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Pages
 Title
 An Overview of Homotopy Type Theory and the Univalent Foundations of Mathematics.
 Creator

Dunn, Lawrence, Department of Mathematics
 Abstract/Description

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

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

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

Grazzini, Courtney, Department of Mathematics
 Abstract/Description

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Liu, Xinyang, Mio, Washington, Kumar, Piyush, Hurdal, Monica K., Quine, Jack, Cogan, Nick, Department of Mathematics, Florida State University
 Abstract/Description

We construct a framework for the analysis of shapes in Euclidean space of any dimension. In this framework, a shape is represented as a continuous map from a reference Riemannian manifold M. To quantify global shape differences, the framework employs a Sobolevtype metric considering information of both the position and the firstorder derivative at each point of the shape. Since firstorder derivatives are very sensitive to small variations, the derivative term in the metric is smoothed out...
Show moreWe construct a framework for the analysis of shapes in Euclidean space of any dimension. In this framework, a shape is represented as a continuous map from a reference Riemannian manifold M. To quantify global shape differences, the framework employs a Sobolevtype metric considering information of both the position and the firstorder derivative at each point of the shape. Since firstorder derivatives are very sensitive to small variations, the derivative term in the metric is smoothed out to reduce noise by a heat operator, which is constructed using the eigenvalues and eigenfunctions of the LaplaceBeltrami operator on M. A preshape space is constructed as an unit sphere of an inner product space, where each shape can be viewed as a point on it. A geodesic shape distance on the preshape space and an extrinsic distance are given based on the proposed metric. Besides global measures, we develop an energy function to quantify local shape divergence. Specifically, it is modified to reflect the magnitude of local shrinkage or expansion. In practical computations, the framework based on continuous representations is discretized using simplicial complex. To address issues in statistical shape analysis for a population of shapes, we present algorithms to calculate the mean shape and to perform principal component analysis on the tangent plane at the mean on the preshape space. The shape models and statistical tools are applied to three data sets of magnetic resonance (MR) scans of the hippocampus to study blindness and Alzheimer's disease. The shape of a hippocampus is represented using either a triangular mesh to represent its contour surface or a cubical mesh to represent its whole volume.
Show less  Date Issued
 2010
 Identifier
 FSU_migr_etd1148
 Format
 Thesis
 Title
 Monte Carlo and QuasiMonte Carlo Methods in Financial Derivative Pricing.
 Creator

Göncü, Ahmet, Okten, Giray, Huffer, Fred, Ewald, Brian, Kercheval, Alec N., Mascagni, Michael, Department of Mathematics, Florida State University
 Abstract/Description

In this dissertation, we discuss the generation of low discrepancy sequences, randomization of these sequences, and the transformation methods to generate normally distributed random variables. Two well known methods for generating normally distributed numbers are considered, namely; BoxMuller and inverse transformation methods. Some researchers and financial engineers have claimed that it is incorrect to use the BoxMuller method with lowdiscrepancy sequences, and instead, the inverse...
Show moreIn this dissertation, we discuss the generation of low discrepancy sequences, randomization of these sequences, and the transformation methods to generate normally distributed random variables. Two well known methods for generating normally distributed numbers are considered, namely; BoxMuller and inverse transformation methods. Some researchers and financial engineers have claimed that it is incorrect to use the BoxMuller method with lowdiscrepancy sequences, and instead, the inverse transformation method should be used. We investigate the sensitivity of various computational finance problems with respect to different normal transformation methods. BoxMuller transformation method is theoretically justified in the context of the quasiMonte Carlo by showing that the same error bounds apply for BoxMuller transformed point sets. Furthermore, new error bounds are derived for financial derivative pricing problems and for an isotropic integration problem where the integrand is a function of the Euclidean norm. Theoretical results are derived for financial derivative pricing problems; such as European call, Asian geometric, and Binary options with a convergence rate of 1/N. A stratified BoxMuller algorithm is introduced as an alternative to BoxMuller and inverse transformation methods, and new numerical evidence is presented in favor of this method. Finally, a statistical test for pseudorandom numbers is adapted for measuring the uniformity of transformed low discrepancy sequences.
Show less  Date Issued
 2009
 Identifier
 FSU_migr_etd4144
 Format
 Thesis
 Title
 Closed Form Solutions of Linear Difference Equations.
 Creator

Cha, Yongjae, Van Hoeij, Mark, Van Engelen, Robert A., Agashe, Amod, Aldrovandi, Ettore, Aluﬃ, Paolo, Department of Mathematics, Florida State University
 Abstract/Description

In this thesis we present an algorithm that finds closed form solutions for homogeneous linear recurrence equations. The key idea is transforming an input operator Linp to an operator Lg with known solutions. The main problem of this idea is how to find a solved equation Lg to which Linp can be reduced. To solve this problem, we use local data of a difference operator, that is invariant under the transformation.
 Date Issued
 2011
 Identifier
 FSU_migr_etd3960
 Format
 Thesis
 Title
 Computational Aeroacoustics Cascade Model of Fan Noise.
 Creator

Lepoudre, Philip P., Tam, Christopher, Shih, Chiang, Gallivan, Kyle, Hussaini, Yousuff, Wang, Xiaoming, Department of Mathematics, Florida State University
 Abstract/Description

A Computational Aeroacoustics [CAA] cascade model has been built to study the generation and propagation mechanisms of noise resulting from the interaction of the fan and outlet guide vanes in a highbypass ratio turbofan engine. Also called rotorstator interaction noise, this noise source is a dominant contributor to the total tone and broadband noise levels produced by the engine, and therefore an improved understanding of the noise generation processes will assist in developing successful...
Show moreA Computational Aeroacoustics [CAA] cascade model has been built to study the generation and propagation mechanisms of noise resulting from the interaction of the fan and outlet guide vanes in a highbypass ratio turbofan engine. Also called rotorstator interaction noise, this noise source is a dominant contributor to the total tone and broadband noise levels produced by the engine, and therefore an improved understanding of the noise generation processes will assist in developing successful noise reduction strategies. The CAA cascade model directly solves the nonlinear compressible NavierStokes equations on a twodimensional linear cascade representation of the fan blade rows. The model incorporates real blade geometry and the rotor and stator blade rows are joined together with a sliding interface method. The fullycoupled aerodynamic flow and acoustic field are directly captured in one high resolution simulation, and therefore the noise production and propagation mechanisms can be visualized and measured in detail. The model includes the fullycoupled physics of the nonlinear sound generation and propagation in swirling wake flow, as well as the transmission and reflection of sound through the blade rows. Previous models of rotorstator interaction noise have typically involved some level of decoupling between the blade rows in order to simplify the noise problem. Stateoftheart CAA methodology is used to produce a high quality numerical solution with minimal dissipation and dispersion of supported waves. The multisizemesh multitimestep Dispersion Relation Preserving [DRP] scheme is used for efficient computation of the wide range of length and time scales in the problem. A conformal mapping technique is used to generate bodyfitted grids around the blade shapes, which are overset on a background grid to create the blade rows. An optimized interpolation scheme is employed for data transfer between the overset grids and also to create the sliding interface between the moving rotorfixed grid and stationary statorfixed grid. A completely new computer program was built for efficient implementation of the cascade model on parallel computers using Message Passing Interface [MPI], and the code was shown to have good parallel performance. The program is a general purpose solver for CAA calculations involving complex flow and geometry, and is a valuable resource for future research. A representative rotorstator cascade with three rotor blades and five stator blades was constructed using real fan and outlet guide vane crosssectional shapes from the NASA Glenn 22in. model fan. A fully developed flow was obtained through the blade rows at the approach condition of the model fan. The performance of the sliding interface method was analyzed by comparing the solution on the rotorfixed and statorfixed grids at the coincident sliding interface mesh line, and the error in grid transfer interpolation was found to be comparable to the low error levels of the underlying DRP scheme. The simulation was used to produce animations of pressure and Mach contour, which provided a wealth of visual information about the flow field and noise generation and propagation behaviour in the cascade. The ability of the CAA cascade model to produce a high fidelity picture of the interaction noise has been demonstrated. In addition, the velocity and pressure fields were measured at various axial locations in the domain to quantify the mean and fluctuating components of the swirling wake flow between the blade rows and after the stator. The tone noise results were compared with interaction tone linear theory. The theory predicted the existence of a small number of propagating spinning wave modes at harmonics of rotor blade passing frequency [BPF]. In particular, the dominant interaction tone at BPF, labelled , was predicted to have two wave fronts in the circumferential domain period and to spin counter to the direction of the rotor. This interaction tone was clearly visible in animations of the pressure contour as an intense shock wave moving at an oblique spiral angle between the blade rows and after the stator. The wave shape was measured using a moving average, and the high amplitude waveform showed characteristic nonlinear steepening, which calls into question the common assumption that the interaction tones can be adequately represented by single linear wave modes. The spinning modes in the solution were measured at various axial locations using a joint temporalspatial modal decomposition of the fluctuating pressure field, and very good agreement was observed with the modal content predicted by linear theory. The relationship of the mode spiral angle to blade stagger angle and the phase velocity of the spinning modes were shown to govern the transmission and reflection behaviour of the modes through the blade rows. The mode was reflected and frequency shifted by the rotor, and the reflected mode propagated through the stator blade row to the outlet. Only corotating modes were able to propagate through the rotor to the inlet, and hence the sound levels in the inlet were significantly lower than in the outlet. This behaviour is in good agreement with the trends observed in experimental studies of fan noise. The unsteady flow and surface pressure fluctuations around a stator blade were also measured. Spectral analysis of the surface pressure fluctuations revealed the highest sound pressure levels occurred near the blade leading edge and on the upper blade surface near the trailing edge. The sound source mechanisms on the stator blade are related to the fluctuating loading on the blade as it cuts through the rotor wake profile and experiences significant variation in the local angle of attack.
Show less  Date Issued
 2011
 Identifier
 FSU_migr_etd3115
 Format
 Thesis
 Title
 Envelopes, Duality, and Multipliers for Certain NonLocally Convex HardyLorentz Spaces.
 Creator

