To the pair (E,c), where E is a countable Borel equivalence relation on a standard Borel space (X,A) and c a normalized Borel T-valued 2-cocycle on E, we associate a sequentially weakly closed Borel *-algebra Br*(E,c), contained in the bounded linear operators on L^2(E). Associated to Br*(E,c) is a natural (Borel) Cartan subalgebra (Definition 6.4.10) L(Bo(X)) isomorphic to the bounded Borel functions on X. Then L(Bo(X)) and its normalizer (the set of the unitaries u in Br*(E,c) such that u*fu in L(Bo(X)), f in L(Bo(X))) countably generates the Borel *-algebra Br*(E,c). In this thesis, we study Br*(E,c) and in particular prove that: i) If E is smooth, then Br*(E,c) is a type I Borel *-algebra (Definition 6.3.10). ii) If E is a hyperfinite, then Br*(E,c) is a Borel AF-algebra (Definition 7.5.1). iii) Generalizing Kumjian's definition, we define a Borel twist G over E and its associated sequentially closed Borel *-algebra Br*(G). iv) Let a Borel Cartan pair (B, Bo) denote a sequentially closed Borel *-algebra B with a Borel Cartan subalgebra Bo, where B is countably Bo-generated. Generalizing Feldman-Moore's result, we prove that any pair (B, Bo) can be realized uniquely as a pair (Br*(E,c), L(Bo(X))). Moreover, we show that the pair (Br*(E,c), L(Bo(X))) is a complete invariant of the countable Borel equivalence relation E. v) We prove a Krieger type theorem, by showing that two aperiodic hyperfinite countable equivalence relations are isomorphic if and only if their associated Borel *-algebras Br*(E1) and Br*(E2) are isomorphic.

To the pair (E,c), where E is a countable Borel equivalence relation on a standard Borel space (X,A) and c a normalized Borel T-valued 2-cocycle on E, we associate a sequentially weakly closed Borel *-algebra Br*(E,c), contained in the bounded linear operators on L^2(E). Associated to Br*(E,c) is a natural (Borel) Cartan subalgebra (Definition 6.4.10) L(Bo(X)) isomorphic to the bounded Borel functions on X. Then L(Bo(X)) and its normalizer (the set of the unitaries u in Br*(E,c) such that u*fu in L(Bo(X)), f in L(Bo(X))) countably generates the Borel *-algebra Br*(E,c). In this thesis, we study Br*(E,c) and in particular prove that: i) If E is smooth, then Br*(E,c) is a type I Borel *-algebra (Definition 6.3.10). ii) If E is a hyperfinite, then Br*(E,c) is a Borel AF-algebra (Definition 7.5.1). iii) Generalizing Kumjian's definition, we define a Borel twist G over E and its associated sequentially closed Borel *-algebra Br*(G). iv) Let a Borel Cartan pair (B, Bo) denote a sequentially closed Borel *-algebra B with a Borel Cartan subalgebra Bo, where B is countably Bo-generated. Generalizing Feldman-Moore's result, we prove that any pair (B, Bo) can be realized uniquely as a pair (Br*(E,c), L(Bo(X))). Moreover, we show that the pair (Br*(E,c), L(Bo(X))) is a complete invariant of the countable Borel equivalence relation E. v) We prove a Krieger type theorem, by showing that two aperiodic hyperfinite countable equivalence relations are isomorphic if and only if their associated Borel *-algebras Br*(E1) and Br*(E2) are isomorphic.

