Applications Of Matrix Function And Group Action

Let ϒ be a three-dimensional Lipschitz polyhedron, and assume that the matrix function μ is piecewise constant on a polyhedral partition of ϒ. Based on regularity results for solutions to two-dimensional anisotropic transmission problems near corner points we obtain conditions on μ and the intersection angles between interfaces and ∂ϒ ensuring that the operator -∇ · μ∇ maps the Sobolev space [Formula: see text] isomorphically onto W-1,q(ϒ) for some q > 3.

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On the matrix versions of q-zeta function, q-digamma function and q-polygamma function

Asian-European Journal of Mathematics ◽

10.1142/s1793557120501429 ◽

2019 ◽

Vol 13 (08) ◽

pp. 2050142

Author(s):

Ravi Dwivedi ◽

Vivek Sahai

Keyword(s):

Zeta Function ◽

Matrix Function ◽

Integral Representations ◽

Digamma Function ◽

Polygamma Function ◽

The Matrix ◽

Matrix Series ◽

Zeta Matrix

This paper deals with the [Formula: see text]-analogues of generalized zeta matrix function, digamma matrix function and polygamma matrix function. We also discuss their regions of convergence, integral representations and matrix relations obeyed by them. We also give a few identities involving digamma matrix function and [Formula: see text]-hypergeometric matrix series.

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Block Toeplitz operators with frequency-modulated semi-almost periodic symbols

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171203107107 ◽

2003 ◽

Vol 2003 (34) ◽

pp. 2157-2176 ◽

Cited By ~ 4

Author(s):

A. Böttcher ◽

S. Grudsky ◽

I. Spitkovsky

Keyword(s):

Toeplitz Operator ◽

Frequency Modulation ◽

Real Line ◽

Matrix Function ◽

Toeplitz Operators ◽

Matrix Functions ◽

Almost Periodic ◽

The Matrix ◽

Matrix Symbol ◽

Block Toeplitz Operators

This paper is concerned with the influence of frequency modulation on the semi-Fredholm properties of Toeplitz operators with oscillating matrix symbols. The main results give conditions on an orientation-preserving homeomorphismαof the real line that ensure the following: ifbbelongs to a certain class of oscillating matrix functions (periodic, almost periodic, or semi-almost periodic matrix functions) and the Toeplitz operator generated by the matrix functionb(x)is semi-Fredholm, then the Toeplitz operator with the matrix symbolb(α(x))is also semi-Fredholm.

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A Numerical Solution Using an Adaptively Preconditioned Lanczos Method for a Class of Linear Systems Related with the Fractional Poisson Equation

Journal of Applied Mathematics and Stochastic Analysis ◽

10.1155/2008/104525 ◽

2008 ◽

Vol 2008 ◽

pp. 1-26 ◽

Cited By ~ 9

Author(s):

M. Ilić ◽

I. W. Turner ◽

V. Anh

Keyword(s):

Linear System ◽

Linear Systems ◽

Poisson Equation ◽

Matrix Function ◽

Fractional Power ◽

Analytic Solutions ◽

Linear System Of Equations ◽

Exact Analytic Solutions ◽

The Matrix ◽

Symmetric Positive Definite Matrices

This study considers the solution of a class of linear systems related with the fractional Poisson equation (FPE) (−∇2)α/2φ=g(x,y) with nonhomogeneous boundary conditions on a bounded domain. A numerical approximation to FPE is derived using a matrix representation of the Laplacian to generate a linear system of equations with its matrix A raised to the fractional power α/2. The solution of the linear system then requires the action of the matrix function f(A)=A−α/2 on a vector b. For large, sparse, and symmetric positive definite matrices, the Lanczos approximation generates f(A)b≈β0Vmf(Tm)e1. This method works well when both the analytic grade of A with respect to b and the residual for the linear system are sufficiently small. Memory constraints often require restarting the Lanczos decomposition; however this is not straightforward in the context of matrix function approximation. In this paper, we use the idea of thick-restart and adaptive preconditioning for solving linear systems to improve convergence of the Lanczos approximation. We give an error bound for the new method and illustrate its role in solving FPE. Numerical results are provided to gauge the performance of the proposed method relative to exact analytic solutions.

