The Fibonomial numbers are defined by \[ \begin{bmatrix}n \\ k \end{bmatrix} = \frac{\prod_{i=n-k+1} ^{n} F_i}{\prod_{j=1}^{k} F_j} \] where $F_i$ is the $i$th Fibonacci number, defined by the recurrence $F_n=F_{n-1}+F_{n-2}$ with initial conditions $F_0=0,F_1=1$. In the past year, Sagan and Savage have derived a combinatorial interpretation for these Fibonomial numbers, an interpretation that relies upon tilings of a partition and its complement in a given grid.In this thesis, I investigate previously proven theorems for the Fibonomial numbers and attempt to reinterpret and reprove them in light of this new combinatorial description. I also present combinatorial proofs for some identities I did not find elsewhere in my research and begin the process of creating a general mapping between the two different Fibonomial interpretations. Finally, I provide a discussion of potential directions for future work in this area.

Le cube de Fibonacci est un sous-graphe isométrique de l'hyper- cube ayant un nombre de Fibonacci de sommets. Le cube de Fibonacci a été initialement introduit par W-J. Hsu comme un réseau d'interconnexion et, comme l'hypercube, il a des propriétés topologiques très attractives, mais avec une croissance plus modérée. Parmi ces propriétés, nous discutons de l'hamiltonicité dans le cube de Fibonacci et aussi dans le cube de Lucas qui est obtenu à partir du cube de Fibonacci en supprimant toutes les chaînes qui commencent et nissent avec 1. Nous trouvons également le nombre de som- mets des cubes de Fibonacci et Lucas ayant une certaine excentricité. En n, nous présentons une étude de deux cubes du point de vue de la domination et du 2-packing.

Submitted by Automa??o e Estat?stica (sst@bczm.ufrn.br) on 2016-08-16T23:13:57Z No. of bitstreams: 1 KarloSergioMedeirosLeopoldino_DISSERT.pdf: 10746375 bytes, checksum: 97d53d2b72cf6a8a121fc1e63b2d8735 (MD5) / Approved for entry into archive by Arlan Eloi Leite Silva (eloihistoriador@yahoo.com.br) on 2016-08-24T19:57:21Z (GMT) No. of bitstreams: 1 KarloSergioMedeirosLeopoldino_DISSERT.pdf: 10746375 bytes, checksum: 97d53d2b72cf6a8a121fc1e63b2d8735 (MD5) / Made available in DSpace on 2016-08-24T19:57:21Z (GMT). No. of bitstreams: 1 KarloSergioMedeirosLeopoldino_DISSERT.pdf: 10746375 bytes, checksum: 97d53d2b72cf6a8a121fc1e63b2d8735 (MD5) Previous issue date: 2016-05-13 / A presente disserta??o tem por objetivo apresentar um estudo relativo ? sequ?ncia de Fibonacci, iniciado a partir de um simples problema de reprodu??o de coelhos e da raz?o ?urea, originada de uma constru??o geom?trica, para aplica??es no ensino b?sico. A ideia principal da disserta??o ? apresentar registros hist?ricos da ocorr?ncia destes conceitos na natureza e na ci?ncia e suas in- flu?ncias nos meios sociais, culturais e cient?ficos. Apresenta-se ainda a identifica??o e caracteriza??o das propriedades b?sicas desses conceitos e de que forma ocorre a liga??o entre os mesmos e, principalmente, as consequ?ncias intrigantes das mesmas. Mostram-se atividades enfatizando constru??es geom?tricas, liga??es com outras ?reas da matem?tica, curiosidades relativas a estes conceitos e an?lise de quest?es presentes em vestibulares e Enem (Exame nacional do ensino m?dio), de forma a mostrar a import?ncia destes temas no ensino b?sico, constituindo-se uma excelente oportunidade de despertar nos alunos novos olhares no campo da ci?ncia e da vida a partir do tema apresentado, bem como promover novas maneiras de pensar a matem?tica como ci?ncia transformadora da sociedade. / 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.

This master thesis deals with the investment recommendation for Czechoslovak Com- mercial Bank, a.s. The recommendation is based on business strategy based on Fibonacci retracement and analysis of the current risks of financial markets. The first part deals with the parameters and the results of the business strategy. Second part proposes investment and non-investment recommendations. At the end of the thesis are summarized all aspects of the research.

No description available.

Los sistemas dinámicos no lineales pueden mostrar una variedad de patrones de comportamiento según los valores de los parámetros físicos y las propiedades intrínsecas del sistema. Entre los sistemas dinámicos, los modelos de población y los sistemas de reacciones químicas han recibido una atención especial debido a su dinámica complicada. Estos sistemas dinámicos no lineales exhiben fenómenos dinámicos complejos, tales como comportamientos con múltiples puntos de equilibrio, bifurcaciones, oscilaciones de ciclo límite y caos. En este estudio, investigamos el comportamiento cualitativo de varios modelos dinámicos de derivadas fraccionarias. Hemos observado que el sistema discreto muestra comportamientos dinámicos mucho más ricos, como bifurcaciones y caos, en comparación con su contraparte de sistema continuo, por lo que primero discretizamos el modelo estudiado. Después, analizamos la estabilidad y las bifurcaciones cerca de los equilibrios del modelo discretizado. Los resultados demuestran que los comportamientos dinámicos del modelo discretizado son sensibles al parámetro de orden fraccional y al parámetro de discretización, entre a otros parámetros. Finalmente, realizamos simulaciones numéricas para explicar y validar nuestros hallazgos teóricos. Además, calculamos los exponentes máximos de Lyapunov para confirmar la presencia de comportamiento caótico en los modelos discretos estudiados. La tesis se divide en cuatro capítulos principales: Capítulo Uno: en este capítulo, presentamos la parte introductoria de la tesis, e incluimos resúmenes de todos los resultados que hemos obtenido. Capítulo Dos: este capítulo incluye los trabajos publicados en revistas científicas clasificadas. Capítulo Tres: este capítulo contiene un trabajo que actualmente está en proceso revisión. Capítulo Cuatro: aquí, resumimos todos los hallazgos principales de esta tesis e incluimos algunas propuestas para trabajos futuros.

