Submitted by Luciana Ferreira (lucgeral@gmail.com) on 2015-10-08T11:31:38Z No. of bitstreams: 2 Dissertação - Luis Henrique Pereira da Silva - 2015.pdf: 1093576 bytes, checksum: 6d4e251c8d5464c6328fb953341355d9 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2015-10-08T12:56:48Z (GMT) No. of bitstreams: 2 Dissertação - Luis Henrique Pereira da Silva - 2015.pdf: 1093576 bytes, checksum: 6d4e251c8d5464c6328fb953341355d9 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Made available in DSpace on 2015-10-08T12:56:48Z (GMT). No. of bitstreams: 2 Dissertação - Luis Henrique Pereira da Silva - 2015.pdf: 1093576 bytes, checksum: 6d4e251c8d5464c6328fb953341355d9 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Previous issue date: 2015-03-27 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This work aims to demonstrate in a practical way the divisibility criteria 2-97 in sieve Eratostenes with cutting the right and the left, based on the method of multiplication and division Egyptian. The entire process is demonstrated using the divisibility to whole numbers, greatest common divisor, prime numbers, decomposition in prime factors and matching. / Este trabalho tem como objetivo demonstrar de modo prático os critérios de divisibilidade de 2 a 97 no crivo de Eratóstenes com os corte a direita e a esquerda, baseando-se no método de multiplicação e divisão egípcia. Todo processo é demostrado utilizando a divisibilidade para números inteiros, máximo divisor comum, números primos, decomposi ção em fatores primos e congruência.

Submitted by Renata Lopes (renatasil82@gmail.com) on 2016-05-10T13:29:13Z No. of bitstreams: 1 leticiavasconcellosdesouza.pdf: 334599 bytes, checksum: ecaf1358f31b66f2a2e8740f4db33535 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2016-06-15T13:12:10Z (GMT) No. of bitstreams: 1 leticiavasconcellosdesouza.pdf: 334599 bytes, checksum: ecaf1358f31b66f2a2e8740f4db33535 (MD5) / Made available in DSpace on 2016-06-15T13:12:10Z (GMT). No. of bitstreams: 1 leticiavasconcellosdesouza.pdf: 334599 bytes, checksum: ecaf1358f31b66f2a2e8740f4db33535 (MD5) Previous issue date: 2015-08-14 / Este trabalho é voltado para professores que atuam nas séries finais do Ensino Fundamental. Tem como objetivo mostrar que é possível introduzir o estudo de Congruência Modular nesse segmento de ensino, buscando facilitar a resolução de diversas situações-problema. A motivação para escolha desse tema é que há a possibilidade de tornar mais simples a resolução de muitos exercícios trabalhados nessa etapa de ensino e que são inclusive cobrados em provas de admissão à escolas militares e em olimpíadas de Matemática para esse nível de escolaridade. Inicialmente é feita uma breve síntese do conjunto dos Números Inteiros, com suas operações básicas, relembrando também o conceito de números primos, onde é apresentado o crivo de Eratóstenes; o mmc (mínimo múltiplo comum) e o mdc (máximo divisor comum), juntamente com o Algoritmo de Euclides. Apresenta-se alguns exemplos de situações-problema e exercícios resolvidos envolvendo restos deixados por uma divisão para então, em seguida, ser dada a definição de congruência modular. Finalmente, são apresentadas sugestões de exercícios para serem trabalhados em sala de aula, com uma breve resolução. / The aims of this work is teachers working in the final grades of elementary school. It aspires to show that it is possible to introduce the study of Modular congruence this educational segment, seeking to facilitate the resolution of numerous problem situations. The motivation for choosing this theme is that there is the possibility to make it simpler to solve many problems worked at this stage of education and are even requested for admittance exams to military schools and mathematical Olympiads for that level of education. We begin with a brief summary about integer numbers, their basic operations, also recalling the concept of prime numbers, where the sieve of Eratosthenes is presented; the lcm (least common multiple) and the gcd (greatest common divisor), along with the Euclidean algorithm. We present some examples of problem situations and solved exercises involving debris left by a division and then, we give the definition of modular congruence . Finally , we present suggestions for exercises to be worked in the classroom, with a short resolution.

