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 Title
 A Riemannian Approach for Computing Geodesics in Elastic Shape Space and Its Applications.
 Creator

You, Yaqing, Gallivan, Kyle A., Absil, PierreAntoine, Erlebacher, Gordon, Ökten, Giray, Sussman, Mark, Florida State University, College of Arts and Sciences, Department of...
Show moreYou, Yaqing, Gallivan, Kyle A., Absil, PierreAntoine, Erlebacher, Gordon, Ökten, Giray, Sussman, Mark, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

This dissertation proposes a Riemannian approach for computing geodesics for closed curves in elastic shape space. The application of two Riemannian unconstrained optimization algorithms, Riemannian Steepest Descent (RSD) algorithm and Limitedmemory Riemannian BroydenFletcherGoldfarbShanno (LRBFGS) algorithm are discussed in this dissertation. The application relies on the definition and computation for basic differential geometric components, namely tangent spaces and tangent vectors,...
Show moreThis dissertation proposes a Riemannian approach for computing geodesics for closed curves in elastic shape space. The application of two Riemannian unconstrained optimization algorithms, Riemannian Steepest Descent (RSD) algorithm and Limitedmemory Riemannian BroydenFletcherGoldfarbShanno (LRBFGS) algorithm are discussed in this dissertation. The application relies on the definition and computation for basic differential geometric components, namely tangent spaces and tangent vectors, Riemannian metrics, Riemannian gradient, as well as retraction and vector transport. The difference between this Riemannian approach to compute closed curve geodesics as well as accurate geodesic distance, the existing PathStraightening algorithm and the existing Riemannian approach to approximate distances between closed shapes, are also discussed in this dissertation. This dissertation summarizes the implementation details and techniques for both Riemannian algorithms to achieve the most efficiency. This dissertation also contains basic experiments and applications that illustrate the value of the proposed algorithms, along with comparison tests to the existing alternative approaches. It has been demonstrated by various tests that this proposed approach is superior in terms of time and performance compared to a stateoftheart distance computation algorithm, and has better performance in applications of shape distance when compared to the distance approximation algorithm. This dissertation applies the Riemannian geodesic computation algorithm to calculate Karcher mean of shapes. Algorithms that generate less accurate distances and geodesics are also implemented to compute shape mean. Test results demonstrate the fact that the proposed algorithm has better performance with sacrifice in time. A hybrid algorithm is then proposed, to start with the fast, less accurate algorithm and switch to the proposed accurate algorithm to get the gradient for Karcher mean problem. This dissertation also applies Karcher mean computation to unsupervised learning of shapes. Several clustering algorithms are tested with the distance computation algorithm and Karcher mean algorithm. Different versions of Karcher mean algorithm used are compared with tests. The performance of clustering algorithms are evaluated by various performance metrics.
Show less  Date Issued
 2018
 Identifier
 2018_Su_You_fsu_0071E_14686
 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
 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
 Symmetric Surfaces and the Character Variety.
 Creator

Leach, Jay, Petersen, Kathleen L., Duke, D. W., Heil, Wolfgang H., Ballas, Samuel A., Florida State University, College of Arts and Sciences, Department of Mathematics
 Abstract/Description

We extend Culler and Shalen's work on constructing essential surfaces in 3manifolds to orbifolds. A consequence of this work is that every valuation on the canonical component that detects an essential surface, detects an essential surface that is preserved by every orientation preserving symmetry on the manifold. This Theorem applies to orientable hyperbolic manifolds, with orientation preserving symmetry group, whose quotient by this group is an orbifold with a flexible cusp, which is the...
Show moreWe extend Culler and Shalen's work on constructing essential surfaces in 3manifolds to orbifolds. A consequence of this work is that every valuation on the canonical component that detects an essential surface, detects an essential surface that is preserved by every orientation preserving symmetry on the manifold. This Theorem applies to orientable hyperbolic manifolds, with orientation preserving symmetry group, whose quotient by this group is an orbifold with a flexible cusp, which is the case for most hyperbolic 3manifolds. We then look at a family of two bridge knots where our theorem shows it is impossible for every essential surface to be detected on the canonical component. We then prove that all surfaces that are preserved by the orientation preserving symmetries of these knots are detected by ideal points on the canonical component of the character variety by calculating the canonical component of the Apolynomial for the family of knots. We then prove that every essential surface in these knot that is not detected on the canonical component of the character variety is detected on another component.
Show less  Date Issued
 2018
 Identifier
 2018_Su_Leach_fsu_0071E_14753
 Format
 Thesis
 Title
 Using Number Talks with Supports to Increase the Early Number Sense Skills of Preschool Students with Autism Spectrum Disorder.
 Creator

Henning, Bonnie Lynne, Whalon, Kelly J., Ke, Fengfeng, Hanline, Mary Frances, Whitacre, Ian Michael, Florida State University, College of Education, School of Teacher Education
 Abstract/Description

This multiple probe across participants design evaluated the effectiveness of teaching early number sense skills (ENS) to young children (age 4) with autism spectrum disorder (ASD) using Number Talks with supports. Following participation in Number Talks with supports, young children with ASD learned the ENS skills of subitizing, onetoone correspondence, number conservation, and magnitude discrimination. This study included a baseline condition, a Number Talks alone condition, and a Number...
Show moreThis multiple probe across participants design evaluated the effectiveness of teaching early number sense skills (ENS) to young children (age 4) with autism spectrum disorder (ASD) using Number Talks with supports. Following participation in Number Talks with supports, young children with ASD learned the ENS skills of subitizing, onetoone correspondence, number conservation, and magnitude discrimination. This study included a baseline condition, a Number Talks alone condition, and a Number Talks with supports condition in order to evaluate how much support young learners with ASD required to learn ENS skills during Number Talks. The Number Talks with support condition combined the socially constructivism learning techniques in Number Talks alone with the direct instruction practices of visual supports, a least to most prompting hierarchy, and explicit modeling. A functional relationship was found between Number Talks with supports and increased ENS skills of all three participants with ASD. The ENS skills were also maintained at near mastery criteria levels by all three participants with ASD. A peer comparison as well as peer pre and posttest data showed that peers also increased their ENS skills from baseline to the end of intervention. This study successfully combined the socially constructed learning technique of Number Talks with direct instruction support, and increased the ENS skills of young children with ASD and peers alike. Implications for practice and future research are discussed.
Show less  Date Issued
 2018
 Identifier
 2018_Su_Henning_fsu_0071E_14690
 Format
 Thesis
 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
 Overcoming Geometric Limitations in the Finite Element Method by Means of Polynomial Extension: Application to Second Order Elliptic Boundary Value and Interface Problems.
 Creator

Cheung, James, Gunzburger, Max D., Steinbock, Oliver, Bochev, Pavel B., Perego, Mauro, Peterson, Janet S., Shanbhag, Sachin, Florida State University, College of Arts and...
Show moreCheung, James, Gunzburger, Max D., Steinbock, Oliver, Bochev, Pavel B., Perego, Mauro, Peterson, Janet S., Shanbhag, Sachin, Florida State University, College of Arts and Sciences, Department of Scientific Computing
Show less  Abstract/Description

