Stellar structure is investigated within the framework of scalar-tensor gravity. Novel perturbative analytical results are obtained for constant-density stars and for Newtonian polytropes in the quadratic model with coupling function A(Φ) = exp(αΦ+1/2βΦ^2). They are compared to full numerical calculations, and possible applications to main-sequence stars, white dwarfs, and the Chandrasekhar mass are indicated. It is found that Buchdahl's theorem is violated in Brans-Dicke theory for stars with exponentially-decaying density profiles. However, the mass-to-radius ratio M/R tends to the constant-density value in a certain limit. It is observed that for β < 0, there exists a maximum value of η = P0/ρo for constant-density stars, where P0 and ρ0 are the central pressure and density, respectively. It is conjectured that if such a maximum value also exists for other equations of state, and is less than the constant-density maximum value, then knowledge of P/ρ in the centre of a star can be used to constrain β. / Thesis / Master of Science (MSc)

The spin structure factor is the dynamical information coming from inelastic neutron scattering. In this work we develop the technology of tensor networks as a numerical tool to be able to compute physical observables reliably for one-dimensional quantum systems. The main technical message of this thesis is that tensor networks provide a controlled way to compute spin structure factors. The algorithms in this thesis are tested on the anisotropic Majumdar--Ghosh model and the results of these simulations are presented and discussed. / Thesis / Master of Science (MSc)

The beta-decay interaction contains two terms which consist of invariant products of two scalars and two tensors respectively. The relative absolute magnitude of these two terms is fairly well established but there has been some controversy over their relative sign. In this thesis the form of the interaction is investigated by means of an analysis of the second-forbidden decay spectrum of Cs137 and it is concluded that the relative sign of the scalar and tensor terms is negative. / Thesis / Master of Science (MS)

In biology and bioinformatics, a variety of data share a common property that challenges numerous cutting-edge research studies: heterogeneities at the individual level with respect to more than one factor. Examples of such heterogeneities include but are not limited to: 1) unequal susceptibility of different patients, and 2) large diversity in gene length, GC content, etc., along with the resulting gene characteristics. For many biological data analysis studies, the critical first step is usually to infer null probability distribution of observed data with the heterogeneities in multiple (confounding) factors taken into account, so that we can further investigate the impact of other factor(s) of interest. Obviously, the heterogeneities heavily influence the potential conclusions that we may draw from statistical analyses of the data. However, modeling such heterogeneities has been challenging, not only due to the inapplicable explicit modeling of all factors with heterogeneous effects on the data, but also because of the non-independence of many factors from one another. Existing methods, either partially/fully neglected the heterogeneity issue at all, or took care of each factor's heterogeneity in isolation. Evidences have shown the insufficiency of such strategies and the errors they may produce in downstream analyses. The emergence of large-scale data sets provides the opportunity to directly and comprehensively learn the heterogeneity from the data without explicitly modeling the mechanisms behind or exerting strong assumptions. The data, as often stored or organized as multidimensional contingency tensors, lead to a natural perspective of modeling heterogeneity with each impact factor of interest being one dimension. The heterogeneity in each factor's impact on the variable of interest can be captured by the marginal property of the data tensor with respect to the corresponding dimension. For instance, in a single-cell sequencing dataset, which can be organized as a matrix with each row representing a gene and each column representing a cell, the heterogeneity caused by both the gene and cell factors can be modeled. In this dissertation, we develop a novel model, Conditional Multifactorial Contingency (CMC), that models the intertwined heterogeneities in all dimensions of the data tensor and infers the probability distribution of each entry of the data tensor jointly conditioned on these heterogeneities. In the proposed CMC model, the problem is formulated as a maximum entropy problem of the contingency tensor's probability distribution subject to the marginal constraints, under the assumption that the individuals within each dimension are independent. The marginal constraints are applied to the expected value instead of observed trial outcomes, which plays a key role in avoiding the innumerable combinations of trial outcomes and leading to an elegant expression form of the entry's probability distribution. The model is first developed for 3D binary data matrix, then extended to multidimensional data tensors and integer data tensors. Furthermore, missing values are taken into account and CMC is extended to be compatible with data with missing values. Being empowered by CMC, we conducted four case studies for real-world bioinformatics research problems: (1) driving transcription factor (TF) identification; (2) scRNA-seq data normalization; (3) cancer-associated gene identification; (4) cell similarity quantification. For each of these case studies, we proposed a whole analysis framework and specific adaptation design for CMC. For the driving-TF identification, compared with traditional methods, we considered the variations in the gene's binding affinity in addition to the typically considered variations in TF's binding affinity. The driving TFs were identified