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Pages
 Title
 The 1Type of Algebraic KTheory as a Multifunctor.
 Creator

Valdes, Yaineli, Aldrovandi, Ettore, Rawling, John Piers, Agashe, Amod S., Aluffi, Paolo, Petersen, Kathleen L., Hoeij, Mark van, Florida State University, College of Arts and...
Show moreValdes, Yaineli, Aldrovandi, Ettore, Rawling, John Piers, Agashe, Amod S., Aluffi, Paolo, Petersen, Kathleen L., Hoeij, Mark van, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

It is known that the category of Waldhausen categories is a closed symmetric multicategory and algebraic Ktheory is a multifunctor from the category of Waldhuasen categories to the category of spectra. By assigning to any Waldhausen category the fundamental groupoid of the 1type of its Ktheory spectrum, we get a functor from the category of Waldhausen categories to the category of Picard groupoids, since stable 1types are classified by Picard groupoids. We prove that this functor is a...
Show moreIt is known that the category of Waldhausen categories is a closed symmetric multicategory and algebraic Ktheory is a multifunctor from the category of Waldhuasen categories to the category of spectra. By assigning to any Waldhausen category the fundamental groupoid of the 1type of its Ktheory spectrum, we get a functor from the category of Waldhausen categories to the category of Picard groupoids, since stable 1types are classified by Picard groupoids. We prove that this functor is a multifunctor to a corresponding multicategory of Picard groupoids.
Show less  Date Issued
 2018
 Identifier
 2018_Sp_Valdes_fsu_0071E_14374
 Format
 Thesis
 Title
 Affine Dimension of Smooth Curves and Surfaces.
 Creator

Williams, Ethan Randy, Oberlin, Richard, Ormsbee, Michael J., Reznikov, Alexander, Bauer, Martin, Florida State University, College of Arts and Sciences, Department of Mathematics
 Abstract/Description

Our aim is to study the affine dimension of some smooth manifolds. In Chapter 1, we review the notions of Minkowski and Hausdorff dimension, and compare them with the lesser studied affine dimension. In Chapter 2, we focus on understanding the affine dimension of curves. In Section 2.1, we review the existing results for the affine dimension of a strictly convex curve in the plane, and in Section 2.2, we classify the smooth curves in ℝn based on affine dimension. In Chapter 3, we classify the...
Show moreOur aim is to study the affine dimension of some smooth manifolds. In Chapter 1, we review the notions of Minkowski and Hausdorff dimension, and compare them with the lesser studied affine dimension. In Chapter 2, we focus on understanding the affine dimension of curves. In Section 2.1, we review the existing results for the affine dimension of a strictly convex curve in the plane, and in Section 2.2, we classify the smooth curves in ℝn based on affine dimension. In Chapter 3, we classify the smooth hypersurfaces in ℝ3 with nonnegative Gaussian curvature based on affine dimension, and in Chapter 4 we provide a lower and upper bound for the affine dimension of smooth, convex hypersurfaces in ℝn.
Show less  Date Issued
 2018
 Identifier
 2018_Sp_Williams_fsu_0071E_14512
 Format
 Thesis
 Title
 Algorithmic Lung Nodule Analysis in Chest Tomography Images: Lung Nodule Malignancy Likelihood Prediction and a Statistical Extension of the Level Set Image Segmentation Method.
 Creator

Hancock, Matthew C. (Matthew Charles), Magnan, Jeronimo Francisco, Duke, D. W., Hurdal, Monica K., Mio, Washington, Florida State University, College of Arts and Sciences,...
Show moreHancock, Matthew C. (Matthew Charles), Magnan, Jeronimo Francisco, Duke, D. W., Hurdal, Monica K., Mio, Washington, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

Lung cancer has the highest mortality rate of all cancers in both men and women in the United States. The algorithmic detection, characterization, and diagnosis of abnormalities found in chest CT scan images can aid radiologists by providing additional medicallyrelevant information to consider in their assessment of medical images. Such algorithms, if robustly validated in clinical settings, carry the potential to improve the health of the general population. In this thesis, we first give an...
Show moreLung cancer has the highest mortality rate of all cancers in both men and women in the United States. The algorithmic detection, characterization, and diagnosis of abnormalities found in chest CT scan images can aid radiologists by providing additional medicallyrelevant information to consider in their assessment of medical images. Such algorithms, if robustly validated in clinical settings, carry the potential to improve the health of the general population. In this thesis, we first give an analysis of publicly available chest CT scan annotation data, in which we determine upper bounds on expected classification accuracy when certain radiological features are used as inputs to statistical learning algorithms for the purpose of inferring the likelihood of a lung nodule as being either malignant or benign. Second, a statistical extension of the level set method for image segmentation is introduced and applied to both syntheticallygenerated and real threedimensional image volumes of lung nodules in chest CT scans, obtaining results comparable to the current stateoftheart on the latter.
Show less  Date Issued
 2018
 Identifier
 2018_Sp_Hancock_fsu_0071E_14427
 Format
 Thesis
 Title
 Algorithms for Solving Linear Differential Equations with Rational Function Coefficients.
 Creator

Imamoglu, Erdal, van Hoeij, Mark, van Engelen, Robert, Agashe, Amod S. (Amod Sadanand), Aldrovandi, Ettore, Aluffi, Paolo, Florida State University, College of Arts and Sciences...
Show moreImamoglu, Erdal, van Hoeij, Mark, van Engelen, Robert, Agashe, Amod S. (Amod Sadanand), Aldrovandi, Ettore, Aluffi, Paolo, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

This thesis introduces two new algorithms to find hypergeometric solutions of second order regular singular differential operators with rational function or polynomial coefficients. Algorithm 3.2.1 searches for solutions of type: exp(∫ r dx) ⋅ ₂F₁ (a₁,a₂;b₁;f) and Algorithm 5.2.1 searches for solutions of type exp(∫ r dx) (r₀ ⋅ ₂F₁(a₁,a₂;b₁;f) + r₁ ⋅ ₂F´₁ (a₁,a₂;b₁;f)) where f, r, r₀, r₁ ∈ ℚ̅(̅x̅)̅ and a₁,a₂,b₁ ∈ ℚ and denotes the Gauss hypergeometric function. The algorithms use modular...
Show moreThis thesis introduces two new algorithms to find hypergeometric solutions of second order regular singular differential operators with rational function or polynomial coefficients. Algorithm 3.2.1 searches for solutions of type: exp(∫ r dx) ⋅ ₂F₁ (a₁,a₂;b₁;f) and Algorithm 5.2.1 searches for solutions of type exp(∫ r dx) (r₀ ⋅ ₂F₁(a₁,a₂;b₁;f) + r₁ ⋅ ₂F´₁ (a₁,a₂;b₁;f)) where f, r, r₀, r₁ ∈ ℚ̅(̅x̅)̅ and a₁,a₂,b₁ ∈ ℚ and denotes the Gauss hypergeometric function. The algorithms use modular reduction, Hensel lifting, rational function reconstruction, and rational number reconstruction to do so. Numerous examples from different branches of science (mostly from combinatorics and physics) showed that the algorithms presented in this thesis are very effective. Presently, Algorithm 5.2.1 is the most general algorithm in the literature to find hypergeometric solutions of such operators. This thesis also introduces a fast algorithm (Algorithm 4.2.3) to find integral bases for arbitrary order regular singular differential operators with rational function or polynomial coefficients. A normalized (Algorithm 4.3.1) integral basis for a differential operator provides us transformations that convert the differential operator to its standard forms (Algorithm 5.1.1) which are easier to solve.
Show less  Date Issued
 2017
 Identifier
 FSU_SUMMER2017_Imamoglu_fsu_0071E_13942
 Format
 Thesis
 Title
 An Analysis of Conjugate Harmonic Components of Monogenic Functions and Lambda Harmonic Functions.
 Creator

BallengerFazzone, Brendon Kerr, Nolder, Craig, Harper, Kristine, Aldrovandi, Ettore, Case, Bettye Anne, Quine, J. R. (John R.), Ryan, John Barry, Florida State University,...
Show moreBallengerFazzone, Brendon Kerr, Nolder, Craig, Harper, Kristine, Aldrovandi, Ettore, Case, Bettye Anne, Quine, J. R. (John R.), Ryan, John Barry, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

Clifford analysis is seen as the higher dimensional analogue of complex analysis. This includes a rich study of Clifford algebras and, in particular, monogenic functions, or Cliffordvalued functions that lie in the kernel of the CauchyRiemann operator. In this dissertation, we explore the relationships between the harmonic components of monogenic functions and expand upon the notion of conjugate harmonic functions. We show that properties of the even part of a Cliffordvalued function...
Show moreClifford analysis is seen as the higher dimensional analogue of complex analysis. This includes a rich study of Clifford algebras and, in particular, monogenic functions, or Cliffordvalued functions that lie in the kernel of the CauchyRiemann operator. In this dissertation, we explore the relationships between the harmonic components of monogenic functions and expand upon the notion of conjugate harmonic functions. We show that properties of the even part of a Cliffordvalued function determine properties of the odd part and vice versa. We also explore the theory of functions lying in the kernel of a generalized Laplace operator, the λLaplacian. We explore the properties of these socalled λharmonic functions and give the solution to the Dirichlet problem for the λharmonic functions on annular domains in Rⁿ.
Show less  Date Issued
 2016
 Identifier
 FSU_2016SP_BallengerFazzone_fsu_0071E_13136
 Format
 Thesis
 Title
 Analysis of Functions of SplitComplex, Multicomplex, and SplitQuaternionic Variables and Their Associated Conformal Geometries.
 Creator

Emanuello, John Anthony, Nolder, Craig, Tabor, Samuel Lynn, Case, Bettye Anne, Quine, J. R. (John R.), Florida State University, College of Arts and Sciences, Department of...
Show moreEmanuello, John Anthony, Nolder, Craig, Tabor, Samuel Lynn, Case, Bettye Anne, Quine, J. R. (John R.), Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

The connections between algebra, geometry, and analysis have led the way for numerous results in many areas of mathematics, especially complex analysis. Considerable effort has been made to develop higher dimensional analogues of the complex numbers, such as Clifford algebras and Multicomplex numbers. These rely heavily on geometric notions, and we explore the analysis which results. This is what is called hypercomplex analysis. This dissertation explores the most prominent of these higher...
Show moreThe connections between algebra, geometry, and analysis have led the way for numerous results in many areas of mathematics, especially complex analysis. Considerable effort has been made to develop higher dimensional analogues of the complex numbers, such as Clifford algebras and Multicomplex numbers. These rely heavily on geometric notions, and we explore the analysis which results. This is what is called hypercomplex analysis. This dissertation explores the most prominent of these higher dimensional analogues and highlights a many of the relevant results which have appeared in the last four decades, and introduces new ideas which can be used to further the research of this discipline. Indeed, the objects of interest are Clifford algebras, the algebra of the Multicomplex numbers, and functions which are valued in these algebras and lie in the kernels of linear operators. These lead to prominent results in Clifford analysis and multicomplex analysis which can be viewed as analogues of complex analysis. Additionally, we explain the link between Clifford algebras and conformal geometry. We explore two low dimensional examples, namely the splitcomplex numbers and splitquaternions, and demonstrate how linear fractional transformations are conformal mappings in these settings.
Show less  Date Issued
 2015
 Identifier
 FSU_migr_etd9326
 Format
 Thesis
 Title
 Analysis of Regularity and Convergence of Discretization Methods for the Stochastic Heat Equation Forced by SpaceTime White Noise.
 Creator

Wills, Anthony Clinton, Wang, Xiaoming, Ewald, Brian D., Reina, Laura, Bowers, Philip L., Case, Bettye Anne, Ökten, Giray, Florida State University, College of Arts and Sciences...
Show moreWills, Anthony Clinton, Wang, Xiaoming, Ewald, Brian D., Reina, Laura, Bowers, Philip L., Case, Bettye Anne, Ökten, Giray, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

We consider the heat equation forced by a spacetime white noise and with periodic boundary conditions in one dimension. The equation is discretized in space using four different methods; spectral collocation, spectral truncation, finite differences, and finite elements. For each of these methods we derive a spacetime white noise approximation and a formula for the covariance structure of the solution to the discretized equation. The convergence rates are analyzed for each of the methods as...
Show moreWe consider the heat equation forced by a spacetime white noise and with periodic boundary conditions in one dimension. The equation is discretized in space using four different methods; spectral collocation, spectral truncation, finite differences, and finite elements. For each of these methods we derive a spacetime white noise approximation and a formula for the covariance structure of the solution to the discretized equation. The convergence rates are analyzed for each of the methods as the spatial discretization becomes arbitrarily fine and this is confirmed numerically. Dirichlet and Neumann boundary conditions are also considered. We then derive covariance structure formulas for the two dimensional stochastic heat equation using each of the different methods. In two dimensions the solution does not have a finite variance and the formulas for the covariance structure using different methods does not agree in the limit. This means we must analyze the convergence in a different way than the one dimensional problem. To understand this difference in the solution as the spatial dimension increases, we find the Sobolev space in which the approximate solution converges to the solution in one and two dimensions. This result is then generalized to n dimensions. This gives a precise statement about the regularity of the solution as the spatial dimension increases. Finally, we consider a generalization of the stochastic heat equation where the forcing term is the spatial derivative of a spacetime white noise. For this equation we derive formulas for the covariance structure of the discretized equation using the spectral truncation and finite difference method. Numerical simulation results are presented and some qualitative comparisons between these two methods are made.
Show less  Date Issued
 2015
 Identifier
 FSU_migr_etd9488
 Format
 Thesis
 Title
 Approximating Nonlocal Diffusion Problems Using Quadrature Rules Generated by Radial Basis Functions.
 Creator

Lyngaas, Isaac Ron, Peterson, Janet S., Gunzburger, Max D., Burkardt, John V., Florida State University, College of Arts and Sciences, Department of Mathematics
 Abstract/Description

