In the study of associated discrete variables, limitations on the range of the possible association measures (Pearson correlation, odds ratio, etc.) arise from the form of the joint probability function between the variables. These limitations are known as the Fréchet bounds. The bounds for cases involving associated binary variables are explored in the context of simulating datasets with a desired correlation and set of marginal probabilities. A new method for creating such datasets is compared to an existing method that uses the multivariate probit. A method for simulating associated binary variables using a desired odds ratio and known marginal probabilities is also presented. The Fréchet bounds for correlation between dependent binomial and negative binomial variables are determined as families of ranges in various cases. An example of a realistic analysis involving the Fréchet bounds in a dependent binomial setting is presented.

Made available in DSpace on 2017-07-21T20:56:33Z (GMT). No. of bitstreams: 1 Pedro Kiochi Kondo.pdf: 1227541 bytes, checksum: daffb8a8bc299356bce288603753944c (MD5) Previous issue date: 2014-09-30 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / In this work some topics of the Discrete or Finite Calculus are developed. In particular, we study difference operators, factorial powers, Stirling numbers of the first and second type, the Newton’s formula of differences, the fundamental theorem of the Finite Calculus, the summation process, and the Bernoulli numbers and Bernoulli polynomials. Then we show the effectiveness of the theory for the calculation of closed formulas for the value of many finite sums. We also study the classical problem of obtaining the polynomials which express the value of the sums of powers of natural numbers. / Neste trabalho desenvolvemos alguns tópicos do Cálculo Discreto ou Finito. Em particular, estudamos operadores de diferenças, potências fatoriais, números de Stirling do primeiro e do segundo tipo, a fórmula de diferenças de Newton, o teorema fundamental do Cálculo Finito, o processo de somação e os números e polinômios de Bernoulli. Mostramos então a eficácia da teoria no cálculo de fórmulas fechadas para o valor de diversas somas finitas. Também estudamos o problema clássico de obter os polinômios que expressam o valor de somas de potências de números naturais.

Submitted by Angela Maria de Oliveira (amolivei@uepg.br) on 2018-11-29T18:07:06Z No. of bitstreams: 2 license_rdf: 811 bytes, checksum: e39d27027a6cc9cb039ad269a5db8e34 (MD5) Maikon Luiz.pdf: 959643 bytes, checksum: aaf472f5b8a9a29532793d01234788a9 (MD5) / Made available in DSpace on 2018-11-29T18:07:06Z (GMT). No. of bitstreams: 2 license_rdf: 811 bytes, checksum: e39d27027a6cc9cb039ad269a5db8e34 (MD5) Maikon Luiz.pdf: 959643 bytes, checksum: aaf472f5b8a9a29532793d01234788a9 (MD5) Previous issue date: 2018-10-19 / Neste trabalho,estudamos os números e os polinomios de Bernoulli,bem como algumas de suas aplicações mais importantes em Teoria dos Números. Com base em uma caracterização ao simples, os polinômios de Bernoulli são introduzidos e, posteriormente, os números de Bernoulli. As séries de Fourier dos polinomios de Bernoulli são utilizadas na demonstração da equação funcional da função teta. Esta equação, por sua vez, é utilizada na demonstração da celebre equação funcional da função zeta, que tem importância central na teoria da distribuição dos números primos. Além das conexões com a funções especiais zeta e teta, discutimos também, em detalhe,conexões entre os números e os polinomios de Bernoulli com a função gama. Essas relações são então exploradas para produzir belas fórmulas para certos valores da função zeta, entre outras aplicações. / In this work we study Bernoulli numbers and Bernoulli polynomials, as well as some of its most important applications in Number Theory. Based on a simple characterization, the Bernoulli polynomials are introduced and, later, the Bernoulli numbers. The Fourier series of the Bernoulli polynomials are used to demonstrate the functional equation of the theta function. This equation, in turn, is used in the proof of the famous functional equation of the zeta function, which is central to the theory of prime number distribution. In addition to the connections with the special functions zeta and theta, we also discuss, in detail, connections between the Bernoulli numbers and Bernoulli polynomials with the gamma function. These relations are then explored to produce beautiful formulas for certain values of the zeta function,among other applications.

