The vector bundles with the basic structure space with affine connection. It is shown that in the present bundles the linear inducts the affine connection, and the curvature of the objects of the present connection is traced. Having defined the concept of the first differential extension of the vector bundles, an indication is made that the linear connection of vector bundles inducts the elongated linear connection of space and linear co-connection, expression form of the linear co-connection components and their interrelation. There are derived commutative formulas of the inducted connection and forms of its components of curvature objects.

Submitted by Renata Lopes (renatasil82@gmail.com) on 2017-03-06T15:14:46Z No. of bitstreams: 1 ederdealmeidaperez.pdf: 749381 bytes, checksum: 7338f694cc850057100e730b520d74eb (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2017-03-06T20:25:35Z (GMT) No. of bitstreams: 1 ederdealmeidaperez.pdf: 749381 bytes, checksum: 7338f694cc850057100e730b520d74eb (MD5) / Made available in DSpace on 2017-03-06T20:25:35Z (GMT). No. of bitstreams: 1 ederdealmeidaperez.pdf: 749381 bytes, checksum: 7338f694cc850057100e730b520d74eb (MD5) Previous issue date: 2012-08-24 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / O reconhecimento de padrões de movimentos tem se tornado um campo de pesquisa muito atrativo nos últimos anos devido, entre outros fatores, à grande massificação de dados em vídeos e a tendência na criação de interfaces homem-máquina que utilizam expressões faciais e corporais. Esse campo pode ser considerado um dos requisitos chave para análise e entendimento de vídeos. Neste trabalho é proposto um descritor de movimentos baseado em tensores de 2a ordem e histogramas de gradientes (HOG - Histogram of Oriented Gradients). O cálculo do descritor é rápido, simples e eficaz. Além disso, nenhum aprendizado prévio é necessário sendo que a adição de novas classes de movimentos ou novos vídeos não necessita de mudanças ou que se recalculem os descritores já existentes. Cada quadro do vídeo é particionado e em cada partição calcula-se o histograma de gradientes no espaço e no tempo. A partir daí calcula-se o tensor do quadro e o descritor final é formado por uma série de tensores de cada quadro. O descritor criado é avaliado classificando-se as bases de vídeos KTH e Hollywood2, utilizadas na literatura atual, com um classificador Máquina Vetor Suporte (SVM). Os resultados obtidos na base KTH são próximos aos descritores do estado da arte que utilizam informação local do vídeo. Os resultados obtidos na base Hollywood2 não superam o estado da arte, mas são próximos o suficiente para concluirmos que o método proposto é eficaz. Apesar de a literatura apresentar descritores que possuem resultados superiores na classificação, suas abordagens são complexas e de alto custo computacional. / The motion pattern recognition has become a very attractive research field in recent years due to the large amount of video data and the creation of human-machine interfaces that use facial and body expressions. This field can be considered one of the key requirements for analysis and understanding in video. This thesis proposes a motion descriptor based on second order tensor and histograms of oriented gradients. The calculation of the descriptor is fast, simple and effective. Furthermore, no prior knowledge of data basis is required and the addition of new classes of motion and videos do not need to recalculate the existing descriptors. The frame of a video is divided into a grid and the histogram of oriented gradients is computed in each cell. After that, the frame tensor is computed and the final descriptor is built by a series of frame tensors. The descriptor is evaluated in both KTH and Hollywood2 data basis, used in the current literature, with a Support Vector Machine classifier (SVM). The results obtained on the basis KTH are very close to the descriptors of the state-of-the-art that use local information of the video. The results obtained on the basis Hollywood2 not outweigh the state-of-the-art but are close enough to conclude that the proposed method is effective. Although the literature presents descriptors that have superior results, their approaches are complex and with computational cost.

