# Browsing by Department "Mathematics"

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Item Open Access A Bargaining Theory of the “Edwards’ Effect” in the 2007-8 Democratic Presidential Primary(2009) Li, Alex S.2008’s Democratic Presidential Primary will go down as one of the most competitive races in recent history. Two candidates, Senators Barack Obama (Illinois) and Hillary Clinton (New York), fought a see-saw battle to obtain enough delegates/vote-shares for the Democratic nomination. Although the race eventually dwelled down to these two players, for a while it was a dynamic three-player-race with Senator John Edwards (North Carolina) in the fold. During that time, many people were puzzled by Edwards’ insisting on staying in the race even when he had no foreseeable chance of becoming the party’s eventual nominee. In this honors thesis, I construct a theoretical model to explain Edwards’ reason for staying in the race. My model found that if Edwards attains a certain amount of vote-shares, depending on the external circumstances, he could have pushed the election into a backroom negotiation phase. In this phase, he would have become the most pivotal player as his relatively low amount of vote-shares would ironically turn him into the player with the greatest negotiating power. This could have allowed him to come out of the backroom negotiation with a final prize value that would have exceeded the efforts he inputted. My paper coins this as “The Edwards Effect” and explores the ramifications and conditions for its existence.Item Open Access A Closer Look at ADC Multivariate GARCH(2009-05-04T17:49:11Z) Haftel, JaredIn the past thirty years, academia and the marketplace have devoted signi cant e ort and resources toward gaining a better understanding of how volatility changes over time in the nancial markets and how changes in one market a ect changes in another. All of these attempts involve modeling the covariance matrix of two or more asset returns using the period-earlier covariance matrix. In this paper, we outline the volatility modeling process for an Antisymmetric Dynamic Covariance (ADC) multivariate Generalized Autoregressive Conditional Heteroskedacity (GARCH) model, explain the math involved, and attempt to estimate the parameters of the model using the Broyden-Fletcher-Goldfarb-Shanno (BFGS) optimization algorithm. We nd several barriers to estimating parameters using BFGS and suggest using alternative algorithms to estimate the ADC multivariate GARCH in the future.Item Open Access A Dynamical Nephrovascular Model of Renal Autoregulation(2014) Sgouralis, IoannisThe main functions of the kidney take place in the nephrons. For their proper operation, nephrons need to be supplied with a stable blood flow that remains constant despite fluctuations of arterial pressure. Such stability is provided by the afferent arterioles, which are unique vessels in the kidney capable of adjusting diameter. By doing so, afferent arterioles regulate blood delivery downstream, where the nephrons are located. The afferent arterioles respond to signals initiated by two mechanisms: the myogenic response which operates to absorb pressure perturbations within the vasculature, and tubuloglomerular feedback which operates to stabilize salt reabsorption.

In this thesis, a mathematical model of the renal nephrovasculature that represents both mechanisms in a dynamical context is developed. For this purpose, de- tailed representations of the myogenic mechanism of vascular smooth muscles and the tubular processes are developed and combined in a single comprehensive model. The resulting model is formulated with a large number of ordinary differential equations that represent the intracellular processes of arteriolar smooth muscles, coupled with a number of partial differential equations, mainly of the advection-diffusion-reaction type, that represent blood flow, glomerular filtration and the tubular processes. Due to its unique activation characteristics, the myogenic response is formulated with a set of delay differential equations.

The model is utilized to assess a verity of physiological phenomena: the conduction of vasomotor responses along the afferent arteriole, autoregulation under physiologic as well as pathophysiologic conditions, and renal oxygenation. A first attempt to model the impact of diabetes mellitus on renal hemodynamics is also made. Further, an application with clinical significance is presented. Namely, renal oxygenation is estimated under conditions that simulate those observed during cardiopulmonary surgery. Results indicate the development of renal hypoxia, which suggests an important pathway for the development of acute kidney injury.