Orientador: Ana Cláudia Moreira Almeida-Verdu / Resumo: A deficiência auditiva sensorioneural, de grau severo-profundo, bilateral e pré-lingual pode comprometer a aquisição de repertórios verbais. O implante coclear (IC) é um dispositivo auditivo que habilita o acesso aos sons da fala e permite a aprendizagem de habilidades de ouvinte e de falante. Estudos entre a Fonoaudiologia e a Análise do Comportamento têm adotado o modelo das relações de equivalência e investigado as condições de ensino sob as quais essa população aprende as habilidades auditivas e as relações que o ouvir estabelece com o falar, de palavras às sentenças. O procedimento de matching to sample (MTS) tem sido utilizado para ensinar relações condicionais entre estímulos. Procedimentos associados ao MTS minimizam a ocorrência de erros durante a aprendizagem do reconhecimento auditivo de sentenças, como o fading out. O objetivo geral desta pesquisa foi controlar o efeito do procedimento de fading out no ensino de discriminações condicionais entre sentença de quatro termos ditadas e figuras de ações, de modo a verificar qual rotina de ensino (fading blocado ou fading randomizado) é mais eficaz. Foram avaliados também os efeitos desses procedimentos sobre a formação de classes de estímulos equivalentes, a precisão da fala na nomeação de figuras e os desempenhos recombinativos. Participaram duas alunas, com idade entre sete e nove anos, com deficiência auditiva sensorioneural, bilateral, de grau moderado a profundo, e usuárias de IC. As sentenças foram organizadas em ... (Resumo completo, clicar acesso eletrônico abaixo) / Abstract: Sensorineural hearing loss, severe-profound, bilateral and pre-lingual, can compromise the acquisition of verbal repertoires. The cochlear implant (CI) is an auditory device that enables access to speech sounds and allows the learning of listening and speaking skills. Studies between Speech-Language Pathology and Behavior Analysis have adopted the equivalence relations model and investigated the teaching conditions under which this population learns the auditory skills and the relationships that listening establishes with speaking, from words to sentences. The matching to sample (MTS) procedure has been used to teach conditional relations between stimuli. Procedures associated with the MTS minimize the occurrence of errors during the learning of auditory recognition of sentences, such as fading out. The general objective of this research was to control the effect of the fading out procedure in the teaching of conditional discriminations between sentences of four dictated terms and action figures, in order to verify which teaching routine (blocked fading or randomized fading) is more effective. The effects of these procedures on the formation of equivalent stimulus classes, the accuracy of speech in the naming of figures and the recombinative performances were also evaluated. Two students participated, aged between seven and nine years, with bilateral sensorineural hearing loss, of moderate to profound degree, and CI users. The sentences were organized in two matrices (Sets 1 ... (Complete abstract click electronic access below) / Mestre

In this paper we explore coloring theorems for the reals, its quotients, cardinals, and their combinations. This work is done under the scope of the axiom of determinacy. We also explore generalizations of Mycielski's theorem and show how these can be used to establish coloring theorems. To finish, we discuss the strange realm of long unions.

O Estudo 1 deste trabalho teve como objetivo avaliar, em participantes típicos, possíveis controles de estímulos envolvidos no processo de aquisição do controle por unidades verbais mínimas durante os treinos de discriminação condicional dos estímulos compostos (palavras), e durante os testes que avaliavam deste repertório. O objetivo do Estudo 2 foi realizar uma replicação sistemática dos procedimentos metodológicos e de análise empregados no Estudo 1 com uma participante com autismo e, propor procedimentos individualizados de correção de controle restrito de estímulos. No Estudo 1 foram analisados dados anteriormente coletados, por meio de um programa de ensino de leitura generalizada, de três participantes e, também, foi proposta uma categorização dos pares de estímulos relacionados às respostas dos participantes e os estímulos modelo apresentados, em cada tentativa de erro, em termos da presença de diferenças críticas ou múltiplas nos elementos dos estímulos compostos. No Estudo 2 foi aplicado o programa para a emergência de controles por unidades verbais mínimas e também foi realizada a análise de erros do Estudo 1 que, em parte, auxiliou na introdução de procedimentos específicos e individualizados para a correção de padrões de controle restrito de estímulos. Os resultados do Estudo 1 e 2 apontaram que, de forma geral, os quatro participantes demonstraram aquisição do controle por unidades verbais mínimas, mas foi observada ocorrência de variabilidade no que se refere à predominância de erros envolvendo diferenças críticas ou múltiplas e no que se refere à necessidade de introdução de procedimentos específicos e individualizados para maximizar o repertório de aquisição esperado ou corrigir controles restritos de estímulos. / Study 1 of this work aimed to evaluate, in children with typical development, possible controls of stimuli involved in the process of acquisition of control by minimal verbal units. Investigation took place during the training sessions of conditional discrimination of compound stimuli (written words), and during the tests that evaluated the emergence of control by minimal verbal units. The purpose of Study 2 was threefold. Firstly, to carry out a systematic replication of the procedures employed in Study 1 with a participant with autism. Secondly, to widen the scope of analysis of Study1. And thirdly, to propose procedures for correction of individual control to restricted stimuli. In Study 1, previously-collected data were analyzed through a program of recombinative reading with three participants. The pairs of stimuli related to the responses of participants and the stimuli model presented in each trial error were categorized. The results were analyzed to see critical and multiple differences of compound stimuli. In Study 2, not only were the same program and categorization applied, but specific and individualized procedures to correct patterns of restricted control of stimuli were introduced. The results of Study 1 and 2 showed that, in general, the four participants demonstrated acquisition of control by minimal verbal units. However, occurrence of variability regarding the prevalence of errors involving multiple or critical differences was observed. There was also variability with regards to the need to introduce specific and individualized procedures to maximize the expected repertoire or correct restricted stimuli control.

