Esta dissertação apresenta, aos alunos e professores do ensino Médio, uma noção elementar da criptografia, através de alguns tipos de cifras, a trinca americana e do método de criptografia RSA. Para que isso fosse possível houve a introdução de conceitos básicos entre eles, conjuntos, funções, divisibilidade, números primos, congruência, teorema de Fermat e teorema de Euler, que garantem o funcionamento de algumas dessas cifras, da trinca americana e do sistema RSA. Com relação à trinca americana, que é um sistema que permite comunicar uma troca de chave, iremos propor uma composição de cifras, para que haja uma troca de mensagens e seja um exemplo motivador que introduza o sistema de RSA. Além disso, esses conceitos básicos podem ser úteis ao serem levados à sala de aula como motivação para o aprendizado dos alunos, seja para calcular com mais agilidade e simplicidade determinados exercícios, seja para resolver uma situação problema ou mesmo para descobrir uma nova maneira de visualizar conteúdos já vistos em sala de aula. / This dissertation presentes, to students and high school teachers, an elementary notion of cryptography through some types of cyphers, the asymmetric key algorithm and the RSA encryption method. To make this possible, we introduce basic concepts among them, set theory, functions, divisibility, primes, congruence, Fermat\'s theorem and Euler\'s theorem, which guarantee the functioning of some of these encryptions. Relating to the asymmetric key algorithm, which is a system that allows you to communicate a key exchange, we will propose a set of cyphers, so that it is possible a secure message exchange, which is also a motivating example to introduce the RSA system. In addition, these basic concepts can be useful when being taken to the classroom as the motivation for the learning of students, whether to calculate with more agility and simplicity certain exercises, whether to resolve a situation-problem or even to discover a new way to discuss subjects usually seen in the classroom.

Representation theory is a field of study within abstract algebra that originated around the turn of the 19th century in the work of Frobenius on representations of finite groups. More recently, Hopf algebras -- a class of algebras that includes group algebras, enveloping algebras of Lie algebras, and many other interesting algebras that are often referred to under the collective name of ``quantum groups'' -- have come to the fore. This dissertation will discuss generalizations of certain results from group representation theory to the setting of Hopf algebras. Specifically, our focus is on the following two areas: Frobenius divisibility and Kaplansky's sixth conjecture, and the adjoint representation and the Chevalley property. / Mathematics

This diploma thesis deals with the topic of divisibility of natural numbers in mathematics in the 6th year of primary schools. The first, theoretical part of the diploma thesis introduces the concept of didactic situations from the Theory of Didactic Situations in Mathematics by G. Brousseau, the theory of multiple intelligences according to H. Gardner and the theory of cognitive development according to J. Piaget. All chapters are projected into the experimental part. The diploma thesis is focused on the topic of divisibility, therefore a chapter of the same title is included in this part. From the history point of view, the sieve of Eratosthenes and Mersenne prime numbers are described here in two chapters. The practical part of the thesis includes a chapter containing an analysis of various didactic methods used in six sets of mathematics textbooks. Attention is paid to similarities and differences in the sets of tasks. In each textbook the topic of divisibility is compared in regard to the introduction of the divisibility rules. The following two chapters are devoted to an experiment designed to help pupils understand the divisibility rules for numbers 2, 5, 10, 8 and 4. The experiment took place over several days in two parallel classes, each of which was pedagogically led for a long time by a...

