Crystallographic texture and its evolution are known to be major sources of anisotropy in polycrystalline metals. Highly simplified phenomenological models cannot usually provide reliable predictions of the materials anisotropy under complex deformation paths, and lack the fidelity needed to optimize the microstructure and mechanical properties during the production process. On the other hand, physics-based models such as crystal plasticity theories have demonstrated remarkable success in predicting the anisotropic mechanical response in polycrystalline metals and the evolution of underlying texture in finite plastic deformation. However, the integration of crystal plasticity models with finite element (FE) simulations tools (called CPFEM) is extremely computationally expensive, and has not been adopted broadly by the advanced materials development community. The current dissertation has mainly focused on addressing the challenges associated with integrating the recently developed spectral database approach with a commercial FE tool to permit computationally efficient simulations of heterogeneous deformations using crystal plasticity theories. More specifically, the spectral database approach to crystal plasticity solutions was successfully integrated with the implicit version of the FE package ABAQUS through a user materials subroutine, UMAT, to conduct more efficient CPFEM simulations on both fcc and bcc polycrystalline materials. It is observed that implementing the crystal plasticity spectral database in a FE code produced excellent predictions similar to the classical CPFEM, but at a significantly faster computational speed. Furthermore, an important application of the CPFEM for the extraction of crystal level plasticity parameters in multiphase materials has been demonstrated in this dissertation. More specifically, CPFEM along with a recently developed data analysis approach for spherical nanoindentation and Orientation Imaging Microscopy (OIM) have been used to extract the critical resolved shear stress of the ferrite phase in dual phase steels. This new methodology offers a novel efficient tool for the extraction of crystal level hardening parameters in any single or multiphase materials.

Plusieurs familles de fonctions spéciales de plusieurs variables, appelées fonctions d'orbites, sont définies dans le contexte des groupes de Weyl de groupes de Lie simples compacts/d'algèbres de Lie simples. Ces fonctions sont étudiées depuis près d'un siècle en raison de leur lien avec les caractères des représentations irréductibles des algèbres de Lie simples, mais également de par leurs symétries et orthogonalités. Nous sommes principalement intéressés par la description des relations d'orthogonalité discrète et des transformations discrètes correspondantes, transformations qui permettent l'utilisation des fonctions d'orbites dans le traitement de données multidimensionnelles. Cette description est donnée pour les groupes de Weyl dont les racines ont deux longueurs différentes, en particulier pour les groupes de rang $2$ dans le cas des fonctions d'orbites du type $E$ et pour les groupes de rang $3$ dans le cas de toutes les autres fonctions d'orbites. / Several families of multivariable special functions, called orbit functions, are defined in the context of Weyl groups of compact simple Lie groups/Lie algebras. These functions have been studied for almost a century now because of their relation to characters of irreducible representations of Lie algebras, their symmetries and orthogonalities. Our main interest is the description of discrete orthogonality relations and their corresponding discrete transforms which allow the applications of orbit functions in the processing of multidimensional data. This description is provided for the Weyl group of different lengths of root, in particular groups of rank 2 for so-called $E-$orbit functions and of rank 3 for all the other families of special functions.