Item Open Access A Generalized Lyapunov Construction for Proving Stabilization by Noise(2012) Kolba, Tiffany NicoleNoise-induced stabilization occurs when an unstable deterministic system is stabilized by the addition of white noise. Proving that this phenomenon occurs for a particular system is often manifested through the construction of a global Lyapunov function. However, the procedure for constructing a Lyapunov function is often quite ad hoc, involving much time and tedium. In this thesis, a systematic algorithm for the construction of a global Lyapunov function for planar systems is presented. The general methodology is to construct a sequence of local Lyapunov functions in different regions of the plane, where the regions are delineated by different behaviors of the deterministic dynamics. A priming region, where the deterministic drift is directed inward, is first identified where there is an obvious choice for a local Lyapunov function. This priming Lyapunov function is then propagated to the other regions through a series of Poisson equations. The local Lyapunov functions are lastly patched together to form one smooth global Lyapunov function.

The algorithm is applied to a model problem which displays finite time blow up in the deterministic setting in order to prove that the system exhibits noise-induced stabilization. Moreover, the Lyapunov function constructed is in fact what we define to be a super Lyapunov function. We prove that the existence of a super Lyapunov function, along with a minorization condition, implies that the corresponding system converges to a unique invariant probability measure at an exponential rate that is independent of the initial condition.

Item Open Access A Geometric Approach to Biomedical Time Series Analysis(2020) Malik, JohnBiomedical time series are non-invasive windows through which we may observe human systems. Although a vast amount of information is hidden in the medical field's growing collection of long-term, high-resolution, and multi-modal biomedical time series, effective algorithms for extracting that information have not yet been developed. We are particularly interested in the physiological dynamics of a human system, namely the changes in state that the system experiences over time (which may be intrinsic or extrinsic in origin). We introduce a mathematical model for a particular class of biomedical time series, called the wave-shape oscillatory model, which quantifies the sense in which dynamics are hidden in those time series. There are two key ideas behind the new model. First, instead of viewing a biomedical time series as a sequence of measurements made at the sampling rate of the signal, we can often view it as a sequence of cycles occurring at irregularly-sampled time points. Second, the "shape" of an individual cycle is assumed to have a one-to-one correspondence with the state of the system being monitored; as such, changes in system state (dynamics) can be inferred by tracking changes in cycle shape. Since physiological dynamics are not random but are well-regulated (except in the most pathological of cases), we can assume that all of the system's states lie on a low-dimensional, abstract Riemannian manifold called the phase manifold. When we model the correspondence between the hidden system states and the observed cycle shapes using a diffeomorphism, we allow the topology of the phase manifold to be recovered by methods belonging to the field of unsupervised manifold learning. In particular, we prove that the physiological dynamics hidden in a time series adhering to the wave-shape oscillatory model can be well-recovered by applying the diffusion maps algorithm to the time series' set of oscillatory cycles. We provide several applications of the wave-shape oscillatory model and the associated algorithm for dynamics recovery, including unsupervised and supervised heartbeat classification, derived respiratory monitoring, intra-operative cardiovascular monitoring, supervised and unsupervised sleep stage classification, and f-wave extraction (a single-channel blind source separation problem).