Chip-firing is a deceptively simple game played on the vertices of a graph, which was independently discovered in probability theory, poset theory, graph theory, and statistical physics. In recent years, chip-firing has been employed in the development of a theory of divisors on graphs analogous to the classical theory for Riemann surfaces. In particular, Baker and Norin were able to use this set up to prove a combinatorial Riemann-Roch formula, whose classical counterpart is one of the cornerstones of modern algebraic geometry. It is now understood that the relationship between divisor theory for graphs and algebraic curves goes beyond pure analogy, and the primary operation for making this connection precise is tropicalization, a certain type of degeneration which allows us to treat graphs as “combinatorial shadows” of curves. The development of this tropical relationship between graphs and algebraic curves has allowed for beautiful applications of chip-firing to both algebraic geometry and number theory. In this thesis we continue the combinatorial development of divisor theory for graphs. In Chapter 1 we give an overview of the history of chip-firing and its connections to algebraic geometry. In Chapter 2 we describe a reinterpretation of chip-firing in the language of partial graph orientations and apply this setup to give a new proof of the Riemann-Roch formula. We introduce and investigate transfinite chip-firing, and chip-firing with respect to open covers in Chapters 3 and 4 respectively. Chapter 5 represents joint work with Arash Asadi, where we investigate Riemann-Roch theory for directed graphs and arithmetical graphs, the latter of which are a special class of balanced vertex weighted graphs arising naturally in arithmetic geometry.

In this thesis we study the computation of the greatest common divisor of two polynomials. Firstly, properties of Sylvester matrices are considered as well as their role in computation. We then note, that this approach can be naturally generalized for several polynomials. In the penultimate section, Bézout matrices are studied as an analogy to the Sylvester ones, providing necessary comparison. Extension for more than polynomials is presented here as well. Algorithms corresponding to the individual approaches are presented as well. Finally, the algorithms are implemented in MATLAB and are compared in numerical experiments. Powered by TCPDF (www.tcpdf.org)

CoordenaÃÃo de AperfeÃoamento de Pessoal de NÃvel Superior / O atual trabalho tem como objetivo principal estruturar estudantes, professores e amantes da matemÃtica para a melhor compreensÃo, interpretaÃÃo e resoluÃÃo de problemas que venham a ser solucionados usando-se as EquaÃÃes Diofantinas. Para isso, foram usadas tÃcnicas como o uso de inequaÃÃes e o mÃtodo paramÃtrico que sÃo conteÃdos estudados pelos professores do Ensino Fundamental e MÃdio. TambÃm foi utilizada para isso a apresentaÃÃo de vÃrios exemplos, todos resolvidos, que servirÃo como objeto de estudo para professores, universitÃrios, estudantes escolares e amantes da matemÃtica. No primeiro capÃtulo abordaremos os fatos histÃricos de grandes matemÃticos que contribuÃram com o desenvolvimento das EquaÃÃes Diofantinas. Jà no segundo capÃtulo, vamos conhecer melhor a essÃncia da Teoria Elementar dos NÃmeros, apresentando, demonstrando e exemplificando as ferramentas matemÃticas que serÃo utilizadas na resoluÃÃo das EquaÃÃes Diofantinas. Por fim, no terceiro capÃtulo, introduziremos as EquaÃÃes Diofantinas e os mÃtodos de determinaÃÃo de soluÃÃes das mesmas, aplicando-as em situaÃÃes-problema do cotidiano. A conclusÃo desse trabalho enfatiza a importÃncia da compreensÃo algÃbrica e geomÃtrica das EquaÃÃes Diofantinas, e que o contato com problemas desta Ãrea contribua para que o leitor desenvolva de modo criativo, suas habilidades cognitivas. à importante ressaltar que a introduÃÃo à resoluÃÃo de problemas dessa natureza nÃo necessita de conhecimentos superiores, podendo ser abordado no Ensino Fundamental e MÃdio. / The current work has as objective main to structuralize students, professors and loving of the mathematics for the best understanding, interpretation and resolution of problems that come to be solved using the Diofantinas Equations. For this, they had been used techniques as the use of inequalities and the parametric method that are contents studied for the professors of Basic and Average Education. Also the presentation of some examples, all decided, that they will serve as object of study for professors, collegeâs student was used for this, pertaining to school and loving students of the mathematics. In the first chapter we will approach the facts historical of great mathematicians who had contributed with the development of the Diofantinas Equations. No longer according to chapter, we go to better know the essence of the Elementary Theory of the Numbers, presenting, demonstrating and exemplifying the mathematical tools that will be used in the resolution of the Diofantinas Equations. Finally, in the third chapter, we will introduce the Diofantinas Equations and the methods of determination of solutions of the same one, applying them in situation-problem of the daily one. The conclusion of this work emphasizes the importance of the algebraic and geometric understanding of the Diofantinas Equations, and that the contact with problems of this area contributes so that the reader develops in creative way, its cognitive abilities. It is important to stand out that the introduction to the resolution of problems of this nature does not need superior knowledge, being able to be boarded in Basic and Average education.