Ce manuscrit regroupe différents travaux explorant les interactions entre les tenseurs et l'apprentissage automatique. Le premier chapitre est consacré à l'extension des modèles de séries reconnaissables de chaînes et d'arbres aux graphes. Nous y montrons que les modèles d'automates pondérés de chaînes et d'arbres peuvent être interprétés d'une manière simple et unifiée à l'aide de réseaux de tenseurs, et que cette interprétation s'étend naturellement aux graphes ; nous étudions certaines propriétés de ce modèle et présentons des résultats préliminaires sur leur apprentissage. Le second chapitre porte sur la minimisation approximée d'automates pondérés d'arbres et propose une approche théoriquement fondée à la problématique suivante : étant donné un automate pondéré d'arbres à n états, comment trouver un automate à m<n états calculant une fonction proche de l'originale. Le troisième chapitre traite de la régression de faible rang pour sorties à structure tensorielle. Nous y proposons un algorithme d'apprentissage rapide et efficace pour traiter un problème de régression dans lequel les sorties des tenseurs. Nous montrons que l'algorithme proposé est un algorithme d'approximation pour ce problème NP-difficile et nous donnons une analyse théorique de ses propriétés statistiques et de généralisation. Enfin, le quatrième chapitre introduit le modèle de mélanges algébriques de distributions. Ce modèle considère des combinaisons affines de distributions (où les coefficients somment à un mais ne sont pas nécessairement positifs). Nous proposons une approche pour l'apprentissage de mélanges algébriques qui étend la méthode tensorielle des moments introduite récemment. . / This thesis tackles several problems exploring connections between tensors and machine learning. In the first chapter, we propose an extension of the classical notion of recognizable function on strings and trees to graphs. We first show that the computations of weighted automata on strings and trees can be interpreted in a natural and unifying way using tensor networks, which naturally leads us to define a computational model on graphs: graph weighted models; we then study fundamental properties of this model and present preliminary learning results. The second chapter tackles a model reduction problem for weighted tree automata. We propose a principled approach to the following problem: given a weighted tree automaton with n states, how can we find an automaton with m<n states that is a good approximation of the original one? In the third chapter, we consider a problem of low rank regression for tensor structured outputs. We design a fast and efficient algorithm to address a regression task where the outputs are tensors. We show that this algorithm generalizes the reduced rank regression method and that it offers good approximation, statistical and generalization guarantees. Lastly in the fourth chapter, we introduce the algebraic mixture model. This model considers affine combinations of probability distributions (where the weights sum to one but may be negative). We extend the recently proposed tensor method of moments to algebraic mixtures, which allows us in particular to design a learning algorithm for algebraic mixtures of spherical Gaussian distributions.

Cette thèse porte sur différents aspects géométriques et morphologiques concernant des problèmes de mouillage et d'empilement de sphères. Nous proposons tout d'abord une nouvelle méthode de simulation pour étudier le mouillage et le glissement d'un liquide sur une surface nanostructurée: un modèle de champ de phase en lien avec la théorie de la fonctionnelle de la densité dynamique. Nous étudions grâce à cette méthode la possibilité de transformer une surface quelconque en surface omniphobe (c'est à dire qui repousse tous les liquides). Nous montrons que contrairement à la théorie classique de Cassie-Baxter-Wenzel, il est possible d'inverser la mouillabilité d'une surface en la texturant, et nous montrons qu'une surface monovaluée, i.e. sans constrictions, peut produire un comportement omniphobe c'est à dire repousser tous les liquides grâce à un effet de pointe. La géométrie a également un effet considérable dans les milieux vitreux ou bloqués. Les empilements aléatoires de sphères conduisent par exemple à des état bloqués ("jamming") et nous montrons que la structure locale de ces systèmes est universelle, c'est à dire indépendante de la méthode de préparation. Pour cela, nous introduisons des paramètres d'ordre - les tenseurs de Minkowski - qui suppriment les problèmes de robustesse qu'ont les paramètres d'ordre utilisés classiquement. Ces nouveaux paramètres d'ordre conduisent à une vision unifiée, basée sur des principes géométriques. Enfin, nous montrons grâce aux tenseurs de Minkowski que les empilements de sphères se mettent à cristalliser au delà du point d'empilement aléatoire le plus dense ("random close packing") / This thesis is devoted to several aspects of geometry and morphology in wetting problems and hard sphere packings. First, we propose a new method to simulate wetting and slip on nanostructured substrates: a phase field model associated with a dynamical density theory approach. We showed omniphobicity, meaning repellency, no matter the chemical properties of the liquid on monovalued surfaces, i.e. surfaces without overhangs, which is in contradiction with the macroscopic Cassie-Baxter-Wenzel theory, can produce so-called We checked systematically the impact of the surface parameters on omniphobic repellency, and we show that the key ingredient are line tensions, which emerge from needle shaped surface structures. Geometrical effects have also an important influence on glassy or jammed systems, for example amorphous hard sphere systems in infinite pressure limit. Such hard sphere packings got stuck in a so-called jammed phase, and we shall demonstrate that the local structure in such systems is universal, i.e. independent of the protocol of the generation. For this, robust order parameters - so-called Minkowski tensors - are developed, which overcome robustness deficiencies of widely used order parameters. This leads to a unifying picture of local order parameters, based on geometrical principles. Furthermore, we find with the Minkowski tensor analysis crystallization in jammed sphere packs at the random closed packing point