In this dissertation, we present a new approach for approximating the solution of second order partial differential equations and interface problems. This approach is based on the classical finite element method, where instead of using geometric manipulations to fit the discrete domain to the curved domain given by the continuous problem, we use polynomial extensions to enforce that a suitably constructed extension of the numerical solution matches the boundary condition given by the...
Show moreIn this dissertation, we present a new approach for approximating the solution of second order partial differential equations and interface problems. This approach is based on the classical finite element method, where instead of using geometric manipulations to fit the discrete domain to the curved domain given by the continuous problem, we use polynomial extensions to enforce that a suitably constructed extension of the numerical solution matches the boundary condition given by the continuous problem in the weak sense. This method is thus aptly named the Polynomial Extension Finite Element Method (PEFEM). Using this approach, we may approximate the solution of elliptic interface problems by enforcing that the extension of the solution on their respective subdomains matches weakly the continuity conditions prescribed by the continuous problem on a curved interface. This method is then called the Method of Virtual Interfaces (MVI), since, while the continuous interface exists in the context of the continuous problem, it is virtual in the context of its numerical approximation. The key benefits of this polynomial extension approach is that it is simple to implement and that it is optimally convergent with respect to the best approximation results given by interpolation. Theoretical analysis and computational results are presented.
Show less  Date Issued
 2018
 Identifier
 2018_Sp_Cheung_fsu_0071E_14328
 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
 Optimal Portfolio Execution under TimeVarying Liquidity Constraints.
 Creator

Lin, HuaYi, Fahim, Arash, Atkins, Jennifer, Kercheval, Alec N., Ökten, Giray, Florida State University, College of Arts and Sciences, Department of Mathematics
 Abstract/Description

The problem of optimal portfolio execution has become one of the most important problems in the area of financial mathematics. Over the past two decades, numerous researchers have developed a variety of different models to address this problem. In this dissertation, we extend the LOB (Limit Order Book) model proposed by Obizhaeva and Wang (2013) by incorporating a more realistic assumption on the order book depth; the amount of liquidity provided by a LOB market is finite at all times. We use...
Show moreThe problem of optimal portfolio execution has become one of the most important problems in the area of financial mathematics. Over the past two decades, numerous researchers have developed a variety of different models to address this problem. In this dissertation, we extend the LOB (Limit Order Book) model proposed by Obizhaeva and Wang (2013) by incorporating a more realistic assumption on the order book depth; the amount of liquidity provided by a LOB market is finite at all times. We use an algorithmic approach to solve the problem of optimal execution under timevarying constraints on the depth of a LOB. For the simplest case where the order book depth stays at a fixed level for the entire trading horizon, we reduce the optimal execution problem into a onedimensional rootfinding problem which can be readily solved by standard numerical algorithms. When the depth of the LOB is monotone in time, we first apply the KKT (KarushKuhnTucker) conditions to narrow down the set of candidate strategies and then use a dichotomybased search algorithm to pin down the optimal one. For the general case that the order book depth doesn't exhibit any particular pattern, we start from the optimal strategy subject to no liquidity constraints and iterate over execution strategy by sequentially adding more constraints to the problem in a specific fashion until primal feasibility is achieved. Numerical experiments indicate that our algorithms give comparable results to those of current existing convex optimization toolbox CVXOPT with significantly lower time complexity.
Show less  Date Issued
 2018
 Identifier
 2018_Sp_Lin_fsu_0071E_14349
 Format
 Thesis
 Title
 Using Mathematical Tools to Investigate the Autoimmune Hair Loss Disease Alopecia Areata.
 Creator

Dobreva, Atanaska, Cogan, Nicholas G., Stroupe, M. Elizabeth, Bertram, R., Hurdal, Monica K., Florida State University, College of Arts and Sciences, Department of Mathematics
 Abstract/Description

Alopecia areata is an autoimmune condition where the immune system attacks hair follicles and disrupts their natural cycle through phases of growth, regression, and rest. The disease manifests with distinct hair loss patterns, and what causes it and how to treat it are open questions. We first construct an ODE model for alopecia areata in follicles which are in stage of growth. The dynamical system describes the behavior of immune cells and signals highlighted by experimental studies as...
Show moreAlopecia areata is an autoimmune condition where the immune system attacks hair follicles and disrupts their natural cycle through phases of growth, regression, and rest. The disease manifests with distinct hair loss patterns, and what causes it and how to treat it are open questions. We first construct an ODE model for alopecia areata in follicles which are in stage of growth. The dynamical system describes the behavior of immune cells and signals highlighted by experimental studies as primarily involved in the disease development. We perform sensitivity analysis and linear stability and bifurcation analysis to investigate the importance of processes in relation to the levels of immune cells. Our findings indicate that the proinflammatory pathway via the messenger protein interferongamma and the immunosuppressive pathway via hair follicle immune privilege agents are crucial. Next, we incorporate follicle cycling into the model and explore what processes have the greatest impact on the duration of hair growth in healthy versus diseased follicles. The results suggest that some processes matter in both cases, but there are differences, as well. Finally, the study presents and analyzes a PDE model which captures patterns characteristic of hair loss in alopecia areata.
Show less  Date Issued
 2018
 Identifier
 2018_Sp_Dobreva_fsu_0071E_14479
 Format
 Thesis
 Title
 Using RBFGenerated Quadrature Rules to Solve Nonlocal Continuum Models.
 Creator

Lyngaas, Isaac R., Peterson, Janet S., Musslimani, Ziad H., Gunzburger, Max D., Quaife, Bryan, Shanbhag, Sachin, Florida State University, College of Arts and Sciences,...
Show moreLyngaas, Isaac R., Peterson, Janet S., Musslimani, Ziad H., Gunzburger, Max D., Quaife, Bryan, Shanbhag, Sachin, Florida State University, College of Arts and Sciences, Department of Scientific Computing
Show less  Abstract/Description

Recently nonlocal continuum models have gained interest as alternatives to traditional PDE models due to their capability of handling solutions with discontinuities and their ease of modeling anomalous diffusion. The typical approach used for approximating timedependent nonlocal integrodifferential models is to use finite element or discontinuous Galerkin methods; however, these approaches can be quite computationally intensive especially when solving problems in more than one dimension due...
Show moreRecently nonlocal continuum models have gained interest as alternatives to traditional PDE models due to their capability of handling solutions with discontinuities and their ease of modeling anomalous diffusion. The typical approach used for approximating timedependent nonlocal integrodifferential models is to use finite element or discontinuous Galerkin methods; however, these approaches can be quite computationally intensive especially when solving problems in more than one dimension due to the approximation of the nonlocal integral. In this work, we propose a novel method based on using radial basis functions to generate accurate quadrature rules for the nonlocal integral appearing in the model and then coupling this with a finite difference approximation to the timedependent terms. The viability of our method is demonstrated through various numerical tests on time dependent nonlocal diffusion, nonlocal anomalous diffusion, and nonlocal advection problems in one and two dimensions. In addition to nonlocal problems with continuous solutions, we modify our approach to handle problems with discontinuous solutions. We compare some numerical results with analogous finite element results and demonstrate that for an equivalent amount of computational work we obtain much higher rates of convergence.
Show less  Date Issued
 2018
 Identifier
 2018_Fall_Lyngaas_fsu_0071E_14886
 Format
 Thesis
 Title
 Trend and VariablePhase Seasonality Estimation from Functional Data.
 Creator

Tai, LiangHsuan, Gallivan, Kyle A., Srivastava, Anuj, Wu, Wei, Klassen, E. (Eric), Ökten, Giray, Florida State University, College of Arts and Sciences, Department of Mathematics
 Abstract/Description

