Mathematics Dissertations

The integration of digital pathology with whole slide image (WSI) represents a transformative advance in the field of medical diagnostics. This dissertation investigates the application of WSI across three pivotal areas: three-dimensional (3D) reconstruction of large tissues at cellular resolution for quantitative analysis, sampling error quantification using liver tissue cases, predictive response analysis, and survival prediction, specifically targeting histopathological evaluations in breast cancer treatment. Our initial study focuses on 3D virtual needle biopsies, utilizing 3D reconstructions of histopathological images to quantify and reduce sampling errors, thus enhancing diagnostic reliability. Subsequently, we introduce NACNet, a novel histology context-aware transformer based graph convolution network to predict responses to neoadjuvant chemotherapy in Triple-Negative Breast Cancer (TNBC) using WSI data. By extracting and integrating graph based spatial Tissue microenvironment (TME) features from WSIs with image features, NACNet surpasses traditional diagnostic methods in predictive accuracy. Our final project extends this approach by developing a multimodal deep learning model that combines WSI features with genomic and clinical data to forecast long-term survival outcomes in breast cancer patients. This model serves not only as a prognostic tool but also aids in the customization of treatment protocols, thereby advancing the practice of precision oncology. Through these studies, this dissertation demonstrates the critical role of WSI in reducing sampling discrepancies, improving accuracy in treatment response prediction, and enhancing survival analysis in clinical settings. The collective findings underscore the significant potential for WSI-assisted diagnostics to inform more precise and personalized patient care strategies in oncology.

This dissertation explores the use of the Frobenius endomorphism to detect the regularity of a ring within the framework of commutative algebra. In commutative local rings of prime characteristic, invariants such as the Hilbert-Kunz multiplicity, F-signature, and Frobenius Betti numbers have been shown to detect regularity. Polstra and Smirnov [PS21] showed that the Frobenius Euler characteristic can be used to determine regularity for the class of strongly F-regular rings. Here, we extend their results by relaxing the condition of strong F-regularity. We show that the Frobenius Euler characteristic can detect regularity for rings with sufficient F-splitting and for rings that are Cohen-Macaulay.

We investigate the usefulness of deep learning when applied to both control theory and partial differential equations (PDEs). We will develop new network architectures and methodologies to approach the solving of high-dimensional problems. Specifically, we develop a network architecture called Lyapunov-Net for approximating Lyapunov functions in high-dimensions and a new methodology called Neural Control for finding solution operators for high-dimensional parabolic PDEs. The theoretical accuracy and numerical efficiency of these approaches will be investigated along with implementation details to use them in practice.

A clear understanding of the actual infection rate is imperative for control and prevention of diseases. This knowledge is essential in formulating effective vaccination strategies and assessing the level of herd immunity required to contain the virus. In recent years, advanced regularization techniques have emerged as a powerful tool aimed at stable estimation of infectious disease parameters that are crucial for the design of adequate public health policies. This dissertation presents a theoretical and numerical study of a novel optimization procedure aimed to achieve a stable estimation of incidence reporting rates and time-dependent effective reproduction numbers using real data on new incidence cases, daily new deaths, and vaccination percentages. The iteratively regularized optimization algorithm introduced here is versatile and applicable to a wide range of data fitting problems constrained by various biological models, particularly those that need to account for under-reporting of cases. To support this, general nonlinear observation operators in real Hilbert spaces are used in the proposed convergence analysis. The theoretical findings are demonstrated through numerical simulations using the susceptible unvaccinated (S), susceptible vaccinated (V ), infected unvaccinated (Is), infected vaccinated (Iv), recovered (R), and deceased (D) compartmental model and real data from the Delta variant of the COVID-19 pandemic in different states of the US. In the second part of this dissertation, a biological model with a flexible choice of control strategies for an emerging virus is considered. In the model, the disease transmission is mitigated using both non-medical control (like social distancing and other behavioral changes) and the control by treatment with antiviral medication.

