Mathematics Theses and Dissertations

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G-INVARIANT REPRESENTATIONS USING COORBITS
(2024) Tsoukanis, Efstratios; Balan, Radu V; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
Consider a finite-dimensional real vector space and a finite group acting unitarily on it. We investigate the general problem of constructing Euclidean stable embeddings of the quotient space of orbits. Our embedding relies on subsets of sorted coorbits with respect to chosen window vectors.Our main injectivity results examine the conditions under which such embeddings are injective. We establish these results using semialgebraic techniques. Furthermore, our main stability result states and demonstrates that any embedding based on sorted coorbits is automatically bi-Lipschitz when injective. We establish this result using geometric function techniques. Our work has applications in data science, where certain systems exhibit intrinsic invariance to group actions. For instance, in graph deep learning, graph-level regression and classification models must be invariant to node labeling.
MATHEMATICS OF THE DYNAMICS AND CONTROL OF THE SARS-COV-2 PANDEMIC
(2024) Pant, Binod; Gumel, Abba B.; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
The pneumonia-like illness that emerged late in 2019, caused by SARS-CoV-2 (and coined COVID-19), became the greatest public health challenge humans have faced since the 1918/1919 influenza pandemic, causing over 670 million confirmed cases and 7 million fatalities globally. This dissertation contributes in providing deep qualitative insights and understanding on the transmission dynamics and control of the pandemic, using mathematical modeling approaches together with data analytics and computation. Specifically, it addresses some of the pertinent challenges associated with modeling the dynamics of the disease, notably the disproportionate effect of the disease on certain (risk and demographic) populations (inducing various heterogeneities) and behavior changes with respect to adherence or lack thereof to interventions. An $m-$group model, which monitors the temporal dynamics of the disease in m heterogeneous populations, was designed and used to study the impact of age heterogeneity and vaccination on the spread of the disease in the United States. For instance, the disease-free equilibrium for the case of the model with m=1 (i.e., the model with a homogeneous population) was shown to be globally-asymptotically stable for two special cases (when vaccine is perfect or when disease-induced mortality is negligible) whenever the associated reproduction number of the homogeneous model is less than one. The homogeneous model has a unique endemic equilibrium whenever the reproduction threshold exceeds unity (this equilibrium was shown to be globally-asymptotically stable for a special case, using a nonlinear Lyapunov function of Goh-Volterra type). The homogeneous model was fitted to the observed cumulative mortality data for the SARS-CoV-2 pandemic in the United States during the period from January to May of 2022 (when Omicron was the predominant variant). It was shown that vaccine-derived herd immunity (needed to eliminate the disease) cannot be attained using the homogeneous model regardless of the proportion of individuals fully vaccinated. Such vaccine-derived immunity can, however, be achieved using the $m$-group heterogeneous model, with $m=2$ (where the total population is split into two groups: those under 65 years of age, and those 65 years and older), if at least 61\% of the susceptible population is fully vaccinated. Thus, this dissertation shows that heterogeneity reduces the level of vaccine coverage needed to eliminate the pandemic (and models that do not account for heterogeneity may be over-estimating the vaccination coverage needed to achieve herd immunity in the community). To quantify the impact of human behavior changes on the spread and control of the pandemic, we designed a novel behavior-epidemiology model which considers numerous metrics for inducing human behavior changes (such as current level of disease burden and intervention adherence fatigue). Unlike the equivalent model without human behavior explicitly incorporated, the behavior-epidemiology model fits the observed cumulative mortality and predicts the observed daily mortality data very well. It was also shown that the behavior metrics related to the level of SARS-CoV-2 mortality and symptomatic transmission were more influential in inducing positive behavior changes than all other behavior metrics considered. Finally, a model was developed to assess the utility of wastewater surveillance to study the transmission dynamics and control of SARS-CoV-2 in a community. Specifically, we developed and calibrated a wastewater-based epidemiology model using wastewater data from Miami-Dade county, Florida, during the third wave of the SARS-CoV-2 pandemic. The model showed a strong correlation between the observed (detected) weekly case data and the corresponding weekly data predicted by the calibrated model. The model's prediction of the week when maximum number of SARS-CoV-2 cases will be recorded in the county during the simulation period precisely matched the time when the maximum observed/reported cases were recorded (August 14, 2021). Furthermore, the model's projection of the maximum number of cases for the week of August 14, 2021 was about 15 times higher than the maximum observed weekly case count for the county on that day (i.e., the maximum case count estimated by the model was 15 times higher than the actual/observed count for confirmed cases). In addition to being in line with other modeling studies, this result is consistent with the CDC estimate that the reported confirmed case data may be 10 times lower than the actual (since the confirmed data did not account for asymptomatic and presymptomatic transmission). Furthermore, the model accurately predicted a one-week lag between the peak in weekly COVID-19 case and hospitalization data during the time period of the study in Miami-Dade, with the model-predicted hospitalizations peaking on August 21, 2021. Detailed time-varying global sensitivity analysis was carried out to determine the parameters (wastewater-based, epidemiological and biological) that have the most influence on the chosen response function (namely, the cumulative viral load in the wastewater). This analysis identified key parameters that significantly affect the value of the response function (hence, they should be targeted for intervention). This dissertation conclusively showed that wastewater surveillance data can be a very powerful indicator for measuring (i.e., providing early-warning signal and current burden) and predicting the future trajectory and burden (e.g., number of cases and hospitalizations) of emerging and re-emerging infectious diseases, such as SARS-CoV-2, in a community.
