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 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
 Peridynamic Multiscale Models for the Mechanics of Materials: Constitutive Relations, Upscaling from Atomistic Systems, and Interface Problems.
 Creator

Seleson, Pablo D, Gunzburger, Max, Rikvold, Per Arne, ElAzab, Anter, Peterson, Janet, Shanbhag, Sachin, Lehoucq, Richard B., Parks, Michael L., Department of Scientific...
Show moreSeleson, Pablo D, Gunzburger, Max, Rikvold, Per Arne, ElAzab, Anter, Peterson, Janet, Shanbhag, Sachin, Lehoucq, Richard B., Parks, Michael L., Department of Scientific Computing, Florida State University
Show less  Abstract/Description

This dissertation focuses on the non local continuum peridynamics model for the mechanics of materials, related constitutive models, its connections to molecular dynamics and classical elasticity, and its multiscale and multimodel capabilities. A more generalized role is defined for influence functions in the statebased peridynamic model which allows for the strength of non local interactions to be modulated. This enables the connection between different peridynamic constitutive models,...
Show moreThis dissertation focuses on the non local continuum peridynamics model for the mechanics of materials, related constitutive models, its connections to molecular dynamics and classical elasticity, and its multiscale and multimodel capabilities. A more generalized role is defined for influence functions in the statebased peridynamic model which allows for the strength of non local interactions to be modulated. This enables the connection between different peridynamic constitutive models, establishing a hierarchy that reveals that some models are special cases of others. Furthermore, this allows for the modulation of the strength of non local interactions, even for a fixed radius of interactions between material points in the peridynamics model. The multiscale aspect of peridynamics is demonstrated through its connections to molecular dynamics. Using higherorder gradient models, it is shown that peridynamics can be viewed as an upscaling of molecular dynamics, preserving the relevant dynamics under appropriate choices of length scales. The statebased peridynamic model is shown to be appropriate for the description of multiscale and multimodel systems. A formulation for nonlocal interface problems involving scalar fields is presented, and derivations of non local transmission conditions are derived. Specializations that describe local, non local, and local/non local transmission conditions are considered. Moreover, the convergence of the non local transmission conditions to their classical local counterparts is shown. In all cases, results are illustrated by numerical experiments.
Show less  Date Issued
 2010
 Identifier
 FSU_migr_etd0273
 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
 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
 The Oneand TwoSample Problem for Data on Hilbert Manifolds with Applications to Shape Analysis.
 Creator

Qiu, Mingfei, Patrangenaru, Victor, Liu, Xiuwen, Slate, Elizabeth H., Barbu, Adrian G. (Adrian Gheorghe), Clickner, Robert Paul, Paige, Robert, Florida State University, College...
Show moreQiu, Mingfei, Patrangenaru, Victor, Liu, Xiuwen, Slate, Elizabeth H., Barbu, Adrian G. (Adrian Gheorghe), Clickner, Robert Paul, Paige, Robert, Florida State University, College of Arts and Sciences, Department of Statistics
Show less  Abstract/Description

This dissertation is concerned with high level imaging analysis. In particular, our focus is on extracting the projective shape information or the similarity shape from digital camera images or Magnetic Resonance Imaging(MRI). The approach is statistical without making any assumptions about the distributions of the random object under investigation. The data is organized as points on a Hilbert manifold. In the case of projective shapes of finite dimensional configuration of points, we...
Show moreThis dissertation is concerned with high level imaging analysis. In particular, our focus is on extracting the projective shape information or the similarity shape from digital camera images or Magnetic Resonance Imaging(MRI). The approach is statistical without making any assumptions about the distributions of the random object under investigation. The data is organized as points on a Hilbert manifold. In the case of projective shapes of finite dimensional configuration of points, we consider testing a onesample null hypothesis, while in the infinite dimensional case, we considered a neighborhood hypothesis testing methods. For 3D scenes, we retrieve the 3D projective shape, and use the Lie group structure of the projective shape space. We test the equality of two extrinsic means, by introducing the mean projective shape change. For 2D MRI of midsections of Corpus Callosum contours, we use an automatic matching technique that is necessary in pursuing a onesample neighborhood hypothesis testing for the similarity shapes. We conclude that the mean similarity shape of the Corpus Callosum of average individuals is very far from the shape of Albert Einstein's, which may explain his geniality. Another application of our Hilbert manifold methodology is twosample testing problem for VeroneseWhitney means of projective shapes of 3D contours. Particularly, our data consisting comparing 3D projective shapes of contours of leaves from the same tree species.
Show less  Date Issued
 2015
 Identifier
 FSU_2015fall_Qiu_fsu_0071E_12922
 Format
 Thesis
 Title
 An investigation of the effect of instruction in the structure of problemsolving strategies on students' performance.
 Creator

