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Esta dissertação tem por objetivo apresentar a Criptografia RSA, que é o método de criptografia mais utilizado no mundo atualmente. Iniciamos a dissertação com um breve histórico sobre a criptografia e em seguida introduzimos a teoria matemática empregada no método pertencente a teoria dos números. Finalizamos a dissertação com a descrição de uma aplicação simples do método levado para uma sala de aula do ensino médio. Este texto pretende introduzir o tema de maneira simples e por esta razão, fazemos uso de muitos exemplos. Esperamos ainda que o leitor compreenda o que torna este método eficiente e seguro. / The main goal of this work is to introduce the RSA Criptography that is the most used method in Criptography nowadays. We begin the dissertation with a brief introduction about criptography and then we discuss concepts from number theory used in the method. Finally we present a description of a simple application of Criptography made in a High school classroom. This text intend to introduce the subject in a simple way for this reason we present several examples. We hope that the reader have the comprehension of the methods and of its security.

The FermaT Transformation Engine is an industrial strength toolset for the migration of Assembler and Cobol based legacy systems to C. It uses an intermediate language and several dozen mathematical proven transformations to raise the abstraction level of a source code or to restructure and simplify it as needed. The actual program transformation process with the aid of this toolset is semi-automated which means that a maintainer has not only to apply one transformation after another but also to evaluate the transformation result. This can be a very difficult task especially if the given program is very large and if a lot of transformations have to be applied. Moreover, it cannot be assured that a transformation target will be achieved because it relies on the decisions taken by the respective maintainer which in turn are based on his personal knowledge. Even a small mistake can lead to a failure of the entire program transformation process which usually causes an extensive and time consuming backtrack. Furthermore, it is difficult to compare the results of different transformation sequences applied on the same program. To put it briefly, the manual approach is inflexible and often hard to use especially for maintainers with little knowledge about transformation theory. There already exist different approaches to solve these well known problems and to simplify the accessibility of the FermaT Transformation Engine. One recently presented approach is based on a particular prediction technique whereas another is based on various search tactics. Both intend to automatise the program transformation process. However, the approaches solve some problems but not without introducing others. On the one hand, the prediction based approach is very fast but often not able to provide a transformation sequence which achieves the defined program transformation targets. The results depend a lot on the algorithms which analyse the given program and on the knowledge which is available to make the right decisions during the program transformation process. On the other hand, the search based approach usually finds suitable results in terms of the given target but only in combination with small programs and short transformation sequences. It is simply not possible to perform an extensive search on a large-scale program in reasonable time. To solve the described problems and to extend the operating range of the FermaT Transformation Engine, this thesis proposes a constraint based program transformation system. The approach is semi-automated and provides the possibility to outline an entire program transformation process on the basis of constraints and transformation schemes. In this context, a constraint is a condition which has to be satisfied at some point during the application of a transformation sequence whereas a transformation scheme defines the search space which consists of a set of transformation sequences. After the constraints and the scheme have been defined, the system uses a unique knowledge-based prediction technique followed by a particular search tactic to reduce the number of transformation sequences within the search space and to find a transformation sequence which is applicable and which satisfies the given constraints. Moreover, it is possible to describe those transformation schemes with the aid of a formal language. The presented thesis will provide a definition and a classification of constraints for program transformations. It will discuss capabilities and effects of transformations and their value to define transformation sets. The modelling of program transformation processes with the aid of transformation schemes which in turn are based on finite automata will be presented and the inclusion of constraints into these schemes will be explained. A formal language to describe transformation schemes will be introduced and the automated construction of these schemes from the language will be shown. Furthermore, the thesis will discuss a unique prediction technique which uses the capabilities of transformations, an evaluation of the transformation sequences on the basis of transformation effects and a particular search tactic which is related to linear and tree search tactics. The practical value of the presented approach will be proven with the aid of three medium-scale case studies. The first one will show how to raise the abstraction level whereas the second one will show how to decrease the complexity of a particular program. The third one will show how to increase the execution speed of a selected program. Moreover, the work will be summarised and evaluated on the basis of the research questions. Its limitations will be disclosed and some suggestion for future work will be made.