The problem of estimating trend and seasonality has been studied over several decades, although mostly using single time series setup. This dissertation studies the problem of estimating these components from a functional data point of view, i.e. multiple curves, in situations where seasonal effects exhibit arbitrary time warpings or phase variability across different observations. Rather than ignoring the phase variability, or using an offtheshelf alignment method to remove phase, we take...
Show moreThe problem of estimating trend and seasonality has been studied over several decades, although mostly using single time series setup. This dissertation studies the problem of estimating these components from a functional data point of view, i.e. multiple curves, in situations where seasonal effects exhibit arbitrary time warpings or phase variability across different observations. Rather than ignoring the phase variability, or using an offtheshelf alignment method to remove phase, we take a modelbased approach and seek Maximum Likelihood Estimators (MLEs) of the trend and the seasonal effects, while performing alignments over the seasonal effects at the same time. The MLEs of trend, seasonality, and phase are computed using a coordinate descent based optimization method. We use bootstrap replication for computing confidence bands and for testing hypothesis about the estimated components. We also utilize loglikelihood for selecting the trend subspace, and for comparisons with other candidate models. This framework is demonstrated using experiments involving synthetic data and three real data (Berkeley growth velocity, U.S. electricity price, and USD exchange fluctuation). Our framework is further applied to another biological problem, significance analysis of gene sets of timecourse gene expression data and outperform the stateoftheart method.
Show less  Date Issued
 2017
 Identifier
 FSU_2017SP_Tai_fsu_0071E_13816
 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
 Scroll Waves and How They Interact with NonReactive Spheres, Tori, and Knots.
 Creator

Weingard, Daniel, Bertram, R. (Richard), Rikvold, Per Arne, Steinbock, Oliver, Hurdal, Monica K., Magnan, Jerry F., Florida State University, College of Arts and Sciences,...
Show moreWeingard, Daniel, Bertram, R. (Richard), Rikvold, Per Arne, Steinbock, Oliver, Hurdal, Monica K., Magnan, Jerry F., Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

Threedimensional reactiondiffusion systems are able to produce scroll waves which rotate around a curve called the filament. A scroll ring is formed when the filament is a closed curve. In isotropic systems where reactants have equal diffusion coefficients, scroll rings will shrink over time and eventually collapse. Chemical experiments and numerical studies have shown that filaments can pin to certain nonreactive objects and prevent scroll ring collapse. With numerical simulation, we study...
Show moreThreedimensional reactiondiffusion systems are able to produce scroll waves which rotate around a curve called the filament. A scroll ring is formed when the filament is a closed curve. In isotropic systems where reactants have equal diffusion coefficients, scroll rings will shrink over time and eventually collapse. Chemical experiments and numerical studies have shown that filaments can pin to certain nonreactive objects and prevent scroll ring collapse. With numerical simulation, we study how new types of objects affect scroll ring behavior. In particular, we explore the properties of random sphere arrangements that prevent scroll ring collapse. In addition, we discover a novel mechanism that causes scroll rings to expand when pinning to a nonreactive trefoil knot.
Show less  Date Issued
 2017
 Identifier
 FSU_2017SP_Weingard_fsu_0071E_13790
 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
 Topology of ngonal Curve Complements.
 Creator

Aktas, Mehmet Emin, Hironaka, Eriko, Mio, Washington, Kumar, Piyush, Heil, Wolfgang, Aluffi, Paolo, Florida State University, College of Arts and Sciences, Department of...
Show moreAktas, Mehmet Emin, Hironaka, Eriko, Mio, Washington, Kumar, Piyush, Heil, Wolfgang, Aluffi, Paolo, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

This thesis has two parts. The first part concerns topological invariants of the ngonal plane curves. Our first result is an application of Krammer representations to the Libgober invariant for plane curve complements. This gives a multivariable invariant that depends only on the fundamental group. Our second results is an algorithm to compute the braid monodromy and Libgober polynomial invariant of ngonal curves. We show that the algorithm improves on existing algorithms. We compare the...
Show moreThis thesis has two parts. The first part concerns topological invariants of the ngonal plane curves. Our first result is an application of Krammer representations to the Libgober invariant for plane curve complements. This gives a multivariable invariant that depends only on the fundamental group. Our second results is an algorithm to compute the braid monodromy and Libgober polynomial invariant of ngonal curves. We show that the algorithm improves on existing algorithms. We compare the information one gets from Alexander and Krammer polynomials. The second and main part of our thesis focuses on properties of dessins d'enfants associated to trigonal curves. Degtyarev first studied dessins d'enfants in this context giving a new method for computing braid monodromies and fundamental groups. Our first result is a classification of all possible combinatorial data that can occur for trigonal curves of low degree, as well as bounds on the number of possibilities for all degree. We also study deformations of trigonal curves and corresponding deformations of their dessins. Of special interest to Degtyarev was the case when the dessins are maximal. Our second result gives a sufficient condition for a trigonal curve to be deformable to one that is maximal.
Show less  Date Issued
 2017
 Identifier
 FSU_2017SP_Aktas_fsu_0071E_13779
 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
 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
 Random Sobol' Sensitivity Analysis and Model Robustness.
 Creator

Mandel, David, Ökten, Giray, Hussaini, M. Yousuff, Huffer, Fred W. (Fred William), Kercheval, Alec N., Fahim, Arash, Florida State University, College of Arts and Sciences,...
Show moreMandel, David, Ökten, Giray, Hussaini, M. Yousuff, Huffer, Fred W. (Fred William), Kercheval, Alec N., Fahim, Arash, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

This work develops both the theoretical foundation and the practical application of random Sobol' analysis with two goals. The first is to provide a more general and accommodating approach to global sensitivity analysis, in which the parameter distribution themselves contain uncertainty, and hence the sensitivity results are random quantities as well. The framework for this approach is motivated by empirical evidence of such behavior, and examples of this behavior in interest rate and...
Show moreThis work develops both the theoretical foundation and the practical application of random Sobol' analysis with two goals. The first is to provide a more general and accommodating approach to global sensitivity analysis, in which the parameter distribution themselves contain uncertainty, and hence the sensitivity results are random quantities as well. The framework for this approach is motivated by empirical evidence of such behavior, and examples of this behavior in interest rate and temperature modeling are provided. The second goal is to compare competing models on their robustness, a notion developed and defined to provide a quantitative solution to model selection based on model uncertainty and sensitivity
Show less  Date Issued
 2017
 Identifier
 FSU_2017SP_Mandel_fsu_0071E_13682
 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
 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
 Third Order AHypergeometric Functions.
 Creator

Xu, Wen, Hoeij, Mark van, Reina, Laura, Agashe, Amod S. (Amod Sadanand), Aldrovandi, Ettore, Aluffi, Paolo, Florida State University, College of Arts and Sciences, Department of...
Show moreXu, Wen, Hoeij, Mark van, Reina, Laura, Agashe, Amod S. (Amod Sadanand), Aldrovandi, Ettore, Aluffi, Paolo, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

To solve globally bounded order $3$ linear differential equations with rational function coefficients, this thesis introduces a partial $_3F_2$solver (Section~\ref{3F2 type solution}) and $F_1$solver (Chapter~\ref{F1 solver}), where $_3F_2$ is the hypergeometric function $_3F_2(a_1,a_2,a_3;b_1,b_2\,\,x)$ and $F_1$ is the Appell's $F_1(a,b_1,b_2,c\,\,x,y).$ To investigate the relations among order $3$ multivariate hypergeometric functions, this thesis presents two multivariate tools:...
Show moreTo solve globally bounded order $3$ linear differential equations with rational function coefficients, this thesis introduces a partial $_3F_2$solver (Section~\ref{3F2 type solution}) and $F_1$solver (Chapter~\ref{F1 solver}), where $_3F_2$ is the hypergeometric function $_3F_2(a_1,a_2,a_3;b_1,b_2\,\,x)$ and $F_1$ is the Appell's $F_1(a,b_1,b_2,c\,\,x,y).$ To investigate the relations among order $3$ multivariate hypergeometric functions, this thesis presents two multivariate tools: compute homomorphisms (Algorithm~\ref{hom}) of two $D$modules, where $D$ is a multivariate differential ring, and compute projective homomorphisms (Algorithm~\ref{algo ProjHom}) using the tensor product module and Algorithm~\ref{hom}. As an application, all irreducible order $2$ subsystems from reducible order $3$ systems turn out to come from Gauss hypergeometric function $_2F_1(a,b;c\,\,x)$ (Chapter~\ref{chapter applications}).
Show less  Date Issued
 2017
 Identifier
 FSU_FALL2017_XU_fsu_0071E_14234
 Format
 Thesis
 Title
 Sorvali Dilatation and Spin Divisors on Riemann and Klein Surfaces.
 Creator

