79 research outputs found
Computably regular topological spaces
This article continues the study of computable elementary topology started by
the author and T. Grubba in 2009 and extends the author's 2010 study of axioms
of computable separation. Several computable T3- and Tychonoff separation
axioms are introduced and their logical relation is investigated. A number of
implications between these axioms are proved and several implications are
excluded by counter examples, however, many questions have not yet been
answered. Known results on computable metrization of T3-spaces from M.
Schr/"oder (1998) and T. Grubba, M. Schr/"oder and the author (2007) are proved
under uniform assumptions and with partly simpler proofs, in particular, the
theorem that every computably regular computable topological space with
non-empty base elements can be embedded into a computable metric space. Most of
the computable separation axioms remain true for finite products of spaces
Products of effective topological spaces and a uniformly computable Tychonoff Theorem
This article is a fundamental study in computable analysis. In the framework
of Type-2 effectivity, TTE, we investigate computability aspects on finite and
infinite products of effective topological spaces. For obtaining uniform
results we introduce natural multi-representations of the class of all
effective topological spaces, of their points, of their subsets and of their
compact subsets. We show that the binary, finite and countable product
operations on effective topological spaces are computable. For spaces with
non-empty base sets the factors can be retrieved from the products. We study
computability of the product operations on points, on arbitrary subsets and on
compact subsets. For the case of compact sets the results are uniformly
computable versions of Tychonoff's Theorem (stating that every Cartesian
product of compact spaces is compact) for both, the cover multi-representation
and the "minimal cover" multi-representation
Computable Jordan Decomposition of Linear Continuous Functionals on
By the Riesz representation theorem using the Riemann-Stieltjes integral,
linear continuous functionals on the set of continuous functions from the unit
interval into the reals can either be characterized by functions of bounded
variation from the unit interval into the reals, or by signed measures on the
Borel-subsets. Each of these objects has an (even minimal) Jordan decomposition
into non-negative or non-decreasing objects. Using the representation approach
to computable analysis, a computable version of the Riesz representation
theorem has been proved by Jafarikhah, Lu and Weihrauch. In this article we
extend this result. We study the computable relation between three Banach
spaces, the space of linear continuous functionals with operator norm, the
space of (normalized) functions of bounded variation with total variation norm,
and the space of bounded signed Borel measures with variation norm. We
introduce natural representations for defining computability. We prove that the
canonical linear bijections between these spaces and their inverses are
computable. We also prove that Jordan decomposition is computable on each of
these spaces
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
Representations of measurable sets in computable measure theory
This article is a fundamental study in computable measure theory. We use the
framework of TTE, the representation approach, where computability on an
abstract set X is defined by representing its elements with concrete "names",
possibly countably infinite, over some alphabet {\Sigma}. As a basic
computability structure we consider a computable measure on a computable
-algebra. We introduce and compare w.r.t. reducibility several natural
representations of measurable sets. They are admissible and generally form four
different equivalence classes. We then compare our representations with those
introduced by Y. Wu and D. Ding in 2005 and 2006 and claim that one of our
representations is the most useful one for studying computability on measurable
functions
Randomness extraction and asymptotic Hamming distance
We obtain a non-implication result in the Medvedev degrees by studying
sequences that are close to Martin-L\"of random in asymptotic Hamming distance.
Our result is that the class of stochastically bi-immune sets is not Medvedev
reducible to the class of sets having complex packing dimension 1
Computable Separation in Topology, from T_0 to T_3
This article continues the study of computable elementary topology started in (Weihrauch, Grubba 2009). We introduce a number of computable versions of the topological to separation axioms and solve their logical relation completely. In particular, it turns out that computable is equivalent to computable . The strongest axiom is used in (Grubba, Schroeder, Weihrauch 2007) to construct a computable metric
Computable Riesz Representation for the Dual of C[0;1]
AbstractBy the Riesz representation theorem for the dual of C[0;1], for every continuous linear operator F:C[0;1]→R there is a function g:[0;1]→R of bounded variation such thatF(f)=∫fdg(f∈C[0;1]). The function g can be normalized such that V(g)=‖F‖. In this paper we prove a computable version of this theorem. We use the framework of TTE, the representation approach to computable analysis, which allows to define natural computability for a variety of operators. We show that there are a computable operator S mapping g and an upper bound of its variation to F and a computable operator S′ mapping F and its norm to some appropriate g
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