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On the Difference Hierarchy in Countably Based T0-Spaces

Victor Selivanov
2008 Electronical Notes in Theoretical Computer Science  
We establish some results on some variants of the Hausdorff difference hierarchy.  ...  We discuss also a broad class of effective topological spaces closely relevant to our study of the difference hierarchy and to computability in topology.  ...  Difference Hierarchies of Sets In this section we improve some previous results on the DH, in particular extend the theory in [19] to the ω-continuous domains.  ... 
doi:10.1016/j.entcs.2008.12.022 fatcat:nhetciwsl5fo5erktj35jlavu4

Type Classes and Filters for Mathematical Analysis in Isabelle/HOL [chapter]

Johannes Hölzl, Fabian Immler, Brian Huffman
2013 Lecture Notes in Computer Science  
The improvements are centered on two primary contributions: a generic theory of limits based on filters, and a new hierarchy of type classes that includes various topological, metric, vector, and algebraic  ...  These let us apply many results in multivariate analysis to types which are not Euclidean spaces, such as the extended real numbers, bounded continuous functions, or finite maps.  ...  Acknowledgements We want to thank John Harrison and his colleagues for the development of HOL Light's multivariate analysis.  ... 
doi:10.1007/978-3-642-39634-2_21 fatcat:d467hfmuxveqpjhqdsi6dubhka

Towards a descriptive set theory for domain-like structures

Victor L. Selivanov
2006 Theoretical Computer Science  
This is a survey of results in descriptive set theory for domains and similar spaces, with the emphasis on the -algebraic domains.  ...  Since the subject is still in its beginning, we discuss in detail several open questions and possible future development. We also mention some relevant facts of (effective) descriptive set theory.  ...  I am grateful to the university for supporting my stays and to Dieter Spreen for making the stays possible. I thank also an anonymous referee for the careful reading and useful remarks.  ... 
doi:10.1016/j.tcs.2006.07.053 fatcat:zw4tsdo5ozahfpeiyuelgg4mom

Total Representations [article]

Victor Selivanov
2013 arXiv   pre-print
Almost all representations considered in computable analysis are partial. We provide arguments in favor of total representations (by elements of the Baire space).  ...  of topological spaces relevant to computable analysis.  ...  I also thank Vasco Brattka, Peter Hertling and Matthias Schröder for stimulating discussions, and to the anonymous referees for many useful suggestions and bibliographical hints.  ... 
arXiv:1304.1239v1 fatcat:3kp275ziknfqfamsvzduxvvwkm

MEASURE-THEORETIC UNIFORMITY IN RECURSION THEORY AND SET THEORY [chapter]

GERALD E. SACKS
1999 World Scientific Series in 20th Century Mathematics  
The recursion-theoretic concepts touched on are : the arithmetic, hyperarithmetic and analytic hierarchies of Kleene [12]; co-models of the hyperarithmetic comprehension axiom of Kreisel [16]; recursive  ...  He shows: if ZF (Zermelo-Fraenkel set theory) is consistent, then ZF+ "there exists a translationinvariant, countably additive extension of Lebesque measure defined on all sets of reals"+ "the dependent  ...  Think of (2N)% as the product of countably many copies of 2N and finitely many one-point spaces, one one-point space {At} for each i e K.  ... 
doi:10.1142/9789812812926_0009 fatcat:4boxavdyqvf5hnqykrxyla6vwq

Measure-Theoretic Uniformity in Recursion Theory and Set Theory

Gerald E. Sacks
1969 Transactions of the American Mathematical Society  
The recursion-theoretic concepts touched on are : the arithmetic, hyperarithmetic and analytic hierarchies of Kleene [12]; co-models of the hyperarithmetic comprehension axiom of Kreisel [16]; recursive  ...  He shows: if ZF (Zermelo-Fraenkel set theory) is consistent, then ZF+ "there exists a translationinvariant, countably additive extension of Lebesque measure defined on all sets of reals"+ "the dependent  ...  Think of (2N)% as the product of countably many copies of 2N and finitely many one-point spaces, one one-point space {At} for each i e K.  ... 
doi:10.2307/1995364 fatcat:tk3hbi5ypjb5tb32m3cszbkgpq

Measure-theoretic uniformity in recursion theory and set theory

Gerald E. Sacks
1969 Transactions of the American Mathematical Society  
The recursion-theoretic concepts touched on are : the arithmetic, hyperarithmetic and analytic hierarchies of Kleene [12]; co-models of the hyperarithmetic comprehension axiom of Kreisel [16]; recursive  ...  He shows: if ZF (Zermelo-Fraenkel set theory) is consistent, then ZF+ "there exists a translationinvariant, countably additive extension of Lebesque measure defined on all sets of reals"+ "the dependent  ...  Think of (2N)% as the product of countably many copies of 2N and finitely many one-point spaces, one one-point space {At} for each i e K.  ... 
doi:10.1090/s0002-9947-1969-0253895-6 fatcat:yhekdmjz2ffete6oat7tu3nrze

Topological Ramsey Spaces Dense in Forcings [article]