by comparing the observed binding state and the estimated binding probability conditioned on TF/gene binding affinities. For the scRNA-seq data normalization, besides gene factor and cell factor, we figured out one more factor impacting the read counts, cDNA length, and applied CMC to comprehensively analyze the three factors. For cancer-associated gene identification, the CMC model is applied to systematically model the patient, gene, and mutation type factors in the mutation count data. As for the last application, to the best of our knowledge, our solution is the first proposed cell-to-cell-type similarity quantification method, thanks to the availability of CMC to systematically model and remove the impact of cell and gene factors. We studied the theoretical properties of the proposed model and validated the effectiveness and efficiency of our method through experiments. The uniqueness of the probability solution and the convergence of the algorithm was proved. In the endeavor to identify true driving TFs, CMC significantly boosted the best record of success rate, which was proved using data with ground truth. Besides, in an exploratory study without ground truth, in addition to the previously known TFs, Olig1 (ranks 2nd), Olig2 (ranks 3rd), and Sox10 (ranks 4th), we successfully identified Ppp1r14b (ranks 1st) and Zfp36l1 (ranks 6th) that function in oligodendrocyte lineage development, which was validated via biological knock-out experiments and, has led to genuine biological discoveries. In the scRNA-seq data normalization, experimental results show that, by taking the cell, gene, and cDNA-length factors into account, the normalized data achieves lower variances for housekeeping genes than the peer methods. Besides, the data normalized by the CMC model leads to better accuracy of downstream DEG detection than that normalized by peer normalization methods. In cancer-associated gene identification, the CMC model is able to eliminate most of the likely artefactual findings resulted by considering the hidden factors separately. In the cell similarity quantification, CMC based model enables the identification of cell types by establishing between-species cell similarity quantification, regardless of contamination in scRNA-seq data. / Doctor of Philosophy / Biological data are complicated and typically influenced by numerous factors, including characteristics of biological subjects, physical or chemical properties of molecules, artifacts created by experimental operations, and so on. The information of real interest in a biology/bioinformatics study can be buried in all sorts of irrelevant factors and their impacts on the data. Consider a simple example where a study is conducted to figure out if an association exists between a specific gene and a cancer. Although this gene shows obviously different frequencies of mutation in two groups of people, patients and the normal, we cannot safely confirm the association from this observation. Such differential mutation levels can also be a result of the diversity among all these people in how easily this gene is mutated in a person (related to many characteristics of this person besides "cancer/not"). We call this diversity "heterogeneity", and it actually can be seen everywhere, in people, in genes, in cells, and in cell types, etc. One needs to take good care of such heterogeneities so as to draw firm statistical hence scientific conclusions. However, handling the heterogeneities is far from trivial. On the one hand, it is generally impossible to fully understand the mechanisms behind those diversities, let alone to explicitly and rigorously formulate them. One the other hand, it is not rare that multiple factors intertwine with one another, in which case all these factors must be considered systematically in order to model the data precisely. Existing methods, either partially/fully neglected the heterogeneity issue at all, or took care of each factor's heterogeneity in isolation. Evidences have shown the insufficiency of such strategies and the errors they may produce in downstream analyses. As the exact mechanisms behind heterogeneities are usually not available, we aim to learn and infer the heterogeneities' effects on data from data itself. A large group of biological data can be stored or organized as multidimensional contingency tensors, with each impact factor of interest being one dimension. The heterogeneity in each factor's impact on the variable of interest can be captured by the marginal property of the data tensor with respect to the corresponding dimension, for example, the row sum and the column sum in a 2D tensor. In this dissertation, under the assumption that the individuals of each dimension are independent, we proposed a novel model, Conditional Multifactorial Contingency (CMC), that models the intertwined heterogeneities in all dimensions of the data tensor and infers the probability distribution of each entry of the data tensor jointly conditioned on these heterogeneities. The eventual and most comprehensive version of CMC can work on multidimensional binary or integer data tensors, even in cases where some values in the tensor are missing. CMC was initiated from elegant and simple statistical principles, derived through rigorous theoretical proofs, but ended up as a powerful tool being widely applicable to real-world biology/bioinformatics studies. Being empowered by CMC, we conducted four case studies for real-world bioinformatics research problems: (1) driving transcription factor (TF) identification; (2) scRNA-seq data normalization; (3) cancer-associated gene identification; (4) cell similarity quantification. For each of these case studies, we proposed a whole analysis framework and specific adaptation design for CMC. In each of them, our method based on CMC outperformed existing methods and provided inspiring clues for biological discoveries, which have been validated by biological experiments.

