Dedekind order completion of C(X) by Hausdorff continuous functions

The concept of Hausdorff continuous interval valued functions, developed within the theory of Hausdorff approximations and originaly defined for interval valued functions of one real variable is extended to interval valued functions defined on a topological space X. The main result is that the set of all finite Hausdorff continuous functions on any topological space X is Dedekind order complete. Hence it contains the Dedekind order completion of the set C(X) of all continuous real functions defined on X as well as the Dedekind order completion of the set C_b(X) of all bounded continuous functions on X. Under some general assumptions about the topological space X the Dedekind order completions of both C(X) and C_b(X) are characterised as subsets of the set of all Hausdorff continuous functions. This solves a long outstanding open problem about the Dedekind order completion of C(X). In addition, it has major applications to the regularity of solutions of large classes of nonlinear PDEs

Multiplicativity Factors for Function Norms

AbstractLet (T, Ω, m) be a measure space; let ρ be a function norm on M = M(T, Ω, m), the algebra of measurable functions on T; and let Lρ be the space {f ∈ M : ρ(f) < ∞} modulo the null functions. If Lρ, is an algebra, then we call a constant μ > 0 a multiplicativity factor for ρ if ρ(fg) ≤ μρ(f) ρ(g) for all f, g ∈ Lρ. Similarly, λ > 0 is a quadrativity factor if ρ(f2) ≤ λρ(f)2 for all f. The main purpose of this paper is to give conditions under which Lρ, is indeed an algebra, and to obtain in this case the best (least) multiplicativity and quadrativity factors for ρ. The first of our two principal results is that if ρ is σ-subadditive, then Lρ is an algebra if and only if Lρ is contained in L∞. Our second main result is that if (T, Ω, m) is free of infinite atoms, ρ is σ-subadditive and saturated, and Lρ, is an algebra, then the multiplicativity and quadrativity factors for ρ coincide, and the best such factor is determined by sup{||f||∞: f ∈ Lρ, ρ(f) ≤ 1}

Multiplicativity Factors for Orlicz Space Function Norms

AbstractLet ρφ be a function norm defined by a Young function φ with respect to a measure space (T, Ω, m), and let Lφ be the Orlicz space determined by ρφ. If Lφ is an algebra, then a constant μ > 0 is called a multiplicativity factor for ρφ, if ρφ,(fg) ≤ μρφ(f) ρφ(g) for all f, g ∈ Lφ. The main objective of this paper is to give conditions under which Lφ is indeed an algebra, and to obtain in this case the best (least) multiplicativity factor for ρφ. The first of our principal results is that Lφ is an algebra if and only if minf ≡ inf{m(A) > 0 : A ∈ Ω} > 0 or x∞(φ) ≡ sup{x ≥ 0 : φ(x) < ∞} < ∞ Our second main result states that if Lφ is an algebra and (T, Ω, m) is free of infinite atoms, then the best multiplicativity factor for ρφ is φ−1(1/minf if minf > 0, and x∞(φ) if minf = 0

Scotland, Catalonia and the “right” to self-determination: a comment suggested by Kathryn Crameri’s “Do Catalans Have the Right to Decide?

No abstract available

Multivariate risks and depth-trimmed regions

We describe a general framework for measuring risks, where the risk measure takes values in an abstract cone. It is shown that this approach naturally includes the classical risk measures and set-valued risk measures and yields a natural definition of vector-valued risk measures. Several main constructions of risk measures are described in this abstract axiomatic framework. It is shown that the concept of depth-trimmed (or central) regions from the multivariate statistics is closely related to the definition of risk measures. In particular, the halfspace trimming corresponds to the Value-at-Risk, while the zonoid trimming yields the expected shortfall. In the abstract framework, it is shown how to establish a both-ways correspondence between risk measures and depth-trimmed regions. It is also demonstrated how the lattice structure of the space of risk values influences this relationship.Comment: 26 pages. Substantially revised version with a number of new results adde

Hypernatural Numbers as Ultrafilters

In this paper we present a use of nonstandard methods in the theory of ultrafilters and in related applications to combinatorics of numbers

An axiomatic approach to the non-linear theory of generalized functions and consistency of Laplace transforms

We offer an axiomatic definition of a differential algebra of generalized functions over an algebraically closed non-Archimedean field. This algebra is of Colombeau type in the sense that it contains a copy of the space of Schwartz distributions. We study the uniqueness of the objects we define and the consistency of our axioms. Next, we identify an inconsistency in the conventional Laplace transform theory. As an application we offer a free of contradictions alternative in the framework of our algebra of generalized functions. The article is aimed at mathematicians, physicists and engineers who are interested in the non-linear theory of generalized functions, but who are not necessarily familiar with the original Colombeau theory. We assume, however, some basic familiarity with the Schwartz theory of distributions.Comment: 23 page