This thesis presents training of an end-to-end autoencoder model using the transformer, with an encoder that can encode sentences into fixed-length latent vectors and a decoder that can reconstruct the sentences using image representations. Encoding and decoding sentences to and from these image representations are central to the model design. This method allows new sentences to be generated by traversing the Euclidean space, which makes vector arithmetic possible using sentences. Machines excel in dealing with concrete numbers and calculations, but do not possess an innate infrastructure designed to help them understand abstract concepts like natural language. In order for a machine to process language, scaffolding must be provided wherein the abstract concept becomes concrete. The main objective of this research is to provide such scaffolding so that machines can process human language in an intuitive manner.

In this thesis we consider an open question of Feige that asks whether there always exists a constantly Lipschitz bijection of an n2 -point subset of Z2 onto a regular grid [n] × [n] for every n ∈ N. We relate this question to an already resolved problem of the existence of a bounded positive measurable density in R2 that is not the Jacobian of any bilipschitz map. This problem was resolved by Burago and Kleiner [1], and independently, by McMullen [12]. We present the work of Burago and Kleiner, analyze its relation to Feige's problem and sug- gest a continuous formulation of Feige's question in a special case. Then we present the Burago-Kleiner density, make several observation about the properties of this density, and after that we construct a density that is everywhere nonrealizable as the Jacobian of a bilipschitz map. Subsequently, we discuss our continuous variant of Feige's question, provide several observation concerning it, and finally, we try to use the everywhere nonrealizable density constructed before to answer our continuous variant of Feige's question. However, this last task still remains incomplete. 1

This work focuses on efficient modelling and adaptive control of friction damping in bladed disks. To efficiently simulate the friction contact, a full-3D time-discrete contact model is reformulated and an analytical expression for the Jacobian matrix is derived that reduces the computation time drastically with respect to the classical finite difference method. The developed numerical solver is applied on bladed disks with shroud contact and the advantage of full-3D contact model compared to a quasi-3D contact model is presented. The developed numerical solver is also applied on bladed disks with strip damper and multiple friction contacts and obtained results are discussed. Furthermore, presence of higher harmonics in the nonlinear contact forces is analyzed and their effect on the excitation of the different nodal diameters of the bladed disk are systematically presented. The main parameters that influence the effectiveness of friction damping in bladed disks are engine excitation order,  contact stiffnesses,  friction coefficient, relative motion at the friction interface and the normal contact load. Due to variation in these parameters during operation, the obtained friction damping in practice may differ from the optimum value. Therefore, to control the normal load adaptively that will lead to an optimum damping in the system despite these variations, use of magnetostrictive actuator is proposed. The magnetostrictive material that develops an internal strain under the influence of an external magnetic field is employed to increase and decrease the normal contact load. A linearized model of the magnetostrictive actuator is used to characterize the magnetoelastic behavior of the actuator.  A nonlinear static contact analysis of the bladed disk reveals that a change of normal load more than 700 N can be achieved using a reasonable size of the actuator. This will give a very good control on friction damping once applied in practice. / <p>QC 20170310</p> / TurboPower

Neste texto é feita uma discussão sobre alguns resultados que fornecem condições suficientes para que um difeomorfismo local, do espaço euclideano n-dimensional nele próprio, seja injetivo. Dentro deste cenário, são exploradas as contribuições destes resultados na tentativa de solucionar conhecidas conjecturas no meio científico como a Conjectura Jacobiana e a Conjectura de Ponto Fixo. Do ponto de vista dinâmico, existem relações entre injetividade global e estabilidade assintótica global. Neste sentido, os resultados também são contextualizados com respeito a importantes conjecturas de estabilidade assintótica: Conjectura de Markus-Yamabe e o Problema de LaSalle / We present some results which give suficient conditions for a local diffeomorphism from the n-dimensional Euclidean space into itself be globally injective. Within this context, we consider some partial results addressed to solve the well known Fixed Point Conjecture and Jacobian Conjecture. From the dynamical point of view, there are connections between global injectivity and global asymptotic stability. In this way, we present a solution of the Markus-Yamabe Conjecture and of the LaSalle Problem

