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Pages
 Title
 A Spectral Element Method to Price Single and MultiAsset European Options.
 Creator

Zhu, Wuming, Kopriva, David A., Huﬀer, Fred, Case, Bettye Anne, Kercheval, Alec N., Okten, Giray, Wang, Xiaoming, Department of Mathematics, Florida State University
 Abstract/Description

We develop a spectral element method to price European options under the BlackScholes model, Merton's jump diffusion model, and Heston's stochastic volatility model with one or two assets. The method uses piecewise high order Legendre polynomial expansions to approximate the option price represented pointwise on a GaussLobatto mesh within each element. This piecewise polynomial approximation allows an exact representation of the nonsmooth initial condition. For options with one asset under...
Show moreWe develop a spectral element method to price European options under the BlackScholes model, Merton's jump diffusion model, and Heston's stochastic volatility model with one or two assets. The method uses piecewise high order Legendre polynomial expansions to approximate the option price represented pointwise on a GaussLobatto mesh within each element. This piecewise polynomial approximation allows an exact representation of the nonsmooth initial condition. For options with one asset under the jump diffusion model, the convolution integral is approximated by high order GaussLobatto quadratures. A second order implicit/explicit (IMEX) approximation is used to integrate in time, with the convolution integral integrated explicitly. The use of the IMEX approximation in time means that only a block diagonal, rather than full, system of equations needs to be solved at each time step. For options with two variables, i.e., two assets under the BlackScholes model or one asset under the stochastic volatility model, the domain is subdivided into quadrilateral elements. Within each element, the expansion basis functions are chosen to be tensor products of the Legendre polynomials. Three iterative methods are investigated to solve the system of equations at each time step with the corresponding second order time integration schemes, i.e., IMEX and CrankNicholson. Also, the boundary conditions are carefully studied for the stochastic volatility model. The method is spectrally accurate (exponentially convergent) in space and second order accurate in time for European options under all the three models. Spectral accuracy is observed in not only the solution, but also in the Greeks.
Show less  Date Issued
 2008
 Identifier
 FSU_migr_etd0513
 Format
 Thesis
 Title
 Modeling the Folding Pattern of the Cerebral Cortex.
 Creator

Striegel, Deborah A., Hurdal, Monica K., Steinbock, Oliver, Quine, Jack, Sumners, DeWitt, Bertram, Richard, Department of Mathematics, Florida State University
 Abstract/Description

The mechanism for cortical folding pattern formation is not fully understood. Current models represent scenarios that describe pattern formation through local interactions and one recent model is the intermediate progenitor model. The intermediate progenitor (IP) model describes a local chemicallydriven scenario, where an increase in intermediate progenitor cells in the subventricular zone (an area surrounding the lateral ventricles) correlates to gyral formation. This dissertation presents...
Show moreThe mechanism for cortical folding pattern formation is not fully understood. Current models represent scenarios that describe pattern formation through local interactions and one recent model is the intermediate progenitor model. The intermediate progenitor (IP) model describes a local chemicallydriven scenario, where an increase in intermediate progenitor cells in the subventricular zone (an area surrounding the lateral ventricles) correlates to gyral formation. This dissertation presents the Global Intermediate Progenitor (GIP) model, a theoretical biological model that uses features of the IP model and further captures global characteristics of cortical pattern formation. To illustrate how global features can effect the development of certain patterns, a mathematical model that incorporates a Turing system is used to examine pattern formation on a prolate spheroidal surface. Pattern formation in a biological system can be studied with a Turing reactiondiffusion system which utilizes characteristics of domain size and shape to predict which pattern will form. The GIP model approximates the shape of the lateral ventricle with a prolate spheroid. This representation allows the capture of a key shape feature, lateral ventricular eccentricity, in terms of the focal distance of the prolate spheroid. A formula relating domain scale and focal distance of a prolate spheroidal surface to specific prolate spheroidal harmonics is developed. This formula allows the prediction of pattern formation with solutions in the form of prolate spheroidal harmonics based on the size and shape of the prolate spheroidal surface. By utilizing this formula a direct correlation between the size and shape of the lateral ventricle, which drives the shape of the ventricular zone, and cerebral cortical folding pattern formation is found. This correlation is illustrated in two different applications: (i) how the location and directionality of the initial cortical folds change with respect to evolutionary development and (ii) how the initial folds change with respect to certain diseases, such as Microcephalia Vera and Megalencephaly Polymicrogyria Polydactyly with Hydrocephalus. The significance of the model, presented in this dissertation, is that it elucidates the consistency of cortical patterns among healthy individuals within a species and addresses interspecies variability based on global characteristics. This model provides a critical piece to the puzzle of cortical pattern formation.
Show less  Date Issued
 2009
 Identifier
 FSU_migr_etd0394
 Format
 Thesis
 Title
 QuasiMonte Carlo and Genetic Algorithms with Applications to Endogenous Mortgage Rate Computation.
 Creator

Shah, Manan, Okten, Giray, Goncharov, Yevgeny, Srinivasan, Ashok, Bellenot, Steve, Case, Bettye Anne, Kercheval, Alec, Kopriva, David, Nichols, Warren, Department of Mathematics...
Show moreShah, Manan, Okten, Giray, Goncharov, Yevgeny, Srinivasan, Ashok, Bellenot, Steve, Case, Bettye Anne, Kercheval, Alec, Kopriva, David, Nichols, Warren, Department of Mathematics, Florida State University
Show less  Abstract/Description