Submitted by JOSIANE DE CARVALHO REZENDE null (josiane.carvalho.rezende@gmail.com) on 2017-02-20T22:45:59Z No. of bitstreams: 1 COMFICHACATALOGRAFICAdissertacaoJOSIANE2017.pdf: 1131643 bytes, checksum: 9fc3ec5860776cf6c5bd4a8267a39302 (MD5) / Approved for entry into archive by Juliano Benedito Ferreira (julianoferreira@reitoria.unesp.br) on 2017-02-23T17:45:13Z (GMT) No. of bitstreams: 1 rezende_jc_me_rcla.pdf: 1131643 bytes, checksum: 9fc3ec5860776cf6c5bd4a8267a39302 (MD5) / Made available in DSpace on 2017-02-23T17:45:13Z (GMT). No. of bitstreams: 1 rezende_jc_me_rcla.pdf: 1131643 bytes, checksum: 9fc3ec5860776cf6c5bd4a8267a39302 (MD5) Previous issue date: 2017-02-03 / A procura pela solução de alguns problemas relevantes, ou ainda, de equações, têm sido uma fonte de inspiração para ampliar os conjuntos numéricos. Quanto ao conjunto dos números complexos, um importante resultado é que todo polinômio de grau n (maior ou igual a 1) e com coeficientes complexos tem n raízes complexas. De modo geral, o presente trabalho tem o objetivo de contextualizar algumas aplicações das raízes da unidade na matemática. Apresentamos sua aplicação em um caso particular do Teorema de Dirichlet, na construção de reticulados, cuja utilidade está ligada a problemas de transmissão de sinal, e na história da resolução do Último Teorema de Fermat. / The search for the solution of some relevant problems, or even of equations, has been a source of inspiration to extend the numerical sets. As for the set of complex numbers, an important result is that every polynomial of degree n (bigger or equal 1) and with complex coefficients has n complex roots. In general, the present work aims to contextualize some applications of the roots of unit in mathematics. We present its application in a particular case of the Dirichlet Theorem, in the construction of lattices, whose utility is linked to signal transmission problems, and in the history of the resolution of the Fermat's Last Theorem.

The technique of mobile location has become a popular research topic since the number of related applications for the location information is growing rapidly. The decision to make the location of mobile phones under the U.S. Federal Communications Commission (FCC) in 1996 is one of the driving forces to research and provide solutions to it. But, in wireless communication systems, non line of sight (NLOS) propagation is a key and difficult issue to improve mobile location estimation. We propose an efficient location algorithm which can mitigate the influence of NLOS error. First, based on the geometric relationship between known positions of the base stations, the theorem of ¡§Fermat Point¡¨ is utilized to collect the candidate positions (CPs) of the mobile station. Then, a set of weighting parameters are computed using a density-based clustering method. Finally, the location of mobile station is estimated by solving the optimal solution of the weighted objective function. Different distributions of NLOS error models are used to evaluate the performance of this method. Simulation results show that the performance of the least range measure (LRM) algorithm is slightly better than density-based clustering algorithm (DCA), and superior to the range based linear lines of position algorithm (LLOP) and range scaling algorithm (RSA) on location accuracy under different NLOS environments. The simulation results also satisfy the location accuracy demand of Enhanced 911 (E-911).

