Matrix interpretation of multiple orthogonality

In this work we give an interpretation of a (s (d + 1 ) + 1)-term recurrence relation in terms of type II multiple orthogonal polynomials. We rewrite this recurrence relation in matrix form and we obtain a three-term recurrence relation for vector polynomials with matrix coefficients. We present a matrix interpretation of the type II multi-orthogonality conditions. We state a Favard type theorem and the expression for the resolvent function associated to the vector of linear functionals. Finally a reinterpretation of the type II Hermite-Padé approximation in matrix form is given.     

Algebraic methods for the study of some linear matrix equations

An algebraic viewpoint permits the formulation of necessary and sufficient conditions for the existence of a unique solution to some linear matrix equations. The theory developed is then used to express the solution in a finite series form and to prove a stability theorem.     

Complementarity forms of theorems of Lyapunov and Stein, and related results

The well-known Lyapunov's theorem in matrix theory / continuous dynamical systems asserts that a (complex) square matrix A is positive stable (i.e., all eigenvalues lie in the open right-half plane) if and only if there exists a positive definite matrix X such that AX+XA^* is positive definite. In this paper, we prove a complementarity form of this theorem: A is positive stable if and only if for any Hermitian matrix Q, there exists a positive semidefinite matrix X such that AX+XA^*+Q is positive semidefinite and X[AX+XA^*+Q]=0. By considering cone complementarity problems corresponding to linear transformations of the form I−S, we show that a (complex) matrix A has all eigenvalues in the open unit disk of the complex plane if and only if for every Hermitian matrix Q, there exists a positive semidefinite matrix X such that X−AXA^*+Q is positive semidefinite and X[X−AXA^*+Q]=0. By specializing Q (to −I), we deduce the well known Stein's theorem in discrete linear dynamical systems: A has all eigenvalues in the open unit disk if and only if there exists a positive definite matrix X such that X−AXA^* is positive definite.     

On necessary and sufficient conditions of the regularity of summation methods for Laguerre-Fourier series

Summary The problem of convergence of linear means is considered for the Laguerre-Fourier series of continuous functions. An upper estimate is obtained for the Laguerre-Lebesgue function in terms of the entries of the matrix which determines the linear summability method in question. This allows us to prove for such series an analogue of the well-known theorem by S. M. Nikol'skii which provides necessary and sufficient conditions for the summability of trigonometric Fourier series. A theorem on the regularity of the summability methods is also established.     

Mean square stability for systems on stochastically generated discrete time scales

We consider dynamic systems which evolve on discrete time domains where the time steps form a sequence of independent, identically distributed random variables. In particular, we classify the mean-square stability of linear systems on these time domains using quadratic Lyapunov functionals. In the case where the system matrix is a function of the time step, our results agree with and generalize stability results found in the Markov jump linear systems literature. In the case where the system matrix is constant, our results generalize, illuminate, and extend to the stochastic realm results in the field of dynamic equations on time scales. In order to help see the factors that contribute to stability, we prove a sufficient condition for the solvability of the Lyapunov equation by appealing to a fixed point theorem of Ran and Reurings. Finally, an example using observer-based feedback control is presented to demonstrate the utility of the results to control engineers who cannot guarantee uniform timing of the system.     

Cartan's conjecture with moving targets of same growth and effective Wirsing's theorem over function fields

We formulate a generalization of the Cartan's theorem and the Cartan's conjecture over function fields in the case where the coefficients of the linear forms can be nonconstant functions. This generalization is used to prove the Wirsing's theorem over function fields. Furthermore, the methods we use here provide effective results. Received September 4, 1998; in final form April 23, 1999 / Published online May 8, 2000     

On a theorem of Hermite and Hurwitz

The Hermite-Hurwitz theorem computes the degree, over $\text{R}$, of a real rational function ? in terms of the signature of an associated quadratic form—known today as the Hankel matrix of ?. This formula, which Hermite was led to by his work on the problem of representing integers as sums of squares, gave rise to striking applications in the theory of equations and in the stability theory of ordinary differential equations. In this paper, this theorem and various generalizations to the matrix-valued case are discussed and described in terms of signature formulae. These include its relation to stability theory and the matrix Hermite-Hurwitz theorem of Bitmead-Anderson as applied to questions of circuit synthesis. This also includes a global form of Hörmander's signature formula for the Maslov index of a rational loop in a Lagrangian Grassmannian, due to Byrnes and Duncan, and applications to the topology of spaces of rational matrix-valued functions, following the work of Brockett, Byrnes, and Duncan. This includes, in particular, a topological proof of the matrix Hermite-Hurwitz theorem.     

关于常系数齐次线性微分方程组的解法研究

利用凯莱-哈密顿定理给出矩阵指数函数eAt的简洁计算方法;同时利用约当标准形推导出求常系数齐次线性微分方程组通解的循环公式.     

Floquet theorem for linear implicit nonautonomous difference systems

The aim of this paper is to develop the Floquet theory for linear implicit difference systems (LIDS). It is proved that any index-1 LIDS can be transformed into its Kronecker normal form. Then the Floquet theorem on the representation of the fundamental matrix of index-1 periodic LIDS has been established. As an immediate consequence, the Lyapunov reduction theorem is proved. Some applications of the obtained results are discussed.     

