Current Search: Research Repository (x) » * (x) » Thesis (x) » Department of Mathematics (x)
Search results
Pages
 Title
 Modeling the Synchronous Behavior of Pancreatic Islets.
 Creator

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

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

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

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

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

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

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

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

Salehy, Seyyed Navid, Kercheval, Alec N., Ewald, Brian, Fahim, Arash, Ökten, Giray, Huffer, Fred W. (Fred William), Florida State University, College of Arts and Sciences,...
Show moreSalehy, Seyyed Navid, Kercheval, Alec N., Ewald, Brian, Fahim, Arash, Ökten, Giray, Huffer, Fred W. (Fred William), Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

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

Sorribes Rodriguez, Inmaculada Concepcion, Jain, Harsh Vardhan, Sang, QingXiang, Cogan, Nicholas G., Moore, Matthew Nicholas J., Florida State University, College of Arts and...
Show moreSorribes Rodriguez, Inmaculada Concepcion, Jain, Harsh Vardhan, Sang, QingXiang, Cogan, Nicholas G., Moore, Matthew Nicholas J., Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

The diagnosis and treatment of gliomas continues to pose a significant challenge for oncologists who not only have to contend with managing acute neurological symptoms, but also the almost inevitable development of resistance to treatment. Indeed, the last 25 years have produced minimal advancements in treatment efficacy, even though significant efforts and resources have been invested in the quest for breakthroughs. This effort has not been restricted only to clinicians or oncologists, with...
Show moreThe diagnosis and treatment of gliomas continues to pose a significant challenge for oncologists who not only have to contend with managing acute neurological symptoms, but also the almost inevitable development of resistance to treatment. Indeed, the last 25 years have produced minimal advancements in treatment efficacy, even though significant efforts and resources have been invested in the quest for breakthroughs. This effort has not been restricted only to clinicians or oncologists, with mathematical modeling also playing an increasingly important role. A variety of models aimed at providing new insights into glioma growth and response to treatment have been proposed. Initially designed to capture fundamental behavior of tumor cells, such as growth and motility, these models quickly became wellestablished and multiple extensions have since been introduced. However, as increasing biological details of how tumor cells respond to treatment at cellular and subcellular levels are revealed, mathematical models need to include this state of the art knowledge. The work presented in this thesis seeks to do this by refocusing our attention back to the most fundamental question: why are gliomas fatal? Biologically, it is known that glioma lethality is driven by a fast growth that increases intracranial pressure resulting in lethal neurological damage, which current treatments fail to prevent due to tumor cell resistance to treatments such as chemotherapy. By creating mathematical models inspired by these key elements of glioma malignancy, the work presented here seeks to elucidate what drives resistance to chemotherapy and how to overcome or mitigate it, as well as how malignancy correlated with intracranial pressure dynamics. Thus, the work comprises two main parts: (1) in silico optimization of treatment strategies using chemotherapy coupled with novel cellrepair inhibitors currently in various stages of the clinical trial; and (2) a study of tumorinduced intracranial pressure and edema in gliomas of grade IIV. A wide variety of mathematical modeling techniques are used, that incorporate biomechanical, biochemical, pharmacokinetics, and pharmacodynamics aspects, and include a level of detail hitherto unconsidered. The proposed models are validated and analyzed by employing a diverse set of mathematical tools that range from structural identifiability, parameter estimation, to global and local sensitivity analysis. As a result of this work, we propose a treatment strategy that showed a 30% improvement in patient survival time over conventional treatment when treating heterogeneous brain tumors in silico. Moreover, the second part of this work demonstrates how the spatiotemporal dynamics of tumorinduced intracranial pressure correlate with cancer grade, providing a better understanding of the mechanisms that underlie increased intracranial pressure onset. Both projects come together as a first step towards a better understanding of the poor survival rates of patients afflicted with gliomas. They raise new questions about what characterizes the malignancy of primary brain tumors and how clinicians can fight it. Continued modeling effort in these directions has the potential to make an impact in the field of brain cancer diagnostics and treatment.
Show less  Date Issued
 2019
 Identifier
 2019_Summer_SorribesRodriguez_fsu_0071E_15274
 Format
 Thesis
 Title
 Mathematical Models of Prostate Cancer Progression and Response to Treatment.
 Creator

Barnaby, Johnna Pauline, Jain, Harsh Vardhan, Sang, QingXiang, Bertram, R. (Richard), Cogan, Nicholas G., Florida State University, College of Arts and Sciences, Department of...
Show moreBarnaby, Johnna Pauline, Jain, Harsh Vardhan, Sang, QingXiang, Bertram, R. (Richard), Cogan, Nicholas G., Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

Prostate cancer is currently a major pubic health issue due to the fact that it is the second most common cancer in men with an estimated 174,650 new cases diagnosed in 2019. While prostate cancer has a survival rate of nearly 100% for localized cancer that number quickly drops once the tumor has metastasized. Treatments for this metastasized state have shown varied response based on the phenotype of the tumor. Additionally, studies have show that the biomarker used to determine response to...
Show moreProstate cancer is currently a major pubic health issue due to the fact that it is the second most common cancer in men with an estimated 174,650 new cases diagnosed in 2019. While prostate cancer has a survival rate of nearly 100% for localized cancer that number quickly drops once the tumor has metastasized. Treatments for this metastasized state have shown varied response based on the phenotype of the tumor. Additionally, studies have show that the biomarker used to determine response to treatment, PSA, is an inadequate indicator of tumor progression and response to treatment. I propose two ODE models of prostate cancer and response to treatment that try to quantify these two diculties in treating prostate cancer. In the rst, I model blood vessel formation within the tumor to better understand the mechanisms by which PSA is able to leak into the blood, and how this may not correlate to tumor burden. I further extend this model to the human case where tumor volume is usually unknown. In the second model I explore combinations of treatments with androgen deprivation therapy and an immunotherapy vaccine. I propose a model of tumor and immune cell dynamics after treatment with androgen deprivation therapy. I then include additional treatment with a vaccine. With the aid of this model I simulate tumor response to various combinations of treatments, and determine an optimal treatment protocol. I also explore how resistance to androgen deprivation therapy may aect the ecacy of the vaccine. These models give a better understanding of tumor response to treatment. Conclusions from these models can aid with optimizing treatment response.
Show less  Date Issued
 2019
 Identifier
 2019_Summer_Barnaby_fsu_0071E_15328
 Format
 Thesis
 Title
 On the Use of Conformal Mappings, Invariants and Warpings in Investigations of the Cortical Surface.
 Creator

Eady, Carolyn M. (Carolyn Marie), Cogan, Nicholas G., Stroupe, M. Elizabeth (Margaret Elizabeth), Bowers, Philip L., Mio, Washington, Florida State University, College of Arts...
Show moreEady, Carolyn M. (Carolyn Marie), Cogan, Nicholas G., Stroupe, M. Elizabeth (Margaret Elizabeth), Bowers, Philip L., Mio, Washington, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