A new direct full variational approach is described. The approach exploits a tensor (Kronecker) product construction of the many-electron Hamiltonian and has a number of computational advantages. Explicit assembly and storage of the Hamiltonian matrix is avoided by using the Kronecker product structure to form matrix-vector products directly from the molecular integrals. Computation-intensive integral transformations and formula tapes are unnecessary. The wavefunction is expanded in terms of spin-free primitive kets rather than Slater determinants or configuration state functions and is equivalent to a full configuration interaction expansion. The approach suggests compact storage schemes and algorithms which are naturally suited to parallel and pipelined machines. Sample calculations for small two- and four-electron systems are presented. The preliminary ground state potential energy surface of the hydrogen molecule dimer is computed by the tensor product method using a small basis set. / Ph. D.

Os P-splines de Eilers e Marx (1996) são métodos de suavização que é uma combinação de bases B-splines e uma penalização discreta sobre os coeficientes das bases utilizados para suavizar dados normais e não normais em uma ou mais dimensões, no caso de várias dimensões utiliza-se como suavização o produto tensor dos P-splines. Também os P-splines são utilizados como representação de modelos mistos Currie et al. (2006) pela presença de características tais como: efeitos fixos, efeitos aleatórios, correlação espacial ou temporal e utilizados em modelos mais generalizados tais como os modelos mistos lineares generalizados e modelos mistos aditivos generalizados. Neste trabalho apresentou-se toda a abordagem, metodologia e descrição dos P-splines como modelos mistos e como componentes das estruturas suavizadoras de variáveis unidimensionais e bidimensionais dos modelos mistos aditivos generalizados, mostrando essa abordagem e propondo seu uso em uma aplicação no comportamento dos níveis médios da produção de cana-de-açúcar sob a influência das alterações das variáveis climáticas como temperatura e precipitação, que foram medidos ao longo de 10 anos em cada mesorregião do Estado de São Paulo. O motivo de usar essa abordagem como método de suavização é que muitas vezes não é conhecido a tendência dessas covariáveis climáticas mas sabe-se que elas influenciam diretamente sobre a variável resposta. Além de permitir essa abordagem inclusão de efeitos fixos e aleatórios nos modelos a serem propostos, permitirá a inclusão do processo autoregressivo AR(1) como estrutura de correlação nos resíduos. / P-splines of Eilers e Marx (1996) are methods of smoothing that is a combination of B-splines bases and penalty the coefficients of the bases used to smooth normal and non-normal data in one or more dimensions; in the case of several dimensions it is used as smoothing the tensor product of the P-splines. Also the P-splines are used as representation of mixed models Currie et al. (2006) by the presence of characteristics such as: fixed effects, random effects, spatial or temporal correlation and used in more generalized models such as generalized linear mixed models and generalized additive mixed models. In this work the whole approach, methodology and description of the P-splines as mixed models and as components of the smoothing structures of one-dimensional and two-dimensional variables of generalized additive mixed models were presented, showing this approach and proposing its application in the behavior of the average levels of sugarcane production, which is influenced by changes in climatic variables such as temperature and precipitation , which were measured over 10 years in each mesoregion of the state of São Paulo. The reason for using this approach as a smoothing method is that the tendency of these climate covariables is not know for the most part, but is known that they influence directly the response variable, besides allowing this approach to include fixed and random effects in the models to be proposed, will allow the inclusion of the autoregressive process AR(1) as a correlation structure in the residuos.