Symbolic computation has been widely applied to Combinatorics, Number Theory, and also other fields. Many reliable and fast algorithms with corresponding implementations now have been established and developed. Using the tool of Experimental Mathematics, especially with the help of mathematical software, in particularly Mathematica, we could visualize the data, manipulate algorithms and implementations. The work presented here, based on symbolic computation, involves the following two parts. The first part introduces a systematic integration method, called the Method of Brackets. It only consists of a small number of simple and direct rules coming from the Schwinger parametrization of Feynman diagrams. Verification of each rule makes this method rigorous. Then it follows a necessary theorem that different series representations of the integrand, though lead to different processes of computations, do not affect the result. Examples of application lead to further discussions on analytic continuation, especially on Pochhammer symbol, divergent series and connection to Mellin transform of the Method of Brackets. In the end, comparison with other integration methods and a Mathematica package manual are presented. The second part provides a symbolic approach on the study of Bernoulli numbers and its generalizations. The Bernoulli symbol $\mathcal{B}$ originally comes from Umbral Calculus, as a formal approach to Sheffer sequences. Recently, a rigorous footing by probabilistic proof makes it also a random variable with its density function a shifted hyperbolic trigonometric function. Such an approach together with general method on random variables gives a variety of results on generalized Bernoulli polynomials, multiple zeta functions, and also other related topics. / Lin Jiu

This thesis is to develop a novel consensus algorithm or protocol for multi-agent systems in the event of communication link failure over the network. The structure or topology of the system is modeled by an algebraic graph theory, and defined as a discrete time-invariant system with a second-order dynamics. The communication link failure is governed by a Bernoulli process. Lyapunov-based methodologies and Linear Matrix Inequality (LMI) techniques are then applied to find an appropriate controller gain by satisfying the sufficient conditions of the error dynamics. Therefore, the controller with the calculated gain is guaranteed to drive the system to reach a consensus. Finally, simulation and experiment studies are carried out by using two Mobile Robot Pioneer 3-DXs and one Pioneer 3-AT as a team to verify the proposed work.

Ce mémoire porte sur l'étude de la distribution du nombre de deux succès consécutifs associé à une expérience aléatoire ou les variables sont de Bernoulli indépendantes, mais pas nécessairement identiquement réparties. D'abord, nous revenons en détails sur un certain nombre de résultats dans le cas unidimensionnel dont ceux de Diaconis, Mori, Joffe et al., Csorgd et Wu, Holst parmi d'autres, ainsi que les travaux de Ait Aoudia et Marchand dans le cas bivarié (l'étude de la loi de la somme de deux distributions marginales du vecteur bivarié ) avec applications pour des modèles de Bernoulli échangeables dont celui du modèle d'urne de Pdlya. Ensuite, on présente une extension multidimensionnelle de ces résultats suivie d'une étude détaillée du cas bivarié (l'étude de la distribution du vecteur bivarié et de sa loi limite).

Thesis (Ph. D.)--Rutgers University, 2010. / "Graduate Program in Statistics and Biostatistics." Includes bibliographical references (p. 55-57).

No description available.

O problema de n?corpos é um dos problemas mais importantes em Sistemas Dinâmicos. Nós estudamos o modelo do problema dos três corpos restrito introduzido por Sitnikov. Nesse modelo os corpos primários tem a mesma massa e o terceiro corpo é de massa muito pequena com respeito aos corpos primários. Usando os métodos de Alekseev, nós mostramos a existência de uma ?ferradura de Smale?como um subsistema da dinâmica do terceiro corpo e concluímos ricas conseqüências probabilísticas. Nós também estudamos o problema pelo método de Melnikov / The n?body problem is one of the most important problems in dynamical systems. We study the model introduced by Sitnikov of restricted three body problem. In this model the primaries are of equal mass and the third body is very small with respect to the primaries. Using methods of Alekseev, we show the existence of ?Smale horseshoe?as a subsystem of the dynamic of the third body and conclude rich probabilistic consequences. We also study the same problem by Melnikov?s method

Neste trabalho, procuramos resolver o problema direto da equação da viga de Euler- Bernoulli bi-engastada com condições iniciais nulas. Estudamos o problema inverso da viga, que consiste em identificar a fonte de vibração, modelada como um elemento em L2, usando como dado a velocidade de um ponto arbitrário da viga, durante um intervalo de tempo arbitrariamente pequeno. A relevância deste trabalho na Engenharia encontra-se, por exemplo, na identificação de danos estruturais em vigas. / In this work, we try to solve the direct problem of the clamped-clamped Euler- Bernoulli beam equation, with zero initial conditions. We study the inverse problem of the beam, consisting in the identification of the source of vibration, shaped as an element in L2, using as data the speed from an arbitrary point of the beam, during a time interval arbitrarily small. The relevance of this work in Engineering, for example, is in the identification of structural damage in beams.