Mathematical Sciences Department Theses and Dissertations

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Mathematical Models of Major Arterial Occlusion
(2025-05) Zhao, Erin; Arciero, Julia; Barber, Jared; Kuznetsov, Alexey; Zhu, Luoding
The occlusion of a major artery constitutes a serious health concern as it can restrict blood flow and oxygen transport to dependent tissue regions. Fortunately, the vasculature surrounding the occlusion has mechanisms by which it can adapt to try to restore and maintain adequate perfusion to these regions, though the details of these compensatory mechanisms are not well understood. The aim of the present study is to use mathematical modeling to investigate the effects of major arterial occlusion in multiple tissues and vascular geometries. A network representing the vasculature of the rat hindlimb is used to study peripheral arterial disease characterized by femoral artery occlusion. This work couples responses that occur on different time scales, namely vessel dilation and constriction on a short time scale and structural changes including arteriogenesis and angiogenesis on a long time scale. In the acute time frame, the responses that contribute most to changes in vascular tone are increases in flow and shear stress in collateral vessels and increases in metabolic signaling in distal arterioles. On the chronic scale, arteriogenesis is found to have a significantly larger effect on flow restoration than angiogenesis. A model of the major arteries and regions of the human brain is used to assess the impact of stroke caused by middle cerebral artery occlusion and the role of leptomeningeal collaterals in restoring flow downstream of the occlusion. The effects of incorporating pulsatile blood flow and arterial distensibility are also examined. The model demonstrates that the leptomeningeal collaterals are critical to restoring blood flow to the middle region, but the degree to which this is successful is highly dependent on conditions such as oxygen demand and arterial pressure. Overall, the results obtained from this study provide valuable insight into the vascular response mechanisms that contribute the most to flow compensation after occlusion and factors that may improve or worsen perfusion deficits. Insight from these models will inform the mechanisms and/or vessels to target in potential new treatments for peripheral arterial disease and stroke.
Early Detection of Treatment's Side Effects: A Sequential Approach
(2025-05) Wang, Jiayue; Boukai, Ben; Li, Fang; Peng, Hanxiang; Sarkar, Jyotirmoy
With the emergence and spread of infectious diseases with pandemic potential, such as COVID-19, the urgency for vaccine development has led to unprecedented compressed and accelerated schedules that shortened the standard development timeline. To monitor the potential side eﬀect(s) of the vaccine during the (initial) vaccination campaign, we developed an optimal sequential test that allows for the early detection of potential side eﬀect(s). This test employs a rule to stop the vaccination process once the observed number of side eﬀect incidents exceeds a certain (pre-determined) threshold. The optimality of the proposed sequential test is justified when compared with the (α,β) optimality of the non-randomized fixed-sample Uniformly Most Powerful (UMP) test. Firstly, we construct an optimal sequential procedure for the case of a single side effect. We study the properties of the sequential test and derive the exact expressions of the Average Sample Number (ASN) curve of the stopping time (and its variance) via the regularized incomplete beta function. Additionally, we derive the asymptotic distribution of the relative ’savings’ in ASN as compared to fixed sample size. Moreover, we construct the post-test parameter estimate and studied its sampling properties, including its asymptotic behavior under local-type alternatives. These limiting behavior results are the consistency and asymptotic normality of the post-test parameter estimator. We conclude this part with a small simulation study illustrating the asymptotic performance of the point and interval estimation and provide a detailed example, based on COVID-19 side eﬀect data (see Beatty et al. (2021)) of our suggested testing procedure. Next, we propose an optimal sequential procedure for the case of two (or more) side eﬀects. While the sequential procedure we employ, simultaneously monitors several of the treatment’s side eﬀects, the (α,β)-optimal test we propose does not require any information about the inter-correlation between these potential side eﬀects to obtain the optimal sample size and critical values. However, in all of the subsequent