Dans cette thèse, nous nous intéressons à l'ensemble $Int\left(\mathcal O _K \right)$ des polynômes à valeurs entières sur l'anneau $\mathcal{O}_K$ des entiers d'un corps de nombres $K$. Selon Pólya, une base $\left(f_{n}\right)_{n\in \mathbb{N}}$ du $\mathcal O _K$-module $Int\left(\mathcal O _K \right)$ est dite régulière si pour tout $n \in \mathbb{N}$, $\deg(f_{n})=n$. Un corps $K$ tel que $ Int \left(\mathcal{O}_K \right)$ possède une base régulière est dit de Pólya et le groupe de Pólya d'un corps de nombres $K$ est un sous-groupe du groupe de classes de $K$ qui peut être considéré comme une mesure de l'écart pour un corps au fait d'être de Pólya. Nous étudions le groupe de Pólya d'un compositum $L= K_1 K_2$ de corps de nombres galoisiens et établissons des liens avec la ramification des nombres premiers dans chacune des extensions $K_1 /\mathbb{Q}$ et $K_2 /\mathbb{Q}$. Nous appliquons ces résultats aux corps de nombres de petit degré afin d'élargir la famille des corps de Pólya quadratiques déjà caractérisés. Par ailleurs, une condition pour qu'un corps de nombres $K$ soit de Pólya est que tous les produits d'idéaux de $K$ de même norme soient principaux. Par analogie avec le problème classique du plongement, on peut se poser la question suivante : tout corps de nombres $K$ peut-il être plongé dans un corps de Pólya? Nous donnons une réponse positive à cette question : pour tout corps $K$, le corps de classes de Hilbert $H_K$ de $K$ est un corps de Pólya . Toujours par analogie avec le problème de plongement où l'on sait que les idéaux de $\mathcal{O}_K$ deviennent principaux dans $\mathcal{O}_{H_K}$, on peut définir la notion d'extension de Pólya d'un corps $K$ : il s'agit de corps $L$ contenant $K$ dans lesquels le groupe de Pólya de $K$ devient trivial par extensions des idéaux, ce sont aussi des corps $L$ tels que le $\mathcal O _L$-module engendré par $Int\left(\mathcal O _K \right)$ possède une base régulière. Outre $H_K$ dans le cas général, dans le cas où $K$ est une extension abélienne, la capitulation des idéaux ambiges de $K$ montre que le corps de genre de $K$ en est une extension de Pólya. Ceci nous amène à des questions de minimalité et d'unicité concernant les corps et extensions de Pólya.

Det finns ett stort behov i Sverige att satsa på matematik i skolan, det visar både PISA och TIMSS-undersökningar. Nationella proven visar även att många elever presterar sämre på uppgifter som bearbetar problemlösningsförmågan. Enligt flera forskare främjar problemlösning elevers matematiska kunskaper och enligt ämnesplanerna ska elever även kunna använda denna förmåga i sitt vardagliga liv efter skoltiden. Syftet med studien är att tillhanda djupare förståelse inom matematisk problemlösning genom studiens frågeställning: Vilka utmaningar finns det i problemlösning för gymnasielever utifrån Polyas lösningsstrategier? För att undersöka detta har en systematisk litteraturstudie tillämpats. Urvalet hämtades från artiklar som fokuserar på problemlösning för high school elever med fokus på Polyas problemlösningsstrategier. En tematisk analys applicerades vid bearbetningen av empirin och resultatet kunde kategoriseras i olika teman. Resultatet visade att de främsta utmaningarna för eleverna var att kunna förstå problemet, att kunna omvandla ett problem i vardagen till matematik, att kunna bearbeta problemlösningsuppgifter på ett korrekt sätt och slutligen att kunna reflektera över svaret. Elever tenderar att arbeta med procedurförmågan i större utsträckning än problemlösning vilket hämmar elevers problemlösningsförmåga då inte tränas i den. Elever kan gynnas av att lära sig om hur man löser problemlösning vilket kan bidra till att eleverna strukturerar sina lösningar och tänkande. Denna studie är relevant för lärare inom matematik eftersom det finns brister i elevers problemlösningsförmåga enligt internationella och nationella undersökningar. Det är ett relevant bidrag till matematikdidaktiken och kan användas i lektionsplanering.

Neste trabalho apresentamos uma demonstração detalhada para um conhecido teorema de I. J. Schoenberg que caracteriza certas funções positivas definidas em esferas. Analisamos ainda um critério para a obtenção de positividade definida de uma função a partir de condições de suavidade e convexidade dela, em uma tentativa de ratificar alguns resultados da literatura conhecidos como critérios de Pólya. / In this work we present a proof for a famous theorem of Schoenberg on positive definite functions on spheres. We analyze some results that deduce positive definiteness from diferentiability and convexity assumption on the function, an attempt to ratify some Pólya type conjectures found in the literature.

