Abstract. We derive the sharp constants for the inequalities on the Heisenberg group H whose analogues on Euclidean space R are the well known Hardy-Littlewood-Sobolev inequalities. Only one special case had been known previously, due to Jerison-Lee more than twenty years ago. From these inequalities we obtain the sharp constants for their duals, which are the Sobolev inequalities for the Lapla...

We consider the network design problem which consists in determining at minimum cost a 2-edge connected network such that the shortest cycle (a “ring”) to which each edge belongs, does not exceed a given length K. We identify a class of valid inequalities, called cycle inequalities, valid for the problem and show that this inequalities together with the so-called cut inequalities yield an integ...

A new class of system of generalized parametric nonlinear quasivariational inequalities involving various classes of mappings is introduced and studied. With the properties of maximal monotone mappings, the equivalence between the class of system of generalized parametric nonlinear quasivariational inequalities and a class of fixed point problems is proved and an iterative algorithm with errors...

MOSER-TRUDINGER AND ADAMS TYPE INEQUALITIES AND THEIRAPPLICATIONSbyNGUYEN LAMAugust 2014Advisor: Dr. Guozhen LuMajor: MathematicsDegree: Doctor of PhilosophyIn this dissertation, we study some variants of the Moser-Trudinger inequalities and Adamsinequalities. The proofs of these inequalities relied crucially on the symmetrization arguments inthe lite...

We study mixed integer conic sets involving a general regular (closed, convex, full dimensional, and pointed) cone K such as the nonnegative orthant, the Lorentz cone or the positive semidefinite cone. In a unified framework, we introduce K-minimal inequalities and show that under mild assumptions, these inequalities together with the trivial cone-implied inequalities are sufficient to describe...

BACKGROUND The magnitude of educational inequalities in mortality avoidable by medical care in 16 European populations was compared, and the contribution of inequalities in avoidable mortality to educational inequalities in life expectancy in Europe was determined. METHODS Mortality data were obtained for people aged 30-64 years. For each country, the association between level of education an...

In this paper, we establish a one-to-one correspondence between information inequalities and group inequalities. The major implication of our result is that we can prove information inequalities by proving the corresponding group inequalities, and vice versa. By giving a group-theoretic proof for all Shannon-type inequalities, we suggest that new inequalities could be discovered by making use o...

This note proves sharp affine Gagliardo-Nirenberg inequalities which are stronger than all known sharp Euclidean Gagliardo-Nirenberg inequalities and imply the affine L−Sobolev inequalities. The logarithmic version of affine L−Sobolev inequalities is verified. Moreover, An alternative proof of the affine Moser-Trudinger and Morrey-Sobolev inequalities is given. The main tools are the equimeasur...

In this paper, we present procedures to obtain facet-defining inequalities for the independence system polytope. These procedures are defined for inequalities which are not necessarily rank inequalities. We illustrate the use of these procedures by deriving strong valid inequalities for the acyclic induced subgraph, triangle free induced subgraph, bipartite induced subgraph, and knapsack polyto...

Despite the great success of quantum mechanics, questions regarding its application still exist and the boundary between quantum and classical mechanics remains unclear. Based on the philosophical assumptions of macrorealism and noninvasive measurability, Leggett and Garg devised a series of inequalities (LG inequalities) involving a single system with a set of measurements at different times. ...