The generating function of the generalized Fibonacci sequence

Using tools of the theory of orthogonal polynomials we obtain the generating function of the generalized Fibonacci sequence established by Petronilho for a sequence of real or complex numbers {Qn} defined by Q0 = 0, Q1 = 1, Qm = ajQm−1 + bjQm−2, m ≡ j (mod k), where k ≥ 3 is a fixed integer, and a0, a1, . . . , ak−1, b0, b1, . . . , bk−1 are 2k given real or complex numbers, with bj #0 for 0 ≤ j ≤ k−1. For this sequence some convergence proprieties are obtained.

Deficient generalized Fibonacci maximum path graphs /

"UILU-ENG 78 1739."

On generalized orlicz sequence spaces of Fourier coefficients for trigonometric gap series. I.

We investigate the operator associating with a function fєLp2π, 1sequence of Fourier coefficients of ƒ with respect to a trigonometric gap system, as well as an operator from a modular space X ρs(ϕ) to the generalized Orlicz sequence space lϕ.

Ondas de spin em quasi-cristais magnônicos

In this paper we investigate the spectra of band structures and transmittance in magnonic quasicrystals that exhibit the so-called deterministic disorders, specifically, magnetic multilayer systems, which are built obeying to the generalized Fibonacci (only golden mean (GM), silver mean (SM), bronze mean (BM), copper mean (CM) and nickel mean (NM) cases) and k-component Fibonacci substitutional sequences. The theoretical model is based on the Heisenberg Hamiltonian in the exchange regime, together with the powerful transfer matrix method, and taking into account the RPA approximation. The magnetic materials considered are simple cubic ferromagnets. Our main interest in this study is to investigate the effects of quasiperiodicity on the physical properties of the systems mentioned by analyzing the behavior of spin wave propagation through the dispersion and transmission spectra of these structures. Among of these results we detach: (i) the fragmentation of the bulk bands, which in the limit of high generations, become a Cantor set, and the presence of the mig-gap frequency in the spin waves transmission, for generalized Fibonacci sequence, and (ii) the strong dependence of the magnonic band gap with respect to the parameters k, which determines the amount of different magnetic materials are present in quasicrystal, and n, which is the generation number of the sequence k-component Fibonacci. In this last case, we have verified that the system presents a magnonic band gap, whose width and frequency region can be controlled by varying k and n. In the exchange regime, the spin waves propagate with frequency of the order of a few tens of terahertz (THz). Therefore, from a experimental and technological point of view, the magnonic quasicrystals can be used as carriers or processors of informations, and the magnon (the quantum spin wave) is responsible for this transport and processing

EIGENVALUES of FIBONACCI STOCHASTIC ADDING MACHINE

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Fibonacci Series, Golden Proportions, and the Human Biology

Pythagoras, Plato and Euclid’s paved the way for Classical Geometry. The idea of shapes that can be mathematically defined by equations led to the creation of great structures of modern and ancient civilizations, and milestones in mathematics and science. However, classical geometry fails to explain the complexity of non-linear shapes replete in nature such as the curvature of a flower or the wings of a Butterfly. Such non-linearity can be explained by fractal geometry which creates shapes that emulate those found in nature with remarkable accuracy. Such phenomenon begs the question of architectural origin for biological existence within the universe. While the concept of a unifying equation of life has yet to be discovered, the Fibonacci sequence may establish an origin for such a development. The observation of the Fibonacci sequence is existent in almost all aspects of life ranging from the leaves of a fern tree, architecture, and even paintings, makes it highly unlikely to be a stochastic phenomenon. Despite its wide-spread occurrence and existence, the Fibonacci series and the Rule of Golden Proportions has not been widely documented in the human body. This paper serves to review the observed documentation of the Fibonacci sequence in the human body.

Sequências de Fibonacci e a Razão Áurea - aplicações no ensino básico

This thesis aims to present a study of the Fibonacci sequence, initiated from a simple problem of rabbits breeding and the Golden Ratio, which originated from a geometrical construction, for applications in basic education. The main idea of the thesis is to present historical records of the occurrence of these concepts in nature and science and their influence on social, cultural and scientific environments. Also, it will be presented the identification and the characterization of the basic properties of these concepts and howthe connection between them occurs,and mainly, their intriguing consequences. It is also shown some activities emphasizing geometric constructions, links to other mathematics areas, curiosities related to these concepts and the analysis of questions present in vestibular (SAT-Scholastic Aptitude Test) and Enem(national high school Exam) in order to show the importance of these themes in basic education, constituting an excellent opportunity to awaken the students to new points of view in the field of science and life, from the presented subject and to promote new ways of thinking mathematics as a transformative science of society.

Vacancy ordered phases and one-dimensional quasiperiodicity

The metastable vacancy ordered phases observed in aluminium transition metal alloys on rapid solidification or vapour deposition can be considered as a periodic arrangement of a truncated quasiperiodic string based on the Fibonacci sequence along the left angle bracket111right-pointing angle bracket stacking direction of the original CsCl cell. Using the projection formalism developed in the context of quasicrystals, the diffraction patterns of the vacancy ordered phases are calculated for both commensurate and incommensurate projection from a periodic cubic cell in four dimensions. These are compared with experimentally observed patterns. It is shown that at increasingly longer periodicity the patterns from commensurate crystals become indistinguishable from the truly quasiperiodic one. It is suggested that there is a strong link between vacancy ordered phases and quasicrystals.

