Graduation date: 1994

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.

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Made available in DSpace on 2015-03-03T15:28:32Z (GMT). No. of bitstreams: 1 ManassesPN_DISSERT.pdf: 2146671 bytes, checksum: a562634d1e686680a598403ed93762dd (MD5) Previous issue date: 2010-05-17 / Two-level factorial designs are widely used in industrial experimentation. However, many factors in such a design require a large number of runs to perform the experiment, and too many replications of the treatments may not be feasible, considering limitations of resources and of time, making it expensive. In these cases, unreplicated designs are used. But, with only one replicate, there is no internal estimate of experimental error to make judgments about the significance of the observed efects. One of the possible solutions for this problem is to use normal plots or half-normal plots of the efects. Many experimenters use the normal plot, while others prefer the half-normal plot and, often, for both cases, without justification. The controversy about the use of these two graphical techniques motivates this work, once there is no register of formal procedure or statistical test that indicates \which one is best". The choice between the two plots seems to be a subjective issue. The central objective of this master's thesis is, then, to perform an experimental comparative study of the normal plot and half-normal plot in the context of the analysis of the 2k unreplicated factorial experiments. This study involves the construction of simulated scenarios, in which the graphics performance to detect significant efects and to identify outliers is evaluated in order to verify the following questions: Can be a plot better than other? In which situations? What kind of information does a plot increase to the analysis of the experiment that might complement those provided by the other plot? What are the restrictions on the use of graphics? Herewith, this work intends to confront these two techniques; to examine them simultaneously in order to identify similarities, diferences or relationships that contribute to the construction of a theoretical reference to justify or to aid in the experimenter's decision about which of the two graphical techniques to use and the reason for this use. The simulation results show that the half-normal plot is better to assist in the judgement of the efects, while the normal plot is recommended to detect outliers in the data / Os experimentos fatoriais 2k s?o muito utilizados na experimenta??o industrial. Contudo, quanto maior o n?mero de fatores considerados maior ser? a quantidade de provas necess?rias para a execu??o de um experimento, e realizar replica??es dos tratamentos pode ser invi?vel, considerando as limita??es de recursos e de tempo, tornando tal experimento dispendioso. Nestes casos, s~ao utilizados os fatoriais 2k n?o replicados. Mas, sem replica??oo, n?o ? poss?vel obter uma estimativa direta da variabilidade do erro experimental para se avaliar a signific^ancia dos efeitos. Uma das poss?veis solu??es para este problema ? utilizar os gr?fificos normal ou semi-normal dos efeitos. Muitos pesquisadores usam o gr?fifico normal, ao passo que outros preferem o semi-normal e, em muitas vezes, para ambos os casos, sem alguma justificativa. A controv?rsia sobre o uso destas duas t?cnicas gr?ficas ? o que motiva a realiza??o do presente trabalho, uma vez que n?o h? registro de procedimento formal ou teste estat?stico que indique \qual delas ? melhor". A escolha entre os dois gr?fificos parece ser uma quest~ao subjetiva. O objetivo central desta disserta??o ?, ent?o, realizar um estudo comparativo experimental dos gr?fificos normal e semi-normal no contexto da an?lise dos experimentos fatoriais 2k n?o replicados. Tal estudo consiste na constru??o de cen?rios simulados, nos quais o desempenho dos gr?fificos em detectar os efeitos significativos e identificar valores discrepantes ? avaliado com o intuito de verificar as seguintes quest?es: Um gr?fifico pode ser melhor que o outro? Em que situa??es? Que informa??es um gr?fifico acrescenta ? an?lise do experimento que possam complementar aquelas fornecidas pelo outro gr?fifico? Quais as restri??es no uso de cada gr?fifico? Com isso, prop?e-se confrontar estas duas t?cnicas; examin?-las simultaneamente a fim de conhecer semelhan?as, diferen?as ou rela??es que possam contribuir para a constru??o de um referencial te?rico que sirva como um subs?dio para justificar ou auxiliar na decis~ao do pesquisador sobre qual das duas t?cnicas gr?fificas utilizar e o porqu^e deste uso. Os resultados das simula??es mostram que o gr?fifico semi-normal ? melhor para auxiliar no julgamento dos efeitos, ao passo que o gr?fifico normal ? recomendado para detectar a presen?a de valores discrepantes nos dados

Les systèmes dynamiques qui évoluent sur un grand intervalle de temps (dynamique moléculaire, prédiction astronomique, turbulence...) occupent une place importante dans le domaine de la science de l'ingénieur. Leur résolution numérique constitue, jusqu'à l'heure actuelle, un défi. En effet, la simulation de la solution nécessite un solveur non seulement rapide mais aussi qui respecte les propriétés physiques du problème, pour garantir la stabilité. Dans cette thèse, on se propose d'étudier, vis-à-vis de cette problématique, un schéma d'intégration temporelle basée sur la décomposition de la solution en série temporelle, suivie de la technique de resommation de Borel des séries divergentes. On analyse alors la rapidité du schéma sur des problèmes modèles. Ensuite, on montre sa capacité à préserver la structure des équations (symplecticité, iso-spectralité, conservation de l'énergie...) à un ordre arbitrairement élevé. Par la suite, on applique le schéma à la résolution d'équations aux dérivées partielles issues de la mécanique, dont les équations de la chaleur, de Burgers et de Navier-Stokes bidimensionnelles. Pour cela, on associe le schéma à une méthode de discrétisation par éléments finis en espace. Enfin, dans le but de rendre l'algorithme plus robuste, on s'intéresse à la représentation de la somme de Borel par une série de factorielle généralisée. / Dynamical systems which evolve in a large time interval (molecular dynamic, astronomical prediction, turbulence…) take an important place in engineering science. Their numerical resolution has so far constituted a challenge. Indeed, the simulation of the solution requires a solver which is not only fast but also respects the physical properties of the problem, to ensure the stability. In this thesis, we propose to study, regarding this issue, a time integration scheme based on the decomposition of the solution into time series, followed by Borel's resummation technique of divergent series. We analyse the speed of scheme on model problems. Next, we show its capability to preserve the structure of the equation (symplecticity, iso-spectrality, conservation of energy…) up to an arbitrary high order. Thereafter, we use the scheme to resolve partial differential equations coming from mechanics, including the two-dimensional heat equation, Burger’s equation and the Navier-Stokes equation. To this aim, we choose a finite element method for space discretisation. Finally, and in order to make the algorithm more robust, we are interested in the representation of the Borel sum by a generalized factorials series.