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Department of Mathematics and Statistics: Dr. Andrew D. Lewis
Selected preprints of Dr. Andrew D. Lewis
Recent Submissions

Noncommutative Independences for Pairs of Faces
In noncommutative probability theories, many different notions of independence arise in various contexts. According to the early classification work, there are only five such notions that are universal/natural in the sense ... 
Information and Estimation Theoretic Approaches to Data Privacy
Warner [145] in 1960s proposed a simple mechanism, now referred to as the randomized response model, as a remedy for what he termed “evasive answer bias” in survey sampling. The randomized response setting is as follows: ... 
Statistical Methods For Biomarker Threshold Models in Clinical Trials
In clinical trials, the main objective is to investigate the treatment effects on patients. However, many molecularly targeted drugs or treatments tend to benefit a subset of patients more, identified by a certain biomarker. ... 
Representation of numbers by quaternary quadratic forms
(20160927) 
An Rpackage for the Estimation and Testing of Multiple Covariates and Biomarker Interactions for Survival Data Based on Local Partial Likelihood
(20160927)When we study the variables that a ffect survival time, we usually estimate their eff ects by the Cox regression model. In biomedical research, e ffects of the covariates are often modi ed by a biomarker variable. This ... 
On The Dirichlet Distribution
(20160927)The Dirichlet distribution is a multivariate generalization of the Beta distribution. It is an important multivariate continuous distribution in probability and statistics. In this report, we review the Dirichlet distribution ... 
Degeneracy of velocity constraints in rigid body systems
(20160927)The equations governing the dynamics of rigid body systems with velocity constraints are singular at degenerate configurations in the constraint distribution. In this report, we describe the causes of singularities in the ... 
An impulsive differential equation model for Marek's disease
(20160922)Many dynamical processes are subject to abrupt changes in state. Often these perturbations can be periodic and of short duration relative to the evolving process. These types of phenomena are described well by what are ... 
Elliptic Functions and Elliptic Interpolation
(20160921) 
Irreducibility of Random Hilbert Schemes
(20160913)We prove that a random Hilbert scheme that parametrizes the closed subschemes with a fixed Hilbert polynomial in some projective space is irreducible and nonsingular with probability greater than $0.5$. To consider the ... 
Estimation of Sample Size and Power For Quantile Regression
(20160824)Quantile regression (QR) was first introduced by Roger Koenker and Gilbert Bassett in 1978. It is robust to outliers which affect least squares estimator on a large scale in linear regression. Instead of modeling mean of ... 
Codomain Rigidity of the Dirichlet to Neumann Operator for the Riemannian Wave Equation
(20160823)We study the Dirichlet to Neumann operator for the Riemannian wave equation on a compact Riemannian manifold. If the Riemannian manifold is modelled as an elastic medium, this operator represents the data available to an ... 
A Review on Repeated Games and Reputations with Incomplete Information
(20160720)In this project we review the effects of reputation within the context of game theory. This is done through a study of two key papers. First, we examine a paper from Fudenberg and Levine: Reputation and Equilibrium ... 
Geometry of Dirac Operators
(20160705)Let $M$ be a compact, oriented, even dimensional Riemannian manifold and let $S$ be a Clifford bundle over $M$ with Dirac operator $D$. Then \[ \textsc{Atiyah Singer: } \quad \text{Ind } \mathsf{D}= \int_M ... 
Computationally Intensive Methods for Spectrum Estimation
(20160427)Spectrum estimation is an essential technique for analyzing time series data. A leading method in the field of spectrum estimation is the multitaper method. The multitaper method has been applied to many scientific fields ... 
Higher rank sieves and applications
(20160425)This thesis focuses on some of the key sieve theoretic ideas behind recent progress on bounded gaps between the primes. One such idea is the notion of higher rank sieve weights, first proposed by Atle Selberg and applied ... 
On the role of regularity in mathematical control theory
(20160408)In this thesis, we develop a coherent framework for studying timevarying vector fields of different regularity classes and their flows. This setting has the benefit of unifying all classes of regularity. In particular, ... 
On the Equivalence Between Maximum Likelihood and Minimum Distance Decoding for Binary Contagion and QueueBased Channels with Memory
(20160217)We study the optimal maximum likelihood (ML) block decoding of general binary codes sent over two classes of binary additive noise channels with memory. Specifically, we consider the infinite and finite memory Polya contagion ... 
The Chowla Problem and its generalizations
(20150915)In this thesis, we study the vanishing of certain $L$series attached to periodic arithmetical functions. Throughout the writeup, we let $f$ be an algebraicvalued (at times even rationalvalued) arithmetical function, ... 
The Saddle Point Method and its Applications to Number Theory
(20150915)In this thesis, we study the classical procedures useful in obtaining asymptotic expansions of functions defined by integrals and their applications to number theory. The final chapter in the thesis reports on a recent ...