In this dissertation, we introduce a genetic algorithm approach to estimate the star discrepancy of a point set. This algorithm allows for the estimation of the star discrepancy in dimensions larger than seven, something that could not be done adequately by other existing methods. Then, we introduce a class of random digitpermutations for the Halton sequence and show that these permutations yield comparable or better results than their deterministic counterparts in any number of dimensions...
Show moreIn this dissertation, we introduce a genetic algorithm approach to estimate the star discrepancy of a point set. This algorithm allows for the estimation of the star discrepancy in dimensions larger than seven, something that could not be done adequately by other existing methods. Then, we introduce a class of random digitpermutations for the Halton sequence and show that these permutations yield comparable or better results than their deterministic counterparts in any number of dimensions for the test problems considered. Next, we use randomized quasiMonte Carlo methods to numerically solve a onefactor mortgage model expressed as a stochastic fixedpoint problem. Finally, we show that this mortgage model coincides with and is computationally faster than Citigroup's MOATS model, which is based on a binomial tree approach.
Show less  Date Issued
 2008
 Identifier
 FSU_migr_etd0297
 Format
 Thesis
 Title
 Variance Gamma Pricing of American Futures Options.
 Creator

Yoo, Eunjoo, Nolder, Craig A., Huﬀer, Fred, Case, Bettye Anne, Kercheval, Alec N., Quine, Jack, Department of Mathematics, Florida State University
 Abstract/Description

In financial markets under uncertainty, the classical BlackScholes model cannot explain the empirical facts such as fat tails observed in the probability density. To overcome this drawback, during the last decade, Lévy process and stochastic volatility models were introduced to financial modeling. Today crude oil futures markets are highly volatile. It is the purpose of this dissertation to develop a mathematical framework in which American options on crude oil futures contracts are priced...
Show moreIn financial markets under uncertainty, the classical BlackScholes model cannot explain the empirical facts such as fat tails observed in the probability density. To overcome this drawback, during the last decade, Lévy process and stochastic volatility models were introduced to financial modeling. Today crude oil futures markets are highly volatile. It is the purpose of this dissertation to develop a mathematical framework in which American options on crude oil futures contracts are priced more effectively than by current methods. In this work, we use the Variance Gamma process to model the futures price process. To generate the underlying process, we use a random tress method so that we evaluate the option prices at each tree node. Through fifty replications of a random tree, the averaged value is taken as a true option price. Pricing performance using this method is accessed using American options on crude oil commodity contracts from December 2003 to November 2004. In comparison with the Variance Gamma model, we price using the BlackScholes model as well. Over the entire sample period, a positive skewness and high kurtosis, especially in the shortterm options, are observed. In terms of pricing errors, the Variance Gamma process performs better than the BlackScholes model for the American options on crude oil commodities.
Show less  Date Issued
 2008
 Identifier
 FSU_migr_etd0691
 Format
 Thesis
 Title
 A Comparison Study of Principal Component Analysis and Nonlinear Principal Component Analysis.
 Creator

Wu, Rui, Magnan, Jerry F., Bellenot, Steven, Sussman, Mark, Department of Mathematics, Florida State University
 Abstract/Description

In the field of data analysis, it is important to reduce the dimensionality of data, because it will help to understand the data, extract new knowledge from the data, and decrease the computational cost. Principal Component Analysis (PCA) [1, 7, 19] has been applied in various areas as a method of dimensionality reduction. Nonlinear Principal Component Analysis (NLPCA) [1, 7, 19] was originally introduced as a nonlinear generalization of PCA. Both of the methods were tested on various...
Show moreIn the field of data analysis, it is important to reduce the dimensionality of data, because it will help to understand the data, extract new knowledge from the data, and decrease the computational cost. Principal Component Analysis (PCA) [1, 7, 19] has been applied in various areas as a method of dimensionality reduction. Nonlinear Principal Component Analysis (NLPCA) [1, 7, 19] was originally introduced as a nonlinear generalization of PCA. Both of the methods were tested on various artificial and natural datasets sampled from: "F(x) = sin(x) + x", the Lorenz Attractor, and sunspot data. The results from the experiments have been analyzed and compared. Generally speaking, NLPCA can explain more variance than a neural network PCA (NN PCA) in lower dimensions. However, as a result of increasing the dimension, the NLPCA approximation will eventually loss its advantage. Finally, we introduce a new combination of NN PCA and NLPCA, and analyze and compare its performance.
Show less  Date Issued
 2007
 Identifier
 FSU_migr_etd0704
 Format
 Thesis
 Title
 Numerical Methods for Portfolio Risk Estimation.
 Creator

Zhang, Jianke, Kercheval, Alec, Huﬀer, Fred, Gallivan, Kyle, Beaumont, Paul, Nichols, Warren, Department of Mathematics, Florida State University
 Abstract/Description