Brain mapping studies the structure and function of the brain through imaging techniques in an effort to relate function to specific anatomical regions. The highly folded nature of the human brain makes visualization difficult; thus neuroscientists desire quantitative methods to assist with characterization of anatomical differences between subjects. As cortical folding patterns are unique to each person, methods for quantifying differences across subjects are lacking. A first step in...
Show moreBrain mapping studies the structure and function of the brain through imaging techniques in an effort to relate function to specific anatomical regions. The highly folded nature of the human brain makes visualization difficult; thus neuroscientists desire quantitative methods to assist with characterization of anatomical differences between subjects. As cortical folding patterns are unique to each person, methods for quantifying differences across subjects are lacking. A first step in quantifying data is to define a normal, healthy brain in terms of both shape and function. Establishing spatial characteristics of a healthy brain, or an aggregate of healthy brains, will allow for better diagnosis and treatment of brain injuries and diseases. The research discussed in this thesis uses a twodimensional, or ``flat", map of the cortical surface based on a threedimensional reconstruction from medical images acquired using magnetic resonance imaging. Such mappings can be used to measure crosssubject differences, as well as the development of abnormalities in longitudinal studies of diseased patient brains. The ``flat" mappings we discuss are constructed using a mathematical approach known as circle packing. Our research makes use of conformal mappings and the properties preserved under them, defining ways to apply them to brain data. These conformally invariant properties take into account aspects of the threedimensional shape of the surface which are not easily visualized, and allow us to quantify them in a twodimensional result. In particular, we develop methods for calculation of extremal length and harmonic measure. We also give various forms of visualization of each invariant, and in some cases visualization of biological data in conjunction with our calculations. After calculating methods of comparison to be used indirectly, we offer two adaptations to an existing program which allow for the direct comparison of cortical surfaces. The data we use does not fit specific criteria of the program, so these adaptations are necessary to generalize the usage of the existing program. We again use the concept of circle packing for ``flat" mappings; however, the ingenuity of our adaptations lies in the connecting between these ``flat" maps. We follow a procedure for transforming the coordinates of a source surface into those of a destination surface, through a system of weighting vertices by barycentric or gyrobarycentric coordinates. Though our research focuses on a small region of the human brain, we proceed with the expectation that all methods can be generalized to other regions of the brain. As many psychological disorders have structural manifestations, the research can be used to investigate anomalies in brain data for such illnesses. Additionally, the methods we provide can be further generalized to surfaces other than the brain, laying groundwork for use in general investigations of two and threedimensional surfaces.
Show less  Date Issued
 2019
 Identifier
 2019_Summer_Eady_fsu_0071E_15275
 Format
 Thesis
 Title
 Global Optimization in Stellar Evolution Applications.
 Creator

Georgiadou, Antigoni, Plewa, Tomasz, Sussman, Mark, MeyerBäse, Anke, Ökten, Giray, Cogan, Nicholas G., Florida State University, College of Arts and Sciences, Department of...
Show moreGeorgiadou, Antigoni, Plewa, Tomasz, Sussman, Mark, MeyerBäse, Anke, Ökten, Giray, Cogan, Nicholas G., Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

Computer modeling is extensively used to probe structure and evolution of stars and planets. These computations allow astrophysicists to connect theoretical models of star formation and evolution to astronomical observations. Because stellar evolution is a highly complex process and long evolutionary timescales only allow one to obtain just a glimpse into the stellar life, finding initial stellar model parameters that lead to specific observed objects requires substantial effort and expert...
Show moreComputer modeling is extensively used to probe structure and evolution of stars and planets. These computations allow astrophysicists to connect theoretical models of star formation and evolution to astronomical observations. Because stellar evolution is a highly complex process and long evolutionary timescales only allow one to obtain just a glimpse into the stellar life, finding initial stellar model parameters that lead to specific observed objects requires substantial effort and expert knowledge. Typically probing parameter space of this problem is done using a trialanderror approach, which is inefficient, incomplete and prone to bias. We propose to decipher this scientific challenge by formulating a suitable constrained global optimization problem in which the optimization problem is solved using either the Controlled Random Search method (CRS) or the MIDACO evolutionary antcolony opti mization algorithm. The MESA stellar evolution code is used to calculate the cost func tion, and problem constraints are a combination of simple ranges of input parameters and observationsdependent penalty function. We use the resulting new package, MESAGO, to identify initial conditions for progenitor stars of young, nearby, and therefore relatively wellobserved supernovae. We present the results for our initial target objects, supernovae SN 1987A and SN 1993J. Luminosity and effective temperature are the observed variables and are part of the cost function. The initial progenitor mass, convective α parameter, semiconvection coef ficient αsc and metallicity Z are the control variables (input variables). The objective is to minimize the difference between an observed state for the star and a model state using control variables (input variables) given the constraint that the model is a reacting system of hyperbolic conservation laws, with bounds on initial conditions.
Show less  Date Issued
 2019
 Identifier
 2019_Fall_Georgiadou_fsu_0071E_15471
 Format
 Thesis
 Title
 Novel Numerical Analysis Methods, Using the WENO and WENOZ Algorithms, for Combining Observational Data with Model Predictions for Improving Forecasts.
 Creator

Liu, Feifan, Sussman, Mark, Huffer, Fred W. (Fred William), Gallivan, Kyle A., Moore, Matthew Nicholas J., Florida State University, College of Arts and Sciences, Department of...
Show moreLiu, Feifan, Sussman, Mark, Huffer, Fred W. (Fred William), Gallivan, Kyle A., Moore, Matthew Nicholas J., Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

“Data assimilation is a mathematical discipline that seeks to optimally combine theory (usually in the form of a numerical model) with observations.” — Wikipedia Strong constraint 4DVariational data assimilation (4DVar) seeks to find optimal initial condition for computing solutions to appropriate dynamical system equations. The optimal initial condition minimizes the weighted mismatch between (i) the computed initial condition and given “background” data and (ii) the computed state...
Show more“Data assimilation is a mathematical discipline that seeks to optimally combine theory (usually in the form of a numerical model) with observations.” — Wikipedia Strong constraint 4DVariational data assimilation (4DVar) seeks to find optimal initial condition for computing solutions to appropriate dynamical system equations. The optimal initial condition minimizes the weighted mismatch between (i) the computed initial condition and given “background” data and (ii) the computed state trajectory and observed data. The optimal initial condition and subsequent prediction are sensitive to the numerical method used to discretize the dynamical system equations. Especially, if the expected forecast has large gradients, the sensitivity of a prediction due to a given selected numerical method is significantly more pronounced. Weighted essential nonoscillatory(WENO) schemes, as a class of finite difference schemes, have been applied extensively in computational fluid dynamics for numerically solving problems with solutions which contain both strong shocks and rich smooth region structures. However, WENO, and its improved version WENOZ, have never been analyzed as discretization techniques for the data assimilation model equations (i.e. data assimilation constraint equations) as compared to existing stateoftheart discretization strategies. In this dissertation, we analyze the properties of the WENO and WENOZ schemes, when used in the “4DVar” process for the linear advection equation, and compare them quantitatively to the following schemes: Upwind Scheme, LaxFriedrichs Scheme, LaxWendroff and Quick Scheme. In our analysis, we also compare the performance of the WENO and WENOZ based 4DVar algorithms when used together with L1 norm total variation regularization technique which reduces Gibb’s phenomena of discontinuous observed data in the data assimilation process.
Show less  Date Issued
 2019
 Identifier
 2019_Fall_Liu_fsu_0071E_15552
 Format
 Thesis
 Title
 A Riemannian Approach for Computing Geodesics in Elastic Shape Space and Its Applications.
 Creator