Ghunaym, Ghunaym, Nichols, Eugene Douglas, Florida State University
 Abstract/Description

"The purpose of this study was to investigate the conjecture that instruction in the strategies of Pattern Discovery, Trial and Error, Working Backward, Contradiction, Substitution, and Use of Diagrams would result in the development of problemsolving ability and that students under this instruction are likely to exhibit better achievement than students who do not receive explicit instruction in problemsolving strategies"Introduction.
 Date Issued
 1985
 Identifier
 FSU_acr1501
 Format
 Thesis
 Title
 Investigating Vesicle Adhesions Using Multiple Phase Field Functions.
 Creator

Gu, Rui, Wang, Xiaoqiang, Gunzburger, Max D., Wang, Xiaoming, Peterson, Janet S., Ye, Ming, Florida State University, College of Arts and Sciences, Department of Scientific...
Show moreGu, Rui, Wang, Xiaoqiang, Gunzburger, Max D., Wang, Xiaoming, Peterson, Janet S., Ye, Ming, Florida State University, College of Arts and Sciences, Department of Scientific Computing
Show less  Abstract/Description

We construct a phase field model for simulating the adhesion of a cell membrane to a substrate. The model features two phase field functions which are used to simulate the membrane and the substrate. An energy model is defined which accounts for the elastic bending energy and the contact potential energy as well as, through a penalty method, vesicle volume and surface area constraints. Numerical results are provided to verify our model and to provide visual illustrations of the interactions...
Show moreWe construct a phase field model for simulating the adhesion of a cell membrane to a substrate. The model features two phase field functions which are used to simulate the membrane and the substrate. An energy model is defined which accounts for the elastic bending energy and the contact potential energy as well as, through a penalty method, vesicle volume and surface area constraints. Numerical results are provided to verify our model and to provide visual illustrations of the interactions between a lipid vesicle and substrates having complex shapes. Examples are also provided for the adhesion process in the presence of gravitational and point pulling forces. A comparison with experimental results demonstrates the effectiveness of the two phase field approach. Similarly to simulating vesiclesubstrate adhesion, we construct a multiphasefield model for simulating the adhesion between two vesicles. Two phase field functions are introduced to simulate each of the two vesicles. An energy model is defined which accounts for the elastic bending energy of each vesicle and the contact potential energy between the two vesicles; the vesicle volume and surface area constraints are imposed using a penalty method. Numerical results are provided to verify the efficacy of our model and to provide visual illustrations of the different types of contact. The method can be adjusted to solve endocytosis problems by modifying the bending rigidity coefficients of the two elastic bending energies. The method can also be extended to simulate multicell adhesions, one example of which is erythrocyte rouleaux. A comparison with laboratory observations demonstrates the effectiveness of the multiphase field approach. Coupled with fluid, we construct a phase field model for simulating vesiclevessel adhesion in a flow. Two phase field functions are introduced to simulate the vesicle and vessel respectively. The fluid is modeled and confined inside the tube by a phase field coupled NavierStokes equation. Both vesicle and vessel are transported by fluid flow inside our computational domain. An energy model regarding the comprehensive behavior of vesiclefluid interaction, vesselfluid interaction, vesiclevessel adhesion is defined. The vesicle volume and surface area constraints are imposed using a penalty method, while the vessel elasticity is modeled under Hooke's Law. Numerical results are provided to verify the efficacy of our model and to demonstrate the effectiveness of our fluidcoupled vesicle vessel adhesion phase field approach by comparison with laboratory observations.
Show less  Date Issued
 2015
 Identifier
 FSU_2015fall_Gu_fsu_0071E_12873
 Format
 Thesis
 Title
 The role of pictures in first grade children's perception of mathematical relationships.
 Creator

Campbell, Patricia F., Nichols, Eugene Douglas, Florida State University
 Abstract/Description

