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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
 Using Boundary ElementBased Nearfield Acoustic Holography to Predict the Source Pressures and Sound Field of an Acoustic Guitar.
 Creator

Goldsberry, Benjamin, Mathematics
 Abstract/Description

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

Israels, Brett, Department of Physics
 Abstract/Description

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

Radzimski, Vanessa, Mathematics
 Abstract/Description

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

Porwoll, Kathryn, School of Teacher Education
 Abstract/Description

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

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

In 1837 Dirichlet proved the infinitude of primes in all arithmetic coprime sequences. This was done by look at Dirichlet Lfunctions, Dirichlet series, Dirichlet characters (modulo k), and Euler Products. In this thesis, the necessary facts, theorems, and properties are shown in order to prove Dirichlet's Theorem, concluding with a proof of Dirichlet's Theorem.
 Date Issued
 2015
 Identifier
 FSU_migr_uhm0560
 Format
 Thesis
 Title
 Construction and Implementation of a BenchTop Aquaponic System as a Context for Teaching Science in Secondary Schools.
 Creator

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Colantuono, Giuseppe, Dewar, William, Chassignet, Eric, Clayson, Carol Anne, Program in Geophysical Fluid Dynamics, Florida State University
 Abstract/Description

This work attempts to express the change, induced by the introduction of stratification, of the Rossby eigenmodes of a closed basin with bottom topography, filled with a uniform fluid in the unperturbed configuration. Such a modification has been found in the output of a set of numerical experiments on the Argentine Basin and then computed analytically: stratification has been introduced in the mathematical form of a perturbation of a homogeneous fluid over a non flat bottom. It has been...
Show moreThis work attempts to express the change, induced by the introduction of stratification, of the Rossby eigenmodes of a closed basin with bottom topography, filled with a uniform fluid in the unperturbed configuration. Such a modification has been found in the output of a set of numerical experiments on the Argentine Basin and then computed analytically: stratification has been introduced in the mathematical form of a perturbation of a homogeneous fluid over a non flat bottom. It has been found that the eigenmodes lose their barotropic character and differences appear in the dynamical fields (velocity and pressure) from upper to lower layer. Moreover, the frequency of the modes changes; an analytical expression of this frequency correction as a function of stratification has been found. The impact on modal frequency of a geostrophic background flow has also been computed analytically. Some simple analytical results for an elementary geometrical setting have been determined.
Show less  Date Issued
 2009
 Identifier
 FSU_migr_etd3567
 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
 The Oneand TwoSample Problem for Data on Hilbert Manifolds with Applications to Shape Analysis.
 Creator

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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