Nonlocal models differ from traditional partial differential equation (PDE) models because they contain no spatial derivatives; instead an appropriate integral is used. Nonlocal models are especially useful in the case where there are issues calculating the spatial derivatives of a PDE model. In many applications (e.g., biological systems, flow through porous media) the observed rate of diffusion is not accurately modeled by the standard diffusion differential operator but rather exhibits so...
Show moreNonlocal models differ from traditional partial differential equation (PDE) models because they contain no spatial derivatives; instead an appropriate integral is used. Nonlocal models are especially useful in the case where there are issues calculating the spatial derivatives of a PDE model. In many applications (e.g., biological systems, flow through porous media) the observed rate of diffusion is not accurately modeled by the standard diffusion differential operator but rather exhibits socalled anomalous diffusion. Anomalous diffusion can be represented in a PDE model by using a fractional Laplacian operator in space whereas the nonlocal approach only needs to slightly modify its integral formulation to model anomalous diffusion. Anomalous diffusion is one such case where approximating the spatial derivative operator is a difficult problem. In this work, an approach for approximating standard and anomalous nonlocal diffusion problems using a new technique that utilizes radial basis functions (RBFs) is introduced and numerically tested. The typical approach for approximating nonlocal diffusion problems is to use a Galerkin formulation. However, the Galerkin formulation for nonlocal diffusion problems can often be difficult to compute efficiently and accurately especially for problems in multiple dimensions. Thus, we investigate the alternate approach of using quadrature rules generated by RBFs to approximate the nonlocal diffusion problem. This work will be split into three major parts. The first will introduce RBFs and give some examples of how they are used. This part will motivate our approach for using RBFs on the nonlocal diffusion problem. In the second part, we will derive RBFgenerated quadrature rules in one dimension and show they can be used to approximate nonlocal diffusion problems. The final part will address how the RBF quadrature approach can be extended to higher dimensional problems. Numerical test cases are shown for both the standard and anomalous nonlocal diffusion problems and compared with standard finite element approximations. Preliminary results show that the method introduced is viable for approximating nonlocal diffusion problems and that highly accurate approximations are possible using this approach.
Show less  Date Issued
 2016
 Identifier
 FSU_FA2016_Lyngaas_fsu_0071N_13512
 Format
 Thesis
 Title
 Arithmetic Aspects of Noncommutative Geometry: Motives of Noncommutative Tori and Phase Transitions on GL(n) and Shimura Varieties Systems.
 Creator

Shen, Yunyi, Marcolli, Matilde, Aluffi, Paolo, Chicken, Eric, Bowers, Philip L., Petersen, Kathleen L., Florida State University, College of Arts and Sciences, Department of...
Show moreShen, Yunyi, Marcolli, Matilde, Aluffi, Paolo, Chicken, Eric, Bowers, Philip L., Petersen, Kathleen L., Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

In this dissertation, we study three important cases in noncommutative geometry. We first observe the standard noncommutative object, noncommutative torus, in noncommutative motives. We work with the category of holomorphic bundles on a noncommutative torus, which is known to be equivalent to the heart of a nonstandard tstructure on coherent sheaves of an elliptic curve. We then introduce a notion of (weak) tstructure in dg categories. By lifting the nonstandard tstructure to the t...
Show moreIn this dissertation, we study three important cases in noncommutative geometry. We first observe the standard noncommutative object, noncommutative torus, in noncommutative motives. We work with the category of holomorphic bundles on a noncommutative torus, which is known to be equivalent to the heart of a nonstandard tstructure on coherent sheaves of an elliptic curve. We then introduce a notion of (weak) tstructure in dg categories. By lifting the nonstandard tstructure to the tstructure that we defined, we find a way of seeing a noncommutative torus in noncommutative motives. By applying the tstructure to a noncommutative torus and describing the cyclic homology of the category of holomorphic bundle on the noncommutative torus, we finally show that the periodic cyclic homology functor induces a decomposition of the motivic Galois group of the Tannakian category generated by the associated auxiliary elliptic curve. In the second case, we generalize the results of Laca, Larsen, and Neshveyev on the GL2ConnesMarcolli system to the GLnConnesMarcolli systems. We introduce and define the GLnConnesMarcolli systems and discuss the existence and uniqueness questions of the KMS equilibrium states. Using the ergodicity argument and Hecke pair calculation, we classify the KMS states at different inverse temperatures β. Specifically, we show that in the range of n − 1 < β ≤ n, there exists only one KMS state. We prove that there are no KMS states when β < n − 1 and β ̸= 0, 1, . . . , n − 1,, while we actually construct KMS states for integer values of β in 1 ≤ β ≤ n − 1. For β > n, we characterize the extremal KMS states. In the third case, we push the previous results to more abstract settings. We mainly study the connected Shimura dynamical systems. We give the definition of the essential and superficial KMS states. We further develop a set of arithmetic tools to generalize the results in the previous case. We then prove the uniqueness of the essential KMS states and show the existence of the essential KMS stats for high inverse temperatures.
Show less  Date Issued
 2017
 Identifier
 FSU_SUMMER2017_Shen_fsu_0071E_13982
 Format
 Thesis
 Title
 Asset Pricing Equilibria for Heterogeneous, LimitedInformation Agents.
 Creator

Jones, Dawna Candice, Kercheval, Alec N., Beaumont, Paul M, Van Winkle, David H., Nichols, Warren, Ökten, Giray, Florida State University, College of Arts and Sciences,...
Show moreJones, Dawna Candice, Kercheval, Alec N., Beaumont, Paul M, Van Winkle, David H., Nichols, Warren, Ökten, Giray, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

The standard general equilibrium asset pricing models typically make two simplifying assumptions: homogeneous agents and the existence of a rational expectations equilibrium. This context sometimes yields outcomes that are inconsistent with the empirical findings. We hypothesize that allowing agent heterogeneity could assist in replicating the empirical results. However, the inclusion of heterogeneity in models where agents are fully rational proves impossible to solve without severe...
Show moreThe standard general equilibrium asset pricing models typically make two simplifying assumptions: homogeneous agents and the existence of a rational expectations equilibrium. This context sometimes yields outcomes that are inconsistent with the empirical findings. We hypothesize that allowing agent heterogeneity could assist in replicating the empirical results. However, the inclusion of heterogeneity in models where agents are fully rational proves impossible to solve without severe simplifying assumptions. The reason for this difficulty is that heterogeneous agent models generate an endogenously complicated distribution of wealth across the agents. The state space for each agent's optimization problem includes the complex dynamics of the wealth distribution. There is no general way to characterize the interaction between the distribution of wealth and the macroeconomic aggregates. To address this issue, we implement an agentbased model where the agents have bounded rationality. In our model, we have a complete markets economy with two agents and two assets. The agents are heterogeneous and utility maximizing with constant coefficient of relative risk aversion [CRRA] preferences. How the agents address the stochastic behaviour of the evolution of the wealth distribution is central to our task since aggregate prices depend on this behaviour. An important component of this dissertation involves dealing with the computational difficulty of dynamic heterogeneousagent models. That is, in order to predict prices, agents need a way to keep track of the evolution of the wealth distribution. We do this by allowing each agent to assume that a priceequivalent representative agent exists and that the representative agent has a constant coefficient of relative risk aversion. In so doing, the agents are able to formulate predictive pricing and demand functions which allow them to predict aggregate prices and make consumption and investment decisions each period. However, the agents' predictions are only approximately correct. Therefore, we introduce a learning mechanism to maintain the required level of accuracy in the agents' price predictions. From this setup, we find that the model, with learning, will converge over time to an approximate expectations equilibrium, provided that the the initial conditions are close enough to the rational expectations equilibrium prices. Two main contributions in our work are: 1) to formulate a new concept of approximate equilibria, and 2) to show how equilibria can be approximated numerically, despite the fact that the true state space at any point in time is mathematically complex. These contributions offer the possibility of characterizing a new class of asset pricing models where agents are heterogeneous and only just slightly limited in their rationality. That is, the partially informed agents in our model are able to forecast and utilitymaximize only just as well as economists who face problems of estimating aggregate variables. By using an exogenously assigned adaptive learning rule, we analyse this implementation in a Lucastype heterogeneous agent model. We focus on the sensitivity of the risk parameter and the convergence of the model to an approximate expectations equilibrium. Also, we study the extent to which adaptive learning is able to explain the empirical findings in an asset pricing model with heterogeneous agents.
Show less  Date Issued
 2015
 Identifier
 FSU_migr_etd9624
 Format
 Thesis
 Title
 Character Varieties of Knots and Links with Symmetries.
 Creator

Sparaco, Leona H., Petersen, Kathleen L., Harper, Kristine, Ballas, Sam, Bowers, Philip L., Hironaka, Eriko, Florida State University, College of Arts and Sciences, Department...
Show moreSparaco, Leona H., Petersen, Kathleen L., Harper, Kristine, Ballas, Sam, Bowers, Philip L., Hironaka, Eriko, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

: Let M be a hyperbolic manifold. The SL2(C) character variety of M is essentially the set of all representations ρ : π1(M) → SL2(C) up to trace equivalence. This algebraic set is connected to many geometric properties of the manifold M. We examine the effect of symmetries of M on its character variety. We compute the SL2(C) and PSL2(C) character varieties for an infinite family of twobridge hyperbolic knots with symmetry. We explore the effect the symmetry has on the character variety and...
Show more: Let M be a hyperbolic manifold. The SL2(C) character variety of M is essentially the set of all representations ρ : π1(M) → SL2(C) up to trace equivalence. This algebraic set is connected to many geometric properties of the manifold M. We examine the effect of symmetries of M on its character variety. We compute the SL2(C) and PSL2(C) character varieties for an infinite family of twobridge hyperbolic knots with symmetry. We explore the effect the symmetry has on the character variety and exploit this symmetry to factor the character variety. We then find the geometric genus of both components of the character variety. We compute the SL2(C) character variety for the Borromean ring complement in S^3. Further, we explore how the symmetries effect this character variety. Finally, we prove some general results about the structure of character varieties of links with symmetries.
Show less  Date Issued
 2017
 Identifier
 FSU_SUMMER2017_Sparaco_fsu_0071E_13851
 Format
 Thesis
 Title
 Characteristic Classes and Local Invariants of Determinantal Varieties and a Formula for Equivariant ChernSchwartzMacPherson Classes of Hypersurfaces.
 Creator

Zhang, Xiping, Aluffi, Paolo, Piekarewicz, Jorge, Aldrovandi, Ettore, Petersen, Kathleen L., Hoeij, Mark van, Florida State University, College of Arts and Sciences, Department...
Show moreZhang, Xiping, Aluffi, Paolo, Piekarewicz, Jorge, Aldrovandi, Ettore, Petersen, Kathleen L., Hoeij, Mark van, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

Determinantal varieties parametrize spaces of matrices of given ranks. The main results of this dissertation are computations of intersectiontheoretic invariants of determinantal varieties. We focus on the ChernMather and ChernSchwartzMacPherson classes, on the characteristic cycles, and on topologically motivated invariants such as the local Euler obstruction. We obtain explicit formulas in both the ordinary and the torusequivariant setting, and formulate a conjecture concerning the...
Show moreDeterminantal varieties parametrize spaces of matrices of given ranks. The main results of this dissertation are computations of intersectiontheoretic invariants of determinantal varieties. We focus on the ChernMather and ChernSchwartzMacPherson classes, on the characteristic cycles, and on topologically motivated invariants such as the local Euler obstruction. We obtain explicit formulas in both the ordinary and the torusequivariant setting, and formulate a conjecture concerning the effectiveness of the ChernSchwartzMacPherson classes of determinantal varieties. We also prove a vanishing property for the ChernSchwartzMacPherson classes of general group orbits. As applications we obtain formulas for the sectional Euler characteristic of determinantal varieties and the microlocal indices of their intersection cohomology sheaf complexes. Moreover, for a close embedding we define the equivariant version of the Segre class and prove an equivariant formula for the ChernSchwartzMacPherson classes of hypersurfaces of projective varieties.
Show less  Date Issued
 2018
 Identifier
 2018_Sp_Zhang_fsu_0071N_14521
 Format
 Thesis
 Title
 Conformal Tilings and Type.
 Creator

Mayhook, Dane, Bowers, Philip L., Riley, Mark A., Heil, Wolfgang H., Klassen, E. (Eric), Florida State University, College of Arts and Sciences, Department of Mathematics
 Abstract/Description

This paper examines a class of geometric tilings known as conformal tilings, first introduced by Bowers and Stephenson in a 1997 paper, and later developed in a series of papers by the same authors. These tilings carry a prescribed conformal structure in that the tiles are all conformally regular, and admit a reflective structure. Conformal tilings are essentially uniquely determined by their combinatorial structure, which we encode as a planar polygonal complex. It is natural to consider not...
Show moreThis paper examines a class of geometric tilings known as conformal tilings, first introduced by Bowers and Stephenson in a 1997 paper, and later developed in a series of papers by the same authors. These tilings carry a prescribed conformal structure in that the tiles are all conformally regular, and admit a reflective structure. Conformal tilings are essentially uniquely determined by their combinatorial structure, which we encode as a planar polygonal complex. It is natural to consider not just a single planar polygonal complex, but its entire local isomorphism class. We present a case study on the local isomorphism class of the discrete hyperbolic plane complex, ultimately providing a constructive description of each of its uncountably many members. Conformal tilings may tile either the complex plane or the Poincaré disk, and answering the type problem motivates the remainder of the paper. Subdivision operators are used to repeatedly subdivide and amalgamate tilings, and Bowers and Stephenson prove that when a conformal tiling admits a combinatorial hierarchy manifested by an expansive, conformal subdivision operator, then that tiling is parabolic and tiles the plane. We introduce a new notion of hierarchya fractal hierarchyand generalize their result in some cases by showing that conformal tilings which admit a combinatorial hierarchy manifested by an expansive, fractal subdivision operator are also parabolic and tile the plane, assuming that two generic conditions for conformal tilings are true. This then answers the problem for certain expansion complexes, showing that expansion complexes for appropriate rotationally symmetric subdivision operators are necessarily parabolic.
Show less  Date Issued
 2016
 Identifier
 FSU_2016SU_Mayhook_fsu_0071E_13406
 Format
 Thesis
 Title
 Developing SRSF Shape Analysis Techniques for Applications in Neuroscience and Genomics.
 Creator

