Characteristic matrices of weakly curved elastic layers

The characteristic matrices of plane-parallel, cylindrical, and spherical layers are investigated. On the basis of these investigations, it is possible to determine formulas for the characteristic matrices of an arbitrary weakly curved elastic layer.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Maternaticheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 99, pp. 74–84, 1980.     

Inverses of -distance matrices of a tree

The determinant and the inverse of the distance matrix of a tree have been investigated in the literature, following the classical formulas due to Graham and Pollak for the determinant, and due to Graham and Lovász for the inverse. We consider two q-analogs of the distance matrix of a tree and obtain formulas for the inverses of the two distance matrices. Yan and Yeh have previously obtained expressions for the determinants of the two distance matrices. Some related results are proved.     

Exact integration of the stiffness matrix of an 8‐node plane elastic finite element by symbolic computation

Computer algebra systems (CAS) are powerful tools for obtaining analytical expressions for many engineering applications in both academic and industrial environments. CAS have been used in this paper to generate exact expressions for the stiffness matrix of an 8‐node plane elastic finite element. The Maple software system was used to identify six basic formulas from which all the terms of the stiffness matrix could be obtained. The formulas are functions of the Cartesian coordinates of the corner nodes of the element, and elastic parameters Young's modulus and Poisson's ratio. Many algebraic manipulations were performed on the formulas to optimize their efficiency. The redaction in CPU time using the exact expressions as opposed to the classical Gauss–Legendre numerical integration approach was over 50%. In an additional study of accuracy, it was shown that the numerical approach could lead to quite significant errors as compared with the exact approach, especially as element distortion was increased.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2007     

Coefficients of reflection and refraction in the case of elastic-fluid systems

Processes of reflection and refraction are investigated in the case of a vertically layered elastic-fluid system. Special matrices characterizing the layers, the boundaries, and the entire system are introduced to find the refraction coefficients. By means of these matrices general formulas for the refraction coefficients are obtained, and the matrix method is generalized to elastic-fluid systems.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeliniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 62, pp. 154–167, 1976.     

Stress state in multilayer cylindrical tubes made of elastically compressible linear viscoelastic material

Through the use of Abel's operators and э operators, the authors obtained an implicit solution for N-layered cylindrical tubes with alternating elastic and viscoelastic layers under conditions of elastic compressibility of the viscoelastic medium. Explicit expressions were written for the reactive pressures in 3- and 2-layered cylindrical tubes with alternating elastic and viscoelastic layers; these are analyzed in detail. Based on the solution obtained earlier for a condition where the viscoelastic operator corresponding to Poisson's coefficient is taken as a constant, it has been shown that the hypotheses discussed will lead in time to qualitatively different stress states. The overall results obtained are illustrated on an example.     

Wave Attenuation in Effective Models Describing Porous and Fractured Media Saturated by a Fluid

Wave attenuation is introduced in the effective model of media that consists of alternating elastic and fluid layers. This attenuation is due to the friction on the boundaries between elastic and fluid layers and is described by additional terms in equations of the effective model. An investigation of these equations allows one to derive expressions of the attenuation coefficients for every body wave propagating along the layers. Bibliography: 9 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 297, 2003, pp. 216–229.     

Electromechanical properties of continuous fibre-reinforced piezoelectric composites

The objective of this paper is to present analytical expressions of the basic effective constants (for transducer applications) of fibrous piezoelectric composites with a periodic structure. These expressions were obtained on the basis of the asymptotic averaging method, following the same procedure used in [1] for the purely elastic case. The averaged electromechanical properties for these piezocomposites can be calculated from the above formulas. The results obtained from the theoretical predictions with the use of homogenization method show good agreement with the existing experimental results.Published in Mekhanika Kompozitnykh Materialov, Vol. 33, No. 5, pp. 670–680, September–October, 1997.     

Extremal Ranks of Some Nonlinear Matrix Expressions with Applications

In this paper, we establish a group of closed-form formulas for the maximal and minimal ranks of a nonlinear matrix expression with respect to two variant matrices by using a linearization method and some known formulas for extremal ranks of linear matrix expressions. In addition, by using some pure algebraic operations of matrices and their generalized inverses, we derive the maximal and minimal ranks of the above nonlinear matrix expression, where the two variant matrices are any solutions of two consistent matrix equations. As an application, we derive some sufficient and necessary conditions for the existence of the solution of a nonlinear matrix function.     

Equalities for orthogonal projectors and their operations

A complex square matrix A is called an orthogonal projector if A ² = A = A*, where A* denotes the conjugate transpose of A. In this paper, we give a comprehensive investigation to matrix expressions consisting of orthogonal projectors and their properties through ranks of matrices. We first collect some well-known rank formulas for orthogonal projectors and their operations, and then establish various new rank formulas for matrix expressions composed by orthogonal projectors. As applications, we derive necessary and sufficient conditions for various equalities for orthogonal projectors and their operations to hold.     

Szegö-Kac-Achiezer formulas in terms of realizations of the symbol

For rational and analytic matrix functions new formulas are obtained for the limits in the Szegö-Kac-Achiezer limit theorems. In the rational case the new expressions are given in terms of finite matrices which come from special representations of the matrix functions. These representations are known as realizations in mathematical systems theory.     

On Effective Models of Porous Stratified Media with a Sliding Contact Between Layers

In order to study wave propagation in porous layered media with a sliding contact between the elastic phases on the interfaces, effective models of these media are investigated. For these models, the front sets of four waves excited by point sources are established and formulas for the wave velocities along the axes are derived. The methods of constructing the front sets applied in this paper allow one to point out special features of these front sets such as loops and juts. The particular case where all of the layers are identical and a sliding contact occurs between layers is also considered. Bibliography: 8 titles.     

