Συνδέουμε τα Συνεχή Κλάσματα με τα Ορθογώνια Πολυώνυμα. Ξεκινώντας από τον Stieltjes και το ομώνυμο "Πρόβλημα Ροπών", φτάνουμε μέχρι τις μέρες μας μελετώντας αυτή τη σχέση με μεθόδους Συναρτησιακής Ανάλυσης. / We study the connection between Continued Fractions and Orthogonal Polynomials. We start from Stieltjes and his "Moment Problem". Then we present Chain sequences, methods of Functional Analysis and the Birth-Death processes.

Submitted by ANA KARLA PEREIRA RODRIGUES (anakarla_@hotmail.com) on 2017-08-30T13:15:08Z No. of bitstreams: 1 arquivototal.pdf: 799210 bytes, checksum: 8de2ace5434a5d92b8604de7573abfc4 (MD5) / Approved for entry into archive by ANA KARLA PEREIRA RODRIGUES (anakarla_@hotmail.com) on 2017-08-30T13:17:30Z (GMT) No. of bitstreams: 1 arquivototal.pdf: 799210 bytes, checksum: 8de2ace5434a5d92b8604de7573abfc4 (MD5) / Made available in DSpace on 2017-08-30T13:17:30Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 799210 bytes, checksum: 8de2ace5434a5d92b8604de7573abfc4 (MD5) Previous issue date: 2016-02-26 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / The study of ontinued fra tions will start with some histori al fa ts, aiming at a better understanding of the subje t. We will bring the de nition of ontinued fra tions for a number α real, with the de nition for α rational and α irrational. The dis ussion will fo us on meaning results for the al ulation of redu ed and good approximations of irrational numbers, also aimed at determining the error between the redu ed and the irrational number. We will bring a study of the periodi ontinued fra tions, with emphasis on Lagrange theorem, whi h relates a periodi ontinued fra tion and a quadrati equation. Finishing with a fo us on problem solving, as the al ulation of ontinued fra tions of irrational numbers of the form √a2 + b, as well as proof of the irrationality of e by al ulating its ontinued. / O estudo das frações ontínuas terá ini io om alguns fatos históri os, visando uma melhor ompreensão do tema. Traremos a de nição de frações ontínuas para um erto número α real, apresentando a de nição para α ra ional e para α irra ional. A dis ussão será entrada em resultados importantes para o ál ulo de reduzidas e boas aproximações de números irra ionais, visando também a determinação do erro entre a reduzida e o número irra ional. Traremos um estudo sobre as frações ontínuas periódi as, om enfase ao teorema de Langrange, que rela iona uma fração ontínua periódi a e uma equação do segundo grau. Finalizando om enfoque na resolução de problemas, omo o ál ulo de frações ontínuas de números irra ionais da forma √a2 + b, assim omo a prova da irra ionalidade de e através do ál ulo de sua fração ontínua.

Orientador: Eliana Xavier Linhares de Andrade / Banca: Vanessa Avansini Botta Pirani / Banca: Cleonice Fátima Bracciali / Resumo: O principal objetivo deste trabalho é estudar métodos para construir os numeradores e denominadores parciais da fração contínua que corresponde a duas expansões em série de potências de uma função analítica f(z); em z =0 e em z = 00. / Abstract: The main purpose of this work is to two series expansions of an analytic function f(z); in z =0 and z =00 simultaneously. Furthermore we considered the case when there are zero coefficients in the series and also whwn there is symmetry in the coefficients of the two series. Some examples are given. / Mestre

CoordenaÃÃo de AperfeÃoamento de Pessoal de NÃvel Superior / Um resgate as fraÃÃes contÃnuas tem seu inÃcio com uma abordagem histÃrica, mostrando aquilo que se sabe hoje sobre esse assunto à fruto de estudos de vÃrios matemÃticos pelo mundo. AlÃm da histÃria, o texto à dividido em mais cinco capÃtulos e um apÃndice, que mostram atravÃs de teoremas e exemplos a vantagem, indiscutÃvel, da aproximaÃÃo de nÃmeros reais atravÃs de nÃmeros racionais, usando o dispositivo das fraÃÃes contÃnuas. / The rescue A continuous fractions got their start with a historical approach, showing what is known today about this issue is the result of studies by various mathematical world. Besides the story, the text is divided into five chapters and an appendix, showing through theorems and examples advantage, indisputable, the approximation of real numbers by rational numbers, using the device of continued fractions.