O Estudo 1 deste trabalho teve como objetivo avaliar, em participantes típicos, possíveis controles de estímulos envolvidos no processo de aquisição do controle por unidades verbais mínimas durante os treinos de discriminação condicional dos estímulos compostos (palavras), e durante os testes que avaliavam deste repertório. O objetivo do Estudo 2 foi realizar uma replicação sistemática dos procedimentos metodológicos e de análise empregados no Estudo 1 com uma participante com autismo e, propor procedimentos individualizados de correção de controle restrito de estímulos. No Estudo 1 foram analisados dados anteriormente coletados, por meio de um programa de ensino de leitura generalizada, de três participantes e, também, foi proposta uma categorização dos pares de estímulos relacionados às respostas dos participantes e os estímulos modelo apresentados, em cada tentativa de erro, em termos da presença de diferenças críticas ou múltiplas nos elementos dos estímulos compostos. No Estudo 2 foi aplicado o programa para a emergência de controles por unidades verbais mínimas e também foi realizada a análise de erros do Estudo 1 que, em parte, auxiliou na introdução de procedimentos específicos e individualizados para a correção de padrões de controle restrito de estímulos. Os resultados do Estudo 1 e 2 apontaram que, de forma geral, os quatro participantes demonstraram aquisição do controle por unidades verbais mínimas, mas foi observada ocorrência de variabilidade no que se refere à predominância de erros envolvendo diferenças críticas ou múltiplas e no que se refere à necessidade de introdução de procedimentos específicos e individualizados para maximizar o repertório de aquisição esperado ou corrigir controles restritos de estímulos. / Study 1 of this work aimed to evaluate, in children with typical development, possible controls of stimuli involved in the process of acquisition of control by minimal verbal units. Investigation took place during the training sessions of conditional discrimination of compound stimuli (written words), and during the tests that evaluated the emergence of control by minimal verbal units. The purpose of Study 2 was threefold. Firstly, to carry out a systematic replication of the procedures employed in Study 1 with a participant with autism. Secondly, to widen the scope of analysis of Study1. And thirdly, to propose procedures for correction of individual control to restricted stimuli. In Study 1, previously-collected data were analyzed through a program of recombinative reading with three participants. The pairs of stimuli related to the responses of participants and the stimuli model presented in each trial error were categorized. The results were analyzed to see critical and multiple differences of compound stimuli. In Study 2, not only were the same program and categorization applied, but specific and individualized procedures to correct patterns of restricted control of stimuli were introduced. The results of Study 1 and 2 showed that, in general, the four participants demonstrated acquisition of control by minimal verbal units. However, occurrence of variability regarding the prevalence of errors involving multiple or critical differences was observed. There was also variability with regards to the need to introduce specific and individualized procedures to maximize the expected repertoire or correct restricted stimuli control.

This paper presents a class of Stochastic Petri Nets with concurrent transition firings. It is assumed that transitions occur in steps and that for every step each enabled transition decides probabilistically whether it wants to participate in the step or not. Among the transitions which what to participate in a step, a maximal number is chosen to perform the firing step. The observable behavior is defined and equivalence relations are introduced. The equivalence relations extend the well-known trace and bisimulation equivalences for systems with step semantics to Stochastik Petri Nets with concurrent transition firing. It is shown that the equivalence notions form a lattice of interrelations.

