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Pages
 Title
 Zeta regularized products and modular constants.
 Creator

Heydari, Shahryar., Florida State University
 Abstract/Description

The purpose of this dissertation is to, first outline a theory of Zeta regularized products which will work for sequences of complex numbers, and second to use this theory to compute Zeta regularized products and modular constants for sequences which are integer combinations of a fixed set of complex numbers., The gamma function $\Gamma(z)$ is represented as the ratio of two Zeta regularized products. This relation is then extended to define multiple gamma functions as the ratio of two...
Show moreThe purpose of this dissertation is to, first outline a theory of Zeta regularized products which will work for sequences of complex numbers, and second to use this theory to compute Zeta regularized products and modular constants for sequences which are integer combinations of a fixed set of complex numbers., The gamma function $\Gamma(z)$ is represented as the ratio of two Zeta regularized products. This relation is then extended to define multiple gamma functions as the ratio of two corresponding Zeta regularized products. A full account of the functional equations associated with multiple gamma functions is also given. The double gamma function is investigated in detail., Some other special functions are also discussed. Namely Jacobi's theta function $\theta\sb1$, the Weierstrass sigma function $\sigma(z),$ and $P(z\vert\tau)$ defined by, The determinant of the Laplacian on an ndimensional flat Torus is computed for $n \geq$ 2, by computing
Show less  Date Issued
 1992, 1992
 Identifier
 AAI9234228, 3087894, FSDT3087894, fsu:76704
 Format
 Document (PDF)
 Title
 Variational data assimilation with twodimensional shallow water equations and threedimensional Florida State University global spectral models.
 Creator

Wang, Zhi., Florida State University
 Abstract/Description

This thesis investigates the feasibility of the 4D variational data assimilation (VDA) applied to realistic situations and improves existing largescale unconstrained minimization algorithms. It first develops the second order adjoint (SOA) theory and applies it to a shallowwater equations (SWE) model on a limitedarea domain to calculate the condition numbers of the Hessian. Then the Hessian/vector product obtained by the SOA approach is applied to one of the most efficient minimization...
Show moreThis thesis investigates the feasibility of the 4D variational data assimilation (VDA) applied to realistic situations and improves existing largescale unconstrained minimization algorithms. It first develops the second order adjoint (SOA) theory and applies it to a shallowwater equations (SWE) model on a limitedarea domain to calculate the condition numbers of the Hessian. Then the Hessian/vector product obtained by the SOA approach is applied to one of the most efficient minimization algorithms, namely the truncatedNewton (TN) algorithm. The newly obtained algorithm is applied here to a limitedarea SWE model with model generated data where the initial conditions serve as control variables., Next, the thesis applies the VDA to an adiabatic version of the Florida State University Global Spectral Model (FSUGSM). The impact of observations distributed over the assimilation period is investigated. The efficiency of the 4D VDA is demonstrated with different sets of observations., In all of the previous experiments, it is assumed that the model is perfect, and so is the data. The solution of the problem will have a perfect fit to the data. This is of course unrealistic., The nudging data assimilation (NDA) technique consists in achieving a compromise between the model and observations by relaxing the model state towards the observations during the assimilation period by adding a nonphysical diffusiontype term to the model equations. Variational nudging data assimilation (VNDA) combines the VDA and NDA schemes in the most efficient way to determine optimally the best conditions and optimal nudging coefficients simultaneously. The humidity and other different parameterized physical processes are not included in the adjoint model integration. Thus the calculation of the gradients by the adjoint model is approximate since the forecast model is used in its fullphysics (diabatic) operational form.
Show less  Date Issued
 1993, 1993
 Identifier
 AAI9413304, 3088278, FSDT3088278, fsu:77082
 Format
 Document (PDF)
 Title
 Variance Reduction Techniques in Pricing Financial Derivatives.
 Creator

Salta, Emmanuel R., Okten, Giray, Srinivasan, Ashok, Case, Bettye Anne, Ewald, Brian, Nolder, Craig, Quine, John R., Department of Mathematics, Florida State University
 Abstract/Description

In this dissertation, we evaluate existing Monte Carlo estimators and develop new Monte Carlo estimators for pricing financial options with the goal of improving precision. In Chapter 2, we discuss the conditional expectation Monte Carlo estimator for pricing barrier options, and show that the formulas for this estimator that are used in the literature are incorrect. We provide a correct version of the formula. In Chapter 3, we focus on importance sampling methods in estimating the price of...
Show moreIn this dissertation, we evaluate existing Monte Carlo estimators and develop new Monte Carlo estimators for pricing financial options with the goal of improving precision. In Chapter 2, we discuss the conditional expectation Monte Carlo estimator for pricing barrier options, and show that the formulas for this estimator that are used in the literature are incorrect. We provide a correct version of the formula. In Chapter 3, we focus on importance sampling methods in estimating the price of barrier options. We show how a simulated annealing procedure can be used to estimate the parameters required in the importance sampling method. We end this chapter by evaluating the performance of the combined importance sampling and conditional expectation method. In Chapter 4, we analyze the estimators introduced by Ross and Shanthikumar in pricing barrier options and present a numerical example to test their performance.
Show less  Date Issued
 2008
 Identifier
 FSU_migr_etd2102
 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
 The value of socialized arithmetic drills and tests as positive factors in personality development.
 Creator