Neste trabalho apresentamos uma demonstração detalhada para um conhecido teorema de I. J. Schoenberg que caracteriza certas funções positivas definidas em esferas. Analisamos ainda um critério para a obtenção de positividade definida de uma função a partir de condições de suavidade e convexidade dela, em uma tentativa de ratificar alguns resultados da literatura conhecidos como critérios de Pólya. / In this work we present a proof for a famous theorem of Schoenberg on positive definite functions on spheres. We analyze some results that deduce positive definiteness from diferentiability and convexity assumption on the function, an attempt to ratify some Pólya type conjectures found in the literature.

Nous nous intéressons au comportement asymptotique de plusieurs urnes de type Polya fortement renforcées et interagissantes. Le principal de notre étude porte sur les renforcements exponentiels ou assimilés ainsi que sur les interactions temporelles, c'est-à-dire lors desquelles les urnes n'interagissent qu'à certains instants aléatoires. Dans ce cas, nous mettons en évidence une transition de phase selon la fréquence des interactions. Si celle ci est supérieure à 1/2, les urnes se fixent toutes sur la même couleur tandis que si elle est inférieure à 1/2, une couleur majoritaire se dégage mais certaines urnes peuvent continuer à tirer une autre couleur aux instants où il n'y a pas interaction. Lorsque le renforcement devient infini, nous pouvons calculer la loi du nombre d'urnes se comportant de cette dernière façon quand le nombre total d'urnes est égal à deux ou est un nombre impair.Quand l'interaction est totale, c'est-à-dire quand toutes les urnes interagissent à tout instant, nous montrons alors qu'un renforcement fort et croissant, mais plus nécessairement exponentiel, suffit à obtenir la fixation de toutes les urnes sur la même couleur.Pour finir, nous discutons brièvement du modèle d'interaction spatiale dans lequel les urnes sont situées sur les sommets d'un graphe et n'interagissent qu'avec leurs voisines. Nous dégageons alors quelques propriétés préliminaires concernant les sous-graphes susceptibles de se fixer sur une couleur avec une probabilité positive. / We study the asymptotic behavior of several Polya-type strongly reinforced interacting urns. The main results deal with exponential or exponential-like reinforcements and temporal interactions, that is when the urns interact only at some random times. In that case, we show the existence of a transition of phase depending on the frequency of interactions. If this frequency is larger than 1/2, all the urns eventually fixate on the same color, while if it is smaller than 1/2, a majority color will be fixed after some finite random time but while some of the urns eventually draw only the majority color, there can be other urns that still draw other colors at times where there is no interactions. When the reinforcement becomes infinite, we can calculate the law of the number of urns of later type when the total number of urns is two or an odd integer greater than two.When the interaction is maximal, that is when all the urns interact at any time, we show that a strong and non-decreasing reinforcement, but not necessarily exponential, suffices to obtain the fixation of all the urns on the same color.At the end, we consider briefly the spatial interaction model in which the urns are located on the vertices of a graph and interact only with their neighbors. In that case, we discuss some properties of sub-graphs that can fixate on one color with positive probability.