Wesolowski, Sergiusz, Wu, Wei, Bertram, R. (Richard), Srivastava, Anuj, Beerli, Peter, Mio, Washington, Florida State University, College of Arts and Sciences, Department of...
Show moreWesolowski, Sergiusz, Wu, Wei, Bertram, R. (Richard), Srivastava, Anuj, Beerli, Peter, Mio, Washington, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

Dissertation focuses on exploring the capabilities of the SRSF statistical shape analysis framework through various applications. Each application gives rise to a specific mathematical shape analysis model. The theoretical investigation of the models, driven by real data problems, give rise to new tools and theorems necessary to conduct a sound inference in the space of shapes. From theoretical standpoint the robustness results are provided for the model parameters estimation and an ANOVA...
Show moreDissertation focuses on exploring the capabilities of the SRSF statistical shape analysis framework through various applications. Each application gives rise to a specific mathematical shape analysis model. The theoretical investigation of the models, driven by real data problems, give rise to new tools and theorems necessary to conduct a sound inference in the space of shapes. From theoretical standpoint the robustness results are provided for the model parameters estimation and an ANOVAlike statistical testing procedure is discussed. The projects were a result of the collaboration between theoretical and applicationfocused research groups: the Shape Analysis Group at the Department of Statistics at Florida State University, the Center of Genomics and Personalized Medicine at FSU and the FSU's Department of Neuroscience. As a consequence each of the projects consists of two aspects—the theoretical investigation of the mathematical model and the application driven by a real life problem. The applications components, are similar from the data modeling standpoint. In each case the problem is set in an infinite dimensional space, elements of which are experimental data points that can be viewed as shapes. The three projects are: ``A new framework for Euclidean summary statistics in the neural spike train space''. The project provides a statistical framework for analyzing the spike train data and a new noise removal procedure for neural spike trains. The framework adapts the SRSF elastic metric in the space of point patterns to provides a new notion of the distance. ``SRSF shape analysis for sequencing data reveal new differentiating patterns''. This project uses the shape interpretation of the Next Generation Sequencing data to provide a new point of view of the exon level gene activity. The novel approach reveals a new differential gene behavior, that can't be captured by the stateofthe art techniques. Code is available online on github repository. ``How changes in shape of nucleosomal DNA near TSS influence changes of gene expression''. The result of this work is the novel shape analysis model explaining the relation between the change of the DNA arrangement on nucleosomes and the change in the differential gene expression.
Show less  Date Issued
 2017
 Identifier
 FSU_FALL2017_Wesolowski_fsu_0071E_14177
 Format
 Thesis
 Title
 Diffuse Interface Method for TwoPhase Incompressible Flows.
 Creator

Han, Daozhi, Wang, Xiaoming, Höflich, Peter, Gallivan, Kyle A., Kopriva, David A., Oberlin, Daniel M., Sussman, Mark, Florida State University, College of Arts and Sciences,...
Show moreHan, Daozhi, Wang, Xiaoming, Höflich, Peter, Gallivan, Kyle A., Kopriva, David A., Oberlin, Daniel M., Sussman, Mark, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

In this contribution, we focus on the study of multiphase flow using the phase field approach. Multiphase flow phenomena are ubiquitous. Common examples include coupled atmosphere and ocean system (air and water), oil reservoir (water, oil and gas), cloud and fog (water vapor, water and air). Multiphase flows also play an important role in many engineering and environmental science applications. For two fluids with matched density, the CahnHilliardNavierStokes system (CHNS) is a well...
Show moreIn this contribution, we focus on the study of multiphase flow using the phase field approach. Multiphase flow phenomena are ubiquitous. Common examples include coupled atmosphere and ocean system (air and water), oil reservoir (water, oil and gas), cloud and fog (water vapor, water and air). Multiphase flows also play an important role in many engineering and environmental science applications. For two fluids with matched density, the CahnHilliardNavierStokes system (CHNS) is a well accepted phase field model. We propose a novel second order in time numerical scheme for solving the CHNS system. The scheme is based on a second order convexsplitting for the CahnHilliard equation and pressureprojection for the NavierStokes equation. We show that the scheme is massconservative, satisfies a modified energy law and is therefore unconditionally stable. Moreover, we prove that the scheme is unconditionally uniquely solvable at each time step by exploring the monotonicity associated with the scheme. Thanks to the simple coupling of the scheme, we design an efficient Picard iteration procedure to further decouple the computation of CahnHilliard equation and NavierStokes equation. We implement the scheme by the mixed finite element method. Ample numerical experiments are performed to validate the accuracy and efficiency of the numerical scheme. In addition, we propose a novel decoupled unconditionally stable numerical scheme for the simulation of twophase flow in a HeleShaw cell which is governed by the CahnHilliardHeleShaw system (CHHS). The temporal discretization of the CahnHilliard equation is based on a convexsplitting of the associated energy functional. Moreover, the capillary forcing term in the Darcy equation is separated from the pressure gradient at the time discrete level by using an operatorsplitting strategy. Thus the computation of the nonlinear CahnHilliard equation is completely decoupled from the update of pressure. Finally, a pressurestabilization technique is used in the update of pressure so that at each time step one only needs to solve a Poisson equation with constant coefficient. We show that the scheme is unconditionally stable. Numerical results are presented to demonstrate the accuracy and efficiency of our scheme. The CHNS system and CHHS system are two widely used phase field models for twophase flow in a single domain (either conduit or HeleShaw cell/porous media). There are applications such as flows in unconfined karst aquifers, karst oil reservoir, proton membrane exchange fuel cell, where multiphase flows in conduits and in porous media must be considered together. Geometric configurations that contain both conduit (or vug) and porous media are termed karstic geometry. We present a family of phase field (diffusive interface) models for two phase flow in karstic geometry. These models, the socalled CahnHilliardStokesDarcy system, together with the associated interface boundary conditions are derived by utilizing Onsager's extremum principle. The models derived enjoy physically important energy laws and are consistent with thermodynamics. For the analysis of the CahnHilliardStokesDarcy system, we show that there exists at least a global in time finite energy solution by the compactness argument. A weakstrong uniqueness result is also established, which says that the strong solution, if exists, is unique in the class of weak solutions. Finally, we propose and analyze two unconditionally stable numerical algorithms of first order and second order respectively, for solving the CHSD system. A decoupled numerical procedure for practical implementation of the schemes are also presented. The decoupling is realized through explicit discretization of the velocity in the CahnHilliard equation and extrapolation in time of the interface boundary conditions. At each time step, one only needs to solve a CahnHilliard type equation in the whole domain, a Darcy equation in porous medium, and a Stokes equation in conduit in a separate and sequential fashion. Two numerical experiments, boundary driven and buoyancy driven flows, are performed to illustrate the effectiveness of our scheme. Both numerical simulations are of physical interest for transport processes of twophase flow in karst geometry.
Show less  Date Issued
 2015
 Identifier
 FSU_migr_etd9609
 Format
 Thesis
 Title
 Diffusion Approximation of a Risk Model.
 Creator

Cheng, Zailei, Zhu, Lingjiong, Niu, Xufeng, Fahim, Arash, Lee, Sanghyun, Florida State University, College of Arts and Sciences, Department of Mathematics
 Abstract/Description

We consider a classical risk process with arrival of claims following a nonstationary Hawkes process. We study the asymptotic regime when the premium rate and the baseline intensity of the claims arrival process are large, and claim size is small. The main goal of the article is to establish a diffusion approximation by verifying a functional central limit theorem and to compute the ruin probability in finitetime horizon. Numerical results will also be given.
 Date Issued
 2018
 Identifier
 2018_Fall_Cheng_fsu_0071E_14916
 Format
 Thesis
 Title
 Distributed Neural Network Models for Birdsong Production.
 Creator

Galvis, Daniel Eduardo, Bertram, R., Trombley, Paul Q., Cogan, Nicholas G., Hyson, Richard Lee, Johnson, Frank, Vo, Theodore, Florida State University, College of Arts and...
Show moreGalvis, Daniel Eduardo, Bertram, R., Trombley, Paul Q., Cogan, Nicholas G., Hyson, Richard Lee, Johnson, Frank, Vo, Theodore, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

Birdsong is a model system for the production of learned, serially ordered motor movements, such as playing a musical instrument or riding a bicycle. To this end, the neural mechanisms underlying birdsong have been studied in great depth, and many tools have been developed for analyzing the spectral and temporal structure of song. In this dissertation, I develop mathematical neural network models to explain how the nuclei in the song system interact to produce song. These models are...
Show moreBirdsong is a model system for the production of learned, serially ordered motor movements, such as playing a musical instrument or riding a bicycle. To this end, the neural mechanisms underlying birdsong have been studied in great depth, and many tools have been developed for analyzing the spectral and temporal structure of song. In this dissertation, I develop mathematical neural network models to explain how the nuclei in the song system interact to produce song. These models are constrained by the structural connectivity of the song system, the signaling properties of individual neurons and circuits, and circuitbreaking behavioral studies. Chapter 1 provides an overview of songbirds as a model system for speech production, outlines the structure of song, and describes the structure and function of the song system. Chapter 2 describes the neurophysiology and mathematical models of a premotor nucleus, called HVC (proper name), that is essential for song learning and production. In Chapter 3, I develop the neural network model for song production and use it to explain the effects of partial lesions of HVC on song. Furthermore, I use the model to make predictions about the behavioral effects of these lesions and reanalyze the data, validating those predictions. Finally, in Chapter 4 I develop a simplified version of the model that sacrifices spiking dynamics of neurons while maintaining the essential higherlevel features of the model. I use this model to study interhemispheric synchronization and the effects of unilateral perturbations of HVC on song. The model captures the effects of these perturbations, particularly unilateral temperature manipulation and electrical stimulation of HVC, and makes predictions about the circuitlevel effects of these perturbations.
Show less  Date Issued
 2018
 Identifier
 2018_Sp_Galvis_fsu_0071E_14310
 Format
 Thesis
 Title
 Effective Methods in Intersection Theory and Combinatorial Algebraic Geometry.
 Creator

Harris, Corey S. (Corey Scott), Chicken, Eric, Aldrovandi, Ettore, Kim, Kyounghee, Petersen, Kathleen L., Florida State University, College of Arts and Sciences, Department of...
Show moreHarris, Corey S. (Corey Scott), Chicken, Eric, Aldrovandi, Ettore, Kim, Kyounghee, Petersen, Kathleen L., Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

This dissertation presents studies of effective methods in two main areas of algebraic geometry: intersection theory and characteristic classes, and combinatorial algebraic geometry. We begin in chapter 2 by giving an effective algorithm for computing Segre classes of subschemes of arbitrary projective varieties. The algorithm presented here comes after several others which solve the problem in special cases, where the ambient variety is for instance projective space. To our knowledge, this...
Show moreThis dissertation presents studies of effective methods in two main areas of algebraic geometry: intersection theory and characteristic classes, and combinatorial algebraic geometry. We begin in chapter 2 by giving an effective algorithm for computing Segre classes of subschemes of arbitrary projective varieties. The algorithm presented here comes after several others which solve the problem in special cases, where the ambient variety is for instance projective space. To our knowledge, this is the first algorithm to be able to compute Segre classes in projective varieties with arbitrary singularities. In chapter 3, we generalize an algorithm by Goward for principalization of monomial ideals in nonsingular varieties to work on any scheme of finite type over a field, proving that the more general class of r.c. monomial subschemes in arbitrarily singular varieties can be principalized by a sequence of blowups at codimension 2 r.c. monomial centers. The main result of chapter 4 is a classification of the monomial Cremona transformations of the plane up to conjugation by certain linear transformations. In particular, an algorithm for enumerating all such maps is derived. In chapter 5, we study the multiview varieties and compute their ChernMather classes. As a corollary we derive a polynomial formula for their Euclidean distance degree, partially addressing a conjecture of Draisma et al. [35]. In chapter 6, we discuss the classical problem of counting planes tangent to general canonical sextic curves at three points. We investigate the situation for real and tropical sextics. In chapter 6, we explicitly compute equations of an Enriques surface via the involution on a K3 surface.
Show less  Date Issued
 2017
 Identifier
 FSU_2017SP_Harris_fsu_0071E_13829
 Format
 Thesis
 Title
 Efficient and Accurate Numerical Schemes for Long Time Statistical Properties of the Infinite Prandtl Number Model for Convection.
 Creator

Woodruff, Celestine, Wang, Xiaoming, Sang, QingXiang Amy, Case, Bettye Anne, Ewald, Brian D., Gunzburger, Max D., Florida State University, College of Arts and Sciences,...
Show moreWoodruff, Celestine, Wang, Xiaoming, Sang, QingXiang Amy, Case, Bettye Anne, Ewald, Brian D., Gunzburger, Max D., Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

In our work we analyze and implement numerical schemes for the infinite Prandtl number model for convection. This model describes the convection that is a potential driving force behind the flow and temperature of the Earth's mantle. There are many schemes available, but most are given with no mention of their ability to adequately capture the long time statistical properties of the model. We investigate schemes with the potential to actually capture these statistics. We further show...
Show moreIn our work we analyze and implement numerical schemes for the infinite Prandtl number model for convection. This model describes the convection that is a potential driving force behind the flow and temperature of the Earth's mantle. There are many schemes available, but most are given with no mention of their ability to adequately capture the long time statistical properties of the model. We investigate schemes with the potential to actually capture these statistics. We further show numerically that our schemes align with current knowledge of the model's characteristics at low Rayleigh numbers.
Show less  Date Issued
 2015
 Identifier
 FSU_2015fall_Woodruff_fsu_0071E_12813
 Format
 Thesis
 Title
 An Electrophysiological and Mathematical Modeling Study of Developmental and Sex Effects on Neurons of the Zebra Finch Song System.
 Creator

