This paper examines concurrency control in databases beginning with some basic terminology and issues that will be used throughout the paper, and an informal intuitive description of serializability. This paper later provides a comprehensive mathematical theory. This theory forms the basis for the development of the various algorithmic tests for serializability which guarantee the consistency of a database which permits concurrency.

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The Theory of Hestenes Ternary Rings and Complex Ternary Algebras were discussed and applications were presented for Hestenes Ternary Rings and *-reciprocal of Complex Ternary Algebras. Specifically, the author presented the traditional algebraic theory of rings and algebra. This presentation was followed by the basic theory of Hestenes Ternary Rings and Hestenes Complex Ternary Algebras. The culmination of the paper was Chapter IV where the author presented applications of the latter theory, the most distinguished being the generalization of the Chevalley-Jacobson Density Theorem to the Theory of Hestenes Ternary Rings and Algebras.

<p> In this work, we study the long time behavior of reaction-diffusion models arising from mathematical biology. First, we study a reaction-diffusion population model with time delay. We establish a comparison principle for coupled upper/lower solutions, prove the existence/uniqueness result for the model, and show the global asymptotic behavior of the model by constructing successive improved upper/lower solutions. Next, we consider a reaction-diffusion equation with continuous delay and spatial variable coefficients. We prove the global attractivity of the positive steady state by showing that the omega limit set is a singleton. Finally, we study an SIS reaction-diffusion model with spatial heterogeneous disease transmission and recovery rates. We define a basic reproduction number and obtain some existence and non-existence results of the endemic equilibrium of the model. We then study the global attractivity of the steady state for two special cases.</p>

<p> Motivated by A. Z. Petrov's classification of four-dimensional Lorentzian metrics, we provide an algebraic classification of the isometry-isotropy pairs of four-dimensional pseudo-Riemannian metrics admitting local slices with five-dimensional isometries contained in the Lorentz algebra. A purely Lie algebraic approach is applied with emphasis on the use of Lie theoretic invariants to distinguish invariant algebra-subalgebra pairs. This method yields an algorithm for identifying isometry-isotropy pairs subject to the aforementioned constraints.</p>

<p> Via novel path-routing techniques we prove a lower bound on the I/O-complexity of all recursive matrix multiplication algorithms computed in serial or in parallel and show that it is tight for all square and near-square matrix multiplication algorithms. Previously, tight lower bounds were known only for the classical &Theta; (<i>n</i>³) matrix multiplication algorithm and those similar to Strassen's algorithm that lack multiple vertex copying. We first prove tight lower bounds on the I/O-complexity of Strassen-like algorithms, under weaker assumptions, by constructing a routing of paths between the inputs and outputs of sufficiently small subcomputations in the algorithm's CDAG. We then further extend this result to all recursive divide-and-conquer matrix multiplication algorithms, and show that our lower bound is optimal for algorithms formed from square and nearly square recursive steps. This requires combining our new path-routing approach with a secondary routing based on the Loomis-Whitney Inequality technique used to prove the optimal I/O-complexity lower bound for classical matrix multiplication.</p>

The Wiener-Hopf technique is a powerful tool for constructing analytic solutions for a wide range of problems in physics and engineering. The key step in its application is solution of the Riemann-Hilbert problem, which consists of finding a piece-wise analytic (vector-) function in the complex plane for a specified behavior of its discontinuities. In this dissertation, the applied theory of vector Riemann-Hilbert problems is reviewed. The analytical solution representing the problem on a Riemann surface, and a numerical solution that reduces the problem to singular integral equations, are considered, as well as a combination of the numerical and analytical techniques (partial Wiener-Hopf factorization) is proposed. In this work, we begin with a brief survey of the Riemann-Hilbert problem: constructing solution of the scalar Riemann-Hilbert problem for a class of Holder continuous functions; considering classes of matrices that admit the closed-form solution of the vector Riemann-Hilbert problem; discussing numerical and analytical techniques of constructing solutions of vector Riemann-Hilbert problems. We continue with applications of the Wiener-Hopf technique to problems of Dynamic fracture mechanics: reviewing well-known solutions to problems on propagation of a semi-infinite crack in an unbounded plane in the cases of a stationary crack, a crack propagating at a constant speed, and a crack propagating at a non-uniform arbitrary speed. Based on those, we derive solutions to new problems on a semi-infinite crack propagation in a half-plane (steady-state and transient problems for subsonic speeds) as well as in a composite strip (for intersonic speeds). These latter results are new and were first derived by Y. Antipov and the author.

Quantization is an important part of signal processing. Several issues influence the performance of a quantization algorithm. One is the âbasisâ we choose to represent the signal, another is how we quantize the âbasisâ coefficients. We desire to explore these two things in this thesis. In the first chapter, we review frame theory and fusion frame theory. In the second chapter, we introduce a popular quantization algorithm known as Sigma-Delta quantization and show how to apply it to finite frames. Then we give the definition and properties of Sobolev duals which are optimized duals associated to Sigma-Delta quantization. The contraction Sobolev duals depends on the frame, and in chapter 3, we prove that for any finite unit-norm frame, the best error bound that can be achieved from the reconstruction with Sobolev duals in rth Sigma-Delta quantization is equal to order O(N????^(-r)), where the error bound can be related to both operator norm and Frobenius norm. In the final chapter, we develop Sigma-Delta quantization for fusion frames. We construct stable first-order and high-order Sigma-Delta algorithms for quantizing fusion frame projections of f onto W_n, where W_n is an M_n dimension subspace of R^d. Our stable 1st-order quantizer uses only log2(Mn+1) bits per subspace. Besides, we give an algorithm to calculate the Kashin representations for fusion frames to improve the performance of the high-order Sigma-Delta quantization algorithm. Then by defining the left inverse and the canonical left inverse for fusion frames, we prove the property that the canonical left inverse has the minimal operator norm and Frobenius norm. Based on this property, we give the idea of Sobolev left inverses for fusion frames and prove it leads to minimal squared error.

Shift-invariant spaces and Gabor systems are frequently used in approximation theory and signal processing. In these settings, it is advantageous for the generators of such spaces to be localized and for the spaces to be representative of a large class of functions. For Gabor systems, the celebrated Balian-Low Theorem shows that if the integer translations and modulations of a function in $L^2(R)$ form an orthonormal basis for $L^2(R)$, then either the function or its fourier transform must be poorly localized. The present work shows that similar results hold in certain shift-invariant spaces. In particular, if the integer translates of a well-localized function in $L^2(R^d)$ form a frame for the shift-invariant space, $V$, generated by the function then $V$ cannot be invariant under any non-integer shift. Similar results are proven under a variety of different basis properties, for finitely many generating functions, and with the extra-invariance property replaced with redundancy in the translates of the generator. Examples are given showing the sharpness of these results.