In the literature, there are well-established models to analyze measurable variables in the open interval (0.1) that represent rates, proportions or index; Among them are the models associated with the Beta and Simplex distributions. In practice, there is a need to have multivariate models, in particular the bivariate case. In this sense, the main objective of this work is to propose the Multivariate Simplex regression model (MRSM) via the copula function. Estimators for the parameters are found via the maximum likelihood (MV) method and, via a simulation study, their asymptotic behavior is studied. A diagnostic analysis such as: residual analysis and global influence (generalized Cook’s distance and likelihood departure) are developed, with the aim of identifying possible atypical and/or influential points and suitability of the model to the data. Finally, the results are applied to a set of real data to exemplify the developed methodology.

In the mid 1960’s, the german mathematician Hans J. Zassenhaus, inspired by
Gram Higman’s thesis and the work of Ian Hughes and Kenneth R. Pearson, stated several
conjectures that changed radically the field of research in group rings. Since then many
papers have been published about these conjectures, where either an affirmative answer
is settled for some specific case, or some counterexamples are determined. Still, it is only
very recently that Florian Eisele and Leo Margolis, based on some joint work with Ángel
del Rı́o, have found a counterexample for the only conjecture which has been left open.
This fact motivated the writing of this dissertation.

There are several applications for metrics on Grassmannians, such as machine learning, wireless communication, and computer vision. However, computing the distances between subspaces of different dimensions presents challenges, especially due to the dimensional asymmetry of these subspaces. Therefore, it is necessary to use asymmetric metrics to deal with this situation. In this work, we extend the Fubini-Study metric as an asymmetric angle, which has useful properties and is easy to compute.

Every joint probability distribution brings with it information about the individual behavior of each variable (marginal distributions) and the dependency structure that guides the relationship between them. Statistical modeling through copula functions provides an individual analysis of these elements, enabling a detailed study of the association structures that guide a random vector. A copula of interest (and object of study of this dissertation) is the Power Variance Function (PVF) copula. It starts from the univariate three-parameter PVF distribution, derived as an extension of the positive stable distribution. The PVF copula class comprises a family of Archimedean copula that includes Clayton, Gumbel and Inverse Gaussian copulas as special cases. Through a methodological approach of literature review, it was possible to present in this work a general review of copulas, containing the main definitions, basic properties, Sklar's theorem, Fréchet-Hoeffding bounds, dependence/association measures and parametric families, as well as identifying the main known results regarding the PVF copula, both in the bivariate and multivariate cases. In addition to the usual dependency measures (Kendall's 𝜏 and Spearman's 𝜌), other measures like Blomqvist's 𝛽 and Gini's 𝛾 were investigated; in this sense, simulations of the PVF copula dependence measures with different values (i.e., different combinations of parameters) were performed. In addition, discussions were developed about the relationship of the BB9 copula with the PVF copula; specifically, we presented a demonstration that the PVF copula is a subclass of the Archimedean copula. Finally, three methods for generating data from the PVF copula and its special cases, as well as simulation and case studies, were provided.

Our goal in this work is to study the classic and celebrated Slice Theorem. Initially proven by Koszul, the Slice Theorem states that given a proper action of a Lie group on a manifold, there exists a slice passing through each point in M , i.e., a submanifold transversal to the orbit passing through the given point with some special properties. This theorem is a fundamental tool in the Theory of Transformation Groups. This result allows us to reduce the study of a Lie group action near an orbit to the study of the
geometry transversal to the orbit.

Several researchers have used the traditional Cox proportional hazards model, which has a simple interpretation and can be extended to incorporate time-dependent covariates; however, it is observed that in the practice, the data set does not always fit this model because they do not satisfy the usual properties of proportionality of failure rates and the effect of the covariate over time cannot be detected. In this work we have studied the modelling of survival data using the time-dependent generalized logistic model (GTDL). The use of this model is a significant alternative proposed by Mackenzie (1996) that satisfy the presupposition of non-proportionality of risks. In order to assess the goodness of fit of the model, we introduced Cox-Snell, modified Cox-Snell, martingale, deviance, randomized quantile, NMSP and NRSP residuals. We conducted a simulation study via Monte Carlo to investigate the asymptotic behaviour of the maximum likelihood estimators of the GTDL model obtained through the direct maximization of the log-likelihood function, for cases where the survival function is proper and when we have a model of cure fraction. Also, another simulation study is conducted to know the empirical distribution of residuals. Finally, we applied the methodologies studied to a set of data available in the literature involving patients with lung cancer.