Long-term behavior of randomly perturbed Hamiltonian systems: large deviations and averaging
(2024) Yan, Shuo; Koralov, Leonid; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
This dissertation concerns various asymptotic problems related to the long-term macroscopic behavior of randomly perturbed Hamiltonian systems, with different types of perturbations and on different time scales. Since the perturbations of the systems are assumed to be small while the systems are observed at large times, non-trivial phenomena can arise due to the interplay between the perturbation size and the temporal and spatial scales, and the systems often demonstrate qualitatively distinct types of behavior depending on the subtle quantitative relation between asymptotic parameters. More specifically, on a natural time scale, i.e., in time required for the dynamics to move distance of order one, we investigate the dynamical systems with fast-oscillating perturbations and obtain precise estimates on the distribution. In particular, we calculate the exact asymptotics of the distribution in the case of linear dynamical systems. This problem also inspires the study of the local limit theorem for time-inhomogeneous functions of Markov processes. The local limit theorem is a significant and widely used tool in problems of pure and applied mathematics as well as statistics. This result has been included in [1] and submitted for publication. On the time scale that is inversely proportional to the effective size of the perturbation, we prove that the evolution of the first integral of the Hamiltonian system with fast-oscillating perturbations converges to a Markov process on the corresponding Reeb graph, with certain gluing conditions specified at the interior vertices. The result is parallel to the celebrated Freildin-Wentzell theory on the averaging principle of additive white-noise perturbations of Hamiltonian systems, and provides a description of the long-term behavior of a system when adopting an alternative approach to modeling random noise. Moreover, the current result provides the first scenario where the motion on a graph and the corresponding gluing conditions appear due to the averaging of a slow-fast system. It allows one to consider, for instance, the long-time diffusion approximation for an oscillator with a potential with more than one well. This result has been submitted for publication [2]. In the last part of the dissertation, we return to the more classical case of additive diffusion-type perturbations, combine the ideas of large deviations and averaging, and establish a large deviation principle for the first integral of the Hamiltonian system on intermediate time scales. Besides representing a new step in large deviations and averaging, this result will have important applications to reaction-diffusion equations and branching diffusions. The latter two concepts concern the evolution of various populations (e.g., in biology or chemical reactions). This result has been published in the journal Stochastics and Dynamics [3].
The Twining Character Formula for Split Groups and a Cellular Paving for Quasi-split Groups
(2024) Hopper, Jackson; Haines, Thomas; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
The dissertation contains two main results. The first is on the twisted Weyl character formula for split groups and the second is a cellular paving result for convolution morphisms in partial affine flag varieties of quasi-split groups. Let G^ be a connected reductive group over an algebraically closed field of characteristic 0 with a pinning-preserving outer automorphism σ. Jantzen’s twining character formula relates the trace of the action of σ on a highest-weight representation of G^ to the character of a corresponding highest-weight representation of a related group. This paper extends the methods of Hong’s geometric proof for the case G^ is adjoint, to prove that the formula holds for all split, connected, reductive groups, and examines the role of additional hypotheses. In particular, it is shown that for a disconnected reductive group G, the affine Grassmannian of G is isomorphic to the affine Grassmannian of its neutral component. In the final section, it is explained how these results can be used to draw conclusions about quasi-split groups over a non-Archimedean local field. This paper thus provides a geometric proof of a generalization of the Jantzen twining character formula, and provides some apparently new results of independent interest along the way. Now we turn to the context of Chapter 3. Let G be a tamely ramified, quasi-split group over a Laurent series field K = k((t)), where k is either finite or algebraically closed. If k is finite of order q and the split adjoint form of G contains a factor of type D4, then we also assume either 3 divides q or 3 divides q-1. Given a sequence of Schubert varieties contained in a fixed partial affine flag variety F for G, consider the convolution morphism m that maps the twisted product of those Schubert varieties into the partial affine flag variety F. We show that the fibers of m are paved by finite products of affine spaces and punctured affine spaces. This generalizes a result of Haines, which proves a similar result in the case G is split and defined over k. A consequence for structure constants of parahoric Hecke algebras is deduced.
