On the Robust Stability of 2D Schur Polynomials

In this note, the problem of the robust stability for a two-dimensional (two-variable) Schur polynomial which is the characteristic polynomial of a discrete-time linear time-invariant system is investigated. A new approach based on the Rouché theorem is adopted. The extension to the robust stability for multidimensional (multivariable) polynomials is also provided. Interesting sufficient conditions for such robust stability are derived. A two-dimensional example is included to support the theoretical result.     

Optimum Radius of Robust Stability for Schur Polynomials

The paper investigates the problem of the robust stability of Schur polynomials. Recently, a new approach based on the Rouche theorem of classical complex analysis has been adopted for the solution of this problem. In this paper, an improvement of the previous solution is presented. This is the optimum solution of the robust stability problem for Schur polynomials, which is obtained by solving a minimization problem and is better than other methods in robust stability literature. Three numerical examples are given to illustrate the proposed method.     

On the Optimum Radius of Robust Stability for Schur Polynomials

A recent paper (Ref. 1) established a new approach to estimate the robust stability radius of a Schur polynomial. This note points out that the approach given in Ref. 1 is not correct and also gives a counterexample to the main result of Ref. 1.     

On Roots and Error Constants of Optimal Stability Polynomials

Optimal stability polynomials are polynomials whose stability region is as large as possible in a certain region, here the negative real axis. We are interested in such polynomials which in addition, obey a certain order condition. An important application of these polynomials is the construction of stabilized explicit Runge-Kutta methods. In this paper we will give some properties of the roots of these polynomials, and prove that their error constant is always positive. Furthermore, for a given order, the error constant decreases as the degree increases.     

Evaluation of Certain Integrals Involving the Product of Classical Hermite's Polynomials Using Laplace Transform Technique and Hypergeometric Approach

In this paper some novel integrals associated with the product of classical Hermite's polynomials ∫_(-∞)~(+∞)(x~2)~mexp(-x~2){Hr (x)}2dx, ∫_0~∞exp(-x~2)H_(2k)(x)H_(2s+1)(x)dx,∫_0~∞exp(-x~2)H_(2k)(x)H_(2s)(x)dx and ∫_0~∞exp(-x~2)H_(2k+1)(x)H_(2s+1)(x)dx,are evaluated using hypergeometric approach and Laplace transform method, which is a different approach from the approaches given by the other authors in the field of special functions. Also the results may be of significant nature, and may yield numerous other interesting integrals involving the product of classical Hermite's polynomials by suitable simplifications of arbitrary parameters.     

单位圆周上正交多项式渐近分析的Riemann-Hilbert方法

在单位圆周上考虑带特定权函数的正交多项式,利用Deift P．和Zhou X．所引进的关于振荡型Riemann-Hilbert问题的最速下降法,建立了这类正交多项式在整个复平面上的强渐近公式,发展和改进了一些经典结果．     

波动方程反演问题的一种新的逼近方法

本文给出了一种求解波动方程反演问题的“多目标函数法”.这种方法简单、有效,并具有明确的物理意义.对于三维问题的程序化它有很强的优越性.     

Asymptotics of Orthogonal Polynomials: Some Old, Some New, Some Identities

We briefly review some asymptotics of orthonormal polynomials. Then we derive the Bernstein–Szeg, the Riemann–Hilbert (or Fokas–Its–Kitaev), and Rakhmanov projection identities for orthogonal polynomials and attempt a comparison of their applications in asymptotics.     

不确定中立型线性随机时滞系统的鲁棒稳定性

建立了中立型随机微分时滞方程的LaSalle不变原理,然后应用LaSalle不变原理讨论了不确定中立型随机时滞系统的随机渐近稳定和几乎必然指数稳定的代数判据, 同时给出示例说明结果的有效性．     

Inversion of Analytic Functions via Canonical Polynomials: A Matrix Approach

An alternative to Lagrange inversion for solving analytic systems is our technique of dual vector fields. We implement this approach using matrix multiplication that provides a fast algorithm for computing the coefficients of the inverse function. Examples include calculating the critical points of the sinc function. Maple procedures are included which can be directly translated for doing numerical computations in Java or C. A preliminary version of this paper has been presented at AISC 2006.     

Integer Roots Chromatic Polynomials of Non-Chordal Graphs and the Prouhet-Tarry-Escott Problem

In this paper, we give an affirmative answer to a question of Dmitriev concerning the existence of a non-chordal graph with a chordless cycle of order n whose chromatic polynomial has integer roots for a few values of n, extending prior work of Dong et al. Received: April, 2003     

高阶Bernoulli多项式和高阶Euler多项式的新计算公式

使用发生函数方法,利用两种第一类Stirling数给出高阶Bernoulli多项式和高阶Euler多项式的简捷计算公式.     