Dans cette thèse nous étudions quelques aspects des certains espaces de fonctions. D’une part nous nous intéressons aux singularités des applications W^{1,n} à valeurs dans la sphère unité S^n, et d’autre part, aux relèvements des applications W^{s,p} à valeurs dans le cercle S^1.La première partie concerne le problème de minimisation d’une énergie de type Dirichlet à poids. Les fonctions admissibles sont les fonctions continues hors d’un ensemble singulier donné prescrit par le bord d’un courant rectifiable. Nous obtenons la formule exacte, résultat qui améliore celui d’Alberto, Baldi et Orlando (2003). Il s’agit aussi d’une généralisation des résultats obtenus précédemment par Brezis, Coron, Lieb (1986), Almgren, Browder, Lieb (1988).La deuxième partie porte sur le meilleur contrôle des phases des applications uni-modulaires et elle se repose sur les travaux de Bourgain, Brezis, Mironescu (2000, 2002). A l’aide de quelques méthodes connues et des méthodes nouvelles, nous étudions des estimations optimales des semi-normes W^{s,p} des relèvements selon les différentes valeurs de s et de p. Nous obtenons aussi une nouvelle caractérisation de W^{s,p} pour s<1 en termes de semi-norme dyadique / In this thesis we study some aspects of certain functional spaces. On the one hand we focus on the singularities of maps W^{1, n} with values in the unit sphere S^n, and secondly, on liftings of maps W^{s, p} with values in the circle S^1.The first part concerns the minimization problem of a weighted Dirichlet energy. Admissible maps are functions which are continuous functions outside a given singular set prescribed by the boundary of a rectifiable current. We obtain the exact formula, which improves the result of Alberto, Baldi and Orlando (2003). In the same time, we generalize some results previously obtained by Brezis, Coron, Lieb (1986), Almgren, Browder, Lieb (1988).The second part focuses on the best control of unimodular maps and it is based on the work of Bourgain, Brezis, Mironescu (2000, 2002). Using some known methods and some new ones, we study optimal estimates of seminorms W^{s, p} of liftings, for different values of s and p. We also obtain a new characterization of the space W^{s, p} for s<1 in terms of dyadic seminorm

This study presents Computer-Aided Fixture Design Verification (CAFDV) - the methods and implementations to define, measure and optimize the quality of fixture designs. CAFDV verifies a fixture for its locating performance, machining surface accuracy, stability, and surface accessibility. CAFDV also optimizes a fixture for its locator layout design, initial clamping forces, and tolerance specification. The demand for CAFDV came from both fixture design engineers and today's supply chain managers. They need such a tool to inform them the quality of a fixture design, and to find potential problems before it is actually manufactured. For supply chain managers, they will also be able to quantitatively measure and control the product quality from vendors, with even little fixture design knowledge. CAFDV uses two models - one geometric and one kinetic - to represent, verify and optimize fixture designs. The geometric model uses the Jacobian Matrix to establish the relationship between workpiece-fixture displacements, and the kinetic model uses the Fixture Stiffness Matrix to link external forces with fixture deformation and workpiece displacement. Computer software for CAFDV has also been developed and integrated with CAD package I-DEAS TM. CAD integration and a friendly graphic user interface allows the user to have easy interactions with 3D models and visual feedback from analysis results.