具无限时滞的非线性积分微分方程的周期解

本文考虑具无限时滞非线性积分微分方程和其中t∈R,T≥0是常数,x∈Rn;A(t,x),C(t,s)为n×n连续的函数矩阵;f(t,x),g(t,x),b(t)是n维连续向量.本文利用线性系统的指数型二分性理论和不动点定理研究此系统,建立了保证其周期解存在性.唯一性的充分条件.得到了一些新的结果,推广了相关文献的主要结果.     

Strong Szegő Asymptotics and Zeros of the Zeta‐Function

Assuming the Riemann hypothesis, we prove the weak convergence of linear statistics of the zeros of the Riemann ζ‐function to a Gaussian field, with covariance structure corresponding to the H ^1/2‐norm of the test functions. For this purpose, we obtain an approximate form of the explicit formula, relying on Selberg's smoothed expression for ζ'/ζ and the Helffer‐Sjöstrand functional calculus. Our main result is an analogue of the strong Szeg? theorem, known for Toeplitz operators and random matrix theory. © 2014 Wiley Periodicals, Inc.     

Laguerre matrix polynomial series expansion: Theory and computer applications

This paper deals with the expansion of matrix functions in series of Laguerre matrix polynomials of a complex matrix parameter. A new asymptotic expression for positive and large x and n is obtained for this family of matrix polynomials that improves previous existing expressions. This expression is applied to improving the conditions imposed on the matrix function that are to be expanded in a series of non-hermitian Laguerre matrix polynomials, providing a new series expansion theorem that is more general than previous works. An application to solving linear differential systems without matrix exponentials is included.     

Abel群的一些分解定理的推广(Ⅲ)

从主理想整环上有界模分解的Prüfer-Baer定理出发,研究(无限维)向量空间的代数的线性变换的几个基本问题,得到了如下结果:设V是域F上的(无限维)向量空间,A是V上的一个代数的线性变换,则有(1)若任何与A可交换的线性变换均与线性变换B可交换,则B=f(A),其中f是F上的多项式.进而线性变换B也是代数的.(2) V中存在一组基,使A在这组基下的矩阵是有理标准型(经典标准型)矩阵.当F是代数闭域时,经典标准型矩阵即为若当标准型矩阵.(3)当F是代数闭域时,A存在相应的Jordan-Chevalley分解.进一步,该结论在完全域上仍成立.这些研究推广了有限维向量空间上线性变换的相关结果.     

Dynamics and interpretation of some integrable systems via matrix orthogonal polynomials

In this work we characterize a high-order Toda lattice in terms of a family of matrix polynomials orthogonal with respect to a complex matrix measure. In order to study the solution of this dynamical system, we give explicit expressions for the Weyl function, generalized Markov function, and we also obtain, under some conditions, a representation of the vector of linear functionals associated with this system. We show that the orthogonality is embedded in these structure and governs the high-order Toda lattice. We also present a Lax-type theorem for the point spectrum of the Jacobi operator associated with a Toda-type lattice.     

一种关于图的关联矩阵秩的定理证明的新方法

关于图的关联矩阵的一个重要的定理是r个结点的连通图G的关联矩阵的秩是r-1.利用一般域上的线性空间理论,给出了无向图的关联矩阵秩的定理证明,该方法结构严谨且利于学生理解和接受.     

输入存贮线性有限自动机的极小化

由输入存贮线性有限自动机的线性系数组成的矩阵得出输入存贮线性有限自动机极小的等价定理,由此定理得出输入存贮线性有限自动机的极小化方法.     

Student reasoning about the invertible matrix theorem in linear algebra

I report on how a linear algebra classroom community reasoned about the invertible matrix theorem (IMT) over time. The IMT is a core theorem that connects many fundamental concepts through the notion of equivalency. As the semester progressed, the class developed the IMT in an emergent fashion. As such, the various equivalences took form and developed meaning as students came to reason about the ways in which key ideas involved were connected. Microgenetic and ontogenetic analyses (Saxe in J Learn Sci 11(2–3):275–300, 2002) framed the structure of the investigation. The results focus on shifts in the mathematical content of argumentation over time and the centrality of span and linear independence in classroom argumentation.     

A Generalization of a Result in Linear Programming

The simplex method of linear programming depends upon a theoremconcerning the maximization of a linear function subject tolinear constraints. In the present paper this theorem is generalizedto one concerning the maximization subject to linear constraintsof a function which is an increasing function in each of a numberof linear functions. A method for deriving further extensionsof the theorem is suggested and a particular example workedout. It is shown also how the theorem leads to a simple algorithmin a particular case and it is suggested that other similarapplications may be found.     

An analytic refinement of the Heyde theorem on linear forms of independent random variables

In terms of the characteristic function of the joint distribution of two linear forms of independent random variables, one refines Heyde's known theorem on the characterization of the Gaussian distribution by the property of symmetry of the conditional distribution of one linear form under a given second form.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 166, pp. 54–59, 1988.