Based on the monografies of Michael Vaughan–Lee [6] and of Charles R. Leedham-Green
and Susan McKay [4], this work focuses on the study of two constructions associating
Lie algebras to p-groups. These are used to solve the Restricted Burnside Problem and
the Coclass Conjectures, with the aim of translating a problem of group theory into the
theory of Lie algebras. To some extent, the two problems are of opposite nature, and
necessarily the associated Lie algebras are distinct. On the other hand, the method shows
great analogies, and in both cases the construction is based on the behaviour of powers
and commutators in a suitable descending central series.

This dissertation will focus on the correlation between the ideas of forcing axioms and
choice principles that makes the similarity of the latter much more evident. We show a
clearer and more objective proof of how the full axiom of choice can be seen as a "global"
axiom of forcing, an original result due to Stevo Todorcevic. At last, we explore ideas for
an original research that studies forcing axioms for certain classes of trees.

This dissertation aims at the construction of the Dawson-Watanabe process as a scaling limit of the branching Brownian motion, the latter being also related here to the solution of a heat equation with a source. The existence of the Dawson-Watanabe process (also called super Brownian motion or superprocess) is a consequence that this process is a solution to a martingal problem obtained through the scale limit of the branching Brownian motion. The proof follows the classic structure of tightness and uniqueness of limit points, and the uniqueness, in this followed path, results from the characterization of the Dawson-Watanabe process as a dual solution of a certain partial differential equation.

Let $\Gamma$ be a finite simple graph, with a set of vertices $V$. Consider the group $A_{\Gamma}=\langle V \,|\, R\rangle$, where $R$ is formed by the elements $aba^{-1}b^{-1}$ for each pair of vertices $a$ and $b$ adjacent in $\Gamma$. So $A_{\Gamma}$ is said group
partially commutative, or ``Right-angled Artin Group''. Partially commutative groups encompass a wide variety of groups, ranging from $F_n$ free groups to free abelian groups
$\mathbb{Z}^n$. We aim to prove that the group $Out(A_\Gamma)$ is residually finite, based mainly on two articles by Charney and Vogtmann who carried out the study of properties already
known and shared between $Out(F_n)$ and $Out(\mathbb{Z}^n)\simeq GL(n,\mathbb{Z})$, among them the property of being residually finite.

Regression models with binary response variables (1 - occurrence of the event of interest or "success'', 0 - non-occurrence of the event of interest or "failure'') have been intensively applied in several areas of knowledge, such as health, finance, industry, among others. Traditionally, the most used model in binary regression has been the logistic regression model. However, it uses the logit link function, which is a symmetric link function and may not be suitable in certain situations, for example, when one of the response variable classes is disproportionate to the other (imbalanced data set). The main aim of this work is to present new binary regression models using symmetric and asymmetric link functions. The parameter estimation of the proposed models (namely, the double Lindley, asymmetric double Lindley, power double Lindley, and reversal power double Lindley binary regression models) is performed with the classical maximum likelihood method. In order to compare and select the "best'' model among the different distributions, information criteria (AIC and BIC) and measures of predictive performance (AUC, balanced accuracy, sensitivity, specificity, positive and negative predictive values, F1-Score, Matthews correlation coefficient, among others) are used. Through the analysis of two real data sets, one on breast cancer, obtained from the University of California, Irvine's (UCI) Machine Learning Repository, and another on a competition promoted by Santander Bank for the Kaggle community, we show that models using the proposed link functions can provide a better fit and predictive ability than models using standard links, such as logit.