Rhoads, Louise Nash, Edwards, W., Florida State University
 Abstract/Description

"This paper is an account of a project which began primarily as an attempt to set up an activity program that would result in a high degree of accuracy in computational operations. It soon became evident that the activity was serving as an instrument for removing emotional blocks and relieving frustrations. A pleasant atmosphere of cooperative interprise took the place of classroom discipline, and the teacher found herself no longer a 'party to,' but a partner in a learning situation. The...
Show more"This paper is an account of a project which began primarily as an attempt to set up an activity program that would result in a high degree of accuracy in computational operations. It soon became evident that the activity was serving as an instrument for removing emotional blocks and relieving frustrations. A pleasant atmosphere of cooperative interprise took the place of classroom discipline, and the teacher found herself no longer a 'party to,' but a partner in a learning situation. The program has completed its fourth year. It has been revised, adapted, and added to, when the situation indicated a need. The classes taking part in the most recently revised activities are the ones whose performances are described in this paper"Introduction.
Show less  Date Issued
 1949
 Identifier
 FSU_historic_AKP4901
 Format
 Thesis
 Title
 Using RBFGenerated Quadrature Rules to Solve Nonlocal Continuum Models.
 Creator

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

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

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

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

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

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

Awosanya, Ayokunle, Jakubowski, Elizabeth, Wills, Herbert, Florida State University
 Abstract/Description

The results reported here are the product of the research titled: Using history in the teaching of mathematics. The subjects are students in two classes of algebra II course at Florida State University High School 36 students makes and females whose ages are mostly 18 and a few 17 and 16 years old. Algebra II is a course that is usually taken by high school seniors in 12th grade and a few 11th or 10th grade students which explains why the ages of the students are mostly 18 and a few 17...
Show moreThe results reported here are the product of the research titled: Using history in the teaching of mathematics. The subjects are students in two classes of algebra II course at Florida State University High School 36 students makes and females whose ages are mostly 18 and a few 17 and 16 years old. Algebra II is a course that is usually taken by high school seniors in 12th grade and a few 11th or 10th grade students which explains why the ages of the students are mostly 18 and a few 17 and 16 years old. In this investigation, both quantitative study and qualitative study were employed. The quantitative study was the main study a teaching experiment using quasiexperimental methodology that involves two groups group 1 and group 2. Group 1 is the control group, where various algebraic/mathematical concepts, or topics were taught or explained to students with the necessary formulas. Group 2 was the experimental group in which the accounts of the historical origin of algebraic/mathematical concepts and the mathematicians (Lewis Carroll, Archimedes, Pythagoras, and Sophie Germain) who brought forward or created the concepts were used to augment pedagogical lessons and exercises used for this study as the main feature of pedagogy. The qualitative study augmented the main quantitative study; it was a followup interview for students to probe further an indepth rationale for the research theme, using history in the teaching of mathematics. The statistical analysis results indicated that there is a significant difference in the mean of score for the control group students and the mean of scores of the experimental group is greater than the mean on scores of student's performance in the control group; and the interview questions responses indeed corroborate the fact that the use of history in teaching mathematics does improve learning and understanding of algebraic/mathematical concepts.
Show less  Date Issued
 2001
 Identifier
 FSU_historic_akx6428
 Format
 Thesis
 Title
 Using Deal.II to Solve Problems in Computational Fluid Dynamics.
 Creator

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

Finite element methods are a common tool to solve problems in computational fluid dynamics (CFD). This thesis explores the finite element package deal.ii and specific applications to incompressible CFD. Some notation and results from finite element theory are summarised, and a brief overview of some of the features of deal.ii is given. Following this, several CFD applications are presented, including the Stokes equations, the NavierStokes equations and the equations for Darcy flow in porous...
Show moreFinite element methods are a common tool to solve problems in computational fluid dynamics (CFD). This thesis explores the finite element package deal.ii and specific applications to incompressible CFD. Some notation and results from finite element theory are summarised, and a brief overview of some of the features of deal.ii is given. Following this, several CFD applications are presented, including the Stokes equations, the NavierStokes equations and the equations for Darcy flow in porous media. Comparison with benchmark problems are provided for the Stokes and NavierStokes equations and a case study looking at foam deformation is provided for Darcy flow. Code is provided where applicable.
Show less  Date Issued
 2016
 Identifier
 FSU_2016SP_Bystricky_fsu_0071N_13237
 Format
 Thesis
 Title
 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
 Unveiling Mechanisms for Electrical Activity Patterns in Neurons and Pituitary Cells Using Mathematical Modeling and Analysis.
 Creator

Sengul, Sevgi, Bertram, R. (Richard), TabakSznajder, Joel, Steinbock, Oliver, Quine, J. R. (John R.), Cogan, Nicholas G., Florida State University, College of Arts and Sciences...
Show moreSengul, Sevgi, Bertram, R. (Richard), TabakSznajder, Joel, Steinbock, Oliver, Quine, J. R. (John R.), Cogan, Nicholas G., Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