Introdução: Os transtornos depressivos constituem um problema de saúde pública na terceira idade, e estima-se que a depressão será uma das três principais causas de sobrecarga de doença no mundo nas próximas decadas. Métodos de neuroimagem têm sido amplamente utilizados em estudos de depressão em idosos, pois são técnicas não invasivas que permitem a detecção de alterações cerebrais estruturais e funcionais. Fração de Anisotropia (FA) e Difusividade Média (MD) são índices indiretos da integridade micro-estrutural da substância branca, mensurados através de imagens de tensor de difusão (DTI). A maioria dos estudos a respeito de depressão e neuroimagem tem focado apenas em possíveis diferenças em regiões de interesse (ROI) previamente determinadas. Pesquisas sobre depressão em idosos e as alterações estruturais por tensor de difusão em todo o cérebro são escassos. O objetivo deste estudo foi investigar a existência de alterações nos parâmetros de FA ou MD em todo o cérebro, sem uma região de interesse previamente definida, comparando idosos deprimidos a idosos sem depressão. Métodos: Exames de imagem cerebral por ressonância magnética foram obtidos de 47 idosos deprimidos (idade média=70,94 ± 6,98), segundo critérios diagnósticos do DSM-IV, e 36 idosos sem depressão (idade média=69,39 ± 7,21) (p=0,32). O exame de neuroimagem dos sujeitos foi realizado em aparelho de ressonância magnética (RM) de 1,5 T, (TE mínimo, TR=10000ms, FOV=26, matriz=128x128, espessura=5mm). Os parâmetros de difusão das imagens de RM foram obtidos a partir de 25 direções não colineares com um b-valor de 1000s/mm2 juntamente com imagem sem gradientes de difusão b=0. Antes da aquisição dos exames de imagem, um psiquiatra administrou os seguintes testes: Mini-Exame do Estado Mental (MEEM), Teste Cognitivo Cambridge (CAMCOG), Escala Montgomery-Aberg de Depressão (MADRS) e Escala de Depressão de Hamilton (HAM-D). Não foram encontradas diferenças significativas nos dados demográficos entre os grupos. A análise estatística baseada no voxel dos dados de FA foi realizada com uso da ferramenta TBSS, parte do programa FSL, que projeta a FA de cada indivíduo em um esqueleto de FA média antes de aplicar a análise estatística baseada no voxel entre os sujeitos da amostra. Diferenças entre os grupos foram controladas para idade. Resultados: Os escores médios da avaliação cognitiva para o grupo de idosos deprimidos foram: CAMCOG=82,94 ± 13,95 e MEEM=25,21 ± 3,74; e para o grupo controle: CAMCOG=90,83 ± 8,88 (p=0,017) e MEEM=27,86 ± 1,99 (p=0,004). Quanto às escalas de sintomatologia depressiva, tem-se no grupo de idosos deprimidos: MADRS=23,23 ± 8,60 HAM-D=18,64 ± 6,17; e no grupo de idosos sem depressão: MADRS=1,39 ± 1,20, HAM-D=2,67 ± 1,57. Após o ajuste por idade, o grupo de idosos deprimidos não apresentou diferenças significativas nos parâmetros de FA e de MD. Os escores da avaliação cognitiva (CAMCOG e MEEM) não se correlacionaram significativamente aos parâmetros de FA nem de MD. Resultados semelhantes foram obtidos após a correlação com escores das escalas de sintomatologia depressiva (MADRS e HAM-D). Conclusão: Não houve diferença significativa, na amostra estudada, dos parâmetros de FA ou MD entre os idosos deprimidos e idosos sem depressão quando o cérebro é analisado sem a utilização de ROI. Não houve correlação, na presente amostra, entre avaliação cognitiva e FA ou MD nem entre gravidade da depressão e estes parâmetros de avaliação de alteração de substância branca / Introduction: Depressive disorders constitute a public health problem in old age, and depression is projected to be one of the three leading causes of burden of disease worldwide in the next decades. Neuroimaging methods have been widely used in studies of depression in the elderly, because they are noninvasive techniques that allow the detection of structural and functional brain changes. Fractional Anisotropy (FA) and Mean Diffusivity (MD) are neuroimaging index of micro-structural white matter integrity, measured using diffusion tensor imaging. Most studies investigating