Quantum Algorithms for Nonconvex Optimization: Theory and Implementation
(2024) Leng, Jiaqi; Wu, Xiaodi XW; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
Continuous optimization problems arise in virtually all disciplines of quantitative research. While convex optimization has been well-studied in recent decades, large-scale nonconvex optimization problems remain intractable in both theory and practice. Quantum computers are expected to outperform classical computers in certain challenging computational problems. Some quantum algorithms for convex optimization have shown asymptotic speedups, while the quantum advantage for nonconvex optimization is yet to be fully understood. This thesis focuses on Quantum Hamiltonian Descent (QHD), a quantum algorithm for continuous optimization problems. A systematic study of Quantum Hamiltonian Descent is presented, including theoretical results concerning nonconvex optimization and efficient implementation techniques for quantum computers. Quantum Hamiltonian Descent is derived as the path integral of classical gradient descent algorithms. Due to the quantum interference of classical descent trajectories, Quantum Hamiltonian Descent exhibits drastically different behavior from classical gradient descent, especially for nonconvex problems. Under mild assumptions, we prove that Quantum Hamiltonian Descent can always find the global minimum of an unconstrained optimization problem given a sufficiently long runtime. Moreover, we demonstrate that Quantum Hamiltonian Descent can efficiently solve a family of nonconvex optimization problems with exponentially many local minima, which most commonly used classical optimizers require super-polynomial time to solve. Additionally, by using Quantum Hamiltonian Descent as an algorithmic primitive, we show a quadratic oracular separation between quantum and classical computing. We consider the implementation of Quantum Hamiltonian Descent for two important paradigms of quantum computing, namely digital (fault-tolerant) and analog quantum computers. Exploiting the product formula for quantum Hamiltonian simulation, we demonstrate that a digital quantum computer can implement Quantum Hamiltonian Descent with gate complexity nearly linear in problem dimension and evolution time. With a hardware-efficient sparse Hamiltonian simulation technique known as Hamiltonian embedding, we develop an analog implementation recipe for Quantum Hamiltonian Descent that addresses a broad class of nonlinear optimization problems, including nonconvex quadratic programming. This analog implementation approach is deployed on large-scale quantum spin-glass simulators, and the empirical results strongly suggest that Quantum Hamiltonian Descent has great potential for highly nonconvex and nonlinear optimization tasks.
CUR Matrix Approximation Through Convex Optimization
(2024) Linehan, Kathryn; Balan, Radu V; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
In this dissertation we present work on the CUR matrix approximation. Specifically, we present 1) an approximation of the proximal operator of the L-infinity norm using a neural network, 2) a novel deterministic CUR formulation and algorithm, and 3) a novel application of CUR as a feature selection method to determine discriminant proteins when clustering protein expression data in a self-organizing map (SOM). The proximal operator of the L-infinity norm arises in our CUR algorithm. Since the computation of the proximal operator of the L-infinity norm requires a sort of the input data (or at least a partial sort similar to quicksort), we present a neural network to approximate the proximal operator. A novel aspect of the network is that it is able to accept vectors of varying lengths due to a feature selection process that uses moments of the input data. We present results on the accuracy of the approximation, feature importance, and computational efficiency of the approach, and present an algorithm to calculate the proximal operator of the L-infinity norm exactly, relate it to the Moreau decomposition, and compare its computational efficiency to that of the approximation. Next, we present a novel deterministic CUR formulation that uses convex optimization to form the matrices C and R, and a corresponding algorithm that uses bisection to ensure that the user selected number of columns appear in C and the user selected number of rows appear in R. We implement the algorithm using the surrogate functional technique of Daubechies et al. [Communications on Pure and Applied Mathematics, 57.11 (2004)] and extend the theory of this approach to apply to our CUR formulation. Numerical results are presented that demonstrate the effectiveness of our CUR algorithm as compared to the singular value decomposition (SVD) and other CUR algorithms. Last, we use our CUR approximation as a feature selection method in the application by Higuera et al. [PLOS ONE, 10(6) (2015)] to determine discriminant proteins when clustering protein expression data in an SOM. This is a novel application of CUR and to the best of our knowledge, this is the first use of CUR on protein expression data. We compare the performance of our CUR algorithm to other CUR algorithms and the Wilcoxon rank-sum test (the original feature selection method in the work).
