Orientador: Plamen Emilov Koshlukov / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatisticas e Computação Cientifica / Made available in DSpace on 2018-08-07T19:30:45Z (GMT). No. of bitstreams: 1 Alves_SergioMota_D.pdf: 708263 bytes, checksum: 1d9ec0b24db06ea81853a6e7d6794b18 (MD5) Previous issue date: 2006 / Resumo: As álgebras verbalmente primas são bem conhecidas em característica 0, já sobre corpos de característica p > 2 pouco sabemos sobre elas. Nesse trabalho vamos discutir algumas diferenças entre estes dois casos de característica sobre corpos infinitos. Iniciamos mostrando que o Teorema do Produto Tensorial de Kemer e duas de suas conseqüências não podem ser transportados para corpos infinitos de característica positiva p > 2. Em seguida, discutiremos algumas propriedades envolvendo as álgebras Aa;b, a saber, mostraremos que as álgebras Aa;b e Ma+b(E) não são PI-equivalentes e que as álgebras Aa;a e Ma;a (E) ­ não são PI-equivalentes, e apresentaremos um resultado que enfatiza a importância dos monômios na determinação do ideal das identidades das álgebras Zn £ Z2-graduadas Aa;b em característica positiva. Por ¯m, apresentaremos modelos genéricos e calcularemos a dimensão de Gelfand-Kirillov para as álgebras relativamente livres de posto m nas variedades determinadas pelas álgebras E ­ E, Aa;b e Ma;a(E) ­ E. Como conseqüência, obteremos a prova da não PI- equivalência entre álgebras importantes para PI-teoria em característica positiva / Abstract: The verbally prime algebras are well understood in characteristic 0 while over a field of characteristic p > 2 little is known about them. In this work we discuss some sharp di®erences between these two cases for the characteristic. First we show that the so-called Kemer's Tensor Product Theorem and two of its consequences cannot be extended for infnite fields of positive characteristic p > 2. Afterwards we prove that the algebras Aa;b and Ma+b(E) are not PI equivalent, while the algebras Aa;a and Ma;a(E) ­ E are PI equivalent. Moreover we obtain a result showing the importance of the monomials in the Zn £ Z2-graded T-ideal of the algebra Aa;b. Finally, we exhibit constructions of generic models. By using these models we compute the Gelfand-Kirillov dimension of the relatively free algebras of rank m in the varieties generated by E ­E, Aa;b, and Ma;a(E)­E. As consequence we obtain the PI non equivalence of important algebras for the PI theory in positive characteristic / Doutorado / Algebra / Doutor em Matemática