Dans cette thèse, nous étudions des transformées de type Radon sur certains espaces symétriques. Une transformée de type Radon associe à toute fonction continue à support compact sur une variété $M$ ses intégrales sur une classe $Xi$ de sous-variétés de $M$. Le problème sur lequel nous nous concentrons est l'inversion d'une telle transformée, c'est-à-dire déterminer la fonction à partir de ses intégrales sur les sous-variétés dans $Xi$. Nous présentons d'abord la solution de ce problème inverse due à Sigurdur Helgason et François Rouvière, entre autres, lorsque $M$ est un espace symétrique riemannien isotrope et $Xi$ une certaine orbite de sous-variétés totalement géodésiques de $M$ sous l'action d'un groupe de transformations de Lie de $M$. La transformée de Radon associée est qualifiée de totalement géodésique.Sur les espaces symétriques pseudo-riemanniens semisimples, nous considérons une autre transformée de type Radon, qui associe à toute fonction continue à support compact ses intégrales orbitales, c'est-à-dire ses intégrales sur les orbites du sous-groupe d'isotropie du groupe des transvections. L'inversion des intégrales orbitales, qui est donnée par une formule-limite, a été obtenue par Sigurdur Helgason sur les espaces symétriques lorentziens à courbure sectionnelle constante et par Jeremy Orloff sur tout espace symétrique pseudo-riemannien semisimple de rang un. Nous résolvons le problème d'inversion des intégrales orbitales sur les espaces de Cahen-Wallach, qui sont les modèles d'espaces symétriques lorentziens indécomposables résolubles.Pour finir, nous nous intéressons aux transformées de type Radon sur les espaces symétriques symplectiques à courbure de type Ricci. L'inversion des orbitales intégrales sur ces espaces lorsqu'ils sont semisimples a déjà été obtenue par Jeremy Orloff. En revanche, lorsque ces espaces ne sont pas semisimples, la transformée donnée par les intégrales orbitales n’est pas inversible. Ensuite, nous déterminons les orbites de sous-variétés totalement géodésiques symplectiques ou lagrangiennes sous l'action d'un groupe de transformations de Lie de l'espace de départ. Dans ce contexte, la méthode d'inversion développée par Sigurdur Helgason et François Rouvière, entre autres, ne fonctionne que pour les transformées de Radon totalement géodésiques symplectiques sur les espaces symétriques kählériens à courbure holomorphe constante. Les formules d'inversion de ces transformées sur les espaces hyperboliques complexes sont dues à François Rouvière. Nous calculons les formules d'inversion de ces transformées sur les espaces projectifs complexes. / In this thesis, we study Radon-type transforms on some symmetric spaces. A Radon-type transform associates to any compactly supported continuous function on a manifold $M$ its integrals over a class $Xi$ of submanifolds of $M$. The problem we address is the inversion of such a transform, that is determining the function in terms of its integrals over the submanifolds in $Xi$. We first present the solution to this inverse problem which is due to Sigurdur Helgason and François Rouvière, amongst others, when $M$ is an isotropic Riemannian symmetric space and $Xi$ a particular orbit of totally geodesic submanifolds of $M$ under the action of a Lie transformation group of $M$. The associated Radon transform is qualified as totally geodesic.On semisimple pseudo-Riemannian symmetric spaces, we consider an other Radon-type transform, which associates to any compactly supported continuous function its orbital integrals, that is its integrals over the orbits of the isotropy subgroup of the transvection group. The inversion of orbital integrals, which is given by a limit-formula, has been obtained by Sigurdur Helgason on Lorentzian symmetric spaces with constant sectional curvature and by Jeremy Orloff on any rank-one semisimple pseudo-Riemannian symmetric space. We solve the inverse problem for orbital integrals on Cahen-Wallach spaces, which are model spaces of solvable indecomposable Lorentzian symmetric spaces.In the last part of the thesis, we are interested in Radon-type transforms on symplectic symmetric spaces with Ricci-type curvature. The inversion of orbital integrals on these spaces when they are semisimple has already been obtained by Jeremy Orloff. However, when these spaces are not semisimple, the orbital integral operator is not invertible. Next, we determine the orbits of symplectic or Lagrangian totally geodesic submanifolds under the action of a Lie transformation group of the starting space. In this context, the technique of inversion that has been developed by Sigurdur Helgason and François Rouvière, amongst others, only works for symplectic totally geodesic Radon transforms on Kählerian symmetric spaces with constant holomorphic curvature. The inversion formulas for these transforms on complex hyperbolic spaces are due to François Rouvière. We compute the inversion formulas for these transforms on complex projective spaces. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

This study proposes a new spectral representation called the Zeros of Z-Transform (ZZT), which is an all-zero representation of the z-transform of the signal. In addition, new chirp group delay processing techniques are developed for analysis of resonances of a signal. The combination of the ZZT representation with the chirp group delay processing algorithms provides a useful domain to study resonance characteristics of source and filter components of speech. Using the two representations, effective algorithms are developed for: source-tract decomposition of speech, glottal flow parameter estimation, formant tracking and feature extraction for speech recognition. The ZZT representation is mainly important for theoretical studies. Studying the ZZT of a signal is essential to be able to develop effective chirp group delay processing methods. Therefore, first the ZZT representation of the source-filter model of speech is studied for providing a theoretical background. We confirm through ZZT representation that anti-causality of the glottal flow signal introduces mixed-phase characteristics in speech signals. The ZZT of windowed speech signals is also studied since windowing cannot be avoided in practical signal processing algorithms and the effect of windowing on ZZT representation is drastic. We show that separate patterns exist in ZZT representations of windowed speech signals for the glottal flow and the vocal tract contributions. A decomposition method for source-tract separation is developed based on these patterns in ZZT. We define chirp group delay as group delay calculated on a circle other than the unit circle in z-plane. The need to compute group delay on a circle other than the unit circle comes from the fact that group delay spectra are often very noisy and cannot be easily processed for formant tracking purposes (the reasons are explained through ZZT representation). In this thesis, we propose methods to avoid such problems by modifying the ZZT of a signal and further computing the chirp group delay spectrum. New algorithms based on processing of the chirp group delay spectrum are developed for formant tracking and feature estimation for speech recognition. The proposed algorithms are compared to state-of-the-art techniques. Equivalent or higher efficiency is obtained for all proposed algorithms. The theoretical parts of the thesis further discuss a mixed-phase model for speech and phase processing problems in detail.

A Constant Proportion Debt Obligation is a credit derivative which has been introduced to generate a surplus return over a riskless market return. The surplus payments should be obtained by synthetically investing in a risky asset (such as a credit index) and using a linear leverage strategy which is capped for bounding the risk. In this thesis, we investigate two approaches for investigation of constant proportion debt obligations. First, we search for an optimal leverage strategy which minimises the mean-square distance between the final payment and the final wealth of constant proportion debt obligation by the use of optimal control methods. We show that the optimal leverage function for constant proportion debt obligations in a mean-square sense coincides with the one used in practice for geometric type diffusion processes. However, the optimal strategy will lead to a shortfall for some cases. The second approach of this thesis is to develop a pricing formula for constant proportion debt obligations. To do so, we consider both the early defaults and the default on the final payoff features of constant proportion debt obligations. We observe that a constant proportion debt obligation can be modelled as a barrier option with rebate. In this respect, given the knowledge on barrier options, the pricing equation is derived for a particular leverage strategy.