analyses, including the derivations of the exact expressions of the ASN, the power function, and the properties of the post-test (or post-detection) estimators, we accounted specifically, for the correlation between the potential side eﬀects. In the real-life application (such as post-marketing surveillance), the number of available observations is large enough to justify asymptotic analyses of the sequential procedure (testing and post-detection estimation) properties. Accordingly, we also derive the consistency and asymptotic normality of our post-test estimators. Moreover, to compare two specific side eﬀects, their relative risk plays an important role. We derive the distribution of the estimated relative risk in the asymptotic framework to provide appropriate inference. To illustrate the theoretical results presented, we provide two detailed examples based on the data of side eﬀects on COVID-19 vaccine collected in Nigeria (see Ilori et al. (2022)). Finally, we consider the Bayesian framework and construct a Bayesian Sequential testing procedure to test the Relative Risk between two specific treatments based on the binary data obtained from the two-arm clinical trial. We noticed that the optimal sequential test mentioned in the first part can be utilized here to test Relative Risk as a one-sided hypothesis testing procedure. To apply the optimal sequential test more straightforward, we introduce the Bayesian framework into our analysis. Since by the Stopping Rule Principle (SRP), the posterior probabilities remain unaffected by the stopping rule used to reach that point with accumulated data. Accordingly, using the Bayesian test, we obtain the corresponding decision on each data point without aﬀecting by the optimal stopping rule as in classical sequential test. Moreover, with the modified Bayesian test, we obtain the corresponding conditional error probabilities on each data point. Specifically, we utilized the data from Silva, Kulldorﬀ, and Katherine Yih (2020) to analyze the results obtained from two tests under several diﬀerent priors and make the conclusion.
Advancement of Inferences for Genetic/Treatment Effects under Semiparametric Models
(2025-05) Cui, Yishan; Wang, Honglang; Li, Fang; Peng, Hanxiang; Sarkar, Jyotirmoy
This dissertation presents three methodological advancements in semiparametric modeling and inference, with applications in longitudinal data analysis and individualized treatment rules (ITRs). First, we enhance the semiparametric profile estimator for analyzing longitudinal data, addressing the challenge of within-subject correlation. By incorporating a nonparametric operator-regularized approach for estimating the covariance function, we develop a refined estimator that significantly improves efficiency over traditional local kernel smoothing methods, which assume an independent correlation structure. We further introduce an Empirical Likelihood (EL)-based inference method and demonstrate, through simulations and an application to the Genetic Analysis Workshop 18 dataset, that our approach attains the semiparametric efficiency bound and outperforms existing methods. Second, we propose a rank-based inference procedure for ITRs under a semiparametric single-index varying coefficient model, where the nonparametric coefficient function is assumed to be monotone increasing. By leveraging maximum rank correlation, our method circumvents direct estimation of the nonparametric function, thereby mitigating potential biases. For hypothesis testing, we derive the asymptotic distribution of the proposed estimator using de-biasing techniques. Monte Carlo simulations and an application to the ACTG175 dataset confirm the effectiveness of our approach. Finally, we develop a jackknife empirical likelihood ratio test to enhance hypothesis testing in the semiparametric single-index varying coefficient model. Existing methods often rely on plug-in variance-covariance estimators that approximate indicator functions using a sigmoid transformation, which are computationally complex and difficult to implement. Our proposed test offers a much simpler computational approach while achieving the same effectiveness. Extensive simulations and real-data analysis using the ACTG175 dataset further demonstrate the efficiency and practicality of our method. Together, these contributions enhance the efficiency and reliability of semiparametric estimation and inference, particularly in the contexts of longitudinal data analysis and individualized treatment decision-making.
A Dynamical Approach to the Potts Model on Cayley Tree
(2024-12) Pannipitiya, Diyath Nelaka; Kitchens, Bruce P.; Roeder, Roland K. W.; Geller, William; Perez, Rodrigo A.