"This study investigated whether there is a relationship between first grade children's ability to tell a story about a dynamic picture or a sequence of three dynamic pictures and their ability to describe the picture(s) by a number sequence. The artistic variables characterizing the pictures were controlled so as to provide information concerning which types of illustrations best facilitated interpretation of the pictures and perception of mathematical relationships. An 8 x 2 design allowed...
Show more"This study investigated whether there is a relationship between first grade children's ability to tell a story about a dynamic picture or a sequence of three dynamic pictures and their ability to describe the picture(s) by a number sequence. The artistic variables characterizing the pictures were controlled so as to provide information concerning which types of illustrations best facilitated interpretation of the pictures and perception of mathematical relationships. An 8 x 2 design allowed analysis of the effects of the form of the drawing, the number of pictures, and the response condition. Ninetysix first grade children were individually tested using an instrument designed by the investigator. Statistical analysis revealed that neither drawing style nor the number of pictures had a significant effect on either the level of assimilation within the stories, the perception of motion, or the number sentence responses. Analysis of the response condition revealed a significant difference favoring the force condition on number sentence responses. Also, initially viewing and interpreting sequences provided a learning experience to significantly effect the interpretation of single pictures"Abstract.
Show less  Date Issued
 1976
 Identifier
 FSU_abj0548
 Format
 Thesis
 Title
 Interactions between spatial and verbal abilities and two methods of presenting modulus seven arithmetic.
 Creator

Hussien, Gaber A, Nichols, Eugene Douglas, Florida State University
 Abstract/Description

"The present investigation was designed to study the effect of two instructional treatments on the achievement of students of different abilitiesVerbal and Spatial. This was achieved by studying the interaction between the two treatments and each of the verbal and the spatial abilities. The instructional treatments were Figural and Verbal programmed units designed to teach concepts related to modulus seven arithmetic. Subjects for the study were 90 students enrolled in the first year...
Show more"The present investigation was designed to study the effect of two instructional treatments on the achievement of students of different abilitiesVerbal and Spatial. This was achieved by studying the interaction between the two treatments and each of the verbal and the spatial abilities. The instructional treatments were Figural and Verbal programmed units designed to teach concepts related to modulus seven arithmetic. Subjects for the study were 90 students enrolled in the first year mathematics course at Elmansoura College of Education in Egypt for the academic year 19781979"Abstract.
Show less  Date Issued
 1979
 Identifier
 FSU_aby7218
 Format
 Thesis
 Title
 A study of the prediction of achievement in some topics in college freshman mathematics from measures of "structureofintellect" factors.
 Creator

Altman, Betty J., Nichols, Eugene Douglas, Florida State University
 Abstract/Description

For several reasons, Guilford's psychological theory, "The StructureofIntellect" (SI), seems a good candidate for relating to the learning of mathematics. The general purposes of this study were to identify SI factors which would be significantly related to achievement in a juniorcollege mathematics course for nonscience, nonmathematics majors and to determine whether semantic factors would be better predictors than symbolic for students classified as having high verbal ability. The two...
Show moreFor several reasons, Guilford's psychological theory, "The StructureofIntellect" (SI), seems a good candidate for relating to the learning of mathematics. The general purposes of this study were to identify SI factors which would be significantly related to achievement in a juniorcollege mathematics course for nonscience, nonmathematics majors and to determine whether semantic factors would be better predictors than symbolic for students classified as having high verbal ability. The two topics in the mathematics course which were selected for study were (1) numeration in other bases and (2) finite systems.
Show less  Date Issued
 1975
 Identifier
 FSU_abd5132
 Format
 Thesis
 Title
 The effect of the knowledge of logic in proving mathematical theorems in the context of mathematical induction.
 Creator

Walter, Robert Lee, Nichols, Eugene Douglas, Florida State University
 Abstract/Description

