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[pt] Subjacente à teoria dos números ordinais transfinitos de Cantor, há uma perspectiva finitista. Segundo tal perspectiva, Deus pode bem ordenar o infinito usando, para tanto, de procedimentos similares ao ato de contar, entendido como o ato de bem ordenar o finito. Desta maneira, um diálogo natural entre Cantor e Dedekind torna-se possível, dado que Dedekind foi o primeiro a tratar o ato de contar como sendo, em sua essência, uma forma de bem ordenar o mundo espáciotemporal pelos números naturais. Nesta tese, o conceito de número ordinal transfinito, de Cantor, é entendido como uma extensão do conceito dedekindiano de número natural. / [en] Underlying Cantor s transfinite ordinal numbers theory, there is a finistic perspective. Accordingly that perspective, God can well order the infinite using, for that, similar procedures to the act of counting, understood as the act of well order the finite. That s why a natural dialog between Cantor and Dedekind becomes possible, since Dedekind was the first to consider the act of counting as being, in its essence, a way of well order the spatial-temporal world by natural numbers. In this thesis, the concept of Cantor´s transfinite ordinal number is understood as an extension of dedekindian concept of natural number.

In the dissertation we study the C(K) spaces focusing on the case when K is countable compact and more specifically, the structure of C() spaces for < ω1 via special type of trees that they contain. The dissertation is composed of three major sections. In the first section we give a detailed proof of the theorem of Bessaga and Pelczynski on the isomorphic classification of C() spaces. In due time, we describe the standard bases for C(ω) and prove that the bases are monotone. In the second section we consider the lattice-trees introduced by Bourgain, Rosenthal and Schechtman in C() spaces, and define rerooting and restriction of trees. The last section is devoted to the main results. We give some lower estimates of the ordinal-indices in C(ω). We prove that if the tree in C(ω) has large order with small constant then each function in the root must have infinitely many big coordinates. Along the way we deduce some upper estimates for c0 and C(ω), and give a simple proof of Cambern's result that the Banach-Mazur distance between c0 and c = C(ω) is equal to 3.

Master of Science / Department of Statistics / Nora Bello / Ordered categorical responses are frequently encountered in many disciplines. Examples of interest in agriculture include quality assessments, such as for soil or food products, and evaluation of lesion severity, such as teat ends status in dairy cattle. Ordered categorical responses are characterized by multiple categories or levels recorded on a ranked scale that, while apprising relative order, are not informative of magnitude of or proportionality between levels. A number of statistically sound models for ordered categorical responses have been proposed, such as logistic regression and probit models, but these are commonly underutilized in practice. Instead, the ordinary least squares linear regression model is often employed with ordered categorical responses despite violation of basic model assumptions. In this study, the inferential implications of this approach are investigated using a simulation study that evaluates robustness based on realized Type I error rate and statistical power. The design of the simulation study is motivated by applied research cases reported in the literature. A variety of plausible scenarios were considered for simulation, including various shapes of the frequency distribution and different number of categories of the ordered categorical response. Using a real dataset on frequency of antimicrobial use in feedlots, I demonstrate the inferential performance of ordinary least squares linear regression on ordered categorical responses relative to a probit model.

Phylogenetics is a field primarily concerned with the reconstruction of the evolutionary history of present day species. Evolutionary history is often modeled by a phylogenetic tree, similar to a family tree. To recreate a phylogenetic tree from information about current species, one needs to make assumptions about the evolutionary process. These assumptions can range from full parametrised models of evolution to simple observations. This thesis looks at the reconstruction of phylogenetic trees under two different assumptions. The first, known as the ordinal assumption, has been previously studied and asserts that as species evolve, they become more dissimilar. The second, the convex assumption, has not previously been studied in this context and asserts that changes species go through to become dissimilar are progressively larger than the current differences between those species. This thesis presents an overview of mathematical results in tree reconstruction from dissimilarity maps (also known as distance matrices) and develops techniques for reasoning about the ordinal and convex assumptions. In particular, three main results are presented: a complete classification of phylogenetic trees with four leaves under the ordinal assumption; a partial classification of phylogenetic trees with four leaves under the convex assumption; and, an independent proof of a result on the relationship between ultrametrics and the ordinal assumption.

Gene expression studies are of growing importance in the field of medicine. In fact, subtypes within the same disease have been shown to have differing gene expression profiles (Golub et al., 1999). Often, researchers are interested in differentiating a disease by a categorical classification indicative of disease progression. For example, it may be of interest to identify genes that are associated with progression and to accurately predict the state of progression using gene expression data. One challenge when modeling microarray gene expression data is that there are more genes (variables) than there are observations. In addition, the genes usually demonstrate a complex variance-covariance structure. Therefore, modeling a categorical variable reflecting disease progression using gene expression data presents the need for methods capable of handling an ordinal outcome in the presence of a high dimensional covariate space. In this research we present a method that combines the stereotype regression model (Anderson, 1984) with an elastic net penalty (Friedman et al., 2010) as a method capable of modeling an ordinal outcome for high-throughput genomic datasets. Results from applying the proposed method to both simulated and gene expression data will be reported and the effectiveness of the proposed method compared to a univariable and heuristic approach will be discussed.