Lengfield, Marc, Oberlin, Daniel M., McKeague, Ian, Bellenot, Steven, Nolder, Craig, Department of Mathematics, Florida State University
 Abstract/Description

This dissertation is a study of the HardyLorentz spaces Hp,q for indices in the range 0 < p < 1, 0 < q ≤ ∞. More precisely, for p as indicated, we describe the dual spaces and Banach envelopes of the spaces Hp,q for finite values of q and do the same for H0p,∞, the closure of the polynomials in Hp,∞. In addition, we determine the sBanach envelopes for the spaces Hp,q in the cases 0 < q < p < s ≤ 1 and 0 < p < q ≤ s ≤ 1. As an application of our results we determine the multiplier spaces (Hp...
Show moreThis dissertation is a study of the HardyLorentz spaces Hp,q for indices in the range 0 < p < 1, 0 < q ≤ ∞. More precisely, for p as indicated, we describe the dual spaces and Banach envelopes of the spaces Hp,q for finite values of q and do the same for H0p,∞, the closure of the polynomials in Hp,∞. In addition, we determine the sBanach envelopes for the spaces Hp,q in the cases 0 < q < p < s ≤ 1 and 0 < p < q ≤ s ≤ 1. As an application of our results we determine the multiplier spaces (Hp,q, ℓs) for 0 < p < 1, 0 < q, s ≤ ∞.
Show less  Date Issued
 2004
 Identifier
 FSU_migr_etd3131
 Format
 Thesis
 Title
 On Picard 2Stacks and Length 3 Complexes of Abelian Sheaves.
 Creator

Tatar, Ahmet Emin, Aldrovandi, Ettore, Capstick, Simon, Agashe, Amod, Aluﬃ, Paolo, Klassen, Eric, Department of Mathematics, Florida State University
 Abstract/Description

In Seminaire de Geometrie Algebrique 4 (SGA4), Expose XVIII, Pierre Deligne proves that to any Picard stack one can associate a complex of abelian sheaves of length 2. He also studies the morphisms between such stacks and shows that such a morphism defines a class of fractions in the derived category of complexes of abelian sheaves of length 2. From these two preliminary results, he finally deduces that the derived category of complexes of abelian sheaves of length 2 is equivalent to the...
Show moreIn Seminaire de Geometrie Algebrique 4 (SGA4), Expose XVIII, Pierre Deligne proves that to any Picard stack one can associate a complex of abelian sheaves of length 2. He also studies the morphisms between such stacks and shows that such a morphism defines a class of fractions in the derived category of complexes of abelian sheaves of length 2. From these two preliminary results, he finally deduces that the derived category of complexes of abelian sheaves of length 2 is equivalent to the category of Picard stacks with morphisms being the isomorphism classes. In this dissertation, we generalize his work, following closely his steps in SGA4, to the case of Picard 2stacks. But this generalization requires first a clear description of a Picard 2category as well as of a 2functor between such 2categories that respects Picard structure. Once this has been done, we can talk about category of Picard 2stacks and prove that the derived category of complexes of abelian sheaves of length 3 is equivalent to the category of Picard 2stacks.
Show less  Date Issued
 2010
 Identifier
 FSU_migr_etd1674
 Format
 Thesis
 Title
 An Asymptotically Preserving Method for Multiphase Flow.
 Creator

Jemison, Matthew, Sussman, Mark, Nof, Doron, Cogan, Nick, Gallivan, Kyle, Wang, Xiaoming, Department of Mathematics, Florida State University
 Abstract/Description

A unified, asymptoticallypreserving method for simulating multiphase flows using an exactly mass, momentum, and energy conserving CellIntegrated SemiLagrangian advection algorithm is presented. The new algorithm uses a semiimplicit pressure update scheme that asymptotically preserves the standard incompressible pressure projection method in the limit of infinite sound speed. The asymptotically preserving attribute makes the new method applicable to compressible and incompressible flows,...
Show moreA unified, asymptoticallypreserving method for simulating multiphase flows using an exactly mass, momentum, and energy conserving CellIntegrated SemiLagrangian advection algorithm is presented. The new algorithm uses a semiimplicit pressure update scheme that asymptotically preserves the standard incompressible pressure projection method in the limit of infinite sound speed. The asymptotically preserving attribute makes the new method applicable to compressible and incompressible flows, including stiff materials, which enables large time steps characteristic of incompressible flow algorithms rather than the small time steps required by explicit methods. Shocks are captured and material discontinuities are tracked, without the aid of any approximate or exact Riemann solvers. The new method enables one to simulate the flow of multiple materials, each possessing a potentially exotic equation of state. Simulations of multiphase flow in one and two dimensions are presented which illustrate the effectiveness of the new algorithm at efficiently computing multiphase flows containing shock waves and material discontinuities with large ''impedance mismatch.'' Additionally, new techniques related to the MomentofFluid interface reconstruction are presented, including a novel, asymptoticallypreserving method for capturing ''filaments,'' and an improved method for initializing the MomentofFluid optimization problem on unstructured, triangular grids.
Show less  Date Issued
 2014
 Identifier
 FSU_migr_etd9012
 Format
 Thesis
 Title
 Asset Market Dynamics of Heterogeneous Agent Models with Learning.
 Creator

Guan, Yuanying, Beaumont, Paul M., Kercheval, Alec N., Marquis, Milton, MestertonGibbons, Mike, Nichols, Warren D., Department of Mathematics, Florida State University
 Abstract/Description

