scholarly journals On effective criterion of stability of partial indices for matrix polynomials

2020 ◽  
Vol 476 (2238) ◽  
pp. 20200012
Author(s):  
N. V. Adukova ◽  
V. M. Adukov
Keyword(s):  
Small Perturbation ◽  
Stability Criterion ◽  
Matrix Function ◽  
Toeplitz Matrices ◽  
Matrix Functions ◽  
Matrix Polynomials ◽  
Partial Indices ◽  
The Stability ◽  
The Given

In the work, we obtain an effective criterion of the stability of the partial indices for matrix polynomials under an arbitrary sufficiently small perturbation. Verification of the stability is reduced to calculation of the ranks for two explicitly defined Toeplitz matrices. Furthermore, we define a notion of the stability of the partial indices in the given class of matrix functions. This means that we will consider an allowable small perturbation such that a perturbed matrix function belong to the same class as the original one. We prove that in the class of matrix polynomials the Gohberg–Krein–Bojarsky criterion is preserved, i.e. new stability cases do not arise. Our proof of the stability criterion in this class does not use the Gohberg–Krein–Bojarsky theorem.

scholarly journals Analysis of stability problems via matrix Lyapunov functions

1990 ◽  
Vol 3 (4) ◽  
pp. 209-226 ◽  
Author(s):  
Anatoly A. Martynyuk
Keyword(s):  
Lyapunov Functions ◽  
Systems Of Nonlinear Equations ◽  
Matrix Functions ◽  
Singularly Perturbed Systems ◽  
Two Component System ◽  
Two Component ◽  
Lyapunov Matrix ◽  
Analysis Of Stability ◽  
The Stability ◽  
The Given

The stability of nonlinear systems is analyzed by the direct Lyapunov's method in terms of Lyapunov matrix functions. The given paper surveys the main theorems on stability, asymptotic stability and nonstability. They are applied to systems of nonlinear equations, singularly-perturbed systems and hybrid systems. The results are demonstrated by an example of a two-component system.

scholarly journals Symmetric spectral factorisation of self-adjoint rational matrix functions

1997 ◽  
Vol 56 (1) ◽  
pp. 95-107
Author(s):  
G.J. Groenewald ◽  
M.A. Petersen
Keyword(s):  
Matrix Function ◽  
Matrix Functions ◽  
Rational Matrix ◽  
The Right ◽  
The Given ◽  
Rational Matrix Function ◽  
Spectral Factor

For a self-adjoint rational matrix function, not necessarily analytic at infinity, the existence of a right (symmetric) spectral factorisation is described in terms of a given left spectral factorisation. The formula for the right spectral factor is given in terms of the formula for the given left spectral factor. All formulas are based on a special realisation of a rational matrix function, which is different from ones that have been used before.

Criterion for the stability of the partial indices of a circular matrix function

1988 ◽  
Vol 44 (4) ◽  
pp. 775-780 ◽  
Author(s):  
P. M. Tishin
Keyword(s):  
Matrix Function ◽  
Partial Indices ◽  
The Stability

scholarly journals On the Extended Hypergeometric Matrix Functions and Their Applications for the Derivatives of the Extended Jacobi Matrix Polynomial

2020 ◽  
Vol 2020 ◽  
pp. 1-8 ◽  
Author(s):  
Fuli He ◽  
Ahmed Bakhet ◽  
M. Abdalla ◽  
M. Hidan
Keyword(s):  
Matrix Function ◽  
Matrix Polynomial ◽  
Jacobi Matrix ◽  
Integral Representations ◽  
Matrix Functions ◽  
Matrix Polynomials ◽  
Special Cases ◽  
Generating Matrix Functions ◽  
Generating Matrix ◽  
Derivatives Of

In this paper, we obtain some generating matrix functions and integral representations for the extended Gauss hypergeometric matrix function EGHMF and their special cases are also given. Furthermore, a specific application for the extended Gauss hypergeometric matrix function which includes Jacobi matrix polynomials is constructed.

scholarly journals Exact conditions for preservation of the partial indices of a perturbed triangular 2 × 2 matrix function

2020 ◽  
Vol 476 (2237) ◽  
pp. 20200099 ◽  
Author(s):  
Victor M. Adukov ◽  
Gennady Mishuris ◽  
Sergei V. Rogosin
Keyword(s):  
Unit Circle ◽  
Matrix Function ◽  
Triangular Matrix ◽  
Specific Class ◽  
Matrix Functions ◽  
Approximate Methods ◽  
Parametric Space ◽  
Partial Indices ◽  
Unstable Set ◽  
Factorization Technique

The possible instability of partial indices is one of the important constraints in the creation of approximate methods for the factorization of matrix functions. This paper is devoted to a study of a specific class of triangular matrix functions given on the unit circle with a stable and unstable set of partial indices. Exact conditions are derived that guarantee a preservation of the unstable set of partial indices during a perturbation of a matrix within the class. Thus, even in this probably simplest of cases, when the factorization technique is well developed, the structure of the parametric space (guiding the types of matrix perturbations) is non-trivial.

scholarly journals Regular approximate factorization of a class of matrix-function with an unstable set of partial indices

2018 ◽  
Vol 474 (2209) ◽  
pp. 20170279 ◽  
Author(s):  
G. Mishuris ◽  
S. Rogosin
Keyword(s):  
Matrix Function ◽  
Stable Set ◽  
Matrix Functions ◽  
Stable Sets ◽  
Approximate Factorization ◽  
Original Matrix ◽  
Partial Indices ◽  
Unstable Set ◽  
The Matrix ◽  
The Difference

