Orientador: Plamen Emilov Kochloukov / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-17T03:42:15Z (GMT). No. of bitstreams: 1 Silva_DiogoDinizPereiradaSilvae_D.pdf: 1168055 bytes, checksum: 49c676076235e3eef6f8a27594f092f7 (MD5) Previous issue date: 2010 / Resumo: Neste trabalho apresentamos um estudo sobre identidades polinomiais graduadas em álgebras não associativas. Mais precisamente estudamos as identidades polinomiais graduadas da álgebra de Lie das matrizes de ordem 2 com traço zero com as três graduações naturais, a Z2-graduação, a Z2 _ Z2-graduação e a Z-graduação, neste caso conseguimos uma nova demonstração baseada em métodos elementares dos resultados de [27] que não se baseia em resultados da Teoria de Invariantes, estes resultados foram publicados em [30]. Estudamos também as identidades graduadas da álgebra de Jordan das matrizes simétricas de ordem 2, neste caso obtivemos bases para as identidades graduadas dessa álgebra de Jordan em todas as possíveis graduações, obtivemos também bases para as identidades fracas para os pares (Bn; Jn) e (B; J), onde Bn e B denotam as álgebras de Jordan de uma forma bilinear simétrica não degenerada nos espaços vetoriais Vn e V respectivamente, onde Vn tem dimensão n e V tem dimensão 1, esses resultados estão no artigo [29], aceito para publicação / Abstract: In this thesis we study graded identities in non associative algebras. Namely we study graded polynomial identities for the Lie algebra of the 2_2 matrices with trace zero with it's three natural gradings, the Z2-grading, the Z2_Z2-grading and the Z-grading, in this case we obtained a new proof of the results of [27] that doesn't involve use of Invariant Theory, this results were published in [30]. We also studied the graded identities of the Jordan algebra of the symmetric matrices of order two, we obtained basis for the graded identities of this Jordan algebra in all possible gradings, we also obtained basis for the weak identities of the pairs (Bn; Jn) and (B; J), where Bn and B are the Jordan algebras of a symmetric bilinear form in a the vector spaces Vn and V respectively, where Vn has dimension n and V has countable dimension, this results are in the article [29], accepted for publication / Doutorado / Álgebra Não-Comutativa / Doutor em Matemática

Made available in DSpace on 2014-10-09T12:49:06Z (GMT). No. of bitstreams: 0 / Made available in DSpace on 2014-10-09T14:06:38Z (GMT). No. of bitstreams: 1 09670.pdf: 10232628 bytes, checksum: 1590f43108c1cbc7b2065202eea72fa8 (MD5) / Dissertacao (Mestrado) / IPEN/D / Instituto de Pesquisas Energeticas e Nucleares - IPEN/CNEN-SP

As relações de dependência entre variáveis aleatórias é um dos assuntos mais discutidos em probabilidade e estatística e a forma mais abrangente de estudar essas relações é por meio da distribuição conjunta. Nos últimos anos vem crescendo a utilização de cópulas para representar a estrutura de dependência entre variáveis aleatórias em uma distribuição multivariada. Contudo, ainda existe pouca literatura sobre cópulas quando as distribuições marginais são discretas. No presente trabalho será apresentada uma proposta não-paramétrica de estimação da distribuição conjunta bivariada de variáveis aleatórias discretas utilizando cópulas e polinômios de Bernstein. / The relations of dependence between random variables is one of the most discussed topics in probability and statistics and the best way to study these relationships is through the joint distribution. In the last years has increased the use of copulas to represent the dependence structure among random variables in a multivariate distribution. However, there is still little literature on copulas when the marginal distributions are discrete. In this work we present a non-parametric approach for the estimation of the bivariate joint distribution of discrete random variables using copulas and Bernstein polynomials.