ASYMPTOTIC AND NUMERICAL ANALYSIS IN KAHLER GEOMETRY: EDGE METRICS, EINSTEIN METRICS AND SOLITONS
(2024) Ji, Yuxiang; Rubinstein, Yanir A; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
This thesis is divided into two parts. The first part focuses on theoretical problems arising from the study of K\"{a}hler edge geometry. The second part introduces a practical method for finding K\"{a}hler--Einstein and soliton metrics that exist on toric Fano surfaces. This method is based on a numerical implementation of the Ricci iteration, which essentially solves a sequence of Monge--Amp\`{e}re equations. Let $M$ be a compact \K manifold and $D=D_1+\dots D_r$ be a simple normal crossing divisor. Given that $K_M+D$ is ample, there exist \KE crossing edge metrics on $M$ with edge singularities of cone angle $\beta_i$ along each component $D_i$ for small $\beta_i$. The first result in the thesis shows that such negatively curved K\"{a}hler--Einstein crossing edge metrics converge to the K\"{a}hler--Einstein mixed cusp and edge metrics smoothly away from the divisor, as some of the cone angles approach $0$. We further show that, near the divisor, a family of appropriately renormalized K\"{a}hler--Einstein crossing edge metrics converges to a mixed cylinder and edge metric in the pointed Gromov--Hausdorff sense as some of the cone angles approach $0$ at (possibly) different speeds. Generalizing $\mathbb{P}^1$, Calabi--Hirzebruch manifolds are constructed by adding an infinite section to the total space of a tensor product of the hyperplane bundle over the projective space, leading to two disjoint divisors: the zero and infinite sections. The second main result in the thesis is the discovery of K\"{a}hler--Einstein edge metrics with singularities along the two divisors on Calabi--Hirzebruch manifolds, and the study on Gromov--Hausdorff limits of these metrics when either cone angle tends to its extreme value. As a very special case, we show that the Eguchi--Hanson metric arises in this way naturally as a Gromov--Hausdorff limit. We also completely describe all other (possibly rescaled) Gromov--Hausdorff limits which exhibit a wide range of behaviors, resolving in this setting a conjecture of Cheltsov--Rubinstein. This gives a new interpretation of both the Eguchi--Hansonspace and Calabi’s Ricci flat spaces as limits of compact singular Einstein spaces. The second part of the thesis focuses on numerical implementation of the Ricci iteration on toric del Pezzo surfaces: $\mathbb{P}^2$, $\mathbb{P}^1\times \mathbb{P}^1$, and blow-up of $\mathbb{P}^2$ at one, two or three distinct points in general position. The Ricci iteration on these surfaces can be reduced to solving a sequence of real Monge--Amp\`{e}re equations in $\mathbb{R}^2$ with the second boundary value condition. As the third contribution of the thesis, we successfully conduct the Ricci iteration on the aforementioned surfaces. We find that the resulting solutions numerically converge to either the unique K\"{a}hler--Einstein metric on $\mathbb{P}^2, \mathbb{P}^1\times \mathbb{P}^1$, and $\dPThr$, or the unique K\"{a}hler--Ricci soliton metric on $\dPOne$ and $\dPTwo$. This provides a novel numerical approach to finding K\"{a}hler--Einstein and soliton metrics on these manifolds. Our numerical results also provide evidence that, in the toric case, the Ricci iteration may converge and produce canonical metrics without the necessity of modifying the metrics obtained during the iterations by automorphisms.
Towards a Classification of Almost Complex and Spin^h Manifolds
(2024) Mills, Keith; Rosenberg, Jonathan; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
We show that all homotopy CP^ns, smooth closed manifolds with the oriented homotopy type of CP^n, admit almost complex structures for 3 ≤ n ≤ 6, and classify these structures by their Chern classes for n=4, 6. Our methods provide a new proof of a result of Libgober and Wood on the classification of almost complex structures on homotopy CP^4s. We also show that all homotopy RP^(2k+1)s admit stably almost complex structures. Spin^h manifolds are the quaternionic analogue to spin^c manifolds. At the prime 2 we compute the spin^h bordism groups by proving a structure theorem for the cohomology of the spin^h bordism spectrum MSpin^h as a module over the mod 2 Steenrod algebra. This provides a 2-local splitting of MSpin^h as a wedge sum of familiar spectra. We also compute the decomposition of H^*(MSpin^h; Z/2Z) explicitly in degrees up through 30 via a counting process.