Causal inference in Statistics considers statistical approaches to establish a causal relationship between manipulable variables (treatment/intervention) and an outcome and it has been vastly developed methodologically in the last few decades. It is well-known that selection bias is likely to lead to incorrect inference in observational studies. Matching methods have been used as an alternative to the definition of comparison groups through the computation of propensity scores. Traditionally, the use of Propensity Score Matching (PSM) involves data analysis of observational studies under assumption of independence of the sample units.The underlying assumption, however, can be rather unreasonable when repeated measures of the outcome are collected in a longitudinal study. Recently, methods have being proposed to incorporate PSM into diverse dependency structures in the statistical model's framework (Li et al, 2013 e Arpino et al, 2016). When the treatment is initiated sequentially during a period of time, a class of longitudinal matching methods have also being proposed in the literature to emulate a random experiment (Li et al, 2001; Lu, 2005; Thomas et al, 2020). The goal of this dissertation is to systematically organize and summarize the statistical methodology involving propensity scores for the estimation of causal effects when analyzing data with longitudinal dependency structure. These methodologies are illustrated with the analysis of three datasets with different longitudinal dependency structure

Cancer is the name given to a set of more than 100 diseases that have in common the disordered growth of cells that invade tissues and organs, according to the INCA – Instituto Nacional de Câncer. In 2018, approximately 18 million new cancer cases were diagnosed worldwide. Among these, breast cancer is the one that most affects women worldwide. In Brazil, between 2018 and 2019, breast cancer incidence estimates are 59,700 new cases, with 29.5% of cancers in women, according to the INCA. One way to diagnose breast cancer is by capturing radiographic images(mammograms) for patients at high risk. Mammography is able to identify suspicious changes in cancer before the onset of symptoms and its analysis is done by radiologists, who check for the existence of breast cancer. One way to help these professionals to classify mammography images is to use some computational techniques. Diagnosis based on computational techniques can help the doctor make a more accurate decision.
A computational technique widely used to analyze images are the CNN - Convolutional Neural Networks, its use can improve the diagnosis of breast cancer by helping radiologists and doctors. This work has the general objective of demonstrating the benefits of implementing CNN’s for cancer prediction in radiographic data.
For this, a database provided by the oncology hospital ACCamargo, which specializes in the diagnosis, treatment and research of cancer, was used. A combination of models with transfer of learning was used to classify these images obtaining an ACC of 84.66%.

This work presents a proof of the C^1-structural stability for C^2-diffeomorphisms which are Axiom A and satisfy the strong transversality condition, defined on a smooth compact and boundaryless manifold.

Let $n\ge 3$. We studied the quotient group $B_n/[P_n, P_n]$ from the Artin braid group $B_n$ by the commutator subgroup of the Artin pure braid group $P_n$. The group $B_n/[P_n, P_n]$ is a crystallographic group that has no even-order element and has infinite elements of odd order. We also showed that there is a one-to-one correspondence between the conjugacy classes of  finite odd-order elements of $B_n/[P_n,P_n]$ with the  the conjugacy classes of  finite odd-order elements of the symmetric group $S_n$ and we realized the abelian subgroups of odd order of $S_n$ in $B_n/[P_n,P_n]$. In the case of $n=3$ we studied crystallographic subgroups of $B_3/[P_3, P_3]$ of dimension $3$. In this work we used as a main reference Gonçalves, Guaschi and Ocampo (2017). In addition to what was done in [Gonçalves, Guaschi and Ocampo 2017] , we studied the conjugacy classes of infinite order elements in $B_3/[P_3, P_3]$ and the Coxeter's quotient in $B_n/[P_n, P_n]$. 

This paper addresses the best known selection principles and uses the topological games associated with them to study results that involve the topological spaces that satisfy these principles. For example, we will show that if $ X $ is a Lindelöf space such that $ | X | <$ cov ($ \ mathcal {M} $), then player $ ONE $ has no winning strategy for $ G_1 (\ mathcal {O} _X, \ mathcal {O} _X) $ '', and this proves that `` Every Lindelöf space with a cardinality smaller than cov $ (\ mathcal {M}) $ is a Rothberger space ' '. In this dissertation we will also study $ D $ - spaces, selectively c.c.c. and star selection principles. We present proofs for relevant results of the literatura, such as ``Every  $T_1$ Menger space is a $ D-space '', which is demonstrated via games (using another result whose proof is presented in detail in the dissertation, which is " $ X $ is a Menger space if, and only if, $ ONE $ does not have a winning strategy for $ G \ textrm {fin}} (\ mathcal {O} _X, \ mathcal {O} _X $ '').  As $T_1$  Menger spaces  are D-spaces, one concludes that a counterexample to the conjecture `` All Lindelöf space and regular is a D-space? '', which remains unanswered since the 1970?s, can not be a Menger space.

