We consider a large class of differential inequalities on complete connected Riemannian manifolds, and provide a sufficient condition in terms of volume growth for the uniqueness of nonnegative solutions to the differential inequalities.

The moment-entropy inequality shows that a continuous random variable with given second moment and maximal Shannon entropy must be Gaussian. Stam’s inequality shows that a continuous random variable with given Fisher information and minimal Shannon entropy must also be Gaussian. The CramérRao inequality is a direct consequence of these two inequalities. In this paper the inequalities above are ...

Logarithmic Sobolev inequalities are a well-studied technique for estimating rates of convergence of Markov chains to their stationary distributions. In contrast to continuous state spaces, discrete settings admit several distinct log Sobolev inequalities, one of which is the subject of this paper. Here we derive modi!ed log Sobolev inequalities for some models of random walk, including the ran...

1.1. Azuma-Hoeffding Inequality. Concentration inequalities are inequalities that bound probabilities of deviations by a random variable from its mean or median. Our interest will be in concentration inequalities in which the deviation probabilities decay exponentially or superexponentially in the distance from the mean. One of the most basic such inequality is the Azuma-Hoeffding inequality fo...

Abstract: In this paper, motivated by the research on qualitative properties of solutions for fractional differential-integro equations, some new Gronwall-Bellman type inequalities in two independent variables are established. Based on these inequalities, explicit bounds for unknown functions concerned are derived. As for applications, we apply the inequalities established to research boundedne...

This paper derives exponential concentration inequalities and polynomial moment inequalities for the spectral norm of a random matrix. The analysis requires a matrix extension of the scalar concentration theory developed by Sourav Chatterjee using Stein’s method of exchangeable pairs. When applied to a sum of independent random matrices, this approach yields matrix generalizations of the classi...

This paper introduces and studies the class of differential variational inequalities (DVIs) in a finite-dimensional Euclidean space. The DVI provides a powerful modeling paradigm for many applied problems in which dynamics, inequalities, and discontinuities are present; examples of such problems include constrained time-dependent physical systems with unilateral constraints, differential Nash g...