Orientador: Marcos Vieira Teixeira / Resumo: A presente investigação teve como objetivo conceber uma cronologia histórica das primeiras noções de independência linear e de base de um espaço vetorial. Um recorte histórico com dezesseis autores, entre matemáticos e físicos, foi delineado. Dois artigos foram motivadores dessa pesquisa: The History of the Concept of a Finite-Dimensional Vector de Jeremy Gray (1980), que questionava o conceito e origem do espaço vetorial, estimulando um estudo mais aprofundado sobre o tema e The Axiomatization of Linear Algebra de Gregory Moore (1995) num trabalho sobre a axiomatização da Álgebra Linear entre 1875 a 1940. Foi elaborada extensa pesquisa em obras originais dos principais matemáticos do século XVII até o século XIX, considerando o relacionamento entre eles, troca de cartas e a conformidade de estudos. Essa busca foi efetuada nas principais bibliotecas digitais como Gallica, E rara, além sites especializados como Euler Archive, à procura de artigos e livros onde se apresentasse elementos que direcionassem para ideias de conjuntos linearmente independente e de base. De Leibniz a Peano, obras originais desses dezesseis matemáticos e físicos foram analisadas, trazendo essas ideias de conjuntos linearmente independentes e de base em diferentes contextos de áreas da Matemática e da Física. Iniciando com Leibniz, que percebeu a necessidade de um ente que representasse magnitude, direção e sentido, essa cronologia passou por Euler e Lagrange com a resolução de equações diferenciais, co... (Resumo completo, clicar acesso eletrônico abaixo) / Abstract: The present investigation aimed to conceive a historical chronology of the first notions of linear and basic independence of a vector space. A historical section with sixteen authors, between mathematicians and physicists, was designed. Two articles motivated this research: The History of the Concept of a Finite-Dimensional Vector by Jeremy Gray (1980), which questioned the concept and origin of the vector space, stimulating a deeper study on the theme and The Axiomatization of Linear Algebra by Gregory Moore (1995) in a work on the axiomatization of Linear Algebra between 1875 and 1940. Extensive research was carried out on original works by leading mathematicians from the 17th to the 19th century, considering the relationship between them, exchange of letters and the conformity of studies . This search was carried out in the main digital libraries such as Gallica, E rara, in addition to specialized sites such as Euler Archive, looking for articles and books where elements were presented that would lead to ideas of linearly independent and basic sets. From Leibniz to Peano, original works of these sixteen mathematicians and physicists were analyzed, bringing these ideas from linearly independent and basic sets in different contexts in the areas of Mathematics and Physics. Starting with Leibniz, who realized the need for an entity that represented magnitude, direction and sense, this chronology went through Euler and Lagrange with the resolution of differential equations, Wes... (Complete abstract click electronic access below) / Doutor

We evaluate the validity of the Adaptive Market Hypothesis (AMH) in a Swedish context by testing for stock return predictability on the OMXS30 stock index between 1986 and 2014 using daily returns and monthly two year moving subsamples. To our knowledge, this is the first study to evaluate the AMH in a Swedish context. Three tests for linear independence based on Lo and MacKinlay (1988) variance ratio test, namely the Chow and Denning joint test as well as Wright (2000) joint rank and sign tests are used. We also test for non-linear independence using the BDS test statistics. Presented in our findings is evidence of time-varying predictability where stock returns go through periods of return predictability and non-predictability. When evaluating the different market conditions (volatility, bull, bear, up, down and normal markets) we find that these different market conditions govern the degree of stock return predictability in different ways. Our findings support the AMH on the OMXS30 stock index and in contrast to previous research regarding market efficiency on the Swedish stock market, we do not find persistent stock return predictability over the short and long term.

Cette thèse s'inscrit dans le prolongement du résultat d'Apéry donnant l'irrationalité de ζ (3) et de celui de Ball-Rivoal prouvant qu'il existe une infinité d'entiers impairs en lesquels la fonction zêta de Riemann prend des valeurs irrationnelles. Un outil crucial dans la démonstration de Ball-Rivoal est le critère d'indépendance linéaire de Nesterenko, qui a été généralisé par Fischler et Zudilin pour exploiter sous des hypothèses très restrictives la présence de diviseurs communs aux coefficients des formes linéaires. Une généralisation ultérieure due à Fischler s'applique lorsqu'on dispose d'approximations simultanées des nombres réels en question (et non plus de combinaisons Z-linéaires petites de ces nombres).Dans cette thèse, on améliore ce dernier résultat en affaiblissant considérablement les hypothèses sur les diviseurs. On démontre aussi un critère d'indépendance linéaire analogue, dans l'esprit de celui de Siegel. Dans une autre partie en commun avec Zudilin, on construit, en utilisant des identités hypergéométriques, des approximations simultanées de ζ (2) et ζ (3) qui permettent de démontrer en même temps l'irrationalité de ces deux nombres. En appliquant essentiellement le critère démontré précédemment, on en déduit une minoration des combinaisons Z-linéaires de 1, ζ 2) et ζ (3), sous des hypothèses de divisibilité très fortes sur les coefficients (si bien que l'indépendance linéaire sur Q de ces trois nombres est toujours conjecturale). / This Ph.D. thesis lies in the path opened by Apéry who proved the irrationality of ζ(3) andalready followed by Ball-Rivoal who proved that there are infinitely many odd integers at which Riemann zeta function takes irrational values. A fundamental tool in the proof of Ball-Rivoal is Nesterenko’s linear independence criterion. This criterion has been generalized by Fischler and Zudilin to use common divisors of the coefficients of linear forms, under some restrictive assumptions. Then Fischler gave another generalization for simultaneous approximations (instead of small Z-linear combinations).In this Ph.D. thesis, we improve this last result by greatly weakening the assumption on thedivisors. We prove also an analogous linear independence criterion in the spirit of Siegel. Inanother part joint with Zudilin, we construct simultaneous linear approximations to ζ(2) and ζ(3) using hypergeometric identitites. These linear approximations allow one to prove at thesame time the irrationality of ζ(2) and that of ζ(3). Then, using a criterion from the previouspart, we deduce a lower bound on Z-linear combinations of 1, ζ(2) and ζ(3), under somestrong divisibility hypotheses on the coefficients (so that the Q-linear independence of thesethree numbers still remains an open problem).

