Representation theory of finite groups in Wiener–Hopf factorization problem

We consider the Wiener–Hopf factorization problem for a matrix function that is completely defined by its first column: the succeeding columns are obtained from the first one by means of a finite group of permutations. The symmetry of this matrix function allows us to reduce the dimension of the problem. In particular, we find some relations between its partial indices and can compute some of the indices. In special cases, we can explicitly obtain the Wiener–Hopf factorization of the matrix function.     

线性不可逆过程的协同效应分析

利用矩阵理论对线性不可逆过程的协同效应进行了分析.与向量空间相类比,定义了热力学流空间中的内积以及协同系数.协同系数的大小反映了两个不可逆过程间的协同程度.由唯象系数矩阵引出了协同矩阵与协同系数矩阵.对于导热不可逆过程,协同矩阵所对应的二次型是耗散函数.对于孤立体系,证明了协同矩阵所对应的二次型对时间的导数为负值,它可以作为体系的一个李雅普诺夫函数.     

Constrained matrix optimization with applications

In this work we study a minimization problem for a matrix-valued function under linear constraints, in the case of a singular matrix. The proposed method differs from others on the restriction of the minimizing matrix to the range of the corresponding quadratic function. Moreover, we present two applications of the proposed minimization method in Linear Regression and B-spline smoothing.     

Rational approximation to the Fermi–Dirac function with applications in density functional theory

We are interested in computing the Fermi–Dirac matrix function in which the matrix argument is the Hamiltonian matrix arising from density functional theory (DFT) applications. More precisely, we are really interested in the diagonal of this matrix function. We discuss rational approximation methods to the problem, specifically the rational Chebyshev approximation and the continued fraction representation. These schemes are further decomposed into their partial fraction expansions, leading ultimately to computing the diagonal of the inverse of a shifted matrix over a series of shifts. We describe Lanczos and sparse direct methods to address these systems. Each approach has advantages and disadvantages that are illustrated with experiments.     

A comparison principle for functions of a uniformly random subspace

This note demonstrates that it is possible to bound the expectation of an arbitrary norm of a random matrix drawn from the Stiefel manifold in terms of the expected norm of a standard Gaussian matrix with the same dimensions. A related comparison holds for any convex function of a random matrix drawn from the Stiefel manifold. For certain norms, a reversed inequality is also valid.     

矩阵损失函数下回归系数矩阵的非齐次线性估计的可容许性

在二次矩阵损失函数下研究了协方差矩阵未知的多元线性模型中回归系数矩阵的可估线性函数的矩阵非齐次线性估计的可容许性,给出了矩阵非齐次线性估计在线性估计类中可容许的一个充要条件.     

Delayed perturbation of Mittag‐Leffler functions and their applications to fractional linear delay differential equations

In this paper, we propose a delayed perturbation of Mittag‐Leffler type matrix function, which is an extension of the classical Mittag‐Leffler type matrix function and delayed Mittag‐Leffler type matrix function. With the help of the delayed perturbation of Mittag‐Leffler type matrix function, we give an explicit formula of solutions to linear nonhomogeneous fractional delay differential equations.     

Stable iterations for the matrix square root

Any matrix with no nonpositive real eigenvalues has a unique square root for which every eigenvalue lies in the open right half-plane. A link between the matrix sign function and this square root is exploited to derive both old and new iterations for the square root from iterations for the sign function. One new iteration is a quadratically convergent Schulz iteration based entirely on matrix multiplication; it converges only locally, but can be used to compute the square root of any nonsingular M-matrix. A new Padé iteration well suited to parallel implementation is also derived and its properties explained. Iterative methods for the matrix square root are notorious for suffering from numerical instability. It is shown that apparently innocuous algorithmic modifications to the Padé iteration can lead to instability, and a perturbation analysis is given to provide some explanation. Numerical experiments are included and advice is offered on the choice of iterative method for computing the matrix square root.     

Skew‐selfadjoint Dirac systems with rational rectangular Weyl functions: explicit solutions of direct and inverse problems and integrable wave equations

In this paper we study direct and inverse problems for discrete and continuous skew‐selfadjoint Dirac systems with rectangular (possibly non‐square) pseudo‐exponential potentials. For such a system the Weyl function is a strictly proper rational matrix function and any strictly proper rational matrix function appears in this way. In fact, extending earlier results, given a strictly proper rational matrix function we present an explicit procedure to recover the corresponding potential using techniques from mathematical system and control theory. We also introduce and study a nonlinear generalized discrete Heisenberg magnet model, extending earlier results for the isotropic case. A large part of the paper is devoted to the related discrete systems of which the pseudo‐exponential potential depends on an additional continuous time parameter. Our technique allows us to obtain explicit solutions for the generalized discrete Heisenberg magnet model and evolution of the Weyl functions.     

Isotropic covariance matrix polynomials on spheres

This article characterizes the covariance matrix function of a Gaussian or elliptically contoured vector random field that is stationary, isotropic, and mean square continuous on the sphere. By applying the characterization to examine the validity of a matrix function whose entries are polynomials of degrees up to 4, we obtain a necessary and sufficient condition for the polynomial matrix to be an isotropic covariance matrix function on the sphere.     

