Complexity properties of recursively enumerable sets andsQ-completeness

The notions of effectively subcreative set and strongly effectively acceleratable set are introduced. It is proved that the notions of effectively subcreative set, strongly effectively acceleratable set, andsQ-complete recursively enumerable set are equivalent. Translated fromMatematicheskie Zametki, Vol. 62, No. 3, pp. 425–429, September, 1997. Translated by V. N. Dubrovsky     

Complexity properties of recursively enumerable sets andbsQ-completeness

The notions of boundedly strongly effectively speedable set and boundedly effectively speedable set are introduced. It is proved that the notions of boundedly strongly effectively speedable set, boundedly effectively speedable set, creative set, andbsQ-complete recursively enumerable set are equivalent. Translated fromMatematicheskie Zametki, Vol. 68, No. 4, pp. 554–559, October, 2000.     

Lipschitz equivalence of self-similar sets and hyperbolic boundaries

Kaimanovich (2003) [9] introduced the concept of augmented tree on the symbolic space of a self-similar set. It is hyperbolic in the sense of Gromov, and it was shown by Lau and Wang (2009)  [12] that under the open set condition, a self-similar set can be identified with the hyperbolic boundary of the tree. In the paper, we investigate in detail a class of simple augmented trees and the Lipschitz equivalence of such trees. The main purpose is to use this to study the Lipschitz equivalence problem of the totally disconnected self-similar sets which has been undergoing some extensive development recently.     

Fuzzy Optimization Problems Based on Ordering Cones

The solution concepts of the fuzzy optimization problems using ordering cone (convex cone) are proposed in this paper. We introduce an equivalence relation to partition the set of all fuzzy numbers into the equivalence classes. We then prove that this set of equivalence classes turns into a real vector space under the settings of vector addition and scalar multiplication. The notions of ordering cone and partial ordering on a vector space are essentially equivalent. Therefore, the optimality notions in the set of equivalence classes (in fact, a real vector space) can be naturally elicited by using the similar concept of Pareto optimal solution in vector optimization problems. Given an optimization problem with fuzzy coefficients, we introduce its corresponding (usual) optimization problem. Finally, we prove that the optimal solutions of its corresponding optimization problem are the Pareto optimal solutions of the original optimization problem with fuzzy coefficients.     

Is the Mandelbrot set computable?

This work is concerned with the question whether the Mandelbrot set is computable. The computability notions that we consider are studied in computable analysis and will be introduced and discussed. We show that the exterior of the Mandelbrot set, the boundary of the Mandelbrot set, and the hyperbolic components satisfy certain natural computability conditions. We conclude that the two‐sided distance function of the Mandelbrot set is computable if the famous hyperbolicity conjecture is true. We also formulate the question whether the distance function of the Mandelbrot set is computable in terms of the escape time. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Constructive equivalence relations on computable probability measures

A central object of study in the field of algorithmic randomness are notions of randomness for sequences, i.e., infinite sequences of zeros and ones. These notions are usually defined with respect to the uniform measure on the set of all sequences, but extend canonically to other computable probability measures. This way each notion of randomness induces an equivalence relation on the computable probability measures where two measures are equivalent if they have the same set of random sequences.In what follows, we study the equivalence relations induced by Martin-Löf randomness, computable randomness, Schnorr randomness and Kurtz randomness, together with the relations of equivalence and consistency from probability theory. We show that all these relations coincide when restricted to the class of computable strongly positive generalized Bernoulli measures. For the case of arbitrary computable measures, we obtain a complete and somewhat surprising picture of the implications between these relations that hold in general.     

Local set theory

In 1945, Eilenberg and MacLane introduced the new mathematical notion of category. Unfortunately, from the very beginning, category theory did not fit into the framework of either Zermelo—Fraenkel set theory or even von Neumann—Bernays—Gödel set-class theory. For this reason, in 1959, MacLane posed the general problem of constructing a new, more flexible, axiomatic set theory which would be an adequate logical basis for the whole of naïve category theory. In this paper, we give axiomatic foundations for local set theory. This theory might be one of the possible solutions of the MacLane problem.Translated from Matematicheskie Zametki, vol. 77, no. 2, 2005, pp. 194–212.Original Russian Text Copyright © 2005 by V. K. Zakharov.This revised version was published online in April 2005 with a corrected issue number.     

Ulm Classification of Analytic Equivalence Relations in Generic Universes

We prove that if every real belongs to a set generic extension of L, then every Σ equivalence relation E on reals either admits a Δ₁ reduction to the equality on the set 2^{< ω1} of all countable binary sequences, or the Vitali equivalence E₀ continuously embeds in E. The proofs are based on a topology generated by OD sets.     

Estimates of the Entropy of the Set of Means for Some Classes of Stationary and Quasistationary Sequences

We consider the set of arithmetic means for some classes of stationary and quasistationary sequences. Order-sharp estimates of the entropy of this set for the classes under consideration are given.__________Translated from Matematicheskie Zametki, vol. 78, no. 1, 2005, pp. 52–58.Original Russian Text Copyright © 2005 by V. F. Gaposhkin.     

