This paper investigates four important methods of solving summation and product problems in mathematics competitions. Chapter 1 presents the basic concepts of sequence and series, including arithmetic sequence (series), geometric sequence (series) and infinite geometric sequence (series). Chapter 2 handles the binomial coefficients and binomial theorem and show they how can be applied to compute series sum. Chapter 3 deals with power series, including interchanging summation and differentiation; interchanging summation and integration; and generating function which expresses a sequence as coefficients arising from a power series in variables. Chapter 4 provides four methods of telescoping sum, including antidifference, partial fractions, trigonometric functions, and factorial functions. Chapter 5 discusses the telescoping product which the main ideas and techniques are analogous to telescoping sum. Two types of telescoping product including difference of two squares and trigonometric functions are investigated.

How to quantify security is a classic question in the security community that until today has had no plausible answer. Unfortunately, current security evaluation models are often either quantitative but too specific (i.e., applicability is limited), or comprehensive (i.e., system-level) but qualitative. The importance of quantifying security cannot be overstated, but doing so is difficult and complex, for many reason: the “physics” of the amount of security is ambiguous; the operational state is defined by two confronting parties; protecting and breaking systems is a cross-disciplinary mechanism; security is achieved by comparable security strength and breakable by the weakest link; and the human factor is unavoidable, among others. Thus, security engineers face great challenges in defending the principles of information security and privacy. This thesis addresses model-based system-level security quantification and argues that properly addressing the quantification problem of security first requires a paradigm shift in security modeling, addressing the problem at the abstraction level of what defines a computing system and failure model, before any system-level analysis can be established. Consequently, we present a candidate computing systems abstraction and failure model, then propose two failure-centric model-based quantification approaches, each including a bounding system model, performance measures, and evaluation techniques. The first approach addresses the problem considering the set of controls. To bound and build the logical network of a security system, we extend our original work on the Information Security Maturity Model (ISMM) with Reliability Block Diagrams (RBDs), state vectors, and structure functions from reliability engineering. We then present two different groups of evaluation methods. The first mainly addresses binary systems, by extending minimal path sets, minimal cut sets, and reliability analysis based on both random events and random variables. The second group addresses multi-state security systems with multiple performance measures, by extending Multi-state Systems (MSSs) representation and the Universal Generating Function (UGF) method. The second approach addresses the quantification problem when the two sets of a computing system, i.e., assets and controls, are considered. We adopt a graph-theoretic approach using Bayesian Networks (BNs) to build an asset-control graph as the candidate bounding system model, then demonstrate its application in a novel risk assessment method with various diagnosis and prediction inferences. This work, however, is multidisciplinary, involving foundations from many fields, including security engineering; maturity models; dependability theory, particularly reliability engineering; graph theory, particularly BNs; and probability and stochastic models.