Orientador: Carina Alves / Banca: Marta Cilene Gadotti / Banca: Cristiano Torezzan / Resumo: A procura pela solução de alguns problemas relevantes, ou ainda, de equações, têm sido uma fonte de inspiração para ampliar os conjuntos numéricos. Quanto ao conjunto dos números complexos, um importante resultado é que todo polinômio de grau n (maior ou igual a 1) e com coeficientes complexos tem n raízes complexas. De modo geral, o presente trabalho tem o objetivo de contextualizar algumas aplicações das raízes da unidade na matemática. Apresentamos sua aplicação em um caso particular do Teorema de Dirichlet, na construção de reticulados, cuja utilidade está ligada a problemas de transmissão de sinal, e na história da resolução do Último Teorema de Fermat / Abstract: The search for the solution of some relevant problems, or even of equations, has been a source of inspiration to extend the numerical sets. As for the set of complex numbers, an important result is that every polynomial of degree n (bigger or equal 1) and with complex coefficients has n complex roots. In general, the present work aims to contextualize some applications of the roots of unit in mathematics. We present its application in a particular case of the Dirichlet Theorem, in the construction of lattices, whose utility is linked to signal transmission problems, and in the history of the resolution of the Fermat's Last Theorem / Mestre

CoordenaÃÃo de AperfeÃoamento de Pessoal de NÃvel Superior / O presente trabalho tem por objetivo motivar os estudantes a se interessarem por problemas de mÃnimos geomÃtricos, em especial relacionados à simetrias, atravÃs do uso do software educacional Geogebra. Fornecendo um guia de atividades que podem ser aplicadas em sala de aula. Trata do Problema de Heron e do Problema de Fermat, apresentando suas respectivas resoluÃÃes e algumas variaÃÃes interessantes. / His paper has the objective of motivate the students to interest for problems of geometric minimuns, specially those related to symmetry, using the educational software Geogebra. Providing a guide of activities that can be used in classes. Treats about Heronâs Problem and Fermatâs Problem, presenting the resolutions and same interesting variations.

Submitted by Leonardo Cavalcante (leo.ocavalcante@gmail.com) on 2018-05-02T18:31:29Z No. of bitstreams: 1 Arquivototal.pdf: 790723 bytes, checksum: 7e439258b05733c014dcd3e7de230c1f (MD5) / Made available in DSpace on 2018-05-02T18:31:29Z (GMT). No. of bitstreams: 1 Arquivototal.pdf: 790723 bytes, checksum: 7e439258b05733c014dcd3e7de230c1f (MD5) Previous issue date: 2017-08-31 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this present work, principles and theorems associated to integers are returned, as well as eigenvalues and eigenvectors problems, highlighting a Hermitian matrix. Then it is emphasized to the Circulating Matrices, through which it is found the association to two well-defined polynomials: the representative and the characteristic. Later a brief account about the history of polynomial equations is made, drafting the Cardano-Tartaglia Formulas associated to them. Afterwards a unification is made in the resolution process of the polynomial equations of smaller degrees than the equal to 4, by means of the circulating matrices. The work is completed by proving a Fermat theorem for n = 3, using the Cardano-Tartaglia Formulas. / No presente trabalho, princípios e teoremas associados aos números inteiros são retomados, bem como problemas de autovalores e autovetores, sendo ressaltada a matriz Hermitiana. Em seguida é dado ênfase às Matrizes Circulantes, através das quais verifica-se a associação a dois polinômios bem definidos: o representante e o característico. Posteriormente realiza-se um breve relato acerca da história das equações polinomiais, destacandose as Fórmulas de Cardano-Tartaglia associadas às mesmas. Logo após é feita uma unificação no processo de resolução das equações polinomiais de graus menores do que o igual a 4, por meio das matrizes circulantes. O trabalho é finalizado, sendo provado o Teorema de Fermat para n = 3, recorrendo-se às Fórmulas de Cardano-Tartaglia.