Cette thèse traite des langages reconnaissables de mots indexés par des ordinaux. Plusieurs classes d'automates qui reconnaissent de tels mots ont été introduites par Büchi. Elles diffèrent par la longueur des mots reconnus par les automates. Nous en utilisons quatre: la classe pour les mots de longueur , celle pour les mots de longueur inférieure à , où n est un entier naturel, celle pour les mots de longueur dénombrable, et celle pour les mots de longueur quelconque. Nous y ajoutons la classe des automates de Kleene traditionnelle, sur les mots finis. Nous remontrons que ces différentes définitions d'automates sont équivalentes, c'est-à-dire que données deux de ces classes et un automate d'une des deux, la restriction du langage reconnu par l'automate aux mots du domaine le plus petit des deux classes est la restriction du langage reconnu par un automate de l'autre classe au même domaine. Nous donnons également une présentation unifiée de la déterminisation pour chacune des classes qui reconnaît au plus des mots de longueur dénombrable. Les semigroupes finis sont un formalisme équivalent aux automates pour définir des ensembles de mots finis. Perrin, Pin et Wilke ont introduits des structures algébriques adaptées à l'étude des langages de mots de longueur , qui, quand elles sont finies, sont équivalentes aux automates. Nous généralisons l'approche algébrique de la théorie des langages reconnaissables de mots de longueur aux mots de longueur inférieure à , puis aux mots de longueur dénombrable. Pour cela, nous définissons deux structures algébriques, les -semigroupes et les -semigroupes, qui, quand elles sont finies, sont équivalentes respectivement aux automates pour les mots de longueur inférieure à et aux automates pour les mots de longueur dénombrable. Comme pour le cas des mots de longueur , une algèbre syntaxique peut être canoniquement associée à chaque langage reconnaissable. Nous définissons le produit de Schützenberger et le produit en couronne sur les -semigroupes. Nous étendons également le théorème des variétés d'Eilenberg aux mots de longueur dénombrable. Finalement, nous remontrons l'équivalence entre langages reconnus par automates et langages définis par énoncés de logique monadique du second ordre quand on s'intéresse aux mots de longueur dénombrable. Le théorème d'équivalence de Schützenberger entre langages sans étoile et semigroupes finis apériodiques est étendu aux mots de longueur inférieure à , et le théorème d'équivalence entre langages sans étoile et langages définis par énoncés de logique du premier ordre de l'ordre linéaire de McNaughton et Papert est étendu aux mots de longueur quelconque.

<p>Does the country people live in affect the probability of them experiencing happiness? Can a country variable in an ordinal regression model be affected when microeconomic and macroeconomic factors are added to the model? The possible outcomes are either that the country variable affects less when the additional predictors are added to the model, or that they stay the same. The micro data is collected from the European Social Survey database, the macro data is collected from the World Bank. The country variable becomes less substantial when additional variables are added to the model. The variable with the most influence over expected happiness apart from the country variable is whether the individual often socializes with friends or not. It’s statistically significant that the supervened variables make the country variable less volatile in some cases.</p>

Does the country people live in affect the probability of them experiencing happiness? Can a country variable in an ordinal regression model be affected when microeconomic and macroeconomic factors are added to the model? The possible outcomes are either that the country variable affects less when the additional predictors are added to the model, or that they stay the same. The micro data is collected from the European Social Survey database, the macro data is collected from the World Bank. The country variable becomes less substantial when additional variables are added to the model. The variable with the most influence over expected happiness apart from the country variable is whether the individual often socializes with friends or not. It’s statistically significant that the supervened variables make the country variable less volatile in some cases.

Self-rated health (SRH) is an outcome commonly studied by demographers, epidemiologists, and sociologists of health, typically measured using an ordinal scale. SRH is analyzed in cross-sectional and longitudinal studies for both descriptive and inferential purposes, and has been shown to have significant validity with regard to predicting mortality. Despite the wide spread use of this measure, only limited attention is explicitly given to its unique attributes in the case of longitudinal studies. While self-rated health is assumed to represent a latent continuous and dynamic process, SRH is actually measured discretely and asymmetrically. Thus, the validity of methods ignoring the scale of measurement remains questionable. We compare three approaches to modeling SRH with repeated measures over time: linear multilevel models (MLM or LGM), including corrections for non-normality; and marginal and conditional ordered-logit models for longitudinal data. The models are compared using simulated data and illustrated with results from the Health and Retirement Study. We find that marginal and conditional models result in very different interpretations, but that conditional linear and non-linear models result in similar substantive conclusions, albeit with some loss of power in the linear case. In conclusion, we suggest guidelines for modeling self-rated health and similar ordinal outcomes in longitudinal studies. / text