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In this dissertation, we provide combinatorial methods to obtain the probabilistic mul-tivariate generating function that counts the occurrences of patterns in a text generated by a Markovian source. The generating function can then be expanded into the Taylor series in which the power of a term gives the size of a text and the coeÿcient provides the proba-bilities of all possible pattern occurrences with the text size. The analysis is on the basis of the inclusion-exclusion principle to pattern counting (Goulden and Jackson, 1979 and 1983) and its application that Bassino et al. (2012) used for obtaining the generating function in the context of the Bernoulli text source. We followed the notations and concepts created by Bassino et al. in the discussion of distinguished patterns and non-reduced pattern sets, with modifications to the Markovian dependence. Our result is derived in the form of a linear matrix equation in which the number of linear equations depends on the size of the alphabet. In addition, we compute the moments of pattern occurrences and discuss the impact of a Markovian text to the moments comparing to the Bernoulli case. The methodology that we use involves the inclusion-exclusion principle, stochastic recurrences, and combinatorics on words including probabilistic multivariate generating functions and moment generating functions.<br>

<p> In second order perturbation theory for nuclear matter, an exact treatment of the Pauli exclusion principle is given from a geometrical point of view. All the kinematic effects of the Pauli exclusion principle are then included in a function K(k,k',q), which is related to the Euler's function through a double integration. With this function K(k,k',q), we can treat the Pauli correction in nuclear matter in a more exact way so that a check to the conventional angular average approximation is obtained. For separable core nuclear potential, this function K(k,k',q) serves as a very convenient apparatus for the perturbation calculation of the binding energy in nuclear matter.</p> / Thesis / Master of Science (MSc)

Arsenic is found in drinking waters in many countries and since maximum allowable concentration is as low as 10 &micro / g/L, there are many research efforts to separate it from water. Membrane methods are used more and more widely in separation operations in recent years. Arsenic is mainly present in water as arsenite [As(III)] and arsenate [As(V)]. As pH of water changes, molecular formulas of As(III) and As(V) change. In this study, the performance of different ultrafiltration membranes for arsenic removal from water was investigated at different pH values, different feed concentrations and presence of other anions (SO42-, HPO42-, NO3-, Cl-). Donnan exclusion effect on separation was discussed since distribution of arsenite and arsenate anions change in water due to change in pH of the solution. Experiments were conducted via batch and continuous modes. For continuous ultrafiltration experiments, 30 kDa of polysulfone and 20 kDa of polyether sulfone membranes were used. Batch ultrafiltration experiments were performed with the usage of 3 kDa of regenerated cellulose membrane. Higher retention values for As(V) were obtained compared to retention values of As(III). When membranes&rsquo / performances were investigated, it was seen that highest As(V) removal was observed with the usage of polysulfone membrane. Increase in feed concentration and presence of other anions caused decrement in separation. Hydride Generation Atomic Absorption Spectrometry was used to perform analyses. Hydride generator part was designed, constructed and optimized to obtain reliable and accurate absorbance values.

Thesis (Ph.D.)--University of Melbourne, School of Physics, 2009. / Typescript. Includes bibliographical references (leaves 171-174)

In this paper, we will review two fundamental counting methods: inclusionexclusion and pigeonhole principles. The inclusion-exclusion principle considers the elements of the sets satisfied some conditions, and avoids repeat counting by disjoint sets. We also use the inclusion-exclusion principle to solve the problems of Euler phi function and the number of onto functions in number theory, and derangement and the number of nonnegative integer solutions of equations in combinatorics. We derive the closed-form formula to those problems. For the forbidden positions problems, we use the rook polynomials to simplify the counting process. We also show the form of the inclusion-exclusion principle in probability, and use it to solve some probability problems. The pigeonhole principle is an easy concept. We can establish some sets and use the pigeonhole principle to discuss the extreme value about the number of elements. Choose the pigeons and pigeonholes, properly, and solve problems by the concept of the pigeonhole principle. We also introduce the Ramsey theorem which is an important application of the pigeonhole principle. This theorem provides a method to solve problems by complete graph. Finally, we give some contest problems about the inclusion-exclusion and pigeonhole principles to show how those principles are used.

In this dissertation I analyze Sir Arthur Eddington's statistical theory as developed in the first six chapters of his posthumously published Fundamental Theory. In particular I look at the mathematical structure, philosophical implications, and relevancy to modern physics. This analysis is the only one of Fundamental Theory that compares it to modern quantum field theory and is the most comprehensive look at his statistical theory in four decades. Several major insights have been made in this analysis including the fact that he was able to derive Pauli's Exclusion Principle in part from Heisenberg's Uncertainty Principle. In addition the most profound general conclusion of this research is that Fundamental Theory is, in fact, an early quantum field theory, something that has never before been suggested. Contrary to the majority of historical reports and some comments by his contemporaries, this analysis shows that Eddington's later work is neither mystical nor was it that far from mainstream when it was published. My research reveals numerous profoundly deep ideas that were ahead of their time when Fundamental Theory was developed, but that have significant applicability at present. As such this analysis presents several important questions to be considered by modern philosophers of science, physicists, mathematicians, and historians. In addition it sheds new light on Eddington as a scientist and mathematician, in part indicating that his marginalization has been largely unwarranted.