Computational neuroscience is a relatively new area that utilizes the computational analyses of neural systems as well as development of mathematical models. Analyses of neural systems help us to gain a deeper understanding of how different dynamical variables contribute to generate a given electrical behavior and modelling helps to explain experimental results or make predictions that can be tested experimentally. Due to the complexity of nervous system behavior, mathematical models often...
Show moreComputational neuroscience is a relatively new area that utilizes the computational analyses of neural systems as well as development of mathematical models. Analyses of neural systems help us to gain a deeper understanding of how different dynamical variables contribute to generate a given electrical behavior and modelling helps to explain experimental results or make predictions that can be tested experimentally. Due to the complexity of nervous system behavior, mathematical models often have many variables, however simpler lowerdimensional models are also important for understanding complex behavior. The work described herein utilizes both approaches in two separate, but related, studies in computational neuroscience. In the first study, we determined the contributions of two negative feedback mechanisms in the HodgkinHuxley model. Hodgkin and Huxley pioneered the use of mathematics in the description of an electrical impulse in a squid axon, developing a differential equation model that has provided a template for the behavior of many other neurons and other excitable cells. The HodgkinHuxley model has two negative feedback variables. The activation of a current (n), subtracts from the positive feedback responsible for the upstroke of an impulse. We call this subtractive negative feedback. Divisive feedback is provided by the inactivation of the positive feedback current (h), which divides the current. Why are there two negative feedback variables when only one type of negative feedback can produce rhythmic spiking? We detect if there is any advantage to having both subtractive and divisive negative feedback in the system and the respective contributions of each to rhythmic spiking by using three different metrics. The first measures the width of a parameter regime within which tonic spiking is a unique and stable limit cycle oscillation. The second metric, contribution analysis, measures how changes in the time scale parameters of the feedback variables affect the durations of the "active phase" during the action potential and the interspike interval "silent phase" of a tonically spiking model. The third metric, dominant scale analysis, measures a sensitivity of the voltage dynamics to each of the ionic currents and ranks their influence. xi In the second study, we used electrophysiology data provided from the collaborating lab of Mike Shipston combined with mathematical modelling to show how two different neurohormones regulate patterns of electrical activity in corticotrophs. Corticotroph cells of the anterior pituitary are electrically excitable cells and are an integral component of the stress the neuroendocrine response to stress. Stress activates neurons in the hypothalamus to release corticotrophinreleasing hormone (CRH) and arginine vasopressin (AVP). These neurohormones act on corticotrophs in the anterior pituitary gland, which secrete another hormone, adrenocorticotropic hormone (ACTH). ACTH enters the general circulation and stimulates the adrenal cortex to secrete corticosteroid (cortisol in humans). Corticotrophs display single spike activity under basal conditions which can be converted to complex bursting behavior after stimulation by the combination of CRH and AVP. Bursting is much more effective at releasing ACTH than is spiking, so this transition is physiologically important. We investigated the underlying mechanisms controlling this transition to bursting by mathematical modelling combined with the experimental data. The significance of the work in this dissertation is that it provides a very good example of how experiments and modelling can complement each other and how the right mathematical tools can increase our understanding of even a very old and much studied model.
Show less  Date Issued
 2014
 Identifier
 FSU_migr_etd9245
 Format
 Thesis
 Title
 Understanding the Induction of a Science Teacher: The Interaction of Identity and Context.
 Creator

Saka, Yavuz, Southerland, Sherry A., Kittleson, Julie, Hunter, Todd
 Abstract/Description

The demanding first years of teaching are a time when many teachers leave the teaching profession or discard the reformminded practice emphasized in teacher preparation. If we are to lessen teacher attrition and more effectively support teachers during their development, a better understanding of what occurs during their induction into the profession is needed. The question that drove this research was what factors influence how a beginning science teacher negotiates entry into teaching?...
Show moreThe demanding first years of teaching are a time when many teachers leave the teaching profession or discard the reformminded practice emphasized in teacher preparation. If we are to lessen teacher attrition and more effectively support teachers during their development, a better understanding of what occurs during their induction into the profession is needed. The question that drove this research was what factors influence how a beginning science teacher negotiates entry into teaching? Specifically, we sought to understand how a beginning science teacher's identities interact with the teaching context, how this interactions shapes his use of reform minded teaching practice, and how the negotiation of identity, context and practice influence a novice teacher's employment decisions. The study involved two years of data collection; data included classroom and school observations, questionnaires, interviews, and teaching artifacts (such as lesson plans and assessments). The results demonstrate how conflicts in identities, institutional expectations, and personal dispositions of this novice influenced his transition in becoming a member of his school community. Implications of these interactions for teacher preparation and support are provided.
Show less  Date Issued
 2013
 Identifier
 FSU_migr_ste_faculty_publications0012, 10.1007/s1116501293105
 Format
 Citation
 Title
 Uncertainty Quantification of Nonlinear Stochastic Phenomena.
 Creator

Jimenez, Edwin, Hussaini, M. Y., Srivastava, Anuj, Sussman, Mark, Kopriva, David, Department of Mathematics, Florida State University
 Abstract/Description

