Made available in DSpace on 2016-06-02T20:02:47Z (GMT). No. of bitstreams: 1 2968.pdf: 7361973 bytes, checksum: b415b87956904ca667a791b2090743ac (MD5) Previous issue date: 2010-03-12 / Secretaria da Educação do Estado de São Paulo / The objective of this dissertation is to describe the work that we carry through in the elaboration of didactic material that became the teaching of logarithms more significant for the students of high school level. For this we constructed a didactic sequence using resources of mathematical education. To organize our research we use principles of Didactic Engineering. Our sequence also includes exposing lessons, activities with the use of the computer and the calculator. This didactic sequence includes Worksheets elaborated with the concern to make the student had the maximum autonomy as possible to decide them. As well as these activities, exposing lessons, activities with the use of computer and calculator are also part of our sequence. The sequence was applied to 42 students of two classrooms of first grade of high school level, in a cooperative school of the city of Araraquara - SP. For that, 15 lessons of 50 minutes each were used. Most of the activities was developed in pairs. The results were gotten through the analysis of the activities solved by the students and also through the comment and the notes that we made during the application of the sequence. According to these results we consider that our objectives were reached, because the activities were developed successfully and a posteriori analysis showed us that the applied sequence contributed for the learning of the students to occur of significant form. / O objetivo desta dissertação é descrever o trabalho que realizamos na elaboração de material didático que tornasse o ensino de logaritmos mais significativo para os estudantes do Ensino Médio. Para isso construímos uma sequência didática utilizando recursos da Educação Matemática. Para organizar nossa pesquisa usamos princípios da Engenharia Didática. Essa sequência didática inclui Folhas de Atividades que foram elaboradas com a preocupação de fazer com que o estudante tivesse o máximo de autonomia possível para resolvê-las. Nossa sequência também inclui aulas expositivas, atividades com o uso do computador e da calculadora. A sequência foi aplicada para 42 estudantes de duas salas de primeira série do Ensino Médio, em uma escola cooperativa da cidade de Araraquara SP. Para isso foram utilizadas 15 aulas de 50 minutos cada. Em sua grande maioria, as atividades foram desenvolvidas em duplas. Os resultados foram obtidos através da análise das atividades resolvidas pelos estudantes e também da observação e das anotações que fizemos durante a aplicação da sequência. De acordo com esses resultados consideramos que nossos objetivos foram alcançados, pois as atividades foram desenvolvidas com sucesso e a análise a posteriori nos mostrou que a sequência aplicada contribuiu para que o aprendizado dos estudantes ocorresse de forma significativa.

Nous étudions ici deux problèmes diophantiens distincts. Le premier concerne les points entiers sur les courbes modulaires associées au normalisateur de sous-groupe de Cartan non déployé. Le deuxième concerne la recherche de point de multiplication complexe sur les droites. Dans les deux cas la méthode de résolution est algorithmique. On utilise la méthode de Baker sur les formes linéaires en logarithmes ainsi que des méthodes de réduction effectives. En particulier cette méthode permet d’obtenir les points entiers sur la courbe associée au normalisateur de sous-groupe de Cartan non déployé pour les niveaux compris entre 7 et 71. / We study here two diophantine problem. The first one deals with integral point on modular curves associated to normalizer of non-split Cartan subgroup. The second one is about finding singular moduli on straight line. In both cases, we solve theproblem in an algorithmic way. We use Baker’s method on linear form in logarithm and some effective technical of reduction. In particular this method gives integral points on the curve associated to normalizer of non-split Cartan subgroup for level between 7 and 71.