Localização eletrônica de sistemas aperiódicos em uma dimensão

Desde a descoberta do estado quasicristalino por Daniel Shechtman et al. em 1984 e da fabricação por Roberto Merlin et al. de uma superrede artificial de GaAs/ AlAs em 1985 com características da sequência de Fibonacci, um grande número de trabalhos teóricos e experimentais tem relatado uma variedade de propriedades interessantes no comportamento de sistemas aperiódicos. Do ponto de vista teórico, é bem sabido que a cadeia de Fibonacci em uma dimensão se constitui em um protótipo de sucesso para a descrição do estado quasicristalino de um sólido. Dependendo da regra de inflação, diferentes tipos de estruturas aperiódicas podem ser obtidas. Esta diversidade originou as chamadas regras metálicas e devido à possibilidade de tratamento analítico rigoroso este modelo tem sido amplamente estudado. Neste trabalho, propriedades de localização em uma dimensão são analisadas considerando-se um conjunto de regras metálicas e o modelo de ligações fortes de banda única. Considerando-se o Hamiltoniano de ligações fortes com um orbital por sítio obtemos um conjunto de transformações relativas aos parâmetros de dizimação, o que nos permitiu calcular as densidades de estados (DOS) para todas as configurações estudadas. O estudo detalhado da densidade de estados integrada (IDOS) para estes casos, mostra o surgimento de plateaux na curva do número de ocupação explicitando o aparecimento da chamada escada do diabo" e também o caráter fractal destas estruturas. Estudando o comportamento da variação da energia em função da variação da energia de hopping, construímos padrões do tipo borboletas de Hofstadter, que simulam o efeito de um campo magnético atuando sobre o sistema. A natureza eletrônica dos auto estados é analisada a partir do expoente de Lyapunov (γ), que está relacionado com a evolução da função de onda eletrônica ao longo da cadeia unidimensional. O expoente de Lyapunov está relacionado com o inverso do comprimento de localização (ξ= 1 /γ), sendo nulo para os estados estendidos e positivo para estados localizados. Isto define claramente as posições dos principais gaps de energia do sistema. Desta forma, foi possível analisar o comportamento autossimilar de cadeias com diferentes regras de formação. Analisando-se o espectro de energia em função do número de geração de cadeias que seguem as regras de ouro e prata foi feito, obtemos conjuntos do tipo-Cantor, que nos permitiu estudar o perfil do calor específico de uma cadeia e Fibonacci unidimensional para diversas gerações

Three-wave Nonlinear Scattering by Quasiperiodic Dielectric Structure

The combinatorial frequency generation by a Fibonacci type quasi-periodic dielectric multilayered structure illuminated by two plane waves has been analysed. The effects of the layer parameters and Fibonacci sequence order on the properties of the combinatorial frequency waves emitted from the stacked nonlinear layers are discussed.

Exercise sheet 4 pdf

Exercises and solutions in PDF

Exercise sheet 4

Exercises and solutions in LaTex

A influência do esqueleto açúcar-fostato no transporte da molécula de DNA

This dissertation analyses the influence of sugar-phosphate structure in the electronic transport in the double stretch DNA molecule, with the sequence of the base pairs modeled by two types of quasi-periodic sequences: Rudin-Shapiro and Fibonacci. For the sequences, the density of state was calculated and it was compared with the density of state of a piece of human DNA Ch22. After, the electronic transmittance was investigated. In both situations, the Hamiltonians are different. On the analysis of density of state, it was employed the Dyson equation. On the transmittance, the time independent Schrödinger equation was used. In both cases, the tight-binding model was applied. The density of states obtained through Rudin-Shapiro sequence reveal to be similar to the density of state for the Ch22. And for transmittance only until the fifth generation of the Fibonacci sequence was acquired. We have considered long range correlations in both transport mechanism

Excitações elementares em super-redes quasiperiódicas com simetria espelho

In this thesis, we investigated the magnonic and photonic structures that exhibit the so-called deterministic disorder. Speci cally, we studied the effects of the quasiperiodicity, associated with an internal structural symmetry, called mirror symmetry, on the spectra of photonics and magnonics multilayer. The quasiperiodicity is introduced when stacked layers following the so-called substitutional sequences. The three sequences used here were the Fibonacci sequence, Thue-Morse and double-period, all with mirror symmetry. Aiming to study the propagation of light waves in multilayer photonic, and spin waves propagation in multilayer magnonic, we use a theoretical model based on transfer matrix treatment. For the propagation of light waves, we present numerical results that show that the quasiperiodicity associated with a mirror symmetry greatly increases the intensity of transmission and the transmission spectra exhibit a pro le self-similar. The return map plotted for this system show that the presence of internal symmetry does not alter the pattern of Fibonacci maps when compared with the case without symmetry. But when comparing the maps of Thue-Morse and double-time sequences with their case without the symmetry mirror, is evident the change in the pro le of the maps. For magnetic multilayers, we work with two di erent systems, multilayer composed of a metamagnetic material and a non-magnetic material, and multilayers composed of two cubic Heisenberg ferromagnets. In the rst case, our calculations are carried out in the magnetostatic regime and calculate the dispersion relation of spin waves for the metamgnetic material considered FeBr2. We show the e ect of mirror symmetry in the spectra of spin waves, and made the analysis of the location of bulk bands and the scaling laws between the full width of the bands allowed and the number of layers of unit cell. Finally, we calculate the transmission spectra of spin waves in quasiperiodic multilayers consisting of Heisenberg ferromagnets. The transmission spectra exhibit self-similar patterns, with regions of scaling well-de ned in frequency and the return maps indicates only dependence of the particular sequence used in the construction of the multilayer

Explorando a matemática do número Ф, o número de ouro

Pós-graduação em Matemática em Rede Nacional - IBILCE