Let A be an n × n real matrix. We define orthogonal similarity-transversality property (OSTP) if the smooth manifold consisting of the real matrices orthogonally similar to A and the smooth manifold Q(sgn(A)) (consisting of all real matrices having the same sign pattern as A), both considered as embedded smooth submanifolds of R^{n × n}, intersect transversally at A. More specifically, with S = [s_ij ] being the n×n generic skew-symmetric matrix whose strictly lower (or upper) triangular entries are regarded as independent free variables, we say that A has the OSTP if the Jacobian matrix of the entries of AS − SA at the zero entry positions of A with respect to the strictly lower (or upper) triangular entries of S has full row rank. We also formulate several properties like OSTP2 and show that if a matrix A has the OSTP, then every superpattern of the sign pattern sgn(A) allows a matrix orthogonally similar to A, and every matrix sufficiently close to A also has the OSTP. This approach provides a theoretical foundation for constructing matrices orthogonally similar to a given matrix while the entries have certain desired signs or zero-nonzero restrictions. In particular, several important classes of zero-nonzero patterns and sign patterns that require or allow the OSTP are identified, and we gives several examples illustrating some applications.

The empirical likelihood (EL), introduced by Owen (1988, 1990), is a powerful tool for constructing confidence intervals in nonparametric settings. Significant developments based on empirical likelihood have been made in recent years. In this dissertation, we investigate the performance of two extensions of EL, sample empirical likelihood (Chen and Kim, 2014) and Bayesian jackknife empirical likelihood (Cheng and Zhao, 2019) approaches, for several statistical problems.

One effective way to conduct statistical disclosure control is to use scrambled responses. Scrambled responses can be generated by using a controlled random device. We propose using the sample empirical likelihood approach to conduct statistical inference (using a Wilk-type confidence interval) under a complex survey design with scrambled responses.

Missing data, which are common in a variety of fields, reduce the representativeness of the sample and can lead to inference problems. We apply the Bayesian jackknife empirical likelihood method for inference with missing data and causal inference. The semiparametric fractional imputation estimator, proposed by Chen and Kim (2017), propensity score weighted estimator, and doubly robust estimator were used for inference with missing data.

The partial area under the receiver operating characteristic curve (pAUC) is a measure of diagnostic test performance. We propose using Bayesian jackknife empirical likelihood for inference for the pAUC and comparison of two tests.

Extensive simulation studies are conducted to compare the performance in terms of the coverage rate and average length of confidence interval between proposed methods and normal approximation/jackknife empirical likelihood methods. Furthermore, we demonstrate the application of the proposed approaches using real datasets.

Let $A$ be an $n\times n$ real matrix. As shown in the recent paper \cite{Fallat22}, if the manifolds $\mathcal{M}_A$$=\{ G^{-1}AG: G\in\text{GL}(n,\mathbb{R})\}$ and $Q($sgn$(A))$ (consisting of all real matrices having the same sign pattern as $A)$, both considered as embedded submanifolds of $\mathbb{R}^{n\times n}$, intersect transversally at $A$, then every superpattern of sgn$(A)$ also allows a matrix similar to $A.$ Such paper introduced a condition on $A$ equivalent to the above transversality, called the \emph{nonsymmetric strong spectral property} (nSSP).

In this dissertation, this transversality property of $A$ is characterized using an alternative, more direct and convenient condition, called the \emph{similarity-transversality property} (STP). Let $X=[x_{ij}]$ be a generic matrix of order $n$ whose entries are independent variables. The STP of $A$ is defned as the full row rank property of the Jacobian matrix of the entries of $AX-XA$ at the zero entry positions of $A$ with respect to the nondiagonal entries of $X.$ This new approach makes it possible to take better advantage of the combinatorial structure of the matrix $A$, and provides theoretical foundation for constructing matrices similar to a given matrix while the entries have certain desired signs. In particular, several important classes of zero-nonzero patterns and sign patterns that require or allow this transversality property are identified. Examples illustrating many possible applications are provided.

