CONSTRUCTION OF POLYNOMIAL MATRIX USING BLOCK COEFFICIENT MATRIX REPRESENTATION AUTO-REGRESSIVE MOVING AVERAGE MODEL FOR ACTIVELY CONTROLLED STRUCTURES

The polynomial matrix using the block coefficient matrix representation auto-regressive moving average(referred to as the PM-ARMA)model is constructed in this paper for actively controlled multi-degree-of-freedom(MDOF)structures with time-delay through equivalently transforming the preliminary state space realization into the new state space realization.The PM-ARMA model is a more general formulation with respect to the polynomial using the coefficient representation auto-regressive moving average(ARMA)model due to its capability to cope with actively controlled structures with any given structural degrees of freedom and any chosen number of sensors and actuators.(The sensors and actuators are required to maintain the identical number.)under any dimensional stationary stochastic excitation.

LEVERRIER-LAGUERRE ALGORITHM FOR THE MATRIX POLYNOMIAL OF DEGREE TWO

An algorithm for simultaneous computation of the adjoint G(s) and determinant d(s) of the matrix polynomial s2J-sA 1-A 2 is presented, where J is a singular matrix. Both G(s) and d(s) are expressed relative to a basis of Laguerre orthogonal polynomials. This algorithm is a new extension of Leverrier-Fadeev algorithm..

On Humbert Matrix Polynomials of Two Variables

In this paper we introduce Humbert matrix polynomials of two variables. Some hypergeometric matrix representations of the Humbert matrix polynomials of two variables, the double generating matrix functions and expansions of the Humbert matrix polynomials of two variables in series of Hermite polynomials are given. Results of Gegenbauer ma-trix polynomials of two variables follow as particular cases of Humbert matrix polynomials of two variables.

Analytic Expression of Arbitrary Matrix Elements for Boson Exponential Quadratic Polynomial Operators

Making use of the transformation relation among usual, normal, and antinormal ordering for the multimode boson exponential quadratic polynomial operators （BEQPO＇s）I we present the analytic expression of arbitrary matrix elements for BEQPO＇s. As a preliminary application, we obtain the exact expressions of partition function about the boson quadratic polynomial system, matrix elements in particle-number, coordinate, and momentum representation, and P representation for the BEQPO＇s.

Reply to Comment on “On Humbert Matrix Polynomials of Two Variables”

The formula subject to comment in Reference [1] is correct.

A Comment on “On Humbert Matrix Polynomials of Two Variables”

In this comment we will demonstrate that one of the main formulas given in Ref. [1] is incorrect.

Volterra Integral Equation of Hermite Matrix Polynomials

The primary purpose of this paper is to present the Volterra integral equa- tion of the two-variable Hermite matrix polynomials. Moreover, a new representation of these matrix polynomials are established here.

Generalized Form of Hermite Matrix Polynomials via the Hypergeometric Matrix Function

The object of this paper is to present a new generalization of the Hermite matrix polynomials by means of the hypergeometric matrix function. An integral representation, differential recurrence relation and some other properties of these generalized forms are established here. Moreover, some new properties of the Hermite and Chebyshev matrix polynomials are obtained. In particular, the two-variable and two-index Chebyshev matrix polynomials of two matrices are presented.

THE CHARACTERISTIC POLYNOMIAL ANDEIGENVALUES OF FINILE RIODAN MATRIXTHE CHARACTERISTIC POLYNOMIAL ANDEIGENVALUES OF FINILE RIODAN MATRIX

In this paper we deal with the characteristic polynomial of finite Riodan matix. We giveseveral forms of its explicit expressions. Its applications to combinatorial identities, specially to F-Lidentities, are stated.

ORTHOGONAL MATRIX POLYNOMIALS WITH RESPECT TO LINEAR MATRIX MOMENT FUNCTION ALS:THEORY AND APPLICATIONS

In this paper orthogonal matrix polynomials with respect to a right matrix moment functional an introduced. Basic results, important examples and applications to the approximation of matrix integrals are studied. Error bounds for the proposed matrix quadrature rules are given.

HERMITE MATRIX POLYNOMIALS AND SECOND ORDER MATRIX DIFFERENTIAL EQUATIONS

In this paper we introduce the class of Hermite's matrix polynomials which appear as finite series solutions of second order matrix differential equations Y'-xAY'+BY=0.An explicit expression for the Hermite matrix polynomials,the orthogonality property and a Rodrigues' formula are given.

PSEUDO-DIVISION ALGORITHM FOR MATRIX MULTIVARIABLE POLYNOMIAL AND ITS APPLICATION

Pseudo-division algorithm for matrix multivariable polynomial are given, thereby with the view of differential algebra, the sufficient and necessary conditions for transforming a class of partial differential equations into infinite dimensional Hamiltonianian system and its concrete form are obtained. Then by combining this method with Wu's method, a new method of constructing general solution of a class of mechanical equations is got, which several examples show very effective.

