Separating inequalities for non-negative covariances

We demonstrate explicit inequalities which separate covariances generated by non-negative stochastic processes from the class consisting of all non-negative covariances. The first class of covariances corresponds to completely positive matrices whereas the second class corresponds to doubly non-negative matrices. A key ingredient in our analysis is a new result the "convexity" of quatratic functions defined on the surface of a sphere.     

Separating inequalities for non-negative covariances

We demonstrate explicit inequalities which separate covariances generated by non-negative stochastic processes from the class consisting of all non-negative covariances. The first class of covariances corresponds to completely positive matrices whereas the second class corresponds to doubly non-negative matrices. A key ingredient in our analysis is a new result the "convexity" of quatratic functions defined on the surface of a sphere.     

Harnack inequalities for graphs with non-negative Ricci curvature

We establish a Harnack inequality for finite connected graphs with non-negative Ricci curvature. As a consequence, we derive an eigenvalue lower bound, extending previous results for Ricci flat graphs.     

Prophet inequalities for averages of independent non-negative random variables

Mathematische Zeitschrift -     

Nash type inequalities for fractional powers of non-negative self-adjoint operators

Assuming that a Nash type inequality is satisfied by a non-negative self-adjoint operator , we prove a Nash type inequality for the fractional powers of . Under some assumptions, we give ultracontractivity bounds for the semigroup generated by .

     

Martingale inequalities for spline sequences

Positivity - We show that D. Lépingle’s $$L_1(ell _2)$$-inequality $$begin{aligned} left| left( sum _n {mathbb {E}}[f_n | {mathscr {F}}_{n-1}]^2 right) ^{1/2}right|...     

Uniqueness result for non-negative solutions of semi-linear inequalities on Riemannian manifolds

We consider certain semi-linear partial differential inequalities on complete connected Riemannian manifolds and provide a simple condition in terms of volume growth for the uniqueness of a non-negative solution. We also show the sharpness of this condition.     

A class of gap functions for variational inequalities

Recently Auchmuty (1989) has introduced a new class of merit functions, or optimization formulations, for variational inequalities in finite-dimensional space. We develop and generalize Auchmuty's results, and relate his class of merit functions to other works done in this field. Especially, we investigate differentiability and convexity properties, and present characterizations of the set of solutions to variational inequalities. We then present new descent algorithms for variational inequalities within this framework, including approximate solutions of the direction finding and line search problems. The new class of merit functions include the primal and dual gap functions, introduced by Zuhovickii et al. (1969a, 1969b), and the differentiable merit function recently presented by Fukushima (1992); also, the descent algorithm proposed by Fukushima is a special case from the class of descent methods developed in this paper. Through a generalization of Auchmuty's class of merit functions we extend those inherent in the works of Dafermos (1983), Cohen (1988) and Wu et al. (1991); new algorithmic equivalence results, relating these algorithm classes to each other and to Auchmuty's framework, are also given.Corresponding author.     

The Gagliardo-Nirenberg inequalities and manifolds of non-negative Ricci curvature

This article is devoted to show that complete non-compact Riemannian manifolds with non-negative Ricci curvature of dimension greater than or equal to two in which some Gagliardo-Nirenberg type inequality holds are not very far from the Euclidean space.     

A class of history-dependent variational-hemivariational inequalities

We consider a new class of variational-hemivariational inequalities which arise in the study of quasistatic models of contact. The novelty lies in the special structure of these inequalities, since each inequality of the class involve unilateral constraints, a history-dependent operator and two nondifferentiable functionals, of which at least one is convex. We prove an existence and uniqueness result of the solution. The proof is based on arguments on elliptic variational-hemivariational inequalities obtained in our previous work [23], combined with a fixed point result obtained in [30]. Then, we prove a convergence result which shows the continuous dependence of the solution with respect to the data. Finally, we present a quasistatic frictionless problem for viscoelastic materials in which the contact is modeled with normal compliance and finite penetration and the elasticity operator is associated to a history-dependent Von Mises convex. We prove that the variational formulation of the problem cast in the abstract setting of history-dependent quasivariational inequalities, with a convenient choice of spaces and operators. Then we apply our general results in order to prove the unique weak solvability of the contact problem and its continuous dependence on the data.     

A Newton method for a class of quasi-variational inequalities

A variant of the Newton method for nonsmooth equations is applied to solve numerically quasivariational inequalities with monotone operators. For this purpose, we investigate the semismoothness of a certain locally Lipschitz operator coming from the quasi-variational inequality, and analyse the generalized Jacobian of this operator to ensure local convergence of the method. A simplified variant of this approach, applicable to implicit complementarity problems, is also studied. Small test examples have been computed.This work has been supported in parts by a grant from the German Scientific Foundation and by a grant from the Czech Academy of Sciences.     

A class of means and related inequalities

The aim of this paper is two-fold: First we describe a certain class of strictly positive, continuous functions u:I defined on certain intervals I]0,+[, and demonstrate that each function u of this class permits the definition of an associated mean M_u (a₁,...,a_n) for any finite number a₁,...,a_n of numbers in I. The arithmetic and the geometric mean are special cases of these u-means M_u. -Thereafter, we improve the classical inequality between the geometric and the arithmetic mean and derive from it a corresponding inequality between certain u-means and the arithmetic mean.     

A class of quasilinear elliptic hemivariational inequalities

1. IntroductionIn this paper we shall deal with the quasilinear elliptic hemivaxiational inequalitieswhere the symbol OJ designates Clarke's generalized gradiellt of a locally Lipschitz functionalJ (see [1, 2]). The discontinuity is assumed to be in the lower order term OJ(u(x)). If Ais a linear operator and the disconiinuity is monotone, the problems have been studied,for example, in [3-6]. If A is a linear opertor and the discolltinuity is nonmonotone, thiskind of problems have been inve…     

A new method for a class of linear variational inequalities

In this paper we introduce a new iterative scheme for the numerical solution of a class of linear variational inequalities. Each iteration of the method consists essentially only of a projection to a closed convex set and two matrix-vector multiplications. Both the method and the convergence proof are very simple.This work is supported by the National Natural Science Foundation of the P.R. China and NSF of Jiangsu.     

A new projection method for a class of variational inequalities

In this paper, we revisit the numerical approach to variational inequality problems involving strongly monotone and Lipschitz continuous operators by a variant of projected reflected gradient method. Contrary to what done so far, the resulting algorithm uses a new simple stepsize sequence which is diminishing and nonsummable. This brings the main advantages of the algorithm where the construction of aproximation solutions and the formulation of convergence are done without the prior knowledge of the Lipschitz and strongly monotone constants of cost operators. The assumptions in the formulation of theorem of convergence are also discussed in this paper. Numerical results are reported to illustrate the behavior of the new algorithm and also to compare with others.