In the mid-1920s, the great Albert Einstein proposed that at extremely low temperatures, a gas of bosonic particles will enter a new phase where a large fraction of them occupy the same quantum state. This state would bring many of the peculiar features of quantum mechanics, previously reserved for small samples consisting only of a few atoms or molecules, up to a macroscopic scale. This is what we today call a Bose-Einstein condensate. It would take physicists almost 70 years to realize Einstein's idea, but in 1995 this was finally achieved. The research on Bose-Einstein condensates has since taken many directions, one of the most exciting being to study their behavior when they are placed in optical lattices generated by laser beams. This has already produced a number of fascinating results, but it has also proven to be an ideal test-ground for predictions from certain nonlinear lattice models. Because on the other hand, nonlinear science, the study of generic nonlinear phenomena, has in the last half century grown out to a research field in its own right, influencing almost all areas of science and physics. Nonlinear localization is one of these phenomena, where localized structures, such as solitons and discrete breathers, can appear even in translationally invariant systems. Another one is the (in)famous chaos, where deterministic systems can be so sensitive to perturbations that they in practice become completely unpredictable. Related to this is the study of different types of instabilities; what their behavior are and how they arise. In this thesis we compare classical and quantum mechanical nonlinear lattice models which can be applied to BECs in optical lattices, and also examine how classical nonlinear concepts, such as localization, chaos and instabilities, can be transfered to the quantum world.

This thesis presents a systematic numerical modeling framework to simulate the stress-deformation and coupled stress-deformation-flow processes by performing uniaxial and biaxial compressive tests on fractured rock models with considering the effects of different loading conditions, different loading directions (anisotropy), and coupled hydro-mechanical processes for evaluating strength and deformability behavior of fractured rocks. By using code UDEC of discrete element method (DEM), a series of numerical experiments were conducted on discrete fracture network models (DFN) at an established representative elementary volume (REV), based on realistic geometrical and mechanical data of fracture systems from field mapping at Sellafield, UK. The results were used to estimate the equivalent Young’s modulus and Poisson’s ratio and to fit the Mohr-Coulomb and Hoek-Brown failure criteria, represented by equivalent material properties defining these two criteria. The results demonstrate that strength and deformation parameters of fractured rocks are dependent on confining pressures, loading directions, water pressure, and mechanical and hydraulic boundary conditions. Fractured rocks behave nonlinearly, represented by their elasto-plastic behavior with a strain hardening trend. Fluid flow analysis in fractured rocks under hydro-mechanical loading conditions show an important impact of water pressure on the strength and deformability parameters of fractured rocks, due to the effective stress phenomenon, but the values of stress and strength reduction may or may not equal to the magnitude of water pressure, due to the influence of fracture system complexity. Stochastic analysis indicates that the strength and deformation properties of fractured rocks have ranges of values instead of fixed values, hence such analyses should be considered especially in cases where there is significant scatter in the rock and fracture parameters. These scientific achievements can improve our understanding of fractured rocks’ hydro-mechanical behavior and are useful for the design of large-scale in-situ experiments with large volumes of fractured rocks, considering coupled stress-deformation-flow processes in engineering practice. / <p>QC 20141111</p>

After reviewing some properties of the two dimensional hyperbolic toral automorphism called Arnold's discrete cat map, including its generalizations with matrices having positive unit determinant, this thesis contains a definition of a novel cat map where the elements of the matrix are found in the sequence of Pell numbers. This mapping is therefore denoted as Pell's cat map. The main result of this thesis is a theorem determining the upper bound for the minimal period of Pell's cat map. From numerical results four conjectures regarding properties of Pell's cat map are also stated. A brief exposition of some applications of Arnold's discrete cat map is found in the last part of the thesis.