The standard Lucas asset pricing model makes two common assumptions of homogeneous agents and rational expectations equilibrium. However, these assumptions are unrealistic for real financial markets. In this work, we relax these assumptions and establish a Lucas type agentbased asset pricing model. We create an artificial economy with a single risky asset and populate it with heterogeneous, boundedly rational, utility maximizing, infinitely lived and forward looking agents. We restrict...
Show moreThe standard Lucas asset pricing model makes two common assumptions of homogeneous agents and rational expectations equilibrium. However, these assumptions are unrealistic for real financial markets. In this work, we relax these assumptions and establish a Lucas type agentbased asset pricing model. We create an artificial economy with a single risky asset and populate it with heterogeneous, boundedly rational, utility maximizing, infinitely lived and forward looking agents. We restrict agents' information by allowing them to use only available information when they make optimal choices. With independent, identically distributed market returns, agents are able to compute their policy functions and the equilibrium pricing function with Duffie's method (Duffie, 1988) without perfect information about the market. When agents are out of equilibrium, they simultaneously compute their policy functions with predictive pricing functions and use adaptive learning schemes to learn the motion of the correct pricing function. Agents are able to learn the correct equilibrium pricing function with certain risk and learning parameters. In some other cases, the market price has excess volatility and the trading volume is very high. Simulations of the market behavior show rich dynamics, including a whole cascade from period doubling bifurcations to chaos. We apply the full families theory (De Melo and Van Strien, 1993) to prove that the rich dynamics do not come from numerical errors but are embedded in the structure of our dynamical system.
Show less  Date Issued
 2011
 Identifier
 FSU_migr_etd3938
 Format
 Thesis
 Title
 Probabilistic Methods in Estimation and Prediction of Financial Models.
 Creator

Nguyen, Nguyet Thi, Okten, Giray, Hawkes, Lois, Case, Bettye Anne, Kim, Kyounghee, Nichols, Warren, Zhang, Jinfeng, Department of Mathematics, Florida State University
 Abstract/Description

Many computational finance problems can be classified into two categories: estimation and prediction. In estimation, one starts with a probability model and expresses the quantity of interest as an expected value or a probability of an event. These quantities are then computed either exactly, or numerically using methods such as numerical PDEs or Monte Carlo simulation. Many problems in derivative pricing and risk management are in this category. In prediction, the main objective is to use...
Show moreMany computational finance problems can be classified into two categories: estimation and prediction. In estimation, one starts with a probability model and expresses the quantity of interest as an expected value or a probability of an event. These quantities are then computed either exactly, or numerically using methods such as numerical PDEs or Monte Carlo simulation. Many problems in derivative pricing and risk management are in this category. In prediction, the main objective is to use methods such as machine learning, neural networks, or Markov chain models, to build a model, train it using historical data, and predict future behavior of some financial indicators. In this dissertation, we consider an estimation method known as the (randomized) quasiMonte Carlo method. We introduce an acceptancerejection algorithm for the quasiMonte Carlo method, which substantially increases the scope of applications where the method can be used efficiently. We prove a convergence result, and discuss examples from applied statistics and derivative pricing. In the second part of the dissertation, we present a novel prediction algorithm based on hidden Markov models. We use the algorithm to predict economic regimes, and stock prices, based on historical data.
Show less  Date Issued
 2014
 Identifier
 FSU_migr_etd9059
 Format
 Thesis
 Title
 Hypergeometric Solutions of Linear Differential Equations with Rational Function Coefficients.
 Creator

Kunwar, Vijay Jung, Van Hoeij, Mark, Van Engelen, Robert A., Agashe, Amod, Aldrovandi, Ettore, Hironaka, Eriko, Petersen, Kathleen, Department of Mathematics, Florida State...
Show moreKunwar, Vijay Jung, Van Hoeij, Mark, Van Engelen, Robert A., Agashe, Amod, Aldrovandi, Ettore, Hironaka, Eriko, Petersen, Kathleen, Department of Mathematics, Florida State University
Show less  Abstract/Description

Let L be a second order linear differential equation with rational function coefficients. We want to find a solution (if that exists) of L in terms of 2F1hypergeometric function. This thesis presents two algorithms to find such solution in the following cases: 1. L has five regular singularities where at least one of them is logarithmic. 2. L has hypergeometric solution of degree three, i.e, L is solvable in terms of 2F1(a,b;c  f) where f is a rational function of degree three.
 Date Issued
 2014
 Identifier
 FSU_migr_etd9021
 Format
 Thesis
 Title
 Constructing NonTrivial Elements of the ShafarevichTate Group of an Abelian Variety.
 Creator

Biswas, Saikat, Agashe, Amod, Aggarwal, Sudhir, Hironaka, Eriko, Van Hoeij, Mark, Aldrovandi, Ettore, Department of Mathematics, Florida State University
 Abstract/Description

The ShafarevichTate group of an elliptic curve is an important invariant of the curve whose conjectural finiteness can sometimes be used to determine the rank of the curve. The second part of the Birch and SwinnertonDyer (BSD) conjecture gives a conjectural formula for the order of the ShafarevichTate group of a elliptic curve in terms of other computable invariants of the curve. Cremona and Mazur initiated a theory that can often be used to verify the BSD conjecture by constructing non...
Show moreThe ShafarevichTate group of an elliptic curve is an important invariant of the curve whose conjectural finiteness can sometimes be used to determine the rank of the curve. The second part of the Birch and SwinnertonDyer (BSD) conjecture gives a conjectural formula for the order of the ShafarevichTate group of a elliptic curve in terms of other computable invariants of the curve. Cremona and Mazur initiated a theory that can often be used to verify the BSD conjecture by constructing nontrivial elements of the ShafarevichTate group of an elliptic curve by means of the MordellWeil group of an ambient curve. In this thesis, we extract a general theorem out of Cremona and Mazur's work and give precise conditions under which such a construction can be made. We then give an extension of our result which provides new theoretical evidence for the BSD conjecture. Finally, we prove a theorem that gives an alternative method to potentially construct nontrivial elements of the ShafarevichTate group of an elliptic curve by using the component groups of a second curve.
Show less  Date Issued
 2011
 Identifier
 FSU_migr_etd3717
 Format
 Thesis
 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
 Modeling, Analysis and Simulation of the StokesDarcy System with BeaversJoseph Interface Condition.
 Creator

Hua, Fei, Gunzburger, Max, Wang, Xiaoming, Hu, Xiaolong, Ewald, Brian, Horne, Rudy, Department of Mathematics, Florida State University
 Abstract/Description

In this dissertation, the coupling phenomenon of porous media flow and free flow is extensively studied, with an application to studying the solute transport in the groundwater system of karst aquifers. This dissertation consists of three major parts. The first part focuses on the modeling and wellposedness of the mathematical equations. The Stokes equations are used for the free flow part and the Darcy's law is used for the porous media flow part. This is called the StokesDarcy system. To...
Show moreIn this dissertation, the coupling phenomenon of porous media flow and free flow is extensively studied, with an application to studying the solute transport in the groundwater system of karst aquifers. This dissertation consists of three major parts. The first part focuses on the modeling and wellposedness of the mathematical equations. The Stokes equations are used for the free flow part and the Darcy's law is used for the porous media flow part. This is called the StokesDarcy system. To couple the two spatially nonoverlapping yet neighboring regions, BeaversJoseph interface condition is used and studied. We show that the transient StokeDarcy system with BeaversJoseph condition is wellposed when a proper scaling parameter is introduced that essentially brings the two physical processes to the same time scale. The steady state problem with BeaversJoseph condition is also studied and the wellpossedness is obtained under some assumptions of the BeaversJoseph parameter α. We then, in the second part, put the StokesDarcy system under finite element analysis. The analysis is conducted under the framework of a Gårding type inequality established in the first part. First, the spatially semidiscretized problem is studied. Then, a Stokes type projection is devised to aid the convergence rate analysis for the fully discretized system. We obtain a rigorous error bound on the L2 norm with suitable regularity assumptions. The rate is then verified by numerical tests using the TaylorHood and quadratic elements pair for the free flow and porous media flow respectively. In the third part, the research is further carried on to compare the StokesDarcy system with conventional models that is used to study the karst aquifer, an important and susceptible groundwater system. In particular, the coupled continuum pipe flow (CCPF) model, the most conventionally used one, is picked as a the counter part to the StokesDarcy model. By using the laboratory experiment results as reference, we compare the two models in the aspects of flow rate, head distribution and ability to predict contaminant transport. We reject the conventional wisdom in choosing the exchange coefficient αex in the CCPF model and propose a new region where the coefficient should fall in. In the new parameter region, we compare the StokesDarcy model and CCFP model with different boundary and geometry setting to motivate the necessity of switching away from the crude and less physically reasonable CCPF model.
Show less  Date Issued
 2009
 Identifier
 FSU_migr_etd3686
 Format
 Thesis
 Title
 Asymptotic Behaviour of Convection in Porous Media.
 Creator