Orientador: Plamen Emilov Koshlukov / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-14T13:33:07Z (GMT). No. of bitstreams: 1 Fidelis_Marcello_D.pdf: 592299 bytes, checksum: dea5983279c32bbe6e1ffd7e1372fcf4 (MD5) Previous issue date: 2005 / Resumo: Neste trabalho estudamos os produtos tensoriais de T-ideais T-primos sobre corpos infinitos. O comportamento destes produtos tensoriais sobre corpos de caracteristica zero foi descrito por Kemer. Primeiramente mostramos, usando os m'etodos introduzidos por Regev, que tal descri¸cao vale se nos restringirmos apenas aos polinomios multilineares. Num segundo momento, aplicando identidades graduadas, mostramos que o Teorema sobre o Produto Tensorial 'e falso para os T-ideais das 'algebras M1,1(E) e E E, onde E 'e a 'algebra de Grassmann com dimensao infinita; M1,1(E) consiste das matrizes 2 × 2 sobre E tendo somente elementos pares (i.e. centrais) de E na diagonal principal, e a outra diagonal consistindo de elementos 'impares (anticomutitativos) de E. Entao voltamos nossa atencao para outros produtos tensoriais e estudamos suas respectivas identidades graduadas. Obtivemos novas demonstracoes de alguns dos casos do Teorema sobre o Produto Tensorial de Kemer. Note que estas demonstracoes nao dependem da teoria sobre a estrutura dos T-ideais, mas sao "elementares". Finalmente, usando outra vez identidades polinomiais graduadas, mostramos que o Teorema sobre o Produto Tensorial nao 'e valido em mais um caso: quando o corpo base possui caracteristica positiva. Isto vem para mostrar novamente que a teoria sobre a estrutura dos T-ideais e, essencialmente, uma teoria sobre identidades polinomiais multilineares. / Abstract: In this work we study tensor products of T-prime T-ideals over infinite fields. The behaviour of these tensor products over a field of characteristic zero was described by Kemer. First we show, using methods due to Regev, that such a description holds if one restricts oneself to multilinear polynomials only. Second, applying graded polynomial identities, we prove that the Tensor Product Theorem fails for the T-ideals of the algebras M1,1(E) and E E where E is the infinite dimensional Grassmann algebra; M1,1(E) consists of the 2×2 matrices over E having even (i.e. central) elements of E in the main diagonal, and the other diagonal consisting of odd (anticommuting) elements of E. Then we pass to other tensor products and study the respective graded identities. We obtain new proofs of some cases of Kemer's Tensor Product Theorem. Note that these proofs do not depend on the structure theory of T-ideals but are "elementary" ones. Finally, using graded polynomial identities once again, we show that the Tensor Product Theorem fails in one more case when the base field is of positive characteristic. All this comes to show once more that the structure theory of T-ideals is essentially about the multilinear polynomial identities / Doutorado / Matematica / Doutor em Matemática

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Durant cette thèse, une méthode d'analyse statistique sur la gerbe d'éclats d’une bombe, en particulier sur leurs masses, a été mise au point. Nous avions à disposition trois échantillons partiels de données expérimentales et un modèle mécanique simulant l'explosion d'un anneau. Dans un premier temps, un modèle statistique a été créé à partir du modèle mécanique fourni, pour générer des données pouvant être similaires à celles d'une expérience. Après cela, la distribution des masses a pu être étudiée. Les méthodes d'analyse classiques ne donnant pas de résultats suffisamment précis, une nouvelle méthode a été mise au point. Elle consiste à représenter la masse par une variable aléatoire construite à partir d'une base de polynômes chaos. Cette méthode donne de bons résultats mais ne permet pas de prendre en compte le lien entre les éclats d'une même charge. Il a donc été décidé ensuite de modéliser la masse par un processus stochastique, et non par une variable aléatoire. La portée des éclats, qui dépend en partie de la masse, a elle aussi été modélisée par un processus. Pour finir, une analyse de sensibilité a été effectuée sur cette portée avec les indices de Sobol. Ces derniers s'appliquant aux variables aléatoires, nous les avons adaptés aux processus stochastiques de manière à prendre en compte les liens entre les éclats. Dans la suite, les résultats de cette dernière analyse pourront être améliorés. Notamment, grâce à des indices présentés en dernière partie qui seraient adaptés aux variables dépendantes, et permettraient l'utilisation de processus stochastiques à accroissements non indépendants. / During this thesis, a method of statistical analysis on sheaf of bomb fragments, in particular on their masses, has been developed. Three samples of incomplete experimental data and a mechanical model which simulate the explosion of a ring were availables. First, a statistical model based on the mechanical model has been designed, to generate data similar to those of an experience. Then, the distribution of the masses has been studied. The classical methods of analysis being not accurate enough, a new method has been developed. It consists in representing the mass by a random variable built from a basis of chaos polynomials. This method gives good results however it doesn't allow to take into account the link between slivers. Therefore, we decided to model the masses by a stochastic process, and not a random variable. The range of fragments, which depends of the masses, has also been modeled by a process. Last, a sensibility analysis has been carried out on this range with Sobol indices. Since these indices are applied to random variables, it was necessary to adapt them to stochastic process in a way that take into account the links between the fragments. In the last part, it is shown how the results of this analysis could be improved. Specifically, the indices presented in the last part are adapted to dependent variables and therefore, they could be suitable to processes with non independent increases.