Abstract Cognitive Radio (CR) improves the efficiency of spectrum utilization by allowing non- licensed users to utilize bands when not occupied by licensed users. In this thesis, we address several challenges currently limiting the wide use of cognitive radios. These challenges include identification of unoccupied bands, energy consumption and other technical challenges. Improved accuracy edge detection techniques are developed for CR to mitigate both noise and estimation error variance effects. Next, a reduced complexity Simplified DFT (SDFT) is proposed for use in CR. Then, a sub-Nyquist rate A to D converter is introduced to reduce energy consumption. Finally, a novel multi-resolution PSD estimation based on expectation-maximization algorithm is introduced that can obtain a more accurate PSD within a specified sensing time.

Of the many existing eukaryotic gene finding software programs, none are able to guarantee accurate identification of genomic protein coding regions and other biological signals central to pathway from DNA to the protein. Eukaryotic gene finding is difficult mainly due to noncontiguous and non-continuous nature of genes. Existing approaches are heavily dependent on the compositional statistics of the sequences they learn from and are not equally suitable for all types of sequences. This thesis firstly develops efficient digital signal processing-based methods for the identification of genomic protein coding regions, and then combines the optimum signal processing-based non-data-driven technique with an existing data-driven statistical method in a novel system demonstrating improved identification of acceptor splice sites. Most existing well-known DNA symbolic-to-numeric representations map the DNA information into three or four numerical sequences, potentially increasing the computational requirement of the sequence analyzer. Proposed mapping schemes, to be used for signal processing-based gene and exon prediction, incorporate DNA structural properties in the representation, in addition to reducing complexity in subsequent processing. A detailed comparison of all DNA representations, in terms of computational complexity and relative accuracy for the gene and exon prediction problem, reveals the newly proposed ?paired numeric? to be the best DNA representation. Existing signal processing-based techniques rely mostly on the period-3 behaviour of exons to obtain one dimensional gene and exon prediction features, and are not well equipped to capture the complementary properties of exonic / intronic regions and deal with the background noise in detection of exons at their nucleotide levels. These issues have been addressed in this thesis, by proposing six one-dimensional and three multi-dimensional signal processing-based gene and exon prediction features. All one-dimensional and multi-dimensional features have been evaluated using standard datasets such as Burset/Guigo1996, HMR195, and the GENSCAN test set. This is the first time that different gene and exon prediction features have been compared using substantial databases and using nucleotide-level metrics. Furthermore, the first investigation of the suitability of different window sizes for period-3 exon detection is performed. Finally, the optimum signal processing-based gene and exon prediction scheme from our evaluations is combined with a data-driven statistical technique for the recognition of acceptor splice sites. The proposed DSP-statistical hybrid is shown to achieve 43% reduction in false positives over WWAM, as used in GENSCAN.

Thesis (MSc (Mathematics))--Stellenbosch University, 2008. / LULU smoothers is a class of nonlinear smoothers and they are compositions of the maximum and minimum operators. By analogy to the discrete Fourier transform and the discrete wavelet transform, one can use LULU smoothers to create a nonlinear multiresolution analysis of a sequence with pulses. This tool is known as the Discrete Pulse Transform (DPT). Some research have been done into the distributional properties of the LULU smoothers. There exist results on the distribution transfers of the basic LULU smoothers, which are the building blocks of the discrete pulse transform. The output distributions of further smoothers used in the DPT, in terms of input distributions, has been a challenging problem. We motivate the use of these smoothers by first considering linear filters as well as the median smoother, which has been very popular in signal and image processing. We give an overview of the attractive properties of the LULU smoothers after which we tackle their output distributions. The main result is the proof of a recursive formula for the output distribution of compositions of LULU smoothers in terms of a given input distribution.