Item Open Access A maximum entropy-based approach for the description of the conformational ensemble of calmodulin from paramagnetic NMR(2016-05-04) Thelot, FrancoisCharacterizing protein dynamics is an essential step towards a better understanding of protein function. Experimentally, we can access information about protein dynamics from paramagnetic NMR data such as pseudocontact shifts, which integrate ensemble-averaged information about the motion of proteins. In this report, we recognize that the relative position of the two domains of calmodulin can be represented as the evolution of one of the domains in the space of Euclidean motions. From this perspective, we suggest a maximum entropy-based approach for finding a probability distribution on SE(3) satisfying experimental NMR measurements. While sampling of SE(3) is performed with the ensemble generator EOM, the proposed framework can be extended to uniform sampling of the space of Euclidean motions. At the end of this study, we find that the most represented protein conformations for calmodulin corresponds to conformations in which both protein domains are in close contact, despite being largely different from each other. Such a representation agrees with the random coil linker model, and sharply differs with the extended crystal structure of calmodulin.Item Open Access A Model of Speculative Attacks and Devaluations in Korea and Indonesia(2009) Lin, Austin YiSince the beginning of the Bretton Woods era, currency crises and speculative attacks have affected the world economy. This paper presents a model, originally derived by Blanco and Garber, that predicts one-period ahead probabilities of a currency devaluation and the expected exchange rate conditional on a devaluation. The analysis is then applied to Korea and Indonesia during the periods of 1960-1980 and 1969-1989, respectively. Despite numerous devaluations during both periods, all of the calculated probabilities of devaluation in the next period are close to zero for both Korea and Indonesia. However, it is promising that rises in predicted probabilities of devaluation are observed before actual devaluations for Indonesia.Item Open Access A Model of the Foot and Ankle in Running(2011-05-11) Waggoner, BoWe present several variations on a model and simulation of the foot and ankle during the course of one running stride. We summarize the motivation behind the model and similar work in the field, then describe the model and the results obtained. In the model, the shin and foot are each represented by thin rods, while two major muscle groups are modeled as exponential springs. The ground is modeled as a network of points connected by damped linear springs. Results on ground impact forces and physiological parameters are presented. In particular, we find that heel striking tends to produce higher peak impact forces than forefoot striking, we search for foot parameters producing the most effective foot strike, we compare force-time data obtained to experimental results, and we compare the effects of different ground and shoe properties on foot strike.Item Open Access A Simple Cardiac Model Exhibiting Stationary Discordant Alternans(2011-04-25) Sae Sue, TanawitThe term alternans refers to a period-doubling bifurcation that occurs in rapidly-paced cardiac cells: although all stimuli are equally spaced, short and long action potentials alternate with one another. Cardiac waves that propagate in extended tissue often su er phase reversals at various locations, a phenomenon that is known as discordant alternans. Even simulations cannot capture the full complexity of this phenomenon with realistic cardiac models in three-dimensional geometry. Significant insight into the phenomenon has been gained from the study of simple cardiac models propagating in just one dimension, and this is the context of the present paper.Item Open Access A Spectral Deferred Correction Method for Solving Cardiac Models(2011) Bowen, Matthew M.Many numerical approaches exist to solving models of electrical activity in the heart. These models consist of a system of stiff nonlinear ordinary differential equations for the voltage and other variables governing channels, with the voltage coupled to a diffusion term. In this work, we propose a new algorithm that uses two common discretization methods, operator splitting and finite elements. Additionally, we incorporate a temporal integration process known as spectral deferred correction. Using these approaches,

we construct a numerical method that can achieve arbitrarily high order in both space and time in order to resolve important features of the models, while gaining accuracy and efficiency over lower order schemes.

Our algorithm employs an operator splitting technique, dividing the reaction-diffusion systems from the models into their constituent parts.

We integrate both the reaction and diffusion pieces via an implicit Euler method. We reduce the temporal and splitting errors by using a spectral deferred correction method, raising the temporal order and accuracy of the scheme with each correction iteration.

Our algorithm also uses continuous piecewise polynomials of high order on rectangular elements as our finite element approximation. This approximation improves the spatial discretization error over the piecewise linear polynomials typically used, especially when the spatial mesh is refined.

As part of these thesis work, we also present numerical simulations using our algorithm of one of the cardiac models mentioned, the Two-Current Model. We demonstrate the efficiency, accuracy and convergence rates of our numerical scheme by using mesh refinement studies and comparison of accuracy versus computational time. We conclude with a discussion of how our algorithm can be applied to more realistic models of cardiac electrical activity.

