京都大学 / 0048 / 新制・課程博士 / 博士(情報学) / 甲第20739号 / 情博第653号 / 新制||情||113(附属図書館) / 京都大学大学院情報学研究科数理工学専攻 / (主査)教授 中村 佳正, 教授 山下 信雄, 教授 西村 直志 / 学位規則第4条第1項該当 / Doctor of Informatics / Kyoto University / DFAM

Submitted by Liliane Ferreira (ljuvencia30@gmail.com) on 2018-04-11T12:39:47Z No. of bitstreams: 2 Dissertação - Carlos Alirio Rico Acevedo - 2018.pdf: 1289579 bytes, checksum: 0b60c803c3d9f6f61772e58e7d624086 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2018-04-12T11:29:32Z (GMT) No. of bitstreams: 2 Dissertação - Carlos Alirio Rico Acevedo - 2018.pdf: 1289579 bytes, checksum: 0b60c803c3d9f6f61772e58e7d624086 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2018-04-12T11:29:32Z (GMT). No. of bitstreams: 2 Dissertação - Carlos Alirio Rico Acevedo - 2018.pdf: 1289579 bytes, checksum: 0b60c803c3d9f6f61772e58e7d624086 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2018-03-16 / Conselho Nacional de Pesquisa e Desenvolvimento Científico e Tecnológico - CNPq / Let $ k \geq 2.$ an integer. The recurrence $ \fk{n} = \sum_ {i = 0}^k \fk{n-i} $ for $ n> k $, with initial conditions $F_{-(k-2)}^{(k)}=F_{-(k-3)}^{(k)}=\cdots=F_{0}^{(k)}=0$ and $F_1^{ (k)} = 1$, which is called the $k$-generalized Fibonacci sequence. When $ k = 2 ,$ we have the Fibonacci sequence $ \{ F_n \}_{n\geq 0}.$ We will show that the equation $F_{n}^{x}+F_{n+1}^x=F_{m}$ does not have no non-trivial integer solutions $ (n, m, x) $ to $ x> 2 $. On the other hand, for $ k \geq 3,$ we will show that the diophantine equation $\epi$ does not have integer solutions $ (n, m, k, x) $ with $ x \geq 2 $. In both cases, we will use initially Matveev's Theorem, for linear forms in logarithms and the reduction method due to Dujella and Pethö, to limit the variables $ n, \; m $ and $ x $ at intervals where the problem is computable. In addition, in the case for $ k\geq 3 $, we will use the fact that the dominant root the $k$-generalized Fibonacci sequence is exponentially close to 2 to bound $k$, a method developed by Bravo and Luca. / Seja $k\geq 2$ inteiro, considere-se a recorrência $\fk{n}=\sum_{i=0}^{k}\fk{n-i}$ para $n>k$, com condições iniciais $F_{-(k-2)}^{(k)}=F_{-(k-3)}^{(k)}=\cdots=F_{0}^{(k)}=0$ e $F_{1}^{(k)}=1$, que é a sequência de Fibonacci $k$-generalizada. No caso quando $k=2$, é dizer, para a sequência de Fibonacci $\{F_n\}_{n\geq 0}$, vai-se mostrar que a equação $F_{n}^{x}+F_{n+1}^x=F_{m}$ não possui soluções inteiras não triviais $(n,m,x)$ para $x>2$. Por outro lado para, $k\geq 3$ se mostrar que a equação diofantina $\epi$ não possui soluções inteiras $(n,m,k,x)$ com $x\geq 2$. Em ambos casos, inicialmente são usados resultados como o Teorema de Matveev, para formas lineares em logaritmos e o método de redução de Dujella e Pethö, para limitar as variáveis $n, \; m$ e $x$ em intervalos onde o problema seja computável. Adicionalmente, no caso para $k\geq 3$ é usado que a raiz dominante da sequência de Fibonacci $k$-generalizada e exponencialmente próxima a 2, para limitar $k$, o que é um método desenvolvido por Bravo e Luca.