This thesis is an educational text for high school students. It aims to teach them how to solve typical problems concerning divisibility found in mathematical correspondence seminars and mathematical olympiad. Basic notions from the theory of divisibility are recalled (e.g. prime numbers, divisors, multiples). Criteria of divisibility by 2 to 20 are introduced, as well as diophantine equations and practical applications of prime numbers in real life. One whole chapter is dedicated to problems and exercises. Powered by TCPDF (www.tcpdf.org)

Made available in DSpace on 2016-04-27T16:57:25Z (GMT). No. of bitstreams: 1 Fernanda Fuentes Azambuja.pdf: 2505770 bytes, checksum: 3bbdb5dea1fcb8bf7c8f1f8044d16c66 (MD5) Previous issue date: 2013-05-22 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / The student s difficulty in High School with the polynomial is famous, specially, about divisibility. Such fact instigated determining, with the purpose of research hereby presented: investigating the effect of the retaking of the divisibility of the natural numbers with a student from High School and its comprehension about polynomial divisibility. An empirical research was held, of qualitative focus, more specifically a study of ethnographical case, according to André (2005), in which the calculator was used not only as a motivating tool for the retaking, but also as a tool for data survey. The analyses were based, mainly, in basic elements of the theory APOS (Dubinsky and MCDonald, 2001). It was concluded that, although subjects have done the correlation between the algorithms of divisibility of natural numbers and of polynomials, and the fake conception of polynomial constructed by them that identified a polynomial as a number that undermined the possibility of the intended analogy. As for the use of the calculator, it was concluded that it was the tool that helped the subjects not to deviate of the focus of the proposed activities, helping them to recover the conceptions of divisibility of the natural numbers / A dificuldade do estudante do Ensino Médio com o conteúdo de Polinômios é notória, em especial, sobre a divisibilidade. Tal fato instigou determinar, como objetivo da pesquisa aqui apresentada: investigar o efeito da retomada da divisibilidade dos números naturais com estudante do Ensino Médio em sua compreensão sobre a divisibilidade de polinômios. Realizou-se uma pesquisa empírica, de cunho qualitativo, mais especificamente um estudo de caso etnográfico, conforme André (2005), na qual a calculadora foi utilizada não só como um instrumento motivador para a retomada, como também como instrumento de coleta de dados. As análises embasaram-se, sobretudo, em elementos básicos da teoria APOS (Dubinsky e MCDonald, 2001). Concluiu-se que, embora os sujeitos tenham feito a correlação entre os algoritmos da divisibilidade dos números naturais e dos polinômios, a falsa concepção de polinômio construída por eles que identificavam um polinômio como um número solapou a possibilidade da analogia pretendida. Quanto ao uso da calculadora, concluiu-se que ela foi um instrumento que auxiliou os sujeitos a não se desviarem do foco das atividades propostas, auxiliando-os a resgatar as concepções de divisibilidade dos números naturais

Submitted by Luciana Ferreira (lucgeral@gmail.com) on 2016-03-17T14:31:15Z No. of bitstreams: 2 Dissertação - Paulo Júnio de Oliveira - 2015.pdf: 1566271 bytes, checksum: a8c1caa1354936dd420024e5bc704cb9 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2016-03-17T14:33:55Z (GMT) No. of bitstreams: 2 Dissertação - Paulo Júnio de Oliveira - 2015.pdf: 1566271 bytes, checksum: a8c1caa1354936dd420024e5bc704cb9 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Made available in DSpace on 2016-03-17T14:33:55Z (GMT). No. of bitstreams: 2 Dissertação - Paulo Júnio de Oliveira - 2015.pdf: 1566271 bytes, checksum: a8c1caa1354936dd420024e5bc704cb9 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Previous issue date: 2015-11-10 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / The aim of this dissertation is to discuss the problem of infinite divisibility of bodies, a problem which was already discussed in the classic period by Aristotle and his analysis of Zeno’s paradoxes. Our working hypothesis is that in the Tractatus-Logico-Philosophicus Wittgenstein has offered a reformulation of this very problem when he discusses the process of analysis of propositions. One of the central thesis in the Tractatus is that all ordinary proposition can be completely analyzed and that this process of analysis has to be finite. Based on that, we argue that it necessarily follows that the elements present in the state of affairs described by the proposition cannot be further divided since the analysis of the proposition which describes such a state is necessarily finite. / O objetivo deste trabalho é discutir o problema da divisibilidade infinita de “corpos”, um problema que era discutido já no período clássico por Aristóteles e sua análise dos paradoxos de Zenão. Nossa hipótese de trabalho é a de que no Tractatus-Logico-Philosophicus Wittgenstein teria apresentado uma possível reformulação desse problema ao tratar da análise de proposições. Uma das teses centrais no Tractatus é a de que toda a proposição tem uma análise lógica completa e esse processo de análise tem de ter um fim. Baseado nisso, nós argumentamos que segue-se necessariamente que os elementos presentes no estado de coisas descritos pela proposição não podem prosseguir sendo subdivididos, uma vez que o processo de análise da proposição que descreve tal estado de coisas é necessariamente finito.