Diaz, Diana Lissett Flores, Bertram, R. (Richard), Fadool, Debra Ann, Hyson, Richard L., Jain, Harsh Vardhan, Johnson, Frank (Professor of Psychology), Mio, Washington, Florida...
Show moreDiaz, Diana Lissett Flores, Bertram, R. (Richard), Fadool, Debra Ann, Hyson, Richard L., Jain, Harsh Vardhan, Johnson, Frank (Professor of Psychology), Mio, Washington, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

Learned motor patterns such as speaking, playing musical instruments and dancing require a defined sequence of movements. The mechanism of acquiring and perfecting these types of learned behaviors involve a highly complex neurological process not exclusive to humans. In fact, vocal learning in songbirds is a wellknown model to study the neural basis of motor learning, particularly human speech acquisition. In this dissertation, I explored differences in the intrinsic physiology of vocal...
Show moreLearned motor patterns such as speaking, playing musical instruments and dancing require a defined sequence of movements. The mechanism of acquiring and perfecting these types of learned behaviors involve a highly complex neurological process not exclusive to humans. In fact, vocal learning in songbirds is a wellknown model to study the neural basis of motor learning, particularly human speech acquisition. In this dissertation, I explored differences in the intrinsic physiology of vocal cortex neurons – which underlie song acquisition and production in the zebra finch (Taeniopygia guttata) – as a function of age, sex, and experience using a combination of electrophysiology and mathematical modeling. Using three developmental time points in male zebra finches, Chapter 3 presents evidence of intrinsic plasticity in vocal cortex neurons during vocal learning. The experimental results in this chapter revealed age and possibly learningrelated changes in the physiology of these neurons, while the mathematical models suggest possible variations in both the expression and kinetics of several ion channels that cause the physiological changes. Exploiting the fact that male zebra finches exhibit auditory and vocal song learning, while females exhibit auditory song learning only, in Chapter 4 I compared the physiology of vocal cortex neurons between sexes. This comparison reveals aspects of the neurons’ physiology specialized for singing (males only) vs. auditory learning of song (both males and females). Finally, in Chapter 4 I explored the effect of auditory learning in the physiology of vocal cortex neurons in females. Experimental results and mathematical models revealed regulation in ion channel expression due to auditory learning. In summary, this dissertation describes the effect of three new variables – age, sex, and experience – now known to influence the physiology of key neurons in vocal learning.
Show less  Date Issued
 2017
 Identifier
 FSU_SUMMER2017_Diaz_fsu_0071E_14037
 Format
 Thesis
 Title
 Ensemble Methods for Capturing Dynamics of Limit Order Books.
 Creator

Wang, Jian, Zhang, Jinfeng, Ökten, Giray, Kercheval, Alec N., Mio, Washington, Simon, Capstick C., Florida State University, College of Arts and Sciences, Department of Mathematics
 Abstract/Description

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 highfrequency limit order books such as predicting price spread crossing opportunities in a future time interval. The paper is more datadriven oriented, so experiments with ve realtime stock data from NASDAQ, measured by nanosecond, are established. The...
Show moreAccording 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 highfrequency limit order books such as predicting price spread crossing opportunities in a future time interval. The paper is more datadriven oriented, so experiments with ve realtime 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 outofsample 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, highfrequency trading, data analysis, ensemble methods, F1 score.
Show less  Date Issued
 2017
 Identifier
 FSU_SUMMER2017_Wang_fsu_0071E_14047
 Format
 Thesis
 Title
 Estimating Sensitivities of Exotic Options Using Monte Carlo Methods.
 Creator

Yuan, Wei, Ökten, Giray, Kim, Kyounghee, Huffer, Fred W. (Fred William), Kercheval, Alec N., Nichols, Warren, Florida State University, College of Arts and Sciences, Department...
Show moreYuan, Wei, Ökten, Giray, Kim, Kyounghee, Huffer, Fred W. (Fred William), Kercheval, Alec N., Nichols, Warren, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

In this dissertation, methods of estimating the sensitivity of complex exotic options, including options written on multiple assets, and have discontinuous payoffs, are investigated. The calculation of the sensitivities (Greeks) is based on the finite difference method, pathwise method, likelihood ratio method and kernel method, via Monte Carlo or quasiMonte Carlo simulation. Direct Monte Carlo estimators for various sensitivities of weather derivatives and mountain range options are given....
Show moreIn this dissertation, methods of estimating the sensitivity of complex exotic options, including options written on multiple assets, and have discontinuous payoffs, are investigated. The calculation of the sensitivities (Greeks) is based on the finite difference method, pathwise method, likelihood ratio method and kernel method, via Monte Carlo or quasiMonte Carlo simulation. Direct Monte Carlo estimators for various sensitivities of weather derivatives and mountain range options are given. The numerical results show that the pathwise method outperforms other methods when the payoff function is Lipschitz continuous. The kernel method and the central finite difference methods are competitive when the payoff function is discontinuous.
Show less  Date Issued
 2015
 Identifier
 FSU_migr_etd9528
 Format
 Thesis
 Title
 Evolutionary Dynamics of Bacterial Persistence under Nutrient/Antibiotic Actions.
 Creator

Ebadi, Sepideh, Cogan, Nicholas G., Beerli, Peter, Bertram, R., Ökten, Giray, Vo, Theodore, Florida State University, College of Arts and Sciences, Department of Mathematics
 Abstract/Description

Diseases such as tuberculosis, chronic pneumonia, and inner ear infections are caused by bacterial biofilms. Biofilms can form on any surface such as teeth, floors, or drains. Many studies show that it is much more difficult to kill the bacteria in a biofilm than planktonic bacteria because the structure of biofilms offers additional layered protection against diffusible antimicrobials. Among the bacteria in planktonicbiofilm populations, persisters is a subpopulation that is tolerant to...
Show moreDiseases such as tuberculosis, chronic pneumonia, and inner ear infections are caused by bacterial biofilms. Biofilms can form on any surface such as teeth, floors, or drains. Many studies show that it is much more difficult to kill the bacteria in a biofilm than planktonic bacteria because the structure of biofilms offers additional layered protection against diffusible antimicrobials. Among the bacteria in planktonicbiofilm populations, persisters is a subpopulation that is tolerant to antibiotics and that appears to play a crucial role in survival dynamics. Understanding the dynamics of persister cells is of fundamental importance for developing effective treatments. In this research, we developed a method to better describe the behavior of persistent bacteria through specific experiments and mathematical modeling. We derived an accurate mathematical model by tightly coupling experimental data and theoretical model development. By focusing on dynamic changes in antibiotic tolerance owing to phenotypic differences between bacteria, our experiments explored specific conditions that are relevant to specifying parameters in our model. We deliver deeper intuitions to experiments that address several current hypotheses regarding phenotypic expression. By comparing our theoretical model to experimental data, we determined a parameter regime where we obtain quantitative agreement with our model. This validation supports our modeling approach and our theoretical predictions. This model can be used to enhance the development of new antibiotic treatment protocols.
Show less  Date Issued
 2018
 Identifier
 2018_Sp_Ebadi_fsu_0071E_14324
 Format
 Thesis
 Title
 Exploration of the Role of Disinfection Timing, Duration, and Other Control Parameters on Bacterial Populations Using a Mathematical Model.
 Creator

Acar, Nihan, Cogan, Nicholas G., Keller, Thomas C. S., Bertram, R., Mio, Washington, Florida State University, College of Arts and Sciences, Department of Mathematics
 Abstract/Description

Tolerant bacteria enmeshed in a biofilm causes several difficult to treat illnesses like tuberculosis, chronic pneumonia, and chronic inner ear infections. These diseases typically respond poorly to antibiotics due to high tolerance. Bacterial tolerance can be genotypic (resistancee.g. MRSA), phenotypic (nonheritable) or environmental (e.g. nutrient gradients). Persister formation is phenotypic tolerance that is highly tolerant to disinfection. Constant dosing is typically ineffective in...
Show moreTolerant bacteria enmeshed in a biofilm causes several difficult to treat illnesses like tuberculosis, chronic pneumonia, and chronic inner ear infections. These diseases typically respond poorly to antibiotics due to high tolerance. Bacterial tolerance can be genotypic (resistancee.g. MRSA), phenotypic (nonheritable) or environmental (e.g. nutrient gradients). Persister formation is phenotypic tolerance that is highly tolerant to disinfection. Constant dosing is typically ineffective in eliminating persister cells. To generate an effective treatment protocol, more research must examine the dynamics of persister cells. This study investigates how manipulating the application of antibiotics and the addition of nutrient may enhance the disinfection of a bacterial population in batch culture. Previous studies focused on the antimicrobial agent as a control variable to eliminate the bacterial population. In addition to antibiotic treatments, we consider the significance of the nutrient in eradicating the susceptible and persister cells since the disinfection of the susceptible population is dependent on nutrient intake. We present a mathematical model that captures the dynamics between susceptible and persister bacteria with antibiotic and nutrient as control variables. We investigate the optimal dosewithdrawal timing of antibiotic in several cases including: constant nutrient in time, dynamic nutrient in time, and piecewise constant nutrient in time. Also a global sensitivity analysis method, Partial Rank Correlation Coefficient (PRCC), is applied to determine the significance of model parameters for a quantity of interest. The highlights of this study are; 1.) Constant dosing is not an effective disinfection protocol. 2.) Nutrient plays a significant role such that in the presence of nutrient, bacterial population is eliminated much faster. 3.) Checking the eigenvalues of the established Poincaré map gives us information on how to choose withdrawdose timing for the nonlinear system. 4.) Periodic dosewithdraw offers a more efficient disinfection provided dose time is longer than withdrawal of antibiotic. 5.) As duration of dose decreases, the elimination of bacteria decreases and the death rate becomes insignificant.
Show less  Date Issued
 2018
 Identifier
 2018_Su_Acar_fsu_0071E_14749
 Format
 Thesis
 Title
 Exponential Convergence Fourier Method and Its Application to Option Pricing with Lévy Processes.
 Creator

Gu, Fangxi, Nolder, Craig, Huffer, Fred W. (Fred William), Kercheval, Alec N., Nichols, Warren D., Ökten, Giray, Florida State University, College of Arts and Sciences,...
Show moreGu, Fangxi, Nolder, Craig, Huffer, Fred W. (Fred William), Kercheval, Alec N., Nichols, Warren D., Ökten, Giray, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

Option pricing by the Fourier method has been popular for the past decade, many of its applications to Lévy processes has been applied especially for European options. This thesis focuses on exponential convergence Fourier method and its application to discrete monitoring options and Bermudan options. An alternative payoff truncating method is derived to compare the benchmark Hilbert transform. A general error control framework is derived to keep the Fourier method out of an overflow problem....
Show moreOption pricing by the Fourier method has been popular for the past decade, many of its applications to Lévy processes has been applied especially for European options. This thesis focuses on exponential convergence Fourier method and its application to discrete monitoring options and Bermudan options. An alternative payoff truncating method is derived to compare the benchmark Hilbert transform. A general error control framework is derived to keep the Fourier method out of an overflow problem. Numerical results verify that the alternative payoff truncating sinc method performs better than the benchmark Hilbert transform method under the error control framework.
Show less  Date Issued
 2016
 Identifier
 FSU_FA2016_Gu_fsu_0071E_13579
 Format
 Thesis
 Title
 Flow Equivalence Classes of PseudoAnosov Surface Homeomorphisms.
 Creator

Billet, Robert, Hironaka, Eriko, Petersen, Kathleen L., Duke, Dennis, Fenley, Sergio, Heil, Wolfgang, Florida State University, College of Arts and Sciences, Department of...
Show moreBillet, Robert, Hironaka, Eriko, Petersen, Kathleen L., Duke, Dennis, Fenley, Sergio, Heil, Wolfgang, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

This dissertation explores pseudoAnosov elements of the mapping class group of an oriented surface from the point of view of fibered face theory. This theory runs dual to the classical way of thought where, rather than fixing a surface S and studying Mod(S), we study the set of all pseudoAnosov mapping classes over all oriented surfaces. Let S be a connected, compact, oriented surface. The mapping class group of S, denoted Mod(S), is the group of orientation preserving homeomorphisms of S...
Show moreThis dissertation explores pseudoAnosov elements of the mapping class group of an oriented surface from the point of view of fibered face theory. This theory runs dual to the classical way of thought where, rather than fixing a surface S and studying Mod(S), we study the set of all pseudoAnosov mapping classes over all oriented surfaces. Let S be a connected, compact, oriented surface. The mapping class group of S, denoted Mod(S), is the group of orientation preserving homeomorphisms of S which act by the identity on θS considered up to isotopy. If no power of a mapping class leaves an essential curve invariant, the mapping class is pseudoAnosov. In this case, the mapping class preserves an expanding and contracting foliation with expansion factor λ. The set of all pseudoAnosov mapping classes admits a natural partition into flow equivalence classes. Such a class can be described as the surface cross sections transverse to a pseudoAnosov flow in a hyperbolic fibered three manifold. Using the operation of Murasugi sum, we systematically study the flow equivalence classes that can be expressed as iterated Hopf plumbings on a disk in the 3sphere. Such a surface is always a fiber surface for its boundary link. The data on how to attach the Hopf bands is conveniently packaged in a graph and, since the Coxeter element of this graph is, up to sign, the monodromy of the fiber surface, such links are called Coxeter links. The investigation splits into three main developments. The first result deals with the overall structure of the flow equivalence classes corresponding to Coxeter links as subspaces of the real vector space H¹(M;[the set of real numbers]), where M is the link exterior in the 3sphere. The second result sheds light on a natural dynamically minimal representative in each class. We then give an algorithm that takes a class as input and outputs a multivariable polynomial which can be used to compute the expansion factor of any element contained in the class. By interpreting the mapping tori of the pseudoAnosov mapping classes as link exteriors in the 3sphere, we are able to identify the meridians of the link components with a basis for H¹(M;[integers]). With a few careful knot theoretic observations, we show that any surface with positive linking number to the original link is a fiber surface. With slightly stronger assumptions on the link, we show that the entire Thurston norm is determined by the norms of spanning surfaces for the individual components. It is easy to construct pseudoAnosov mapping classes with small expansion factor on surfaces with high Euler characteristic. One way this can be achieved is by composing a periodic mapping class with a pseudoAnosov map that is supported on a small subsurface. Since the flow equivalence class of a pseudoAnosov homeomorphism contains maps supported on surfaces of arbitrarily high Euler characteristic, we consider the function λ❘x(S)❘. Using properties of this function and the above results, we find a natural minimizing element with respect to this function. The third result amounts to computing the Teichmüller polynomial for the fibered face in question. This can be a difficult process in general. Perhaps the most notable issues are explicitly computing the fixed cohomology and a traintrack for a surface automorphism. After finding ways around these problems and others, we give the full algorithm to compute the Teichmüller polynomial.
Show less  Date Issued
 2016
 Identifier
 FSU_FA2016_Billet_fsu_0071E_13563
 Format
 Thesis
 Title
 A GameTheoretic Analysis of Competition over Indivisible Resources.
 Creator