Orientadora: Profa. Dra. Ana Carolina Boero / Dissertação (mestrado) - Universidade Federal do ABC, Programa de Pós-Graduação em Mestrado Profissional em Matemática em Rede Nacional, 2017. / A Razão Áurea possui uma longa história e atualmente é muito mistificada. Nesse trabalho, são apresentadas relações matemáticas e propriedades da Razão Áurea e da Sequência de Fibonacci, sendo constatado que se tratam apenas de casos particulares que podem ser obtidos através de uma recorrência linear de segunda ordem homogênea de onde surge um conjunto de números irracionais com características semelhantes. Foram mostradas, ainda, possibilidades de atividades que de fato contemplam a Razão Áurea e a Sequência de Fibonacci e os cuidados necessários com informações equivocadas e manipuladas. / The Golden Ratio has a long story and currently it¿s very mystified. In this paper, mathematical relations and properties of the Golden Ratio and the Fibonacci Sequence are introduced, stating that they are only particular cases, which can be obtained through a second homogeneous order linear recurrence from where comes a set of irrational numbers with similar characteristics. We explained, as well, possibilities of activities that actually contemplate the Golden Ratio and the Fibonacci Sequence, and the necessary cares with wrong and manipulated information.

Due to the system does not recognize equations and formulas the resumo and abstract can be found in the PDF file. / Devido ao sistema não reconhecer equações e fórmulas o resumo e abstract encontra-se no arquivo em PDF.

Submitted by isabela.moljf@hotmail.com (isabela.moljf@hotmail.com) on 2016-08-18T13:58:42Z No. of bitstreams: 1 leandrodeoliveirasodre.pdf: 681870 bytes, checksum: 2aa85f9c6534a3e3fbb9b9999b6dc538 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2016-08-19T11:52:31Z (GMT) No. of bitstreams: 1 leandrodeoliveirasodre.pdf: 681870 bytes, checksum: 2aa85f9c6534a3e3fbb9b9999b6dc538 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2016-08-19T11:52:50Z (GMT) No. of bitstreams: 1 leandrodeoliveirasodre.pdf: 681870 bytes, checksum: 2aa85f9c6534a3e3fbb9b9999b6dc538 (MD5) / Made available in DSpace on 2016-08-19T11:52:50Z (GMT). No. of bitstreams: 1 leandrodeoliveirasodre.pdf: 681870 bytes, checksum: 2aa85f9c6534a3e3fbb9b9999b6dc538 (MD5) Previous issue date: 2013-03-09 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Neste trabalho são apresentadas curiosidades, propriedades matemáticas, aplicações além do campo puramente matemático e um pouco da história de dois números: o número 142857 e o Número de Ouro. Além disso, são propostas algumas atividades didáticas para o estudo desses números em aulas de Matemática. O número 142857 é chamado de cíclico porque 142857x2 = 285714, 142857x3 = 428571, 142857x4 = 571428, 142857x5 = 714285 e 142857x6 = 857142 e o Número de Ouro tem aplicações na Botânica, Zoologia, Artes, Engenharia de Materiais e tem muitas relações com a sequência de Fibonacci. Palavras-chaves: números cíclicos, Número de Ouro, sequência de Fibonacci, atividades didáticas, curiosidades matemáticas. / This work presents curiosities, mathematical properties, applications beyond the purely mathematical field and some of the history of two numbers: the number 142857 and the golden number. In addition, some educational activities for the study of these numbers in mathematics classes are proposed. The number 142857 is called of cyclic because 142857x2 = 285714, 142857x3 = 428571, 142857x4 = 571428, 142857x5 = 714285 e 142857x6 = 857142 and the golden number is applied in botany, zoology, art, materials engineering and has many relationships with the Fibonacci sequence. Keywords: cyclic numbers, golden number, Fibonacci sequence, educational activities, mathematical curiosities.