Ce manuscrit de thèse porte sur l’analyse mathématique de modèles intégro-différentiels issus de la génétique des populations. Les deux modèles étudiés sont des équations de réaction-dispersion de type ∂tp(t,m) = UD[p](t,m) + f[p](t,m). Ils décrivent la dynamique de la distribution de la fitness (ou valeur sélective) dans une population asexuée sous l’effet des mutations et de la sélection représentées respectivement par les termes non locaux UD[p](t,m) et par f[p](t,m). La différence entre les deux modèles se situe au niveau du terme de mutation. En effet, dans le premier modèle, les effets des mutations sur la fitness ne dépendent pas de la fitness du parent, cela se traduit donc par un terme de convolution classique : D[p](t,m) =RR J(m−y)p(t,y)dy−p(t,m). Lorsqu’une mutation a lieu, la fonction J(m−y) représente la densité de probabilité pour un individu de fitness y d’avoir un descendant de fitness m. Le taux de mutation est donné par la constante U. Dans le second modèle, les effets des mutations sur la fitness dépendent aussi de la fitness du parent. Dans ce cas, un individu de fitness y a un descendant de fitness m avec la densité de probabilité Jy(m−y). Ce type de dépendance apparaît naturellement lorsque l’on suppose qu’il existe une fitness optimale (ou encore un optimum phénotypique). Pour chacun des deux modèles, nous établissons dans un premier temps des résultats d’existence et d’unicité ainsi que des propriétés de décroissance de la solution. Cette décroissance permet de définir la fonction génératrice des cumulants (CGF) associée à la distribution de fitness. La CGF est la solution d’une équation de transport non locale. Pour le premier modèle, l’étude de cette équation permet d’obtenir une solution analytique et donc d’obtenir une description complète de la distribution p(t,m) via ses moments. Nous étudions ensuite les états stationnaires pour chacun des deux modèles, et établissons des conditions suffisantes pour l’existence et la non-existence de phénomènes de concentration, correspondant à une accumulation d’individus de phénotypes optimaux. Nos résultats sont comparés à des sorties de modèles stochastiques individu-centrés représentant le même type de dynamiques évolutives. / This manuscript is devoted to the mathematical analysis of integro-differential models from population genetics. Both models are reaction-dispersion equations of the form ∂tp(t,m) = UD[p](t,m)+ f[p](t,m). They describe the dynamics of fitness distribution in an asexual population under the effect of mutation and selection. These two processes are represented by the nonlocal terms UD[p](t,m) and by f[p](t,m) respectively. The difference between the models rests on the mutation term. Indeed, in the first model, the mutation effects on fitness do not depend on the fitness of the parent. Thus, the mutation term is a standard convolution product: D[p](t,m) =RR J(m−y)p(t,y)dy −p(t,m). When a mutation occurs, the function J(m − y) represents the density of probability for an individual with fitness y to have an offspring with fitness m. The mutation rate is given by the constant U. In the second model, the mutation effects on fitness depend on the fitness of the parent. In this case, an individual with fitness y has an offspring with fitness m with a probability density Jy(m−y). This type of dependence naturally arises when the existence of an optimal fitness (or a phenotypic optimum) is assumed. For both models, we first establish existence and uniqueness results as well as decay properties of the solution. The decay property allows us to define the cumulant generating function (CGF). The CGF obeys a nonlocal transport equation. In the first model, we compute the analytical solution of this transport equation and thus, we obtain a complete description of the distribution p(t,m) through its moments. Then, we study the stationary states for both models, and establish sufficient conditions for the existence and non-existence of a concentration phenomenon corresponding to an accumulation of individuals with best possible phenotype. The results are compared to the results of stochastic individual based models which represent the same kind of evolutionary dynamics.

Ce mémoire étudie deux familles de fonctions orthogonales, soit les fonctions d'orbite de Weyl et les fonctions d'orbite de Hartley. Chacune de ces familles est associée à une algèbre de Lie simple et cette recherche se limite aux algèbres A₂, C₂ et G₂ de rang 2. Les fonctions d'orbite de Weyl ont été largement étudiées depuis des années en raison de leurs propriétés exceptionnelles. Nouvellement, elles ont été utilisées pour générer des polynômes de Chebyshev généralisés et calculer les fonctions génératrices de ces polynômes pour les algèbres de Lie simples de rang 2. Les fonctions d'orbite de Hartley, quant à elles, ont été récemment introduites par Hrivnák et Juránek et l'étude de ces dernières ne fait que débuter. L'objectif de ce mémoire est de définir des polynômes de Chebyshev généralisés associés aux fonctions de Hartley et de calculer les fonctions génératrices de ceux-ci pour les algèbres A₂, C₂ et G₂. Le premier chapitre introduit les systèmes de racines et le groupe de Weyl, original et affine, ainsi que leurs domaines fondamentaux, afin que le lecteur ait les notations et définitions pour comprendre les chapitres suivants. Le deuxième chapitre présente et étudie les fonctions de Weyl. Il définit également leurs polynômes de Chebyshev généralisés et se termine en présentant les différentes fonctions génératrices de ces polynômes pour les algèbres de Lie simples de rang 2. Finalement, le troisième chapitre contient les résultats originaux; il expose les fonctions de Hartley et certaines de leurs propriétés. Il définit les polynômes de Chebyshev généralisés de celles-ci et énonce également leurs relations d'orthogonalité discrète. Il conclut en calculant les fonctions génératrices de ces polynômes pour les algèbres A₂, C₂ et G₂. / This master's thesis studies two families of orthogonal functions, the Weyl orbit functions and the Hartley orbit functions. Each of these families is associated to a simple Lie algebra and the present work is limited to the algebras A₂, C₂ and G₂ of rank 2. Weyl orbit functions have been widely studied for years because of their exceptional properties. Recently, these properties have been used to generate generalized Chebyshev polynomials and to compute the generating functions of these polynomials for the simple Lie algebras of rank 2. Hartley orbit functions, on the other hand, were recently introduced by Hrivnák and Juránek and the study of the latter has only begun. The objective of this thesis is to define the generalized Chebyshev polynomials of Hartley orbit functions and to compute their generating functions for the algebras A₂, C₂ and G₂. The first chapter introduces root systems and the Weyl group, original and affine, and their fundamental domains, so that the reader has the notations and definitions at hand to read the following chapters. The second chapter introduces and studies Weyl orbit functions. It also defines their generalized Chebyshev polynomials and ends by presenting the different generating functions of these polynomials for simple Lie algebras of rank 2. Finally, the third chapter contains the original contribution; it presents the Hartley functions and some of their properties. It defines the generalized Chebyshev polynomials of these and also states their discrete orthogonality relations. It concludes by computing the generating functions of these polynomials for the algebras A₂, C₂ and G₂.