Almalki, Yahya Ahmed, Nolder, Craig, Huffer, Fred W. (Fred William), Klassen, E. (Eric), Klassen, E. (Eric), van Hoeij, Mark, Florida State University, College of Arts and...
Show moreAlmalki, Yahya Ahmed, Nolder, Craig, Huffer, Fred W. (Fred William), Klassen, E. (Eric), Klassen, E. (Eric), van Hoeij, Mark, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

We review the Sorvali dilatation of isomorphisms of covering groups of Riemann surfaces and extend the definition to groups containing glidereflections. Then we give a bound for the distance between two surfaces, one of them resulting from twisting the other at a decomposing curve. Furthermore, we study spin structures on Riemann and Klein surfaces in terms of divisors. In particular, we take a closer look at spin structures on hyperelliptic and pgonal surfaces defined by divisors supported...
Show moreWe review the Sorvali dilatation of isomorphisms of covering groups of Riemann surfaces and extend the definition to groups containing glidereflections. Then we give a bound for the distance between two surfaces, one of them resulting from twisting the other at a decomposing curve. Furthermore, we study spin structures on Riemann and Klein surfaces in terms of divisors. In particular, we take a closer look at spin structures on hyperelliptic and pgonal surfaces defined by divisors supported on branch points. Moreover, we study invariant spin divisors under automorphisms and antiholomorphic involutions of Riemann surfaces.
Show less  Date Issued
 2017
 Identifier
 FSU_SUMMER2017_ALMALKI_fsu_0071E_14064
 Format
 Thesis
 Title
 QuasiMonte Carlo and Markov Chain QuasiMonte Carlo Methods in Estimation and Prediction of Time Series Models.
 Creator

Tzeng, YuYing, Ökten, Giray, Beaumont, Paul M., Srivastava, Anuj, Kercheval, Alec N., Kim, Kyounghee (Professor of Mathematics), Florida State University, College of Arts and...
Show moreTzeng, YuYing, Ökten, Giray, Beaumont, Paul M., Srivastava, Anuj, Kercheval, Alec N., Kim, Kyounghee (Professor of Mathematics), Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

Randomized quasiMonte Carlo (RQMC) methods were first developed in mid 1990’s as a hybrid of Monte Carlo and quasiMonte Carlo (QMC) methods. They were designed to have the superior error reduction properties of lowdiscrepancy sequences, but also amenable to the statistical error analysis Monte Carlo methods enjoy. RQMC methods are used successfully in applications such as option pricing, high dimensional numerical integration, and uncertainty quantification. This dissertation discusses the...
Show moreRandomized quasiMonte Carlo (RQMC) methods were first developed in mid 1990’s as a hybrid of Monte Carlo and quasiMonte Carlo (QMC) methods. They were designed to have the superior error reduction properties of lowdiscrepancy sequences, but also amenable to the statistical error analysis Monte Carlo methods enjoy. RQMC methods are used successfully in applications such as option pricing, high dimensional numerical integration, and uncertainty quantification. This dissertation discusses the use of RQMC and QMC methods in econometric time series analysis. In time series simulation, the two main problems are parameter estimation and forecasting. The parameter estimation problem involves the use of Markov chain Monte Carlo (MCMC) algorithms such as MetropolisHastings and Gibbs sampling. In Chapter 3, we use an approximately completely uniform distributed sequence which was recently discussed by Owen et al. [2005], and an RQMC sequence introduced by O ̈kten [2009], in some MCMC algorithms to estimate the parameters of a Probit and SVlogAR(1) model. Numerical results are used to compare these sequences with standard Monte Carlo simulation. In the time series forecasting literature, there was an earlier attempt to use QMC by Li and Winker [2003], which did not provide a rigorous error analysis. Chapter 4 presents how RQMC can be used in time series forecasting with its proper error analysis. Numerical results are used to compare various sequences for a simple AR(1) model. We then apply RQMC to compute the valueatrisk and expected shortfall measures for a stock portfolio whose returns follow a highly nonlinear Markov switching stochastic volatility model which does not admit analytical solutions for the returns distribution. The proper use of QMC and RQMC methods in Monte Carlo and Markov chain Monte Carlo algorithms can greatly reduce the computational error in many applications from sciences, en gineering, economics and finance. This dissertation brings the proper (R)QMC methodology to time series simulation, and discusses the advantages as well as the limitations of the methodology compared the standard Monte Carlo methods.
Show less  Date Issued
 2017
 Identifier
 FSU_SUMMER2017_Tzeng_fsu_0071E_13607
 Format
 Thesis
 Title
 Comparison of Different Noise Forcings, Regularization of Noise, and Optimal Control for the Stochastic NavierStokes Equations.
 Creator

Zhao, Wenju, Gunzburger, Max D., Sussman, Mark, Peterson, Janet S., Quaife, Bryan, Huang, Chen (Professor of Scientific Computing), Florida State University, College of Arts and...
Show moreZhao, Wenju, Gunzburger, Max D., Sussman, Mark, Peterson, Janet S., Quaife, Bryan, Huang, Chen (Professor of Scientific Computing), Florida State University, College of Arts and Sciences, Department of Scientific Computing
Show less  Abstract/Description

Stochastic NavierStokes equations have been widely applied in various computational fluid dynamics (CFD) fields in recent years. It can be considered as another milestone in CFD. Our work focuses on exploring some theoretical and numerical properties of the stochastic NavierStokes equations and related optimal control problems. In particular, we consider: a numerical comparison of solutions of the stochastic NavierStokes equations perturbed by a large range of random noises in time and...
Show moreStochastic NavierStokes equations have been widely applied in various computational fluid dynamics (CFD) fields in recent years. It can be considered as another milestone in CFD. Our work focuses on exploring some theoretical and numerical properties of the stochastic NavierStokes equations and related optimal control problems. In particular, we consider: a numerical comparison of solutions of the stochastic NavierStokes equations perturbed by a large range of random noises in time and space; effective Martingale regularized methods for the stochastic NavierStokes equations with additive noises; and the stochastic NavierStokes equations constrained stochastic boundary optimal control problems. We systemically provide numerical simulation methods for the stochastic NavierStokes equations with different types of noises. The noises are classified as colored or white based on their autocovariance functions. For each type of noise, we construct a representation and a simulation method. Numerical examples are provided to illustrate our schemes. Comparisons of the influence of different noises on the solution of the NavierStokes system are presented. To improve the simulation accuracy, we impose a Martingale correction regularized method for the stochastic NavierStokes equations with additive noise. The original systems are split into two parts, a linear stochastic Stokes equations with Martingale solution and a stochastic modified NavierStokes equations with smoother noise. In addition, a negative fractional Laplace operator is introduced to regularize the noise term. Stability and convergence of the pathwise modified NavierStokes equations are proved. Numerical simulations are provided to illustrate our scheme. Comparisons of nonregularized and regularized noises for the NavierStokes system are presented to further demonstrate the efficiency of our numerical scheme. As a consequence of the above work, we consider a stochastic optimal control problem constrained by the NavierStokes equations with stochastic Dirichlet boundary conditions. Control is applied only on the boundary and is associated with reduced regularity, compared to interior controls. To ensure the existence of a solution and the efficiency of numerical simulations, the stochastic boundary conditions are required to belong almost surely to H¹(∂D). To simulate the system, state solutions are approximated using the stochastic collocation finite element approach, and sparse grid techniques are applied to the boundary random field. Oneshot optimality systems are derived from Lagrangian functionals. Numerical simulations are then made, using a combination of Monte Carlo methods and sparse grid methods, which demonstrate the efficiency of the algorithm.
Show less  Date Issued
 2017
 Identifier
 FSU_SUMMER2017_Zhao_fsu_0071E_14002
 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
 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
 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
 SpaceTime Spectral Element Methods in Fluid Dynamics and Materials Science.
 Creator

