Let k be an algebraically closed field of characteristic 0. This thesis develops techniques used to determine the structure of a finite dimensional Hopf algebra over k. The Hopf algebras of dimension $\leq$11 are classified. / Let p be a prime number, r a positive integer, and n = p$\sp{\rm r}-1$. Let GF(p$\sp{\rm r}$) be the Galois field of order p$\sp{\rm r}$. Let G = GF(p$\sp{\rm r}$) $\times$ $\sb\varphi$ GF(p$\sp{\rm r}$)$\sp\cdot$ be the semidirect product of GF(p$\sp{\rm r}$) and GF(p$\sp{\rm r}$)$\sp\cdot$ relative to the homomorphism $\varphi$:GF(p$\sp{\rm r}$)$\sp\cdot$ $\to$ AutGF(p$\sp{\rm r}$) defined by $\varphi$(x)(v) = xv for v$\in$ GF(p$\sp{\rm r}$) and x$\in$ GF(p$\sp{\rm r}$)$\sp\cdot$. A Hopf algebra H of dimension n$\sp2$(n + 1) is constructed which contains a Hopf subalgebra isomorphic to (kG)*. H is shown to be isomorphic to its linear dual. / Source: Dissertation Abstracts International, Volume: 50-02, Section: B, page: 0606. / Major Professor: Warren D. Nichols. / Thesis (Ph.D.)--The Florida State University, 1988.

In recent years the problem of evaluating the determinants of Laplacians on Riemannian manifolds has received considerable attention. The theory of multiple gamma functions play an important role in computations of determinants of Laplacians on manifolds of constant curvature. These functions were introduced by E. W. Barnes in about 1900. / We are particularly interested in the functional determinant for the n-sphere S$\sp{n}$ with the standard metric. For all n we give a factorization it into multiple gamma functions and use this factorization to compute nice closed form expressions for the determinant in cases n = 1, 2 and 3. / In the course of this investigation we give a new proof of the multiplication formulas for the simple and double gamma functions. / Source: Dissertation Abstracts International, Volume: 52-03, Section: B, page: 1476. / Major Professor: J. Quine. / Thesis (Ph.D.)--The Florida State University, 1991.

All punctured Klein bottle bundles over S$\sp1$ are classified. For each of those, all their two-sided incompressible surfaces are described, up to isotopy. This is used to obtain information on Dehn fillings of the bundles. For example, there is a manifold M with a nontrivial knot k in it, so that infinitely many Dehn surgeries on k yield M. / Source: Dissertation Abstracts International, Volume: 51-09, Section: B, page: 4386. / Major Professor: Wolfgang H. Heil. / Thesis (Ph.D.)--The Florida State University, 1990.

It has been shown that quasitriangular Hopf algebras (QTHAs) have been increasingly playing important roles in many areas of mathematics and physics. Some people believe that the theory of quantum groups will be the group theory of next century. The main goal of this thesis is to develop methods to determine the quasitriangular structures (R-matrices) of a finite dimensional Hopf algebra over a field. The primary research that I have done in this thesis touched quantum groups from several directions. We prove the main results in this thesis that are stated as follows. Let H be a finite dimensional Hopf algebra over a field k. If H is unimodular, then the R-matrices of H can be embedded in the center of the quantum double D(H), a QTHA associated to H that was discovered by Drinfel'd. If H is cosemisimple (equivalently, if the dual algebra of H is semisimple), then the R-matrices of H correspond to central idempotents in D(H). Hence, for a finite dimensional cosemisimple Hopf algebra H (such as the group algebra of a finite group), one can possibly locate all the R-matrices among the set of central idempotents of D(H), which is a finite set in many general contexts. We will see that there are many non-trivial R-matrices arising from finite non-abelian groups. Non-trivial R-matrices of non-abelian group algebras allow us to use groups to construct quantum groups. / Source: Dissertation Abstracts International, Volume: 57-03, Section: B, page: 1851. / Major Professor: Warren Nichols. / Thesis (Ph.D.)--The Florida State University, 1996.

