非时齐马氏过程的对数Sobolev不等式

本文将时齐马氏过程的经典对数Sobolev不等式推广到非时齐马氏过程,建立了非时齐马氏过程的转移半群与对数Sobolev不等式之间的关系.     

流形上无穷维扩散过程的遍历性

本文使用耦合方法研究流形上无穷维扩散过程的遍历性。所得结果在许多情形下优于Deuschel和Stroock使用对数Sobolev不等式所获得的结果。     

对称扩散过程的对数Sobolev不等式,指数可积性和熵的指数衰减性

本文讨论了对称扩散过程的指数可积性，熵的指数衰减性与Ｓｏｂｏｌｅｖ不等式之间的等价关系，并给出了对称扩散过程的对数Ｓｏｂｏｌｅｖ不等式成立的一个充分条件。     

圈上Moebius测度的Sobolev不等式（英文）

本文考虑单位圈上的Moebius测度的问题.利用文献[1]和[2]中的方法,把圈上的Moebius测度的估计转化为对一维扩散过程的相关估计上,得到了单位圈上的Moebius测度的Poincar不等式,对数Sobolev不等式和Sobolev不等式的最佳常数的双边估计.     

带有不同权函数的退化拟线性椭圆方程弱解的Holder正则性

我们研究了一类系数依赖于两不同权函数的退化拟线性椭圆方程．利用加权Sobolev不等式,加权Poincare不等式及Moser迭代技巧,得到了非负弱解的Harnack不等式,证明了弱解的H61der连续性．     

分布时滞反应扩散Hopfield神经网络的全局指数稳定性

利用拓扑度理论和广义Halanay不等式研究了分布时滞反应扩散Hopfield神经网络的平衡点的存在性及全局指数稳定性.给出的判别指数稳定性的代数判据易于验证,具有广泛适用性.     

带自扩散的N个种群的Lotka-Volterra竞争模型整体解的一致有界性

本文运用Gagliardo-Nirenberg不等式和Sobolev嵌入定理证明了带有自扩散的n个种群的Lotka-Volterra竞争模型整体解的一致有界性．     

环上含参变量的Boltzmann测度的对数Sobolev不等式

本文主要研究环上的含参变量h的Boltzmann测度μ_h的对数Sobolev不等式.通过降维方法以及对该不等式最佳常数C_LS(μ_h)的估计,证明了该测度关于h满足一致的对数Sobolev不等式,且对数Sobolev最佳常数C_LS(μ_h)在h>0时是具有常数阶的.结合已有的结果,再次佐证对数Sobolev不等式严格强于Talagrand传输不等式以及Poincaré不等式.     

三维L_p和L_∞模的离散Sobolev不等式

离散的Sobolev不等式在差分方法理论中特别是在证明差分格式稳定性和收敛性时是重要的工具.在[1—3]中,讨论了一维离散的不等式和插值公式;[4]证明了一些L_p模的离散不等式.为了研究非线性偏微分方程解法,需要多维L_∞模的离散Sobolev不等式.本文在L_ρ模不等式的基础上证明了三维L_∞模Sobolev不等式.     

三维L_p和L_∞模的离散Sobolev不等式

离散的Sobolev不等式在差分方法理论中特别是在证明差分格式稳定性和收敛性时是重要的工具.在[1—3]中,讨论了一维离散的不等式和插值公式;[4]证明了一些L_p模的离散不等式.为了研究非线性偏微分方程解法,需要多维L_∞模的离散Sobolev不等式.本文在L_ρ模不等式的基础上证明了三维L_∞模Sobolev不等式.     

One-Relator Groups of Exponential Growth Have Uniformly Exponential Growth

Mathematical Notes -     

Exponential independence

For a set $S$ of vertices of a graph $G$, a vertex $u$ in $V\left(G\right)?S$, and a vertex $v$ in $S$, let ${\mathrm{dist}}_{(G,S)}(u,v)$ be the distance of $u$ and $v$ in the graph $G?(S?\left\{v\right\})$. Dankelmann et al. (2009) define $S$ to be an exponential dominating set of $G$ if $w_{(G,S)}\left(u\right)\ge 1$ for every vertex $u$ in $V\left(G\right)?S$, where $w_{(G,S)}\left(u\right)={\sum }_{v\in S}{\left(\frac{1}{2}\right)}^{{\mathrm{dist}}_{(G,S)}(u,v)?1}$. Inspired by this notion, we define $S$ to be an exponential independent set of $G$ if $w_{(G,S?\left\{u\right\})}\left(u\right)<1$ for every vertex $u$ in $S$, and the exponential independence number ${\alpha }_e\left(G\right)$ of $G$ as the maximum order of an exponential independent set of $G$.Similarly as for exponential domination, the non-local nature of exponential independence leads to many interesting effects and challenges. Our results comprise exact values for special graphs as well as tight bounds and the corresponding extremal graphs. Furthermore, we characterize all graphs $G$ for which ${\alpha }_e\left(H\right)$ equals the independence number $\alpha \left(H\right)$ for every induced subgraph $H$ of $G$, and we give an explicit characterization of all trees $T$ with ${\alpha }_e\left(T\right)=\alpha \left(T\right)$.     

Exponential epigroups

We show that an exponential epigroup is a band of unipotent exponential epigroups (in which the set of idempotents forms a subsemigroup). Also, we investigate the structure of unipotent exponential epigroups. In addition, a characterization of the least normal band of groups congruence is given for an arbitrary medial epigroup. It is also shown that k-exponential eventually regular semigroups are necessarily epigroups.     

Exponential Function

We give the meaning to expressions of the form a^r
Where the base is a a positive real number and the exponent is r a positive real number.     

Exponential Renormalization

Moving beyond the classical additive and multiplicative approaches, we present an “exponential” method for perturbative renormalization. Using Dyson’s identity for Green’s functions as well as the link between the Faà di Bruno Hopf algebra and the Hopf algebras of Feynman graphs, its relation to the composition of formal power series is analyzed. Eventually, we argue that the new method has several attractive features and encompasses the BPHZ method. The latter can be seen as a special case of the new procedure for renormalization scheme maps with the Rota–Baxter property. To our best knowledge, although very natural from group-theoretical and physical points of view, several ideas introduced in the present paper seem to be new (besides the exponential method, let us mention the notions of counter-factors and of order n bare coupling constants).     

Exponential Structures

Generating functions of the type occur throughout enumerative combinatorics. We give a general setting for obtaining such generating functions in which the numbers M (n) have an explicit combinatorial meaning. Applications are given to the computation of Möbius functions and related invariants of certain posets.     

Exponential nonnegativity

Let be a Banach algebra, , the spectrum of and the spectral abscissa of . If , then we show that there exists an algebra cone such that is exponentially nonnegative with respect to and the spectral radius is increasing on .

     

Exponential statistical manifold

We consider the non-parametric statistical model ε(p) of all positive densities q that are connected to a given positive density p by an open exponential arc, i.e. a one-parameter exponential model p(t), t ∈ I, where I is an open interval. On this model there exists a manifold structure modeled on Orlicz spaces, originally introduced in 1995 by Pistone and Sempi. Analytic properties of such a manifold are discussed. Especially, we discuss the regularity of mixture models under this geometry, as such models are related with the notion of e- and m-connections as discussed by Amari and Nagaoka.     

Exponential integrability of martingales

We obtain the exponential integrability of the maximal function, the quadratic variation and the conditional quadratic variation of bounded martingales and exponential integrable martingales.