From the classic work of Gohberg & Krein (1958 Uspekhi Mat. Nauk. XIII , 3–72. (Russian).), it is well known that the set of partial indices of a non-singular matrix function may change depending on the properties of the original matrix. More precisely, it was shown that if the difference between the largest and the smallest partial indices is larger than unity then, in any neighbourhood of the original matrix function, there exists another matrix function possessing a different set of partial indices. As a result, the factorization of matrix functions, being an extremely difficult process itself even in the case of the canonical factorization, remains unresolvable or even questionable in the case of a non-stable set of partial indices. Such a situation, in turn, has became an unavoidable obstacle to the application of the factorization technique. This paper sets out to answer a less ambitious question than that of effective factorizing matrix functions with non-stable sets of partial indices, and instead focuses on determining the conditions which, when having known factorization of the limiting matrix function, allow to construct another family of matrix functions with the same origin that preserves the non-stable partial indices and is close to the original set of the matrix functions.

scholarly journals The Nevanlinna parametrization for a matrix moment problem

2001 ◽  
Vol 89 (2) ◽  
pp. 245 ◽  
Author(s):  
Pedro Lopez-Rodriguez
Keyword(s):  
Half Plane ◽  
Moment Problem ◽  
Matrix Function ◽  
Unitary Matrix ◽  
Matrix Functions ◽  
Matrix Polynomials ◽  
Analytic Matrix ◽  
The Matrix ◽  
Upper Half Plane ◽  
Matrix Moment

We obtain the Nevanlinna parametrization for an indeterminate matrix moment problem, giving a homeomorphism between the set $V$ of solutions to the matrix moment problem and the set $\mathcal V$ of analytic matrix functions in the upper half plane such that $V(\lambda )^*V(\lambda )\le I$. We characterize the N-extremal matrices of measures (those for which the space of matrix polynomials is dense in their $L^2$-space) as those whose corresponding matrix function $V(\lambda )$ is a constant unitary matrix.

A fixed point approach to the stability of sextic Lie *-derivations

Filomat ◽  
2017 ◽  
Vol 31 (15) ◽  
pp. 4933-4944
Author(s):  
Dongseung Kang ◽  
Heejeong Koh
Keyword(s):  
Functional Equation ◽  
Fixed Point ◽  
General Solution ◽  
Single Case ◽  
Fixed Point Method ◽  
Point Method ◽  
Fixed Point Approach ◽  
The Stability ◽  
The Given ◽  
Lie Derivations

We obtain a general solution of the sextic functional equation f (ax+by)+ f (ax-by)+ f (bx+ay)+ f (bx-ay) = (ab)2(a2 + b2)[f(x+y)+f(x-y)] + 2(a2-b2)(a4-b4)[f(x)+f(y)] and investigate the stability of sextic Lie *-derivations associated with the given functional equation via fixed point method. Also, we present a counterexample for a single case.

Pressure dependent ultrasonic properties of hcp hafnium metal

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Ramanshu P. Singh ◽  
Shakti Yadav ◽  
Giridhar Mishra ◽  
Devraj Singh
Keyword(s):  
Elastic Constants ◽  
Pressure Range ◽  
Room Temperature ◽  
Ultrasonic Attenuation ◽  
Coupling Constants ◽  
Ultrasonic Properties ◽  
Acoustic Coupling ◽  
Third Order Elastic Constants ◽  
The Stability ◽  
The Given

Abstract The elastic and ultrasonic properties have been evaluated at room temperature between the pressure 0.6 and 10.4 GPa for hexagonal closed packed (hcp) hafnium (Hf) metal. The Lennard-Jones potential model has been used to compute the second and third order elastic constants for Hf. The elastic constants have been utilized to calculate the mechanical constants such as Young’s modulus, bulk modulus, shear modulus, Poisson’s ratio, and Zener anisotropy factor for finding the stability and durability of hcp hafnium metal within the chosen pressure range. The second order elastic constants were also used to compute the ultrasonic velocities along unique axis at different angles for the given pressure range. Further thermophysical properties such as specific heat per unit volume and energy density have been estimated at different pressures. Additionally, ultrasonic Grüneisen parameters and acoustic coupling constants have been found out at room temperature. Finally, the ultrasonic attenuation due to phonon–phonon interaction and thermoelastic mechanisms has been investigated for the chosen hafnium metal. The obtained results have been discussed in correlation with available findings for similar types of hcp metals.

scholarly journals Some remarks on the stability of the Cauchy equation and completeness

2021 ◽  
Author(s):  
Harald Fripertinger ◽  
Jens Schwaiger
Keyword(s):  
Normed Space ◽  
Additive Function ◽  
Functional Equations ◽  
Constant Function ◽  
Cauchy Equation ◽  
Cauchy Difference ◽  
International Conference ◽  
The Difference ◽  
The Stability ◽  
The Given

AbstractIt was proved in Forti and Schwaiger (C R Math Acad Sci Soc R Can 11(6):215–220, 1989), Schwaiger (Aequ Math 35:120–121, 1988) and with different methods in Schwaiger (Developments in functional equations and related topics. Selected papers based on the presentations at the 16th international conference on functional equations and inequalities, ICFEI, Bȩdlewo, Poland, May 17–23, 2015, Springer, Cham, pp 275–295, 2017) that under the assumption that every function defined on suitable abelian semigroups with values in a normed space such that the norm of its Cauchy difference is bounded by a constant (function) is close to some additive function, i.e., the norm of the difference between the given function and that additive function is also bounded by a constant, the normed space must necessarily be complete. By Schwaiger (Ann Math Sil 34:151–163, 2020) this is also true in the non-archimedean case. Here we discuss the situation when the bound is a suitable non-constant function.

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