We consider orthogonal polynomials P_n satisfying orthogonality relations where the measure of orthogonality is, in general, a complex-valued Borel measure supported on subsets of the complex plane. In our consideration we will focus on measures of the form d\mu(z) = \rho(z) dz where the function \rho may depend on other auxiliary parameters. Much of the asymptotic analysis is done via the Riemann-Hilbert problem and the Deift-Zhou nonlinear steepest descent method, and relies heavily on notions from logarithmic potential theory.

Cette thèse concerne principalement deux objets classiques étroitement liés: d'une part la variété des polynômes complexes unitaires de degré $d>1$ à une variable, et à racines simples (donc de discriminant différent de zéro), et d'autre part, les groupes de tresses d'Artin avec d brins. Le travail présenté dans cette thèse propose une nouvelle approche permettant des calculs cohomologiques explicites à coefficients dans n'importe quel faisceau. En vue de calculs cohomologiques explicites, il est souhaitable d'avoir à sa disposition un bon recouvrement au sens de Čech. L'un des principaux objectifs de cette thèse est de construire un tel recouvrement basé sur des graphes (appelés signatures) qui rappellent les `dessins d'enfant' et qui sont associées aux polynômes complexes classifiés par l'espace de polynômes. Cette décomposition de l'espace de polynômes fournit une stratification semi-algébrique. Le nombre de composantes connexes de chaque strate est calculé dans le dernier chapitre ce cette thèse. Néanmoins, cette partition ne fournit pas immédiatement un recouvrement adapté au calcul de la cohomologie de Čech (avec n'importe quels coefficients) pour deux raisons liées et évidentes: d'une part les sous-ensembles du recouvrement ne sont pas ouverts, et de plus ils sont disjoints puisqu'ils correspondent à différentes signatures. Ainsi, l'objectif principal du chapitre 6 est de ``corriger'' le recouvrement de départ afin de le transformer en un bon recouvrement ouvert, adapté au calcul de la cohomologie Čech. Cette construction permet ensuite un calcul explicite des groupes de cohomologie de Čech à valeurs dans un faisceau localement constant. / This thesis mainly concerns two closely related classical objects: on the one hand, the variety of unitary complex polynomials of degree $ d> 1 $ with a variable, and with simple roots (hence with a non-zero discriminant), and on the other hand, the $d$ strand Artin braid groups. The work presented in this thesis proposes a new approach allowing explicit cohomological calculations with coefficients in any sheaf. In order to obtain explicit cohomological calculations, it is necessary to have a good cover in the sense of Čech. One of the main objectives of this thesis is to construct such a good covering, based on graphs that are reminiscent of the ''dessins d'enfants'' and which are associated to the complex polynomials. This decomposition of the space of polynomials provides a semi-algebraic stratification. The number of connected components in each stratum is counted in the last chapter of this thesis. Nevertheless, this partition does not immediately provide a ''good'' cover adapted to the computation of the cohomology of Čech (with any coefficients) for two related and obvious reasons: on the one hand the subsets of the cover are not open, and moreover they are disjoint since they correspond to different signatures. Therefore, the main purpose of Chapter 6 is to ''correct'' the cover in order to transform it into a good open cover, suitable for the calculation of the Čech cohomology. It is explicitly verified that there is an open cover such that all the multiple intersections are contractible. This allows an explicit calculation of cohomology groups of Čech with values in a locally constant sheaf.

Graph polynomials are polynomials associated to graphs that encode the number of subgraphs with given properties. We list different frameworks used to define graph polynomials in the literature. We present the edge elimination polynomial and introduce several graph polynomials equivalent to it. Thereby, we connect a recursive definition to the counting of colorings and to the counting of (spanning) subgraphs. Furthermore, we define a graph polynomial that not only generalizes the mentioned, but also many of the well-known graph polynomials, including the Potts model, the matching polynomial, the trivariate chromatic polynomial and the subgraph component polynomial. We proof a recurrence relation for this graph polynomial using edge and vertex operation. The definitions and statements are given in such a way that most of them are also valid in the case of hypergraphs.