A Stability Estimate for a Solution in the Inverse Problem of Finding the Coefficients of Lower Derivatives

We consider the problem of finding the coefficients of the first derivatives in a second-order hyperbolic equation. The additional information is the trace of a solution and its normal derivative on the lateral surface of the cylindrical domain of some direct problem for the original equation. The impulse point source lies outside the domain in which the sought coefficients are determined and is a parameter of the problem. We suppose that the number of sources for which the trace of a solution is given coincides with the number of the coefficients to be determined. The main result of this article is a stability estimate for a solution to the inverse problem under consideration.     

A Stability Estimate for a Solution of the Problem of Determining the Dielectric Permittivity

We consider the problem of determining the dielectric permittivity for a nonconducting and nonmagnetic medium. As information we take the traces of the tangential components of the electromagnetic field on the lateral surface of a cylindrical domain. These traces correspond to a solution to some direct problem for the Maxwell system. The impulse source of the current flux lies outside the domain in which the coefficient is sought. The main result of the article is a stability estimate for a solution to the inverse problem in question.     

Von Neumann-Wigner theorem: Level repulsion and degenerate eigenvalues

We investigate the spectral properties of Schrödinger operators with point interactions, focusing attention on the interplay between level repulsion (von Neumann-Wigner theorem) and the symmetry of the confi.guration of point interactions. The explicit solution of the problem allows observing level repulsion for two centers. For a large number of centers, we investigate the families of point interactions leading to the maximum degeneracy.     

A Stability Estimate for a Solution to the Problem of Determination of Two Coefficients of a Hyperbolic Equation

We consider the problem of determination of two coefficients (x) and q(x) in a hyperbolic equation. Here (x) is the coefficient of the first derivative with respect to t and q(x) is the coefficient of the solution itself. We suppose that these coefficients are small in some norm and supported in a disk D. Oscillations are excited by the impulse function (t)(x·) supported on the straight line t=0, x·=0. Here is a unit vector playing the role of a parameter of the problem. The acoustic field generated by this source lying outside D is measured at the points of the boundary of D together with the normal derivative on some time interval of a fixed length T for two different values of the parameter . We prove that, for a sufficiently large T, the given information determines the sought coefficients uniquely. We obtain a stability estimate for a solution to the problem.     

匹配对策模型的核心稳定性

本文研究匹配合作对策模型的核心稳定性。基于线性规划对偶理论和图论的相关知识，我们首先证明了匹配对策有稳定核心当且仅当其基础二部图有完美匹配。其次我们讨论了几个与核心稳定性密切相关的性质（核心的包容性、对策的精确性和可扩性）并证明了它们的等价性。基于这些结果，我们还讨论了相应问题的算法。     

一般Lurie系统的鲁棒绝对稳定新准则(英文)

本文讨论了具多非线性模不确定的一般Lurie系统的鲁棒问题.用Lyapunov函数方法,得到了一些新的鲁棒绝对稳定代数准则,这些准则减少了现有的结果的保守性.     

最优稳定多项式的根及误差常数

本文研究了最优稳定多项式的根的一些性质 ,并证明了误差常数总是正的 ,而且 ,对给定的阶 ,误差常数随着多项式次数的增加而减小 .     

On the Stability of One Characterization of Stable Distributions

In 1923, G. Polya proved that if X ₁ and X ₂ are independent identically distributed random variables (i.i.d.r.v.) with finite variance, then the distributions of X ₁ and (X ₁+X ₂)/ $\sqrt 2$ are coincidental iff X ₁ has the normal distribution with zero mean. Is an analogous theorem possible for an couple of statistics X ₁ and (X ₁+X ₂)/2^1/α if α<2? P. Lévy constructed an example that denies that hypothesis. However, having supplemented the condition of coincidence of the distributions of X ₁ and (X ₁+X ₂)/2^1/α with a similar condition, namely, requiring, in addition, for the distributions of X ₁ and (X ₁+X ₂+X ₃)/3^1/α to be coincident (here X ₁,X ₂ and X ₃ are i.i.d.r.v.), P. Lévy has proved that X ₁ and X ₂ have a strictly stable distribution. The stability of this characterization in a metric λ₀ (that is defined in the class of characteristic functions by analogy with a uniform metric defined in the class of distributions) without an additional symmetry assumption as well as the stability in a Lévy metric L are analizied in this paper.