On considère R une relation d’équivalence borélienne standard de type I I1 sur un espace de probabilités (X, µ). On étudie une certaine propriété d’homogénéité pour un graphage fixé de la relation R : on suppose que les feuilles du graphage sont toutes isomorphes à un certain graphe transitif (connexe, infini, localement fini) Γ. Que peut-on dire sur la relation ? Dans ce cas, en considérant une action "à la Mackey", on montre qu’il existe (Z ,η) un revêtement standard probabilisé de (X, µ), une action libre (qui préserve η) sur Z du groupe G (localement compact, à base dénombrable d’ouverts) des automorphismes du graphe et un isomorphisme stable des groupoïdes mesurés associés. On fait le lien entre les propriétés du groupe G et celles de la relation de départ ; en particulier la propriété (T), (H) et la moyennabilité "passent" du graphe à la relation et réciproquement. On déduit aussi de la construction quelques couplages d’équivalence mesurée (ou plus généralement des "randembeddings") entre certains sous-groupes des automorphismes de Γ et tout groupe qui contient orbitalement la relation R. Dans un deuxième chapitre, on aborde le cas particulier de la propriété (T) relative pour les paires de groupes (ΓxZ^2, Z^2), où Γ est un sous-groupe non moyennable de SL(2,Z). Cette propriété a d’abord été prouvée par Marc Burger, puis "re-démontrée" plus "visuellement" quelques années plus tard dans le cas de SL(2,Z)xZ^2 par Y. Shalom, en utilisant des découpages du plan. On reprend cette technique dans le cas général du théorème de Burger afin d’obtenir par un algorithme des constantes de Kazhdan explicites pour toute paire (ΓxZ^2, Z^2). / We consider a measure preserving standard borel equivalence relation R on a standard probability space (X,µ). We study a particular property of homogeneity for a fixed graphing of the relation R : We assume that the leaves of the graphing are all isomorphic to a given transitive graph Γ (connected, infinite, locally finite). What can be known about the relation ?In this case, considering a « Mackey action », we show that there exists a standard covering of (X,µ) i.e. a standard space Z; a probability measure η; a free, measure-preserving action on Z of G the (locally compact, second countable) group of all graph automorphisms of Γ and a stable isomorphism of the associated measured groupoid with R. We investigate some links between properties of G (resp. of the graph Γ) and those of R. In particular, Kazhdan property (T), Haagerup property (H) and amenability are preserved from the graph to the relation and conversely. We also deduce from the construction some couplings of measured equivalence (more generally some randembeddings) between subgroups of G and any group orbitally containing R. In a second chapter, we deal with the relative property (T) for the pairs (ΓxZ^2,Z^2), where Γ is a non-amenable subgroup of SL(2,Z). This property was first proved by M. Burger. Later on, Y. Shalom gave a more geometrical proof in the case of SL(2,Z)xZ^2, by using partitions of the plane. Following the same techniques in the general case of Burger's theorem, we develop an algorithm producing explicit constants for all pairs (ΓxZ^2,Z^2).

M.Sc. / In this thesis we study some of the propertie of a clas called automatic structures. Automatic structures are structures that can be encoded (in some defined way) into a set of regular languages. This encoding allows one to prove many interesting properties about automatic structures, including decidabilty results.

This dissertation deals with three topics in descriptive set theory. First, the order topology is a natural topology on ordinals. In Chapter 2, a complete classification of order topologies on ordinals up to Borel isomorphism is given, answering a question of Benedikt Löwe. Second, a map between separable metrizable spaces X and Y preserves complete metrizability if Y is completely metrizable whenever X is; the map is resolvable if the image of every open (closed) set in X is resolvable in Y. In Chapter 3, it is proven that resolvable maps preserve complete metrizability, generalizing results of Sierpiński, Vainštein, and Ostrovsky. Third, an equivalence relation on a Polish space has the Laczkovich-Komjáth property if the following holds: for every sequence of analytic sets such that the limit superior along any infinite set of indices meets uncountably many equivalence classes, there is an infinite subsequence such that the intersection of these sets contains a perfect set of pairwise inequivalent elements. In Chapter 4, it is shown that every coanalytic equivalence relation has the Laczkovich-Komjáth property, extending a theorem of Balcerzak and Głąb.