In this dissertation, three major topics in graph theory and computational neuroscience are explored: linear arboricity of graphs, $ f $-density parameter in fractional graph edge coloring, and quadruped locomotion gait with sensory feedback and central interactions. A linear forest is a disjoint union of path graphs. The \textit{linear arboricity} of a simple graph $ G $, denoted by $ \operatorname{la}(G) $, is the least number of linear forests into which the graph can be partitioned. The long-standing \textit{Linear Arboricity Conjecture} (LAC) from 1981 asserts that $ \operatorname{la}(G) \le \lceil (\Delta(G)+1)/2 \rceil $. In Chapter~\ref{chap:LAC}, two methods are introduced and the LAC is proved for several different graph classes, especially degenerate graphs with relatively large maximum degree. Let $f$ be a function from $V(G)$ to $\mathbb{Z}_+$. The {\em fractional $f$-density} $\mathcal{W}^*_{f}$ of a loopless multigraph $G$ is defined as: $$ \mathcal{W}^*_{f}(G)=\max _{U \subseteq V,|U| \geq 2}\frac{|E(U)|}{\lfloor f(U) / 2\rfloor}, $$ where $ f(U)=\sum_{v\in U} f(v) $. It is a generalization of the density parameter in graph edge coloring problem which plays an important role in determining the $ f $-chromatic index. In Chapter~\ref{chap:density}, a polynomial-time algorithm is given for calculating $\mathcal{W}^*_{f}(G)$ in terms of the number of vertices of $G$. \textit{Quadrupeds locomotion} is a complex process involving specific interactions between the central neural controller and the mechanical components of the system. In Chapter~\ref{chap:locomotion}, a tractable mathematical model of mouse locomotion are created and analyzed using biomechanical data and recent findings on the organization of neural interactions within the spinal locomotor circuitry. Several model versions are investigated and compared to existing experimental data on mouse locomotion. The results highlight the specific roles of sensory feedback and some central propriospinal interactions between circuits controlling fore and hind limbs for speed-dependent gait expression. The models suggest that postural imbalance feedback may be critically involved in the control of swing-to-stance transitions in each limb and the stabilization of walking direction.

This thesis presents several novel results on the nonlinear and emergent collective dynamics of crowds and populations in complex systems. Though, historically, the list of suspension bridges destabilized by pedestrian collective motion is long, the phenomenon still needs to be fully understood, especially regarding the effect of human-to-human interactions on the structure, and often incorrectly explained using synchronization theory. We present a simple general formula that quantifies the effect of pedestrian effective damping of a suspension bridge and illustrate it by simulating three mathematical models, including one with a strong propensity for synchronization. Despite the subtle effects of gait strategies in determining precise instability thresholds, our results show that average negative damping is always the trigger of pedestrian-induced high-amplitude lateral vibration of suspension bridges. Furthermore, we show that human-to-human interactions of heterogeneous pedestrians can trigger the instability of a bridge more effectively than crowds of identical pedestrians. We will also discuss the role of crowd heterogeneity in possible phase pulling between pedestrians and bridge motion. We also develop a model for the evolution of toxic memes on 4chan and report a significant influence on Twitter’s anti-vaccine conspiracy discourse over a nine-year period. We show that 4chan topics evolve according to an emergent process mathematically similar to classic reinforcement learning methods, tending to maximize the expected toxicity of future discourse. We demonstrate that these topics can invade Twitter and persist in an endemic state corresponding to the associated spreading rate and initial distribution of post rates and coexisting with a higher-traffic regime of dynamics. We discuss the implication of this result for preventing large-scale disinformation campaigns.

