# Alumni Dissertations and Theses

• ### Forms of Generic Common Knowledge

Author:
Evangelia Antonakos
Year of Dissertation:
2013
Program:
Mathematics
Sergei Artemov
Abstract:

In multi-agent epistemic logics, common knowledge has been a central consideration of study. A generic common knowledge (G.C.K.) system is one that yields iterated knowledge I(φ): ‘any agent knows that any agent knows that any agent knows…φ’ for any number of iterations. Generic common knowledge yields iterated knowledge, G.C.K.(φ)→I(φ), but is not necessarily logically equivalent to it. This contrasts with the most prevalent formulation of common knowledge C as equivalent to iterated knowledge. A spectrum of systems may satisfy the G.C.K. condition, of which C is just one. It has been shown that in the usual epistemic scenarios, G.C.K. can replace conventional common knowledge and Artemov has noted that such standard sources of common knowledge as public announcements of atomic sentences generally yield G.C.K. rather than C.

In this dissertation we study mathematical properties of generic common knowledge and compare them to the traditional common knowledge notion. In particular, we contrast the modal G.C.K. logics of McCarthy (e.g. M4) and Artemov (e.g. S4nJ) with C-systems (e.g. S4nC) and present a joint C/G.C.K. implicit knowledge logic S4nCJ as a conservative extension of both. We show that in standard epistemic scenarios in which common knowledge of certain premises is assumed, whose conclusion does not concern common knowledge (such as Muddy Children, Wise Men, Unfaithful Wives, etc.), a lighter G.C.K. can be used instead of the traditional, more complicated, common knowledge. We then present the first fully explicit G.C.K. system LPn(LP). This justification logic realizes the corresponding modal system S4nJ so that G.C.K., along with individual knowledge modalities, can always be made explicit.

• ### An algorithmic approach to the differential Galois theory of second-order linear differential equations with differential parameters

Author:
Carlos Arreche Aguayo
Year of Dissertation:
2014
Program:
Mathematics
Alexey Ovchinnikov
Abstract:

We present algorithms to compute the differential Galois group G associated via the parameterized Picard-Vessiot theory to a parameterized second-order linear differential equation with respect to d/dx, with coefficients in the field of rational functions F(x) over a differential field F, where we think of the derivations on F as being derivations with respect to parameters. We build on an earlier procedure, developed by Dreyfus, that computes G when the equation is unimodular, assuming either that G is reductive, or else that its maximal reductive quotient is differentially constant. We first show how to modify the space of parametric derivations to reduce the computation of the unipotent radical of G to the case when the reductive quotient is differentially constant in the unimodular case. For non-unimodular equations, we reinterpret a classical change-of-variables procedure in Galois-theoretic terms in order to reduce the computation of G to the computation of an associated unimodular differential Galois group H. We establish a parameterized version of the Kolchin-Ostrowski theorem and apply it to give more direct proofs than those found in the literature of the fact that the required computations can be performed effectively. We then extract from these algorithms a complete set of criteria to decide whether any of the solutions to a parameterized second-order linear differential equation is differentially transcendental with respect to the parametric derivations. We give various examples of computation and some applications to differential transcendence

• ### Geometric Interpretation of the Two Dimensional Poisson Kernel And Its Applications.

Author:
Sergei Artamoshin
Year of Dissertation:
2011
Program:
Mathematics
Jozef Dodziuk
Abstract:

Hermann Schwarz, while studying complex analysis, introduced the geometric interpretation for the Poisson kernel in 1890. We shall see here that the geometric interpretation can be useful to develop a new approach to some old classical problems as well as to obtain several new results, mostly related to hyperbolic geometry. For example, we obtain One Radius Theorem saying that any two radial eigenfunctions of a Hyperbolic Laplacian assuming the value 1 at the origin can not assume any other common value within some interval [0, p], where the length of this interval depends only on the location of the eigenvalues on the complex plane and does not depend on the distance between them.

• ### The Geometry of Gauss' Composition Law

Author:
Yelena Baishanski
Year of Dissertation:
2010
Program:
Mathematics
Lucien Szpiro
Abstract:

Gauss' identification of a composition law for primitive integral binary quadratic forms of given discriminant D--which provides the set FD<\sub> of SL2<\sub>(Z) equivalence classes of such forms with a group structure--essentially amounts to the discovery of the class group of an order in a quadratic number field. We consider quadratic extensions of the field of rational functions k(u), where k is an algebraically closed field, and seek an analogue of Gauss composition in this context. A quadratic extension of k(u) corresponds to the function field of a curve C with affine model t2<\super> = D(u) for some polynomial D = D(u) in k[u], which is of odd degree if and only if C has a smooth ramified point at infinity. Focusing on this case--the analogue of quadratic number fields with one complex place at infinity--we extend the notion of the degree of a Weil divisor on a curve to Cartier divisors on C, and find a bijection between the set of SL2<\sub>(k[u])-equivalence classes of primitive forms with coefficients in k[u] of discriminant D, and the group Pic0<\super>(C) of isomorphism classes of degree zero lines bundles on C. In parallel fashion, we reinterpret the arithmetic case using Arakelov's invention of metrics associated to the infinite places of a number field. Given an invertible R-module L for R a quadratic ring of discriminant D and fraction field K, we have for each infinite place v of K a corresponding one-dimensional C-vector space Lv<\sub>, with a positive non-degenerate hermitian metric. Using a notion of degree of an invertible metrized module--which mirrors the notion of degree used in the geometric case, yielding in both cases a "product formula" deg(f) = 0 for a principal divisor (f)--we establish for D < 0 a bijection between FD<\sub> and the compactified Picard group Picc<\sub>0<\super>(R) of isometry classes of degree zero invertible R-modules.

