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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.

In this dissertation we study free boundary problems that model the evolution of interfaces in the presence of elasticity, such as thin film profiles and material void boundaries. These problems are characterized by the competition between the elastic bulk energy and the anisotropic surface energy. First, we consider the evolution equation with curvature regularization that models the motion of a two-dimensional thin film by evaporation-condensation on a rigid substrate. The film is strained due to the mismatch between the crystalline lattices of the two materials and anisotropy is taken into account. We present the results contained in [62] where the author establishes short time existence, uniqueness and regularity of the solution using De Giorgi’s minimizing movements to exploit the L2 -gradient flow structure of the equation. This seems to be the first analytical result for the evaporation-condensation case in the presence of elasticity. Second, we consider the relaxed energy introduced in [20] that depends on admissible pairs (E, u) of sets E and functions u defined only outside of E. For dimension three this energy appears in the study of the material voids in solids, where the pairs (E, u) are interpreted as the admissible configurations that consist of void regions E in the space and of displacements u of the atoms of the crystal. We provide the precise mathematical framework that guarantees the existence of minimal energy pairs (E, u). Then, we establish that for every minimal configuration (E, u), the function u is C 1,γ loc -regular outside an essentially closed subset of E. No hypothesis of starshapedness is assumed on the voids and all the results that are contained in [18] hold true for every dimension d ≥ 2.

Dumb holes (also known as acoustic black holes) are fluid flows which include an "acoustic horizon:" a surface, analogous to a gravitational horizon, beyond which sound may pass but never classically return. Soundwaves in these flows will therefore experience "effective geometries" which are identical to black hole spacetimes up to a conformal factor. By adjusting the parameters of the fluid flow, it is possible to create an effective geometry which is conformal to the Anti-de Sitter black hole spacetime- a geometry which has recieved a great deal of attention in recent years due to its conjectured holographic duality to Conformal Field Theories. While we would not expect an acoustic analogue of the AdS-CFT correspondence to exist, this dumb hole provides a means, at least in principle, of experimentally testing the theoretical properties of the AdS spacetime. In particular, I have calculated the quasinormal mode spectrum of this acoustic geometry. / Graduate / 0986 / 0753 / jpbabb@yahoo.ca

Most strength grading machines on the European market use an averagemodulus of elasticity (MOE), estimated on a relatively large distance along awood member, as the indicating property (IP) to bending strength. Theaccuracy of such grading machines in terms of coefficient of determination israther low at R2 ≈ 0.5. This research is motivated by a desire to increase theaccuracy of the strength grading in the industry today. The aim of the presentstudy is to contribute knowledge of local variation in bending stiffness/MOEwith high resolution and thus locate weak sections due to stiffness reducingfeatures (the most important is knots) for structural timber.The present study introduces three methods that involve structural dynamics,classical beam theory and optical measurement to assess local wood stiffness.Specifically: The dynamic method, in which a wood member is treated as an ordinaryphysical structure and the local stiffness is studied by exploring itsdynamic properties. In Method II, a bending MOE profile is established based on local fibre angle information. The local fibre orientation is detected through highresolution laser scanning based on the tracheid effect.  For Method III, a bending MOE profile is established using surfacestrain information under four-point bending. A high resolution strainfield is obtained using the digital image correlation (DIC) technique. From the present study, the two latter methods are more favourable inevaluating the local stiffness within a piece of structural timber. Moreover, thestudy reveals that the established bending MOE profiles using the two lattermethods, i.e. based on information of the local fibre angle and surface strain,agree reasonably well. However, for some patterns of knot clusters, the localbending MOE, calculated on the basis of fibre angles, is significantly higherthan the local bending MOE estimated on the basis of surface strain. / De flesta av de utrustningar för hållfasthetssortering som utnyttjas på deneuropeiska marknaden använder ett medelvärde på elasticitetsmodulen(MOE), beräknat på en relativt stor längd av en sågad planka, som indikativparameter (IP). Sådan hållfasthetssortering ger en noggrannhet i termer avförklaringsgrad på R2 ≈ 0.5, vilket är ganska lågt. Arbetet i denna studiemotiveras av en önskan att öka noggrannheten i hållfasthetssorteringen. Syftetmed denna studie är att bidra med kunskap om lokala variationer iböjstyvhet/MOE med hög upplösning och att lokalisera veka snitt (där kvistarär den viktigaste försvagande faktorn) för konstruktionsvirke.Den aktuella studien introducerar tre metoder som omfattar strukturdynamik,klassisk balkteori och optisk mätning vid bedömningen av lokal styvhet imaterialet. Specifikt:  Metod I, där den lokala böjstyvheten studerades genom de dynamiskaegenskaperna såsom egenfrekvens och modform.  Metod II, där en MOE profil beräknas på basis av information om lokalafibervinklar på ett virkesstyckes ytor. Den lokala fiberorienteringen mätsmed högupplöst laserskanning baserad på den så kallade trakeideffekten.  Metod III, där en MOE-profil fastställdes med hjälp avtöjningsinformation för en hel flatsida av en planka belastad med konstantböjmoment. Det högupplösta töjningsfältet erhölls med hjälp av teknikför Digital Image Correlation (DIC). Studien visar att de två sistnämnda metoderna är mycket lämpade för attutvärdera den lokala styvheten i ett virkesstycke. Dessutom visar studien att deMOE-profiler som togs fram med hjälp av de två sistnämnda metoderna,vilka baseras på information om lokala fibervinklar och töjningsfältet på ytan,stämde överens för större delen av virkesstycket. För visa kvistgrupper kan dock den lokala böjstyvheten högre med metoden baserad på fibervinklar.