Made available in DSpace on 2016-04-27T16:58:28Z (GMT). No. of bitstreams: 1 Fernando Tavares da Silva.pdf: 4680921 bytes, checksum: 745c13dd17b022406b1dfc18f8f102f2 (MD5) Previous issue date: 2007-10-25 / The objective of this research is to investigate the approach used in proofs and demonstrations of the logarithmic mathematical object, in a collection of textbooks adopted in the Brazilian Secondary School; as well as to conceive and apply a didactic sequence to introduce the student into the deductive mathematical thought. The research intends to answer the following questions: (1) How does the author of the textbooks approach the process of proving, regarding the subject logarithms in his work? Are the readers stimulated to find out proofs in the suggested activities? (2) Which difficulties do first grade students of the Brazilian Secondary School present during a process of proving? In order to answer the first question, we have analysed the collection Matemática for Secondary School by Luiz Roberto Dante, making use of the criteria of the National Book Catalogue for Secondary School (CNLEM). For the second question, we have used some elements of the didactic engineering methodology , making use of the types of proofs by Balacheff. The results of our analysis bring forward that the author of the collection is always concerned about presenting some kind of justification or demonstration for each element introduced. However, there are few activities that stimulate the reader to produce proofs. Regarding the didactic sequence, the research presents some difficulties observed in the process of production of proofs. Furthermore, it shows that the sequence provided the students development from empiric validations to deductive validations / O objetivo desta pesquisa é investigar a abordagem conferida a provas e demonstrações do objeto matemático logaritmo, numa coleção de livros didáticos para o Ensino Médio, bem como conceber e aplicar uma seqüência didática para introduzir o aluno da primeira série do Ensino Médio ao pensamento matemático dedutivo. A pesquisa procura responder às seguintes questões de pesquisa: (1) Como o autor de livros didáticos aborda o processo de prova em relação ao tema logaritmo na sua coleção? Os alunos leitores são estimulados a realizar provas em atividades propostas? (2) Quais dificuldades os alunos da primeira série do Ensino Médio apresentam durante um processo de produção de provas? Para responder à primeira questão analisamos a coleção Matemática do Ensino Médio de autoria de Luiz Roberto Dante utilizando para isso os critérios do Catálogo Nacional do livro para o Ensino Médio (CNLEM). Para a segunda questão, adotamos alguns elementos da metodologia engenharia didática. Empregamos para essa análise a tipologia de provas de Balacheff. Os resultados das nossas análises aduzem que o autor da coleção se preocupa em oferecer sempre algum tipo de justificativa ou demonstração para cada elemento novo apresentado. Entretanto, há poucas atividades que estimulam o leitor a produzir provas. No tocante à seqüência didática, a pesquisa aponta algumas dificuldades verificadas no processo de produção de provas e mostra que apesar disso, a seqüência permitiu um avanço por parte dos alunos de validações empíricas para as validações dedutivas