EFFICIENT COMPUTATIONAL ALGORITHMS FOR MAGNETIC EQUILIBRIUM IN A FUSION REACTOR
(2024) Liang, Jiaxing; Elman, Howard C.; Sanchez-Vizuet, Tonatiuh; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
In a magnetic confinement fusion reactor, like a Tokamak, hydrogen isotopes are injected into the chamber and heated to form a plasma. The hot plasma tends to expand, so it's crucial to confine it within the center of the reactor to prevent contact with the reactor walls. This confinement is achieved through magnetic fields generated by currents running through external coils surrounding the reactor. However, these currents may suffer from uncertainties, which could arise from factors like temperature fluctuations and material impurities. These variables introduce a level of unpredictability in the plasma's behavior and the overall stability of the fusion process. This thesis aims to investigate the impact that stochasticity in the current intensities has on the confinement properties of the plasma and to estimate the expected behavior of the magnetic field. While the focus is on the variability in current intensities, the tools developed can be applied to other sources of uncertainty, such as the positioning of coils and the source term parameters. To quantify the variability in model predictions and to evaluate the statistical properties of solutions over a range of parameter values, traditional sampling methods like Monte Carlo, often require intensive and expensive nonlinear computations. To tackle this challenge, we propose three approaches. Firstly, we focus on the development and application of a surrogate function, constructed via a stochastic collocation approach on a sparse grid in the parameter space. This surrogate function is employed to replace the nonlinear solution in Monte Carlo sampling processes. For our numerical experiments, we evaluate the efficiency and accuracy of the outcomes produced by the surrogate, in comparison with those obtained through direct nonlinear solutions. Our findings indicate that a carefully selected surrogate function reduces the sampling cost -- achieving acceleration factors ranging from 7 to over 30 -- while preserving the accuracy of the results. The second part of the thesis explores the multilevel Monte Carlo approach, investigating its potential for cost savings compared to simple Monte Carlo. This method involves conducting the majority of computations on a sequence of coarser spatial grids compared to what a simple Monte Carlo simulation would typically use. We examine this approach with non-linear computation, using both uniformly refined meshes and adaptively refined grids guided by a discrete error estimator. Numerical experiments reveal substantial cost reductions achieved through multilevel methods, typically ranging from a factor of 60 to exceeding 200. Adaptive gridding results in more accurate computation of relevant geometric parameters. In the last part of this work, we explore hybridmethods that integrate surrogates with multilevel Monte Carlo to further reduce the sampling cost. We establish the optimal construction and sampling costs for the surrogate-based multilevel Monte Carlo. Numerical results demonstrate that surrogate-based multilevel Monte Carlo remarkably reduces the computational burden, requiring only 0.1 to 14 seconds for a target relative mean square error ranging from $8\times 10^{-3}$ to $2\times10^{-4}$, reducing the cost of direct computation by factors of 50 to 300. In terms of accuracy, the surrogate-based sampling results exhibit close congruence with those obtained via direct computation, both in plasma boundary and geometric descriptors. The primary contributions of our work entail the application of stochastic collocation techniques and multilevel Monte Carlo methods to analyze plasma behavior under uncertainties in current within fusion reactors. Furthermore, we establish the universal sampling cost for the surrogate-enhanced multilevel Monte Carlo approach. Our methodology presents a paradigm in how we approach and manage computational challenges in this field.
Structured discovery in graphs: Recommender systems and temporal graph analysis
(2024) Peyman, Sheyda Do'a; Lyzinski, Vince V.; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
Graph-valued data arises in numerous diverse scientific fields ranging from sociology, epidemiology and genomics to neuroscience and economics.For example, sociologists have used graphs to examine the roles of user attributes (gender, class, year) at American colleges and universities through the study of Facebook friendship networks and have studied segregation and homophily in social networks; epidemiologists have recently modeled Human-nCov protein-protein interactions via graphs, and neuroscientists have used graphs to model neuronal connectomes. The structure of graphs, including latent features, relationships between the vertex and importance of each vertex are all highly important graph properties that are main aspects of graph analysis/inference. While it is common to imbue nodes and/or edges with implicitly observed numeric or qualitative features, in this work we will consider latent network features that must be estimated from the network topology.The main focus of this text is to find ways of extracting the latent structure in the presence of network anomalies. These anomalies occur in different scenarios: including cases when the graph is subject to an adversarial attack and the anomaly is inhibiting inference, and in the scenario when detecting the anomaly is the key inference task. The former case is explored in the context of vertex nomination information retrieval, where we consider both analytic methods for countering the adversarial noise and also the addition of a user-in-the-loop in the retrieval algorithm to counter potential adversarial noise. In the latter case we use graph embedding methods to discover sequential anomalies in network time series.
STATISTICAL DATA FUSION WITH DENSITY RATIO MODEL AND EXTENSION TO RESIDUAL COHERENCE
(2024) Zhang, Xuze; Kedem, Benjamin; Mathematical Statistics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
Nowadays, the statistical analysis of data from diverse sources has become more prevalent. The Density Ratio Model (DRM) is one of the methods for fusing and analyzing such data. The population distributions of different samples can be estimated basedon fused data, which leads to more precise estimates of the probability distributions. These probability distributions are related by assuming the ratios of their probability density functions (PDFs) follow a parametric form. In the previous works, this parametric form is assumed to be uniform for all ratios. In Chapter 1, an extension is made to allow this parametric form to vary for different ratios. Two methods of determining the parametric form for each ratio are developed based on asymptotic test and Akaike Information Criterion (AIC). This extended DRM is applied to Radon concentration and Pertussis rates to demonstrate the use of this extension in univariate case and multivariate case, respectively. The above analysis is made possible when data in each sample are independent and identically distributed (IID). However, in many cases, statistical analysis is entailed for time series in which data appear to be sequentially dependent. In Chapter 2, an extension is made for DRM to account for weakly dependent data, which allows us to investigate the structure of multiple time series on the strength of each other. It is shown that the IID assumption can be replaced by proper stationarity, mixing and moment conditions. This extended DRM is applied to the analysis of air quality data which are recorded in chronological order. As mentioned above, DRM is suitable for the situation where we investigate a single time series based on multiple alternative ones. These time series are assumed to be mutually independent. However, in time series analysis, it is often of interest to detect linear and nonlinear dependence between different time series. In such dependent scenario, coherence is a common tool to measure the linear dependence between two time series, and residual coherence is used to detect a possible quadratic relationship. In Chapter 3, we extend the notion of residual coherence and develop statistical tests for detecting linear and nonlinear associations between time series. These tests are applied to the analysis of brain functional connectivity data.