The first goal of this Thesis is to contribute to the study of principally polarized abelian varieties (ppav), especially to the Schottky and the Torelli problems. Ppav admit a duality theory analogous to that of projective spaces, where the role played by hyperplanes in projective spaces is played by divisors representing the principal polarization. Thus, given a subvariety Y of a ppav, we can define its thetadual T(Y) as the set of divisors representing the principal polarization that contain this subvariety. This set admits a natural schematic structure (as defined by Pareschi and Popa). Jacobian and Prym varieties are classical examples of ppav constructed from curves. Besides, they are interesting because some properties of the curves involved in their construction are reflected in their geometry or in the geometry of some special subvarieties. For example, in the case of Jacobians we have the BrillNoether loci Wd ( W1 corresponds to the AbelJacobi curve) and in the case of Pryms we have the AbelPrym curve C. In chapter III, we study the schematic structure of the thetadual of the BrillNoether loci Wd and the AbelPrym curve. In the first case, we obtain with different methods, the result of Pareschi and Popa T(Wd)= Wgd1. In the case of the AbelPrym curve C, we get that T(C)=V², where V² is the second PrymBrillNoether locus with the schematic structure defined by Welters. Pareschi and Popa have proved a result for ppavs analogous to the Castelnuovo Lemma for projective spaces. That is, if (A,Θ) is a ppav of dimension g, then g+2 distinct points in general position with respect to Θ, but in special position with respect to 2Θ, have to be contained in a curve of minimal degree in A, i.e. an AbelJacobi curve. In particular, they obtain a Schottky result because A has to be a Jacobian variety and a Torelli result, because the curve is the intersection of all the divisors in |2Θ| that contain the g+2 points. In chapter IV, as Eisenbud and Harris have done in the projective Castelnuovo Lemma, we extend this result to possibly nonreduced finite schemes. The second goal of this thesis is the study of varieties of general type. Almost by definition, pluricanonical maps are the essential tool to study them. One of the main problems in this area is to find geometric or numerical conditions to guarantee that the mth pluricanonical map (for low m) induces a birational equivalence with its image. The classification of surfaces whose bicanonical map is nonbirational has attracted considerable interest among algebraic geometers. In chapter V, we give a sufficient numerical condition for the birationality of the bicanonical map of irregular varieties of arbitrary dimension. We also prove that, if X is a primitive variety, then it only admits very special fibrations to other irregular varieties. For primitive varieties we get that the following are equivalent: X is birational to a divisor Θ in an indecomposable ppav, the irregularity q(X) > dim X and the bicanonical map is nonbirational. When X is a primitive variety of general type and q(X) = dim X we prove, under certain conditions over the Stein factorization of the Albanese map, that the only possibility for the bicanonical map being nonbirational is that X is a double cover branched along a divisor in |2Θ|. These results extend to arbitrary dimension, wellknown theorems in the case of surfaces and curves. / El primer objectiu d'aquesta tesi és contribuir a l'estudi de les varietats abelianes principalment polaritzades (vapp), especialment als problemes de Schottky i Torelli. Les vapp admeten una teoria de dualitat anàloga a la dualitat dels espais projectius, on el paper que juguen els hiperplans de l'espai projectiu és substituït pels divisors que representen la polarització principal. Així doncs, donada una subvarietat Y d'una vapp, podem definir el seu thetadual T(Y) com el conjunt dels divisors que representen la polarització principal i contenen aquesta subvarietat. Aquest conjunt admet una estructura esquemàtica natural (tal i com la defineixen Pareschi i Popa). Les varietats Jacobianes i de Prym són exemples clàssics de vapp construïdes a partir de corbes. A més, són interessants perquè certes propietats de les corbes involucrades es veuen reflectides en elles o en algunes subvarietats especials. Per exemple, en el cas de les Jacobianes tenim els llocs de BrillNoether Wd ( W1 correspon a la corba d'AbelJacobi) i en el cas de les Pryms tenim la corba d'AbelPrym C. Al capítol III de la tesi s'estudia l'estructura esquemàtica del thetadual dels llocs de BrillNoether Wd i de la corba d'AbelPrym. En el primer cas, es reobté amb uns altres mètodes, el resultat de Pareschi i Popa T(Wd)= Wgd1. En el cas de la corba d'AbelPrym C, s'obté que T(C)=V², onV² és el segon lloc de PrymBrillNoether amb l'estructura esquemàtica definida per Welters. Pareschi i Popa han demostrat un resultat anàleg per les vapp al Lemma de Castelnuovo pels espais projectius. És a dir, si (A,Θ) és una vapp de dimensió g, aleshores g+2 punts en posició general respecte Θ, però en posició especial respecte 2Θ, han d'estar continguts en una corba de grau minimal a A, i.e. una corba d'AbelJacobi. En particular, s'obté un resultat de Schottky ja que A ha de ser una Jacobiana i un resultat de Torelli, ja que la corba és la intersecció de tots els divisors de |2Θ| que contenen els g+2 punts. Al capítol IV, tal i com Eisenbud i Harris van fer en el cas projectiu, s'estén aquest resultat a esquemes finits possiblement no reduïts. El segon objectiu d'aquesta tesi és contribuir a l'estudi de les varietats de tipus general. Pràcticament per definició, les aplicacions pluricanòniques són essencials pel seu estudi. Un dels problemes principals de l'àrea és donar condicions geomètriques o numèriques per assegurar que la mèsima aplicació pluricanònica (per m baix) indueix una equivalència biracional amb la imatge. La classificació de les superfícies que tenen l'aplicació bicanònica no biracional ha atret l'atenció de molts geòmetres algebraics. Al capítol V, es dóna un criteri numèric suficient per assegurar la biracionalitat de l'aplicació bicanònica de les varietats irregulars de dimensió arbitrària. També es demostra que si X és una varietat primitiva, aleshores només admet fibracions molt especials a altres varietats irregulars. Per aquestes varietats s'obté que és equivalent que X sigui biracional a un divisor Θ en una vapp indescomponible, a què la irregularitat q(X) > dim X i l'aplicació bicanònica sigui no biracional. Quan X és una varietat primitiva de tipus general i q(X) = dim X es demostra sota certes condicions de la descomposició de Stein del morfisme d'Albanese, que l'única possibilitat per tal que l'aplicació bicanònica sigui no biracional és que X sigui un recobriment doble sobre una vapp ramificat al llarg d'un divisor a |2Θ|. Aquest resultats estenen a dimensió arbitrària, teoremes ben coneguts en el cas de superfícies i corbes.

