WWW.DISSERS.RU

БЕСПЛАТНАЯ ЭЛЕКТРОННАЯ БИБЛИОТЕКА

   Добро пожаловать!


Pages:     | 1 | 2 || 4 | 5 |   ...   | 29 |

Московский государственный университет им. М.В. Ломоносова, Москва E-mail:helenbunina@yandex.ru Мальцевские чтения 2009 Пленарные доклады Развитие идей А. И. Мальцева в теории вычислимых моделей С. С. Гончаров В докладе обсуждаются проблемы инициированные в работах А. И. Мальцева по исследованию проблем существования вычислимых моделей, существования вычислимых представлений для различных алгебраических структур и вопросы автоустойчивости и алгоритмической размерности алгебраических систем и новые результаты в этих направлениях. А также связи теории вычислимых нумераций и вычислимых нумераций для различных иерархий с проблемами теории вычислимых моделей.

Список литературы [1] Mal’tsev A. I. Constructible algebras. I. Usp. Mat. Nauk, 16 (1961), No. 3, 3–60.

[2] Mal’tsev A. I. Algebraic Systems. Berlin: Springer Verlag, 1976.

[3] Mal’tsev A. I. Algorithms and Recursive Functions. Groningen: Wolters-Noordoff, 1970.

[4] Ash C. J., Knight J. F. Computable Structures and the Hyperarithmetical Hierarchy. Elsevier Science, 2000.

[5] Ershov Yu. L., Goncharov S. S. Constructive Models. In: Siberian school of algebra and logic, vol. 6, New York: Kluwer Academic/Plenum Press, Consultants bureau, 2000.

[6] Goncharov S. S., Harizanov V. S., Knight J. F., McCoy C., Miller R. G., Solomon R. Enumerations in computable structure theory. Annals of Pure and Applied Logic, 136 (2005), N 3, 219–246.

[7] Goncharov S. S. Computability and Models. In: Mathematical Problems from Applied Logic II. International Mathematical Series, Springer. 2007. V. 5, P. 99–216.

[8] Soare R. Recursively Enumerable Sets and Degrees. New York: Springer-Verl., 1987.

Новосибирский государственный университет, Институт математики СО РАН, Новосибирск E-mail:gonchar@math.nsc.ru Мальцевские чтения 2009 Пленарные доклады По дороге от логики к алгебре Ю. Л. Ершов В докладе обсуждается влияние следующих трех статей А. И. Мальцева на развитие исследований в мировой математике:

I. Untersuchungen aus dem Gebiete der mathematischen Logik. — Мат. сб., 1936, т. 1, № 3, 323–335.

II. Об одном общем методе получения локальных теорем теории групп. —Учен.

зап. Ивановского пед. ин-та, 1941, т. 1, № 1, 3–9.

III. О неразрешимости элементарных теорий некоторых полей. —Сиб. мат. журн., 1960, т. 1, № 1, 71–77.

Институт математики СО РАН, Новосибирск E-mail:ershov@math.nsc.ru Мальцевские чтения 2009 Пленарные доклады Предельные алгебраические системы и алгебро-геометрическая граница В. Н. Ремесленников Фиксируем алгебраическую систему (алгебру) A языка L (без предикатов). В статье [1] доказана следующая так называемая объединяющая теорема.

Теорема. Пусть L — язык без предикатов, A и C — алгебраические системы языка L, причём A нётерова по бескоэффициентным уравнениям, а C конечно порождена.

Тогда следующие свойства алгебраической системы C равносильны:

(1) C — координатная алгебраическая система неприводимого алгебраического множества над A для системы бескоэффициентных уравнений;

(2) Th(A) Th(C), то есть C (A);

(3) Th(A) Th(C);

(4) алгебраическая система C вложима в ультрастепень алгебраической системы A;

(5) алгебраическая система C дискриминируется алгебраической системой A;

(6) C — предельная над A алгебраическая система;

(7) C — алгебраическая система, определяемая полным атомарным типом теории Th(A) языка L.

