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

An Asymptotically Optimal TwoPart FixedRate Coding Scheme for Networked Control
It is known that under fixedrate information constraints, adaptive quantizers can be used to stabilize an openloopunstable linear system on $\mathbb{R}^n$ driven by unbounded noise. These adaptive schemes can be designed ... 
A Superlacunary ItoKawada Theorem, and Applications to the Equidistribution of Generalized RudinShapiro Polynomials
The ItoKawada Theorem is a classical result in the theory of random walks on compact groups. It states that essentially the only obstruction to the equidistribution of a random walk with independent and identically ... 
A Morse IndexMaslov Index Theorem for Discrete Lagrangian Systems
We develop a discrete analog to Morse theory for discrete Lagrangian systems, including a discrete Morse index theorem and a correspondence theorem between the Morse indices of critical trajectories in discrete Lagrangian ... 
Information Bottleneck Methods for Fairness and Privacy in Machine Learning
Designing machine learning algorithms that are accurate yet fair, not discriminating based on any sensitive attribute, and also private, not revealing users’ personal information, has become of paramount importance for ... 
A Quantification of Long Transient Dynamics
We present a systematic study of transient dynamics starting with a technical definition of transient points which are initial data of an autonomous system of ordinary differential equations that can lead to “long transient ... 
Examining the Burnside Problem on Diff∞ ω (S2)
Let $G$ be a group. $G$ is called \emph{periodic} if, for every element $g \in G$, there is a positive integer $n \in \N$ such that $g^n = \id_G$, and \emph{periodic of bounded exponent} if one such $n$ works for all $g ... 
Marked Length Spectrum Rigidity of Anosov Manifolds
We provide a proof that, for compact Riemannian manifolds with Anosov geodesic flow and nonpositive curvature, two sufficiently close metrics with equal marked length spectrum will necessarily be isometric. To be precise, ... 
A Study of the Minimum pvalue and Related Methods for the Identification of TreatmentSensitive Groups
In clinical practices, a fundamentally important problem is to identify a subgroup of patients who may benefit more in terms of a clinical outcome from a certain treatment based on a specific clinical variable or biomarker, ... 
Tail Asymptotics for the Limiting Distribution of Theta Sums
We define theta sums to be exponential sums of the form S_{N}(x; \alpha, \beta) := \sum_{n =1}^{N} e((\tfrac{1}{2} n^2 + \beta n)x + \alpha n), where e(z) = e^{2 \pi i z}. If \alpha and \beta are fixed rational numbers, ... 
The Exponential Map for TimeVarying Vector Fields
The exponential map that characterises the flows of vector fields is the key in understanding the basic structural attributes of control systems in geometric control theory. However, this map does not exist due to the lack ... 
Comparison and Continuity Properties of Equilibrium Values in Information Structures for Stochastic Games
In stochastic games where players measure a costrelevant exogenous state variable through measurement channels, an information structure is the joint probability measure induced on the state space and player measurement ... 
Online Learning in Control Theory
In this thesis, we study two classes of problems in optimal control theory involving unknown parameters, with focus on LinearQuadraticGaussian systems. In the first problem, the control system is known and linear, and ... 
Optimal Transport in Mechanism Design, Differential Games, and Ensemble Control
In this thesis, we study optimal transport problems arising in game and controltheoretic settings. The thesis has three parts; in part one we study a variation of the matching problem over a continuum of consumers and ... 
Multivariate Hilbert Polynomials For Smooth Projective Toric Varieties
Generalizing Hilbert polynomials for varieties embedded in projective space, we develop the theory of multivariate Hilbert polynomials for varieties embedded in a smooth projective toric variety. These polynomials can be ... 
Theory and Application of Kernel Estimation Methods for CovariateVarying Effects in Survival Analysis
Study of prognostic and predictive biomarkers play an important role in the design and analysis of clinical trials. The development of personalized medicine calls for the investigation of the biomarker effect on the ... 
Numerical Restrictions on Seshadri Curves, with Applications to P^1 x P^1
We study rpoint Seshadri constants on smooth surfaces. The Seshadri constant is bounded above by the Nagata bound, and in cases where it is not maximal this fact is witnessed by a set of curves which we call Nagata curves. ... 
Decentralized Stochastic Control with Many Decision Makers and their MeanField Limit
In this thesis, we study large stochastic team problems (known also as decentralized stochastic control problems) with finite as well as countably infinite number of decision makers. In the first part of the thesis, ... 
Statistical Models for Identification of Treatmentsensitive Subgroups Based on Longitudinal Outcomes in Clinical Trials
In randomized clinical trials, an average effect of a treatment is often evaluated over all patients enrolled. But in the era of personalized medicine, there is an increasing interest in identification of patients who may ... 
ZeroDelay Lossy Coding of Linear Vector Markov Sources with Applications to Networked Control
In this thesis we study the optimal zerodelay coding (quantization) of $\mathbb{R}^d$valued linearly generated Markov sources under quadratic distortion. The structure and existence of deterministic and stationary ... 
Infinitesimal Probability Theory with Amalgamation
We introduce the notion of operatorvalued infinitesimal independence for the free, Boolean, and monotone cases. We show that operatorvalued infinitesimal free (respectively Boolean, monotone) independence is equivalent ...