Pei, Chaoxu, Sussman, Mark, Hussaini, M. Yousuff, Dewar, William K., Cogan, Nicholas G., Wang, Xiaoming, Florida State University, College of Arts and Sciences, Department of...
Show morePei, Chaoxu, Sussman, Mark, Hussaini, M. Yousuff, Dewar, William K., Cogan, Nicholas G., Wang, Xiaoming, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

In this manuscript, we propose spacetime spectral element methods to solve problems arising from fluid dynamics and materials science. Many engineering applications require one to solve complex problems, such as flows containing multiscale structure in either space or time or both. It is straightforward that highorder methods are always more accurate and efficient than loworder ones for solving smooth problems. For example, spectral element methods can achieve a given level of accuracy...
Show moreIn this manuscript, we propose spacetime spectral element methods to solve problems arising from fluid dynamics and materials science. Many engineering applications require one to solve complex problems, such as flows containing multiscale structure in either space or time or both. It is straightforward that highorder methods are always more accurate and efficient than loworder ones for solving smooth problems. For example, spectral element methods can achieve a given level of accuracy with significantly fewer degrees of freedom compared to methods with algebraic convergence rates, e.g., finite difference methods. However, when it comes to complex problems, a high order method should be augmented with, e.g., a level set method or an artificial viscosity method, in order to address the issues caused by either sharp interfaces or shocks in the solution. Complex problems considered in this work are problems with solutions exhibiting multiple scales, i.e., the Stefan problem, nonlinear hyperbolic problems, and problems with smooth solutions but forces exhibiting disparate temporal scales, such as advection, diffusion and reaction processes. Correspondingly, two families of spacetime spectral element methods are introduced in order to achieve spectral accuracy in both space and time. The first category of spacetime methods are the fully implicit spacetime discontinuous Galerkin spectral element methods. In the fully implicit spacetime methods, time is treated as an additional dimension, and the model equation is rewritten into a spacetime formulation. The other category of spacetime methods are specialized for problems exhibiting multiple time scales: multiimplicit spacetime spectral element methods are developed. The method of lines approach is employed in the multiimplicit spacetime methods. The model is first discretized by a discontinuous spectral element method in space, and the resulting ordinary differential equations are then solved by a new multiimplicit spectral deferred correction method. A novel fully implicit spacetime discontinuous Galerkin (DG) spectral element method is presented to solve the Stefan problem in an Eulerian coordinate system. This method employs a level set procedure to describe the timeevolving interface. To deal with the prior unknown interface, a backward transformation and a forward transformation are introduced in the spacetime mesh. By combining an Eulerian description with a Lagrangian description, the issue of dealing with the implicitly defined arbitrary shaped spacetime elements is avoided. The backward transformation maps the unknown timevarying interface in the fixed frame of reference to a known stationary interface in the moving frame of reference. In the moving frame of reference, the transformed governing equations, written in the spacetime framework, are discretized by a DG spectral element method in each spacetime slab. The forward transformation is used to update the level set function and then to project the solution in each phase onto the new corresponding timedependent domain. Two options for calculating the interface velocity are presented, and both options exhibit spectral accuracy. Benchmark tests in one spatial dimension indicate that the method converges with spectral accuracy in both space and time for the temperature distribution and the interface velocity. The interrelation between the interface position and the temperature makes the Stefan problem a nonlinear problem; a Picard iteration algorithm is introduced in order to solve the nonlinear algebraic system of equations and it is found that just a few iterations lead to convergence. We also apply the fully implicit spacetime DG spectral element method to solve nonlinear hyperbolic problems. The spacetime method is combined with two different approaches for treating problems with discontinuous solutions: (i) spacetime dependent artificial viscosity is introduced in order to capture discontinuities/shocks, and (ii) the sharp discontinuity is tracked with spacetime spectral accuracy, as it moves through the grid. To capture the discontinuity whose location is initially unknown, an artificial viscosity term is strategically introduced, and the amount of artificial viscosity varies in time within a given spacetime slab. It is found that spectral accuracy is recovered everywhere except in the "troublesome element(s)'' where the unresolved steep/sharp gradient exists. When the location of a discontinuity is initially known, a spacetime spectrally accurate tracking method has been developed so that the spectral accuracy of the position of the discontinuity and the solution on either side of the discontinuity is preserved. A Picard iteration method is employed to handle nonlinear terms. Within each Picard iteration, a linear system of equations is solved, which is derived from the spacetime DG spectral element discretization. Spectral accuracy in both space and time is first demonstrated for the Burgers' equation with a smooth solution. For tests with discontinuities, the present spacetime method enables better accuracy at capturing the shock strength in the element containing shock when higher order polynomials in both space and time are used. Moreover, the spectral accuracy of the shock speed and location is demonstrated for the solution of the inviscid Burgers' equation obtained by the shock tracking method, and the sensitivity of the number of Picard iterations to the temporal order is discussed. The dynamics of many physical and biological systems involve two or more processes with a wide difference of characteristic time scales, e.g., problems with advection, diffusion and reaction processes. The computational cost of solving a coupled nonlinear system of equations is expensive for a fully implicit (i.e., "monolithic") spacetime method. Thus, we develop another type of a spacetime spectral element method, which is referred to as the multiimplicit spacetime spectral element method. Rather than coupling space and time together, the method of lines is used to separate the discretization of space and time. The model is first discretized by a discontinuous spectral element method in space and the resulting ordinary differential equations are then solved by a new multiimplicit spectral deferred correction method. The present multiimplicit spectral deferred correction method treats processes with disparate temporal scales independently, but couples them iteratively by a series of deferred correction steps. Compared to lower order operator splitting methods, the splitting error in the multiimplicit spectral deferred correction method is eliminated by exploiting an iterative coupling strategy in the deferred correction procedure. For the spectral element discretization in space, two advective flux reconstructions are proposed: extended elementwise flux reconstruction and nonextended elementwise flux reconstruction. A loworder Istable building block time integration scheme is introduced as an explicit treatment for the hyperbolic terms in order to obtain a stable and efficient building block for the spectrally accurate spacetime scheme along with these two advective flux reconstructions. In other words, we compare the extended elementwise reconstruction with Istable building block scheme with the nonextended elementwise reconstruction with Istable building block scheme. Both options exhibit spectral accuracy in space and time. However, the solutions obtained by extended elementwise flux reconstruction are more accurate than those yielded by nonextended elementwise flux reconstruction with the same number of degrees of freedom. The spectral convergence in both space and time is demonstrated for advectiondiffusionreaction problems. Two different coupling strategies in the multiimplicit spectral deferred correction method are also investigated and both options exhibit spectral accuracy in space and time.
Show less  Date Issued
 2017
 Identifier
 FSU_SUMMER2017_Pei_fsu_0071E_13972
 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
 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
 A longitudinal study on predictors of early calculation development among young children at risk for learning difficulties.
 Creator