In portfolio risk management, a global covariance matrix forecast often needs to be adjusted by changing diagonal blocks corresponding to specific submarkets. Unless certain constraints are obeyed, this can result in the loss of positive definiteness of the global matrix. Imposing the proper constraints while minimizing the disturbance of offdiagonal blocks leads to a nonconvex optimization problem in numerical linear algebra called the Weighted Orthogonal Procrustes Problem. We analyze...
Show moreIn portfolio risk management, a global covariance matrix forecast often needs to be adjusted by changing diagonal blocks corresponding to specific submarkets. Unless certain constraints are obeyed, this can result in the loss of positive definiteness of the global matrix. Imposing the proper constraints while minimizing the disturbance of offdiagonal blocks leads to a nonconvex optimization problem in numerical linear algebra called the Weighted Orthogonal Procrustes Problem. We analyze and compare two local minimizing algorithms and offer an algorithm for global minimization. Our methods are faster and more effective than current numerical methods for covariance matrix revision.
Show less  Date Issued
 2007
 Identifier
 FSU_migr_etd0542
 Format
 Thesis
 Title
 An Analysis of Conjugate Harmonic Components of Monogenic Functions and Lambda Harmonic Functions.
 Creator

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

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

Achuthan, Srisairam, Quine, John R., Cross, Timothy A., Sumners, DeWitt, Bertram, Richard, Department of Mathematics, Florida State University
 Abstract/Description

Of late, pathbreaking advances are taking place and flourishing in the field of solidstate Nuclear Magnetic Resonance (ssNMR)spectroscopy. One of the major applications of ssNMR techniques is to high resolution threedimensional structures of biological molecules like the membrane proteins. An explicit example of this is PISEMA (Polarization Inversion Spin Exchange at Magic Angle). This dissertation studies and analyzes the use of the orientational restraints in general, and particularly...
Show moreOf late, pathbreaking advances are taking place and flourishing in the field of solidstate Nuclear Magnetic Resonance (ssNMR)spectroscopy. One of the major applications of ssNMR techniques is to high resolution threedimensional structures of biological molecules like the membrane proteins. An explicit example of this is PISEMA (Polarization Inversion Spin Exchange at Magic Angle). This dissertation studies and analyzes the use of the orientational restraints in general, and particularly the restraints measured through PISEMA. Here, we have applied our understanding of orientational restraints to briefly investigate the structure of Amantadine bound M2TMD, a membrane protein in Influenza A Virus. We model the protein backbone structure as a discrete curve in space with atoms represented by vertices and covalent bonds connecting them as the edges. The oriented structure of this curve with respect to an external vector is emphasized. The map from the surface of the unit sphere to the PISEMA frequency plane is examined in detail. The image is a powder pattern in the frequency plane. A discussion of the resulting image is provided. Solutions to PISEMA equations lead to multiple orientations for the magnetic field vector for a given point in the frequency plane. These are duly captured by sign degeneracies for the vector coordinates. The intensity of NMR powder patterns is formulated in terms of a probability density function for 1d spectra and a joint probability density function for the 2d spectra. The intensity analysis for 2d spectra is found to be rather helpful in addressing the robustness of the PISEMA data. To build protein structures by gluing together diplanes, certain necessary conditions have to be met. We formulate these as continuity conditions to be realized for diplanes. The number of oriented protein structures has been enumerated in the degeneracy framework for diplanes. Torsion angles are expressed via sign degeneracies. For aligned protein samples, the PISA wheel approach to modeling the protein structure is adopted. Finally, an atomic model of the monomer structure of M2TMD with Amantadine has been elucidated based on PISEMA orientational restraints. This is a joint work with Jun Hu and Tom Asbury. The PISEMA data was collected by Jun Hu and the molecular modeling was performed by Tom Asbury.
Show less  Date Issued
 2006
 Identifier
 FSU_migr_etd0109
 Format
 Thesis
 Title
 Deterministic and Stochastic Aspects of Data Assimilation.
 Creator

Akella, Santharam, Navon, Ionel Michael, O'Brien, James J., Erlebacher, Gordon, Wang, Qi, Sussman, Mark, Department of Mathematics, Florida State University
 Abstract/Description

The principles of optimal control of distributed parameter systems are used to derive a powerful class of numerical methods for solutions of inverse problems, called data assimilation (DA) methods. Using these DA methods one can efficiently estimate the state of a system and its evolution. This information is very crucial for achieving more accurate long term forecasts of complex systems, for instance, the atmosphere. DA methods achieve their goal of optimal estimation via combination of all...
Show moreThe principles of optimal control of distributed parameter systems are used to derive a powerful class of numerical methods for solutions of inverse problems, called data assimilation (DA) methods. Using these DA methods one can efficiently estimate the state of a system and its evolution. This information is very crucial for achieving more accurate long term forecasts of complex systems, for instance, the atmosphere. DA methods achieve their goal of optimal estimation via combination of all available information in the form of measurements of the state of the system and a dynamical model which describes the evolution of the system. In this dissertation work, we study the impact of new nonlinear numerical models on DA. High resolution advection schemes have been developed and studied to model propagation of flows involving sharp fronts and shocks. The impact of high resolution advection schemes in the framework of inverse problem solution/ DA has been studied only in the context of linear models. A detailed study of the impact of various slope limiters and the piecewise parabolic method (PPM) on DA is the subject of this work. In 1D we use a nonlinear viscous Burgers equation and in 2D a global nonlinear shallow water model has been used. The results obtained show that using the various advection schemes consistently improves variational data assimilation (VDA) in the strong constraint form, which does not include model error. However, the cost functional included efficient and physically meaningful construction of the background cost functional term, J_b, using balance and diffusion equation based correlation operators. This was then followed by an indepth study of various approaches to model the systematic component of model error in the framework of a weak constraint VDA. Three simple forms, decreasing, invariant, and exponentially increasing in time forms of evolution of model error were tested. The inclusion of model error provides a substantial reduction in forecasting errors, in particular the exponentially increasing form in conjunction with the piecewise parabolic high resolution advection scheme was found to provide the best results. Results obtained in this work can be used to formulate sophisticated forms of model errors, and could lead to implementation of new VDA methods using numerical weather prediction models which involve high resolution advection schemes such as the van Leer slope limiters and the PPM.
Show less  Date Issued
 2006
 Identifier
 FSU_migr_etd0145
 Format
 Thesis
 Title
 Discontinuous Galerkin Spectral Element Approximations on Moving Meshes for Wave Scattering from Reflective Moving Boundaries.
 Creator

