Closure and commutability results for Gamma-limits and the geometric linearization and homogenization of multi-well energy functionals

Under a suitable notion of equivalence of integral densities we prove a $\Gamma$-closure theorem for integral functionals: The limit of a sequence of $\Gamma$-convergent families of such functionals is again a $\Gamma$-convergent family. Its $\Gamma$-limit is the limit of the $\Gamma$-limits of the original problems. This result not only provides a common basic principle for a number of linearization and homogenization results in elasticity theory. It also allows for new applications as we exemplify by proving that geometric linearization and homogenization of multi-well energy functionals commute

Reverse Mathematics and parameter-free Transfer

Recently, conservative extensions of Peano and Heyting arithmetic in the spirit of Nelson's axiomatic approach to Nonstandard Analysis, have been proposed. In this paper, we study the Transfer axiom of Nonstandard Analysis restricted to formulas without parameters. Based on this axiom, we formulate a base theory for the Reverse Mathematics of Nonstandard Analysis and prove some natural reversals, and show that most of these equivalences do not hold in the absence of parameter-free Transfer.Comment: 22 pages; to appear in Annals of Pure and Applied Logi

Exhaustible sets in higher-type computation

We say that a set is exhaustible if it admits algorithmic universal quantification for continuous predicates in finite time, and searchable if there is an algorithm that, given any continuous predicate, either selects an element for which the predicate holds or else tells there is no example. The Cantor space of infinite sequences of binary digits is known to be searchable. Searchable sets are exhaustible, and we show that the converse also holds for sets of hereditarily total elements in the hierarchy of continuous functionals; moreover, a selection functional can be constructed uniformly from a quantification functional. We prove that searchable sets are closed under intersections with decidable sets, and under the formation of computable images and of finite and countably infinite products. This is related to the fact, established here, that exhaustible sets are topologically compact. We obtain a complete description of exhaustible total sets by developing a computational version of a topological Arzela--Ascoli type characterization of compact subsets of function spaces. We also show that, in the non-empty case, they are precisely the computable images of the Cantor space. The emphasis of this paper is on the theory of exhaustible and searchable sets, but we also briefly sketch applications

Dixmier Traces as Singular Symmetric Functionals and Applications to Measurable Operators

This paper introduces a new approach to the non-normal Dixmier and Connes-Dixmier traces (introduced by Dixmier and adapted to non-commutative geometry by Connes) on a general Marcinkiewicz space associated with an arbitrary semifinite von Neumann algebra. By unifying various constructions, and translating the situation of Dixmier traces into the theory of singular symmetric functionals on Marcinkiewicz function/operator spaces, we obtain the results (i) and (ii) below. The results are stated here, for the reader, in terms of the ideal $L^{(1,\infty)}$ of compact operators whose partial sums of singular values are of logarithmic divergence. (i) a positive compact operator $x$ in $L^{(1,\infty)}$ yields the same value for an arbitrary Connes-Dixmier trace (ie. $x$ is measurable in the sense of Connes) if and only if $\lim_{N\to\infty} \frac{1}{Log N}\sum_{n=1}^N s_n(x)$ exists, where $s_n(x)$ are the singular values of the compact operator $x$; (ii) the set of Dixmier traces and the set of Connes-Dixmier traces are norming sets (up to equivalence) for the space $L^{(1,\infty)}/L^{(1,\infty)}_0$, where the space $L^{(1,\infty)}_0$ is the closure of all finite rank operators in the norm $||.||_{(1,\infty)}$.Comment: 31 pages, LaTex source, to appear in J. Funct. Ana