Contemporary research in digital signal processing (DSP) is focused on issues of computational complexity, very high data rate and large quantum of data. Thus, the success in newer applications and areas hinge on handling these issues. Conventional ways to address these challenges are to develop newer structures like Multirate signal processing, Multiple Input Multiple Output(MIMO), bandpass sampling, compressed domain sensing etc. In the implementation domain, the approach is to look at floating point over fixed point representation and / or longer wordlength etc., related to number representations and computations. Of these, a simple approach is to look at number representation, perhaps with a simple integer. This automatically guarantees accuracy and zero quantization error as well as longer wordlength. Thus, it is necessary and interesting to explore viable DSP alternatives that can reduce complexity and yet match the required performance. The main aim of this work is to highlight the importance, use and analysis of integer sequences. Firstly, the thesis explores the use of integer sequences as windowing functions. The results of these investigations show that integer sequences and their convolution, indeed, outperform many of the classical real valued window functions in terms of mainlobe width, sidelobe attenuation etc. Secondly, the thesis proposes techniques to approximate discrete Gaussian distribution using integer sequences. The key idea is to convolve symmetrized integer sequences and examine the resulting profiles. These profiles are found to approximate discrete Gaussian distribution with a mean square error of the order of 10−8 or less. While looking at integer sequences to approximate discrete Gaussian, Fibonacci sequence was found to exhibit some interesting properties. The third part of the thesis proves certain fascinating optimal probabilistic limit properties (mean and variance) of Fibonacci sequence. The thesis also provides complete generalization of these properties to probability distributions generated by second order linear recurrence relation with integer coefficients and any kth order linear recurrence relation with unit coefficients. In addition to the above, the thesis also throws light on possible architectural implications of using integer sequences in DSP applications and ideas for further exploration.

This thesis will focus on generating the probability mass function using Fibonacci sequenceas the coefficient of the power series. The discrete probability, named Fibonacci distribution,was formed by taking into consideration the recursive property of the Fibonacci sequence,the radius of convergence of the power series, and additive property of mutually exclusiveevents. This distribution satisfies the requisites of a legitimate probability mass function. It's cumulative distribution function and the moment generating function are then derived and the latter are used to generate moments of the distribution, specifically, the mean and the variance. The characteristics of some convergent sequences generated from the Fibonacci sequenceare found useful in showing that the limiting form of the Fibonacci distribution is a geometricdistribution. Lastly, the paper showcases applications and simulations of the Fibonacci distribution using MATLAB. / <p></p><p></p><p></p>

No description available.

Made available in DSpace on 2015-05-08T14:45:04Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 1848162 bytes, checksum: 8d8c6d6ea96038f73be05f042425a488 (MD5) Previous issue date: 2010-09-17 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / The prices of securities traded on stock exchanges, as well as any other commodity in the financial market fluctuate naturally with the demand for these products. These oscillations, along with the asymmetry of information about the prices of these products generate volatility processes. Charles Dow in the early twentieth century created sector indexes, in which papers met the same area of activity, according to him, several indicators point to the same direction would be a sign that this really would be a tendency to drive the market, thus characterizing the Dow Theory. Ralph Nelson Elliott (1871-1948) studied the average prices of the Dow Jones Industrial and realized repetitions in the market changes, their observations were summarized in what became known as "The Wave Principle." Elliott developed his theory based on so-called Fibonacci sequence, discovered by Leonardo Pizza (Fibonacci) around 1200. In addition to the Dow Theory and the Theory of waves in this work was done using the Theory of Rationality of the agents as a complementary way to explain the decision process of investors, as happens in situations of uncertainty. A rational decision involves selecting the choice which has the largest expected return for a given level of risk. / Os preços dos ativos negociados em bolsas de valores, assim como qualquer outro tipo de commodity do mercado financeiro, oscilam naturalmente com a procura por esses produtos. Essas oscilações, juntamente com a assimetria das informações acerca dos preços desses produtos geram processos de volatilidade. Charles Dow, no início do século XX criou índices setoriais, nos quais reunia papéis da mesma área de atividade, segundo ele, se vários índices apontassem para a mesma direção seria um sinal de que realmente essa seria uma tendência de movimentação do mercado, caracterizando assim a Teoria de Dow. Ralph Nelson Elliott (1871-1948) estudou as cotações médias dos índices Dow Jones Industrial e percebeu repetições nas alterações do mercado, suas observações foram resumidas no que ficou conhecido como O Princípio da Onda . Elliott desenvolveu a sua teoria com base na denominada Sequência de Fibonacci, descoberta por Leonardo de Pizza (Fibonacci) por volta de 1200. Além da Teoria de Dow e da Teoria das Ondas, nesse trabalho, fez-se uso da Teoria da Racionalidade dos agentes como uma forma complementar para se explicar o processo de decisão dos investidores, dado que acontecem em situações de incerteza. Uma decisão racional implica em selecionar a escolha que apresente o maior retorno esperado para um dado nível de risco.