This thesis introduces a new mathematical object: collection of arithmetic progressions with elements satisfying the inverse property, \j-th terms of i-th and (i+1)-th progressions are multiplicative inverses of each other modulo (j+1)-th term of i-th progression". Such a collection is uniquely de ned for any pair (a; d) of co-prime integers. The progressions of the collection are ordered. Thus we call it a sequence rather than a collection. The results of the thesis are on the following number theoretic, computational and cryptographic aspects of the defined sequence and its generalizations. The sequence is closely connected to the classical Euclidean algorithm. Precisely, certain consecutive progressions of the sequence form \groupings". The difference between the common differences of any two consecutive progressions of a grouping is same. The number of progressions in a grouping is connected to the quotient sequence of the Euclidean algorithm on co-prime input pairs. The research community has studied extensively the behavior of the Euclidean algorithm. For the rst time in the literature, the connection (proven in the thesis) shows what the quotients of the algorithm signify. Further, the leading terms of progressions within groupings satisfy a mirror image symmetry property, called \symmetricity". The property is subject to the quotient sequence of the Euclidean algorithm and divisors of integers of the form x2 y2 falling in specific intervals. The integers a, d are the primary quantities of the defined sequence in a computational sense. Given the two, leading term and common difference of any progression of the sequence can be computed in time quadratic in the binary length of d. On the other hand, the inverse computational question of finding (a; d), given information on some terms of the sequence, is interesting. This problem turns out to be hard as it requires finding solutions to an nearly-determined system of multivariate polynomial equations. Two sub-problems arising in this context are shown to be equivalent to the problem of factoring integers. The reduction to the factoring problem, in both cases, is probabilistic. Utilizing the computational difficulty of solving the inverse problem, and the sub-problems (mentioned above), we propose a symmetric-key cryptographic scheme (SKCS), and a public key cryptographic scheme (PKCS). The PKCS is also based on the hardness of the problem of finding square-roots modulo composite integers. Our proposal uses the same algorithmic and computational primitives for effecting both the PKCS and SKCS. In addition, we use the notion of the sequence of arithmetic progressions to design an entity authentication scheme. The proof of equivalence between one of the inverse computational problems (mentioned above) and integer factoring led us to formulate and investigate an independent problem concerning the largest divisor of integer N bounded by the square-root of N. We present some algorithmic and combinatorial results. In the course of the above investigations, we are led to certain open questions of number theoretic, combinatorial and algorithmic nature. These pertain to the quotient sequence of the Euclidean algorithm, divisors of integers of the form x2 y2 p in specific intervals, and the largest divisor of integer N bounded by N.

