Throughout, H is a finite dimensional Hopf algebra with antipode S over an algebraically closed field k satisfying (dimH)1 $\ne$ 0. G(H) denotes the group of grouplikes. A Hopf algebra is said to be involutory if the square of its antipode is the identity. We study the mathematics related to Kaplansky's 5$\sp{th}$ conjecture, (1), that if the Hopf algebra H or its dual H* is semisimple as an algebra, then H is involutory. / Suppose that B is a sub Hopf algebra of H. We investigate the question under which assumptions B involutory implies H involutory. We succeed in showing that if H is cosemisimple and $rank\sb{B}H \leq$ 4, then H is semisimple and cosemisimple and $S\sp2$ = id. / If char(k) $\not=$ 2, we establish a relationship between $S\sp2$ and grouplikes $g\in G$(H) acting on simple subcoalgebras. It is shown that if A is a simple subcoalgebra A of dimension $n\sp2$ and $g\in$ G(H) is a grouplike of order n such that gA = A then $S\sp2\vert\sb{A} = id\sb{A}$. This enables us to verify Kaplansky's conjecture for cosemisimple Hopf algebras all of whose simple subcoalgebras have dimension 1 or $q\sp2$ with q prime. / The main part of our thesis concentrates on cosemisimple Hopf algebras all of whose simple subcoalgebras have dimension at most 9. It is shown that if A is a simple subcoalgebra A of dimension 9 and char(k) $\not=$ 2,3, then $S\sp2\vert\sb{A}$ has order 1, 2 or 3. If the characteristic of the field k is p $>$ dimH, then H is semisimple and cosemisimple. As a consequence, we are able to verify Kaplansky's conjecture for cosemisimple Hopf algebras with "small" simple subcoalgebras, if the characteristic of the field k is p $>$ (dimH)$\sp2$. / Source: Dissertation Abstracts International, Volume: 53-07, Section: B, page: 3520. / Major Professor: Warren D. Nichols. / Thesis (Ph.D.)--The Florida State University, 1992.

The purpose of this dissertation is to, first outline a theory of Zeta regularized products which will work for sequences of complex numbers, and second to use this theory to compute Zeta regularized products and modular constants for sequences which are integer combinations of a fixed set of complex numbers. / The gamma function $\Gamma(z)$ is represented as the ratio of two Zeta regularized products. This relation is then extended to define multiple gamma functions as the ratio of two corresponding Zeta regularized products. A full account of the functional equations associated with multiple gamma functions is also given. The double gamma function is investigated in detail. / Some other special functions are also discussed. Namely Jacobi's theta function $\theta\sb1$, the Weierstrass sigma function $\sigma(z),$ and $P(z\vert\tau)$ defined by / The determinant of the Laplacian on an n-dimensional flat Torus is computed for $n \geq$ 2, by computing / Source: Dissertation Abstracts International, Volume: 53-07, Section: B, page: 3523. / Major Professor: John R. Quine. / Thesis (Ph.D.)--The Florida State University, 1992.

We develop an adaptive spectral element method to price American options, whose solutions contain a moving singularity, automatically and to within prescribed errors. The adaptive algorithm uses an error estimator to determine where refinement or de-refinement is needed and a work estimator to decide whether to change the element size or the polynomial order. We derive two local error estimators and a global error estimator. The local error estimators are derived from the Legendre coefficients and the global error estimator is based on the adjoint problem. One local error estimator uses the rate of decay of the Legendre coefficients to estimate the error. The other local error estimator compares the solution to an estimated solution using fewer Legendre coefficients found by the Tau method. The global error estimator solves the adjoint problem to weight local error estimates to approximate a terminal error functional. Both types of error estimators produce meshes that match expectations by being fine near the early exercise boundary and strike price and coarse elsewhere. The produced meshes also adapt as expected by de-refining near the strike price as the solution smooths and staying fine near the moving early exercise boundary. Both types of error estimators also give solutions whose error is within prescribed tolerances. The adjoint-based error estimator is more flexible, but costs up to three times as much as using the local error estimate alone. The global error estimator has the advantages of tracking the accumulation of error in time and being able to discount large local errors that do not affect the chosen terminal error functional. The local error estimator is cheaper to compute because the global error estimator has the added cost of solving the adjoint problem. / 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 17, 2011. / Error Estimation, Adjoint, Adaptive, Spectral Element Methods / Includes bibliographical references. / David Kopriva, Professor Directing Dissertation; Paul Eugenio, University Representative; Bettye Anne Case, Committee Member; Kyle Gallivan, Committee Member; Craig Nolder, Committee Member; Giray Okten, Committee Member.