Este trabalho apresenta um estudo comparativo entre diferentes famílias de transformada Wavelet aplicadas à análise cepstral de sinais digitais de fala humana, com o objetivo específico de determinar o período de pitch dos mesmos e, ao final, propõe um algoritmo diferencial para realizar tal operação, levando-se em consideração aspectos importantes do ponto de vista computacional, tais como: desempenho, complexidade do algoritmo, plataforma utilizada, dentre outros. São apresentados também, os resultados obtidos através da implementação da nova técnica (baseada na transformada wavelet) em comparação com a abordagem tradicional (baseada na transformada de Fourier). A implementação da técnica foi testada em linguagem C++ padrão ANSI sob as plataformas Windows XP Professional SP3, Windows Vista Business SP1, Mac OSX Leopard e Linux Mandriva 10. / This work presents a comparative study between different family of wavelets applied on cepstral analysis of the digital speech human signal with specific objective for determining of pitch period of the same and in the end, proposes an differential algorithm to make such a difference operation take into consideration important aspects of computational point of view, such as: performance, algorithm complexity, used platform, among others. They are also present, the results obtained through of the technique implementation compared with the traditional approach. The technique implementation was tested in C++ language standard ANSI under the platform Windows XP Professional SP3 Edition, Windows Vista Business SP1, MacOSX Leopard and Linux Mandriva 10.

This thesis analyses the impact of granular aggregates with structures using experiments and numerical simulations. Original contributions include an insight into multiple factors affecting the loading and damage to the structures, along with study of numerical parameters important for realistic prediction of the interaction between the granular media and structures. It extends the current understanding related to such interactions, with an underlying motivation to guide strategies in order to reduce the structural damage. The response of structures impacted by granular media (sand or soil) is of significant research interest for many applications. One of the applications is for landmine explosions which causes ejection of soil from ground and damage to structures impacted by this ejected soil. Experimentation is done in a laboratory setting where the cylindrical sand slugs are generated at high speed using an impulse provided by a piston. This induces a velocity gradient along the slug, because of which the slug expands during the flight before impacting the target. Deformable as well as rigid flat targets are considered in two orientations relative to the incoming slug: perpendicular (i.e. normal orientation) and inclined at an angle of 45°. The targets are supported by force transducers to capture the loading from the slug. Simulations are performed using a combination of discrete particle and finite element schemes, which enables the analysis of the fully coupled interaction between the flowing granular media and the structure. A contact model involving multiple parameters is used for inter-particle and particle-target contact. Firstly, a numerical analysis is performed to characterise the temporal evolution of slugs and their impact on monolithic beams constrained at the ends. Out of all the parameters used for inter-particle contact definition in discrete particle method, only the contact stiffness is found to effect the velocity gradient in the slug before it impacts the target. Other factor influencing the gradient is the acceleration provided by the piston. A strong dependence of beam deflection on the stand-off distance is observed due to the velocity gradient in the slugs. As the second step, the effect of target surface properties on the transmitted momentum is analysed. Experiments are done by applying coatings of different hardness and roughness on the target surface impacted by sand slugs. For normally oriented targets, the transmitted momentum is observed to be insensitive to the change in surface coating. In contrast, for inclined targets, a significant influence of coatings is observed. Additionally, the momentum transmitted to the inclined targets is always less than that for normal targets. Numerical analysis of this surface effect reveals that assuming the slug particles to be spherical shape in simulations does not capture the particle/target interactions accurately and under-predicts the frictional loading on the target. Following this, a detailed numerical study is done to understand the effect of the shape of particles in the slug. Simple shaped non-spherical particles are constructed by combining spherical sub-particles. With increasing angularity of particles in the slug, the frictional loading on the target is shown to increase. This results in an increase of momentum transmitted to inclined targets. For normally oriented targets however, the particle shape does not affect the overall transmitted momentum, which is a behaviour similar to that observed when studying the effect of target surface properties. In addition, effect of fracture of particles in the slug is analysed by using beam connections between sub-particles that break during the impact with the target. If the fracture results in increasing particle angularity, the transmitted momentum increases, whereas the situation reverses if fracture results on more spherical shaped particles. Lastly, a strategy to reduce the loading on the targets is analysed by using sacrificial coating on the target surface. In experiments, this coating is placed on the rigid target surface using a lubricant at their interface. When impacted by the slug, this coating slides on the target surface, resulting in a reduction of frictional loading on the target. If the friction at the coating/target interface vanishes, the transmitted momentum approaches the theoretical minimum value. Simulations are used to first validate the experimental observations and then to extend the concept of sliding coatings using deformable targets. Both the transmitted momentum and deflections depended on the thickness of the target and coating. When a coating is used, the deflections increase due to reduction in target thickness. It is found that the best strategy to reduce the damage to the target is to use least possible thickness of the coating and minimise the friction at the interface between the coating and the target. The presented work examines many of the factors that affect the loading on the target impacted by granular slugs, in addition to characterising the expansion of slugs before the target impact. The analysed factors include those already known such as target stand-off distance, inclination and unveils others such as target surface properties and granular properties. The numerical analysis discloses important parameters and shows the effect of particle shape, highlighting the shortcomings of widely used spherical particle assumption in the numerical studies. A strategy using a sacrificial coating to reduce damage to the target is also analysed.