"Let P(n) be a statement for every positive integer n. We denote the set of all positive integers by N and consider G = {n [is an element of] N [such that] P(n) is true}. The principle of mathematical induction can now be stated as follows: If [(i) 1 [is an element of] G and, (ii) for all k [is an element of] N if k [is an element of] G, then k + 1 [is an element of] G], then G = N. Now symbolize this statement as follows: P: 1 [is an element of] G. R: k [is an element of] G. S: k + 1 [is an...
Show more"Let P(n) be a statement for every positive integer n. We denote the set of all positive integers by N and consider G = {n [is an element of] N [such that] P(n) is true}. The principle of mathematical induction can now be stated as follows: If [(i) 1 [is an element of] G and, (ii) for all k [is an element of] N if k [is an element of] G, then k + 1 [is an element of] G], then G = N. Now symbolize this statement as follows: P: 1 [is an element of] G. R: k [is an element of] G. S: k + 1 [is an element of] G. Q: G = N. Therefore the statement of the principle of mathematical induction can be seen in the following form. If [P and, [for all] k [is an element of] N (if R, then S)], then Q. One strategy for teaching this principle is to explain that in order to apply the principle of mathematical induction and assert Q, one must appeal to the logical rule of modus ponens (the law of detachment). That is, we must affirm the antecedent [P and, [for all] k [is an element of] N (if R, then S)], and then we can assert Q. Therefore the research hypothesis for this study was that if people have the prerequisite knowledge of logic, and that if they are taught the principle of mathematical induction in terms of logic, then they will perform better on a criterion test over the principle of mathematical induction than people who are not taught in terms of logic"Introduction.
Show less  Date Issued
 1972
 Identifier
 FSU_agg0249
 Format
 Thesis
 Title
 Inductive discovery learning, reception learning, and formal verbalization of mathematical concepts.
 Creator

Hanson, Lawrence Eugene, Nichols, Eugene Douglas, Florida State University
 Abstract/Description

Theoretical speculations abound on all sides of the following two questions: 1. What are the relative merits of the reception and discovery modes of learning? 2. What effect does forcing a student to immediately verbalize his newly discovered concept have on his ability to retain and transfer this concept? The purpose of the present study is to seek answers to these questions on the basis of experimental evidence.
 Date Issued
 1967
 Identifier
 FSU_agh3055
 Format
 Thesis
 Title
 An exploratory study of the effectiveness of computer graphic and simulations in a computerstudent interactive environment in illustrating random sampling and the central limit theorem.
 Creator

Myers, Kitty Neel, Denmark, E. T., Florida State University
 Abstract/Description

"The purposes of this study were: (1) to investigate the effectiveness of the computerstudent interactive method in presenting statistical concepts and in instructing students in the applications of these concepts, and (2) to develop instruments that test for the understanding of these concepts and the mastery of these application skills"Abstract.
 Date Issued
 1990
 Identifier
 FSU_afs7567
 Format
 Thesis
 Title
 A comparison of verbal and nonverbal instruction in elementary school mathematics.
 Creator

Hollingsworth, Caroline Dean, Nichols, Eugene Douglas, Florida State University
 Abstract/Description

This study assessed the relative effectiveness of verbal and nonverbal teaching methods in facilitating the learning of mathematics. The two treatments differed only in that nonverbal instruction did not permit oral communication or use of written words. Chalkboard instruction was characterized by complete silence in nonverbal classes. In verbal classes, new terminology was introduced by writing the terms on the board and using them thorughout the lesson. Four fourthgrade classes consisting...
Show moreThis study assessed the relative effectiveness of verbal and nonverbal teaching methods in facilitating the learning of mathematics. The two treatments differed only in that nonverbal instruction did not permit oral communication or use of written words. Chalkboard instruction was characterized by complete silence in nonverbal classes. In verbal classes, new terminology was introduced by writing the terms on the board and using them thorughout the lesson. Four fourthgrade classes consisting of 88 students in one school were randomly assigned to treatment groups so that two were taught nonverbally, and two by the conventional verbal method. Two teachers were assigned one class of each type. Treatment and teacher factors were crossed in a pretestposttest control group design. The demonstrated comparability of the two teaching methods not only points to nonverbal instruction as an alternate mode, but also seriously questions the effectiveness of conventional teacher talk in enhancing learning. Teachers with a creative bent should be encouraged to experiment with nonverbal instruction and design activities for all levels of development. The technique could be used effectively to break the routine of conventional instruction. The importance of nonverbal components should be stressed in methods courses for pre and inservice teachers. Techniques of nonverbal instruction should be practiced in student teaching practices.
Show less  Date Issued
 1973
 Identifier
 FSU_afa8702
 Format
 Thesis
 Title
 How Geogebra Contributes to Middle Grade Algebra I Students' Conceptual Understanding of Functions.
 Creator