The present work quantifies uncertainty in two nonlinear problems using efficient sampling methods and polynomial chaos expansions. The first application is to the Rothermel wildland fire spread model. This model consists of a nonlinear system of algebraic and transcendental equations that relates environmental variables (input parameter groups) such as fuel type, fuel moisture, terrain, and wind to describe the fire environment. The second application quantifies aeroacoustic uncertainty of a...
Show moreThe present work quantifies uncertainty in two nonlinear problems using efficient sampling methods and polynomial chaos expansions. The first application is to the Rothermel wildland fire spread model. This model consists of a nonlinear system of algebraic and transcendental equations that relates environmental variables (input parameter groups) such as fuel type, fuel moisture, terrain, and wind to describe the fire environment. The second application quantifies aeroacoustic uncertainty of a Joukowski airfoil in stochastic vortical gusts. The stochastic gusts are described by random variables that model the gust amplitudes and frequency. The quantification of uncertainty is measured in terms of statistical moments. We construct moment estimates using a variance reduction procedure as well as an efficient stochastic collocation method.
Show less  Date Issued
 2009
 Identifier
 FSU_migr_etd3511
 Format
 Thesis
 Title
 Uncertainty Quantification and Data Fusion Based on DempsterShafer Theory.
 Creator

He, Yanyan, Hussaini, M. Yousuff, Oates, William S., Kopriva, David A., Sussman, Mark, Department of Mathematics, Florida State University
 Abstract/Description

Quantifying uncertainty in modeling and simulation is crucial since the parameters of the physical system are inherently nondeterministic and knowledge of the system embodied in the model is incomplete or inadequate. The most welldeveloped nonadditivemeasure theory  the DempsterShafer theory of evidence  is explored for uncertainty quantification and propagation. For ''uncertainty quantification," we propose the MinMax method to construct belief functions to represent uncertainty in...
Show moreQuantifying uncertainty in modeling and simulation is crucial since the parameters of the physical system are inherently nondeterministic and knowledge of the system embodied in the model is incomplete or inadequate. The most welldeveloped nonadditivemeasure theory  the DempsterShafer theory of evidence  is explored for uncertainty quantification and propagation. For ''uncertainty quantification," we propose the MinMax method to construct belief functions to represent uncertainty in the information (data set) involving the inseparably mixed type of uncertainties. Using the principle of minimum uncertainty and the concepts of entropy and specificity, the MinMax method specifies a partition of a finite interval on the real line and assigns belief masses to the uniform subintervals. The method is illustrated in a simple example and applied to the total uncertainty quantification in flight plan of two actual flights. For ''uncertainty propagation," we construct belief/probability density functions for the output or the statistics of the output given the belief/probability density functions for the uncertain input variables. Different approaches are introduced for aleatory uncertainty propagation, epistemic uncertainty propagation, and mixed type of uncertainty propagation. The impact of the uncertain input parameters on the model output is studied using these approaches in a simple example of aerodynamic flow: quasionedimensional nozzle flow. In the situation that multiple models are available for the same quantity of interest, the combination rules in the DempsterShafer theory can be utilized to integrate the predictions from the different models. In the present work, we propose a robust and comprehensive procedure to combine multiple bodies of evidence. It is robust in that it can combine multiple bodies of evidence, consistent or otherwise. It is comprehensive in the sense that it examines the bodies of evidence strongly conflicted with others, reconstructs the basic belief mass functions by discounting, and then fuses all the bodies of evidence using an optimally parametrized combination rule. The proposed combination procedure is applied to radiotherapy dose response outcome analysis.
Show less  Date Issued
 2013
 Identifier
 FSU_migr_etd8563
 Format
 Thesis
 Title
 Trend and VariablePhase Seasonality Estimation from Functional Data.
 Creator

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

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

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

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

Wen, John Fengping., Florida State University
 Abstract/Description

It has been shown that quasitriangular Hopf algebras (QTHAs) have been increasingly playing important roles in many areas of mathematics and physics. Some people believe that the theory of quantum groups will be the group theory of next century. The main goal of this thesis is to develop methods to determine the quasitriangular structures (Rmatrices) of a finite dimensional Hopf algebra over a field. The primary research that I have done in this thesis touched quantum groups from several...
Show moreIt has been shown that quasitriangular Hopf algebras (QTHAs) have been increasingly playing important roles in many areas of mathematics and physics. Some people believe that the theory of quantum groups will be the group theory of next century. The main goal of this thesis is to develop methods to determine the quasitriangular structures (Rmatrices) of a finite dimensional Hopf algebra over a field. The primary research that I have done in this thesis touched quantum groups from several directions. We prove the main results in this thesis that are stated as follows. Let H be a finite dimensional Hopf algebra over a field k. If H is unimodular, then the Rmatrices of H can be embedded in the center of the quantum double D(H), a QTHA associated to H that was discovered by Drinfel'd. If H is cosemisimple (equivalently, if the dual algebra of H is semisimple), then the Rmatrices of H correspond to central idempotents in D(H). Hence, for a finite dimensional cosemisimple Hopf algebra H (such as the group algebra of a finite group), one can possibly locate all the Rmatrices among the set of central idempotents of D(H), which is a finite set in many general contexts. We will see that there are many nontrivial Rmatrices arising from finite nonabelian groups. Nontrivial Rmatrices of nonabelian group algebras allow us to use groups to construct quantum groups.
Show less  Date Issued
 1996, 1996
 Identifier
 AAI9622873, 3088888, FSDT3088888, fsu:77687
 Format
 Document (PDF)
 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
 Theoretical, Computational, and Experimental Topics in Anterior Pituitary Cell Signaling.
 Creator