Item Open Access A Stochastic Spatial Model for Tumor Growth(2014-04-29) Dheeraj, AashiqEvolutionary game theory can be used to study the interactions of different cell phenotypes and describe tumor population dynamics. Instead of killing tumor cells, clinical treatment could aim to change the nature of the evolutionary game-- enabling healthy cells to outcompete malignant cells. Most applications of evolutionary game theory to tumor growth have considered the tumor as a homogeneously mixing population that is governed by the replicator equation. We model the tumor population as an interacting particle system (IPS), with discrete individuals, stochastic local interactions, and explicit spatial consideration. Using this model, we see how predictions are changed when space is taken into account. In particular, we consider Basanta's work on glioma progression, the analysis of multiple myeloma proposed by Dingli et al., and Tomlinson's model for tumors containing cytotoxin-producing cells. Our model agrees with Basanta's in that we should have coexistence between the three tumor phenotypes, but the spatial model allows coexistence in a significantly wider region of parameter space. Dingli's tumor population exhibits bistability in a certain parameter regime. Our spatial model predicts a transition between the two stable states at a critical parameter value, so there is no bistability. In Tomlinson's game, the IPS does not allow for coexistence between cell types.Item Open Access A Strategy and Honesty Based Comparison of Preferential Ballot Voting Methods(2013-05-02) Mallernee, JamesThis paper presents an analysis of various preferential ballot voting systems based on the idea that voters should be encouraged to vote honestly and independently of the other votes cast. Random votes are simulated in three and four candidate elections with N voters, while a block of votes of size b, all of which are all the same, represents the votes of a subset of the electorate with a given preference. Given b and N, we examine the likelihood P that, for a variety of voting methods, it benefits this body of voters to cast a block of votes that does not represent their true preferences. We then view P as a function of the single variable b/ ￼ N, and compare the function P for various preferential ballot voting methods, noting which methods are more likely to encourage dishonest or strategic voting under different circumstances.Item Open Access A Study of Edge Toric Ideals using Associated Graphs(2012-04-26) Shen, YingyiThis thesis studies properties of edge toric ideals and resolutions by analyzing the associated graphs of algebraic structures. It mainly focused on proving that the repeated edges in a graph wouldn't change some properties of its underlying algebraic structure. An application of this result is that when we study multi-edge graphs, we can simplify in nite numbers of graphs to a simple one by deleting all the repeated edges.Item Open Access A Theory of Optimal Sick Pay(2009) Tutt, AndrewIllness significantly reduces worker productivity, yet how employers respond to the possibility of illness and its effects on work performance is not well understood. The 2003 American Productivity Audit pegged the cost to employers of lost productive time due to illness at 225.8 billion US dollars/year. More importantly, 71% of that loss was explained by reduced performance while at work. Studies of worker illness have been up to this point empirical, focused primarily on characteristics which co-vary with worker illness and absenteeism. This paper seeks to understand how employers mitigate the impact of illness on profits through a microeconomic model, elucidating how employers influence workers through salary-based incentives to mitigate its associated costs, providing firms and policy makers with a comprehensive theoretical method for formulating optimal sick pay policies.Item Open Access A Third Order Numerical Method for Doubly Periodic Electromegnetic Scattering(2007-07-31) Nicholas, Michael JWe here developed a third-order accurate numerical method for scattering of 3D electromagnetic waves by doubly periodic structures. The method is an intuitively simple numerical scheme based on a boundary integral formulation. It involves smoothing the singular Green's functions in the integrands and finding correction terms to the resulting smooth integrals. The analytical method is based on the singular integral methods of J. Thomas Beale, while the scattering problem is motivated by the 2D work of Stephanos Venakides, Mansoor Haider, and Stephen Shipman. The 3D problem was done with boundary element methods by Andrew Barnes. We present a method that is both more straightforward and more accurate. In solving these problems, we have used the M\"{u}ller integral equation formulation of Maxwell's equations, since it is a Fredholm integral equation of the second kind and is well-posed. M\"{u}ller derived his equations for the case of a compact scatterer. We outline the derivation and adapt it to a periodic scatterer. The periodic Green's functions found in the integral equation contain singularities which make it difficult to evaluate them numerically with accuracy. These functions are also very time consuming to evaluate numerically. We use Ewald splitting to represent these functions in a way that can be computed rapidly.We present a method of smoothing the singularity of the Green's function while maintaining its periodicity. We do local analysis of the singularity in order to identify and eliminate the largest sources of error introduced by this smoothing. We prove that with our derived correction terms, we can replace the singular integrals with smooth integrals and only introduce a error that is third order in the grid spacing size. The derivation of the correction terms involves transforming to principal directions using concepts from differential geometry. The correction terms are necessarily invariant under this transformation and depend on geometric properties of the scatterer such as the mean curvature and the differential of the Gauss map. Able to evaluate the integrals to a higher order, we implement a \mbox{GMRES} algorithm to approximate solutions of the integral equation. From these solutions, M\"{u}ller's equations allow us to compute the scattered fields and transmission coefficients. We have also developed acceleration techniques that allow for more efficient computation.We provide results for various scatterers, including a test case for which exact solutions are known. The implemented method does indeed converge with third order accuracy. We present results for which the method successfully resolves Wood's anomaly resonances in transmission.Item Open Access Absolute Continuity of Singular SPDEs and Bayesian Inference on Dynamical Systems(2023) Su, LangxuanWe explore the interplay among probability, stochastic analysis, and dynamical systems through two lenses: (1) absolute continuity of singular stochastic partial differential equations (SPDEs); (2) Bayesian inference on dynamical systems.