Submitted by ANA KARLA PEREIRA RODRIGUES (anakarla_@hotmail.com) on 2017-09-05T16:47:02Z No. of bitstreams: 1 arquivototal.pdf: 1153647 bytes, checksum: a384e4d5e2acf05cec52ece972237c23 (MD5) / Approved for entry into archive by Viviane Lima da Cunha (viviane@biblioteca.ufpb.br) on 2017-09-06T10:49:22Z (GMT) No. of bitstreams: 1 arquivototal.pdf: 1153647 bytes, checksum: a384e4d5e2acf05cec52ece972237c23 (MD5) / Made available in DSpace on 2017-09-06T10:49:22Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 1153647 bytes, checksum: a384e4d5e2acf05cec52ece972237c23 (MD5) Previous issue date: 2016-08-17 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / The process of teaching and learning of mathematics is closely related to the resolution of theoretical and practical problems, which often involve situations of everyday life in our society. This work aims to present the Inclusion and Exclusion Principle as a tool for solving many problems involving counting elements, especially those that appear double, triple counting, among others. It also seeks to relate it with the determination of prime numbers of a number and the Sieve of Eratosthenes, use it to systematize the Formula of the function Fi ( Phi) Euler, as well as for determining the number of permutations Chaotic and number of Sobrejetoras functions. / O processo de ensino aprendizagem da Matemática está intimamente relacionado com a resolução de problemas teóricos e práticos, os quais geralmente envolvem situações do cotidiano de nossa sociedade. Esse trabalho tem como objetivo apresentar o Princípio da Inclusão e Exclusão como ferramenta para resolução de vá- rios modelos de problemas que envolvem a contagem de elementos, principalmente aquelas que aparecem contagem duplas, triplas, dentre outras. Além disso, busca relacioná-lo com a determinação de números primos de um número e com o Crivo de Eratóstenes, utilizá-lo para sistematizar a Fórmula da Função Fi ( ) de Euler, bem como para a determinação do Número de Permutações Caóticas e do Número de Funções Sobrejetoras.

The thesis includes delatiled explanation of combinatorial principles used in school mathematics. The single principles are explained in details and practicised. The tasks at the end of the chapter serve readers for testing acquired knoledge.

<p align="justify">Les transformations de supersymétrie (ou de Darboux) sont appliquées à l'étude du problème inverse, c'est à dire à la construction d'un potentiel d'interaction à partir de données de collisions, en mécanique quantique. En effet, ces transformations permettent de construire de nouveaux potentiels à partir d'un potentiel donné. Leur formalisme est étudié en détail, ainsi que celui correspondant à l'itération de deux telles transformations (paires de transformations).</p><p><p align="justify">La présence d'états liés rend le problème inverse ambigu :plusieurs potentiels ayant des spectres liés différents peuvent avoir les mêmes propriétés pour la description des collisions; de tels potentiels sont dits équivalents en phase. Une décomposition originale du problème inverse est proposée pour gérer efficacement cette ambiguïté :dans un premier temps, un potentiel est construit à partir des données de collision (ce qui constitue le problème inverse proprement dit); dans un second temps, tous les potentiels équivalents en phase au potentiel ainsi obtenu sont construits. Avant ce travail, il était connu que ces deux aspects du problème inverse pouvaient être traités à l'aide de paires de transformations de supersymétrie.</p><p><p align="justify">En ce qui concerne la construction de potentiels équivalents, nous étendons les méthodes existantes à des catégories de potentiels très utilisées en physique nucléaire, à savoir les potentiels optiques (ou complexes), les potentiels en voies couplées et les potentiels dépendant linéairement de l'énergie. En utilisant une paire de transformations permettant d'enlever un état lié, nous comparons les propriétés physiques des potentiels nucléaires profonds (c'est à dire possédant des états liés interdits par le principe de Pauli) et peu profonds. Des calculs dans des modèles à trois corps du noyau à halo d'6He et de la collision 16O+17O à basse énergie n'ont pas révélé d'importantes différences entre ces familles de potentiels. D'autres types de transformations permettent d'ajouter des états liés à énergie et normalisation arbitraires. Cependant, dans le cas à plusieurs voies, leur utilisation est compliquée par la possibilité d'avoir des états liés dégénérés et non dégénérés. Une étude préliminaire à deux voies montre que ces deux types d'états peuvent être traités par supersymétrie.</p><p><p align="justify">En ce qui concerne le problème inverse proprement dit, nous montrons que l'utilisation de transformations simples (plutôt que de paires) permet une meilleure compréhension des méthodes existantes, tant pour l'inversion à moment cinétique orbital fixe que pour l'inversion à énergie fixe. De plus, l'utilisation de transformations simples mène dans certains cas à de nouvelles catégories de potentiels. Ainsi, nous construisons un nouveau potentiel d'interaction nucléon nucléon pour l'onde 1S; ce potentiel possède une singularité en r 2 à l'origine. La possibilité de construire des potentiels profonds par inversion est brièvement discutée. Pour les voies couplées, une étude bibliographique révèle certaines propriétés contradictoires des méthodes existantes, mais une analyse complète reste à faire.</p><p> / Doctorat en sciences appliquées / info:eu-repo/semantics/nonPublished