Morita equivalence of dual operator algebras

We consider notions of Morita equivalence appropriate to weak* closed algebras of Hilbert space operators. We obtain new variants, appropriate to the dual algebra setting, of the basic theory of strong Morita equivalence, and new nonselfadjoint analogues of aspects of Rieffel’s W^∗-algebraic Morita equivalence.     

Some similarity measures of intuitionistic fuzzy sets and their applications to multiple attribute decision making

Atanassov (1986) defined the notion of intuitionistic fuzzy set, which is a generalization of the notion of Zadeh’ fuzzy set. In this paper, we first develop some similarity measures of intuitionistic fuzzy sets. Then, we define the notions of positive ideal intuitionistic fuzzy set and negative ideal intuitionistic fuzzy set. Finally, we apply the similarity measures to multiple attribute decision making under intuitionistic fuzzy environment.     

Generating coset representatives for permutation groups

A group of symmetries can divide the group of all permutations of n marks into equivalence classes of isomorphs; each class is a coset of the group of symmetries. An algorithm for generating a set of isomorph, or coset, representatives is discussed. It can be used when n! is too large for all permutations to be useful, but the number of equivalence classes is manageable. The group of symmetries can be varied to match many natural notions of similarity. A computation on permutations that is restricted to a set of representatives can then concentrate on the essentially different cases.     

集合上的递归函数

本文在Ｊｅｎｓｅｎ和Ｋａｒｐ工作的基础上引进了集合上的递归函数的概念．研究了递归集函数的初步性质，讨论了递归集函数与Ｊｅｎｓｅｎ和Ｋａｒｐ定义的原始递归集函数及递归数论函数之间的关系，并给出了ＺＦＣ的可定义集模型上递归集函数的范式定理．     

Banach空间中的弱凸集和W-太阳集

本文在光滑Banach空间的框架下,引进弱凸集和W-太阳集的概念,研究它们性质,并给出了在逼近问题中的应用.     

Finite-tight sets

We introduce two notions of tightness for a set of measurable functions — the finite-tightness and the Jordan finite-tightness with the aim to extend certain compactness results (as biting lemma or Saadoune-Valadier’s theorem of stable compactness) to the unbounded case. These compactness conditions highlight their utility when we look for some alternatives to Rellich-Kondrachov theorem or relaxed lower semicontinuity of multiple integrals. Finite-tightness locates the great growths of a set of measurable mappings on a finite family of sets of small measure. In the Euclidean case, the Jordan finite-tight sets form a subclass of finite-tight sets for which the finite family of sets of small measure is composed by d-dimensional intervals. The main result affirms that each tight set H ⊆ W ^1,1 for which the set of the gradients ∇H is a Jordan finite-tight set is relatively compact in measure. This result offers very good conditions to use fiber product lemma for obtaining a relaxed lower semicontinuity condition.      

Small subsets of the reals and tree forcing notions

We discuss the question of which properties of smallness in the sense of measure and category (e.g. being a universally null, perfectly meager or strongly null set) imply the properties of smallness related to some tree forcing notions (e.g. the properties of being Laver-null or Miller-null).

     

Relations on words

In the first part of this survey, we present classical notions arising in combinatorics on words: growth function of a language, complexity function of an infinite word, pattern avoidance, periodicity and uniform recurrence. Our presentation tries to set up a unified framework with respect to a given binary relation.In the second part, we mainly focus on abelian equivalence, $k$-abelian equivalence, combinatorial coefficients and associated relations, Parikh matrices and $M$-equivalence. In particular, some new refinements of abelian equivalence are introduced.     

About the Admissible Predicates on Admissible Sets

We study the admissible predicates, i.e., the predicates having the property that their addition to the signature of an admissible set preserves the property “to be an admissible set.” We show that the family of these predicates is much wider than the family of Δ-predicates. We also construct a family of admissible predicates of cardinality 2^ω such that the addition of an arbitrary pair of predicates of this family to the signature of an admissible set violates the admissibility of the latter as well as other examples of families of admissible predicates.Original Russian Text Copyright © 2005 Morozov A. S.The author was supported by the International Russian-German Program (Grant RFRB-DFG 01-01-04003), the Russian Science Support Foundation, and the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grant 2112.2003.1).__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 841–850, July–August, 2005.     

Slice-Continuous Sets in Reflexive Banach Spaces: Some Complements

In this paper we study a class of closed convex sets introduced recently by Ernst et al. (J Funct Anal 223:179–203, 2005) and called by these authors slice-continuous sets. This class, which plays an important role in the strong separation of convex sets, coincides in ℝ ⁿ with the well known class of continuous sets defined by Gale and Klee in the 1960s. In this article we achieve, in the setting of reflexive Banach spaces, two new characterizations of slice-continuous sets, similar to those provided for continuous sets in ℝ ⁿ by Gale and Klee. Thus, we prove that a slice-continuous set is precisely a closed and convex set which does not possess neither boundary rays, nor flat asymptotes of any dimension. Moreover, a slice-continuous set may also be characterized as being a closed and convex set of non-void interior for which the support function is continuous except at the origin. Dedicated to Boris Mordukhovich in honour of his 60th birthday.