The Ising model is one of the most important theoretical models in statistical physics, which was originally developed to describe ferromagnetism. A system of magnetic particles, for example, can be modeled as a linear chain in one dimension or a lattice in two dimensions, with one particle at each lattice point. Then each particle is assigned a spin σi ∈ {±1}. The q-state Potts model is a generalization of the Ising model, where each spin σi may take on q ≥3 number of states {0,··· ,q−1}. Both models have temperature T and an externally applied magnetic field h as parameters. Many statistical and physical properties of the q- state Potts model can be derived by studying its partition function. This includes phase transitions as T and/or h are varied. The celebrated Lee-Yang Theorem characterizes such phase transitions of the 2-state Potts model (the Ising model). This theorem does not hold for q > 2. Thus, phase transitions for the Potts model as h is varied are more complicated and mysterious. We give some results that characterize the phase transitions of the 3-state Potts model as h is varied for constant T on the binary rooted Cayley tree. Similarly to the Ising model, we show that for fixed T >0the 3-state Potts model for the ferromagnetic case exhibits a phase transition at one critical value of h or not at all, depending on T. However, an interesting new phenomenon occurs for the 3-state Potts model because the critical value of h can be non-zero for some range of temperatures. The 3-state Potts model for the antiferromagnetic case exhibits a phase transition at up to two critical values of h. The recursive constructions of the (n + 1)st level Cayley tree from two copies of the nth level Cayley tree allows one to write a relatively simple rational function relating the Lee-Yang zeros at one level to the next. This allows us to use techniques from dynamical systems.
Robust Inference for Heterogeneous Treatment Effects With Applications to NHANES Data
(2024-12) Mo, Ran; Wang, Honglang; Li, Fang; Tan, Fei; Peng, Hanxiang
Estimating the conditional average treatment effect (CATE) using data from the National Health and Nutrition Examination Survey (NHANES) provides valuable insights into the heterogeneous impacts of health interventions across diverse populations, facilitating public health strategies that consider individual differences in health behaviors and conditions. However, estimating CATE with NHANES data face challenges often encountered in observational studies, such as outliers, heavy-tailed error distributions, skewed data, model misspecification, and the curse of dimensionality. To address these challenges, this dissertation presents three consecutive studies that thoroughly explore robust methods for estimating heterogeneous treatment effects. The first study introduces an outlier-resistant estimation method by incorporating M-estimation, replacing the \(L_2\) loss in the traditional inverse propensity weighting (IPW) method with a robust loss function. To assess the robustness of our approach, we investigate its influence function and breakdown point. Additionally, we derive the asymptotic properties of the proposed estimator, enabling valid inference for the proposed outlier-resistant estimator of CATE. The method proposed in the first study relies on a symmetric assumption which is commonly required by standard outlier-resistant methods. To remove this assumption while maintaining unbiasedness, the second study employs the adaptive Huber loss, which dynamically adjusts the robustification parameter based on the sample size to achieve optimal tradeoff between bias and robustness. The robustification parameter is explicitly derived from theoretical results, making it unnecessary to rely on time-consuming data-driven methods for its selection. We also derive concentration and Berry-Esseen inequalities to precisely quantify the convergence rates as well as finite sample performance. In both previous studies, the propensity scores were estimated parametrically, which is sensitive to model misspecification issues. The third study extends the robust estimator from our first project by plugging in a kernel-based nonparametric estimation of the propensity score with sufficient dimension reduction (SDR). Specifically, we adopt a robust minimum average variance estimation (rMAVE) for the central mean space under the potential outcome framework. Together with higher-order kernels, the resulting CATE estimation gains enhanced efficiency. In all three studies, the theoretical results are derived, and confidence intervals are constructed for inference based on these findings. The properties of the proposed estimators are verified through extensive simulations. Additionally, applying these methods to NHANES data validates the estimators' ability to handle diverse and contaminated datasets, further demonstrating their effectiveness in real-world scenarios.