Made available in DSpace on 2016-04-27T16:59:01Z (GMT). No. of bitstreams: 1 Juliana de Lima Gregorutti.pdf: 3173463 bytes, checksum: e4d3df5954602eb3e569881010726ade (MD5) Previous issue date: 2009-12-17 / Secretaria da Educação do Estado de São Paulo / An analysis was carried out for this research to get to know how 5th graders mobilize their knowledge about the subject: divisibility and natural numbers, in which the aim is to build a new concept, the Divisibility Criteria for the numbers two, three and five. It is expected that this knowledge serves them as a way to comprehend the division bearing in mind that the researches from Gregolin (2002), Castela (2005) and Fonseca (2005) show division as the mathematical operation that students present greater difficult. Thus, a Research Tool made of four activities was preferred, being two formal questions, as the questions commonly proposed in text books, and the two others were more recreational, proposed by means of games. The research was built, according to the methodological referential of the Didactic Engineering, proposed by Artigue (1996). Previously to our Research Tool, the experiencing of our Pilot was made, developed in three sessions, with 25 students graduating from 4th grade. The results collected in the Pilot Helped outline the Research Tool that was built based on the Theory of Register of Semiotic Representation, proposed by Duval (2003). By means of the analysis of the data collected, it was observed that while establishing the Divisibility Criteria by 2, 3 and 5, the students quickly observed the numeric pattern for the Divisibility Criteria by 2 and 5, concluding that, it was only possible to divide even numbers by 2 and by only possible to divide by 5 the numbers ending in zero or five. However, they couldn t, in an autonomous way, observe when the number can be divided by three. Besides the expectations that were predicted, with this experience there were some very interesting positive surprises, as the importance of the students interaction among themselves and with the teacher for the building of new knowledge, and also the students satisfaction while becoming leaders of their knowledge / Nesta pesquisa, foi feita uma análise para conhecer como seis alunos da 5° série do Ensino Fundamental mobilizam seus conhecimentos a respeito do assunto: divisibilidade de números naturais, em que se visa a construção de um novo conceito, os Critérios de Divisibilidade para os números dois, três e cinco. Espera-se que este conhecimento sirva-lhes como um caminho para a compreensão da divisão, tendo em vista que as pesquisas de Gregolin (2002), Castela (2005) e Fonseca (2005) apontam a divisão como a operação que os alunos apresentam maior dificuldade. Desta maneira, privilegiou-se um Instrumento de Pesquisa composto de quatro atividades, sendo duas questões formais, como as comumente propostas nos livros didáticos, e as outras duas mais lúdicas, propostas por meio de jogos. A pesquisa foi moldada, seguindo o referencial metodológico da Engenharia Didática, proposto por Artigue (1996). Previamente a nosso Instrumento de Pesquisa, foi feita a experimentação de nosso Piloto, desenvolvido em três sessões, com 25 alunos concluintes da 4º série do Ensino Fundamental. Os resultados coletados no Piloto auxiliaram a delinear o Instrumento de Pesquisa que foi construído embasado na Teoria dos Registros de Representações Semióticas, proposta por Duval (2003). Por meio da análise de dados coletados, notou-se que ao estabelecer os Critérios de Divisibilidade por 2, 3 e 5, os alunos rapidamente notaram o padrão numérico para os Critérios de Divisibilidade por 2 e 5, concluindo que, por 2 só era possível dividi-los por números pares e por 5 pelos números que terminam em zero ou cinco. No entanto, não conseguiram, de maneira autônoma, notar quando o número pode ser dividido por três. Além das expectativas que havíamos previsto, com esta experimentação tivemos algumas surpresas positivas muito interessantes, como a importância da interação dos alunos entre si e com o professor para a construção de novos conhecimentos, e também a satisfação dos alunos ao tornarem-se protagonistas de seus conhecimentos