We study two "disjoint disks properties" in dimension 3 due to H. W. Lambert and R. B. Sher (Pacific. J. Math. 24 (1968) 511-518), the Dehn lemma property (DLP) and the map separation property (MSP). Theorem 1. Let G be a cell-like closed 0-dimensional upper semicontinuous decomposition of a 3-manifold M (possibly with boundary) with N(,G)(L-HOOK) int M. Then the following statements are equivalent: (i) M/G has the DLP; (ii) M/G has the MSP; (iii) M/G is a 3-manifold. Theorem 2. Let C be the class of all compact generalized 3-manifolds X with dim S(X) (LESSTHEQ) 0 and let C(,0)(L-HOOK) C be the subclass of all X(ELEM)C with S(X) (L-HOOK) {pt} and X (TURNEQ) S('3). Then the following statements are equivalent: (i) The Poincare conjecture in dimension three is true; (ii) If X(ELEM)C has the DLP or the MSP then S(X) = (SLASHCIRC); (iii) If X(ELEM)C(,0) has the DLP or the MSP then S(X) = (SLASHCIRC). / We also study neighborhoods of peripherally 1-acyclic compacta in nonorientable 3-manifolds. We prove a finiteness and a neighborhood theorem for such compacta and as an application extend a result of J. L. Bryant and R. C. Lacher concerning resolutions of almost (,2)-acyclic images of orientable 3-manifolds (Math. Proc. Camb. Phil. Soc. 88 (1980) 311-320), to nonorientable 3-manifolds. Theorem 3. Let f be a closed, monotone mapping from a 3-manifold M onto a locally simply connected (,2)-homology 3-manifold X. Suppose that there is a 0-dimensional set Z(L-HOOK)X such that H('1)(f('-1)(x); (,2)) = 0 for all x (ELEM) X - Z. Then the set C = {x(ELEM)X(VBAR)f('-1)(x) is not cell-like} is locally finite in X. Moreover X has a resolution. / Included is an investigation of the basic properties of generalized 3-manifolds with boundary, a topics on which little study has been done so far, as well as some results on regular neighborhoods of compacta in 3-manifolds with applications to homotopic PL embeddings of compact polyhedra into 3-manifolds. / Source: Dissertation Abstracts International, Volume: 44-02, Section: B, page: 0519. / Thesis (Ph.D.)--The Florida State University, 1983.

The main result in this thesis bounds the combinatorial modulus of a ring in a triangulation graph in terms of the modulus of a related ring. The bounds depend only on how the rings are related and not on the rings themselves. This may be used to solve the combinatorial type problem in a variety of situation, most significant in graphs with unbounded degree. Other results regarding the type problem are presented along with several examples illustrating the limits of the results. / A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Degree Awarded: Summer Semester, 2006. / Date of Defense: July 6, 2006. / Graph Theory, Circle Packing, Discrete Conformal Geometry, Conformal Type / Includes bibliographical references. / Philip Bowers, Professor Directing Dissertation; Lois Hawkes, Outside Committee Member; Steve Bellenot, Committee Member; Eric Klassen, Committee Member; Craig Nolder, Committee Member; Jack Quine, Committee Member.

Throughout, H is a Hopf algebra over a field k of characteristic p, G(H) is the group of grouplikes of H and L is any subgroup of G(H). We denote the antipode of H by S. We investigate the freeness of Hopf algebras as modules over their group algebras of grouplikes. / In chapter II we consider semisimple group algebras kL. We prove that for finite dimensional H all nonzero objects in the category (' )(, ) of left (H,kL)-Hopf modules are free kL-modules. We also prove this in the case when S('2) = id. Hence, for a finite dimensional H, the number of one-dimensional ideals divides the dimension of H and the order of S divides 4(.)dimension of H. / In chapter III we prove that a finite dimensional H is a free k-module for any g (ELEM) G(H), even if p divides the order of g. Further we establish that a finite dimensional H is a free kL-module if and only if H is a free kA-module for any elementary abelian p-subgroup A of L. / In chapter IV we prove that a finite dimensional H over an algebraically closed field k of characteristic p is a free kL-module, if H does not contain any simple subcoalgebra of dimension (lp)('2) for any natural number 1 (GREATERTHEQ) 2. Further we construct an example of an infinite dimensional H showing that not all objects in (' )(, ) are free kL-modules. Finally we show that any infinite dimensional H is a free kL-module, if L is an infinite group which contains no nontrivial finite subgroup. Also, if the dimension of H equals the dimension of the coradical of H, then H is a free module over any of its semisimple group algebras k where g (ELEM) G(H). / Source: Dissertation Abstracts International, Volume: 47-01, Section: B, page: 0246. / Thesis (Ph.D.)--The Florida State University, 1985.

Let M be a closed non-orientable 3-manifold with a Heegaard splitting of genus two. We show that, if M has a non-separating essential Klein bottle, then there is a non-separating essential Klein bottle (or torus) K such that the intersection of K and one of the handlebodies in the Heegaard splitting is an essential disk. Also, if every essential Klein bottle (or torus) is separating in M and if M has a non-trivial canonical system of 2-sided tori and Klein bottles, then there is a canonical system such that the intersection of this system with one of the handlebodies in the Heegaard splitting consists of at most two essential disks. We use these results to give a complete list of all the non-orientable 3-manifolds with a Heegaard splitting of genus two which are either not P('2)-irreducible or contain an incompressible torus or Klein bottle. / Source: Dissertation Abstracts International, Volume: 48-03, Section: B, page: 0784. / Thesis (Ph.D.)--The Florida State University, 1987.