In 1878, Lucas \cite{lucas} investigated the sequences $(\ell_n)_{n=0}^\infty$ where $$\ell_n=\frac{\alpha^n-\beta^n}{\alpha-\beta},$$ $\alpha \beta$ and $\alpha+\beta$ are coprime integers, and where $\beta/\alpha$ is not a root of unity. Lucas sequences are divisibility sequences; if $m|n$, then $\ell_m|\ell_n$, and more generally, $\gcd(\ell_m,\ell_n)=\ell_{\gcd(m,n)}$ for all positive integers $m$ and $n$. Matijasevic utilised this divisibility property of Lucas sequences in order to resolve Hilbert's 10th problem. \noindent In 1930, Lehmer \cite{lehmer} introduced the sequences $(u_n)_{n=0}^\infty$ where \begin{eqnarray*} u_n& = & \frac{\alpha^{n}-\beta^n}{\alpha^{\epsilon(n)}-\beta^{\epsilon(n)}},\\ \epsilon(n)&=&\left\{\begin{array}{ll} 1, \hspace{.1in}\mbox{if}\hspace{.1in}n\equiv 1 \pmod 2;\\ 2, \hspace{.1in}\mbox{if}\hspace{.1in}n\equiv 0\pmod 2;\end{array}\right. \end{eqnarray*} $\alpha \beta$ and $(\alpha +\beta)^2$ are coprime integers, and where $\beta/\alpha$ is not a root of unity. The sequences $(u_n)_{n=0}^\infty$ are known as Lehmer sequences, and the terms of these sequences are known as Lehmer numbers. Lehmer showed that his sequences had similar divisibility properties to those of Lucas sequences, and he used them to extend the Lucas test for primality. \noindent We define a prime divisor $p$ of $u_n$ to be a primitive divisor of $u_n$ if $p$ does not divide $$(\alpha^2-\beta^2)^2u_3\cdots u_{n-1}.$$ Note that in the list of prime factors of the first $n-1$ terms of the sequence $(u_n)_{n=0}^\infty$, a primitive divisor of $u_n$ is a new prime factor. \noindent We let \begin{eqnarray*} \kappa& = & k(\alpha \beta\max\{(\alpha-\beta)^2,(\alpha+\beta)^2\}),\\ \eta & = & \left\{\begin{array}{ll}1\hspace{.1in}\mbox{if}\hspace{.1in}\kappa\equiv 1\pmod 4,\\ 2\hspace{.1in}\mbox{otherwise},\end{array}\right. \end{eqnarray*} where $k(\alpha \beta \max\{(\alpha-\beta)^2,(\alpha+\beta)^2\})$ is the squarefree kernel of $\alpha \beta \max\{(\alpha-\beta)^2,(\alpha+\beta)^2\}$. On the one hand, building on the work of Schinzel \cite{schinzelI}, we prove that if $n>4$, $n\neq 6$, $n/(\eta \kappa)$ is an odd integer, and the triple $(n,\alpha,\beta)$, in case $(\alpha-\beta)^2>0$, is not equivalent to a triple $(n,\alpha,\beta)$ from an explicit table, then the $n$th Lehmer number $u_n$ has at least two primitive divisors. Moreover, we prove that if $n\geq 1.2\times 10^{10}$, and $n/(\eta \kappa)$ is an odd integer, then the $n$th Lehmer number $u_n$ has at least two primitive divisors. On the other hand, building on the work of Stewart \cite{stewart77}, we prove that there are only finitely many triples $(n,\alpha,\beta)$, where $n>6$, $n\neq 12$, and $n/(\eta \kappa)$ is an odd integer, such that the $n$th Lehmer number $u_n$ has less than two primitive divisors, and that these triples may be explicitly determined. We determine all of these triples $(n,\alpha,\beta)$ up to equivalence explicitly when $6<n\leq 30$, $n\neq 12$, and $n/(\eta \kappa)$ is an odd integer, and we tabulate the triples $(n,\alpha,\beta)$ we discovered, up to equivalence, for $30<n\leq 500$. Finally, we show that the conditions $n>6$, $n\neq 12$, are best possible, subject to the truth of two plausible conjectures.