Two problems are studied. First, the analytic continuation of the real periodic solutions of van der Pol's equation to complex values of the damping parameter $\varepsilon$ are discussed. This continuation shows the existence of an infinite family of singular complex periodic solutions associated with values of $\varepsilon$ lying on two curves $\Gamma$ and $\bar\Gamma$ (see Figure 8) located symmetrically in the $\varepsilon$-plane. These singular solutions are found to cause the existence of the moving singularities of the Poincare-Lindstedt series for the real limit cycle which were developed at great length by Andersen and Geer (7), and were analyzed, using Pade approximants, by Dadfar et al. (10). A numerical method for the computation of these singular solutions is described. In addition, an asymptotic description of them for large values of $\vert\varepsilon\vert$ is obtained using the method of matched asymptotic expansions. Our results suggest that the existence of the complex singular solutions may, in general, play an important role in the utility of computer-generated perturbation expansions at moderate or large values of the perturbation parameter. / Our second study involves a model, due to Lagerstrom, of the steady flow of a viscous incompressible fluid past an object in (m + 1) dimensions. The model is a nonlinear boundary-value problem in the range $\varepsilon \leq x $ 0 and $m >$ 0. Our results suggest that a similar iteration may be an effective method of approximation of viscous flows at moderate Reynolds numbers. / Source: Dissertation Abstracts International, Volume: 51-07, Section: B, page: 3416. / Major Professor: Christopher Hunter. / Thesis (Ph.D.)--The Florida State University, 1990.

The main purpose of this dissertation is to study some basic properties of Riemann surfaces. The Jacobian of a Riemann surface is one of the most important algebraic and analytic characteristics for the surface. Related to Jacobian of a Riemann surface are Riemann Period Matrix, Jacobian Lattice, and Jacobian Variety. There are algebraic and analytic aspects of study of Jacobians. / In Chapter 2, we will establish a number of algebraic properties of complex lattices and tori that are fundamental to the study of Jacobians of Riemann surfaces. In Chapter 3, we will consider an extremal problem of Riemann surface. The problem was first studied by Buser and Sarnak (BS), who introduced the concept of maximal minimal norms of the Jacobian lattices of Riemann surfaces, and obtained a number of properties for a Riemann surface of large genus. We will answer to their conjecture that Klein's surface would be an absolute extremal Riemann surface in the case of genus 3, and prove that their conjecture is not really true. To complete our proof, we will first give out a sufficient, possibly necessary, condition for a Riemann surface to have a maximal minimal norm of its Jacobian lattice, and then prove that, based on the results of Quine (Q2), there exists a local extremal Riemann surface in the case of genus 3 that has a bigger minimal norm than Klein's surface does. It seems to be extremely difficult to find out all the extremal Riemann surfaces no matter how one nontrivially defines the extremality. / Among all the essential work to this paper are the results obtained by Rauch and Lewittes (RL), Quine (Q1) (Q2), and the classic discussions of perfect and eutactic forms introduced by Voronoi (VO) and studied extensively by Barnes (BA3) and many other mathematicians (CS2). / Buser and Sarnak in their paper (BS) have obtained a number of interesting characteristics for surfaces of large genus. Our discussion is very computational and probably not applicable to higher genus case. We will also provide some information on Riemann surfaces of genus 3 such as Klein's surface and Fermat's surface, to hopefully help further study in this area. / Source: Dissertation Abstracts International, Volume: 56-08, Section: B, page: 4369. / Major Professor: John R. Quine. / Thesis (Ph.D.)--The Florida State University, 1995.

Source: Dissertation Abstracts International, Volume: 40-06, Section: B, page: 2701. / Thesis (Ph.D.)--The Florida State University, 1979.

Of late, path-breaking advances are taking place and flourishing in the field of solid-state Nuclear Magnetic Resonance (ssNMR)spectroscopy. One of the major applications of ssNMR techniques is to high resolution three-dimensional structures of biological molecules like the membrane proteins. An explicit example of this is PISEMA (Polarization Inversion Spin Exchange at Magic Angle). This dissertation studies and analyzes the use of the orientational restraints in general, and particularly the restraints measured through PISEMA. Here, we have applied our understanding of orientational restraints to briefly investigate the structure of Amantadine bound M2-TMD, a membrane protein in Influenza A Virus. We model the protein backbone structure as a discrete curve in space with atoms represented by vertices and covalent bonds connecting them as the edges. The oriented structure of this curve with respect to an external vector is emphasized. The map from the surface of the unit sphere to the PISEMA frequency plane is examined in detail. The image is a powder pattern in the frequency plane. A discussion of the resulting image is provided. Solutions to PISEMA equations lead to multiple orientations for the magnetic field vector for a given point in the frequency plane. These are duly captured by sign degeneracies for the vector coordinates. The intensity of NMR powder patterns is formulated in terms of a probability density function for 1-d spectra and a joint probability density function for the 2-d spectra. The intensity analysis for 2-d spectra is found to be rather helpful in addressing the robustness of the PISEMA data. To build protein structures by gluing together diplanes, certain necessary conditions have to be met. We formulate these as continuity conditions to be realized for diplanes. The number of oriented protein structures has been enumerated in the degeneracy framework for diplanes. Torsion angles are expressed via sign degeneracies. For aligned protein samples, the PISA wheel approach to modeling the protein structure is adopted. Finally, an atomic model of the monomer structure of M2-TMD with Amantadine has been elucidated based on PISEMA orientational restraints. This is a joint work with Jun Hu and Tom Asbury. The PISEMA data was collected by Jun Hu and the molecular modeling was performed by Tom Asbury. / A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Degree Awarded: Fall Semester, 2006. / Date of Defense: October 4, 2006. / Intensity analysis, Frenet frames, Orientational restraints, Protein structures / Includes bibliographical references. / John R. Quine, Professor Directing Dissertation; Timothy A. Cross, Outside Committee Member; DeWitt Sumners, Committee Member; Richard Bertram, Committee Member.

