As classes de equivalência de estímulos são produto das contingências de reforço. No entanto, as pesquisas têm utilizado apenas contingências de reforço positivo e investigado, na sua maioria, como eventos relacionados a um mesmo reforçador podem passar a compor uma mesma classe. O objetivo do presente trabalho foi avaliar o efeito de diferentes contingências de reforço, positivo e negativo, no estabelecimento de discriminações condicionais na formação de classes de equivalência em humanos. Foram realizados dois experimentos. No Experimento I, 12 participantes treinaram simultaneamente 12 relações condicionais (AB e BC) utilizando três arranjos de contingência de reforçamento diferentes: reforçamento positivo (Ganhar/Manter), reforçamento negativo (Manter/Perder) e uma contingência mista de reforçamento negativo e positivo (Ganhar/Perder), seguidos de testes de formação de classes de equivalência. No Experimento I, os participantes expostos aos testes de equivalência acertaram todas as tentativas de teste. Estes resultados demonstram a possibilidade de formação de classes de equivalência em uma contingência de reforço negativo, porém um possível efeito de teto impede a comparação de cada contingência na formação de classes. No Experimento II, sete participantes foram expostos ao treino de 12 relações condicionais, com classes com um maior número de nódulos (AB, BC e CD) utilizando apenas duas contingências de reforço (Ganhar/Manter e Manter/Perder). Em ambos os experimentos, a sequência de aquisição das discriminações e o viés inicial produzido pelas contingências de reforço negativo sugerem um impacto diferencial da punição em relação ao reforçamento. Os resultados nos testes de equivalência sugerem que a formação de classes é menos provável quando utilizada uma contingência de reforço negativo em relação a de reforço positivo. Foi discutida a generalidade da formulação sobre formação de classes de equivalência, priorizando os padrões de controle de estímulos produzido pelas diferentes contingências. Ressalta-se a necessidade de maior investigação sobre quais os efeitos do uso de contingências aversivas no controle da resposta e mais especificamente no controle de estímulos / Equivalence classes are produced by reinforcement contingencies, however, research have used exclusively positive reinforcement and investigated mostly how events related to a same reinforcer may result in the formation a same class. The aim of this study was to evaluate how different reinforcement contingencies affect the acquisition of conditional discriminations and equivalence class formation in humans. Two experiments were conducted. In Experiment I, 12 participants learned 12 conditional relations (AB BC) under three different contingencies: positive reinforcement (Gain/Maintain), negative reinforcement (Maintain/Lose) and a mixed contingency (Gain/Lose). Equivalence class test were applied subsequently. In Experiment I, participants that underwent the equivalence tests performed correctly in all trials, although a roof effect might have prevented a proper comparison of the contingencies effect. In Experiment II, seven participants learned 12 conditional relations with more nodes (AB, BC e CD) and two reinforcement contingencies (Gain/Maintain and Maintain/Lose). In both experiments, the conditional discrimination acquisition sequence and the initial bias produced by the negative reinforcement contingency suggest a differential effect of punishment in relation to reinforcement. The equivalence test results suggest the possibility of equivalence class formation using a negative reinforcement contingency and indicate that equivalence classes are less probable using a negative reinforcement contingency than a positive reinforcement one. The generality of the equivalence class formulation was discussed emphasizing the investigation of stimuli control patterns produced by the contingencies. The lack of knowledge about stimulus control produced by aversive contingencies was highlighted

As classes de equivalência de estímulos são produto das contingências de reforço. No entanto, as pesquisas têm utilizado apenas contingências de reforço positivo e investigado, na sua maioria, como eventos relacionados a um mesmo reforçador podem passar a compor uma mesma classe. O objetivo do presente trabalho foi avaliar o efeito de diferentes contingências de reforço, positivo e negativo, no estabelecimento de discriminações condicionais na formação de classes de equivalência em humanos. Foram realizados dois experimentos. No Experimento I, 12 participantes treinaram simultaneamente 12 relações condicionais (AB e BC) utilizando três arranjos de contingência de reforçamento diferentes: reforçamento positivo (Ganhar/Manter), reforçamento negativo (Manter/Perder) e uma contingência mista de reforçamento negativo e positivo (Ganhar/Perder), seguidos de testes de formação de classes de equivalência. No Experimento I, os participantes expostos aos testes de equivalência acertaram todas as tentativas de teste. Estes