This thesis explores one possible method of preventing helicopter crashes caused by brownout using the noise generated by the helicopter rotor as an altimeter. The hypothesis under consideration is that the helicopter's height, velocity, and obstacle locations with respect to the helicopter, can be determined by comparing incident and reflected rotor noise signals, provided adequate bandwidth and signal to noise ratio. Heights can be determined by measuring the cepstrum of the reflected helicopter noise. The velocity can be determined by measuring small amounts of Doppler distortion using the Mellin-Scale Transform. Height and velocity detection algorithms are developed, optimized for this application, and tested using a microphone array. The algorithms and array are tested using a hemianechoic chamber and outside in Georgia Tech's Burger Bowl. Height and obstacle detection are determined to be feasible with the existing array. Velocity detection and surface mapping are not successfully accomplished.

This Master degree paper analyses hyperbolic image filtering in spaces of different dimensionality. It investigates the problem of optimal filtering space selection. Several popular image compression methods (both lossless and lossy) are reviewed. This paper analyses the problems of image smoothness parameter discovering, image dimensionality changing, hyperbolic image filtering and filtering efficiency evaluation and provides the solution methods of the problems. Schemes for the experimental examination of theoretical propositions and hypotheses are prepared. This paper comprehensively describes experiments with one-, two- and threedimensional images and the results of the experiments. Conclusions about the efficiency of hyperbolic image filtering in other than "native" image space are based on the results of the experiments. The criterion for the selection of optimal image filtering space is evaluated. Guidelines for further research are also discussed. The presentation Specific Features of Hyperbolic Image Filtering, which was based on this Master degree paper, was made at the conference Mathematics and Mathematical Modeling (KTU – 2004). This text is available in appendixes.

Sakykime, kad F yra normuota parabolinė tikrinė forma pilnosios modulinės grupės atžvilgiu, L(s, F) yra susieta su L funkcijos sąsūka L(s, F, χ) su Dirichlė charakteriu moduliu q, kai q yra pirminis skaičius. Bakalauro darbe įrodyta ribinė teorema L funkcijų sąsūkų argumentui arg L(s, F, χ). / Let F(z) a holomorfic normalized Hecke eigen cups form of weight κ for the full modular group, L(s, F), s = σ + it, be the L-function attached to the form F. Let L(s, F, χ) denote a twist of L(s, F) with a Dirichlet character χ modulo q, by the Dirichlet series and can be analytically continued to an entire function. It has an Euler product over prime numbers. We obtain the weak dinvergence for probability measures μQ(arg L(s, F, χ) ∈ A), A ∈ B(γ), where γ is the unit circle on the complex plane, as Q → ∞. For the proof, the method of characteristic transforms and the limit measures in limit theorems obtained are defined characteristic transforms.

The finite affine group is a matrix group whose entries come from a finite field. A natural subgroup consists of those matrices whose entries all come from a subfield instead. In this paper, I will introduce intermediate sub- groups with entries from both the field and a subfield. I will also examine the representations of these intermediate subgroups as well as the branch- ing diagram for the resulting subgroup chain. This will allow us to create a fast Fourier transform for the group that uses asymptotically fewer opera- tions than the brute force algorithm.

Plusieurs familles de fonctions spéciales de plusieurs variables, appelées fonctions d'orbites, sont définies dans le contexte des groupes de Weyl de groupes de Lie simples compacts/d'algèbres de Lie simples. Ces fonctions sont étudiées depuis près d'un siècle en raison de leur lien avec les caractères des représentations irréductibles des algèbres de Lie simples, mais également de par leurs symétries et orthogonalités. Nous sommes principalement intéressés par la description des relations d'orthogonalité discrète et des transformations discrètes correspondantes, transformations qui permettent l'utilisation des fonctions d'orbites dans le traitement de données multidimensionnelles. Cette description est donnée pour les groupes de Weyl dont les racines ont deux longueurs différentes, en particulier pour les groupes de rang $2$ dans le cas des fonctions d'orbites du type $E$ et pour les groupes de rang $3$ dans le cas de toutes les autres fonctions d'orbites. / Several families of multivariable special functions, called orbit functions, are defined in the context of Weyl groups of compact simple Lie groups/Lie algebras. These functions have been studied for almost a century now because of their relation to characters of irreducible representations of Lie algebras, their symmetries and orthogonalities. Our main interest is the description of discrete orthogonality relations and their corresponding discrete transforms which allow the applications of orbit functions in the processing of multidimensional data. This description is provided for the Weyl group of different lengths of root, in particular groups of rank 2 for so-called $E-$orbit functions and of rank 3 for all the other families of special functions.