Decoupling inequalities with exponential constants

A Remark on the Inequalities of Bernstein - Markov Type in Exponential Orlicz and Lorentz Spaces

We prove in this article the generalizations on the exponential Orlicz spaces Markov's - Bernstein's inequalities for algebraic polynomials and rational functions.

Nikolskii Type Inequalities for Entire Functions of Exponential Type in Lorentz Zygmund Spaces

Nikolskii type inequalities for entire functions of exponential type on Rn for the Lorentz Zygmund spaces are obtained. Some new limiting cases are examined. Application to Besov type spaces of logarithmic smoothness is given.

Sharp exponential inequalities for the Ornstein-Uhlenbeck operator

The optimal constants in a class of exponential type inequalities for the Ornstein-Uhlenbeck operator in the Gauss space are detected. The existence of extremal functions in the relevant inequalities is also established. Our results disclose analogies and dissimilarities in comparison with Adams' inequality for the Laplace operator, a companion of our inequalities in the Euclidean space.

Exponential inequalities for positive linear mappings

In this article, we present exponential-type inequalities for positive linear mappings and Hilbert space operators, by means of convexity and the Mond-Pe\v carić method. The obtained results refine and generalize some known results. As an application, we present extensions for operator-like geometric and harmonic means.

Decoupling Inequalities for the Tail Probabilities of Multivariate U-statistics

In this paper the following result, which allows one to decouple U-Statistics in tail probability, is proved in full generality. Theorem 1. Let $X_i$ be a sequence of independent random variables taking values in a measure space $S$, and let $f_{i_1...i_k}$ be measurable functions from $S^k$ to a Banach space $B$. Let $(X_i^{(j)})$ be independent copies of $(X_i)$. The following inequality holds for all $t \ge 0$ and all $n\ge 2$, $$ P(||\sum_{1\le i_1 \ne ... \ne i_k \le n} f_{i_1 ... i_k}(X_{i_1},...,X_{i_k}) || \ge t) \qquad\qquad$$ $$ \qquad\qquad\le C_k P(C_k||\sum_{1\le i_1 \ne ... \ne i…

Hitting times, functional inequalities, lyapunov conditions and uniform ergodicity

The use of Lyapunov conditions for proving functional inequalities was initiated in [5]. It was shown in [4, 30] that there is an equivalence between a Poincar{é} inequality, the existence of some Lyapunov function and the exponential integrability of hitting times. In the present paper, we close the scheme of the interplay between Lyapunov conditions and functional inequalities by $\bullet$ showing that strong functional inequalities are equivalent to Lyapunov type conditions; $\bullet$ showing that these Lyapunov conditions are characterized by the finiteness of generalized exponential momen…

用于时滞神经网络系统改进指数稳定性分析的加权积分不等式的进一步分析

Further analysis of weighted integral inequalities for improved exponential stability analysis of time delay neural networks systems

该研究针对时滞神经网络系统的指数稳定性分析，提出带加权正交多项式的新型加权积分不等式与加权互凸不等式，结合Lyapunov-Krasovskii泛函和线性矩阵不等式得到新的稳定性判据，经数值算例验证有效。

The proof of three power-exponential inequalities

In this paper we prove three power-exponential inequalities for positive real numbers. In particular, we conclude that this proofs give affirmatively answers to three, until now, open problems (conjectures~4.4, 2.1 and 2.2) posed by C{î}rtoaje in the following two works: "{\it J. Inequal. Pure Appl. Math.} 10, Article 21, 2009" and "{\it J. Nonlinear Sci. Appl.} 4:2:130-137, 2011". Moreover, we present a new proof of the inequality $a^{ra}+b^{rb}\ge a^{rb}+b^{ra}$ for all positive real numbers $a$ and $b$ and $r\in [0,e]$. In addition, three new conjectures are presented.

Exponential integrability in the spirit of Moser-Trudinger's inequalities of functions with finite non-local, non-convex energy

Let $d \ge 1$, $p \ge d$, and let $Ω$ be a smooth bounded open subset of $\mathbb{R}^d$. We prove some exponential integrability in the spirit of Moser-Trudinger's inequalities for measurable functions $u$ defined in $Ω$ such that $$ \mathop{\int_Ω \int_Ω}_{|u(x) - u(y)| > δ} \frac{1}{|x-y|^{d+p}} \, dx \, dy < + \infty, $$ for some $δ> 0$. This double integral appeared in characterizations of Sobolev spaces and involved in improvements of the Sobolev inequaliies, Poincaré inequalities, and Hardy inequalities.

非交换鞅的大偏差不等式

Large Deviation Inequalities for Noncommutative Martingales

本文研究非交换随机变量的大偏差不等式，建立非交换独立情况下的均匀指数可积性刻画，推导非交换鞅差的两种偏差不等式，并提出非交换Gordin分解以推导非交换遍历定理。

相依二元随机变量的矩阵集中不等式

Matrix concentration inequalities for dependent binary random variables

本文针对满足特定独立性假设的二元随机变量建立矩阵集中不等式，并利用解耦不等式将随机子矩阵算子范数估计推广到一般强 Rayleigh 抽样方案。

Bernstein-Nikolskii and Plancherel-Polya inequalities in $L_{p}$-norms on non-compact symmetric spaces

By using Bernstein-type inequality we define analogs of spaces of entire functions of exponential type in $L_{p}(X), 1\leq p\leq \infty$, where $X$ is a symmetric space of non-compact. We give estimates of $L_{p}$-norms, $1\leq p\leq \infty$, of such functions (the Nikolskii-type inequalities) and also prove the $L_{p}$- Plancherel-Polya inequalities which imply that our functions of exponential type are uniquely determined by their inner products with certain countable sets of measures with compact supports and can be reconstructed from such sets of "measurements" in a stable way.

Exponential decay for sc-gradient flow lines

In this paper we introduce the notion of sc-action functionals and their sc-gradient flow lines. Our approach is inspired by Floer's unregularized gradient flow. The main result of this paper is that under a Morse condition sc-gradient flow lines have uniform exponential decay towards critical points. The ultimate goal for the future is to construct a M-polyfold bundle over a M-polyfold such that the space of broken sc-gradient flow lines is the zero set of a appropriate sc-section. Here uniform exponential decay is essential. Of independent interest is that we derive exponential decay estimat…