Stuttgart, Univ., Thesis, 2003.

Stuttgart, Univ., Diss., 2004.

Univ., Diplomarb., 2003--Kassel.

Thesis (Ph. D.)--University of Missouri-Columbia, 2007. / The entire dissertation/thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file (which also appears in the research.pdf); a non-technical general description, or public abstract, appears in the public.pdf file. Title from title screen of research.pdf file (viewed on September 28, 2007) Vita. Includes bibliographical references.

Tech. Universiẗat, Diss., 2003--Aachen.

Universiẗat, Diss., 1994--Saarbrücken.

Universiẗat, Diss., 2006--Kassel.

The present dissertation consists of four interconnected projects. In the first, we introduce and study what we call factorizable module algebras. These are $U_q(\mathfrak{g})$-module algebras $A$ which factor, potentially after localization, as the tensor product of the subalgebra $A^+$ of highest weight vectors of $A$ and a copy of the quantum coordinate algebra $\mathcal{A}_q[U]$, where $U$ is a maximal unipotent subgroup of $G$, a semisimple Lie group whose Lie algebra is $\mathfrak{g}$. The class of factorizable module algebras is surprisingly rich, in particular including the quantum coordinate algebras $\mathcal{A}_q[Mat_{m,n}]$, $\mathcal{A}_q[G]$ and $\mathcal{A}_q[G/U]$. It is closed under the braided tensor product and, moreover, the subalgebra $A^+$ of each such $A$ is naturally a module algebra over the quantization of $\mathfrak{g}^*$, the Lie algebra of the Poisson dual group $G^*$. The aforementioned examples of factorizable module algebras all possess dual canonical bases which behave nicely with respect to factorization $A=A^+\otimes \mathcal{A}_q[U]$. We expect the same is true for many other members of this class, including braided tensor products of such. To facilitate such a construction in tensor products, we propose an axiomatic framework of based modules which, in particular, vastly generalizes Lusztig's notion of based modules. We argue that all of the aforementioned $U_q(\mathfrak{g})$-module algebras (and many others) with their dual canonical bases are included, along with their tensor products. One of the central objects of study emerging from our generalization of Lusztig's based modules is a new (very canonical) basis $\mathcal{B}^{\diamond n}$ in the $n$-th braided tensor power $\mathcal{A}_q[G/U]$. We argue (yet conjecturally) that $\mathcal{A}_q[G/U]^{\underline{\otimes}n}$ has a quantum cluster structure and conjecture that the expected cluster structure structure on $\mathcal{A}_q[G/U]^{\underline{\otimes}n}$ is completely controlled by the real elements of our canonical basis $\mathcal{B}^{\diamond n}$. Finally, in order to partially explain the monoidal structures appearing above, we provide an axiomatic framework to construct examples of bialgebroids of Sweedler type. In particular, we describe a bialgebroid structure on $\mathfrak{u}_q(\mathfrak{g})\rtimes\mathbb{Q} C_2$, where $\mathfrak{u}_q(\mathfrak{g})$ is the small quantum group and $C_2$ is the cyclic group of order two. This dissertation contains previously published co-authored material.

Submitted by Amanda Silva (amanda.osilva2@ufpe.br) on 2015-03-03T14:16:54Z No. of bitstreams: 2 Dissertação Mariel Cadena da Matta.pdf: 2355898 bytes, checksum: 9c0530af680cf965137a2385d949b799 (MD5) license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) / Made available in DSpace on 2015-03-03T14:16:54Z (GMT). No. of bitstreams: 2 Dissertação Mariel Cadena da Matta.pdf: 2355898 bytes, checksum: 9c0530af680cf965137a2385d949b799 (MD5) license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) Previous issue date: 2013 / FACEPE / A Dosimetria citogenética empregando análise de cromossomos dicêntricos é o “padrão ouro” para estimativas da dose absorvida após exposições acidentais às radiações ionizantes. Todavia, este método é laborioso e dispendioso, o que torna necessária a introdução de ferramentas computacionais que dinamizem a contagem dessas aberrações cromossômicas radioinduzidas. Os atuais softwares comerciais, utilizados no processamento de imagens em Biodosimetria, são em sua maioria onerosos e desenvolvidos em sistemas dedicados, não podendo ser adaptados para microscópios de rotina laboratorial. Neste contexto, o objetivo da pesquisa foi o desenvolvimento do software ChromoSomeClassification para processamento de imagens de metáfases de linfócitos (não irradiados e irradiados) coradas com Giemsa a 5%. A principal etapa da análise citogenética automática é a separação correta dos cromossomos do fundo, pois a execução incorreta desta fase compromete o desenvolvimento da classificação automática. Desta maneira, apresentamos uma proposta para a sua resolução baseada no aprimoramento da imagem através das técnicas de mudança do sistema de cores, subtração do background e aumento do contraste pela modificação do histograma. Assim, a segmentação por limiar global simples, seguida por operadores morfológicos e pela técnica de separação de objetos obteve uma taxa de acerto de 88,57%. Deste modo, os cromossomos foram enfileirados e contabilizados, e assim, a etapa mais laboriosa da Dosimetria citogenética foi realizada. As características extraídas dos cromossomos isolados foram armazenadas num banco de dados para que a classificação automática fosse realizada através da Rede Neural com Funções de Ativação de Base Radial (RBF). O software proposto alcançou uma taxa de sensibilidade de 76% e especificidade de 91% que podem ser aprimoradas através do acréscimo do número de objetos ao banco de dados e da extração de mais características dos cromossomos.