• ### Sensor Strip Cover: Maximizing Network Lifetime on an Interval

Author:
Benjamin Baumer
Year of Dissertation:
2012
Program:
Mathematics
Amotz Bar-Noy
Abstract:

Suppose that n sensors are deployed on a one-dimensional region (a strip, or interval) that we wish to cover with a wireless sensor network. Each sensor is equipped with a finite battery, and has an adjustable sensing range, which we control. If each sensor's battery drains in inverse linear proportion to its sensing radius, which schedule will maximize the lifetime of the resulting network? We study this Sensor Strip Cover problem and several related variants. For the general Sensor Strip Cover problem, we analyze performance in both the worst-case and average-case for several algorithms, and show that the simplest algorithm, in which the sensors take turns covering the entire line, has a tight 3/2-approximation ratio. Moreover, we demonstrate a more sophisticated algorithm that achieves an expected lifetime of within 12% of the theoretical maximum against uniform random deployment of the sensors. We show that if the sensing radii can be set only once, then the resulting Set Once Strip Cover problem is NP-hard. However, if all sensors must be activated immediately, then we provide a polynomial time algorithm for the resulting Set Radius Strip Cover problem. Finally, we consider the imposition of a duty cycling restriction, which forces disjoint subsets of the sensors (called shifts) to act in concert to cover the entire interval. We provide a polynomial-time solution for the case in which each shift contains at most two sensors. For shifts of size k, we provide worst-case and average-case analysis for the performance of several algorithms.

• ### Normal Families and Mondromies of Holomorphic Motions

Author:
Michael Beck
Year of Dissertation:
2012
Program:
Mathematics
Yunping Jiang
Abstract:

We explore some generalizations of results in holomorphic motions that result from Earle's infinite-dimensional generalization of Montel's Theorem. We then investigate topological obstructions to extending holomorphic motions. We finish with some miscellaneous facts.

• ### Points of Canonical Height Zero on Projective Varieties

Author:
Anupam Bhatnagar
Year of Dissertation:
2010
Program:
Mathematics
Lucien Szpiro
Abstract:

Let k be an algebraically closed field of characteristic zero, C a smooth connected projective curve defined over k, K =k(C) the function field of C. Let Y be a projective K-variety, L a very ample line bundle on Y and α : Y &rarr Y a K-morphism such that α *<\super>L = L × d. We prove that a projective integral C-scheme Y is isotrivial when it is covered by a projective integral k-scheme X= X0<\sub> × C, where X0<\sub> is a k-scheme. This result provides a setup for a conjecture of L. Szpiro on parametrization of points of canonical height zero of the dynamical system (Y,L, α).

• ### On the Arithmetic and Geometry of Quaternion Algebras: a spectral correspondence for Maass waveforms

Author:
Terrence Blackman
Year of Dissertation:
2011
Program:
Mathematics
Stefan Lemurell
Abstract:

Abstract

## ABSTRACT

Let be an indefinite rational division quaternion algebra with discriminant d equal to pq where p and q are primes such that p,q > 2 and let pq be a maximal order in . Further, let pq,p2rq2s,r,s 1 be an order of index p2rq2s in pq with Eichler invariant equal to negative one at p and at q . Finally, let pq,p2rq2s1 be the cocompact Fuchsian group given as the group of units of norm one in pq,p2rq2s. Using the classical Selberg trace formula, we show that the positive Laplace eigenvalues, including multiplicities, for Maass forms on pq,p2rq2s1 coincide with the Laplace spectrum for Maass newforms defined on the Hecke congruence group Γ0(M) where, M, the level of the congruence group, is equal to p2r+1q2s+1, i.e., the discriminant of pq,p2rq2s.

• ### Late Points of Projections of Planar Symmetric Random Walks on the Lattice Torus

Author:
Michael Carlisle
Year of Dissertation:
2012
Program:
Mathematics
Jay Rosen
Abstract:

We examine the cover time and set of late points of a symmetric random walk on Z2 projected onto the torus Z2K. This extends the work done for the simple random walk in [Late Points, DPRZ, 2006] to a large class of random walks. The approach uses comparisons between planar and toral hitting times and distributions on annuli, and uses only random walk methods. There are also generalizations of Green's functions, hitting times, and hitting distributions on Z2 and Z2K which are of independent interest.

• ### Uniqueness Theorems for Some Nonlinear Parabolic Equations

Author:
Yimao Chen
Year of Dissertation:
2012
Program:
Mathematics
Leon Karp
Abstract:

We study the uniqueness of solutions of the Cauchy problem of two nonlinear parabolic equations in this thesis. We first study the uniqueness of the solutions of the initial value problem associated with the infinity-Laplacian operators. We prove the uniqueness of solutions of the Cauchy problem for the infinity-Laplacian heat equation in a class of functions with exponential growth. We also study the uniqueness of the solutions of the evolution associated with the minimal surface equation. We obtain a new uniqueness class of solutions of the Cauchy problem for the parabolic minimal surface equation, which is reminiscent of the classical results for the heat equation.