O Brasil é um dos maiores produtores de leite do mundo, porém é necessário que se produza não só em quantidade, mas com qualidade adequada ao consumo e ao beneficiamento. Com a entrada em vigor da Instrução Normativa 51 (2002), a qualidade do leite nacional passou a ser monitorada, sendo exigido um padrão mínimo. Dentre os aspectos analisados, estão os teores de proteína e gordura e a contagem de células somáticas. Diante disso, o objetivo desse trabalho foi de estimar componentes de variância, coeficientes de herdabilidade e comparar modelos de diferentes ordens de ajuste por meio de funções polinomiais de Legendre, sob modelos de regressão aleatória, com a finalidade de predizer o modelo mais adequado para descrever as mudanças nas variâncias associadas aos teores de proteína, gordura e à contagem de células somáticas de vacas holandesas de primeira lactação. Foi utilizado um banco de dados com 27.988 dados de teores de gordura e proteína e 27.883 de escore de células somáticas, referentes a 4.945 vacas e a matriz de parentesco continha 30.843 animais. Foram utilizados quatro modelos, com polinômios ortogonais de Legendre de ordens de 3 a 6 e variância residual homogênea. Os modelos que melhor se ajustaram para gordura foram o de 5ª e 6ª ordens, para proteína, o de 4ª ordem e para escore de células somáticas foram os de 4ª e 6ª ordens. As estimativas de herdabilidade variaram de 0,07 a 0,56 para teor de gordura; de 0,13 a 0,66 para teor de proteína e de 0,08 a 0,50 para escore de células somáticas, nos diferentes modelos estudados. De acordo com os resultados, modelos de regressão aleatória são adequados para descrever variações no teor de gordura e proteína e no escore de células somáticas em função do estágio de lactação em que a vaca se encontra. / Brazil is one of the largest milk producers in the world, but it is necessary to produce not only in quantity but in quality suitable for consumption and processing. With the entry into force of the Federal Normative Instruction 51 (IN-51), the national quality of milk started to be monitored, with a required minimum standard. Among the aspects studied are the protein and fat contents and somatic cell count. Thus, the aim of this study was to estimate variance components, heritability coefficients and compare models with different orders of adjustment of Legendre polynomials, by random regression models in order to predict the most appropriate model to describe variances associated with changes in levels of protein, fat and somatic cell count of first lactation Holstein cows. We used a database with 27,988 data from fat and protein content and a database with 27,883 of somatic cell score, relative to 4,945 cows and the relationship matrix contained 30,843 animals. We used four models with orthogonal Legendre polynomials of orders 3-6 and homogeneous residual variance. The models that best adjusted for fat were of the 5th and 6th orders, for protein was of the 4th order and somatic cell score were of the 4th and 6th order. The heritabilities estimated ranged from 0.07 to 0.56 for fat, 0.13 to 0.66 for protein and 0.08 to 0.50 for somatic cell score in the different models studied. According to the results, random regression models are suitable to describe variations in fat and protein contents and somatic cell score according to the stage of lactation.