Robots moving about in complex environments must be capable of determining and performing difficult motion sequences to accomplish tasks. As the tasks become more complicated, robots with greater dexterity are required. An increase in the number of degrees of freedom and a desire for autonomy in uncertain environments with real-time requirements leaves much room for improvement in the current popular robot motion planning algorithms. In this thesis, state of the art robot motion planning techniques are surveyed. A solution to the general movers problem in the context of motion planning for robots is presented. The proposed robot motion planner solves the general movers problem using a sample-based tree planner combined with an incremental simulator. The robot motion planner is demonstrated both in simulation and the real world. Experiments are conducted and the results analyzed. Based on the results, methods for tuning the robot motion planner to improve the performance are proposed.

La simulation numérique de systèmes d'équations différentielles raides ordinaires ou algébriques est devenue partie intégrante dans le processus de conception des systèmes mécaniques à dynamiques complexes. L'objet de ce travail est de développer des méthodes numériques pour réduire les temps de calcul par le parallélisme en suivant deux axes : interne à l'intégrateur numérique, et au niveau de la décomposition de l'intervalle de temps. Nous montrons l'efficacité limitée au nombre d'étapes de la parallélisation à travers les méthodes de Runge-Kutta et DIMSIM. Nous développons alors une méthodologie pour appliquer le complément de Schur sur le système linéarisé intervenant dans les intégrateurs par l'introduction d'un masque de dépendance construit automatiquement lors de la mise en équations du modèle. Finalement, nous étendons le complément de Schur aux méthodes de type "Krylov Matrix Free". La décomposition en temps est d'abord vue par la résolution globale des pas de temps dont nous traitons la parallélisation du solveur non-linéaire (point fixe, Newton-Krylov et accélération de Steffensen). Nous introduisons les méthodes de tirs à deux niveaux, comme Parareal et Pita dont nous redéfinissons les finesses de grilles pour résoudre les problèmes raides pour lesquels leur efficacité parallèle est limitée. Les estimateurs de l'erreur globale, nous permettent de construire une extension parallèle de l'extrapolation de Richardson pour remplacer le premier niveau de calcul. Et nous proposons une parallélisation de la méthode de correction du résidu.

This thesis is concerned with the efficient computation of Jacobian matrices of nonlinear vector maps using automatic differentiation (AD). Specifically, we propose the use of two directed edge separator methods, the weighted minimum separator and natural order separator methods, to exploit the structure of the computational graph of the nonlinear system.This allows for the efficient determination of the Jacobian matrix using AD software. We will illustrate the promise of this approach with computational experiments.