## Ph.D. Theses

**April 23, 2015 Nonparametric Bayesian Quantile RegressionChao ChangAdvisor: Nan LinAbstract:** We propose new nonparametric Bayesian approaches to quantile regression using Dirichlet process mixture (DPM) models. All the existing quantile regression methods based on DPMs require the kernel density to satisfy the quantile constraint, hence the kernel densities are themselves usually in the form of mixtures. One innovation of our approaches is that we impose no constraint on the kernel, thus a wide range of densities can be chosen as the kernels of the DPM model. The quantile constraint is satisfied by a post-processing of the DPM by a suitable location shift. As a result, our proposed models use simpler kernels and yet possess great flexibility by mixing over both the location parameter and the scale parameter. The posterior consistency of our proposed model is studied carefully. And Markov chain Monte Carlo algorithms are provided for posterior inference. The performance of our approaches is evaluated using simulated data and real data. Moreover, we are able to incorporate random effects into our models such that our approaches can be extended to handle longitudinal data.

**April 23, 2015 Determining Fractional Conversion for a Class of Reaction-Diffusion SystemsMatthew WallaceAdvisor: Renato FeresAbstract:** We use probabilistic methods to study a class of first-order reaction-diffusion systems in which gas reactants and products diffuse in a domain and reaction takes place at relatively small catalytic sites. The central problem is to determine as explicitly as possible the probability of reaction -- equivalently, the yield of the reaction -- in terms of geometric parameters of the system and the reaction constant.

**April 16, 2015 Incompatibility of Diophantine Equations Arising from the Strong Factorial ConjectureBrady RocksAdvisor: David WrightAbstract:** The Strong Factorial conjecture was recently formulated by Arno van den Essen and Eric Edo. The problem is motivated by several outstanding problems including the Jacobian, Image, and Vanishing conjectures. In this defense, we discuss how the conjecture can be reformulated in terms of systems of integer polynomials and we present several special cases in which the conjecture holds.

**April 6, 2015 Regularity of the Bergman Projection on Variants of the Hartogs TriangleLiwei ChenAdvisor: Steven KrantzAbstract: **The Bergman projection is initially defined on $L^2$ space. However, its behavior on other function spaces, e.g. $L^p$, Sobolev, and H\"{o}lder spaces, is of considerable interest. In this defense, we focus on variants of the Hartogs triangle. We study the $L^p$ mapping properties of the weighted Bergman projections on these domains. As applications, we obtain the $L^p$ regularity of the twisted-weighted Bergman projections and the $L^p$ Sobolev regularity of the ordinary Bergman projection on the corresponding domains.

**March 20, 2015 Wavelet Factorizations and Related PolynomialsDavid MeyerAdvisor:** Victor Wickerhauser

Abstract: The lifting scheme, in particular nearest neighbor lifting, is a way to implement the discrete wavelet transform for finite impulse response filters. The connection between the nearest neighbor factorizations and the normality of the associated polynomial remainder sequence will be discussed. Asymptotic behavior of both Daubechies filter coefficients and nearest neighbor factorizations will also be given.

**February 17, 2015 Application of Machine Learning to Mapping and Simulating Gene Regulatory NetworksHien-haw LiowAdvisor: Ed SpitznagelAbstract: **This dissertation explores, proposes, and examines methods of applying modernmachine learning and Bayesian statistics in the quantitative and qualitative modeling of gene regulatory networks using high-throughput gene expression data. A semi-parametric Bayesian model based on random forest is developed to infer quantitative aspects of gene regulation relations; a parametric model is developed to predict geneexpression levels solely from genotype information. Simulation of network behavior is shown to complement regression analysis greatly in capturing the dynamics of gene regulatory networks. Finally, as an application and extension of novel approaches in gene expression analysis, new methods of discovering topological structure of gene regulatory networks are developed and shown to provide improvement over existing methods.

**October 24, 2014 Applications of Nonlinear OptimizationYao XieAdvisor: Victor WickerhauserAbstract: **Three applications of nonlinear optimization will be presented. The first application is about learning low dimensional manifolds in high dimensional data. The second application is to derive robust treatment plans in radiotherapy against patient motion uncertainties. The third application is to devise an approximate convex functional alternative for an important metric in radiotherapy.

## M.A. Theses

**March 31, 2015 Projected Composition Operators on the Dirichlet SpacePatrick LopattoAdvisor: John McCarthyM.A. in Mathematics with ThesisView at http://openscholarship.wustl.edu/art_sci_etds/379/Abstract:** A projected composition operator is a composition operator composed with a projection. Rochberg has proven several results concerning projected composition operators on the Hardy space. Building on his work, I prove analogous results for projected composition operators on the Dirichlet space. This talk will feature explicit examples, d-bar techniques, and probabilistic methods.