Orientador: Walter Alexandre Carnielli / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Filosofia e Ciências Humanas / Made available in DSpace on 2018-08-23T18:22:31Z (GMT). No. of bitstreams: 1 Matulovic_Mariana_D.pdf: 1191409 bytes, checksum: 5228f60f9fdb9f3a9df31d448de09ca2 (MD5) Previous issue date: 2013 / Resumo: O presente trabalho tem por objetivo explorar, em diversas vertentes, o caráter universal de uma ferramenta poderosa de prova, apta a ser utilizada em lógicas clássicas e não clássicas, em particular em lógicas multivaloradas proposicionais (determinísticas e não-determinísticas), em lógicas paraconsistentes, em lógicas modais e na Lógica de Primeira Ordem. Trata-se do Método de Prova de Anéis de Polinômios, que também pode, em princípio, ser visto do ponto de vista da semântica algébrica, desenvolvido inicialmente em (Carnielli 2005b). O método traduz fórmulas de uma lógica específica em polinômios (em geral finitos, mas podendo ser infinitos) com coeficientes em corpos finitos, e transforma o problema de se encontrar demonstrações no correlato algébrico da busca de soluções de sistemas de equações polinomiais. Esta universalidade do método possibilita a abertura de diversas linhas de pesquisa, sendo a questão da verofuncionalidade e suas generalizações uma delas. Outras linhas de pesquisa são: possibilidades de se investigar enfoques alternativos da complexidade computacional, prova automática de teoremas, métodos heurísticos em lógica e correlações entre álgebra e lógica. Este trabalho analisa e compara sistemas de anéis de polinômios para sistemas com verofuncionalidade generalizada, como no caso das semânticas não-determinísticas, e ainda em sistemas onde a verofuncionalidade é perdida, tais como em sistemas multivalorados reduzidos a bivalorados através da conhecida redução de Suszko. O método de anéis de polinômios, além de poderoso e elegante em sua aparente simplicidade, constitui ainda um ótimo instrumento pedagógico. Em relação á lógica clássica, definimos um anel de polinômios para a Lógica de Primeira Ordem, fundamentado em um novo domínio que opera com somas e produtos infinitos, o qual se denomina domínio de séries generalizadas fechado por produtos. Finalmente, procuramos avaliar todas as potencialidades do método, principalmente no aspecto inerente á questão de se poder pensar em uma característica unificadora na medida que utiliza o mesmo viés matemático para traduzir diferentes sistemas lógicos em variedades algébricas similares. Além disso, analisamos as interrelações do método com respeito a lógica algébrica (ou álgebra da lógica), e avaliamos suas perspectivas / Abstract: This investigation aims to explore, in various aspects, the universal character of a powerful proof method, able to be used in classical and non-classical logics, in particular in propositional many-valued logics (deterministic and non- deterministic) in paraconsistent logics, in modal logics and in First Order Logic. This is the Method of Polynomial Rings, which can also be considered as an algebraic semantics, initially developed in (Carnielli 2005b). The method translates logical formulas into specific polynomials (usually finite, but sometimes infinite) with coefficients infinite fields, and transforms the problem of finding proofs in the search for solutions of systems of polynomial equations. This universality of the method enables the opening of several research lines, in particular the issue of truth-functionality and its generalizations. Other lines of research are: the possibilities of investigating alternative approaches of computational complexity, automatic theorem proving, heuristic methods in logic and correlations between algebra and logic. This study compares and analyzes the polynomial ring systems for systems with generalized truth-functionality, as in the case of non- deterministic semantic and even in systems where truth-functionality is lost, such as those many-valued systems reduced to bivalued by means of the so-called Suszko reduction. The method of polynomial rings, besides being a powerful and elegant apparatus in its apparent simplicity, is still a great teaching tool. Regarding classical logic, we define the polynomial ring for First Order Logic , based on a new domain that operates on sums and infinite products, called domain of generalized series closed under products. Finally, we evaluate the full potential of the method, especially in what concerns the question of obtaining a unifying feature that uses the same mathematical basis to translate different logical systems on similar algebraic varieties. Furthermore, we address the connections of the method with respect to algebraic logic (algebra of logic), and evaluate their perspectives / Doutorado / Filosofia / Doutora em Filosofia