AcostaMinoli, Cesar Augusto, Kopriva, David, Srivastava, Anuj, Hussaini, M. Yousuﬀ, Sussman, Mark, Ewald, Brian, Department of Mathematics, Florida State University
 Abstract/Description

This dissertation develops and evaluates a high order method to compute wave scattering from moving boundaries. Specifically, we derive and evaluate a Discontinuous Galerkin Spectral elements method (DGSEM) with Arbitrary Lagrangian Eulerian (ALE) mapping to compute conservation laws on moving meshes and numerical boundary conditions for Maxwell's equations, the linear Euler equations and the nonlinear Euler gasdynamics equations to calculate the numerical flux on reflective moving...
Show moreThis dissertation develops and evaluates a high order method to compute wave scattering from moving boundaries. Specifically, we derive and evaluate a Discontinuous Galerkin Spectral elements method (DGSEM) with Arbitrary Lagrangian Eulerian (ALE) mapping to compute conservation laws on moving meshes and numerical boundary conditions for Maxwell's equations, the linear Euler equations and the nonlinear Euler gasdynamics equations to calculate the numerical flux on reflective moving boundaries. We use one of a family of explicit time integrators such as AdamsBashforth or low storage explicit RungeKutta. The approximations preserve the discrete metric identities and the Discrete Geometric Conservation Law (DGCL) by construction. We present timestep refinement studies with moving meshes to validate the moving mesh approximations. The test problems include propagation of an electromagnetic gaussian plane wave, a cylindrical pressure wave propagating in a subsonic flow, and a vortex convecting in a uniform inviscid subsonic flow. Each problem is computed on a timedeforming mesh with three methods used to calculate the mesh velocities: From exact differentiation, from the integration of an acceleration equation, and from numerical differentiation of the mesh position. In addition, we also present four numerical examples using Maxwell's equations, one example using the linear Euler equations and one more example using nonlinear Euler equations to validate these approximations. These are: reflection of light from a constantly moving mirror, reflection of light from a constantly moving cylinder, reflection of light from a vibrating mirror, reflection of sound in linear acoustics and dipole sound generation by an oscillating cylinder in an inviscid flow.
Show less  Date Issued
 2011
 Identifier
 FSU_migr_etd0111
 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
 Developing SRSF Shape Analysis Techniques for Applications in Neuroscience and Genomics.
 Creator

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

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

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

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

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

In our work we analyze and implement numerical schemes for the infinite Prandtl number model for convection. This model describes the convection that is a potential driving force behind the flow and temperature of the Earth's mantle. There are many schemes available, but most are given with no mention of their ability to adequately capture the long time statistical properties of the model. We investigate schemes with the potential to actually capture these statistics. We further show...
Show moreIn our work we analyze and implement numerical schemes for the infinite Prandtl number model for convection. This model describes the convection that is a potential driving force behind the flow and temperature of the Earth's mantle. There are many schemes available, but most are given with no mention of their ability to adequately capture the long time statistical properties of the model. We investigate schemes with the potential to actually capture these statistics. We further show numerically that our schemes align with current knowledge of the model's characteristics at low Rayleigh numbers.
Show less  Date Issued
 2015
 Identifier
 FSU_2015fall_Woodruff_fsu_0071E_12813
 Format
 Thesis
 Title
 A Mathematical Model of Cerebral Cortical Folding Development Based on a Biomechanical Hypothesis.
 Creator

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

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

Willyard, Matthew, Kopriva, David, Eugenio, Paul, Case, Bettye Anne, Gallivan, Kyle, Nolder, Craig, Okten, Giray, Department of Mathematics, Florida State University
 Abstract/Description