In this dissertation, we investigate jackknife empirical likelihood methods motivated by recent statistics research and other related fields. Computational intensity of empirical likelihood can be significantly reduced by using jackknife empirical likelihood methods without losing computational accuracy and stability. We demonstrate that proposed jackknife empirical likelihood methods are able to handle several challenging and open problems in terms of elegant asymptotic properties and accurate simulation result in finite samples. These interesting problems include ROC curves with missing data, the difference of two ROC curves in two dimensional correlated data, a novel inference for the partial AUC and the difference of two quantiles with one or two samples. In addition, empirical likelihood methodology can be successfully applied to the linear transformation model using adjusted estimation equations. The comprehensive simulation studies on coverage probabilities and average lengths for those topics demonstrate the proposed jackknife empirical likelihood methods have a good performance in finite samples under various settings. Moreover, some related and attractive real problems are studied to support our conclusions. In the end, we provide an extensive discussion about some interesting and feasible ideas based on our jackknife EL procedures for future studies.

Given a loopless multigraph $G$ and a vertex-function $f\ : \ V(G) \rightarrow \mathbb{N}\backslash \{0\}$, an {\bf $f$-(edge)-coloring} of $G$ is an assignment of colors to the edges of $G$ such that each color appears at each vertex $v\in V(G)$ at most $f(v)$ times. The {\bf $f$-chromatic index} of a graph $G$, denoted by $\chi'_f(G)$, is the least integer $k$ such that $G$ admits an $f$-coloring using $k$ colors. Clearly, a proper graph edge-coloring is a special $f$-coloring, where $f\equiv 1$. By definition, the chromatic index $\chi'(G)$ is exactly $\chi_1'(G)$.

The {\bf total chromatic number} $\chi''(G)$ of a graph $G$ is the least number of colors assigned to the edges and vertices of $G$ such that no two adjacent edges receive the same color, no two adjacent vertices receive the same color and no edge has the same color as its two endpoints. The full text will be divided into the following three chapters.

In the first chapter, we first give some basic definitions, notation, and related terminologies that this dissertation needs. Then we introduce the research background and research significance of this dissertation, as well as the main results of this paper.

In the second Chapter, we mainly study the relationship $\chi''(G)$ and $\chi'(G)$. By definition, $\chi'(G) \le \chi''(G)$ for every graph $G$.

In 1984, Goldberg conjectured that for any multigraph $G$,

if $\chi'(G) \ge \D(G) +3$ then $\chi''(G) = \chi'(G)$.

In this chapter, we show that Goldberg's conjecture is asymptotically true. More specifically, we prove that for a multigraph $G$ with maximum degree $\D$ sufficiently large, if $\chi'(G) \ge \D + 10\D^{35/36}$, then we have $\chi''(G) = \chi'(G)$.

In the third Chapter, we confirm the Goldberg-Seymour conjecture for$f$-coloring, which states that $\chi'_f(G) \le \max\{ \D_f(G) +1, \omega_f(G)\}$, where $f$-maximum degree $\D_f(G)$ and $f$-density $\omega_f(G)$ of a weighted graph $(G, f)$ are defined as $\max_{v\in V(G)} \left\lceil{\frac{d(v)}{f(v)}}\right\rceil$ and

$ \max \left\lceil{ \frac{|E(H)|} {\lfloor \sum_{v\in V(H)}f(v)/2}\rfloor}\right\rceil$, respectively. Our result

implies that $\chi_f'(G)$ can only assume one of two consecutive integers :

$\max\{\D_f(G), \omega_f(G)\}$ and $\max\{\D_f(G) +1, \omega_f(G)\}$.

So an analog to Vizing's theorem on proper edge-colorings of simple

graphs holds for $f$-coloring of all multigraphs.