Обозначим через = (A) универсальное замыкание A и через — конечно порождённую часть.

Как показано в [1], все координатные алгебры неприводимых алгебраических множеств над A, определяемыми системами уравнений без коэффициентов, содержатся в. Более того, если алгебра A нётерова по уравнениям, то в силу приведённой теоремы верно и обратное включение. Поэтому проблема классификации неприводимых алгебраических множеств над A эквивалентна проблеме классификации алгебр из класса.

Понятие предельной алгебры (а их несколько) —это понятия, возникшие при изучении алгебр из топологическими методами. Мы даём три основных определения предельной алгебры для A. И в каждом из этих определений алгебры из и только они будут предельными. С помощью одной из трёх предложенных топологических конструкций будет введена алгебро-геометрическая граница для A и, по определению, это будет семейство топологических пространств AGBn для всех целых чисел n 1.

Список литературы [1] Daniyarova E., Miasnikov A., Remeslennikov V. Unification theorems in algebraic geometry. Algebra and Discrete Mathematics, 1 (2008).

Омский филиал Института математики СО РАН, Омск E-mail:remesl@ofim.oscsbras.ru Мальцевские чтения 2009 Пленарные доклады Burnside problem on periodic groups and related topics S. I. Adian The Burnside problem. Fix m 2 and consider the group B(m, n) = a1,..., am | Xn =with the identity Xn =1. Is the group B(m, n) always finite In the monograph “History of combinatorial group theory” (Springer-Verlag, 1982), Prof. Wilhelm Magnus characterized the Burnside problem as follows:

Very much like Fermat’s Last Theorem in number theory, Burnside’s problem has acted as a catalyst for research in group theory. The fascination exerted by a problem with an extremely simple formulation which then turns out to be extremely difficult has something irresistible about it to the mind of the mathematician.

A negative solution of the Burnside problem on periodic groups was given in joint works by P. S. Novikov and S. I. Adian published in “Izvestiya AN SSSR”, 1968, vol.32. The infiniteness of free periodic groups B(m, n) was proved for all exponents n = kl with odd k 4381 and m>1 generators.

For a solution of the Burnside problem Novikov and Adian created a new method based on a classification of periodic words and a corresponding system of defining relations for the group B(m, n) by simultaneous induction on a natural parameter called rank. Later in the monograph “The Burnside problem and identities in groups” (Nauka, 1975) the result was improved for any odd k 665.

Investigations of periodic groups and solutions of many other old complicated problems in the group theory on the base of various modifications of the created method were continued during the last 40 years by S. I. Adian, his students and successors.

A survey of these results and some historical details will be given in the talk.

Steklov Mathematical Institute, Moscow E-mail:sia@mi.ras.ru Мальцевские чтения 2009 Пленарные доклады Models-theoretic aspects of the non-computable universe M. Arslanov Main focus of my talk will the n-c.e. Turing degrees (the degree theoretic counterpart of the finite levels of the Ershov hierarchy). These degree structures has been closely investigated for almost forty years. Much research has been centered on the comparison between the structures of the m-c.e. and n-c.e. degrees for 1 m < n, and various similarities and differences have been found. Nevertheless, the most principal questions on the interior arrangement of these degree structures remain unanswered. The main open questions in this list are the following:

• Definability of the various levels of the n-c.e. hierarchy, both relatively and within wider local structures; more specifically, questions related to the definability of the relations of ‘computably enumerable’ and ‘computably enumerable in’;

• Decidability of the restricted fragments of theories of these structures;

• Model theoretic similarities and differences between n-c.e. degree structures for n>2.

In my talk I will give a survey of the current status of these questions and discuss a number of related open questions.

General questions of definability and the role of splitting and nonsplitting, and also the description of new relationships between information content and degree theoretic structure also will be considered.