Dayi, Guner, Jakubowski, Elizabeth M., Berry, Frances Stokes, Rice, Diana Claries, Davis F., Angela, Florida State University, College of Education, School of Teacher Education
 Abstract/Description

The current study examined how GeoGebra contributed to middle grade Algebra I students' conceptual understanding of functions. In order to gain a deeper understanding a case study approach was utilized. Vinner (1983), and Vinner and Dreyfus' (1989) concept definition and concept image framework was used to analyze the students' function definition. O'Callaghan's (1994) component of translating was used to analyze the students' comparison of different function representations, and his...
Show moreThe current study examined how GeoGebra contributed to middle grade Algebra I students' conceptual understanding of functions. In order to gain a deeper understanding a case study approach was utilized. Vinner (1983), and Vinner and Dreyfus' (1989) concept definition and concept image framework was used to analyze the students' function definition. O'Callaghan's (1994) component of translating was used to analyze the students' comparison of different function representations, and his component of modeling and interpreting was used to analyze the students' use of functions to model relationships between quantities. The following results were derived from the analyses. Having more correct concept images of functions through GeoGebra could also bring about a more correct definition. The dependency upon the concept definition to verify if a given example was a function could not contribute to the concept image. In order to gain correct concept images more integration of technology into algebra instructions was crucial to explore and interact with more function models. GeoGebra was an ideal environment to perform a transition among the representations. All three cases were able to understand how the given realworld problems transformed to GeoGebra simulator and the reverse procedure. The role of instructor was very important to guide and facilitate the learning. The results indicated that verification and exploration of more functions on GeoGebra contributed to a better conceptual understanding of a function definition. The advantages of GeoGebra were obvious for the translating component. The realworld problem scenario could be better modeled and interpreted via a simulator on GeoGebra and the need for algebraic symbolic manipulations could disappear.
Show less  Date Issued
 2015
 Identifier
 FSU_2015fall_Dayi_fsu_0071E_12946
 Format
 Thesis
 Title
 The development and testing of a teachtest instrument for prediction of success in college freshman mathematics.
 Creator

Smith, Joe Kelly, Heimer, Ralph T., Florida State University
 Abstract/Description

"The purpose of this research is the development and testing of an instrument to be used in prediction of success in college freshman mathematics courses"Introduction.
 Date Issued
 1967
 Identifier
 FSU_ahm6748
 Format
 Thesis
 Title
 An experiment to compare the effectiveness of instruction versus discovery in generalizing the strategy of a simple game.
 Creator

Page, Robert Leroy, Nichols, Eugene Douglas, Florida State University
 Abstract/Description

"The purpose of the study was to determine whether there is a difference in the ability of two equally capable groups of subjects to generalize the winning strategy of a simple game when one group learns the perfect strategy for one form of the game by the discovery method and the other group learns it by reading an explanation of the strategy"Introduction.
 Date Issued
 1970
 Identifier
 FSU_ahk1593
 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
 The Effects of GameBased Learning in an OpensimSupported Virtual Environment for Mathematical Performance.
 Creator

Kim, Heesung, Ke, Fengfeng, Kim, YoungSuk, Jeong, Allan C., Paek, Insu, Florida State University, College of Education, Department of Educational Psychology and Learning Systems
 Abstract/Description

This experimental study was intended to examine whether gamebased learning (GBL) that encompasses four particular game characteristics (challenges, a storyline, rewards, and the integration of gameplay with learning content) in the OpenSimulatorsupported virtual reality (VR) learning environment can improve mathematical achievement and motivation for elementary school students toward math learning. In this pre and posttest experimental comparison study, data were collected from 132...
Show moreThis experimental study was intended to examine whether gamebased learning (GBL) that encompasses four particular game characteristics (challenges, a storyline, rewards, and the integration of gameplay with learning content) in the OpenSimulatorsupported virtual reality (VR) learning environment can improve mathematical achievement and motivation for elementary school students toward math learning. In this pre and posttest experimental comparison study, data were collected from 132 fourth graders through an achievement test, and a Short Instructional Materials Motivational Survey (SIMMS). The same tasks were provided to the experimental and control groups. Tasks for the experimental group involved the following four game characteristics: (1) challenges, (2) a storyline, (3) rewards, and (4) the integration of gameplay with learning content. The control group was given the same tasks and learning environment setting (OpenSimulatorsupported VR) that was used for the experimental group. The exception was that the control group tasks did not include the game characteristics: (1) challenges, (2) a storyline, (3) rewards, and (4) the integration of gameplay with learning content. Analysis of covariance (ANCOVA) using a treatment (treatment vs. control) on the achievement indicated a significant effect of GBL in the VR environment on math knowledge test performance. For motivation, the results indicated that there was no significant difference on the posttest scores for the perceived motivational quality of the learning activity (MQLA) between the experimental group and the control group.
Show less  Date Issued
 2015
 Identifier
 FSU_2015fall_Kim_fsu_0071E_12885
 Format
 Thesis
 Title
 A study of interactions between "StructureofIntellect" factors and two methods of presenting concepts of modulus seven arithemetic.
 Creator