Correlation asymptotics for non‐translation invariant lattice spin systems

We obtain asymptotic expressions for the Green kernels of certain non‐translation invariant transition matrices using methods of semiclassical and microlocal analysis. Combined with a result by Bach and Møller this yields asymptotic formulas for the truncated two‐point correlation functions of certain non‐translation invariant lattice models of real‐valued spins. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

横观各向同性层状弹性场地格林函数的数值解

采用横观各向同性层状弹性模型模拟半空间之上的层状场地,用人工透射边界代替半空间以吸收能量。利用薄层元素法给出任意节面上环形垂直及水平简谐荷载作用下场地的位移公式,并由此导出官类场地任意节面上在垂直及水平简谐集中荷作用下的位移响应显式解。算例表明本文方法与理论解相近,横观各向同性性质对格林函数有明显的影响。     

A matricial computation of rational quadrature formulas on the unit circle

A matricial computation of quadrature formulas for orthogonal rational functions on the unit circle, is presented in this paper. The nodes of these quadrature formulas are the zeros of the para-orthogonal rational functions with poles in the exterior of the unit circle and the weights are given by the corresponding Christoffel numbers. We show how these nodes can be obtained as the eigenvalues of the operator Möbius transformations of Hessenberg matrices and also as the eigenvalues of the operator Möbius transformations of five-diagonal matrices, recently obtained. We illustrate the preceding results with some numerical examples.     

约束矩阵方程的解的表示及行列式公式

本文利用Ｂｏｔｔ－Ｄｕｆｆｉｎ逆导出约束矩阵方程的唯一解或通解的表示及其较为简洁的行列式公式，并由之给出了某些特殊矩阵方程与约束性方程组的解以及几个重要广义逆矩阵的表示及其较是洁的行列式公式，我们的结果改进与推广了文（２－１１，１４，１５）中的有关结果。     

Distributions of eigenvalues and inertias of some block Hermitian matrices consisting of orthogonal projectors

A complex square matrix A is called an orthogonal projector if A ²?=?A?=?A*, where A* is the conjugate transpose of A. In this article, we first give some formulas for calculating the distributions of real eigenvalues of a linear combination of two orthogonal projectors. Then, we establish various expansion formulas for calculating the inertias, ranks and signatures of some 2?×?2 and 3?×?3, as well as k?×?k block Hermitian matrices consisting of two orthogonal projectors. Many applications of the formulas are presented in characterizing interval distributions of numbers of eigenvalues, and nonsingularity of these block Hermitian matrices. In addition, necessary and sufficient conditions are given for various equalities and inequalities of these block Hermitian matrices to hold.     

Generalized cauchy-vandermonde matrices

Matrices of the form [C V] consisting of a generalized Cauchy matrix and a generalized Vandermonde matrix are considered. Using the displacement structure of these matrices, inversion formulas and criteria are presented. The interpretation of linear systems with such a coefficient matrix as tangential interpolation problems leads to the concept of fundamental matrix, which is basic in this approach. For fundamental matrices recursion formulas are established. From them, fast inversion algorithms emerge that work for arbitrary nonsingular matrices of this kind.     

Generalized Runge-Kutta methods for coupled systems of hyperbolic differential equations

Runge-Kutta formulas are discussed for the integration of systems of differential equations. The parameters of these formulas are square matrices with component-dependent values. The systems considered are supposed to originate from hyperbolic partial differential equations, which are coupled in a special way. In this paper the discussion is concentrated on methods for a class of two coupled systems. For these systems first and second order formulas are presented, whose parameters are diagonal matrices. These formulas are further characterized by their low storage requirements, by a reduction of the computational effort per timestep, and by their relatively large stability interval along the imaginary axis. The new methods are compared with stabilized Runge-Kutta methods having scalar-valued parameters. It turns out that a gain factor of 2 can be obtained.     

The Generating Function for the Airy Point Process and a System of Coupled Painlevé II Equations

For a wide class of Hermitian random matrices, the limit distribution of the eigenvalues close to the largest one is governed by the Airy point process. In such ensembles, the limit distribution of the k th largest eigenvalue is given in terms of the Airy kernel Fredholm determinant or in terms of Tracy–Widom formulas involving solutions of the Painlevé II equation. Limit distributions for quantities involving two or more near‐extreme eigenvalues, such as the gap between the k th and the ℓth largest eigenvalue or the sum of the k largest eigenvalues, can be expressed in terms of Fredholm determinants of an Airy kernel with several discontinuities. We establish simple Tracy–Widom type expressions for these Fredholm determinants, which involve solutions to systems of coupled Painlevé II equations, and we investigate the asymptotic behavior of these solutions.     

Equalities and inequalities for inertias of hermitian matrices with applications

The inertia of a Hermitian matrix is defined to be a triplet composed of the numbers of the positive, negative and zero eigenvalues of the matrix counted with multiplicities, respectively. In this paper, we show some basic formulas for inertias of 2×2 block Hermitian matrices. From these formulas, we derive various equalities and inequalities for inertias of sums, parallel sums, products of Hermitian matrices, submatrices in block Hermitian matrices, differences of outer inverses of Hermitian matrices. As applications, we derive the extremal inertias of the linear matrix expression A-BXB^∗ with respect to a variable Hermitian matrix X. In addition, we give some results on the extremal inertias of Hermitian solutions to the matrix equation AX=B, as well as the extremal inertias of a partial block Hermitian matrix.