Die vorliegende Dissertation befasst sich mit der ramanspektroskopischen Charakterisierung von Galliumnitrid (GaN). Der Zusammenhang zwischen Waferkrümmung und mechanischer Restspannungen wird diskutiert. Mit Hilfe konfokaler Mikro-Ramanmessungen wurden Dotierprofile nachgewiesen sowie die Ladungsträgerkonzentration und -beweglichkeit ermittelt. Sämtliche Ramantensorelemente von wz-GaN wurden erstmals durch die Anwendung verschiedener Streugeometrien bestimmt. Eine neu entwickelte Vorwärtsstreuanordnung ermöglichte die Beobachtung von Phonon-Polaritonen. Es konnte gezeigt werden, dass von der theoretischen und experimentellen Betrachtung der Ramanstreuintensitäten dieser Elementaranregungen eindeutig das Vorzeichen der Faust-Henry-Koeffizienten von wz-GaN abgeleitet werden kann. Im Rahmen dieser Arbeit wurden alle Faust-Henry-Koeffizienten für GaN experimentell bestimmt. / This thesis focuses on special aspects of the Raman spectroscopical characterization of wurtzite gallium nitride (wz-GaN). The correlation between wafer curvature and residual stress is discussed. By means of confocal micro-Raman measurements doping profiles were detected as well as the density and mobility of free charge carriers were deduced. All Raman scattering cross sections of wz-GaN were determined the first time using different scattering configurations. A novel method for near-forward scattering was developed in order to observe phonon polaritons with pure symmetry. It is shown that the theoretical and experimental consideration of the Raman scattering efficiency of these elementary excitations allow for determining the sign of the Faust-Henry coefficients of wz-GaN unambiguously. The Faust-Henry coefficients of GaN were deduced from Raman scattering efficiencies of corresponding TO and LO phonons.

Cosmology is the study of the universe as a whole, and attempts to describe the behaviour of the universe mathematically. The simplest relativistic cosmological models are derived from Einstein's field equations with the assumptions of isotropy and homogeneity. In this thesis, a few simple cosmological models will be derived and evaluated with respect to their description of our universe i.e., how well they match observational data from e.g., the cosmic background radiation and redshift from distant supernovae. The models are derived from Einstein's field equations, which is why a large portion of the thesis will lay the ground work for the field equations by introducing and explaining the language of tensors. / Kosmologi är läran om universum i stort samt dess matematiska beskrivning. De enklaste relativistiska kosmologiska modellerna kan härledas från Einsteins fältekvationer med hjälp av antaganden om isotropi och homogenitet. I denna rapport kommer ett par av de enklaste modellerna att härledas, samt evalueras baserat på hur väl de beskriver vårt universum, det vill säga hur bra de passar de observationer som gjorts på exempelvis den kosmiska bakgrundsstrålningen och rödskifte från avlägsna supernovor. Modellerna härleds utifrån Einsteins fältekvationer, varför en stor del av rapporten består av en introduktion till tensoranalys.

No description available.