Ce travail de thèse concerne la prédiction des structures en béton soumises à des chargements extrêmes. Il s'intéresse plus particulièrement au comportement du béton sous fort confinement où la contrainte peut atteindre des niveaux de l'ordre du giga Pascal. La modélisation de ce comportement doit être capable de reproduire la compaction irréversible. Pour ce faire, deux lois de comportement élasto-plastique - endommageable ont été développées et implantées dans un code aux éléments discrets. Les paramètres utilisés dans ces lois sont calibrés par les simulations des essais de traction/compression uniaxial, des essais hydrostatiques et triaxiaux. Une fois les paramètres calibrés, la loi montrant le meilleur agrément avec l'expérience a été choisie pour la prédiction de la réponse du béton sous différents niveaux de confinement. Les résultats du modèle sont analysés non seulement à l'échelle macroscopique mais également à l'échelle de l'élément discret. La nécessité de prendre en compte une loi d'interaction de type élasto-plastique-endommageable est aussi montrée. La deuxième partie du travail de thèse développe une méthode de couplage entre le modèle éléments discrets et un modèle d'écoulement compressible en tenant compte des mécanismes physiques fondamentaux d'interaction entre l'écoulement interne et les particules solides d'un matériau poreux. Le problème d'écoulement est résolu par une méthode en volumes finis, où le volume est discrétisé en tétraèdres issus d'une triangulation régulière de Delaunay. Notre modèle est une adaptation aux fluides compressibles d'un modèle développé initialement pour les écoulements incompressibles. Ce couplage a été utilisé pour simuler le comportement triaxial des bétons humides et saturés sous différents niveaux de confinement. Les résultats nous montrent une bonne reproduction du comportement non-drainé du béton saturé sous faible confinement. Pour fort confinement, les simulations ne se rapprochent des résultats expérimentaux qu'au prix d'une compressibilité du fluide plus faible que celle de l'eau. Par ailleurs, la contrainte effective était une variable pertinente pour décrire le comportement du béton humide par un état limite intrinsèque indépendant du degré de saturation. / This thesis work deals with the predicting of concrete structures submitted to some extreme loadings, and, more particularly, focuses on behavior of concrete under a high-confining pressure. At this range of pressures, irreversible compaction of the material occurs and needs to be considered. Doing so, two elasto-plastic-damaged constitutive laws have been developed and implanted into a discrete element numerical code. Local parameters to be used in these constitutive laws are identified by simulating reference uniaxial traction/compression tests and triaxial compression tests. Once these parameters have been obtained, the law showing the best agreement with the experimental data has been chosen to predict the reponse of concrete sample for triaxial compressive tests at different levels of confinement. The numerical results have been analyzed not only at macroscopic scale but also at discrete element scale. The need of a constitutive law taking into account the elasto-plastic-damaged behavior has been also proved. The second objective of the thesis work was to develop a fluid flow – coupled discrete element model by considering fundamental physical mechanisms of the interaction between the internal fluide flow and the solid particles of a porous material. The flow problem is solved by the finite volume method, where the volume is discretized into tetrahedra issue of a regular Delaunay triangulation. Our model is an adaptation for elastic fluids of a model originally developed for incompressible flows. The developed fluid-flow coupled discrete element has been used to simulate the undrained triaxial behavior of concrete under different levels of confinement. The results show a good reproduction of undrained behavior of saturated concrete under low confinement. For high confinement, the simulations only resemble the experimental results when the fluid compressibility is lower than that of water. Moreover, the effective stress was a relevant variable to describe the behavior of the wet concrete by an intrinsic limit state independent of the degree of saturation.