Submitted by ANA KARLA PEREIRA RODRIGUES (anakarla_@hotmail.com) on 2017-08-29T15:44:42Z No. of bitstreams: 1 arquivototal.pdf: 691607 bytes, checksum: 68ddd45857d5c0c6e60229a957089adf (MD5) / Approved for entry into archive by Fernando Souza (fernandoafsou@gmail.com) on 2017-08-29T15:47:36Z (GMT) No. of bitstreams: 1 arquivototal.pdf: 691607 bytes, checksum: 68ddd45857d5c0c6e60229a957089adf (MD5) / Made available in DSpace on 2017-08-29T15:47:36Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 691607 bytes, checksum: 68ddd45857d5c0c6e60229a957089adf (MD5) Previous issue date: 2014-08-01 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This work presents a study of prime numbers, how they are distributed, how many prime numbers are there between 1 and a real number x, formulas that generate primes, and a generalization to Bertrand's Postulate. Six proofs that there are in nitely many primes using reductio ad absurdum, Fermat numbers, Mersenne numbers, Elementary Calculus and Topology are discussed. / Este trabalho apresenta um estudo sobre os números primos, como estão distribu ídos, quantos números primos existem entre 1 e um número real x qualquer, fórmulas que geram primos, além de uma generalização para o Postulado de Bertrand. São abordadas seis demonstrações que mostram que existem in nitos números primos usando redução ao absurdo, Números de Fermat, Números de Mersenne, Cálculo Elementar e Topologia.

CoordenaÃÃo de AperfeÃoamento de Pessoal de NÃvel Superior / Ao longo das Ãltimas dÃcadas, a transmissÃo do conhecimento matemÃtico na EducaÃÃo BÃsica sofreu diversas mudanÃas. âO Ensino Tradicionalâ da matemÃtica era baseado na memorizaÃÃo de fÃrmulas, havendo assim uma mecanizaÃÃo no processo de resoluÃÃo de problemas, onde o discente era visto como um ser passivo. A nova visÃo de ensino, que busca significar o que conteÃdo exposto em sala, motivou a escolha desse tema, visto que situaÃÃes problemas envolvendo equaÃÃes diofantinas podem ser facilmente percebidas em nosso cotidiano. O objetivo deste trabalho à oportunizar a realizaÃÃo de uma leitura consultiva para o professor do Ensino BÃsico, e asseverar que essas equaÃÃes podem ser aplicadas na EducaÃÃo BÃsica como uma ferramenta que instiga o pensamento lÃgico, o raciocÃnio, a compreensÃo e a interpretaÃÃo matemÃtica. A formulaÃÃo desse material que està dividido em cinco capÃtulos se deu atravÃs de levantamento bibliogrÃfico por meio de pesquisas descritivas. A introduÃÃo compÃe o primeiro capÃtulo. O segundo capÃtulo versa sobre o Legado de Diofanto: vida e obras, ressaltando sua obra titulada âArithmeticaâ que contribuiu significativamente para o desenvolvimento da teoria dos nÃmeros. O terceiro capÃtulo trata das equaÃÃes diofantinas lineares de n variÃveis. O quarto capÃtulo aborda as ternas itagÃricas, o MÃtodo das Secantes e Tangentes de Fermat na busca de soluÃÃes racionais para quaÃÃes, com coeficientes racionais, da forma ax2+by2 = c, e um caso particular do Ãltimo Teorema de Fermat. O quinto capÃtulo à composto de problemas sobre equaÃÃes diofantinas lineares. / Over the past decades, the transmission of mathematical knowledge in basic education has undergone several changes. The âTeaching Traditionalâ math was based on memorizing formulas, so there mechanization in problem solving where the student was seen as a liability to be process. The new vision of education that seeks to signify exposed to room content, motivated the choice of this theme, as diophantine equations involving situations problems can be easily noticed in our daily lives. The objective of this work is an opportunity for a realization of an advisory reading for the teacher of basic education, and assert that these equations can be applied in basic education as a tool that encourages the logical thinking, reasoning, understanding and mathematical interpretation. The formulation of this material which is divided into five chapters was through literature review through descriptive research. The introduction comprises the first chapter. The second chapter deals with the Legacy of Diophantus: life and works, emphasizing his work entitled âArithmeticaâ which contributed significantly to the development of number theory. The third chapter deals with linear Diophantine equations in n variables. The fourth chapter discusses the Pythagorean tender, Fermatâs of secants and Tangents method, in finding rational solutions to equations with rational coefficients, of the form ax2 + by2 = c and a particular case Fermatâs Last Theorem. The fifth chapter is composed of problems on linear diophantine equations.