稳定多项式

The Electrostatic Properties of Zeros of Exceptional Laguerre and Jacobi Polynomials and stable interpolation

We will examine the electrostatic properties of exceptional and regular zeros of $X_m$-Laguerre and $X_m$-Jacobi polynomials. Since there is a close connection between the electrostatic properties of the zeros and the stability of interpolation on the system of zeros, we can deduce an Egerváry-Tur\' an type result as well. The limit of the energy on the regular zeros are also investigated.

Algebraic and qualitative aspects of quadratic vector fields related with classical orthogonal polynomials

This paper is a sequel of the reference \cite[§4.2, p.p. 1782--1783]{almp}, in where some families of quadratic polynomial vector fields related with orthogonal polynomials were studied. We extend such results that contain some details related with differential Galois Theory as well the inclusion of Darboux theory of integrability and qualitative theory of dynamical systems.

正交多项式

Orthogonal Polynomials

在这项调查中，不同方面的理论正交多项式的一个（实或复）变量进行审查。未讨论单位圆上的正交多项式。

Generalized Hurwitz polynomials

We describe a wide class of polynomials, which is a natural generalization of Hurwitz stable polynomials. We also give a detailed account of so-called self-interlacing polynomials, which are dual to Hurwitz stable polynomials but have only real and simple zeroes. All proofs are given using properties of rational functions mapping the upper half-plane of the complex plane to the lower half-plane. Matrices with self-interlacing spectra and other applications of generalized Hurwitz polynomials are discussed.

$σ$-Stable Matrices

$σ$-Stable matrices are introduced and it is shown that the real roots of the polynomials comprising the coefficients of the characteristic polynomial indicate the coefficient sign changes. A proof of Obrechkoff is then used to show that the largest real root from the coefficients is the point of stability when the maximal eigenvalue of the $σ$-stable matrix is in $\mathbb{R}$. Some implications of the coefficient behaviour for a scaling relation are then discussed.

Global bounds on stable polynomials

A classical inequality of Szász bounds polynomials with no zeros in the upper half plane entirely in terms of their first few coefficients. Borcea-Brändén generalized this result to several variables as a piece of their characterization of linear maps on polynomials preserving stability. In this paper, we improve Szász's original inequality, use determinantal representations to prove Szász type inequalities in two variables, and then prove that one can use the two variable inequality to prove an inequality for several variables.

指数稳定的非线性系统在有界区域上具有多项式李雅普诺夫函数

Exponentially Stable Nonlinear Systems have Polynomial Lyapunov Functions on Bounded Regions

证明了在有界集上充分光滑的非线性常微分方程指数稳定的充分必要条件是存在多项式李雅普诺夫函数.主要结果是：如果存在一个n次连续可微的李雅普诺夫函数证明在R^n的一个有界子集上指数稳定，则存在一个多项式李雅普诺夫函数证明在同一区域上指数稳定.例如，如果微分方程的右侧是多项式或至少n次连续可微，则这种连续李雅普诺夫函数将存在。证明是基于推广的维尔斯特拉斯逼近定理的可微函数在几个变量。具体地说，我们展示了如何使用多项式来逼近Sobolev范数W^{1，\infty }中的可微函数以达到任何期望的精度。我们将这个近似结果与二阶泰勒级数展开相结合，发现多项式李雅普诺夫函数可以在有界集上任意好地逼近连续李雅普诺夫函数。我们的调查是出于使用多项式优化算法来构造多项式李雅普诺夫函数。

q-Charlier多项式的组合数学

The combinatorics of q-Charlier polynomials

我们描述了各个方面的铝萨拉姆-Carlitz $q$ -Charlier多项式。这些包括组合描述的时刻，正交关系，和线性化系数。

乘积域中的稳定多项式与容许分子

Stable polynomials and admissible numerators in product domains

本文研究了在多圆盘内无零点的多项式 $p$ 的容许分子理想，即在边界零点附近使 $q/p$ 有界的多项式 $q$ 的理想，并在零点集光滑非退化及孤立零点情形下给出了理想描述。