In the first part, we prove that up to a certain singular regime, the law of the stochastic Burgers equation at a fixed time is absolutely continuous with respect to the corresponding stochastic heat equation with the nonlinearity removed. The results follow from an extension of the Girsanov Theorem to handle less spatially regular solutions while only proving absolute continuity at a fixed time. To deal with the singularity, we introduce a novel decomposition in the spirit of Da Prato-Debussche and Gaussian chaos decomposition in singular SPDEs, by separating out the noise into different levels of regularity, along with a number of renormalization techniques. The number of levels in this decomposition diverges to infinite as we move to the stochastic Burgers equation associated with the KPZ equation. This result illustrates the fundamental probabilistic structure of a class of singular SPDEs and a notion of ``ellipticity'' in the infinite-dimensional setting.

In the second part, we establish connections between large deviations and a class of generalized Bayesian inference procedures on dynamical systems. We show that posterior consistency can be derived using a combination of classical large deviation techniques, such as Varadhan's lemma, conditional/quenched large deviations, annealed large deviations, and exponential approximations. We identified the divergence term as the Donsker-Varadhan relative entropy rate, also related to the Kolmogorov-Sinai entropy in ergodic theory. As an application, we prove new quenched/annealed large deviation asymptotics and a new Bayesian posterior consistency result for a class of mixing stochastic processes. In the case of Markov processes, one can obtain explicit conditions for posterior consistency, when estimates for log-Sobolev constants are available, which makes our framework essentially a black box. We also recover state-of-the-art posterior consistency on classical dynamical systems with a simple proof. Our approach has the potential of proving posterior consistency for a wide range of Bayesian procedures in a unified way.

Item Open Access Accelerating the Computation of Density Functional Theory's Correlation Energy under Random Phase Approximations(2019) Thicke, KyleWe propose novel algorithms for the fast computation of density functional theory's exchange-correlation energy in both the particle-hole and particle-particle random phase approximations (phRPA and ppRPA). For phRPA, we propose a new cubic scaling algorithm for the calculation of the RPA correlation energy. Our scheme splits up the dependence between the occupied and virtual orbitals in the density response function by use of Cauchy's integral formula. This introduces an additional integral to be carried out, for which we provide a geometrically convergent quadrature rule. Our scheme also uses the interpolative separable density fitting algorithm to further reduce the computational cost in a way analogous to that of the resolution of identity method.