This dissertation develops and evaluates a high order method to compute wave scattering from moving boundaries. Specifically, we derive and evaluate a Discontinuous Galerkin Spectral elements method (DGSEM) with Arbitrary Lagrangian- Eulerian (ALE) mapping to compute conservation laws on moving meshes and numerical boundary conditions for Maxwell's equations, the linear Euler equations and the nonlinear Euler gas-dynamics equations to calculate the numerical flux on reflective moving boundaries. We use one of a family of explicit time integrators such as Adams-Bashforth or low storage explicit Runge-Kutta. The approximations preserve the discrete metric identities and the Discrete Geometric Conservation Law (DGCL) by construction. We present time-step refinement studies with moving meshes to validate the moving mesh approximations. The test problems include propagation of an electromagnetic gaussian plane wave, a cylindrical pressure wave propagating in a subsonic flow, and a vortex convecting in a uniform inviscid subsonic flow. Each problem is computed on a time-deforming mesh with three methods used to calculate the mesh velocities: From exact differentiation, from the integration of an acceleration equation, and from numerical differentiation of the mesh position. In addition, we also present four numerical examples using Maxwell's equations, one example using the linear Euler equations and one more example using nonlinear Euler equations to validate these approximations. These are: reflection of light from a constantly moving mirror, reflection of light from a constantly moving cylinder, reflection of light from a vibrating mirror, reflection of sound in linear acoustics and dipole sound generation by an oscillating cylinder in an inviscid flow. / A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Degree Awarded: Spring Semester, 2011. / Date of Defense: March 15, 2011. / Discontinuous Galerkin Spectral Element Method, Moving Boundary Conditions, DGSEM, Moving Mesh, Arbitrary Lagrangian-Eulerian, ALE, Discrete Geometric Conservation Law (DGCL) / Includes bibliographical references. / David Kopriva, Professor Directing Thesis; Anuj Srivastava, University Representative; M. Yousuff Hussaini, Committee Member; Mark Sussman, Committee Member; Brian Ewald, Committee Member.

The principles of optimal control of distributed parameter systems are used to derive a powerful class of numerical methods for solutions of inverse problems, called data assimilation (DA) methods. Using these DA methods one can efficiently estimate the state of a system and its evolution. This information is very crucial for achieving more accurate long term forecasts of complex systems, for instance, the atmosphere. DA methods achieve their goal of optimal estimation via combination of all available information in the form of measurements of the state of the system and a dynamical model which describes the evolution of the system. In this dissertation work, we study the impact of new nonlinear numerical models on DA. High resolution advection schemes have been developed and studied to model propagation of flows involving sharp fronts and shocks. The impact of high resolution advection schemes in the framework of inverse problem solution/ DA has been studied only in the context of linear models. A detailed study of the impact of various slope limiters and the piecewise parabolic method (PPM) on DA is the subject of this work. In 1-D we use a nonlinear viscous Burgers equation and in 2-D a global nonlinear shallow water model has been used. The results obtained show that using the various advection schemes consistently improves variational data assimilation (VDA) in the strong constraint form, which does not include model error. However, the cost functional included efficient and physically meaningful construction of the background cost functional term, J_b, using balance and diffusion equation based correlation operators. This was then followed by an in-depth study of various approaches to model the systematic component of model error in the framework of a weak constraint VDA. Three simple forms, decreasing, invariant, and exponentially increasing in time forms of evolution of model error were tested. The inclusion of model error provides a substantial reduction in forecasting errors, in particular the exponentially increasing form in conjunction with the piecewise parabolic high resolution advection scheme was found to provide the best results. Results obtained in this work can be used to formulate sophisticated forms of model errors, and could lead to implementation of new VDA methods using numerical weather prediction models which involve high resolution advection schemes such as the van Leer slope limiters and the PPM. / A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Degree Awarded: Spring Semester, 2006. / Date of Defense: April 3, 2006. / Finite Volume Methods, Data Assimilation, Numerical Weather Prediction, Optimal Control, High Resolution Schemes / Includes bibliographical references. / Ionel Michael Navon, Professor Directing Dissertation; James J. O'Brien, Outside Committee Member; Gordon Erlebacher, Committee Member; Qi Wang, Committee Member; Mark Sussman, Committee Member.