Parshad, Rana Durga, Wang, Xiaoming, Ye, Ming, Case, Bettye Anne, Ewald, Brian, N.Kercheval, Alec, Nolder, Craig, Department of Mathematics, Florida State University
 Abstract/Description

This dissertation investigates asymptotic behaviour of convection in a fluid saturated porous medium. We analyse the DarcyBoussinesq system under perturbation of the DarcyPrandtl number parameter. In very tightly packed media this parameter is of very large order and can be driven to infinity to yield the infinite DarcyPrandtl number model. We show convergence of global attractors and invariant measures of the DarcyBoussinesq system to that of the infinite DarcyPrandtl number model with...
Show moreThis dissertation investigates asymptotic behaviour of convection in a fluid saturated porous medium. We analyse the DarcyBoussinesq system under perturbation of the DarcyPrandtl number parameter. In very tightly packed media this parameter is of very large order and can be driven to infinity to yield the infinite DarcyPrandtl number model. We show convergence of global attractors and invariant measures of the DarcyBoussinesq system to that of the infinite DarcyPrandtl number model with respect to perturbation of the DarcyPrandtl number parameter.
Show less  Date Issued
 2009
 Identifier
 FSU_migr_etd2182
 Format
 Thesis
 Title
 Level Set and Conservative Level Set Methods on Dynamic Quadrilateral Grids.
 Creator

Simakhina, Svetlana, Sussman, Mark, Roper, Michael, Kopriva, David, Ewald, Brian, Peterson, Janet, Department of Mathematics, Florida State University
 Abstract/Description

The work in this thesis is motivated by the application of spray combustion. If one develops algorithms to simulate spray generation, for example the primary breakup of a liquid jet in a gas crossflow, then a bodyfitted or Lagrangian methods would require "surgery" in order to continue a simulation beyond the point at which a droplet is torn into multiple droplets. The liquid volume must also be conserved in simulating spray generation. In this thesis, an Eulerian front tracking method...
Show moreThe work in this thesis is motivated by the application of spray combustion. If one develops algorithms to simulate spray generation, for example the primary breakup of a liquid jet in a gas crossflow, then a bodyfitted or Lagrangian methods would require "surgery" in order to continue a simulation beyond the point at which a droplet is torn into multiple droplets. The liquid volume must also be conserved in simulating spray generation. In this thesis, an Eulerian front tracking method with conserved fluid volume is developed to represent and update an interface between two fluids. It's a level set (LS) method with global volume fix, and the underlying grid is a structured, dynamic, curvilinear grid. We compared our newly developed method to the coupled level set and volume of fluid method (CLSVOF) for two strategic test problems. The first problem, the rotation of a notched disk, tests for robustness. The second problem (proposed in this thesis), the deformation of a circular interface in an incompressible, deforming, velocity field, tests for order of accuracy. We found that for the notched disk problem, the CLSVOF method is superior to the new combined level set method/curvilinear grid method. For a given number of grid points, the CLSVOF method always outperforms the combined level set/curvilinear grid method. On the other hand, for the deformation of a circular interface problem, the combined level set/curvilinear grid method gives better accuracy than the CLSVOF method, for a given number of grid points. Unfortunately the new method is more expensive because a new mesh must be generated periodically. We note that the volume error of the new level set/curvilinear grid algorithm is comparable to that of the CLSVOF method for all test cases tried. We prove that the conservative level set (CLS) method has O(1) local truncation error in an advection scheme. The following developments of the conservative level set (CLS) method are presented in the thesis: new CLS function remapping algorithm and new CLS reinitialization algorithm. The new developments allow one to implement the CLS method on a dynamic quadrilateral grid but don't remedy the order of the method. A new algorithm for quasicubic interpolation is presented. Quasicubic interpolation has been used for local polynomial interpolation on an orthogonal mesh before, but never on a general, nonorthogonal curvilinear mesh. The new (tunnel quasicubic) algorithm enables one to find a global piecewise polynomial interpolation of degree three on an orthogonal mesh, and to find a local polynomial interpolation of degree three on a curvilinear mesh.
Show less  Date Issued
 2010
 Identifier
 FSU_migr_etd1724
 Format
 Thesis
 Title
 Constant Proportions Portfolio Strategies in an Evolutionary Context under a Dividend Factor Model.
 Creator

Mavroudis, Konstantinos, Nolder, Craig, Schlagenhauf, Don, Beaumont, Paul, Case, Bettye Anne, Kercheval, Alec, Sumners, De Witt, Department of Mathematics, Florida State University
 Abstract/Description

In this dissertation we explore the impact of various constantproportions investment strategies in an economic evolutionary market. Dividends are generated according to a new Dividend Factor Model. Furthermore, Dividends were estimated and calibrated from data using Principal Component Analysis and Factor Analysis. Moreover, we perform simulations to study the longrun outcome of an evolutionary competition with several well diversified constantproportions strategies, among them some...
Show moreIn this dissertation we explore the impact of various constantproportions investment strategies in an economic evolutionary market. Dividends are generated according to a new Dividend Factor Model. Furthermore, Dividends were estimated and calibrated from data using Principal Component Analysis and Factor Analysis. Moreover, we perform simulations to study the longrun outcome of an evolutionary competition with several well diversified constantproportions strategies, among them some innovative strategies. We present and compare a variety of simulations with dividends being artificially generated according to the many different versions of our model. Our simulation results are important for both theoretical and practical reasons. In theoretical terms we have a model where, although the true rational strategy is the only probable dominant strategy, it is also possible for some "behavioral" rules to perform better under specific circumstances. In practical terms we suggest new constantproportions strategies that could be superior for investors at least in the short run.
Show less  Date Issued
 2008
 Identifier
 FSU_migr_etd2654
 Format
 Thesis
 Title
 Finite Abelian Group Actions on Orientable Circle Bundles over Surfaces.
 Creator

Ibrahim, Caroline Maher Boulis, Heil, Wolfgang, Hollander, Myles, Hironaka, Eriko, Klassen, Eric, Department of Mathematics, Florida State University
 Abstract/Description