Fletcher, Patrick Allen, Bertram, R. (Richard), TabakSznajder, Joel, Yang, Wei, Cogan, Nicholas G., Mascagni, Michael V., Mio, Washington, Florida State University, College of...
Show moreFletcher, Patrick Allen, Bertram, R. (Richard), TabakSznajder, Joel, Yang, Wei, Cogan, Nicholas G., Mascagni, Michael V., Mio, Washington, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

The neuroendocrine system presents a diversity of patterns in hormone secretion across a large range of temporal scales. These patterns of secretion are crucial to many aspects of animal survival and behavior. This offers a wealth of important and interesting topics of study amenable to theoretical, experimental, and computational approaches. The pituitary gland plays a large role in neuroendocrine physiology by decoding signals from the brain, translating those signals and relaying them to...
Show moreThe neuroendocrine system presents a diversity of patterns in hormone secretion across a large range of temporal scales. These patterns of secretion are crucial to many aspects of animal survival and behavior. This offers a wealth of important and interesting topics of study amenable to theoretical, experimental, and computational approaches. The pituitary gland plays a large role in neuroendocrine physiology by decoding signals from the brain, translating those signals and relaying them to the glands and tissues of the body via the circulatory system. The cells of the anterior pituitary gland use a diverse set of biochemical processes to accomplish this task. The first step is the activation of a hormone receptor, which convert an extracellular signal to an intracellular one. These receptors in turn trigger cascades of chemical reactions involving enzymes and intracellular messengers, as well as changes in membrane potential dynamics and intracellular Ca2+ ion concentrations, all of which are important in controlling the production and secretion of hormones at appropriate levels. The first portion of this dissertation studies how such biochemical systems respond to patterned inputs, such as pulsatile patterns that are characteristic of hormones in the reproductive system, growth hormone, and stressresponsive hormones. We examine how models of some common intracellular signaling components such as receptor binding and protein phosphorylation respond to pulsed inputs. We then study how a preference for a specific input pulse frequency may arise from the interactions between transcription factors, which are molecules regulating the expression of genes and in this case responsible for production of a hormone. The next portion of this work describes experiments performed to demonstrate the direct actions of a hormone, oxytocin, on three pituitary cell types. Intracellular Ca2+ ion concentration was measured in primary cultured female rat pituitary cells in vitro using video microscopy and a fluorescent \casensitive dye. Oxytocin triggered increases in intracellular Ca2+ concentration as well as secretion of hormone from gonadotrophs, somatotrophs, and lactotrophs in a manner consistent with a direct action via the oxytocin receptor. Finally, computational tools are presented for harnessing the computational power of graphics processing units to rapidly compute numerical solutions to initial value problems such as those that arise in the study of pituitary cell electrophysiology. This provides tools for more rapid model exploration by computing the ensembles of parameter combinations required in parameter sweep and parameter space sampling computations in parallel. This allows rapid computations to be performed on inexpensive modern desktop or laptop computers. As a case study, we use a model of membrane potential dynamics of pituitary lactotroph cells, which is produces spiking and bursting patterns. We compare the actions of three K+ channel conductances known to be increased by the action of the hormone dopamine in lactotrophs. Paradoxically, low levels of dopamine stimulate Ca2+ increases despite only increasing these typically inhibitory conductances, a result previously hypothesized to be due to a transition from spiking to bursting. We compare the mechanisms by which one of the K+ channels is able to promote this stimulatory effect while the other two are not, and we find that this effect is robust in a population of model cells with randomized background parameters.
Show less  Date Issued
 2015
 Identifier
 FSU_migr_etd9598
 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
 Symmetry Solutions of the Multiphase Model with Biological Applications.
 Creator

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

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

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

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

Cannon, Ruby R., Curtis, H. A., Florida State University
 Abstract/Description

This paper is a study of the general mathematics program in the secondary school. The purpose of this study is to help the inexperienced teacher plan his program of work for the year. The teacher will not find a definite program that he may follow stepbystep, but suggestions that will be of help in developing an effective program.
 Date Issued
 1950
 Identifier
 FSU_historic_akv1201
 Format
 Thesis
 Title
 A STUDY OF STRONG SRINGS AND PRUEFER VMULTIPLICATION DOMAINS.
 Creator

MALIK, SAROJ BALA., The Florida State University
 Abstract/Description

In this work two types of rings have been studied, strong Srings and Prufer vmultiplication domains. Let R be a Prufer domain then R{X} is a strong Sring. For an integrally closed domain R, each tideal is a finite type videal if and only if each prime tideal is a finite type videal. The semigroup ring R{X;S} is a Prufer vmultiplication domain if and only if R and K{X;S} are. A PVMD is an Sdomain.
 Date Issued
 1979, 1979
 Identifier
 AAI8017668, 2989587, FSDT2989587, fsu:74094
 Format
 Document (PDF)
 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
 A Stock Market AgentBased Model Using Evolutionary Game Theory and Quantum Mechanical Formalism.
 Creator