Peng, Peng, Namkung, Jessica M, Fuchs, Douglas, Fuchs, Lynn S, Patton, Samuel, Yen, Loulee, Compton, Donald L, Zhang, Wenjuan, Miller, Amanda, Hamlett, Carol
 Abstract/Description

The purpose of this study was to explore domaingeneral cognitive skills, domainspecific academic skills, and demographic characteristics that are associated with calculation development from first grade to third grade among young children with learning difficulties. Participants were 176 children identified with reading and mathematics difficulties at the beginning of first grade. Data were collected on working memory, language, nonverbal reasoning, processing speed, decoding, numerical...
Show moreThe purpose of this study was to explore domaingeneral cognitive skills, domainspecific academic skills, and demographic characteristics that are associated with calculation development from first grade to third grade among young children with learning difficulties. Participants were 176 children identified with reading and mathematics difficulties at the beginning of first grade. Data were collected on working memory, language, nonverbal reasoning, processing speed, decoding, numerical competence, incoming calculations, socioeconomic status, and gender at the beginning of first grade and on calculation performance at four time points: the beginning of first grade, the end of first grade, the end of second grade, and the end of third grade. Latent growth modeling analysis showed that numerical competence, incoming calculation, processing speed, and decoding skills significantly explained the variance in calculation performance at the beginning of first grade. Numerical competence and processing speed significantly explained the variance in calculation performance at the end of third grade. However, numerical competence was the only significant predictor of calculation development from the beginning of first grade to the end of third grade. Implications of these findings for early calculation instructions among young atrisk children are discussed.
Show less  Date Issued
 20161201
 Identifier
 FSU_pmch_27572520, 10.1016/j.jecp.2016.07.017, PMC5052117, 27572520, 27572520, S00220965(16)301059
 Format
 Citation
 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
 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
 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
 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
 Using Deal.II to Solve Problems in Computational Fluid Dynamics.
 Creator

Bystricky, Lukas, Peterson, Janet C., Shanbhag, Sachin, Burkardt, John V., Florida State University, College of Arts and Sciences, Department of Scientific Computing
 Abstract/Description

Finite element methods are a common tool to solve problems in computational fluid dynamics (CFD). This thesis explores the finite element package deal.ii and specific applications to incompressible CFD. Some notation and results from finite element theory are summarised, and a brief overview of some of the features of deal.ii is given. Following this, several CFD applications are presented, including the Stokes equations, the NavierStokes equations and the equations for Darcy flow in porous...
Show moreFinite element methods are a common tool to solve problems in computational fluid dynamics (CFD). This thesis explores the finite element package deal.ii and specific applications to incompressible CFD. Some notation and results from finite element theory are summarised, and a brief overview of some of the features of deal.ii is given. Following this, several CFD applications are presented, including the Stokes equations, the NavierStokes equations and the equations for Darcy flow in porous media. Comparison with benchmark problems are provided for the Stokes and NavierStokes equations and a case study looking at foam deformation is provided for Darcy flow. Code is provided where applicable.
Show less  Date Issued
 2016
 Identifier
 FSU_2016SP_Bystricky_fsu_0071N_13237
 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
 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
 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
 Symmetry Solutions of the Multiphase Model with Biological Applications.
 Creator

Ekrut, David, Cogan, Nicholas G., Keller, Thomas C. S., Quine, J. R. (John R.), Hurdal, Monica K., Jain, Harsh Vardhan, Florida State University, College of Arts and Sciences,...
Show moreEkrut, David, Cogan, Nicholas G., Keller, Thomas C. S., Quine, J. R. (John R.), Hurdal, Monica K., Jain, Harsh Vardhan, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

The multiphase model has given keen insights into many aspects of biology, from crawling cells to biogel morphology and tumor angiogenesis [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. Phases are averaged fractions of a control volume, treated as viscoelastic fluidlike or viscoelastic solids. Derived by conservation laws, the governing equations of the multiphase model are nonlinear partial dierential equations (PDEs). Nonlinear PDEs are often difficult to solve. For this reason, asymptotic and...
Show moreThe multiphase model has given keen insights into many aspects of biology, from crawling cells to biogel morphology and tumor angiogenesis [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. Phases are averaged fractions of a control volume, treated as viscoelastic fluidlike or viscoelastic solids. Derived by conservation laws, the governing equations of the multiphase model are nonlinear partial dierential equations (PDEs). Nonlinear PDEs are often difficult to solve. For this reason, asymptotic and numerical methods have been the predominant tool to analyze these problems, but this is not the only approach to yield results. Reduction transformations can take a system of nonlinear PDEs and reduce it to a system of ordinary differential equations (ODEs), which are typically less difficult to solve. Finding and using such reductions was the focus of this thesis. In this work, we provide a framework for producing analytic solutions to the multiphase model by way of transformations. We begin by deriving exact solutions to a free boundary problem with a sharp interface, a sharp interface being the discontinuity occurring in gel dynamics where a boundary layer separates a mixture of phases from a region of pure solvent. The main focus of study is to track the dynamic interface between the gel and the pure solvent as the gels swell and deswell. These questions arise in designing drug delivery methods, for example [11]. In the event of no mass production, we are able to use this analytic solution to replicate the numerical results provided by others with a closed form solution. Further, this solution yields additional information not recovered by other methods. In addition to being able to track the front velocity, we recover the time dependency lost by asymptotic methods. Next, we explore the multiphase system with sources and sinks. Once more, the reduction transformation we develop provides closed form solutions for the multiphase problem. Assessing the characteristic curves for a smooth boundary, we find shocks and rarefactions arise with logistic growth. Shocks occur when multivalued solutions are present for certain boundary conditions, and rarefactions occur when solutions vanish on the boundary. In gel dynamics, this "disturbance'' occurs when there is a disruption from phases are attempting to occupy the same space. Finally, we develop a general reduction transformation for an arbitrary number of phases and dimensions. We show the effectiveness of this transformation by reducing two known biological examples. The first model has spatially one dimension with three phases and describes tumor encapsulation and transcapular spread. The second has two spatial dimensions with four phases and describes vascular tumor growth. In addition to the novel contribution of mapping the multiphase system in j spatial dimensions of n phases, there are many directions for this research. We have made progress in the formal analysis of these complicated, nonlinear PDEs. More interestingly, we have opened several directions that appear to be fruitful avenues for future study. During the analysis of the twophase model, we discovered many exact solutions without clear significance. We could explore the nature of these solutions and attempt to uncover biological relevance. When we added the growth function, we discovered shock and rarefaction waves for logistic growth. We could pursue alternative forms for growth. After designing a general reduction transformation, we show the reduction of two multiphase systems. We could derive solutions from either of the above examples by solving the reduced ODE systems. Also, now that we have a better understanding of the analytical solution structure of the multiphase system, it is possible to seek compatible forms for mass redistribution and pressure terms in the equations. Essentially, we can seek to derive mathematical forms for hydrostatic pressure, a phenomenon which cannot be captured empirically, by imposing specific source/sinks functions seen in numerical models.
Show less  Date Issued
 2016
 Identifier
 FSU_2016SP_Ekrut_fsu_0071E_13019
 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
 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
 Statistical Analysis on Object Spaces with Applications.
 Creator

Yao, Kouadio David, Patrangenaru, Victor, Kercheval, Alec N., Liu, Xiuwen, Mio, Washington, Wang, Xiaoming, Florida State University, College of Arts and Sciences, Department of...
Show moreYao, Kouadio David, Patrangenaru, Victor, Kercheval, Alec N., Liu, Xiuwen, Mio, Washington, Wang, Xiaoming, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