Orientador: Sueli Irene Rodrigues Costa / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-23T07:10:07Z (GMT). No. of bitstreams: 1 Ferreira_Jamil_D.pdf: 1528200 bytes, checksum: a1ca349425c4bcf544a36d17d3157b3c (MD5) Previous issue date: 2013 / Resumo: O número de classes de divisores de grau zero, h, de corpos de funções algébricas elípticos e hiperelípticos desempenha papel importante nos esquemas criptográficos baseados em curvas elípticas e hiperelípticas. Nesse contexto, h é um número grande e é usualmente procurado por meio de algoritmos (baby step - giant step, por exemplo) em um intervalo de números reais obtido após um truncamento no produto infinito de Euler da função zeta do corpo de funções. Tendo a desigualdade de Hasse-Weil como motivação, encontramos identidades finitas para h que são também explícitas no sentido de que seus custos computacionais são diretamente deduzíveis dessas identidades. Como consequência, obtivemos também identidades finitas e explícitas para os coeficientes ai do L-polinômio da função zeta. Ferramentas fundamentais nesta pesquisa foram as L-séries de Artin e outros resultados envolvendo os símbolos polinomiais de Legendre / Abstract: The divisor class number of degree zero, h, of elliptic and hyperelliptic function fields plays an important role in cryptographic schemes based on elliptic and hyperelliptic curves. In this context, h is a large number and it is usually searched by means of algorithms (baby step - giant step, for example) in an interval of real numbers obtained after truncating the infinit Euler product coming from the zeta function of the function field. Taking the Hasse-Weil inequality as motivation, we derived finite identities for h which are also explicit in the sense that their computational costs are straightforwardly derivable from these identities. We also obtained finite and explicit identities for the coefficients ai of the L-polynomialof the zeta function. Fundamental tools for this research were the Artin L-series and other results involving the Legendre polynomial symbols / Doutorado / Matematica / Doutor em Matemática

Motivated by results of linear algebra over fields, rings and tropical semirings, we present a systematic way to understand the behaviour of matrices with entries in an arbitrary semiring. We focus on three closely related problems concerning the row and column spaces of matrices. This allows us to isolate and extract common properties that hold for different reasons over different semirings, yet also lets us identify which features of linear algebra are specific to particular types of semiring. For instance, the row and column spaces of a matrix over a field are isomorphic to each others' duals, as well as to each other, but over a tropical semiring only the first of these properties holds in general (this in itself is a surprising fact). Instead of being isomorphic, the row space and column space of a tropical matrix are anti-isomorphic in a certain order-theoretic and algebraic sense. The first problem is to describe the kernels of the row and column spaces of a given matrix. These equivalence relations generalise the orthogonal complement of a set of vectors, and the nature of their equivalence classes is entirely dependent upon the kind of semiring in question. The second, Hahn-Banach type, problem is to decide which linear functionals on row and column spaces of matrices have a linear extension. If they all do, the underlying semiring is called exact, and in this case the row and column spaces of any matrix are isomorphic to each others' duals. The final problem is to explain the connection between the row space and column space of each matrix. Our notion of a conjugation on a semiring accounts for the different possibilities in a unified manner, as it guarantees the existence of bijections between row and column spaces and lets us focus on the peculiarities of those bijections. Our main original contribution is the systematic approach described above, but along the way we establish several new results about exactness of semirings. We give sufficient conditions for a subsemiring of an exact semiring to inherit exactness, and we apply these conditions to show that exactness transfers to finite group semirings. We also show that every Boolean ring is exact. This result is interesting because it allows us to construct a ring which is exact (also known as FP-injective) but not self-injective. Finally, we consider exactness for residuated lattices, showing that every involutive residuated lattice is exact. We end by showing that the residuated lattice of subsets of a finite monoid is exact if and only if the monoid is a group.

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

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.