Abstract Let G ( D ) be a linear partial differential operator on $${\mathbb {R}}^n$$ R n , with constant coefficients. Moreover let $$\Omega \subset {\mathbb Ω ⊂ open and $$F\in L^1_{\text {loc}} (\Omega {C}}^N)$...

In intuitionistic analysis, Brouwer's Continuity Principle implies, together with an Axiom of Countable Choice, that the positively Borel sets form a genuinely growing hierarchy: every level of the hierarchy contains sets that do not occur at any lower level. ?0. Introduction. 0.1. ?. Borel, H. Lebesgue, R. Baire, N. Lusin, A. Souslin and others, the founding fathers of descriptive set theory, ...

We study some natural sets arising in the theory of ordinary differential equations in one variable from the point of view of descriptive set theory and in particular classify them within the Borel hierarchy. We prove that the set of Cauchy problems for ordinary differential equations which have a unique solution is Π2-complete and that the set of Cauchy problems which locally have a unique sol...

We describe circumstances under which non-empty small sets can be defined from families of infinite sets. In the process, we establish generalizations of Mansfield’s perfect set theorem for κ-Souslin sets and the Lusin-Novikov uniformization theorem. An intersecting family is a collection of sets, any two of which have non-empty intersection. These have been the subject of much study in combina...

Let L be the divergence form elliptic operator with complex bounded mea-surable coefficients, ω the positive concave function on (0,∞) of strictly critical lowertype pω ∈ (0, 1] and ρ(t) = t/ω(t) for t ∈ (0,∞). In this paper, the authorsstudy the Orlicz-Hardy spaceHω,L(R) and its dual spaceBMOρ,L∗(R), where L∗denotes the adjoint operator of L in L(R). Several characteriz...

All spaces considered are metric separable and are denoted usually by the letters X, Y , or Z. ω stands for the set of all natural numbers. If a metric separable space is additionally complete, we call it Polish; if it is a continuous image of ω or, equivalently, of a Polish space, it is called Souslin. The main subject of the present paper is the structure of Baire class 1 functions. Recent de...

Introduction In 12] is presented a constructive deenition of Borel subsets of Cantor space and their inclusion relation. They are seen as symbolic expressions built from the Boolean algebra of simple (closed and open) subsets of by formal countable disjunction and conjunction. If X is such a symbolic expression, it is clear for instance how to deene by induction the formal complement X 0 of X: ...

In this paper we introduce a new stable mathematical model for locating and measuring the medial axis of geometric objects, called the quadratic multiscale medial axis map of scale λ, and provide a sharp regularity result for the squared-distance function to any closed nonempty subset K of Rn. Our results exploit properties of the function Cl λ(dist 2(·; K)) obtained by applying the quadratic l...

It is a great pleasure for me to greet Professor Wade on the occasion of his sixtieth birthday. I am greeting him on behalf of my colleagues in Moscow University who, like myself, have vivid and happy memories of his visit to Russia. It was 26 years ago when I met Bill Wade in Moscow where he spent the academic year 1977-78 as a visiting professor at the Department of Mathematics of Moscow Stat...