The classical n-bugs problem has attracted considerable attention from researchers. This problem stems from the study of movement of a group of animals. In the context of multi- agent systems the problem has been modelled as cyclic pursuit. Under this paradigm, every agent, indexed i, chases its unique leader, agent i + 1 (modulo n), with n being the total number of agents. In the existing literature, cyclic pursuit has been studied for homogeneous agents where each agent’s velocity is proportional to the distance separating it from its leader and is directed along the line joining it to its leader. The constant of proportionality, initially chosen to be the same for all the agents, resulted in consensus in position, without the need for any centralized controller. Later, the constant of proportionality, alternately called the gain, was allowed to be heterogeneous and positional consensus was still achieved. Moreover, it was shown that the point of convergence, where the agents rendezvous, could be chosen at will, except for some diagnostic cases. In this thesis, besides admitting heterogeneous gains, the agents are assumed to pursue their respective leaders with an angle of deviation from the line joining them to their corresponding leaders. This expands the reachability set (set of points where the agents can rendezvous) for the system of agents to include points that were hitherto unreachable. Sufficient conditions for stability of such systems have been derived in this thesis. Detailed analysis of the reachability set has also been carried out. Some researchers have also investigated hierarchical cyclic pursuit, where there are multiple levels of pursuit. For instance, in the two level hierarchical pursuit, the agents are divided into m groups of n agents each, where each agent in a group chases its leader within the group as well as a similarly indexed agent in its leading group. Thus, groups of agents are also in cyclic pursuit. So far, only homogeneous gains were considered under this paradigm. The present thesis admits heterogeneous gains and establishes necessary and sufficient conditions for the stability of heterogeneous hierarchical cyclic pursuit, that generalize existing results. Reachable sets are also derived for this case. It is proved that the existing results can be derived as special cases of the ones considered in this thesis. As an extension to a realistic application, the importance of expansion in reachable set vis-a`-vis capturing a moving target is highlighted in this thesis. It has been shown that if the target’s initial position is reachable, then using a control law proposed in the thesis, the target can be captured. This control law is essentially an augmented cyclic pursuit law with the target’s velocity information fed to each agent in addition to the conventional cyclic pursuit command. Analysis has been carried out for agents with double integrator dynamics as well. A control law in conjunction with an algorithm is proposed that helps ensure global reachability of agents, with double integrator dynamics, in cyclic pursuit. Another application, in which cyclic pursuit and a closely related topology called platooning have been coupled together to track the boundaries of unknown regions and constantly monitor them, is addressed in this thesis. This problem is especially important in monitoring forest fire, marine contamination, volcanic ash eruptions, etc., and can protect human life by cordoning off unsafe regions using multiple autonomous agents. Lastly, discrete time cyclic pursuit laws are analyzed to obtain results similar to the continuous time counterparts that exist in the literature. Moreover, heterogeneous gains and deviations are admitted similar to the continuous time version considered in this thesis. Gershgorin’s theorem is used extensively to arrive at sufficient conditions for the stability of such discrete time deviated cyclic pursuit systems. Reachability sets are also derived. In case of discrete time systems, loss of synchronization due to no common clock for autonomous agents is a very realistic scenario. This thesis obtains some results on the stability of such asynchronous cyclic pursuit systems and indicates that special precautions are needed for dealing with heterogeneous cyclic pursuit systems even when one gain is negative, since the system may not converge, depending on the initial positions of the agents and the sequence of updates.