Karabiyik, Tugba, MestertonGibbons, Mike, Ryvkin, Dmitry, Cogan, Nicholas G., Ökten, Giray, Florida State University, College of Arts and Sciences, Department of Mathematics
 Abstract/Description

In this dissertation, we build several gametheoretic models to explore animal contest behavior. Classical game theory predicts that respect for ownership or "Bourgeois" behavior can arise as an arbitrary convention to avoid costly fights. The same theory also predicts that disrespect for ownership or "antiBourgeois" behavior can evolve under the same conditions. However, Bourgeois is very common in nature, while antiBourgeois is very rare. In order to explain the rarity of antiBourgeois...
Show moreIn this dissertation, we build several gametheoretic models to explore animal contest behavior. Classical game theory predicts that respect for ownership or "Bourgeois" behavior can arise as an arbitrary convention to avoid costly fights. The same theory also predicts that disrespect for ownership or "antiBourgeois" behavior can evolve under the same conditions. However, Bourgeois is very common in nature, while antiBourgeois is very rare. In order to explain the rarity of antiBourgeois behavior, we create a singleround HawkDove model with four pure strategies: Hawk or H, Bourgeois or B, antiBourgeois or X and Dove or D. We show that if intruders sometimes believe themselves to be owners, then the resulting confusion can broaden the conditions under which Bourgeois behavior is evolutionarily stable in the singleround HawkDove game. We also develop a multiperiod HawkDove model that includes the effect of confusion over ownership. We determine the effect of ownership uncertainty on Bourgeois vs. antiBourgeois strategies, and we show how this effect can allow a fighting population to evolve to a nonfighting population under increasing costs of fighting. Another possible explanation for the rarity of antiBourgeois behavior in nature is that two Xstrategists would exchange roles repeatedly over many rounds in a costly "infinite regress." Here we further analyze an existing infiniteregress model to allow for polymorphic ESSs, and thus explore conditions that favor partial respect for ownership. We identify a pathway through which respect for ownership can evolve from total disrespect under increasing costs of fighting.
Show less  Date Issued
 2015
 Identifier
 FSU_migr_etd9368
 Format
 Thesis
 Title
 GameTheoretic Models of Animal Behavior Observed in Some Recent Experiments.
 Creator

Dai, Yao, MestertonGibbons, Mike, Hurdal, Monica K., Kercheval, Alec N., Quine, J. R. (John R.), Florida State University, College of Arts and Sciences, Department of Mathematics
 Abstract/Description

In this dissertation, we create three theoretical models to answer questions raised by recent experiments that lie beyond the scope of current theory. In the landmarkeffect model, we determine size, shape and location for a territory that is optimal in the sense of minimizing defense costs, when a given proportion of the boundary is landmarked and its primary benefit in terms of fitness is greater ease of detecting intruders across it. In the subjectiveresourcevalue model, we develop a...
Show moreIn this dissertation, we create three theoretical models to answer questions raised by recent experiments that lie beyond the scope of current theory. In the landmarkeffect model, we determine size, shape and location for a territory that is optimal in the sense of minimizing defense costs, when a given proportion of the boundary is landmarked and its primary benefit in terms of fitness is greater ease of detecting intruders across it. In the subjectiveresourcevalue model, we develop a gametheoretic model based on the WarofAttrition game. Our results confirm that allowing players to adapt their subjective resource value based on their experiences can generate strong winner effects with weak or even no loser effects, which is not predicted by other theoretical models. In the rearguardaction model, we develop two versions of a gametheoretic model with different hypotheses on the function of volatile chemical emissions in animal contests, and we compare their results with observations in experiments. The two hypotheses are whether volatile chemicals are released to prevent the winner of the current round of contest from translating its victory into permanent possession of a contested resource, or are used to prevent a winner from inflicting costs on a fleeing loser.
Show less  Date Issued
 2017
 Identifier
 FSU_2017SP_Dai_fsu_0071E_13762
 Format
 Thesis
 Title
 GPU Computing in Financial Engineering.
 Creator

Xu, Linlin, Ökten, Giray, Sinha, Debajyoti, Bellenot, Steven F., Gallivan, Kyle A., Kercheval, Alec N., Florida State University, College of Arts and Sciences, Department of...
Show moreXu, Linlin, Ökten, Giray, Sinha, Debajyoti, Bellenot, Steven F., Gallivan, Kyle A., Kercheval, Alec N., Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

GPU computing has become popular in computational finance and many financial institutions are moving their CPU based applications to the GPU platform. We explore efficient implementations for two main financial problems on GPU: pricing, and computing sensitivities (Greeks). Since most Monte Carlo algorithms are embarrassingly parallel, Monte Carlo has become a focal point in GPU computing. GPU speedup examples reported in the literature often involve Monte Carlo algorithms, and there are...
Show moreGPU computing has become popular in computational finance and many financial institutions are moving their CPU based applications to the GPU platform. We explore efficient implementations for two main financial problems on GPU: pricing, and computing sensitivities (Greeks). Since most Monte Carlo algorithms are embarrassingly parallel, Monte Carlo has become a focal point in GPU computing. GPU speedup examples reported in the literature often involve Monte Carlo algorithms, and there are software tools commercially available that help migrate Monte Carlo financial pricing models to GPU. We present a survey of Monte Carlo and randomized quasiMonte Carlo methods, and discuss existing (quasi) Monte Carlo sequences in NVIDIA's GPU CURAND libraries. We discuss specific features of GPU architecture relevant for developing efficient (quasi) Monte Carlo methods. We introduce a recent randomized quasiMonte Carlo method, and compare it with some of the existing implementations on GPU, when they are used in pricing caplets in the LIBOR market model and mortgage backed securities. We then develop a cacheaware implementation of a 3D parabolic PDE solver on GPU. We apply the wellknown CraigSneyd scheme and derive the corresponding discretization. We discuss memory hierarchy of GPU and suggest a data structure that is suitable for GPU's caching system. We compare the performance of the PDE solver on CPU and GPU. Finally, we consider sensitivity analysis for financial problems via Monte Carlo and PDE methods. We review three commonly used methods and point out their advantages and disadvantages. We present a survey of automatic differentiation (AD), and show the challenges faced in memory consumption when AD is applied in financial problems. We discuss two optimization techniques that help reduce memory footprint significantly. We conduct the sensitivity analysis for the LIBOR market model and suggest an optimization for its AD implementation on GPU. We also apply AD to a 3D parabolic PDE and use GPU to reduce the execution time.
Show less  Date Issued
 2015
 Identifier
 FSU_migr_etd9526
 Format
 Thesis
 Title
 High Order LongTime Accurate Methods for the StokesDarcy System and Uncertainty Quantification of Contaminant Transport.
 Creator

Sun, Dong, Wang, Xiaoming, Gunzburger, Max D., Wang, Xiaoqiang, Ewald, Brian D., Cogan, Nicholas G., Florida State University, College of Arts and Sciences, Department of...
Show moreSun, Dong, Wang, Xiaoming, Gunzburger, Max D., Wang, Xiaoqiang, Ewald, Brian D., Cogan, Nicholas G., Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

The dissertation includes two parts. The first part consists of designing and analyzing high order longtime accurate numerical methods for StokesDarcy system. We propose second and thirdorder efficient and longtime accurate numerical methods, called IMplicitEXplicit methods (IMEX) for the coupled StokesDarcy system. Although the original continuum StokesDarcy PDE system is fully coupled, our algorithm is capable of decoupling the system into two subsystems so that a single Stokes and...
Show moreThe dissertation includes two parts. The first part consists of designing and analyzing high order longtime accurate numerical methods for StokesDarcy system. We propose second and thirdorder efficient and longtime accurate numerical methods, called IMplicitEXplicit methods (IMEX) for the coupled StokesDarcy system. Although the original continuum StokesDarcy PDE system is fully coupled, our algorithm is capable of decoupling the system into two subsystems so that a single Stokes and a single Darcy system can be computed in a parallel fashion without iteration. All the schemes we proposed are proven to be unconditionally stable and longtime stable. The bound on the error is uniformintime, which is among the first of this kind for second and thirdorder methods of StokesDarcy system. Error estimates for the second order BackwardDifferentiation scheme are proved. The second part concerns the Uncertainty of Quantification (UQ) of the contaminant transport. We compute the convectiondiffusion equation with Streamline Upwind PetrovGalerkin (SUPG) method. The quantity of interest is acquired using Monte Carlo and Sparse Grid methods in order to study the sensitivity with respect to the random parameters.
Show less  Date Issued
 2015
 Identifier
 FSU_migr_etd9692
 Format
 Thesis
 Title
 HighOrder, Efficient, Numerical Algorithms for Integration in Manifolds Implicitly Defined by Level Sets.
 Creator

Khanmohamadi, Omid, Sussman, Mark, Plewa, Tomasz, Moore, M. Nicholas J. (Matthew Nicholas J.), Ökten, Giray, Florida State University, College of Arts and Sciences, Department...
Show moreKhanmohamadi, Omid, Sussman, Mark, Plewa, Tomasz, Moore, M. Nicholas J. (Matthew Nicholas J.), Ökten, Giray, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

New numerical algorithms are devised for highorder, efficient quadrature in domains arising from the intersection of a hyperrectangle and a manifold implicitly defined by level sets. By casting the manifold locally as the graph of a function (implicitly evaluated through a recurrence relation for the zero level set), a recursion stack is set up in which the interface and integrand information of a single dimension after another will be treated. Efficient means for the resulting dimension...
Show moreNew numerical algorithms are devised for highorder, efficient quadrature in domains arising from the intersection of a hyperrectangle and a manifold implicitly defined by level sets. By casting the manifold locally as the graph of a function (implicitly evaluated through a recurrence relation for the zero level set), a recursion stack is set up in which the interface and integrand information of a single dimension after another will be treated. Efficient means for the resulting dimension reduction process are developed, including maps for identifying lowerdimensional hyperrectangle facets, algorithms for minimal coordinateflip vertex traversal, which, together with our multilinearformbased derivative approximation algorithms, are used for checking a proposed integration direction on a facet, as well as algorithms for detecting interfacefree subhyperrectangles. The multidimensional quadrature nodes generated by this method are inside their respective domains (hence, the method does not require any extension of the integrand) and the quadrature weights inherit any positivity of the underlying singledimensional quadrature method, if present. The accuracy and efficiency of the method are demonstrated through convergence and timing studies for test cases in spaces of up to seven dimensions. The strengths and weaknesses of the method in high dimensional spaces are discussed.
Show less  Date Issued
 2017
 Identifier
 FSU_SUMMER2017_Khanmohamadi_fsu_0071E_14013
 Format
 Thesis
 Title
 The Impact of Competition on Elephant Musth Strategies: A Gametheoretic Model.
 Creator

Wyse, J. Maxwell (John Maxwell), MestertonGibbons, Mike, Huffer, Fred W. (Fred William), Hurdal, Monica K., Cogan, Nicholas G., Florida State University, College of Arts and...
Show moreWyse, J. Maxwell (John Maxwell), MestertonGibbons, Mike, Huffer, Fred W. (Fred William), Hurdal, Monica K., Cogan, Nicholas G., Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

Mature male African elephants are known to periodically enter a temporary state of heightened aggression called "musth," often linked with increased androgens, particularly testosterone. Sexually mature males are capable of entering musth at any time of year, and will often travel long distances to find estrous females. When two musth bulls or two nonmusth bulls encounter one another, the agonistic interaction is usually won by the larger male. When a smaller musth bull encounters a larger...
Show moreMature male African elephants are known to periodically enter a temporary state of heightened aggression called "musth," often linked with increased androgens, particularly testosterone. Sexually mature males are capable of entering musth at any time of year, and will often travel long distances to find estrous females. When two musth bulls or two nonmusth bulls encounter one another, the agonistic interaction is usually won by the larger male. When a smaller musth bull encounters a larger nonmusth bull, however, the smaller musth male can win. The relative mating success of musth males is due partly to this fighting advantage, and partly to estrous females' general preference for musth males. Though musth behavior has long been observed and documented, the evolutionary advantages of musth remain poorly understood. Here we develop a gametheoretic model of male musth behavior which assumes musth duration as a parameter, and distributions of small, medium and large musth males are predicted in both time and space. The predicted results are similar to the observed timing strategies in the Amboseli National Park elephant population. We discuss small male musth behavior, musthestrus coincidence, the effects of estrous female spatial heterogeneity on musth timing, conservation applications, the assumptions underpinning the model and possible modifications to the model for the purpose of determining musth duration.
Show less  Date Issued
 2017
 Identifier
 FSU_2017SP_Wyse_fsu_0071E_13713
 Format
 Thesis
 Title
 Insulin Secretion Rhythms: Calcium Regulation of BetaCell Metabolism and Rescue of Islet Oscillations.
 Creator