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On the matrix function Ax+X′A′

Archive for Rational Mechanics and Analysis ◽

10.1007/bf00253335 ◽

1962 ◽

Vol 9 (1) ◽

pp. 93-96 ◽

Cited By ~ 24

Author(s):

O. Taussky ◽

H. Wielandt

Keyword(s):

Matrix Function ◽

The Matrix

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A New Algorithm to Approximate Bivariate Matrix Function via Newton-Thiele Type Formula

Journal of Applied Mathematics ◽

10.1155/2013/642818 ◽

2013 ◽

Vol 2013 ◽

pp. 1-10

Author(s):

Rongrong Cui ◽

Chuanqing Gu

Keyword(s):

Continued Fractions ◽

Matrix Function ◽

Continued Fraction ◽

Generalized Inverse ◽

New Method ◽

Rectangular Grid ◽

Type Formula ◽

The Matrix ◽

Rational Interpolants ◽

Better Than

A new method for computing the approximation of bivariate matrix function is introduced. It uses the construction of bivariate Newton-Thiele type matrix rational interpolants on a rectangular grid. The rational interpolant is of the form motivated by Tan and Fang (2000), which is combined by Newton interpolant and branched continued fractions, with scalar denominator. The matrix quotients are based on the generalized inverse for a matrix which is introduced by C. Gu the author of this paper, and it is effective in continued fraction interpolation. The algorithm and some other important conclusions such as divisibility and characterization are given. In the end, two examples are also given to show the effectiveness of the algorithm. The numerical results of the second example show that the algorithm of this paper is better than the method of Thieletype matrix-valued rational interpolant in Gu (1997).

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Gradient estimates for Neumann boundary value problem of Monge–Ampère type equations

Communications in Contemporary Mathematics ◽

10.1142/s0219199716500413 ◽

2016 ◽

Vol 19 (04) ◽

pp. 1650041 ◽

Cited By ~ 1

Author(s):

Feida Jiang ◽

Ni Xiang ◽

Jinju Xu

Keyword(s):

Matrix Function ◽

Symmetric Matrix ◽

Type Equation ◽

Neumann Boundary ◽

Gradient Estimates ◽

Neumann Boundary Value Problem ◽

The Matrix ◽

Elliptic Solutions ◽

Monge Ampere ◽

Gradient Bound

This paper concerns the gradient estimates for Neumann problem of a certain Monge–Ampère type equation with a lower order symmetric matrix function in the determinant. Under a one-sided quadratic structure condition on the matrix function, we present two alternative full discussions of the global gradient bound for the elliptic solutions.

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Approximation of Irrational Matrix Functions and Its Application to Continuous-Discrete Model Conversion

Journal of Dynamic Systems Measurement and Control ◽

10.1115/1.3153029 ◽

1989 ◽

Vol 111 (2) ◽

pp. 142-145 ◽

Cited By ~ 5

Author(s):

Muh-Yang Chen ◽

Chyi Hwang

Keyword(s):

Matrix Function ◽

Discrete Model ◽

Matrix Functions ◽

Rational Matrix ◽

Improved Method ◽

Matrix Logarithm ◽

The Matrix ◽

Model Conversion ◽

Irrational Function ◽

Rational Matrix Function

In this paper, an improved method of rational approximation is presented for evaluating the irrational matrix function f(A), where A is a square matrix and f(s) is a scalar irrational function which is analytic on the spectrum of A. The improvement in the accuracy of the approximation off (A) by a rational matrix function is achieved by using the multipoint Pade approximants to f(s). An application example to model conversion involving the evaluations of the matrix exponential exp (AT) and the matrix logarithm ln(F) is provided to illustrate the superiority of the method.

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Polynomials for crystal frameworks and the rigid unit mode spectrum

Philosophical Transactions of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rsta.2012.0030 ◽

2014 ◽

Vol 372 (2008) ◽

pp. 20120030 ◽

Cited By ~ 10

Author(s):

S. C. Power

Keyword(s):

Explicit Formula ◽

Algebraic Variety ◽

Matrix Function ◽

Singular Points ◽

Cubic Symmetry ◽

Long Wavelength ◽

Wavelength Limit ◽

Mode Spectrum ◽

Harmonic Excitations ◽

The Matrix

To each discrete translationally periodic bar-joint framework in , we associate a matrix-valued function defined on the d -torus. The rigid unit mode (RUM) spectrum of is defined in terms of the multi-phases of phase-periodic infinitesimal flexes and is shown to correspond to the singular points of the function and also to the set of wavevectors of harmonic excitations which have vanishing energy in the long wavelength limit. To a crystal framework in Maxwell counting equilibrium, which corresponds to being square, the determinant of gives rise to a unique multi-variable polynomial . For ideal zeolites, the algebraic variety of zeros of on the d -torus coincides with the RUM spectrum. The matrix function is related to other aspects of idealized framework rigidity and flexibility, and in particular leads to an explicit formula for the number of supercell-periodic floppy modes. In the case of certain zeolite frameworks in dimensions two and three, direct proofs are given to show the maximal floppy mode property (order N ). In particular, this is the case for the cubic symmetry sodalite framework and some other idealized zeolites.

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