You, Yaqing, Gallivan, Kyle A., Absil, PierreAntoine, Erlebacher, Gordon, Ökten, Giray, Sussman, Mark, Florida State University, College of Arts and Sciences, Department of...
Show moreYou, Yaqing, Gallivan, Kyle A., Absil, PierreAntoine, Erlebacher, Gordon, Ökten, Giray, Sussman, Mark, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

This dissertation proposes a Riemannian approach for computing geodesics for closed curves in elastic shape space. The application of two Riemannian unconstrained optimization algorithms, Riemannian Steepest Descent (RSD) algorithm and Limitedmemory Riemannian BroydenFletcherGoldfarbShanno (LRBFGS) algorithm are discussed in this dissertation. The application relies on the definition and computation for basic differential geometric components, namely tangent spaces and tangent vectors,...
Show moreThis dissertation proposes a Riemannian approach for computing geodesics for closed curves in elastic shape space. The application of two Riemannian unconstrained optimization algorithms, Riemannian Steepest Descent (RSD) algorithm and Limitedmemory Riemannian BroydenFletcherGoldfarbShanno (LRBFGS) algorithm are discussed in this dissertation. The application relies on the definition and computation for basic differential geometric components, namely tangent spaces and tangent vectors, Riemannian metrics, Riemannian gradient, as well as retraction and vector transport. The difference between this Riemannian approach to compute closed curve geodesics as well as accurate geodesic distance, the existing PathStraightening algorithm and the existing Riemannian approach to approximate distances between closed shapes, are also discussed in this dissertation. This dissertation summarizes the implementation details and techniques for both Riemannian algorithms to achieve the most efficiency. This dissertation also contains basic experiments and applications that illustrate the value of the proposed algorithms, along with comparison tests to the existing alternative approaches. It has been demonstrated by various tests that this proposed approach is superior in terms of time and performance compared to a stateoftheart distance computation algorithm, and has better performance in applications of shape distance when compared to the distance approximation algorithm. This dissertation applies the Riemannian geodesic computation algorithm to calculate Karcher mean of shapes. Algorithms that generate less accurate distances and geodesics are also implemented to compute shape mean. Test results demonstrate the fact that the proposed algorithm has better performance with sacrifice in time. A hybrid algorithm is then proposed, to start with the fast, less accurate algorithm and switch to the proposed accurate algorithm to get the gradient for Karcher mean problem. This dissertation also applies Karcher mean computation to unsupervised learning of shapes. Several clustering algorithms are tested with the distance computation algorithm and Karcher mean algorithm. Different versions of Karcher mean algorithm used are compared with tests. The performance of clustering algorithms are evaluated by various performance metrics.
Show less  Date Issued
 2018
 Identifier
 2018_Su_You_fsu_0071E_14686
 Format
 Thesis
 Title
 Riemannian Optimization Methods for Averaging Symmetric Positive Definite Matrices.
 Creator

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

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

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

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

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

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

Zhang, Xiping, Aluffi, Paolo, Piekarewicz, Jorge, Aldrovandi, Ettore, Petersen, Kathleen L., Hoeij, Mark van, Florida State University, College of Arts and Sciences, Department...
Show moreZhang, Xiping, Aluffi, Paolo, Piekarewicz, Jorge, Aldrovandi, Ettore, Petersen, Kathleen L., Hoeij, Mark van, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

Determinantal varieties parametrize spaces of matrices of given ranks. The main results of this dissertation are computations of intersectiontheoretic invariants of determinantal varieties. We focus on the ChernMather and ChernSchwartzMacPherson classes, on the characteristic cycles, and on topologically motivated invariants such as the local Euler obstruction. We obtain explicit formulas in both the ordinary and the torusequivariant setting, and formulate a conjecture concerning the...
Show moreDeterminantal varieties parametrize spaces of matrices of given ranks. The main results of this dissertation are computations of intersectiontheoretic invariants of determinantal varieties. We focus on the ChernMather and ChernSchwartzMacPherson classes, on the characteristic cycles, and on topologically motivated invariants such as the local Euler obstruction. We obtain explicit formulas in both the ordinary and the torusequivariant setting, and formulate a conjecture concerning the effectiveness of the ChernSchwartzMacPherson classes of determinantal varieties. We also prove a vanishing property for the ChernSchwartzMacPherson classes of general group orbits. As applications we obtain formulas for the sectional Euler characteristic of determinantal varieties and the microlocal indices of their intersection cohomology sheaf complexes. Moreover, for a close embedding we define the equivariant version of the Segre class and prove an equivariant formula for the ChernSchwartzMacPherson classes of hypersurfaces of projective varieties.
Show less  Date Issued
 2018
 Identifier
 2018_Sp_Zhang_fsu_0071N_14521
 Format
 Thesis
 Title
 Symmetric Surfaces and the Character Variety.
 Creator

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

We extend Culler and Shalen's work on constructing essential surfaces in 3manifolds to orbifolds. A consequence of this work is that every valuation on the canonical component that detects an essential surface, detects an essential surface that is preserved by every orientation preserving symmetry on the manifold. This Theorem applies to orientable hyperbolic manifolds, with orientation preserving symmetry group, whose quotient by this group is an orbifold with a flexible cusp, which is the...
Show moreWe extend Culler and Shalen's work on constructing essential surfaces in 3manifolds to orbifolds. A consequence of this work is that every valuation on the canonical component that detects an essential surface, detects an essential surface that is preserved by every orientation preserving symmetry on the manifold. This Theorem applies to orientable hyperbolic manifolds, with orientation preserving symmetry group, whose quotient by this group is an orbifold with a flexible cusp, which is the case for most hyperbolic 3manifolds. We then look at a family of two bridge knots where our theorem shows it is impossible for every essential surface to be detected on the canonical component. We then prove that all surfaces that are preserved by the orientation preserving symmetries of these knots are detected by ideal points on the canonical component of the character variety by calculating the canonical component of the Apolynomial for the family of knots. We then prove that every essential surface in these knot that is not detected on the canonical component of the character variety is detected on another component.
Show less  Date Issued
 2018
 Identifier
 2018_Su_Leach_fsu_0071E_14753
 Format
 Thesis
 Title
 The 1Type of Algebraic KTheory as a Multifunctor.
 Creator

