Thin and thick sets in locally convex spaces

We introduce in this work some normed space notions such as norming, thin and thick sets in general locally convex spaces. We also study some effects of thick sets on the uniform boundedness-like principles in locally convex spaces such as “weak*-bounded sets are strong*-bounded if and only if the space is a Banach–Mackey space”. It is proved that these principles occur under some weaker conditions by means of thick sets. Further, we show that the thickness is a duality invariant, that is, all compatible topologies for some locally convex space have the same thick sets.     

A Characterization of Self-similar Symbolic Spaces

In this paper we use fractal structures to study self-similar sets and self-similar symbolic spaces. We show that these spaces have a natural fractal structure, justifying the name of fractal structure, and we characterize self-similar symbolic spaces in terms of fractal structures. We prove that self-similar symbolic spaces can be characterized in a similar way, in the form, to the definition of classical self-similar sets by means of iterated function systems. We also study when a self-similar symbolic space is a self-similar set. Finally, we study relations between fractal structures with “pieces” homeomorphic to the space and different concepts of self-homeomorphic spaces. Along the paper, we propose several methods in order to construct self-similar sets and self-similar symbolic spaces from a geometrical approach. This allows to construct these kind of spaces in a very easy way.     

Special elements of the lattice of epigroup varieties

We describe right-hand skew Boolean algebras in terms of a class of presheaves of sets over Boolean algebras called Boolean sets, and prove a duality theorem between Boolean sets and étalé spaces over Boolean spaces.     

Trivariate Cr Polynomial Macroelements

Trivariate C^r macroelements defined in terms of polynomials of degree 8r + 1 on tetrahedra are analyzed. For r = 1,2, these spaces reduce to well-known macroelement spaces used in data fitting and in the finite-element method. We determine the dimension of these spaces, and describe stable local minimal determining sets and nodal minimal determining sets. We also show that the spaces approximate smooth functions to optimal order.     

De rham homology for networks of manifolds

This paper describes in terms of differential forms the real homology of a certain class of spaces, which we call networks. Networks include, besides smooth manifolds, singular sets of toral actions, classifying spaces of Lie groups, etc. A generalized Thom isomorphism theorem is also proved in this context.     

Convex functions with continuous epigraph or continuous level sets

Convex functions with continuous epigraph in the sense of Gale and Klée have been studied recently by Auslender and Coutat in a finite-dimensional setting. Here, we provide characterizations of such functionals in terms of the Legendre-Fenchel transformation in general locally convex spaces. Also, we show that the concept of continuous convex sets is of interest in these spaces. We end with a characterization of convex functions on Euclidean spaces with continuous level sets.     

The De Giorgi measure and an obstacle problem related to minimal surfaces in metric spaces

We study the existence of a set with minimal perimeter that separates two disjoint sets in a metric measure space equipped with a doubling measure and supporting a Poincaré inequality. A measure constructed by De Giorgi is used to state a relaxed problem, whose solution coincides with the solution to the original problem for measure theoretically thick sets. Moreover, we study properties of the De Giorgi measure on metric measure spaces and show that it is comparable to the Hausdorff measure of codimension one. We also explore the relationship between the De Giorgi measure and the variational capacity of order one. The theory of functions of bounded variation on metric spaces is used extensively in the arguments.     

On the aronszajn property for a differential equation of fractional order in Banach spaces

In this paper we investigate some topological properties of solution sets of a differential equation of fractional order in Banach spaces. Our assumptions and proofs are expressed in terms of measures of noncompactness.     

Borel Measure Extensions of Measures Defined on Sub-σ-Algebras

We develop a new approach to the measure extension problem, based on nonstandard analysis. The class of thick topological spaces, which includes all locally compact and all K-analytic spaces, is introduced in this paper, and measure extension results of the following type are obtained: If (X,  ) is a regular, Lindelöf, and thick space, σ[ ] is a σ-algebra, and ν is a finite measure on , inner regular with respect to the closed sets in , then ν has a Radon extension. The methods developed here allow us to improve on previously known extension results.     

Sufficiency of broken lines in the modulus method and removable sets

We establish the sufficiency of the family of broken lines in calculating the modulus of a condenser. We extend the Ahlfors-Beurling definition of removable sets basing on rectangles to weighted Sobolev spaces with a Muckenhoupt weight. We obtain exact characteristics of removable sets in terms of girth by broken lines. We prove the invariance of weighted Sobolev spaces under quasi-isometric mappings.     

Characterizations of Sets of Finite Perimeter Using Heat Kernels in Metric Spaces

The overarching goal of this paper is to link the notion of sets of finite perimeter (a concept associated with N^1,1-spaces) and the theory of heat semigroups (a concept related to N^1,2-spaces) in the setting of metric measure spaces whose measure is doubling and supports a 1-Poincaré inequality. We prove a characterization of sets of finite perimeter in terms of a short time behavior of the heat semigroup in such metric spaces. We also give a new characterization of BV functions in terms of a near-diagonal energy in this general setting.     