Modelo de predicción de la geometría final de una pieza de chapa, radio y ángulo de doblado final, producida mediante un proceso de doblado al aire. / Prediction model of final geometry of sheet metal part, radius and final bending angle, manufactured by air free -V bending process.

This study is an attempt to simulate aspects of human visual perception by automating the detection of specific types of objects in digital images. The success of the methods attempted here was measured by how well results of experiments corresponded to what a typical human’s assessment of the data might be. The subject of the study was images of live fish taken underwater by digital video or digital still cameras. It is desirable to be able to automate the processing of such data for efficient stock assessment for fisheries management. In this study some well known statistical pattern classification techniques were tested and new syntactical/ structural pattern recognition techniques were developed. For testing of statistical pattern classification, the pixels belonging to fish were separated from the background pixels and the EM algorithm for Gaussian mixture models was used to locate clusters of pixels. The means and the covariance matrices for the components of the model were used to indicate the location, size and shape of the clusters. Because the number of components in the mixture is unknown, the EM algorithm has to be run a number of times with different numbers of components and then the best model chosen using a model selection criterion. The AIC (Akaike Information Criterion) and the MDL (Minimum Description Length) were tested.The MDL was found to estimate the numbers of clusters of pixels more accurately than the AIC, which tended to overestimate cluster numbers. In order to reduce problems caused by initialisation of the EM algorithm (i.e. starting positions of mixtures and number of mixtures), the Dynamic Cluster Finding algorithm (DCF) was developed (based on the Dog-Rabbit strategy). This algorithm can produce an estimate of the locations and numbers of clusters of pixels. The Dog-Rabbit strategy is based on early studies of learning behaviour in neurons. The main difference between Dog-Rabbit and DCF is that DCF is based on a toroidal topology which removes the tendency of cluster locators to migrate to the centre of mass of the data set and miss clusters near the edges of the image. In the second approach to the problem, data was extracted from the image using an edge detector. The edges from a reference object were compared with the edges from a new image to determine if the object occurred in the new image. In order to compare edges, the edge pixels were first assembled into curves using an UpWrite procedure; then the curves were smoothed by fitting parametric cubic polynomials. Finally the curves were converted to arrays of numbers which represented the signed curvature of the curves at regular intervals. Sets of curves from different images can be compared by comparing the arrays of signed curvature values, as well as the relative orientations and locations of the curves. Discrepancy values were calculated to indicate how well curves and sets of curves matched the reference object. The total length of all matched curves was used to indicate what fraction of the reference object was found in the new image. The curve matching procedure gave results which corresponded well with what a human being being might observe.