Modeling and Simulation of Osteocyte-Fluid Interaction in a Lacuno-Canalicular Network in Three Dimensions
(2024-12) Karimli, Nigar; Barber, Jared; Zhu, Luoding; Arciero, Julia; Na, Sungsoo
Bone health relies on its cells' ability to sense and respond to mechanical forces, a process primarily managed by osteocytes embedded within the bone matrix. The cells reside in the lacuno-canalicular network (LCN), a complex structure, comprised of lacunae (small cavities) and canaliculi (microscopic channels), through which they communicate and receive nutrients. The mechanotransduction (MT) process, by which osteocytes convert mechanical signals from mechanical loading into biochemical responses, is essential for bone remodeling but remains poorly understood. Both in-vitro and in-vivo studies present challenges in directly measuring the cellular stresses and strains involved, making computational modeling a valuable tool for studying osteocyte mechanics. In this dissertation, we present a coarse-grained, integrative model designed to simulate stress and strain distributions within an osteocyte and its microenvironment. Our model features the osteocyte membrane represented as a network of viscoelastic springs, with six slender, arm-like osteocytic processes extending from the membrane. The osteocyte is immersed in interstitial fluid and encompassed by the rigid extracellular matrix (ECM). The cytosol and interstitial fluid are both modeled as water-like, viscous incompressible fluids, allowing us to capture the fluid-structure interactions crucial to understanding the MT. To simulate these interactions, we employ the Lattice Boltzmann - Immersed Boundary (LB-IB) method. This approach couples the Lattice Boltzmann method, which numerically solves fluid equations, with the immersed boundary method, which handles the interactions between the osteocyte structures and the surrounding fluids. This framework consists of a system of integro-partial differential equations describing both fluid and solid dynamics, enabling a detailed examination of force, strain, and stress distribution within the osteocyte. Major results include 1) increased incoming flow routes results in increased stress and strain, 2) regions of higher stress and strain are concentrated near the junctions where the osteocytic processes meet the main body.
Sample Size Determination for Subsampling in the Analysis of Big Data, Multiplicative Models for Confidence Intervals and Free-Knot Changepoint Models
(2024-05) Zhang, Sheng; Peng, Hanxiang; Tan, Fei; Sarkar, Jyoti; Boukai, Ben
The dissertation consists of three parts. Motivated by subsampling in the analysis of Big Data and by data-splitting in machine learning, sample size determination for multidimensional parameters is presented in the first part. In the second part, we propose a novel approach to the construction of confidence intervals based on improved concentration inequalities. We provide the missing factor for the tail probability of a random variable which generalizes Talagrand’s (1995) result of the missing factor in Hoeffding’s inequalities. We give the procedure for constructing confidence intervals and illustrate it with simulations. In the third part, we study irregular change-point models using free-knot splines. The consistency and asymptotic normality of the least squares estimators are proved for the irregular models in which the linear spline is not differentiable. Simulations are carried out to explore the numerical properties of the proposed models. The results are used to analyze the US Covid-19 data.
Efficient Inference and Dominant-Set Based Clustering for Functional Data
(2024-05) Wang, Xiang; Wang, Honglang; Boukai, Benzion; Tan, Fei; Peng, Hanxiang
This dissertation addresses three progressively fundamental problems for functional data analysis: (1) To do efficient inference for the functional mean model accounting for within-subject correlation, we propose the refined and bias-corrected empirical likelihood method. (2) To identify functional subjects potentially from different populations, we propose the dominant-set based unsupervised clustering method using the similarity matrix. (3) To learn the similarity matrix from various similarity metrics for functional data clustering, we propose the modularity guided and dominant-set based semi-supervised clustering method. In the first problem, the empirical likelihood method is utilized to do inference for the mean function of functional data by constructing the refined and bias-corrected estimating equation. The proposed estimating equation not only improves efficiency but also enables practically feasible empirical likelihood inference by properly incorporating within-subject correlation, which has not been achieved by previous studies. In the second problem, the dominant-set based unsupervised clustering method is proposed to maximize the within-cluster similarity and applied to functional data with a flexible choice of similarity measures between curves. The proposed unsupervised clustering method is a hierarchical bipartition procedure under the penalized optimization framework with the tuning parameter selected by maximizing the clustering criterion called modularity of the resulting two clusters, which is inspired by the concept of dominant set in graph theory and solved by replicator dynamics in game theory. The advantage offered by this approach is not only robust to imbalanced sizes of groups but also to outliers, which overcomes the limitation of many existing clustering methods. In the third problem, the metric-based semi-supervised clustering method is proposed with similarity metric learned by modularity maximization and followed by the above proposed dominant-set based clustering procedure. Under semi-supervised setting where some clustering memberships are known, the goal is to determine the best linear combination of candidate similarity metrics as the final metric to enhance the clustering performance. Besides the global metric-based algorithm, another algorithm is also proposed to learn individual metrics for each cluster, which permits overlapping membership for the clustering. This is innovatively different from many existing methods. This method is superiorly applicable to functional data with various similarity metrics between functional curves, while also exhibiting robustness to imbalanced sizes of groups, which are intrinsic to the dominant-set based clustering approach. In all three problems, the advantages of the proposed methods are demonstrated through extensive empirical investigations using simulations as well as real data applications.