In 1878, Lucas \cite{lucas} investigated the sequences $(\ell_n)_{n=0}^\infty$ where $$\ell_n=\frac{\alpha^n-\beta^n}{\alpha-\beta},$$ $\alpha \beta$ and $\alpha+\beta$ are coprime integers, and where $\beta/\alpha$ is not a root of unity. Lucas sequences are divisibility sequences; if $m|n$, then $\ell_m|\ell_n$, and more generally, $\gcd(\ell_m,\ell_n)=\ell_{\gcd(m,n)}$ for all positive integers $m$ and $n$. Matijasevic utilised this divisibility property of Lucas sequences in order to resolve Hilbert's 10th problem. \noindent In 1930, Lehmer \cite{lehmer} introduced the sequences $(u_n)_{n=0}^\infty$ where \begin{eqnarray*} u_n& = & \frac{\alpha^{n}-\beta^n}{\alpha^{\epsilon(n)}-\beta^{\epsilon(n)}},\\ \epsilon(n)&=&\left\{\begin{array}{ll} 1, \hspace{.1in}\mbox{if}\hspace{.1in}n\equiv 1 \pmod 2;\\ 2, \hspace{.1in}\mbox{if}\hspace{.1in}n\equiv 0\pmod 2;\end{array}\right. \end{eqnarray*} $\alpha \beta$ and $(\alpha +\beta)^2$ are coprime integers, and where $\beta/\alpha$ is not a root of unity. The sequences $(u_n)_{n=0}^\infty$ are known as Lehmer sequences, and the terms of these sequences are known as Lehmer numbers. Lehmer showed that his sequences had similar divisibility properties to those of Lucas sequences, and he used them to extend the Lucas test for primality. \noindent We define a prime divisor $p$ of $u_n$ to be a primitive divisor of $u_n$ if $p$ does not divide $$(\alpha^2-\beta^2)^2u_3\cdots u_{n-1}.$$ Note that in the list of prime factors of the first $n-1$ terms of the sequence $(u_n)_{n=0}^\infty$, a primitive divisor of $u_n$ is a new prime factor. \noindent We let \begin{eqnarray*} \kappa& = & k(\alpha \beta\max\{(\alpha-\beta)^2,(\alpha+\beta)^2\}),\\ \eta & = & \left\{\begin{array}{ll}1\hspace{.1in}\mbox{if}\hspace{.1in}\kappa\equiv 1\pmod 4,\\ 2\hspace{.1in}\mbox{otherwise},\end{array}\right. \end{eqnarray*} where $k(\alpha \beta \max\{(\alpha-\beta)^2,(\alpha+\beta)^2\})$ is the squarefree kernel of $\alpha \beta \max\{(\alpha-\beta)^2,(\alpha+\beta)^2\}$. On the one hand, building on the work of Schinzel \cite{schinzelI}, we prove that if $n>4$, $n\neq 6$, $n/(\eta \kappa)$ is an odd integer, and the triple $(n,\alpha,\beta)$, in case $(\alpha-\beta)^2>0$, is not equivalent to a triple $(n,\alpha,\beta)$ from an explicit table, then the $n$th Lehmer number $u_n$ has at least two primitive divisors. Moreover, we prove that if $n\geq 1.2\times 10^{10}$, and $n/(\eta \kappa)$ is an odd integer, then the $n$th Lehmer number $u_n$ has at least two primitive divisors. On the other hand, building on the work of Stewart \cite{stewart77}, we prove that there are only finitely many triples $(n,\alpha,\beta)$, where $n>6$, $n\neq 12$, and $n/(\eta \kappa)$ is an odd integer, such that the $n$th Lehmer number $u_n$ has less than two primitive divisors, and that these triples may be explicitly determined. We determine all of these triples $(n,\alpha,\beta)$ up to equivalence explicitly when $6<n\leq 30$, $n\neq 12$, and $n/(\eta \kappa)$ is an odd integer, and we tabulate the triples $(n,\alpha,\beta)$ we discovered, up to equivalence, for $30<n\leq 500$. Finally, we show that the conditions $n>6$, $n\neq 12$, are best possible, subject to the truth of two plausible conjectures.