A finite group G acts freely on an orientable manifold M if each element of G is a homeomorphism of M, without fixed points, and the multiplication in G is the composition of homeomorphisms. The map from M to M/G of M to the orbit space is a regular cover map. Algebraically, associated with the Gaction is a surjective homomorphism from the fundamental group of M into G. Two Gactions are equivalent if there exists an orientation preserving homeomorphism on M, inducing the identity on G, that...
Show moreA finite group G acts freely on an orientable manifold M if each element of G is a homeomorphism of M, without fixed points, and the multiplication in G is the composition of homeomorphisms. The map from M to M/G of M to the orbit space is a regular cover map. Algebraically, associated with the Gaction is a surjective homomorphism from the fundamental group of M into G. Two Gactions are equivalent if there exists an orientation preserving homeomorphism on M, inducing the identity on G, that preserves the group action. This topological definition is translated to an algebraic definition as two Gactions are equivalent if and only if the associated surjections into G are equivalent via an automorphism of the fundamental group of M. For the manifolds M considered in this dissertation every automorphism of the fundamental group of M can be realized by a homeomorphism of M. Hence there is a onetoone correspondence between the topological and algebraic equivalence. The problem of classifying fixedpoint free finite abelian group actions on surfaces had been investigated by, among others, Nielsen, Smith and Zimmermann. Nielsen classifies cyclic actions on surfaces. He gives a list of automorphisms which he uses in his classification. Smith does the classification for special abelian groups. His approach is different from Nielsen's in the algebraic methods he uses. Zimmermann gives an algebraic solution to the classification of any finite abelian group action on closed surfaces. His technique is to get every surjective homomorphism from the fundamental group of the surface into G in normal form and then differentiate between the normal forms. In this dissertation we classify fixedpoint free finite abelian group actions on circle bundles. By results of Waldhausen every homeomorphism of M is isotopic to a fiber preserving homeomorphism; that is, it preserves the S1 factor of the bundle. This corresponds to the algebraic condition that any automorphism on the fundamental group of M preserves the center of the group. We use the same approach as that of Nielsen on surfaces. We give algorithms to bring every surjective homomorphism from the fundamental group of the bundle into the group G to normal form. From there we differentiate between the normal forms based on Nielsen's results. The results obtained are for circle bundles over surfaces of genus g greater than or equal to 2. A complete classification is given for the case that the circle bundle is a product bundle and G is a finite abelian group. We also obtain a complete classification of cyclic group actions and finite abelian group actions on circle bundles that are not product bundles.
Show less  Date Issued
 2004
 Identifier
 FSU_migr_etd3887
 Format
 Thesis
 Title
 Quantifying Phenotypic Variation Through Local Persistent Homology and Imaging.
 Creator

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

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

Gutierrez, Juan B. (Juan Bernardo), Hurdal, Monica K., Travis, Joseph, Case, Bettye Anne, Quine, Jack, Sumners, DeWitt, Bertram, Richard, Cogan, Nick G., Department of...
Show moreGutierrez, Juan B. (Juan Bernardo), Hurdal, Monica K., Travis, Joseph, Case, Bettye Anne, Quine, Jack, Sumners, DeWitt, Bertram, Richard, Cogan, Nick G., Department of Mathematics, Florida State University
Show less  Abstract/Description

This dissertation presents and evaluates a theoretical method of eradication of invasive species through the use of Trojan Y chromosomes. The mathematical analysis of the Trojan Y chromosome eradication strategy is presented for the ODE case and the PDE case in R. It is shown that is possible to cause local extinction of species that have XY sex determination systems as long as they are susceptible to sex reversal. The existence of global attractors is shown for this system, and global...
Show moreThis dissertation presents and evaluates a theoretical method of eradication of invasive species through the use of Trojan Y chromosomes. The mathematical analysis of the Trojan Y chromosome eradication strategy is presented for the ODE case and the PDE case in R. It is shown that is possible to cause local extinction of species that have XY sex determination systems as long as they are susceptible to sex reversal. The existence of global attractors is shown for this system, and global attractors are proposed as descriptors of the dynamics of the infinite dimensional system. The case of Poecilia formosa is studied as a natural case of Trojan X chromosomes; it is shown in this case that the combination of stochasticdependent dissipation and high sensitivity to perturbations can lead to coexistence of P. formosa and P. mexicana. Similarities between the Trojan X chromosomes and Trojan Y chromosomes cases indicate that local extinction could occur in practice for the latter.
Show less  Date Issued
 2009
 Identifier
 FSU_migr_etd3892
 Format
 Thesis
 Title
 Analysis and Approximation of a TwoBand GinzburgLandau Model of Superconductivity.
 Creator

Chan, WanKan, Gunzburger, Max, Peterson, Janet, Manousakis, Efstratios, Wang, Xiaoming, Department of Mathematics, Florida State University
 Abstract/Description

In 2001, the discovery of the intermetallic compound superconductor MgB2 having a critical temperature of 39K stirred up great interest in using a generalization of the GinzburgLandau model, namely the twoband timedependent GinzburgLandau (2BTDGL) equations, to model the phenomena of twoband superconductivity. In this work, various mathematical and numerical aspects of the twodimensional, isothermal, isotropic 2BTDGL equations in the presence of a timedependent applied magnetic field...
Show moreIn 2001, the discovery of the intermetallic compound superconductor MgB2 having a critical temperature of 39K stirred up great interest in using a generalization of the GinzburgLandau model, namely the twoband timedependent GinzburgLandau (2BTDGL) equations, to model the phenomena of twoband superconductivity. In this work, various mathematical and numerical aspects of the twodimensional, isothermal, isotropic 2BTDGL equations in the presence of a timedependent applied magnetic field and a timedependent applied current are investigated. A new gauge is proposed to facilitate the inclusion of a timedependent current into the model. There are three parts in this work. First, the 2BTDGL model which includes a timedependent applied current is derived. Then, assuming sufficient smoothness of the boundary of the domain, the applied magnetic field, and the applied current, the global existence, uniqueness and boundedness of weak solutions of the 2BTDGL equations are proved. Second, the existence, uniqueness, and stability of finite element approximations of the solutions are shown and error estimates are derived. Third, numerical experiments are presented and compared to some known results which are related to MgB2 or general twoband superconductivity. Some novel behaviors are also identified.
Show less  Date Issued
 2007
 Identifier
 FSU_migr_etd3923
 Format
 Thesis
 Title
 A Computational Study of Ion Conductance in the KcsA K⁺ Channel Using a NernstPlanck Model with Explicit Resident Ions.
 Creator

Jung, Yong Woon, Mascagni, Michael A., Huﬀer, Fred, Bowers, Philip, Klassen, Eric, Cogan, Nick, Department of Mathematics, Florida State University
 Abstract/Description

In this dissertation, we describe the biophysical mechanisms underlying the relationship between the structure and function of the KcsA K+ channel. Because of the conciseness of electrodiffusion theory and the computational advantages of a continuum approach, NernstPlanck (NP) type models such as the GoldmanHodgkinKatz (GHK) and PoissonNernstPlanck (PNP) models have been used to describe currents in ion channels. However, the standard PNP (SPNP) model is known to be inapplicable to...
Show moreIn this dissertation, we describe the biophysical mechanisms underlying the relationship between the structure and function of the KcsA K+ channel. Because of the conciseness of electrodiffusion theory and the computational advantages of a continuum approach, NernstPlanck (NP) type models such as the GoldmanHodgkinKatz (GHK) and PoissonNernstPlanck (PNP) models have been used to describe currents in ion channels. However, the standard PNP (SPNP) model is known to be inapplicable to narrow ion channels because it cannot handle discrete ion properties. To overcome this weakness, we formulated the explicit resident ions NernstPlanck (ERINP) model, which applies a local explicit model where the continuum model fails. Then we tested the effects of the ERI Coulomb potential, the ERI induced potential, and the ERI dielectric constant for ion conductance were tested in the ERINP model. Using the currentvoltage (IV ) and currentconcentration (IC) relationships determined from the ERINP model, we discovered biologically significant information that is unobtainable from the traditional continuum model. The mathematical analysis of the K+ ion dynamics revealed a tight structurefunction system with a shallow well, a deep well, and two K+ ions resident in the selectivity filter. We also demonstrated that the ERINP model not only reproduced the experimental results with a realistic set of parameters, it also reduced CPU costs.
Show less  Date Issued
 2010
 Identifier
 FSU_migr_etd3741
 Format
 Thesis
 Title
 The Fractal Nature of Lightning: An Investigation of the Fractal Relationship of the Structure of Lightning to Terrain.
 Creator

GrahamJones, Brian Clay, Hunter, Christopher, Elsner, James B., Bellenot, Steve, Department of Mathematics, Florida State University
 Abstract/Description