Behr, Merlyn J., Nichols, Eugene Douglas, Florida State University
 Abstract/Description

"In general terms, the purposes of this study were two in number: (1) to suggest whether unique mental factors as identified by methods of factor analysis are correlated with success in usual school learning situations and (2) to suggest whether it is possible to design instructional materials in a way which would suit the learner's mental ability profile"Introduction.
 Date Issued
 1967
 Identifier
 FSU_ahp9230
 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
 Comparison of Different Noise Forcings, Regularization of Noise, and Optimal Control for the Stochastic NavierStokes Equations.
 Creator

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

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

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

According to rapid development in information technology, limit order books(LOB) mechanism has emerged to prevail in today's nancial market. In this paper, we propose ensemble machine learning architectures for capturing the dynamics of highfrequency limit order books such as predicting price spread crossing opportunities in a future time interval. The paper is more datadriven oriented, so experiments with ve realtime stock data from NASDAQ, measured by nanosecond, are established. The...
Show moreAccording to rapid development in information technology, limit order books(LOB) mechanism has emerged to prevail in today's nancial market. In this paper, we propose ensemble machine learning architectures for capturing the dynamics of highfrequency limit order books such as predicting price spread crossing opportunities in a future time interval. The paper is more datadriven oriented, so experiments with ve realtime stock data from NASDAQ, measured by nanosecond, are established. The models are trained and validated by training and validation data sets. Compared with other models, such as logistic regression, support vector machine(SVM), our outofsample testing results has shown that ensemble methods had better performance on both statistical measurements and computational eciency. A simple trading strategy that we devised by our models has shown good prot and loss(P&L) results. Although this paper focuses on limit order books, the similar frameworks and processes can be extended to other classication research area. Keywords: limit order books, highfrequency trading, data analysis, ensemble methods, F1 score.
Show less  Date Issued
 2017
 Identifier
 FSU_SUMMER2017_Wang_fsu_0071E_14047
 Format
 Thesis
 Title
 On the Multidimensional Default Threshold Model for Credit Risk.
 Creator

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

This dissertation is based on the structural model framework for default risk that was first introduced by garreau2016structural (henceforth: the "GK model"). In this approach, the time of default is defined as the first time the logreturn of the firm's stock price jumps below a (possibly stochastic) "default threshold'' level. The stock price is assumed to follow an exponential L\'evy process and, in the multidimensional case, a multidimensional L\'evy process. This new structural model is...
Show moreThis dissertation is based on the structural model framework for default risk that was first introduced by garreau2016structural (henceforth: the "GK model"). In this approach, the time of default is defined as the first time the logreturn of the firm's stock price jumps below a (possibly stochastic) "default threshold'' level. The stock price is assumed to follow an exponential L\'evy process and, in the multidimensional case, a multidimensional L\'evy process. This new structural model is mathematically equivalent to an intensitybased model where the intensity is parameterized by a L\'evy measure. The dependence between the default times of firms within a basket is the result of the jump dependence of their respective stock prices and described by a L\'evy copula. To extend the previous work, we focus on generalizing the joint survival probability and related results to the ddimensional case. Using the link between L\'evy processes and multivariate exponential distributions, we derive the joint survival probability and characterize correlated default risk using L\'evy copulas. In addition, we extend our results to include stochastic interest rates. Moreover, we describe how to use the default threshold as the interface for incorporating additional exogenous economic factors, and still derive basket credit default swap (CDS) prices in terms of expectations. If we make some additional modeling assumptions such that the default intensities become affine processes, we obtain explicit formulas for the single name and firsttodefault (FtD) basket CDS prices, up to quadrature.
Show less  Date Issued
 2017
 Identifier
 FSU_SUMMER2017_Zhou_fsu_0071E_14012
 Format
 Thesis
 Title
 Algorithms for Solving Linear Differential Equations with Rational Function Coefficients.
 Creator

