We show that any Borel action on a standard Borel space of a group which is topologically isomorphic to the sum of a countable abelian group with a countable sum of lines and circles induces an orbit equivalence relation which is hypersmooth. We also show that any Borel action of a second countable locally compact abelian group on a standard Borel space induces an orbit equivalence relation which is essentially hyperfinite, generalizing a result of Gao and Jackson for the countable abelian groups.

Consider an m x zero-one matrix A. An s x t submatrix of A is said to be even if the sum of its entries is even. In this paper, we focus on the case m = n and s = t = 2. The maximum number M(n) of even 2 x 2 submatrices of A is clearly ( 2n) 2, and corresponds to the matrix A having, e.g., all ones (or zeros). A more interesting question, motivated by Turán numbers and Hadamard matrices, is that of the minimum number m(n) of such matrices. It has recently been shown that m(n) ≥ 1/2 ( 2n) 2 - Bn 3 for some constant B. In this paper we show that if the matrix A = A n is considered to be induced by an infinite zero one matrix obtained at random, then P(E n ≤1/2( 2n) 2 - Cn 2 log n infinitely often) = 0, where E n denotes the number of even 2 x 2 submatrices of A n. Results such as these provide us with specific information about the tightness of the concentration of E n around its expected value of 1/2 ( 2n) 2.

In this thesis, we discuss three main projects which are related to Polish groups and their actions on standard Borel spaces. In the first part, we show that the complexity of the classification problem of continua is Borel bireducible to a universal orbit equivalence relation induce by a Polish group on a standard Borel space. In the second part, we compare the relative complexity of various types of classification problems concerning subspaces of [0,1]^n for all natural number n. In the last chapter, we give a topological characterization theorem for the class of locally compact two-sided invariant non-Archimedean Polish groups. Using this theorem, we show the non-existence of a universal group and the existence of a surjectively universal group in the class.

Pratima Sadangi in a Ph.D. thesis submitted to Utkal University proved results on degree of approximation of functions by operators associated with their Fourier series. In this dissertation, we consider degree of approximation of functions in Hα,ρ by different operators. In Chapter 1 we mention basic definitions needed for our work. In Chapter 2 we discuss different methods of summation. In Chapter 3 we define the Hα,ρ metric and present the degree of approximation problem relating to Fourier series and conjugate series of functions in the Hα,ρ metric using Karamata (Κλ) means. In Chapter 4 we present the degree of approximation of an integral associated with the conjugate series by the Euler, Borel and (e,c) means of a series analogous to the Hardy-Littlewood series in the Hα,ρ metric. In Chapter 5 we propose problems to be solved in the future.

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Mémoire numérisé par la Division de la gestion de documents et des archives de l'Université de Montréal

This thesis deals with study of mappings preserving Borel classes or absolute Borel classes. We prove a theorem which shows that under some assumptions there exists a (selection) function with certain properties. Using this theorem we obtain several results on preservation of Borel classes. Moreover, thanks to that theorem we prove a theorem on preservation of absolute Borel classes under a perfect mapping. Next, we show an assertion which implies that a piecewise closed mapping has a restriction that is "piecewise perfect" and its image is equal to the image of the original mapping. Under certain additional assumptions we prove a similar assertion for an Fσ-mapping instead of a piecewise closed mapping. Using these assertions and the theorem on preservation of absolute Borel classes under a perfect mapping we obtain further results on preservation of absolute Borel classes, in particular, for piecewise closed mappings and Fσ- -mappings. In the last chapter we study mappings such that the inverse image of an open set under these mappings is of a particular additive class. 1

Neste trabalho,nos apresentamos a K-teoria algébrica a qual é um ramo da álgebra que associa para cada anel comutativo comunidade R, uma sequencia de grupos abelianos ditos de n-ésimos K-grupos do anel R, denotada por Kn(R) . A meados da década de 1950,Alexander Grothendieck da a definição do K0(R) de um anel R. Em 1962, Hyman Bass e Stephen Schanuel apresenta a primeira definição adequada do K1(R) de um anel R. Em 1970, Daniel Quillen da uma definição geral dos K-grupos de um anel R a partir da +- construção do espaço classificante BGL(R). Nosso interesse é o estudo dos K-grupos sobre o anel de inteiros OF sobre um corpo numérico F. Usando alguns resultados de homologia dos grupos lineares, neste trabalho daremos a definição do mapa regulador de Borel. / In this paper,we present the algebraic K-theory,which is a branch of algebra that associates to any ring with unit R a sequence of abelian groups called n-th K-groups of R, denoted by Kn(R). The mid-1950s, Alexander Grothendieck gave a definition of the K0(R) of any ring R. In1962, Hyman Bass and Stephen Schanuel gave the first adequate definition of K1 of any ring R. In 1970, Daniel Quillen gave a general definition of K-groups of any ring R using the +- construction of the classifying space BGL(R). Our interest is the study of the K-groups on the ring of integers OF over a number field F. Using some results of homology of linear groups, this work will give the definition of Borel\'s regulator map.

Neste trabalho,nos apresentamos a K-teoria algébrica a qual é um ramo da álgebra que associa para cada anel comutativo comunidade R, uma sequencia de grupos abelianos ditos de n-ésimos K-grupos do anel R, denotada por Kn(R) . A meados da década de 1950,Alexander Grothendieck da a definição do K0(R) de um anel R. Em 1962, Hyman Bass e Stephen Schanuel apresenta a primeira definição adequada do K1(R) de um anel R. Em 1970, Daniel Quillen da uma definição geral dos K-grupos de um anel R a partir da +- construção do espaço classificante BGL(R). Nosso interesse é o estudo dos K-grupos sobre o anel de inteiros OF sobre um corpo numérico F. Usando alguns resultados de homologia dos grupos lineares, neste trabalho daremos a definição do mapa regulador de Borel. / In this paper,we present the algebraic K-theory,which is a branch of algebra that associates to any ring with unit R a sequence of abelian groups called n-th K-groups of R, denoted by Kn(R). The mid-1950s, Alexander Grothendieck gave a definition of the K0(R) of any ring R. In1962, Hyman Bass and Stephen Schanuel gave the first adequate definition of K1 of any ring R. In 1970, Daniel Quillen gave a general definition of K-groups of any ring R using the +- construction of the classifying space BGL(R). Our interest is the study of the K-groups on the ring of integers OF over a number field F. Using some results of homology of linear groups, this work will give the definition of Borel\'s regulator map.