This study focuses on the relationship between the structure of lightning and how it may or may not be related to the topography below it.
 Date Issued
 2006
 Identifier
 FSU_migr_etd4055
 Format
 Thesis
 Title
 All Speed MultiPhase Flow Solvers.
 Creator

Kadioglu, Samet Y., Sussman, Mark, Telotte, John, Hussaini, Yousuﬀ, Wang, Qi, Erlebacher, Gordon, Department of Mathematics, Florida State University
 Abstract/Description

A new second order primitive preconditioner technique (an all speed method) for solving all speed single/multiphase flow is presented. With this technique, one can compute both compressible and incompressible flows with Machuniform accuracy and efficiency (i.e., accuracy and efficiency of the method are independent of Mach numbers). The new primitive preconditioner (all speed/Mach uniform) technique can handle both strong and weak shocks, providing highly resolved shock solutions together...
Show moreA new second order primitive preconditioner technique (an all speed method) for solving all speed single/multiphase flow is presented. With this technique, one can compute both compressible and incompressible flows with Machuniform accuracy and efficiency (i.e., accuracy and efficiency of the method are independent of Mach numbers). The new primitive preconditioner (all speed/Mach uniform) technique can handle both strong and weak shocks, providing highly resolved shock solutions together with correct shock speeds. In addition, the new technique performs very well at the zero Mach limit. In the case of multiphase flow, the new primitive preconditioner technique enables one to accurately treat phase boundaries in which there is a large impedance mismatch. When solving multidimensional all speed multiphase flows, we introduce adaptive solution techniques which exploit the advantages of Machuniform methods. We compute a variety of problems from low (low speed) to high Mach number (high speed) flows including multiphase flow tests, i.e, computing the growth and collapse of adiabatic bubbles for study of underwater explosions
Show less  Date Issued
 2005
 Identifier
 FSU_migr_etd3391
 Format
 Thesis
 Title
 Efficient and Accurate Numerical Schemes for Long Time Statistical Properties of the Infinite Prandtl Number Model for Convection.
 Creator

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

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

Jones, Deborah, Aluffi, Paolo, Aldrovandi, Ettore, Hironaka., Eriko, Klassen, Eric, Reina, Laura, Department of Mathematics, Florida State University
 Abstract/Description

This dissertation studies certain intersection numbers of exceptional divisions arising from blowing up subspaces of lattices associated to graphs. These permit the computation of the Segre class of a scheme associated to the graph/lattice. Explicit formulas are provided for lattices associated to trees and several patterns among these numbers are explored. The problem can be related to the study of socalled Cremona transformations. It is shown that the geometry of such transformations...
Show moreThis dissertation studies certain intersection numbers of exceptional divisions arising from blowing up subspaces of lattices associated to graphs. These permit the computation of the Segre class of a scheme associated to the graph/lattice. Explicit formulas are provided for lattices associated to trees and several patterns among these numbers are explored. The problem can be related to the study of socalled Cremona transformations. It is shown that the geometry of such transformations explain a certain symmetry pattern we discovered.
Show less  Date Issued
 2003
 Identifier
 FSU_migr_etd3426
 Format
 Thesis
 Title
 GPU Computing in Financial Engineering.
 Creator

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

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

Celestino, Christian Edgar Laing, Sumners, De Witt, Greenbaum, Nancy, Mio, Washington, Hurdal, Monica, Department of Mathematics, Florida State University
 Abstract/Description

Given an edgeoriented polygonal graph in R3, we describe a method for computing the writhe as the average of weighted directional writhe numbers of the graph in a few directions. These directions are determined by the graph and the weights are determined by areas of pathconnected open regions on the unit sphere. Within each open region, the directional writhe is constant. We developed formulas for the writhe of polygons on Bravais lattices and a few crystallographic groups, and discuss...
Show moreGiven an edgeoriented polygonal graph in R3, we describe a method for computing the writhe as the average of weighted directional writhe numbers of the graph in a few directions. These directions are determined by the graph and the weights are determined by areas of pathconnected open regions on the unit sphere. Within each open region, the directional writhe is constant. We developed formulas for the writhe of polygons on Bravais lattices and a few crystallographic groups, and discuss applications to ring polymers. In addition, we obtained a closed formula for the writhe for graphs which extends the formula for the writhe of a polygon in R3, including the important special case of writhe of embedded open arcs. Additionally, we have developed shape descriptors based on a family of geometric measures for the purpose of classification and identification of shape differences for graphs. These shape descriptors involve combinations of writhe and average crossing numbers of curves, as well as total curvature, ropelength and thickness. We have applied these shape descriptors to RNA tertiary structures and families of sulcal curves from human brain surfaces. Preliminary results give an automatic method to distinguish RNA motifs. Clear differentiation among tRNA and/or ribozymes, and a distinction among mesophilic and thermophilic tRNA is shown. In addition, we notice a direct correlation between the length of an RNA backbone and its mean average crossing number which is described accurately by a power function. As a neuroscience application, human brain surfaces were extracted from MRI scans of human brains. In our preliminary results, an automatic differentiation between sulcal paths from the left or right hemispheres, an age differentiation and a malefemale classification were achieved.
Show less  Date Issued
 2007
 Identifier
 FSU_migr_etd3314
 Format
 Thesis
 Title
 A Mathematical Model of Cerebral Cortical Folding Development Based on a Biomechanical Hypothesis.
 Creator

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

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

Hu, Wenbo, Kercheval, Alec, Huﬀer, Fred, Case, Bettye, Nichols, Warren, Nolder, Craig, Department of Mathematics, Florida State University
 Abstract/Description

The distributions of many financial quantities are wellknown to have heavy tails, exhibit skewness, and have other nonGaussian characteristics. In this dissertation we study an especially promising family: the multivariate generalized hyperbolic distributions (GH). This family includes and generalizes the familiar Gaussian and Student t distributions, and the socalled skewed t distributions, among many others. The primary obstacle to the applications of such distributions is the numerical...
Show moreThe distributions of many financial quantities are wellknown to have heavy tails, exhibit skewness, and have other nonGaussian characteristics. In this dissertation we study an especially promising family: the multivariate generalized hyperbolic distributions (GH). This family includes and generalizes the familiar Gaussian and Student t distributions, and the socalled skewed t distributions, among many others. The primary obstacle to the applications of such distributions is the numerical difficulty of calibrating the distributional parameters to the data. In this dissertation we describe a way to stably calibrate GH distributions for a wider range of parameters than has previously been reported. In particular, we develop a version of the EM algorithm for calibrating GH distributions. This is a modification of methods proposed in McNeil, Frey, and Embrechts (2005), and generalizes the algorithm of Protassov (2004). Our algorithm extends the stability of the calibration procedure to a wide range of parameters, now including parameter values that maximize loglikelihood for our real market data sets. This allows for the first time certain GH distributions to be used in modeling contexts when previously they have been numerically intractable. Our algorithm enables us to make new uses of GH distributions in three financial applications. First, we forecast univariate ValueatRisk (VaR) for stock index returns, and we show in outofsample backtesting that the GH distributions outperform the Gaussian distribution. Second, we calculate an efficient frontier for equity portfolio optimization under the skewedt distribution and using Expected Shortfall as the risk measure. Here, we show that the Gaussian efficient frontier is actually unreachable if returns are skewed t distributed. Third, we build an intensitybased model to price Basket Credit Default Swaps by calibrating the skewed t distribution directly, without the need to separately calibrate xi the skewed t copula. To our knowledge this is the first use of the skewed t distribution in portfolio optimization and in portfolio credit risk.
Show less  Date Issued
 2005
 Identifier
 FSU_migr_etd3694
 Format
 Thesis
 Title
 Mathematical Models of Dengue Fever and Measures to Control It.
 Creator

