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In the ¡Krst chapter, separation of variables is used to derive the explicit particular solutions for a class of singularly perturbed di¤erential equations with constant coe¢ cients on a rectangular domain. Although only the Dirichlet boundary condition is taken into account; it can be similarly extended to other boundary conditions. Based on these results, the behavior of the solutions and their derivatives can be easily illustrated. Moreover, we have proposed a model with exact solution, which can be used to explore the behavior of layer and to test numerical methods. Hence, these analytic solutions are very important to the study in this ¡Keld. In the second chapter, we study the model of Shi¤ et al. [20]. It is a biharmonic equation on the rectangular domain [¡ a; a]£ [0; b] with clamped boundary condition. We compute its most accurate numerical solution by boundary approximation method (BAM), which is a special version of spectral method or collocation method. Its convergence unfortunately is not as good as the usual spectral method with exponential decay rate. We discover that the slowdown is due to the very mild singularity at two corners not considered by BAM. We further simplify the basis functions and their partial derivatives. Using these functions we can construct several models useful for testing numerical methods. We also explore how the stress intensity factor depends on the sizes of domain a and b, and the load ¸ by reducing the original problem with three parameters lambda, a, b to that with only one parameter t.

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Two different perturbation estimation methodologies are compared in this thesis. One is time delay method (TDM), the other is sliding mode techniqueli (SMT). When analyzing the TDM, Pade approximation is used to approximate a function with time-delay argument. The comparisons of perturbation estimation accuracy resulted from TDM and SMT both in time domain and frequency domain are addressed, and two numerical examples are also given for demonstration.

Recently, the stochastic approximation Monte Carlo algorithm has been proposed by Liang et al. (2005) as a general-purpose stochastic optimization and simulation algorithm. An annealing version of this algorithm was developed for real small protein folding problems. The numerical results indicate that it outperforms simulated annealing and conventional Monte Carlo algorithms as a stochastic optimization algorithm. We also propose one method for the use of secondary structures in protein folding. The predicted protein structures are rather close to the true structures. Phylogenetic trees have been used in biology for a long time to graphically represent evolutionary relationships among species and genes. An understanding of evolutionary relationships is critical to appropriate interpretation of bioinformatics results. The use of the sequential structure of phylogenetic trees in conjunction with stochastic approximation Monte Carlo was developed for phylogenetic tree reconstruction. The numerical results indicate that it has a capability of escaping from local traps and achieving a much faster convergence to the global likelihood maxima than other phylogenetic tree reconstruction methods, such as BAMBE and MrBayes.

Vertex shader, one of the core parts in 3D graphics systems, is to speed up the operations of coordinate transformation and lighting in 3D graphics pipeline, and vector ALU is the key part of a vertex shader. This thesis proposes several unified architectures that integrate the floating-point vector arithmetic unit and special function unit in order to share some hardware resource. We propose three different architectures for the design of the unified vector ALU. The first architecture includes a single-instruction-multiple-data (SIMD) vector arithmetic unit, and uses table-based method with first-order approximation to calculate some special functions. The second architecture use higher-order approximation to reduce the table sizes and share the floating-point multipliers in the SIMD vector unit. The proposed third architecture has two copies of hardware that can compute two dot-product operations in parallel and thus increase the throughput of the matrix computation by a factor of two. Furthermore, the two dot-product units can be used to perform the interpolation for special function calculation.

Allocation problems have been central to the development of the theory of algorithms and also find applications in several realms of computer science and economics. In this thesis we initiate a systematic study of these problems in situations with limited information. Towards this end we explore several modes by which data may be obfuscated from the algorithm. We begin by investigating temporal constraints where data is revealed to the algorithm over time. Concretely, we consider the online bipartite matching problem in the unknown distribution model and present the first algorithm that breaches the 1-1/e barrier for this problem. Next we study issues arising from data acquisition costs that are prevalent in ad-systems and kidney exchanges. Motivated by these constraints we introduce the query-commit model and present constant factor algorithms for the maximum matching and the adwords problem in this model. Finally we assess the approximability of several classical allocation problems with multiple agents having complex non-linear cost functions. This presents an additional obstacle since the support for the cost functions may be extremely large entailing oracle access. We show tight information theoretic lower bounds for the general class of submodular functions and also extend these results to get lower bounds for a subclass of succinctly representable non-linear cost functions.

Thesis (Ph.D.)--York University, 2007. Graduate Programme in Physics and Astronomy. / Typescript. Includes bibliographical references (leaves 99-102). Also available on the Internet. MODE OF ACCESS via web browser by entering the following URL: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&res_dat=xri:pqdiss&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&rft_dat=xri:pqdiss:NR29318

Thesis (Ph. D. in Physics)--Vanderbilt University, 2003. / Title from PDF title screen. Includes bibliographical references.