Cette thèse apporte une contribution aux méthodes numériques pour la simulation moléculaire ab initio, et plus spécifiquement pour le calcul de structures électroniques par l’équation de Schrödinger, ou par des formalismes comme la théorie de Hartree-Fock ou la théorie de la fonctionnelle de la densité. Elle propose une stratégie pour construire des bases mixtes ondelettes-gaussiennes dans l’approximation de Galerkin, combinant les qualités respectives de ces deux types de bases avec l’objectif de mieux capturer les points de rebroussement de la fonction d’onde. Les nombreux logiciels actuellement disponibles à l’usage des chimistes dans ce domaine (VASP, Gaussian, ABINIT...) se différencient par divers choix méthodologiques, notamment celui des fonctions de base pour exprimer les orbitales atomiques. Nouvel arrivant sur le marché, le code massivement parallèle BigDFT a opté pour les bases d’ondelettes. Comme le nombre de niveaux de résolution y est limité pour des raisons de performance,ces fonctions ne peuvent déployer pleinement leur puissance. La question posée est alors de savoir comment accroître la précision des calculs tous électrons au voisinage des singularités de type cusp de la solution, sans augmenter excessivement la complexité de BigDFT. La réponse que nous suggérons consiste à enrichir la base de fonctions d’échelles (niveau de résolution bas des bases d’ondelettes) par des fonctions gaussiennes centrées sur chaque position de noyau. La difficulté principale dans la construction d’une telle base mixte réside dans la détermination optimale du nombre de gaussiennes requises et de leurs écarts-types, de sorte que ces gaussiennes supplémentaires soient le mieux possible compatibles avec la base existante sous la contrainte d’un seuil d’erreur donné à l’avance. Nous proposons pour cela l’utilisation conjointe d’un estimateur a posteriori sur la diminution du niveau d’énergie et d’un algorithme glouton, ce qui aboutit à une suite incrémentale quasi-optimale de gaussiennes supplémentaires. Cette idée est directement inspirée des techniques de bases réduites. Nous développons les fondements théoriques de cette stratégie sur deux modèles 1-D linéaires qui sont des simplifications de l’équation de Schrödinger pour un électron,posée en domaine infini ou domaine périodique. Ces modèles prototypes sont étudiés en profondeur dans la première partie. La définition de l’estimateur a posteriori de type norme duale du résidu, ainsi que la déclinaison de la philosophie glouton en différents algorithmes concrets, sont présentées en seconde partie, accompagnées de résultats numériques. Les algorithmes proposés vont dans le sens d’une économie croissante du temps de calcul.Ils sont aussi de plus en plus empiriques, au sens où ils reposent de plus en plus sur les intuitions avec lesquelles les chimistes sont familiers. En particulier, le dernier algorithme pour plusieurs noyaux s’appuie en partie sur la validité du transfert atome/molécule et rappelle dans une certaine mesure les bases d’orbitales atomiques. / This thesis aims to be a contribution to numerical methods for ab initio molecular simulation, and more specifically for electronic structure calculations by means of the Schrödingerequation or formalisms such as the Hartree-Fock theory or the Density Functional Theory. It puts forward a strategy to build mixed wavelet-Gaussian bases for the Galerkinapproximation, combining the respective advantages of these two types of bases in orderto better capture the cusps of the wave function.Numerous software programs are currently available to the chemists in this field (VASP,Gaussian, ABINIT... ) and differ from each other by various methodological choices,notably that of the basis functions used for expressing atomic orbitals. As a newcomer tothis market, the massively parallel BigDFT code has opted for a basis of wavelets. Dueto performance considerations, the number of multiresolution levels has been limited andtherefore users cannot benefit from the full potential of wavelets. The question is thus howto improve the accuracy of all-electron calculations in the neighborhood of the cusp-typesingularities of the solution, without excessively increasing the complexity of BigDFT.The answer we propose is to enrich the scaling function basis (low level of resolutionof the wavelet basis) by Gaussian functions centered on each nucleus position. The maindifficulty in constructing such a mixed basis lies in the optimal determination of the numberof Gaussians required and their standard deviations, so that these additional Gaussiansare compatible in the best possible way with the existing basis within the constraint of anerror threshold given in advance. We advocate the conjunction of an a posteriori estimateon the diminution of the energy level and a greedy algorithm, which results in a quasi-optimal incremental sequence of additional Gaussians. This idea is directly inspired bythe techniques of reduced bases.We develop the theoretical foundations of this strategy on two 1-D linear models thatare simplified versions of the Schrödinger equation for one electron in an infinite domainor a periodic domain. These prototype models are investigated in depth in the firstpart. The definition of the a posteriori estimate as a residual dual norm, as well as theimplementation of the greedy philosophy into various concrete algorithms, are presented inthe second part, along with extensive numerical results. These algorithms allow for moreand more saving of CPU time and become more and more empirical, in the sense thatthey rely more and more on the intuitions with which chemists are familiar. In particular,the last proposed algorithm partly assumes the validity of the atom/molecule transfer andis somehow reminiscent of atomic orbitals bases.