Risk managers currently seek new advances in statistical methodology to better forecast and quantify uncertainty. This thesis comprises a collection of new Bayesian models and computational methods which collectively aim to better estimate parameters and predict observables when data arise from stochastic failure processes. Such data commonly arise in reliability theory and survival analysis to predict failure times of mechanical devices, compare medical treatments, and to ultimately make well-informed risk management decisions. The collection of models proposed in this thesis advances the quality of those forecasts by providing computational modeling methodology to aid quantitative based decision makers. Through these models, a reliability expert will have the ability: to model how future decisions affect the process; to impose his prior beliefs on hazard rate shapes; to efficiently estimate parameters with MCMC methods; to incorporate exogenous information in the form of covariate data using Cox proportional hazard models; to utilize nonparametric priors for enhanced model flexibility. Managers are often forced to make decisions that affect the underlying distribution of a stochastic process. They regularly make these choices while lacking a mathematical model for how the process may itself depend significantly on their decisions. The first model proposed in this thesis provides a method to capture this decision dependency; this is used to make an optimal decision policy in the future, utilizing the interactions of the sequences of decisions. The model and method in this thesis is the first to directly estimate decision dependency in a stochastic process with the flexibility and power of the Bayesian formulation. The model parameters are estimated using an efficient Markov chain Monte Carlo technique, leading to predictive probability densities for the stochastic process. Using the posterior distributions of the random parameters in the model, a stochastic optimization program is solved to determine the sequence of decisions that minimise a cost-based objective function over a finite time horizon. The method is tested with artificial data and then used to model maintenance and failure time data from a condenser system at the South Texas Project Nuclear Operating Company (STPNOC). The second and third models proposed in this thesis offer a new way for survival analysts and reliability engineers to utilize their prior beliefs regarding the shape of hazard rate functions. Two generalizations of Weibull models have become popular recently, the exponentiated Weibull and the modified Weibull densities. The popularity of these models is largely due to the flexible hazard rate functions they can induce, such as bathtub, increasing, decreasing, and unimodal shaped hazard rates. These models are more complex than the standard Weibull, and without a Bayesian approach, one faces difficulties using traditional frequentist techniques to estimate the parameters. This thesis develops stylized families of prior distributions that should allow engineers to model their beliefs based on the context. Both models are first tested on artificial data and then compared when modeling a low pressure switch for a containment door at the STPNOC in Bay City, TX. Additionally, survival analysis is performed with these models using a famous collection of censored data about leukemia treatments. Two additional models are developed using the exponentiated and modified Weibull hazard functions as a baseline distribution to implement Cox proportional hazards models, allowing survival analysts to incorporate additional covariate information. Two nonparametric methods for estimating survival functions are compared using both simulated and real data from cancer treatment research. The quantile pyramid process is compared to Polya tree priors and is shown to have a distinct advantage due to the need for choosing a distribution upon which to center a Polya tree. The Polya tree and the quantile pyramid appear to have effectively the same accuracy when the Polya tree has a very well-informed choice of centering distribution. That is rarely the case, however, and one must conclude that the quantile pyramid process is at least as effective as Polya tree priors for modeling unknown situations. / text

Syftet med detta arbete var att undersöka vad elever reflekterar över vid problemlösning. För att finna svar på detta genomfördes observationer, där elever i årskurs 8, fick lösa ett matematiskt problem. Eleverna arbetade enligt Polyas problemlösningsmetod. Denna metod är indelad i fyra faser där den sista fasen är att se tillbaka och reflektera över det lösta problemet. Resultatet visar på en ovana bland eleverna att reflektera efter löst problem. När eleverna fick anvisningar vad de skulle reflektera över, ökade både motivationen och förmågan. Studien pekar på att motivation är en faktor av betydelse för förmågan att nå framgång vid reflektionen.

The Poisson distribution is commonly used in analyzing count data, and many insurance companies are interested in studying the related risk models and ruin probability theory. Over the past century, many different bivariate models have been developed in the literature. The bivariate Poisson distribution was first introduced by Campbell (1934) for modelling bivariate accident data. However, in some situations, a given dataset may possess over-dispersion compared to Poisson distribution which moti- vated researchers to develop alternative models to handle such situations. In this regard, Minkova and Balakrishnan (2014a) developed the Type I bivariate Polya- Aeppli distribution by using compounding with Geometric random variables and the trivariate reduction method. Inference for this Type I bivariate Polya-Aeppli distribution is the topic of this thesis. The parameters in a model are used to describe and summarize a given sample within a specific distribution. So, their estimation becomes important and the goal of estimation theory is to seek a method to find estimators for the parameters of interest that have some good properties. There exist many methods of finding estimators such as Method of Moments, Bayesian estimators, Least Squares, and Maximum Likelihood Estimators (MLEs). Each method of estimation has its own strength and weakness (Casella and Berger (2008)). Minkova and Balakrishnan (2014a) discussed the moment estimation of the parameters of the Type I bivariate Polya-Aeppli dis- tribution. In this thesis, we develop the likelihood inference for this model. A simulation study is carried out with various parameter settings. The obtained results show that the MLEs require more computational time compared to Moment estimation. However, Method of Moments (MoM) did not result in good estimates for all the simulation settings. In terms of mean squared error and bias, we observed that MLEs performed, in most of the settings, better than MoM. Finally, we apply the Type I bivariate Polya-Aeppli model to a real dataset containing the frequencies of railway accidents in two subsequent six year periods. We also carry out some hypothesis tests using the Wald test statistic. From these results, we conclude that the two variables belong to the same univariate Polya-Aeppli distribution but are correlated. / Thesis / Master of Science (MSc)

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Time-dependent queueing models are important since most of real-life problems are time-dependent. We develop a numerical approximation algorithm for the mean, variance and higher-order moments of the number of entities in the system at time t for the Ph(t)/Ph(t)/s/c queueing model. This model can be thought as a reparameterization to the G(t)/GI(t)/s. Our approach is to partition the state space into known and identifiable structures, such as the M(t)/M(t)/s/c or M(t)/M(t)/1 queueing models. We then use the Polya-Eggenberger distribution to approximate certain unknown probabilities via a two-moment matching algorithm. We describe the necessary steps to validate the approximation and measure the accuracy of the model. / Master of Science