We develop an adaptive spectral element method to price American options, whose solutions contain a moving singularity, automatically and to within prescribed errors. The adaptive algorithm uses an error estimator to determine where refinement or derefinement is needed and a work estimator to decide whether to change the element size or the polynomial order. We derive two local error estimators and a global error estimator. The local error estimators are derived from the Legendre...
Show moreWe develop an adaptive spectral element method to price American options, whose solutions contain a moving singularity, automatically and to within prescribed errors. The adaptive algorithm uses an error estimator to determine where refinement or derefinement is needed and a work estimator to decide whether to change the element size or the polynomial order. We derive two local error estimators and a global error estimator. The local error estimators are derived from the Legendre coefficients and the global error estimator is based on the adjoint problem. One local error estimator uses the rate of decay of the Legendre coefficients to estimate the error. The other local error estimator compares the solution to an estimated solution using fewer Legendre coefficients found by the Tau method. The global error estimator solves the adjoint problem to weight local error estimates to approximate a terminal error functional. Both types of error estimators produce meshes that match expectations by being fine near the early exercise boundary and strike price and coarse elsewhere. The produced meshes also adapt as expected by derefining near the strike price as the solution smooths and staying fine near the moving early exercise boundary. Both types of error estimators also give solutions whose error is within prescribed tolerances. The adjointbased error estimator is more flexible, but costs up to three times as much as using the local error estimate alone. The global error estimator has the advantages of tracking the accumulation of error in time and being able to discount large local errors that do not affect the chosen terminal error functional. The local error estimator is cheaper to compute because the global error estimator has the added cost of solving the adjoint problem.
Show less  Date Issued
 2011
 Identifier
 FSU_migr_etd0892
 Format
 Thesis
 Title
 Combinatorial Type Problems for Triangulation Graphs.
 Creator

Wood, William E., Bowers, Philip, Hawkes, Lois, Bellenot, Steve, Klassen, Eric, Nolder, Craig, Quine, Jack, Department of Mathematics, Florida State University
 Abstract/Description

The main result in this thesis bounds the combinatorial modulus of a ring in a triangulation graph in terms of the modulus of a related ring. The bounds depend only on how the rings are related and not on the rings themselves. This may be used to solve the combinatorial type problem in a variety of situation, most significant in graphs with unbounded degree. Other results regarding the type problem are presented along with several examples illustrating the limits of the results.
 Date Issued
 2006
 Identifier
 FSU_migr_etd0794
 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 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
 Riemannian Optimization Methods for Averaging Symmetric Positive Definite Matrices.
 Creator

Yuan, Xinru, Gallivan, Kyle A., Absil, PierreAntoine, Erlebacher, Gordon, Ökten, Giray, Bauer, Martin, Florida State University, College of Arts and Sciences, Department of...
Show moreYuan, Xinru, Gallivan, Kyle A., Absil, PierreAntoine, Erlebacher, Gordon, Ökten, Giray, Bauer, Martin, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

Symmetric positive definite (SPD) matrices have become fundamental computational objects in many areas. It is often of interest to average a collection of symmetric positive definite matrices. This dissertation investigates different averaging techniques for symmetric positive definite matrices. We use recent developments in Riemannian optimization to develop efficient and robust algorithms to handle this computational task. We provide methods to produce efficient numerical representations of...
Show moreSymmetric positive definite (SPD) matrices have become fundamental computational objects in many areas. It is often of interest to average a collection of symmetric positive definite matrices. This dissertation investigates different averaging techniques for symmetric positive definite matrices. We use recent developments in Riemannian optimization to develop efficient and robust algorithms to handle this computational task. We provide methods to produce efficient numerical representations of geometric objects that are required for Riemannian optimization methods on the manifold of symmetric positive definite matrices. In addition, we offer theoretical and empirical suggestions on how to choose between various methods and parameters. In the end, we evaluate the performance of different averaging techniques in applications.
Show less  Date Issued
 2018
 Identifier
 2018_Su_Yuan_fsu_0071E_14736
 Format
 Thesis
 Title
 Neural Rule Ensembles: Encoding Feature Interactions into Neural Networks.
 Creator

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

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

Dunn, Lawrence, Department of Mathematics
 Abstract/Description

Homotopy type theory, the basis of ''univalent foundations'' of mathematics, is a formal system with intrinsic connections to computer science, homotopy theory, and higher category theory. Rooted in type theory, the theoretical basis of most modern proof assistants, the system admits an interpretation as a logical calculus for homotopy theory and suggests a foundational system for which abstract ''spaces''  not unstructured sets  are the most primitive objects. This perspective offers...
Show moreHomotopy type theory, the basis of ''univalent foundations'' of mathematics, is a formal system with intrinsic connections to computer science, homotopy theory, and higher category theory. Rooted in type theory, the theoretical basis of most modern proof assistants, the system admits an interpretation as a logical calculus for homotopy theory and suggests a foundational system for which abstract ''spaces''  not unstructured sets  are the most primitive objects. This perspective offers both a computational foundational for mathematics and a direct method for reasoning about homotopy theory. We present here a broad contextual overview of homotopy type theory, including a sufficiently thorough examination of the classical foundations which it replaces as to make clear the extent of its innovation. We will explain that homotopy type theory is, loosely speaking and among other things, a programming language for mathematics, especially one with native support for homotopy theory.
Show less  Date Issued
 2014
 Identifier
 FSU_migr_uhm0304
 Format
 Thesis
 Title
 Dirichlet's Theorem and Analytic Number Theory.
 Creator