Let $G$ be a graph, $V (G)$ and $E(G)$ be the vertex set and edge set of $G$, respectively. A perfect matching of $G$ is a set of edges, $M\subseteq E(G)$, such that each vertex in $G$ is incident with exactly one edge in $M$. An $r$-regular graph is said to be an $r$-graph if $|\partial(X)| \geq r$ for each odd set $X \subseteq V(G)$, where $|\partial(X)|$ denotes the set of edges with precisely one end in $X$. One of the most famous conjectures in Matching Theory, due to Berge, states that every 3-graph $G$ has five perfect matchings such that each edge of $G$ is contained in at least one of them. Likewise, generalization of the Berge Conjecture given, by Seymour, asserts that every $r$-graph $G$ has $2r-1$ perfect matchings that covers each $e \in E(G)$ at least once. In the first part of this thesis, I will provide a lower bound to number of perfect matchings needed to cover the edge set of an $r$-graph. I will also present some new conjectures that might shade a light towards the generalized Berge conjecture. In the second part, I will present a proof of a conjecture stating that there exists a pair of graphs $G$ and $H$ with $H\supset G$, $V(H)=V(G)$ and $|E(H)| = |E(G)| +k$ such that mean subtree order of $H$ is smaller then mean subtree order of $G$.

Nonparametric statistical inference methods have become very popular in recent years and are the preferred method of inference among many statisticians as the data are not assumed to come from any particular statistical distribution. Empirical likelihood (EL) is one of these methods and has been popular since published by Owen (1988). While it is inspired by the usual maximum likelihood methods, it also has some flaws: heavy computations, low accuracy for the small samples and skewed data. In this dissertation, we investigate some EL's extensions, i.e., the jackknife empirical likelihood (JEL), the adjusted jackknife empirical likelihood (AJEL), the mean jackknife empirical likelihood (MJEL), the mean adjusted jackknife empirical likelihood (MAJEL),the adjusted mean jackknife empirical likelihood (AMJEL), and the transformed jackknife empirical likelihood (TJEL) in constructing confidence intervals (CI) for particular parameters of interest such as the correlation coefficient in different statistical problems. These methods increase the length of the confidence intervals for the correlation coefficient resulting in better coverage probabilities and perform better than EL methods for small sample sizes. We propose a new plug-in approach of JEL to reduce the computational cost in estimating the symmetry of various statistical distributions. One of the main results of the JEL is the nonparametric extension of Wilks' theorem for parametric likelihood ratios. We explore JEL, AJEL, MJEL, AMJEL, and MAJEL to construct a confidence interval for the correlation coefficient for data with multiplicative distortion errors. Further, we explore JEL, AJEL, MJEL, and TJEL for the k-th correlation coefficient in estimating measures of symmetry for data with multiplicative distortion errors. Finally, using exponential calibration, we develop JEL methods for the correlation coefficient between two variables with additive distortion measurement errors.

Atrial fibrillations and heart failure are among the leading cardiovascular diseases in the world. Understanding the development and progression of these diseases requires a thorough knowledge of the electrophysiological mechanisms in a healthy and diseased cardiac myocyte. This goal can be achieved by using mathematical modeling along with experimental investigations. Here, we developed two new comprehensive mathematical models of the mouse atrial and ventricular myocytes. The first one is a novel compartmentalized mathematical model of mouse atrial myocytes. This model combines the action potential, [Ca²⁺]_i dynamics, and β-adrenergic signaling cascade for a subpopulation of right atrial myocytes with a developed transverse-axial tubule system. The model consists of three compartments related to β-adrenergic signaling (caveolae, extracaveolae, and cytosol) and employs local control of Ca²⁺-release. It also simulates the mechanisms of action potential generation and describes atrial-specific Ca²⁺ handling and frequency dependences of the action potential and [Ca²⁺]_i transients. The model showed that the T-type Ca²⁺ current significantly affects the later stage of the action potential with little effect on [Ca²⁺]_i transients. Blocking the small-conductance Ca²⁺-activated K⁺ current leads to the prolongation of the action potential at high intracellular Ca²⁺ concentrations. Simulation results obtained from the atrial cell model were compared to those from ventricular myocytes. The developed model presents a valuable tool for studying complex electrical properties in the mouse atria and could be applied to understand atrial physiology and arrhythmogenesis. The second model is a novel compartmentalized mathematical model of failing mouse ventricular myocytes after TAC procedure. The model effectively describes the cell geometry, action potentials, [Ca²⁺]_i transients, and β₁- and β₂-adrenergic signaling in the failing cells. Simulation results obtained from a failing cells’ model were compared to those from the normal ventricular myocytes. Exploration of the model revealed the sarcoplasmic reticulum Ca²⁺ load mechanisms in failing ventricular myocytes. We also described a proarrhythmic behavior of Ca²⁺ dynamics upon stimulation with isoproterenol and mechanisms of the proarrhythmic behavior suppression. The developed model can be used to explain the existing experimental data on failing mouse ventricular myocytes and make experimentally testable predictions of the failing myocyte behavior.