In this thesis we study several properties of finitely presented groups, through the unifying paradigm of encoding sought-after group properties into presentations and detecting group properties from presentations, in the context of Geometric Group Theory. A group law is said to be detectable in power subgroups if, for all coprime m and n, a group G satisfies the law if and only if the power subgroups G(^m) and G(ⁿ) both satisfy the law. We prove that for all positive integers c, nilpotency of class at most c is detectable in power subgroups, as is the k-Engel law for k at most 4. In contrast, detectability in power subgroups fails for solvability of given derived length: we construct a finite group W such that W(²) and W(³) are metabelian but W has derived length 3. We analyse the complexity of the detectability of commutativity in power subgroups, in terms of finite presentations that encode a proof of the result. We construct a census of two-generator one-relator groups of relator length at most 9, with complete determination of isomorphism type, and verify a conjecture regarding conditions under which such groups are automatic. Furthermore, we introduce a family of one-relator groups and classify which of them act properly cocompactly on complete CAT(0) spaces; the non-CAT(0) examples are counterexamples to a variation on the aforementioned conjecture. For a subclass, we establish automaticity, which is needed for the census. The deficiency of a group is the maximum over all presentations for that group of the number of generators minus the number of relators. Every finite group has non-positive deficiency. For every prime p we construct finite p-groups of arbitrary negative deficiency, and thereby complete Kotschick's proposed classification of the integers which are deficiencies of Kähler groups. We explore variations and embellishments of our basic construction, which require subtle Schur multiplier computations, and we investigate the conditions on inputs to the construction that are necessary for success. A well-known question asks whether any two non-isometric finite volume hyperbolic 3-manifolds are distinguished from each other by the finite quotients of their fundamental groups. At present, this has been proved only when one of the manifolds is a once-punctured torus bundle over the circle. We give substantial computational evidence in support of a positive answer, by showing that no two manifolds in the SnapPea census of 72 942 finite volume hyperbolic 3-manifolds have the same finite quotients. We determine examples of sizeable graphs, as required to construct finitely presented non-hyperbolic subgroups of hyperbolic groups, which have the fewest vertices possible modulo mild topological assumptions.

Dans cette thèse, nous nous intéressons à différents opérateurs d'inf-convolutions et à leurs applications à une classe d'inégalités de transport générales, plus spécifiquement sur les graphes. Notre objet de recherche s'inscrit donc dans les théories du transport de mesure et de l'analyse fonctionnelle. En introduisant une notion de gradient adapté au cadre discret (et plus généralement à tout espace métrique dont les boules sont compactes), nous prouvons que certains opérateurs d'inf-convolution sont solutions d'une inéquation d'Hamilton Jacobi sur les graphes. Ce résultat nous permet d'étendre au cadre discret un théorème classique de Bobkov, Gentil et Ledoux. Plus précisément nous montrons que des inégalités de transport faible (adaptées au cadre discret) sont équivalentes, sur un graphe, à l'hypercontractivité des opérateurs d'inf-convolutions. On en déduit plusieurs résultats concernant différentes inégalités fonctionnelles, dont celle de Sobolev logarithmique et de transport faible. Nous étudions par ailleurs les propriétés générales de différents opérateurs d'inf-convolutions, incluant le précédent, mais aussi un opérateur relié à un modèle issu de la physique (et au phénomène de grande déviation), toujours sur les graphes (dérivabilités, convexité, points extremum etc.). Dans un deuxième temps, nous nous intéressons aux liens entre différentes notions de courbure de Ricci sur les graphes -- proposées récemment par plusieurs auteurs -- et les inégalités fonctionnelles de type transport-entropie, ou transport-information associées à une chaîne de Markov. Nous obtenons également une borne supérieure sur le diamètre d'un graphe dont la courbure, en un certain sens, est minorée, un résultat à la Bonnet-Myers. Enfin, en nous restreignant au cas de la dimension 1, sur la droite réelle, nous obtenons une caractérisation d'une inégalité de transport faible et de l'inégalité de Sobolev logarithmique restreinte aux fonctions convexes. Ces résultats utilisent des propriétés géométriques liés à l'ordre convexe. / In this thesis, we interest in different inf-convolution operators and their applications to a class of general transportation inequalities, more specifically in the graphs. Therefore, our research topic fits in the theories of transportation and functional analysis. By introducing a gradient notion adapting to a discrete space (more generally to all space in which all closed balls are compact), we prove that some inf-convolution operators are solutions of a Hamilton-Jacobi's inequation. This result allows us to extend a classical theorem from Bobkov, Gentil and Ledoux. More precisely, we prove that, in a graph, some weak transport inequalities are equivalent to the hypercontractivity of inf-convolution operators. Thanks to this result, we deduce some properties concerning different functional inequalities, including Log-Sobolev inequalities and weak-transport inequalities. Besides, we study some general properties (differentiability, convexity, extreme points etc.) of different inf-convolution operators, including the one before, but also an operator related to a physical model (and to a large deviation phenomenon). We stay always in a graph. Secondly, we interest in connections between different notions of discrete Ricci curvature on the graphs which are proposed by several authors in the recent years, and functional inequalities of type transport-entropy, or transport-information related to a Markov chain. We also obtain an extension of Bonnet-Myers' result: an upper bound on the diameter of a graph of which the curvature is floored in some ways. Finally, restricting in the real line, we obtains a characterisation of a weak transport inequality and a log-Sobolev inequality restricted to convex functions. These results are from the geometrical properties related to the convex ordering.