Montin, Benoit S., Nolder, Craig A., Huﬀer, Fred W., Case, Bettye Anne, Beaumont, Paul M., Kercheval, Alec N., Sumners, DeWitt L., Department of Mathematics, Florida State...
Show moreMontin, Benoit S., Nolder, Craig A., Huﬀer, Fred W., Case, Bettye Anne, Beaumont, Paul M., Kercheval, Alec N., Sumners, DeWitt L., Department of Mathematics, Florida State University
Show less  Abstract/Description

The financial market is modelled as a complex selforganizing system. Three economic agents interact in a simplified economy and seek the maximization of their wealth. Replicator dynamics are used as a myopic behavioral rule to describe how agents learn and benefit from their experiences. Stock price fluctuations result from interactions between economic agents, budget constraints and conservation laws. Time is discrete. Invariant distributions over the state space, that is to say probability...
Show moreThe financial market is modelled as a complex selforganizing system. Three economic agents interact in a simplified economy and seek the maximization of their wealth. Replicator dynamics are used as a myopic behavioral rule to describe how agents learn and benefit from their experiences. Stock price fluctuations result from interactions between economic agents, budget constraints and conservation laws. Time is discrete. Invariant distributions over the state space, that is to say probability measures that remain unchanged by the oneperiod transition rule, form stochastic equilibria for our composite system. When agents make mistakes, there is a unique stochastic steady state which reflects the average and limit behavior. Convergence of the iterates occurs at a geometric rate in the total variation norm. Interestingly, when the probability of making a mistake tends to zero, the invariant distribution converges weakly to a stochastic equilibrium for the model without mistakes. Most agentbased computational economies heavily rely on simulations. Having adopted a simple representation of financial markets, we have been able to prove the above theoretical results and gain intuition on complexity economics. The impact of simple monetary policies on the limit stock price distribution, such as a decrease of the riskfree rate of interest, has been analyzed. Of interest as well, the limit stock log return distribution presents realworld features (skewed and leptokurtic) that more traditional models usually fail to explain or consider. Our artificial market is incomplete. The bid and ask prices of a vanilla Call option have been computed to illustrate option pricing in our setting.
Show less  Date Issued
 2004
 Identifier
 FSU_migr_etd2331
 Format
 Thesis
 Title
 Stochastic Volatility Extensions of the Swap Market Model.
 Creator

Tzigantcheva, Milena G. (Milena Gueorguieva), Nolder, Craig, Huﬀer, Fred, Case, Bettye Anne, Kercheval, Alec, Quine, Jack, Sumners, De Witt, Department of Mathematics, Florida...
Show moreTzigantcheva, Milena G. (Milena Gueorguieva), Nolder, Craig, Huﬀer, Fred, Case, Bettye Anne, Kercheval, Alec, Quine, Jack, Sumners, De Witt, Department of Mathematics, Florida State University
Show less  Abstract/Description

Two stochastic volatility extensions of the Swap Market Model, one with jumps and the other without, are derived. In both stochastic volatility extensions of the Swap Market Model the instantaneous volatility of the forward swap rates evolves according to a squareroot diffusion process. In the jumpdiffusion stochastic volatility extension of the Swap Market Model, the proportional lognormal jumps are applied to the swap rate dynamics. The speed, the flexibility and the accuracy of the fast...
Show moreTwo stochastic volatility extensions of the Swap Market Model, one with jumps and the other without, are derived. In both stochastic volatility extensions of the Swap Market Model the instantaneous volatility of the forward swap rates evolves according to a squareroot diffusion process. In the jumpdiffusion stochastic volatility extension of the Swap Market Model, the proportional lognormal jumps are applied to the swap rate dynamics. The speed, the flexibility and the accuracy of the fast fractional Fourier transform made possible a fast calibration to European swaption market prices. A specific functional form of the instantaneous swap rate volatility structure was used to meet the observed evidence that volatility of the instantaneous swap rate decreases with longer swaption maturity and with larger swaption tenors.
Show less  Date Issued
 2008
 Identifier
 FSU_migr_etd1762
 Format
 Thesis
 Title
 Stochastic Modeling of Financial Derivatives.
 Creator

Huang, Wanwan, Okten, Giray, Ewald, Brian, Huffer, Fred, Kercheval, Alec, Tang, Qihe, Kim, Kyounghee, Department of Mathematics, Florida State University
 Abstract/Description

The Coupled Additive Multiplicative Noises (CAM) model is introduced as a stochastic volatility process to extend the classical BlackScholes model. The fast Fourier transform (FFT) method is used to compute the values of the probability density function of the underlying assets under the CAM model, as well as the price of European call options. We discuss four dierent discretization schemes for the CAM model: the Euler scheme, the simplied weak Euler scheme, the order 2 weak Taylor scheme...
Show moreThe Coupled Additive Multiplicative Noises (CAM) model is introduced as a stochastic volatility process to extend the classical BlackScholes model. The fast Fourier transform (FFT) method is used to compute the values of the probability density function of the underlying assets under the CAM model, as well as the price of European call options. We discuss four dierent discretization schemes for the CAM model: the Euler scheme, the simplied weak Euler scheme, the order 2 weak Taylor scheme and the stochastic AdamsBashforth scheme. A martingale control variate method for pricing European call options is developed, and its advantages in terms of variance reduction are investigated numerically. We also develop Monte Carlo methods for estimating the sensitivities of the European call options under the CAM model.
Show less  Date Issued
 2013
 Identifier
 FSU_migr_etd7429
 Format
 Thesis
 Title
 Steady Dynamics in Shearing Flows of Nematic Liquid Crystalline Polymers.
 Creator