Frey, Thomas W., Department of Mathematics
 Abstract/Description

In 1837 Dirichlet proved the infinitude of primes in all arithmetic coprime sequences. This was done by look at Dirichlet Lfunctions, Dirichlet series, Dirichlet characters (modulo k), and Euler Products. In this thesis, the necessary facts, theorems, and properties are shown in order to prove Dirichlet's Theorem, concluding with a proof of Dirichlet's Theorem.
 Date Issued
 2015
 Identifier
 FSU_migr_uhm0560
 Format
 Thesis
 Title
 An Oblate Spheroid Model of Cortical Folding.
 Creator

Grazzini, Courtney, Department of Mathematics
 Abstract/Description

In previous work, Striegel and Hurdal have developed a mathematical model for cortical folding pattern formation during development (Striegel). A Turing reactiondiffusion system and a prolate spheroid domain were used to model the shape of the ventricle during development. They assumed a chemical hypothesis for cortical folding development. The chemical hypothesis suggests that a radial glial cell duplicates into an intermediate progenitor (IP) cell and a new radial glial cell only if it is...
Show moreIn previous work, Striegel and Hurdal have developed a mathematical model for cortical folding pattern formation during development (Striegel). A Turing reactiondiffusion system and a prolate spheroid domain were used to model the shape of the ventricle during development. They assumed a chemical hypothesis for cortical folding development. The chemical hypothesis suggests that a radial glial cell duplicates into an intermediate progenitor (IP) cell and a new radial glial cell only if it is activated. In turn, the IP cell duplicates into one or two neuroblasts. These cells form the cortical layer. The amplifications due to activated radial glial cells create gyral walls, and the inhibited cells create sulcal valleys. In this research, we modify Striegel and Hurdal's model to use an oblate spheroid domain. We develop mathematical equations using this new domain and investigate the role of various parameters through numerical stimulations. We suggest how these results can be applied to diseases, such as ventriculomegaly and holoprosencephaly, which alter the shape and size of the brain.
Show less  Date Issued
 2015
 Identifier
 FSU_migr_uhm0487
 Format
 Thesis
 Title
 Evolutionary Dynamics of Bacterial Persistence under Nutrient/Antibiotic Actions.
 Creator

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

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

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

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

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

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

Manakov, Andrey, Magnan, Jeronimo Francisco, Duke, Dennis, Beaumont, Paul, Case, Bettye Anne, Nolder, Craig, Florida State University, College of Arts and Sciences, Department...
Show moreManakov, Andrey, Magnan, Jeronimo Francisco, Duke, Dennis, Beaumont, Paul, Case, Bettye Anne, Nolder, Craig, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

In this work, we research several aspects of creating a general trading strategy by developing a forecasting model that uses an Artificial Neural Network (ANN) model that is based on the Convolutional Neural Network (CNN). In particular, we introduce inverted inputs and demonstrate that they reduce directional bias and reduce correlation with respect to the buyandhold strategy (for the underlying instrument). We empirically address issues of applying an ANN to create a trading strategy that...
Show moreIn this work, we research several aspects of creating a general trading strategy by developing a forecasting model that uses an Artificial Neural Network (ANN) model that is based on the Convolutional Neural Network (CNN). In particular, we introduce inverted inputs and demonstrate that they reduce directional bias and reduce correlation with respect to the buyandhold strategy (for the underlying instrument). We empirically address issues of applying an ANN to create a trading strategy that does not use the ANN output to predict price (or its change) but provides a specific trading allocation of the underlying security for the next day of trading by using a global Sharperatiodependent cost function, instead of the oftenused sum of local (or individual) squared prediction errors. The importance of the Sharpedependent cost function and Sharpe ratio being an appropriate measure of trading strategy is addressed and discussed. We propose a method of comparison of the trading results to random trading that employs the Sharperatio distribution. We also discuss the uniqueness of the trained solution and ways to make it more independent of the initialization of the ANN's weights, either by averaging, or by the sharing of markets when pretraining the convolutional layers. The proposed method tested well in the controlled environment of artificially generated time series with different properties, extracting signal where present. It is applied to real market time series, and compared with the performance of more traditional methods, and shows promise for creating a less risky, profitable, trading strategy for a portfolio consisting of alternative investments together with the buyandhold of underlying securities.
Show less  Date Issued
 2018
 Identifier
 2019_Spring_Manakov_fsu_0071E_14890
 Format
 Thesis
 Title
 Diffusion Approximation of a Risk Model.
 Creator

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

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

Vinson, Ryan M., Bertram, R. (Richard), Miller, Brian G., Jain, Harsh Vardhan, Magnan, Jeronimo Francisco, Roper, Michael Gabriel, Florida State University, College of Arts and...
Show moreVinson, Ryan M., Bertram, R. (Richard), Miller, Brian G., Jain, Harsh Vardhan, Magnan, Jeronimo Francisco, Roper, Michael Gabriel, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