Sequences often occur in many branches of applied mathematics. Recurrence relation is a powerful tool to characterize and study sequences. Some commonly used methods for solving recurrence relations will be investigated. Many examples with applications in algorithm, combination, algebra, analysis, probability, etc, will be discussed. Finally, some well-known contest problems related to recurrence relations will be addressed.

Multivariate mixed Poisson processes are special multivariate counting processes whose coordinates are, in general, dependent. The first part of this thesis is devoted to properties which multivariate counting processes may possess. Such properties are, for example, the Markov property, the multinomial property and regularity. With regard to regularity we study the properties of transition probabilities and intensities. The second part of this thesis restricts the class of all multivariate counting processes by additional assumptions leading to different types of multivariate mixed Poisson processes which, however, are connected with each other. Using a multivariate version of the Bernstein-Widder theorem, it is shown that multivariate mixed Poisson processes are characterized by the multinomial property. Furthermore, regularity of multivariate mixed Poisson processes and properties of their moments are studied in detail. Throughout this thesis, two types of stability of properties of multivariate counting processes are studied: It is shown that most properties of a multivariate counting process are stable under certain linear transformations including the selection of single coordinates and summation of all coordinates. It is also shown that the different types of multivariate mixed Poisson processes under consideration are in a certain sense stable in time.

Le mouvement Brownien réfléchi de manière oblique dans le quadrant, introduit par Harrison, Reiman, Varadhan et Williams dans les années 80, est un objet largement analysé dans la littérature probabiliste. Cette thèse, qui présente l’étude complète de la mesure invariante de ce processus dans tous les cônes du plan, a pour objectif plus global d’étendre au cadre continu une méthode analytique développée initialement pour les marches aléatoires dans le quart de plan par Fayolle, Iasnogorodski et Malyshev dans les années 70. Cette approche est basée sur des équations fonctionnelles, reliant des fonctions génératrices dans le cas discret et des transformées de Laplace dans le cas continu. Ces équations permettent de déterminer et de résoudre des problèmes frontière satisfaits par ces fonctions génératrices. Dans le cas récurrent, cela permet de calculer explicitement la mesure invariante du processus avec rebonds orthogonaux, dans le chapitre 2, et avec rebonds quelconques, dans le chapitre 3. Les transformées de Laplace des mesures invariantes sont prolongées analytiquement sur une surface de Riemann induite par le noyau de l’équation fonctionnelle. L’étude des singularités et l’application de méthodes du point col sur cette surface permettent de déterminer l’asymptotique complète de la mesure invariante selon toutes les directions dans le chapitre 4. / Obliquely reflected Brownian motion in the quadrant, introduced by Harrison, Reiman, Varadhan and Williams in the eighties, has been studied a lot in the probabilistic literature. This thesis, which presents the complete study of the invariant measure of this process in all the cones of the plan, has for overall aim to extend to the continuous framework an analytic method initially developped for random walks in the quarter plane by Fayolle, Iasnogorodski and Malyshev in the seventies. This approach is based on functional equations which link generating functions in the discrete case and Laplace transform in the continuous case. These equations allow to determine and to solve boundary value problems satisfied by these generating functions. In the recurrent case, it permits to compute explicitly the invariant measure of the process with orthogonal reflexions, in the chapter 2, and with any reflexions, in the chapter 3. The Laplace transform of the invariant measure is analytically extended to a Riemann surface induced by the kernel of the functional equation. The study of singularities and the use of saddle point methods on this surface allows to determine the full asymptotics of the invariant measure along every directions in the chapter 4.

Multivariate mixed Poisson processes are special multivariate counting processes whose coordinates are, in general, dependent. The first part of this thesis is devoted to properties which multivariate counting processes may possess. Such properties are, for example, the Markov property, the multinomial property and regularity. With regard to regularity we study the properties of transition probabilities and intensities. The second part of this thesis restricts the class of all multivariate counting processes by additional assumptions leading to different types of multivariate mixed Poisson processes which, however, are connected with each other. Using a multivariate version of the Bernstein-Widder theorem, it is shown that multivariate mixed Poisson processes are characterized by the multinomial property. Furthermore, regularity of multivariate mixed Poisson processes and properties of their moments are studied in detail. Throughout this thesis, two types of stability of properties of multivariate counting processes are studied: It is shown that most properties of a multivariate counting process are stable under certain linear transformations including the selection of single coordinates and summation of all coordinates. It is also shown that the different types of multivariate mixed Poisson processes under consideration are in a certain sense stable in time.