Metric spaces of fuzzy sets

Two classes of metrics are introduced for spaces of fuzzy sets. Their equivalence is discussed and basic properties established. A characterisation of compact and locally compact subsets is given in terms of boundedness and p-mean equileft-continuity, and the spaces shown to be locally compact, complete and separable metric spaces.     

Hardy inequalities and Assouad dimensions

We establish both sufficient and necessary conditions for weighted Hardy inequalities in metric spaces in terms of Assouad (co)dimensions. Our sufficient conditions in the case where the complement is thin are new, even in euclidean spaces, while in the case of a thick complement, we give new formulations for previously known sufficient conditions which reveal a natural duality between these two cases. Our necessary conditions are rather straight-forward generalizations from the unweighted case but, together with some examples, indicate the essential sharpness of our results. In addition, we consider the mixed case in which the complement may contain both thick and thin parts.     

Weak regularity of functions and sets in Asplund spaces

In this paper, we study a new concept of weak regularity of functions and sets in Asplund spaces. We show that this notion includes prox-regular functions, functions whose subdifferential is weakly submonotone and amenable functions in infinite dimension. We establish also that weak regularity is equivalent to Mordukhovich regularity in finite dimension. Finally, we give characterizations of the weak regularity of epi-Lipschitzian sets in terms of their local representations.     

Join-continuous frames,Priestley's duality and biframes

The primary purpose of this paper is to study join-continuous frames. We present two representation theorems for them: one in terms of -subframes of complete Boolean algebras and the other in terms of certain Priestley spaces. This second representation is used to prove that the topological spaces whose frame of open sets is join-continuous are characterized by a condition which says that certain intersections of open sets are open. Finally, we show that Priestley's duality can be viewed as a partialization of the dual adjunction between the categories of, respectively, bitopological spaces and biframes, stated by B. Banaschewski, G. C. L. Brümmer and K. A. Hardie in [5].This work was partially supported by Centro de Matemáíica da Universidade de Coimbra.     

Definability and textures

This paper aims to give a new perspective for definability in rough set theory. First, a counterpart of definability is introduced in textural approximation spaces. Then a complete field of sets for texture spaces is defined and using textural arguments, some new results are obtained for rough sets. It is shown that definability can be also discussed in terms of a complete field of fuzzy sets on a fuzzy lattice for the various fuzzy approximation spaces. It is also given a partial affirmative answer to an open problem posed by Wei-Zhi Wu in On some mathematical structures of T-fuzzy rough set algebras in infinite universes of discourse in Fundamenta Informaticae 108 (3–4) 2011 337–369.     

On non-enlargeable positive sets

The theory of positive sets on SSD spaces and Banach SNL spaces has been introduced by S. Simons. Monotone sets can be considered as a special case of positive sets. The characterizations of enlargement of positive sets in SSD spaces have been studied by Bo? and Csetnek. In this paper, we present characterizations of non-enlargeable positive sets in SSD spaces and Banach SNL spaces.     

Inequalities for Positive Definite Functions

A characterization of bounded sets in Banach spaces in terms of asymptotic cones and the Hausdorff deviations of sets from them homothetic images is obtained. Similar results for generalizations of the notion of boundedness are presented. Boundedness criteria have previously been known only for recessively compact sets.

     

Sampling in Paley-Wiener spaces on combinatorial graphs

A notion of Paley-Wiener spaces on combinatorial graphs is introduced. It is shown that functions from some of these spaces are uniquely determined by their values on some sets of vertices which are called the uniqueness sets. Such uniqueness sets are described in terms of Poincare-Wirtinger-type inequalities. A reconstruction algorithm of Paley-Wiener functions from uniqueness sets which uses the idea of frames in Hilbert spaces is developed. Special consideration is given to the -dimensional lattice, homogeneous trees, and eigenvalue and eigenfunction problems on finite graphs.

     

Calmness of Set-Valued Mappings Between Asplund Spaces and Application to Equilibrium Problems

The paper deals with the calmness of two classes of nonconvex set-valued mappings in Asplund spaces and its application to equilibrium problems. Its main part is devoted to establish new sufficient conditions for calmness, which are derived in terms of coderivatives and w* boundaries of normal cones to constraint sets. In order to achieve this goal, a new concept so-called “sequential normal smoothness” for the sets in Asplund spaces is introduced and compared with two well-known notions of convexity and semismoothness. Finally, the results are applied to prove necessary optimality conditions for nonparametric equilibrium problems under new weak constraint qualifications.