Der Spigot-Ansatz ist eine elegante Vorgehensweise, spezielle numerische Werte zuverlässig, effizient und mit beliebiger Genauigkeit zu berechnen. Die Stärke des Ansatzes ist seine Effizienz, seine totale Korrektheit und seine mathematisch exakt begründete Sicherstellung einer gewünschten absoluten Genauigkeit. Seine Schwäche ist möglicherweise die eingeschränkte Anwendbarkeit. Es gibt in der Literatur Spigot-Berechnung für e und pi. Wurzelberechnung und Logarithmenberechnung gehören zu den Hauptergebnissen der Dissertation. In Kombination mit den Methoden der Reihentransformation von Zeilberger und Wilf bzw. von Gosper ist der Einsatz zur Berechnung von hypergeometrischen Reihen sehr Erfolg versprechend. Eine interessante offene Frage ist die Berechnung der Feigenbaumkonstanten mit dem Ansatz. 'Spigot' bedeutet 'sukzessive Extraktion von Wertanteilen': die Wertanteile werden extrahiert, als ob sie durch einen Hahn (englisch: spigot) gepumpt werden. Es ist dabei besonders interessant, dass in bestimmten Fällen ein Wert-Anteil mit einer Ziffer der Kodierung des Ergebnisses versehen werden kann. Der Spigot-Ansatz steht damit im Gegensatz zu den konventionellen Iterationsverfahren: in einem Schritt des Spigot-Ansatzes wird jeweils ein Wert-Anteil 'extrahiert' und das gesamte Ergebnis ist die Summe der Wert-Anteile; während ein Schritt in einem Iterationsverfahren die Berechnung eines besseren gesamten Ergebnisses aus dem des vorigen Schritt beinhaltet. Das Grundschema der Berechnung mit dem Spigot-Ansatz sieht folgendermaßen aus: zuerst wird für den zu berechnenden numerischen Wert eine gut konvergierende Reihe mit rationalen Gliedern durch symbolisch-algebraische Methoden hergeleitet; dann wird für eine gewünschte Genauigkeit eine Teilsumme ausgewählt; anschließend werden aus der Teilsumme Wert-Anteile iterativ extrahiert. Die Extraktion von Wert-Anteilen aus der Teilsumme geschieht mit dem Spigot-Algorithmus, der auf Sale zurück geht, nur Integer-Arithmetik benötigt und sich als eine verallgemeinerte Form der Basis-Konvertierung dadurch auffassen lässt, dass die Teilsumme als die Kodierung des Wertes in einer inhomogenen Basis interpretiert wird. Die Spigot-Idee findet auch in der Überführung einer Reihe in eine besser konvergierende Reihe auf der Art und Weise Anwendung, dass Wert-Anteile aus der Reihe extrahiert werden, um die originale Reihe werttreu zur Reihe der Wert-Anteile zu transformieren. Dies geschieht mit den Methoden der Reihentransformation von Gosper bzw. Wilf. Die Dissertation umfasst im Wesentlichen die Formalisierung des Spigot-Algorithmus und der Gosperschen Reihentransformation, eine systematische Darstellung der Ansätze, Methoden und Techniken der Reihenentwiclung und Reihentransformation (die Herleitung von Reihen mit Hilfe charakteristischer Funktionalgleichungen; Methoden der Reihentransformation von Euler, Kummer, Markoff, Gosper, Zeilberger und Wilf) sowie die Methoden zur Berechnung von Wurzeln und Logarithmen mit dem Spigot-Ansatz. Es ist interessant zu sehen, wie sich die Grundideen des Spigot-Algorithmus, der Wurzelberechnung und der Logarithmenberechnung jeweils im Wesentlichen durch eine Gleichung ausdrücken lassen. Es ist auch interessant zu sehen, wie sich verschiedene Methoden der Reihentransformation auf einige einfache Grundideen zurückführen lassen. Beispiele für den Beweis von totalen Korrektheit (bei iterativer Berechnung von Wurzeln) könnte auch von starkem Interesse sein. Um die Zuverlässigkeit anderer Methoden zur Berechnung von natürlichen Logarithmen zu überprüfen, scheint der Vergleich der Ergebnisse mit den des Spigot-Ansatzes die beste Methode zu sein. Anders als bei Wurzelberechnung kann hierbei zur Überprüfung die inverse Berechnung nicht angewandt werden. / spigot, total correctness, acceleration of series, computation of roots, computation of logarithms Reliable numerical computations with spigot approach Spigot approach is an elegant way to compute special numerical values reliably, efficiently and with arbitrary accuracy. The advantage of this way are its efficiency and its total correctness including the bounding of the absolute error. The disadvantage is perhaps its restricted applicableness. There are spigot computation for e an pi. The computation of roots and logarithms belongs to the main results of this thesis. In combination with the methods for acceleration of series by Gosper as well as by Zeilberger and Wilf is the use for numerical summation of hypergeometric series very promising. An interesting open question is the computation of the Feigenbaum constant by this way. ‘Spigot’ means ‘successive extraction of portions of value’: the portions of value are ‘extracted’ as if they were pumped through a spigot. It is very interesting in certain case, where these portions can be interpreted