Valdes, Yaineli, Aldrovandi, Ettore, Rawling, John Piers, Agashe, Amod S., Aluffi, Paolo, Petersen, Kathleen L., Hoeij, Mark van, Florida State University, College of Arts and...
Show moreValdes, Yaineli, Aldrovandi, Ettore, Rawling, John Piers, Agashe, Amod S., Aluffi, Paolo, Petersen, Kathleen L., Hoeij, Mark van, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

It is known that the category of Waldhausen categories is a closed symmetric multicategory and algebraic Ktheory is a multifunctor from the category of Waldhuasen categories to the category of spectra. By assigning to any Waldhausen category the fundamental groupoid of the 1type of its Ktheory spectrum, we get a functor from the category of Waldhausen categories to the category of Picard groupoids, since stable 1types are classified by Picard groupoids. We prove that this functor is a...
Show moreIt is known that the category of Waldhausen categories is a closed symmetric multicategory and algebraic Ktheory is a multifunctor from the category of Waldhuasen categories to the category of spectra. By assigning to any Waldhausen category the fundamental groupoid of the 1type of its Ktheory spectrum, we get a functor from the category of Waldhausen categories to the category of Picard groupoids, since stable 1types are classified by Picard groupoids. We prove that this functor is a multifunctor to a corresponding multicategory of Picard groupoids.
Show less  Date Issued
 2018
 Identifier
 2018_Sp_Valdes_fsu_0071E_14374
 Format
 Thesis
 Title
 Affine Dimension of Smooth Curves and Surfaces.
 Creator

Williams, Ethan Randy, Oberlin, Richard, Ormsbee, Michael J., Reznikov, Alexander, Bauer, Martin, Florida State University, College of Arts and Sciences, Department of Mathematics
 Abstract/Description

Our aim is to study the affine dimension of some smooth manifolds. In Chapter 1, we review the notions of Minkowski and Hausdorff dimension, and compare them with the lesser studied affine dimension. In Chapter 2, we focus on understanding the affine dimension of curves. In Section 2.1, we review the existing results for the affine dimension of a strictly convex curve in the plane, and in Section 2.2, we classify the smooth curves in ℝn based on affine dimension. In Chapter 3, we classify the...
Show moreOur aim is to study the affine dimension of some smooth manifolds. In Chapter 1, we review the notions of Minkowski and Hausdorff dimension, and compare them with the lesser studied affine dimension. In Chapter 2, we focus on understanding the affine dimension of curves. In Section 2.1, we review the existing results for the affine dimension of a strictly convex curve in the plane, and in Section 2.2, we classify the smooth curves in ℝn based on affine dimension. In Chapter 3, we classify the smooth hypersurfaces in ℝ3 with nonnegative Gaussian curvature based on affine dimension, and in Chapter 4 we provide a lower and upper bound for the affine dimension of smooth, convex hypersurfaces in ℝn.
Show less  Date Issued
 2018
 Identifier
 2018_Sp_Williams_fsu_0071E_14512
 Format
 Thesis
 Title
 Metric Learning for Shape Classification: A Fast and Efficient Approach with Monte Carlo Methods.
 Creator

Cellat, Serdar, Mio, Washington, Ökten, Giray, Aggarwal, Sudhir, Cogan, Nicholas G., Jain, Harsh Vardhan, Florida State University, College of Arts and Sciences, Department of...
Show moreCellat, Serdar, Mio, Washington, Ökten, Giray, Aggarwal, Sudhir, Cogan, Nicholas G., Jain, Harsh Vardhan, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

Quantifying shape variation within a group of individuals, identifying morphological contrasts between populations and categorizing these groups according to morphological similarities and dissimilarities are central problems in developmental evolutionary biology and genetics. In this dissertation, we present an approach to optimal shape categorization through the use of a new family of metrics for shapes represented by a finite collection of landmarks. We develop a technique to identify...
Show moreQuantifying shape variation within a group of individuals, identifying morphological contrasts between populations and categorizing these groups according to morphological similarities and dissimilarities are central problems in developmental evolutionary biology and genetics. In this dissertation, we present an approach to optimal shape categorization through the use of a new family of metrics for shapes represented by a finite collection of landmarks. We develop a technique to identify metrics that optimally differentiate and categorize shapes using Monte Carlo based optimization methods. We discuss the theory and the practice of the method and apply it to the categorization of 62 mice offsprings based on the shape of their skull. We also create a taxonomic classification tree for multiple species of fruit flies given the shape of their wings. The results of these experiments validate our method.
Show less  Date Issued
 2018
 Identifier
 2018_Sp_Cellat_fsu_0071E_14295
 Format
 Thesis
 Title
 Optimal Portfolio Execution under TimeVarying Liquidity Constraints.
 Creator

Lin, HuaYi, Fahim, Arash, Atkins, Jennifer, Kercheval, Alec N., Ökten, Giray, Florida State University, College of Arts and Sciences, Department of Mathematics
 Abstract/Description

The problem of optimal portfolio execution has become one of the most important problems in the area of financial mathematics. Over the past two decades, numerous researchers have developed a variety of different models to address this problem. In this dissertation, we extend the LOB (Limit Order Book) model proposed by Obizhaeva and Wang (2013) by incorporating a more realistic assumption on the order book depth; the amount of liquidity provided by a LOB market is finite at all times. We use...
Show moreThe problem of optimal portfolio execution has become one of the most important problems in the area of financial mathematics. Over the past two decades, numerous researchers have developed a variety of different models to address this problem. In this dissertation, we extend the LOB (Limit Order Book) model proposed by Obizhaeva and Wang (2013) by incorporating a more realistic assumption on the order book depth; the amount of liquidity provided by a LOB market is finite at all times. We use an algorithmic approach to solve the problem of optimal execution under timevarying constraints on the depth of a LOB. For the simplest case where the order book depth stays at a fixed level for the entire trading horizon, we reduce the optimal execution problem into a onedimensional rootfinding problem which can be readily solved by standard numerical algorithms. When the depth of the LOB is monotone in time, we first apply the KKT (KarushKuhnTucker) conditions to narrow down the set of candidate strategies and then use a dichotomybased search algorithm to pin down the optimal one. For the general case that the order book depth doesn't exhibit any particular pattern, we start from the optimal strategy subject to no liquidity constraints and iterate over execution strategy by sequentially adding more constraints to the problem in a specific fashion until primal feasibility is achieved. Numerical experiments indicate that our algorithms give comparable results to those of current existing convex optimization toolbox CVXOPT with significantly lower time complexity.
Show less  Date Issued
 2018
 Identifier
 2018_Sp_Lin_fsu_0071E_14349
 Format
 Thesis
 Title
 Using Mathematical Tools to Investigate the Autoimmune Hair Loss Disease Alopecia Areata.
 Creator