The Shuffling Effect: Vertex Label Error’s Impact on Hypothesis Testing, Classification, and Clustering in Graph Data
(2024) Saxena, Ayushi; Lyzinski, Vince; Mathematical Statistics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
The increasing prevalence of graph and network-valued data across various disciplines has prompted significant interest and research in recent years. This dissertation explores the impact of vertex shuffling, or vertex misalignment, on the statistical network inference tasks of hypothesis testing, classification, and clustering. Our focus is within the framework of multiple network inference, where existing methodologies often assume known vertex correspondence across networks. This assumption frequently does not hold in practice. Through theoretical analyses, simulations, and experiments, we aim to reveal the effects of vertex shuffling on different types of performance.Our investigation begins with an examination of two-sample network hypothesis testing, focusing on the decrease in statistical power resulting from vertex shuffling. In this work, our analysis focuses on the random dot product and stochastic block model network settings. Subsequent chapters delve into the effects of shuffling on graph classification and clustering, showcasing how misalignment negatively impacts accuracy in categorizing and clustering graphs (and vertices) based on their structural characteristics. Various machine learning algorithms and clustering methodologies are explored, revealing a theme of consistent performance degradation in the presence of vertex shuffling. We also explore how graph matching algorithms can potentially mitigate the effects of vertex misalignment and recover the lost performance. Our findings also highlight the risk of graph matching as a pre-processing tool, as it can induce artificial signal. These findings highlight the difficulties and subtleties of addressing vertex shuffling across multiple network inference tasks and suggest avenues for future research in order to enhance the robustness of statistical inference methodologies in complex network environments.
A Data Assimilation System for Lake Erie Based on the Local Ensemble Transform Kalman Filter
(2024) Russell, David Scott; Ide, Kayo; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
Data assimilation (DA) is the process by which a model forecast is adjusted to account for recent observations, taking into account both forecast and observation uncertainties. Although DA is common in numerical weather prediction (NWP) and other applications at global and regional scales, DA for large lakes such as North America's Great Lakes is still at an early stage of research and is not yet used operationally. In particular, the use of an ensemble-based approach to DA has scarcely been explored for large lakes, despite its growing popularity in operational NWP centers worldwide due to its dynamic estimation of the forecast covariance. Using Lake Erie as a test case, this study investigates the potential of ensemble DA to i) propagate forecast improvements throughout the lake and across forecast variables, and ii) inform the design of in-situ observing systems. The local ensemble transform Kalman filter (LETKF) is an efficient, localized, flexible variant of the ensemble Kalman filter (EnKF) that is used in multiple operational NWP centers. This work presents the development of a DA system for Lake Erie, which uses the LETKF to adjust forecasts of temperatures, currents, and water levels throughout the lake, using only lake surface temperature (LST) and temperature profile (TP) observations. The impact of both types of observations on all three forecast variables is evaluated within the framework of observing system simulation experiments (OSSEs), in which a DA system attempts to reconstruct a nature run (NR) by assimilating simulated observations of the NR. Observing system design questions are explored by comparing three different TP configurations. Assimilation of LST observations alone produces strong improvements to temperatures throughout the epilimnion (upper layer), while assimilation of TP observations extends these improvements to the hypolimnion (lower layer) near each profile. TP assimilation also shows improved representation of strong gyre currents and associated changes to thermocline depth and surface height, particularly when profiles sample from locations and depths where the thermal stratification in the forecast has been strongly affected by erroneous gyre currents. This work shows that the LETKF can be an efficient and effective tool for improving both forecasts and observing systems for large lakes, two essential ingredients in predicting the onset and development of economically and ecologically important phenomena such as harmful algal blooms (HABs) and hypoxia.