Les dernières décennies ont donné lieux à d'énormes progrès dans la simulation numérique des phénomènes physiques. D'une part grâce au raffinement des méthodes de discrétisation des équations aux dérivées partielles. Et d'autre part grâce à l'explosion de la puissance de calcul disponible. Pourtant, de nombreux problèmes soulevés en ingénierie tels que les simulations multi-physiques, les problèmes d'optimisation et de contrôle restent souvent hors de portée. Le dénominateur commun de ces problèmes est le fléau des dimensions. Un simple problème tridimensionnel requiert des centaines de millions de points de discrétisation auxquels il faut souvent ajouter des milliers de pas de temps pour capturer des dynamiques complexes. L'avènement des supercalculateurs permet de générer des simulations de plus en plus fines au prix de données gigantesques qui sont régulièrement de l'ordre du pétaoctet. Malgré tout, cela n'autorise pas une résolution ``exacte'' des problèmes requérant l'utilisation de plusieurs paramètres. L'une des voies envisagées pour résoudre ces difficultés est de proposer des représentations ne souffrant plus du fléau de la dimension. Ces représentations que l'on appelle séparées sont en fait un changement de paradigme. Elles vont convertir des objets tensoriels dont la croissance est exponentielle $n^d$ en fonction du nombre de dimensions $d$ en une représentation approchée dont la taille est linéaire en $d$. Pour le traitement des données tensorielles, une vaste littérature a émergé ces dernières années dans le domaine des mathématiques appliquées.Afin de faciliter leurs utilisations dans la communauté des mécaniciens et en particulier pour la simulation en mécanique des fluides, ce manuscrit présente dans un vocabulaire rigoureux mais accessible les formats de représentation des tenseurs et propose une étude détaillée des algorithmes de décomposition de données qui y sont associées. L'accent est porté sur l'utilisation de ces méthodes, aussi la bibliothèque de calcul texttt{pydecomp} développée est utilisée pour comparer l'efficacité de ces méthodes sur un ensemble de cas qui se veut représentatif. La seconde partie de ce manuscrit met en avant l'étude de l'écoulement dans une cavité entraînée à haut nombre de Reynolds. Cet écoulement propose une physique très riche (séquence de bifurcation de Hopf) qui doit être étudiée en amont de la construction de modèle réduit. Cette étude est enrichie par l'utilisation de la décomposition orthogonale aux valeurs propres (POD). Enfin une approche de construction ``physique'', qui diffère notablement des développements récents pour les modèles d'ordre réduit, est proposée. La connaissance détaillée de l'écoulement permet de construire un modèle réduit simple basé sur la mise à l'échelle des fréquences d'oscillation (time-scaling) et des techniques d'interpolation classiques (Lagrange,..). / Numerical simulation has experienced tremendous improvements in the last decadesdriven by massive growth of computing power. Exascale computing has beenachieved this year and will allow solving ever more complex problems. But suchlarge systems produce colossal amounts of data which leads to its own difficulties.Moreover, many engineering problems such as multiphysics or optimisation andcontrol, require far more power that any computer architecture could achievewithin the current scientific computing paradigm. In this thesis, we proposeto shift the paradigm in order to break the curse of dimensionality byintroducing decomposition and building reduced order models (ROM) for complexfluid flows.This manuscript is organized into two parts. The first one proposes an extendedreview of data reduction techniques and intends to bridge between appliedmathematics community and the computational mechanics one. Thus, foundingbivariate separation is studied, including discussions on the equivalence ofproper orthogonal decomposition (POD, continuous framework) and singular valuedecomposition (SVD, discrete matrices). Then a wide review of tensor formats andtheir approximation is proposed. Such work has already been provided in theliterature but either on separate papers or into a purely applied mathematicsframework. Here, we offer to the data enthusiast scientist a comparison ofCanonical, Tucker, Hierarchical and Tensor train formats including theirapproximation algorithms. Their relative benefits are studied both theoreticallyand numerically thanks to the python library texttt{pydecomp} that wasdeveloped during this thesis. A careful analysis of the link between continuousand discrete methods is performed. Finally, we conclude that for mostapplications ST-HOSVD is best when the number of dimensions $d$ lower than fourand TT-SVD (or their POD equivalent) when $d$ grows larger.The second part is centered on a complex fluid dynamics flow, in particular thesingular lid driven cavity at high Reynolds number. This flow exhibits a seriesof Hopf bifurcation which are known to be hard to capture accurately which iswhy a detailed analysis was performed both with classical tools and POD. Oncethis flow has been characterized, emph{time-scaling}, a new ``physics based''interpolation ROM is presented on internal and external flows. This methodsgives encouraging results while excluding recent advanced developments in thearea such as EIM or Grassmann manifold interpolation.