Stochastic Navier-Stokes equations have been widely applied in various computational fluid dynamics (CFD) fields in recent years. It can be considered as another milestone in CFD. Our work focuses on exploring some theoretical and numerical properties of the stochastic Navier-Stokes equations and related optimal control problems. In particular, we consider: a numerical comparison of solutions of the stochastic Navier-Stokes equations perturbed by a large range of random noises in time and space; effective Martingale regularized methods for the stochastic Navier-Stokes equations with additive noises; and the stochastic Navier-Stokes equations constrained stochastic boundary optimal control problems. We systemically provide numerical simulation methods for the stochastic Navier-Stokes equations with different types of noises. The noises are classified as colored or white based on their autocovariance functions. For each type of noise, we construct a representation and a simulation method. Numerical examples are provided to illustrate our schemes. Comparisons of the influence of different noises on the solution of the Navier-Stokes system are presented. To improve the simulation accuracy, we impose a Martingale correction regularized method for the stochastic Navier-Stokes equations with additive noise. The original systems are split into two parts, a linear stochastic Stokes equations with Martingale solution and a stochastic modified Navier-Stokes equations with smoother noise. In addition, a negative fractional Laplace operator is introduced to regularize the noise term. Stability and convergence of the path-wise modified Navier-Stokes equations are proved. Numerical simulations are provided to illustrate our scheme. Comparisons of non-regularized and regularized noises for the Navier-Stokes system are presented to further demonstrate the efficiency of our numerical scheme. As a consequence of the above work, we consider a stochastic optimal control problem constrained by the Navier-Stokes equations with stochastic Dirichlet boundary conditions. Control is applied only on the boundary and is associated with reduced regularity, compared to interior controls. To ensure the existence of a solution and the efficiency of numerical simulations, the stochastic boundary conditions are required to belong almost surely to H¹(∂D). To simulate the system, state solutions are approximated using the stochastic collocation finite element approach, and sparse grid techniques are applied to the boundary random field. One-shot optimality systems are derived from Lagrangian functionals. Numerical simulations are then made, using a combination of Monte Carlo methods and sparse grid methods, which demonstrate the efficiency of the algorithm. / A Dissertation submitted to the Department of Scientific Computing in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Summer Semester 2017. / July 13, 2017. / Stochastic Navier-Stokes equations, Stochastic optimal control / Includes bibliographical references. / Max Gunzburger, Professor Directing Dissertation; Mark Sussman, University Representative; Janet Peterson, Committee Member; Bryan Quaife, Committee Member; Chen Huang, Committee Member.

According to rapid development in information technology, limit order books(LOB) mechanism has emerged to prevail in today's nancial market. In this paper, we propose ensemble machine learning architectures for capturing the dynamics of high-frequency limit order books such as predicting price spread crossing opportunities in a future time interval. The paper is more data-driven oriented, so experiments with ve real-time stock data from NASDAQ, measured by nanosecond, are established. The models are trained and validated by training and validation data sets. Compared with other models, such as logistic regression, support vector machine(SVM), our out-of-sample testing results has shown that ensemble methods had better performance on both statistical measurements and computational eciency. A simple trading strategy that we devised by our models has shown good prot and loss(P&L) results. Although this paper focuses on limit order books, the similar frameworks and processes can be extended to other classication research area. Keywords: limit order books, high-frequency trading, data analysis, ensemble methods, F1 score. / A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Summer Semester 2017. / July 18, 2017. / Includes bibliographical references. / Jinfeng Zhang, Professor Co-Directing Dissertation; Giray Okten, Professor Co-Directing Dissertation; Alec Kercheval, Committee Member; Washington Mio, Committee Member; Capstick Simon, University Representative.

This dissertation is based on the structural model framework for default risk that was first introduced by garreau2016structural (henceforth: the "G-K model"). In this approach, the time of default is defined as the first time the log-return of the firm's stock price jumps below a (possibly stochastic) "default threshold'' level. The stock price is assumed to follow an exponential L\'evy process and, in the multidimensional case, a multidimensional L\'evy process. This new structural model is mathematically equivalent to an intensity-based model where the intensity is parameterized by a L\'evy measure. The dependence between the default times of firms within a basket is the result of the jump dependence of their respective stock prices and described by a L\'evy copula. To extend the previous work, we focus on generalizing the joint survival probability and related results to the d-dimensional case. Using the link between L\'evy processes and multivariate exponential distributions, we derive the joint survival probability and characterize correlated default risk using L\'evy copulas. In addition, we extend our results to include stochastic interest rates. Moreover, we describe how to use the default threshold as the interface for incorporating additional exogenous economic factors, and still derive basket credit default swap (CDS) prices in terms of expectations. If we make some additional modeling assumptions such that the default intensities become affine processes, we obtain explicit formulas for the single name and first-to-default (FtD) basket CDS prices, up to quadrature. / A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Summer Semester 2017. / June 15, 2017. / credit default swap, credit risk, L\'evy processes / Includes bibliographical references. / Alec N. Kercheval, Professor Directing Dissertation; Wei Wu, University Representative; Giray Okten, Committee Member; Arash Fahim, Committee Member.