Using Fermat’s Little Theorem, it can be shown that Σmi=1 i m−1 ≡ −1 (mod m) if m is prime. It has been conjectured that the converse is true as well. Namely, that Σmi=1 i m−1 ≡ −1 (mod m) only if m is prime. We shall present some necessary and sufficient conditions for the conjecture to hold, and we will demonstrate that no counterexample exists for m ≤ 1012 .

In this work we are concerned with the characterization of certain classes of stochastic processes via duality formulae. First, we introduce a new formulation of a characterization of processes with independent increments, which is based on an integration by parts formula satisfied by infinitely divisible random vectors. Then we focus on the study of the reciprocal classes of Markov processes. These classes contain all stochastic processes having the same bridges, and thus similar dynamics, as a reference Markov process. We start with a resume of some existing results concerning the reciprocal classes of Brownian diffusions as solutions of duality formulae. As a new contribution, we show that the duality formula satisfied by elements of the reciprocal class of a Brownian diffusion has a physical interpretation as a stochastic Newton equation of motion. In the context of pure jump processes we derive the following new results. We will analyze the reciprocal classes of Markov counting processes and characterize them as a group of stochastic processes satisfying a duality formula. This result is applied to time-reversal of counting processes. We are able to extend some of these results to pure jump processes with different jump-sizes, in particular we are able to compare the reciprocal classes of Markov pure jump processes through a functional equation between the jump-intensities.

This dissertation investigates the theory of quantum stochastic processes and its applications in quantum many-body physics. The main goal is to analyse complexity-theoretic aspects of both static and dynamic properties of physical systems modelled by quantum stochastic processes. The thesis consists of two parts: the first one addresses the computational complexity of certain quantum and classical divisibility questions, whereas the second one addresses the topic of Hamiltonian complexity theory. In the divisibility part, we discuss the question whether one can efficiently sub-divide a map describing the evolution of a system in a noisy environment, i.e. a CPTP- or stochastic map for quantum and classical processes, respectively, and we prove that taking the nth root of a CPTP or stochastic map is an NP-complete problem. Furthermore, we show that answering the question whether one can divide up a random variable $X$ into a sum of $n$ iid random variables $Y_i$, i.e. $X=\sum_{i=1}^n Y_i$, is poly-time computable; relaxing the iid condition renders the problem NP-hard. In the local Hamiltonian part, we study computation embedded into the ground state of a many-body quantum system, going beyond "history state" constructions with a linear clock. We first develop a series of mathematical techniques which allow us to study the energy spectrum of the resulting Hamiltonian, and extend classical string rewriting to the quantum setting. This allows us to construct the most physically-realistic QMAEXP-complete instances for the LOCAL HAMILTONIAN problem (i.e. the question of estimating the ground state energy of a quantum many-body system) known to date, both in one- and three dimensions. Furthermore, we study weighted versions of linear history state constructions, allowing us to obtain tight lower and upper bounds on the promise gap of the LOCAL HAMILTONIAN problem in various cases. We finally study a classical embedding of a Busy Beaver Turing Machine into a low-dimensional lattice spin model, which allows us to dictate a transition from a purely classical phase to a Toric Code phase at arbitrarily large and potentially even uncomputable system sizes.