The representer theorem from the reproducing kernel Hilbert space theory is the origin of many kernel-based machine learning and signal modelling techniques that are popular today. Most kernel functions used in practical applications behave in a homogeneous manner across the domain of the signal of interest, and they are called stationary kernels. One open problem in the literature is the specification of a non-stationary kernel that is computationally tractable. Some recent works solve large-scale optimization problems to obtain such kernels, and they often suffer from non-identifiability issues in their optimization problem formulation. Many practical problems can benefit from using application-specific prior knowledge on the signal of interest. For example, if one can adequately encode the prior assumption that edge contours are smooth, one does not need to learn a finite-dimensional dictionary from a database of sampled image patches that each contains a circular object in order to up-convert images that contain circular edges. In the first portion of this thesis, we present a novel method for constructing non-stationary kernels that incorporates prior knowledge. A theorem is presented that ensures the result of this construction yields a symmetric and positive-definite kernel function. This construction does not require one to solve any non-identifiable optimization problems. It does require one to manually design some portions of the kernel while deferring the specification of the remaining portions to when an observation of the signal is available. In this sense, the resultant kernel is adaptive to the data observed. We give two examples of this construction technique via the grayscale image up-conversion task where we chose to incorporate the prior assumption that edge contours are smooth. Both examples use a novel local analysis algorithm that summarizes the p-most dominant directions for a given grayscale image patch. The non-stationary properties of these two types of kernels are empirically demonstrated on the Kodak image database that is popular within the image processing research community. Tensors and tensor decomposition methods are gaining popularity in the signal processing and machine learning literature, and most of the recently proposed tensor decomposition methods are based on the tensor power and alternating least-squares algorithms, which were both originally devised over a decade ago. The algebraic approach for the canonical polyadic (CP) symmetric tensor decomposition problem is an exception. This approach exploits the bijective relationship between symmetric tensors and homogeneous polynomials. The solution of a CP symmetric tensor decomposition problem is a set of p rank-one tensors, where p is fixed. In this thesis, we refer to such a set of tensors as a rank-one decomposition with cardinality p. Existing works show that the CP symmetric tensor decomposition problem is non-unique in the general case, so there is no bijective mapping between a rank-one decomposition and a symmetric tensor. However, a proposition in this thesis shows that a particular space of rank-one decompositions, SE, is isomorphic to a space of moment matrices that are called quasi-Hankel matrices in the literature. Optimization over Riemannian manifolds is an area of optimization literature that is also gaining popularity within the signal processing and machine learning community. Under some settings, one can formulate optimization problems over differentiable manifolds where each point is an equivalence class. Such manifolds are called quotient manifolds. This type of formulation can reduce or eliminate some of the sources of non-identifiability issues for certain optimization problems. An example is the learning of a basis for a subspace by formulating the solution space as a type of quotient manifold called the Grassmann manifold, while the conventional formulation is to optimize over a space of full column rank matrices. The second portion of this thesis is about the development of a general-purpose numerical optimization framework over SE. A general-purpose numerical optimizer can solve different approximations or regularized versions of the CP decomposition problem, and they can be applied to tensor-related applications that do not use a tensor decomposition formulation. The proposed optimizer uses many concepts from the Riemannian optimization literature. We present a novel formulation of SE as an embedded differentiable submanifold of the space of real-valued matrices with full column rank, and as a quotient manifold. Riemannian manifold structures and tangent space projectors are derived as well. The CP symmetric tensor decomposition problem is used to empirically demonstrate that the proposed scheme is indeed a numerical optimization framework over SE. Future investigations will concentrate on extending the proposed optimization framework to handle decompositions that correspond to non-symmetric tensors.

Le polynôme caractéristique d'une variété abélienne sur un corps fini est défini comme étant celui de son endomorphisme de Frobenius. La première partie de cette thèse est consacrée à l'étude des polynômes caractéristiques de variétés abéliennes de petite dimension. Nous décrivons l'ensemble des polynômes intervenant en dimension 3 et 4, le problème analogue pour les courbes elliptiques et surfaces abéliennes ayant été résolu par Deuring, Waterhouse et Rück.Dans la deuxième partie, nous établissons des bornes supérieures et inférieures sur le nombre de points rationnels des variétés abéliennes sur les corps finis. Nous donnons ensuite des bornes inférieures spécifiques aux variétés jacobiennes. Nous déterminons aussi des formules exactes pour les nombres maximum et minimum de points rationnels sur les surfaces jacobiennes. / The characteristic polynomial of an abelian variety over a finite field is defined to be the characteristic polynomial of its Frobenius endomorphism. The first part of this thesis is devoted to the study of the characteristic polynomials of abelian varieties of small dimension. We describe the set of polynomials which occur in dimension 3 and 4; the analogous problem for elliptic curves and abelian surfaces has been solved by Deuring, Waterhouse and Rück.In the second part, we give upper and lower bounds on the number of points on abelian varieties over finite fields. Next, we give lower bounds specific to Jacobian varieties. We also determine exact formulas for the maximum and minimum number of points on Jacobian surfaces.