Submitted by Cássia Santos (cassia.bcufg@gmail.com) on 2015-10-27T14:48:51Z No. of bitstreams: 2 Dissertação - Ronaldo Pires da Silva - 2015.pdf: 1653286 bytes, checksum: 63a72d8fbcc7390f80fb41dbadaaa9fe (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2015-10-27T14:53:07Z (GMT) No. of bitstreams: 2 Dissertação - Ronaldo Pires da Silva - 2015.pdf: 1653286 bytes, checksum: 63a72d8fbcc7390f80fb41dbadaaa9fe (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Made available in DSpace on 2015-10-27T14:53:07Z (GMT). No. of bitstreams: 2 Dissertação - Ronaldo Pires da Silva - 2015.pdf: 1653286 bytes, checksum: 63a72d8fbcc7390f80fb41dbadaaa9fe (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Previous issue date: 2015-07-06 / Our work had as main objective to study the intersection of integer sequences, denominated polygonal numbers, through Pell's equation. In this context, the solution of two equations will be treated: x2 􀀀 Dy2 = 1 and x2 􀀀 Dy2 = N, jNj > 1. For the rst one we have used results from the theory of continued fractions. For the last one, we have used the method of solution delineated in literature. Besides, propositions referring to the intersection of polygonal numbers for some particular cases are presented and demonstrated. Also, the proposition of the general case is presented and demonstrated. Finally, we have performed the solution of some of Pell's equations in order to determine the intersection of some polygonal numbers. / Nosso trabalho teve como objetivo central estudar a interseção de sequências de inteiros, denominadas números geométricos, através da equação de Pell. Neste contexto, a resolução de duas equações serão tratadas: x2 􀀀 Dy2 = 1 e x2 􀀀 Dy2 = N com jNj > 1. Para a primeira utilizamos importantes resultados presentes na teoria das frações contínuas. Para última, utilizamos o método de resolução delineado na literatura. Além disso, proposições referentes a interseção de números geométricos para alguns casos particulares são apresentadas e demonstradas. Também a proposição do caso geral é apresentada e demonstrada. Por m, realizamos a resolução de algumas equações de Pell para determinarmos a interseção de alguns números geométricos.

Nós estudaremos o comportamento de difeomorfismos do círculo, tanto do ponto de vista combinatório quanto do ponto de vista topológico e da teoria da medida, seguindo os trabalhos de Michael Herman. A cada homeomorfismo do círculo podemos associar um número real positivo, denominado número de rotação. Mostraremos que existe um conjunto de números irracionais de medida de Lebesgue total na reta tal que, se f é um difeomorfismo do círculo de classe \'C POT. r \' que preserva a orientação, com r maior ou igual a 3 e com número de rotação nesse conjunto, então f é pelo menos \'C POT. r - 2\' -conjugada a uma translação irracional. Além disso, mostraremos que dado um caminho \'f IND. t\' de classe \'C POT. 1\' definido em um intervalo [a;b] no conjunto dos difeomorfismos do círculo de classe \'C POT. r\' que preservam a orientação, com r maior ou igual a 3, o conjunto dos parâmetros em que \'f IND. t\' é \'C POT. r - 2\' -conjugada a uma translação irracional tem medida de Lebesgue positiva, desde que os números de rotação em \'f IND. a\' e \'f IND. b\' sejam distintos / We will study the generic behavior of circle diffeomorphisms, in the combinatorial, topological and measure-theoretical sense, following the work of Michael Herman. To each order preserving homeomorphism of the circle we can associate a positive real number, called rotation number, which is invariant under conjugacy. We will show that there is a set of irrational numbers with full Lebesgue measure on R such that, if f is a circle diffeomorphism of class \'C POT. r\', with r greater or equal 3 and with rotation number in that set, then f is at least \'C POT. r - 2\' -conjugated to an irrational translation. Moreover, we will show that if ft is a \'C POT. 1\' -path defined on a interval [a;b] over the set of the circle diffeomorphisms orientation preserving, with r \'> or =\' 3, then the set of parameters where \'f IND. t\' is \'C POT. r - 2\' -conjugated to a irrational translation has positive Lebesgue measure, since the rotation numbers of \'f IND. a\' and \'f IND. b\' are distinct

