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Combinatorial Proofs of CongruencesRouse, Jeremy 01 May 2003 (has links)
Combinatorial techniques can frequently provide satisfying “explanations” of various mathematical phenomena. In this thesis, we seek to explain a number of well known number theoretic congruences using combinatorial methods. Many of the results we prove involve the Fibonacci sequence and its generalizations.
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A razão áurea e a sequência de Fibonacci / The golden ratio and the Fibonacci sequenceMarcelo Manechine Belini 16 September 2015 (has links)
O presente trabalho irá abordar dois temas matemáticos de diferentes contextos históricos mas que apresentam uma relação intrínseca com o número Φ, mais conhecido como número de ouro. Partiremos de uma breve descrição dos conjuntos numéricos N, Z, Q e algumas propriedades dos números racionais para, em seguida, deduzirmos os números irracionais Π e, enfim, os números reais R. Na sequência vamos trabalhar com dois problemas muito antigos: o primeiro aparece na coletânea de livros Os Elementos do matemático grego Euclides, 300 anos a.C., e diz respeito à divisão de um segmento em média e extrema razão e, o segundo, foi publicado no livro Liber Abaci do matemático italiano Leonardo Fibonacci, século XIII, e trata da reprodução de coelhos e a sequência a qual ela origina. Veremos que o número de ouro aparece em ambos os problemas e vem ao longo dos séculos desencadeando muitas teorias que tratam de padrões e beleza. Abordaremos situações do passado e do presente que fazem uso desses padrões, além de fenômenos da natureza. Também apresentaremos um conjunto de atividades para orientar professores do ensino médio de como trabalhar, numa perspectiva interdisciplinar com vários conteúdos da matemática, e o número Φ. / This work addresses two mathematical topics from different historical contexts but that have an intrinsic relationship with the number Φ, better known as the golden number. We start with a brief description of the numerical sets N, Z, Q and some properties of rational numbers, and then deduct the set of irrational numbers π and, finally, the set of real numbers R. In the sequence we work with two very old problems: the first appears in the collection of books The elements of the Greek mathematician Euclid, 300 years BC, and concerns the division of a segment in extreme and mean ratio, and the second, published in the book Liber Abaci of the Italian mathematician Leonardo Fibonacci, in the thirteenth century, and deals with the breeding of rabbits and the sequence which it originates. We will see that the golden number appears on both problems and has over the centuries triggering many theories dealing with standards and beauty. We discuss situations of past and present that makes use of these standards, as well as natural phenomena. We also present a set of activities to guide middle school teachers on how to work in an interdisciplinary perspective with various mathematical content, and the number Φ.
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The Golden Ratio and Fibonacci Sequence in MusicBlankenship, Ryan A. 04 May 2021 (has links)
No description available.
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N-parameter Fibonacci AF C*-AlgebrasFlournoy, Cecil Buford, Jr. 01 July 2011 (has links)
An n-parameter Fibonacci AF-algebra is determined by a constant incidence matrix K of a special form. The form of the matrix K is defined by a given n-parameter Fibonacci sequence. We compute the K-theory of certain Fibonacci AF-algebra, and relate their K-theory to the K-theory of an AF-algebra defined by incidence matrices that are the transpose of K.
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Periodic Coefficients and Random Fibonacci SequencesMcLellan, Karyn Anne 20 August 2012 (has links)
The random Fibonacci sequence is defined by t_1 = t_2 = 1 and
t_n = ± t_{n–1} + t_{n–2} ,
for n ? 3, where each ± sign is chosen at random with P(+) = P(–) = 1/2. We can think of all possible such sequences as forming a binary tree T. Viswanath has shown that almost all random Fibonacci sequences grow exponentially at the rate 1.13198824.... He was only able to find 8 decimal places of this constant through the use of random matrix theory and a fractal measure, although Bai has extended the constant by 5 decimal places. Numerical experimentation is inefficient because the convergence is so slow. We will discuss a new computation of Viswanath's constant which is based on a formula due to Kalmár-Nagy, and uses an interesting reduction R of the tree T developed by Rittaud.