as the digits of the result. With respect to that the spigot approach is the opposition to the iterative approach, where in each step the new better result is computed from the result of the previous step. The schema of spigot approach is characterised as follows: first a series for the value to be computed is derived, then a partial sum of the series is chosen with respect to an desired accuracy, afterwards the portions of value are extracted from the sum. The extraction of potions of value is carried by the spigot algorithm which is due to Sale an requires only integer arithmetic. The spigot algorithm can be understood as a generalisation of radix-conversion if the sum is interpreted as an encoding of the result in a mixed-radix (inhomogeneous) system. The spigot idea is also applied in transferring a series into a better convergent series: portions of value are extracted successively from the original series in order to form the series of extracted potions which should have the same value as the original series. This transfer is carried with the methods for acceleration of series by Gosper and Wilf. The thesis incorporates essentially the formalisation of the spigot algorithm and the method of Gosper for acceleration of series, a systematisation of methods and techniques for derivation and acceleration of series (derivation of series for functions by using their characteristic functional equations; methods for acceleration of series by Euler, Kummer, Markov, Gosper Zeilberger and Wilf) as well as the methods for computation of roots and logarithms by spigot approach. It is interesting to see how to express the basic ideas for spigot algorithm, computation of roots and computation of logarithm respectively in some equations. It is also interesting to see how to build various methods for acceleration of series from some simple basic ideas. Examples for proof of total correctness (for iterative computation of roots) can be of value to read. Comparing with the results produced by spigot approach is possibly the best way for verifying the reliability of other methods for computation of natural logarithms, because (as opposed to root computing) the verification by inverse computation is inapplicable.

Orientador: prof. Dr. Welington Vieira Assunção / Este trabalho ressalta a importancia de um estudo qualitativo de logaritmos, tratando desde as difculdades em ensinar o conteudo ate as limitações dos alunos para compreendê-los. Apos uma breve citação sobre o comportamento dos alunos do Ensino Medio, foi feita uma proposta acerca da preparação das aulas contemplando a historia do assunto abordado para contribuir com esta preparação Para que o aluno possa ter um conhecimento solido sobre os logaritmos, foi pedido uma busca sobre o tema, que continha uma apresentação de um historico sobre o surgimento dos logaritmos e de suas tabelas. Relacionar os logaritmos com equações exponenciais, progressões aritmeticas e geometricas é primordial e este trabalho apresenta o envolvimento de alguns alunos para demonstrar tais relações. Para um grupo de estudos específico, foi iniciado o estudo de cálculo diferencial e integral e feita a apresentação e demonstração dos logaritmos utilizando o conceito de cálculo. Partindo das aplicações dos logaritmos e com base nos exercícios resolvidos e nos questionários respondidos pelos alunos, foi elaborada uma proposta metodologica para minimizar as difculdades de alunos e professores no ensino de logaritmos. Em geral, o aluno do Ensino Medio, alem dos conhecimentos adquiridos ao termino do curso, tem uma nova meta, o vestibular. Entretanto, mesmo que o aluno nao pretenda continuar seus estudos na area de exatas, cabera ao professor conduzir estes conhecimentos novos, não so para o vestibular, mas, sobretudo, para que o mesmo compreenda a essencia do estudo de logaritmos. Para isso, foi imprescindivel relacionar o estudo de logaritmos com demais areas do conhecimento como a Fisica, a Biologia e a Quimica, demonstrando sua aplicabilidade. / Dissertação (mestrado) - Universidade Federal do ABC, Programa de Pós-Graduação em Mestrado Profissional em Matemática em Rede Nacional, 2017. / This research excels the importance of a quality study about logarithms, treating since the teachers teaching diculties to the students comprehension limitations. After a short quotation about the high school students behavoier, the teacher had proposed a concerning through the classes plans and in what the history about the subject in study can contribute with this plannings preparation. So, for a student to have a solid knowledge about logarithm, it had asked a researching about the theme in which have to have an apresentation of the historical logarithm appearance and its index. Its primordial to relate the logarithms with the exponencial equations, the