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

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

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

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

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

We consider a classical risk process with arrival of claims following a nonstationary Hawkes process. We study the asymptotic regime when the premium rate and the baseline intensity of the claims arrival process are large, and claim size is small. The main goal of the article is to establish a diffusion approximation by verifying a functional central limit theorem and to compute the ruin probability in finitetime horizon. Numerical results will also be given.
 Date Issued
 2018
 Identifier
 2018_Fall_Cheng_fsu_0071E_14916
 Format
 Thesis
 Title
 Algorithmic Lung Nodule Analysis in Chest Tomography Images: Lung Nodule Malignancy Likelihood Prediction and a Statistical Extension of the Level Set Image Segmentation Method.
 Creator

Hancock, Matthew C. (Matthew Charles), Magnan, Jeronimo Francisco, Duke, D. W., Hurdal, Monica K., Mio, Washington, Florida State University, College of Arts and Sciences,...
Show moreHancock, Matthew C. (Matthew Charles), Magnan, Jeronimo Francisco, Duke, D. W., Hurdal, Monica K., Mio, Washington, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

Lung cancer has the highest mortality rate of all cancers in both men and women in the United States. The algorithmic detection, characterization, and diagnosis of abnormalities found in chest CT scan images can aid radiologists by providing additional medicallyrelevant information to consider in their assessment of medical images. Such algorithms, if robustly validated in clinical settings, carry the potential to improve the health of the general population. In this thesis, we first give an...
Show moreLung cancer has the highest mortality rate of all cancers in both men and women in the United States. The algorithmic detection, characterization, and diagnosis of abnormalities found in chest CT scan images can aid radiologists by providing additional medicallyrelevant information to consider in their assessment of medical images. Such algorithms, if robustly validated in clinical settings, carry the potential to improve the health of the general population. In this thesis, we first give an analysis of publicly available chest CT scan annotation data, in which we determine upper bounds on expected classification accuracy when certain radiological features are used as inputs to statistical learning algorithms for the purpose of inferring the likelihood of a lung nodule as being either malignant or benign. Second, a statistical extension of the level set method for image segmentation is introduced and applied to both syntheticallygenerated and real threedimensional image volumes of lung nodules in chest CT scans, obtaining results comparable to the current stateoftheart on the latter.
Show less  Date Issued
 2018
 Identifier
 2018_Sp_Hancock_fsu_0071E_14427
 Format
 Thesis
 Title
 Distributed Neural Network Models for Birdsong Production.
 Creator

Galvis, Daniel Eduardo, Bertram, R., Trombley, Paul Q., Cogan, Nicholas G., Hyson, Richard Lee, Johnson, Frank, Vo, Theodore, Florida State University, College of Arts and...
Show moreGalvis, Daniel Eduardo, Bertram, R., Trombley, Paul Q., Cogan, Nicholas G., Hyson, Richard Lee, Johnson, Frank, Vo, Theodore, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

Birdsong is a model system for the production of learned, serially ordered motor movements, such as playing a musical instrument or riding a bicycle. To this end, the neural mechanisms underlying birdsong have been studied in great depth, and many tools have been developed for analyzing the spectral and temporal structure of song. In this dissertation, I develop mathematical neural network models to explain how the nuclei in the song system interact to produce song. These models are...
Show moreBirdsong is a model system for the production of learned, serially ordered motor movements, such as playing a musical instrument or riding a bicycle. To this end, the neural mechanisms underlying birdsong have been studied in great depth, and many tools have been developed for analyzing the spectral and temporal structure of song. In this dissertation, I develop mathematical neural network models to explain how the nuclei in the song system interact to produce song. These models are constrained by the structural connectivity of the song system, the signaling properties of individual neurons and circuits, and circuitbreaking behavioral studies. Chapter 1 provides an overview of songbirds as a model system for speech production, outlines the structure of song, and describes the structure and function of the song system. Chapter 2 describes the neurophysiology and mathematical models of a premotor nucleus, called HVC (proper name), that is essential for song learning and production. In Chapter 3, I develop the neural network model for song production and use it to explain the effects of partial lesions of HVC on song. Furthermore, I use the model to make predictions about the behavioral effects of these lesions and reanalyze the data, validating those predictions. Finally, in Chapter 4 I develop a simplified version of the model that sacrifices spiking dynamics of neurons while maintaining the essential higherlevel features of the model. I use this model to study interhemispheric synchronization and the effects of unilateral perturbations of HVC on song. The model captures the effects of these perturbations, particularly unilateral temperature manipulation and electrical stimulation of HVC, and makes predictions about the circuitlevel effects of these perturbations.
Show less  Date Issued
 2018
 Identifier
 2018_Sp_Galvis_fsu_0071E_14310
 Format
 Thesis
 Title
 Exploration of the Role of Disinfection Timing, Duration, and Other Control Parameters on Bacterial Populations Using a Mathematical Model.
 Creator

Acar, Nihan, Cogan, Nicholas G., Keller, Thomas C. S., Bertram, R., Mio, Washington, Florida State University, College of Arts and Sciences, Department of Mathematics
 Abstract/Description

Tolerant bacteria enmeshed in a biofilm causes several difficult to treat illnesses like tuberculosis, chronic pneumonia, and chronic inner ear infections. These diseases typically respond poorly to antibiotics due to high tolerance. Bacterial tolerance can be genotypic (resistancee.g. MRSA), phenotypic (nonheritable) or environmental (e.g. nutrient gradients). Persister formation is phenotypic tolerance that is highly tolerant to disinfection. Constant dosing is typically ineffective in...
Show moreTolerant bacteria enmeshed in a biofilm causes several difficult to treat illnesses like tuberculosis, chronic pneumonia, and chronic inner ear infections. These diseases typically respond poorly to antibiotics due to high tolerance. Bacterial tolerance can be genotypic (resistancee.g. MRSA), phenotypic (nonheritable) or environmental (e.g. nutrient gradients). Persister formation is phenotypic tolerance that is highly tolerant to disinfection. Constant dosing is typically ineffective in eliminating persister cells. To generate an effective treatment protocol, more research must examine the dynamics of persister cells. This study investigates how manipulating the application of antibiotics and the addition of nutrient may enhance the disinfection of a bacterial population in batch culture. Previous studies focused on the antimicrobial agent as a control variable to eliminate the bacterial population. In addition to antibiotic treatments, we consider the significance of the nutrient in eradicating the susceptible and persister cells since the disinfection of the susceptible population is dependent on nutrient intake. We present a mathematical model that captures the dynamics between susceptible and persister bacteria with antibiotic and nutrient as control variables. We investigate the optimal dosewithdrawal timing of antibiotic in several cases including: constant nutrient in time, dynamic nutrient in time, and piecewise constant nutrient in time. Also a global sensitivity analysis method, Partial Rank Correlation Coefficient (PRCC), is applied to determine the significance of model parameters for a quantity of interest. The highlights of this study are; 1.) Constant dosing is not an effective disinfection protocol. 2.) Nutrient plays a significant role such that in the presence of nutrient, bacterial population is eliminated much faster. 3.) Checking the eigenvalues of the established Poincaré map gives us information on how to choose withdrawdose timing for the nonlinear system. 4.) Periodic dosewithdraw offers a more efficient disinfection provided dose time is longer than withdrawal of antibiotic. 5.) As duration of dose decreases, the elimination of bacteria decreases and the death rate becomes insignificant.
Show less  Date Issued
 2018
 Identifier
 2018_Su_Acar_fsu_0071E_14749
 Format
 Thesis
 Title
 Monte Carlo Scheme for a Singular Control Problem: InvestmentConsumption under Proportional Transaction Costs.
 Creator