Also, we will focus on the growth rate of the expected value of a random Fibonacci sequence, which was studied by Rittaud. This differs from the almost sure growth rate in that we first find an expression for the average of the n^th term in a sequence, and then calculate its growth. We will derive this growth rate using a slightly different and more simplified method than Rittaud, using the tree R and a Pascal-like array of numbers, for which we can further give an explicit formula.
We will also consider what happens to random Fibonacci sequences when we remove the randomness. Specifically, we will choose coefficients which belong to the set {1, –1} and form periodic cycles. By rewriting our recurrences using matrix products, we will analyze sequence growth and develop criteria based on eigenvalue, trace and order, for determining whether a given sequence is bounded, grows linearly or grows exponentially. Further, we will introduce an equivalence relation on the coefficient cycles such that each equivalence class has a common growth rate, and consider the number of such classes for a given cycle length. Lastly we will look at two ways to completely characterize the trace, given the coefficient cycle, by breaking the matrix product up into blocks.
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Linear recurrence sequences of composite numbers / Tiesinės rekurenčiosios sekos sudarytos iš sudėtinių skaičiųŠiurys, Jonas 15 October 2013 (has links)
The main objects studied in this thesis are linear recurrence sequences of composite numbers. We have studied the second order (binary) linear recurrence, tribonacci – like and higher order sequences. Many examples have been given. / Disertacijoje nagrinėjamos tiesinės rekurenčiosios sekos. Ieškoma tokių pradinių narių, kurie generuotų rekurenčiąsias sekas sudarytas iš sudėtinių skaičių. Pilnai išnagrinėtos antros eilės tiesinės rekurenčiosos ir tribonačio tipo sekos, patiekiami pavyzdžiai. Gauti rezultatai ir k-bonačio tipo sekoms.
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Tiesinės rekurenčiosios sekos sudarytos iš sudėtinių skaičių / Linear recurrence sequences of composite numbersŠiurys, Jonas 15 October 2013 (has links)
Disertacijoje nagrinėjamos tiesinės rekurenčiosios sekos. Ieškoma tokių pradinių narių, kurie generuotų rekurenčiąsias sekas sudarytas iš sudėtinių skaičių. Pilnai išnagrinėtos antros eilės tiesinės rekurenčiosos ir tribonačio tipo sekos, patiekiami pavyzdžiai. Gauti rezultatai ir k-bonačio tipo sekoms. / The main objects studied in this thesis are linear recurrence sequences of composite numbers. We have studied the second order (binary) linear recurrence, tribonacci – like and higher order sequences. Many examples have been given.
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Matematika okolo nás - problematika zlatého řezu / Math around us - issue of the Golden sectionKAŇKOVÁ, Jana January 2015 (has links)
Diploma thesis is intended as material for general public. The thesis includes construction methods of the Golden Section and calculation of the Golden Number and its properties. It makes aquainted with the history of the Golden Section and its shows occurrence in a plane geometry. The thesis describes the connection between the Golden Number and the Fibonacci Sequence, occurrence of the Golden Section, Fibonacci Sequnce and the logarithmic spirals in nature in both proportions of living organisms and in plants and on the human body. The text is completed with illustractive figures drawn mostly in GeoGebra. The Golden Number properties and the plane constructions are given. The Golden Section is completed with photos.
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Razão áurea: como motivação ao estudo de conteúdos matemáticos / Golden ratio as a motivation to study mathematics contentSilva, Renato Rodrigues 19 November 2014 (has links)
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Previous issue date: 2014-11-19 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This work goal to show a possible relationship between the Golden Ratio with nature,
animals, architecture, music and also as a motivation to study mathematics content,
such as: ratio, proportion and arithmetic average, making the teaching learning more
enjoyable. The realization of it proceeded from the literature and field research. The
literature describes the history of the Golden Mean and the Fibonacci ratio with the
Golden Ratio. The Fibonacci sequence was known for the problem of pairs of rabbits
(coniculorum Paia) that is found in the book Liber Abacci (Liber Abaci). Also
highlights the relationship between the golden ratio and the nature, proposing that it
can be widely used in daily life of the student, promoting a differentiated learning.
