423 research outputs found
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
and we compute the descriptive set-theoretic complexity of the set of all
for which is continuous at . We give conditions under which the
latter set is either a set or the countable union of
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 is
Borel i.e., we show that if we drop some of the previous conditions then there
is a multi-valued function whose graph is a Borel set and the set of points
of continuity of 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 XNĆX. We prove that the set of sequences (ei)iāN in X which are weakly convergent to some eāX and Q((ei)iāN,e) is a coanalytic subset of XN. The proof applies methods of effective descriptive set theory to Banach space theory. Using Silverās Theorem [J. Silver, Every analytic set is Ramsey, J. Symbolic Logic 35 (1970) 60ā64], this result leads to the following dichotomy theorem: if X is a Banach space, (aij)i,jāNĀ is a regular method of summability and (ei)iāN is a bounded sequence in X, then there exists a subsequence (ei)iāL such that either (I) there exists eāX such that every subsequence (ei)iāH of (ei)iāL is weakly summable w.r.t. (aij)i,jāN to e and Q((ei)iāH,e); or (II) for every subsequence (ei)iāH of (ei)iāL and every eāX with Q((ei)iāH,e)the sequence (ei)iāH is not weakly summable to e w.r.t. (aij)i,jāN. This is a version for weak convergence of an ErdƶsāMagidor result, see [P. Erdƶs, M. Magidor, A note on Regular Methods of Summability, Proc. Amer. Math. Soc. 59 (2) (1976) 232ā234]. 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 Ī±-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 Ī±-convergence. Finally we give conditions under which the pointwise limit of a sequence of arbitrary functions is a continuous function. Ā© 2008 Elsevier B.V. All rights reserved
- ā¦