Ensemble Kalman Inverse Parameter Estimation for Human and Nature Dynamics Two
(2023) Karpovich, Maia; Kalnay, Eugenia; Mote, Safa; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
Since the widespread development of agriculture 10,000 years ago and particularly since the Industrial Revolution beginning in the 18th century, the coupled Earth and Human systems have experienced transformative change. The world's population and Gross Domestic Product have each increased by factors of at least eight in the last two centuries, powered by the intensive use of fossil energy and fossil water. This has had dramatic repercussions on the Earth system's stability, threatened by Human system activities such as habitat destruction, global warming, and depletion of Regenerating and Nonrenewable energy resources that increasingly alter environmental feedbacks. To analyze these changes, we have developed the second generation of the Human and Nature Dynamics model, HANDY2. HANDY2 is designed to simulate the dynamics of energy resources and population over the Industrial era from 1700 to 2200, flexibly incorporating real-world observations of population and energy consumption in an expanded suite of mechanisms that track capital investment, labor force allocation, class mobility, and extraction and production technologies. The use of automated Ensemble Kalman Inversion (EnKI) estimation for HANDY2's parameters allows us to accurately capture the rapid 20th-century rise in the use of phytomass and fossil fuels, as well as the global enrichment of Elites that puts pressure on natural resources and Commoners. EnKI-derived HANDY2 ensembles project that current world policies may lead to a collapse in the world's population by 2200 caused by rapid depletion of resources. However, this collapse can be prevented by a combination of actions taken to support voluntary family planning, lower economic inequality, and most importantly, invest in the rapid expansion of Renewable energy extraction.
TWO-PHASE FLOW OF COMPRESSIBLE VISCOUS DIBLOCK COPOLYMER FLUID
(2023) Ye, Anqi; Trivisa, Konstantina; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
A diblock copolymer is a linear-chain molecule consisting of two types of monomer.Mathematical models for diblock copolymers can aid researchers in studying the material properties of products as upholstery foam, adhesive tape and asphalt additive. Such models incorporate a variety of factors, including concentration difference, connectivity of the subchains, and chemical potential. We consider a flow of two macroscopically immiscible, viscous compressible diblock copolymer fluids. We first give the derivation of this model on the basis of a local dissipation inequality. Second, we prove that there exist weak solutions to this model. The proof of existence relies on constructing an approximating system by means of time-discretization and vanishing dissipation. We then prove that the solutions to these approximating schemes converge to a solution to the original problem. We also cast thought on the large-time behavior with regularity assumption on the limit.
Student Choice Among Large Group, Small Group, and Individual Learning Environments in a Community College Mathematics Mini-Course
(1986) Baldwin, Eldon C.; Davidson, Neil; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, MD)
This study describes the development and implementation of a model for accommodation of preferences for alternative instructional environments. The study was stimulated by the existence of alternative instructional modes, and the absence of a procedure for accommodation of individual student differences which utilized these alternative modes. The Choice Model evolved during a series of pilot studies employing three instructional modes; individual (JM), small group (SGM), and large group (LGM). Three instructors were each given autonomy in designing one learning environment, each utilizing her/his preferred instructional mode. One section of a mathematics course was scheduled for one hundred students. On the first day the class was divided alphabetically into three orientation groups, each assigned to a separate class room. During the first week, the instructors described their respective environments to each group, using video taped illustrations from a previous semester. Environmental preferences were then assessed using take-home student questionnaires. In the final pilot, fifty-five students were oriented to all three environments. Each student was then assigned to his/her preferred learning environment. The distribution of environmental preferences was 24% for IM, 44% for SGM, and 33% for LGM. The following student characteristics were also investigated: 1)sex, 2)age, 3)academic background, 4)mathematics achievement, 5)mathematics attitude, 6)mathematics interest, 7)self-concept, 8)communication apprehension. and 9)interpersonal relations orientation. This investigation revealed several suggestive preference patterns: 1)Females and students with weak academic backgrounds tended to prefer the SGM environment. 2)Students with higher levels of communication apprehension tended to avoid the SGM environment. 3)New college students and students with negative mathematics attitudes tended to avoid the IM environment. 4)Students with higher grades in high school tended to prefer the LGM environment. Student preferences were successfully accommodated, and student evaluations of the Choice Model were generally positive. The literature suggests that opportunities to experience choice in education tend to enhance student growth and development; adaptation and institutionalization of the Model were addressed from this perspective. Additional studies with larger samples were recommended to further investigate environmental preferences with respect t o student and instructor characteristics of gender, age, race, socioeconomic background, academic background, and learning style.
