Mathematical Sciences Department Theses and Dissertations

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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.
A-Optimal Subsampling For Big Data General Estimating Equations
(2019-08) Cheung, Chung Ching; Peng, Hanxiang; Rubchinsky, Leonid; Boukai, Benzion; Lin, Guang; Al Hasan, Mohammad
A significant hurdle for analyzing big data is the lack of effective technology and statistical inference methods. A popular approach for analyzing data with large sample is subsampling. Many subsampling probabilities have been introduced in literature (Ma, \emph{et al.}, 2015) for linear model. In this dissertation, we focus on generalized estimating equations (GEE) with big data and derive the asymptotic normality for the estimator without resampling and estimator with resampling. We also give the asymptotic representation of the bias of estimator without resampling and estimator with resampling. we show that bias becomes significant when the data is of high-dimensional. We also present a novel subsampling method called A-optimal which is derived by minimizing the trace of some dispersion matrices (Peng and Tan, 2018). We derive the asymptotic normality of the estimator based on A-optimal subsampling methods. We conduct extensive simulations on large sample data with high dimension to evaluate the performance of our proposed methods using MSE as a criterion. High dimensional data are further investigated and we show through simulations that minimizing the asymptotic variance does not imply minimizing the MSE as bias not negligible. We apply our proposed subsampling method to analyze a real data set, gas sensor data which has more than four millions data points. In both simulations and real data analysis, our A-optimal method outperform the traditional uniform subsampling method.
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.
Asymptotic Analysis of Structured Determinants via the Riemann-Hilbert Approach
(2019-08) Gharakhloo, Roozbeh; Its, Alexander; Bleher, Pavel; Yattselev, Maxim; Eremenko, Alexandre
In this work we use and develop Riemann-Hilbert techniques to study the asymptotic behavior of structured determinants. In chapter one we will review the main underlying definitions and ideas which will be extensively used throughout the thesis. Chapter two is devoted to the asymptotic analysis of Hankel determinants with Laguerre-type and Jacobi-type potentials with Fisher-Hartwig singularities. In chapter three we will propose a Riemann-Hilbert problem for Toeplitz+Hankel determinants. We will then analyze this Riemann-Hilbert problem for a certain family of Toeplitz and Hankel symbols. In Chapter four we will study the asymptotics of a certain bordered-Toeplitz determinant which is related to the next-to-diagonal correlations of the anisotropic Ising model. The analysis is based upon relating the bordered-Toeplitz determinant to the solution of the Riemann-Hilbert problem associated to pure Toeplitz determinants. Finally in chapter ve we will study the emptiness formation probability in the XXZ-spin 1/2 Heisenberg chain, or equivalently, the asymptotic analysis of the associated Fredholm determinant.
Asymptotics of the Fredholm determinant corresponding to the first bulk critical universality class in random matrix models
(2013-11-06) Bothner, Thomas Joachim; Its, Alexander R.; Bleher, Pavel, 1947-; Tarasov, Vitaly; Eremenko, Alexandre; Mukhin, Evgeny
We study the one-parameter family of determinants $det(I-\gamma K_{PII}),\gamma\in\mathbb{R}$ of an integrable Fredholm operator $K_{PII}$ acting on the interval $(-s,s)$ whose kernel is constructed out of the $\Psi$-function associated with the Hastings-McLeod solution of the second Painlev\'e equation. In case $\gamma=1$, this Fredholm determinant describes the critical behavior of the eigenvalue gap probabilities of a random Hermitian matrix chosen from the Unitary Ensemble in the bulk double scaling limit near a quadratic zero of the limiting mean eigenvalue density. Using the Riemann-Hilbert method, we evaluate the large $s$-asymptotics of $\det(I-\gamma K_{PII})$ for all values of the real parameter $\gamma$.