Most of the data encountered is bounded nonlinear data. The Universe is bounded, planets are sphere like shaped objects, and life growing on Earth comes in various shapes and colors that can hardly be represented as points on a linear space, and even if the object space they sit on is embedded in a Euclidean space, their mean vector can not be represented as a point on that object space, except for the case when such space is convex. To address this misgiving, since the mean vector is the...
Show moreMost of the data encountered is bounded nonlinear data. The Universe is bounded, planets are sphere like shaped objects, and life growing on Earth comes in various shapes and colors that can hardly be represented as points on a linear space, and even if the object space they sit on is embedded in a Euclidean space, their mean vector can not be represented as a point on that object space, except for the case when such space is convex. To address this misgiving, since the mean vector is the minimizer of the expected square distance, following Fr\'echet (1948), on a compact metric space, one may consider both minimizers and maximizers of the expected square distance to a given point on the object space as mean, respectively {\bf antimean} of a given random point. Of all distances on a object space, one considers here the chord distance associated with an embedding of the object space, since for such distances one can give a necessary and sufficient condition for the existence of a unique Fr\'echet mean (respectively Fr\'echet antimean). For such distributions these location parameters are called extrinsic mean (respectively extrinsic antimean), and the corresponding sample statistics are consistent estimators of their population counterparts. Moreover one derives the limit distribution of such estimators around a mean located at a smooth extrinsic antimean. Extrinsic analysis is thus a general framework that allows one to run object data analysis on nonlinear object spaces that can be embedded in a numerical space. In particular one focuses on VeroneseWhitney (VW) means and antimeans of 3D projective shapes of configurations extracted from digital camera images. The 3D data extraction is greatly simplified by an RGB based algorithm followed by the FaugerasHartleyGuptaChen 3D reconstruction method. In particular one derives two sample tests for face analysis based on projective shapes, and more generally a MANOVA on manifolds method to be used in 3D projective shape analysis. The manifold based approach is also applicable to financial data analysis for exchange rates.
Show less  Date Issued
 2016
 Identifier
 FSU_FA2016_Yao_fsu_0071E_13605
 Format
 Thesis
 Title
 Modeling and Simulating Vortex Pinning and Transport Currents for High Temperature Superconductors.
 Creator

Sockwell, K. Chadwick (Kenneth Chadwick), Gunzburger, Max D., Peterson, Janet S., Burkardt, John V., Florida State University, College of Arts and Sciences, Department of...
Show moreSockwell, K. Chadwick (Kenneth Chadwick), Gunzburger, Max D., Peterson, Janet S., Burkardt, John V., Florida State University, College of Arts and Sciences, Department of Scientific Computing
Show less  Abstract/Description

Superconductivity is a phenomenon characterized by two hallmark properties, zero electrical resistance and the Meissner effect. These properties give great promise to a new generation of resistance free electronics and powerful superconducting magnets. However this possibility is limited by the extremely low critical temperature the superconductors must operate under, typically close to 0K. The recent discovery of high temperature superconductors has brought the critical temperature closer to...
Show moreSuperconductivity is a phenomenon characterized by two hallmark properties, zero electrical resistance and the Meissner effect. These properties give great promise to a new generation of resistance free electronics and powerful superconducting magnets. However this possibility is limited by the extremely low critical temperature the superconductors must operate under, typically close to 0K. The recent discovery of high temperature superconductors has brought the critical temperature closer to room temperature than ever before, making the realization of room temperature superconductivity a possibility. Simulations of superconducting technology and materials will be necessary to usher in the new wave of superconducting electronics. Unfortunately these new materials come with new properties such as effects from multiple electron bands, as is the case for magnesium diboride. Moreover, we must consider that all high temperature superconductors are of a Type II variety, which possess magnetic tubes of flux, known as vortices. These vortices interact with transport currents, creating an electrical resistance through a process known as flux flow. Thankfully this process can be prevented by placing impurities in the superconductor, pinning the vortices, making vortex pinning a necessary aspect of our model. At this time there are no other models or simulations that are aimed at modeling vortex pinning, using impurities, in twoband materials. In this work we modify an existing GinzburgLandau model for twoband superconductors and add the ability to model normal inclusions (impurities) with a new approach which is unique to the twoband model. Simulations in an attempt to model the material magnesium diboride are also presented. In particular simulations of vortex pinning and transport currents are shown using the modified model. The qualitative properties of magnesium diboride are used to validate the model and its simulations. One main goal from the computational end of the simulations is to enlarge the domain size to produce more realistic simulations that avoid boundary pinning effects. In this work we also implement the numerical software library Trilinos in order to parallelize the simulation to enlarge the domain size. Decoupling methods are also investigated with a goal of enlarging the domain size as well. The OneBand GinzburgLandau model serves as a prototypical problem in this endeavor and the methods shown that enlarge the domain size can be easily implemented in the twoband model.
Show less  Date Issued
 2016
 Identifier
 FSU_FA2016_Sockwell_fsu_0071N_13577
 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
 The Effects of Representation Format in Problem Representation on Qualitative Understanding and Quantitative Proficiency in a Learning Game Context.
 Creator

Lee, Sungwoong, Ke, Fengfeng, Erlebacher, Gordon, Shute, Valerie J. (Valerie Jean), Dennen, Vanessa P., Florida State University, College of Education, Department of Educational...
Show moreLee, Sungwoong, Ke, Fengfeng, Erlebacher, Gordon, Shute, Valerie J. (Valerie Jean), Dennen, Vanessa P., Florida State University, College of Education, Department of Educational Psychology and Learning Systems
Show less  Abstract/Description

