Spectral density functions of bivariable stable polynomials

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。

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。

非稳定范围内标量值多项式的变量分离

Separation of Variables for Scalar-valued Polynomials in the Non-stable Range

本文研究了非稳定范围内标量值多项式的变量分离问题，证明了条件 $n \geq 2k-1$ 是分离唯一性的充要条件，并给出了非稳定范围内乘法映射核的生成元最高权公式、Hilbert 级数及生成向量列表。

关于行列式和多项式的问题

Questions about determinants and polynomials

本文讨论了系数为实根多项式系数行列式的多项式的实根性的几个问题。我们还考虑了矩阵推广为全正矩阵的一些问题，即全稳定矩阵和全上矩阵。

Approximating real-rooted and stable polynomials, with combinatorial applications

Let $p(x)=a_0 + a_1 x + \ldots + a_n x^n$ be a polynomial with all roots real and satisfying $x \leq -δ$ for some $0<δ<1$. We show that for any $0 < ε<1$, the value of $p(1)$ is determined within relative error $ε$ by the coefficients $a_k$ with $k \leq {c \over \sqrtδ} \ln {n \over ε\sqrt{ δ}}$ for some absolute constant $c > 0$. Consequently, if $m_k(G)$ is the number of matchings with $k$ edges in a graph $G$, then for any $0 < ε< 1$, the total number $M(G)=m_0(G)+m_1(G) + \ldots $ of matchings is determined within relative error $ε$ by the numbers $m_k(G)$ with $k \leq c \sqrtΔ \ln (v /ε)$…

度量空间的爆破多项式：与稳定多项式、图及其距离谱的联系

The blowup-polynomial of a metric space: connections to stable polynomials, graphs and their distance spectra

本文研究了度量空间的爆破多项式，将其与稳定多项式、图及其距离谱联系起来，证明了其多项式性质，并提出新的delta-模体。

Matrix-valued Hermitian Positivstellensatz, lurking contractions, and contractive determinantal representations of stable polynomials

We prove that every matrix-valued rational function $F$, which is regular on the closure of a bounded domain $\mathcal{D}_\mathbf{P}$ in $\mathbb{C}^d$ and which has the associated Agler norm strictly less than 1, admits a finite-dimensional contractive realization $$F(z)= D + C\mathbf{P}(z)_n(I-A\mathbf{P}(z)_n)^{-1} B. $$ Here $\mathcal{D}_\mathbf{P}$ is defined by the inequality $\|\mathbf{P}(z)\|<1$, where $\mathbf{P}(z)$ is a direct sum of matrix polynomials $\mathbf{P}_i(z)$ (so that appropriate Archimedean and approximation conditions are satisfied), and $\mathbf{P}(z)_n=\bigoplus_{i=1}…

调和平均、随机多项式与随机阵

Harmonic mean, random polynomials and stochastic matrices

受学习理论中一个问题的启发，我们研究了一类随机矩阵的优势特征值。这与随机多项式的导数的根有关（通过在区间[0，1]中随机均匀地选取它们的根来生成，尽管我们的结果扩展到其他分布）。这反过来又要求研究上述随机变量的调和平均的统计行为，这反过来又导致我们对稳定律的收敛速度收敛速率和稳定律的尾部估计的微妙问题。