In this paper, we will show how one is able to construct a lattice on the completion of an algebra and to obtain an isomorphism to its Caratheodory Extension. In addition, it will be shown that the lattice form a σ-algebra and a complete Heyting algebra of countable type.

Here and in what follows we assume S is a finite multiset in R d. This means tha t the points in S have "multiplicity". Strict ly speaking, a multiset is a map S : F --, R d and in our case F is finite. Then corekS = M{convS(E) : E C_ F, IF \ El _< k}. From now on we do not say explicitly tha t the sets in question are multisets. This will make the no ta t ion simpler and will not cause confusi...

We consider the multi-marginal optimal transport of aligning several compactly supported marginals on Heisenberg group to minimize total cost, which we take be sum squared Carnot-Caratheodory distances from marginal points their barycenter. Under certain technical hypotheses, prove existence and uniqueness maps. also point out related open questions.

An important problem in quantum information theory is the quantification of entanglement in multipartite mixed quantum states. In this work, a connection between the geometric measure of entanglement and a distance measure of entanglement is established. We present a new expression for the geometric measure of entanglement in terms of the maximal fidelity with a separable state. A direct applic...

This is one of a series of papers examining the interplay between differentiation theory for Lipschitz maps, X → V , and bi-Lipschitz nonembeddability, where X is a metric measure space and V is a Banach space. Here, we consider the case V = L, where differentiability fails. We establish another kind of differentiability for certain X , including R and H, the Heisenberg group with its Carnot-Ca...

Two-dimensional almost-Riemannian structures are generalized Riemannian structures on surfaces for which a local orthonormal frame is given by a Lie bracket generating pair of vector fields that can become collinear. We consider the Carnot–Caratheodory distance canonically associated with an almost-Riemannian structure and study the problem of Lipschitz equivalence between two such distances on...

The aim of this paper is to study the existence of solutions for a quasilinear elliptic system where the nonlinear term is a Caratheodory function on a bounded domain of R , by proving the well known unique continuation property for elliptic system in all dimensions: 1, 2, 3, . . . and the strict monotonocity of eigensurfaces. These properties let us to consider the above problem as a nonresona...