Reports and surveys by the U.S. government and international organizations have repeatedly acknowledged the achievement problem in math in K12 regardless of various efforts (e.g., by the U.S. Department of Education) to diminish it. To address the problem in math achievement in K12, teachers, scholars, and the U.S. government have developed various materials and intervention tools. As a potential platform to address the problem in math achievement, video games generate a large variety of...
Show moreReports and surveys by the U.S. government and international organizations have repeatedly acknowledged the achievement problem in math in K12 regardless of various efforts (e.g., by the U.S. Department of Education) to diminish it. To address the problem in math achievement in K12, teachers, scholars, and the U.S. government have developed various materials and intervention tools. As a potential platform to address the problem in math achievement, video games generate a large variety of perspectives on their value. Along with the debate on the game's inherent good or bad features, there is also a debate on the effectiveness of video games as a learning tool. Regarding these debates and the ambiguous results on video games as learning tools, Greitemeyer and Mügge (2014) postulated that games can provide both positive and negative impacts according to their content (i.e., violent and prosocial games). However, recent literature investigating the use of video games in varied learning contexts shows that the learning effectiveness of games is still inconclusive. A potential reason is that video games mostly facilitate implicit qualitative understanding. Video games consist of rich interactive experiences that help to foster understanding of qualitative relationships in gameplay more than quantitative proficiency that is required in the formal school system (Clark et al. 2011; Squire, Barnett, Grant, & Higginbotham, 2004). Another reason is that educational game designers have paid little attention to designing and developing learning supports in educational games. Therefore, the current study aims to address a comprehensive question  How does an educational game, through the use of learning supports, promote the application of acquired qualitative understanding to math problem solving in formal educational contexts? A promising method to address the aforementioned problem is to externalize cognitive and metacognitive processes (Lajoie, 2009). Externalizing Problem Representation (EPR) refers to a cognitive behavior in which a learner constructs her own representations overtly (Cox, 1999). The processes of EPR are to reorder information in problem solving, to clarify ambiguous parts of the problem, and to modify and enact mental representations including mental animations and images. EPR helps to make missing and implicit information or representations explicit. There are several synonyms of Externalizing Problem Representation (EPR), such as external representation (Zhang, 1997), externalized cognition (Cox & Brna, 1995), and rerepresentation (Ainsworth & Th Loizou, 2003). From the semiotics perspective, EPR can be categorized into two forms by its sign: Iconic and symbolic. Although the potential benefits of externalizing problem representation was claimed in prior research, little attention was paid to investigating the design of EPR in video games. Compared to the studies of mental problem representation, few empirical studies on external representation have been conducted. Hence, it is warranted to examine the efficacy of learning support that promotes externalizing problem representation in two formats (i.e., iconic and symbolic) in the videogamebased learning setting. In light of this, the purpose of this study is to investigate whether EPRpromoting scaffolds (in iconic vs. symbolic formats) enhance qualitative understanding and quantitative proficiency in ratios and proportional relationships in a learning game context. Specifically, the learning game will request players to respond to either iconic or symbolic learning probes that help to externalize the mental representations of the math problems in the game. In this study, quantitative proficiency refers to the problem solving proficiency in both game and formal education context. The current study involves two levels of task complexity (i.e., low complexity vs. high complexity) as a moderating variable. The study addresses the following research questions: 1. Will iconic learning probes promoting EPR enhance qualitative understanding and quantitative proficiency in ratios and proportional reasoning, with the task complexity controlled in the educational game? 2. Will symbolic learning probes promoting EPR enhance qualitative understanding and quantitative proficiency in ratios and proportional reasoning, with task complexity controlled in the educational game? 3. Will iconic learning probes promoting EPR, in comparison to symbolic learning probes promoting EPR, be more effective in enhancing qualitative understanding and quantitative proficiency in ratio and proportional reasoning, with task complexity controlled in the educational game? To accomplish the purpose of this study, learning probes that prompt learners to externalize their internal problem representation were developed in two different formats, iconic and symbolic, based on Mayer's math problem representation model. In the experiment, fortyfive participants in this study processed either iconic or symbolic learning probes during their gameplay. Finally, qualitative understanding and quantitative proficiency were measured three times: before this study, after playing the shipping container episode with a low complexity task, and after playing the shipping container episode with a high complexity task. Regarding Research Question 1, the result of repeatedmeasures ANOVA indicates that, for participants in the Iconic Learning Probe (ILP) group, the difference in qualitative understanding between the pretest, posttest, and posttest 2 was not statistically significant whereas the difference in quantitative proficiency between the pretest, posttest 1, and posttest 2 was statistically significant. Regarding Research Question 2, the result of repeatedmeasures ANOVA indicates that, for participants in the Symbolic Learning Probe (SLP) group, the difference in qualitative understanding between the pretest, posttest 1, and posttest 2 was statistically significant whereas the difference in quantitative understanding between the pretest, posttest 1, and posttest 2 was not statistically significant. Regarding Research Question 3, since there was a significant interaction between the times of measurement and the types of EPR in regard to both qualitative understanding and quantitative proficiency, pairwise comparisons using the Bonferroni method were drawn. There were significant differences in participants' qualitative understanding between ILP and SLP groups in posttest 1 and posttest 2 whereas there was no significant difference in participants' qualitative understanding between ILP and SLP groups in the pretest. Regarding the quantitative proficiency, there were significant differences in participants' quantitative proficiency between ILP and SLP groups in posttest 1 whereas there was no significant difference in participants' quantitative proficiency between ILP and SLP groups in the pretest and posttest 2. In the final chapter, I discussed major research findings of this study based on the theoretical research reviewed in Chapter 2. Then I described the implications of this study and suggestions for future study.
Show less  Date Issued
 2016
 Identifier
 FSU_2016SP_Lee_fsu_0071E_12944
 Format
 Thesis
 Title
 Quantifying Phenotypic Variation Through Local Persistent Homology and Imaging.
 Creator

Li, Mao, Mio, Washington, Aggarwal, Sudhir, Bertram, R. (Richard), MestertonGibbons, Mike, Florida State University, College of Arts and Sciences, Department of Mathematics
 Abstract/Description

Understanding the genetic basis of phenotypic variation in organisms is a central problem in developmental and evolutionary biology. In plant science, to gain insights on such problems as how plants respond to environmental changes and how to breed the next generation of crops, a sound quantification of the variation in complex plant phenotypes is crucial. For example, the shape of leaves, the architecture of root systems, and the morphology of pollen grains are all important and interesting...
Show moreUnderstanding the genetic basis of phenotypic variation in organisms is a central problem in developmental and evolutionary biology. In plant science, to gain insights on such problems as how plants respond to environmental changes and how to breed the next generation of crops, a sound quantification of the variation in complex plant phenotypes is crucial. For example, the shape of leaves, the architecture of root systems, and the morphology of pollen grains are all important and interesting phenotypic traits that require mathematical informed methods to model their variation comprehensively. In this dissertation, we develop topological methods and algorithms based on persistent homology, which let us construct informative summaries of the shape of data. We propose a localized form of persistent homology represented by a continuous persistence diagram field. We prove that such fields are stable and robust to noise and outliers. This technique lets us produce compact, and yet rich summaries of global and local morphology useful for modeling and quantifying variation in complex shapes. This enables statistical approaches such as quantitative trait loci (QTL) analysis, time series analysis of dynamical traits, and the investigation of correlations between morphological traits to study their evolution and developmental constraints. We apply the methods to: (i) QTL analysis of multiple tomato introgression lines through a study of leaf shape and root architecture; (ii) time series analysis of dynamic growing maize root systems; (iii) quantitative analysis of morphology of grass pollen grains; and (iv) an analysis of the complexity of dryland spatial vegetation patterns.
Show less  Date Issued
 2016
 Identifier
 FSU_2016SP_Li_fsu_0071E_13155
 Format
 Thesis
 Title
 Reduced Order Modeling for a Nonlocal Approach to Anomalous Diffusion Problems.
 Creator

Witman, David, Gunzburger, Max D., Peterson, Janet C., Stagg, Scott, Shanbhag, Sachin, Burkardt, John V., Florida State University, College of Arts and Sciences, Department of...
Show moreWitman, David, Gunzburger, Max D., Peterson, Janet C., Stagg, Scott, Shanbhag, Sachin, Burkardt, John V., Florida State University, College of Arts and Sciences, Department of Scientific Computing
Show less  Abstract/Description

With the recent advances in using nonlocal approaches to approximate traditional partial differential equations(PDEs), a number of new research avenues have been opened that warrant further study. One such path, that has yet to be explored, is using reduced order techniques to solve nonlocal problems. Due to the interactions between the discretized nodes or particles inherent to a nonlocal model, the system sparsity is often significantly less than its PDE counterpart. Coupling a reduced...
Show moreWith the recent advances in using nonlocal approaches to approximate traditional partial differential equations(PDEs), a number of new research avenues have been opened that warrant further study. One such path, that has yet to be explored, is using reduced order techniques to solve nonlocal problems. Due to the interactions between the discretized nodes or particles inherent to a nonlocal model, the system sparsity is often significantly less than its PDE counterpart. Coupling a reduced order approach to a nonlocal problem would ideally reduce the computational cost without sacrificing accuracy. This would allow for the use of a nonlocal approach in large parameter studies or uncertainty quantification. Additionally, because nonlocal problems inherently have no spatial derivatives, solutions with jump discontinuities are permitted. This work seeks to apply reduced order nonlocal concepts to a variety of problem situations including anomalous diffusion, advection, the advectiondiffusion equation and solutions with spatial discontinuities. The goal is to show that one can use an accurate reduced order approximation to formulate a solution at a fraction of the cost of traditional techniques.
Show less  Date Issued
 2016
 Identifier
 FSU_2016SP_Witman_fsu_0071E_13130
 Format
 Thesis