arithmatics an geometrics maths progressions and this research shows some studentsinvolvement to demonstrate these relations. For a group of specific studies, the study of diferential and integral calculus was started, and a presentation and a demonstration of logarithms were made, using the diferential and integral calculus concept. Starting from the aplications of logarithms and with the exercises that have been made and with the answered students questionaires, a methodologic proposal had made to minimize the students and teachers diculties in teaching logarithms. In general, the high school student, beyond the knowledge acquired at the end of the course, he/she has a new goal, the vestibular exam. However, if the student doesnt want to continue his/her study in the area, the teachers duty is to conduct this new knowledges not only for whom will do the vestibular exam, but also, to compreend the essence of the logarithms. For this, it was essential to relate the study of logarithms with the others knowedge area, such as physics, biology and chemistry, showing its applicability.

Submitted by Viviane Lima da Cunha (viviane@biblioteca.ufpb.br) on 2015-11-04T10:36:45Z No. of bitstreams: 2 arquivototal.pdf: 4819014 bytes, checksum: b89987c92ac5294da134e67b82d09cd2 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Approved for entry into archive by Viviane Lima da Cunha (viviane@biblioteca.ufpb.br) on 2015-11-04T11:40:46Z (GMT) No. of bitstreams: 2 arquivototal.pdf: 4819014 bytes, checksum: b89987c92ac5294da134e67b82d09cd2 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Made available in DSpace on 2015-11-04T11:40:46Z (GMT). No. of bitstreams: 2 arquivototal.pdf: 4819014 bytes, checksum: b89987c92ac5294da134e67b82d09cd2 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Previous issue date: 2013-04-15 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this paper we are studying cryptography’s evolution throughout history; analyzing the difference between symmetric and asymmetric cryptographies; enunciating definitions and theorems about binary relations, group theories, primitive roots and discrete logarithms; understanding the procedure of Diffie-Hellman’s key change protocol. In the last part in this work, we are proposing three activities to be applied in classroom. / Neste trabalho, vamos estudar a evolução da criptografia ao longo da história; analisar a diferença entre as criptografias simétricas e assimétricas; enunciar definições e teoremas sobre relações binárias, teoria dos grupos, raízes primitivas e logaritmos discretos; entender o procedimento do protocolo da troca de chaves de Diffie-Hellman; e, na parte final deste trabalho, iremos propor três atividades para serem aplicadas em sala de aula.

Orientador: Prof. Dr. Armando Caputi / Dissertação (mestrado) - Universidade Federal do ABC, Programa de Pós-Graduação em Mestrado Profissional em Matemática em Rede Nacional, 2017. / Neste trabalho,utilizamos a Espiral Logarítmica como motivação para a aprendizagem do logaritmo,um assunto em que os educandos costumam ter muita dificuldade. Apresentamos uma breve história sobre o estudo da Espiral Logarítmica e os Logaritmos, além de algumas manifestações da Espiral Logarítmica na natureza e na arte. No tratamento matemático,conceituamos o logaritmo como a integral da função 1=x - isto é, como áreas ob a hipérbole equilátera-e utilizamos o cálculo diferencial e integral para provar suas principais propriedades. Fazemos um comparativo com a conceituação do logaritmo como inversa da exponencial, que é a abordagem usual no Ensino Médio. Após explorarmos algumas das propriedades mais interessantes da Espiral Logarítmica, concluimos o trabalho propondo uma sequência didática que busca aplicar os conceitos vistos em contextos que façam parte da realidade do aluno,de modo a estimular o interesse pelo conhecimento do assunto tratado. / In this paper we use the Logarithmic Spiral as a motivation to learn the logarithm, an issue that the students always have a lot of difficulties.We present a brief story about the study of Logarithmic Spiral and the Logarithms as well as some Logarithmic Spiral manifestations in the nature and art.In the Math handling we conceptualize the logarithm as the integral of the function 1=x - it means,as the area beneath the equilateral hyperbole and we use the integral and diferential calculus to prove its main effects. We compare the logarithm concept as the reverse of the exponencial, which is the usual approach in High School. After exploring some very interesting effects of the Logarithmic Spiral we accomplish the paper purposing a following teaching that demands to apply the viewed concepts in contexts which make part of the student reality in a way that it can encourage the interest in knowing the given subject.