Tsai, WanYu, Fahim, Arash, Atkins, Jennifer, Zhu, Lingjiong, Florida State University, College of Arts and Sciences, Department of Mathematics
 Abstract/Description

Nowadays free boundary problems are considered as one of the most important directions in the mainstream of partial differential equations (PDEs) analysis, with an abundance of applications in various sciences and real world problems. Free boundary problems on finance have been extended in many areas, such as optimal portfolio selection, control credit risks, and different American style products etc. To modelling these financial problems in the real world, the qualitative and quantitative...
Show moreNowadays free boundary problems are considered as one of the most important directions in the mainstream of partial differential equations (PDEs) analysis, with an abundance of applications in various sciences and real world problems. Free boundary problems on finance have been extended in many areas, such as optimal portfolio selection, control credit risks, and different American style products etc. To modelling these financial problems in the real world, the qualitative and quantitative behaviors of the solution to a free boundary problem are still not well understood and also numerical solutions to free boundary problems remain a challenge. Stochastic control problems reduce to freeboundary problems in partial differential equations while there are no bounds on the rate of control. In a free boundary problem, the solution as well as the domain to the PDE need to be determined simultaneously. In this dissertation, we concern the numerical solution of a fully nonlinear parabolic double obstacle problem arising from a finite time portfolio selection problem with proportional transaction costs. We consider optimal allocation of wealth among multiple stocks and a bank account in order to maximize the finite horizon discounted utility of consumption. The problem is mainly governed by a timedependent HamiltonJacobiBellman equation with gradient constraints. We propose a numerical method which is composed of Monte Carlo simulation to take advantage of the highdimensional properties and finite difference method to approximate the gradients of the value function. Numerical results illustrate behaviors of the optimal trading strategies and also satisfy all qualitative properties proved in Dai et al. (2009) and Chen and Dai (2013).
Show less  Date Issued
 2017
 Identifier
 FSU_FALL2017_Tsai_fsu_0071E_14174
 Format
 Thesis
 Title
 Developing SRSF Shape Analysis Techniques for Applications in Neuroscience and Genomics.
 Creator

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

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

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

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

Li, Jian, Cogan, Nicholas G., Chicken, Eric, Gallivan, Kyle A., Hurdal, Monica K., Florida State University, College of Arts and Sciences, Department of Mathematics
 Abstract/Description

Biofilms are thin layers of microorganisms in which cells adhere to each other and stick to a surface. They are resistant to antibiotics and disinfectants due to the protection from extracellular polymeric substance (EPS), which is a gel like selfproduced matrix, consists of polysaccharide, proteins and nucleic acids. Biofilms play significant roles in many applications. In this document, we provide analysis about effects and influences of biofilms in microfiltration and dental plaque...
Show moreBiofilms are thin layers of microorganisms in which cells adhere to each other and stick to a surface. They are resistant to antibiotics and disinfectants due to the protection from extracellular polymeric substance (EPS), which is a gel like selfproduced matrix, consists of polysaccharide, proteins and nucleic acids. Biofilms play significant roles in many applications. In this document, we provide analysis about effects and influences of biofilms in microfiltration and dental plaque removing process. Differential equations are used for modelling the microfiltration process and the optimal control method is applied to analyze the efficiency of the filtration. The multiphase fluid system is introduced to describe the dental plaque removing process and results are obtained by numerical schemes.
Show less  Date Issued
 2017
 Identifier
 FSU_FALL2017_Li_fsu_0071E_13839
 Format
 Thesis
 Title
 Third Order AHypergeometric Functions.
 Creator

Xu, Wen, Hoeij, Mark van, Reina, Laura, Agashe, Amod S. (Amod Sadanand), Aldrovandi, Ettore, Aluffi, Paolo, Florida State University, College of Arts and Sciences, Department of...
Show moreXu, Wen, Hoeij, Mark van, Reina, Laura, Agashe, Amod S. (Amod Sadanand), Aldrovandi, Ettore, Aluffi, Paolo, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

To solve globally bounded order $3$ linear differential equations with rational function coefficients, this thesis introduces a partial $_3F_2$solver (Section~\ref{3F2 type solution}) and $F_1$solver (Chapter~\ref{F1 solver}), where $_3F_2$ is the hypergeometric function $_3F_2(a_1,a_2,a_3;b_1,b_2\,\,x)$ and $F_1$ is the Appell's $F_1(a,b_1,b_2,c\,\,x,y).$ To investigate the relations among order $3$ multivariate hypergeometric functions, this thesis presents two multivariate tools:...
Show moreTo solve globally bounded order $3$ linear differential equations with rational function coefficients, this thesis introduces a partial $_3F_2$solver (Section~\ref{3F2 type solution}) and $F_1$solver (Chapter~\ref{F1 solver}), where $_3F_2$ is the hypergeometric function $_3F_2(a_1,a_2,a_3;b_1,b_2\,\,x)$ and $F_1$ is the Appell's $F_1(a,b_1,b_2,c\,\,x,y).$ To investigate the relations among order $3$ multivariate hypergeometric functions, this thesis presents two multivariate tools: compute homomorphisms (Algorithm~\ref{hom}) of two $D$modules, where $D$ is a multivariate differential ring, and compute projective homomorphisms (Algorithm~\ref{algo ProjHom}) using the tensor product module and Algorithm~\ref{hom}. As an application, all irreducible order $2$ subsystems from reducible order $3$ systems turn out to come from Gauss hypergeometric function $_2F_1(a,b;c\,\,x)$ (Chapter~\ref{chapter applications}).
Show less  Date Issued
 2017
 Identifier
 FSU_FALL2017_XU_fsu_0071E_14234
 Format
 Thesis
 Title
 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
 Sorvali Dilatation and Spin Divisors on Riemann and Klein Surfaces.
 Creator