Shen, Yingyun, MestertonGibbons, Mike, Schwartz, Daniel, Okten, Giray, Cogan, Nick, Ewald, Brian, Department of Mathematics, Florida State University
 Abstract/Description

In this dissertation, we build a compartment model to investigate the dynamics of spread of dengue fever in both human and mosquito populations. We study the demographic factors that influence equilibrium prevalence, and perform a sensitivity analysis on the basic reproduction number. Among several intervention measures, the effects of two potential control methods for dengue fever are estimated: introducing Wolbachia to the mosquito population and introducing vaccines to the human population...
Show moreIn this dissertation, we build a compartment model to investigate the dynamics of spread of dengue fever in both human and mosquito populations. We study the demographic factors that influence equilibrium prevalence, and perform a sensitivity analysis on the basic reproduction number. Among several intervention measures, the effects of two potential control methods for dengue fever are estimated: introducing Wolbachia to the mosquito population and introducing vaccines to the human population. A stochastic model for transmission of dengue fever is also built to explore the effect of some demographic factors.
Show less  Date Issued
 2014
 Identifier
 FSU_migr_etd9093
 Format
 Thesis
 Title
 Radically Elementary Stochastic Summation with Applications to Finance.
 Creator

Zhu, Ming, Nichols, Warren D., Kim, Kyounghee, Huﬀer, Fred W., Ewald, Brian, Kercheval, Alec N., Okten, Giray, Department of Mathematics, Florida State University
 Abstract/Description

This dissertation develops a nonstandard approach to probability, stochastic calculus and financial modeling, within the framework of the Radically Elementary Probability Theory of Edward Nelson. The fundamental objects of investigation are stochastic sums with respect to a martingale, defined on a finite probability space and indexed by a finite set. We study the external (nonstandard) properties of these sums, such as almost sure continuity of trajectories, the Lp property, and the...
Show moreThis dissertation develops a nonstandard approach to probability, stochastic calculus and financial modeling, within the framework of the Radically Elementary Probability Theory of Edward Nelson. The fundamental objects of investigation are stochastic sums with respect to a martingale, defined on a finite probability space and indexed by a finite set. We study the external (nonstandard) properties of these sums, such as almost sure continuity of trajectories, the Lp property, and the Lindeberg condition; we also study external properties of related processes, such as quadratic variation and proper time. Using the tools so developed, we obtain an ItôDoeblin formula for change of variable and a Girsanov theorem for change of measure in a quite general setting. We also obtain results that will aid us in the comparison of certain of the processes we investigate to conventional ones. We illustrate the theory by using general techniques to build stock models driven by Wiener walks, Poisson walks and their combinations, and show in each case that when our parameter processes are constant we recover the prices for European calls of the corresponding models that use conventional stochastic calculus. Finally, we exhibit a model driven by a nonstandard Wiener process that produces different prices for European calls than are given by the conventional BlackScholes model.
Show less  Date Issued
 2014
 Identifier
 FSU_migr_etd9125
 Format
 Thesis
 Title
 DNA Knotting: Occurrences, Consequences & Resolution.
 Creator

Mann, Jennifer Katherine, Sumners, De Witt L., Zechiedrich, E. Lynn, Greenbaum, Nancy L., Heil, Wolfgang, Quine, Jack, Department of Mathematics, Florida State University
 Abstract/Description

This dissertation applies knot theory, DNA topology, linear algebra, statistics, probability theory and statistical mechanics to address questions about knotted, doublestranded DNA. The three main investigations are the cellular effects of knotting, the biophysics of knotting/unknotting and the unknotting mechanism of human topoisomerase IIá. The cellular effects of knotting were done in collaboration with Rick Deibler. The statistical mechanics were done in collaboration with Zhirong Liu...
Show moreThis dissertation applies knot theory, DNA topology, linear algebra, statistics, probability theory and statistical mechanics to address questions about knotted, doublestranded DNA. The three main investigations are the cellular effects of knotting, the biophysics of knotting/unknotting and the unknotting mechanism of human topoisomerase IIá. The cellular effects of knotting were done in collaboration with Rick Deibler. The statistical mechanics were done in collaboration with Zhirong Liu and Hue Sun Chan. Cellular DNA knotting is driven by DNA compaction, topoisomerization, replication, supercoilingpromoted strand collision, and DNA selfinteractions resulting from transposition, sitespecific recombination, and transcription (Spengler, Stasiak, and Cozzarelli 1985; Heichman, Moskowitz, and Johnson 1991; Wasserman and Cozzarelli 1991; Sogo, Stasiak, MartinezRobles et al. 1999). Type II topoisomerases are ubiquitous, essential enzymes that interconvert DNA topoisomers to resolve knots. These enzymes pass one DNA helix through another by creating an enzymebridged transient break. Explicitly how type II topoisomerases recognize their substrate and decide where to unknot DNA is unknown. What are the biological consequences of unresolved cellular DNA knotting? We investigated the physiological consequences of the wellaccepted propensity of cellular DNA to collide and react with itself by analyzing the effects of plasmid recombination and knotting in E. coli using a sitespecific recombination system. Fluctuation assays were performed to determine mutation rates of the strains used in these experiments (Rosche and Foster 2000). Our results show that DNA knotting: (i) promotes replicon loss by blocking DNA replication, (ii) blocks gene transcription, (iii) increases antibiotic sensitivity and (iv) promotes genetic rearrangements at a rate which is four orders of magnitude greater than of an unknotted plasmid. If unresolved, DNA knots can be lethal and may help drive genetic evolution. The faster and more efficiently type II topoisomerase unknots, the less chance for these disastrous consequences. How do type II topoisomerases unknot, rather than knot? If type II topoisomerases act randomly on juxtapositions of two DNA helices, knots are produced with probability depending on the length of the circular DNA substrate. For example, random strand passage is equivalent to random cyclization of linear substrate, and random cyclization of 10.5 kb substrate produces about 3% DNA knots, mostly trefoils (Rybenkov, Cozzarelli, and Vologodskii 1993; Shaw and Wang 1993). However, experimental data show that type II topoisomerases unknot at a level up to 90fold the level achieved by steadystate random DNA strand passage (Rybenkov, Ullsperger, and Vologodskii et al. 1997). Various models have been suggested to explain these results and all of them assume that the enzyme directs the process. In contrast, our laboratory proposed (Buck and Zechiedrich 2004) that type II topoisomerases recognize the curvature of the two DNA helices within a juxtaposition and the resulting angle between the helices. Furthermore, the values of curvature and angle lie within their respective bounds, which are characteristic of DNA knots. Thus, our model uniquely proposes unknotting is directed by the DNA and not the protein. We used statistical mechanics to test this hypothesis. Using a lattice polymer model, we generated conformations from preexisting juxtaposition geometries and studied the resulting knot types. First we determined the statistical relationship between the local geometry of a juxtaposition of two chain segments and whether the loop is knotted globally. We calculated the HOMFLY (Freyd, Yetter, and Hoste et al. 1985) polynomial of each conformation to identify knot types. We found that hooked juxtapositions are far more likely to generate knots than free juxtapositions. Next we studied the transitions between initial and final knot/unknot states that resulted from a type II topoisomeraselike segment passage at the juxtaposition. Selective segment passages at free juxtapositions tended to increase knot probability. In contrast, segment passages at hooked juxtapositions caused more transitions from knotted to unknot states than vice versa, resulting in a steadystate knot probability much less than that at topological equilibrium. In agreement with experimental type II topoisomerase results, the tendency of a segment passage at a given juxtaposition to unknot is strongly correlated with the tendency of that segment passage to decatenate. These quantitative findings show that there exists discriminatory topological information in local juxtaposition geometries that could be utilized by the enzyme to unknot rather than knot. This contrasts with prior thought that the enzyme itself directs unknotting and strengthens the hypothesis proposed by our group that type II topoisomerases act on hooked rather than free juxtapositions. Will a type II topoisomerase resolve a DNA twist knot in one cycle of action? The group of knots known as twist knots is intriguing from both knot theoretical and biochemical perspectives. A twist knot consists of an interwound region with any number of crossings and a clasp with two crossings. By reversing one of the crossings in the clasp the twist knot is converted to the unknot. However, a crossing change in the interwound region produces a twist knot with two less nodes. Naturally occurring knots in cells are twist knots. The unknotting number, the minimal number of crossing reversals required to convert a knot to the unknot, is equal to one for any twist knot. Each crossing reversal performed by a type II topoisomerase requires energy. Within the cell, DNA knots might be pulled tight by forces such as those which accompany transcription, replication and segregation, thus increasing the likelihood of DNA damage. Therefore, it would be advantageous for type II topoisomerases to act on a crossing in the clasp region of a DNA twist knot, thus, resolving the DNA knot in a single step. The mathematical unknotting number corresponds to the smallest number of topoisomerase strand passage events needed to untie a DNA knot. In order to study unknotting of DNA knots by a type II topoisomerase, I used sitespecific recombination systems and a benchtop fermentor to isolate large quantities of knotted DNA. My data show that purified five and sevennoded twist knots are converted to the unknot by human topoisomerase IIá with no appearance of either trefoils or fivenoded twist knots which are possible intermediates if the enzyme acted on one of the interwound nodes. Consequently, these data suggest that type II topoisomerase may preferentially act upon the clasp region of a twist knot. We have uniquely combined biology, chemistry, physics and mathematics to gain insight into the mechanism of type II topoisomerases, which are an important class of drug targets. Our results suggest that DNA knotting alters DNA structure in a way that may drive type II topoisomerase resolution of DNA knots. Ultimately, the knowledge gained about type II topoisomerases and their unknotting mechanism may lead to the development of new drugs and treatments of human infectious diseases and cancer.
Show less  Date Issued
 2007
 Identifier
 FSU_migr_etd2754
 Format
 Thesis
 Title
 Solutions of Second Order Recurrence Relations.
 Creator