Liu, Fangyu, Wang, Qi, Sussman, Mark, Song, Kaisheng, Department of Mathematics, Florida State University
 Abstract/Description

The biaxiality of the steady state solutions and their stability to inplane disturbances in shearing flows of nematic liquid crystalline polymers are studied by using simplified Wang (2002) model. We obtain all the steady states of Wang model exhibit biaxial symmetry in which two directors are confined to the shearing plane and analysis their stability with respect to inplane disturbances at isolated Debra numbers and polymer concentration values.
 Date Issued
 2004
 Identifier
 FSU_migr_etd1190
 Format
 Thesis
 Title
 Statistical Analysis on Object Spaces with Applications.
 Creator

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

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

Pawlak, Kelly, Department of Physics
 Abstract/Description

In 1924 Bose and Einstein predicted that certain types of atomic gases, when cooled down to almost 0K, tend to condense (while remaining a gas) and form a "super atom" that behaves like a single wave rather than an assembly of particles. This phenomenon, known as BoseEinstein condensation (BEC), is counterintuitive as gases usually solidify at very low temperatures. Over the years, many scientists have failed trying to directly observe this phenomenon in laboratory experiments until 1995....
Show moreIn 1924 Bose and Einstein predicted that certain types of atomic gases, when cooled down to almost 0K, tend to condense (while remaining a gas) and form a "super atom" that behaves like a single wave rather than an assembly of particles. This phenomenon, known as BoseEinstein condensation (BEC), is counterintuitive as gases usually solidify at very low temperatures. Over the years, many scientists have failed trying to directly observe this phenomenon in laboratory experiments until 1995. Using a new experimental technique called laser cooling, two groups led by Wolfgang Ketterle and Eric Cornell (MIT) and Carl Wieman (CU) finally observed the formation of BECs and were awarded the physics Nobel Prize in 2001. BECs are now a very active topic in theoretical and experimental physics, having potential use in dozens of applications. Theoretically, the dynamics of the condensate are accurately modeled by the Gross—Pitaevskii equation (GPE). The analysis of the GPE is formidable due to its nonlinearity and therefore numerical simulations are necessary to survey basic BEC dynamics. As a result, there are many open questions regarding the behavior of BEC's and their dynamics. My research looks to answer the question of stability of the condensate. Given a certain experimental configuration, will the condensate remain stable so that data can be collected? Certain kinds of experimental variations are accurately modeled by a low frequency random potential (i.e. "noise"). By including this noise into mathematical workups of common experimental configurations, we can theoretically test the stability of the condensate. We use a 1D mathematical model with the assumption that the gas is dilute and noninteracting sans infrequent elastic collisions between the particles. The results are nontrivial, and show that the condensate favors periodicity.
Show less  Date Issued
 2013
 Identifier
 FSU_migr_undergradresearch0007
 Format
 Citation
 Title
 Spectral Methods for Morphometry.
 Creator

Bates, Jonathan R., Mio, Washington, Patrangenaru, Victor, Bertram, Richard, Liu, Xiuwen, Quine, Jack, Department of Mathematics, Florida State University
 Abstract/Description

Methods from shape analysis are used in morphometry, which is the quantitative analysis of macroscopic anatomical features. We assume that anatomy is flexible, and this brings us to the first problem of resolving how ``shape'' should be represented if it is allowed to bend. We are motivated to use representations of intrinsic geometry, which, for example, does not distinguish a flat sheet of paper from a rolled sheet. The spectral embedding (``heat kernel representation'') as a representation...
Show moreMethods from shape analysis are used in morphometry, which is the quantitative analysis of macroscopic anatomical features. We assume that anatomy is flexible, and this brings us to the first problem of resolving how ``shape'' should be represented if it is allowed to bend. We are motivated to use representations of intrinsic geometry, which, for example, does not distinguish a flat sheet of paper from a rolled sheet. The spectral embedding (``heat kernel representation'') as a representation of intrinsic geometry has many desirable features for computational anatomy and other areas of shape and data analysis. Several breakthroughs are made toward understanding and applying this representation. A novel shape representation is also considered and used for classification of control vs. affected groups. One goal of morphometry is to make statistically objective comparisons. Hence, once a suitable representation of shape is chosen, the second problem is to compare shapes. Shape comparison may occur at many levels of scale. The simplest comparisons are made with global features: volume, length, etc. Finer comparisons may occur at regional levels. A finest level of comparison can be made after matching all homologous points, that is, after finding a oneone correspondence between points on shapes. A point correspondence is found by a registration algorithm. A method for unsupervised shape registration is presented and applied to localize differences between control and affected groups. We focus on the 3D case, where imaging has made anatomical surface data readily available, yet the analysis challenging. Structural MRI of living persons is currently used to study the macroscopic effects on anatomy by neurodegenerative disease (e.g. Alzheimer's). In the earliest stages of Alzheimer's disease (AD), certain brain structures have been observed to have reduced volume, in autopsy and in vivo, including the hippocampus, putamen, and thalamus. Our methods will be applied to these surfaces.
Show less  Date Issued
 2013
 Identifier
 FSU_migr_etd7291
 Format
 Thesis
 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
 Sparse Grid Stochastic Collocation Techniques for the Numerical Solution of Partial Differential Equations with Random Input Data.
 Creator