Weighted Curvatures in Finsler Geometry
(2023-08) Zhao, Runzhong; Shen, Zhongmin; Buse, Olguta; Ramras, Daniel; Roeder, Roland
The curvatures in Finsler geometry can be defined in similar ways as in Riemannian geometry. However, since there are fewer restrictions on the metrics, many geometric quantities arise in Finsler geometry which vanish in the Riemannian case. These quantities are generally known as non-Riemannian quantities and interact with the curvatures in controlling the global geometrical and topological properties of Finsler manifolds. In the present work, we study general weighted Ricci curvatures which combine the Ricci curvature and the S-curvature, and define a weighted flag curvature which combines the flag curvature and the T -curvature. We characterize Randers metrics of almost isotropic weighted Ricci curvatures and show the general weighted Ricci curvatures can be divided into three types. On the other hand, we show that a proper open forward complete Finsler manifold with positive weighted flag curvature is necessarily diffeomorphic to the Euclidean space, generalizing the Gromoll-Meyer theorem in Riemannian geometry.
Values of Ramanujan's Continued Fractions Arising as Periodic Points of Algebraic Functions
(2023-08) Akkarapakam, Sushmanth Jacob; Morton, Richard Patrick; Klimek, Slawomir D.; Roeder, Roland K. W.; Geller, William A.
The main focus of this dissertation is to find and explain the periodic points of certain algebraic functions that are related to some modular functions, which themselves can be represented by continued fractions. Some of these continued fractions are first explored by Srinivasa Ramanujan in early 20th century. Later on, much work has been done in terms of studying the continued fractions, and proving several relations, identities, and giving different representations for them. The layout of this report is as follows. Chapter 1 has all the basic background knowledge and ingredients about algebraic number theory, class field theory, Ramanujan’s theta functions, etc. In Chapter 2, we look at the Ramanujan-Göllnitz-Gordon continued fraction that we call v(τ) and evaluate it at certain arguments in the field K = Q(√−d), with −d ≡ 1 (mod 8), in which the ideal (2) = ℘2℘′2 is a product of two prime ideals. We prove several identities related to itself and with other modular functions. Some of these are new, while some of them are known but with different proofs. These values of v(τ) are shown to generate the inertia field of ℘2 or ℘′2 in an extended ring class field over the field K. The conjugates over Q of these same values, together with 0, −1 ± √2, are shown to form the exact set of periodic points of a fixed algebraic function ˆF(x), independent of d. These are analogues of similar results for the Rogers-Ramanujan continued fraction. See [1] and [2]. This joint work with my advisor Dr. Morton, is submitted for publication to the New York Journal. In Chapters 3 and 4, we take a similar approach in studying two more continued fractions c(τ) and u(τ), the first of which is more commonly known as the Ramanujan’s cubic continued fraction. We show what fields a value of this continued fraction generates over Q, and we describe how the periodic points for described functions arise as values of these continued fractions. Then in the last chapter, we summarise all these results, give some possible directions for future research as well as mentioning some conjectures.

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