Turning Borel sets into clopen sets effectively

We present the effective version of the theorem about turning Borel sets in Polish spaces into clopen sets while preserving the Borel structure of the underlying space. We show that under some conditions the emerging parameters can be chosen in a hyperarithmetical way and using this we prove a uniformity result

The descriptive set-theoretic complexity of the set of points of continuity of a multi-valued function

In this article we treat a notion of continuity for a multi-valued function $F$ and we compute the descriptive set-theoretic complexity of the set of all $x$ for which $F$ is continuous at $x$. We give conditions under which the latter set is either a $G_\delta$ set or the countable union of $G_\delta$ sets. Also we provide a counterexample which shows that the latter result is optimum under the same conditions. Moreover we prove that those conditions are necessary in order to obtain that the set of points of continuity of $F$ is Borel i.e., we show that if we drop some of the previous conditions then there is a multi-valued function $F$ whose graph is a Borel set and the set of points of continuity of $F$ is not a Borel set. Finally we give some analogous results regarding a stronger notion of continuity for a multi-valued function. This article is motivated by a question of M. Ziegler in [{\em Real Computation with Least Discrete Advice: A Complexity Theory of Nonuniform Computability with Applications to Linear Algebra}, {\sl submitted}].Comment: 22 page

The descriptive set-theoretic complexity of the set of points of continuity of a multi-valued function (Extended Abstract)

In this article we treat a notion of continuity for a multi-valued function F and we compute the descriptive set-theoretic complexity of the set of all x for which F is continuous at x. We give conditions under which the latter set is either a G_\delta set or the countable union of G_\delta sets. Also we provide a counterexample which shows that the latter result is optimum under the same conditions. Moreover we prove that those conditions are necessary in order to obtain that the set of points of continuity of F is Borel i.e., we show that if we drop some of the previous conditions then there is a multi-valued function F whose graph is a Borel set and the set of points of continuity of F is not a Borel set. Finally we give some analogue results regarding a stronger notion of continuity for a multi-valued function. This article is motivated by a question of M. Ziegler in "Real Computation with Least Discrete Advice: A Complexity Theory of Nonuniform Computability with Applications to Linear Algebra", (submitted)

A dichotomy result for a pointwise summable sequence of operators

AbstractLet X be a separable Banach space and Q be a coanalytic subset of XN×X. We prove that the set of sequences (ei)i∈N in X which are weakly convergent to some e∈X and Q((ei)i∈N,e) is a coanalytic subset of XN. The proof applies methods of effective descriptive set theory to Banach space theory. Using Silver’s Theorem [J. Silver, Every analytic set is Ramsey, J. Symbolic Logic 35 (1970) 60–64], this result leads to the following dichotomy theorem: if X is a Banach space, (aij)i,j∈N is a regular method of summability and (ei)i∈N is a bounded sequence in X, then there exists a subsequence (ei)i∈L such that either (I) there exists e∈X such that every subsequence (ei)i∈H of (ei)i∈L is weakly summable w.r.t. (aij)i,j∈N to e and Q((ei)i∈H,e); or (II) for every subsequence (ei)i∈H of (ei)i∈L and every e∈X with Q((ei)i∈H,e)the sequence (ei)i∈H is not weakly summable to e w.r.t. (aij)i,j∈N. This is a version for weak convergence of an Erdös–Magidor result, see [P. Erdös, M. Magidor, A note on Regular Methods of Summability, Proc. Amer. Math. Soc. 59 (2) (1976) 232–234]. Both theorems obtain some considerable generalizations

Aflatoxicosis in Poultry

The term aflatoxicosis is applied to the toxicosis which is produced by aflatoxin , a poisonous metabolite produced by certain strains of Aspergillus flavus. This toxin is a part of a series of toxins produced by fungi, which may be ingested by livestock, poultry and humans to produce toxigenic effects--the so-called mycotoxicoses. The occurrence of mycotoxicoses are frequent enough considering the prevalence of fungi and their proliferation in nature. They develop under favorable environmental conditions in foods and feeds of plant origin before and after harvest

The notion of exhaustiveness and Ascoli-type theorems

In this paper we introduce the notion of exhaustiveness which applies for both families and nets of functions. This new notion is close to equicontinuity and describes the relation between pointwise convergence for functions and α-convergence (continuous convergence). Using these results we obtain some Ascoli-type theorems dealing with exhaustiveness instead of equicontinuity. Also we deal with the corresponding notions of separate exhaustiveness and separate α-convergence. Finally we give conditions under which the pointwise limit of a sequence of arbitrary functions is a continuous function. © 2008 Elsevier B.V. All rights reserved