McKenna, Joseph P., Bertram, R. (Richard), Roper, Michael Gabriel, Muslimani, Ziad H., Moore, M. Nicholas J. (Matthew Nicholas J.), Miller, Brian G., Florida State University,...
Show moreMcKenna, Joseph P., Bertram, R. (Richard), Roper, Michael Gabriel, Muslimani, Ziad H., Moore, M. Nicholas J. (Matthew Nicholas J.), Miller, Brian G., Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

Pancreatic islet betacells play a vital role in regulating blood glucose levels by releasing insulin into the bloodstream. Insulin is released in pulses that parallel interacting betacell rhythms, including oscillatory glucose metabolism and periodic calcium influx. We present concurrent time series records of metabolic variables and intracellular calcium levels in glucosestimulated betacells that support regulation of mitochondrial dehydrogenases is the dominant calcium feedback effect...
Show morePancreatic islet betacells play a vital role in regulating blood glucose levels by releasing insulin into the bloodstream. Insulin is released in pulses that parallel interacting betacell rhythms, including oscillatory glucose metabolism and periodic calcium influx. We present concurrent time series records of metabolic variables and intracellular calcium levels in glucosestimulated betacells that support regulation of mitochondrial dehydrogenases is the dominant calcium feedback effect onto metabolism in the insulin secretory pathway. We include this effect into the betacell Dual Oscillator Model to reconcile model simulations with experimental data, then we determine the oscillation mechanism in the modified model. Islets lose the rhythms that govern insulin pulses when glucose is elevated to hyperglycemic levels. We demonstrate with modeling and experiments that oscillations lost to elevated glucose can be recovered by converting the elevated glucose stimulus to a sinusoidal wave. We predict with modeling which periodic glucose stimuli can recover islet oscillations.
Show less  Date Issued
 2017
 Identifier
 FSU_2017SP_McKenna_fsu_0071E_13864
 Format
 Thesis
 Title
 Investigating Persistent Infections Using Mathematical Modeling and Analyses.
 Creator

Jarrett, Angela Michelle, Cogan, Nicholas G., Hussaini, M. Yousuff, Bass, Hank W., Bertram, R. (Richard), Case, Bettye Anne, Hurdal, Monica K., Florida State University, College...
Show moreJarrett, Angela Michelle, Cogan, Nicholas G., Hussaini, M. Yousuff, Bass, Hank W., Bertram, R. (Richard), Case, Bettye Anne, Hurdal, Monica K., Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

While the immune system is extraordinarily complex and powerful, and medical advancements are more spectacular than ever, in recent history we have seen the unfortunate failure of both processes (immune system and drugs) in the increasing levels of persistent infections. This work is an example of a collaborative effort to study multiple forms of persistent infections using mathematical tools and analyses. We will discuss the biological backgrounds for the immune system's components and...
Show moreWhile the immune system is extraordinarily complex and powerful, and medical advancements are more spectacular than ever, in recent history we have seen the unfortunate failure of both processes (immune system and drugs) in the increasing levels of persistent infections. This work is an example of a collaborative effort to study multiple forms of persistent infections using mathematical tools and analyses. We will discuss the biological backgrounds for the immune system's components and functions, the bacterial and viral resistance mechanisms for methicillinresistant Staphylococcus aureus and the human immunodeficiency virus, respectively, and some of the current methods for treating these diseases. Then, using ordinary and partial differential equations we present the results of models that were created to study specific infections—namely, methicillinresistant Staphylococcus aureus osteomyelitis, Staphylococcus aureus nasal carriage, and human immunodeficiency virus prophylactic gel. These models are shown to be in good agreement with the biology by looking at, when possible, their analytical solutions and numerical results when compared to experimental evidence. We further explore these models using several different computational analyses that can be classified as at least one of the following methods: uncertainty quantification, sensitivity analysis, and data assimilation. We give an overview of each of these topics and delve into the technicalities and caveats of each of the analyses we apply. We show that all of the methods presented, individually and in concert, are valuable tools for not only revealing details about the model structure and verifying model robustness, but they can also bring to light elements of the biological phenomena that the model represents. While considering all these details, throughout the manuscript we consider the philosophical perspective of biological modeling and modeling in general.
Show less  Date Issued
 2016
 Identifier
 FSU_2016SP_Jarrett_fsu_0071E_13046
 Format
 Thesis
 Title
 Keeping Pace with the Times: Quantifying Variation of Newly Emerging Biological Shape Data.
 Creator

Xu, Qiuping, Mio, Washington, Kumar, Piyush, Bertram, R. (Richard), Liu, Xiuwen, Quine, J. R. (John R.), Florida State University, College of Arts and Sciences, Department of...
Show moreXu, Qiuping, Mio, Washington, Kumar, Piyush, Bertram, R. (Richard), Liu, Xiuwen, Quine, J. R. (John R.), Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

Shape represents a complex and rich source of biological information that is fundamentally linked to underlying mechanisms and functions. Many fields of biology employ mathematical tools for the statistical analysis of shape variation. However, difficulties in reliably quantifying biological shape, especially for newly emerging shape data, still present an obstacle for researchers to understand how shape variation relates to biological functions and development processes. To overcome these...
Show moreShape represents a complex and rich source of biological information that is fundamentally linked to underlying mechanisms and functions. Many fields of biology employ mathematical tools for the statistical analysis of shape variation. However, difficulties in reliably quantifying biological shape, especially for newly emerging shape data, still present an obstacle for researchers to understand how shape variation relates to biological functions and development processes. To overcome these difficulties, it is desirable to build efficient ways to quantify shapes. Having a quantitative tool in hand, we can further design methods to correlate shape with biological information. The integration of these models with machine learning and statistical inference methods will allow biologists to explore how morphological variation correlates to biological variates and to help advance various areas of research. One goal of this dissertation is to construct new type of shape representation to quantify gene expression data. Advances in microscopy and techniques such as Optical Projection Tomography (OPT) allow researchers to visualize and to study 3D morphological patterns of gene expression domains. Quantitative analysis of gene expression domains and investigation of relationships between gene expression and developmental and phenotypic outcomes are central to advancing our understanding of the genotypephenotype map. However, quantification of shape variation in gene expression domains poses particularly challenging problems, as these domains typically have no clearly defined forms, often appearing seemingly amorphous. Those properties of the gene expression domains make it difficult to analyze shape variation with the tools of landmarkbased geometric morphometrics. In addition, 3D image acquisition and processing introduce many artifacts that further exacerbate the problem. To overcome these difficulties, we present a method that combines OPT scanning, a shape regularization technique and a landmarkfree approach to quantify variation in the morphology of sonic hedgehog expression domains in the frontonasal ectodermal zone (FEZ) of avians and investigate relationships with embryonic craniofacial shape. The landmarkfree approach quantifies variation in shape of amorphous gene expression domains, enhancing their most salient morphological characteristics and being robust to uninformative local shape variation and irregularities associated with image acquisition. The correlation analysis reveals axes in FEZ and embryonichead morphospaces along which variation exhibits a sharp linear relationship at high statistical significance. Combined with qualitative findings, these results have the potential to benefit biologists in exploring the gene expression pathway and in understanding the underlying expression mechanisms. The techniques we used to deal with FEZ meshes should be applicable to analyses of other 3D surfacelike biological structures that have illdefined shape and are relevant to understanding developmental processes and phenotypic variation. Existing biological shape models, such as those based on landmarks, rely on sparse landmarks on the shapes to model shape variations. However, on softtissue surfaces as the face there are few such landmarks. Across the cheek and forehead, for instance, there are no points that have exact biological correspondence and yet aspects of their shape contains useful biological information. The analysis based on the sparse landmarks will compromise the deep and comprehensive morphological information collected by advanced image processing technologies. Thus, instead of using only the limited number of landmarks, we propose to use the spline method to construct dense surface model which covers the entire shape. This brings another goal of this dissertation  to develop such a spline method to build a dense correspondence across all shapes. Although, spline is an active area in shape analysis and also in many other disciplines for interpolation, approximation and regression. Most results have been focused on Euclidean domain. However, data living on manifold occurs often, especially when dealing with shape surfaces, so constructing spline with manifold domain and providing effective computation method for such spline are desirable in reallife problems. To fulfill this goal, we present a general theoretical framework of spline in which the Euclidean domain can be extended to manifold domain. Additionally, we provide computationally effective algorithm to compute such spline function based on bounded rectangular domain. We demonstrate the advantages of this framework by using examples on closed and open manifold domains and by comparing performance with other spline methods. The computation framework shows comparable result with the spline directly constructed on the manifold and also shows clear improvement respect to the thin plate spline method. This manifold spline method has been applied to construct dense surface models of avian embryos shapes. Those dense surface models can establish a correspondence of thousands of points across each 3D image and provide dramatic visualization of shape variation.
Show less  Date Issued
 2015
 Identifier
 FSU_migr_etd9496
 Format
 Thesis
 Title
 Lagrangian Specialization via Log Resolutions and SchwartzMacPherson Chern Classes.
 Creator

Adams, William J. L. (William James Louis), Aluffi, Paolo, Rawling, J. Piers, Aldrovandi, Ettore, Kim, Kyounghee, Agashe, Amod Sadanand, Florida State University, College of...
Show moreAdams, William J. L. (William James Louis), Aluffi, Paolo, Rawling, J. Piers, Aldrovandi, Ettore, Kim, Kyounghee, Agashe, Amod Sadanand, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

This dissertation covers several topics around the idea of the SchwartzMacPherson Chern classes, which were independently constructed by M.H. Schwartz around 1965 and R. MacPherson in the early 1970's. First we review a more recent construction of SchwartzMacPherson Chern class due to G. Kennedy using projective Lagrangian conormals and make explicit some details not found in that work. Around 1980, J.L. Verdier obtained a specialization formula for SchwartzMacPherson Chern classes, which...
Show moreThis dissertation covers several topics around the idea of the SchwartzMacPherson Chern classes, which were independently constructed by M.H. Schwartz around 1965 and R. MacPherson in the early 1970's. First we review a more recent construction of SchwartzMacPherson Chern class due to G. Kennedy using projective Lagrangian conormals and make explicit some details not found in that work. Around 1980, J.L. Verdier obtained a specialization formula for SchwartzMacPherson Chern classes, which was recovered by Kennedy again using the Lagrangian setting and FultonMacPherson intersection theory, elaborating on work of C. Sabbah. A verbatim reading of this approach reveals several subtleties. We introduce an alternate blowup construction to address some of these subtleties, as well as an alternate definition based upon the work of C. Sabbah. We then prove that either of these constructions indeed provides a complete alternative proof of Verdier's specialization formula. In the early 2010's P. Aluffi revisited Verdier's work and developed a more calculable approach to specialization utilizing constructible functions and the Weak Factorization Theorem of Abramovich, Karu, Matsuki, and Wlodarczyk. Aluffi's construction gives an explicit formula when in the case of a divisor with normal crossings and nonsingular components. We take those ideas to the Lagrangian setting to define a new Langrangian cycle, the Asp cycle. All of the proofs in that section utilize Langrangians. We prove that the Asp cycle agrees with the specialization cycle introduced by Sabbah and Kennedy in the special case in which the subvariety is the central fiber of a family over a smooth curve. Thus the Asp cycle may be viewed as a generalization of the KennedySabbah cycle. We give a corresponding generalization of Verdier's specialization formula.
Show less  Date Issued
 2015
 Identifier
 FSU_migr_etd9535
 Format
 Thesis
 Title
 Local and Global Bifurcations in FiniteDimensional Center Manifold Equations of DoubleDiffusive Convection.
 Creator

Eilertsen, Justin, Magnan, Jeronimo Francisco, Duke, D. W. (Dennis W.), Bertram, R. (Richard), Wang, Xiaoming, Musslimani, Ziad H., Florida State University, College of Arts and...
Show moreEilertsen, Justin, Magnan, Jeronimo Francisco, Duke, D. W. (Dennis W.), Bertram, R. (Richard), Wang, Xiaoming, Musslimani, Ziad H., Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

A finite dimensional amplitude equation model of 2dimensional doublediffusive convection near a quadruplezero (codimension 4) bifurcation point is derived using center manifold reduction. The derivation employs small perturbationtheory to obtain an asymptomatic solution to the 2dimensional NavierStokes equations. The coefficients of the amplitude equations are derived for two parameter regimes corresponding to high and moderate thermal Rayleigh numbers. By numerically approximating the...
Show moreA finite dimensional amplitude equation model of 2dimensional doublediffusive convection near a quadruplezero (codimension 4) bifurcation point is derived using center manifold reduction. The derivation employs small perturbationtheory to obtain an asymptomatic solution to the 2dimensional NavierStokes equations. The coefficients of the amplitude equations are derived for two parameter regimes corresponding to high and moderate thermal Rayleigh numbers. By numerically approximating the Poincare map of the amplitude equations, local and global bifurcations are detected that lead to birth of strange attractors. Specifically, strange attractors are generated by homoclinic explosions in the Poincare map. For high thermal Rayleigh numbers, this route to chaos in the Poincare map is analogous to that route present in the continuous ShimizuMorioka and Rucklidge models, where the bifurcation to periodic convection is supercritical. For low thermal Rayleigh numbers, the route to chaos in the Poincare map is shown to be analogous to the route observed in the Lorenz equations. Additionally, the bifurcations of the strange attractors of the Poincare map are studied, and numerical simulations reveal the presence of period doubling regimes and intermittency, as well as exotic bifurcations which include splitting, and interior crises, of strange attractors.
Show less  Date Issued
 2016
 Identifier
 FSU_2016SU_Eilertsen_fsu_0071E_13410
 Format
 Thesis
 Title
 LowRank Riemannian Optimization Approach to the Role Extraction Problem.
 Creator