Kazan State University, Department of Mathematics, Kremlevskaja 18, Kazan 420008, Russia E-mail:Marat.Arslanov@ksu.ru Мальцевские чтения 2009 Пленарные доклады Turing computable embeddings J. F. Knight There are approaches from different branches of logic, letting us compare classes of countable structures and to say that one is more difficult to classify, up to isomorphism. One model theoretic approach involves cardinality, up to isomorphism. For example, Q-vector spaces are easier to classify, up to isomorphism, than undirected graphs, because there are only countably many non-isomorphic Q-vector spaces, while there are 2 non-isomorphic graphs. Friedman and Stanley [1] developed an approach that involves Borel embeddings and “Borel cardinality”. This approach let them say that classification is strictly simpler for fields of characteristic 0 and finite transcendence degree, and for Abelian p-groups, than for graphs, even though each of these classes has 2 non-isomorphic structures.

Borel cardinality does not allow us to distinguish among classes with just 0 nonisomorphic structures. In [2], there is a related approach, involving Turing computable embeddings and “effective cardinality”. This approach lets us make some distinctions. For example, the effective cardinality for number fields is strictly smaller than that for Q-vector spaces. We can make further distinctions. For a completion T of PA, Mod(T ) has maximal Borel cardinality, but it does not have maximal effective cardinality.

Results in [3] suggest a uniform way of looking for non-embeddability results. To locate a given class in relation to others, we consider the form of the sentences that distinguish among non-isomorphic members. In particular, we may re-work in this way the result of Friedman and Stanley showing non-embeddability of graphs in Abelian p-groups [4]. We say that a class K has “Ulm type” if non-isomorphic members of K can be distinguished by infinitary sentences lying in the thinnest admissible set that contains the ordinals computable from the structures.

References [1] Friedman H., Stanley L. A Borel reducibility theory for classes of countable structures. J. Symb. Logic, 54 (1989), 894–914.

[2] Calvert W., Cummins D., Knight J. F., Miller S. Comparing classes of finite structures. Algebra and Logic, 43 (2004), 365–373.

[3] Knight J. F., Miller S., Vanden Boom M. Turing computable embeddings. J. Symb. Logic, 73 (2007), 901–918.

[4] Fokina E., Knight J. F., Maher C., Melnikov A., Quinn S. M. Classes of Ulm type, and relations between the class of rank-homogeneous trees and other classes, preprint.

Mathematics Department, University of Notre Dame, USA E-mail:knight.1@nd.edu Мальцевские чтения 2009 Пленарные доклады Recent applications of proof theory to core mathematics U. Kohlenbach In recent years specially designed forms and novel extensions of functional interpretations have been used to establish general logical metatheorems ([3, 4, 1]) that guarantee the extractability of highly uniform effective bounds from large classes of proofs in nonlinear analysis, metric fixed point theory, ergodic theory and topological dynamics. ‘Uniform’ here means that the bounds do not depend on parameters from

Abstract

metric, hyperbolic, normed or Hilbert spaces as long as local metric bounds are given whereas no compactness is assumed. For such strong uniformity results to hold it is crucial that the underlying abstract spaces are not assumed to be separable.

The approach has led to numerous applications with new (both qualitative as well as quantitative) results in the above mentioned areas of mathematics. In particular, one obtains explicit uniform versions of so-called metastability results in the sense of T. Tao.

We will report on recent new applications to fixed point theory ([6]) and ergodic theory ([7]).

Most recently, it has been shown that proofs in nonseparable Hilbert space theory that are heavily based on weak compactness arguments are covered as well ([5]). As an example we will report on an analysis of a proof due to F. Browder ([2]) which might well serve as a model for the treatment of many other uses of weak compactness.

References [1] Briseid E. M. Logical aspects of of rates of convergence in metric spaces. To appear in: J. Symbolic Logic.

[2] Browder F. E. Convergence of approximants to fixed points of nonexpansive nonlinear mappings in Banach spaces. Arch. Rational Mech. Anal., 24 (1967), 82–90.

[3] Gerhardy P., Kohlenbach U. General logical metatheorems for functional analysis. Trans. Amer. Math.

Soc., 360 (2008), 2615–2660.

[4] Kohlenbach U. Applied Proof Theory: Proof Interpretations and their Use in Mathematics. Springer Monographs in Mathematics. Springer Heidelberg-Berlin, 2008.

