In this thesis, we present three projects. First, we investigate the numerical approximation of Hamilton-Jacobi equations with the Caputo time-fractional derivative. We introduce an explicit in time discretization of the Caputo derivative and a finite-difference scheme for the approximation of the Hamiltonian. We show that the approximation scheme so obtained is stable under an appropriate condition on the discretization parameters and converges to the unique viscosity solution of the Hamilton-Jacobi equation. Also, we study the numerical approximation of a system of PDEs which arises from an optimal control problem for the time-fractional Fokker-Planck equation with time-dependent drift. The system is composed of a backward time-fractional Hamilton-Jacobi-Bellman equation and a forward time-fractional Fokker-Planck equation. We approximate Caputo derivatives in the system by means of L1 schemes and the Hamiltonian by finite differences. The scheme for the Fokker-Planck equation is constructed in such a way that the duality structure of the PDE system is preserved on the discrete level. We prove the well-posedness of the scheme and the convergence to the solution of the continuous problem. Finally, we study a particle approximation for one-dimensional first-order Mean-Field-Games with local interactions with planning conditions. Our problem comprises a system of a Hamilton-Jacobi equation coupled with a transport equation. As we are dealing with the planning problem, we prescribe initial and terminal distributions for the transport equation. The particle approximation builds on a semi-discrete variational problem. First, we address the existence and uniqueness of the semi-discrete variational problem. Next, we show that our discretization preserves some conserved quantities. Finally, we prove that the approximation by particle systems preserves displacement convexity. We use this last property to establish uniform estimates for the discrete problem. All results for the discrete problem are illustrated with numerical examples.

The NAD glycohydrolase (NADase) from Bungarus fasciatus venom was purified over 1000-fold to electrophoretic homogeneity through a 3-step procedure which included affinity chromatography on Cibacron Blue agarose. The enzyme exhibited a broad pH profile with the optimum range between 7-8. Studies on the substrate specificity of B. fasciatus venom NADase demonstrated that alterations in the purine ring were less pronounced then alterations in the pyridinium moiety of NAD. Product inhibition studies indicated nicotinamide to be a noncompetitive inhibitor with a K_i = 1.4 mM and ADP-ribose to be a competitive inhibitor with a K_i =0.4 mM. The purified enzyme was inactivated by both 2,4-pentane dione and Woodward's Reagent K suggesting the involvement of a lysine and carboxyl group in the catalytic process. In contrast to other known NADases, the snake venom enzyme did not self-inactivate. The purified B. fasciatus venom NADase catalyzed a transglycosidation reaction (ADP-ribose transfer) with a number of acceptor molecules. The functioning of a variety of substituted pyridine bases as acceptor molecules was demonstrated through the formation of the corresponding NAD analogs. The enzyme also catalyzed the transfer of ADP-ribose to aliphatic alcohols (methanol to hexanol, inclusive) and a positive chainlength effect was observed in the functioning of these acceptors. Kinetic studies of transglycosidation reactions were consistent with the partitioning of an enzyme-ADP-ribose intermediate between water and nucleophilic acceptors as has been proposed in earlier studies of mammalian NADases. The partitioning of this intermediate between water and pyridine bases can be correlated with the basicity of the ring nitrogen of the pyridine derivative. The K_i of pyridine bases in the hydrolytic reaction did not equate to the K_m of these bases in the pyridine base exchange reaction suggesting two forms of the NADase with varying affinity for the pyridine bases. This implys the pyridine base exchange reaction to be more complicated than originally proposed. / Master of Science

The process of fractional distillation has been performed for many years but has only within the past half century become a true science. However, because of the many variables involved and the uncertainty of their effect upon fractionating column efficiency, the design of columns has long been a major engineering problem. Plate and column efficiencies, as given in the literature, vary widely and it was thought possible that the inconsistencies of the results reported might be due to poor equilibrium within the column, caused in part by faulty sampling. In order to determine the operating characteristics of a fifteen plate bubble-cap column (8-3/8" I.D., one 3-7/8" bubble-cap per plate, and plate spacing of 5-7/8"), the effects of the operating variables, and the distillation characteristics of two different types of binary mixtures, the column assembly was redesigned for continuous distillation and provisions made for the introduction of feed at its boiling point to any one of the bottom eight plates. The binary — mixtures distilled were isopropyl alcohol — water (distillation rates 157 - 376 gms./min., feed concentrations 3.1 - 10.8 mol per cent isopropyl alcohol, reflux ratio 3:1, and feed rate of 330 gms./min.), and toluene - ethylene dichloride (distillation rates 203 and 196 gms./min., feed concentrations 41.4 and 39.7 mol percent ethylene dichloride, reflux ratio 4:1, and feed rate of 138 gms./min.) In order to reduce the possibility of disturbances within the column by removal of large plate samples, the refractive index method of analysis was used which required maximum samples of only 4 ml. Results of the experiments made indicated that the assumption of the McCabe - Thiele operating line is probably incorrect. In the case of the isopropyl alcohol — water fractionation, a considerable divergence (largest between 10 and 40 mol per cent isopropyl alcohol) between the McCabe - Thiele and the actual operating line was noted, the actual operating line being a curve approximately the shape of the equilibrium curve. Murphree Plate Efficiencies for the mixture varied from 0 to 100 per cent, depending primarily on the relative deviations of the two operating lines from the equilibrium curve. The McCabe - Thiele operating line for the system toluene — ethylene dichloride closely approximated the actual operating line. Murphree Plate Efficiencies varied from 44.5 to 121 percent, but the individual plate efficiencies were more consistent with the average efficiency. The rate of distillation over the range of 157 to 376 gms./min. and feed concentration over the range of 3.1 to 10.8 mol percent isopropyl alc1ohol, at a reflux ratio of 3:1 had negligible effect on product purity which ranged from approximately 62 to 67 mol per cent isopropyl alcohol. / Master of Science