Webster, Clayton G. (Clayton Garrett), Gunzburger, Max D., Gallivan, Kyle, Peterson, Janet, Tempone, Raul, Department of Mathematics, Florida State University
 Abstract/Description

The objective of this work is the development of novel, efficient and reliable sparse grid stochastic collocation methods for solving linear and nonlinear partial differential equations (PDEs) with random coefficients and forcing terms (input data of the model). These techniques consist of a Galerkin approximation in the physical domain and a collocation, in probability space, on sparse tensor product grids utilizing either ClenshawCurtis or Gaussian abscissas. Even in the presence of...
Show moreThe objective of this work is the development of novel, efficient and reliable sparse grid stochastic collocation methods for solving linear and nonlinear partial differential equations (PDEs) with random coefficients and forcing terms (input data of the model). These techniques consist of a Galerkin approximation in the physical domain and a collocation, in probability space, on sparse tensor product grids utilizing either ClenshawCurtis or Gaussian abscissas. Even in the presence of nonlinearities, the collocation approach leads to the solution of uncoupled deterministic problems, just as in the Monte Carlo method. The full tensor product spaces suffer from the curse of dimensionality since the dimension of the approximating space grows exponentially in the number of random variables. When this number is moderately large, we combine the advantages of isotropic sparse collocation with those of anisotropic full tensor product collocation: the first approach is effective for problems depending on random variables which weigh equally in the solution; the latter approach is ideal when solving highly anisotropic problems depending on a relatively small number of random variables. We also include a priori and a posteriori procedures to adapt the anisotropy of the sparse grids to each problem. These procedures are very effective for the problems under study. This work also provides a rigorous convergence analysis of the fully discrete problem and demonstrates: (sub)exponential convergence in the asymptotic regime and algebraic convergence in the preasymptotic regime, with respect to the total number of collocation points. Numerical examples illustrate the theoretical results and compare this approach with several others, including the standard Monte Carlo. For moderately large dimensional problems, the sparse grid approach with a properly chosen anisotropy is very efficient and superior to all examined methods. Due to the high cost of effecting each realization of the PDE this work also proposes the use of reducedorder models (ROMs) that assist in minimizing the cost of determining accurate statistical information about outputs from ensembles of realizations. We explore the use of ROMs, that greatly reduce the cost of determining approximate solutions, for determining outputs that depend on solutions of stochastic PDEs. One is then able to cheaply determine much larger ensembles, but this increase in sample size is countered by the lower fidelity of the ROM used to approximate the state. In the contexts of proper orthogonal decompositionbased ROMs, we explore these counteracting effects on the accuracy of statistical information about outputs determined from ensembles of solutions.
Show less  Date Issued
 2007
 Identifier
 FSU_migr_etd1223
 Format
 Thesis
 Title
 Sparse Approximation and Its Applications.
 Creator

Li, Qin, Erlebacher, Gordon, Wang, Xiaoming, Hart, Robert, Peterson, Janet, Sussman, Mark, Gallivan, Kyle A., Department of Mathematics, Florida State University
 Abstract/Description

In this thesis, we tackle the fundamental problem of how to effectively and reliably calculate sparse solutions to underdetermined systems of equations. This class of problems is found in applied mathematics, electrical engineering, statistics, geophysics, just to name a few. This dissertation concentrates on developing efficient and robust solution algorithms, and applies them in several applications in the field of signal/image processing. The first contribution concerns the development of...
Show moreIn this thesis, we tackle the fundamental problem of how to effectively and reliably calculate sparse solutions to underdetermined systems of equations. This class of problems is found in applied mathematics, electrical engineering, statistics, geophysics, just to name a few. This dissertation concentrates on developing efficient and robust solution algorithms, and applies them in several applications in the field of signal/image processing. The first contribution concerns the development of a new Iterative Shrinkage algorithm based on Surrogate Function, ISSFK, for finding the best Kterm approximation to an image. In this problem, we seek to represent an image with K elements from an overcomplete dictionary. We present a proof that this algorithm converges to a local minimum of the NP hard sparsity constrained optimization problem. In addition, we choose curvelets as the dictionary. The approximation obtained by our approach achieves higher PSNR than that of the best Kterm wavelet (CohenDaubechiesFauraue 97) approximation. We extends ISSF to the application of Morphological Component Analysis, which leads to the second contribution, a new algorithm MCAISSF with an adaptive thresholding strategy. The adaptive MCAISSF algorithm approximates the problem from the synthesis approach, and it is the only algorithm that incorporate an adaptive strategy to update its algorithmic parameter. Compared to the existent MCA algorithms, our method is more efficient and is parameter free in the thresdholding update. The third contribution concerns the nonconvex optimization problems in Compressive Sensing (CS), which is an important extension of sparse approximation. We propose two new iterative reweighted algorithms based on Alternating Direction Method of Multiplier, IR1ADM and IR2ADM, to solve the ellp,0.
Show less  Date Issued
 2011
 Identifier
 FSU_migr_etd1399
 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