Pancreatic islets of Langerhans are responsible for the release of the hormone insulin. This release is pulsatile, and proper insulin levels are necessary to maintain glucose homeostasis. In order to achieve the requisite insulin levels, the many islets of Langerhans must release the hormone in phase with one another. However, islets are not connected to each other in a physical way, so the cause of this synchronization is unclear. One hypothesis is that acetylcholine (ACh) release from...
Show morePancreatic islets of Langerhans are responsible for the release of the hormone insulin. This release is pulsatile, and proper insulin levels are necessary to maintain glucose homeostasis. In order to achieve the requisite insulin levels, the many islets of Langerhans must release the hormone in phase with one another. However, islets are not connected to each other in a physical way, so the cause of this synchronization is unclear. One hypothesis is that acetylcholine (ACh) release from intrapancreatic ganglia can give rise to these synchronized insulin signals. We test the nature of these ACh pulses, and find that their application need not be periodic to achieve synchronization. We also challenge previous results which suggest that ACh pulses may not be the underlying cause of synchronization, due to glucose's ability to override their effects. We find that the two chemical signals can not only coexist, but actually reinforce each other. Finally, we explore how islets may be able to maintain synchronicity through the effects of a coupling agent produced within the islets themselves.
Show less  Date Issued
 2019
 Identifier
 2019_Spring_Vinson_fsu_0071E_15046
 Format
 Thesis
 Title
 Belief Function Theory: Monte Carlo Methods and Application to Stock Markets.
 Creator

Salehy, Seyyed Nima, Ökten, Giray, Srivastava, Anuj, Cogan, Nicholas G., Fahim, Arash, Florida State University, College of Arts and Sciences, Department of Mathematics
 Abstract/Description

Belief function theory, also known as DempsterShafer theory or evidence theory, gives a general framework for quantifying, representing, and managing uncertainty, and it is widely used in several applications from artificial intelligence to accounting. The belief function theory provides tools to combine several sources' opinions (belief functions), among which, Dempster's rule of combination is the most commonly used. The main drawback of using Dempster's rule to combine belief functions is...
Show moreBelief function theory, also known as DempsterShafer theory or evidence theory, gives a general framework for quantifying, representing, and managing uncertainty, and it is widely used in several applications from artificial intelligence to accounting. The belief function theory provides tools to combine several sources' opinions (belief functions), among which, Dempster's rule of combination is the most commonly used. The main drawback of using Dempster's rule to combine belief functions is its computational complexity, which limits the application of Dempster's rule to small number of belief functions. We introduce a family of new Monte Carlo and quasiMonte Carlo algorithms aimed at approximating Dempster's rule of combination. Then, we present numerical results to show the superiority of the new methods over the existing ones. The algorithms are then used to implement some stock investment strategies based on DempsterShafer theory. We will introduce a new strategy, and apply it to the U.S. stock market over a certain period of time. Numerical results suggest the strategies based on the belief function theory outperform the S&P 500 index, with our new strategy giving the best returns.
Show less  Date Issued
 2019
 Identifier
 2019_Spring_SALEHY_fsu_0071E_15151
 Format
 Thesis
 Title
 Mathematical Modeling and Sensitivity Analysis for Biological Systems.
 Creator

Aggarwal, Manu, Cogan, Nicholas G., Hussaini, M. Yousuff, Chicken, Eric, Jain, Harsh Vardhan, Bertram, R. (Richard), Mio, Washington, Florida State University, College of Arts...
Show moreAggarwal, Manu, Cogan, Nicholas G., Hussaini, M. Yousuff, Chicken, Eric, Jain, Harsh Vardhan, Bertram, R. (Richard), Mio, Washington, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

In this work, we propose a framework to develop testable hypotheses for the effects of changes in the experimental conditions on the dynamics of a biological system using mathematical models. We discuss the uncertainties present in this process and show how information from different experiment regimes can be used to identify a region in the parameter space over which subsequent mathematical analysis can be conducted. To determine the significance of variation in the parameters due to varying...
Show moreIn this work, we propose a framework to develop testable hypotheses for the effects of changes in the experimental conditions on the dynamics of a biological system using mathematical models. We discuss the uncertainties present in this process and show how information from different experiment regimes can be used to identify a region in the parameter space over which subsequent mathematical analysis can be conducted. To determine the significance of variation in the parameters due to varying experimental conditions, we propose using sensitivity analysis. Using our framework, we hypothesize that the experimentally observed decrease in the survivability of bacterial populations of Xylella fastidiosa (causal agent of Pierce’s Disease) upon addition of zinc, might be because of starvation of the bacteria in the biofilm due to an inhibition of the diffusion of the nutrients through the extracellular matrix of the biofilm. We also show how sensitivity is related to uncertainty and identifiability; and how it can be used to drive analysis of dynamical systems, illustrating it by analyzing a model which simulates bursting oscillations in pancreatic βcells. For sensitivity analysis, we use Sobol’ indices for which we provide algorithmic improvements towards computational efficiency. We also provide insights into the interpretation of Sobol’ indices, and consequently, define a notion of the importance of parameters in the context of inherently flexible biological systems.
Show less  Date Issued
 2019
 Identifier
 2019_Spring_Aggarwal_fsu_0071E_15070
 Format
 Thesis
 Title
 Surface Subgroups of 3Manifold Groups.
 Creator

Rasheed, Mohammad Aamir, Heil, Wolfgang H., Wahl, Horst, Bowers, Philip L., Ballas, Samuel A., Florida State University, College of Arts and Sciences, Department of Mathematics
 Abstract/Description