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

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

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

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

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

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

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

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

Goldsberry, Benjamin, Mathematics
 Abstract/Description

In recording studios, the placement of microphones to record an acoustic guitar is very much subjected to trial and error and audio engineer preference. In order to make more informed microphone placement decisions, Nearfield Acoustic Holography is used to study the sound pressures of the guitar. This technique involves solving the integral formulation of the Helmholtz equation over the surface of the guitar. By measuring the acoustic pressures surrounding the guitar, an inverse problem can...
Show moreIn recording studios, the placement of microphones to record an acoustic guitar is very much subjected to trial and error and audio engineer preference. In order to make more informed microphone placement decisions, Nearfield Acoustic Holography is used to study the sound pressures of the guitar. This technique involves solving the integral formulation of the Helmholtz equation over the surface of the guitar. By measuring the acoustic pressures surrounding the guitar, an inverse problem can be solved to derive the pressures on the surface of the guitar. Then, the surface pressures are used to study the pressure propagations in the farfield. Using the superposition of waves principle, chords played on the guitar can be studied by summing the pressure waves of the three notes that make a chord. Studying the wave fields are then used to either validate current microphone techniques, or require new microphone placements and patterns.
Show less  Date Issued
 2012
 Identifier
 FSU_migr_uhm0088
 Format
 Thesis
 Title
 Network Theoretical Approaches to Partitioning of Red Power Grids.
 Creator

Israels, Brett, Department of Physics
 Abstract/Description

Power grids are innately susceptible to electrical faults. Here we present divisive and agglomerative networktheoretical approaches to achieve intentional intelligent islanding of a power grid in order to limit cascading power failures in case a fault occurs. A power grid is modeled here as a network consisting of nodes and edges. The various methods we use are designed to partition a power grid network into smaller communities of noes with local generating capacity (islands). Here we...
Show morePower grids are innately susceptible to electrical faults. Here we present divisive and agglomerative networktheoretical approaches to achieve intentional intelligent islanding of a power grid in order to limit cascading power failures in case a fault occurs. A power grid is modeled here as a network consisting of nodes and edges. The various methods we use are designed to partition a power grid network into smaller communities of noes with local generating capacity (islands). Here we discuss results of using spectral matrix methods along with Monte Carlo methods to analyze and partition an illustrative example network, as well as the Floridian power grid, and the power distribution system for a conceptual allelectric naval vessel. We also contrast the effects of approximating the generating capacity of generators according to their degrees versus using their actual generating capacities. Finally, we propose an implementation strategy as well as possible directions for future study.
Show less  Date Issued
 2012
 Identifier
 FSU_migr_uhm0110
 Format
 Thesis
 Title
 Tame Symbols and Reciprocity Laws in Number Theory and Geometry.
 Creator

Radzimski, Vanessa, Mathematics
 Abstract/Description

The tame symbol serves many purposes in mathematics, and is of particular value when we use it to evaluate curves over certain number _elds. A wellknown example is that of the Hilbert symbol, which gives us insight into the existence of a rational solution to a conic of the form ax2 + by2 = c for a; b; c 2 Q_. In order to properly examine this symbol, we must gain a solid understanding into the padic completion of the rationals, Qp. We will explore the algebraic construction of the subring...
Show moreThe tame symbol serves many purposes in mathematics, and is of particular value when we use it to evaluate curves over certain number _elds. A wellknown example is that of the Hilbert symbol, which gives us insight into the existence of a rational solution to a conic of the form ax2 + by2 = c for a; b; c 2 Q_. In order to properly examine this symbol, we must gain a solid understanding into the padic completion of the rationals, Qp. We will explore the algebraic construction of the subring of padic integers, Zp, whose _eld of fractions is Qp itself. In general, we may look at a type of tame symbol, which we call a local symbol, that we take over an algebraic curve defined over a field into some abelian group G. The properties of these local symbols correspond directly to those of the Hilbert symbol. We then examine if it is possible to de_ne a type of local symbol over a degree 2 extension of Z, the Gaussian Integers Z[i]. The construction of this symbol is analogous to one for a degree 2 extension of Z which is a Euclidean domain. Central extensions of groups are explored to give a foundation for viewing the tame symbol in terms of determinates as viewed by Parshin.
Show less  Date Issued
 2012
 Identifier
 FSU_migr_uhm0063
 Format
 Thesis
 Title
 Florida Middle School Teachers' Perspective on and Preparedness for the Common Core State Standards in Mathematics.
 Creator