Natasha Dobrinen
2018 arXiv   pre-print
In cases where a forcing turned out to be equivalent to a topological Ramsey space, the strong Ramsey-theoretic techniques have aided in a fine-tuned analysis of the Rudin-Keisler and Tukey structures  ...  Each topological Ramsey space is endowed with a partial ordering which can be modified to a σ-closed 'almost reduction' relation analogously to the partial ordering of 'mod finite' on [ω]^ω.  ...  An ultrafilter U on a countable base set B is a filter such that each subset of B or its complement is in U.  ... 
arXiv:1702.02668v2 fatcat:64gtar3mt5g5ppcpoivl5dlhk4

RNA Shape Space Topology

Jan Cupal, Stephan Kopp, Peter F. Stadler
2000 Artificial Life  
one phenotype is accessible from another through changes at the genotypic level.  ...  The shape space of RNA secondary structures is used to illustrate this approach. We show that evolutionary trajectories are continuous if and only if they follow connected paths in phenotype space.  ...  This article was supported in part by Austrian Fonds zur Förderung der Wissenschaftlichen Forschung Proj. No. P13093-GEN. This paper is LA-UR 99:1324.  ... 
doi:10.1162/106454600568294 pmid:10943663 fatcat:xp6r5ejwjbfofo3snlk4j4hb6y

On the complexity of Borel equivalence relations with some countability property [article]

Dominique Lecomte
2018 arXiv   pre-print
In order to do this, we prove some extensions of the Louveau-Saint Raymond theorem which itself generalized the Hurewicz theorem characterizing when a Borel subset of a Polish space is G δ .  ...  In particular , we characterize when a Borel equivalence relation with countable equivalence classes is Σ 0 ξ (or Π 0 ξ).  ...  The most classical hierarchy of topological complexity in descriptive set theory is the one given by the Borel classes.  ... 
arXiv:1805.09167v2 fatcat:ntq7lxsjozgbjosddhiv5pbecq

Hierarchies of effective descriptive set theory

Peter G. Hinman
1969 Transactions of the American Mathematical Society  
The roots of modern hierarchy theory lie in the investigations of Baire, Borel, Lebesgue, and others around the turn of the century.  ...  Since each Borel set can be constructed by iteration of the elementary operations of countable union and complementation over a countable well-ordering, it was in general agreed that these sets are constructively  ...  In this section we give a second characterization of sets of the form R(XsAP(s))(F) in terms of a set derivative based on P.  ... 
doi:10.1090/s0002-9947-1969-0265161-3 fatcat:wipfsls3sbhdhlv2fgryjkuuva

Hierarchies of Effective Descriptive Set Theory

Peter G. Hinman
1969 Transactions of the American Mathematical Society  
The roots of modern hierarchy theory lie in the investigations of Baire, Borel, Lebesgue, and others around the turn of the century.  ...  Since each Borel set can be constructed by iteration of the elementary operations of countable union and complementation over a countable well-ordering, it was in general agreed that these sets are constructively  ...  In this section we give a second characterization of sets of the form R(XsAP(s))(F) in terms of a set derivative based on P.  ... 
doi:10.2307/1995348 fatcat:ftex5al2djhyvaamm6gt6b7eay

Concepts of quantum non-Markovianity: A hierarchy

Li Li, Michael J.W. Hall, Howard M. Wiseman
2018 Physics reports  
In this report we compare the descriptions of classical stochastic processes and quantum stochastic processes (as arising in OQSs), and show that there are inherent differences that lead to the non-trivial  ...  In fact, quantum non-Markovianity is highly context-dependent. The results in this report, summarized as a hierarchy figure, bring clarity to the nature of quantum non-Markovianity.  ...  Acknowledgements 58 Acknowledgements On the connection between dynamical decoupling and quantum Markovianity, we acknowledge motivating discussions with Daniel Burgarth and Lorenza Viola.  ... 
doi:10.1016/j.physrep.2018.07.001 fatcat:xpcgmg2cwbe6xkmlfa2zooxk7a

Definable Operations On Weakly Recognizable Sets of Trees

Jacques Duparc, Alessandro Facchini, Filip Murlak, Marc Herbstritt
2011 Foundations of Software Technology and Theoretical Computer Science  
In order to prevent this, alternation needs to be restricted to the choice of direction in the tree.  ...  Alternating automata on infinite trees induce operations on languages which do not preserve natural equivalence relations, like having the same Mostowski-Rabin index, the same Borel rank, or being continuously  ...  Acknowledgements The second author is supported by a grant from the SNFS, n. PBLAP2-132006, while the third author is supported by the Polish government grant no. N N206 567840.  ... 
doi:10.4230/lipics.fsttcs.2011.363 dblp:conf/fsttcs/DuparcFM11 fatcat:lhnv3khkwjc7nhxbmhavbctrja

Normal Numbers and the Borel Hierarchy [article]

Verónica Becher and Pablo Ariel Heiber and Theodore A. Slaman
2013 arXiv   pre-print
Similarly, the set of absolutely normal numbers is Π^0_3-complete in the effective Borel hierarchy.  ...  We show that the set of absolutely normal numbers is Π^0_3-complete in the Borel hierarchy of subsets of real numbers.  ...  This research received support from CONICET and Agencia Nacional de Promoción Científica y Tecnológica, Argentina, from the National Science  ... 
arXiv:1311.0335v1 fatcat:4vu3ksqrhjgz3afnhvefynnlca
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