depression and neuroimaging have focused only in possible differences in regions of interesting (ROI) previously selected. Studies investigating correlation between elderly depression and structural alterations measured by diffusion tensor in the whole brain are scarce. The aim of this study was to investigate the existence of FA or MD differences in the whole brain, without a region of interest previously determined, between elderly depressed and elderly without depression. Methods: Brain magnetic resonance imaging scans were obtained on 47 elderly depressed subjects (mean age=70.9 ± 6.9), according to DSM-IV criteria, and 36 healthy elderly controls (mean age=69.4 ± 7.2) (p=0.32). Scanning of subjects was performed on a 1,5T MRI scanner (TE minimum, TR=10000ms, FOV=26, matrix=128x128, section thickness=5mm). Diffusion MR images were obtained from 25 non-colinear directions with a b-value of 1000 s/mm2 along with a b=0 image with no diffusion gradients. Before MRI acquisition, a psychiatrist administered the following psychiatric tests: Cambridge Cognitve Test Examination (CAMCOG), Mini-Mental State Examination (MMSE), Montgomery-Aberg Depression Rating Scale (MADRS), and Hamilton Rating Scale of Depression (HAM-D). No significant differences were found on demographic data between groups. Voxelwise statistical analysis of FA data was carried out using Tract-Based Spatial Statistics (TBSS), part of FSL program. TBSS projects all subjects\' FA data onto a mean FA tract skeleton, before applying voxelwise cross-subject statistics. Differences between groups were assessed controlling for age. Results: The mean score from cognitive assessment for the whole depression group was: CAMCOG=82,94 ± 13,95 and MMSE=25,21 ± 3,74; and for controls: CAMCOG=90,83 ± 8,88 (p=0,017) and MMSE=27,86 ± 1,99 (p=0,004). Results of depressive symptom assessment for the patient group were MADRS=23.23 ± 8.60 HAM-D=18.64 ± 6.17 and MADRS=1.39 ± 1.20, HAM-D=2.67 ± 1.57 for control group. After controlling for age, geriatric depressed subjects had no significant differences on FA and on MD parameters. No significant correlations were found between scores from cognitive tests (CAMCOG and MMSE), and FA or MD parameters. Similar results were obtained after correlating scores from scales measuring depressive symptoms (MADRS and HAM-D) and FA or MD parameters. Conclusions: There was no significant difference in FA or MD values between elderly depressed and elderly without depression when the brain is analyzed without a ROI previously determined. There was no correlation, in the present sample, between cognitive assessment and FA or MD, neither between severity of depression and these brain white matter parameters

O objetivo deste trabalho é o desenvolvimento de um pós-processador para um software baseado no método dos elementos finitos destinado a problemas de eletromagnetismo. Três métodos de determinação de torque de origem eletromagnética são abordados e comparados do ponto de vista da precisão e implementação, quais sejam: variação da energia magnética, integração do tensor de Maxwell e princípio do trabalho virtual (derivada do jacobiano local). Dois métodos são propostos para a determinação de reatâncias de eixo direto e em quadratura de máquinas síncronas. A metodologia é aplicada na análise de um motor síncrono de relutância de 15 hp, 380v, 4 pólos, cujos resultados são apresentados no capítulo IV. / The aim of this work is to develop a post-processor for use with an electromagnetism-oriented software based upon the Finite Elements Method. Three methods for determining electromagnetic torque are presented and compared in terms of precision and implementation: (i) Variation of Magnetic Energy, (ii) Maxwell Stress Method and (iii) Virtual-work Principle (Local Jacobian Derivative). Two methods for calculating direct-axis and quadrature-axis reactances of synchronous machines are proposed. The methodology is applied in the analysis of a 15 HP, 380 v, 4-pole reluctance synchronous motor and the results are presented in Chapter IV.

CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / Esta tese teve dois objetos de estudo: propriedades estocÃsticas em uma variedade Riemanniana, a saber, Completude EstocÃstica, Parabolicidade e propriedade Feller, e a geometria do tensor de Bakry-Emery. Na primeira parte da tese estudamos tais propriedades estocÃsticas no contexto de submersÃes Riemannianas e imersÃes isomÃtricas, tendo como ponto de partida o trabalho de Pigola e Setti [28] sobre a propriedade Feller. No nosso primeiro resultado, provamos que se uma imersÃo isomÃtrica em uma variedade Cartan-Hadamard possui vetor curvatura mÃdia com norma limitada entÃo a imersÃo à Feller. Um anÃlogo desse resultado jà era conhecido para o caso de completude estocÃstica [30]. Em seguida estabelecemos condiÃÃes necessÃrias e suficientes para que uma submersÃo seja estocasticamente completa (respec. parabÃlica), a saber, se uma submersÃo Riemanniana tem fibra mÃnima e o espaÃo total à estocasticamente completo (respec. parabÃlico) entÃo a base à estocasticamente completa (respec. parabÃlica). Reciprocamente, se a submersÃo Riemanniana tem fibra mÃnima e compacta e a base à estocasticamente completa (respec. parabÃlica) entÃo o espaÃo total à estocasticamente completo (respec. parabÃlico). Finalmente provamos que uma submersÃo Riemanniana tem fibra mÃnima e compacta entÃo o espaÃo total Âe Feller, se, e somente se, a base à Feller. Na segunda parte desta tese estudamos o tensor de Bakry-Emery Ricci, Ricf, que à uma extensÃo, no caso de variedades ponderadas, do tensor de Ricci. Estudamos a seguinte situaÃÃo: Ricci &#8805; -cG, onde c à uma constante positiva e G &#8805; O à uma funÃÃo suave. Esta limitaÃÃo nos permitiu obter algumas consequencias geomÃtricas e topolÃgicas, que passamos a descrever. Seja Mf uma variedade Riemanniana ponderada e po &#1028; Mf fixado. Nosso primeiro resultado à uma estimativa superior, fora da bola geodÃsica de raio ro, para o Laplaciano ponderado da funÃÃo distÃncia r ao ponto po, mf, em termos da integral da funÃÃo G. A primeira consequÃncia dessa estimativa à uma estimativa para o volume ponderado Volf (B(R)) de uma bola geodÃsica de raio R em termos da integral da funÃÃo G. A estimativa de mf, juntamente com a hipÃtese de Æ ser radial e &#1240;r Æ &#8805; -a,a &#8805; 0 (ou | Æ|&#8804; k) tambÃm nos permite demonstrar um teorema de comparaÃÃo entre mf e maG, Laplaciano da funÃÃo distÃncia no modelo de curvatura aG, bem como um teorema de comparaÃÃo entre o volume ponderado de uma bola geodÃsica de raio R em Mf, VolÆ(B(R)), e o volume da bola geodÃsica de raio R no modelo MaG, de curvatura aG. Utilizando uma versÃo ponderada da fÃrmula de Bochner provamos que, se Ricci &#8805; Gâ entÃo Mf satisfaz o princÃpio do mÃximo de Omori-Yau, onde G à funÃÃo suave, positiva, nÃo decrescente e tal que G-1 nÃo à integrÃvel. Em particular concluÃmos que Mf à estocasticamente completa. O prÃximo resultado que obtivemos estende, para o tensor Ricf, um teorema de Myers devido a Ambrose [1]. Para tanto, uma hipÃtese sobre a funÃÃo Æ foi necessÃria. Como aplicaÃÃo, estendemos um resultado de compacidade de Ricci solitons de Fernando-Lopes e Garcia-Rio [15]. Em 1976, Yau [36] provou uma estimativa para o gradiente de uma funÃÃo u, positiva, harmÃnica em B(2R), no caso de M ser completa e Ricf &#8805; -k, k &#8805; 0. Tal estimativa depende apenas de R e k e foi estendida, no caso ponderado, para funÃÃes f harmÃnicas positivas, supondo Ricf &#8805; -k e Ric &#8805; -H, k, H &#8805; 0. Bringhton [9] obteve estimativas para o gradiente de uma funÃÃo *-harmÃnica positiva utilizando somente a hipÃtese Ricf &#8805; -k. As estimativas que obtivemos estendem as estimativas citas acima e, no caso em que Æ=G=0 resultam na estimativa original de Yau. Finalmente, provamos um teorema de comparaÃÃo entre o primeiro autovalor de Dirichlet da bola geodÃsica de raio R em Mf e o primeiro autovalor de Dirichlet da bola geodÃsica de raio MG. Tal resultado estende, para o caso ponderado, um resultado de Bessa e Montenegro [4]. / In this thesis we studied two objects(?): properties in Riemannian manifolds, more precisely stochastic completeness, parabolicity and the Feller property and geometric properties of Bakry Emery Ricci tensor. First, we studied such stochastic properties on Riemannian and isometric immersions. The initial motivation was the work of Pigola and Setti [30] about the Feller property. In our first result, we proved that if a isometric immersion on a Cartan-Hadamard manifold has bounded mean curvature vector then the immersion is Feller. An analogous result was know for stochastic completeness. After we stabilish necessary and sufficient conditions to a