This dissertation is concerned with the study of how various properties such as malnormality and maximality of surface groups embedded in a 3manifold group give us information about the topology of a 3manifold. In this direction we show that the malnormality of certain surface groups is sufficient to detect whether or not there are any Seifert fibered pieces in the JSJ decomposition of a 3manifold. On the other hand topology itself imposes a strong constraint on what properties a surface...
Show moreThis dissertation is concerned with the study of how various properties such as malnormality and maximality of surface groups embedded in a 3manifold group give us information about the topology of a 3manifold. In this direction we show that the malnormality of certain surface groups is sufficient to detect whether or not there are any Seifert fibered pieces in the JSJ decomposition of a 3manifold. On the other hand topology itself imposes a strong constraint on what properties a surface group might have. For example, we show that a surface group associated with an essential embedding must be maximal among all surface groups. The first chapter starts with an overview and introduction to the material along with some of the background material needed to understand this dissertation. Here we provide all the appropriate definitions as well as the statements of the theorems and lemmas that are used in this dissertation. All the theorems stated in chapter 1 are standard and well known results in 3manifold theory and all we have done is provide a brief exposition. We have made an effort to provide appropriate references whenever we could. In the second chapter we study the relationship between malnormal subgroups corresponding to incompressible tori and Klein bottles and the absence of Seifert pieces in the JSJ decomposition. In particular, we show that a rank two free abelian subgroup correspond ing to an embedded incompressible torus in an orientable Haken manifold is a malnormal subgroup if and only if the JSJ piece that contains the torus is nonSeifert. We further generalize this result to any embedded Klein bottle and answer the question of when a maximal abelian subgroup in a Haken manifold group is malnormal. We also explore other conditions that guaranty that there are no Seifert pieces in the JSJ decomposition. Some other results regarding the malnormality of peripheral groups corresponding to higher genus surfaces are also found. The third chapter is concerned with the study of properly embedded incompressible surfaces (closed or otherwise) in a Haken manifold. Here we give a sufficient condition for two embedded surfaces to be isotopic. We show that given two embedded 2sided incompressible surfaces such that the subgroup associated to one is contained in the subgroup associated to the other, then it must be that case that the surfaces are isotopic. This, in particular, shows that it is impossible to embed two surfaces of different genus in an orientable Haken manifold such that one is a subgroup of the other. In the fourth chapter we generalize the results of the third chapter to immersed π1 injective surfaces. We show that any two immersed surfaces satisfying an analogous condi tions on their associated subgroups can always be deformed so that one immersed surface is a covering onto the other immersed surface. In particular, this shows that embedded surface groups are maximal among all surface groups.
Show less  Date Issued
 2019
 Identifier
 2019_Spring_Rasheed_fsu_0071E_15146
 Format
 Thesis
 Title
 Random Walks over Point Processes and Their Application in Finance.
 Creator

Salehy, Seyyed Navid, Kercheval, Alec N., Ewald, Brian, Fahim, Arash, Ökten, Giray, Huffer, Fred W. (Fred William), Florida State University, College of Arts and Sciences,...
Show moreSalehy, Seyyed Navid, Kercheval, Alec N., Ewald, Brian, Fahim, Arash, Ökten, Giray, Huffer, Fred W. (Fred William), Florida State University, College of Arts and Sciences, Department of Mathematics
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In continuoustime models in finance, it is common to assume that prices follow a geometric Brownian motion. More precisely, it is assumed that the price at time t ≥ 0 is given by Zt = Z₀exp(σBt + mt) where Z₀ is the initial price, B is standard Brownian motion, σ is the volatility, and m is the drift. We discuss how Z can be viewed as the limit of a sequence of discrete price models based on random walks. We note that in the usual random walks, jumps can only happen at deterministic times....
Show moreIn continuoustime models in finance, it is common to assume that prices follow a geometric Brownian motion. More precisely, it is assumed that the price at time t ≥ 0 is given by Zt = Z₀exp(σBt + mt) where Z₀ is the initial price, B is standard Brownian motion, σ is the volatility, and m is the drift. We discuss how Z can be viewed as the limit of a sequence of discrete price models based on random walks. We note that in the usual random walks, jumps can only happen at deterministic times. We first construct a natural simple model for price by considering a random walk in which jumps can happen at random times following a counting process N. We then develop a sequence of discrete price models using random walks over point processes. The limit process gives the new price model: Zt = Z₀exp(σBΛt + mΛt), where Λ is the compensator for the counting process N. We note that if N is a Poisson process with intensity 1, then this model coincides with the geometric Brownian motion model for the price. But this new model provides more flexibility as we can choose N to be many other wellknown counting processes. This includes not only homogeneous and inhomogeneous Poisson processes which have deterministic compensators but also Hawkes processes which have stochastic compensators. We also discuss and prove many properties for the process BΛ. For example, we show that BΛ is a continuous square integrable martingale. Moreover, we discuss when BΛ has uncorrelated increments and when it has independent increments. Moreover, we investigate how the BlackScholes pricing formula will change if the price of the risky asset follows this new model when N is an inhomogeneous Poisson process. We show that the usual BlackScholes formula is obtained when the counting process N is a Poisson process with intensity 1.
Show less  Date Issued
 2019
 Identifier
 2019_Spring_Salehy_fsu_0071E_15152
 Format
 Thesis