法

定义于球面的浅水方程能够很好地描述浅齐次的不可压缩非黏滞流体层的性状,它在全球大气模型、海洋数字模型和天气预报的数值计算中都有广泛的应用,浅水方程的一般形式如下：     

On Nonstationary Stokes Problem and Navier–Stokes Problem in a Half-Space with Initial Data Nondecreasing at Infinity

We prove the solvability of the Stokes problem and the local (in time) solvability of the Navier–Stokes problem in the half-space under the condition that the initial velocity is only bounded and continuous. The proof is based on estimates for the entries of the Green matrix for the Stokes problem. Bibliography: 8 titles.     

实对称带状矩阵逆特征值问题

研究了一类实对称带状矩阵逆特征值问题：给定三个互异实数λ,μ和v及三个非零实向量x,y和z,分别构造实对称五对角矩阵T和实对称九对角矩阵A,使其都具有特征对(λ,x),(μ,y)和(v,z)．给出了此类问题的两种提法,研究了问题的可解性以及存在惟一解的充分必要条件,最后给出了数值算法和数值例子．     

Real valued iterative methods for solving complex symmetric linear systems

Complex valued systems of equations with a matrix R + 1S where R and S are real valued arise in many applications. A preconditioned iterative solution method is presented when R and S are symmetric positive semi‐definite and at least one of R, S is positive definite. The condition number of the preconditioned matrix is bounded above by 2, so only very few iterations are required. Applications when solving matrix polynomial equation systems, linear systems of ordinary differential equations, and using time‐stepping integration schemes based on Padé approximation for parabolic and hyperbolic problems are also discussed. Numerical comparisons show that the proposed real valued method is much faster than the iterative complex symmetric QMR method. Copyright © 2000 John Wiley & Sons, Ltd.     

完全域上矩阵多项式方程的可解性

本文利用一般域上的λ-矩阵理论，研究了矩阵多项式方程的可解性，证明了完全域上矩阵多项式方程有解的充要条件，这些条件同时提供了解此类矩阵方程的方法。     

On estimates for solutions of elliptic boundary value problems in a layer

We consider boundary value problems for elliptic operators with constant coefficients in a layer, i.e., in a domain between two parallel planes. We assume that the Lopatinskii condition and the condition of the unique solvability of an auxiliary problem for an ordinary differential operator are satisfied. We prove theorems on the solvability and smoothness of solutions in Sobolev spaces with weight of exponential type.     

Interaction between thermoelastic and scalar oscillation fields

Three-dimensional mathematical problems of the interaction between thermoelastic and scalar oscillation fields in a general physically anisotropic case are studied by the boundary integral equation methods. Uniqueness and existence theorems are proved by the reduction of the original interface problems to equivalent systems of boundary pseudodifferential equations. In the non-resonance case the invertibility of the corresponding matrix pseudodifferential operators in appropriate functional spaces is shown on the basis of the generalized Sommerfeld-Kupradze type thermoradiation conditions for anisotropic bodies. In the resonance case the co-kernels of the pseudodifferential operators are analysed and the efficient conditions of solvability of the original interface problems are established.