Marchand, Melissa Sue, Gallivan, Kyle A., Dooren, Paul van, Erlebacher, Gordon, Sussman, Mark, Florida State University, College of Arts and Sciences, Department of Mathematics
 Abstract/Description

This dissertation uses Riemannian optimization theory to increase our understanding of the role extraction problem and algorithms. Recent ideas of using the lowrank projection of the neighborhood pattern similarity measure and our theoretical analysis of the relationship between the rank of the similarity measure and the number of roles in the graph motivates our proposal to use Riemannian optimization to compute a lowrank approximation of the similarity measure. We propose two indirect...
Show moreThis dissertation uses Riemannian optimization theory to increase our understanding of the role extraction problem and algorithms. Recent ideas of using the lowrank projection of the neighborhood pattern similarity measure and our theoretical analysis of the relationship between the rank of the similarity measure and the number of roles in the graph motivates our proposal to use Riemannian optimization to compute a lowrank approximation of the similarity measure. We propose two indirect approaches to use to solve the role extraction problem. The first uses the standard twophase process. For the first phase, we propose using Riemannian optimization to compute a lowrank approximation of the similarity of the graph, and for the second phase using kmeans clustering on the lowrank factor of the similarity matrix to extract the role partition of the graph. This approach is designed to be efficient in time and space complexity while still being able to extract good quality role partitions. We use basic experiments and applications to illustrate the time, robustness, and quality of our twophase indirect role extraction approach. The second indirect approach we propose combines the two phases of our first approach into a onephase approach that iteratively approximates the lowrank similarity matrix, extracts the role partition of the graph, and updates the rank of the similarity matrix. We show that the use of Riemannian rankadaptive techniques when computing the lowrank similarity matrix improves robustness of the clustering algorithm.
Show less  Date Issued
 2017
 Identifier
 FSU_FALL2017_Marchand_fsu_0071E_14046
 Format
 Thesis
 Title
 A Mathematical Model of Cerebral Cortical Folding Development Based on a Biomechanical Hypothesis.
 Creator

Kim, Sarah, Hurdal, Monica K., Steinbock, Oliver, Bertram, R. (Richard), Cogan, Nicholas G., Florida State University, College of Arts and Sciences, Department of Mathematics
 Abstract/Description

The cerebral cortex is a thin folded sheet of neural tissue forming the outmost layer of the cerebrum (brain). Several biological hypotheses have suggested dierent mechanisms involved the development of its folding pattern into sulci (inward valleys) and gyri (outward hills). One hypothesis suggests that mechanical tension along corticocortical connections is the principal driving force for cortical folding development. We propose a new mathematical model based on the tensionbased...
Show moreThe cerebral cortex is a thin folded sheet of neural tissue forming the outmost layer of the cerebrum (brain). Several biological hypotheses have suggested dierent mechanisms involved the development of its folding pattern into sulci (inward valleys) and gyri (outward hills). One hypothesis suggests that mechanical tension along corticocortical connections is the principal driving force for cortical folding development. We propose a new mathematical model based on the tensionbased hypothesis surrounding the 26th week of gestational age when the human brain cortex noticeably begins to fold. In our model, the deformation of a twodimensional semicircular domain is analyzed through the theory of elasticity. The governing coupled partial differential equations are implemented computationally using a finite element formulation. Plausible brain tissue elasticity parameters with reasonable brain domain size parameters were used in our simulation. Gyrication index which is a measure of cortical foldings is employed to compare the degree of folding between the simulation results. The proposed model provides an approach for studying the connections between two different biological hypotheses by determining the magnitude of the applied tension force from the previous mathematical models of cortical folding which are based on a biochemical hypothesis. It allows our model to explain the mechanisms behind disorders occurring in all stages of development. In addition, the ability to freely set the directions and magnitudes of the applied forces allows to analysis of various abnormal cortical foldings by comparing MR imaging features of human brain cortical disorders. Our simulation results show that the unveiled mechanisms underlying the abnormal cortical folding development are well captured by our proposed model.
Show less  Date Issued
 2015
 Identifier
 FSU_2015fall_Kim_fsu_0071E_12872
 Format
 Thesis
 Title
 Mathematical Modeling and Analysis of Gene Knockout Compensation in Pancreatic BCells.
 Creator

Yildirim, Vehpi, Bertram, R. (Richard), Fadool, Debra Ann, Mio, Washington, Ökten, Giray, Roper, Michael Gabriel, Florida State University, College of Arts and Sciences,...
Show moreYildirim, Vehpi, Bertram, R. (Richard), Fadool, Debra Ann, Mio, Washington, Ökten, Giray, Roper, Michael Gabriel, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

Living systems consist of several complex interacting components. Depending on the complexity of the organism, these components can span from molecules to tissues and organs. Systems biology is the interdisciplinary field of study that uses mathematical and computational tools to describe and investigate the roles these components play in biological systems and the way their interactions result in functionality. The collaborative work between biological and mathematical sciences brings deeper...
Show moreLiving systems consist of several complex interacting components. Depending on the complexity of the organism, these components can span from molecules to tissues and organs. Systems biology is the interdisciplinary field of study that uses mathematical and computational tools to describe and investigate the roles these components play in biological systems and the way their interactions result in functionality. The collaborative work between biological and mathematical sciences brings deeper insights into understanding living systems because, even with the recent advancements in technology, it is impossible to acquire all types of empirical data on many living systems. Technical restrictions together with the complexity of the system components usually give rise to this limitation. Hence, this interdisciplinary field of study makes great contributions to both clinical and basic research by solving these complexities and helping to better interpret the acquired data. Besides, biological experiments can be expansive and time consuming. Therefore, testing biological hypotheses with mathematical models can be significantly beneficial. In this regard, mathematical models can be thought as microscopes developed for specific living systems and inexpensive and fast ways of simulating experiments. Insulin secreting pancreatic betacells are very good examples of such complex systems. Activity of these cells is controlled by extremely complex metabolic and electrophysiological pathways. Therefore, mathematical modeling approaches are proven to be very effective in the study of pancreatic betacells. Impairments in the activity of these cells lead to impaired insulin secretion, which can have life threatening complications in the body. Thus, understanding the mechanisms underlying betacell activity and insulin secretion is crucial. Pancreatic betacells are excitable cells and they produce electrical activity with the ion channels they express in their plasma membranes. In pancreatic betacells, insulin secretion is regulated through pathways that link cellular metabolism to the membrane potential through ion channels they express in their plasma membranes. In the initiation and modulation of the insulin secretion ATPsensitive K+ channels (K(ATP) channels) play a significant role by coupling cell metabolism to the membrane potential. Defects in the expression of K(ATP) channels lead to hypoglycemia associated with excessive insulin secretion in humans. However, mice seem to be able to overcome these defects by employing alternative mechanisms. In this dissertation, we investigate the pathological conditions associated with ATPsensitive K+ channel deficiency in betacells and, with a systems biology approach, we propose mechanisms through which mice can compensate for these defects. Using mathematical modeling we explain the dynamics of these compensatory mechanisms and make predictions to test their plausibility. We also demonstrate the results of the in vitro experiments performed in accordance with our model predictions. One of the longterm goals of this study is helping to identify possible therapeutic targets for the treatment of the congenital hypoglycemia that results from K(ATP) channel deficiency. The overall aim of this dissertation is using mathematical modeling and analysis techniques to better understand the experimental data on pancreatic betacells and guide future research by making testable predictions.
Show less  Date Issued
 2017
 Identifier
 FSU_2017SP_Yildirim_fsu_0071E_13685
 Format
 Thesis
 Title
 Mathematical Modeling of Biofilms with Applications.
 Creator

Li, Jian, Cogan, Nicholas G., Chicken, Eric, Gallivan, Kyle A., Hurdal, Monica K., Florida State University, College of Arts and Sciences, Department of Mathematics
 Abstract/Description

Biofilms are thin layers of microorganisms in which cells adhere to each other and stick to a surface. They are resistant to antibiotics and disinfectants due to the protection from extracellular polymeric substance (EPS), which is a gel like selfproduced matrix, consists of polysaccharide, proteins and nucleic acids. Biofilms play significant roles in many applications. In this document, we provide analysis about effects and influences of biofilms in microfiltration and dental plaque...
Show moreBiofilms are thin layers of microorganisms in which cells adhere to each other and stick to a surface. They are resistant to antibiotics and disinfectants due to the protection from extracellular polymeric substance (EPS), which is a gel like selfproduced matrix, consists of polysaccharide, proteins and nucleic acids. Biofilms play significant roles in many applications. In this document, we provide analysis about effects and influences of biofilms in microfiltration and dental plaque removing process. Differential equations are used for modelling the microfiltration process and the optimal control method is applied to analyze the efficiency of the filtration. The multiphase fluid system is introduced to describe the dental plaque removing process and results are obtained by numerical schemes.
Show less  Date Issued
 2017
 Identifier
 FSU_FALL2017_Li_fsu_0071E_13839
 Format
 Thesis
 Title
 Metric Learning for Shape Classification: A Fast and Efficient Approach with Monte Carlo Methods.
 Creator

Cellat, Serdar, Mio, Washington, Ökten, Giray, Aggarwal, Sudhir, Cogan, Nicholas G., Jain, Harsh Vardhan, Florida State University, College of Arts and Sciences, Department of...
Show moreCellat, Serdar, Mio, Washington, Ökten, Giray, Aggarwal, Sudhir, Cogan, Nicholas G., Jain, Harsh Vardhan, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

Quantifying shape variation within a group of individuals, identifying morphological contrasts between populations and categorizing these groups according to morphological similarities and dissimilarities are central problems in developmental evolutionary biology and genetics. In this dissertation, we present an approach to optimal shape categorization through the use of a new family of metrics for shapes represented by a finite collection of landmarks. We develop a technique to identify...
Show moreQuantifying shape variation within a group of individuals, identifying morphological contrasts between populations and categorizing these groups according to morphological similarities and dissimilarities are central problems in developmental evolutionary biology and genetics. In this dissertation, we present an approach to optimal shape categorization through the use of a new family of metrics for shapes represented by a finite collection of landmarks. We develop a technique to identify metrics that optimally differentiate and categorize shapes using Monte Carlo based optimization methods. We discuss the theory and the practice of the method and apply it to the categorization of 62 mice offsprings based on the shape of their skull. We also create a taxonomic classification tree for multiple species of fruit flies given the shape of their wings. The results of these experiments validate our method.
Show less  Date Issued
 2018
 Identifier
 2018_Sp_Cellat_fsu_0071E_14295
 Format
 Thesis
 Title
 Modeling Credit Risk in the Default Threshold Framework.
 Creator

Chiu, ChunYuan, Kercheval, Alec N., Chicken, Eric, Ökten, Giray, Fahim, Arash, Florida State University, College of Arts and Sciences, Department of Mathematics
 Abstract/Description

The default threshold framework for credit risk modeling developed by Garreau and Kercheval [SIAM Journal on Financial Mathematics, 7:642673, 2016] enjoys the advantages of both the structural form models and the reduced form models, including excellent analytical tractability. In their paper, the closed form default time distribution of a company is derived when the default threshold is a constant or a deterministic function. As for stochastic default threshold, it is shown that the...
Show moreThe default threshold framework for credit risk modeling developed by Garreau and Kercheval [SIAM Journal on Financial Mathematics, 7:642673, 2016] enjoys the advantages of both the structural form models and the reduced form models, including excellent analytical tractability. In their paper, the closed form default time distribution of a company is derived when the default threshold is a constant or a deterministic function. As for stochastic default threshold, it is shown that the survival probability can be derived as an expectation. How to specify the stochastic default threshold so that this expectation can be obtained in closed form is however left unanswered. The purpose of this thesis is to fulfill this gap. In this thesis, three credit risk models with stochastic default thresholds are proposed, under each of which the closed form default time distribution is derived. Unlike Garreau and Kercheval's work where the logreturn of a company's stock price is assumed to be independent and identically distributed and the interest rate is assumed constant, in our new proposed models the random interest rate and the stochastic volatility of a company's stock price are taken into consideration. While in some cases the defaultable bond price, the credit spread and the CDS premium are derived in closed form under the new proposed models, in others it seems not so easy. The difficulty that stops us from getting closed form formulas is also discussed in this thesis. Our new models involve the Heston model, which has a closed form characteristic function. We found the common characteristic function formula used in the literature not always applicable for all input variables. In this thesis the safe region of the formula is analyzed completely. A new formula is also derived that can be used to find the characteristic function value in some cases when the common formula is not applicable. An example is given where the common formula fails and one should use the new formula.
Show less  Date Issued
 2016
 Identifier
 FSU_FA2016_Chiu_fsu_0071E_13584
 Format
 Thesis
 Title
 Modelling Limit Order Book Dynamics Using Hawkes Processes.
 Creator

Chen, Yuanda, Kercheval, Alec N., Beaumont, Paul M., Ewald, Brian D., Zhu, Lingjiong, Florida State University, College of Arts and Sciences, Department of Mathematics
 Abstract/Description