<p>A survival distribution is developed for exponential two-component systems that can survive as long as at least one of the two components in the system function. It is assumed that the two components are initially independent and non-identical. If one of the two components fail (repair is impossible), the surviving component is subject to a different failure rate due to the stress caused by the failure of the other.</p> <p>In this paper, we consider such an exponential two-component system failure model when the observed failure time data are (1) complete, (2) Type-I censored, (3) Type-I censored with partial information on component failures, (4) Type-II censored and (5) Type-II censored with partial information on component failures. In these situations, we discuss the maximum likelihood estimates (MLEs) of the parameters by assuming the lifetimes to be exponentially distributed. The exact distributions (whenever possible) of the MLEs of the parameters are then derived by using the conditional moment generating function approach. Construction of confidence intervals for the model parameters are discussed by using the exact conditional distributions (when available), asymptotic distributions, and two parametric bootstrap methods. The performance of these four confidence intervals, in terms of coverage probabilities are then assessed through Monte Carlo simulation studies. Finally, some examples are presented to illustrate all the methods of inference developed here.</p> <p>In the case of Type-I and Type-II censored data, since there are no closed-form expressions for the MLEs, we present an iterative maximum likelihood estimation procedure for the determination of the MLEs of all the model parameters. We also carry out a Monte Carlo simulation study to examine the bias and variance of the MLEs.</p> <p>In the case of Type-II censored data, since the exact distributions of the MLEs depend on the data, we discuss the exact conditional confidence intervals and asymptotic confidence intervals for the unknown parameters by conditioning on the data observed.</p> / Thesis / Doctor of Philosophy (PhD)

In this thesis, we investigate several inferential issues regarding the lifetime data from exponential distribution under different censoring schemes. For reasons of time constraint and cost reduction, censored sampling is commonly employed in practice, especially in reliability engineering. Among various censoring schemes, progressive Type-I censoring provides not only the practical advantage of known termination time but also greater flexibility to the experimenter in the design stage by allowing for the removal of test units at non-terminal time points. Hence, we first consider the inference for a progressively Type-I censored life-testing experiment with k uniformly spaced intervals. For small to moderate sample sizes, a practical modification is proposed to the censoring scheme in order to guarantee a feasible life-test under progressive Type-I censoring. Under this setup, we obtain the maximum likelihood estimator (MLE) of the unknown mean parameter and derive the exact sampling distribution of the MLE through the use of conditional moment generating function under the condition that the existence of the MLE is ensured. Using the exact distribution of the MLE as well as its asymptotic distribution and the parametric bootstrap method, we discuss the construction of confidence intervals for the mean parameter and their performance is then assessed through Monte Carlo simulations. Next, we consider a special class of accelerated life tests, known as step-stress tests in reliability testing. In a step-stress test, the stress levels increase discretely at pre-fixed time points and this allows the experimenter to obtain information on the parameters of the lifetime distributions more quickly than under normal operating conditions. Here, we consider a k-step-stress accelerated life testing experiment with an equal step duration τ. In particular, the case of progressively Type-I censored data with a single stress variable is investigated. For small to moderate sample sizes, we introduce another practical modification to the model for a feasible k-step-stress test under progressive censoring, and the optimal τ is searched using the modified model. Next, we seek the optimal τ under the condition that the step-stress test proceeds to the k-th stress level, and the efficiency of this conditional inference is compared to the preceding models. In all cases, censoring is allowed at each change stress point iτ, i = 1, 2, ... , k, and the problem of selecting the optimal Tis discussed using C-optimality, D-optimality, and A-optimality criteria. Moreover, when a test unit fails, there are often more than one fatal cause for the failure, such as mechanical or electrical. Thus, we also consider the simple stepstress models under Type-I and Type-II censoring situations when the lifetime distributions corresponding to the different risk factors are independently exponentially distributed. Under this setup, we derive the MLEs of the unknown mean parameters of the different causes under the assumption of a cumulative exposure model. The exact distributions of the MLEs of the parameters are then derived through the use of conditional moment generating functions. Using these exact distributions as well as the asymptotic distributions and the parametric bootstrap method, we discuss the construction of confidence intervals for the parameters and then assess their performance through Monte Carlo simulations. / Thesis / Doctor of Philosophy (PhD)