Ce travail, à l'interface entre les mathématiques appliquées et la physique, s'articule autour de la modélisation numérique des vésicules biologiques, un modéle pour les globules rouges du sang. Pour cela, le modéle de Canham et Helfrich est adopté pour décrire le comportement des vésicules. La modélisation numérique utilise la méthode des fonctions de niveau dans un cadre éléments finis. Un nouvel algorithme de résolution numérique combinant une technique de multiplicateurs de Lagrange avec une adaptation automatique de maillages garantit la conservation exacte des volumes et des surfaces. Cet algorithme permet donc de dépasser une limitation cruciale actuelle de la méthode des fonctions de niveau, à savoir les pertes de masse couramment observées dans ce type de problémes. De plus, les propriétés de convergence de la méthode des fonctions de niveau se trouvent ainsi grandement améliorées, comme l'indiquent de nombreux tests numériques. Ces tests comprennent notamment des problémes d'advection élémentaires, des mouvements par courbure moyenne ainsi que des mouvements par diffusion de surface. Concernant l'équilibre statique des vésicules, une condition générale d'équilibre d'Euler-Lagrange est obtenue à l'aide d'outils de dérivation de forme. En dynamique, le mouvement d'une vésicule sous l'action d'un écoulement de cisaillement est étudié dans le cadre des nombres de Reynolds élevés. L'effet du confinement est considéré, et les régimes classiques de chenille de char et de basculement sont retrouvés. Finalement, pour la premiére fois, l'effet des termes inertiels est étudié et on montre qu'au delà d'une valeur critique du nombre de Reynolds, la vésicule passe d'un mouvement de basculement à un mouvement de chenille de char. / This work, at the interface between the Applied Mathematics and Physics is connected about the numerical modelisation of biological vesicles, a pattern for the red blood cells. For this reason, the pattern of Canham and Helfrich is adopted to describe the behaviour of the vesicles. The numerical modelisation uses the Level Set method in finite element framework. A new algorithm of numerical resolution combining one technique of Lagrange multipliers with an automatic mesh adaptation ensures the accurate conservation of volumes and surfaces. Thus this algorithm enables to exceed an existing crucial restriction of the Level Set method, that's to say, the wastes of mass usually noticed in this kind of problems. Moreover, the proprieties of convergence of the Level Set method are thus much more improved, as shown in many numerical tests. Those tests chiefly include elementary problems of advection, motions by mean curvature just as motions by spread of surface. Concerning the static equilibrum of the vesicles, a mechanical equilibrum equation (Euler-Lagrange equation) of a vesicle membrane under a generalized elastic bending energy is obtained and the approach is based on shape optimization tools. In dynamics, the motion of a vesicle under the effect of a shear flow is elaborated in the frames of reference of high Reynolds numbers. The effect of confinement is respected, and the standard regimes of tank-treading and of tumbling motion are found again. Finally, for the first time, the effect of the inertia terms is elaborated and we show that beyond a critical value of the number of Reynolds the vesicle passes from a tumbling motion to a tank-treading motion.