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

We review the Sorvali dilatation of isomorphisms of covering groups of Riemann surfaces and extend the definition to groups containing glidereflections. Then we give a bound for the distance between two surfaces, one of them resulting from twisting the other at a decomposing curve. Furthermore, we study spin structures on Riemann and Klein surfaces in terms of divisors. In particular, we take a closer look at spin structures on hyperelliptic and pgonal surfaces defined by divisors supported...
Show moreWe review the Sorvali dilatation of isomorphisms of covering groups of Riemann surfaces and extend the definition to groups containing glidereflections. Then we give a bound for the distance between two surfaces, one of them resulting from twisting the other at a decomposing curve. Furthermore, we study spin structures on Riemann and Klein surfaces in terms of divisors. In particular, we take a closer look at spin structures on hyperelliptic and pgonal surfaces defined by divisors supported on branch points. Moreover, we study invariant spin divisors under automorphisms and antiholomorphic involutions of Riemann surfaces.
Show less  Date Issued
 2017
 Identifier
 FSU_SUMMER2017_ALMALKI_fsu_0071E_14064
 Format
 Thesis
 Title
 HighOrder, Efficient, Numerical Algorithms for Integration in Manifolds Implicitly Defined by Level Sets.
 Creator

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

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

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

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

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

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

Chen, Yuanda, Kercheval, Alec N., Beaumont, Paul M., Ewald, Brian D., Zhu, Lingjiong, Florida State University, College of Arts and Sciences, Department of Mathematics
 Abstract/Description

The Hawkes process serves as a natural choice for modeling selfexciting dynamics, such as the behavior of an electronic exchangehosted limit order book (LOB). However, due to the lack of analytical solutions, probability estimates of future events often must rely on Monte Carlo simulation. Although Monte Carlo simulation is known to be good at solving pathdependent problems, it has the limitation that a high computation time is often required to get good accuracy. This is a concern in...
Show moreThe Hawkes process serves as a natural choice for modeling selfexciting dynamics, such as the behavior of an electronic exchangehosted limit order book (LOB). However, due to the lack of analytical solutions, probability estimates of future events often must rely on Monte Carlo simulation. Although Monte Carlo simulation is known to be good at solving pathdependent problems, it has the limitation that a high computation time is often required to get good accuracy. This is a concern in fields like algorithmic trading where fast calculation is essential. In this dissertation we propose the use of a 4dimensional Hawkes process to model the LOB and to forecast midprice movement probabilities using Monte Carlo simulation. We study the feasibility of making this prediction quickly enough to be applicable in practice. We show that fast predictions are feasible, and show in tests on real data that the model has some trading value in forecasting midprice movements. This dissertation also compares the performance of several popular computer languages, Python, MATLAB, Cython and C, in singlecore experiments, and examines the scalability for parallel computing using Cython and C.
Show less  Date Issued
 2017
 Identifier
 FSU_2017SP_Chen_fsu_0071E_13187
 Format
 Thesis
 Title
 Trend and VariablePhase Seasonality Estimation from Functional Data.
 Creator

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

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

Wyse, J. Maxwell (John Maxwell), MestertonGibbons, Mike, Huffer, Fred W. (Fred William), Hurdal, Monica K., Cogan, Nicholas G., Florida State University, College of Arts and...
Show moreWyse, J. Maxwell (John Maxwell), MestertonGibbons, Mike, Huffer, Fred W. (Fred William), Hurdal, Monica K., Cogan, Nicholas G., Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

Mature male African elephants are known to periodically enter a temporary state of heightened aggression called "musth," often linked with increased androgens, particularly testosterone. Sexually mature males are capable of entering musth at any time of year, and will often travel long distances to find estrous females. When two musth bulls or two nonmusth bulls encounter one another, the agonistic interaction is usually won by the larger male. When a smaller musth bull encounters a larger...
Show moreMature male African elephants are known to periodically enter a temporary state of heightened aggression called "musth," often linked with increased androgens, particularly testosterone. Sexually mature males are capable of entering musth at any time of year, and will often travel long distances to find estrous females. When two musth bulls or two nonmusth bulls encounter one another, the agonistic interaction is usually won by the larger male. When a smaller musth bull encounters a larger nonmusth bull, however, the smaller musth male can win. The relative mating success of musth males is due partly to this fighting advantage, and partly to estrous females' general preference for musth males. Though musth behavior has long been observed and documented, the evolutionary advantages of musth remain poorly understood. Here we develop a gametheoretic model of male musth behavior which assumes musth duration as a parameter, and distributions of small, medium and large musth males are predicted in both time and space. The predicted results are similar to the observed timing strategies in the Amboseli National Park elephant population. We discuss small male musth behavior, musthestrus coincidence, the effects of estrous female spatial heterogeneity on musth timing, conservation applications, the assumptions underpinning the model and possible modifications to the model for the purpose of determining musth duration.
Show less  Date Issued
 2017
 Identifier
 FSU_2017SP_Wyse_fsu_0071E_13713
 Format
 Thesis
 Title
 Scroll Waves and How They Interact with NonReactive Spheres, Tori, and Knots.
 Creator

Weingard, Daniel, Bertram, R. (Richard), Rikvold, Per Arne, Steinbock, Oliver, Hurdal, Monica K., Magnan, Jerry F., Florida State University, College of Arts and Sciences,...
Show moreWeingard, Daniel, Bertram, R. (Richard), Rikvold, Per Arne, Steinbock, Oliver, Hurdal, Monica K., Magnan, Jerry F., Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

Threedimensional reactiondiffusion systems are able to produce scroll waves which rotate around a curve called the filament. A scroll ring is formed when the filament is a closed curve. In isotropic systems where reactants have equal diffusion coefficients, scroll rings will shrink over time and eventually collapse. Chemical experiments and numerical studies have shown that filaments can pin to certain nonreactive objects and prevent scroll ring collapse. With numerical simulation, we study...
Show moreThreedimensional reactiondiffusion systems are able to produce scroll waves which rotate around a curve called the filament. A scroll ring is formed when the filament is a closed curve. In isotropic systems where reactants have equal diffusion coefficients, scroll rings will shrink over time and eventually collapse. Chemical experiments and numerical studies have shown that filaments can pin to certain nonreactive objects and prevent scroll ring collapse. With numerical simulation, we study how new types of objects affect scroll ring behavior. In particular, we explore the properties of random sphere arrangements that prevent scroll ring collapse. In addition, we discover a novel mechanism that causes scroll rings to expand when pinning to a nonreactive trefoil knot.
Show less  Date Issued
 2017
 Identifier
 FSU_2017SP_Weingard_fsu_0071E_13790
 Format
 Thesis
 Title
 Insulin Secretion Rhythms: Calcium Regulation of BetaCell Metabolism and Rescue of Islet Oscillations.
 Creator

McKenna, Joseph P., Bertram, R. (Richard), Roper, Michael Gabriel, Muslimani, Ziad H., Moore, M. Nicholas J. (Matthew Nicholas J.), Miller, Brian G., Florida State University,...
Show moreMcKenna, Joseph P., Bertram, R. (Richard), Roper, Michael Gabriel, Muslimani, Ziad H., Moore, M. Nicholas J. (Matthew Nicholas J.), Miller, Brian G., Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