Localization for random perturbations of anisotropic periodic media

We prove localization for random perturbations of periodic divergence form operators of the form ∇ · a_ω · ∇ near the band edges. Here a_ω is a matrix function which results from an Anderson type perturbation of a periodic matrix function.     

The norm of a matrix after a similarity transformation

The norm of a similarity transformation of a matrix is studied considered as a function of the transformation matrix, and conditions for stationary points of this function are given in terms of properties of the transformed matrix.     

Robust factor analysis

Our aim is to construct a factor analysis method that can resist the effect of outliers. For this we start with a highly robust initial covariance estimator, after which the factors can be obtained from maximum likelihood or from principal factor analysis (PFA). We find that PFA based on the minimum covariance determinant scatter matrix works well. We also derive the influence function of the PFA method based on either the classical scatter matrix or a robust matrix. These results are applied to the construction of a new type of empirical influence function (EIF), which is very effective for detecting influential data. To facilitate the interpretation, we compute a cutoff value for this EIF. Our findings are illustrated with several real data examples.     

Rational matrix approximation with a priori error bounds for non-symmetric matrix Riccati equations with analytic coefficients

In this paper the exact solution of the non-symmetric matrixRiccati equation with analytic coefficients is approximatedby a rational matrix function with a prefixed accuracy. Thisrational matrix function is locally defined as the exact solutionof a Riccati problem with matrix polynomial coefficients obtainedby truncation of the Taylor expansions of the matrix coefficientsof the original problem.     

混合Boussinesq 方程的有限亏格解

借助于Lenard 递推方程和定态零曲率方程, 我们给出与3 × 3 矩阵谱问题相联系的一族混合Boussinesq 方程. 利用Lax 矩阵的特征多项式, 引入一条三角曲线K_m-1, 由此构造出相应的Baker-Akhiezer 函数、亚纯函数和Dubrovin- 型方程. 混合Boussinesq 流在Abel 映射下被拉直. 基于三角曲线和三类Abel 微分的理论, 我们得到了Baker-Akhiezer 函数、亚纯函数的Riemann θ 函数表示, 特别地, 给出了混合Boussinesq 方程的有限亏格解.     

The Generalized Matrix Sector Function and the Separation of Matrix Eigenvalues

The matrix sector function of A is introduced and generalizedto the matrix sector function of g(A), where the complex matrixA may have a real or complex characteristic polynomial and g(A)is a matrix function of a conformal mapping. The generalizedmatrix sector function of A is employed to separate the matrixeigenvalues relative to a sector, a circle, and a sector ofa circle in the complex plane without actually seeking the characteristicpolynomial and the matrix eigenvalues relative to a sector,a circle, and a sector of a circle in the complex plane withoutactually seeking the characteristic polynomial and the matrixeigenvalues themselves. Also, the generalized matrix sectorfunction of A is utilized to carry out the block-diagonalizationand block-triangularization of a system matrix, which are usefulin developing applications to mathematical science and control-systemproblems.     

Time Varying Isotropic Vector Random Fields on Spheres

For a vector random field that is isotropic and mean square continuous on a sphere and stationary on a temporal domain, this paper derives a general form of its covariance matrix function and provides a series representation for the random field, which involve the ultraspherical polynomials. The series representation is somehow an imitator of the covariance matrix function, but differs from the spectral representation in terms of the ordinary spherical harmonics, and is useful for modeling and simulation. Some semiparametric models are also illustrated.     

Finite-Band Solutions for the Hierarchy of Coupled Toda Lattices

Based on the characteristic polynomial of Lax matrix for the hierarchy of coupled Toda lattices associated with a \(3\times3\) discrete matrix spectral problem, we introduce a trigonal curve with two infinite points, from which we establish the associated Dubrovin-type equations. The asymptotic properties of the meromorphic function and the Baker-Akhiezer function are studied near two infinite points on the trigonal curve. Finite-band solutions of the entire hierarchy of coupled Toda lattices are obtained in terms of the Riemann theta function.     

A procedure based on finite elements for the solution of nonlinear problems in the kinematic analysis of mechanisms

In the present paper the kinematic analysis of mechanisms is based on the application of finite elements is discussed. It is shown how the kinematic properties of the rigid-body motions of a mechanism can be obtained from an analysis of the stiffness matrix of a simple model comprising rod-type elements in the case of planar mechanisms. In the event that there is also a more complex finite element model of the mechanism, onemay in addition obtain thenode values from the results achieved with the simple model. Special attention is given to nonlinear position problems, i.e. initial, successive, deformed, and static equilibrium. An error function is provided that is valid in each case. This function is derived from the elastic potential function, and uses Laggrange multipliers and penalty functions. The result is an application of the primal-dual method, or augmented Lagraange multipliers (ALM) method. This function is minimized by means of Newtons' method, which leads in simple form to the vector gradient as a force vector. The second-derivative matrix is derived from the stifness matrix, to which a complementary matrix owing toe nonlinearity introduced by the large displacements is added.

This method can be easily implemented on a computer. The computer program will be able to perform a wide variety of kinematic analyses of any planar mechanism with lower pairs. The models of the mechanism are very simple, and need only a few tens of degrees of freedom even for the most complex mechanisms. The CPU time is also very low due to the simplicity of the method and its good convergence properties.