The purpose of this study was to identify and analyze the observable cognitive processes of experts in mathematics while they work on proof-construction activities using the Principle of Mathematical Induction (PMI). Graduate student participants in the study worked on ``nonstandard" mathematical induction problems that did not involve algebraic identities or finite sums. This study identified some of the problem solving-strategies used by the participants during a Cognitive Task Analysis (Feldon, 2007) as well as epistemological obstacles they encountered while working with PMI. After the Cognitive Task Analysis, the graduate students participated in two semi-structured interviews. These interviews explored graduate students' beliefs about proofs and proof techniques and situates their use of PMI within the contexts of these beliefs.

Two primary theoretical frameworks were used to analyze participant cognition and the qualitative data collected. First, the study used Action, Process, Object, Schema (APOS) Theory (Asiala et al., 1996) to to study and analyze the participants' conceptual understanding of the technique of mathematical induction and to test a preliminary genetic decomposition adapted from previous studies on PMI (Dubinsky \& Lewin 1996, 1999; Garcia-Martinez & Parraguez, 2017). Second, an Expert Knowledge Framework (Bransford, Brown, & Cocking, 1999; Shepherd & Sande, 2014) was used to classify the participants' responses to the semi-structured interview questions according to several characteristics of expertise. The study identified several results which (1) give insight to the mental constructions used by mathematical experts when solving problem involving PMI; (2) offer some implications for improving the instruction of PMI in introductory proofs classrooms; and (3) provide results that allow for future comparison between expert and novice mathematical learners.

A signed graph is an ordered pair (G,Σ), where G = (V, E) is a graph and Σ ⊆ E. The edges in Σ are called odd, and the edges in E \ Σ are called even. The family of matrices S(G,Σ) is defined such that if [a_i,j] = A ∈ S(G,Σ), then a_i,j < 0 if there is at least one edge between i and j and if all edges between i and j are even; a_i,j > 0 if there is at least one edge between i and j and if all edges between i and j are odd; a_i,j ∈ R if there is at least one even edge and at least one odd edge between i and j; and a_i,j = 0 if there are no edges between i and j. The maximum nullity of a signed graph M(G,Σ) is the largest corank(A) for A ∈ S(G, Σ). The matrix A ∈ S(G, Σ) has the Strong Arnold Property with respect to (G,Σ) if X = 0 is the only matrix such that AX = 0, and x_i,j = 0 if i is adjacent to j or i = j. The stable maximum nullity of a signed graph ξ(G,Σ) is the largest corank(A) for A ∈ S(G,Σ) where A has the Strong Arnold Property. Here, we present a combinatorial characterization of signed graphs with maximum nullity at most two, extending a result of Johnson, Loewy, and Smith. We also find the forbidden minors for signed graphs with stable maximum nullity at most two, extending a result of Hogben and van der Holst. We generalize the notion of zero forcing to signed graphs. We find the zero forcing number of signed graphs with maximum nullity at most two, extending a result of Row.