[5] Kohlenbach U. On the logical analysis of proofs based on nonseparable Hilbert space theory. To appear in: Festschrift for Grigori Mints.

[6] Kohlenbach U., Leustean L. Asymptotically nonexpansive mappings in uniformly convex hyperbolic spaces. To appear in: J. European Math. Soc.

[7] Kohlenbach U., Leutean L. A quantitative mean ergodic theorem for uniformly convex Banach spaces.

To appear in: Ergodic Theory and Dynamical Systems.

Dept. of Mathematics, Technische Universitt Darmstadt, Darmstadt (Germany) E-mail:kohlenbach@mathematik.tu-darmstadt.de Мальцевские чтения 2009 Пленарные доклады Algebraic approach to non-classical logics L. L. Maksimova Study of non-classical logics in Novosibirsk started in 60-th due to initiative and by supervision of A. I. Maltsev. His interest to this area was stimulated by the existence of an adequate algebraic semantics for the most known non-classical logics.

In the present paper inter-connections of syntactic properties of non-classical logics and categorial properties of appropriate classes of algebras are investigated. We consider such fundamental properties as the interpolation property, the Beth definability property, joint consistency property and their variants. In the case of modal, superintuitionistic and related logics the mentioned properties of logics proved to be equivalent to appropriate variants of the amalgamation property or epimorphism surjectivity [1, 3]. It allows to solve, for instance, interpolation problem for some important classes of logical calculi and, at the same time, amalgamation problem for varieties. In particular, the following problems are decidable:

• Craig’s interpolation property and deductive interpolation property for superintuitionistic and positive calculi and for modal calculi over the modal S4 logic, • amalgamation and super-amalgamation properties for subvarieties of Heyting algebras, implicative lattices and closure algebras, • projective Beth property and restricted interpolation property over the intuitionistic logic and over the Grzegorczyk logic, • strong epimorphisms surjectivity for subvarieties of Heyting algebras, implicative lattices and of Grzegorczyk algebras, • weak interpolation property over the modal K4 logic [2], • weak amalgamation property for varieties of transitive modal algebras.

References [1] Gabbay D. M., Maksimova L. Interpolation and Definability: Modal and Intuitionistic Logics. Oxford:

Clarendon Press, 2005.

[2] Karpenko A. V. Weak interpolation property in extensions of the logics S4 and K4. Algebra and Logic, 47 (2008), no. 6, 705–722.

[3] Maksimova L. L. Definability and interpolation in non-classical logics. Studia Logica, 82 (2006), 271291.

Sobolev Institute of Mathematics, 630090, Novosibirsk (Russia) E-mail:lmaksi@math.nsc.ru Мальцевские чтения 2009 Пленарные доклады Automorphism spectra of computable structures A. S. Morozov This is a joint work together with Valentina Harizanov (George Washington University, Washington DC) and Russell Miller (CUNY, New York). We define and study the automorphism spectra of computable structures, prove their general properties, and prove that certain sets of Turing degrees can be realized as automorphism spectra, while certain others cannot.

Sobolev Institute of Mathematics, Novosibirsk, Russia E-mail:morozov@math.nsc.ru Мальцевские чтения 2009 Пленарные доклады Computations and functionals of finite types D. Normann The Church—Turing Thesis suggests that there is an absolute concept of computability for (partial) functions taking natural numbers or words in a finite alphabet as arguments.

If we extend the idea of computations to arguments from other domains, the picture is not so clear. In particular, if we want to study computations relative to functionals of finite types, there seems to be a freedom both in which functionals we want to consider as inputs and in which principles of computation we may permit.

Pages:     | 1 | 2 || 4 | 5 |   ...   | 29 |






© 2011 www.dissers.ru - «Бесплатная электронная библиотека»

Материалы этого сайта размещены для ознакомления, все права принадлежат их авторам.
Если Вы не согласны с тем, что Ваш материал размещён на этом сайте, пожалуйста, напишите нам, мы в течении 1-2 рабочих дней удалим его.