In recent years, combinatorial properties of finite and infinite words have become increasingly important in fields of physics, biology, mathematics, and computer science. In particular, the fascinating family of Sturmian words has become an extremely active subject of research. These infinite binary sequences have numerous applications in various fields of mathematics, such as symbolic dynamics, the study of continued fraction expansion, and also in some domains of physics ( quasicrystal modelling ) and computer science ( pattern recognition, digital straightness ). There has also been a recent surge of interest in a natural generalization of Sturmian words to more than two letters - the so - called episturmian words, which include the well - known Arnoux - Rauzy sequences. This thesis represents a significant contribution to the study of Sturmian and episturmian words, and related objects such as generalized Thue - Morse words and substitutions on a finite alphabet. Specifically, we prove some new properties of certain palindromic factors of the infinite Fibonacci word; establish generalized ' singular ' decompositions of suffixes of certain morphic Sturmian words; completely describe where palindromes occur in characteristic Sturmian words; explicitly determine all integer powers occurring in a certain class of k-strict episturmian words ( including the k-bonacci word ) ; and prove that certain episturmian and generalized Thue - Morse continued fractions are transcendental. Lastly, we begin working towards a proof of a characterization of invertible substitutions on a finite alphabet, which generalizes the fact that invertible substitutions on two letters are exactly the Sturmian morphisms. / Thesis (Ph.D.)--School of Mathematical Sciences, 2006.

In this thesis we study the isoperimetric problem on trees and graphs with bounded treewidth. Let G = (V,E) be a finite, simple and undirected graph. For let δ(S,G)= {(u,v) ε E : u ε S and v ε V – S }be the edge boundary of S. Given an integer i, 1 ≤ i ≤ | V| , let the edge isoperimetric value of G at I be defined as be(i,G)= mins v;|s|= i | δ(S,G)|. For S V, let φ(S,G) = {u ε V – S : ,such that be the vertex boundary of S. Given an integer i, 1 ≤ i ≤ | V| , let the vertex isoperimetric value of G at I be defined as bv(i,G)= The edge isoperimetric peak of G is defined as be(G) =. Similarly the vertex isoperimetric peak of G is defined as bv(G)= .The problem of determining a lower bound for the vertex isoperimetric peak in complete k-ary trees of depth d,Tdkwas recently considered in[32]. In the first part of this thesis we provide lower bounds for the edge and vertex isoperimetric peaks in complete k-ary trees which improve those in[32]. Our results are then generalized to arbitrary (rooted)trees. Let i be an integer where . For each i define the connected edge isoperimetric value and the connected vertex isoperimetric value of G at i as follows: is connected and is connected A meta-Fibonacci sequence is given by the reccurence a(n)= a(x1(n)+ a1′(n-1))+ a(x2(n)+ a2′(n -2)), where xi: Z+ → Z+ , i =1,2, is a linear function of n and ai′(j)= a(j) or ai′(j)= -a(j), for i=1,2. Sequences belonging to this class have been well studied but in general their properties remain intriguing. In the second part of the thesis we show an interesting connection between the problem of determining and certain meta-Fibonacci sequences. In the third part of the thesis we study the problem of determining be and bv algorithmically for certain special classes of graphs. Definition 0.1. A tree decomposition of a graph G = (V,E) is a pair where I is an index set, is a collection of subsets of V and T is a tree whose node set is I such that the following conditions are satisfied: (For mathematical equations pl see the pdf file)

Dans cette thèse nous étudions la caractérisation et l'énumération de polyominos définis par des contraintes de convexité et ou d'évitement de motifs. Nous nous intéressons à l'énumération des polyominos k-convexes selon le semi périmètre, qui n'était connue que pour k=1,2. Nous énumérons une sous classe, les polyominos k-parallélogrammes, grâce à une décomposition récursive dont nous déduisons la fonction génératrice qui est rationnelle. Cette fonction génératrice s'exprime à l'aide des polynômes de Fibonacci, ce qui nous permet d'en déduire une bijection avec les arbres planaires ayant une hauteur inférieure ou égale à k+2. Dans la deuxième partie, nous examinons la notion d'évitement de motif, qui a été essentiellement étudiée pour les permutations. Nous introduisons ce concept dans le contexte de matrices de permutations et de polyominos. Nous donnons des définitions analogues à celles données pour les permutations et nous explorons ses propriétés ainsi que celles du poste associé. Ces deux approches peuvent être utilisées pour traiter des problèmes ouverts sur les polyominos ou sur d'autres objets combinatoires.

After reviewing some properties of the two dimensional hyperbolic toral automorphism called Arnold's discrete cat map, including its generalizations with matrices having positive unit determinant, this thesis contains a definition of a novel cat map where the elements of the matrix are found in the sequence of Pell numbers. This mapping is therefore denoted as Pell's cat map. The main result of this thesis is a theorem determining the upper bound for the minimal period of Pell's cat map. From numerical results four conjectures regarding properties of Pell's cat map are also stated. A brief exposition of some applications of Arnold's discrete cat map is found in the last part of the thesis.