For ppRPA, we propose an algorithm based on stochastic trace estimation. A contour integral is used to break up the dependence between orbitals. The logarithm is expanded into a polynomial, and a variant of the Hutchinson algorithm is proposed to find the trace of the polynomial. This modification of the Hutchinson algorithm allows us to use the structure of the problem to compute each Hutchinson iteration in only quadratic time. This is a large asymptotic improvement over the previous state-of-the-art quartic-scaling method and over the naive sextic-scaling method.

Item Open Access Adaptive Data Representation and Analysis(2018) Xu, JierenThis dissertation introduces and analyzes algorithms that aim to adaptively handle complex datasets arising in the real-world applications. It contains two major parts. The first part describes an adaptive model of 1-dimensional signals that lies in the field of adaptive time-frequency analysis. It explains a current state-of-the-art work, named the Synchrosqueezed transform, in this field. Then it illustrates two proposed algorithms that use non-parametric regression to reveal the underlying os- cillatory patterns of the targeted 1-dimensional signal, as well as to estimate the instantaneous information, e.g., instantaneous frequency, phase, or amplitude func-

tions, by a statistical pattern driven model.

The second part proposes a population-based imaging technique for human brain

bundle/connectivity recovery. It applies local streamlines as novelly adopted learn- ing/testing features to segment the brain white matter and thus reconstruct the whole brain information. It also develops a module, named as the streamline diffu- sion filtering, to improve the streamline sampling procedure.

Even though these two parts are not related directly, they both rely on an align- ment step to register the latent variables to some coordinate system and thus to facilitate the final inference. Numerical results are shown to validate all the pro- posed algorithms.

Item Open Access Algebraic Data Structure for Decomposing Multipersistence Modules(2020-11-12) Li, JoeySingle-parameter persistent homology techniques in topological data analysis have seen increasing usage in recent years. These techniques have found particular success because of the existence of a complete, discrete, efficiently computable invariant to describe persistence modules in the single-parameter case: the barcode. Attempts to develop an equally robust theory of multiparameter persistent homology, however, have been slow to progress because there is no natural multiparameter analogue to the barcode. Relatively little is known about the structure of decompositions of multiparameter persistence (multipersistence) modules or how to classify their indecomposables. In fact, even for the problem of computing decompositions, there currently is no generalization to multiple parameters of the decomposition algorithm from single-parameter persistent homology. In this paper, we define a new algebraic data structure, the QR code, which was first proposed in https://arxiv.org/abs/1709.08155 but was formulated somewhat erroneously. Additionally, we prove a theorem stating that the QR code recovers all the information of the module it encodes. We suggest that this new data structure, which seeks to encode a module using births and deaths rather than births and relations, may be the correct language in which to solve the problem of decomposing arbitrary finitely generated multipersistence modules.Item Open Access Algebraic De Rham Theory for Completions of Fundamental Groups of Moduli Spaces of Elliptic Curves(2018) Luo, MaTo study periods of fundamental groups of algebraic varieties, one requires an explicit algebraic de Rham theory for completions of fundamental groups. This thesis develops such a theory in two cases. In the first case, we develop an algebraic de Rham theory for unipotent fundamental groups of once punctured elliptic curves over a field of characteristic zero using the universal elliptic KZB connection of Calaque-Enriquez-Etingof and Levin-Racinet. We use it to give an explicit version of Tannaka duality for unipotent connections over an elliptic curve with a regular singular point at the identity. In the second case, we develop an algebriac de Rham theory for relative completion of the fundamental group of the moduli space of elliptic curves with one marked point. This allows the construction of iterated integrals involving modular forms of the second kind, whereas previously Brown and Manin only studied iterated integrals of holomorphic modular forms.