Dans cette thèse, nous appliquons des outils issus de la théorie d’Arakelov à l’étude de problèmes de géométrie diophantienne. Une notion centrale dans notre étude est la théorie des pentes des fibrés vectoriels hermitiens, introduite par Bost dans les années 90. Nous travaillons plus précisément avec sa généralisation dans le cadre adélique, inspirée par Zhang et développée par Gaudron. Ce mémoire s’articule autour de deux axes principaux. Le premier consiste en l’étude d’un remarquable théorème de géométrie diophantienne dû à Faltings etWüstholz, qui généralise le théorème du sous-espace de Schmidt. Nous commencerons par retranscrire la démonstration de Faltings et Wüstholz dans le langage de la théorie des pentes. Dans un second temps, nous établirons des variantes effectives de ce théorème, que nous appliquerons pour démontrer une généralisation effective du théorème de Liouville valable pour les points fermés d’une variété projective fixée. Ce résultat fournit en particulier une majoration explicite de la hauteur des points satisfaisant une inégalité analogue à celle du théorème de Liouville classique. Dans la seconde partie de cette thèse, nous établirons de nouvelles mesures d’indépendance linéaire de logarithmes dans un groupe algébrique commutatif, dans le cas dit rationnel.Notre approche utilise les arguments de la méthode de Baker revisitée par Philippon et Waldschmidt, combinés avec des outils de la théorie des pentes. Nous y intégrons un nouvel argument, inspiré par des travaux antérieurs de Bertrand et Philippon, nous permettant de contourner les difficultés introduites par le cas périodique. Cette approche évite le recours à une extrapolation sur les dérivations à la manière de Philippon et Waldschmidt. Nous parvenons ainsi à supprimer une hypothèse technique contraignante dans plusieurs théorèmes de Gaudron, tout en précisant les mesures obtenues. / In this thesis, we study diophantine geometry problems on projective varieties and commutative algebraic groups, by means of tools from Arakelov theory. A central notion in this work is the slope theory for hermitian vector bundles, introduced by Bost in the 1990s. More precisely, we work with its generalization in an adelic setting, inspired by Zhang and developed by Gaudron. This dissertation contains two major lines. The first one is devoted to the study of a remarkable theorem due to Faltings and Wüstholz, which generalizes Schmidt’s subspace theorem. We first reformulate the proof of Faltings and Wüstholz using the formalism of adelic vector bundles and the adelic slope method. We then establish some effective variants of the theorem, and we deduce an effective generalization of Liouville’s theorem for closed points on a projective variety defined over a number field. In particular, we give an explicit upper bound for the height of the points satisying a Liouville-type inequality. In the second part, we establish new measures of linear independence of logarithms over a commutative algebraic group. We focus our study on the rational case. Our approach combines Baker’s method (revisited by Philippon and Waldschmidt) with arguments from the slope theory. More importantly, we introduce a new argument to deal with the periodic case, inspired by previous works of Bertrand and Philippon. This method does not require the use of an extrapolation on derivations in the sense of Philippon-Waldschmidt. In this way, we are able to remove an important hypothesis in several theorems of Gaudron establishing lower bounds for linear forms in logarithms.