resultados demonstram a possibilidade de formação de classes de equivalência em uma contingência de reforço negativo, porém um possível efeito de teto impede a comparação de cada contingência na formação de classes. No Experimento II, sete participantes foram expostos ao treino de 12 relações condicionais, com classes com um maior número de nódulos (AB, BC e CD) utilizando apenas duas contingências de reforço (Ganhar/Manter e Manter/Perder). Em ambos os experimentos, a sequência de aquisição das discriminações e o viés inicial produzido pelas contingências de reforço negativo sugerem um impacto diferencial da punição em relação ao reforçamento. Os resultados nos testes de equivalência sugerem que a formação de classes é menos provável quando utilizada uma contingência de reforço negativo em relação a de reforço positivo. Foi discutida a generalidade da formulação sobre formação de classes de equivalência, priorizando os padrões de controle de estímulos produzido pelas diferentes contingências. Ressalta-se a necessidade de maior investigação sobre quais os efeitos do uso de contingências aversivas no controle da resposta e mais especificamente no controle de estímulos / Equivalence classes are produced by reinforcement contingencies, however, research have used exclusively positive reinforcement and investigated mostly how events related to a same reinforcer may result in the formation a same class. The aim of this study was to evaluate how different reinforcement contingencies affect the acquisition of conditional discriminations and equivalence class formation in humans. Two experiments were conducted. In Experiment I, 12 participants learned 12 conditional relations (AB BC) under three different contingencies: positive reinforcement (Gain/Maintain), negative reinforcement (Maintain/Lose) and a mixed contingency (Gain/Lose). Equivalence class test were applied subsequently. In Experiment I, participants that underwent the equivalence tests performed correctly in all trials, although a roof effect might have prevented a proper comparison of the contingencies effect. In Experiment II, seven participants learned 12 conditional relations with more nodes (AB, BC e CD) and two reinforcement contingencies (Gain/Maintain and Maintain/Lose). In both experiments, the conditional discrimination acquisition sequence and the initial bias produced by the negative reinforcement contingency suggest a differential effect of punishment in relation to reinforcement. The equivalence test results suggest the possibility of equivalence class formation using a negative reinforcement contingency and indicate that equivalence classes are less probable using a negative reinforcement contingency than a positive reinforcement one. The generality of the equivalence class formulation was discussed emphasizing the investigation of stimuli control patterns produced by the contingencies. The lack of knowledge about stimulus control produced by aversive contingencies was highlighted

The random Fibonacci sequence is defined by t_1 = t_2 = 1 and t_n = ± t_{n–1} + t_{n–2} , for n ? 3, where each ± sign is chosen at random with P(+) = P(–) = 1/2. We can think of all possible such sequences as forming a binary tree T. Viswanath has shown that almost all random Fibonacci sequences grow exponentially at the rate 1.13198824.... He was only able to find 8 decimal places of this constant through the use of random matrix theory and a fractal measure, although Bai has extended the constant by 5 decimal places. Numerical experimentation is inefficient because the convergence is so slow. We will discuss a new computation of Viswanath's constant which is based on a formula due to Kalmár-Nagy, and uses an interesting reduction R of the tree T developed by Rittaud. Also, we will focus on the growth rate of the expected value of a random Fibonacci sequence, which was studied by Rittaud. This differs from the almost sure growth rate in that we first find an expression for the average of the n^th term in a sequence, and then calculate its growth. We will derive this growth rate using a slightly different and more simplified method than Rittaud, using the tree R and a Pascal-like array of numbers, for which we can further give an explicit formula. We will also consider what happens to random Fibonacci sequences when we remove the randomness. Specifically, we will choose coefficients which belong to the set {1, –1} and form periodic cycles. By rewriting our recurrences using matrix products, we will analyze sequence growth and develop criteria based on eigenvalue, trace and order, for determining whether a given sequence is bounded, grows linearly or grows exponentially. Further, we will introduce an equivalence relation on the coefficient cycles such that each equivalence class has a common growth rate, and consider the number of such classes for a given cycle length. Lastly we will look at two ways to completely characterize the trace, given the coefficient cycle, by breaking the matrix product up into blocks.