Blood circulation and aqueous humor flow in the eye : multi-scale modeling and clinical applications
(2016-06-14) Cassani, Simone; Guidoboni, Giovanna; Arciero, Julia Concetta; Harris, Alon
Glaucoma is a multi-factorial ocular disease associated with death of retinal ganglion cells and irreversible vision loss. Many risk factors contribute to glaucomatous damage, including elevated intraocular pressure (IOP), age, genetics, and other diseases such as diabetes and systemic hypertension. Interestingly, alterations in retinal hemodynamics have also been associated with glaucoma. A better understanding of the factors that contribute to these hemodynamic alterations could lead to improved and more appropriate clinical approaches to manage and hopefully treat glaucoma patients. In this thesis, we develop several mathematical models aimed at describing ocular hemodynamics and oxygenation in health and disease. Precisely we describe: (i) a time-dependent mathematical model for the retinal circulation that includes macrocirculation, microcirculation, phenomenological vascular regulation, and the mechanical effect of IOP on the retinal vasculature; (ii) a steady-state mathematical model for the retinal circulation that includes macrocirculation, microcirculation, mechanistic vascular regulation, the effect of IOP on the central retinal artery and central retinal vein, and the transport of oxygen in the retinal tissue using a Krogh cylinder type model; (iii) a steady-state mathematical model for the transport of oxygen in the retinal microcirculation and tissue based on a realistic retinal anatomy; and (iv) a steady-state mathematical model for the production and drainage of aqueous humor (AH). The main objective of this work is to study the relationship between IOP, systemic blood pressure, and the functionality of vascular autoregulation; the transport and exchange of oxygen in the retinal vasculature and tissue; and the production and drainage of AH, that contributes to the level of IOP. The models developed in this thesis predict that (i) the autoregulation plateau occurs for different values of IOP in hypertensive and normotensive patients. Thus, the level of blood pressure and functionality of autoregulation affect the changes in retinal hemodynamics caused by IOP and might explain the inconsistent outcomes of clinical studies; (ii) the metabolic and carbon dioxide mechanisms play a major role in the vascular regulation of the retina. Thus, the impairment of either of these mechanisms could cause ischemic damage to the retinal tissue; (iii) the multi-layer description of transport of oxygen in the retinal tissue accounts for the effect of the inner and outer retina, thereby improving the predictive ability of the model; (iv) a greater reduction in IOP is obtained if topical medications target AH production rather that AH drainage and if IOP-lowering medications are administrated to patients that exhibit a high initial level of IOP. Thus, the effectiveness of IOP-lowering medications depend on a patient’s value of IOP. In conclusion, the results of this thesis demonstrate that the insight provided by mathematical modeling alongside clinical studies can improve the understanding of diseases and potentially contribute to the clinical development of new treatments.
Certain Aspects of Quantum and Classical Integrable Systems
(2022-08) Kosmakov, Maksim; Tarasov, Vitaly; Its, Alexander; Mukhin, Evgeny; Ramras, Daniel
We derive new combinatorail formulas for vector-valued weight functions for the evolution modules over the Yangians Y (gl_n). We obtain them using the Nested Algebraic Bethe ansatz method. We also describe the asymptotic behavior of the radial solutions of the negative tt* equation via the Riemann-Hilbert problem and the Deift-Zhou nonlinear steepest descent method.
Commutants of composition operators on the Hardy space of the disk
(2013-11-06) Carter, James Michael; Cowen, Carl C.; Klimek, Slawomir; Perez, Rodrigo A.; Chin, Raymond; Bell, Steven R.; Mukhin, Evgeny
The main part of this thesis, Chapter 4, contains results on the commutant of a semigroup of operators defined on the Hardy Space of the disk where the operators have hyperbolic non-automorphic symbols. In particular, we show in Chapter 5 that the commutant of the semigroup of operators is in one-to-one correspondence with a Banach algebra of bounded analytic functions on an open half-plane. This algebra of functions is a subalgebra of the standard Newton space. Chapter 4 extends previous work done on maps with interior fixed point to the case of the symbol of the composition operator having a boundary fixed point.
Connection Problem for Painlevé Tau Functions
(2019-08) Prokhorov, Andrei; Its, Alexander; Bleher, Pavel; Eremenko, Alexandre; Tarasov, Vitaly
We derive the diﬀerential identities for isomonodromic tau functions, describing their monodromy dependence. For Painlev´e equations we obtain them from the relation of tau function to classical action which is a consequence of quasihomogeneity of corresponding Hamiltonians. We use these identities to solve the connection problem for generic solution of Painlev´e-III(D8) equation, and homogeneous Painlev´e-II equation. We formulate conjectures on Hamiltonian and symplectic structure of general isomonodromic deformations we obtained during our studies and check them for Painlev´e equations.
D-bar and Dirac Type Operators on Classical and Quantum Domains
(2012-08-29) McBride, Matthew Scott; Klimek, Slawomir; Cowen, Carl C.; Ji, Ron; Dadarlat, Marius
I study d-bar and Dirac operators on classical and quantum domains subject to the APS boundary conditions, APS like boundary conditions, and other types of global boundary conditions. Moreover, the inverse or inverse modulo compact operators to these operators are computed. These inverses/parametrices are also shown to be bounded and are also shown to be compact, if possible. Also the index of some of the d-bar operators are computed when it doesn't have trivial index. Finally a certain type of limit statement can be said between the classical and quantum d-bar operators on specialized complex domains.

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