Pancreatic islet betacells play a vital role in regulating blood glucose levels by releasing insulin into the bloodstream. Insulin is released in pulses that parallel interacting betacell rhythms, including oscillatory glucose metabolism and periodic calcium influx. We present concurrent time series records of metabolic variables and intracellular calcium levels in glucosestimulated betacells that support regulation of mitochondrial dehydrogenases is the dominant calcium feedback effect...
Show morePancreatic islet betacells play a vital role in regulating blood glucose levels by releasing insulin into the bloodstream. Insulin is released in pulses that parallel interacting betacell rhythms, including oscillatory glucose metabolism and periodic calcium influx. We present concurrent time series records of metabolic variables and intracellular calcium levels in glucosestimulated betacells that support regulation of mitochondrial dehydrogenases is the dominant calcium feedback effect onto metabolism in the insulin secretory pathway. We include this effect into the betacell Dual Oscillator Model to reconcile model simulations with experimental data, then we determine the oscillation mechanism in the modified model. Islets lose the rhythms that govern insulin pulses when glucose is elevated to hyperglycemic levels. We demonstrate with modeling and experiments that oscillations lost to elevated glucose can be recovered by converting the elevated glucose stimulus to a sinusoidal wave. We predict with modeling which periodic glucose stimuli can recover islet oscillations.
Show less  Date Issued
 2017
 Identifier
 FSU_2017SP_McKenna_fsu_0071E_13864
 Format
 Thesis
 Title
 Random Sobol' Sensitivity Analysis and Model Robustness.
 Creator

Mandel, David, Ökten, Giray, Hussaini, M. Yousuff, Huffer, Fred W. (Fred William), Kercheval, Alec N., Fahim, Arash, Florida State University, College of Arts and Sciences,...
Show moreMandel, David, Ökten, Giray, Hussaini, M. Yousuff, Huffer, Fred W. (Fred William), Kercheval, Alec N., Fahim, Arash, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

This work develops both the theoretical foundation and the practical application of random Sobol' analysis with two goals. The first is to provide a more general and accommodating approach to global sensitivity analysis, in which the parameter distribution themselves contain uncertainty, and hence the sensitivity results are random quantities as well. The framework for this approach is motivated by empirical evidence of such behavior, and examples of this behavior in interest rate and...
Show moreThis work develops both the theoretical foundation and the practical application of random Sobol' analysis with two goals. The first is to provide a more general and accommodating approach to global sensitivity analysis, in which the parameter distribution themselves contain uncertainty, and hence the sensitivity results are random quantities as well. The framework for this approach is motivated by empirical evidence of such behavior, and examples of this behavior in interest rate and temperature modeling are provided. The second goal is to compare competing models on their robustness, a notion developed and defined to provide a quantitative solution to model selection based on model uncertainty and sensitivity
Show less  Date Issued
 2017
 Identifier
 FSU_2017SP_Mandel_fsu_0071E_13682
 Format
 Thesis
 Title
 Effective Methods in Intersection Theory and Combinatorial Algebraic Geometry.
 Creator

Harris, Corey S. (Corey Scott), Chicken, Eric, Aldrovandi, Ettore, Kim, Kyounghee, Petersen, Kathleen L., Florida State University, College of Arts and Sciences, Department of...
Show moreHarris, Corey S. (Corey Scott), Chicken, Eric, Aldrovandi, Ettore, Kim, Kyounghee, Petersen, Kathleen L., Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

This dissertation presents studies of effective methods in two main areas of algebraic geometry: intersection theory and characteristic classes, and combinatorial algebraic geometry. We begin in chapter 2 by giving an effective algorithm for computing Segre classes of subschemes of arbitrary projective varieties. The algorithm presented here comes after several others which solve the problem in special cases, where the ambient variety is for instance projective space. To our knowledge, this...
Show moreThis dissertation presents studies of effective methods in two main areas of algebraic geometry: intersection theory and characteristic classes, and combinatorial algebraic geometry. We begin in chapter 2 by giving an effective algorithm for computing Segre classes of subschemes of arbitrary projective varieties. The algorithm presented here comes after several others which solve the problem in special cases, where the ambient variety is for instance projective space. To our knowledge, this is the first algorithm to be able to compute Segre classes in projective varieties with arbitrary singularities. In chapter 3, we generalize an algorithm by Goward for principalization of monomial ideals in nonsingular varieties to work on any scheme of finite type over a field, proving that the more general class of r.c. monomial subschemes in arbitrarily singular varieties can be principalized by a sequence of blowups at codimension 2 r.c. monomial centers. The main result of chapter 4 is a classification of the monomial Cremona transformations of the plane up to conjugation by certain linear transformations. In particular, an algorithm for enumerating all such maps is derived. In chapter 5, we study the multiview varieties and compute their ChernMather classes. As a corollary we derive a polynomial formula for their Euclidean distance degree, partially addressing a conjecture of Draisma et al. [35]. In chapter 6, we discuss the classical problem of counting planes tangent to general canonical sextic curves at three points. We investigate the situation for real and tropical sextics. In chapter 6, we explicitly compute equations of an Enriques surface via the involution on a K3 surface.
Show less  Date Issued
 2017
 Identifier
 FSU_2017SP_Harris_fsu_0071E_13829
 Format
 Thesis
 Title
 GameTheoretic Models of Animal Behavior Observed in Some Recent Experiments.
 Creator

Dai, Yao, MestertonGibbons, Mike, Hurdal, Monica K., Kercheval, Alec N., Quine, J. R. (John R.), Florida State University, College of Arts and Sciences, Department of Mathematics
 Abstract/Description

In this dissertation, we create three theoretical models to answer questions raised by recent experiments that lie beyond the scope of current theory. In the landmarkeffect model, we determine size, shape and location for a territory that is optimal in the sense of minimizing defense costs, when a given proportion of the boundary is landmarked and its primary benefit in terms of fitness is greater ease of detecting intruders across it. In the subjectiveresourcevalue model, we develop a...
Show moreIn this dissertation, we create three theoretical models to answer questions raised by recent experiments that lie beyond the scope of current theory. In the landmarkeffect model, we determine size, shape and location for a territory that is optimal in the sense of minimizing defense costs, when a given proportion of the boundary is landmarked and its primary benefit in terms of fitness is greater ease of detecting intruders across it. In the subjectiveresourcevalue model, we develop a gametheoretic model based on the WarofAttrition game. Our results confirm that allowing players to adapt their subjective resource value based on their experiences can generate strong winner effects with weak or even no loser effects, which is not predicted by other theoretical models. In the rearguardaction model, we develop two versions of a gametheoretic model with different hypotheses on the function of volatile chemical emissions in animal contests, and we compare their results with observations in experiments. The two hypotheses are whether volatile chemicals are released to prevent the winner of the current round of contest from translating its victory into permanent possession of a contested resource, or are used to prevent a winner from inflicting costs on a fleeing loser.
Show less  Date Issued
 2017
 Identifier
 FSU_2017SP_Dai_fsu_0071E_13762
 Format
 Thesis
 Title
 Topology of ngonal Curve Complements.
 Creator

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

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