A SEMI-CONJUGATE MATRIX BOUNDARY VALUE PROBLEM FOR GENERAL ORTHOGONAL POLYNOMIALS ON AN ARBITRARY SMOOTH JORDAN CURVE

In this article, the author characterizes orthogonal polynomials on an arbitrary smooth Jordan curve by a semi-conjugate matrix boundary value problem, which is different from the Riemann-Hilbert problems that appear in the theory of Riemann -Hilbert approach to asymptotic analysis for orthogonal polynomials on a real interval introduced by Fokas, Its, and Kitaev and on the unit circle introduced by Baik, Deift, and Johansson. The author hopes that their characterization may be applied to asymptotic analysis for general orthogonal polynomials by combining with a new extension of steepest descent method which we are looking for.

An explicit solution to polynomial matrix right coprime factorization with application in eigenstructure assignment

在这篇论文，输入状态转移功能的多项式矩阵权利 coprimefactorization 的一个明确的答案以 Krylov 矩阵和系数矩阵的对的 thePseudo 可控制性索引被获得。建议途径仅仅需要解决一系列线性方程。generalizedSylvester 矩阵方程和由州的反馈的参量的 eigenstructure 赋值的问题的一种类型的这个答案的应用程序被调查。这些新答案简单，他们拥有更好结构的性质和方便的空想使用。一个例子显示出建议结果的效果。

Hermite Matrix Polynomial Collocation Method for Linear Complex Differential Equations and Some Comparisons

In this paper, we introduce a Hermite operational matrix collocation method for solving higher-order linear complex differential equations in rectangular or elliptic domains. We show that based on a linear algebra theorem, the use of different polynomials such as Hermite, Bessel and Taylor in polynomial collocation methods for solving differential equations leads to an equal solution, and the difference in the numerical results arises from the difference in the coefficient matrix of final linear systems of equations. Some numerical examples will also be given.

Bounds for Polynomial’s Roots from Fiedler and Sparse Companion Matrices for Submultiplicative Matrix Norms

We use submultiplicative companion matrix norms to provide new bounds for roots for a given polynomial P(X) over the field C[X]. From a n×n Fiedler companion matrix C, sparse companion matrices and triangular Hessenberg matrices are introduced. Then, we identify a special triangular Hessenberg matrix Lr, supposed to provide a good estimation of the roots. By application of Gershgorin’s theorems to this special matrix in case of submultiplicative matrix norms, some estimations of bounds for roots are made. The obtained bounds have been compared to known ones from the literature precisely Cauchy’s bounds, Montel’s bounds and Carmichel-Mason’s bounds. According to the starting formel of Lr, we see that the more we have coefficients closed to zero with a norm less than 1, the more the Sparse method is useful.

ORTHOGONAL MATRIX POLYNOMIALS WITH RESPECT TO A CONJUGATE BILINEAR MATRIX MOMENT FUNCTIONAL: BASIC THEORY

In this paper basic results for a theory of orthogonal matrix polynomials with respect to a conjugate bilinear matrix moment functional are proposed. Properties of orthogonal matrix polynomial sequences including a three term matrix relationship are given. Positive definite conjugate bilinear matrix moment functionals are introduced and a characterization of positive definiteness in terms of a block Haenkel moment matrix is established. For each positive definite conjugate bilinear matrix moment functional an associated matrix inner product is defined.

Novel stability criteria for fuzzy Hopfield neural networks based on an improved homogeneous matrix polynomials technique

The global stability problem of Takagi-Sugeno（T-S） fuzzy Hopfield neural networks（FHNNs） with time delays is investigated.Novel LMI-based stability criteria are obtained by using Lyapunov functional theory to guarantee the asymptotic stability of the FHNNs with less conservatism.Firstly,using both Finsler＇s lemma and an improved homogeneous matrix polynomial technique,and applying an affine parameter-dependent Lyapunov-Krasovskii functional,we obtain the convergent LMI-based stability criteria.Algebraic properties of the fuzzy membership functions in the unit simplex are considered in the process of stability analysis via the homogeneous matrix polynomials technique.Secondly,to further reduce the conservatism,a new right-hand-side slack variables introducing technique is also proposed in terms of LMIs,which is suitable to the homogeneous matrix polynomials setting.Finally,two illustrative examples are given to show the efficiency of the proposed approaches.

A NOTE ON SOBOLEV ORTHOGONALITY FOR LAGUERRE MATRIX POLYNOMIALS

Abstract. Let {L（Ln^（A,λ）（x）}n≥0 be the sequence of monic Laguerre matrix polynomials defined on [0,∞） byLn^（A,λ）（x）=n!/（-λ）^n ∑nk-0（-λ）^k/k!（n-k）!（A＋I）n[（A＋I）k]^-1x^k,where A ∈ C^r×r. It is known that {Ln^（A,λ）（x）}n≥0 is orthogonal with respect to a matrix moment functional when A satisfies the spectral condition that Re（z） 〉 -1 for every z E or（a）. In this note we show that forA such that σ（A） does not contain negative integers, the Laguerre matrix polynomials Ln^（A,λ）（x） are orthogonal with respect to a non-diagonal SobolevLaguerre matrix moment functional, which extends two cases： the above matrix case and the known scalar case.

Pseudo Laguerre Matrix Polynomials, Operational Identities and Quasi-Monomiality

The main purpose of this paper is to introduce the matrix extension of the pseudo Laguerre matrix polynomials and to explore the formal properties of the operational rules and the principle of quasi-monomiality to derive a number of properties for pseudo Laguerre matrix polynomials.