Multilayered metallo-dielectric nanoparticles are increasingly considered in various applications to control the spatial and temporal behavior of electromagnetic fields. In particular, the surface mode excitation by photons or electrons in metal nanorings finds significant applications because of the implied field distribution and electromagnetic energy confinement. However, most solid nanorings that are multilayered and/or embedded in a medium have non-simply connected geometry resulting in surface modes which are not linearly independent. That is, unlike particle plasmon eigenmodes in other geometries, the amplitudes of the eigenmodes of tori exhibit a distinct forward and backward coupling. We investigate the surface modes of such toroidal nano-structures and obtain the canonical plasmon dispersion relations and resonance modes for arbitrarily layered nanorings. When seeking the nonretarded surface modes for a stratified solid torus, we obtain a three-term difference equation which plays an important role in obtaining the needed dispersion relations. The obtained dispersion relations are investigated in depth in terms of the involved matrix continued fractions and their convergence properties including their determinant forms for computing the plasmon eigenmodes. The numerical solutions of the dispersion relations in case of a solid ring are presented for comparison and the resonance frequencies for the first few dominant modes of a ring composed of plasmon supporting materials such as gold, silver, and aluminum are provided and compared to those for a silicon ring. The mode complementarity and hybridization in multilayered toroidal structures is discussed and different ring configurations are simulated in the quasistatic limit by selecting number of layers modeled by their local dielectric functions. A generalized Green’s function with derivation intricacies addressed for multilayer tori is obtained from which one may calculate and study the scattering behavior of any of the modes that may exist in the many layer system. In particular, the electric potential distribution corresponding to individual poloidal and toroidal modes in response to an arbitrarily polarized external field and the field of electrons is obtained. The results are applied to obtain the local density of states and decay rate of a dipole near the center of the torus. Finally, two new types of toroidal particles in the form of janus nanorings are introduced.

The PhD thesis considers four topics in dynamical systems and is based on one paper and three manuscripts. In Paper I we apply methods of interval analysis in order to compute the rigorous enclosure of rotation number. The described algorithm is supplemented with a method of proving the existence of periodic points which is used to check rationality of the rotation number. In Manuscript II we provide a numerical algorithm for computing critical points of the multiplier map for the quadratic family (i.e., points where the derivative of the multiplier with respect to the complex parameter vanishes). Manuscript III concerns continued fractions of quadratic irrationals. We show that the generating function corresponding to the sequence of denominators of the best rational approximants of a quadratic irrational is a rational function with integer coefficients. As a corollary we can compute the Lévy constant of any quadratic irrational explicitly in terms of its partial quotients. Finally, in Manuscript IV we develop a method for finding rigorous enclosures of all odd periodic solutions of the stationary Kuramoto-Sivashinsky equation. The problem is reduced to a bounded, finite-dimensional constraint satisfaction problem whose solution gives the desired information about the original problem. Developed approach allows us to exclude the regions in L2, where no solution can exist.

This thesis aims to study a connection between indecomposable elements in the cubic fields and the Jacobi-Perron algorithm (JPA). JPA is a multidimensional generalization of the usual continued fractions algorithm. We work in the family of Ennola's cubic fields and we examine how the indecomposable elements are related to elements originating from this algorithm and whether some of these elements generate all indecomposable elements in the fields. We formulate conjectures on how to determine which elements will generate the indecomposable elements. We also prove some necessary conditions that have to hold for elements originating from this algorithm to generate indecomposable elements. 1

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