The field research was the application of the proposed activities presented
throughout the study in a rural school of the Federal District, with the purpose to
promote the recognition that it is possible to understand the relationship between
math and everyday living. Initially the diagnosis 1 (ATTACHMENT A), containing
socio-cultural issues and also the diagnosis 2 (ATTACHMENT B) containing specific
questions of reason, proportion, arithmetic mean and golden ratio was applied. After
applying the diagnosis twelve o'clock classes were taught using contextualized and
interdisciplinary methodologies where activities (ATTACHMENT C, D, E, F) were
applied seeking to respond to the objectives of this study. In closing the interventions
took place applying the same initial diagnosis in order to determine whether
interventions have provided new results. In analyzing the results of the second
application of diagnosis was realized a significant increase in students' understanding
about the content worked. The results show that when there is an understanding of
the relationship between mathematics learning and everyday life, students can define
new knowledge and relate school learning and their daily lives, which facilitates
learning. / Este trabalho tem por objetivo, mostrar uma possível relação da Razão Áurea com a
natureza, os animais, a arquitetura, a música e também como motivação ao estudo
de conteúdos de Matemática, tais como: razão, proporção e média aritmética,
tornando o ensino-aprendizagem mais prazeroso. A realização do mesmo procedeu
a partir da pesquisa bibliográfica e de campo. A pesquisa bibliográfica descreve a
história do Número de Ouro e a relação da Sequência de Fibonacci com a Razão
Áurea. A Sequência de Fibonacci ficou conhecida pelo problema dos pares de
coelhos (paia coniculorum) que é encontrado no livro Liber Abacci (Líber Ábacos).
Destaca ainda a relação entre a razão áurea e a natureza, propondo-se que esta
pode ser amplamente utilizada no cotidiano do discente, buscando promover uma
melhor aprendizagem. A pesquisa de campo consistiu na aplicação das atividades
propostas apresentadas ao longo do estudo em uma escola da zona rural do Distrito
Federal, tendo como fim promover o reconhecimento de que é possível
compreender a relação entre o ensino de matemática e a vivência cotidiana.
Inicialmente foi aplicado o diagnóstico 1 (ANEXO A), contendo questões
socioculturais e também o diagnóstico 2 (ANEXO B) contendo questões específicas
de razão, proporção, média aritmética e razão áurea. Após a aplicação do
diagnóstico foram ministradas doze aulas utilizando-se metodologias
contextualizadas e interdisciplinares em que foram aplicadas atividades (ANEXO C,
D, E, F) buscando responder aos objetivos deste estudo. Ao encerrar as
intervenções realizou-se a aplicação do mesmo diagnóstico inicial com o intuito de
averiguar se as intervenções propiciaram novos resultados. Na análise dos
resultados da segunda aplicação do diagnóstico foi percebido um aumento
significativo na compreensão dos discentes em relação ao conteúdo trabalhado. Os
resultados evidenciam que quando há uma compreensão da relação entre
aprendizagem matemática e a vida cotidiana, os discentes conseguem delimitar
novos saberes e relacionar a aprendizagem escolar e sua vivência diária, o que
facilita a aprendizagem.
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Predikce vývoje pohybu kurzu na forexu / Prediction of Exchange Rate Movements on ForexBalog, Miroslav January 2015 (has links)
The thesis deals with the possibility of prediction of the exchange rate on forex. The combination of Elliott wave principle and Fibonacci numbers examines to what extent and in what time periods it is possible to predict exchange rate. The thesis use fundamental analysis and MACD oscillator to confirm the accuracy of this prediction.
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