DISTRIBUTION OF PRIME ORDER IDEAL CLASSES OF QUADRATIC CLASS GROUPS
(2023) Wedige, Melanka Saroad; Ramachandran, Niranjan; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
The Cohen-Lenstra heuristics predicts that for any odd prime k, the k-part of quadraticclass groups occur randomly on the space of k-groups with respect to a natural probability measure. However, apart from the first moments of the 3-torsion part of quadratic class groups, consequences of these heuristics still remain highly conjectural. The quadratic ideal classes have geometric representations on the modular curve as CM-points in the case of negative discriminants and as closed primitive geodesics in the case of positive discriminants. We mainly focus on the asymptotic distribution of these geometrical objects. As motivation, it is seen that in the case of imaginary quadratic fields, knowledge on the (joint) distribution of k-order CM-points leads in-general to the resolution of the Cohen-Lenstra conjectures on moments of the k-part of class groups. As a first step, inspired by the works of Duke, Hough conjectures that the k-order CM-points are equidistributed on the modular curve. Although the case with k = 3 was resolved by Hough himself, k > 3 remains unresolved. In this dissertation, we revisit Hough’s conjectures, with empirical evidence. We were able to reprove the conjecture for k = 3, and even stronger to show that the result holds along certain subfamilies of imaginary quadratic fields defined by local behaviors of their discriminants. In addition, we study the case for k > 3. We introduce a heuristics model, and show that this model agrees with Hough’s conjectures. We also show that the difference between the actual asymptotics and the heuristic model reduces down to the distribution of solutions to certain quadratic congruences. We, then again inspired by Duke’s work, investigate an analog for the real quadratic fields. Backed by empirical evidence, we go on to conjecture the asymptotic behavior of the length of k-order geodesics on the modular curve. In addition, based on a theorem and its proof by Siegel, we prove certain results that may shed light on a probable proof direction of these conjectures.
GRAPH-BASED DATA FUSION WITH APPLICATIONS TO MAGNETIC RESONANCE IMAGING
(2023) Emidih, Jeremiah; Czaja, Wojciech; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
This thesis is concerned with development and applications of data fusion methods in thecontext of Laplacian eigenmaps. Multimodal data can be challenging to work with using classical statistical and signal processing techniques. Graphs provide a reference frame for the study of otherwise structure-less data. We combine spectral methods on graphs and geometric data analysis in order to create a novel data fusion model. We also provide examples of applications of this model to bioinformatics, color transformation and superresolution, and magnetic resonance imaging.
A Multifaceted Quantification of Bias in Large Language Models
(2023) Sotnikova, Anna; Daumé III, Hal; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
Language models are rapidly developing, demonstrating impressive capabilities in comprehending, generating, and manipulating text. As they advance, they unlock diverse applications across various domains and become increasingly integrated into our daily lives. Nevertheless, these models, trained on vast and unfiltered datasets, come with a range of potential drawbacks and ethical issues. One significant concern is the potential amplification of biases present in the training data, generating stereotypes and reinforcing societal injustices when language models are deployed. In this work, we propose methods to quantify biases in large language models. We examine stereotypical associations for a wide variety of social groups characterized by both single and intersectional identities. Additionally, we propose a framework for measuring stereotype leakage across different languages within multilingual large language models. Finally, we introduce an algorithm that allows us to optimize human data collection in conditions of high levels of human disagreement.
Modeling the fracture of polymer networks
(2023) Tao, Manyuan; Cameron, Maria; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
This dissertation is devoted to modeling the fracture of highly elastic materials that consist of polymer networks, such as elastomers and hydrogels. These polymer materials are composed of long polymer chains of repeating molecular units, which are crosslinked to form a three-dimensional network structure. A polymer network fractures by breaking covalent bonds, but the experimentally measured strength of a polymer network is orders of magnitude lower than the strength of covalent bonds. In this dissertation, we develop mesoscale models to understand what are the necessary ingredients leading to a large reduction in the strength of polymer networks observed in experiments. We hypothesize that the large reduction in strength is caused by statistical variation in lengths of polymer chains and a J-shaped stress-stretch relationship. The polymer chain carries entropic forces for most of the extension and carries covalent forces only for a narrow range of the extension. As a result, the statistical distribution of chain lengths causes only a small fraction of polymer chains to be highly stressed when the network is near fracture. We test this hypothesis using two mesoscale models: an idealized parallel chain model and a two-dimensional network model. Both models assume a statistical distribution for the lengths of polymer chains. Polymer chains are represented by freely-jointed chains that feature a nonlinear J-shaped stress-stretch relationship. The parallel chain model allows for simple calculations and is amenable for analysis by analytical tools. The network model accounts for the effect of stress concentration and is amenable for numerical simulations. Our models show that the combination of a J-shaped stress-stretch relationship and a distribution of chain lengths leads to a large reduction in strength, while keeping the variability in strength small from sample to sample. The large scatter in chain lengths causes a reduction in strength by up to two orders of magnitude, which explains a portion of the giant discrepancy between the experimentally measured strength of hydrogels and the strength of covalent bonds. Furthermore, our models demonstrate a power law relationship between the strength and the scatter in chain lengths. We provide an analytical derivation of the power law by taking advantage of the simplicity of the parallel chain model. In addition to studying macroscopic fracture properties, we further investigate the microscopic characteristics and the breaking mechanism of the polymer network, using the network model. By examining the characteristics of shortest paths, we find that the links traversed by a large number of shortest paths are more likely to break. Finally, we connect the microstructure of the network to the macroscopic mechanical properties. It is observed that the strength of the network correlates with the growth of holes during deformation.

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