Boolean functions are an important area of study for cryptography. These functions, consisting merely of one's and zero's, are the heart of numerous cryptographic systems and their ability to provide secure communication. Boolean functions have application in a variety of such systems, including block ciphers, stream ciphers and hash functions. The continued study of Boolean functions for cryptography is therefore fundamental to the provision of secure communication in the future. This thesis presents an investigation into the analysis of Boolean functions and in particular, analysis of affine transformations with respect to both the design and application of Boolean functions for cryptography. Past research has often been limited by the difficulties arising from the magnitude of the search space. The research presented in this thesis will be shown to provide an important step towards overcoming such restrictions and hence forms the basis for a new analysis methodology. The new perspective allows a reduced view of the Boolean space in which all Boolean functions are grouped into connected equivalence classes so that only one function from each class need be established. This approach is a significant development in Boolean function research with many applications, including class distinguishing, class structures, self mapping analysis and finite field based s-box analysis. The thesis will begin with a brief overview of Boolean function theory; including an introduction to the main theme of the research, namely the affine transformation. This will be followed by the presentation of a fundamental new theorem describing the connectivity that exists between equivalence classes. The theorem of connectivity will form the foundation for the remainder of the research presented in this thesis. A discussion of efficient algorithms for the manipulation of Boolean functions will then be presented. The ability of Boolean function research to achieve new levels of analysis and understanding is centered on the availability of computer based programs that can perform various manipulations. The development and optimisation of efficient algorithms specifically for execution on a computer will be shown to have a considerable advantage compared to those constructed using a more traditional approach to algorithm optimisation. The theorem of connectivety will be shown to be fundamental in the provision many avenues of new analysis and application. These applications include the first non-exhaustive test for determining equivalent Boolean functions, a visual representation of the connected equivalence class structure to aid in the understanding of the Boolean space and a self mapping constant that enables enumeration of the functions in each equivalence class. A detailed survey of the classes with six inputs is also presented, providing valuable insight into their range and structure. This theme is then continued in the application Boolean function construction. Two important new methodologies are presented; the first to yield bent functions and the second to yield the best currently known balanced functions of eight inputs with respect to nonlinearity. The implementation of these constructions is extremely efficient. The first construction yields bent functions of a variety of algebraic order and inputs sizes. The second construction provides better results than previously proposed heuristic techniques. Each construction is then analysed with respect to its ability to produce functions from a variety of equivalence classes. Finally, in a further application of affine equivalence analysis, the impact to both s-box design and construction will be considered. The effect of linear redundancy in finite field based s-boxes will be examined and in particular it will be shown that the AES s-box possesses complete linear redundancy. The effect of such analysis will be discussed and an alternative construction to s-box design that ensures removal of all linear redundancy will be presented in addition to the best known example of such an s-box.

Made available in DSpace on 2016-06-02T20:30:07Z (GMT). No. of bitstreams: 1 5064.pdf: 2133437 bytes, checksum: 91011d8e6a06f6e7760589cf0b30918d (MD5) Previous issue date: 2013-03-13 / Universidade Federal de Minas Gerais / Studies about the effects of contextual control over equivalence classes have been conducted since the 1980s and a relevant question in this area, both in terms of experimental and applied research, is whether changes in contextual variables could result in changes in classes originally formed under different contextual variables. This research investigated, in four studies, the effects of procedures for establishing the contextual control of conditional relations and equivalence classes. The participants were undergraduate students. The stimuli used as sample and comparison were abstract figures and the contextual stimuli were represented by the colors of the background on which the stimuli were presented. In Study 1, we conducted a procedure for the formation of equivalence classes on backgrounds of red and blue colors. Later we conducted a test that showed the equivalence relations in the presence of contextual stimuli blue, red and green. The results showed that participants had responses consistent with the last equivalence class trained, independent of contextual stimulus. Study 2 replicated the previous study, but tests also showed baseline relations. The results obtained were similar to Study 1. Study 3, initially, partially replicated the procedure of previous studies, only for conditional relations AB and then tested the effects of a conditional discrimination training procedure interspersing the colors of the background. The results revealed that this procedure was most effective in setting the control contextual than those conducted in Studies 1 and 2. Study 4 investigated the effects of a interpersed training procedure for the establishment of the contextual control of recombined conditional relations. The procedure consisted of six phases. Phase1 was conducted a conditional discrimination training interspersing the colors of the background. In Phase 2, we carried out the test of relations trained in Phase1. In Phase 3, tests were conducted in which trials were included with a green background. Phase 4 included the symetry relations. Phase 5 presented the same relations were tested in Step 4, however the configuration of stimulus presentation on the computer screen was different. Finally, in Step 6, a test aimed to verify the abstraction of responding under contextual control, with new stimulus and background colors. The results suggest that the contextual control was established for some of the participants. Investigations about the baseline relations training, specifically the interspersed training , for establishing the contextual control is required to produce more robust results. / Estudos acerca dos efeitos do controle contextual sobre classes