The purpose of this project was to see why projective geometry is related to the sort of sensors that machines and humans use for vision.

Dans certains phénomènes physiques modélisés par des EDP, les coefficients intervenant dans les équations ne sont pas des fonctions déterministes fixées, et dépendent de paramètres qui peuvent varier. Ceci se produit par exemple dans le cadre de la modélisation des écoulements en milieu poreux lorsqu’on décrit le champ de perméabilité par un processus stochastique pour tenir compte de l’incertitude sur ce champs. Dans d’autres cadres, il peut s’agir de paramètres déterministes que l’on cherche à ajuster, par exemple pour optimiser un certain critère sur la solution. La solution u dépend donc non seulement de la variable x d’espace/temps mais aussi d’un vecteur y = (yj) de paramètres potentiellement nombreux, voire en nombre infinis. L’approximation numérique en y de l’application (x,y)-> u(x, y) est donc impossible par les méthodes classiques de type éléments finis, et il faut envisager des approches adaptées aux grandes dimensions. Cette thèse est consacrée à l’étude théorique et l’approximation numérique des EDP paramétriques en grandes dimensions. Pour une large classe d’EDP avec une certaine dépendance anisotrope en les paramètres yj, on étudie de la régularité en y de l’application u et on propose des méthodes d’approximation numérique dont les performances ne subissent pas les détériorations classiquement observées en grande dimension. On cherche en particulier à évaluer la complexité de la classe des solutions {u(y)}, par exemple au sens des épaisseurs de Kolmogorov, afin de comprendre les limites inhérentes des méthodes numériques. On analyse en pratique les propriétés de convergences de diverses méthodes d’approximation avec des polynômes creux. / For certain physical phenomenon that are modelled by PDE, the coefficients intervening in the equations are not fixed deterministic functions, but depend on parameters that may vary.

Nous proposons des méthodes simples et efficaces pour manipuler des expressions trigonométriques de la forme $F=\sum_{k} f_k\cos\tfrac{k\pi}{n}, f_k\in Z$ où $d<n$ fixé. Nous utilisons les polynômes unitaires de Chebyshev qui forment une base de $Z[x]$ avec laquelle toutes les opérations arithmétiques peuvent être exécutées aussi rapidement qu'avec la base de monômes, mais également déterminer le signe et une approximation de $F$, calculer le polynôme minimal de $F$. Dans ce cadre nous calculons efficacement le polynôme minimal de $2\cos\frac{\pi}{n}$ et aussi le polynôme cyclotomique $\Phi_n$. Nous appliquons ces méthodes au calcul des diagrammes de nœuds de Chebyshev $C(a,b,c,\varphi) : x=T_a(t), y=T_b(t), z=T_c(t+\varphi)$, ce qui permet de tester si une courbe donnée est un nœud, et aussi lister tous les nœuds de Chebyshev possibles quand un triple $(a,b,c)$ fixé en bonne complexité. / We propose a set of simple and fast algorithms for evaluating and using trigonometric expressions in the form $F=\sum_{k}f_k\cos\frac{k\pi}{n}$, $f_k\in Z$ where $d<n$ fixed. We make use of the monic Chebyshev polynomials as a basis of $Z[x]$. We can perform arithmetic operations (multiplication, division, gcd) on polynomials expressed in a Chebyshev basis (with the same bit-complexity as in the monomial basis), compute the sign of $F$, evaluate it numerically and compute its minimal polynomial in $Q[x]$. We propose simple and efficient algorithms for computing the minimal polynomial of $2\cos\frac{\pi}{n}$ and also the cyclotomic polynomial $\Phi_n$. As an application, we give a method to determine the Chebyshev knot's diagrams $C(a,b,c,\varphi) : x=T_a(t),y=T_b(t), z=T_c(t+\varphi)$ which allows to test if a given curve is a Chebyshev knot, and point out all the possible Chebyshev knots coressponding a fixed triple $(a,b,c)$, all of these computings can be done with a good bit complexity.