An n × n matrix factorization of a polynomial f is a pair of n × n matrices (P, Q) such that PQ = f In, where In is the n × n identity matrix. In this dissertation, we study matrix factorizations of an arbitrary element in a given unital ring. This study is motivated on the one hand by the construction of the unit object in the bicategory LGK of Landau-Ginzburg models (of great utility in quantum physics) whose 1−cells are matrix factorizations of polynomials over a commutative ring K, and on the other hand by the existing tensor product of matrix factorizations b⊗. We observe that the pair of n × n matrices that appear in the matrix factorization of an element in a unital ring is not unique. Next, we propose a new operation on matrix factorizations denoted e⊗ which is such that if X is a matrix factorization of an element f in a unital ring (e.g. the power series ring K[[x1, ..., xr]] f) and Y is a matrix factorization of an element g in a unital ring (e.g. g ∈ K[[y1, ..., ys]]), then Xe⊗Y is a matrix factorization of f g in a certain unital ring (e.g. in case f ∈ K[[x1, ..., xr]] and g ∈ K[[y1, ..., ys]], then f g ∈ K[[x1, ..., xr , y1, ..., ys]]). e⊗ is called the multiplicative tensor product of X and Y. After proving that this product is bifunctorial, many of its properties are also stated and proved. Furthermore, if MF(1) denotes the category of matrix factorizations of the constant power series 1, we define the concept of one-step connected category and prove that there is a one-step connected subcategory of (MF(1),e⊗) which is semi-unital semi-monoidal. We also define the concept of right pseudo-monoidal category which generalizes the notion of monoidal category and we prove that (MF(1),e⊗) is an example of this concept. Furthermore, we define a summand-reducible polynomial to be one that can be written in the form f = t1 + · · · + ts + g11 · · · g1m1 + · · · + gl1 · · · glml under some specified conditions where each tk is a monomial and each gji is a sum of monomials. We then use b⊗ and e⊗ to improve the standard method for matrix factorization of polynomials on this class and we prove that if pji is the number of monomials in gji, then there is an improved version of the standard method for factoring f which produces factorizations of size 2 Qm1 i=1 p1i+···+ Qml i=1 pli−( Pm1 i=1 p1i+···+ Pml i=1 pli) times smaller than the size one would normally obtain with the standard method. Moreover, details are given to elucidate the intricate construction of the unit object of LGK. Thereafter, a proof of the naturality of the right and left unit maps of LGK with respect to 2−morphisms is presented. We also prove that there is no direct inverse for these (right and left) unit maps, thereby justifying the fact that their inverses are found only up to homotopy. Finally, some properties of matrix factorizations are exploited to state and prove a necessary condition to obtain a Morita context in LGK.

Pólya and Schur completely characterized all real-rootedness preserving linear operators acting on the standard monomial basis in their famous work from 1914. The corresponding eigenvalues are from then on known as multiplier sequences. In 2009 Borcea and Br\"and\'en gave a complete characterization for general linear operators preserving real-rootedness (and stability) via the symbol. Relying heavily on these results, in this thesis, we are able to completely characterize multiplier sequences for generalized Laguerre bases. We also apply our methods to reprove the characterization of Hermite multiplier sequences achieved by Piotrowski in 2007.

In 1991, A. Máté precisely calculated the first-order asymptotic behavior of the sequence of Christoffel functions associated with Szego measures on the unit circle. Our principal goal is the abstraction of his result in two directions: We compute the translated asymptotics, limn λn(µ, x + a/n), and obtain, as a corollary, a universality limit for the fairly broad class of Szego weights. Finally, we prove Máté’s result for measures supported on smooth curves in the plane. Our proof of the latter derives, in part, from a precise estimate of certain weighted means of the Faber polynomials associated with the support of the measure. Finally, we investigate a variety of applications, including two novel applications to ill-posed problems in Hilbert space and the mean ergodic theorem.