A lacunary series is a Taylor series with large gaps between its non-zero coefficients. In this thesis we exploit these gaps to obtain results of linear independence of values of lacunary series at integer points. As well, we will study different methods found in Diophantine approximation which we use to study arithmetic properties of values of lacunary series at algebraic points. Among these methods will be Mahler's method and a new approach due to Jean-Paul Bézivin.

A lacunary series is a Taylor series with large gaps between its non-zero coefficients. In this thesis we exploit these gaps to obtain results of linear independence of values of lacunary series at integer points. As well, we will study different methods found in Diophantine approximation which we use to study arithmetic properties of values of lacunary series at algebraic points. Among these methods will be Mahler's method and a new approach due to Jean-Paul Bézivin.

No description available.

In this thesis we discuss linear dependence of translations which is intimately related to the zero divisor conjecture. We also discuss the square integrable representations of the generalized Wyle-Heisenberg group in 𝑛² dimensions and its relations with Gabor's question from Gabor Analysis in the light of the time-frequency equation. We study the zero divisor conjecture in relation to the reduced 𝐶*-algebras and operator norm 𝐶*-algebras. For certain classes of groups we address the zero divisor conjecture by providing an isomorphism between the the reduced 𝐶*-algebra and the operator norm 𝐶*-algebra. We also provide an isomorphism between the algebra of weak closure and the von Neumann algebra under mild conditions. Finally, we prove some theorems about the injectivity of some spaces as ℂ𝐺 modules for some groups 𝐺. / Master of Science

當我們解一個遞迴關係的特徵方程式時，不管解得的根是相異根或者是重根，皆視這些跟產生的解之線性獨立為理所當然，因此很容易寫出此遞迴關係的通解乃是這些解的線性組合。在本文中，我們將透過廣義的范德蒙行列式(Generalized Vandermonde Determinant)的計算，很清楚地看出這些解之間的線性獨立。 / When we solve a characteristic equation of a recurrence relation, no matter what the roots are distinct or not, we take the linear independence of these solutions producing by each root for granted. Basing on this fact, we can easily write out the general solutions of this recurrence relation by using linear combination of these solutions. In this paper, we will see the linear independence of these solutions very clearly through the calculation of Generalized Vandermonde Determinant.

Made available in DSpace on 2016-04-27T16:57:53Z (GMT). No. of bitstreams: 1 dissertacao_andre_lucio_grande.pdf: 4293121 bytes, checksum: 4477bafc1f5e2bb6d741991b929a5d16 (MD5) Previous issue date: 2006-08-14 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / This research aims to investigate which semiotic representation registers are most used in Linear Algebra textbooks in the presentation and study of the topic linear independence. The mathematics textbook is a resource much used by the teachers in its work. They present many representation registers, which can to be analyzed qualitatively to assess their potential contribution to the teaching-learning process. To this end, five Linear Algebra textbooks were selected for an analysis which involved the diagnosis of the representation registers used in the definition of linear independence, as well as the examples and exercises proposed for its study. The theory used in the analysis of these textbooks was Raymond Duval´s theory of Registers of Semiotic Representation, which highlights the importance of changes between registers in the learning of mathematical objects. In each textbooks, the registers present were classified and analyzed, along with the possible transformations of registers as carried out in the resolution of the examples and considered exercises. As result, scarcity of the use of some registers of representation it was evidenced in the chapters that approach the slight knowledge of independence and linear dependence, as for example, the geometric one to determine if a set of vectors is linearly independent or not, and the register of the natural language, in the resolution of examples and considered exercises that involve demonstrations. The analysis also indicated a lack of attention to conversions between registers in the examples and proposed exercises of analyzed textbooks / Esta pesquisa tem como objetivo investigar nos livros didáticos de Álgebra Linear quais são os registros de representação semiótica mais utilizados no estudo das noções e atividades propostas sobre independência linear. O livro didático é um recurso muito utilizado pelos professores no seu trabalho, o qual apresenta muitos registros de representação que podem ser analisados qualitativamente para a contribuição no processo de ensino-aprendizagem. Para isso, foram selecionados cinco livros didáticos de Álgebra Linear que se diagnosticou os registros de representação utilizados na definição, exemplos e exercícios propostos a respeito de independência linear. A teoria utilizada na análise dos livros didáticos foi a dos Registros de Representação Semiótica de Raymond Duval, que destaca a importância da mudança de registros na aprendizagem dos objetos matemáticos. Em cada obra, os registros foram classificados e analisados, bem como as possíveis transformações de registros realizadas na resolução dos exemplos e exercícios propostos. Como resultado, constatou-se nos capítulos que abordam as noções de independência e dependência linear uma escassez da utilização de alguns registros de representação, como por exemplo, o geométrico para determinar se um conjunto de vetores é linearmente independente ou não, e o registro da língua natural, na resolução de exemplos e exercícios propostos que envolvem demonstrações. Além disso, verificou-se a falta de conversão de registros nas definições, exemplos e exercícios propostos dos livros didáticos analisados