The Hawkes process serves as a natural choice for modeling selfexciting dynamics, such as the behavior of an electronic exchangehosted limit order book (LOB). However, due to the lack of analytical solutions, probability estimates of future events often must rely on Monte Carlo simulation. Although Monte Carlo simulation is known to be good at solving pathdependent problems, it has the limitation that a high computation time is often required to get good accuracy. This is a concern in...
Show moreThe Hawkes process serves as a natural choice for modeling selfexciting dynamics, such as the behavior of an electronic exchangehosted limit order book (LOB). However, due to the lack of analytical solutions, probability estimates of future events often must rely on Monte Carlo simulation. Although Monte Carlo simulation is known to be good at solving pathdependent problems, it has the limitation that a high computation time is often required to get good accuracy. This is a concern in fields like algorithmic trading where fast calculation is essential. In this dissertation we propose the use of a 4dimensional Hawkes process to model the LOB and to forecast midprice movement probabilities using Monte Carlo simulation. We study the feasibility of making this prediction quickly enough to be applicable in practice. We show that fast predictions are feasible, and show in tests on real data that the model has some trading value in forecasting midprice movements. This dissertation also compares the performance of several popular computer languages, Python, MATLAB, Cython and C, in singlecore experiments, and examines the scalability for parallel computing using Cython and C.
Show less  Date Issued
 2017
 Identifier
 FSU_2017SP_Chen_fsu_0071E_13187
 Format
 Thesis
 Title
 Monte Carlo Scheme for a Singular Control Problem: InvestmentConsumption under Proportional Transaction Costs.
 Creator

Tsai, WanYu, Fahim, Arash, Atkins, Jennifer, Zhu, Lingjiong, Florida State University, College of Arts and Sciences, Department of Mathematics
 Abstract/Description

Nowadays free boundary problems are considered as one of the most important directions in the mainstream of partial differential equations (PDEs) analysis, with an abundance of applications in various sciences and real world problems. Free boundary problems on finance have been extended in many areas, such as optimal portfolio selection, control credit risks, and different American style products etc. To modelling these financial problems in the real world, the qualitative and quantitative...
Show moreNowadays free boundary problems are considered as one of the most important directions in the mainstream of partial differential equations (PDEs) analysis, with an abundance of applications in various sciences and real world problems. Free boundary problems on finance have been extended in many areas, such as optimal portfolio selection, control credit risks, and different American style products etc. To modelling these financial problems in the real world, the qualitative and quantitative behaviors of the solution to a free boundary problem are still not well understood and also numerical solutions to free boundary problems remain a challenge. Stochastic control problems reduce to freeboundary problems in partial differential equations while there are no bounds on the rate of control. In a free boundary problem, the solution as well as the domain to the PDE need to be determined simultaneously. In this dissertation, we concern the numerical solution of a fully nonlinear parabolic double obstacle problem arising from a finite time portfolio selection problem with proportional transaction costs. We consider optimal allocation of wealth among multiple stocks and a bank account in order to maximize the finite horizon discounted utility of consumption. The problem is mainly governed by a timedependent HamiltonJacobiBellman equation with gradient constraints. We propose a numerical method which is composed of Monte Carlo simulation to take advantage of the highdimensional properties and finite difference method to approximate the gradients of the value function. Numerical results illustrate behaviors of the optimal trading strategies and also satisfy all qualitative properties proved in Dai et al. (2009) and Chen and Dai (2013).
Show less  Date Issued
 2017
 Identifier
 FSU_FALL2017_Tsai_fsu_0071E_14174
 Format
 Thesis
 Title
 Multiscale Summaries of Probability Measures with Applications to Plant and Microbiome Data.
 Creator

Díaz Martínez, Diego Hernán, Mio, Washington, Tschinkel, Walter R. (Walter Reinhart), MestertonGibbons, Mike, Florida State University, College of Arts and Sciences, Department...
Show moreDíaz Martínez, Diego Hernán, Mio, Washington, Tschinkel, Walter R. (Walter Reinhart), MestertonGibbons, Mike, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

Traditional descriptors such as the mean and the covariance matrix give useful global summaries of data and probability measures. Nevertheless, when distributions with more complex topological and geometrical behaviors arise, these methods fall short in accurately describing them. This dissertation explores and develops new methods that provide more informative summaries of complex probability measures using multiscale analogs of the Fréchet function and the covariance tensor which encode...
Show moreTraditional descriptors such as the mean and the covariance matrix give useful global summaries of data and probability measures. Nevertheless, when distributions with more complex topological and geometrical behaviors arise, these methods fall short in accurately describing them. This dissertation explores and develops new methods that provide more informative summaries of complex probability measures using multiscale analogs of the Fréchet function and the covariance tensor which encode variation of data with respect to any point in the domain. These multiscale methods are developed using kernel functions and diffusion distances and are helpful in obtaining more information on localtoregionaltoglobal behavior of probability measures, unlike the traditional take which only gives global summaries. We applied the methods to the analysis of climatic data of the Fabaceae plant family (legumes) and to microbiome data related to the Clostridium difficile infection in the human gut. Our studies reveal patterns of climatological adaptation of various legume taxa and changes in the interactions of microbial communities in the presence of infection which are helpful in monitoring the resolution of the disease.
Show less  Date Issued
 2016
 Identifier
 FSU_2016SP_DiazMartinez_fsu_0071E_13067
 Format
 Thesis
 Title
 Myrberg's Numerical Uniformization.
 Creator

Li, Xin, Seppälä, Mika, Van Hoeij, Mark, Reina, Laura, Nolder, Craig, Quine, J. R. (John R.), Florida State University, College of Arts and Sciences, Department of Mathematics
 Abstract/Description

In 1920, Myrberg [23] presented a method for the numerical uniformization of hyperelliptic curves. His ingenious method is based on iterating opening certain slots. He obtains the uniformization as a limit of such a process. Myrberg considered a central part of the process, the convergence of the sequence approximating the uniformizing mapping, as obvious. Later, Seppala [27] proved the convergence by normal family techniques. This work reviews the construction of Myrberg's [23], and the...
Show moreIn 1920, Myrberg [23] presented a method for the numerical uniformization of hyperelliptic curves. His ingenious method is based on iterating opening certain slots. He obtains the uniformization as a limit of such a process. Myrberg considered a central part of the process, the convergence of the sequence approximating the uniformizing mapping, as obvious. Later, Seppala [27] proved the convergence by normal family techniques. This work reviews the construction of Myrberg's [23], and the convergence results of Seppala's[27]. Then we extended this uniformization algorithm to hyperelliptic curves defined by polynomial with odd degree. The most important contribution of this work is providing an optimized iteration algorithm, since the original method cannot provide accurate approximation. In the end, we implemented this optimized algorithm to build an applet, MyrbergUniform, which offers fast computation of numerical uniformization.
Show less  Date Issued
 2015
 Identifier
 FSU_migr_etd9380
 Format
 Thesis
 Title
 Neural Rule Ensembles: Encoding Feature Interactions into Neural Networks.
 Creator

Dawer, Gitesh, Barbu, Adrian G., Gallivan, Kyle A., Erlebacher, Gordon, Ökten, Giray, Sussman, Mark, Florida State University, College of Arts and Sciences, Department of...
Show moreDawer, Gitesh, Barbu, Adrian G., Gallivan, Kyle A., Erlebacher, Gordon, Ökten, Giray, Sussman, Mark, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

Artificial Neural Networks form the basis of very powerful learning methods. It has been observed that a naive application of fully connected neural networks often leads to overfitting. In an attempt to circumvent this issue, a prior knowledge pertaining to feature interactions can be encoded into these networks. This defines a taskspecific structure on an underlying representation and helps in reducing the number of learnable parameters. Convolutional Neural Network is such an adaptation of...
Show moreArtificial Neural Networks form the basis of very powerful learning methods. It has been observed that a naive application of fully connected neural networks often leads to overfitting. In an attempt to circumvent this issue, a prior knowledge pertaining to feature interactions can be encoded into these networks. This defines a taskspecific structure on an underlying representation and helps in reducing the number of learnable parameters. Convolutional Neural Network is such an adaptation of artificial neural networks for image datasets which exploits the spatial relationship among the features and explicitly encodes the translational equivariance. Similarly, Recurrent Neural Networks are designed to exploit the temporal relationship inherent in sequential data. However, for tabular datasets, any prior structure on feature relationships is not apparent. In this work, we use decision trees to capture such feature interactions for this kind of datasets and define a mapping to encode extracted relationships into a neural network. This addresses the initialization related concerns of fully connected neural networks and enables learning of compact representations compared to state of the art treebased approaches. Empirical evaluations and simulation studies show the superiority of such an approach over fully connected neural networks and treebased approaches.
Show less  Date Issued
 2018
 Identifier
 2018_Su_Dawer_fsu_0071E_14670
 Format
 Thesis
 Title
 Nonlinear SchrödingerType Systems: Complex Lattices and NonParaxiality.
 Creator

Cole, Justin, Musslimani, Ziad H., Höflich, Peter A., Wang, Xiaoming, Moore, M. Nicholas J. (Matthew Nicholas J.), Florida State University, College of Arts and Sciences,...
Show moreCole, Justin, Musslimani, Ziad H., Höflich, Peter A., Wang, Xiaoming, Moore, M. Nicholas J. (Matthew Nicholas J.), Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

This thesis investigates nonlinear systems that are dispersive and conservative in nature and wellapproximated by the nonlinear Schrödinger (NLS) equation. The NLS equation is the prototypical equation for describing such phenomena and it has been utilized in a large number of physical systems. This work considers novel applications and exotic parameter regimes that fall inside the class of solutions well described by nonlinear Schrödingertype systems. A brief historical, physical, and...
Show moreThis thesis investigates nonlinear systems that are dispersive and conservative in nature and wellapproximated by the nonlinear Schrödinger (NLS) equation. The NLS equation is the prototypical equation for describing such phenomena and it has been utilized in a large number of physical systems. This work considers novel applications and exotic parameter regimes that fall inside the class of solutions well described by nonlinear Schrödingertype systems. A brief historical, physical, and mathematical introduction to deriving the NLS equation and its variants is presented. The topics considered in detail cover optical systems in various media and are naturally divided into two parts: nonparaxiality through the inclusion of higherorder dispersion/diffraction and beam propagation in the presence of complex lattices. The higherorder dispersion/diffraction effects on soliton solutions are considered in detail. The propagation of a short soliton pulse as it travels down a fiber optic in the presence of a linear timeperiodic potential is considered. Due to the short duration of the pulse fourthorder dispersive effects are relevant. The band gap structure is determined using FloquetBloch theory and the shape of its dispersion curves as a function of the fourthorder dispersion coupling constant β is discussed. Several features not observed in the absence of highorder dispersion (β=0) are highlighted, such as a nonzero threshold value of potential strength below which there is no band gap and the formation of novel localized modes at large potential amplitudes. A higher order two band tight binding model is introduced that captures and intuitively explains most of the numerical results related to the spectral bands. Lattice solitons corresponding to spectral eigenvalues lying in the semiinfinite and first band gaps are constructed. Stability of various localized lattice modes is studied via linear stability analysis and direct numerical simulation. Next the spectral transverse instabilities of onedimensional solitary wave solutions to the twodimensional NLS equation with biharmoinc diffraction and subject to higherdimensional perturbations are studied. Physically, the inclusion of the biharmonic term corresponds to spatial beams with a narrow width in comparison to their wavelength. A linear boundary value problem governing the evolution of the transverse perturbations is derived. The eigenvalues of the perturbations are numerically computed and a finite band of unstable transverse modes is found to exist. In the long wavelength limit an asymptotic formula for the perturbation growth rate that agrees well with the numerical findings. Using a variational formulation based on Lagrangian model reduction, an approximate expression for the perturbation eigenvalues is obtained and its validity is compared with both the asymptotic and numerical results. The dynamics of a onedimensional soliton stripe in the presence of a transverse perturbation is studied using direct numerical simulations. The second half of the dissertation is concerned with beam propagation in the presence of complex lattices, in particular lattices that possess paritytime (PT) symmetries. A new family of nonhermitian optical potentials given in terms of double exponential periodic functions is introduced. The center of PTsymmetry is not around zero and the potential satisfies a shifted PTsymmetry relation at two distinct locations. These novel refractive index modulations are examined from the perspective of optical lattices that are homogeneous along the propagation direction. The diffraction dynamics, abrupt phase transitions in the eigenvalue spectrum and exceptional points in the band structure are studied in detail. In addition, the nonlinear properties of coherent structures in Kerr media is probed. The spatial symmetries of such lattice solitons follow the shifted PTsymmetric relations. Furthermore, such lattice solitons have a power threshold and their linear and nonlinear stability is critically dependent on their spatial symmetry point. In the final chapter a class of exact multicomponent constant energy solutions to a Manakov system in the presence of an external PTsymmetric complex potential is constructed. This type of uniform wave pattern displays a nontrivial phase whose spatial dependence is induced from the lattice structure. In this regard, light can propagate without scattering while retaining its original form despite the presence of inhomogenous gain and loss. The constantintensity continuous waves are then used to perform a modulational instability analysis in the presence of both nonhermitian media and cubic nonlinearity. A linear stability eigenvalue problem is formulated that governs the dynamical evolution of the periodic perturbation and its spectrum is numerically determined using FourierFloquetBloch theory. The stability of the selffocusing and defocusing cases is considered and compared to the zeropotential results. Our linear stability results are supplemented with direct (nonlinear) numerical simulations.
Show less  Date Issued
 2016
 Identifier
 FSU_2016SP_Cole_fsu_0071E_13102
 Format
 Thesis
 Title
 On the Multidimensional Default Threshold Model for Credit Risk.
 Creator

Zhou, Chenchen, Kercheval, Alec N., Wu, Wei, Ökten, Giray, Fahim, Arash, Florida State University, College of Arts and Sciences, Department of Mathematics
 Abstract/Description

This dissertation is based on the structural model framework for default risk that was first introduced by garreau2016structural (henceforth: the "GK model"). In this approach, the time of default is defined as the first time the logreturn 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...
Show moreThis dissertation is based on the structural model framework for default risk that was first introduced by garreau2016structural (henceforth: the "GK model"). In this approach, the time of default is defined as the first time the logreturn 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 intensitybased 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 ddimensional 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 firsttodefault (FtD) basket CDS prices, up to quadrature.
Show less  Date Issued
 2017
 Identifier
 FSU_SUMMER2017_Zhou_fsu_0071E_14012
 Format
 Thesis