An optimization algorithm has been developed to study the effectiveness of substituting time tested ODEs with FDEs as applied to ship motions, specifically with an eye toward modeling different forms of roll damping. Relations between the order of differentiation a and damping coefficient b in the FDEs have been drawn for changing damping, added moment of inertia, and initial roll angle. A pitch model has also been studied and compared to the roll model. The error at each of these a and b pairs has also been calculated using an L2-norm. An initial effort was made to correlate the FDE coefficients to differing mechanisms of roll damping as characterized by Himeno. / Master of Science

A theorem due to Scarf in 1967 is examined in detail. Several versions of this theorem exist, some which appear at first unrelated. Two versions can be shown to be equivalent to a result due to Sperner in 1928: for a proper labelling of the vertices in a simplicial subdivision of an n-simplex, there exists at least one elementary simplex which carries all labels {0,1,..., n}. A third version is more akin to Dantzig's simplex method and is also examined. In recent years many new applications in combinatorics have been found, and we present several of them. Two applications are in the area of fair division: cake cutting and rent partitioning. Two others are graph theoretic: showing the existence of a fractional stable matching in a hypergraph and the existence of a fractional kernel in a directed graph. For these last two, we also show the second implies the first.

Fractional calculus enables the possibility of using real number powers or complex number powers of the differentiation operator. The fundamental connection between fractional calculus and subordination processes is explored and affords a physical interpretation for a fractional trajectory, that being an average over an ensemble of stochastic trajectories. With an ensemble average perspective, the explanation of the behavior of fractional chaotic systems changes dramatically. Before now what has been interpreted as intrinsic friction is actually a form of non-Markovian dissipation that automatically arises from adopting the fractional calculus, is shown to be a manifestation of decorrelations between trajectories. Nonlinear Langevin equation describes the mean field of a finite size complex network at criticality. Critical phenomena and temporal complexity are two very important issues of modern nonlinear dynamics and the link between them found by the author can significantly improve the understanding behavior of dynamical systems at criticality. The subject of temporal complexity addresses the challenging and especially helpful in addressing fundamental physical science issues beyond the limits of reductionism.

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A theorem due to Scarf in 1967 is examined in detail. Several versions of this theorem exist, some which appear at first unrelated. Two versions can be shown to be equivalent to a result due to Sperner in 1928: for a proper labelling of the vertices in a simplicial subdivision of an n-simplex, there exists at least one elementary simplex which carries all labels {0,1,..., n}. A third version is more akin to Dantzig's simplex method and is also examined. In recent years many new applications in combinatorics have been found, and we present several of them. Two applications are in the area of fair division: cake cutting and rent partitioning. Two others are graph theoretic: showing the existence of a fractional stable matching in a hypergraph and the existence of a fractional kernel in a directed graph. For these last two, we also show the second implies the first.

In this paper, we focus on the goodness of fit test for self-similar property of two well-known processes: the fractional Brownian motion and the fractional autoregressive integrated moving average process. The Hurst parameter of the self-similar process is estimated by the embedding branching process method proposed by Jones and Shen (2004). The goodness of fit test for self-similarity is based on the Pearson chi-square test statistic. We approximate the null distribution of the test statistic by a scaled chi-square distribution to correct the size bias problem of the conventional chi-square distribution. The scale parameter and degrees of freedom of the test statistic are determined via regression method. Simulations are performed to show the finite sample size and power of the proposed test. Empirical applications are conducted for the high frequency financial data and human heart rate data.

This dissertation considers a subclass of sum-of-ratios fractional combinatorial optimization problems (FCOPs) whose linear versions admit polynomial-time exact algorithms. This topic lies in the intersection of two scarcely researched areas of fractional programming (FP): sum-of-ratios FP and combinatorial FP. Although not extensively researched, the sum-of-ratios problems have a number of important practical applications in manufacturing, administration, transportation, data mining, etc. Since even in such a restricted research domain the problems are numerous, the main focus of this dissertation is a mathematical programming study of the three, probably, most classical FCOPs: Minimum Multiple Ratio Spanning Tree (MMRST), Minimum Multiple Ratio Path (MMRP) and Minimum Multiple Ratio Cycle (MMRC). The first two problems are studied in detail, while for the other one only the theoretical complexity issues are addressed. The dissertation emphasizes developing solution methodologies for the considered family of fractional programs. The main contributions include: (i) worst-case complexity results for the MMRP and MMRC problems; (ii) mixed 0-1 formulations for the MMRST and MMRC problems; (iii) a global optimization approach for the MMRST problem that extends an existing method for the special case of the sum of two ratios; (iv) new polynomially computable bounds on the optimal objective value of the considered class of FCOPs, as well as the feasible region reduction techniques based on these bounds; (v) an efficient heuristic approach; and, (vi) a generic global optimization approach for the considered class of FCOPs. Finally, extensive computational experiments are carried out to benchmark performance of the suggested solution techniques. The results confirm that the suggested global optimization algorithms generally outperform the conventional mixed 0{1 programming technique on larger problem instances. The developed heuristic approach shows the best run time, and delivers near-optimal solutions in most cases.