de equivalência vêm sendo conduzidos desde a década de 1980 e uma questão relevante nesta área, tanto do ponto de vista experimental quanto aplicado, é se mudanças em variáveis contextuais poderiam resultar em mudanças em classes originalmente formadas sob variáveis contextuais diferentes. A presente pesquisa investigou, em quatro estudos, os efeitos de variáveis de procedimentos para o estabelecimento do controle contextual de relações condicionais e classes de equivalência. Os participantes dos estudos foram estudantes universitários. Os estímulos utilizados como modelo e comparação foram figuras abstratas e os estímulos contextuais foram representados pelas cores do plano de fundo sobre os quais os estímulos eram apresentados. No Estudo 1, foi conduzido um procedimento para a formação de classes de equivalência diferentes na presença das cores azul e vermelha e posteriormente foi conduzido um teste que apresentava as relações de equivalência na presença dos estímulos contextuais azul, vermelho e verde. Os resultados revelaram que os participantes apresentaram respostas consistentes com a última classe de equivalência treinada, independente do estímulo contextual. O Estudo 2 replicou o estudo anterior, incluindo nos testes, além de relações de equivalência, relações de linha de base. Os resultados obtidos foram semelhantes ao do Estudo 1. O Estudo 3, inicialmente, replicou parcialmente o procedimento dos estudos anteriores apenas para relações condicionais AB e, em seguida, testou-se os efeitos de um procedimento de treino de discriminação condicional AB intercalando-se as cores do plano de fundo. Os resultados revelaram que este procedimento foi mais eficaz no estabelecimento do controle contextual que os conduzidos nos Estudos 1 e 2. O Estudo 4 avaliou os efeitos de um procedimento de treino intercalado para o estabelecimento do controle contextual de relações condicionais recombinadas. O procedimento contou com seis etapas, sendo que na Etapa 1 foi conduzido um treino de discriminação condicional intercalando-se as cores do plano de fundo. Na Etapa 2, realizou-se o teste das relações treinadas na Etapa1. Na Etapa 3, foram conduzidos testes nos quais foram incluídas tentativas com um plano de fundo da cor verde. A Etapa 4 incluiu as relações de simetria BA. Na Etapa 5, foram testadas as mesmas relações da Etapa 4, no entanto a configuração de apresentação dos estímulos na tela do computador foi diferente. Finalmente, na Etapa 6, conduziu-se um teste cujo objetivo foi verificar a abstração do responder sob controle contextual, com estímulos e cores do plano de fundo inéditas no experimento. Os resultados sugerem que o controle contextual foi estabelecido para parte dos participantes. Investigações acerca dos treinos das relações de linha de base, especificamente do treino intercalado, para o estabelecimento do controle contextual são necessárias para a produção de resultados mais robustos.

This thesis examines the relation between classical and quantum mechanics from philosophical, mathematical and physical standpoints. It first presents arguments in support of "conjectural realism" in scientific theories distinguished by explicit contextual structure and empirical testability; and it analyses intertheoretic reduction in terms of weakly equivalent theories over a domain of applicability. Familiar formulations of classical and quantum mechanics are shown to follow from a general theory of mechanics based on pure states with an intrinsic probability structure. This theory is developed to the stage where theorems from quantum logic enable expression of the state geometry in Hilbert space. Quantum and classical mechanics are then elaborated and applied to subsystems and the measurement process. Consideration is also given to spacetime geometry and the constraints this places on the dynamics. Physics and Mathematics, it is argued, are growing apart; the inadequate treatment of approximations in general and localization in quantum mechanics in particular are seen as contributing factors. In the description of systems, the link between localization and lack of knowledge shows that quantum mechanics should reflect the domain of applicability. Restricting the class of states provides a means of achieving this goal. Localisation is then shown to have a mathematical expression in terms of compactness, which in tum is applied to yield a topological theory of bound and scattering states: Finally, the thesis questions the validity of "classical limits" and "quantisations" in intertheoretic reduction, and demonstrates that a widely accepted classical limit does not constitute a proof of reduction. It proposes a procedure for determining whether classical and quantum mechanics are weakly equivalent over a domain of applicability, and concludes that, in this restricted sense, classical mechanics reduces to quantum mechanics.

Par la présente recherche, nous avons voulu explorer les avenues possibles offertes par l'enrichissement de l'idée de fonction mathématique conçue de façon active, notamment par les notions de composition de fonctions et de cheminements équivalents. Pour ce faire, nous avons développé un ensemble de problèmes et quelques outils d'exploration reflétant la conception active que nous entendions, puis nous les avons mis entre les mains d'élèves du secondaire regroupés en équipes de deux ou trois. Les résolutions proposées par les participants et les expressions utilisées par ces derniers pour les décrire montrent des manifestations d'une compréhension géométrique de la fonction favorable à la recherche de solutions, qui se distingue de celle pouvant émerger de l'interprétation de sa représentation graphique dans le plan cartésien. Il semble également que l'activité favorisée par la résolution des problèmes ou par la manipulation des outils puisse contribuer à l'évolution du rapport à la preuve mathématique d'une manière suffisamment significative pour encourager d'éventuelles explorations dans cette direction. / Through this research, we wanted to explore the possible avenues offered by the enrichment of the idea of mathematical function conceived in an active manner, in particular by the notions of composition of functions and equivalent walks. To achieve this, we developed a set of problems and some exploration tools reflecting the active conception that we intended, then we put them in the hands of secondary school students grouped into teams of two or three. The solutions proposed by the participants as well as the expressions that they used to describe them show manifestations of a geometric understanding of the function favorable to the search for solutions, which is distinct from the one that can be derived from its graph in the Cartesian plane. In addition, students' reasoning in solving the proposed problems or manipulating the exploration tools showed some movement between different conceptions of mathematical proof ; this could warrant further research in that direction.