In several cases, it is necessary to predict continuous time-dependent variables. As a particular case of this work, we have an interest in predicting the price of cryptocurrencies, which are increasingly popular, where it is seen a high growth and high price variability. In this sense, we aim to forecast the daily closing price of bitcoin, etherium and dash using the support vector machine theory, as well as an adaptation of this methodology to the case of time series forecasting, called recurrent SRV . The results obtained show a substantial gain in the daily forecast, surpassing traditional methods.

In this work we will discuss the notion of curvature in the particular context of the Kähler varieties. We will study Ricci's curvature and show that in the case of the Kähler varieties this can be described by a particularly simple formula in complex coordinates.
We will also study the holomorphic sectional curvature, which is an analog of the sectional curvature in the real case, and which, as in the real case, characterizes the curvature tensor of the Kähler variety considered.
In addition, we will comment on the case where the holomorphic sectional curvature is constant and the corresponding classification.

Recently there has been an increase in the use of methodology involving propensity score to minimize selection bias in observational studies which the aim is to observe causal relationships. The definition of propensity score is the probability one has to receive a treatment, given the covariates measured at the baseline. Also, a propensity score can be used to adjust treatment effect through pairing or weighing, using the inverse of this probability. Propensity score matching demands several implementation steps such as propensity score estimation, selecting a matching algorithm, and evaluating the intervention effects. A great portion of researches that use propensity score as an approach, assumes independence of observations. However, in studies of many fields of knowledge such as education, social science, and even health, the research design has a hierarchical structure with aggregated individuals. When it comes to multilevel structured data, the propensity score estimation must take into account the effect different levels have on an observation. For this multilevel context, the quality os the matching between the treated and control group and the estimate of the treatment were evaluated through a simulation study proceded from different models for the propensity score estimation. At last, the methodology used in this paper was applied to a real data set exploring the effect of the Bolsa Família program on nutritional status through BMI in a multilevel household survey conducted in Camaçari - Bahia, from October 2011 to January 2012.

In this dissertation, the problem of Bayesian networks (BNs) structure estimation from data is addressed. This structure represents, in a clear and graphical way, the joint probability distribution of the variables involved in a problem, through its conditional independence relations among the considered space. Due to its huge search space, structure estimation from data has been intensively studied and is considered the most challenging task related to working with BNs. In this context, this work started with a systematic literature review (SLR) with the goals of understanding how Bayesian network’s studies have been developed through the last two decades and in which fields and how this technique has been applied. In a second step, a new BNs structure estimation algorithm is proposed, based on a methodology applied to the context of genetic interaction. Finally, the generated structures and algorithm’s performance are compared to that of two commonly used algorithms in the literature, through simulation in four data sets, two artificial and two real, related to finance and real estate fields.

In this work we used Survival Analysis techniques in two sets of financial data in the presence of covariables and censored observations. The data represent the times until the occurrence of the event of interest (churn for the first set of data and default for the second set of data) in different credit products. Some models of non-default and non-default survival were proposed and assumed for marginal distributions in the presence of competitive causes with different activation mechanisms (Odd Weibull-Poisson models). Copulation functions (Clayton, Clayton Rotated 90 degrees, Frank, Gumbel, Farlie-Gumbel-Morgenstern, Plackett, Gaussian and Student t) were used in the study of dependence between failure times of the same individual. For inferential purposes, a classical approach was considered using the Inference Function for Margins (IFM), proposed by Joe and Xu (1996), whose results were compared to those obtained using the method of maximum likelihood, which makes the estimation of the parameters in a single stage

The goal of this work is to present an introduction to the study of isoparametric submanifolds of the hyperbolic space. To this end, give an introduction to the theory of isometric immersions between semi-Riemannian manifolds, and deduce the fundamental equations of first and second order. In addition, we study basic notions of the geometry of isoparametric submanifolds in Lorentz space like the curvature normals, parallel and focal manifolds, and Weyl group. Finally theorem about decomposition of isoparametric submanifolds of hyperbolic space by adapting the in point of view of [12] (where the 1-forms language is used) to the language of tangent and normal sections.

The main objective of this work is to present the Mobile Referential Method, which consists of choosing a reference that moves with the observer along a "trajectory". That is, for each point of a variety M, it is possible to define an appropriate orthonormal referential, that allows to define differential forms invariant wij in M, that is, forms that do not change with the action of a group of LIE G. These forms differentials will allow to decide when two submanifolds of M are locally isometric, that is, differ only locally, by a transformation of G.