Riemannian submersion be stochastically complete (parabolic). More precisely if a Riemannian submersion has minimal fiber and the total space is stochastically complete (parabolic ) then the basis is also stochastically complete ( parabolic ). Conversely, if the Riemannian submersion has compact minimal fiber and the basis is stochastically complete ( parabolic, Feller ) then the total space also is. We also proved that if a Riemannian submersion has compact minimal fiber then the total space is Feller if, and only if the the basis is Feller. In the second part we studied the Barkry Emery Ricci tensor Ricf, wich is a natural extension of the Ricci tensor in the context of weighted manifolds. We studied the following: suppose that Ricf has a lower bound âcG where G is a smooth nonnegative function and c a positive constant. Such lower bound allow us to obtain some geometric and topological consequences as we describe below. Consider Mf a weighted Riemannian manifold. The first consequence is an upper estimate, outside a geodesic ball of radius r0, for the weighted Laplacian of the Riemannian distance in terms of the function G. Let Mf be a weighted Riemannian manifold and po &#1028; Mf fixed. Our first result is an upper bound, outside of a geodesic ball of radius R centered in po, for the weighted Laplacian os the Riemannian distance function from po in terms od the function G. The first consequence of this estimate is an estimate for the weighted volume Volf (B(R)) of a geodesic ball with radius R in terms of the integral of G. This estimate together the assumption of f be radial and &#1240; f &#8805; - a, a&#8805; 0 (or | f | &#8804;k ) allow us to prove a comparison theorem for mf e mag, the Laplacian of distance function of the Riemannian model fo curvature aG, as such as a comparison theoremfor the weighted volume of a geodesic ball with radius R on the Riemannian model MaG, with curvature aG. Using a weighted version of the Bochner formula we proved that Ricf &#8805; Gâ then Mf satisfies the Omori-Yau Maximum Principle, where G is a positive, nondecreasing smooth function, such that G-1 does not belong to L1(Mf). In particular we conclude that Mf is stochastically complete. The next result we proved extends, for the tensor Ricf, a type Myers theorem due to Ambrose [1]. For this an additional assumption on f was required. As an aplication of this result we extended a result about compacity of Ricci solitons due to Fernandez-Lopez e GarcÃa-Rio [15]. In 1976, Yau [36] proved an estimate for the gradient of a positive harmonic funcion u, defined on B(2R), when M is complete and Ric &#8805; -k, k&#8805; 0. Such estimate depends only on R and k and was extended, to the weighted, to the case, to f-harmonic positive functions, when Ricf &#8805; - k and Ric &#8805; - H, k, H &#8805; 0. Brighton [9] obtained estimates for the gradient of a positive f-harmonic function assuming only Ricf &#8805; -k. We obtained estimates for the case Ricf &#8805; -G where G is a smooth nonnegative function and when f= G = 0 we recover the original estimate of Yau. Finally we proved a comparison theorem between the first eigenvalue of the geodesic ball of radius r on Mf and the first eigenvalue of the geodesic ball of radius r of the model MG. Such result extends, to the weighted case, a result due to Bessa e Montenegro [4].

O objetivo deste trabalho é o desenvolvimento de um pós-processador para um software baseado no método dos elementos finitos destinado a problemas de eletromagnetismo. Três métodos de determinação de torque de origem eletromagnética são abordados e comparados do ponto de vista da precisão e implementação, quais sejam: variação da energia magnética, integração do tensor de Maxwell e princípio do trabalho virtual (derivada do jacobiano local). Dois métodos são propostos para a determinação de reatâncias de eixo direto e em quadratura de máquinas síncronas. A metodologia é aplicada na análise de um motor síncrono de relutância de 15 hp, 380v, 4 pólos, cujos resultados são apresentados no capítulo IV. / The aim of this work is to develop a post-processor for use with an electromagnetism-oriented software based upon the Finite Elements Method. Three methods for determining electromagnetic torque are presented and compared in terms of precision and implementation: (i) Variation of Magnetic Energy, (ii) Maxwell Stress Method and (iii) Virtual-work Principle (Local Jacobian Derivative). Two methods for calculating direct-axis and quadrature-axis reactances of synchronous machines are proposed. The methodology is applied in the analysis of a 15 HP, 380 v, 4-pole reluctance synchronous motor and the results are presented in Chapter IV.