In many fields of science, engineering, and economics large amounts of data are stored and there is a need to analyze these data in order to extract information for various purposes. Data mining is a general concept involving different tools for performing this kind of analysis. The development of mathematical models and efficient algorithms is of key importance. In this thesis we discuss algorithms for the reduced rank regression problem and algorithms for the computation of the best multilinear rank approximation of tensors. The first two papers deal with the reduced rank regression problem, which is encountered in the field of state-space subspace system identification. More specifically the problem is \[ \min_{\rank(X) = k} \det (B - X A)(B - X A)\tp, \] where $A$ and $B$ are given matrices and we want to find $X$ under a certain rank condition that minimizes the determinant. This problem is not properly stated since it involves implicit assumptions on $A$ and $B$ so that $(B - X A)(B - X A)\tp$ is never singular. This deficiency of the determinant criterion is fixed by generalizing the minimization criterion to rank reduction and volume minimization of the objective matrix. The volume of a matrix is defined as the product of its nonzero singular values. We give an algorithm that solves the generalized problem and identify properties of the input and output signals causing a singular objective matrix. Classification problems occur in many applications. The task is to determine the label or class of an unknown object. The third paper concerns with classification of handwritten digits in the context of tensors or multidimensional data arrays. Tensor and multilinear algebra is an area that attracts more and more attention because of the multidimensional structure of the collected data in various applications. Two classification algorithms are given based on the higher order singular value decomposition (HOSVD). The main algorithm makes a data reduction using HOSVD of 98--99 \% prior the construction of the class models. The models are computed as a set of orthonormal bases spanning the dominant subspaces for the different classes. An unknown digit is expressed as a linear combination of the basis vectors. The resulting algorithm achieves 5\% in classification error with fairly low amount of computations. The remaining two papers discuss computational methods for the best multilinear rank approximation problem \[ \min_{\cB} \| \cA - \cB\| \] where $\cA$ is a given tensor and we seek the best low multilinear rank approximation tensor $\cB$. This is a generalization of the best low rank matrix approximation problem. It is well known that for matrices the solution is given by truncating the singular values in the singular value decomposition (SVD) of the matrix. But for tensors in general the truncated HOSVD does not give an optimal approximation. For example, a third order tensor $\cB \in \RR^{I \x J \x K}$ with rank$(\cB) = (r_1,r_2,r_3)$ can be written as the product \[ \cB = \tml{X,Y,Z}{\cC}, \qquad b_{ijk}=\sum_{\lambda,\mu,\nu} x_{i\lambda} y_{j\mu} z_{k\nu} c_{\lambda\mu\nu}, \] where $\cC \in \RR^{r_1 \x r_2 \x r_3}$ and $X \in \RR^{I \times r_1}$, $Y \in \RR^{J \times r_2}$, and $Z \in \RR^{K \times r_3}$ are matrices of full column rank. Since it is no restriction to assume that $X$, $Y$, and $Z$ have orthonormal columns and due to these constraints, the approximation problem can be considered as a nonlinear optimization problem defined on a product of Grassmann manifolds. We introduce novel techniques for multilinear algebraic manipulations enabling means for theoretical analysis and algorithmic implementation. These techniques are used to solve the approximation problem using Newton and Quasi-Newton methods specifically adapted to operate on products of Grassmann manifolds. The presented algorithms are suited for small, large and sparse problems and, when applied on difficult problems, they clearly outperform alternating least squares methods, which are standard in the field.

Made available in DSpace on 2014-07-29T16:02:18Z (GMT). No. of bitstreams: 1 Dissertacao Lidiane Mayumi Hieda.pdf: 465165 bytes, checksum: a5ce3ff47770899f6a4edcca3e40ed69 (MD5) Previous issue date: 2011-05-13 / This work was based on papers On Compact Minimal Surfaces with non-negative Gaussian Curvature in a Space of Constant Curvature: I and Minimal Surfaces with Constant Curvature in 4-dimensional Space Forms, by Katsuei Kenmotsu, consisting in the classification of minimal surfaces with constant Gaussian curvature K in a 4-dimensional space form without any global assumption. We will show that an isometric minimal immersion x: M2(K) &#8594; M4(c), where c is sectional curvature, is either totally geodesic, or locally Clifford Torus, or locally a Veronese surface. As a corollary, we have that there is not isometric minimal immersions with constant negative Gaussian curvature into unit sphere S4(1) even locally. / Este trabalho foi baseado nos artigos On CompactMinimal Surfaces with non-negative Gaussian Curvature in a Space of Constant Curvature: I e Minimal Surfaces with Constant Curvature in 4-dimensional Space Forms de Katsuei Kenmotsu que consistem em classificar superfícies mínimas com curvatura Gaussiana constante K nas formas espaciais 4-dimensionais, sem alguma hipótese global. Mostraremos que uma imersão isométrica mínima x : M2(K) &#8594; M4(c), onde c é a curvatura seccional, ou é totalmente geodésica, ou localmente um Toro de Clifford, ou localmente uma superfície de Veronese. Como corolário, temos que não existe uma imersão isométrica mínima com curvatura Gaussiana constante negativa numa esfera unitária S4(1) mesmo que localmente.