Levy, Giles, Van Hoeij, Mark, Van Engelen, Robert A., Aldrovandi, Ettore, Aluﬃ, Paolo, Department of Mathematics, Florida State University
 Abstract/Description

This thesis presents three algorithms each of which returns a transformation from a base equation to the input using transformations that preserve order and homogeneity (referred to as gttransformations). The first and third algorithm are new and the second algorithm is an improvement over prior algorithms for the second order case. The first algorithm `Find 2F1' finds a gttransformation to a recurrence relation satisfied by a hypergeometric series u(n) = hypergeom([a+n, b],[c],z), if such...
Show moreThis thesis presents three algorithms each of which returns a transformation from a base equation to the input using transformations that preserve order and homogeneity (referred to as gttransformations). The first and third algorithm are new and the second algorithm is an improvement over prior algorithms for the second order case. The first algorithm `Find 2F1' finds a gttransformation to a recurrence relation satisfied by a hypergeometric series u(n) = hypergeom([a+n, b],[c],z), if such a transformation exists. The second algorithm `Find Liouvillian' finds a gttransformation to a recurrence relation of the form u(n+2) + b(n)u(n) = 0 for some b(n) in C(n), if such a transformation exists. The third algorithm `Database Solver' takes advantage of a large database of sequences, `The OnLine Encyclopedia of Integer Sequences' maintained by Neil A. J. Sloane at AT&T Labs Research. It employs this database by using the recurrence relations that they satisfy as base equations from which to return a gttransformation, if such a transformation exists.
Show less  Date Issued
 2010
 Identifier
 FSU_migr_etd3099
 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
 Singleand MultipleObjective Stochastic Programming Models with Applications to Aerodynamics.
 Creator

Croicu, AnaMaria, Hussaini, M. Yousuﬀ, Srivastava, Anuj, Kopriva, David, Wang, Qi, Department of Mathematics, Florida State University
 Abstract/Description

Deterministic design assumes that there is no uncertainty in the modeling parameters, and as a consequence, there is no variability in the simulation outputs. Therefore, deterministic optimal designs that are obtained without taking into account uncertainty are usually unreliable. This is the case with transonic shape optimization, where the randomness in the cruise Mach number might have significant impact on the optimal geometric design. In this context, a stochastic search turns out to be...
Show moreDeterministic design assumes that there is no uncertainty in the modeling parameters, and as a consequence, there is no variability in the simulation outputs. Therefore, deterministic optimal designs that are obtained without taking into account uncertainty are usually unreliable. This is the case with transonic shape optimization, where the randomness in the cruise Mach number might have significant impact on the optimal geometric design. In this context, a stochastic search turns out to be more appropriate. Approaches to stochastic optimization have followed a variety of modeling philosophies, but little has been done to systematically compare different models. The goal of this thesis is to present a comparison between two stochastic optimization algorithms, with the emphasis on applications, especially on the airfoil shape optimization. Singleobjective and multiobjective optimization programs are analyzed as well. The relationship between the expected minimum value (EMV) criterion and the minimum expected value (MEV) criterion is explored, and it is shown that, under favorable conditions, a better optimal point could be obtained via the EMV approach. Unfortunately, the advantages of using the EMV approach are far outweighed by the prohibitive exorbitant computational cost.
Show less  Date Issued
 2005
 Identifier
 FSU_migr_etd3027
 Format
 Thesis
 Title
 Massively Parallel Algorithms for CFD Simulation and Optimization on Heterogeneous ManyCore Architectures.
 Creator

Duffy, Austen C., Sussman, Mark, Hussaini, M. Yousuﬀ, Van Engelen, Robert, Cogan, Nick, Gallivan, Kyle, Department of Mathematics, Florida State University
 Abstract/Description

In this dissertation we provide new numerical algorithms for use in conjunction with simulation based design codes. These algorithms are designed and best suited to run on emerging heterogenous computing architectures which contain a combination of traditional multicore processors and new programmable manycore graphics processing units (GPUs). We have developed the following numerical algorithms (i) a new Multidirectional Search (MDS) method for PDE constrained optimization that utilizes a...
Show moreIn this dissertation we provide new numerical algorithms for use in conjunction with simulation based design codes. These algorithms are designed and best suited to run on emerging heterogenous computing architectures which contain a combination of traditional multicore processors and new programmable manycore graphics processing units (GPUs). We have developed the following numerical algorithms (i) a new Multidirectional Search (MDS) method for PDE constrained optimization that utilizes a Multigrid (MG) strategy to accelerate convergence, this algorithm is well suited for use on GPU clusters due to its parallel nature and is more scalable than adjoint methods (ii) a new GPU accelerated point implicit solver for the NASA FUN3D code (unstructured NavierStokes) that is written in the Compute Unified Device Architecture (CUDA) language, and which employs a novel GPU sharing model, (iii) novel GPU accelerated smoothers (developed using PGI Fortran with accelerator compiler directives) used to accelerate the multigrid preconditioned conjugate gradient method (MGPCG) on a single rectangular grid, and (iv) an improved pressure projection solver for adaptive meshes that is based on the MGPCG method which requires fewer grid point calculations and has potential for better scalability on hetergeneous clusters. It is shown that a multigrid  multidirectional search (MGMDS) method can run up to 5.5X faster than the MDS method when used on a one dimensional data assimilation problem. It is also shown that the new GPU accelerated point implicit solver of FUN3D is up to 5.5X times faster than the CPU version and that the solver can perform up to 40% faster on a single GPU being shared by four CPU cores. It is found that GPU accelerated smoothers for the MGPCG method on uniform grids can run over 2X faster than the nonaccelerated versions for 2D problems, and that the new MGPCG pressure projection solver for adaptive grids is up to 4X faster than the previous MG algorithm.
Show less  Date Issued
 2011
 Identifier
 FSU_migr_etd0651
 Format
 Thesis