**September 2, 2014 Estimating the Risk of Getting Alzheimer's Disease by a Two-Stage ModelTian WangM.A. in Statistics with ThesisAdvisor: Jimin DingAbstract: **The population of patients suffering from Alzheimer's Disease (AD) has increased significantly these years. A dataset of patients who were at the risk of getting AD with their demographic information and answers to cognitive function questions is investigated. Generalized linear mixed effect model is the first model fit to the data to extract the information for each patient. A proportional Hazard model and a times-dependent Cox model are proposed afterwards by the information obtained.

## Undergraduate Theses

**March 30, 2015 A Comparison of the Lasso and Dantzig Selector in Linear Regression ModelsAlexander Dong, Latin HonorsAdvisor: Todd KuffnerAbstract: **We introduce two variable section techniques, the Lasso and the Dantzig Selector, then compares properties of both of them. We present and analyze the results of the Lasso and Dantzig Selector on three data sets. Special attention will be paid to the conditions on equivalence of the both methods.

**March 27, 2015 $N$-Complexes and $S_n$ RepresentationsNati Friedenberg, ARTU & Latin HonorsAdvisor: John ShareshianAbstract: **Abstract: We introduce the notion of $N$-complexes a generalization of chain complexes defined by Kapranov. We discuss an infinite class of $N$-complexes that yield interesting $S_n$ representations. Specifically, they give us sequences of (Brauer) characters $(\chi_n:S_n\rightarrow\mathbb{C})_{n\in\mathbb{N}}$ that address an open question posed by Richard Stanley. We then discuss a conjecture of ours regarding the concentration of the homology of these $N$-complexes which would imply explicit constructions of these interesting $S_n$ representations.

**March 26, 2015 Computing the Dispersion Relation of Periodic Quantum GraphsAlan Talmage, ARTU & Latin HonorsAdvisor: Renato FeresAtstract: **We present methods of Kuchment and Post for calculating the dispersion relation of periodic quantum graphs and reformulate them in terms of a matrix-valued function on such graphs. We show how this leads to simpler methods for computing the dispersion relations for many periodic quantum graphs. We also reduce the computation of the dispersion relation for a periodic extension of an undirected Cayley graph of a finite group to finding the eigenvalues of the adjacency matrix of the undirected Cayley graph of the finite group.

**March 25, 2015 The Hermitian Symplectic Group, Neumann Extension Theory, and Scattering on Quantum GraphTyler Ellison, ARTU & Latin HonorsAdvisor: Renato FeresAbstract: **We will begin with the definition of a skew-Hermitian form and the corresponding Hermitian symplectic group. We will motivate these definitions with a brief discussion of quantum theory and von Neumann's self-adjoint extension theory. The necessary tools from Hermitian symplectic linear algebra will be developed in order to analyze the self-adjoint extensions of Hamiltonian operators on metric graphs. Finally, with the concrete quantum graph problem on hand, we will consider the physical implications of the Hermitian symplectic group.

**March 24, 2015 Polynomial AutomorphismsBowei Zhao, Latin HonorsAdvisor: David WrightAbstract: **We will begin with a few definitions about polynomial automorphisms, general automorphism groups, stable tameness, and how they relate to the famous Jacobian conjecture. Then we will prove a few significant result on polynomial automorphisms over a regular ring. These results will lead to the important theorem that two dimensional polynomial automorphisms over a regular ring are stably tame.

**March 24, 2015 Modelling Air Pollution with GISDoyeong Yu, Latin HonorsAdvisor: Ed SpitznagelAbstract: **We use a regression based approach to model Nitrogen(IV) Oxide concentrations in the contiguous United States with Geographic Information System (GIS). Based on the pollution, traffic, population and land cover data in 2011, we present a multivariate regression model which can be used to interpolate the concentrations in non-measurement locations.

**February 16, 2015 Schatten-class Truncated Toeplitz OperatorsPatrick Lopatto, ARTU & Latin HonorsAdvisor: John McCarthyAbstract:** Truncated Toeplitz operators are a generalization of the classical Toeplitz operators introduced by Sarason in 2007. I will discuss the problem of determining when these operators lie in a given Schatten ideal. This is joint work with Professor Richard Rochberg.

Mathematics Theses may be posted at http://openscholarship.wustl.edu ⇨