Porwoll, Kathryn, School of Teacher Education
 Abstract/Description

Abstract: (Florida, Common Core State Standards, Mathematics Teachers, 2013) This thesis was designed to determine middle school teachers' perceptions of their state of readiness to enact the Common Core Standards in mathematics in the State of Florida. The descriptive study employed a survey of 100 middle school mathematics teachers throughout the State of Florida in an effort to understand how the state, counties, and administrations can best serve educators through the critical transition...
Show moreAbstract: (Florida, Common Core State Standards, Mathematics Teachers, 2013) This thesis was designed to determine middle school teachers' perceptions of their state of readiness to enact the Common Core Standards in mathematics in the State of Florida. The descriptive study employed a survey of 100 middle school mathematics teachers throughout the State of Florida in an effort to understand how the state, counties, and administrations can best serve educators through the critical transition from Next Generation Sunshine State Standards to the Common Core State Standards. The survey developed for this study included twentyseven questions. In order to determine how Florida compares to a similar survey of a generalized sample of United States teachers, this thesis compares the results of the Floridian survey to the results from the national sample. The results suggest that Florida teachers' perceptions of preparedness lag behind that of the rest of the country. The implications of these results are discussed.
Show less  Date Issued
 2013
 Identifier
 FSU_migr_uhm0274
 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
 Construction and Implementation of a BenchTop Aquaponic System as a Context for Teaching Science in Secondary Schools.
 Creator

Fernandez, Sofia, Goldsby, Kenneth A., Department of Biological Science
 Abstract/Description

Aquaponics is an integrated biological system that essentially combines a soilless garden with an aquarium. It is important because it uses less water than commercial farming, is ecofriendly, and provides a local source of food for its practitioners. Aquaponics is also important because of its capacity to serve as an authentic teaching tool in science classrooms. This thesis is divided into three components. First we will describe the construction and implementation of our Benchtop...
Show moreAquaponics is an integrated biological system that essentially combines a soilless garden with an aquarium. It is important because it uses less water than commercial farming, is ecofriendly, and provides a local source of food for its practitioners. Aquaponics is also important because of its capacity to serve as an authentic teaching tool in science classrooms. This thesis is divided into three components. First we will describe the construction and implementation of our Benchtop Aquaponics System (BAS). Next, the results of an experiment that compares two methods of establishing bacteria–culture in a fishless system will be presented. Finally, the potential for use of the BAS in STEM classrooms will be discussed. The goals of this project are to (1) create an Aquaponics system that has a small ecological footprint and not take up too much room in the classroom, (2) further the current body of research on applied aquaponic systems, and (3) provide a pedagogical tool that involves students in building equipment and solving authentic problems as a gateway for learning. The BAS is assembled in 3 separate compartments, a plant tray, an aquarium, and a bacteria reservoir, with PVC piping connecting the three. It is designed around a wooden frame that is smaller than 18 ft3. This design allows for students (and teachers) to easily access and see the different compartments of the system. Many of the problems we encountered came from plumping issues related to the fountain pump or the bell siphon; these were solved using applied physics principles. Other problems we faced, including biological were solved using more consistent testing and chemical reagents to stabilize our BAS. We learned ultimately that time is the key component in establishing a bacteria colony in any aquaponic system. We also learned that establishing bacteria is the most important step in setting up a successful aquaponic system whether on a large or miniaturized scale. Some aspects of this project that need further investigation include the importance of changing out the water of the system, whether dissolved oxygen is necessary for bacteria, and how/why consistently adding bacteria may stunt the ability of a bacteria colony to form. Conclusively we have found that it is not only possible to establish such an aquaponic system that is built by students, but it is also possible to maintain it. Further research is needed to estimate the Benchtop Aquaponic System's teaching potential within STEM classrooms.
Show less  Date Issued
 2015
 Identifier
 FSU_migr_uhm0507
 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